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HISTORY 
 
 OF 
 
 SCIENTIFIC IDEAS. 
 
 VOLUME I. 
 
Cambridge ; 
 
 PRINTED BY C. J. CLAY, M. 
 AT THE UNIVERSITY PllESS. 
 
HISTORY 
 
 OF 
 
 SCIENTIFIC IDEAS. 
 
 BY WILLIAM WHEWELL, D.D., 
 
 MASTER OF TRINITY COLLEGE, CAMBRIDGE, AND 
 CORRESPONDING MEMBER OF THE INSTITUTE OF FRANCE. 
 
 BEING THE FIRST PAKT OF THE PHILOSOPHY 
 OF THE INDUCTIVE SCIENCES. 
 
 THE THIRD EDITION. 
 IN TWO VOLUMES. 
 
 AAMIIAAIA EXONTES AIAAO20Y2IN AAAHAOI 
 
 VOLUME I. 
 
 LONDON: 
 JOHN W. PARKER AND kON, WEST STRAND. 
 
 1858. 
 
 UNI VKUM'i '.- ,, 
 
VJT 
 Y,f 
 
 PREFACE TO THIS EDITION. 
 
 THE Chapters now offered to the Reader were 
 formerly published as a portion of The Philosophy 
 of the Inductive Sciences, founded upon their History : 
 but the nature and subject of these Chapters are more 
 exactly described by the present title, The History 
 of Scientific Ideas. For this part of the work is 
 mainly historical, and was, in fact, collected from the 
 body of scientific literature, at the same time that 
 the History of the Inductive Sciences was so collected. 
 The present work contains the history of Science so 
 far as it depends on Ideas; the former work contains 
 the same history so far as it is derived from Obser- 
 vation. The leading features in that were Theories 
 inferred from Facts; the leading features of this are 
 Discussions of Theories tending to make them con- 
 sistent with the conditions of human thought. 
 
 The Ideas of which the History is here given are 
 mainly the following: 
 
 Space, Time, Number, Motion, Cause, Force, Mat- 
 ter, Medium, Intensity, Scale, Polarity, Element, Affi- 
 nity, Substance, Atom, Symmetry, Likeness, Natural 
 Classes, Species, Life, Function, Vital Forces, Final 
 
VI PREFACE. 
 
 Causes, Historical Causation, Catastrophe and Uni- 
 formity, First Cause. 
 
 The controversies to which the exact fixation of 
 these Ideas and their properties have given occasion 
 form a large and essential part of the History of 
 Science: but they also form an important part of 
 the Philosophy of Science, for no Philosophy of Sci- 
 ence can be complete which does not solve the 
 difficulties, antitheses, and paradoxes on which such 
 controversies have turned. I have given a survey 
 of such controversies, generally carried from their 
 earliest origin to their latest aspect; and have stated 
 what appeared to me the best solution of each pro- 
 blem. This has necessarily involved me in much 
 thorny metaphysics; but such metaphysics is a neces- 
 sary part of the progress of Science. The human mind 
 deriving its knowledge of Truth from the observation 
 of nature, cannot evade the task of determining at 
 every step how Truth is consistent with itself. This 
 is the Metaphysics of Progressive Knowledge, and this 
 is the matter of this present History. 
 
 Of the remaining part of what was formerly pub- 
 lished as the Philosophy of the Inductive Sciences, 
 an additional part, described in the Introduction to 
 the present work, will shortly be published. 
 
 TRINITY LODGE, 
 
 May 24, 1858. 
 
 ERRATUM, p. 157, 1. n from top, for sciences read science. 
 
CONTENTS 
 
 OF 
 
 THE FIRST VOLUME. 
 
 TAGB 
 
 PREFACE v 
 
 PART I. 
 OF IDEAS. 
 
 INTRODUCTION 
 
 3 
 
 BOOK I. 
 
 OF IDEAS IN GENERAL. 
 
 CHAP. I. OP THE FUNDAMENTAL ANTITHESIS OF PHILO- 
 SOPHY ....... 23 
 
 Sect. i. Thoughts and Things 
 
 2. Necessary and Experiential Truths . . 25 
 
 3. Deduction and Induction . . . .27 
 
 4. Theories and facts ..... 29 
 
 5. Ideas and Sensations . . . ' . .30 
 
 6. Reflexion and Sensation . . . . 33 
 
 7. Objective and Objective . . . . 35 
 
 8. Matter and Form ..... 38 
 
 9. Man the Interpreter of Nature . .41 
 
 10. The Fundamental Antithesis is inseparable . 43 
 
 11. Successive Generalization . . . .49 
 
Vlll CONTENTS OF 
 
 PAOB 
 
 CHAP. II. OP TECHNICAL TERMS . . . . 54 
 
 Art. i. Examples. 
 
 2. Use of Terms. 
 
 CHAP. III. OF NECESSARY TRUTHS . . . .57 
 
 Art. i. The two Elements of Knowledge, 
 
 2. Shown by necessary Truths. 
 
 3. Examples of necessary Truths in numbers. 
 
 4. The opposite cannot be distinctly conceived. 
 
 5. Other Examples. 
 
 6. Universal Truths. 
 
 CHAP. IV. OF EXPERIENCE 65 
 
 Art. i. Experience cannot prove necessary Truths, 
 2. Except when aided by Ideas. 
 
 CHAP. V. OF THE GROUNDS OF NECESSARY TRUTHS . 69 
 
 Art. i. 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. VI. THE FUNDAMENTAL IDEAS ARE NOT DERIVED 
 
 FROM EXPERIENCE .... 76 
 
 Art. i. No connexion observed. 
 
 2. Faculties implied in observation. 
 
 3. We are to examine our Faculties. 
 
 CHAP. VII. OF THE PHILOSOPHY OF THE SCIENCES . .81 
 Sciences arranged according to Ideas. 
 
THE FIRST VOLUME. IX 
 
 BOOK II. 
 
 THE PHILOSOPHY OF THE PUEE SCIENCES. 
 
 PAGB 
 
 CHAP. I. Or THE PUKE SCIENCES 88 
 
 Art. I. Geometry, Arithmetic, Algebra, 
 
 1. Are not Inductive Sciences : 
 
 3. Are Mathematical Sciences. 
 
 4. Mixed Mathematics. 
 
 5. Space, Time, Number. 
 
 CHAP. II. OF THE IDEA OF SPACE . . . . 91 
 Art. i. Space is an Idea, 
 
 2. Not derived from Experience, 
 
 3. As Geometrical Truth shows. 
 
 4. Space is a Form of Experience. 
 
 5. The phrase not essential. 
 
 CHAP. III. OF SOME PECULIARITIES OF THE IDEA OF 
 
 SPACE 95 
 
 Art. i. Space is not an Abstract Notion. 
 
 1. Space is infinite. 
 
 3. Space is real. 
 
 4. Space is a Form of Intuition. 
 
 5. Figure. 
 
 6. Three Dimensions. 
 
 CHAP. IV. OF THE DEFINITIONS AND AXIOMS WHICH 
 
 KELATE TO SPACE 98 
 
 Art. i. 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 107 
 
 Art. i. How is Geometry hypothetical ? 
 2. What was Stewart's view? 
 
CONTENTS OF 
 
 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 
 
 Art. i. 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. 
 
 11. Perception includes Activity. 
 
 12. Perception of the Skyey Dome. 
 
 13. Reid's Idomenians. 
 
 14. Motion of the Eye. 
 
 15. Searching Motion. 
 
 1 6. Sensible Spot. 
 
 17. Expressions implying Motion. 
 
 CHAP. VII. OF THE IDEA OF TIME 
 
 Art. i. Time an Idea not derived from Experience. 
 
 2. Time is a Form of Experience. 
 
 3. Number. 
 
 4. Is Time derived from Motion ? 
 
 CHAP. VIII. OF SOME PECULIARITIES IN THE IDEA OP 
 
 TIME 
 
 Art. i. 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. 
 
 117 
 
 134 
 
THE FIRST VOLUME. XI 
 
 PAGE 
 
 CHAP. IX. OP THE AXIOMS WHICH RELATE TO NUMBER . 138 
 
 Art. i. Grounds of Arithmetic, 
 
 2. Intuition. 
 
 3. Arithmetical Axioms, 
 
 4. Are Conditions of Numerical Eeasoning 
 
 5. In all Arithmetical Operations. 
 
 6. Higher Numbers. 
 
 CHAP. X. OP THE PERCEPTION OF TIME AND NUMBER . 141 
 Art. i. Memory. 
 
 2. Sense of Successiveness 
 
 3. Implies Activity. 
 
 4. Number also does so. 
 
 5. And apprehension of Rhythm. 
 
 Note to Chapter X 146 
 
 CHAP. XI. OF MATHEMATICAL REASONING . . 147 
 
 Art. i. Discursive Reasoning. 
 
 2. Technical Terms of Reasoning. 
 
 3. Geometrical Analysis and Synthesis. 
 
 CHAP. XII. OF THE FOUNDATIONS OF THE HIGHER MA- 
 THEMATICS 151 
 
 Art. i. The Idea of a Limit. 
 
 2. The use of General Symbols. 
 
 3. Connexion- of Symbols and Analysis. 
 
 CHAP. XIII. THE DOCTRINE OF MOTION . . * .156 
 Art. i. Pure Mechanism. 
 
 2. Formal Astronomy. 
 CHAP. XIV. OF THE APPLICATION OF MATHEMATICS TO 
 
 THE INDUCTIVE SCIENCES . . .159 
 Art. i. The Ideas of Space and Number are clear from 
 the first. 
 
 2. Their application in Astronomy. 
 
 3. Conic Sections, &c. 
 
 4. Arabian Numerals. 
 
 5. Newton's Lemmas. 
 
 6. Tides. 
 
 7. Mechanics. 
 
 8. Optics. 
 
 9. Conclusion. 
 
Xll CONTENTS OP 
 
 BOOK III. 
 
 THE PHILOSOPHY OF THE MECHANICAL 
 
 SCIENCES. 
 
 "PAGE 
 CHAP. I. OF THE MECHANICAL SCIENCES . . . 171 
 
 CHAP. II. OF THE IDEA OF CAUSE . . . .173 
 Art. i. Not derived from Observation, 
 
 1. 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 178 
 
 Art. i. 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 184 
 
 Art. i. 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. 
 
 OF THE ORIGIN OF OUR CONCEPTIONS OF FORCE 
 
 AND MATTER 205 
 
 Art. i. Force. 
 
 2. Matter. 
 
 3. Solidity. 
 
 4. Inertia. 
 
 5. Application. 
 
THE FIRST VOLUME. Xlll 
 
 PAGE 
 
 CHAP. VI. OF THE ESTABLISHMENT OF THE PRINCIPLES 
 
 OF STATICS 212 
 
 Art. i. 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 PBINCIPLES 
 
 OF DYNAMICS 235 
 
 Art. i. 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 PABADOX OF UNIVERSAL PROPO- 
 SITIONS OBTAINED FROM EXPERIENCE . 263 
 Art. i. 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 . . .272 
 Art. i. General course of the History. 
 
XIV CONTENTS OP 
 
 FA.OB 
 
 Art. 2. Particulars as to the Law. 
 
 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 ME- 
 CHANICAL IDEAS 279 
 
 Art. i. 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. 
 
 Ji. Secondary Mechanical Sciences. 
 
 BOOK IV. 
 
 THE PHILOSOPHY OF THE SECONDARY 
 MECHANICAL SCIENCES. 
 
 CHAP. I. OF THE IDEA OF A MEDIUM AS COMMONLY 
 
 EMPLOYED 293 
 
 Art. i. 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 DIFFERENT SENSES . . . .302 
 Art. i. Difference of Senses. ' 
 Sect. I. Prerogatives of Sight. 
 Art. 2. Position. 
 3. Distance. 
 
THE FIRST VOLUME. XV 
 
 PAGH 
 
 Sect. II. Prerogatives of Hearing. 
 
 Art. 4. Musical Intervals. 
 
 5. Chords. 
 
 6. Khythm. 
 
 Sect. III. TAe 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. ii. Brown's Opinion. 
 
 CHAP. III. SUCCESSIVE ATTEMPTS AT THE SCIENTIFIC 
 
 APPLICATION OF THE IDEA OF A MEDIUM 322 
 
 Art. i. Introduction. 
 
 1. Sound. 
 3- Light. ' 
 4. Heat. 
 
 CHAP. IV. OF THE MEASURE OF SECONDARY QUALITIES 333 
 Sect. I. Scales of Qualities in General. 
 
 Art. i. 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. g. Photometer. 
 
 10. Cyanometer. 
 
 Sect. V. Scales of Heat. 
 
 Art. ii. Thermometers. 
 
 12. Their progress. 
 
 13. Fixed Points. 
 
 14. Concordance of Thermometers. 
 
 15. Natural Measure. 
 
 1 6. Law of Cooling. 
 
XVI CONTENTS OF THE FIRST VOLUME. 
 
 PAGE 
 
 Art. 17. Theory of Exchanges. 
 
 1 8. Air Thermometer. 
 
 19. Theory of Heat. 
 
 10. Other Instruments. , 
 Sect. VI. Scales of other Quantities. 
 
 Art. a i. 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 . . , .359 
 Art. i. 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 . . 371 
 
 Art. i. 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. 
 jo. 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. 
 
THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 INDUCTIVE SCIENCES, 
 
 INTRODUCTION. 
 
 VOL. I. 
 
LI Ii U A I,' 
 
 UN 5 V KKS JT *' 
 
 CALIFORNIA. 
 
 INTEODUCTION. 
 
 THE PHILOSOPHY OF SCIENCE, if the phrase were to 
 be understood in the comprehensive sense which 
 most naturally offers itself to our thoughts, would 
 imply nothing less than a complete insight into the 
 essence and conditions of all real knowledge, and an 
 exposition of the best methods for the discovery of 
 new truths. We must narrow and lower this concep- 
 tion, 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 Philosophy, 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 examining how truths, now universally re- 
 cognized, 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 commonly term Sciences; and of these bodies 
 of exact and enduring knowledge, we have within our 
 
 B 2 
 
4 PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 reach so large and varied a collection, that we may 
 examine them, and the history of their . formation, 
 with a good prospect of deriving from 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 con- 
 cerned 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 ob- 
 jects, 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 pro- 
 vinces of knowledge, so long as this process is duly 
 and legitimately performed, the results will be real 
 substantial truth. And although this process, with the 
 conditions under which it is legitimate, and the gene- 
 ral 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 PHI- 
 LOSOPHY OF THE INDUCTIVE SCIENCES. 
 
 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 de- 
 rived 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 condi- 
 
INTRODUCTION. 5 
 
 tions. On one subject of human speculation after an- 
 other, man's knowledge assumes that exact and substan- 
 tial character which leads us to term it Science; and in 
 all these cases, whether inert matter or living bodies, 
 whether permanent relations or successive occurrences, 
 be the subject of our attention, we can point out cer- 
 tain 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 speculation 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 unin- 
 telligent bodies, we shall still find some general analo- 
 gies Avhich 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 Sci- 
 ences may throw light upon the nature and extent of 
 our knowledge in every department of human specula- 
 tion. 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 sem- 
 blances ; by examining the steps by which such acqui- 
 sitions have been made, we may discover the conditions 
 under which truth is to be obtained; by tracing the 
 boundary-line between our knowledge and our ignor- 
 ance, 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 em- 
 ployment of speculative men from the earliest to the 
 most recent times. To inquire into the original, cer- 
 tainty, and compass of man's knowledge, the limits of 
 his capacity, the strength and weakness of his reason, 
 has been the professed purpose of many of the most 
 conspicuous and valued labours of the philosophers of 
 
PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 all periods up to our own day. It may appear, there- 
 fore, 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 accu- 
 mulated 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 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 development 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. Al- 
 though the terms which describe our imdertaking may 
 be the same which have often been employed by pre- 
 vious 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 ap- 
 propriate 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. 
 
INTRODUCTION. 7 
 
 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 re- 
 vised and corrected or extended. Erroneous and par- 
 tial 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 exagge- 
 rated 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 supplying what is wanting. In 
 such cases, all the merits and excellencies of the labours 
 of the preceding times do 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 pro- 
 gress 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 circum- 
 stances 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 
 
8 PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 of which I speak has already been announced; it is 
 this : that we purpose to collect our doctrines concern- 
 ing 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 universally 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. 
 
 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 preca- 
 rious 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 v/as the true analy- 
 sis of the Doctrines thus adduced ? and, Whether they 
 might safely be taken as types of real Knowledge 1 
 
 This has usually been the case when there have 
 been adduced, as standard examples of the formation 
 of human knowledge, doctrines belonging to supposed 
 sciences 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 
 sanctioned by universal reception, and formed into a 
 connected system, and traced through the steps of their 
 gradual discovery and establishment, so as to make 
 
INTRODUCTION". 9 
 
 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 phi- 
 lological, and other knowledge, is governed by the same 
 laws as that of physical science. But since, at present, 
 the 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 Physi- 
 cal 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 conclu- 
 sions stated in the present work, respecting knowledge 
 and discovery, are drawn from a connected and syste- 
 matic survey of the whole range of Physical Science and 
 its History ; whereas, hitherto, philosophers have con- 
 tented 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 philosophical 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, phy- 
 sical and physiological doctrines, the sciences of classifi- 
 
IO PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 cation and of causation, must alike be taken into our 
 account, in order that we may learn what are the gene- 
 ral characters of real knowledge. When our conclusions 
 assume so comprehensive a shape that they apply to a 
 range of subjects so vast and varied as these, we may 
 feel some confidence 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 philosophy. We may, for in- 
 stance, render our views of truth and its evidence so 
 rigid and confined as to be quite worthless, by founding 
 them too much on the contemplation 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 mechanical 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 
 sciences. As a preliminary to this undertaking, a sur- 
 vey 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 of the Inductive Sciences. 
 
 In that work I have endeavoured to trace the steps 
 by which men acquired each main portion of that 
 knowledge on which they now look with so much 
 confidence and satisfaction. The events which that 
 History relates, the speculations and controversies 
 
INTRODUCTION. 1 1 
 
 which are there described, 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 historical 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 discovery 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 examina- 
 tion of the domain of human knowledge was conducted 
 rather with the view of discovering 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 1 , 'to make a general and faithful perambulation 
 of learning, with an inquiry what parts thereof lie 
 fresh and waste, and not improved and converted by 
 the industry of man ; to the end that such a plot made 
 and recorded to memory, may both minister light to 
 any public designation, and also serve to excite volun- 
 tary endeavours.' Nor will it be foreign to our scheme 
 also hereafter to examine with a like purpose the fron- 
 tier-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 
 
 1 Advancement of Learning, b. i. p. 74. 
 
12 PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 of culture, conspiring with the native forces of the 
 bounteous 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 
 knowledge was thenceforth to be cultivated, have little 
 reference 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 histori- 
 cal 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 Astro- 
 nomy. The rules which Bacon gives for the conduct 
 of scientific researches are obtained, as it were, by divi- 
 nation, from the contemplation of subjects 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 our notice so many sciences, 
 of such wide extent, so well established; a Philoso- 
 phy of the Sciences ought, it must seem, to be found- 
 ed, 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 veri- 
 fied. 
 
 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 
 
INTRODUCTION. 13 
 
 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, with 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 
 casually and arbitrarily. They took for their moral 
 any which the story might suggest. But such a pro- 
 ceeding as this cannot suffice for us, whose aim is to 
 obtain a consistent body of philosophy from a contem- 
 plation of the whole of Science and its History. For 
 our purpose it is necessary to resolve scientific truths 
 into their conditions and ingredients, in order that we 
 may see in what manner each of these has been and is 
 to be provided, in the cases which we may have to 
 consider. This accordingly is necessarily the first part 
 of our task : to analyse Scientific Truth into its Ele- 
 ments. This attempt will occupy the earlier portion 
 of the present work ; and will necessarily be somewhat 
 long, and perhaps, in many parts, abstruse and unin- 
 viting. The risk of such an inconvenience is inevit- 
 able; 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 analy- 
 sis 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 ap- 
 pear to him obscure or repulsive. 
 
14 PHILOSOPHY OP INDUCTIVE SCIENCES. 
 
 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 
 or astronomy, is of a more limited and less animated 
 kind. 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 know- 
 ledge. 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 emi- 
 nently certain, clear, and definite. The portions of 
 knowledge to which I 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, Bo- 
 tany and Physiology, are each recognized as large and 
 substantial collections of undoubted truths. Men are 
 
INTRODUCTION. 1 5 
 
 wont to dwell with pride and triumph on the acquisi- 
 tions 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 in- 
 quiry 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 re- 
 cognize 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 
 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 magnificence of the pile. The certainty 
 and the constant progress of science are things so un- ' 
 questioned, that we are at least engaged in an intel- 
 ligible inquiry, when we are examining the grounds 
 and nature of that certainty, the causes and laws of 
 that progress. 
 
 To this inquiry, 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 at- 
 tempts, 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 
 
1 6 PHILOSOPHY OF INDUCTIVE SCIENCES. 
 
 words of hopeful exhortation, with which the greatest 
 of those who have trodden this path encouraged him- 
 self 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 
 disposed to what shall advantage themselves, and put- 
 ting 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 wander- 
 ings 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 themselves this our Reform as something of infinite 
 dimension, and beyond the grasp of mortal man, when 
 in truth it is the end and true limit of infinite errour; 
 and is by 110 means unmindful of the condition of mor- 
 tality 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.' 
 
 [The Philosophy of the Inductive Sciences, accord- 
 ing to our view, must be founded upon the History of 
 such Sciences ; which history we have attempted in a 
 former work. The events of that history may be de- 
 scribed generally as the rise of Theories out of Facts. 
 But besides this, which we may term the external his- 
 tory of Theories, there is an internal history of Theories, 
 namely, the series of steps by which the human mind 
 becomes capable of forming each Theory. Hence to 
 complete the History of the Sciences as derived from 
 Facts, we require a history of the Ideas by which such 
 derivation has been made possible : and thus, the Firbt 
 Part of our Philosophy must be a History of Scientific 
 Ideas; a labour no less historical than our former 
 work, and concerned with the same events; but which 
 has been purposely kept separate during the composi- 
 
INTRODUCTION. 1 7 
 
 tion, in order that it might be afterwards presented in 
 a more systematic form, which I have here attempted 
 to do. 
 
 Scientific Ideas are the Conditions of the derivation 
 of Sciences from Facts : but can any method or methods 
 be given by which such a Derivation can be ensured, or 
 at least, aided? Many such methods have been pro- 
 posed; of which the most celebrated is the Novum 
 Organon of Bacon, of which the title was intended to 
 imply that its scope goes much beyond the Organon of 
 Aristotle. With the experience of the formation of 
 Science which the world has had since Bacon's time, it 
 does not appear presumptuous to suppose that we can 
 now improve or correct his methods; nor to term such 
 an attempt Novum Organon Renovatum. 
 
 The Philosophy of the Inductive Sciences, then, 
 contains these two parts, The History of Scientific Ideas, 
 and the Novum Organon RenovatumJ\ 
 
 VOL. I. 
 
THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 INDUCTIVE SCIENCES 
 
 PART I. 
 HISTORY OF SCIENTIFIC IDEAS. 
 
 [We have just spoken of Theories and Facts, of Ideas 
 and Facts, and of Inductive Sciences, which imply the 
 opposition of Induction and Deduction. The explana- 
 tion of these antitheses must be the starting point of 
 our Philosophy.] 
 
 C 2 
 
[Knowledge grows, and] through the ages one increasing pur- 
 pose runs, 
 
 And the thoughts of men are widen'd with the process of the 
 Suns. 
 
BOOK I. 
 
 OF IDEAS IN GENERAL. 
 
Quse adhuc inventa sunt in Scientiis, ea hujusmodi sunt ut 
 Notionibus Vulgaribus fere subjaceant : ut vero ad interiora et 
 remotiora Naturae penetretur, necesse est ut tarn NOTIONES 
 quam AXIOMATA niagis certa et munitS, viS, a particularibus 
 abstrahantur ; atque omnino melior et certior intellectus adope- 
 ratio in usum veniat. 
 
 BACON, Nov. Org., Lib. i. Aphor. xviii. 
 
BOOK I. 
 
 Li i; i; A a 
 
 UN1VKUSIT V 
 
 OF IDEAS IN GENERAL. 
 
 CHAPTER I. 
 
 OF THE FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 
 
 Sect. i. Thoughts and Things. 
 
 IN order that we may do something towards deter- 
 mining the nature and conditions of human know- 
 ledge, (which I have already stated as the purpose of this 
 work,) I shall have to refer to an antithesis or opposi- 
 tion, which is familiar and generally recognized, and in 
 which the distinction of the things opposed to each 
 other is commonly 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 contrast, yet shew that the things which are 
 contrasted cannot be separated : I must explain that 
 the antithesis is constant and essential, but yet that 
 there is no fixed and permanent line dividing its mem- 
 bers. 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 shew the connexion of these 
 different modes of expression, and I will begin with that 
 form which is the simplest and most idiomatic. 
 
24 OF IDEAS IN GENERAL. 
 
 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 we think ; they are ac- 
 tions of our minds. Things, on the contraiy, are some- 
 thing different from ourselves and independent of us ; 
 something 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 inter- 
 nal 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 something about the sun or the moon ; namely, 
 that those objects perform certain revolutions and go 
 through certain 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 pro- 
 gress of time as marked by them. And however regu- 
 lar were the motions 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 con- 
 ceived as a human being, possessing the human powers 
 of thinking and reckoning, but kept in a dark world 
 with nothing to mark the progress of existence. The 
 latter is the case of brute animals, 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. 
 
FUNDAMENTAL ANTITHESIS OP PHILOSOPHY. 25 
 
 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 disap- 
 pears 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 an- 
 other way of looking at the antithesis of which I have 
 spoken ; and which I may, for the reasons which I have 
 just mentioned, call the FUNDAMENTAL ANTITHESIS OF 
 PHILOSOPHY. 
 
 Sect. 2. Necessary and Experiential Truths. 
 
 MOST persons are familiar with the distinction of neces- 
 sary and contingent truths. The former kind are 
 Truths which cannot but be true; as that 19 and n 
 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 anything 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 con- 
 trast 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 
 
26 OF IDEAS IN GENERAL. 
 
 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 con- 
 cern numbers are necessary truths : 1 9 and 1 1 not only 
 do make 30, but must make that number, and cannot 
 make anything else. In the same manner, it is a neces- 
 sary 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 other- 
 wise. 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 ex- 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 27 
 
 periential 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 and Things. Necessary Truths are derived 
 from our own Thoughts : Experiential truths are de- 
 rived from our observation of Things about us. The 
 opposition of Necessary and Experiential Truths is 
 another aspect of the Fundamental Antithesis of Phi- 
 losophy. 
 
 Sect. 3. Deduction and Induction. 
 
 I HAVE already stated that geometrical truths are esta- 
 blished by demonstrations deduced from definitions 
 and axioms. The term Deduction is specially applied 
 to such a course of demonstration of truths from defi- 
 nitions 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 parallelograms, 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 pro- 
 cess of successive deduction is the scheme of all geome- 
 trical 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 rea- 
 soning from these, other truths which are demonstra- 
 tively 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 
 
28 OP IDEAS IN GENERAL. 
 
 which are applicable to the Things about us ; as for 
 instance, the propositions respecting circles and spheres 
 applicable to the motions of the heavenly bodies. This 
 is Deduction, or Deductive Reasoning. 
 
 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 obtained 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 pro- 
 cess is obvious. In the former, we proceed at each step 
 from general truths to particular applications of them ; 
 in the latter, from particular observations to a general 
 truth which includes them. In the former case we 
 mav be said to reason downwards, in the latter case, 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 29 
 
 upwards; for general notions are conceived as stand- 
 ing above particiilars. 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. Demonstration 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 
 shewn. Deductive truths are the results of relations 
 among our own Thoughts. Inductive truths are rela- 
 tions 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 obser- 
 vations 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 
 mt)ves 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- 
 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 
 elliptical 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 
 
3O OF IDEAS IN GENERAL. 
 
 are the particular observations from which, as I have 
 said, such Propositions are inferred by Induction. The 
 Antithesis of Theory and Fact implies the fundamen- 
 tal Antithesis of Thoughts and Things; for a Theory 
 (that is, a true Theory) may be described as a Thought 
 which is contemplated 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 Sedations. 
 
 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 of expression also, which involvethe same Fun- 
 damental 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 contem- 
 plated distinct from the things to which they are ap- 
 plied; and so contemplated, I term them Ideas. And 
 
FUNDAMENTAL ANTITHESIS OP PHILOSOPHY. 3! 
 
 the other element, the impressions upon our senses 
 which they connect, are called Sensations. 
 
 I term space, time, cause, &c., Ideas, because they 
 are general relations among our sensations, apprehended 
 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 magnet, we do not see the attraction or force 
 which produces 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 ex- 
 ternal things. 
 
 But though our perceptions of external things in- 
 volve some act of the mind, they must involve something 
 else besides an act of the mind. If we must exercise 
 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 something 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 interpreted 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 
 
32 OF IDEAS IN GENERAL. 
 
 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 sensations 
 are generally considered more distinct than its opera- 
 tions. The world without is held to be more clearly 
 real than the faculties within. That there is something 
 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 appre- 
 ciating the impressions made upon them. Most per- 
 sons are more likely to doubt whether the mind be 
 always actively applying Ideas to the objects which it 
 perceives, than whether it perceive them passively by 
 means of Sensations. 
 
 But yet a little consideration will show us that an 
 activity of the mind, and an activity according to cer- 
 tain 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 in- 
 ferred 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 Number. 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 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 33 
 
 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 knowedge; 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 
 some other terms, besides the phrases already noticed, 
 which have a reference, more or less direct, to the 
 Fundamental 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 per- 
 plexed as to their bearing upon the view here presented 
 to him. 
 
 The celebrated doctrine of Locke, that all our 
 1 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 Sensa- 
 tion 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 connexions and relations which are not Sensa- 
 tions. 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 
 
 VOL. I. D 
 
34 OF IDEAS IN GENERAL. 
 
 to point out what those Ideas are, and to show the 
 mode in which each of them enters into our know- 
 ledge. His purpose was, to prove that there are no 
 Ideas, except the reflex acts of the mind: our endea- 
 vour 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 knowledge, 
 those who have added to our knowledge in every age 
 have referred to principles which the mind itself sup- 
 plies. 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 Reflexion, 
 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 suflice to observe 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 what- 
 ever they contain of truth. But our use of the term 
 Idea will be more fully explained hereafter. 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 35 
 
 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 
 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 sub- 
 ject 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 objec- 
 tive 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 view- 
 ing their structure by which he explained, in a very 
 striking and beautiful manner, the relations of the dif- 
 ferent parts of a plant to each other; as has been nar- 
 rated 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 
 
 D 2 
 
36 OF IDEAS IN GENERAL. 
 
 merely embody the images which a fertile genius sug- 
 gests, 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 sys- 
 tem 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 to- 
 wards 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 observation, 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." J Gothe then goes on to say, that he had been 
 grieved to the very soul by 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 observation, he had been led 
 to look with repugnance and hostility upon anything 
 which professed to depend upon ideas. ' Yet,' he ob- 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 37 
 
 serves, ' 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 became 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 our- 
 selves 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 gene- 
 ral sense in which we propose to use the term, but is a 
 particular modification of the general Ideas of symme- 
 try, developement, and the like ; and we shall hereafter 
 see, according to the phraseology which we shall ex- 
 plain in the next chapter, how such a diagram might 
 express the ideal conception of a plant. 
 
 The antithesis of subjective and objective is very fami- 
 liar 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 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 be- 
 tween 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; Deduc- 
 tion and Induction; Theory and Fact; Ideas and 
 Sensations; Reflexion and Sensation; Subjective and 
 
38 OF IDEAS IN GENERAL. 
 
 Objective; we may add others, by which distinctions 
 depending more or less upon the fundamental anti- 
 thesis have been denoted. Thus we speak of the in- 
 ternal 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 
 Not-me (Ich and Nicht-ich). Upon such phraseology 
 we may observe, that to have the fundamental anti- 
 thesis of which we speak really understood, is of the 
 highest consequence to philosophy, but that little ap- 
 pears 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 dif- 
 ferent 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 the matter, and a cube the form, of a 
 die. This comparison 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 geometrical concep- 
 tion; 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 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 39 
 
 from others, cannot exist. Yet the distinction of Mat- 
 ter and Form is real; and, as a subject of contempla- 
 tion, 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 philo- 
 sophy on such subjects. In like manner, though Sen- 
 sations 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 speculators 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 modifications 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 trans- 
 forming 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 
 
40 OF IDEAS IN GENERAL. 
 
 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 architecture. In like manner, in order that we 
 may make any progress in the philosophy of know- 
 ledge, 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 
 transformed sensations, is erroneous as well as frivo- 
 lous. For it conveys, and is intended to convey, the 
 opinion that our sensations have one form which pro- 
 perly belongs to them; and that, in order to become 
 ideas, they are converted into some other form. But 
 the truth is, that our sensations, of themselves, with- 
 out 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 resemblance 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. With- 
 out the relations of thought which we here term 
 Ideas, the sensations are matter without form. Matter 
 without form cannot exist : and in like manner sensa- 
 tions 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 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 4! 
 
 invested with form. Thus Virgil 1 speaks of the thun- 
 derbolt as informed by the hands of Brontes, and Ste- 
 ropes, 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 sig- 
 nificance and purpose. And hence the term is again 
 used 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 condi- 
 tion, is looked upon as matter, when viewed with re- 
 ference to education and knowledge, by which it is 
 afterwards moulded; and hence these are, in our lan- 
 guage, termed information. If we confine ourselves 
 to the first of these three uses of the term, we may 
 correct the erroneous 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 Inter- 
 pretation. And this image also is, in many respects, 
 
 Ferrum exercebant vasto Cyclopes in Antro 
 Brontesque Steropesque et nudus membra Pyracmon ; 
 His informatum manibus, jam parte polita 
 Fulmen erat. JEn. viii. 424. 
 
42 OF IDEAS IN GENERAL. 
 
 instructive. 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 presented to the Interpreter are 
 really objects of sensation : the notion of letters as 
 signs of words, the notion of 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 sensation 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 character of the 
 two elements of knowledge, and their necessary com- 
 bination, 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 de- 
 grees 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 pro- 
 bably 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 con- 
 jectured meaning, on the one hand, and the observed 
 marks on the other, were distinct things, though these 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 43 
 
 two things would become united as elements of one 
 act of knowledge when we had hit upon the right con- 
 jecture. 
 
 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 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 1 ? Is it a Fact or a Theory that the stars re- 
 volve 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 a, 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 can- 
 not rest any reasoning upon this distinction of Theory 
 
44 OF IDEAS IN GENERAL. 
 
 and Fact; and we cannot avoid asking whether there 
 is any real distinction in this antithesis, and if so, what 
 it is. 
 
 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, really, a result of his own Ideas. A per- 
 son 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 be- 
 lieving 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 mecha- 
 nical 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 simplest 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 opposition. 
 But the Facts are Facts, so far as the Ideas have 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 consi- 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 45 
 
 dered 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 to- 
 wards 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 interpreta- 
 tion of the phenomena ; although we may be conscious 
 of the act of interpretation, and may then regard the 
 attraction as a Fact. 
 
 Nor is it in such cases only that we interpret pheno- 
 mena 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 precon- 
 ceived 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 
 locality, 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 appear- 
 ance. And in these mixtures of observation and in- 
 ference, 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 
 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 solidity 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 
 
46 OF IDEAS IN GENERAL. 
 
 a painted representation of a cube. The scene of nature 
 is a picture without 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 trans- 
 lating as they read. The draughtsman, indeed, is com- 
 pelled, 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, spacious 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 pheno- 
 mena 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 
 expressed cannot be separated. Ideas and Sensations, 
 Thoughts and Things, Subject and Object, cannot in 
 any case be applied absolutely and exclusively. Our 
 Sensations 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 op- 
 posed to Sensations and Things. We have termed 
 them Ideas. It may be said they are Relations of 
 Things, or of Sensations. But granting this form of 
 expression, still a Relation is not a Thing or a Sensa- 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 47 
 
 tion; and therefore we must still liave 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 exclusively 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 apprehension 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 
 Observation implies Reasoning ; so, on the other hand, 
 Ideas cannot exist where Sensation has not been ; Rea- 
 soning cannot go on when there has not 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- 
 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 
 employment. By being so employed, our Ideas are 
 unfolded and defined ; and such developement and defi- 
 nition cannot 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 conceiving things which are num- 
 bered; 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 pro- 
 cesses which at present go on in our minds. We can- 
 not say that Objects create Ideas; for to perceive 
 Objects we must already have Ideas. But we may 
 
48 OF IDEAS IN GENERAL. 
 
 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 percep- 
 tion of Objects. 
 
 We cannot say of any Ideas, as of the Idea of space, 
 or time, or number, that they are absolutely and ex- 
 clusively Ideas. We cannot conceive what space, or 
 time, or number, would be in our minds, if we had 
 never perceived any Thing or 'Things in space or time. 
 We cannot 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 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 include 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 tiling 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 previous information, what 
 Ideas does it imply, to conceive the Fact as a Fact? 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 49 
 
 Does not the apprehension of the Fact imply assump- 
 tions 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 ancients 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 emphatically than this 
 had? 
 
 These remarks are by no means urged in order to 
 show that no Fact can be certainly known to be true ; 
 but only, to show that no Fact can be certainly shown 
 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. u. Successive Generalization. 
 
 IN the preceding pages we have been led to the doc- 
 trine, 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 phe- 
 nomena without us; yet that we cannot distinguish 
 and define the members of this antithesis separately. 
 Theories become Facts, by becoming certain and fami- 
 liar: 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 pro- 
 gress of human knowledge respecting any branch of 
 speculation, there may be several such steps in succes- 
 sion, each depending upon and including the preceding. 
 
 VOL. I. E 
 
5O OF IDEAS IN GENERAL. 
 
 The theoretical views which one generation of dis- 
 coverers establishes, become the facts from which the 
 next generation 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, may be found to be but one of 
 many circles, comprehended within the circuit of some 
 wider generalization. 
 
 This remark has already been made, and illustrated, 
 in the History of the Inductive Sciences 2 ; and, in truth, 
 the whole of the history of science is full of suggestions 
 and exemplifications of this course of tilings. It may 
 be convenient, however, to select a few instances which 
 may further explain and confirm this view of the pro- 
 gress 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 epi- 
 cycles, thus truly representing the apparent motions of 
 the heavenly bodies, had accumulated to an inconve- 
 nient 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 
 
 Hist. Itidudivc Sciences, b. viL c. ii. sect. 5. 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 51 
 
 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, 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 3 ; '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, ac- 
 cording to the law of the inverse square of the dis- 
 tance. The same law was shown to prevail in the 
 gravitation of the earth. It was shown, too, by induc- 
 tion 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 induc- 
 tions which we have first mentioned made by one in- 
 dividual, but the new advance, the higher flight, the 
 closing victory, fell to the lot of the same extraordi- 
 nary person. 
 
 The attraction of the sun upon the planets, of the 
 moon iipon the earth, of the planets on each other, of 
 the parts of the earth on themselves, of the sun and 
 
 3 Hist. Indiwtive Sciences, b. v. c. iv. sect 3. 
 
 E 2 
 
52 OF IDEAS IN GENERAL. 
 
 moon upon the ocean; all these truths, each of itself 
 a great discovery, were included by Newton in the 
 higher generalization, of the universal gravitation of 
 matter, by which each particle is drawn to every other 
 according to the law of the inverse square : and thus 
 this long advance 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 Astronomy. 
 
 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 subor- 
 dination and connexion is, in this case, less familiar to 
 men's minds than in Astronomy, and is, perhaps, more 
 difficult to present in a clear and definite shape. Syl- 
 vius 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 effects 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 4 , who introduced the conception of the Chemi- 
 cal Affinity or Elective Attraction, by which certain 
 elements select other elements, as if by preference. 
 That combustion, whether a chemical union or a chemi- 
 cal separation of ingredients, is of the same nature with 
 acidification, was the doctrine of Beccher 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 Caven- 
 dish. And all these truths, thus brought to light by 
 
 Hist. Inductive Sciences, b. xiv. c. iii. 
 
FUNDAMENTAL ANTITHESIS OF PHILOSOPHY. 53 
 
 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 seve- 
 ral 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 gradu- 
 ally climb to the highest forms of human knowledge, 
 and obtain from experience and observation a vast col- 
 lection 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 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. f i i > I > \ 
 
 J.J I I > lA *\ I * 
 
 UNIVKKS1TY 
 
 CALIFOUN 
 
CHAPTER II. 
 
 OF TECHNICAL TERMS. 
 
 i. TT has already been stated that we gather know- 
 JL ledge from the external world, when we are able 
 to apply, to the facts which we observe, some ideal con- 
 ception, 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 connected 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 mean- 
 ing, and appropriated to the service of science, we may 
 call Technical Terms. It is in a great measure by in- 
 venting 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 rea- 
 dily use them in advancing 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 em- 
 ploy many of these. Thus we have had instances of 
 technical Terms expressing geometrical conceptions, as 
 Ellipsis, Radius Vector, Axis, Plane, the Proportion of 
 the Inverse Square, and the like. Other Terms have 
 described mechanical conceptions, as A ccelerating Force 
 and Attraction. Again, chemistry exhibits (as do all 
 sciences) a series of Terms which mark the steps of our 
 
OF TECHNICAL TERMS. 55 
 
 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 esta- 
 blishment of Dalton'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 ex- 
 pound his electro-chemical theory, to introduce such 
 terms as Anode and Cathode, An'ion and GatJiion. 
 
 2. I need not adduce any further examples, for my 
 object at present is only to point out the use and in- 
 fluence of such language : its rules and principles I 
 shall hereafter 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 current 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 pre- 
 cision than belongs to any terms which express feel- 
 ings ; 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 gene- 
 rations that follow; and laden with this, their precious 
 freight, they sail safely across gulfs of time in which 
 empires have suffered shipwreck, and 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 medi- 
 cine of the Greeks, and the algebra and chemistry of 
 the Arabians. And we can in an instant, by means of 
 a few words, call to our own recollection, or convey to 
 the apprehension of another person, phenomena and 
 relations of phenomena in optics, mineralogy, chemistry, 
 which are so complex and abstruse, that it might seem 
 to require the utmost subtlety of the human mind to 
 
56 OF IDEAS IN GENERAL. 
 
 grasp them, even if that were made the sole object of 
 its efforts. By this remarkable effect of Technical Lan- 
 guage, 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 in- 
 strument) 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 Pro- 
 cess, and by the aid of Technical Terms, man has 
 been constantly 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 III. 
 
 OF NECESSARY TRUTHS. 
 
 i. TjWERY advance in human knowledge consists, as 
 Pj we have seen, in adapting new ideal conceptions 
 to ascertained 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 acquire ; is a 
 task on which we must now enter, and which will em- 
 ploy 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 knowledge acts of the mind as well as impressions 
 of sense; and that our knowledge derives, from these 
 acts, a generality, 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 
 knowledge possesses characters which neither of these 
 
58 OF IDEAS IN GENERAL. 
 
 elements alone could bestow. "Without our ideas, our 
 sensations could have no connexion ; without external 
 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 necessary, 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 knowledge. I have already stated briefly the differ- 
 ence 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 uni- 
 versally: 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 in- 
 volve; 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 consi- 
 dered as expressing, in some measure, the essential cha- 
 racters of our ideas. These fundamental principles I 
 shall afterwards 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. 
 
 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 
 
OF NECESSARY TRUTHS. 59 
 
 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 
 definition ; 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 evident? 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 definition that 3 and 2 make 5, it must be a matter 
 of definition 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 1 5 1 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 according 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 shown that the 
 process 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 is itself evidently identical, however it may 
 be possible to prove it by reducing it to evidently iden- 
 tical propositions. And the same is the case with all 
 other numerical propositions ; 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 
 
60 OF IDEAS IN GENERAL. 
 
 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 
 finding examples of these merely observed truths. We 
 find that sugar dissolves in water, and forms a trans- 
 parent 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 sub- 
 stances contract in the same circumstances; but can 
 any one know that this will be so otherwise than by 
 observation? We have here propositions 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 determined 
 by some unknown selection, not demonstrable con- 
 sequences of the essence of things, inevitable and per- 
 ceived to be inevitable. According to the phraseology 
 which has been frequently used by philosophical wri- 
 ters, they are contingent, not necessary truths. 
 
 It is requisite to insist upon this opposition, because 
 no insight can be obtained into the true nature of 
 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 contin- 
 gent truths, he will not be able to go along with us in 
 our researches into the foundations of human know- 
 ledge ; nor, indeed, to pursue with success any specu- 
 lation on the subject. But, in fact, this distinction is 
 one that can hardly fail to be at once understood. It 
 
OF NECESSARY TRUTHS. 6 1 
 
 is insisted upon by almost all the best modern, as well 
 as ancient, metaphysicians 1 , 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 neces- 
 sary truth for their subject, as geometry, or the pro- 
 perties of numbers, so as to obtain a familiar acquaint- 
 ance 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 can- 
 not know without looking at them; that regular hexa- 
 gons 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 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 
 
 Aristotle, Dr Whately, Dugalcl Stewart, &c. 
 
62 OF IDEAS IN GENERAL. 
 
 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 con- 
 ceive the contrary of the proposition just mentioned 
 respecting parallelograms. 
 
 But it is necessary, in applying this distinction, to 
 bear in mind the terms of it; that we cannot dis- 
 tinctly conceive the contrary of a necessary truth. For 
 in a certain loose, indistinct way, persons conceive the 
 contrary of necessary geometrical truths, when they 
 erroneously conceive false propositions to be true. 
 Thus, Hobbes erroneously held that he had discovered 
 a means of geometrically 'doubling the cube,' as it is 
 called, that is, finding two mean proportionals between 
 two given lines ; a problem which cannot be solved by 
 plane geometry. Hobbes not only proposed a con- 
 struction for this purpose, but obstinately maintained 
 that it was right, when it had been proved to be 
 wrong. But then, the discussion showed how indis- 
 tinct 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 supposed, 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 opposite of a geometrical truth, forms 
 no exception to the assertion, that this opposite cannot 
 be distinctly conceived. 
 
 In like manner, the indistinct conceptions of chil- 
 dren and of rude savages do not invalidate the distinc- 
 tion of necessary and experiential truths. Children 
 and savages make mistakes even with regard to num- 
 bers; and might easily happen to assert that 27 and 
 38 are equal to 63 or 64. But such mistakes cannot 
 
OF NECESSARY TRUTHS. 63 
 
 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 something 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 1 1 ; for children and savages might 
 be found so unfamiliar with numbers as not to reject 
 the assertion that 7 and 4 are i o, 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 experience. 
 
 5. 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 dis- 
 tinctly, 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 construction 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 propo- 
 sitions 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 impossi- 
 ble theoretically to make a perpetual motion by mere 
 mechanism as to make the diagonal of a square com- 
 mensurable with the side. 
 
64 OF IDEAS IN GENERAL. 
 
 6. Necessary truths must be universal truths. If 
 any property belong to a right-angled triangle necessa- 
 rily, it must belong to all right-angled triangles. And 
 it shall be proved in the following Chapter, that truths 
 possessing these two characters, of Necessity and Uni- 
 versality, cannot possibly be the mere results of ex- 
 perience. 
 
 [Necessary truths are not considered as a portion of 
 the Inductive Sciences. They are Deductions from our 
 Ideas. Thus the necessary truths which constitute the 
 Science of Geometry are Deductions from our Idea of 
 Space : the necessary truths which constitute the 
 Science of Arithmetic are Deductions from our notions 
 of Number; which perhaps involves necessarily the 
 Idea of Time. But though we do not call those 
 Sciences Inductive which involve properties of Space, 
 Number and Time alone, the properties of Space, Time 
 and Number enter in many very important ways into 
 the Inductive Sciences ; and therefore the Ideas of 
 Space, Time and Number require to be considered in 
 the first place. And moreover the examination of 
 these Ideas is an essential step towards the exami- 
 nation of other Ideas : and the conditions of the pos- 
 sibility and certainty of truth, which are exemplified 
 in Geometry and Arithmetic, open to us important 
 views respecting the conditions of the possibility and 
 certainty of all Scientific Truth. We shall therefore 
 in the next Book examine the Ideas on which the Pure 
 Sciences, Geometry and Arithmetic, are founded. But 
 we must first say a little more of Ideas in general.] 
 
CHAPTER IV. 
 
 OF EXPERIENCE. 
 
 i. T HERE employ the term Experience in a more 
 JL definite 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 experi- 
 ence, is of a clear and precise nature; and the pas- 
 sions and feelings and interests, which make the lessons 
 of experience in practical 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 know- 
 ledge. We know by experience that animals which 
 ruminate are cloven-hoofed ; and we know this in no 
 other manner. 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 por- 
 tions of our knowledge might be mentioned. 
 
 Now what we have here to remark is thisj that 
 in no case can experience prove a proposition to be 
 necessarily or universally true. However many in- 
 stances we may have observed of the truth of a propo- 
 sition, 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 
 VOL. i. F 
 
66 OF IDEAS IN GENERAL. 
 
 still cannot be sure that some creature will not here- 
 after be discovered which has the first of these attri- 
 butes 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 satellites 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 
 cannot 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 num- 
 ber of observations. And, however numerous these 
 may be, they can show nothing with regard to the in- 
 finite number of cases in which the experiment has not 
 been made. Experience being thus unable to prove a 
 feet to be universal, is, as will readily be seen, still 
 more incapable of proving a truth to be necessary. 
 Experience 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 in any accumulation of cases, any reason for 
 what must 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 succes- 
 sion; but the succession supplies, in its occurrence, no 
 
OF EXPERIENCE. 67 
 
 reason for its recurrence. She contemplates external 
 objects; but she cannot detect any internal bond, which 
 indissolubly connects the future with the past, the pos- 
 sible with the real. To learn a proposition by experi- 
 ence, and to see it to be necessarily true, are two alto- 
 gether different processes 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 proper- 
 ties of light, the general principles of chemistry, so 
 learnt 1 ? 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 neces- 
 sity of which she has no comprehension. If these doc- 
 trines 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 observa- 
 tion truths of a wide and comprehensive kind, which 
 approach to the simplicity and universality of the 
 truths of pure science; she gives them this character 
 by throwing upon them the light of her own Funda- 
 mental Ideas. 
 
 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 doc- 
 trine 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 de- 
 viation from the law. It is doubted still, by some 
 
 F2 
 
68 OF IDEAS IN GENERAL. 
 
 persons, with respect to the double stars. But sup- 
 pose all these doubts to be banished, and the law^ to be 
 universal; is it then proved to be necessary? Mani- 
 festly 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 reason- 
 ing 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 sug- 
 gestion of Bessel, that the intensity 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 speculations 
 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 sur- 
 vey of both hemispheres of the sky. And since the 
 extent of this truth is thus to be determined by refer- 
 ence to observed facts, it is clear that no mere accu- 
 mulation of 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 im- 
 portant 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 univer- 
 sal, the question still recurs, from what source these 
 characters of universality and necessity are derived. 
 
 The answer to this question we will attempt to give 
 in the next chapter. 
 
, ; nit 
 
 ^CALlFu 
 
 CHAPTER V. 
 
 OF THE GROUNDS OF NECESSARY TRUTHS. 
 
 i. rpO the question just stated, I reply, that the 
 JL necessity 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 active operations of our minds, with- 
 out which our passive sensations do not become know- 
 ledge. 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 knowledge by our Ideas may 
 be traced to their consequences in the most remote 
 ramifications of scientific truth. 
 
 These enunciations of the necessary and evident r 
 conditions imposed upon our knowledge by the Fun- 
 damental 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 fun- 
 damental 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 pos- 
 sible without an act of the mind, regulated by certain 
 
70 OF IDEAS IN GENERAL. 
 
 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 demonstra- 
 tive 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 de- 
 monstrative or deductive reasoning in any science, (as 
 in geometry, for instance,) consists entirely in com- 
 bining 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 com- 
 plexity; and so on, till we advance to the most recon- 
 dite demonstrable truths ; these last, however intricate 
 and unexpected, still involving no principles except 
 the original definitions and axioms. Thus, by com- 
 bining 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 demonstrate the equality of 
 triangles, under certain assumed conditions. Again, 
 by combining this result with the Definition of paral- 
 lelograms, 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 proposition, again, we prove 
 the equality of the square on the hypotenuse of a tri- 
 angle 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 rea- 
 soning, the object of which is to connect those first 
 principles, and to exhibit the effects of their combina- 
 tion in the shape of demonstration. 
 
GROUNDS OF NECESSARY TRUTHS. 71 
 
 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 collusiveness, 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 straight 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 respec- 
 tively, the major premiss, the minor premiss, and the 
 conclusion; or, more briefly, the major, the minor, and 
 the conlcusion. 
 
 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 uni- 
 versally 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 consequence ; and thus we proceed from 
 syllogism to syllogism, in each one employing some 
 general truth in some particular instance. Any proof 
 which occurs in geometry, or any other science of de- 
 monstration, may thus be reduced to a series of pro- 
 cesses, in each of which we pass from some general 
 proposition to the narrower and more special proposi- 
 tions which it includes. And this process of deriving 
 truths by the mere combination of general principles, 
 applied in particular hypothetical cases, is called de- 
 duction; 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 
 
72 OF IDEAS IN GENERAL. 
 
 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 
 general 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 first principles, it may, nevertheless, con- 
 tain the result of the combination of several such 
 principles, and may thus assume a great degree of 
 complexity, and may appear 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 respecting 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 parallels; and, perhaps, 
 others. The conclusion is complicated 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 1 .' 
 ' 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- 
 
 Whately's Logic, pp. 237, 238. 
 
GROUNDS OF NECESSARY TRUTHS. 73 
 
 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 principle, such as we have supposed Axioms to 
 be, is absolutely required. It has been asserted that 
 in geometry, 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 geometry 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 mag- 
 nitude ; consequently, it cannot contain anything equi- 
 valent to ' shortest.' Thus the propositions of geometry 
 are not merely identical propositions ; nor have we in 
 their general character anything to countenance the 
 assertion, that they are the results of definitions 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 fundamental 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 properly to exhibit 
 the grounds of necessary truth. 
 
 7. Thus the real logical basis of every body of de- 
 monstrated truths are the Definitions and Axioms 
 
74 OF IDEAS IN GENERAL. 
 
 which are the first principles of the reasonings. But 
 when we are 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 demonstra- 
 ble 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 be- 
 longing 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 Me- 
 chanics 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 
 Fundamental Ideas. And the Definitions are not ar- 
 bitrary definitions, but are determined by a necessity 
 no less rigorous than the Axioms themselves. We 
 could not think of geometrical truths without conceiv- 
 ing a circle ; and we could not reason concerning such 
 truths without defining a circle in some mode equiva- 
 lent to that which is commonly adopted. The Defi- 
 nitions of parallels, of right angles, and the like, are 
 quite as necessarily prescribed 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 of a straight line. 
 But in whatever shape the principle appear, as Defi- 
 nition or as Axiom, it has about it nothing casual or 
 
GROUNDS OF NECESSARY TRUTHS. 75 
 
 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 developments of a fun- 
 damental 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 principle, by itself, exhibits only one 
 aspect of this body, taken together they convey a suf- 
 ficient 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 
 Definitions and Axioms, even taken all together, ex- 
 press the Idea incompletely. Thus the Definitions and 
 Axioms of Geometry, as they are stated in our ele- 
 mentary works, do not fully express the Idea of Space 
 as it exists in our minds. For, in addition to these, 
 other Axioms, independent of these, and no less evi- 
 dent, can be stated; and are in fact stated when we 
 come to the Higher Geometry. 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 revealed, 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 YI. 
 
 THE FUNDAMENTAL IDEAS ARE NOT DERIVED FROM 
 EXPERIENCE. 
 
 i. T3 Y the course of speculation contained in the 
 J-J 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 propositions 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 fundamental 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 develop- 
 ments 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. Geometry in every proposition speaks a lan- 
 guage which experience never dares to utter; and in- 
 deed 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. 
 
FUNDAMENTAL IDEAS ARE NOT DERIVATIVE. 77 
 
 She unhesitatingly assents to the laws which geometry 
 delivers, but she does not pretend to see the origin of 
 their obligation. She is always ready to acknowledge 
 the sway of pure scientific 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 herself the source of that authority. 
 
 David Hume asserted 1 , 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 observa- 
 tion is, not the truth of his conclusion, but the false- 
 hood of his premises; not that, therefore, we can 
 know nothing of natural connexion, but that, there- 
 fore, we have some other source of knowledge than ex- 
 perience : 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 
 important for us to examine their nature and proper- 
 ties, 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 reply, by referring to a parallel instance. 
 Without light we should have no perception of visible 
 
 Essays, voL ii. p. 70. 
 
78 OF IDEAS IN GENERAL. 
 
 figure; yet the power of perceiving visible figure can- 
 not 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 pos- 
 sessed 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 existence) 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 external 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 facul- 
 ties 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 employ- 
 ment of our ideas, does not make it the less essential 
 for us to examine those ideas in their most intimate 
 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 con- 
 templated 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 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 multiplication of elements in man's nature, 
 without any advantage to our knowledge of his true 
 
FUNDAMENTAL IDEAS ARE NOT DERIVATIVE. 79 
 
 constitution. We may, it is said, assert man to have a 
 faculty of walking, of standing, of breathing, of speak- 
 ing; but what, it is asked, is gained by such assertions? 
 To this I reply, that we undoubtedly have such facul- 
 ties as those just named; that it is by no means unim- 
 portant to consider them; and that the main question 
 in such cases is, whether they are separate and inde- 
 pendent faculties, or complex and derivative 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, breathing, 
 for instance, a great part of the operation can be re- 
 duced 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 ad- 
 dition to the voluntary agency which they transmit, 
 which is essential 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 de- 
 serving 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 in- 
 stance, man has a faculty of apprehending time, and a 
 faculty of reckoning numbers : are these distinct, or is 
 one faculty derived from the other 1 ? To analyze the 
 various combinations of our ideas and observations into 
 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 
 
8O OF IDEAS IN GENERAL. 
 
 of forming the idea of space, we at least, by such a 
 commencement, circumscribe a certain portion of the 
 field of our investigations, which, we shall afterwards 
 endeavour to show, requires and rewards a special ex- 
 amination. And though we shall thus have to sepa- 
 rate 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 VII. 
 
 OF THE PHILOSOPHY OF THE SCIENCES. 
 
 ~Ylf 7E proceed, in the ensuing Books, to the closer 
 V V examination of a considerable number of those 
 Fundamental Ideas on which the sciences, hitherto 
 most successfully cultivated, are founded. In this 
 task, oui' 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 speculative knowledge is sub- 
 jected. I shall also try to prove, for some of these 
 Ideas in particular, what has been already urged re- 
 specting them in general, that they are not derived 
 from observation, but necessarily impose their condi- 
 tions upon that knowledge of which observation sup- 
 plies 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 development 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 OF 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, 
 
 VOL. I. G 
 
82 OF IDEAS IN GENERAL. 
 
 treat of the Philosophy of the Pure Sciences, the Philo- 
 sophy 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 1 . 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 specu- 
 lation, others are common to more. A previous enu- 
 meration of Ideas thus collected 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 (Book n.) : 
 
 Of the Ideas on which the Mechanical Sciences (as 
 Mechanics, Hydrostatics, Physical Astronomy) more 
 peculiarly rest; the ideas of Force and Matter, or 
 rather the idea of Cause, which is the basis of these 
 (Book m.) : 
 
 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 
 (Book iv.) : 
 
 Of the Ideas which are the basis of Mechanico- 
 chemical and Chemical Science; Polarity, Chemical 
 Affinity, and Substance; and the Idea of Symmetry, a 
 necessary part of the Philosophy of Crystallography 
 (Books v. vi.) : 
 
 Of the Ideas on which the Classificatory Sciences 
 proceed (Mineralogy, Botany, and Zoology); namely, 
 
 History of the Inductive Sciences. 
 
PHILOSOPHY OF SCIENCES. 83 
 
 the Ideas of Resemblance, and of its gradations, and of 
 Natural Affinity (Books vn. vin.) : 
 
 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 (Book ix.) : 
 
 We have, besides these, the Palsetiological Sciences, 
 which proceed mainly on the conception of Historical 
 Causation (Book x.) : 
 
 It is plain that when we have proceeded so far as 
 this, we have advanced to the verge of those specula- 
 tions 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 im- 
 portant 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 inevitably 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 sug- 
 gest. 
 
 G 2 
 
BOOK II. 
 
 THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 PURE SCIENCES. 
 
The way in which we are led to regard human knowledge is 
 like the way in which Copernicus was led to regard the heavens. 
 When the explanation of the celestial motions could not be made 
 to go right so long as he assumed that all the host of stars turns 
 round the spectator, he tried whether it would not answer better 
 if he made the spectator turn, and left the stars at rest. We 
 may make a similar trial in Metaphysics, as to our way of look- 
 ing at objects. If our view of them must be governed alto- 
 gether by the properties of the objects themselves, I see not how 
 man can know anything about them a priori. But if the thing, 
 as an object of the senses, is regulated by the constitution of our 
 power of knowing, I can very readily represent to myself this 
 possibility. 
 
 KANT, Kritik d. R. V. Pref. 
 
BOOK II. 
 
 THE PHILOSOPHY OF THE PUKE SCIENCES. 
 
 [The principal question discussed in the last Book 
 was this (see chaps, v. and vi.) : How are necessary and 
 universal truths possible 1 ? And the answer then given 
 was : that the necessity and universality of truths are 
 derived from the Fundamental Ideas which they involve. 
 And we proceed in this Book to exemplify this doc- 
 trine in the case of the truths of Geometry and Arith- 
 metic, which derive their necessity and universality 
 from the Fundamental Ideas of Space, and Time, or 
 Number. 
 
 The question thus examined is that which Kant 
 undertook to deal with in his celebrated work, Kritik 
 der reinen Vernunft (Examination of the Pure Rea- 
 son) : and our solution of the Problem, so far as the 
 Ideas of Space and Time are concerned, agrees in the 
 main with his. The arguments contained in chapters 
 ii. and vi. of this Book, are the leading arguments 
 respecting Space and Time, in Kant's Kritik. Kant, 
 however, instead of calling Space and Time Ideas, 
 calls them the necessary Forms of our experience, as I 
 have stated in the text. 
 
 But though I have adopted Kant's arguments as to 
 Space and Time, all that follows in the succeeding 
 Books, with regard to other Ideas, has no resemblance 
 to any doctrines of Kant or his school (with the excep- 
 tion, perhaps, of some of the views on the Idea of 
 Cause). The nature and character of the other Scientific 
 Ideas which I have examined, in the succeeding Books, 
 have been established by an analysis of the history of 
 the several Sciences to which those Ideas are essential, 
 and an examination of the writings of the principal 
 discoverers in those Sciences.] 
 
CHAPTER I. 
 
 OF THE PURE SCIENCES. 
 
 i. ALL external objects and events which we can 
 JL\. contemplate 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 occurrences. The special laws of nature, 
 considered under the various aspects which constitute 
 the different sciences, are obtained by a mixed refer- 
 ence 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 men- 
 tioned, Space, Time, and Number, constitute 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 refer- 
 ence to the phenomena of matter, are hence termed 
 Pure Sciences. The principal sciences of this class 
 are Geometry, Theoretical Arithmetic, and Algebra 
 considered in its most general sense, as the investiga- 
 tion of the relations of space and number by means of 
 general symbols. 
 
 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 con- 
 sisted 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 
 
OF THE PURE SCIENCES. 89 
 
 among such, conceptions as are derivable from the 
 ideas. These sciences have no principles besides defi- 
 nitions and axioms, and no process of proof but deduc- 
 tion; this process, however, assuming here a most re- 
 markable character; and exhibiting a combination 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 
 intellectual faculties, in a form peculiarly free from 
 admixture of extraneous elements. They were strenu- 
 ously cultivated by the Greeks, both with a view to 
 such a discipline, and from the love of speculative 
 truth which prevailed among that people: and the 
 name mathematics) by which they are designated, in- 
 dicates 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 phenomena and laws of external nature, as Astro- 
 nomy, 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 obser- 
 vation; while Geometry, Algebra, and the like sub- 
 jects, which involve no result of experience, are called 
 Pure Mathematics. 
 
 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 For- 
 mal Sciences. In this point of view, they are distin- 
 guished 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, 
 
QO PHILOSOPHY OF THE PURE SCIENCES. 
 
 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. 
 
It A i 
 
 CHAPTER II. 
 
 OF THE IDEA OF SPACE. 
 
 i. "D Y speaking of space as an Idea, I intend to 
 J3 imply, as has already been stated, that the 
 apprehension of objects as existing in space, and of the 
 relations of position, &c., prevailing among them, is not 
 a consequence of experience, but a result of a peculiar 
 constitution and activity of the mind, which is inde- 
 pendent of all experience in its origin, though con- 
 stantly 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 
 in general : but it may be useful to illustrate the doc- 
 trine further in this particular case. 
 
 I assert, then, that spacje 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 observa- 
 tion is possible any otherwise than by beginning with 
 such a representation, 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 
 necessary. Propositions of such a kind could not pos- 
 sibly be obtained from experience; for experience can 
 
92 PHILOSOPHY OF THE PURE SCIENCES. 
 
 only teach us by a limited number of examples, and 
 therefore can never securely establish a universal pro- 
 position : 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 triangles ; it is true in such a way that the 
 contrary cannot be conceived. Experience could not 
 prove such a proposition. 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 
 fact, 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 of the nature of geometrical 
 demonstration. We have heard of persons who con- 
 vinced themselves by measurement that the geometri- 
 cal 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 development of the idea of space 
 had been stifled by other employments. The practical 
 trial of the rule may illustrate, but cannot prove it. 
 The rule will of course be confirmed 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 experi- 
 ence, as the assertion that sixteen and nine make 
 twenty-five. At the same time, the truth of the geo- 
 metrical 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 
 
OF THE IDEA OF SPACE. 93 
 
 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 per- 
 ception 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 be, matter of experience. 
 Our sensations can each convey no information except 
 about itself; each can contain no trace of another 
 additional sensation; and thus no relation and con- 
 nexion 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 condition 
 of perception in the mind, we can conceive the existence 
 of necessary truths, which apply to all perceived 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 Siform 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, however, 
 
94 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 
 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 par- 
 ticular. 
 
CHAPTER III. 
 
 OF SOME PECULIARITIES OF THE IDEA OF SPACE. 
 
 T. OjOME of the Ideas which we shall have to exa- 
 )O mine 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 
 everywhere-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 essen- 
 tially one ; and the complication which exists in it, and 
 the conception of various spaces, depends merely upon 
 boundaries. Space must, therefore, be, as we have 
 said, not a general conception abstracted from particu- 
 lars, but a universal mode of representation, altogether 
 independent of experience. 
 
 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 num- 
 ber 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. 
 
96 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 conform the impressions of sense. And its pecu- 
 liarities appear 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 objects in space, side by side, exterior to each 
 other; space, and objects in so far as they occupy 
 space, have 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 applicable only to that mode of contemplation 
 in which we thus look at objects as made iip 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 
 
PECULIARITIES OF THE IDEA OF SPACE. 97 
 
 gives rise to various conceptions which we have often 
 to consider. Thus limited, space assumes form or 
 figure; and the variety of conceptions thus brought 
 under our notice is infinite. We have eveiy possible 
 form of line, straight line, and curve ; and of curves an 
 endless number; circles, parabolas, hyperbolas, spirals, 
 helices. We have plane surfaces of various shapes, 
 parallelograms, polygons, 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 properties 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 
 fewer. 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 thickness or depth. The origin of such a 
 difference of dimensions 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 other- 
 wise it would not be a boundary. A point has no 
 dimensions. A line has but one dimension, the dis- 
 tance 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 dimension 
 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 ex- 
 tended above or below the plane; or thickness, if we 
 omit all consideration 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. 
 
 VOL. I. H 
 
CHAPTER IV. 
 
 OF THE DEFINITIONS AND AXIOMS WHICH RELATE TO 
 
 SPACE. 
 
 i. npHE relations of space have been apprehended 
 JL 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 in- 
 tuition. 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 terra implies. The 
 Greeks of the time of Plato, had, however, not only 
 possessed themselves of many of the most remarkable 
 elementary theorems of the science ; but had, in seve- 
 ral instances, reached the boundary of the science in 
 its elementary form; as when they proposed to them- 
 selves 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 
 principles 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, 
 
DEFINITIONS RELATING TO SPACE. 99 
 
 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 
 geometry 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 hypothe- 
 ses 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 conse- 
 quences of the conceptions respecting which they are 
 asserted ; and the definitions are no less necessary 
 limitations of conceptions; not requisite in order to 
 arrive at this or that 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 staff, 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 concep- 
 tion 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, with- 
 out some reference to that peculiar case in which the 
 angles are equal : and thus it becomes necessary to 
 
 K 2 
 
IOO PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 charac- 
 ter; for instance, that two straight lines cannot cut 
 each other in more than one point, and an axiom con- 
 cerning 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 geometrical 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 1 . 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 demonstration. And if we 
 were to take any other definition of two parallel straight 
 lines, (as that they are both perpendicular to a third 
 straight line,) we should still, at some point or other of 
 our progress, fall in with the same difficulty of demon- 
 stratively establishing their properties without some 
 further assumption. 
 
 i This axiom is simpler and more convenient than that of Euclid. It is 
 employed by the late Professor Playfair in his Geometry. 
 
DEFINITIONS RELATING TO SPACE. IOI 
 
 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 
 hypotheses 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, which words cannot fully convey. It is 
 not possible to resolve these truths into certain hypo- 
 theses, of which all the rest shall be the necessary 
 logical consequence. The necessity is not hypotheti- 
 cal, 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 reasoning : 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 implicitly in- 
 volved 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 
 necessity of the truths which regard the circle, that 
 we could not in this way unfold a definition into pro- 
 portions, without possessing an intuition of the rela- 
 tions 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 
 
IO2 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 defi- 
 nition of a straight line could be given which rendered 
 the axiom concerning 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 2 . 
 
 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, can- 
 not 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 spokon. 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 there- 
 
 2 I formerly stated views similar to 135 of the Edinburgh Review. As an 
 
 these in some 'Remarks' appended examination of the reviewer's objec- 
 
 to a work which I termed The Me- tions may serve further to illustrate 
 
 chemical Euclid, published in 1837. the subject, I shall annex to this chap- 
 
 These Remarks, so far as they bear ter an answer to the article to which 
 
 upon the question here discussed, I have referred, 
 were noticed and controverted in No. 
 
DEFINITIONS RELATING TO SPACE. 1 03 
 
 fore there cannot be two straight lines, such as the 
 axiom, describes, 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 can- 
 not inclose a space. 
 
 If the separation of the two parts of space were 
 made by any other than a straight line; if, for in- 
 stance, 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 
 Doints, 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 all other forms, are of so vague a nature, that 
 they cannot fitly be made the basis of our elementary 
 geometry; and they are far more conveniently re- 
 placed, as they have been in almost all treatises of 
 
104 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 or fiat : for otherwise, 
 by folding it, we might obtain a line not straight. 
 Now this assumption of a plane appears to take for 
 granted that very conception of a straight line which 
 the sheet was employed 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 por- 
 tions 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 sepa- 
 rates are not similar, one having a convex and the 
 other a concave boundary. But there is a certain in- 
 termediate position of the surface, in which position 
 the two portions of space which it divides have their 
 boundaries exactly similar. In this position, the sur- 
 face 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 inter- 
 mediate form among all convex and concave surfaces 
 by which solid space can be divided, and of its sepa- 
 rating such space into two portions, of which the boun- 
 daries, though they are the same surface in two oppo- 
 site positions, are exactly similar. 
 
DEFINITIONS RELATING TO SPACE. 1 05 
 
 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 interminable space, capable of imaginary di- 
 vision in every direction. 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 portions by a point, which is 
 the common boundary of the two 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 reason- 
 ing upon these subjects. And thus, the views which 
 we have presented of the nature of space being ver- 
 bally expressed by means of certain definitions and 
 axioms, become the groundwork of a long series of de- 
 ductive 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 those laws are written. But besides the interest 
 and importance 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 unexpected results. 
 
IO6 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 ap- 
 plicable 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 doc- 
 trines are applicable : and how rich and various a 
 harvest of knowledge springs from these seemingly 
 scanty principles. 
 
 But before we quit the subject now under our con- 
 sideration, we shall endeavour to answer some objec- 
 tions which have been made to the views here pre- 
 sented; 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 1 . 
 
 Edinburgh Review, No. cxxxv., contains a cri- 
 JL 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 
 hereafter in Chapter xi. 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 preceding chapters (except, perhaps, one 
 or two remarks which have reference to mechanical 
 ideas), it may serve to illustrate the subject if I reply 
 to the objections directly, taking them as the reviewer 
 has stated them. 
 
 i. I had dissented from Stewart's assertion that 
 mathematical truth is hypothetical, or depends upon 
 arbitray definitions ; since we understand by an hypo- 
 
 1 In order to render the present distinct conception in the mind ; that 
 
 chapter more intelligible, it may be the definitions which we employ in 
 
 proper to state briefly the arguments mathematics are not arbitrary or hy- 
 
 which gave occasion to the review, pothetical, but necessary definitions ; 
 
 After noticing Stewart's assertions, that if Stewart had taken as his ex- 
 
 that the certainty of mathematical rea- amples of axioms the peculiar geome- 
 
 soning arises from its depending upon trical axioms, his assertions would 
 
 definitions, and that mathematical have been obviously erroneous ; and 
 
 truth is hypothetical; I urged, that that the real foundation of the truths 
 
 no one has yet been able to construct of mathematics is the Idea of Space, 
 
 a system of mathematical truths by which may be expressed (for pur- 
 
 the aid of definitions alone; that a poses of demonstration) partly by 
 
 definition would not be admissible or definitions and partly by axioms, 
 applicable except it agreed with a 
 
IO8 PHILOSOPHY OP THE PURE SCIENCES. 
 
 thesis a supposition, not only which we may make, but 
 may abstain from making, or may replace by a differ- 
 ent supposition ; whereas the definitions and hypotheses 
 of geometry are necessarily such as they are, and can- 
 not be altered or excluded. The reviewer (p. 84) in- 
 forms us that he understands Stewart, when he speaks 
 of hypotheses and definitions being the foundation of 
 geometry, to speak of the hypothesis that real objects 
 correspond to our geometrical definitions. ' Jf a crys- 
 tal be an exact hexahedron, the geometrical properties 
 of the hexahedron 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 foun- 
 dation 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 obser- 
 vation and measurement, not of definition. 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 propo- 
 sition 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 an- 
 swers, 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 rectangle. 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, observed or hypothetical, affect a universal and 
 necessary truth, which would be equally true if the 
 fact were false? If there be nothing arbitrary or hypo- 
 thetical 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. 
 
ANSWER TO OBJECTIONS. 1 09 
 
 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 con- 
 fuses 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 founda- 
 tions 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 dis- 
 cuss. 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. ii. p. 38) by contesting Reid's opinion 
 on this subject, which is stated generally; 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 con- 
 test this assertion, which left my thesis untouched. 
 I had asserted that the proper geometrical axioms 
 (that two straight lines cannot inclose a space, and the 
 axiom about parallel lines) are indispensable in geome- 
 try. 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 sur- 
 rendered. 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 
 
IIO PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 
 geometry 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 
 considering as a perfect example of intellectual build- 
 ing? 
 
 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 inde- 
 monstrable 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, ' legiti- 
 mate filiations' of the definitions. They would be in- 
 volved in the definitions. 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 perform that which it under- 
 takes 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 can- 
 not be evolved out of them; and that rather than 
 admit that they have any other origin than the defini- 
 tions, 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 
 
ANSWER TO OBJECTIONS. Ill 
 
 phrase do 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 ' legiti- 
 mate filiation' by a genealogical table in a satisfactory 
 form. When this cannot be done, to assert, notwith- 
 standing, that the propositions are involved in the 
 definitions, is a mere begging the question; and to 
 excuse this defect by saying that geometry fails to per- 
 form what she has promised, is to calumniate the 
 character of that science which we profess to make our 
 standard, rather than abandon an arbitrary and un- 
 proved 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 demonstra- 
 tion to the satisfaction of any school of reasoners, the 
 defect must arise from their erroneous views of the 
 nature of demonstrations, and the grounds of mathe- 
 matical certainty. 
 
 4. I conceive, then, that the reviewer has failed 
 altogether to disprove the doctrine that the axioms of 
 geometry 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 logi- 
 cally 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 defini- 
 tion for the name wherever it occurs. I reply, that 
 my doctrine, that a definition expresses a part, but 
 not the whole, of the essential characters of an idea, is 
 certainly at variance with an opinion sometimes main- 
 tained, 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 
 
112 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 obscu- 
 rity, when in the discussion of general principles, we 
 thus substitute definitions 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 conceptions, and not the arbitrary 
 equivalence of two forms of expression, are the foun- 
 dations of our reasoning. 
 
 5. The reviewer conceives that some of the so- 
 called axioms are really definitions. The axiom, that 
 'magnitudes which coincide with each other, that is, 
 which fill 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 win- 
 ter,' does this assertion define anything, though the 
 proposition is perfectly intelligible and distinct? 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 pos- 
 sibility of such a circle. To this I might reply, that it 
 is not at all incumbent upon me to assent to such a 
 change ; for I have all along stated that it is indifferent 
 
ANSWER TO OBJECTIONS. 113 
 
 whether the fundamental properties from which we 
 reason be exhibited as definitions or as axioms, pro- 
 vided the 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 defini- 
 tion 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 definition, as 
 follows : 
 
 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 re- 
 viewer 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 un- 
 derstood, 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 observation 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 
 VOL. i. I 
 
114 PHILOSOPHY OF THE PURE SCIENCES. 
 
 willing to assent; but I would add, that not only 
 these, but all the principles which express the funda- 
 mental conditions of our knowledge, may with equal 
 propriety be termed laws of thought; for these prin- 
 ciples depend upon our ideas, and regulate the active 
 operations of the mind, by which coherence and con- 
 nexion 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 quan- 
 tity, is by no means true. The whole of arithmetic, 
 for instance, the rules for the multiplication and divi- 
 sion of large numbers, the rule 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 re- 
 viewer, and limited to geometrical truths, it is hardly 
 tenable (although, in fact, it matters little to our argu- 
 ment whether it is or no). For the greater part of the 
 Seventh Book of Euclid's Elements, (on Commensur- 
 able and Incommensurable Quantities,) and the Fifth 
 Book, (on Proportion,) depend upon these axioms, with 
 the addition only of the definition or ^xiom (for it may 
 be stated either way) which expresses the idea of pro- 
 portionality in numbers. So that the attempt to dis- 
 prove the necessity 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 ex- 
 ample, 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 prin- 
 ciples 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 
 
ANSWER TO OBJECTIONS. 115 
 
 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 analyse the demonstrations 
 which we have obtained. In every mental process 
 many principles are combined and abbreviated, and 
 thus in some measure concealed and obscured. In 
 analysing these processes, the combination 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 ex- 
 hibit 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 neces- 
 sary for conviction; but they are necessary to display 
 the conditions under which conviction becomes inevit- 
 able. The application of a single one of the axioms 
 just spoken of is so minute a step in the proof, that it 
 appears pedantic 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 ac- 
 cumulation of such apparently imperceptible advances, 
 we can in the end make so vast and so sure a progress. 
 The completeness 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 ele- 
 ments of truth, of axioms for instance, does not pre- 
 vent 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 unphilo- 
 sophical. Axioms and definitions 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 
 
 I 2 
 
Il6 PHILOSOPHY OF THE PURE SCIENCES. 
 
 by asserting the latter to be derived from the former. 
 This bond of connexion exists in the mind of the rea- 
 soner, in his conception of that to which both defini- 
 tion and axiom refer, and consequently in the general 
 Fundamental Idea of which that conception is a modi- 
 fication. 
 
CHAPTER VI. 
 
 OF THE PERCEPTION OF SPACE. 
 
 i. A CCORDING to the views above explained, cer- 
 J\. tain of the impressions of our senses .convey to 
 us the perception of objects as existing in space; inas- 
 much 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 ex- 
 perience, suggest to us the conception of a space sur- 
 rounding 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 mathema- 
 tical reasoning. Again, we see before us a solid ob- 
 ject, a cubical box for instance; we see that it is within 
 reach; we stretch out the hand and perceive by the 
 touch that it has sides, edges, corners, which we had 
 already perceived by vision. 
 
Il8 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 imme- 
 diate. 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 110 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 misleads 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 interior of a building is represented. In 
 these cases, the perspective representations of the vari- 
 ous 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 mani- 
 
OF THE PERCEPTION OF SPACE. 1 19 
 
 festly 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 inter- 
 preting 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 
 mastered is truly admirable; for we forget that our 
 conclusion is obtained indirectly, and mistake a judg- 
 ment 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 to be square, however obliquely it be presented 
 to us. Thisj however, by no means throws any doubt 
 or difficulty on the doctrine that in all these cases we 
 do interpret 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 under- 
 stand 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 
 
I2O PHILOSOPHY OF THE PURE SCIENCES. 
 
 expanse, and collect from it a space occupied by solid 
 objects. In both cases the act of interpretation is be- 
 come 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 pro- 
 cess 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 difficulty 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 
 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 out- 
 line 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 1 : ' 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, 
 
 Phillips On Painting. 
 
OF THE PERCEPTION OF SPACE. 121 
 
 the first and the simplest means we resort to is this 
 picture, by which we separate the form of each object 
 from those that surround 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 w space, 
 the impressions 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 con- 
 vexity will be transferred 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 decision 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 sponta- 
 neous. If we draw black outlines, such as represent 
 the edges of a cube seen in perspective, certain of the 
 lines will cross each other; and we may make this 
 cube appear to assume two different positions, by de- 
 termining 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 distance 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. 
 
122 PHILOSOPHY OF THE PURE SCIENCES. 
 
 Thus we have these abundant and various manifes- 
 tations 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 2 , 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 
 been thrown upon it in a degree which is very remark- 
 able, when we consider that the philosophy of percep- 
 tion 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 physiologist, 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 impor- 
 tant bearing upon the principle which we are en- 
 deavouring 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 
 conveyed 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 
 
 - Theory of Vision. 
 
OF THE PERCEPTION OF SPACE. 123 
 
 shape with, no chance of errour. 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 1 ? Against such an opi- 
 nion Dr. Brown, the metaphysician of whom I speak, 
 urges 3 that the feeling of touch alone, when any ob- 
 ject is applied 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, referred our limbs to their 
 positions; which is to suppose 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 con- 
 scious exertions of the various muscles by which we 
 move our limbs. When we feel out the form and po- 
 sition 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 an- 
 other. 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 conscious 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 con- 
 ditions of space, no less than one who has the use of 
 his eyes. And the muscular consciousness which re- 
 veals to us the position of objects and parts of objects., 
 
 Lectures, Vol. i. p. 459, (1824). 
 
124 PHILOSOPHY OF THE PURE SCIENCES. 
 
 when we feel them out by means of the hand, shows 
 itself in a thousand other ways, and in all our limbs : 
 for our habits of standing, walking, and all other atti- 
 tudes and motions, are regulated by our feeling of our 
 position and that of surrounding objects. And thus, 
 we cannot touch any object without learning something 
 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. 
 
 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 for- 
 cible reasons. Perhaps one of the most striking evi- 
 dences in its favour is that, as I have already intimated, 
 it is the opinion to which another distinguished phi- 
 losopher, Sir Charles Bell, has been led, reasoning 
 entirely upon physiological principles. From his re- 
 searches it resulted that besides the nerves which con- 
 vey the impulse of the will from the brain to the 
 muscle, by which every motion of our limbs is pro- 
 duced, there is another set of nerves which carry back 
 to the brain a sense of the condition of the muscle, 
 and thus regulate its activity; and give us the con- 
 sciousness of our position and relation to surrounding 
 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 con- 
 sciousness of muscular exertion, which he has called a 
 sixth sense 4 , is our guide, Sir C. Bell shows, in the 
 common practical government of our motions; and he 
 states that having given this explanation of perception 
 as a physiological doctrine, he had afterwards with 
 satisfaction seen it confirmed by Dr. Brown's specu- 
 lations. 
 
 1 1. Thus it appears that our consciousness of the 
 relations of space is inseparably and fundamentally 
 connected with our own actions in space. We perceive 
 
 Bridgewater Treatise, p. 195. Phil. Trans. 1826, Pt. ii. p. 167. 
 
OF THE PEECEPTION OF SPACE. 125 
 
 only while we act } our sensations require to be inter- 
 preted by our volitions. The apprehension of exten- 
 sion and figure is far from being a process in which we 
 are inert and passive. We draw lines with our fingers ; 
 we construct surfaces by curving our hands ; we gene- 
 rate spaces by the motion of our arms. When the 
 geometer bids us form lines, or surfaces, or solids by 
 motion, he intends his injunction to be taken as hypo- 
 thetical only; we need only conceive such motions. 
 But yet this hypothesis represents 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 pas- 
 sive 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 hemisphere. 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 circum- 
 stances, we should have perceived only what was pre- 
 sented to the eye in one position ; and if different ap- 
 pearances had been presented in succession, we could 
 
126 PHILOSOPHY OF THE PURE SCIENCES. 
 
 not have connected them as parts of the same picture, 
 for want of any perception of their relative position. 
 They would have been so many detached and inco- 
 herent visual sensations. The muscular sense connects 
 their parts into a whole, making them to be only dif- 
 ferent portions of one universal scene 5 . 
 
 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 
 subject, 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 geometry 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 ap- 
 pears that the notion of such a sphere, and such a 
 connexion of visible objects which are seen in different 
 
 5 It has been objected to this view as a repetition of the picture. That 
 that we might obtain a conception of sight, of itself, .can give us only a 
 the sky as a hemisphere, by being plane picture, the doctrine of Berke- 
 ourselves turned round, (as on a ley, appears to be indisputable; and, 
 music-stool, for instance,) and thus no less so, the doctrine that it is 
 seeing in succession all parts of the the consciousness of our own action 
 sky. But this assertion I conceive to in space which puts together these 
 be erroneous. By being thus turned pictures so that they cover the sur- 
 round, we should see a number of face of a solid body. We can see 
 pictures which we should put toge- length and breadth with our eyes, 
 ther as parts of a plane picture ; and but we must thrust out our arm 
 when we came round to the original towards the flat surface, in order 
 point, we should have no possible that we may, in our thoughts, corn- 
 means of deciding that it was the bine a third dimension with the other 
 same point: it would appear only two. 
 
OF THE PERCEPTION OF SPACE. 127 
 
 directions, cannot be arrived at by sight alone. When 
 the spectator combines in his conception the relations 
 of long-drawn lines and large figures, as he sees them 
 by turning his head to the right and to the left, up- 
 wards and downwards, he ceases to be an Idomenian. 
 And thus our conceptions of the properties of space, 
 derived through the exercise of one mode of percep- 
 tion, 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 boun- 
 dary of them with the eye. While the head is held 
 perfectly still, the eye can rove along the outlines of 
 visible objects, scrutinize each point in succession, and 
 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 6 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 rela- 
 tion in space, which it is not the office of the retina to 
 give, by our consciousness of the efforts of the voluntary 
 
 6 Phil. Trans. 1823. On the Motions of the Eye. 
 
128 PHILOSOPHY OF THE PURE SCIENCES. 
 
 muscles of the eye: and lie 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 7 , is the means by which we be- 
 come aware of the position of objects about us. 
 
 1 6. 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 sensi- 
 bility diminishes on every side by degrees. And the 
 searching motion of the eye arises from the desire 
 which we instinctively feel of receiving upon the sensible 
 spot the image of the object to which the attention is 
 directed. "We are uneasy and 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 
 
 7 Bridgewater Treatise, p. 282. I not because the point on which the 
 
 have adopted, in writing the above, image falls in direct vision is the most 
 
 the views and expressions of Sir sensible point, but that it is the point 
 
 Charles Bell. The essential part of of greatest distinctness of vision. They 
 
 the doctrine there presented is, that urge that a small star, which disap- 
 
 the eye constantly makes efforts to pears when the eye is turned full upon 
 
 turn, so that the image of an object to it, may often be seen by looking a 
 
 which our attention is drawn, shall little away from it : and hence, they 
 
 fall upon a certain particular point of infer that the parts of the retina re- 
 
 the retina ; and that when the image moved from the spot of direct vision, 
 
 falls upon any other point, the eye are more sensible than it is. The 
 
 turns away from this oblique into the facts are very curious, however they 
 
 direct position. Other writers have be explained, but they do not disturb 
 
 maintained that the eye thus turns the doctrine delivered in the text. 
 
OF THE PERCEPTION OF SPACE. 1 29 
 
 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, 
 
 Boughly 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 8 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 depending on its own nature; so it now 
 appears that our perceptions of visible figure are not 
 obtained without an act performed under the same 
 conditions. The sensations of touch and sight are sub- 
 ordinated to an idea which is the basis of our specula- 
 tive knowledge concerning space and its relations; and 
 this same idea is disclosed to our consciousness by its 
 practically regulating our intercourse with the external 
 world. 
 
 By considerations such as have been adduced and 
 referred to, it is proved beyond doubt, that in a great 
 
 8 Byron, Ch. Ear. vL st. 75- 
 VOL. I. 
 
I3O PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 9 . 
 
 We now quit the consideration of the properties of 
 Space, and consider the Idea of Time. 
 
 9 The expression in the first edition 
 was 'large objects and extensive 
 spaces.' In the text as now given, I 
 state a definite size and extent, with- 
 in 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 .figu re. 
 
 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 explanation of 
 that power by which the eye becomes 
 acquainted with the position and re- 
 lation of objects, if the sense of mus- 
 cular activity be excluded which ac- 
 companies the motion of the eyeball.' 
 But surely we should have no diffi- 
 culty 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. ii. 
 
CHAPTER VII. 
 
 OP THE IDEA OF TIME. 
 
 i. TJ ESPECTING the Idea of Time, we may make 
 
 Xt* several of the same remarks which we made 
 concerning the idea of space, in order to show that it 
 is not borrowed from experience, but is a bond of con- 
 nexion among the impressions of sense, derived from a 
 peculiar activity of the mind, and forming a founda- 
 tion both of our experience and of our speculative 
 knowledge. 
 
 Time is not a notion obtained by experience. Ex- 
 perience, 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 succes- 
 siveness, 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 presenta- 
 tion of all occurrences to our minds. We cannot con- 
 ceive this condition to be taken away. We can con- 
 ceive time to go on while nothing- happens in it; but 
 we cannot 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 
 
 K 2 
 
132 PHILOSOPHY OF THE PURE SCIENCES. 
 
 present, and even in our thoughts we cannot form the 
 contrary supposition. 
 
 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 
 contemplation to the mind. Time is one of the neces- 
 sary conditions under which we apprehend the infor- 
 mation which our senses and consciousness give us. 
 By considering 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 conceiving 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 condition impressed upon our 
 knowledge by the constitution of the mind itself ; in- 
 volving 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 ex- 
 perience, 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 absolutely necessary, entirely independent 
 of experience and anterior to it; and so far as the con- 
 ception 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 
 
OF THE IDEA OF TIME. 133 
 
 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 presupposes the necessary conditions that exist- 
 ence in time implies. And thus we see that it is pos- 
 sible there should be necessary truths concerning all 
 Motion, and consequently, 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 re- 
 siding in the mind, afford a foundation for necessary 
 truths. 
 
CHAPTER YIII. 
 
 OF SOME PECULIARITIES OF THE IDEA OF TIME, 
 
 i. mHE Idea of Time, like the Idea of Space, offers 
 JL to our notice some characters which do not be- 
 long 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 con- 
 template any series of occurrences. All actual and 
 possible times exist as Parts, in this original and gene- 
 ral Time. And since all particular times are con- 
 sidered 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 therefore not a general conception 
 gathered from experience. 
 
 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 endj but the beginning 
 and the end of every other existence takes place in it. 
 
 3. Time, like space, is not only a form of per- 
 ception, but of intuition. We contemplate events as 
 
SOME PECULIARITIES OF THE IDEA OF TIME. 135 
 
 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 in- 
 tuition of the external world. 
 
 4. Time is conceived as a quantity of one dimen- 
 sion ; 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, and hence nothing which has any obvious ana- 
 logy 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 combination which might give rise to figures 
 of any kind. The analogy between time and space, 
 which in many circumstances is so clear, here disap- 
 pears altogether. Spaces of two and of three dimen- 
 sions, planes and solids, have nothing to which we can 
 compare them in the conceptions arising out of time. 
 
 6. As figure is a conception solely appropriate to 
 space, there is also a conception which peculiarly be- 
 longs to time, namely, the conception of recurrence of 
 times similarly marked ; or, as it may be termed, 
 
136 PHILOSOPHY OF THE PURE SCIENCES. 
 
 rhythm, using this word in a general sense. The term 
 rhythm is most commonly used to designate the recur- 
 rence of times marked by the syllables of a verse, or 
 the notes of a melody : but it is easy to see that the 
 general conception of such a recurrence does not de- 
 pend on the mode in which it is impressed upon the 
 sense. The forms of such recurrence are innumerable. 
 Thus in such a line as 
 
 Qua'drupeda'nte putre*in sonitti quatit tingula ca"mpum, 
 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 cldse of the da"y when the ha"mlet is still, 
 
 we have two light and one strong syllable repeated 
 four times over. Such repetition is the essence of ver- 
 sification. 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 pur- 
 poses of rhythm by their length only or principally j for 
 example, if either of the above lines were imitated by 
 a melody in the most 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 versi- 
 fication 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 
 
SOME PECULIARITIES OF THE IDEA OF TIME. 137 
 
 relations 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 necessa- 
 rily occur in figures, is quite essential in order to im- 
 press 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 
 syllables. For instance, 
 
 Aw^ke, arise, or b for dver fgll'n. 
 
 Or without any subordination, as when we reckon num- 
 bers, and call them in succession, odd, even, odd, even. 
 
 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, 
 however, we are led to consider each unit with refer- 
 ence to all that have preceded ; and thus the series one, 
 one, one, and so forth, becomes one, two, three, four, 
 five, 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 exten- 
 sive 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 Number, which is, as we have said, the 
 simplest result of repetition. This science is Theoreti- 
 cal Arithmetic, or the speculative doctrine of the pro- 
 perties and relations of numbers; and we must say a 
 few words concerning the principles which it is requi- 
 site to assume as the basis of this science. 
 
CHAPTER IX. 
 
 OF THE AXIOMS WHICH RELATE TO NUMBER. 
 
 i . rr\HE foundations of our speculative knowledge of 
 JL 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 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 princi- 
 ples which we assume as true, without being aware 
 that we have made any assumption; 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 asser- 
 tion is involved in our being able to conceive the 
 number five at all. We perceive this truth by intui- 
 tion, 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 
 principle 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 
 
THE AXIOMS WHICH RELATE TO NUMBER. 139 
 
 five, this is four and five. Mathematical reasoners 
 justify the first inference (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 
 general form in which we might more adequately ex- 
 press this condition of intuition would be this; that 
 ' Two magnitudes are equal when they can be divided 
 into parts which are equal, each to each.' Thus in 
 the above example, 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 con- 
 temptuously of, as empty and barren assertions. In 
 fact, however, although they cannot supply the defi- 
 ciency of the clear intuition 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 pro- 
 perly at the beginning of our treatises on the science 
 of quantity; since they express, as simply as words 
 can express, those conditions of the intuition of magni- 
 tudes on which all reasoning concerning quantity must 
 be based ; and are necessary when we want, not only 
 to see the truth of the elementary reasonings on these 
 subjects, but to put such reasonings in a formal and 
 logical shape. 
 
140 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 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 twelvr; that is, 7 tens, 
 i 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 prin- 
 ciples just noticed, the foundation of the whole of 
 Theoretical Arithmetic. 
 
CHAPTER X. 
 OF THE PERCEPTION OP TIME AND NUMBER. 
 
 i. /^VTJE, perception of the passage of time involves 
 \J 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 requisite 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 j for it will hardly be maintained that 
 the process by which we contemplate 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 j though in this case the 
 effort may be smaller, and consequently more easily 
 overlooked. In common cases, we are unconscious of 
 the act of thought by which we recollect the preceding 
 instant, though we perceive the effort when we recol- 
 lect 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 
 intervals 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 
 
142 PHILOSOPHY OF THE PURE SCIENCES. 
 
 end of each instant as it passes, may very fitly be dis- 
 tinguished in common cases from manifest acts of re- 
 collection, 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 
 successiveness; 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 mus- 
 cular sense we perceive ourselves existing in space. 
 And both our 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 passing. 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 successiveness, like the muscular 
 sense with which we have compared it, implies acti- 
 vity 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 ex- 
 ample, we look at a horse, and see that his legs are 
 four; and this we seem to do at once, without reckon- 
 ing 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 articulating our 
 words. And this is the more manifest, since we begin 
 our acquaintance with number by counting even the 
 
PERCEPTION OF TIME AND NUMBER. 143 
 
 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 persons 
 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, 
 therefore, that when we appear to catch a small num- 
 ber by a single glance of the eye, we do in fact count 
 the units of it in a regular, though very brief succes- 
 sion. To count requires an act of memory. Of this 
 we are sensible 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 l . 
 
 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 accompa- 
 nied by an act of the mind, the apprehension of 
 rhythm is so still more clearly. All the forms of ver- 
 sification and the measures of melodies are the creations 
 of man, who thus realizes in words and sounds the 
 
 i I have considered Number as in- and those cases of small numbers, in 
 
 volving the exercise of the sense of which we seem to see the number at 
 
 succession, because I cannot draw one glance. But if any one holds 
 
 any line between those cases of large Number to be apprehended by a direct 
 
 numbers, in which, the process of act of intuition, as Space and Time 
 
 counting being performed, there is a are, this view will not disturb the 
 
 manifest apprehension of succession ; other doctrines delivered in the text. 
 
144 PHILOSOPHY OF THE PURE SCIENCES. 
 
 forms of recurrence which rise within his own mind. 
 When we hear in a 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 insen- 
 sibly 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 construction that the return- 
 ing 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, obscure 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 requi- 
 site 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 appearances and collections of appear- 
 ances. 
 
NOTE TO CHAPTER X. 
 
 I HAVE in the last ten chapters described Space, Time, and Num- 
 ber by various expressions, all intended to point out their office 
 as exemplifying the Ideal Element of human knowledge. I have 
 called them Fundamental Ideas; Forms of Perception; Forms of 
 Intuition; and perhaps 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 ex- 
 traneous 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 sci- 
 ences, 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 the facts in our appre- 
 hension, 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 contem- 
 plate special cases of two lines inclosing a space, in which it re- 
 mains 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 per- 
 ceives 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 hu- 
 man existence in which the powers have not been awakened and 
 VOL. I. L 
 
146 PHILOSOPHY OF THE PURE SCIENCES. 
 
 developed by such exercise. In this way, experience and obser- 
 vation are necessary conditions and prerequisites of our appre- 
 hension of geometrical (and other) axioms. We cannot see the 
 truth of these axioms without some experience, because we can- 
 not 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. i. sect. 3, and therefore cannot 
 be the ground of necessary truths. 
 
 Another expression which may be used to describe the Fun- 
 damental Ideas here spoken of is suggested by the language of a 
 very profound and acute Review of the former edition. The Re- 
 viewer holds that we pass from special experiences to universal 
 truths in virtue of ' the inductive propensity the irresistible im- 
 pulse of the mind to generalize ad infinitum.' I have already 
 given reasons why I cannot adopt the former expression; but I 
 do not see why space, time, number, cause, and the rest, may 
 not be termed different forms of the impulse of the mind to genera- 
 lize. But if we put together all the Fundamental Ideas as results 
 of the Generalizing Impulse, we must still separate them as dif- 
 ferent 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. 
 
e, 
 
 CHAPTER XI. 
 
 OF MATHEMATICAL REASONING. 
 
 i. Discursive Reasoning. WE have thus seen that 
 our notions of space, time, and their modifications, 
 necessarily 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 
 reasonings 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, 
 intuitive. 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 is intuition, but by passing from one view to 
 another, like those who run from place to place (discur- 
 sus). 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 
 
 I, 2 
 
148 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 
 principles 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 conclu- 
 sion. 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 designates a syllogism in which one 
 of the premises is understood, 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 cen- 
 ter of a circle to the circumference are equal ; or intro- 
 ducing 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 anything is 
 required to be done, as to draw a line or a circle 
 under given conditions, this proposition is a Problem. 
 A theorem requires demonstration ; a problem, solution. 
 And for both purposes the mathematician usually 
 makes a Construction. 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. Sometimes, too, he establishes some 
 Lemma, or preparatory proposition, before he proceeds 
 to his main task; and often he deduces -from his de- 
 monstration 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 
 
OF MATHEMATICAL REASONING. 149 
 
 strange and unintelligible if we should have occasion 
 to use some of them. There is, however, one technical 
 distinction more peculiar, and more important. 
 
 3. Geometrical Analysis and Synthesis. In geome- 
 trical reasoning such as we have described, we intro- 
 duce at every step some new consideration ; and it is 
 by combining all these considerations, that we arrive 
 at the conclusion, that is, the demonstration of the pro- 
 position. Each step tends to the final result, by exhi- 
 biting 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 contributions from different quarters; 
 like a river fed and augmented 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 
 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 
 geometrical Analysis. Having succeeded in this ana- 
 lytical 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 
 
I5O PHILOSOPHY OF THE PURE SCIENCES. 
 
 exhaust the unexplored valleys, and finally obtain a 
 clear knowledge of the place whence the waters flow. 
 Analytical 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 Mathe- 
 matics hold so important a place in relation to the in- 
 ductive sciences, that I shall briefly notice the leading 
 ideas which the ulterior progress of the subject in- 
 volves. 
 
CHAPTER XII. 
 
 OF THE FOUNDATIONS OF THE HIGHER MATHEMATICS. 
 
 i. The Idea of a Limit. THE general truths con- 
 cerning 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 re- 
 specting the lengths of any portions of curve lines; 
 for these cannot be obtained by means of the princi- 
 ples 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 1 
 
 The Idea of a Limit supplies a new mode of esta- 
 blishing 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 car- 
 ried. And thus, we may do what is equivalent to 
 measuring the curve itself; for by multiplying the 
 
152 PHILOSOPHY OF THE PURE SCIENCES. 
 
 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 
 complex 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 com- 
 bination of other hypotheses and definitions 1 . The 
 axiom just noticed, 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, whether proved by the considera- 
 tion 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 
 
 i This assertion cannot be fully axiom. For if we take the supposed 
 proved and illustrated without a re- axiom, that a curve is always less 
 ference to mathematical reasonings than the including broken line, this is 
 which would not be generally intelli- not true, except with a condition ; 
 gible. I have shown the truth of the and in tracing the import of this con- 
 assertion in my Hioughts on the Study dition, we find its necessity becomes 
 of Mathematics, annexed to the Prin- evident only when we introduce a 
 ciples of English University Educa- reference to a Limit. And the same 
 tion. The proof is of this kind : is the case if we attempt to supersede 
 The ultimate equality of an arc of a the notion of a Limit in proving any 
 curve and the corresponding peri- other simple and evident proposition 
 phery of a polygon, when the sides of in which that notion is involved, 
 the polygon are indefinitely increased Therefore these evident truths are 
 in number, is evident. But this truth self-evident, in virtue of the Idea of a 
 cannot be proved from any other Limit. 
 
FOUNDATIONS OF HIGHER MATHEMATICS. 153 
 
 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 ob- 
 scure apprehension of principles equivalent to those of 
 the Method of Limits. Yet the necessary fundamental 
 principle not having, in their time, been clearly de- 
 veloped, their reasonings were both needlessly intricate 
 and imperfectly satisfactory. Moreover they were led 
 to put in the place of axioms, assumptions which were 
 by no means self-evident ; as when Archimedes as- 
 sumed, for the basis of his measure of the circumfer- 
 ence of the circle, the proposition that a circular arc 
 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 Caval- 
 leri, 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 intro- 
 duces, with a distinctness and exactness which re- 
 quired little improvement to make it as unimpeachable 
 as the demonstrations of geometry. And when such 
 processes as Newton thus deduced from the conception 
 of a Limit, are represented by means of general alge- 
 braical symbols instead of geometrical diagrams, we 
 have then before us the Method of Fluxions, 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 
 
154 PHILOSOPHY OF THE PURE SCIENCES. 
 
 results depend for their evidence upon a fundamental 
 conception, the conception of arbitrary symbols as the 
 Signs of quantity and its relations; and upon a cor- 
 responding 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 re- 
 present numbers considered with regard to their nu- 
 merical 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 ques- 
 tion proposed. This was the case, for example, when 
 the answer was a negative quantity ; for when Descartes 
 had introduced the mode of representing 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 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 impos- 
 sible quantities occur. Such processes may usually be 
 confirmed upon other principles, and the truth in 
 question may be established by means of a demonstra- 
 tion in which no such seeming fallacies defeat the 
 reasoning. But it has also been shown in many such 
 cases, that the process in which some of the steps ap- 
 pear 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 sym- 
 bolical 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 
 
FOUNDATIONS OF HIGHER MATHEMATICS. 155 
 
 some definite operation upon quantity. The absolute 
 universality of the interpretation of symbols is the fun- 
 damental 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 principle : 
 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 general in 
 their nature.' And thus, this universality of symbols 
 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 rea- 
 soning, go on adding carefully one known truth to 
 another, till we reach the desired result. On the con- 
 trary, if we have a theorem to prove or a problem to 
 solve which can be 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 just stated, or 
 what the unknown symbols must . signify ; resolving 
 step by step the symbolical 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 con- 
 nexion is so familiar, that the term analysis is fre- 
 quently used to designate symbolical reasoning. 
 
CHAPTER XIII. 
 
 THE DOCTRINE OF MOTION. 
 
 i. 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 
 of Mechanism, independent of force, and considered as 
 the solution of a problem which may be thus enun- 
 ciated: '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 problem 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 Mathematicians ; for example, 
 the various investigations and demonstrations which 
 have been published respecting the form of the Teeth 
 
THE DOCTRINE OF MOTION. 157 
 
 of Wheels, and Mr. Babbage's memoir l on the Lan- 
 guage of Machinery. There are also several works 
 which contain collections of the mechanical contriv- 
 ances 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 sciences, Mechanics Proper, or 
 Machinery, which considers the effect of force trans- 
 mitted by Mechanism from one part of a material com- 
 bination 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 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 Philosophie 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 
 considerations 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 example, as that of Carnot upon Motion con- 
 sidered Geometrically, and the essay of Lanz and Be- 
 tancourt upon the Composition of Machines.' He then 
 proceeds to describe this science nearly as we have 
 
 i On a Method of expressing by Signs the action of Machinery. Phil. Trans. 
 1826, p. 250. 
 
158 PHILOSOPHY OF THE PURE SCIENCES. 
 
 done, and proposes to term it Kinematics (Cinematique), 
 from /aV7ju,a, 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 2 , 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 Kepler endeavoured in vain to conjoin 
 with the knowledge of the motions of the heavenly 
 bodies, those true mechanical conceptions which con- 
 verted formal into physical astronomy 3 . 
 
 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 ex- 
 tended 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 de- 
 scribed, 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 re- 
 specting their bearing on Inductive Science in general. 
 
 8 P. 528. Hist. Indue. Sc. ii. 130. 
 
CHAPTER XIV. 
 
 OF THE APPLICATION OF MATHEMATICS TO THE 
 INDUCTIVE SCIENCES. 
 
 i. A LL objects in the world which can be made the 
 JL\_ 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 de- 
 partment of our investigations of nature. And there 
 is a peculiarity in these Ideas, which has caused the 
 mathematical sciences to be, in all cases, the first suc- 
 cessful efforts of the awakening speculative powers of 
 nations at the commencement of their intellectual pro- 
 gress. Conceptions 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 inquiries. 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 contemporaries, and only became clear enough 
 for purposes of sound scientific reasoning in the suc- 
 ceeding century : the just conception of chemical com- 
 position 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 worth- 
 less. The writers did not see any evidence in what we 
 now consider as the axioms of the science ; nor any in- 
 consistency where we now see self-contradiction. But 
 this was never the case with speculations concerning 
 
l6o PHILOSOPHY OF THE PURE SCIENCES. 
 
 space and number. From their first rise, these were 
 true as far as they went. The Geometry and Arith- 
 metic of the Greeks and Indians, even in their first 
 and most scanty form, contained none but true propo- 
 sitions. Men's intuitions upon these subjects never al- 
 lowed them to slide into error and confusion ; 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 
 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 1 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 2 , he de- 
 clares mathematical knowledge .to be the first and main 
 requisite for the astronomer, and describes the portions 
 of it which he holds necessary for astronomical specu- 
 lators to cultivate. These seem to be, Plane Geometry, 
 Theoretical Arithmetic, the Application of Arithmetic 
 
 Hist. Ind. Sc. b. iii. c. ii. - Epinomis, p. 99' 
 
INDUCTIVE APPLICATION OF MATHEMATICS. l6l 
 
 to planes and to solids, and finally the doctrine of Har- 
 monics. Indeed the bias of Plato appears to be rather 
 to consider mathematics as the essence of the science 
 of astronomy, than as its instrument; and he seems 
 disposed, in this as in other things, to disparage obser- 
 vation, and to aspire after a science founded upon de- 
 monstration 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 set- 
 tings and risings of the stars ; but he must be one who 
 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 di- 
 rected towards Astronomy. Besides many curious pro- 
 positions of plane and solid Geometry, to which their 
 astronomers were led, their Arithmetic, though very 
 inconvenient in its fundamental assumptions (as being 
 sexagesimal not decimal), 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 establish- 
 
 VOL. I. M 
 
1 62 PHILOSOPHY OF THE PURE SCIENCES. 
 
 ing that system of the celestial motions which suc- 
 ceeded the Platonic scheme of cycles and epicycles. 
 If the properties of the conic sections had not been 
 demonstrated by the Greeks, and thus rendered fami- 
 liar to the mathematicians of succeeding ages, Kepler 
 would probably not have been able to discover those 
 laws respecting the orbits and motions of the planets 
 which were the occasion 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, i, 2, 3, 4, 5, 6, 7, 8, 9, o, 
 and by the method of Local Values, instead of the 
 cumbrous sexagesimal arithmetic of the Greeks, was 
 an improvement by which the convenience and facility 
 of numerical calculations were immeasurably aug- 
 mented. 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 im- 
 provement which Albategnius appears to claim for 
 himself 3 ; and by employing also the Tangents of arcs, 
 or, as they called them 4 , 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. 
 Newton's Method of Prime and Ultimate Ratios, 
 which we have already noticed as the first correct ex- 
 position of the doctrine of a Limit, is stated in a series 
 of Lemmas, or preparatory theorems, prefixed to his 
 Treatise on the System oj the World. Both the proper- 
 ties of curve lines and the doctrines concerning force 
 and motion, which he had to establish, required that 
 the common mathematical processes should be metho- 
 dized and extended. If Newton had not been a most 
 
 Delambre, Ast., M. A,, p. 12. * Ibid. p. 17. 
 
INDUCTIVE APPLICATION OF MATHEMATICS. 163 
 
 expert and inventive mathematician, as well as a pro- 
 found and philosophical thinker, he could never have 
 made any one of those vast strides in discovery of 
 which the rapid succession in his work strikes us with 
 wonder 5 . And if we see that the great task begun by 
 him, goes on more slowly in the hands of his imme- 
 diate 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 synthetical modes of investiga- 
 tion, as we have elsewhere observed, were an instru- 
 ment 6 , 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 Continent, the ad- 
 vantages offered by a familiar use of symbols, and by 
 attention to their symmetry and other relations, were 
 accepted without reserve. In this manner the Differ- 
 ential 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 physi- 
 cal astronomy were reduced to difficulties of symbolical 
 calculation, these, indeed, being often sufficiently stub- 
 born. Clairaut, Euler, and D'Alembert employed the 
 increased 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 
 
 Hist. Ind. Sc. b. vii. c. ii. Ibid. p. 175. 
 
 M 2 
 
164 PHILOSOPHY OF THE PURE SCIENCES. 
 
 are still labouring up to the present day. This course 
 was not without its checks and perplexities. We have 
 elsewhere quoted 7 Clairaut's expression when he had 
 obtained the very complex differential equations which 
 contain the solution of the problem of the moon's mo- 
 tion: 'Now integrate them who can!' But in no 
 very long time they were integrated, at least approxi- 
 mately; and the methods of approximation have since 
 then been improved ; so that now, with a due expendi- 
 ture of labour, they may be carried to any extent which 
 is thought desirable. If the methods of astronomical 
 observation should hereafter 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 universal 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 observa- 
 tions. 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 Pro- 
 fessor Airy has calculated the law and amount of an 
 inequality depending upon the mutual attraction of 
 the Earth and Yenus ; 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 pro- 
 gress of astronomy is thus seen in the cases in which 
 theory and observation confirm each other, so this in- 
 fluence appears in another way, in the very few cases 
 in which the 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 re- 
 sult of the attraction of the sun and moon. Nor can 
 there be any doubt that this is true, as a general state- 
 ment; yet the subject is up to the present time a blot 
 
 Hist. Ind. Sc. b. vi. c. vi. sect. 7- 
 
INDUCTIVE APPLICATION OF MATHEMATICS. 165 
 
 on the perfection of the theory of universal gravita- 
 tion ; 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 phenomenon at every place 
 on the earth's surface. And what is the portion of our 
 mathematics which is connected with this solitary sig- 
 nal 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 ma- 
 thematicians, Newton, Maclaurin, Bernoulli, Clairaut, 
 Laplace, to master such questions, all involve some 
 gratuitous 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 probably precisely the absence of a theory which 
 could be reasonably expected to agree with the obser- 
 vations, 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 obser- 
 vations have been pursued, and their results have been 
 resolved into empirical laws, so that the rules of the 
 phenomena have been ascertained, although the de- 
 pendence of these rules upon the lunar and solar forces 
 has not been shown. Here then we have a portion of 
 our knowledge relating to facts undoubtedly dependent 
 upon universal gravitation, in which Observation has 
 outstripped Theory in her progress, 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 
 
1 66 PHILOSOPHY OF THE PURE SCIENCES. 
 
 extension and generalization of the Laws of Motion 
 could never have been effected, unless the Differential 
 and Integral Calculus 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 8 , if it did not prove 
 the law, at least reduced the application of it to analy- 
 tical processes which could be performed in most of 
 those cases in which they were needed. In many in- 
 stances the demands of mechanical science suggested 
 the extension of the methods of pure analysis. The 
 problem of Yibrating Strings gave rise to the Calculus 
 of Partial Differences, which was still further stimu- 
 lated 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 re- 
 markable of new analytical methods, to which me- 
 chanical 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 
 followed 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 
 endowments, 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 
 mathematics may bear upon physical science, by call- 
 ing to mind a circumstance already noticed in the 
 History of Science 9 ; that Fresnel obtained one of the 
 
 8 Hist. Ind. Sc. b. vi c. vi. sect. 7. 
 Hist. Ind. Sc. b. ix. c. xiiL sect. 2. 
 
INDUCTIVE APPLICATION OP MATHEMATICS. l6/ 
 
 most curious confirmations of the theory (the laws of 
 Circular Polarization by reflection) through an inter- 
 pretation of an algebraical expression, which, according 
 to the original conventional meaning of the symbols, 
 involved an impossible quantity. We have already 
 remarked, that in virtue of the principle of the gene- 
 rality 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 mathema- 
 tics, that we can hardly be surprised if some persons 
 have considered mathematical reasoning to be the most 
 essential part of such sciences; and have overlooked 
 the other elements which enter into their formation. 
 How erroneous this view is we shall best see by turn- 
 ing our attention to the other Ideas besides those of 
 space, number, 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 development of such Ideas is quite as neces- 
 sary to the progress of exact and real knowledge as an 
 acquaintance with arithmetic and geometry. 
 
BOOK III. 
 
 THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 MECHANICAL SCIENCES. 
 
IT is only because we subject trains of phenomena, that is, 
 all change whatever, to the law of causality to the relation of 
 cause and effect that experience or empirical knowledge becomes 
 possible. 
 
 KANT, Kr. d. R. V. n Th. i Abth. n Buch. 2 Haupt. 
 
 Quicquid premit vel trahit alterum, tantundem ab eo premitur 
 vel trahitur...Si corpus aliquod in corpus aliud impingens motum 
 ejus vi sua" quomodocunque mutaverit, idem quoque vicissim in 
 motu proprio eandern mutationeni in partem contrariam vi alte- 
 rius (ob sequalitatem pressionis, mutuse) subibit...0btinet etiam 
 hsec Lex in attract: onibus. 
 
 NEWTON, Princip. ad init. 
 
n .\ 
 
 BOOK in. ' 
 
 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 import- 
 ant place; coming into notice immediately after those 
 parts of Astronomy which require for their cultivation 
 merely the ideas of space, time, motion, and number. 
 It appears from our History, that certain truths con- 
 cerning the equilibrium 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. More- 
 over, 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 Hydrostatics. And the 
 doctrines of Mechanics have been found to have a 
 most remarkable bearing upon the motions of the hea- 
 venly bodies; with reference to which, indeed, they 
 were at first principally studied. The explanation 
 of those cosmical facts by means of mechanical princi- 
 
IJ2 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 pies and their consequences, forms the science of Phy- 
 sical Astronomy. These are the principal examples of 
 mechanical science; although some other portions of 
 Physics, as Magnetism and Electrodynamics, introduce 
 mechanical doctrines very largely into their specula- 
 tions. 
 
 Now in all these sciences we have to consider 
 .Forces. In all mechanical reasonings forces enter, 
 either as producing 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 pressure and attractive force. Our business at 
 present is only with such causes as immediately ope- 
 rate upon matter. We shall nevertheless, in the first 
 place, consider 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. 
 
 OP THE IDEA OF CAUSE. 
 
 i. ~V^7E see i n * ne world around us a constant suc- 
 T f 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. 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 experience 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 
 Cause of Motion. That this Idea of Cause is not 
 derived from experience, we prove (as in former cases) 
 by this consideration: 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 proposition we know to be true, not 
 only probably, and generally, and as far as we can see : 
 
174 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 pro- 
 duce what effects ; what is the cause of any particu- 
 lar event; what will be the effect of any peculiar 
 process; these are points on which experience may 
 enlighten us. Observation and experience may be 
 requisite, to enable us to judge respecting such matters. 
 But that every event has some cause, Experience can- 
 not 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 observation. 
 
 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 obser- 
 vation can detect in these appearances the power or 
 quality which we call Cause. Cause is that which con- 
 nects one event with another; but no sense or percep- 
 tion discloses to us, or can disclose, any connexion 
 
OF -THE IDEA OF CAUSE. 175 
 
 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 fol- 
 low another. We have already noticed 1 , that this 
 truth has been urged by metaphysicians 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 neces- 
 sity; 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 therefore 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 example of such a connexion. 
 There is, then, no case in which events can be ob- 
 served to be necessarily connected : our idea of causa- 
 tion, which implies that the event is necessarily con- 
 nected with the cause, cannot be derived from observa- 
 tion. 
 
 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. 
 
 Book i. chap. xiL 
 
176 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 We see nothing but the succession of events ; and by 
 cause we mean nothing but a certain succession of 
 events; namely, a constant, unvarying succession. 
 Cause and effect are only two events of which the 
 second invariably follows the first. We delude our- 
 selves when we imagine that our idea of causation 
 involves anything more. 
 
 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 lan- 
 guage 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 necessity? Why must like events always be pre- 
 ceded 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 equivalent operation. All will allow that in this 
 sense, every kind of event must have some other spe- 
 cific kind of event preceding it. But this turn of 
 men's thoughts shows that they see in events a con- 
 nexion 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 occasion; it is a power, an efficacy, 
 which has a real operation. 
 
 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 experience, and which involves more than mere 
 succession. Similar arguments might be derived from 
 any other maxims of universal and necessary validity, 
 which we can obtain concerning Cause: as, for ex- 
 ample, the maxims that Causes are measured by their 
 Effects, and that Reaction is equal and opposite to 
 
OF THE IDEA OF CAUSE. 177 
 
 Action. These maxims we shall soon have to ex- 
 amine; 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 j while their meaning, including, as it does, 
 something quite different from the mere conception of 
 succession of events, proves that such a conception 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. 
 
 VOL. I. 
 
CHAPTER III. 
 
 MODERN OPINIONS RESPECTING THE IDEA OP CAUSE. 
 
 i. rTIO WARDS the end of the seventeenth century 
 _L there existed in the minds of many of the most 
 vigorous and active speculators of the European lite- 
 rary world, a strong tendency to ascribe the whole of 
 our Knowledge to the teaching of Experience. This 
 tendency, with its consequences, including among them 
 the reaction which was produced when the tenet had 
 been pushed to a length manifestly absurd, has exer- 
 cised a very powerful influence upon the progress of 
 metaphysical doctrines up to the present time. I pro- 
 ceed to notice some of the most prominent of the 
 opinions which have thus obtained 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 ex- 
 clusive dependence of our knowledge upon experience. 
 Agreeably to this general system, he taught 1 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 
 reasoning 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 2 . 
 
 Hume noticed this obvious inconsistency; but de- 
 clared 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 suc- 
 cession of cause and effect, nothing but the habit of 
 associating in our minds what had often been asso- 
 
 1 Essay on the Human Understanding, b. ii. c. xxvi 2 B. iv. c. x. 
 
OPINIONS RESPECTING THE IDEA OF CAUSE. 179 
 
 elated 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 connexion; and 
 therefore we have no means of inferring cause from 
 effect, or effect from cause 3 . The only way in which 
 we recognize Cause and Effect in the field of our ex- 
 perience, is as an unfailing Sequence : we look in vain 
 for anything which can assure us of an infallible Con- 
 sequence. 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 skepticism of its author on religious points, 
 produced a considerable fermentation in the specula- 
 tive 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 familiar arguments con- 
 cerning the necessity of an original cause of all that is 
 and happens. Accordingly these hostile and appa- 
 rently irreconcilable doctrines, the indispensable ne- 
 cessity 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 con- 
 stantly 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 
 
 3 Hume's Phil of the Human Mind, voL i. p. 94. 
 
180 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 maxim that "Every event must have a cause," to be 
 an instinctive law of belief, or a fundamental principle 
 of the human mind 4 . 
 
 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 
 speculations 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. 
 
 Kant's solution of the difficulties just mentioned 
 differs materially from that above stated. According 
 to Brown 5 , succession observed and cause inferred, 
 the memory of past conjunctions of events and the 
 belief of similar future conjunctions, are facts, inde- 
 pendent, 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 connexion in events is requisite to our 
 apprehending them as events. Future occurrences 
 must be connected by causation 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 Causa- 
 tion is a condition under which we think of events, 
 as the relations of space are a condition under which 
 we see objects. 
 
 4. On a subject so abstruse, it is not easy to make 
 our distinctions very clear. Some of Brown's illustra- 
 tions appear to approach very near to the doctrine of 
 Kant. Thus he says 6 , l 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 notwithstanding such approximations in expres- 
 sion, the Kantian doctrine appears to be different from 
 
 4 Stewart's Active Powers, vol. L p. 347. Browne's Lectures, vol. L p. 115. 
 5 Lectures, voL i. p. 114. 6 Lectures, voL i. p. 127. 
 
OPINIONS RESPECTING THE IDEA OF CAUSE. l8l 
 
 the views of Stewart and Brown, as commonly under- 
 stood. According 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 ab- 
 sent, 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 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 repre- 
 sents cause as a relation of the same kind with form, 
 do not appear exactly to fit his opinions. Can the re- 
 lations of figure be properly said to be connected with 
 each other by a law of our nature, or a tendency of our 
 mental constitution? 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 
 constitution 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 
 
1 82 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 make something different from five; so that an in- 
 closed 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 numbers 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 inappropriata 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 
 existence 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 motion 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 know- 
 ledge. 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 know- 
 ledge. We have, in the Mechanical Sciences, certain 
 universal and necessary truths on the subject of causes. 
 Now any view which refers our belief in causation to 
 mere experience or habit, cannot explain the possi- 
 bility of such necessary truths, since experience and 
 habit can never lead to a perception of necessary con- 
 nexion. But a view which teaches us to acknowledge 
 axioms concerning cause, as we acknowledge axioms 
 
OPINIONS RESPECTING THE IDEA OF CAUSE. 183 
 
 concerning space, will lead us to look upon the science 
 of mechanics as equally certain and universal with the 
 science of geometry; and will thus materially 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 
 number or time. And thus the propositions which 
 constitute the science of Mechanics prove that we pos- 
 sess an idea of cause, in the same sense in which the 
 propositions of geometry and arithmetic prove our pos- 
 session 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 ap- 
 prehension of cause by reference to habit, or to a sup- 
 posed 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 ac- 
 tivity, a source of necessary truths ; or, to sum all this 
 in one phrase, a Fundamental Idea. 
 
CHAPTER IV. 
 
 OF THE AXIOMS WHICH RELATE TO THE IDEA OF 
 
 CAUSE. 
 
 i. Causes are abstract Conceptions. WE have now 
 to express, as well as we can, the fundamental charac- 
 ter 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 eflicacy, 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 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 185 
 
 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 supposi- 
 tions which we cannot for a moment admit. That the 
 connexion of effect with cause is universal and neces- 
 sary, is a universal and constant conviction of man- 
 kind. 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 apprehended, the axiom 
 requires no proof, and indeed admits of none which 
 makes it more evident. That notwithstanding its sim- 
 plicity, it is of use in our speculations, we shall here- 
 after see ; but in the first place, we must consider 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 
 no way fall directly under the cognizance of the senses. 
 Cause, of whatever kind, is not apprehended as in- 
 cluding objects and events which share its nature by 
 being co-extensive with certain portions of it, as space 
 and time are. It cannot therefore, like them, be 
 
1 86 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 measured by repetition 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 in- 
 stance, the force of a man is greater than the force of 
 a child. But how much is the one greater than the 
 other 1 ? How are we to compare the abstract concep- 
 tion, 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 things operated on, (a ques- 
 tion 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 modi- 
 fication of the effect corresponding to it. Hence the 
 effect is an unfailing index of the amount of the cause ; 
 and if it be a measurable 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 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 com- 
 pare, any of the effects might be taken. The sun's 
 heat on different days might be measured by the ex- 
 pansion 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 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 187 
 
 support, or by the spaces through which a body 
 would fall in a given time. All these measures are 
 consistent 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 ascer- 
 tained that the whole weight bends it through a space 
 equal to the sum of the inflections 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 beforehand, 
 be more or less than twice as great as the inflection 
 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 
 produce 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 measure whatever. By this means, and by this 
 
1 88 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 means only, we have a complete and consistent mea- 
 sure 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 conceived as capable of addition; and what has 
 been said of the measure of causes in such cases, ap- 
 plies 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 direction 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 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 appli- 
 cation in that science. At present we have, perhaps, 
 sufficiently explained the axiom that causes are mea- 
 sured 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 
 

 AXIOMS RELATING TO THE IDEA OF CAUSE. 189 
 
 the first. In these cases the operation is mutual ; the 
 Action is accompanied 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 press- 
 ing 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, deter- 
 mine each other by a common rule. But this con- 
 sideration leads necessarily to their equality : for since 
 the rule is mutual, if we could for an instant suppose 
 the Reaction to be less than the Action, we must, by 
 the same rule, suppose 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. Accordingly, 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 Questions concerning the 
 communication of motion, that ' Reaction is equal and 
 
190 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 contrary to Action.' The same principle was taken 
 for granted by all parties, in all the controversies con- 
 cerning 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 interpretations of 
 the measure of the term * Action' we were to take. 
 
 6. Extent of the Third Axiom. It may naturally 
 be asked whether this third Axiom respecting causa- 
 tion extends to any other cases than those of mechani- 
 cal action, since the notion of Cause in general has 
 certainly a much wider extent. For instance, when 
 a hot body 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 ex- 
 erted is an abstract term which may be variously 
 understood. The hot hand may melt a cool body, or 
 may warm it : which kind of effect is to be taken as 
 action 1 ? This remains to be determined by other con- 
 siderations. 
 
 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 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 191 
 
 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 
 manifestly mechanical. 
 
 The three axioms of causation which we have now 
 stated are the fundamental maxims of all reasoning 
 concerning 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, ex- 
 tent, 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. 
 
 [2d Ed.] [The Axiom that Reaction is equal and 
 opposite to Action, may appear to be at variance with 
 a maxim concerning Cause which is commonly current ; 
 namely, that the ' Cause precedes Effect, and Effect fol- 
 lows Cause.' For it may be said, if A, the Action, and 
 jR, the Reaction, can be considered as mutually the cause 
 of each other, A must precede ./, and yet must follow 
 it, which is impossible. But to this I reply, that in 
 those 'cases of direct Causation to which the maxim 
 applies, the Cause and Effect are not successive, but 
 simultaneous. If I press against some obstacle, the 
 obstacle resists and returns the pressure at the instant 
 it is exerted, not after any interval of time, however 
 small. The common maxim, that the effect follows the 
 cause, has arisen from the practice of considering, as 
 examples of cause and effect, not instantaneous forces 
 or causes, and the instantaneous changes which they 
 produce; but taking, instead of this latter, the cumu- 
 lative effects produced in the course of time, and com- 
 pared with like results occurring without the action of 
 the cause. Thus, if we alter the length of a clock-pen- 
 dulum, this change produces, as its effect, a subsequent 
 change of rate in the clock : because the rate is mea- 
 sured by the accumulated effects of the pendulum's 
 gravity, before and after the change. But the pendu- 
 lum produces its mechanical effect upon the escape- 
 ment, at the moment of its contact, and each wheel 
 upon the next, at the moment of its contact. As has 
 
PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 been said in a Review of this work, l The time lost in 
 cases of indirect physical causation is consumed in the 
 movements which take place among the parts of the 
 mechanism in action, by which the active forces so 
 transformed into momentum are transported over in- 
 tervals of space to new points of action, the motion of 
 matter in such cases being regarded as a mere carrier 
 of force.' (Quarterly Rev. No. cxxxv. p. 212.) 
 
 This subject I have further treated in the Memoirs 
 of the Cambridge Philosophical Society, vol. vii. part iii.] 
 [In this Third Edition I add this discussion.] 
 
 Discussion of the Question: Are Cause and Effect successive 
 or simultaneous? 
 
 I HAVE at various times laid before this Society dissertations 
 on the metaphysical grounds and elements of our knowledge, and 
 especially on the foundations of the science of mechanics. As these 
 speculations have not failed to excite some attention, both here 
 and elsewhere, I am tempted to bring forward in the same manner 
 some additional disquisitions of the same kind. Indeed, the im- 
 mediate occasion of the present memoir is of itself an evidence 
 that such subjects are not supposed to be without their interest 
 for the general reader ; for I am led to the views and reasonings 
 which I am now about to lay before the Society, by some remarks 
 in one of our most popular Reviews, (The Quarterly Review, 
 Article on the History and Philosophy of the Inductive Sciences, 
 June 1841). A writer of singular acuteness and comprehensive- 
 ness of view has there made remarks upon the doctrines which 
 I had delivered in the Philosophy of the Inductive Sciences, 
 which remarks appear to me in the highest degree instructive 
 and philosophical. I am not, however, going here to discuss 
 fully the doctrines contained in this critique. With respect to its 
 general tendency, I will only observe, that the author does not 
 accept, in the form in which I had given it, the account of the 
 origin and ground of necessary and universal truths. I had 
 stated that our knowledge is derived from Sensations and Ideas ; 
 and that Ideas, which are the conditions of perception, such as 
 space, time, likeness, cause, make universal and necessary know- 
 ledge possible ; whereas, if knowledge were derived from Sensa- 
 tion alone, it could not have those characters. I have moreover 
 
I 
 
 AXIOMS RELATING TO THE IDEA OF CAUSE. 193 
 
 enumerated a long series of Fundamental Ideas as the bases of a 
 corresponding series of sciences, of which sciences I have shown 
 also, by an historical survey, that they claim to possess universal 
 truths, and have their claims allowed. I have gone further : for 
 I have stated the Axioms which flow from these Fundamental 
 Ideas, and which are the logical grounds of necessity and univer- 
 sality in the truths of each science, when the science is presented 
 in the form of a demonstrated system. The Reviewer does not 
 assent to this doctrine, nor to the argument by which it is sup- 
 ported ; namely, that Expeiience cannot lead to universal truths, 
 except by means of a universal Idea supplied by the mind, and 
 infused into the particular facts which observation ministers. 
 He considers that the existence of universal truths in our know- 
 ledge may be explained otherwise. He holds that it is a suffici- 
 ent account of the matter to say that we pass from special expe- 
 rience to universal truth in virtue of 'the inductive propensity 
 the irresistible impulse of the mind to generalize ad infinitum.' 
 I shall not here dwell upon very strong reasons which may be 
 assigned, as I conceive, for not accepting this as a full and satis- 
 factory explanation of the difficulty. Instead of doing so, I 
 shall here content myself with remarking, that even if we adopt 
 the Reviewer's expressions, we must still contend that there are 
 different forms of the impulse of the mind to generalize, correspond- 
 ing to each of the Fundamental Ideas of our system. These 
 Fundamental Ideas, if they be nothing else, must at least be 
 accepted as a classification of the modes of action of the Induc- 
 tive Propensity, as so many different paths and tendencies of 
 the Generalizing Impulse : and the Axioms which I have stated 
 as the express results of the Fundamental Ideas, and as the steps 
 by which those Ideas make universal truths possible, are still no 
 less worthy of notice, if they are stated as the results of our 
 Generalizing Impulse ; and as the steps by which that Impulse, 
 in its many various forms, makes universal truths possible. The 
 Generalizing Impulse in that operation by which it leads us to 
 the Axioms of Geometry, and to those of Mechanics, takes very 
 different courses ; and these courses may well deserve to be sepa- 
 rately studied. And perhaps, even if we accept this view of 
 the philosophy of our knowledge, no simpler or clearer way can 
 be found of describing and distinguishing these fundamentally 
 different operations of the 'Inductive Propensity, than by saying, 
 VOL. I. O 
 
IQ4 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 that in the one case it proceeds according to the Idea of Space, 
 in another according to the Idea of Mechanical Cause ; and the 
 like phraseology may be employed for all the other cases. 
 
 This then being understood, my present object is to consider 
 some very remarkable, and, as appears to me, novel views of the 
 Idea of Cause which the Reviewer propounds. And these may 
 be best brought under our discussion by considering them as an 
 attempt to solve the question, Whether, according to our funda- 
 mental apprehensions of the relation of Cause and Effect, effect 
 follows cause in the order of time, or is simultaneous with it. 
 
 At first sight, this question may seem to be completely decided 
 by our fundamental convictions respecting cause and effect, and 
 by the axioms which have been propounded by almost all writers, 
 and have obtained universal currency among reasoners on this 
 subject. That the cause must precede the effect, that the effect 
 must follow the cause, are, it" might seem, self-evident truths, 
 assumed and assented to by all persons in all reasonings in which 
 those notions occur. Such a doctrine is commonly asserted in 
 general terms, and seems to be verified in all the applications 
 of the idea of cause. A heavy body produces motion by its 
 weight ; the motion produced is subsequent in time to the pres- 
 sure which the weight exerts. In a machine, bodies push or 
 strike each other, and so produce a series of motions ; each motion, 
 in this case, is the result of the motions and configurations which 
 have preceded it. The whole series of such motions employs 
 time; and this time is filled up and measured by the series of 
 causes and effects, the effects being, in their turn, causes of other 
 effects. This is the common mode of apprehending the universal 
 course of events, in which the chain of causation, and the pro- 
 gress of time, are contemplated as each the necessary condition 
 and accompaniment of the other. 
 
 But this, the Critic remarks, is not true in direct causation. ' If 
 the antecedence and consequence in question be understood as the 
 interposition of an interval of time, however small, between the 
 action of the cause and the production of the effect, we regard it 
 as inadmissible. In the production of motion by force, for in- 
 stance, though the effect be cumulative .with continued exertion 
 of the cause, yet each elementary or individual action is, to our 
 apprehension, instanter accompanied with its corresponding incre- 
 ment of momentum in the body moved. In all dynamical rea- 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 195 
 
 sonings no one has ever thought of interposing an instant of time 
 between the action and its resulting momentum; nor does it 
 appear necessary.' This is so evident, that it appears strange it 
 should have the air of novelty ; yet, so far as I am aware, the 
 matter has never before been put in the same point of view. But 
 this being the case, the question occurs, how it is that time seems 
 to be employed in the progress from cause to effect? How is it 
 that the opinion of the effect being subsequent to the cause has 
 generally obtained ? And to this the Critic's answer is obvious : 
 it is so in cases of indirect or of cumulative effect. If a. ball A 
 strikes another, B, and puts it in motion, and B strikes *C, and 
 puts it in motion, A's impact may be considered as the cause, 
 though not the direct cause, of C's motion. Now time, namely 
 the time of B's motion after it is struck by A, and before it 
 strikes C, intervenes between ^4's impact and the beginning of 
 (7's motion : that is, between the cause and its effect. In this 
 sense, the effect is subsequent to the cause. Again, if a body be 
 put in motion by a series of impulses acting at finite intervals of 
 time, all in the same direction, the motion at the end of all these 
 intervals is the effect of all the impulses, and exists after they 
 have all acted. It is the accumulated effect, and subsequent to 
 each separate action of the cause. But in this case, each impulse 
 produces its effect instantaneously, and the time is employed, not 
 in the transition from any cause to its effect, but in the intervals 
 between the action of the several causes, during which intervals 
 the body goes on with the velocity already communicated to it. 
 In each impulse, force produces motion : and the motion goes on 
 till a new change takes place, by the same kind of action. The 
 force may be said, in the language employed by the Critic, to be 
 transformed into momentum; and in the successive impulses, 
 successive portions of force are thus transformed; while in the 
 intervening intervals, the force thus transformed into momentum 
 is carried by the body from one place to another, where a new 
 change awaits it. ' The cause is absorbed and transformed into 
 effect, and therein treasured up.' Hence, as the Writer says, 'The 
 time lost in cases of indirect physical causation is that consumed 
 in the movements which take place among the parts of the 
 mechanism set in action, by which the active forces , so trans- 
 formed into mechanism are transported over intervals of space to 
 new points of action, the motion of matter in such cases being 
 
 O2 
 
196 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 regarded as a mere carrier of force' : and when force is directly 
 counteracted by force, their mutual destruction must be con- 
 ceived, as the Eeviewer says, to be instantaneous. We can 
 therefore hardly resist his conclusion, that men have been misled 
 in assuming sequence as a feature in the relation of cause and 
 effect ; and we may readily assent to his suggestion, that sequence, 
 when observed, is to be held as a sure indication of indirect 
 action, accompanied with a movement of parts. 
 
 But yet if we turn for a moment to other kinds of causation, 
 we seem to be compelled at every step to recognize the truth of 
 the usual maxim upon this subject, that effects are subsequent to 
 causes. Is not poison, taken at a certain moment, the cause of 
 disorder and death which follow at a subsequent period ? Is not a 
 man's early prudence often the cause of his prosperity in later 
 life, and his folly, though for a moment it may produce gratifica- 
 tion, finally the cause of his ruin? And even in the case of 
 mechanism, if, in a clock which goes rightly, we alter the length 
 of the pendulum, is not this alteration the cause of an alteration 
 which afterwards takes place in the rate of the clock's going? 
 Are not all these, and innumerable other cases, instances in which 
 the usual notion of the effect following the cause is verified ? and 
 are they not irreconcileable with the new doctrine of cause and 
 effect being simultaneous? 
 
 In order to disentangle this apparent confusion, let us first con- 
 sider the case last mentioned, of a clock, in wuich some alteration 
 is made which affects the rate of going. 
 
 So long as the parts of the clock remain unaltered, its rate will 
 remain unaltered ; and any part which is considered as capable of 
 alteration, may be considered as, if we please, the cause of the 
 unaltered rate, by being itself unaltered. But we do not usually 
 introduce the positive idea of cause, to correspond with this nega- 
 tion of change. If we speak of the rate as unaltered, we may 
 also say that it is so because there is no cause of alteration. The 
 steady rate is the indication of the absence of any cause of 
 alteration ; and the rate of going measures the progress of time, 
 in a state of things in which causes of change are thus excluded. 
 If an alteration takes place in any part of the clock, once for 
 all, the rate is altered ; but the new rate is steady as the old rate 
 was, and, like it, measures the uniform progress of time. But 
 the difference between the new rate and the old is occasioned by 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 197 
 
 the difference of the parts of the clock ; and the new rate may 
 very properly be said to be caused by the change of the parts, 
 and to be subsequent to it : for it does prevail after the change, 
 and does not prevail before. 
 
 But how is this view to be reconciled with the one just quoted 
 from the Keviewer, and, as it appeared, satisfactorily proved 
 by him ; according to which all mechanical effects are simul- 
 taneous with their causes, and not subsequent to them? We 
 have here the two views in close contact, and in seeming oppo- 
 sition. 
 
 In the going of a clock, the parts are in motion ; and these 
 motions are determined by forces arising from the form and 
 connexion of the parts of the mechanism. Each of the forces 
 thus exerted at any instant produces its effect at the same 
 instant; and thus, so far as the term cause refers to such 
 instantaneous forces, the cause and the effect are simultaneous. 
 But if such instantaneous forces act at successive intervals of 
 time, the motion during each interval is unaltered, and by it3 
 uniform progress measures the progress of time. And thus the 
 motion of the machine consists of a series of intervals, during 
 each of which the motion is uniform, and measures the time; 
 separated from each other by a series of changes, at each of 
 which the change measures the instantaneous force, and is simul- 
 taneous with it. And if, in this case, we suppose, at any point 
 of time, the instantaneous forces to cease, the succession of them 
 being terminated, from that point of time the motion would be 
 uniform. And since the rate of the motion in each interval 
 of time is determined by the instantaneous force which last 
 acted and by the preceding motion, the rate of the motion in 
 each interval of time is determined by all the preceding instan- 
 taneous forces. Hence, when the series of instantaneous forces 
 stops, the rate at which the motion goes on permanently, from 
 that point of time, is determined by the antecedent series of 
 such forces, which series may be considered as an aggregate 
 cause ; and hence it appears, that the permanent effect is deter- 
 mined by the aggregate cause; and in this sense the effect is 
 subsequent to the cause. 
 
 Thus we obtain, in this case, a solution of the difficulty which 
 is placed before us. The instantaneous effect or change is simul- 
 taneous with the instantaneous force or cause by which it is 
 
198 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 produced. But if we consider a series of such instantaneous 
 forces as a single aggregate cause, and the final condition as 
 a permanent effect of this cause, the effect is subsequent to 
 the cause. In this case, the cause is immediately succeeded 
 by the effect. The cause acts in time: the effect goes on in 
 time. The times occupied by the cause and by the effect succeed 
 each other, the one ending 'at the point of time at which the 
 other begins. But the time which the cause occupies is really 
 composed of a series of instants of uniform motion interposed 
 between instantaneous forces; and during the time that this 
 series of causes is going on, to make up the aggregate cause, 
 a series of effects is going on to make up the final effect. There 
 is a progressive cause and a progressive effect which go on 
 together, and occupy the same finite time; and this simultaneous 
 progression is composed of all the simultaneous instantaneous 
 steps of cause and effect. The aggregate cause is the sum of 
 the progression of causes; the final effect is the last term of 
 the progression of effects. At each step, as the Reviewer says, 
 cause is transformed into effect; and it is treasured up in the 
 results during the intermediate intervals ; and the time occupied 
 is not the time which intervenes between cause and effect at each 
 step, but the time which intervenes between these transforma- 
 tions. 
 
 I have supposed forces to act at distinct instants, and to cease 
 to act in the intervals between ; and then, the aggregate of such 
 intervals to make up a finite time, during which an aggregate 
 force acts. But if the action of the force be rigorously con- 
 tinuous, it will easily be seen that all the consequences as to 
 cause and effect will be the same ; the discontinuous action being 
 merely the usual artifice by which, in mathematical reasonings, 
 we obtain results respecting continuous changes. It will still be 
 true, that the uniform motion which takes place after a continu- 
 ous force has acted, is the effect subsequent to the cause ; while 
 the change which takes place at any instant by the action of the 
 force, is the instantaneous effect simultaneous with the cause. 
 
 It may be objected, that this solution does not appear im- 
 mediately to apply: for the motion of a clock is not uniform 
 during any portion of the time. The parts move by intervals of 
 varied motion and of rest; or by oscillations backwards and 
 forwards; and the succession of forces which acts during any 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 199 
 
 oscillation, or any cycle of motion, is repeated during the suc- 
 ceeding oscillation or cycle, and so on indefinitely; and if an 
 alteration be made in the parts, it is not a change once for all, 
 but recurs in its operation in every cycle of the motion. 
 
 But it will be found that this circumstance does not prevent 
 ihe same explanation from being still applicable with a slight 
 modification. Instead of uniform motion in the intervals of 
 causation, we shall have to speak of steady going: and instead 
 of considering all the forces which affect the motion as causes 
 of change of uniform motion, we shall have to speak of changes 
 in the parts of the mechanism as causes of change of rate of 
 going. With this modification, it will still be true, that any 
 instantaneous cause produces its instantaneous effect simul- 
 taneously, while the permanent effect is subsequent to the 
 change which is its cause. The steady going of the clock is 
 assumed as a normal condition, in which it measures the progress 
 of time ; and in this assumption, the notion of cause and effect 
 is not brought into view. But a steady rate thus denoting the 
 mean passage of time, a change in the rate indicates a cause 
 of change. The cliange of rate, as an instantaneous transition 
 from one rate to another, is simultaneous with the change in 
 the parts. But then the changed rate as a continued condition- 
 in which, no new change supervening, the rate again measures 
 the progress of time, is subsequent to the change of parts, for 
 it begins when that ends, and continues when the progress of 
 that has ceased. 
 
 If, however, this be a satisfactory solution of the difficulty in 
 the case of mechanism, how shall we apply the same views to the 
 other cases ? Growth, the effect of food, is subsequent to the act 
 of taking food; disorder, the effect of poison, is subsequent to the 
 introduction of poison into the system. Can we say that the 
 animal would continue unchanged if it were not to take food; 
 and that food is the cause of a change, namely, of growth? 
 This is manifestly false ; for if the animal were not to take food, 
 it would soon perish. But the analogy of the former case, of the 
 clock, will enable us to avoid this perplexity. As we assumed a 
 steady rate of going in the clock to be the measure of time when 
 we considered the effect of mechanism, so we assume a steady 
 rate of action in the animal functions to be the measure of the 
 progress of time when we consider the causes which act upon the 
 
200 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 development and health of animals. Digestion, and of course 
 nutrition, are a part of this normal condition ; they are involved 
 in the steady going of the animal mechanism, and we must sup- 
 pose these functions to go regularly on, in order that the animal 
 may preserve its character of animal. Food and digestion may 
 be considered as causes of the continued existence of the animal, 
 in the same way in which the form of the parts of a clock is the 
 cause of the steady going of a clock. And when we come to cpn- 
 sider causes of change, this kind of causation, which produces 
 a normal condition of things, merely measuring the flow of time, 
 is left out of our account. We can conceive an uniform condition 
 of animal existence, the animal neither growing nor wasting. 
 This being taken as the normal condition, any deviation from 
 this condition indicates a cause, and is taken as the evidence and 
 measure of the cause of change. And thus, in a growing animal, 
 the food partly keeps the animal in continued animal existence, 
 and partly, and in addition to this, causes its growth. Food, 
 in the former view, is always circulating in the system, and 
 is supposed to be uniformly administered ; the cycles of nutrition 
 being merged in the notion of uniform existence, as the oscilla- 
 tions of the pendulum in a clock are merged in the notion of 
 uniform going; and the elementary steps of nutrition which are, 
 in this view, supposed to take place at each instant, produce their 
 instantaneous effect, for they are requisite in the cycle of animal 
 processes which goes on from instant to instant. But on the 
 other hand, in considering growth, we compare the state of 
 an animal with a preceding state, and consider the nutriment 
 taken in the intervening time as the cause of the change : hence 
 this nutriment, as an aggregate, is considered as the cause 
 of growth of the animal; and in this view the effect is sub- 
 sequent to the cause. But yet here, as in the case of mechanism, 
 the progressive effect is simultaneous, step by step, with the 
 progressive cause. There is a series of operations; as for in- 
 stance, intussusception, digestion, assimilation, growth: each of 
 these is a progressive operation; and in the progress of each 
 operation, the steps of the effect and the instantaneous forces 
 are simultaneous. But the end of one operation is the beginning 
 of the next, or at least in part, and hence we have time occupied 
 by the succession. The end of intussusception is the beginning 
 of digestion, the end of digestion the beginning of assimilation, 
 
AXIOMS RELATING TO THE IDEA OF CAUSE. 2OI 
 
 and so on. These aggregate effects succeed each other; and 
 hence growth is subsequent to the taking of food ; though each 
 instantaneous force of animal life, no less than of mechanism, 
 produces an effect simultaneous with its action. Each of these 
 separate operations is an aggregate operation, and occupies time ; 
 and each aggregate effect is a condition of the action of the cause 
 in the next operation. 
 
 Again ; if an animal in a permanent condition, neither waxing 
 nor wasting, may be taken as the normal state in which the func- 
 tions of life measure time, in order that we may consider growth 
 as an effect, to be referred to food as cause ; we may, for other 
 purposes, consider, as the normal condition, an animal waxing 
 and then wasting, according to the usual law of animal life : and 
 we must take this, the healthy progress of an animal, as our 
 normal condition, if we have to consider causes which produce 
 disease. If we have to refer the morbid condition of an animal 
 to the influence of poison, for example, we must consider how 
 far the condition deviates from what it would have been if the 
 poison had not been taken into the frame. The usual progress 
 of the animal functions including its growth, is the measure 
 of time; the deviation from this usual progress is the indication 
 of cause; and the effect of the poison is subsequent to the cause, 
 because the poison acts through the cycle of the animal functions 
 just mentioned, which occupies time ; and because the taking the 
 poison into the system, not any subsequent action of the animal 
 forces in the system, is considered as the event which we must 
 contemplate as a cause. To resume the analogy of the clock : 
 the rate of the clock is altered by altering the parts ; but this 
 alteration itself may occupy time ; as if we alter the rate of 
 a clock by applying a drop of acid, which gradually 'eats off 
 a part of the pendulum, the corrosion, as an aggregate effect, 
 occupies time; and the rates before and after the change are 
 separated by this time. But the application of the drop is the 
 cause ; and thus, in this case the final effect is subsequent to the 
 cause, though here, as in the case of mechanism, the instan- 
 taneous forces always produce a simultaneous effect. 
 
 Thus we have in every case a uniform state, or a state which 
 is considered as uniform, or at least normal; and which is taken 
 as the indication and measure of time; and we have also change, 
 
2O2 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 which is contemplated as a deviation from uniformity, and is 
 taken as the indication and measure of cause. The uniform state 
 may be one which never exists, being purely imaginary ; as the 
 case in which no forces act; and the case in which animal 
 functions go on permanently, the animal neither growing nor 
 wasting. The normal state may also be a state in which change 
 is constantly taking place, as, in fact, even a state of motion is 
 a state of change ; such states also are, in a further sense, that 
 of a clock going by starts, and that of an animal constantly 
 growing: in these cases the changes are all merged in a wider 
 view of uniformity, so that these are taken as the normal states. 
 And in all these cases, successive changes which take place are 
 separated by intervals of time, measured by the normal progress ; 
 and each change is produced by some simultaneous instantaneous 
 cause. But taking the cause in a larger sense, we group these 
 instantaneous causes, and perhaps omit in our contemplation 
 some of the intervening intervals ; and thus assign the cause to 
 a preceding, and the effect to a succeeding time. 
 
 I may observe further, as a corollary from what has been 
 said, that the measure of time is different, when we consider 
 different kinds of causation; and in each case, is homogeneous 
 with the changes which causation effects. In the consideration 
 of mechanical causes, we measure time by mechanical changes ; 
 by uniform motion, or uniform succession of cycles of motion ; by 
 the rotation of a wheel, or the oscillation of a pendulum. But 
 if we have to consider physiological changes, the progress of 
 time is physiologically measured; by the normal progress of 
 vital operations; by the circulation, digestion or development 
 of the organized body ; by the pulse, or by the growth. These 
 different measures of time give to time, so far as it is exhibited 
 by facts and events, a different character in the different cases. 
 Phenomenal time has a different nature and essence according to 
 the kind of the changes which we consider, and which gives us 
 our sole phenomenal indication of cause. 
 
 I fear that I am travelling into matters too abstruse and 
 metaphysical for the occasion: but before I conclude, I will 
 present one other aspect of the subject. 
 
 In stating the difficulty, I referred to cases of moral as well 
 as physical causation ; as when prudence produces prosperity, or 
 
AXIOMS EELATING TO THE IDEA OF CAUSE. 2 03 
 
 when folly produces ruin. It may be asked, whether we are 
 here to apply the same explanation ; whether we are to assume 
 a normal condition of human existence, in which neither pru- 
 dence nor folly are displayed, neither prosperity nor adversity 
 produced; whether we are to conceive the progress of such 
 a state to measure the progress of time, and deviations from 
 it to denote causes of the kind mentioned. It may be asked 
 further, whether, if we do make this supposition, we can resolve 
 the influence of such causes as prudence or imprudence into 
 instantaneous acts, which produce their effects immediately : and 
 which occupy time only by being separated by intervals of the 
 inactive normal moral condition. To this I must here reply, that 
 the discussion of such questions would carry me too far, and 
 would involve speculations not included within the acknowledged 
 domain of this Society, from which I therefore abstain. But 
 I may say, before quitting the subject, that I do not think 
 the suppositions above suggested are untenable; and that in 
 order to include moral causation under the maxims of causation 
 in general, we must necessarily make some such hypothesis. 
 The peculiarity of that kind of causation which the will and 
 the character exert, and which is exerted upon the will and 
 the character, would make this case far more complex and 
 difficult than those already considered ; but, at the same time, 
 would offer us the means of explaining what may seem harsh, in 
 the above analogy. For instance, we should have to assume 
 such a maxim as this: that in moral causation, time is not 
 to be measured by the flow of mechanical or physiological events ; 
 not by the clock, or by' the pulse. Moral causation has its 
 own clock, its own pulse, in the progress of man's moral being ; 
 and by this measure of time is the relation of moral cause and 
 effect to be defined. 
 
 That in estimating moral causation, the progress of time 
 is necessarily estimated by moral changes, and not by machinery, 
 by the progress of events, and not by the going of the clock, 
 is a truth familiar as a practical maxim to all who give their 
 thoughts to dramatic or narrative fictions. Who feels any thing 
 incongruous or extravagantly hurried in the progress of events 
 in that great exhibition of moral causation, the tragedy of 
 Othello? If we were asked what time those vast and terrible 
 
204 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 and complex changes of the being and feelings of the characters 
 occupy, we should say, that, measured on its own scale, the event 
 is of great extent ; that the transaction is of considerable magni- 
 tude in all ways. But if, with previous critics, we look into the 
 progress of time by the day and the hour what is the measure 
 of this history? Forty-eight hours. 
 
CHAPTER Y. 
 
 OF THE ORIGIN OF OUR CONCEPTIONS OF FORCE AND 
 MATTER. 
 
 i. 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 produced, 
 changed, or prevented. This conception is, it would 
 seem, mainly and primarily suggested by our conscious- 
 ness of the exertions by which we put bodies in 
 motion. The Latin and Greek words for Force, Vis, 
 Ft?, 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 mus- 
 cular force : 
 
 Aevrepos our' Aifas TTO\I) [telfrova \aav det/sas 
 ^K' eiriSivfjo-as, tTrtpeiffe S FlN' 
 
 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 exert upon them with similar effect : and thus the 
 labouring horse, the rushing torrent, the descending 
 weight, the elastic bow, were said to exert force. 
 
2O6 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 Homer 1 speaks of the force of the river, Fts Trora/xoto; 
 and Hesiod 2 of the force of the north wind, Fts avtpov 
 jSopeao. 
 
 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 di- 
 rection 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 under- 
 stood. 
 
 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 commentators in the middle ages. By de- 
 grees, in modern times, we see the scientific faculty 
 revive. The conception of Force becomes so far dis- 
 tinct and precise that it can be reasoned upon in a 
 consistent manner, with demonstrated consequences; 
 and a genuine science of Mechanics comes into exist- 
 ence. The foundations of this science are to be found 
 in the Axioms concerning causation 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, arises with our consciousness of our own muscular 
 exertions. But we cannot imagine such exertions 
 without also imagining some bodily substance against 
 which they are exercised. If we press, we press some- 
 thing : if we thrust or throw, there must be something 
 
 i II xxi. a Op. et D. 
 
CONCEPTIONS OF FORCE AND MATTER. 2O/ 
 
 to resist the thrust or to receive the impulse. Without 
 body, muscular 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 
 texture 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 ob- 
 jects 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 pictures of the regions inhabited by disem- 
 bodied 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 impres- 
 sions, so neither do they obey them. The spectator 
 tries in vain to strike or to grasp them. 
 
 Et ni docta comes tenues sine corpore vitas 
 Admoneat volitare cav& 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. 
 
 Neque ilium 
 
 Prensantem nequicquam umbras et multa volentem 
 Dicere, 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 
 
208 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 rela- 
 tions of space. Thus the same organ gives us the per- 
 ception of body as resisting force, and as occupying 
 space ; and by combining 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 exten- 
 sion. 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 resist- 
 ance is precisely that which is requisite in order to 
 make us conscious of cur muscular effort 3 . Neither 
 touch, nor any other mere passive sensation, could pro- 
 duce the perception of extent, as we have already 
 urged: nor could the muscular sense lead to such 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 con- 
 tinue in their places. By fluid bodies, we mean those 
 
 3 Brown's Lectures, L 466. 
 
CONCEPTIONS OF FORCE AND MATTER. 20Q 
 
 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 pressure, 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 
 convictions 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 4 , that 'the bodies which we daily handle hinder 
 by an insurmountable force the approach of the parts 
 of our hands that press them.' Now it is manifest that 
 our observation 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 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 ex- 
 erted 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 insurmount- 
 able 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 
 
 4 Essay, b. ii. c. 4. 
 VOL. I. 
 
2IO PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 resists; for the operation requires an effort. By in- 
 creasing 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 pre- 
 venting the motion, but in a reciprocal force, exerted 
 backwards upon the agent by which the motion is pro- 
 duced. 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 the body, must be the subject of our consideration 
 hereafter. 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 prin- 
 ciple inseparable from the notion of matter itself. 
 
 Hermann says that Kepler first introduced this 
 ' most significant' 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 mechanics. He assumes the slowness of the 
 motions of the planets to increase, (other causes re- 
 maining the same,) as the inertia increases ; and though, 
 even in this assumption, 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 express- 
 ing those laws of motion which depend on the funda- 
 mental principle of the equality of action and reaction. 
 
CONCEPTIONS OF FORCE AND MATTER. 211 
 
 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 muscular frame, which is the main instrument 
 of our perceptions 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 universal truths to be 
 hereafter established respecting the conceptions thus 
 obtained. 
 
 We must now endeavour to trace in what manner 
 these fundamental principles and conceptions are un- 
 folded by means of observation and reasoning, till they 
 become an extensive yet indisputable science. 
 
 p 2 
 
CHAPTER VI. 
 
 OF THE ESTABLISHMENT OF THE PRINCIPLES OF 
 
 STATICS. 
 
 i. 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 
 fundamental mechanical principles, which are of evi- 
 dent 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 
 fundamental 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, fundamental principles of axiomatic evi- 
 dence and necessity ; that these principles derive their 
 axiomatic character from the Idea which they involve, 
 namely, the Idea of Cause; and that through the 
 combination of principles of this kind, the whole sci- 
 ence of Mechanics, including its most complex and 
 remote results, exists as a body of solid and universal 
 truths. 
 
OF THE PRINCIPLES OF STATICS. 213 
 
 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 equilibrium. The two forces are evidently equal, 
 and are a statical exemplification of action and re- 
 action, such as are spoken of in the third axiom 
 concerning causes. Now the same exemplification oc- 
 curs in every case of equilibrium. 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 downwards is opposed 
 and balanced by an equal pressure of the floor up- 
 wards. If the stone rests on a slope, its tendency to 
 slide is counteracted by some equal and opposite lorce, 
 arising, it may be, from the resistance which the slop- 
 ing ground opposes to any motion along its surface. 
 Every case of rest is a case of equilibrium : 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 ba- 
 lance 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 
 
214 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 Causa- 
 tion, 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 stati- 
 cal force produces no other effect than this, that it 
 balances some other statical force; and hence the mea- 
 sure 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 in- 
 volved 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 sup- 
 ported 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 each other are equal, each force is measured 
 by that which it balances ; and since forces are capable 
 of addition, a force of any magnitude is measured by 
 adding together 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 
 
OF THE PRINCIPLES OF STATICS. 215 
 
 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 subj ect 
 to this condition, that they are susceptible of being 
 added : and therefore we cannot take the deflexion of 
 the bough for our measure, 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 obviously nor necessarily 
 true. In this, as in all other cases, although causes 
 must be measured by their effects, we learn from expe- 
 rience only how the effects are to be interpreted, so as 
 to give a true and consistent measure. 
 
 With regard, however, to the measure of statical 
 force, and of weight, no difficulty really occurred to 
 philosophers from the time when they first began to 
 speculate 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 im- 
 plied 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 establish by reasoning the conditions under 
 which weights would be in equilibrium, some other 
 principles were needed in addition to the mere mea- 
 sure of forces. The principles introduced for this pur- 
 pose still resulted from the conception of equal action 
 and reaction; but it required no small clearness of 
 thought to select them rightly, and to employ them 
 
2l6 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 Archi- 
 medes 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 center of gravity in the point midway be- 
 tween 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 collected 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 tlie 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. 
 
OF THE PRINCIPLES OF STATICS. 
 
 The above principle, that the pressure on the sup- 
 port 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 con- 
 ception of equal action and reaction, which equilibrium 
 always implies, what is the origin of its certainty? 
 The principle is never for an instant denied or ques- 
 tioned : 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? Experience 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 de- 
 grees 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 cer- 
 tainty; 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 under- 
 standing 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 mecha- 
 nical 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 
 
2l8 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 experiment were shown which seemed to contradict the 
 principle, his conclusion would be, not that the princi- 
 ple 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 man- 
 ner that they shall be exemplifications of the principle. 
 A mechanical pressure not accompanied by an equal 
 and opposite pressure, can no more be given by expe- 
 rience, than two unequal right angles. With the sup- 
 position 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. 
 
 [2d ed.] [To the doctrine that mechanical principles, 
 such as the one here under consideration (that the pres- 
 sure on the point of support is equal to the sum of the 
 weights), are derived from our Ideas, and do not flow 
 from but regulate our experience, objections are natu- 
 rally made by those who assert all our knowledge to 
 be derived from experience. How, they ask, can we 
 know the properties of pressures, levers and the like, 
 except from experience? What but experience can 
 possibly inform us that a force applied transversely to 
 a lever will have any tendency to turn the lever on its 
 center? This cannot be, except we suppose in the 
 lever tenacity, rigidity and the like, which are quali- 
 ties known only by experience. And it is obvious 
 that this line of argument might be carried on through 
 the whole subject. 
 
 My answer to this objection is a remark of the 
 same kind as one which I have made respecting 
 the Ideas of Space, Time, and Number, in the last 
 Book. The mind, in apprehending events as causes 
 
OF THE PRINCIPLES OF STATICS. 2 19 
 
 and effects, is governed by Laws of its own Ac- 
 tivity; and these Laws govern the results of the 
 mind's action; and make these results conform to 
 the Axioms of Causation. But this activity of the 
 mind is awakened and developed by being exercised; 
 and in dealing with the examples of cause and effect 
 here spoken of, (namely, pressure and resistance, 
 force and motion,) the mind's activity is necessarily 
 governed also by the bodily powers of perception and 
 action. We are human beings only in so far as we 
 have existed in space and time; and of our human 
 faculties, developed by our existence in space and time, 
 space and time are necessary conditions. In like man- 
 ner, we are human beings only in so far as we have 
 bodies, and bodily organs; and our bodies necessarily 
 imply material objects external to us. And hence our 
 human faculties, developed by our bodily existence in 
 a material world, have the conditions of matter for 
 their necessary Laws. I have already said (chap, vi) 
 that our conception of Force arises with our conscious- 
 ness of our own muscular exertions; that Force can- 
 not be conceived without Resistance to exercise itself 
 upon ; and that this resistance is supplied by Matter. 
 And thus the conception of Matter, and of the most 
 general modes in which Matter receives, resists, and 
 transmits force, are parts of our constitution which, 
 though awakened and unfolded by our being in a ma- 
 terial world, are not distinguishable from the original 
 structure of the mind. I do not ascribe to the mind 
 innate Ideas Ideas which it would have, even if it 
 had no intercourse with the world of space, time, and 
 matter; because we cannot imagine a mind in such a 
 state. But I attempt to point out and classify those 
 Conditions of all Experience, to which the intercourse 
 of all minds with the material world has necessarily 
 given rise in all. Truths thus necessarily acquired in 
 the course of all experience, cannot be said to be learnt 
 from experience, in the same sense in which particular 
 facts, at definite times, are learnt from experience 
 learnt by some persons and not by others learnt with 
 more or less of certainty. These latter special truths of 
 
22O PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 experience will be very important subjects of our con- 
 sideration; but our whole chance of discussing them 
 with any profit depends upon our keeping them distinct 
 from the necessary and universal conditions of experi- 
 ence. Here, as everywhere, we must keep in view the 
 fundamental antithesis of Ideas and Facts.] 
 
 6. Oblique Forces. By the aid of the above 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 mathema- 
 tician mastered till modern times. The unpublished 
 manuscripts of Leonardo da Yinci, written in the fif- 
 teenth 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 from that period, 
 mathematicians, having thus become possessed of all 
 the mechanical principles which are requisite in pro- 
 blems respecting equilibrium, soon framed a complete 
 science of Statics. Succeeding writers presented this 
 science in forms variously modified ; for it was found, 
 in Mechanics as in Geometry, that various proposi- 
 tions might be taken as the starting points; and that 
 the collection of truths which it was the mechanician's 
 business to include in his course, might thus be tra- 
 versed by various routes, each path offering a series of 
 satisfactory demonstrations. The fundamental concep- 
 tions 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 
 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 Fundamental 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 demon- 
 strated science. 
 
OF THE PRINCIPLES OF STATICS. 221 
 
 7. A Force may be supposed to act at any Point of 
 its Direction. It has been stated in the history of Me- 
 chanics 1 , that Leonardo da Vinci and Galileo obtained 
 the true measure of the effect of oblique forces, by 
 reasonings 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 ac- 
 customed ourselves 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 ex- 
 ample, one of the wheels by means of which the large 
 bells of a church are rung,) this weight may be sup- 
 ported by means of a rope (not passing along the cir- 
 cumference 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 center; 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 center 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 
 direction meets this arm ; and thus we reduce the case 
 
 Hist. Ind. Sc. b. vl c. i. sect. 2. 
 
222 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 
 impression 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 direc- 
 tion 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 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 com- 
 plete the list of the elementary principles of Statics, we 
 ought also to lay down axioms respecting the other 
 kinds of rigidity 2 . 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 funda- 
 mental conception of matter; they are not the results 
 of any experience, but are the hypotheses to which we 
 are irresistibly led, when we would liberate our rea- 
 sonings concerning force and matter from a dependence 
 on the special results of experience. We cannot even 
 
 2 Such axioms are given in a little work (The MecJianical Euclid) which I 
 published on the Elements of Mechanics. 
 
OF THE PRINCIPLES OF STATICS. 223 
 
 conceive (that is, if we have any clear mechanical con- 
 ceptions 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 
 reasonings, 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 trans- 
 ferred from one point to another; we may suppose dif- 
 ferent forces to act at the same point, and may thus 
 commence our reasonings with a case in which we 
 have to contemplate force, without having to take into 
 our account 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 of forces; 
 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 propositions 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 
 
224 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 parallelogram 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 claim a demonstrative cer- 
 tainty. Daniel Bernoulli, in 1726, gave the first proof 
 of this important proposition on pure statical princi- 
 ples; and thus, as he says 3 , 'proved that statical theo- 
 rems are not less necessarily true than geometrical are.' 
 If we examine this proof of Bernoulli, in order to dis- 
 cover 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 be taken away other forces 
 which are in equilibrium at the same point, the re- 
 mainder 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 Bernoulli's proof, 
 the two forces to be compounded are represented by p 
 and Q; P is resolved into two other forces, x and u; 
 and Q into two others, Y and v, under certain condi- 
 tions. 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 
 
 Comm. Petrop. voL L 
 
OF THE PRINCIPLES OF STATICS. 225 
 
 space, equivalent to its parts added together in any 
 order. The apprehension of forces as having magni- 
 tude, 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 foundations, are independent of experi- 
 ence 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- 
 strations, however, have been given of the same propo- 
 sition. Jacobi, a German mathematician, has collected 
 and examined eighteen of these 4 . 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 conceptions, 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, 
 
 4 These are by the following ma- Salimbeni; Duchayla; two different 
 
 thematicians ; D. Bernoulli (1726) ; proofs by Foncenex (1760) ; three by 
 
 Lambert (1771); Scarella (1756); Ve- D'Alembert; and those of Laplace 
 
 nini (1764) ; Araldi (1806) ; Wachter and M. Poisson. 
 (1815); Kaestner; Marini; Eytelwein; 
 
 VOL. I. O 
 
226 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 constantly been confounded 
 with the composition of force. But though it has 
 been done by very eminent mathematicians, 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 equilibrium of two 
 forces on a lever, or of three forces at a point, can be 
 established without any reference whatever to any 
 motions which the forces might, under other circum- 
 stances, 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 
 combination 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 
 descending. It is easily seen that this maxim carries 
 to a much greater extent the principle which the 
 Greek mathematicians 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 
 
OF THE PRINCIPLES OF STATICS. 22/ 
 
 single rigid mass only; whereas, in the maxim now 
 under our notice, it is asserted of any masses, con- 
 nected by strings, rods, joints, or in any manner. We 
 have already seen that more modern writers on me- 
 chanics, desirous of assuming as fundamental no wider 
 principles than are absolutely necessary, 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 principle 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 em- 
 ployed 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 
 philosophers have reduced the solution of such prob- 
 lems to a clear dependence upon the most simple 
 axioms. 
 
 ii. 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 
 distances, to hang over the Inclined Plane; and his 
 reasoning proceeds upon this assumption, That such 
 a loop so hanging will find a certain position in which 
 it will rest: for otherwise, says he 5 , its motion must 
 go on for ever, which is absurd. It may be asked how 
 
 5 Stevin. Stalique, livre i. prop. 19. 
 
 Q2 
 
228 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 this absurdity of a perpetual motion appears ; and it 
 will perhaps be added, that although the impossibility 
 of a machine with such a condition may be proved as 
 a remote result of mechanical principles, this impossi- 
 bility can hardly be itself recognized as a self-evident 
 truth. But to this we may reply, that the impossi- 
 bility is really evident in the case contemplated by 
 Stevinus ; for we cannot conceive a loop of chain to go 
 on through all eternity, sliding 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 posi- 
 tion 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 de- 
 monstrations, the earliest real advances in the science. 
 
 12. Principle of Virtual Velocities. We have re- 
 ferred above to an assertion often made, that we may, 
 from the simple principles of Mechanics, demonstrate 
 the impossibility of a perpetual motion. In reality, 
 
OF THE PRINCIPLES OF STATICS. 22Q 
 
 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 
 proposition. 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 condi- 
 tions of equilibrium, and is called the Principle of 
 Virtual Velocities 6 , 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 mechanical combinations be 
 complete. Accordingly, some writers have attempted 
 independent and general proofs of the Principle of 
 Virtual Yelocities ; and these proofs rest upon assump- 
 tions 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 num- 
 ber 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, appre- 
 hended at first in a complex and derivative form, have 
 afterwards been reduced to their simpler elements; 
 in which, also, sagacious and inventive men have fixed 
 upon those truths as self-evident, which now appear to 
 us only certain 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 of arbitrary selection, but by 
 having a keener and deeper insight than other persons 
 
 See Hist. Inci Be. b. vL c. ii, sect. 4. 
 
230 PHILOSOPHY OP THE MECHANICAL SCIENCES. 
 
 into the relations which were the object of their con- 
 templation; and in the science now spoken of, they 
 were led to their assumptions by possessing clearly and 
 distinctly the conceptions of mechanical cause and 
 effect, action and reaction, force, and the nature of 
 its operation. 
 
 13. Fluids 2>ress Equally in all Directions. The 
 doctrines which concern the equilibrium of fluids de- 
 pend 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 treatise of Archimedes On Floating 
 Bodies, which contains correct solutions of several 
 problems belonging to this subject, and of some which 
 axe by no means easy. In this treatise, the funda- 
 mental 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 pres- 
 sure exerted upon a fluid in one direction 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 containing ves- 
 sel. 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 involved in the reasoning of Archimedes, is still 
 to the present day the basis of all hydrostatical trea- 
 tises, and is 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 en- 
 deavour to illustrate this. There are many elementary 
 propositions in physics, our knowledge of which indis- 
 
OF THE PRINCIPLES OF STATICS. 23] 
 
 putably depends upon experience; and in these cases 
 there is no difficulty in seeing the evidence of this de- 
 pendence. 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 errours were avoided : 
 the experiments of more recent times have either ren- 
 dered 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 l 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' 1 ; 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- 
 phers 8 , to ascertain the truth or falsehood of this sus- 
 picion. 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 proba- 
 bility, but is rigorously tested by experience. In this 
 case, the assertion, that the law depends upon experi- 
 ence, contains a reference to plain and notorious pas- 
 sages in the history of science. 
 
 Now with regard to the principle that fluids press 
 equally in all directions, the case is altogether different. 
 
 7 Shaw's Boyle, Vol. ii. p. 671. between the observed and calculated 
 
 8 The members were Prony, Arago, elasticity amount to one-hundredth 
 Amp6re, Girard, and Dulong. The of the whole ; nor did the difference 
 experiments were extended to a pres- appear to increase with the increase 
 sure of twenty-seven atmospheres; of pressure. Feclmer, Repertorium, 
 and in no instance did the difference L no. 
 
232 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 It is, indeed, often asserted in works on hydrostatics, 
 that the principle is collected from experience, and 
 sometimes a few experiments are described as exhibit- 
 ing its effect; but these are such as to illustrate and 
 explain, rather than to prove, the truth of the princi- 
 ple: they are never related to have been made with 
 that exactness of precaution and measurement, or that 
 frequency of repetition, which are necessary to esta- 
 blish a purely experimental truth. Nor did such ex- 
 periments 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 ex- 
 actly the same shape as the axioms which he employs 
 in statics, and even in geometry; namely, as an as- 
 sumption. Nor does any intelligent student of the 
 subject find any difficulty in assenting to this funda- 
 mental principle of hydrostatics as soon as it is pro- 
 pounded to him. Experiment was not requisite for its 
 discovery ; experiment is not necessary for its proof at 
 present ; and we may add, that experiment, though it 
 may make the proposition the more readily intelligible, 
 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 
 lluid 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 pressure 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 differ- 
 ent from the original one, is not altered in intensity, 
 
OF THE PRINCIPLES OF STATICS. 233 
 
 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 directions were supposed. If, for 
 instance, the lateral pressure were less than the ver- 
 tical, this could be conceived no other way than as in- 
 dicating 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 ex- 
 actly balance each other : they are the action and the 
 reaction ; and must hence be equal by the same neces- 
 sity 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 rnoveable among each other, leads us 
 necessarily to the principle of the equality of hydro- 
 static pressure in all directions, still this conception 
 itself is obtained from experience, or suggested by 
 observation. And to this we may reply, that the con- 
 ception 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 experi- 
 ence. For if we imagine a vessel full of small, smooth 
 spherical balls, such a collection of balls would ap- 
 proach 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 
 
234 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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, ac- 
 quainted 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 determined; as, for instance, the explanation of 
 what is called the Hydrostatic Paradox. The necessity 
 of such a discipline in order that the reader may enter 
 fully into this part of our speculations, naturally ren- 
 ders them less popular; but this disadvantage is in- 
 evitable 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 established in those sciences. This prepa- 
 ration for philosophizing may be somewhat laborious ; 
 but such labour is necessary if we would pursue specu- 
 lative truth with all the advantages which the present 
 condition of human knowledge places within our reach. 
 
 We may add, that the consequences to which we 
 are directed by the preceding opinions, are of very 
 great importance in their bearing upon our general 
 views respecting human knowledge. I trust to be 
 able to show, that some important distinctions are 
 illustrated, some perplexing paradoxes solved, and some 
 large anticipations of the future extension of our 
 knowledge suggested, by means of the conclusions to 
 which the preceding discussions have conducted us. 
 But before I proceed to these general topics, I must 
 consider the foundations of some of the remaining 
 portions of the science of Mechanics. 
 
CHAPTER YIL 
 
 OF THE ESTABLISHMENT OF THE PRINCIPLES OF 
 DYNAMICS. 
 
 i. TN the History of Mechanics, I have traced the 
 JL 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 application. We have, in these laws, exam- 
 ples of principles which were, historically speaking, 
 obtained by reference to experience. Bearing in mind 
 the object and the result 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 obtained, and those truths which 
 we have already contemplated ; 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 
 motion 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 Gali- 
 leo 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 entertained by most persons, till 
 their attention is fixed upon the arguments by which 
 the first law of motion is established. It is, however, 
 not difficult to lead any person of a speculative habit 
 
236 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 of thought to see that the retardation 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 resist- 
 ance of the air gradually diminish its motion, and not 
 because its motion has any internal principle 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 discussion, 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 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 exter- 
 nal impediments to continued motion, and although, 
 consequently, there will always be some retardation; 
 and an end of the motion of a body left to itself, how- 
 ever long it may be delayed, must at last come; yet 
 we can establish a conviction that if all resistance 
 could be 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 experiment, and that motion 
 
OF THE PRINCIPLES OF DYNAMICS. 237 
 
 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 suppose that there can be none but an external 
 cause of change. But in order to establish the law, 
 it was necessary further to be assured that there is no 
 internal cause 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 reference to observation and experi- 
 ment; 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 mathemati- 
 cians 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 1 , 
 who offers what is called an a 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. 
 
 1 Dynamiqvc. 
 
238 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 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 mathema- 
 tical form 2 . 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 dimi- 
 nish as the time increases, this may be expressed by 
 stating the velocity 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 coefficients, 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 of, 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 sixteen 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 cer- 
 tain magnitude. It would be easy to assign a mathe- 
 matical expression for the velocity of a body, implying 
 that one-hundredth of the velocity, or any other frac- 
 
 Outlines of Natural Philosophy, p. 26. 
 
OF THE PRINCIPLES OF DYNAMICS. 239 
 
 tion, is lost in each second 3 : and where is the absurdity 
 of supposing such an expression really to represent the 
 velocity 1 
 
 Most modern writers on mechanics have embraced 
 the opposite opinion, and have ascribed our know- 
 ledge of this first law of motion to experience. Thus 
 M. Poisson, one of the most eminent of the mathema- 
 ticians who have written on this subject, says 4 , "We 
 cannot afiirm d, 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 show 
 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 experiments 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 our 
 very conception of them, occasion, there are no longer 
 any alterations. Instead of having to guide ourselves 
 
 3 This would be the case, if, t being the number of seconds elapsed, and C 
 some constant quantity, the velocity were expressed by this mathematical 
 formula, C (^V. 
 
 \IOO/ 
 
 * Poisson, Dynamiquc, ed. 2, art. 113. 
 
240 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 re- 
 sult. 
 
 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 according to which bodies fall downwards by 
 the common action of gravity, has already been noticed 
 in the History of Mechanics 5 , as one of the earliest 
 positive advances in the doctrine of motion. These 
 laws were first rightly stated by Galileo, and esta- 
 
 H-ist. Ind. Sc. b. vi. c. ii. sect. 2. 
 
OF THE PRINCIPLES OF DYNAMICS. 24! 
 
 blished by reasoning and by experiment, not without 
 dissent and controversy. The amount of these doc- 
 trines is this : That gravity is a uniform accelerating 
 force; such a uniform force having this for its cha- 
 racter, that it makes the velocity increase in exact pro- 
 portion 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 deduc- 
 tion 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 dis- 
 covery. In defining accelerating force, reference, tacit 
 or express, 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. 236,) any cause which, acting from 
 without, changes the motion of a body. It must, 
 therefore, in this acceptation, 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 causa- 
 tion, from which otherwise we could not deduce any 
 real knowledge. We may assume, in virtue of our 
 general conceptions of force, that under the same 
 circumstances, a greater change of motion implies 
 a greater force producing it; but what are we to 
 expect when the 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 terrestrial 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 proportioned to the velocity which 
 already exists? Force acting on a body at rest, and 
 on a body in motion, appears under very different 
 
 VOL. I. B 
 
242 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 conceivable 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 manner, that although gravity be always the 
 same force, its effect depends upon the velocity which 
 the body under its influence already possesses 1 
 
 Observation and reasoning combined, as we have 
 said, enabled Galileo to answer these questions. He 
 asserted 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. 
 
 If it be asked whether we could have known, an- 
 terior to, or independent of, experiment, that gravity 
 is a uniform 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 completely, though 
 
OF THE PRINCIPLES OF DYNAMICS. 243 
 
 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 can- 
 not, 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 uni- 
 formly accelerated; 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 ac- 
 celerated 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 6 . 
 And the latter supposition is rejected, the author 
 showing, 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 care- 
 fully,' he says, ' we shall find 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 uniformity 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 
 
 Dialogo, iii p. 95. 
 
 2 
 
244 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 accelerated, consists, in the first place, in adducing 
 the maxim that nature always employs the most 
 simple means 7 . But he is far from considering this a 
 decisive argument. ' I,' says one of his speakers, 
 1 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, con- 
 ceived 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 pre- 
 vail in vertical motion, it will also hold when a body 
 is compelled to describe an oblique rectilinear path. 
 This proposition may be shown to be true, if (assuming 
 by anticipation the Third Law of Motion, of which 
 we shall shortly have to speak,) we introduce the con- 
 ception of a uniform statical force as the cause of 
 uniform acceleration. For the force on the inclined 
 plane bears a constant proportion to the vertical force, 
 and this proportion is known from statical considera- 
 tions. But in the work of which we arc speaking, Galileo 
 does not introduce this abstract conception of force as 
 the foundation of his doctrines. Instead of this, he 
 proposes, 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 8 . 
 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 
 likelihood, I wish to augment the probability so far, 
 that it shall be almost as complete as a necessary 
 demonstration.' He then proceeds to describe a very 
 ingenious and simple experiment, which shows that 
 when a body is made to swing upwards at the end of 
 
 Dialogo, UL p. 91. 8 Dialogo, iii. p. 36. 
 
OF THE PRINCIPLES OF DYNAMICS. 245 
 
 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 experimentally confirmed, so far as the 
 force of gravity can be taken as an example of the 
 forces which the postulate contemplates: and con- 
 versely, 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 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 confirmed by the experiment above men- 
 tioned; as it also is, on like grounds, by many other 
 experiments, made upon inclined planes and pendu- 
 lums. 
 
 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 ex- 
 periments. We may make here two remarks; First, 
 that the conception, when established and rightly 
 stated, appears so simple as hardly to require experi- 
 mental 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 propositions 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, in- 
 stead of falling downwards from rest, is thrown in any 
 direction, 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 
 
246 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 mea- 
 sured 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 of gravity under those circumstances. 
 If this effect vary with the varying velocity and direc- 
 tion 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 subjected to its in- 
 fluence : 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 com- 
 pounded with one another, represents, it is proved, the 
 manner in which the motion produced by gravity 
 modifies any other motion which the body may previ- 
 ously have had. 
 
 The discussions which terminated in the general re- 
 ception of this Second Law of Motion among mechani- 
 cal 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 im- 
 pressed upon it, as well as that motion which would 
 
OF THE PRINCIPLES OF DYNAMICS. 247 
 
 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. Galileo 
 further urged, as a presumption in favour of the opinion 
 that the two motions, the circular motion arising 
 from the rotation of the earth, and the downward 
 motion arising from the gravity of the stone, would be 
 compounded in the way we have described, (neither of 
 them disturbing or diminishing the other,) that the 
 first motion was in its own nature not liable to any 
 change or diminution 9 , 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 question 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 want- 
 ing which might confirm the law as true to the last 
 degree of accuracy. Laplace relates (Syst. du Monde, 
 livre iv. chap. 16,) 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 
 
 Dialogo, ii p. 114. 
 
248 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 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 obser- 
 vation, 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 transverse to the direction 
 of the force which the body may have at the same 
 time : or, to express it otherwise, thttt 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 Mecanique 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 sub til ty of thought on such subjects, as 
 well as of power of mathematical generalization 10 And 
 
 10 I may observe that the rule that of resolving them; which is done in 
 we may compound motions, as the the passage to which I refer. (Jtf&. 
 Law supposes, is involved in the step Analyt. ptie. L sect. L art. 3. p. 225.) 
 
OF THE PRINCIPLES OF DYNAMICS. 249 
 
 thus we have here another example of that circum- 
 stance 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 atten- 
 tion to this feature in various portions of the history of 
 science, inasmuch as it will be found to lead to some 
 very extensive and important 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 exer- 
 tion by which we 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 1 ? The 
 answer to this question is contained in the Third Law 
 
 ' Si on conceit que le mouvement mens doivent suivre, chacun en par- 
 d'un corps et les forces qui le sollici- ticulier, les lois des mouvemens rec- 
 tent soient decomposes suivant trois tilignes acce'le're's ou retardes par les 
 lignes droites perpendiculaires entre forces donnges.' Laplace makes the 
 elles, on pourra considgrer sgparg- same assumption in effect, (Mfa C<!1. 
 ment les mouvemens et les forces p. L liv. i art. 7), by resolving the 
 relatives a chacun de ces trois di- forces which act upon a point in 
 rections. Car a cause de la perpen- three rectangular directions, and rea- 
 dicularite des directions il est visible soning separately concerning each 
 que chacun de ces mouvemens par- direction. But in his mode of treat- 
 tiels peut 6tre regarde comme in- ing the subject is involved a prin- 
 dgpendant des deux autres, et qu'il ciple which belongs to the Third 
 ne peut recevoir d'altSration que de Law of Motion, namely, the doc- 
 la part de la force qui agit dans la trine that the velocity is as the 
 direction de ce mouvement ; Ton force,'of which we shall have to speak 
 peut conclure que ces trois mouve- elsewhere. 
 
250 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 of Motion, and it is to this effect : that the Momentum 
 which any pressure produces 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 pressure 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 produce an equal velocity in addition to the 
 former? Or, stating the matter otherwise, the ques- 
 tion is, why each of the two forces will produce its 
 separate effect, unaltered 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 different meaning in this case and in that ; 
 in this place it signifies pressure; in the statement of 
 the Second Law 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 
 
OF THE PRINCIPLES OF DYNAMICS. 251 
 
 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 11 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 con- 
 siderations,) obvious as it may seem; but assumed, as 
 we have already seen, a proposition 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 
 motion of bodies down inclined planes was at first 
 presented in Galileo's Dialogues on the Science of 
 Motion. But his disciple Yiviani was dissatisfied with 
 the assumption thus introduced; and in succeeding 
 editions of the Dialogues, the apparent chasm in the 
 reasoning was much narrowed, by making the proof 
 depend upon a principle 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 12 , 
 ' that in a moving body the impetus, energy, momen- 
 tum, 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 principle so stated 
 amounts to this ; that the velocity produced is as the 
 statical force. And thus this law of motion appears, 
 
 " Dial, della Sc. Nuov. iii. p. 96. See Hist. Ind. Sci. b. vi. c. ii. sect. 5. 
 13 Dialoyo, p. 104. 
 
252 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 in the school of Galileo, to have been suggested and 
 established at first by experiment, but afterwards con- 
 firmed 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 stati- 
 cal 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, propor- 
 tional to the mass of the body. This part of the third 
 law of motion follows from our conception of matter 
 in general as consisting 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 disturbing the effects. 
 
 The measure of the quantity of matter of a body 
 considered as affecting the velocity which pressure 
 produces in the body, is termed its inertia, as we have 
 already stated (c. v.) Inertia is the property by 
 which a 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 another way of expressing this third 
 law of motion, so far as quantity of matter is con- 
 cerned. 
 
 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 
 proportional 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 velocities by gravity, when the known causes of 
 difference are removed. The observations of falling 
 
OF THE PRINCIPLES OF DYNAMICS. 253 
 
 bodies, indeed, are not susceptible of much exactness : 
 but experiments leading to the same result, and capa- 
 ble 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 accu- 
 racy : and thus the weight and the inertia are propor- 
 tional 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 re- 
 action ; 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 con- 
 sistent 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 
 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 13 , 'should we have recourse to this principle em- 
 ployed, at the present day, by everybody, that the 
 force is proportional 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 necessa- 
 rily true ; we shall only avow that the proofs which 
 have hitherto been adduced do not appear to us unex- 
 ceptionable : nor shall we, with some geometers, adopt 
 it as a purely contingent truth; which would be to 
 ruin the certainty of mechanics, 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 use- 
 less 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 
 
 i 3 Dynamique, Pref. p. x. 
 
254 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 equivalent import, which appears to him to possess 
 axiomatic certainty; and this procedure is in consist- 
 ence 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 
 says' 4 , 'proportioned to the velocity? This,' he re- 
 plies, 'we cannot know ct, 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 result of observation.' And again he says 15 , 
 1 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 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 Mecha- 
 nics borrows from experience.' 
 
 It will appear, I think, from the views given in this 
 and several other parts of the present work, that we 
 cannot with justice say that we have very * few data 
 respecting the nature of things,' in speculating con- 
 cerning the laws of the universe ; since all the conse- 
 quences which flow from the relations of our funda- 
 mental ideas, necessarily regulate our knowledge of 
 things, so far as we have any such knowledge. Nor 
 can we say that the nature of matter is unknown to 
 us, in any sense in which we can conceive knowledge 
 as possible. The nature of matter is no more un- 
 known 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 anything which is 
 independent of these relations, is not unknown, but 
 inconceivable. 
 
 Mdc. Cd. p. 15. i fi p. 18. 
 
OF THE PRINCIPLES OF DYNAMICS. 255 
 
 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 
 consideration of what we mean by observation and 
 experience, nature and matter. But it will probably 
 be generally allowed, that, taking into account the ex- 
 planations 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 altogether self-evident. 
 
 6. Action and Reaction are Equal in Moving 
 Bodies. When we have to consider bodies as acting 
 upon one 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 principle 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 
 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 pres- 
 sures 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 
 
256 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 
 reasoners 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 mo- 
 mentum lost and gained applies to this as well as to 
 the other cases. 
 
 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. J)' 'Alembert 's 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 direct; and except the indirect 
 action introduce some modification of the law, they 
 must have the same measure 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 
 
OF THE PRINCIPLES OF DYNAMICS. 257 
 
 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 momen- 
 tum gained and lost, balance each other according 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 D'Alem- 
 bert's 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 con- 
 nexion of propositions, but for a different purpose j 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 j yet it 
 appears to be by no means easy to supersede the neces- 
 sity of independent proofs of the last two of these prin- 
 ciples. Both may be proved or illustrated by experi- 
 ment : 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, 
 
 VOL. I. S 
 
258 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 
 circumstances, 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 ex- 
 periments, 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 
 inclined planes of the same height, was afterwards 
 reduced to the third law of motion by Galileo himself. 
 In the History 16 , 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 Herman 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 early 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 discover- 
 ers by whom it was made. On the contrary, it is evi- 
 dence of the deep sagacity and clear thought which were 
 
 16 B. vi. c. v. sect. 2. 
 
OF THE PRINCIPLES OF DYNAMICS. 259 
 
 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 probably present, though undeveloped, in 
 the minds of the discoverers. We are told of geo- 
 metrical 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 apti- 
 tude for mechanical reasonings, which, existing in the 
 minds of Stevinus, Galileo, Newton, 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 
 evidence which the laws of mechanics have at length 
 assumed, are much favoured by the usage of words 
 among the best writers 011 such subjects. Terms which 
 originally, 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 assertions which at first appear identical propo- 
 sitions become distinct and important principles. Thus 
 force, motion, momentum, are terms which were em- 
 ployed, though in a loose manner, from the very 
 outset of mechanical speculation. And so long as 
 these words retained the vagueness of common lan- 
 guage, it would have been a useless and barren truism 
 to say that 'the momentum is proportional to the 
 force, 5 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 
 product of mass and velocity, these two propositions 
 immediately become distinct statements of the third 
 law of motion and its consequences. In like manner, 
 the assertion that ' gravity is a uniform force' was 
 assented to, 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 com- 
 municated to the same body their effects are com- 
 
 8 2 
 
260 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 pounded,' 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 knowledge ; so that truths which were at first 
 unexpected and recondite, became familiar phrases, 
 and after a few generations sound, even to common 
 ears, like identical propositions. 
 
 i o. Controversy of the Measure of Force. In the 
 History of Mechanics 17 , we have given an account of 
 the controversy which, for some time, occupied the 
 mathematicians 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 history, 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 mo- 
 mentum and of vis viva. It is, in one sense, as 
 D'Alembert has said 18 , a dispute about words; but 
 we are not to infer that, on that account, it was 
 frivolous or useless; for such disputes are one prin- 
 cipal means of reducing the principles of our know- 
 
 1 7 B. vi c. v. sect. 2. that causes may "be justly measured by 
 
 1 8 D'Alembert has also remarked their effects, even when very different 
 (Dynamique, Pref. xxi.) that this con- kinds of effects are taken. That the 
 troversy* shows how little justice and axiom does not point out one pre- 
 precision there is in the pretended else measure, till illustrated by ex- 
 axiom that causes are proportional to perience or by other considerations, 
 their effects.' But this reflection is by we grant: but the same thing occurs 
 no means well founded. For since in the application of other axioms 
 both measures are true, it appears also. 
 
OF THE PRINCIPLES OF DYNAMICS. 261 
 
 ledge 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 mathe- 
 maticians, 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 
 controversy was exactly that tendency which we have 
 been noticing; the disposition of man to assume in 
 his speculations 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 concerning a definition com- 
 bined with a general proposition. Such a question 
 may be readily conceived to have been by no means 
 unimportant; and we may remark, in passing, that 
 such controversies, although they are commonly after- 
 wards 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 funda- 
 mental 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 knowledge, as exemplified in the history of Me- 
 chanics. The general transformation of our views 
 from vague to definite, from complex to simple, from 
 unexpected discoveries to self-evident truths, from 
 seeming contradictions to identical propositions, is 
 very remarkable, but it is by no means peculiar to 
 our subject. The same circumstances, more or less 
 prominent, more or less developed, 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 prospects, the 
 
262 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 tin's a convenient place for 
 introducing them to the reader. 
 
CHAPTER VIII. 
 
 OF THE PARADOX OP UNIVERSAL PROPOSITIONS 
 OBTAINED FROM EXPERIENCE. 
 
 i. TT was formerly stated 1 that experience cannot 
 JL establish any universal or necessary truths. The 
 number of trials which we can make of any proposi- 
 tion is necessarily limited, and observation alone cannot 
 give us any ground of extending the inference to 
 untried cases. Observed 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. 
 
 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 approaches 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 mathema- 
 ticians 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 princi- 
 ples of Mechanics will allow that some at least of the 
 primary laws of motion approach very near to this 
 character of necessary truth j and will confess that it 
 would be difficult to imagine any other consistent 
 
 1 B. L c. iv. Of Experience 
 
264 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 reasoning concerning the double stars; al- 
 though 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? 
 
 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 present 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. Experience, 
 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 contradiction, 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 unsuspected 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 
 
PARADOX OF UNIVERSAL PROPOSITIONS. 265 
 
 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 precarious 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 obser- 
 vation must be limited, and cannot testify concerning 
 anything which is beyond its limits. A general view 
 of our faculties appears to prove 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 approach- 
 ing 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 Causa- 
 tion. Those axioms, being exhibitions of the Idea of 
 Cause under various aspects, are of the most rigorous 
 universality 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 observation 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 
 
266 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 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 experiment 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 experience, pro- 
 pounded only such laws as were vague and inapplicable. 
 But yet these persons manifested the indestructible 
 conviction, belonging to man's speculative nature, that 
 there exist Laws of motion, that is, universal 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 cer- 
 tain 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 2 erred in interpreting the equa- 
 lity of action and reaction upon a lever in motion. In 
 each of these instances, true doctrines, already esta- 
 blished, (whether by experiment or otherwise,) were 
 erroneously applied. And the error was corrected by 
 further reflection, which pointed out that another mode 
 of interpretation was requisite, in order that the axiom 
 
 2 Hist. Ind. Sc. b. vL c. v. sect 2. 
 
PARADOX OF UNIVERSAL PROPOSITIONS. 267 
 
 which, was appealed to in each case might retain its 
 force in the most general sense. And in the reason- 
 ings which avoided or corrected such errors, and which 
 led to substantial general truths, the object of the 
 speculator always was to give to the acknowledged 
 maxims which the Idea of Cause suggested, such a sig- 
 nification as should be consistent with their universal 
 validity. The rule was not accepted as particular at 
 the outset, and afterwards generalized more and more 
 widely; but from the very first, the universality of 
 the rule was assumed, and the question was, how it 
 should be understood so as to be - universally true. 
 At every stage of speculation, the law was regarded as 
 a general law. This was not an aspect which it gradu- 
 ally acquired, by the accumulating contributions of 
 experience, but a feature of its original and native 
 character. What should happen universally, experience 
 might be needed to show: but that what happened 
 should happen universally, was implied in the nature 
 of knowledge. The universality of the laws of motion 
 was not gathered from experience, however much the 
 laws themselves might be so. 
 
 3. Thus we obtain the solution of our Paradox, so 
 far as the case before us is concerned. The laws of 
 motion borrow their form from the Idea of Causation, 
 though their matter may be given by experience : and 
 hence they possess a universality which experience can- 
 not give. They are certainly and universally valid; 
 and the only question for observation to decide is, how 
 they are to be understood. They are like general 
 mathematical formulae, which are known to be true, 
 even while we are ignorant what are the unknown 
 quantities which they involve. It must be allowed, 
 on the other hand, that so long as these formulae are 
 not interpreted by a real study of nature, they are not 
 only useless but prejudicial ; filling men's minds with 
 vague general terms, empty maxims, and unintelligible 
 abstractions, which they mistake for knowledge. Of 
 such perversion of the speculative propensities of man's 
 nature, the world has seen too much in all ages. Yet 
 we must not, on that account, despise these forms of 
 
268 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 truth, since without them, no general knowledge is 
 possible. Without general terms, and maxims, and ab- 
 stractions, we can have no science, no speculation ; hardly, 
 indeed, consistent thought or the exercise of reason. 
 The course of real knowledge is, to obtain from thought 
 and experience the right interpretation of our general 
 terms, the real import of our maxims, the true gene- 
 ralizations which our abstractions involve. 
 
 4. If it be asked, How Experience is able to teach 
 us to interpret aright the general terms which the 
 Axioms of Causation involve ; whence she derives the 
 light which she is to throw on these general notions ; 
 
 , the answer is obvious; namely, that the relations of 
 causation are the conditions of Experience; that the 
 general notions are exemplified in the particular cases 
 of which she takes cognizance. The events which 
 take place about us, and which are the objects of our 
 observation, we cannot conceive otherwise than as sub- 
 ject to the laws of cause and effect. Every event must 
 have a cause; Every effect must be determined by 
 its cause; these maxims are true of the phenomena 
 which form the materials of our experience. It is pre- 
 cisely to them, that these truths apply. It is in the 
 world which we have before our eyes, that these pro- 
 positions are universally verified; and it is therefore 
 by the observation of what we see, that we must learn 
 how these propositions are to be understood. Every 
 fact, every experiment, is an example of these state- 
 ments; and it is therefore by attention to and fa- 
 miliarity with facts and experiments, that we learn 
 the signification of the expressions in which the state- 
 ments are made; just as in any other case we learn 
 the import of language by observing the manner in 
 which it is applied in known cases. Experience is 
 the interpreter of nature ; it being understood that she 
 is to make her interpretation in that comprehensive 
 phraseology which is the genuine language of science. 
 
 5. We may return for an instant to the objection, 
 that experience cannot give us general truths, since, 
 after any number of trials confirming a rule, we may 
 for aught we can foresee, have one which violates the 
 
PARADOX OF UNIVERSAL PROPOSITIONS. 269 
 
 rule. When we have seen a thousand stones fall to the 
 ground, we may see one which does not fall under the 
 same apparent circumstances. How then, it is asked, 
 can experience teach us that all stones, rigorously 
 speaking, will fall if unsupported? And to this we 
 reply, that it is not true that we can conceive one 
 stone to be suspended in the air, while a thousand 
 others fall, without believing some peculiar cause to 
 support it; and that, therefore, such a supposition 
 forms no exception to the law, that gravity is a force 
 by which all bodies are urged downwards. Undoubt- 
 edly we can conceive a body, when dropt or thrown, to 
 move in a line quite different from other bodies : thus 
 a certain missile 3 used by the natives of Australia, and 
 lately brought to this country, when thrown from the 
 hand in a proper manner, describes a curve, and returns 
 to the place from whence it was thrown. But did any 
 one, therefore, even for an instant suppose that the 
 laws of motion are different for this and for other 
 bodies'? On the contrary, was not every person of a 
 speculative turn immediately led to inquire how it was 
 that the known causes which modify motion, the resist- 
 ance of the air and the other causes, produced in this 
 instance so peculiar an effect? And if the motion had 
 been still more unaccountable, it would not have occa- 
 sioned any uncertainty whether it were consistent with 
 the agency of gravity and the laws of motion. If a 
 body suddenly alter its direction, or move in any other 
 unexpected manner, we never doubt that there is a 
 cause of the change. We may continue quite ignorant 
 of the nature of this cause, but this ignorance never 
 occasions a moment's doubt that the cause exists and 
 is exactly suited to the effect. And thus experience 
 can prove or discover to us general rules, but she can 
 never prove that general rules do not exist. Anoma- 
 lies, exceptions, unexplained phenomena, may remind 
 us that we have much still to learn, but they can 
 never make us suppose that truths are not universal. 
 We may observe facts that show us we have not fully 
 
 Called the Bo-me-rang. 
 
2/O PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 understood the meaning of our general laws, but we 
 can never find facts which show our laws to have no 
 meaning. Our experience is bound in by the limits of 
 cause and effect, and can give us no information con- 
 cerning any region where that relation does not prevail. 
 The whole series of external occurrences and objects, 
 through all time and space, exists only, and is con- 
 ceived only, as subject to this relation; and therefore 
 we endeavour in vain to imagine to ourselves when and 
 where and how exceptions to this relation may occur. 
 The assumption of the connexion of cause and effect is 
 essential to our experience, as the recognition of the 
 maxims which express this connexion is essential to our 
 knowledge. 
 
 6. I have thus endeavoured to explain in some mea- 
 sure how, at least in the field of our mechanical 
 knowledge, experience can discover universal truths, 
 though she cannot give them their universality; and 
 how such truths, though borrowing their form from 
 our ideas, cannot be understood except by the actual 
 study of external nature. And thus with regard to 
 the laws of motion, and other fundamental principles 
 of Mechanics, the analysis of our ideas and the history 
 of the progress of the science well illustrate each other. 
 
 If the paradox of the discovery of universal truths 
 by experience be thus solved in one instance, a much 
 wider question offers itself to us ; How far the diffi- 
 culty, and how far the solution, are applicable to other 
 subjects. It is easy to see that this question involves 
 most grave and extensive doctrines with regard to the 
 whole compass of human knowledge : and the views to 
 which we have been led in the present Book of this 
 work are, we trust, fitted to throw much light upon 
 the general aspect of the subject. But after discus- 
 sions so abstract, and perhaps obscure, as those in 
 which we have been engaged for some chapters, I 
 willingly postpone to a future occasion an investiga- 
 tion which may perhaps appear to most readers more 
 recondite and difficult still. And we have, in fact, 
 many other special fields of knowledge to survey, 
 before we are led by the order of our subject, to 
 
PARADOX OF UNIVERSAL PROPOSITIONS. 2JI 
 
 those general questions and doctrines, those antitheses 
 brought into view and again resolved, which a view of 
 the whole territory of human knowledge suggests, and 
 by which the nature and conditions of knowledge are 
 exhibited. 
 
 Before we quit the subject of mechanical science we 
 shall make a few remarks on another doctrine which 
 forms part of the established truths of the science, 
 namely, the doctrine of universal gravitation. 
 
CHAPTER IX. 
 
 OF THE ESTABLISHMENT OF THE LAW OF UNIVERSAL 
 GRAVITATION. 
 
 fTlHE doctrine of universal gravitation is a feature 
 JL of so much importance in the history of science 
 that we shall not pass it by without a few remarks on 
 the nature and evidence of the doctrine. 
 
 i. To a certain extent the doctrine of the attrac- 
 tion of bodies according to the law of the inverse 
 square of the distance, exhibits in its progress among 
 men the same general features which we have noticed 
 in the history of the laws of motion. This doctrine 
 was maintained & priori on the ground of its sim- 
 plicity, and was asserted positively, even before it was 
 clearly understood: notwithstanding this anticipa- 
 tion, its establishment on the ground of facts was 
 a task of vast labour and sagacity: when it had 
 been so established in a general way, there occurred 
 at later periods, an occasional suspicion that it might 
 be approximately true only : these suspicions led to 
 further researches, which showed the rule to be 
 rigorously exact: and at present there are mathe- 
 maticians who maintain, not only that it is true, 
 but that it is a necessary property of matter. A very 
 few words on each of these points will suffice. 
 
 2. I have shown in the History of Science 1 , that the 
 attraction of the sun according to the inverse square 
 of the distance, had been divined by Bullialdus, 
 Hooke, Halley, and others, before it was proved 
 by Newton. Probably the reason which suggested 
 this conjecture was, that gravity might be considered 
 
 B. vii. c. L 
 
LAW OF UNIVERSAL GRAVITATION. 273 
 
 as a sort of emanation; and that thus, like light or 
 any other effect diffused from a center, it must follow 
 the law just stated, the efficacy of the force being 
 weakened in receding from the center, exactly in 
 proportion to the space through which it is diffused. 
 It cannot be denied that such a view appears to be 
 strongly recommended by analogy. 
 
 When it had been proved by Newton that the 
 planets were really retained in their elliptical orbits 
 by a central force, his calculations also showed that 
 the above-stated law of the force must be at least 
 very approximately correct, since otherwise the aphelia 
 of the orbits could not be so nearly at rest as they 
 were. Yet when it seemed as if the motion of the 
 moon's apogee could not be accounted for without 
 some new supposition, the a priori argument in favour 
 of the inverse square did not prevent Clairaut from 
 trying the hypothesis of a small term added to that 
 which expressed the ancient law : but when, in order to 
 test the accuracy of this hypothesis, the calculation of the 
 motion of the moon's apogee was pushed to a greater 
 degree of exactness than had been obtained before, it 
 was found that the new term vanished of itself; and 
 that the inverse square now accounted for the whole of 
 the motion. And thus, as in the case of the second law 
 of motion, the most scrupulous examination terminated 
 in showing the simplest rule to be rigorously true. 
 
 3. Similar events occurred in the history of another 
 part of the law of gravitation: namely, that the at- 
 traction is proportional to the quantity of matter 
 attracted. This part of the law may also be thus 
 stated, That the weight of bodies arising from gravity 
 is proportional to their inertia; and thus, that the 
 accelerating force on all bodies under the same cir- 
 cumstances is the same. Newton made experiments 
 which proved this with regard to terrestrial bodies; 
 for he found that, at the end of equal strings, balls of 
 all substances, gold, silver, lead, glass, wood, &c., oscil- 
 lated in equal times 2 . But a few years ago, doubts 
 
 2 Prin. lib. iii. prop. 6. 
 VOL. I. 
 
274 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 arose among the German astronomers whether this 
 law was rigorously true with regard to the planetary 
 bodies. Some calculations appeared to prove, that the 
 attraction of Jupiter as shown by the perturbations 
 which he produces in the small planets Juno, Yesta, 
 and Pallas, was different from the attraction which he 
 exerts on his own satellites. Nor did there appear to 
 these philosophers anything inconceivable in the suppo- 
 sition that the attraction of a planet might be thus 
 elective. But when Mr. Airy obtained a more exact 
 determination of the mass of Jupiter, as indicated by 
 his effect on his satellites, it was found that this sus- 
 picion was unfounded; and that there was, in this 
 case, no exception to the universality of the rule, that 
 this cosmical attraction is in the proportion of the 
 attracted mass. 
 
 4. Again: when it had thus been shown that a 
 mutual attraction of parts, according to the law above 
 mentioned, prevailed throughout the extent of the solar 
 system, it might still be doubted whether the same law 
 extended to other regions of the universe. It might 
 have been perhaps imagined that each fixed star had 
 its peculiar law of force. But the examination of the 
 motions of double stars about each other, by the two 
 Herschels and others, appears to show that these bodies 
 describe ellipses as the planets do; and thus extends 
 the law of the inverse squares to parts of the uni- 
 verse immeasurably distant from the whole solar sys- 
 tem. 
 
 5. Since every doubt which has been raised with 
 regard to the universality and accuracy of the law of 
 gravitation, has thus ended in confirming the rule, it is 
 not surprizing that men's minds should have returned 
 with additional force to those views which had at first 
 represented the law as a necessary truth, capable of 
 being established by reason alone. When it had been 
 proved by Newton that gravity is really a universal 
 attribute of matter as far as we can learn, his pupils 
 were not content without maintaining it to be an essen- 
 tial quality. This is the doctrine held by Cotes in the 
 preface to the second edition of the Principia (1712) : 
 
LAW OF UNIVERSAL GRAVITATION. 2/5 
 
 ' Gravity,' he says, ' is a primary quality of bodies, as 
 extension, mobility, and impenetrability are.' But 
 Newton himself by no means went so far. In his 
 second Letter to Bentley (1693), he says, 'You some- 
 times speak of gravity as essential and inherent to mat- 
 ter; pray do not ascribe that notion to me. The cause 
 of gravity,' he adds, 'I do not pretend to know, and 
 would take more time to consider of it.' 
 
 Cotes maintains his opinion by urging, that we learn 
 by experience that all bodies possess gravity, and that 
 we do not learn in any other way that they are ex- 
 tended, moveable, or solid. But we have already seen, 
 that the ideas of space, time, and reaction, on which 
 depend extension, mobility, and solidity, are not re- 
 sults, but conditions, of experience. We cannot con- 
 ceive a body except as extended; we cannot conceive 
 it to exert mechanical action except with some kind 
 of solidity. But so far as our conceptions of body 
 have hitherto been developed, we find no difficulty in 
 conceiving two bodies which do not attract each other. 
 
 6. Newton lays down, in the second edition of the 
 Principia, this 'Rule of Philosophizing' (book iii.); 
 that ' The qualities of bodies which cannot be made 
 more or less intense, and which belong to all bodies on 
 which we are able to make experiments, are to be held 
 to be qualities of all bodies in general.' And this Rule 
 is cited in the sixth Proposition of the Third Book of 
 the Principia, (Cor. 2,) in order to prove that gravity, 
 proportional to the quantity of matter, may be asserted 
 to be a quality of all bodies universally. But we may 
 remark that a Rule of Philosophizing, itself of preca- 
 rious authority, cannot authorize us in ascribing uni- 
 versality to an empirical result. Geometrical and 
 statical properties are seen to be necessary, and there- 
 fore universal : but Newton appears disposed to assert 
 a like universality of gravity, quite unconnected with 
 any necessity. It would be a very inadequate state- 
 ment, indeed a false representation, of statical truth, if 
 we were to say, that because every body which has 
 hitherto been tried has been found to have a center of 
 gravity, we venture to assert that all bodies whatever 
 
 T 2 
 
2/6 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 have a center of gravity. And if we are ever able to 
 assert the absolute universality of the law of gravita- 
 tion, we shall have /to rest this truth upon the clearer 
 development of our ideas of matter and force ; not upon 
 a Rule of Philosophizing, which, till otherwise proved, 
 must be a mere rule of prudence, and which the oppo- 
 nent may refuse to admit. 
 
 7. Other persons, instead of asserting gravity to be 
 in its own nature essential to matter, have made hypo- 
 theses concerning some mechanism or other, by which 
 this mutual attraction of bodies is produced 3 . Thus 
 the Cartesians ascribed to a vortex the tendency of 
 bodies to a center ; Newton himself seems to have been 
 disposed to refer this tendency to the elasticity of an 
 ether; Le Sage propounded a curious hypothesis, in 
 which this attraction is accounted for by the impulse 
 of infinite streams of particles flowing constantly 
 through the universe in all directions. In these specu- 
 lations, the force of gravity is resolved into the pressure 
 or impulse of solids or fluids. On the other hand, 
 hypotheses have been propounded, in which the solidity, 
 and other physical qualities of bodies, have been ex- 
 plained by representing the bodies as a collection of 
 points, from which points, repulsive, as well as attrac- 
 tive, forces emanate. This view of the constitution of 
 bodies was maintained and developed by Boscovich, 
 and is hence termed ' Boscovich's Theory :' and the dis- 
 cussion of it will more properly come under our review 
 at a future period, when we speak of the question 
 whether bodies are made up of atoms. But we may 
 observe, that Newton himself appears to have inclined, 
 as his followers certainly did, to this mode of COD tern- 
 plating the physical properties of bodies. In his Pre- 
 face to the Principia, after speaking of the central 
 forces which are exhibited in cosmical phenomena, he 
 says: 'Would that we could derive the other pheno- 
 mena of Nature from mechanical principles by the 
 same mode of reasoning. For many things move me 
 
 3 See Vince, Observations on the Hypothesis respecting Gravitation, and the 
 Critique of that work, Edinb. Rev. voL xiii 
 
LAW OF UNIVERSAL GRAVITATION. 277 
 
 so that I suspect all these phenomena may depend upon 
 certain forces, by which the particles of bodies, through 
 causes not yet known, are either impelled to each other 
 and cohere according to regular figures, or are repelled 
 and recede from each other: which forces being un- 
 known, philosophers have hitherto made their attempts 
 upon nature in vain.' 
 
 8. But both these hypotheses; that by which 
 cohesion and solidity are reduced to attractive and 
 repulsive forces, and that by which attraction is reduced 
 to the impulse and pressure of media; are hitherto 
 merely modes of representing mechanical laws of 
 nature ; and cannot, either of them, be asserted as pos- 
 sessing any evident truth or peremptory authority to 
 the exclusion of the other. This consideration may 
 enable us to estimate the real weight of the difficulty 
 felt in assenting to the mutual attraction of bodies not 
 in contact with each other; for it is often urged that 
 this attraction of bodies at a distance is an absurd sup- 
 position. 
 
 The doctrine is often thus stigmatized, both by 
 popular and by learned writers. It was long received 
 as a maxim in philosophy (as Monboddo informs us 4 ), 
 that a body cannot act where it is not, any more than 
 when it is not. But to this we reply, that time is a 
 necessary condition of our conception of causation, in a 
 different manner from space. The action of force can 
 only be conceived as taking place in a succession of 
 moments, in each of which cause and effect immedi- 
 ately succeed each other: and thus the interval of 
 time between a cause and its remote effect is filled up 
 by a continuous succession of events connected by the 
 same chain of causation. But in space, there is no 
 such visible necessity of continuity; the action and 
 reaction may take place at a distance from each other; 
 all that is necessary being that they be equal and 
 opposite. 
 
 Undoubtedly the existence of attraction is rendered 
 more acceptable to common apprehension by supposing 
 
 Ancient Metaphysics, vol. ii. p. 175. 
 
2/8 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 some intermediate machinery, a cord, or rod, or fluid, 
 by which the forces may be conveyed from one point 
 to another. But such images are rather fitted to satisfy 
 those prejudices which arise from the earlier application 
 of our ideas of force, than to exhibit the real nature of 
 those ideas. If we suppose two bodies to pull each other 
 by means of a rod or cord, we only suppose, in addi- 
 tion to those equal and opposite forces acting upon the 
 two bodies, (which forces are alone essential to mutual 
 attraction) a certain power of resisting transverse pres- 
 sure at every point of the intermediate line : which 
 additional supposition is entirely useless, and quite 
 unconnected with the essential conditions of the case. 
 When , the Newtonians were accused of introducing 
 into philosophy an unknown cause which they termed 
 attraction, they justly replied that they knew as much 
 respecting attraction as their opponents did about im- 
 pulse. In each case we have a knowledge of the con- 
 ception in question so far as we clearly apprehend it 
 under the conditions of those axioms of mechanical 
 causation which form the basis of our science on such 
 subjects. 
 
 Having thus examined the degree of certainty and 
 generality to which our knowledge of the law of uni- 
 versal gravitation has been carried, by the progress of 
 mechanical discovery and speculation up to the pre- 
 sent time, we might proceed to the other branches of 
 science, and examine in like manner their grounds and 
 conditions. But before we do this, it will be worth 
 our while to attend for a moment to the effect which 
 the progress of mechanical ideas among mathemati- 
 cians and mechanical philosophers has produced upon 
 the minds of other persons, who share only in an 
 indirect and derivative manner in the influence of 
 science. 
 
CHAPTER X. 
 
 OF THE GENERAL DIFFUSION OF CLEAR 
 MECHANICAL IDEAS. 
 
 T. ~\T7"E have seen how the progress of knowledge 
 VY upon the subject of motion and force has 
 produced, in the course of the world's history, a great 
 change in the minds of acute and speculative men; so 
 that such persons can now reason with perfect steadi- 
 ness and precision upon subjects on which, at first, 
 their thoughts were vague and confused; and can ap- 
 prehend, as truths of complete certainty and evidence, 
 laws which it required great labour and time to dis- 
 cover. This complete development and clear manifes- 
 tation of mechanical ideas has taken place only among 
 mathematicians and philosophers. But yet a progress 
 of thought upon such subjects, an advance from the 
 obscure to the clear, and from errour to truth, may 
 be traced in the world at large, and among those who 
 have not directly cultivated the exact sciences. This 
 diffused and collateral influence of science manifests 
 itself, although in a wavering and fluctuating manner, 
 by various indications, at various periods of literary 
 history. The opinions and reasonings which are put 
 forth upon mechanical subjects, and above all, the 
 adoption, into common language, of terms and phrases 
 belonging to the prevalent mechanical systems, exhibit 
 to us the most profound discoveries and speculations 
 of philosophers in their effect upon more common 
 and familiar trains of thought. This effect is by no 
 means unimportant, and we shall point out some 
 examples of such indications as we have mentioned. 
 
 2. The discoveries of the ancients in speculative 
 mechanics were, as we have seen, very scanty; and 
 
28O PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 hardly extended their influence to the unmathematical 
 world. Yet the familiar use of the term ' center of 
 gravity' preserved and suggested the most important 
 part of what the Greeks had to teach. The other 
 phrases which they employed, as momentum, energy, 
 virtue, force, and the like, never had any exact mean- 
 ing, even among mathematicians ; and therefore never, 
 in the ancient world, became the means of suggesting 
 just habits of thought. I have pointed out, in the 
 History of Science, several circumstances which appear 
 to denote the general confusion of ideas which prevailed 
 upon mechanical subjects during the times of the Ro- 
 man empire. I have there taken as one of the examples 
 of this confusion, the fable narrated by Pliny and 
 others concerning the echinei's, a small fish, which was 
 said to stop a ship merely by sticking to it 1 . This 
 story was adduced as betraying the absence of any 
 steady apprehension of the equality of action and re- 
 action ; since the fish, except it had some immoveable 
 obstacle to hold by, must be pulled forward by the 
 ship, as much as it pulled the ship backward. If the 
 waiters who speak of this wonder had shown any per- 
 ception of the necessity of a reaction, either produced 
 by the rapid motion of the fish's fins in the water, or 
 in any other way, they would not be chargeable with 
 this confusion of thought; but from their expressions 
 it is, I think, evident that they saw no such necessity 2 . 
 Their idea of mechanical action was not sufficiently 
 distinct to enable them to see the absurdity of suppos- 
 
 1 Hist. Ind. Sc. b. iv. c. L sect. 2. muscular power acting on the water, 
 
 2 See Prof. Powell, On the Nature we may take what Pliny says, Nat. 
 and Evidence of the Laws of Motion. Hist, xxxii. i, 'Domat mundi rabiem, 
 Reports of the Ashmolean Society, nullo suo labore ; non retinendo, aut 
 Oxford. 1837. Professor Powell has alio modo quam adhserendo :' and 
 made an objection to my use of this also what he states in another place 
 instance of confusion of thought ; the (ix. 41), that when it is preserved in 
 remark in the text seems to me to pickle, it may be used in recovering 
 justify what I said in the History, gold which has fallen into a deep 
 As an evidence that the fish was not well All this implies adhesion alone, 
 supposed to produce its effect by its with no conception of reaction. 
 
DIFFUSION OF CLEAR MECHANICAL IDEAS. 28 1 
 
 ing an intense pressure with no obstacle for it to exert 
 itself against. 
 
 3. We may trace, in more modern times also, indi- 
 cations of a general ignorance of mechanical truths. 
 Thus the phrase of shooting at an object 'point-blank,' 
 implies the belief that a cannon-ball describes a path 
 of which the first portion is a straight line. This 
 errour was corrected by the true mechanical principles 
 which Galileo and his followers brought to light; but 
 these principles made their way to popular notice, 
 principally in consequence of their application to the 
 motions of the solar system, and to the controversies 
 which took place respecting those motions. Thus by 
 far the most powerful argument against the reception 
 of the Copernican system of the universe, was that of 
 those who asked, Why a stone dropt from a tower was 
 not left behind by the motion of the earth? The 
 answer to this question, now universally familiar, in- 
 volves a reference to the true doctrine of the compo- 
 sition of motions. Again ; Kepler's persevering and 
 strenuous attempts 3 to frame a physical theory of the 
 universe were frustrated by his ignorance of the first 
 law of motion, which informs us that a body will 
 retain its velocity without any maintaining force. He 
 proceeded upon the supposition that the sun's force 
 was requisite to keep up the motion of the planets, as 
 well as to deflect and modify it; and he was thus led 
 to a system which represented the sun as carrying 
 round the planets in their orbits by means of a vortex, 
 produced by his revolution. The same neglect of the 
 laws of motion presided in the formation of Descartes' 
 system of vortices. Although Descartes had enunci- 
 ated in words the laws of motion, he and his followers 
 showed that they had not the practical habit of refer- 
 ring to these mechanical principles; and dared not 
 trust the planets to move in free space without some 
 surrounding machinery to support them 4 . 
 
 3 Hid. Ind. Sc. b. v. c. iv. and b. vii. Descartes the character which Bacon 
 
 c. i. gives to Aristotle, 'Audax simul et 
 
 * I have, in the History, applied to pavidus :' though he was bold enotigh 
 
282 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 i 
 
 4. When at last mathematicians, following Newton, 
 had ventured to consider the motion of each planet as 
 a mechanical problem not different in its nature from 
 the motion of a stone cast from the hand ; and when 
 the solution of this problem and its immense conse- 
 quences had become matters of general notoriety and 
 interest; the new views introduced, as is usual, new 
 terms, which soon became extensively current. We 
 meet with such phrases as ' flying off in the tangent,' 
 and ' deflexion from the tangent \ ' with antitheses be- 
 tween ' centripetal ' and ' centrifugal force,' or between 
 1 projectile' and 'central force.' 'Centers of force,' 
 ' disturbing forces,' ' perturbations,' and ' perturbations 
 of higher orders,' are not unfrequeiitly spoken of: and 
 the expression ' to gravitate,' and the term ' universal 
 gravitation,' acquired a permanent place in the lan- 
 guage. 
 
 Yet for a long time, and even up to the present day, 
 we find many indications that false and confused ap- 
 prehensions on such subjects are by no means extir- 
 pated. Arguments are urged against the mechanical 
 system of the universe, implying in the opponents an 
 absence of all clear mechanical notions. Many of this 
 class of writers retrograde to Kepler's point of view. 
 This is, for example, the case with Lord Monboddo, 
 who, arguing on the assumption that force is requisite 
 to maintain, as well as to deflect motion, produced a 
 series of attacks upon the Newtonian philosophy ; 
 which he inserted in his Ancient Metaphysics, pub- 
 lished in 1779 and the succeeding years. This writer 
 (like Kepler), measures force by the velocity which the 
 body has 5 , not by that which it gains. Such a use of 
 language would prevent our obtaining any laws of mo- 
 tion at all. Accordingly, the author, in the very next 
 page to that which I have just quoted, abandons this 
 measure of force, and, in curvilinear motion, measures 
 
 to enunciate the laws of motion with- describe their orbits by the agency of 
 out knowing them aright, he had not those laws, without the machinery of 
 the courage to leave the planets to contact. 
 
 6 Anc. Met. voL ii. b. v. c. vL p. 413. 
 
DIFFUSION OF CLEAR MECHANICAL IDEAS. 283 
 
 force by 'the fall from the extremity of the arc.' 
 Again; in his objections to the received theory, he de- 
 nies that curvilinear motion is compounded, although 
 his own mode of considering such motion assumes this 
 composition in the only way in which it was ever 
 intended by mathematicians. Many more instances 
 might be adduced to show that a want of cultivation 
 of the mechanical ideas rendered this philosopher in- 
 capable of judging of a mechanical system. 
 
 The following extract from the Ancient Metaphysics, 
 may be sufficient to show the value of the author's 
 criticism on the subjects of which we are now speak- 
 ing. His object is to prove that there do not exist a 
 centripetal and a centrifugal force in the case of ellip- 
 tical motion. 'Let any man move in a circular or 
 elliptical line described to him; and he will find no 
 tendency in himself either to the center or from it, 
 much less both. If indeed he attempt to make the 
 motion with great velocity, or if he do it carelessly and 
 inattentively, he may go out of the line, either towards 
 the center or from it : but this is to be ascribed, not to 
 the nature of the motion, but to our infirmity; or per- 
 haps to the animal form, which is more fitted for pro- 
 gressive motion in a right line than for any kind of 
 curvilinear motion. But this is not the case with a 
 sphere or spheroid, which is equally adapted to motion 
 in all directions 6 .' We need hardly remind the reader 
 that the manner in which a man running round a 
 small circle, finds it necessary to lean inwards, in order 
 that there may be a centripetal inclination to counter- 
 act the centrifugal force, is a standard example of our 
 mechanical doctrines; and this fact (quite familiar in 
 practice as well as theory) is in direct contradiction of 
 Lord Monboddo's assertion. 
 
 5. A similar absence of distinct mechanical thought 
 appears in some of the most celebrated metaphysicians 
 of Germany. I have elsewhere noted 7 the opinion ex- 
 pressed by Hegel, that the glory which belongs to 
 
 6 Anc. Met. vol. i. b. ii c. 19, p. 264. 
 
 7 Hist. Ind. Sc. b. vil c. ii. sect. 5. 
 
284 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 Kepler has been unjustly transferred to Newton ; and 
 I have suggested, as the explanation of this mode of 
 thinking, that Hegel himself, in the knowledge of me- 
 chanical truth, had not advanced beyond Kepler's 
 point of vie,w. Persons who possess conceptions of 
 space and number, but who have not learnt to deal 
 with ideas of force and causation, may see more value 
 in the discoveries of Kepler than in those of Newton. 
 Another exemplification of this state of mind may be 
 found in Professor Schelling's speculations; for instance, 
 in his Lectures on the Method of Academical Study. In 
 the twelfth Lecture, on the study of Physics and Che- 
 mistry, he says, (p. 266,) 'What the mathematical 
 natural philosophy has done for the knowledge of the 
 laws of the universe since the time that they were dis- 
 covered by his (Kepler's) godlike genius, is, as is well 
 known, this : it has attempted a construction of those 
 laws which, according to its foundations, is altogether 
 empirical. We may assume it as a general rule, that 
 in any proposed construction, that which is not a pure 
 general form cannot have any scientific import or 
 truth. The foundation from which the centrifugal mo- 
 tion of the bodies of the world is derived, is no neces- 
 sary form, it is an empirical fact. The Newtonian 
 attractive force, even if it be a necessary assumption 
 for a merely reflective view of the subject, is still of no 
 significance for the Reason, which recognizes only 
 absolute relations. The grounds of the Keplerian laws 
 can be derived, without any empirical appendage, 
 purely from the doctrine of Ideas, and of the two Uni- 
 ties, which are in themselves one Unity, and in virtue 
 of which each being, while it is absolute in itself, is at 
 the same time in the absolute, and reciprocally.' 
 
 It will be observed, that in this passage our mecha- 
 nical laws are objected to because they are not neces- 
 sary results of our ideas ; which, however, as we have 
 seen, according to the opinion of some eminent me- 
 chanical philosophers, they are. But to assume this 
 evident necessity as a condition of every advance in 
 science, is to mistake the last, perhaps unattainable 
 step, for the first, which lies before our feet. And, 
 
DIFFUSION OF CLEAR MECHANICAL IDEAS. 285 
 
 "without inquiring further about 'the Doctrine of the 
 two Unities,' or the manner in which from that doc- 
 trine we may deduce the Keplerian laws, we may be 
 well convinced that such a doctrine cannot supply any 
 sufficient reason to induce us to quit the inductive 
 path by which all scientific truth up to the present 
 time has been acquired. 
 
 6. But without going to schools of philosophy op- 
 posed to the Inductive School, we may find many loose 
 and vague habits of thinking on mechanical subjects 
 among the common classes of readers and reasoners. 
 And there are some familiar modes of employing the 
 phraseology of mechanical science, which are, in a cer- 
 tain degree, chargeable with inaccuracy, and may pro- 
 duce or perpetuate confusion. Among such cases we 
 may mention the way in which the centripetal and 
 centrifugal forces, and also the projectile and central 
 forces of the planets, are often compared or opposed. 
 Such antitheses sometimes proceed upon the false no- 
 tion that the two members of these pairs of forces are 
 of the same kind : whereas on the contrary the projec- 
 tile force is a hypothetical impulsive force which may, 
 at some former period, have caused the motion to 
 begin; while the central force is an actual force, which 
 must act continuously and during the whole time of 
 the motion, in order that the motion may go on in the 
 curve. In the same manner the centrifugal force is 
 not a distinct force in a strict sense, but only a certain 
 result of the first law of motion, measured by the por- 
 tion of centripetal force which counteracts it. Com- 
 parisons of quantities so heterogeneous imply confusion 
 of thought, and often suggest baseless speculations and 
 imagined reforms of the received opinions. 
 
 7. I might point out other terms and maxims, in 
 addition to those already mentioned, which, though 
 formerly employed in a loose and vague manner, are 
 now accurately understood and employed by all just 
 thinkers; and thus secure and diffuse a right under- 
 standing of mechanical truths. Such are momentum, 
 inertia, quantity of matter, quantity of motion; that 
 force is proportional to its effects; that action and 
 
286 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 reaction are equal; that what is gained in force by 
 machinery is lost in time; that the quantity of motion 
 in the world cannot be either increased or diminished. 
 When the expression of the truth thus becomes easy 
 and simple, clear and convincing, the meanings given 
 to words and phrases by discoverers glide into the 
 habitual texture of men's reasonings, and the effect of 
 the establishment of true mechanical principles is felt 
 far from the school of the mechanician. If these terms 
 and maxims are understood with tolerable clearness, 
 they carry the influence of truth to those who have no 
 direct access to its sources. Many an extravagant pro- 
 ject in practical machinery, and many a wild hypo- 
 thesis in speculative physics, has been repressed by 
 the general currency of such maxims as we have just 
 quoted. 
 
 8. Indeed so familiar and evident are the ele- 
 mentary truths of mechanics when expressed in this 
 simple form, that they are received as truisms; and 
 men are disposed to look back with surprise and scorn 
 at the speculations which were carried on in neglect of 
 them. The most superficial reasoner of modern times 
 thinks himself entitled to speak with contempt and 
 ridicule of Kepler's hypothesis concerning the physical 
 causes of the celestial motions : and gives himself credit 
 for intellectual superiority, because he sees, as self- 
 evident, what such a man could not discover at all. It 
 is well for such a person to recollect, that the real cause 
 of his superior insight is not the pre-eminence of his 
 faculties, but the successful labours of those who have 
 preceded him. The language which he has learnt to 
 use unconsciously, has been adapted to, and moulded 
 on, ascertained truths. When he talks familiarly of 
 "accelerating forces" and "deflexions from the tan- 
 gent," he is assuming that which Kepler did not know, 
 and which it cost Galileo and his disciples so much 
 labour and thought to establish. Language is often 
 called an instrument of thought; but it is also the 
 nutriment of thought; or rather, it is the atmosphere 
 in which thought lives: a medium essential to the 
 activity of our speculative power, although invisible 
 
DIFFUSION OF CLEAR MECHANICAL IDEAS. 287 
 
 and imperceptible in its operation; and an element 
 modifying, by its qualities and changes, the growth and 
 complexion of the faculties which it feeds. In this way 
 the influence of preceding discoveries upon subsequent 
 ones, of the past upon the present, is most penetrating 
 and universal, though most subtle and difficult to trace. 
 The most familiar words and phrases are connected by 
 imperceptible ties with the reasonings and discoveries 
 of former men and distant times. Their knowledge is 
 an inseparable part of ours ; the present generation in- 
 herits and uses the scientific wealth of all the past. 
 And this is the fortune, not only of the great and rich 
 in the intellectual world: of those who have the key 
 to the ancient storehouses, and who have accumulated 
 treasures of their own; but the humblest inquirer, 
 while he puts his reasonings into words, benefits by the 
 labours of the greatest discoverers. When he counts 
 his little wealth, he finds that he has in his hands coins 
 which bear the image and superscription of ancient and 
 modern intellectual dynasties; and that in virtue of 
 this possession, acquisitions are in his power, solid 
 knowledge within his reach, which none could ever 
 have attained to, if it were not that the gold of truth, 
 once dug out of the mine, circulates more and more 
 widely among mankind. 
 
 9. Having so fully examined, in the preceding in- 
 stances, the nature of the progress of thought which 
 science implies, both among the peculiar cultivators of 
 science, and in that wider world of general culture 
 which receives only an indirect influence from scien- 
 tific discoveries, we shall not find it necessary to go 
 into the same extent of detail with regard to the other 
 provinces of human knowledge. In the case of the 
 Mechanical Sciences, we have endeavoured to show, 
 not only that Ideas are requisite in order to form into 
 a science the Facts which nature offers to us, but that 
 we can advance, almost or quite, to a complete identi- 
 fication of the Facts with, the Ideas. In the sciences 
 to which we now proceed, we shall not seek to fill up 
 the chasm by which Facts and Ideas are separated; 
 but we shall endeavour to detect the Ideas which our 
 
288 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 knowledge involves, to show how essential these are ; 
 and in some respects to trace the mode in which they 
 have been gradually developed among men. 
 
 i o. The motions of the heavenly bodies, their laws, 
 their causes, are among the subjects of the first divi- 
 sion of the Mechanical Sciences ; and of these sciences 
 we formerly sketched the history, and have now endea- 
 voured to exhibit the philosophy. If we were to 
 take any other class of motions, t/ieir laws and causes 
 might give rise to sciences which would be mechanical 
 sciences in exactly the same sense in which Physical 
 Astronomy is so. The phenomena of magnets, of elec- 
 trical bodies, of galvanical apparatus, seem to form 
 obvious materials for such sciences ; and if they were so 
 treated, the philosophy of such branches of knowledge 
 would naturally come under our consideration at this 
 point of our progress. 
 
 But on looking more attentively at the sciences of 
 Electricity, Magnetism, and Galvanism, we discover 
 cogent reasons for transferring them to another part of 
 our arrangement ; we find it advisable to associate them 
 with Chemistry, and to discuss their principles when 
 we can connect them with the principles of chemical 
 science. For though the first steps and narrower gene- 
 ralizations of these sciences depend upon mechanical 
 ideas, the highest laws and widest generalizations which 
 we can reach respecting them, involve chemical rela- 
 tions. The progress of these portions of knowledge is 
 in some respects opposite to the progress of Physical 
 Astronomy. In this, we begin with phenomena which 
 appear to indicate peculiar and various qualities in 
 the bodies which we consider, (namely, the heavenly 
 bodies,) and we find in the end that all these qualities 
 resolve themselves into one common mechanical pro- 
 perty, which exists alike in all bodies and parts of 
 bodies. On the contrary, in studying magnetical and 
 electrical laws, we appear at first to have a single ex- 
 tensive phenomenon, attraction and repulsion: but in 
 our attempts to generalize this phenomenon, we find 
 that it is governed by conditions depending upon some- 
 thing quite separate from the bodies themselves, upon 
 
DIFFUSION OF CLEAR MECHANICAL IDEAS. 289 
 
 the presence and distribution of peculiar and transitory- 
 agencies; and, so far as we can discover, the general 
 laws of these agencies are of a chemical nature, and are 
 brought into action by peculiar properties of special 
 substances. In cosmical phenomena, everything, in pro- 
 portion as it is referred to mechanical principles, tends 
 to simplicity, to permanent uniform forces, to one 
 common, positive, property. In magnetical and elec- 
 trical appearances, on the contrary, the application of 
 mechanical principles leads only to a new complexity, 
 which requires a new explanation; and this explana- 
 tion involves changeable and various forces, grada- 
 tions and oppositions of qualities. The doctrine of the 
 universal gravitation of matter is a simple and ultimate 
 truth, in which the mind can acquiesce and repose. 
 We rank gravity among the mechanical attributes of 
 matter, and we see no necessity to derive it from any 
 ulterior properties. Gravity belongs to matter, inde- 
 pendent of any conditions. But the conditions of 
 magnetic or electrical activity require investigation as 
 much as the laws of their action. Of these conditions 
 no mere mechanical explanation can be given; we are 
 compelled to take along with us chemical properties 
 and relations also: and thus magnetism, electricity, 
 galvanism, are meckanico-chemical sciences. 
 
 ii. Before considering these, therefore, I shall 
 treat of what I shall call Secondary Mechanical 
 Sciences; by which expression I mean the sciences 
 depending upon certain qualities which our senses dis- 
 cover to us in bodies ; Optics, which has visible phe- 
 nomena for its subject; Acoustics, the science of hear- 
 ing; the doctrine of Heat, a quality which our touch 
 recognizes : to this last science I shall take the liberty 
 of sometimes giving the name Thermotics, analogous to 
 the names of the other two. If our knowledge of the 
 phenomena of Smell and Taste had been successfully 
 cultivated and systematized, the present part of our 
 work would be the place for the philosophical discus- 
 sion of those sensations as the subjects of science. 
 
 The branches of knowledge thus grouped in one 
 class involve common Fundamental Ideas, from which 
 VOL. i. u 
 
290 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 their principles are derived in a mode analogous, at 
 least in a certain degree, to the mode in which the 
 principles of the mechanical sciences are derived from 
 the fundamental ideas of causation and reaction. We 
 proceed now to consider these Fundamental Ideas, 
 their nature, development, and consequences. 
 
BOOK IV. 
 
 THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 SECONDARY 
 MECHANICAL SCIENCES. 
 
 U 2 
 
ydp TI TOU aia-drjriKov ylverai TO opav vir' avrov 
 ovv TOU 6pwfj.&ov xpc6/AaTos dSiWrov, \direrai 8 viro TOV 
 i5, wcrr' &vayKai6v TI elvai /zeratf* KCVOV 8 yevofj.fr ov ou% 
 s, d\X' 6Xws 01)^^ 6<pdif]<reTcu. Si' rjv ptv o5v alrlav TO 
 dj/a7K:aroj' ^y <j)url 6pa<rdcu, etprjTai. irvp 8^ h d^otv 
 oparai, Kal v ffrirei Kal kv (purl, Kal TOVTO dvdyKrjs' TO yap 
 dicupavts viro TOVTOV ylverai 5ia<pav&. 6 8' ai/ros X670S Kal irepl 
 \f/6(f>ov Kal dew* effriv' ovdkv yap O.VT&V aiTTbuevov T 
 Troiet TV afodT)<ru>, dXX' UTTO fj.fr 6cr/i?js Kal \{/6<pov TO 
 KtveiTai, UTTO 8^ TOI/TOU TWV ala-drjTrjpiw eKdTepof OTav 8' ir avT6 
 TIS eTridfj TO" aiffO-rjT-ripiov TO \f/o^>ovv TJ TO 6ov, oi>8efjdav 
 
 ARISTOT. De Anima, u. 7. 
 
BOOK IV. 
 
 THE PHILOSOPHY OF THE SECONDARY 
 MECHANICAL SCIENCES. 
 
 CHAPTER I. 
 
 OF THE IDEA OF A MEDIUM AS COMMONLY 
 EMPLOYED. 
 
 i. Of Primary and Secondary Qualities. IN the " 
 same way in which the mechanical sciences depend 
 upon the Idea of Cause, and have their principles 
 regulated by the development of that Idea, it will be 
 found that the sciences which have for their subject 
 Sound, Light, and Heat, depend for their principles 
 upon the Funcjamental Idea of Media by means of 
 which we perceive those qualities. Like the idea of 
 cause, this idea of a medium is unavoidably employed, 
 more or less distinctly, in the common, unscientific 
 operations of the understanding; and is recognized as 
 an express principle in the earliest speculative essays 
 of man. But here also, as in the case of the mechani- 
 cal sciences, the development of the idea, and the esta- 
 blishment of the scientific truths which depend upon 
 it, was the business of a succeeding period, and was 
 only executed by means of long and laborious researches, 
 conducted with a constant reference to experiment and 
 observation. 
 
 Among the most prominent manifestations of the in- 
 fluence of the idea of a medium of which we have now to 
 speak, is the distinction of the qualities of bodies into 
 primary, and secondary qualities. This distinction has 
 
294 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 been constantly spoken of in modern times : yet it has 
 often been a subject of discussion among metaphy- 
 sicians whether there be really such a distinction, and 
 what the true difference is. Locke states it thus 1 : 
 Original or Primary qualities of bodies are 'such as 
 are utterly inseparable from the body in what estate 
 soever it may be, such as sense constantly finds in 
 every particle of matter which has bulk enough to be 
 perceived, and the mind finds inseparable from every 
 particle of matter, though less than to make itself 
 singly perceived by our senses:' and he enumerates 
 them as solidity, extension, figure, motion or rest, and 
 number. Secondary qualities, on the other hand, are 
 such * which in truth are nothing in the objects them- 
 selves, but powers to produce various sensations in us 
 by their primary qualities, i.e. by the bulk, figure, 
 texture, and motion of their insensible parts, as 
 colours, sounds, tastes, &c.' 
 
 Dr. Reid 2 , reconsidering this subject, puts the differ- 
 ence in another way. There is, he says, a real foun- 
 dation for the distinction of Primary and Secondary 
 qualities, and it is this: 'That our senses give us a 
 direct and distinct notion of the primary qualities, and 
 inform us what they are in themselves; but of the 
 secondary qualities, our senses give us only a relative 
 and obscure notion. They inform us only that they 
 are qualities that affect us in a certain manner, that is, 
 produce in us a certain sensation; but as to what 
 they are in themselves, our senses leave us in the 
 dark.' 
 
 Dr. Brown 3 states the distinction somewhat other- 
 wise. We give the name of Matter, he observes, to that 
 which has extension and resistance: these, therefore, 
 are Primary qualities of matter, because they compose 
 our definition of it. All other qualities are Secondary, 
 since they are ascribed to bodies only because we find 
 them associated with the primary qualities which form 
 our notion of those bodies. 
 
 Essay, b. iL ch. viii s. g, 10. 2 Essays, b. ii. c. xviL 
 
 3 Lectures, ii. 12. 
 
OF THE IDEA OF A MEDIUM. 295 
 
 It is not necessary to criticize very strictly these 
 various distinctions. If it were, it would be easy to 
 find objections to them. Thus Locke, it may be ob- 
 served, does not point out any reason for believing 
 that his secondary qualities are produced by the pri- 
 mary. How are we to learn that the colour of a rose 
 arises from the bulk, figure, texture, and motion of its 
 particles ? Certainly our senses do not teach us this ; 
 and in what other way, on Locke's principles, can we 
 learn it ? Reid's statement is not more free from the 
 same objection. How does it appear that our notion 
 of Warmth is relative to our own sensations more than 
 our notion of Solidity? And if we take Brown's ac- 
 count, we may still ask whether our selection of certain 
 qualities to form our idea and definition of Matter 
 be arbitrary and without reason? If it be, how can 
 it make a real distinction? if it be not, what is the 
 reason? 
 
 I do not press these objections, because I believe 
 that any of the above accounts of the distinction of 
 Primary and Secondary qualities is right in the main, 
 however imperfect it may be. The difference between 
 such qualities as Extension and Solidity on the one 
 hand, and Colour or Fragrance on the other, is as- 
 sented to by all, with a conviction so firm and inde- . 
 structible, that there must be some fundamental prin- 
 ciple at the bottom of the belief, however difficult it 
 may be to clothe the principle in words. That suc- 
 cessive efforts to express the real nature of the differ- 
 ence were made by inen so clear-sighted and acute 
 as those whom I have quoted, even if none of them 
 are satisfactory, shows how strong and how deeply- 
 seated is the perception of truth which impels us to 
 such attempts. 
 
 The most obvious mode of stating the difference of 
 Primary and Secondary qualities, as it naturally offers 
 itself to speculative minds, appears to be that em- 
 ployed by Locke, slightly modified. Certain of the 
 qualities of bodies, as their bulk, figure, and motion, 
 are perceived immediately in the bodies themselves. 
 Certain other qualities as sound, colour, heat, are per- 
 
2p6 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 ceived by means of some medium. Our conviction 
 that this is the case is spontaneous and irresistible; 
 and this difference of qualities immediately and medi- 
 ately perceived is the distinction of Primary and 
 Secondary qualities. We proceed further to examine 
 this conviction. 
 
 2. The Idea of Externality. In reasoning concern- 
 ing the Secondary Qualities of bodies, we are led to 
 assume the bodies to be external to us, and to be per- 
 ceived by means of some Medium intermediate between 
 us and them. These assumptions are fundamental 
 conditions of perception, inseparable from perception 
 even in thought. 
 
 That objects are external to us, that they are without 
 us, that they have outness, is as clear as it is that 
 these words have any meaning at all. This conviction 
 is, indeed, involved in the exercise of that faculty by 
 which we perceive all things as existing in space ; for 
 by this faculty we place ourselves and other objects in 
 one common space, and thus they are exterior to us. 
 It may be remarked that this apprehension of objects 
 as external to us, although it assumes the idea of 
 space, is far from being implied in the idea of space. 
 The objects which we contemplate are considered as 
 . existing in space, and by that means become invested 
 with certain mutual relations of position; but when 
 we consider them as existing without us, we make the 
 additional step of supposing ourselves and the objects to 
 exist in one common space. The question respecting 
 the Ideal Theory of Berkeley has been mixed up with 
 the recognition of this condition of the externality of 
 objects. That philosopher maintained, as is well known, 
 that the perceptible qualities of bodies have no exist- 
 ence except in a perceiving mind. This system has 
 often been understood as if he had imagined the world 
 to be a kind of optical illusion, like the images which 
 we see when we shut our eyes, appearing to be with- 
 out us, though they are only in our organs ; and thus 
 this Ideal System has been opposed to a belief in an 
 external world. In truth, however, no such opposition 
 exists. The Ideal System is an attempt to. explain the 
 
OF THE IDEA OF A MEDIUM. 297 
 
 mental process of perception, and to get over the diffi- 
 culty of mind being affected by matter. But the author 
 of that system did not deny that objects were per- 
 ceived under the conditions of space and mechanical 
 causation; that they were external and material so 
 far as those words describe perceptible qualities. Berke- 
 ley's system, however visionary or erroneous, did not 
 prevent his entertaining views as just, concerning optics 
 or acoustics, as if he had held any other doctrine of 
 the nature of perception. 
 
 But when Berkeley's theory was understood as a 
 denial of the existence of objects without us, how was 
 it answered? If we examine the answers which are 
 given by Reid and other philosophers to this hypothe- 
 sis, it will be found that they amount to this: that 
 objects are without us, since we perceive that they are 
 so ; that we perceive them to be external, by the same 
 act by which we perceive them to be objects. And 
 thus, in this stage of philosophical inquiry, the ex- 
 ternality of objects is recognized as one of the inevit- 
 able conditions of our perception of them ; and hence 
 the Idea of Externality is adopted as one of the neces- 
 sary foundations of all reasoning concerning all objects 
 whatever. 
 
 3. Sensation by a Medium. Objects, as we have 
 just seen, are necessarily apprehended as without us; 
 and in general, as removed from us by a great or small 
 distance. Yet they affect our bodily senses; and this 
 leads us irresistibly to the conviction that they are 
 perceived by means of something intermediate. Vision, 
 or hearing, or smell, or the warmth of a fire, must be 
 communicated to us by some Medium of Sensation. 
 This unavoidable belief appears in all attempts, the 
 earliest and the latest alike, to speculate upon such 
 subjects. Thus, for instance, Aristotle says 4 , * Seeing 
 takes place in virtue of some action which the sen- 
 tient organ suffers : now it cannot suffer action from 
 the colour of the object directly: the only remain- 
 ing possible case then is, that it is acted upon by an 
 
 Hep! *vxns- ii. 7. See the motto to this Book. 
 
298 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 intervening Medium ; there must then be an interven- 
 ing Medium.' ' And the same may be said,' he adds, 
 1 concerning sounding and odorous bodies; for these do 
 not produce sensation by touching the sentient organ, 
 but the intervening Medium is acted on by the sound 
 or the smell, and the proper organ, by the Medium... 
 In sound the Medium is air; in smell we have no 
 name for it.' In the sense of taste, the necessity of a 
 Medium is not at first so obviously seen, because the 
 object tasted is brought into contact with the organ; 
 but a little attention convinces us that the taste of a 
 solid body can only be perceived when it is conveyed 
 in some liquid vehicle. Till the fruit is crushed, and 
 till its juices are pressed out, we do not distinguish its 
 flavour. In the case of heat, it is still more clear that 
 we are compelled to suppose some invisible fluid, or 
 other means of communication, between the distant 
 body which warms us and ourselves. 
 
 It may appear to some persons that the assumption 
 of an intermedium between the object perceived and 
 the sentient organ results from the principles which 
 form the basis of our mechanical reasonings, that 
 every change must have a cause, and that bodies can 
 act upon each other only by contact. It cannot be 
 denied that this principle does offer itself very naturally 
 as the ground of our belief in media of sensation ; and 
 it appears to be referred to for this purpose by Aris- 
 totle in the passage quoted above. But yet we cannot 
 but ask, Does the principle, that matter produces its 
 effect by contact only, manifestly apply here 1 ? When 
 we so apply it, we include sensation among the effects 
 which material contact produces; a case so different 
 from any merely mechanical effect, that the principle, 
 so employed, appears to acquire a new signification. 
 May we not, then, rather say that we have here a new 
 axiom, That sensation implies a material cause im- 
 mediately acting on the organ, than a new applica- 
 tion of our former proposition, That all mechanical 
 change implies contact? 
 
 The solution of this doubt is not of any material 
 consequence to our reasonings; for whatever be the 
 
OF THE IDEA OF A MEDIUM. 299 
 
 ground of the assumption, it is certain that we do 
 assume the existence of media by which the sensations 
 of sight, hearing, and the like, are produced; and it 
 will be seen shortly that principles inseparably con- 
 nected with this assumption are the basis of the 
 sciences now before us. 
 
 This assumption makes its appearance in the physi- 
 cal doctrines of all the schools of philosophy. It is 
 exhibited perhaps most prominently in the tenets of 
 the Epicureans, who were materialists, and extended 
 to all kinds of causation the axiom of the existence of 
 a corporeal mechanism by which alone the effect is 
 produced. Thus, according to them, vision is pro- 
 duced by certain images or material films which flow 
 from the object, strike upon the eyes, and so become 
 sensible. This opinion is urged with great detail and 
 earnestness by Lucretius, the poetical expositor of the 
 Epicurean creed among the Romans. His fundamen- 
 tal conviction of the necessity of a material medium is 
 obviously the basis of his reasoning, though he attempts 
 to show the existence of such a medium by facts. Thus 
 he argues 5 , that by shouting loud we make the throat 
 sore; which shows, he says, that the voice must be 
 material, so that it can hurt the passage in coming out. 
 
 Haud igitur dubium est quin voces verbaque constent 
 
 Corporeis e principiis ut laedere possint. 
 
 4. The Process of Perception of Secondary Quali- 
 ties. The likenesses or representatives of objects by 
 which they affect our senses were called by some 
 writers species, or sensible species, a term which con- 
 tinued in use till the revival of science. It may 
 be observed that the conception of these species as 
 films cast off from the object, and retaining its shape, 
 was different, as we have seen, from the view which 
 Aristotle took, though it has sometimes been called 
 the Peripatetic doctrine 6 . We may add that the 
 expression was latterly applied to express the suppo- 
 sition of an emanation of any kind, and implied little 
 
 5 De Rerum Naturd, Lib. iv. 529. 6 Brown, voL ii. p. 98. 
 
30O PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 more than that supposition of a Medium of which we 
 are now speaking. Thus Bacon, after reviewing the 
 phenomena of sound, says 7 , ' Videntur motus soni fieri 
 per species spirituales : ita enim loquendum donee cer- 
 tius quippiam inveniatur.' 
 
 Though the fundamental principles of several sciences 
 depend upon the assumption of a Medium of Perception, 
 these principles do not at all depend upon any special 
 view of the Process of our perceptions. The mechanism 
 of that process is a curious subject of consideration; 
 but it belongs to physiology, more properly than either 
 to metaphysics, or to those branches of physics of which 
 we are now speaking. The general nature of the pro- 
 cess is the same for all the senses. The object affects 
 the appropriate intermedium; the medium, through 
 the proper organ, the eye, the ear, the nose, affects the 
 nerves of the particular sense ; and, by these, in some 
 way, the sensation is conveyed to the mind, But to 
 treat the impression upon the nerves as the act of sen- 
 sation which we have to consider, would be to mistake 
 our object, which is not the constitution of the human 
 body, but of the human mind. It would be to mistake 
 one link of the chain for the power which holds the 
 end of the chain. No anatomical analysis of the cor- 
 poreal conditions of vision, or hearing, or feeling warm, 
 is necessary to the sciences of Optics, or Acoustics, or 
 Thermotics. 
 
 Not only is this physiological research an extraneous 
 part of our subject, but a partial pursuit of such a re- 
 search may mislead the inquirer. We perceive objects 
 by 'means of certain media, and by means of certain 
 impressions on the nerves: but we cannot with pro- 
 priety say that we perceive either the media or the 
 impressions on the nerves. What person in the act of 
 seeing is conscious of the little coloured spaces on the 
 retina? or of the motions of the bones of the auditory 
 apparatus whilst he is hearing? Surely, no one. This 
 may appear obvious en6ugh, and yet a writer of no 
 common acuteness, Dr. Brown, has put forth several 
 
 Hist. Son. et Aud. vol. ix. p. 87. 
 
OF THE IDEA OF A MEDIUM. 30 1 
 
 very strange opinions, all resting upon the doctrine 
 that the coloured spaces on the retina are the objects 
 which we perceive; and there are some supposed diffi- 
 culties and paradoxes on the same subject which have 
 become quite celebrated (as upright vision with 4n- 
 verted images), arising from the same confusion of 
 thought. 
 
 As the consideration of the difficulties which have 
 arisen respecting the Philosophy of Perception may 
 serve still further to illustrate the principles on which 
 we necessarily reason respecting the secondary quali- 
 ties of bodies, I shall here devote a few pages to that 
 subject. 
 
CHAPTER II. 
 
 ON PECULIARITIES IN THE PERCEPTIONS OF THE 
 DIFFERENT SENSES. 
 
 i. ~V^7"E cannot doubt that we perceive all secondary 
 V V qualities by means of immediate impressions 
 made, through the proper medium of sensation, upon 
 our organs. Hence all the senses are sometimes 
 vaguely spoken of as modifications of the sense of 
 feeling. It will, however, be seen, on reflection, that 
 this mode of speaking identifies in words things which 
 in our conceptions have nothing in common. No im- 
 pression on the organs of touch can be conceived as 
 having any resemblance to colour or smell. No effort, 
 no ingenuity, can enable us to describe the impressions 
 of one sense in terms borrowed from another. 
 
 The senses have, however, each its peculiar powers, 
 and these powers may be in some respects compared, so 
 as to show their leading resemblances and differences, 
 and the characteristic privileges and laws of each. 
 This is what we shall do as briefly as possible. 
 
 SECT. I. Prerogatives of Sight. 
 
 THE sight distinguishes colours, as the hearing distin- 
 guishes tones j the sight estimates degrees of brightness, 
 the ear, degrees of loudness j but with several resem- 
 blances, there are most remarkable differences between 
 these two senses. 
 
 2. Position. The sight has this peculiar preroga- 
 tive, that it apprehends the place of its objects directly 
 and primarily. We see where an object is at the same 
 instant that we see what it is. If we see two objects, 
 we see their relative position. We cannot help per- 
 
PECULIARITIES OF THE PERCEPTIONS. 303 
 
 ceiving that one is above or below, to the right or to 
 the left of the other, if we perceive them at all. 
 
 There is nothing corresponding to this in sound. 
 When we hear a noise, we do not necessarily assign a 
 place to it. It may easily happen that we cannot tell 
 from which side a thunder-clap comes. And though 
 we often can judge in what direction a voice is heard, 
 this is a matter of secondary impression, and of infer- 
 ence from concomitant circumstances, not a primary 
 fact of sensation. The judgments which we form con- 
 cerning the position of sounding bodies are obtained 
 by the conscious or unconscious comparison of the im- 
 pressions made on the two ears, and on the bones of 
 the head in general; they are not inseparable con- 
 ditions of hearing. We may hear sounds, and be 
 uncertain whether they are * above, around, or under- 
 neath ! ' but the moment anything visible appears, how- 
 ever unexpected, we can say, 'see where it comes!' 
 
 Since we can see the relative position of things, we 
 can see figure, which is but the relative position of the 
 different parts of the boundary of the object. And 
 thus the whole visible world exhibits to us a scene of 
 various shapes, coloured and shaded according to their 
 form and position, but each having relations of position 
 to all the rest; and altogether, entirely filling up the 
 whole range which the eye can command. 
 
 3. Distance. The distance of objects from us is 
 no matter of immediate perception, but is a judgment 
 and inference formed from our sensations, in some- 
 thing of the same way as our judgment of position 
 by the ear, though more precise. That this is so, 
 was most distinctly shown by Berkeley, in his New 
 Theory of Vision. The elements on which we form our 
 judgment are, the effort by which we fix both eyes on 
 the same object, the effort by which we adjust each eye 
 to distinct vision, and the known forms, colours, and 
 parts of objects, as compared with their appearance. 
 The right interpretation of the information which 
 these circumstances give us respecting the true dis- 
 tances and forms of things, is gradually learnt by ex- 
 perience, the lesson being begun in our earliest infancy, 
 and inculcated upon us every hour during which we 
 
304 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 use our eyes. The completeness with which the lesson 
 is learnt is truly admirable; for we forget that our 
 conclusion is obtained indirectly, and mistake a judg- 
 ment on evidence for an intuitive perception. This, 
 however, is not more surprizing than the rapidity and 
 unconsciousness of effort with which we understand 
 the meaning of the speech that we hear, or the book 
 that we read. In both cases, the habit of interpreta- 
 tion is become as familiar as the act of perception. 
 And this is the case with regard to vision. We see 
 the breadth of the street as clearly and readily as we 
 see the house on the other side of it. We see the 
 house to be square, however obliquely it be presented 
 to us. Indeed the difficulty is, to recover the con- 
 sciousness of our real and original sensations ; to 
 discover what is the apparent relation of the lines 
 which appear before us. As we have already said, 
 (book ii. chap. 6) in the common process of vision 
 we suppose ourselves to see that which cannot be 
 seen; and when we would make a picture of an object, 
 the difficulty is to represent what is visible and no 
 more. 
 
 But perfect as is our habit of interpreting what we 
 perceive, we could not interpret if we did not perceive. 
 If the eye did not apprehend visible position, it could 
 not infer actual position, which is collected from Visible 
 position as a consequence : if we did not see apparent 
 figure, we could not arrive at any opinion concerning 
 real form. The perception of place, which is the 
 prerogative of the eye, is the basis of all its other 
 superiority. 
 
 The precision with which the eye can judge of appa- 
 rent position is remarkable. If we had before us two 
 stars distant from each other by one-twentieth of the 
 moon's diameter, we could easily decide the apparent 
 direction of the one from the other, as above or below, 
 to the right or left. Yet eight millions of stars might 
 be placed in the visible hemisphere of the sky at such 
 distances from each other; and thus the eye would 
 recognize the relative position in a portion of its range 
 not greater than one eight-millionth of the whole. 
 Such is the accuracy of the sense of vision in tins 
 
PECULIARITIES OF THE PERCEPTIONS. 305 
 
 respect ; and, indeed, we might with truth have stated 
 it much higher. Our judgment of the position of dis- 
 tant objects in a landscape depends upon features far 
 more minute than the magnitude we have here de- 
 scribed. 
 
 As our object is to point out principally the differ- 
 ences of the senses, we do not dwell upon the delicacy 
 with which we distinguish tints and shades, but pro- 
 ceed to another sense. 
 
 SECT. II. Prerogatives of Hearing. 
 
 THE sense of hearing has two remarkable prerogatives ; 
 it can perceive a definite and peculiar relation between 
 certain tones, and it can clearly perceive two tones to- 
 gether; in both these circumstances it is distinguished 
 from vision, and from the other senses. 
 
 4. Musical intervals. We perceive that two tones 
 have, or have not, certain definite relations to each 
 other, which we call Concords : one sound is a Fifth, 
 an Octave, &c., above the other. And when this is the 
 case, our perception of the relation is extremely pre- 
 cise. It is easy to perceive when a fifth is out of tune 
 by one-twentieth of a tone ; that is, by one-seventieth 
 of itself. To this there is nothing analogous in vision. 
 Colours have certain vague relations to one another; 
 they look well together, by contrast or by resemblance ; 
 but this is an indefinite, and in most cases a casual and 
 variable feeling. The relation of complementary co- 
 lours to one another, as of red to green, is somewhat 
 more definite; but still, has nothing of the exactness 
 and peculiarity which belongs to a musical concord. 
 In the case of the two sounds, there is an exact point 
 at which the relation obtains; when by altering one 
 note we pass this point, the concord does not gradually 
 fade away, but instantly becomes a discord ; and if we 
 go further still, we obtain another concord of quite a 
 different character. 
 
 "We learn from the theory of sound that concords 
 occur when the times of vibration of the notes have 
 exact simple ratios; an octave has these times as i to 
 VOL. i. x 
 
306 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 2 ; a fifth, as 2 to 3. According to the undulatory 
 theory of light, such ratios occur in colours, yet the 
 eye is not affected by them in any peculiar way. The 
 times of the undulations of certain red and certain 
 violet rays are as 2 to 3, but we do not perceive any 
 peculiar harmony or connexion between those colours. 
 
 5. Chords. Again, the ear has this prerogative, 
 that it can apprehend two notes together, yet distinct. 
 If two notes, distant by a fifth from each other, are 
 sounded on two wind instruments, both they and their 
 musical relation are clearly perceived. There is not a 
 mixture, but a concord, a musical interval. In colours, 
 the case is otherwise. If blue and yellow fall on the 
 same spot, they form green ; the colour is simple to the 
 eye ; it can no more be decomposed by the vision than 
 if it were the simple green of the prismatic spectrum : 
 it is impossible for us, by sight, to tell whether it is so 
 or not. 
 
 These are very remarkable differences of the two 
 senses : two colours can be compounded into an appa- 
 rently simple one; two sounds cannot: colours pass 
 into each other by gradations and intermediate tints ; 
 sounds pass from one concord to another by no grada- 
 tions: the most intolerable discord is that which is 
 near a concord. We shall hereafter see how these 
 differences affect the scales of sound and of colour. 
 
 6. Ehythm. We might remark, that as we see 
 objects in space, we hear sounds in time; and that we 
 thus introduce an arrangement among sounds which 
 has several analogies with the arrangement of objects 
 in space. But the conception of time does not seem 
 to be peculiarly connected with the sense of hearing ; 
 a faculty of apprehending tone and time, or in musical 
 phraseology tune and rhythm, are certainly very dis- 
 tinct. I shall not, therefore, here dwell upon such 
 analogies. 
 
 The other Senses have not any peculiar prerogatives, 
 at least none which bear on the formation of science. 
 I may, however, notice, in the feeling of heat, this 
 circumstance; that it presents us with two opposites, 
 heat and cold, which graduate into each other. This 
 
PECULIARITIES OF THE PERCEPTIONS. 307 
 
 is not quite peculiar, for vision also exhibits to us 
 white and black, which are clearly opposites, and 
 which pass into each other by the shades of gray. 
 
 SECT. III. The Paradoxes of Vision. 
 
 7. First Paradox of Vision. Upright Vision. 
 All our senses appear to have this in common ; That 
 they act by means of organs, in which a bundle of 
 nerves receives the impression of the appropriate 
 medium of the sense. In the construction of these 
 organs there are great differences and peculiarities, 
 corresponding, in part at least, to the differences in 
 the information given. Moreover, in some cases, as 
 we have noted in the case of audible position and 
 visible distance, that which seems to be a perception 
 is really a judgment founded on perceptions of which 
 we are not directly aware. It will be seen, therefore, 
 that with respect to the peculiar powers of each sense, 
 it may be asked ; whether they can be explained by 
 the construction of the peculiar organ; whether they 
 are acquired judgments and not direct perceptions; 
 or whether they are inexplicable in either of these 
 ways, and cannot, at present at least, be resolved into 
 anything but conditions of the intellectual act of 
 perception. 
 
 Two of these questions with regard to vision, have 
 been much discussed by psychological writers: the 
 cause of our seeing objects upright by inverted images 
 on the retina; and of our seeing single with two such 
 images. 
 
 Physiologists have very completely explained the 
 exquisitely beautiful mechanism of the eye, considered 
 as analogous to an optical instrument; and it is in- 
 disputable that by means of certain transparent lenses 
 and humours, an inverted image of the objects which 
 are looked at is formed upon the retina, or fine net- 
 work of nerve, with which the back of the eye is 
 lined. We cannot doubt that the impression thus 
 produced on these nerves is essential to the act of 
 vision; and so far as we consider the nerves them- 
 
 X2 
 
308 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 selves to feel or perceive by contact, we may say that 
 they perceive this image, or the affections of light 
 which it indicates. But we cannot with any propriety 
 say that we perceive, or that our mind perceives, this 
 image; for we are not conscious of it, and none but 
 anatomists are aware of its existence : we perceive by 
 means of it. 
 
 A difficulty has been raised, and dwelt upon in a 
 most unaccountable manner, arising from the neglect 
 of this obvious distinction. It has been asked, how 
 is it that we see an object, a man for instance, upright, 
 when the immediate object of our sensation, the image 
 of the man on our retina, is inverted 1 To this we 
 must answer, that we see him upright because the 
 image is inverted; that the inverted image is the 
 necessary means of seeing an upright object. This is 
 granted, and where then is the difficulty 1 ? Perhaps it 
 may be put thus : How is it that we do not judge the 
 man to be inverted, since the sensible image is so? To 
 this we may reply, that we have no notion of upright 
 or inverted, except that which is founded on experience, 
 and that all our experience, without exception, must 
 have taught us that such a sensible image belongs to a 
 man who is in an upright position. Indeed, the con- 
 trary judgment is not conceivable; a man is upright 
 whose head is upwards and his feet downwards. But 
 what are the sensible images of upwards and down- 
 wards ? Whatever be our standard of up and down, 
 the sensible representation of up will be an image 
 moving on the retina towards the lower side, and the 
 sensible representation of down will be a motion to- 
 wards the upper side. The head of the man's image 
 is towards the image of the sky, its feet are towards 
 the image of the ground; how then should it appear 
 otherwise than upright? Do we expect that the 
 whole world should appear inverted 1 Be it so : but 
 if the whole be inverted, how is the relation of the 
 parts altered? Do we expect that we should think 
 our own persons in particular inverted 1 This cannot 
 be, for we look at them as we do at other objects. 
 Do we expect that things should appear to fall up- 
 
PECULIARITIES OF THE PERCEPTIONS. 309 
 
 wards ? Surely not. For what do we know of up- 
 wards, except that it is the direction in which bodies 
 do not fall ] In short, the whole of this difficulty, 
 though it has in no small degree embarrassed meta- 
 physicians, appears to result from a very palpable 
 confusion of ideas; from an attempt at comparison 
 of what we see, with that which the retina feels, as 
 if they were separately presentable. It is a sufficient 
 explanation to say, that we do not see the image on 
 the retina, but see by means of it. The perplexity 
 does not require much more skill to disentangle, than 
 it does to see that a word written in black ink, may 
 signify white 1 . 
 
 8. Second Paradox of Vision. Single Vision. 
 (i.) Small or Distant Objects. The other difficulty, 
 why with two images on the retina we see only one 
 object, is of a much more real and important kind. 
 This effect is manifestly limited by certain circum- 
 stances of a very precise nature ; for if we direct our 
 eyes at an object which is very near the eye, we see 
 
 i The explanation of our seeing on a black surface. Indeed some 
 objects erect when the image is in- persons have contrived to perplex 
 verted has been put very simply, by themselves with these latter ques- 
 saying, 'We call that the lower end tions, as well as the first. 
 of an object which is next the The above explanation is not at all 
 ground.' The observer cannot look affected, as to its substance, if we 
 into his own eye; he knows by ex- adopt Sir David Brewster's expres- 
 perience what kind of image corre- sion, and say that the line of visible 
 spends to a man in an upright posi- direction is a line passing through 
 tion. The anatomist tells him that the center of the spherical surface of 
 this image is inverted : but this does the retina, and therefore of course 
 not disturb the process of judging by perpendicular to the surface. In 
 experience. It does not appear why speaking of ' the inverted image,' it 
 any one should be perplexed at the has always been supposed to be de- 
 notion of seeing objects erect by termined by such lines ; and though 
 means of inverted images, rather the point where they intersect may 
 than at the notion of seeing objects not have been ascertained with exact- 
 large by means of small images ; or ness by previous physiologists, the 
 cubical and pyramidal, by means of philosophical view of the matter was 
 images on a spherical surface; or not in any degree vitiated by this 
 green and red, by means of images imperfection. 
 
310 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 all other objects double. The fact is not, therefore, 
 that we are incapable of receiving two impressions 
 from the two images, but that, under certain con- 
 ditions, the two impressions form one. A little atten- 
 tion shows us that these conditions are, that with both 
 eyes we should look at the same object; and again, we 
 find that to look at an object with either eye, is to 
 direct the eye so that the image falls on or near a 
 particular point about the middle of the retina. Thus 
 these middle points in the two retinas correspond, and 
 we see an image single when the two images fall on 
 the corresponding points. 
 
 Again, as each eye judges of position, and as the 
 two eyes judge similarly, an object will be seen in the 
 same place by one eye and by the other, when the two 
 images which it produces are similarly situated with 
 regard to the corresponding points of the retina 2 . 
 
 This is the Law of Single Vision, at least so far as 
 regards small objects; namely, objects so small that in 
 contemplating them we consider their position only, 
 
 2 The explanation of single vision image of an object absorbs the 
 with two eyes may be put in another weaker, and the object is seen single ; 
 form. Each eye judges immediately yet modified by the combination, as 
 of the relative position of all objects will be seen when we speak of the 
 within the field of its direct vision, single vision of near objects. When 
 Therefore when we look with both the two images of an object are con- 
 eyes at a distant prospect (so distant siderably apart, we see it double, 
 that the distance between the eyes This explanation is not different in 
 is small in comparison) the two pros- substance from the one given in the 
 pects, being similar collections of text ; but perhaps it is better to avoid 
 forms, will coincide altogether, if the assertion that the law of corre- 
 a corresponding point in one and spending points is 'a distinct and 
 in the other coincide. If this be original principle of our constitu- 
 the case, the two images of every tion,' as I had stated in the first 
 object will fall upon corresponding edition. The simpler mode of stating 
 points of the retina, and will appear the law of our constitution appears 
 single. to be to say, that each eye determines 
 
 If the two prospects seen by the similarly the position of objects ; and 
 
 two eyes do not exactly coincide, in that when the positions of an object, 
 
 consequence of nearness of the ob- as seen by the two eyes, coincide (or 
 
 jects, or distortion of the eyes, but nearly coincide) the object is seen 
 
 if they nearly coincide, the stronger single. 
 
PECULIARITIES OF THE PERCEPTIONS. 311 
 
 and not their solid dimensions. Single vision in such 
 cases is a result of the law of vision simply : and it is 
 a mistake to call in, as some have done, the influence of 
 habit and of acquired judgments, in order to determine 
 the result in such cases. 
 
 To ascribe the apparent singleness of objects to the 
 impressions of vision corrected by the experience of 
 touch 3 , would be to assert that a person who had not 
 been in the habit of handling what he saw, would see 
 all objects double; and also, to assert that a person 
 beginning with the double world which vision thus 
 offers to him, would, by the continued habit of han- 
 dling objects, gradually and at last learn to see them 
 single. But all the facts of the case show such suppo- 
 sitions to be utterly fantastical. No one can, in this 
 case, go back from the habitual judgment of the single- 
 ness of objects, to the original and direct perception 
 of their doubleness, as the draughtsman goes back 
 from judgments to perception, in representing solid 
 distances and forms by means of perspective pictures. 
 No one can point out any case in which the habit is 
 imperfectly formed; even children of the most tender 
 age look at an object with both eyes, and see it as one. 
 
 In cases when the eyes are distorted (in squinting), 
 one eye only is used, or if both are employed, there is 
 double vision; and thus any derangement of the cor- 
 respondence of motion in the two eyes will produce 
 double-sigh tedness. 
 
 Brown is one of those 4 who assert that two images 
 suggest a single object because we have always found 
 two images to belong to a single object. He urges as 
 an illustration, that the two words 'he conquered/ 
 by custom excite exactly the same notion as the one 
 Latin word 'vicit;' and thus that two visual images, 
 by the effect of habit, produce the same belief of a 
 single object as one tactual impression. But in order 
 to make this pretended illustration of any value, it 
 ought to be true that when a person has thoroughly 
 learnt the Latin language, he can no longer distinguish 
 
 3 See Brown, voL iL p. 8r. * Lectures, vol. iL p. 81. 
 
312 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 any separate meaning in 'he' and in 'conquered.' We 
 can by no effort perceive the double sensation, when 
 we look at the object with the two eyes. Those who 
 squint, learn by habit to see objects single : but the 
 habit which they acquire is that of 'attending to the 
 impressions of one eye only at once, not of combining 
 the two impressions. It is obvious, that if each eye 
 spreads before us the same visible scene, with the same 
 objects and the same relations of place, then, if one 
 object in each scene coincide, the whole of the two 
 visible impressions will be coincident. And here the 
 remarkable circumstance is, that not only each eye 
 judges for itself of the relations of position which come 
 within its field of view; but that there is a superior 
 and more comprehensive faculty which combines and 
 compares the two fields of view; which asserts or de- 
 nies their coincidence; which contemplates, as in a 
 relative position to one another, these two visible 
 worlds, in which all other relative position is given. 
 This power of confronting two sets of visible images 
 and figured spaces before a purely intellectual tribunal, 
 is one of the most remarkable circumstances in the 
 sense of vision. 
 
 9. (2.) Near Objects. "We have hitherto spoken 
 of the singleness of objects whose images occupy corre- 
 sponding positions on the retina of the two eyes. But 
 here occurs a difficulty. If an object of moderate size, 
 a small thick book for example, be held at a little dis- 
 tance from the eyes, it produces an image on the retina 
 of each eye ; and these two images are perspective re- 
 presentations of the book from different points of view, 
 (the positions of the two eyes,) and are therefore of dif- 
 ferent forms. Hence the two images cannot occupy 
 corresponding points of the retina throughout their 
 whole extent. If the central parts of the two images 
 occupy corresponding points, the boundaries of the two 
 wall not correspond. How is it then consistent with 
 the law above stated that in this case the object ap- 
 pears single? 
 
 It may be observed, that the two images in such a 
 case will differ most widely when the object is not a 
 
PECULIARITIES OF THE PERCEPTIONS. 313 
 
 mere surface, but a solid. If a book, for example, be 
 held with one of its upright edges towards the face, 
 the right eye will see one side more directly than the 
 left eye, and the left eye will see another side more 
 directly, and the outline of the two images upon the 
 two retinas will exhibit this difference. And it may be 
 further observed, that this difference in the images 
 received by the two eyes, is a plain and demonstrative 
 evidence of the solidity of the object seen j since 
 nothing but a solid object could (without some special 
 contrivance) produce these different forms of the images 
 in the two eyes. 
 
 Hence the absence of exact coincidence in the two 
 images on the retina is the necessary condition of the 
 solidity of the object seen, and must be one of the indi- 
 cations by means of which our vision apprehends an 
 object as solid. And that this is so, Mr. Wheatstone 
 has proved experimentally, by means of some most 
 ingenious and striking contrivances. He has devised 5 
 an instrument (the stereoscope) by which two images 
 (drawn in outline) differing exactly as much as the two 
 images of a solid body seen near the face would differ, 
 are conveyed, one to one eye, and the other to the other. 
 And it is found that when this is effected, the object 
 which the images represent is not only seen single, but 
 is apprehended as solid with a clearness and reality of 
 conviction quite distinct from any impression which a 
 mere perspective representation can give. 
 
 At the same time it is found that the object is then 
 only apprehended as single when the two images are 
 such as are capable of being excited by one single 
 object placed in solid space, and seen by the two 
 eyes. If the images differ more or otherwise than this 
 condition allows, the result is, that both are seen, their 
 lines crossing and interfering with one another. 
 
 It may be observed, too, that if an object be of such 
 large size as not to be taken in by a single glance of 
 the eyes, it is no longer apprehended as single by a 
 direct act of perception; but its parts are looked at 
 
 5 Phil. Trans. 1839. 
 
314 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 separately and successively, and the impressions thus 
 obtained are put together by a succeeding act of the 
 mind. Hence the objects which are directly seen as 
 solid, will be of moderate size; in which case it is not 
 difficult to show that the outlines of the two images 
 will differ from each other only slightly. 
 
 Hence we are led to the following, as the Law of 
 Single Vision for near objects : When the two images 
 in the two eyes are situated (part for part) nearly, but 
 not exactly, upon corresponding points, the object is 
 apprehended as single, if the two images are such as 
 are or would be given by a single solid object seen by 
 the two eyes separately : and in this case the object is 
 necessarily apprehended as solid. 
 
 This law of vision does not contradict that stated 
 above for distant objects : for when an object is removed 
 to a considerable distance, the images in the two eyes 
 coincide exactly, and the object is seen as single, though 
 without any direct apprehension of its solidity. The 
 first law is a special case of the second. Under the 
 condition of exactly corresponding points, we have the 
 perception of singleness, but no evidence of solidity. 
 Under the condition of nearly corresponding points, 
 we may have the perception of singleness, and with it, 
 of solidity. 
 
 We have before noted it as an important feature in 
 our visual perception, that while we have two distinct 
 impressions upon the sense, which we can contemplate 
 separately and alternately, (the impressions on the two 
 eyes,) we have a higher perceptive faculty which can 
 recognize these two impressions, exactly similar to each 
 other, as only two images of one and the same assem- 
 blage of objects. But we now see that the faculty by 
 which we perceive visible objects can do much more 
 than this : it can not only unite two impressions, and 
 recognize them as belonging to one object in virtue of 
 their coincidence, but it can also unite and identify 
 them, even when they do not exactly coincide. It can 
 correct and adjust their small difference, so that they 
 are both apprehended as representations of the same 
 figure. It can infer from them a real form, not agree- 
 
PECULIARITIES OF THE PERCEPTIONS. 315 
 
 ing with either of them ; and a solid space, which they 
 are quite incapable of exemplifying. The visual faculty 
 decides whether or not the two ocular images can be 
 pictures of the same solid object, and if they can, it 
 undoubtingly and necessarily accepts them as being so. 
 This faculty operates as if it had the power of calling 
 before it all possible solid figures, and of ascertaining 
 by trial whether any of those will, at the same time, 
 fit both the outlines which are given by the sense. It 
 assumes the reality of solid space, and, if it be possible, 
 reconciles the appearances with that reality. And thus 
 an activity of the mind of a very remarkable and 
 peculiar kind is exercised in the most common act of 
 
 10. It may be said that this doctrine, of such a 
 visual faculty as has been described, is very vague and 
 obscure, since we are not told what are its limits. It 
 adjusts and corrects figures which nearly coincide, so 
 as to identify them. But how nearly, it may be asked, 
 must the figures approach each other, in order that 
 this adjustment may be possible? What discrepance 
 renders impossible the reconcilement of which we 
 speak? Is it not impossible to give a definite answer 
 to these questions, and therefore impossible to lay 
 down definitely such laws of vision as we have stated? 
 To this I reply, that the indefiniteness thus objected 
 to us, is no new difficulty, but one with which philo- 
 sophers are familiar, and to which they are already 
 reconciled. It is, in fact, no other than the indefinite- 
 ness of the limits of distinct vision. How near to the 
 face must an object be brought, so that we shall cease 
 to see it distinctly? The distance, it will be answered, 
 is indefinite : it is different for different persons ; and 
 for the same person, it varies with the degree of effort, 
 attention, and habit. But this indefiniteness is only 
 the indefiuiteness, in another form, of the deviation of 
 the two ocular images from one another : and in reply 
 to the question concerning them we must still say, as 
 before, that in doubtful cases, the power of apprehend- 
 ing an object as single, when this can be done, will 
 vary with effort, attention, and habit. The assumption 
 
316 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 that the apparent object exists as a real figure, in real 
 space, is to be verified, if possible; but, in extreme 
 cases, from the unfitness of the point of view, or from 
 any other cause of visual confusion or deception, the 
 existence of a real object corresponding to the appear- 
 ance may be doubtful ; as in any other kind of percep- 
 tion it may be doubtful whether our senses, under dis- 
 advantageous circumstances, give us true information. 
 The vagueness of the limits, then, within which this 
 visual faculty can be successfully exercised, is no valid 
 argument against the existence of the faculty, or the 
 truth of the law which we have stated concerning its 
 action. 
 
 SECT. IY. The Perception of Visible Figure. 
 
 ii. Visible Figure. There is one tenet on the 
 subject of vision which appears to me so extravagant 
 and unphilosophical, that I should not have thought it 
 necessary to notice it, if it had not been recently pro- 
 mulgated by a writer of great acuteness in a book which 
 has obtained, for a metaphysical work, considerable cir- 
 culation. I speak of Brown's opinion 6 that we have no 
 immediate perception of visible figure. I confess myself 
 unable to comprehend fully the doctrine which he would 
 substitute in the place of the one commonly received. 
 He states it thus 7 : 'When the simple affection of sight 
 is blended with the ideas of suggestion [those arising 
 from touch, &c.] in what are termed the acquired per- 
 ceptions of vision, as, for example, in the perception of 
 a sphere, it is colour only which is blended with the 
 large convexity, and not a small coloured plane.' The 
 doctrine which Brown asserts in this and similar pas- 
 sages, appears to be, that we do not by vision perceive 
 both colour &ndi figure; but that the colour which we see 
 is blended with the figure which we learn the existence 
 of by other means, as by touch. But if this were pos- 
 sible when we can call in other perceptions, how is it 
 possible when we cannot or do not touch the object? 
 
 6 Lectures, voL ii. p. 82. 7 J6. voL ii p. 90. 
 
PECULIARITIES OF THE PERCEPTIONS. 317 
 
 Why does the moon appear round, gibbous, or horned? 
 What sense besides vision suggests to us the idea of her 
 figure? And even in objects which we can reach, what 
 is that circumstance in the sense of vision which sug- 
 gests to us that the colour belongs to the sphere, ex- 
 cept that we see the colour where we see the sphere? 
 If we do not see figure, we do not see position; for 
 figure is the relative position of the parts of a boun- 
 dary. If we do not see position, why do we ascribe 
 the yellow colour to the sphere on our left, rather than 
 to the cube on our right? We associate the colour 
 with the object, says Dr. Brown; but if his opinion 
 were true, we could not associate two colours with two 
 objects, for we could not apprehend the colours as 
 occupying two different places. 
 
 The whole of Brown's reasoning on this subject is 
 so irreconcileable with the first facts of vision, that it 
 is difficult . to conceive how it could proceed from a 
 person who has reasoned with great acuteness concern- 
 ing touch. In order to prove his assertion, he under- 
 takes to examine the only reasons which, he says 8 , he 
 can imagine for believing the immediate perception of 
 visible figure : (i) That it is absolutely impossible, in 
 our present sensations of sight, to separate colour from 
 extension ; and (2) That there are, in fact, figures on 
 the retina corresponding to the apparent figures of 
 objects. 
 
 On the subject of the first reason, he says, that the 
 figure which we perceive as associated with colour, is the 
 real, and not the apparent figure. ' Is there,' he asks, 
 1 the slightest consciousness of a perception of visible 
 figure, corresponding to the affected portion of the 
 retina?' To which, though he seems to think an affirm- 
 ative answer impossible, we cannot hesitate to reply, 
 that there is undoubtedly such a consciousness; that 
 though obscured by being made the ground of habitual 
 inference as to the real figure, this consciousness is 
 constantly referred to by the draughtsman, and easily 
 recalled by any one. We may separate colour, he says 
 
 8 Lectures, voL ii. p. 83. 
 
3l8 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 again 9 , from the figures on the retina, as we may 
 separate it from length, breadth, and thickness, which 
 we do not see. But this is altogether false : we can- 
 not separate colour from length, breadth, and thick- 
 ness, in any other way, than by transferring it to the 
 visible figure which we do see. He cannot, he allows, 
 separate the colour from the visible form of the trunk 
 of a large oak; but just as little, he thinks, can he sepa- 
 rate it from the convex mass of the trunk, which (it is 
 allowed on all hands) he does not immediately see. 
 But in this he is mistaken : for if he were to make a 
 picture of the oak, he would separate the colour from 
 the convex shape, which he does not imitate, but he 
 could not separate it from the visible figure, which he 
 does imitate; and he would then perceive that the 
 fact that he has not an immediate perception of the 
 convex form, is necessarily connected with the fact 
 that he has an immediate perception of the apparent 
 figure; so far is the rejection of immediate perception 
 in the former case from being a reason for rejecting it 
 in the latter. 
 
 Again, with regard to the second argument. It does 
 not, he says, follow, that because a certain figured por- 
 tion of the retina is affected by light, we should see 
 such a figure; for if a certain figured portion of the 
 olfactory organ were affected by odours, we should not 
 acquire by smell any perception of such figure 10 . This 
 is merely to say, that because we do not perceive posi- 
 tion and figure by one sense, we cannot do so by an- 
 other sense. But this again is altogether erroneous. It 
 is an office of our sight to inform us of position, and 
 consequently of figure ; for this purpose, the organ is 
 so constructed that the position of the object deter- 
 mines the position of the point of the retina affected. 
 There is nothing of this kind in the organ of smell; 
 objects in different positions and of different forms do 
 not affect different parts of the olfactory nerve, or por- 
 tions of different shape. Different objects, remote 
 from each other, if perceived by smell, affect the same 
 
 9 Lectures, vol. ii. p. 84. 10 Ib. p. 87. 
 
PECULIARITIES OF THE PERCEPTIONS. 319 
 
 part of the olfactory organs. This is all quite intelli- 
 gible; for it is not the office of smell to inform us 
 of position. Of what use or meaning would be the 
 curious and complex structure of the eye, if it gave us 
 only such vague and wandering notions of the colours 
 and forms of the flowers in a garden, as we receive 
 from their odours when we walk among them blind- 
 fold 1 ? It is, as we have said, the prerogative of vision 
 to apprehend position: the places of objects on the 
 retina give this information. We do not suppose 
 that the affection of a certain shape of nervous expanse 
 will necessarily and in all cases give us the impression 
 of figure ; but we know that in vision it does ; and it is 
 clear that if we did not acquire our acquaintance with 
 visible figure in this way, we could not acquire it in 
 any way 11 . 
 
 The whole of this strange mistake of Brown's ap- 
 pears to arise from the fault already noticed; that of 
 considering the image on the retina as the object in- 
 stead of the means of vision. This indeed is what he 
 says : ' the true object of vision is not the distant body 
 itself, but the light that has reached the expansive ter- 
 mination of the optic nerve 12 .' Even if this were so, 
 we do not see why we should not perceive the position 
 of the impression on this expanded nerve. But as we 
 have already said, the impression on the nerve is the 
 means of vision, and enables us to assign a place, or at 
 least a direction, to the object from which the light 
 proceeds, and thus makes vision possible. Brown, in- 
 deed, pursues his own peculiar view till he involves 
 the subject in utter confusion. Thus he says 13 , l Ac- 
 cording to the common theory [that figure can be 
 perceived by the eye,] a visible sphere is at once to my 
 perception convex and plane ; ' and if the sphere be a 
 one, it is perceived at once to be a sphere- of 
 
 11 When Brown says further (p. 87), his drift is. Does he doubt that there 
 
 that we can indeed show the image is an image formed in the living as 
 
 in the dissected eye ; but that ' it is completely as in the dissected eye ? 
 
 not in the dissected eye that vision J2 Lectures, voL ii. p. 57. 
 
 takes place ;' it is difficult to see what i 3 Ib. voL ii. p. 89. 
 
320 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 many feet in diameter, and a plane circular surface of 
 the diameter of a quarter of an inch.' It is easy to 
 deduce these and greater absurdities, if we proceed on 
 his strange and baseless supposition that the object and 
 the image on the retina are both perceived. But who 
 is conscious of the image on the retina in any other 
 way than as he sees the object by means of it? 
 
 Brown seems to have imagined that he was ana- 
 lysing the perception of figure 'in the same manner in 
 which Berkeley had analyzed the perception of distance. 
 He ought to have recollected that such an undertaking, 
 to be successful, required him to show what elements 
 he analyzed it into. Berkeley analyzed the perception 
 of real figure into the interpretation of visible figure 
 according to certain rules which he distinctly stated. 
 Brown analyzes the perception of visible figure into 
 no elements. Berkeley says, that we do not directly 
 perceive distance, but that we perceive something else, 
 from which we infer distance, namely, visible figure 
 and colour, and our own efforts in seeing; Brown says, 
 that we do not see figure, but infer it; what then do 
 we see, which we infer it from? To this he offers no 
 answer. He asserts the seeming perception of visible 
 figure to be a result of ' association ;' of ' suggestion.' 
 But what meaning can we attach to this? Suggestion 
 requires something which suggests ; and not a hint is 
 given what it is which suggests position. Association 
 implies two things associated; what is the sensation 
 which we associate with form? What is that visual 
 perception which is not figure, and which we mistake 
 for figure? What perception is it that suggests a square 
 to the eye? What impressions are those which have 
 been associated with a visible triangle, so that the 
 revival of the impressions revives the notion of the 
 triangle? Brown has nowhere pointed out such per- 
 ceptions and impressions; nor indeed was it possible 
 for him to do so ; for the only visual perceptions which 
 he allows to remain, those of colour, most assuredly do 
 not suggest visible figures by their differences; red is 
 not associated with square rather than with round, or 
 with round rather than square. On the contrary, the 
 
PECULIAEITIES OF THE PERCEPTIONS. 321 
 
 eye, constructed in a very complex and wonderful 
 manner in order that it may give to us directly the 
 perception of position as well as of colour, has it for 
 one of its prerogatives to give us this information ; and 
 the perception of the relative position of each part of 
 the visible boundary of an object constitutes the per- 
 ception of its apparent figure ; which faculty we can- 
 not deny to the eye without rejecting the plain and 
 constant evidence of our senses, making the mechanism 
 of the eye unmeaning, confounding the object with the 
 means of vision, and rendering the mental process of 
 vision utterly unintelligible. 
 
 Having sufficiently discussed the processes of per- 
 ception, I now return to the consideration of the Ideas 
 which these processes assume. 
 
 VOL. I. 
 
CHAPTER III. 
 
 SUCCESSIVE ATTEMPTS AT THE SCIENTIFIC APPLICA- 
 TION OF THE IDEA OF A MEDIUM. 
 
 i. TN what precedes, we have shown by various con- 
 JL siderations that we necessarily and universally 
 assume the perception of secondary qualities to take 
 place by means of a medium interjacent between the 
 object and the person perceiving. Perception is 
 affected by various peculiarities, according to the 
 nature of the quality perceived: but in all cases a 
 medium is equally essential to the process. 
 
 This principle, which, as we have seen, is accepted 
 as evident by the common understanding of mankind, 
 is confirmed by all additional reflection and discipline 
 of the mind, and is the foundation of all the theories 
 which have been proposed concerning the processes by 
 which the perception takes place, and concerning the 
 modifications of the qualities thus perceived. The 
 medium, and the mode in which the impression is con- 
 veyed through the medium, seem to be different foi* 
 different qualities; but the existence of the medium 
 leads to certain necessary conditions or alternatives, 
 which have successively made their appearance in 
 science, in the course of the attempts of men to theo- 
 rize concerning the principal secondary qualities, sound, 
 light, and heat. We must now point out some of the 
 ways, at first imperfect and erroneous, in which the 
 consequences of the fundamental assumption were 
 traced. 
 
 2. Sound. In all cases the medium of sensation, 
 whatever it is, is supposed to produce the effect of con- 
 veying secondary qualities to our perception by means 
 of its primary qualities. It was conceived to operate 
 
APPLICATION OF THE IDEA OF A MEDIUM. 323 
 
 by the size, form, and motion of its parts. This is 
 a fundamental principle of the class of sciences of 
 which we have at present to speak. 
 
 It was assumed from the first, as we have seen in 
 the passage lately quoted from Aristotle 1 , that in the 
 conveyance of sound, the medium of communication 
 was the air. But although the first theorists were right 
 so far, that circumstance did not prevent their going 
 entirely wrong when they had further to determine the 
 nature of the process. It was conceived by Aristotle 
 that the air acted after the manner of a rigid body; 
 like a staff, which, receiving an impulse at one end, 
 transmits it to the other. Now this is altogether an 
 erroneous view of the manner in which the air con- 
 veys the impulse by which sound is perceived. An 
 approach was made to the true view of this process, 
 by assimilating it to the diffusion of the little circular 
 waves which are produced on the surface of still water 
 when a stone is dropt into it. These little waves 
 begin from the point thus disturbed, and run outwards, 
 expanding on every side, in concentric circles, till 
 they are lost. The propagation of sound through the 
 air from the point where it is produced, was compared 
 by Vitruvius to this diffusion of circular waves in 
 water; and thus the notion of 'a propagation of im- 
 pulse by the waves of a fluid was introduced, in the 
 place of the former notion of the impulse of an un- 
 yielding body. 
 
 But though, taking an enlarged view of the nature 
 of the progress of a wave, this is a just representation 
 of the motion of air in conveying sound, we cannot 
 suppose that the process was, at the period of which 
 we speak, rightly understood. For the waves of water 
 were contemplated only as affecting the surface of the 
 water; and as the air has no surface, the communica- 
 tion must take place by means of an internal motion, 
 which can bear only a remote and obscure resemblance 
 to the waves which we see. And even with regard to 
 the waves of water, the mechanism by which they are 
 
 Supr, p. 297. 
 
 Y 2 
 
324 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 produced and transferred was not at all understood; so 
 that the comparison employed by Vitruvius must be 
 considered rather as a loose analogy than as an exact 
 scientific explanation. 
 
 No correct account of such motions was given, till 
 the formation of the science of Mechanics in modern 
 times had enabled philosophers to understand more 
 distinctly the mode in which motion is propagated 
 through a fluid, and to discern the forces which the 
 process calls into play, so as to continue the motion 
 once begun. Newton introduced into this subject the 
 exact and rigorous conception of an Undulation, which 
 is the true key to the explanation of impulses con- 
 veyed through a fluid. 
 
 Even at the present day, the right apprehension of 
 the nature of an Undulation transmitted through a 
 fluid is found to be very difficult for all persons except 
 those whose minds have been duly disciplined by mathe- 
 matical studies. When we see a wave run along the 
 surface of water, we are apt to imagine at first that a 
 portion of the fluid is transferred bodily from one place 
 to another. But with a little consideration we may 
 easily satisfy ourselves that this is not so : for if we 
 look at a field of standing corn, when a breeze blows 
 over it, we see waves like those of water run along its 
 surface. Yet it is clear that in this case the separate 
 stalks of corn only bend backwards and forwards, and 
 no portion of the grain is really conveyed from one 
 part of the field to the other. This is obvious even to 
 popular apprehension. The poet speaks of 
 
 The rye, 
 
 That stoops its head when whirlwinds rave 
 And springs again in eddying wave 
 As each wild gust sweeps by. 
 
 Each particle of the mass in succession has a small 
 motion backwards and forwards ; and by this means a 
 large ridge made by many such particles runs along the 
 mass to any distance. This is the true conception of an 
 undulation in general. 
 
 Thus, when an Undulation is propagated in a fluid, 
 it is not matter, but form, which is transmitted from 
 
APPLICATION OF THE IDEA OF A MEDIUM. 325 
 
 one place to another. The particles along the line of 
 each wave assume a certain arrangement, and this ar- 
 rangement passes from one part to another, the particles 
 changing their places only within narrow limits, so as 
 to lend themselves successively to the arrangements by 
 which the successive waves, and the intervals between 
 the waves, are formed. 
 
 When such an Undulation is propagated through air, 
 the wave is composed, not, as in water, of particles 
 which are higher than the rest, but of particles wliich 
 are closer to each other than the rest. The wave is not 
 a ridge of elevation, but a line of condensation; and 
 as in water we have alternately elevated and depressed 
 lines, we have in air lines alternately condensed and 
 rarefied. And the motion of the particles is not, as in 
 water, up and down, in a direction transverse to that 
 of the wave which runs forwards ; in the motion of an 
 undulation through air the motion of each particle is 
 alternately forwards and backwards, while the motion 
 of the undulation is constantly forwards. 
 
 This precise and detailed account of the Undulatory 
 Motion of air by which sound is transmitted was first 
 given by Newton. He further attempted to determine 
 the motions of the separate particles, and to point out 
 the force by which each particle affects the next, so as 
 to continue the progress of the undulation once begun. 
 The motions of each particle must be oscillatory; he 
 assumed the oscillations to be governed by the simplest 
 law of oscillation which had come under the notice of 
 mathematicians, (that of small vibrations of a pendu- 
 lum;) and he proved that in this manner the forces 
 which are called into play by the contraction and 
 expansion of the parts of the elastic fluid are such as 
 the continuance of the motion requires. 
 
 Newton's proof of the exact law of Oscillatory Motion 
 of the aerial particles was not considered satisfactory 
 by succeeding mathematicians; for it was found that 
 the same result, the development of forces adequate to 
 continue the motion, would follow if any other law of 
 the motion were assumed. Cramer proved this by a 
 sort of parody on Newton's proof, in which, by the 
 
326 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 alteration of a few phrases in this formula of demon- 
 stration, it was made to establish an entirely different 
 conclusion. 
 
 But the general conception of an Undulation as pre- 
 sented by Newton was, as from its manifest mechanical 
 truth it could not fail to be, accepted by all mathema- 
 ticians; and in proportion as the methods of calcu- 
 lating the motions of fluids were further improved, the 
 necessary consequences of this conception, in the com- 
 munication of sound through air, were traced by un- 
 exceptionable reasoning. This was especially done by 
 Euler and Lagrange, whose memoirs on such motions 
 of fluids are some of the most admirable examples 
 which exist, of refined mathematical methods applied 
 to the solution of difficult mechanical problems. 
 
 But the great step in the formation of the theoiy of 
 sound was undoubtedly that which we have noticed, 
 the introduction of the Conception of an Undulation 
 such as we have attempted to describe it: a state, 
 condition, or arrangement of the particles of a fluid, 
 which is transferred from one part of space to another 
 by means of small motions of the particles, altogether 
 distinct from the movement of the Undulation itself. 
 This is a conception which is not obvious to common 
 apprehension. It appears paradoxical at first sight to 
 speak of a large wave (as the tide-wave) running up a 
 river at the rate of twenty miles an hour, while the 
 stream of the river is all the while flowing downwards. 
 Yet this is a very common fact. And the conception 
 of such a motion must be fully mastered by all who 
 would reason rightly concerning the mechanical trans- 
 mission of impressions through a medium. 
 
 We have described the motion of sound as produced 
 by small motions of the particle forwards and back- 
 wards, while the waves, or condensed and rarefied lines, 
 move constantly forwards. It may be asked what right 
 we have to suppose the motion to be of this kind, since 
 when sound is heard, no such motions of the particles 
 of air can be observed, even by refined methods of 
 observation. Thus Bacon declares himself against the 
 hypothesis of such a vibration, since, as he remarks, it 
 
APPLICATION OF THE IDEA OF A MEDIUM. 327 
 
 cannot be perceived in any visible impression upon 
 the flame of a candle. And to this we reply, that the 
 supposition of this Vibration is made in virtue of a 
 principle which is involved in the original assumption 
 of a medium; namely, That a Medium, in conveying 
 Secondary qualities, operates by means of its Primary 
 qualities, the bulk, figure, motion, and other mechanical 
 properties of its parts. This is an Axiom belonging to 
 the Idea of a Medium. In virtue of this axiom it is 
 demonstrable that the motion of the air, when any how 
 disturbed, must be such as is supposed in our acous- 
 tical reasonings. For the elasticity of the parts of the 
 air, called into play by its expansion and contraction, 
 lead, by a mechanical necessity, to such a motion as 
 we have described. We may add that, by proper con- 
 trivances, this motion may be made perceptible in its 
 visible effects. Thus the theory of sound, as an im- 
 pression conveyed through air, is established upon 
 evident general principles, although the mathema- 
 tical calculations which are requisite to investigate 
 its consequences are, some of them, of a very recon- 
 dite kind. 
 
 3. Light. The early attempts to explain Vision 
 represented it as performed by means of material rays 
 proceeding from the eye, by the help of which the eye 
 felt out the form and other visible qualities of an 
 object, as a blind man might do with his staff. But 
 this opinion could not keep its ground long : for it did 
 not even explain the fact that light is necessary to 
 vision. Light, as a peculiar medium, was next assumed 
 as the machinery of vision; but the mode in which 
 the impression was conveyed through the medium was 
 left undetermined, and no advance was made towards 
 sound theory, on that subject, by the ancients. 
 
 In modern times, when the prevalent philosophy 
 began to assume a mechanical turn (as in the theories 
 of Descartes), light was conceived to be a material sub- 
 stance which is emitted from luminous bodies, and 
 which is also conveyed from all bodies to the eye, so as 
 to render them visible. The various changes of direc- 
 tion by which the rays of light are affected, (reflection, 
 
328 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 refraction, &c.,) Descartes explained, by considering the 
 particles of light as small globules, which change their 
 direction when they impinge upon other bodies, ac- 
 cording to the laws of Mechanics. Newton, with a 
 much more profound knowledge of Mechanics than 
 Descartes possessed, adopted, in the most mature of 
 his speculations, nearly the same view of the nature of 
 light; and endeavoured to show that reflection, refrac- 
 tion, and other properties of light, might be explained 
 as the effects which certain forces, emanating from the 
 particles of bodies, produce upon the luminiferous 
 globules. 
 
 But though some of the properties of light could 
 thus be accounted for by the assumption of particles 
 emitted from luminous bodies, and reflected or refracted 
 by forces, other properties came into view which would 
 not admit of the same explanation. The phenomena 
 of diffraction (the fringes which accompany shadows) 
 could never be truly represented by such an hypothesis, 
 in spite of many attempts which were made. And the 
 colours of thin plates, which show the rays of light 
 to be affected by an alternation of two different con- 
 ditions at small intervals along their length, led New- 
 ton himself to incline, often and strongly, to some 
 hypothesis of undulation. The double refraction of 
 Iceland spar, a phenomenon in itself very complex, 
 could, it was found by Huyghens, be expressed with 
 great simplicity by a certain hypothesis of undula- 
 tions. 
 
 Two hypotheses of the nature of the luminiferous 
 medium were thus brought under consideration; the 
 one representing Light as Matter emitted from the 
 luminous object, the other, as Undulations propagated 
 through a fluid. These two hypotheses remained in 
 presence of each other during the whole of the last 
 century, neither of them gaining any material advan- 
 tage over the other, though the greater part of mathe- 
 maticians, following Newton, embraced the emission 
 theory. But at the beginning of the present century, 
 an additional class of phenomena, those of the inter- 
 ference of two rays of light, were brought under con- 
 
APPLICATION OF THE IDEA OF A MEDIUM. 329 
 
 sideration by Dr Young; and these phenomena were 
 strongly in favour of the undulatory theory, while they 
 were irreconcilable with the hypothesis of emission. 
 If it had not been for the original bias of Newton and 
 his school to the other side, there can be little doubt 
 that from this period light as well as sound would 
 have been supposed to be propagated by undulations ; 
 although in this case it was necessary to assume as the 
 vehicle of such undulations a special medium or ether. 
 Several points of the phenomena of vision no doubt 
 remained unexplained by the undulatory theory, as 
 absorption, and the natural colours of bodies ; but such 
 facts, though they did not confirm, did not evidently 
 contradict the theory of a Luminiferous Ether; and 
 the facts which such a theory did explain, it explained 
 with singular happiness and accuracy. 
 
 But before this Undulatory Theory could be gene- 
 rally accepted, it was presented in an entirely new 
 point of view by being combined with the facts of 
 polarization. The general idea of polarization must be 
 illustrated hereafter; but we may here remark that 
 Young and Fresnel, who had adopted the undulatory 
 theory, after being embarrassed for some time by the 
 new facts which were thus presented to their notice, 
 at last saw that these facts might be explained by con- 
 ceiving the vibrations to be transverse to the ray, the 
 motions of the particles being not backwards and for- 
 wards in the line in which the impulse travels, but to 
 the right and left of that line. This conception of 
 transverse vibrations, though quite unforeseen, had 
 nothing in it which was at all difficult to reconcile 
 with the general notion of an undulation. We have 
 described an undulation, or wave, as a certain condi- 
 tion or arrangement of the particles of the fluid succes- 
 sively transferred from one part of space to another : 
 and it is easily conceivable that this arrangement or 
 wave may be produced by a lateral transfer of the par- 
 ticles from their quiescent positions. This conception 
 of transverse vibrations being accepted, it was found 
 that the explanation of the phenomena of polarization 
 and of those of interference led to the same theory 
 
33O PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 with a correspondence truly wonderful ; and this coin- 
 cidence in the views, collected from two quite distinct 
 classes of phenomena, was justly considered as an 
 almost demonstrative evidence of the truth of this 
 undulatory theory. 
 
 It remained to be considered whether the doctrine 
 of transverse vibrations in a fluid could be reconciled 
 with the principles of Mechanics. And it was found 
 that by making certain suppositions, in which no in- 
 herent improbability existed, the hypothesis of trans- 
 verse vibrations would explain the laws, both of inter- 
 ference and of polarization of light, in air and in 
 crystals of all kinds, with a surprizing fertility and 
 fidelity. 
 
 Thus the Undulatory Theory of Light, like the Un- 
 dulatory Theory of Sound, is recommended by its con- 
 formity to the fundamental principle of the Secondary 
 Mechanical Sciences, that the medium must be sup- 
 posed to transmit its peculiar impulses according to the 
 laws of Mechanics. Although no one had previously 
 dreamt of qualities being conveyed through a medium 
 by such a process, yet when it is once suggested as 
 the only mode of explaining some of the phenomena, 
 there is nothing to prevent our accepting it entirely, 
 as a satisfactory theory for all the known laws of 
 Light. 
 
 4. Heat. With regard to Heat as with regard to 
 Light, a fluid medium was necessarily assumed as the 
 vehicle of the property. During the last century, this 
 medium was supposed to be an emitted fluid. And 
 many of the ascertained Laws of Heat, those which 
 prevail with regard to its radiation more especially, 
 were well explained by this hypothesis 2 . Other effects 
 of heat, however, as for instance latent heat 3 , and the 
 change of consistence of bodies 4 , were not satisfactorily 
 brought into connexion with the hypothesis ; while con- 
 
 2 See the Account of the Theory of Exchanges, Hist. Ind. Sc. b. x. c. i. 
 sect. 2. 
 
 3 Ib. c. ii sect. 3. * Ib. c. ii. sect. 2. 
 
APPLICATION OF THE IDEA OF A MEDIUM. 331 
 
 duction 5 , which at first did not appear to result from 
 the fundamental assumption, was to a certain extent 
 explained as internal radiation. 
 
 But it was by no means clear that an Undulatory 
 Theory of Heat might not be made to explain these 
 phenomena equally well. Several philosophers inclined 
 to such a theory ; and finally, Ampere showed that the 
 doctrine that the heat of a body consists in the undula- 
 tions of its particles propagated by means of the undu- 
 lations of a medium, might be so adjusted as to explain 
 all which the theory of emission could explain, and 
 moreover to account for facts and laws which were out 
 of the reach of that theory. About the same time it 
 was discovered by Prof. Forbes and M. Nobili that 
 radiant heat is, under certain circumstances, polarized. 
 Now polarization had been most satisfactorily explained 
 by means of transverse undulations in the case of light ; 
 while all attempts to modify the emission theory so as 
 to include polarization in it, had been found ineffectual 
 Hence this discovery was justly considered as lending 
 great countenance to the opinion that Heat consists in 
 the vibrations of its proper medium. 
 
 But what is this medium ? Is it the same by which 
 the impressions of Light are conveyed 1 This is a diffi- 
 cult question; or rather it is one which we cannot at 
 present hope to answer with certainty. No doubt the 
 connexion between Light and Heat is so intimate and 
 constant, that we can hardly refrain from considering 
 them as affections of the same medium. But instead 
 of attempting to erect our systems on such loose and 
 general views of connexion, it is rather the business of 
 the philosophers of the present day to determine the 
 laws of the operation of heat, and its real relation to 
 light, in order that we may afterwards be able to con- 
 nect the theories of the two qualities. Perhaps in a 
 more advanced state of our knowledge we may be able 
 to state it as an Axiom, that two Secondary Quali- 
 ties, which are intimately connected in their causes 
 and effects, must be affections of the same Medium. 
 
 6 Ib. c. L sect. 7. 
 
332 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 But at present it does not appear safe to proceed 
 upon such a principle, although many writers, in their 
 speculations both concerning Light and Heat, and con- 
 cerning other properties, have not hesitated to do so. 
 
 Some other consequences follow from the Idea of 
 a Medium which must be the subject of another 
 chapter. 
 
CHAPTER IV. 
 
 OF THE MEASURE OF SECONDARY QUALITIES. 
 
 SECT. I. Scales of Qualities in general. 
 
 riHHE ultimate object of our investigation in each of 
 JL the Secondary Mechanical Sciences, is the nature 
 of the processes by which the special impressions of 
 sound, light, and heat, are conveyed, and the modifi- 
 cations of which these processes are susceptible. And 
 of this investigation, as we have seen, the necessary 
 basis is the principle, that these impressions are trans- 
 mitted by means of a medium. But before we arrive 
 at this ultimate object, we may find it necessary to 
 occupy ourselves with several intermediate objects: 
 before we discover the cause, it may be necessary to 
 determine the laws of the phenomena. Even if we cannot 
 immediately ascertain the mechanism of light or heat, 
 it may still be interesting and important to arrange 
 and measure the effects which we observe. 
 
 The idea of a Medium affects our proceeding in this 
 research also. We cannot measure Secondary qualities 
 in the same manner in which we measure Primary 
 qualities, by a mere addition of parts. There is this 
 leading and remarkable difference, that while both 
 classes of qualities are susceptible of changes of magni- 
 tude, primary qualities increase by addition of exten- 
 sion, secondary, by augmentation of intensity. A space 
 is doubled when another equal space is placed by its 
 side; one weight joined to another makes up the sum 
 of the two. But when one degree of warmth is com- 
 bined with another, or one shade of red colour with 
 another, we cannot in like manner talk of the sum. 
 The component parts do not evidently retain their 
 
334 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 separate existence; we cannot separate a strong green 
 colour into two weaker ones, as we can separate a large 
 force into two smaller. The increase is absorbed into 
 the previous amount, and is no longer in evidence as a 
 part of the whole. And this is the difference which 
 has given birth to the two words extended, and in- 
 tense. That is extended which has 'partes extra 
 partes,' parts outside of parts : that is intense which 
 becomes stronger by some indirect and unapparent 
 increase of agency, like the stretching of the internal 
 springs of a machine, as the term intense implies. 
 Extended magnitudes can at will be resolved into the 
 parts of which they were originally composed, or any 
 other which the nature of their extension admits ; their 
 proportion is apparent; they are directly and at once 
 subject to the relations of number. Intensive magni- 
 tudes cannot be resolved into smaller magnitudes ; we 
 can see that they differ, but we cannot tell in what 
 proportion ; we have no direct measure of their quan- 
 tity. How many times hotter than blood is boiling 
 water 1 ? The answer cannot be given by the aid of our 
 feelings of heat alone. 
 
 The difference, as we have said, is connected with 
 the fundamental principle that we do not perceive 
 Secondary qualities directly, but through a Medium. 
 We have no natural apprehension of light, or sound, 
 or heat, as they exist in the bodies from which they 
 proceed, but only as they affect our organs. We can 
 only measure them, therefore, by some /Scale supplied 
 by their effects. And thus while extended magnitudes, 
 as space, time, are measurable directly and of them- 
 selves; intensive magnitudes, as brightness, loudness, 
 heat, are measurable only by artificial means and con- 
 ventional scales. Space, time, measure themselves: 
 the repetition of a smaller space, or time, while it com- 
 poses a larger one, measures it. But for light and 
 heat we must have Photometers and Thermometers, 
 which measure something which is assumed to be an 
 indication of the quality in question. In the one case, 
 the mode of applying the measure, and the meaning of 
 the number resulting, are seen by intuition; in the 
 
MEASURE OF SECONDARY QUALITIES. 335 
 
 other, they are consequences of assumption and reason- 
 ing. In the one case, they are Units, of which the 
 extension is made up ; in the other, they are Degrees 
 by which the intensity ascends. 
 
 2. When we discover any property in a sensible 
 quality, which at once refers us to number or space, 
 we readily take this property as a measure; and thus 
 we make a transition from quality to quantity. Thus 
 Ptolemy in the third chapter of the First Book of his 
 Harmonics begins thus : l As to the differences which 
 exist in sounds both in quality and in quantity, if we 
 consider that difference which refers to the acuteness 
 and graveness, we cannot at once tell to which of the 
 above two classes it belongs, till we have considered 
 the causes of such symptoms.' But at the end of the 
 chapter, having satisfied himself that grave sounds 
 result from the magnitude of the string or pipe, other 
 things being equal, he infers, 'Thus the difference 
 of acute and grave appears to be a difference of 
 quantity.' 
 
 In the same manner, in order to form Secondary 
 Mechanical Sciences respecting any of the other pro- 
 perties of bodies, we must reduce these properties to a 
 dependence upon quantity, and thus make them sub- 
 ject to measurement. We cannot obtain any sciential 
 truths respecting the comparison of sensible qualities, 
 till we have discovered measures and scales of the 
 qualities which we have to consider ; and accordingly, 
 some of the most important steps in such sciences have 
 been the establishment of such measures and scales, 
 and the invention of the requisite instruments. 
 
 The formation of the mathematical sciences which 
 rest upon the measures of the intensity of sensible 
 qualities took place mainly in the course of the last 
 century. Perhaps we may consider Lambert, a mathe- 
 matician who resided in Switzerland, and published 
 about 1750, as the person who first clearly felt the 
 importance of establishing such sciences. His Photo- 
 metry, Pyrometry, and Hygrometry, are examples of 
 the systematic reduction of sensible qualities (light, 
 heat, moisture) to modes of numerical measurement. 
 
336 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 We now proceed to speak of such modes of measure- 
 ment with regard to the most obvious properties of 
 bodies. 
 
 SECT. II. The Musical Scale. 
 
 3. THE establishment of the Harmonic Canon, that 
 is, of a Scale and .Measure of the musical place of notes, 
 in the relation of high and low, was the first step in the 
 science of Harmonics. The perception of the differ- 
 ences and relations of musical sounds is the office of 
 the sense of hearing ; but these relations are fixed, and 
 rendered accurately recognizable by artificial means. 
 ' Indeed, in all the senses,' as Ptolemy truly says in 
 the opening of his Harmonics, l the sense discovers 
 what is approximately true, and receives accuracy 
 from another quarter : the reason receives the approxi- 
 mately-true from another quarter, and discovers the 
 accurate truth.' We can have no measures of sensible 
 qualities which do not ultimately refer to the sense ; 
 whether they do this immediately, as when we refer 
 Colours to an assumed Standard ; or mediately, as when 
 we measure Heat by Expansion, having previously 
 found by an appeal to sense that the expansion in- 
 creases with the heat. Such relations of sensible 
 qualities cannot be described in words, and can only 
 be apprehended by their appropriate faculty. The 
 faculty by which the relations of sounds are appre- 
 hended is a musical ear in the largest acceptation 
 of the term. In this signification the faculty is nearly 
 universal among men; for all persons have musical 
 ears sufficiently delicate to understand and to imitate 
 the modulations corresponding to various emotions in 
 speaking; which modulations depend upon the succes- 
 sion of acuter and graver tones. These are the rela- 
 tions now spoken of, and these are plainly perceived 
 by persons who have very imperfect musical ears, 
 according to the common use of the phrase. But the 
 relations of tones which occur in speaking are some- 
 what indefinite; and in forming that musical scale 
 which is the basis of our science upon the subject, we 
 
MEASURE OF SECONDARY QUALITIES. 337 
 
 take the most definite and marked of such relations of 
 notes ; such as occur, not in speaking but in singing. 
 Those musical relations of two sounds which we call 
 the octave, the fifth, the fourth, the third, are recognized 
 after a short familiarity with them. These chords or 
 intervals are perceived to have each a peculiar charac- 
 ter, which separates them from the relations of two 
 sounds taken at random, and makes it easy to know 
 them when sung or played on an instrument ; and for 
 most persons, not difficult to sing the sounds in succes- 
 sion exactly, or nearly correct. These musical relations, 
 or concords, then, are the groundwork of our musical 
 series of sounds. But how are we to name these in- 
 describable sensible characters? how to refer, with un- 
 erring accuracy, to a type which exists only in our 
 own perceptions? We must have for this purpose a 
 /Scale and a Standard. 
 
 The Musical Scale is a series of eight notes, ascend- 
 ing by certain steps from the first or key-note to the 
 octave above it, each of the notes being fixed by such 
 distinguishable musical relations as we have spoken of 
 above. We may call these notes c, D, E, F, G, A, B, c; 
 and we may then say that G is determined by its being 
 a fifth above c ; D by its being a fourth below G ; E by 
 its being a third above c; and similarly of the rest. 
 It will be recollected that the terms & fifth, a fourth, a 
 third, have hitherto been introduced as expressing 
 certain simple and indescribable musical relations 
 among sounds, which might have been indicated by 
 any other names. Thus we might call the fifth the 
 dominant, and the fourth the subdominant, as is done 
 in one part of musical science. But the names we 
 have used, which are the common ones, are in fact 
 derived from the number of notes which these inter- 
 vals include in the scale obtained in the above manner. 
 The notes, c, D, E, r, G, being five, the interval from c 
 to G is a fifth, and so of the rest. The fixation of this 
 scale gave the means of describing exactly any note 
 which occurs in the scale, and the method is easily 
 applicable to notes above and below this range; for in 
 a series of sounds higher or lower by an octave than 
 VOL. i. Z 
 
338 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 this standard series, the ear discovers a recurrence of 
 the same relations so exact, that a person may some- 
 times imagine he is producing the same notes as an- 
 other when he is singing the same air an octave higher. 
 Hence the next eight notes may be conveniently de- 
 noted by a repetition of the same letters, as the first ; 
 thus, c, D, E, F, G, A, B, c, d, e, f, g, a, b; and it is easy 
 to devise a continuation of such cycles. And other ad- 
 missible notes are designated by a further modification 
 of the standard ones, as by making each note flat or 
 sharp ; which modification it is not necessary here to 
 consider, since our object is only to show how a standard 
 is attainable, and how it serves the ends of science. 
 
 We may observe, however, that the above is not an 
 exact account of the first, or early Greek scale ; for 
 that scale was founded on a primary division of the 
 interval of two octaves (the extreme range which it 
 admitted) into five tetrachords, each tetrachord in- 
 cluding the interval of a fourth. All the notes of 
 this series had different names borrowed from this 
 division 1 ; thus mese was the middle or key-note; the 
 note below it was lichanos meson, the next below was 
 parypate mesdn, the next lower, hypate meson. The 
 fifth above mese was nete diazeugmendn, the octave was 
 nete hyperbolcedn. 
 
 4. But supposing a complete system of such deno- 
 minations established, how could it be with certainty 
 and rigour applied 1 ? The human ear is fallible, the 
 organs of voice imperfectly obedient; if this were not 
 so, there would be no such thing as a good ear or a 
 good voice. What means can be devised of finding at 
 will a perfect concord, a fifth or a fourth? Or sup- 
 posing such concords fixed by an acknowledged autho- 
 rity, how can they be referred to, and the authority 
 adduced ? How can we enact a Standard of sounds ? 
 
 A Standard was discovered in the Monochord. A 
 musical string properly stretched, may be made to pro- 
 duce different notes, in proportion as we intercept a 
 longer or shorter portion, and make this portion vi- 
 
 Burney's History of Music, vol. i p. 28. 
 
MEASURE OF SECONDARY QUALITIES. 339 
 
 brate. The relation of the length of the strings which 
 thus sound the two notes G and c is fixed and constant, 
 and the same is true of all other notes. Hence the 
 musical interval of any notes of which we know the 
 places in the musical scale, may be reproduced by mea- 
 suring the lengths of string which are known to give 
 them. If c be of the length 180, D is 160, E is 144, p 
 is 135, G is 120; and thus the musical relations are 
 reduced to numerical relations, and the monochord is 
 a complete and perfect Tonometer. 
 
 We have here taken the length of the string as the 
 measure of the tone : but we may observe that there is 
 in us a necessary tendency to assume that the ground 
 of this measure is to be sought in some ulterior cause; 
 and when we consider the matter further, we find this 
 cause in the frequency of these vibrations of the string. 
 The truth that the same note must result from the 
 same frequency of vibration is readily assented to on 
 a slight suggestion of experience. Thus Mersenne 2 , 
 when he undertakes to determine the frequency of 
 vibrations of a given sound, says ' Supponendum est 
 quoscunque nervos et quaslibet chordas unisonum faci- 
 entes eundem efficere numerum recursuum eodem vel 
 equali tempore, quod perpetua constat experientia.' 
 And he proceeds to apply it to cases where experience 
 could not verify this assertion, or at least had not 
 verified it, as to that of pipes. 
 
 The pursuit of these numerical relations of tones 
 forms the science of Harmonics ; of which here we do 
 not pretend to give an account, but only to show, how 
 the invention of a Scale and Nomenclature, a Standard 
 and Measure of the tone of sounds, is its necessary 
 basis. We will therefore now proceed to speak of 
 another subject; colour. 
 
 SECT. III. Scales of Colour. 
 
 5. The Prismatic Scale of Colour. A SCALE of 
 Colour must depend originally upon differences dis- 
 
 Harmonia, lib. ii. prop. ig. 
 
 Z 2 
 
340 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 cernible by the eye, as a scale of notes depends on 
 differences perceived by the ear. In one respect the 
 difficulty is greater in the case of the visible qualities, 
 for there are no relations of colour which the eye pecu- 
 liarly singles out and distinguishes, as the ear selects 
 and distinguishes an octave or a fifth. Hence we are 
 compelled to take an arbitrary scale ; and we have to 
 find one which is fixed, and which includes a proper 
 collection of colours. The prismatic spectrum, or co- 
 loured image produced when a small beam of light passes 
 obliquely through any transparent surface (as the sur- 
 face of a prism of glass,) offers an obvious Standard as 
 far as it is applicable. Accordingly colours have, for 
 various purposes, been designated by their place in the 
 spectrum, ever since the time of Newton ; and we have 
 thus a means of referring to such colours as are in- 
 cluded in the series red, orange, yellow, green, blue, 
 violet, indigo, and the intermediate tints. 
 
 But this scale is not capable of numerical precision. 
 If the spectrum could be exactly defined as to its ex- 
 tremities, and if these colours occupied always the 
 same proportional part of it, we might describe any 
 colour in the above series by the measure of its posi- 
 tion. But the fact is otherwise. The spectrum is too 
 indefinite in its boundaries to afford any distinct point 
 from which we may commence our measures; and 
 moreover the spectra produced by different transparent 
 bodies differ from each other. Newton had supposed 
 that the spectrum and its parts were the same, so long 
 as the refraction was the same; but his successors 
 discovered that, with the same amount of refraction in 
 different kinds of glass, there are different magnitudes 
 of the spectrum; and what is still worse with reference 
 to our present purpose, that the spectra from different 
 glasses have the colours distributed in different propor- 
 tions. In order, therefore, to make the spectrum the 
 scale of colour, we must assume some fixed substance ; 
 for instance, we may take water, and thus a series 
 approaching to the colours of the rainbow will be our 
 standard. But we should still have an extreme diffi- 
 culty in applying such a rule. The distinctions of 
 
MEASURE OF SECONDARY QUALITIES. 34! 
 
 colour which the terms of common language express, 
 are not used with perfect unanimity or with rigorous 
 precision. What one person calls bluish green another 
 calls greenish blue. Nobody can say what is the precise 
 boundary between red and orange. Thus the prismatic 
 scale of colour was incapable of mathematical exactness, 
 and this inconvenience was felt up to our own times. 
 
 But this difficulty was removed by a curious dis- 
 covery of Wollaston and Fraunhofer; who found that 
 there are, in the solar spectrum, certain fine black 
 Lines which occupy a definite place in the series of 
 colours, and can be observed with perfect precision. 
 We have now no uncertainty as to what coloured 
 light we are speaking of, when we describe it as that 
 part of the spectrum in which Fraunhofer's Line c or 
 D occurs. And thus, by this discovery, the prismatic 
 spectrum of sunlight became, for certain purposes, an 
 exact Chromatometer. 
 
 6. Newton's Scale of Colours. Still, such a stand- 
 ard, though definite, is arbitrary and seemingly ano- 
 malous. The lines A, B, c, D, &c., of Fraunhofer's 
 spectrum are distributed without any apparent order 
 or law; and we do not, in this way, obtain numerical 
 measures, which is what, in all cases, we desire to 
 have. Another discovery of Newton, however, gives 
 us a spectrum containing the same colours as the pris- 
 matic spectrum, but produced in another way, so that 
 the colours have a numerical relation. I speak of the 
 laws of the Colours of Thin Plates. The little rainbows 
 which we sometimes see in the cracks of broken glass 
 are governed by fixed and simple laws. The kind of 
 colour produced at any point depends on the thickness 
 of the thin plate of air included in the fissure. If the 
 thickness be eight-millionths of an inch, the colour is 
 orange, if fifteen-millionths of an inch, we have green, 
 and so on ; and thus these numbers, which succeed each 
 other in a regular order from red to indigo, give a 
 numerical measure of each colour; which measure, when 
 we pursue the subject, we find is one of the bases of all 
 optical theory. The series of colours obtained from 
 plates of air of gradually increasing thickness is called 
 
342 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 Newtoris Scale of Colours; but we may observe that 
 this is not precisely what we are here speaking of, a 
 scale of simple colours; it is a series produced by 
 certain combinations, resulting from the repetition of 
 the first spectrum, and is mainly useful as a standard 
 for similar phenomena, and not for colour in general. 
 The real scale of colour is to be found, as we have said, 
 in the numbers which express the thickness of the 
 producing film; in the length of a fit in Newton's 
 phraseology, or the length of an undulation in the 
 modern theory. 
 
 7. Scales of Impure Colours. The standards just 
 spoken of include (mainly at least) only pure and simple 
 colours ; and however complete these standards may be 
 for certain objects of the science of optics, they are in- 
 sufficient for other purposes. They do not enable us to 
 put in their place mixed and impure colours. And 
 there is, in the case of colour, a difficulty already 
 noticed, which does not occur in the case of sound; 
 two notes, when sounded together, are not necessarily 
 heard as one; they are recognized as still two, and as 
 forming a concord or a discord. But two colours form 
 a single colour ; and the eye cannot, in any way, dis- 
 tinguish between a green compound of blue and yellow, 
 and the simple, undecomposable green of the spectrum. 
 By composition of three or more colours, innumerable 
 new colours may be generated which form no part of 
 the prismatic series; and by such compositions is 
 woven the infinitely varied web of colour which forms 
 the clothing of nature. How are we to classify and 
 arrange all the possible colours of objects, so that each 
 shall have a place and name 1 ? How shall we find a 
 chromatometer for impure as well as for pure colour? 
 
 Though no optical investigations have depended on 
 a scale of impure colours, such a scale has been wanted 
 and invented for other purposes ; for instance, in order 
 to identify and describe objects of natural history. 
 Not to speak of earlier essays, we may notice Werner's 
 Nomenclature of Colours, devised for the purpose of 
 describing minerals. This scale of colour was far supe- 
 rior to any which had previously been promulgated. 
 
MEASURE OF SECONDARY QUALITIES. 343 
 
 It was, indeed, arbitrary in the selection of its degrees, 
 and in a great measure in their arrangement; and the 
 colours were described by the usual terms, though 
 generally with some added distinction; as blackish 
 green, bluish green, apple-green, emerald-green. But 
 the great merit of the scale was its giving a, fixed con- 
 ventional meaning to these terms, so that they lost 
 much of their usual vagueness. Thus apple-green did 
 not mean the colour of any green apple casually taken ; 
 but a certain definite colour which the student was to 
 bear in mind, whether or not he had ever seen an 
 apple of that exact hue. The words were not a descrip- 
 tion, but a record of the colour : the memory was to 
 retain a sensation, not a name. 
 
 The imperfection of the system (arising from its 
 arbitrary form) was its incompleteness : however well 
 it served for the reference of the colours which it did 
 contain, it was applicable to no others ; and thus though 
 Werner's enumeration extended to more than a hun- 
 dred colours, there occur in nature a still greater 
 number which cannot be exactly described by means 
 of it. 
 
 In such cases the unclassed colour is, by the Werne- 
 rians, defined by stating it as intermediate between 
 two others: thus we have an object described as be- 
 tween emerald-green and grass-green. The eye is 
 capable of perceiving a gradation from one colour to 
 another ; such as may be produced by a gradual mix- 
 ture in various ways. And if we image to ourselves 
 such a mixture, we can compare with it a given colour. 
 But in employing this method we have nothing to tell 
 us in what part of the scale we must seek for an approxi- 
 mation to our unclassed colour. We have no rule for 
 discovering where we are to look for the boundaries of 
 the definition of a colour which the Wernerian series 
 does not supply. For it is not always between con- 
 tiguous members of the series that the undescribed 
 colour is found. If we place emerald-green between 
 apple-green and grass-green, we may yet have a colour 
 intermediate between emerald-green and leek-green; 
 and, in fact, the Wernerian series of colours is destitute 
 
344 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 of a principle of self-arrangement and gradation ; and 
 is thus necessarily and incurably imperfect. 
 
 8. We should have a complete Scale of Colours, 
 if we could form a series including all colours, and 
 arranged so that each colour was intermediate in its 
 tint between the adjacent terms of the series; for then, 
 whether we took many or few of the steps of the 
 series for our standard terms, the rest could be sup- 
 plied by the law of continuity ; and any given colour 
 would either correspond to one of the steps of our scale 
 or fall between two intermediate ones. The invention 
 of a Chromatometer for Impure Colours, therefore, 
 requires that we should be able to form all possible 
 colours by such intermediation in a systematic man- 
 ner; that is, by the mixture or combination of certain 
 elementary colours according to a simple rule: and 
 we are led to ask whether such a process has been 
 shown to be possible. 
 
 The colours of the prismatic spectrum obviously do 
 form a continuous series; green is intermediate be- 
 tween its neighbours yellow and blue, orange between 
 red and yellow ; and if we suppose the two ends of the 
 spectrum bent round to meet each other, so that the 
 arrangement of the colours may be circular, the violet 
 and indigo will find their appropriate place between 
 the blue and red. And all the interjacent tints of 
 the spectrum, as well as the ones just named, will 
 result from such an arrangement. Thus all the pure 
 colours are produced by combinations two and two of 
 three primary colours, Red, Yellow, and Blue : and the 
 question suggests itself whether these three are not 
 really the only Primary Colours, and whether all the 
 impure colours do not arise from mixtures of the three 
 in various proportions. There are various modes in 
 which this suggestion may be applied to the construc- 
 tion of a scale of colours; but the simplest, and the one 
 which appears really to verify the conjecture that all 
 possible colours may be so exhibited, is the following. 
 A certain combination of red, yellow, and blue, will 
 produce black, or pure grey, and when diluted, will 
 give all the shades of grey which intervene between 
 
MEASURE OF SECONDARY QUALITIES. 345 
 
 black and white. By adding various shades of grey, 
 then, to pure colours, we may obtain all the possible 
 ternary combinations of red, yellow, and blue ; and in 
 this way it is found that we exhaust the range of 
 colours. Thus the circle of pure colours of which we 
 have spoken may be accompanied by several other 
 circles, in which these colours are tinged with a less or 
 greater shade of grey; and in this manner it is found 
 that we have a perfect chromatometer; every possible 
 colour being exhibited either exactly or by means of 
 approximate and contiguous limits. The arrangement 
 of colours has been brought into this final and com- 
 plete form by M. Merimee, whose Chromatic Scale is 
 published by M. Mirbel in his Elements of Botany. 
 We may observe that such a standard affords us a 
 numerical exponent for every colour by means of the 
 proportions of the three primary colours which com- 
 pose it; or, expressing the same result otherwise, by 
 means of the pure colour which is involved, and the 
 proportion of grey by which it is rendered impure. In 
 such a scale the fundamental elements would be the 
 precise tints of red, yellow, and blue which are found 
 or assumed to be primary; the numerical exponents of 
 each colour would depend upon the arbitrary number 
 of degrees which we interpose between each two pri- 
 mary colours; and between each pure colour and abso- 
 lute blackness. No such numerical scale has, however, 
 as yet, obtained general acceptation 3 . 
 
 3 The reference to Fraurihofer's of course the one here meant. It is 
 
 Lines, as a means of determining the an usual practice of optical experi- 
 
 place of a colour in the prismatic menters to refer to the colours of 
 
 series, has been objected to, because, such a spectrum, denning them by 
 
 as is asserted, the colours which are Fraunhofer's Lines. 
 
 in the neighbourhood of each line I do not know whether it needs 
 
 vary with the position of the sun, explanation that the ' first spectrum ' 
 
 state of the atmosphere and the like, in Newton's rings is a ring of the 
 
 It is very evident that coloured light prismatic colours, 
 
 refracted by the prism will not give I have not had an opportunity of 
 
 the same spectrum as white light, consulting Lambert's Photomctria, 
 
 The spectrum given by white light is sive de mensura et gradibus luminis, 
 
346 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 SECT. IY. Scales of Light. 
 
 9. Photometer. ANOTHER instrument much needed 
 .in optical researches is a Photometer, a measure of the 
 intensity of light. In this case, also, the organ of sense, 
 the eye, is the ultimate judge ', nor has any effect of 
 light, as light, yet been discovered which we can sub- 
 stitute for such a judgment. All instruments, such as 
 that of Leslie, which employ the heating effect of light, 
 or at least all that have hitherto been proposed, are 
 inadmissible as photometers. But though the eye can 
 judge of two surfaces illuminated by light of the same 
 colour, and can determine when they are equally 
 bright, or which is the brighter, the eye can by no 
 means decide at sight the proportion of illumination. 
 How much in such judgments we are affected by con- 
 trast, is easily seen when we consider how different is 
 the apparent brightness of the moon at mid-day and at 
 midnight, though the light which we receive from her 
 is, in fact, the same at both periods. In order to apply 
 a scale in this case, we must take advantage of the 
 known numerical relations of light. We are certain 
 that if all other illumination be excluded, two equal 
 luminaries, under the same circumstances, will produce 
 an illumination twice as great a? one does; and we 
 can easily prove, from mathematical considerations, 
 that if light be not enfeebled by the medium through 
 which it passes, the illumination on a given surface 
 will diminish as the square of the distance of the lumi- 
 nary increases. If, therefore, we can by taking a frac- 
 tion thus known of the illuminating effect of one 
 luminary, make it equal to the total effect of another, 
 of which equality the eye is a competent judge, we 
 compare the effects of the two luminaries. In order to 
 make this comparison we may, with Rumford, look at 
 the shadows of the same object made by the two lights, 
 
 colorum, et umbrae, published in 1760, present work is not intended to be 
 
 nor Mayer's Commentatio de Affinitate . complete as a history ; and I hope I 
 
 Colorum, (1758), in which, I believe, have given sufficient historical detail 
 
 he describes a chromatometer. The to answer its philosophical purpose. 
 
MEASURE OF SECONDARY QUALITIES. 347 
 
 or with Ritchie, we may view the brightness produced 
 on two contiguous surfaces, framing an apparatus so 
 that the equality may be brought about by proper 
 adjustment; and thus a measure will become practica- 
 ble. Or we may employ other methods as was done 
 by Wollaston 4 , who reduced the light of the sun by 
 observing it as reflected from a bright globule, and 
 thus found the light of the sun to be 10,000,000,000 
 times that of Sirius, the brightest fixed star. All these 
 methods are inaccurate, even as methods of compari- 
 son ; and do not offer any fixed or convenient numeri- 
 cal standard; but none better have yet been devised 5 . 
 
 10. Cyanometer. As we thus measure the bright- 
 ness of a colourless light, we may measure the intensity 
 of any particular colour in the same way; that is, by 
 applying a standard exhibiting the gradations of the 
 colour in question till we find a shade which is seen to 
 agree with the proposed object. Such an instrument 
 we have in the Cyanometer^ which was invented by 
 Saussure for the purpose of measuring the intensity of 
 the blue colour of the sky. We may introduce into 
 such an instrument a numerical scale, but the numbers 
 in such a scale will be altogether arbitrary. 
 
 SECT. V. Scales of Heat. 
 
 1 1. Thermometers. WHEN we proceed to the sensa- 
 tion of heat, and seek a measure of that quality, we 
 find, at first sight, new difficulties. Our sensations of 
 this kind are more fluctuating than those of vision ; for 
 we know that the same object may feel warm to one 
 hand and cold to another at the same instant, if the 
 hands have been previously cooled and warmed respec- 
 tively. Nor can we obtain here, as in the case of light, 
 self-evident numerical relations of the heat communi- 
 cated in given circumstances; for we know that the 
 
 4 Phil. Trans. 1820, p. 19. 
 
 5 Improved Photometers have been heil ; but they depend upon prin- 
 devised by Professor Wheatstone, ciples similar to those mentioned in 
 Professor Potter, and Professor Stein- the text. 
 
348 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 effect so produced will depend on the warmth of the 
 body to be heated, as well as on that of the source of 
 heat; the summer sun, which warms our bodies, will 
 not augment the heat of a red-hot iron. The cause of 
 the difference of these cases is, that bodies do not 
 receive the whole of their heat, as they receive the 
 whole of their light, from the immediate influence of 
 obvious external agents. There is no readily-discovered 
 absolute cold, corresponding to the absolute darkness 
 which we can easily produce or imagine. Hence we 
 should be greatly at a loss to devise a Thermometer, if 
 we did not find an indirect effect of heat sufficiently 
 constant and measurable to answer this purpose. We 
 discover, however, such an effect in the expansion of 
 bodies by the effect of heat. 
 
 1 2. Many obvious phenomena show that air, under 
 given circumstances, expands by the effect of heat ; the 
 same is seen to be true of liquids, as of water, and 
 spirit of wine ; and the property is found to belong also 
 to the metallic fluid, quicksilver. A more careful 
 examination showed that the increase of bulk in some 
 of these bodies by increase of Heat was a fact of a 
 nature sufficiently constant and regular to afford a 
 means of measuring that previously intangible quality ; 
 and the Thermometer was invented. There were, how- 
 ever, many difficulties to overcome, and many points 
 to settle, before this instrument was fit for the purposes 
 of science. 
 
 An explanation of the way in which this was done 
 necessarily includes an important chapter of the history 
 of Thermotics. We must now, therefore, briefly notice 
 historically the progress, of the Thermometer. The 
 leading steps of this progress, after the first invention 
 of the instrument, were= The establishment of fixed 
 points in the thermometric scale The comparison of 
 the scales of different substances And the reconcile- 
 ment of these differences by some method of inter- 
 preting them as indications of the absolute quantity of 
 heat. 
 
 13. It would occupy too much space to give in 
 detail the history of the successive attempts by which 
 
MEASURE OF SECONDARY QUALITIES. 349 
 
 these steps were effected. A thermometer is described 
 by Bacon under the title Vitrum Calendars; this was 
 an air thermometer. Newton used a thermometer of 
 linseed oil, and he perceived that the first step requi- 
 site to give value to such an instrument was to fix its 
 scale; accordingly he proposed his Scala Graduum 
 Caloris a . But when thermometers of different liquids 
 were compared, it appeared, from their discrepancies, 
 that this fixation of the scale of heat was more difficult 
 than had been supposed. It was, however, effected. 
 Newton had taken freezing water, or rather thawing 
 snow, as the zero of his scale, which is really a fixed 
 point; Halley and Amontons discovered (in 1693 and 
 1702) that the heat of boiling water is another fixed 
 point; and Daniel Gabriel Fahrenheit, of Dantzig, by 
 carefully applying these two standard points, produced, 
 about 1714, thermometers, which were constantly con- 
 sistent with each other. This result was much admired 
 at the time, and was, in fact, the solution of the 
 problem just stated, the fixation of the scale of heat. 
 
 14. But the scale thus obtained is a conventional 
 not a natural scale. It depends upon the fluid employed 
 for the thermometer. The progress of expansion from 
 the heat of freezing to that of boiling water is differ- 
 ent for mercury, oil, water, spirit of wine, air. A 
 degree of heat which is half-way between these two 
 standard points according to a mercurial thermometer, 
 will be below the half-way point in a spirit thermome- 
 ter, and above it in an air thermometer. Each liquid 
 has its own march in the course of its expansion. Deluc 
 and others compared the marches of various liquids, 
 and thus made what we may call a concordance of ther- 
 mometers of various kinds. 
 
 15. Here the question further occurs : Is there not 
 some natural measure of the degrees of heat? It ap- 
 pears certain that there must be such a measure, and 
 that by means of it all the scales of different liquids 
 must be reconciled. Yet this does not seem to have 
 occurred at once to men's minds. Deluc, in speaking 
 
 6 Phil. Trans. 1701. 
 
350 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 of the researches which we have just mentioned, says 7 , 
 ' When I undertook these experiments, it never once 
 came into my thoughts that they could conduct me 
 with any probability to a table of real degrees of heat. 
 But hope grows with success, and desire with hope.' 
 Accordingly he pursued this inquiry for a long course 
 of years. 
 
 What are the principles by which we are to be 
 guided to the true measure of heat? Here, as in all 
 the sciences of this class, we have the general princi- 
 ple, that the secondary quality, Heat, must be supposed 
 to be perceived in some way by a material Medium or 
 Fluid. If we take that which is, perhaps, the simplest 
 form of this hypothesis, that the heat depends upon 
 the quantity of this fluid, or Caloric, which is present, 
 we shall find that we are led to propositions which 
 may serve as a foundation for a natural measure of 
 heat. The Method of Mixtures is one example of such 
 a result. If we mix together two pints of water, one 
 hot and one cold, is it not manifest that the tempera- 
 ture of the mixture must be midway between the two ? 
 Each of the two portions brings with it its own heat. 
 The whole heat, or caloric, of the mixture is the sum 
 of the two; and the heat of each half must be the 
 half of this sum, and therefore its temperature must 
 be intermediate between the temperatures of the equal 
 portions which were mixed. Deluc made experiments 
 founded upon this principle, and was led by them to 
 conclude that 'the dilatations of mercury follow an 
 accelerated march for successive equal augmentations 
 of heat.' 
 
 But there are various circumstances which prevent 
 this method of mixtures from being so satisfactory as 
 at first sight it seems to promise to be. The different 
 capacities for heat of different substances, and even of 
 the same substance at different temperatures, introduce 
 much difficulty into the experiments; and this path of 
 inquiry has not yet led to a satisfactory result. 
 
 Modif. de I'Atmosph. 1782, p. 303. 
 
MEASURE OF SECONDARY QUALITIES. 351 
 
 1 6. Another mode of inquiring into the natural 
 measure of heat is to seek it by researches on the law 
 of cooling of hot bodies. If we assume that the pro- 
 cess of cooling of hot bodies consists in a certain ma- 
 terial heat flying off, we may, by means of certain 
 probable hypotheses, determine mathematically the law 
 according to which the temperature decreases as time 
 goes on j and we may assume that to be the true mea- 
 sure of temperature which gives to the experimental 
 law of cooling the most simple and probable form. 
 
 It appears evident from the most obvious concep- 
 tions which we can form of the manner in which a 
 body parts with its superabundant heat, that the hotter 
 a body is, the faster it cools j though it is not clear 
 without experiment, by what law the rate of cooling 
 will depend upon the heat of the body. Newton 
 took for granted the most simple and seemingly natu- 
 ral law of this dependence: he supposed the rate of 
 cooling to be proportional to the temperature, and 
 from this supposition he could deduce the temperature 
 of a hot iron, calculating from the original temperature 
 and the time during which it had been cooling. By 
 calculation founded on such a basis, he graduated his 
 thermometer. 
 
 17. But a little further consideration showed that 
 the rate of cooling of a hot body depended upon the 
 temperature of the surrounding bodies, as well as upon 
 its own temperature. Prevost's Theory of Exchanges* 
 was propounded with a view of explaining this depend- 
 ence, and was generally accepted. According to this 
 theory, all bodies radiate heat to one another, and are 
 thus constantly giving and receiving heat ; and a body 
 which is hotter than surrounding bodies, cools itself, 
 and warms the surrounding, bodies, by an exchange of 
 heat for heat, in which they are the gainers. Hence if 
 be the temperature of the bodies, or of the space, by 
 which the hot body is surrounded, and + 1 the tempe- 
 rature of the hot body, the rate of cooling will depend 
 
 8 Recherches sur la Chaleur, 1791. Hist. Ind. Sc. b. x. c. i sect. 2. 
 
352 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 upon the excess of the radiation for a temperature 6 + 1, 
 above the radiation for a temperature 0. 
 
 Accordingly, in the admirable researches of MM. 
 Dulong and Petit upon the cooling of bodies, it was 
 assumed that the rate of cooling of the hot body was 
 represented by the excess of F (9 + 1) above F(0) ; where 
 F represented some mathematical function, that is, 
 some expression obtained by arithmetical operations 
 from the temperatures + 1 and ; although what these 
 operations are to be, was left undecided, and was in 
 fact determined by the experiments. And the result 
 of their investigations was, that the function is of this 
 kind : when the temperature increases by equal inter- 
 vals, the function increases in a continued geometric 
 proportion 9 . This was, in fact, the same law which 
 had been assumed by Newton and others, with this 
 difference, that they had neglected the term which 
 depends upon the temperature of the surrounding 
 space. 
 
 1 8. This law falls in so well with the best concep- 
 tions we can form of the mechanism of cooling upon 
 the supposition of a radiant fluid caloric, that it gives 
 great probability to the scale of temperature on which 
 the simplicity of the result depends. Now the tempe- 
 ratures in the formulae just referred to were expressed 
 by means of the air thermometer. Hence MM. Du- 
 long and Petit justly state, that while all different sub- 
 stances employed as thermometers give different laws 
 of thermotical phenomena, their own success in obtain- 
 ing simple and general laws by means of the air ther- 
 mometer, is a strong recommendation of that as the 
 natural scale of heat. They add 10 , 'The well-known 
 uniformity of the principal physical properties of all 
 gases, and especially the perfect identity of their laws 
 of dilatation by heat, [a very important discovery of 
 
 9 The formula for the rate of cool- the body, the state of its surface, and 
 ing is ma e+t - m a e , where the quan- other circumstances.-^. Chim. vii. 
 tity m depends upon the nature of I ^' 
 
 10 Annaks de Chimie, vii 153. 
 
MEASURE OF SECONDARY QUALITIES. 353 
 
 Dalton and Gay Lussac 11 ,] make it very probable that 
 in this class of bodies the disturbing causes have not 
 the same influence as in solids and liquids; and conse- 
 quently that the changes of bulk produced by the 
 action of heat are here in a more immediate depend- 
 ence on the force which produces them.' 
 
 19. Still we cannot consider this point as settled 
 till we obtain a more complete theoretical insight into 
 the nature of heat itself. If it be true that heat con- 
 sists in the vibrations of a fluid, then, although, as 
 Ampere has shown 12 , the laws of radiation will, on 
 mathematical grounds, be the same as they are on the 
 hypothesis of emission, we cannot consider the natural 
 scale of heat as determined, till we have discovered 
 some means of measuring the caloriferous vibrations 
 as we measure luminiferous vibrations. We shall only 
 know what the quantity of heat is when we know 
 what heat itself is j when we have obtained a theory 
 which satisfactorily explains the manner in which the 
 substance or medium of heat produces its effects. When 
 we see how radiation and. conduction, dilatation and 
 liquefaction, are all produced by mechanical changes of 
 the same fluid, we shall then see what the nature of 
 that change is which dilatation really measures, and 
 what relation it bears to any more proper standard of 
 heat. 
 
 We may add, that while our thermotical theory is 
 still so imperfect as it is, all attempts to divine the 
 true nature of the relation between light and heat are 
 premature, and must be in the highest degree insecure 
 and visionary. Speculations in which, from the general 
 assumption of a caloriferous and luminiferous medium, 
 and from a few facts arbitrarily selected and loosely 
 analysed, a general theory of light and heat is as- 
 serted, are entirely foreign to the course of inductive 
 science, and cannot lead to any stable and substantial 
 truth. 
 
 20. Other Instruments for measuring Heat. It 
 does not belong to our present purpose to speak of 
 
 11 Hist. Ind. Sc. b. x. c. ii. sect. i. * 2 Ib. c. iv. 
 
 VOL. I. A A 
 
354 PHILOSOPHY OF THE MECHANICAL SCIENCES. 
 
 instruments of which the object is to measure, not sen- 
 sible qualities, but some effect or modification of the 
 cause by which such qualities are produced : such, for 
 instance, are the Calorimeter, employed by Lavoisier 
 and Laplace, in order to compare the Specific Heat of 
 different substances; and the Actinometer, invented by 
 Sir John Herschel, in order to determine the effect of 
 the Sun's Rays by means of the heat which they com- 
 municate in a given time; which effect is, as may 
 readily be supposed, very different under different cir- 
 cumstances of atmosphere and position. The laws of 
 such effects may be valuable contributions to our 
 knowledge of heat, but the interpretation of them 
 must depend on a previous knowledge of the relations 
 which temperature bears to heat, according to the 
 views just explained. 
 
 SECT. VI. Scales of other Qualities. 
 
 21. BEFOKE quitting the subject of the measures of 
 sensible qualities, we may observe that there are seve- 
 ral other such qualities for which it would be neces- 
 sary to have scales and means of measuring, in order 
 to make any approach to science on such subjects. 
 This is true, for instance, of Tastes and Smells. Indeed 
 some attempts have been made towards a classifica- 
 tion of the Tastes of sapid substances, but these have 
 not yet assumed any satisfactory or systematic charac- 
 ter; and I am not aware that any instrument has 
 been suggested for measuring either the Flavour or 
 the Odour of bodies which possess such qualities. 
 
 22. Quality of Sounds. The same is true of that 
 kind of difference in sounds which is peculiarly termed 
 their Quality; that character by which, for instance, 
 the sound of a flute differs from that of a hautbois, 
 when the note is the same; or a woman's voice from 
 a boy's. 
 
 23. Articulate Sounds. There is also in sounds 
 another difference, of which the nature is still obscure, 
 but in reducing which to rule, and consequently to 
 measure, some progress has nevertheless been made. 
 
MEASURE OF SECONDARY QUALITIES. 355 
 
 I speak of the differences of sound considered as arti- 
 culate. Classifications of the sounds of the usual alpha- 
 bets have been frequently proposed ,' for instance, that 
 which arranges the Consonants in the following groups,: 
 
 Sharp. Flat. Sharp Aspirate. Flat Aspirate. Nasal. 
 
 p b ph (/) bh (v) m 
 
 k g (hard) kh gh ng 
 
 t d th (sharp) th (flat) n 
 
 s z sh zh 
 
 It is easily perceived that the relations of the sounds 
 in each of these horizontal lines are analogous; and 
 accordingly the rules of derivation and modification of 
 words in several languages proceed upon such ana- 
 logies. In the same manner the Vowels may be arranged 
 in an order depending on their sound. But to make 
 such arrangements fixed and indisputable, we ought to 
 know the mechanism by which such modifications are 
 caused. Instruments have been invented by which 
 some of these sounds can be imitated; and if such 
 instruments could be made to produce the above 
 series of . articulate sounds, by connected and regular 
 processes, we should find, in the process, a measure of 
 the sound produced. This has been in a great degree 
 effected for the Vowels by Professor Willis's artificial 
 mode of imitating them. For he finds that if a musical 
 reed be made to sound through a cylindrical pipe, we 
 obtain by gradually lengthening the cylindrical pipe, 
 the series of vowels I, E, A, o, U, with intermediate 
 sounds 13 . In this instrument, then, the length of the 
 pipe would determine the vowel, and might be used 
 numerically to express it. Such an instrument so 
 employed would be a measure of vowel quality, and 
 might be called a Phthongometer. 
 
 Our business at present, however, is not with instru- 
 ments which might be devised for measuring sensible 
 qualities, but with those which have been so used, 
 and have thus been the basis of the sciences in which 
 
 Camb. Trans. voL iii. p. 239. 
 
 AA 2 
 
356 PHILOSOPHY OP THE MECHANICAL SCIENCES. 
 
 such qualities are treated of; and this we have now 
 done sufficiently for our present purpose. 
 
 24. There is another Idea which, though hitherto 
 very vaguely entertained, has had considerable influ- 
 ence in the formation, both of the sciences spoken of 
 in the present Book, and on others which will here- 
 after come under our notice: namely, the Idea of 
 Polarity. This Idea will be the subject of the ensuing 
 Book. And although this Idea forms a part of the 
 basis of various other extensive portions of science, as 
 Optics and Chemistry, it occupies so peculiarly con- 
 spicuous a place in speculations belonging to what I 
 have termed the Mechanico-Chemical Sciences, (Mag- 
 netism and Electricity,) that I shall designate the dis- 
 cussion of the Idea of Polarity as the Philosophy of 
 those Sciences. 
 
BOOK V. 
 
 THE 
 
 PHILOSOPHY 
 
 OF THE 
 
 MECHANICO-CHEMICAL SCIENCES. 
 
En dormant k ces c6ts le nom depots, j'appelerai polarisation 
 la modification qui donne a la lumiere des proprie'te's relatives a 
 ces poles. J'ai tarde" jusqu'a present a admettre ce terme dans 
 la description des phe'nomenes physiques dont il est question ; je 
 n'ai pas ose" 1'introduire dans leg mdmoires ou j'ai public" mes 
 dernieres experiences ; mais les vari^t^s qu'offre ce nouveau 
 ph^nomene, et la difficult^ de les ddcrire, me forcent a admettre 
 cette nouvelle expression, qui signifie simplement la modification 
 que la lumiere a subie en acquerant de nouvelles propri^t^s qui 
 ne sont pas relatives a la direction du rayon, mais seulement a 
 ses c6te's considdres k angles droits et dans un plan perpendicu- 
 laire a sa direction. 
 
 Malus (1811), Mem. de Inst. torn. xi. p. 106. 
 
THE PHILOSOPHY OF THE MECHANICO-CHEMICAL 
 SCIENCES. 
 
 CHAPTER I. 
 
 ATTEMPTS AT THE SCIENTIFIC APPLICATION OF THE 
 IDEA OF POLARITY. 
 
 i. TN" some of the mechanical sciences, as Magnet- 
 _L ism and Optics, the phenomena are found to de- 
 pend upon position (the position of the magnet, or of 
 the ray of light,) in a peculiar alternate manner. This 
 dependence, as it was first apprehended, was repre- 
 sented by means of certain conceptions of space and 
 force, as for instance by considering the two Poles of a 
 magnet. But in all such modes of representing these 
 alternations by the conceptions borrowed from other 
 ideas, a closer examination detected something super- 
 fluous and something defective ; and in proportion as 
 the view which philosophers took of this relation was 
 gradually purified from these incongruous elements, 
 and was rendered more general and abstract by the 
 discovery of analogous properties in new cases, it was 
 perceived that the relation could not be adequately 
 apprehended without considering it as involving a 
 peculiar and independent Idea, which we may designate 
 by the term Polarity. 
 
 We shall trace some of the forms in which this Idea 
 has manifested itself in the history of science. In 
 doing so we shall not begin, as in other Books of this 
 work we have done, by speaking of the notion as it is 
 
360 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 employed in common use : for the relation of Polarity 
 is of so abstract and technical a nature, that it is not 
 employed, at least in any distinct and obvious manner, 
 on any ordinary or practical occasions. The idea 
 belongs peculiarly to the region of speculation : in 
 persons of common habits of thought it is probably 
 almost or quite undeveloped ; and even most of those 
 whose minds have been long occupied by science, find 
 a difficulty in apprehending it in its full generality 
 and abstraction, and stript of all irrelevant hypothesis. 
 
 2. Magnetism. The name and the notion of Poles 
 were first adopted in the case of a magnet. If we 
 have two magnets, their extremities attract and repel 
 each other alternatively. If the first end of the one 
 attract the first end of the other, it repels the second 
 end, and conversely. In order to express this rule 
 conveniently, the two ends of each magnet are called 
 the north pole and the south pole respectively, the 
 denominations being borrowed from the*poles of the 
 earth and heavens. 'These poles,' as Gilbert says 1 , 
 1 regulate the motions of the celestial spheres and of 
 the earth. In like manner the magnet has its poles, 
 a northern and a southern one ; certain and determined 
 points constituted by nature in the stone, the primary 
 terms of its motions and effects, the limits and go- 
 vernors of many actions and virtues.' 
 
 The nature of the opposition of properties of which 
 we speak may be stated thus : 
 
 The North pole of one magnet attracts the South 
 pole of another magnet. 
 
 The North pole of one magnet repels the North pole 
 of another magnet. 
 
 The South pole of one magnet repels the South pole 
 of another magnet. 
 
 The South pole of one magnet attracts the North 
 pole of another magnet. 
 
 It will be observed that the contrariety of position 
 which is indicated by putting the South pole for the 
 North pole in either magnet, is accompanied by the 
 
 De Magn. lib. i. c. iii. 
 
APPLICATION OF THE IDEA OF POLARITY. 361 
 
 opposition of mechanical effect which is expressed by 
 changing attraction into repulsion and repulsion into 
 attraction: and thus we have the general feature of 
 Polarity, A contrast of properties corresponding to a 
 contrast of positions. 
 
 3. Electricity. When the phenomena of Electri- 
 city came to be studied, it appeared that they involved 
 relations in some respects Analogous to those of mag- 
 netism. 
 
 Two kinds of electricity were distinguished, the 
 positive and the negative; and it appeared that two 
 bodies electrized positively, or two electrized nega- 
 tively, repelled each other, like two north or two south 
 magnetic poles; while a positively and a negatively 
 electrized body attracted each other, like the north 
 and south poles of two magnets. In conductors of an 
 oblong form, the electricity could easily be made to 
 distribute itself so that one end should be positively 
 and one end negatively electrized; and then such 
 conductors acted on each other exactly as magnets 
 would do. 
 
 But in conductors, however electrized, there is no 
 peculiar point which can permanently be considered as 
 the pole. The distribution of electricity in the con- 
 ductor depends upon external circumstances : and thus, 
 although the phenomena offer the general character of 
 polarity alternative results corresponding to alterna- 
 tive positions, they cannot be referred to poles. Some 
 other mode of representing the forces must be adopted 
 than that which makes them emanate from permanent 
 points as in a magnet. 
 
 The phenomena of attraction and repulsion in elec- 
 trized bodies were conveniently represented by means 
 of the hypothesis of two electric fluids, a positive and 
 a negative one, which were supposed to be distributed 
 in the bodies. Of these fluids, it was supposed that 
 each repelled its own parts and attracted those of the 
 opposite fluid : and it was found that this hypothesis 
 explained all the obvious laws of electric action. Here 
 then we have the phenomena of polarization explained 
 by a new kind of machinery: two opposite fluids 
 
362 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 distributed in bodies, and supplying them, so to speak, 
 with their polar forces. This hypothesis not only ex- 
 plains electrical attraction, but also the electrical spark : 
 namely, thus : when two bodies, of which the neighbour- 
 ing surfaces are charged with the two opposite fluids, 
 approach near to each other, the mutual attraction of 
 the fluids becomes more and more intense, till at last 
 the excess of fluid on the one body breaks through the 
 air and rushes to the other body, in a form accompa- 
 nied by light and noise. When this transfer has taken 
 place, the attraction ceases, the positive and the nega- 
 tive fluid having neutralized each other. Their effort 
 was to unite; and this union being effected, there is 
 no longer any force in action. Bodies in their natural 
 unexcited condition may be considered as occupied by 
 a combination of the two fluids : and hence we see 
 how the production of either kind of electricity is 
 necessarily accompanied with the production of an equi- 
 valent amount of the opposite kind. 
 
 4. Voltaic JZlectricity. Such is the case in Frank- 
 linic electricity, that which is excited by the common 
 electrical machine. In studying Voltaic electricity, we 
 are led to the conviction that the fluid which is in a 
 condition of momentary equilibrium in electrized con- 
 ductors, exists in the state of a Current in the voltaic 
 circuit. And here we find polar relations of a new 
 kind existing among the forces. Two voltaic Currents 
 attract each other when they are moving in the same, 
 and repel each other when they are moving in oppo- 
 site, directions. 
 
 But we find, in addition to these, other polar rela- 
 tions of a more abstruse kind, and which the supposi- 
 tion of two fluids does not so readily explain. For in- 
 stance, if such fluids existed, distinct from each other, 
 it might be expected that it would be possible to ex- 
 hibit one of them separate from the other. Yet in all 
 the phenomena of electromotive currents, we attempt 
 in vain to obtain one kind of electricity separately. ' I 
 have not,' says Mr. Faraday 2 , ' been able to find a 
 
 2 Researches, 516. 
 
APPLICATION OF THE IDEA OF POLARITY. 363 
 
 single fact which could be adduced to prove the theory 
 of two electricities rather than one, in electric cur- 
 rents ; or, admitting the hypothesis of two electricities, 
 have I been able to perceive the slightest grounds that 
 one electricity can be more powerful than the other, 
 or that it can be present without the other, or that 
 it can be varied or in the slightest degree affected 
 without a corresponding variation in the other.' 'Thus,' 
 he adds, ' the polar character of the powers is rigorous 
 and complete.' Thus, we too may remark, all the 
 superfluous and precarious parts gradually drop off" from 
 the hypothesis which we devise in order to represent 
 polar phenomena; and the abstract notion of Polarity 
 of equal and opposite powers called into existence 
 by a common condition remains unincumbered with 
 extraneous machinery. 
 
 5. Light. Another very important example of the 
 application of the Idea of Polarity is that supplied by 
 the discovery of the polarization of light. 'A ray of 
 light may, by various processes, be modified, so that it 
 has different properties according to its different sides, 
 although this difference is not perceptible by any com- 
 mon effects. If, for instance, a ray thus modified, pass 
 perpendicularly through a circular glass, and fall upon 
 the eye, we may turn the glass round and round in its 
 frame, and we shall make no difference in the bright- 
 ness of the spot which we see. But if, instead of a 
 glass, we look through a longitudinal slice of tourma- 
 line, the spot is alternately dark and bright as we turn 
 the crystal through successive quadrants. Here we 
 have a contrast of Properties (dark and bright) corre- 
 sponding to a contrast of positions, (the position of a 
 line east and west being contrasted with the position 
 north and south,) which, as we have said, is the gene- 
 ral character of Polarity. It was with a view of express- 
 ing this character that the term Polarization was 
 originally introduced. Malus was forced by his disco- 
 veries into the use of this expression. f We find,' he 
 says, in 1811, 'that light acquires properties which are 
 relative only to the sides of the ray, which are the 
 same for the north and south sides of the ray, (using 
 
364 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 the points of the compass for description's sake only,) 
 and which are different when we go from the north 
 and south to the east or to the west sides of the ray. 
 I shall give the name of poles to these sides of the ray, 
 and shall call polarization the modification which gives 
 to light these properties relative to these poles. I 
 have put off hitherto the admission of this term into 
 the description of the physical phenomena with which 
 we have to do : I did not dare to introduce it into the 
 Memoirs in which I published my last observations : 
 but the variety of forms in which this new phenome- 
 non appears, and the difficulty of describing them, 
 compel me to admit this new expression ; which signi- 
 fies simply the modification which light has undergone 
 in acquiring new properties which are not relative to 
 the direction of the ray, but only to its sides considered 
 at right angles to each other, and in a plane perpendi- 
 cular to its direction.' 
 
 The theory which represents light as an emission of 
 particles was in vogue at the time when Malus pub- 
 lished his discoveries; and some of his followers in 
 optical research conceived that the phenomena which 
 he thus described rendered it necessary to ascribe poles 
 and an axis to each particle of light. On this hypo- 
 thesis, light would be polarized when the axes of all 
 the particles were in the same direction : and, mak- 
 ing such a supposition, it may easily be conceived 
 capable of transmission through a crystal whose axis is 
 parallel to that of the luminous particles, and intrans- 
 missible when the axis of the crystal is in a position 
 transverse to that of the particles. 
 
 The hypothesis of particles possessing poles is a rude 
 and arbitrary assumption, in this as in other cases; 
 but it serves to convey the general notion of polarity, 
 which is the essential feature of the phenomena. The 
 term ' polarization of light ' has sometimes been com- 
 plained of in modern times as hypothetical and obscure. 
 But the real cause of obscurity was, that the Idea of 
 Polarity was, till lately, very imperfectly developed in 
 men's minds. As we have seen, the general notion of 
 Polarity, opposite properties in opposite directions, 
 
APPLICATION OF THE IDEA OF POLARITY. 365 
 
 exactly describes the character of the optical phe- 
 nomena to which the term is applied. 
 
 It is to be recollected that in optics we never speak 
 of the poles, but of the plane of polarization of a ray. 
 The word sides, which Newton and Mains have nsed, 
 neither of them appears to have been satisfied with; 
 Newton, in employing it, had recourse to the strange 
 Gallicism of speaking of the coast of usual and of un- 
 usual refraction of a crystal. 
 
 The modern theory of optics represents the plane of 
 polarization of light as depending, not on the position 
 in which the axes of the luminiferous particles lie, but 
 on the direction of those transverse vibrations in which 
 light consists. This theory is, as we have stated in the 
 History, recommended by an extraordinary series of 
 successes in accounting for the phenomena. And this 
 hypothesis of transverse vibrations shows us another 
 mechanical mode, (besides the hypothesis of particles 
 with axes,) by which we may represent the polarity of 
 a ray. But we may remark that the general notion of 
 Polarity, as applied to light in such cases, would sub- 
 sist, even if the undulatory theory were rejected. 
 The idea is, as we have before said, independent of 
 all hypothetical machinery. 
 
 I need not here refer to the various ways in which 
 light may be polarized; as, for instance, by being re- 
 flected from the surface of water, or of glass, at certain 
 angles, by being transmitted, through crystals, and in 
 other ways. In all cases the modification produced, the 
 polarization, is identically the same property. Nor need 
 I mention the various kinds of phenomena which ap- 
 pear as contrasts in the result ; for these are not merely 
 light and dark, or white and black, but red and green, 
 and generally, a colour and its complementary colour, 
 exhibited in many complex and varied configurations. 
 These multiplied modes in which polarized light pre- 
 sents itself add nothing to the original conception of 
 Polarization : and I shall therefore pass on to another 
 subject. 
 
 6. Crystallization. Bodies which are perfectly 
 crystallized exhibit the most complete regularity and 
 
3 66 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 symmetry of form ; and this regularity not only appears 
 in their outward shape, but pervades their whole tex- 
 ture, and manifests itself in their cleavage, their 
 transparency, and in the uniform and determinate 
 optical properties which exist in every part, even in the 
 smallest fragment of the mass. If we conceive crystals 
 as composed of particles, we must suppose these par- 
 ticles to be arranged in the most regular manner; for 
 example, if we suppose each particle to have an axis, 
 we must suppose all these axes to be parallel ; for the 
 direction of the axis of the particles is indicated by 
 the physical and optical properties of the crystal, and 
 therefore this direction must be the same for every 
 portion of the crystal. This parallelism of the axes 
 of the particles may be conceived to result from the 
 circumstance of each particle having poles, the opposite 
 poles attracting each other. In virtue of forces acting 
 as this hypothesis assumes, a collection of small mag- 
 netic particles would arrange themselves in parallel 
 positions; and such a collection of magnetic particles 
 offers a sort of image of a crystal. Thus we are led 
 to conceive the particles of crystals as polarized, and 
 as determined in their crystalline positions by polar 
 forces. This mode of apprehending the constitution of 
 crystals has been adopted by some of our most emi- 
 nent philosophers. Thus Berzelius says 3 , 'It is de- 
 monstrated, that the regular forms of bodies presuppose 
 an effort of their atoms to touch each other by pre- 
 ference in certain points; that is, they are founded 
 upon a Polarity;' he adds, ' a polarity which can be 
 no other than an electric or magnetic polarity.' In 
 this latter clause we have the identity of different kinds 
 of polarity asserted ; a principle which we shall speak 
 of in the next chapter. But we may remark, that 
 even without dwelling upon this connexion, any 
 notion which we can form of the structure of Crystals 
 necessarily involves the idea of Polarity. Whether 
 this polarity necessarily requires us to believe crystals 
 to be composed of Atoms which exert an effort to touch 
 
 3 Essay on the Theory of Chemical Properties, 1820, p. 113. 
 
APPLICATION OF THE IDEA OF POLARITY. 367 
 
 each other in certain points by preference, is another 
 question. And, in agreement with what has been 
 said respecting other kinds of polarity, we shall 
 probably find, on a more profound examination of the 
 subject, that while the Idea of Polarity is essential, the 
 machinery by which it is thus expressed is precarious 
 and superfluous. 
 
 7. Chemical Affinity. We shall have, in the next 
 Book, to speak of Chemical Affinity at some length; 
 but since the ultimate views to which philosophers 
 have been led, induce them to consider the forces 
 of Affinity as Polar Forces, we must enumerate these 
 among the examples of Polarity. In chemical pro- 
 cesses, opposites tend to unite, ajid to neutralize each 
 other by their union. Thus an acid or an alkali 
 combine with vehemence, and form a compound, a 
 neutral salt, which is neither acid nor alkaline. 
 
 This conception of contrariety and mutual neutrali- 
 zation, involves the Idea of Polarity. In the concep- 
 tion as entertained by the earlier chemists, the Idea 
 enters very obscurely : but in the attempts which have 
 more recently been made to connect this relation (of 
 acid and base), with other relations, the chemical 
 elements have been conceived as composed of particles 
 which possess poles j like poles repelling, and unlike 
 attracting each other, as they do in magnetic and 
 electric phenomena. This is, however, a rude and ar- 
 bitrary way of expressing Polarity, and, as may be 
 easily shown, involves many difficulties which do not 
 belong to the Idea itself. Mr. Faraday, who has been 
 led by his researches to a conviction of the polar nature 
 of the forces of chemical affinity, has expressed their 
 character in a more general manner, and without any 
 of the machinery of particles indued with poles. Ac- 
 cording to his view, chemical synthesis and analysis 
 must always be conceived as taking place in virtue of 
 equal and opposite forces, by which the particles are 
 united 1 or separated. These forces, by the very cir- 
 cumstance of their being polar, may be transferred 
 from point to point. For if we conceive a string of 
 particles, and if the positive force of the first particle 
 
368 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 be liberated and brought into action, its negative force 
 also must be set free: this negative force neutralizes 
 the positive force of the next particle, and therefore 
 the negative force of this particle (before employed in 
 neutralizing its positive force) is set free : this is in 
 the same way transferred to the next particle, and so 
 on. And thus we have a positive force active at one 
 extremity of 'a line of particles, corresponding to a 
 negative force at the other extremity, all the interme- 
 diate particles reciprocally neutralizing each other's 
 action. This conception of the transfer of chemical 
 action was indeed at an earlier period introduced by 
 Grotthus 4 , and confirmed by Davy. But in Mr. Fara- 
 day's hands we see it divested of all that is super- 
 fluous, and spoken of, not as a line of particles, but as 
 1 an axis of power, having [at every point] contrary 
 forces, exactly equal, in opposite directions.' 
 
 8. General Remarks. Thus, as we see, the notion 
 of Polarity is applicable to many large classes of phe- 
 nomena. Yet the Idea in a distinct and general form 
 is only of late growth among philosophers. It has 
 gradually been abstracted and refined from many ex- 
 traneous hypotheses which were at first supposed to be 
 essential to it. We have noticed some of these hypo- 
 theses; as the poles of a body; the poles of the parti- 
 cles of a fluid; two opposite fluids; a single fluid in 
 excess and defect; transverse vibrations. To these 
 others might be added. Thus Dr. Prout 5 assumes that 
 the polarity of molecules results from their rotation on 
 their axes, the opposite motions of contiguous molecules 
 being the cause of opposite (positive and negative) 
 polarities. 
 
 But none of these hypotheses can be proved by the 
 fact of Polarity alone ; and they have been in succession 
 rejected when they had been assumed on that ground. 
 Thus Davy, in 1826, speaking of chemical forces says 6 , 
 'In assuming the idea of two ethereal, subtile, elastic 
 
 4 DUMAS, Lefons sur la PMlosophie Chimique, p. 401. 
 6 Bridgewater Treatise, p. 559. 6 Phil Tr. 1826, p. 415. 
 
APPLICATION OF THE IDEA OF POLARITY. 369 
 
 fluids, attractive of the particles of each other, and 
 repulsive as to their own particles, capable of combin- 
 ing in different proportions with bodies, and according 
 to their proportions giving them their specific qualities 
 and rendering them equivalent masses, it would be 
 natural to refer the action of the poles to the repul- 
 sions of the substances combined with the excess of 
 one fluid, and the attractions of those united to the 
 excess of the other fluid; and a history of the pheno- 
 mena, not unsatisfactory to the reason, might in this 
 way be made out. But as it is possible likewise to 
 take an entirely different view of the subject, on the 
 idea of the dependence of the results upon the primary 
 attractive powers of the parts of the combination on 
 a single subtile fluid, I shall not enter into any discus- 
 sion on this obscure part of the theory.' Which of 
 these theories will best represent the case, will depend 
 upon the consideration of other facts, in combination 
 with the polar phenomena, as we see in the history of 
 optical theory. In like manner Mr. Faraday proved 
 by experiment 7 the errour of all theories which ascribe 
 electro-chemical decomposition to the attraction of the 
 poles of the voltaic battery. 
 
 In order that they may distinctly image to them- 
 selves the Idea of Polarity, men clothe it in some of 
 the forms of machinery above spoken of; yet every 
 new attempt shows them the unnecessary difficulties 
 in which they thus involve themselves. But on the 
 other hand it is difficult to apprehend this Idea di- 
 vested of all machinery; and to entertain it in such a 
 form that it shall apply at the same time to magnetism 
 and electricity, galvanism and chemistry, crystalline 
 structure and light. The Idea of Polarity becomes 
 most pure and genuine, when we entirely reject the 
 conception of Poles, as Faraday has taught us to do in 
 considering electro-chemical decomposition; but it is 
 only by degrees and by effort that we can reach this 
 point of abstraction and generality. 
 
 7 Researches, p. 495, &c. 
 VOL. I. B B 
 
37 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 9. There is one other remark which we may here 
 make. It was a maxim commonly received in the an- 
 cient schools of philosophy, that 'Like attracts Like:' 
 but as we have seen, the universal maxim of Polar Phe- 
 nomena is, that Like repels Like, and attracts Unlike. 
 The north pole attracts the south pole, the positive 
 fluid attracts the negative fluid; opposite elements 
 rush together; opposite motions reduce each other to 
 rest. The permanent and stable course of things is 
 that which results from the balance and neutralization 
 of contrary tendencies. Nature is constantly labouring 
 after repose by the effect of such tendencies j and so 
 far as Polar Forces enter into her economy, she seeks 
 harmony by means of discord, and unity by opposition. 
 
 Although the Idea of Polarity is as yet somewhat 
 vague and obscure, even in the minds of the cultivators 
 of physical science, it has nevertheless given birth to 
 some general principles which have been accepted as 
 evident, and have had great influence on the progress 
 of science. These we shall now consider. 
 
CHAPTER II. 
 
 OF THE CONNEXION OF POLARITIES. 
 
 i. TT has appeared in the preceding chapter that in 
 JL cases in which the phenomena suggest to us the 
 idea of Polarity, we are also led to assume some mate- 
 rial machinery as the mode in which the polar forces 
 are exerted. We assume, for instance, globular parti- 
 cles which possess poles, or the vibrations of a fluid, or 
 two fluids attracting each other; in every case, in 
 short, some hypothesis by which the existence and 
 operation of the Polarity is embodied in geometrical 
 and mechanical properties of a medium; nor is it pos- 
 sible for us to avoid proceeding upon the conviction 
 that some such hypothesis must be true; although the 
 nature of the connexion between the mechanism and 
 the phenomena must still be indefinite and arbitrary. 
 
 But since each class of Polar Phenomena is thus 
 referred to an ulterior cause, of which we know no 
 more than that it has a polar character, it follows that 
 different Polarities may result from the same cause 
 manifesting its polar character under different aspects. 
 Taking, for example, the hypothesis of globular par- 
 ticles, if electricity result from an action dependent 
 upon the poles of each globule, magnetism may depend 
 upon an action in the equator of each globule; or 
 taking the supposition of transverse vibrations, if polar- 
 ized light result directly from such vibrations, crystal- 
 lization may have reference to the axes of the elasticity 
 of the medium by which the vibrations are rendered 
 transverse, so far as the polar character only of the 
 phenomena is to be accounted for. I say this may be 
 so, in so far only as the polar character of the phe- 
 nomena is .concerned ; for whether the relation of 
 
 BB 2 
 
372 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 electricity to magnetism, or of crystalline forces to light, 
 can really be explained by such hypotheses, remains to 
 be determined by the facts themselves. But since the 
 first necessary feature of the hypothesis is, that it shall 
 give polarity, and since an hypothesis which does this, 
 may, by its mathematical relations, give polarities of 
 different kinds and in different directions, any two 
 co-existent kinds of polarity may result from the same 
 cause, manifesting itself in various manners. 
 
 The conclusion to which we are led by these general 
 considerations is, that two co-existing classes of polar 
 phenomena may be effects of the same cause. But 
 those who have studied such phenomena more deeply 
 and attentively have, in most or in all cases, arrived 
 at the conviction that the various kinds of Polarity in 
 such cases must be connected and fundamentally iden- 
 tical. As this conviction has exercised a great influ- 
 ence, both upon the discoveries of new facts and upon 
 the theoretical speculations of modern philosophers, 
 and has been put forward by some writers as a univer- 
 sal principle of science, I will consider some of the 
 cases in which it has been thus applied. 
 
 2. Connexion of Magnetic and Electric Polarity. 
 The polar phenomena of electricity and magnetism are 
 clearly analogous in their laws: and obvious facts 
 showed at an early period that there was some con- 
 nexion between the two agencies. Attempts were 
 made to establish an evident and definite relation 
 between the two kinds of force, which attempts pro- 
 ceeded upon the principle now under consideration ; 
 namely, that in such cases, the two kinds of Polarity 
 must be connected. Professor CErsted, of Copenhagen, 
 was one of those who made many trials founded upon 
 this conviction : yet all these were long unsuccessful. 
 At length, in 1820, he discovered that a galvanic cur- 
 rent, passing at right angles near to a magnetic needle, 
 exercises upon it a powerful deflecting force. The con- 
 nexion once detected between magnetism and galva- 
 nism was soon recognized as constant and universal. 
 It was represented in different hypothetical modes by 
 different persons; some considering the galvanic cur- 
 
OF THE CONNEXION OF POLARITIES. 373 
 
 rent as the primitive axis, and the magnet as consti- 
 tuted of galvanic currents passing round it at right 
 angles to the magnetic axis; while others conceived 
 the magnetic axis as the primitive one, and the electric 
 current as implying a magnetic current round the 
 wire. So far as many of the general relations of these 
 two kinds of force were concerned, either mode of 
 representation served to express them; and thus the 
 assumption that the two Polarities, the magnetic and 
 the electric, were fundamentally identical, was verified, 
 so far as the phenomena of magnetic attraction, and the 
 like, were concerned. 
 
 I need not here mention how this was further con- 
 firmed by the experiments in which, by means of the 
 forces thus brought into view, a galvanic wire was 
 made to revolve round a magnet, and a magnet round 
 a galvanic wire; in which artificial magnets were 
 constructed of coils of galvanic wire; and finally, in 
 which the galvanic spark was obtained from the mag- 
 net. The identity which sagacious speculators had 
 divined even before it was discovered, and which 
 they had seen to be universal as soon as it was brought 
 to light, was completely manifested in every imagina- 
 ble form. 
 
 The relation of the electric and magnetic Polarities 
 was found to be, that they were transverse to each 
 other, and this relation exhibited under various condi- 
 tions of form and position of the apparatus, gave rise 
 to very curious and unexpected perplexities. The de- 
 gree of complication which this relation may occasion, 
 may be judged of from the number of constructions 
 and modes of conception oifered by CErsted, Wollaston, 
 Faraday, and others, for the purpose of framing a 
 technical memory of the results. The magnetic polarity 
 gives us the north and south poles of the needle ; the 
 electric polarity makes the current positive and nega- 
 tive; and these pairs of opposites are connected by 
 relations of situation, as above and below, right and 
 left ; and give rise to the resulting motion of the needle 
 one way or the other. 
 
374 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 3. Ampere, by framing his hypotheses of the action 
 of voltaic currents and the constitution of magnets, 
 reduced all these technical rules to rigorous deductions 
 from one general principle. And thus the vague and 
 obscure persuasion that there must be some connexion 
 between Electricity and Magnetism, so long an idle and 
 barren conjecture, was unfolded into a complete theory, 
 according to which magnetic and electromotive actions 
 are only two different manifestations of the same forces j 
 and all the above-mentioned complex relations of pola- 
 rities are reduced to one single polarity, that of the 
 electro-dynamic current. 
 
 4. As the Idea of Polarity was thus firmly esta- 
 blished and clearly developed, it became an instrument 
 of reasoning. Thus it led Ampere to maintain that 
 the original or elementary forces in electro-dynamic 
 action could not be as M. Biot thought they were, a 
 statical couple, but must be directly opposite to each 
 other. The same idea enabled Mr. Faraday to carry 
 on with confidence such reasonings as the following 1 : 
 ' No other known power has like direction with that 
 exerted between an electric current and a magnetic 
 pole ; it is tangential, while all other forces acting at 
 a distance are direct. Hence if a magnetic pole on 
 one side of a revolving plate follow its course by rea- 
 son of its obedience to the tangential force exerted 
 upon it by the very current of electricity which it has 
 itself caused; a similar pole on the other side of the 
 plate should immediately set it free from this force ; for 
 the currents which have to be formed by the two poles 
 are in contrary directions.' And in Article 1114 of 
 his Researches, the same eminent philosopher infers 
 that if electricity and magnetism are considered as the 
 results of a peculiar agent or condition, exerted in 
 determinate directions perpendicular to each other, one 
 must be by some means convertible into the other j and 
 this he was afterwards able to prove to be the case in 
 fact. 
 
 Researches, 244. 
 
OF THE CONNEXION OF POLARITIES. 375 
 
 Thus the principle that the Co-existent Polarities of 
 magnetism and electricity are connected and funda- 
 mentally identical, is not only true, but is far from 
 being either vague or barren. It has been a fertile 
 source both of theories which have, at present, a very 
 great probability, and of the discovery of new and 
 striking facts. We proceed to consider other similar 
 cases. 
 
 5. Connexion of Electrical and Chemical Polari- 
 ties. The doctrine that the chemical forces by which 
 the elements of bodies are held together or separated, 
 are identical with the polar forces of electricity, is a 
 great discovery of modern times; so great and so 
 recent, indeed, that probably men of science in general 
 have hardly yet obtained a clear view and firm hold of 
 this truth. This doctrine is now, however, entirely 
 established in the minds of the most profound and 
 philosophical chemists of our time. The complete 
 development and confirmation of this as of other great 
 truths, was preceded by more vague and confused opi- 
 nions gradually tending to this point; and the progress 
 of thought and of research was impelled and guided, in 
 this as in similar cases, by the persuasion that these 
 co-existent polarities could not fail to be closely con- 
 nected with each other. "While the ultimate and exact 
 theory to which previous incomplete and transitory 
 theories tended is still so new and so unfamiliar, it 
 must needs be a matter of difficulty and responsibility 
 for a common reader to describe the steps by which 
 truth has advanced from point to point. I shall, there- 
 fore, in doing this, guide myself mainly by the histori- 
 cal sketches of the progress of this great theory, which, 
 fortunately for us, have been given us by the two phi- 
 losophers who have played by far the most important 
 parts in the discovery, Davy and Faraday. 
 
 It will be observed that we are concerned here with 
 the progress of theory, and not of experiment, except 
 so far as it is confirmatory of theory. In Davy's 
 Memoir 2 of 1826, on the Relations of Electrical and 
 
 2 Phil. Trans. 1826, p. 383. 
 
376 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 Chemical Changes, he gives the historical details to 
 which I have alluded. Already in 1802 he had con- 
 jectiired that all chemical decompositions might be 
 polar. In 1806 he attempted to confirm this conjec- 
 ture, and succeeded, to his own satisfaction, in estab- 
 lishing 3 that the combinations and decompositions by 
 electricity were referable to the law of electrical attrac- 
 tions and repulsions; and advanced the hypothesis (as 
 he calls it), that chemical and electrical attractions were 
 produced by the same cause, acting in one case on 
 particles, in the other on masses. This hypothesis was 
 most strikingly confirmed by the author's being able to 
 use electrical agency as a more powerful means of 
 chemical decomposition than any which had yet been 
 applied. ' Believing,' he adds, ' that our philosophical 
 systems are exceedingly imperfect, I never attached 
 much importance to this hypothesis; but having 
 formed it after a copious induction of facts, and having 
 gained by the application of it a number of practical 
 results, and considering myself as much the author of 
 it as I was of the decomposition of the alkalies, and 
 having developed it in an elementary work as far as 
 the present state of chemistry seemed to allow, I have 
 never,' he says, ' criticised or examined the manner 
 in which different authors have adopted or explained 
 it, contented, if in the hands of others, it assisted the 
 arrangements of chemistry or mineralogy, or became 
 an instrument of discovery.' When the doctrine had 
 found an extensive acceptance among chemists, at- 
 tempts were made to show that it had been asserted 
 by earlier writers : and though Davy justly denies all 
 value to these pretended anticipations, they serve to 
 show, however dimly, the working of that conviction 
 of the Connexion of Co-existent Properties which all 
 along presided in men's minds during this course of 
 investigation. ' Ritter and Winterl have been quoted,' 
 Davy says 4 , ' among other persons, as having imagined 
 or anticipated the relation between electrical powers 
 and chemical affinities before the discovery of the pile 
 
 3 Phil Trans. 1826, p. 389. 4 Hid. p. 384. 
 
OF THE CONNEXION OF POLARITIES. 377 
 
 of Volta. But whoever will read with attention 
 Bitter's " Evidence that Galvanic action exists in or- 
 ganised nature," and Winter's Prolusiones ad Chemiam 
 sceculi decimi noni, will find nothing to justify this 
 opinion.' He then refers to the Queries of Newton at 
 the end of his Optics. ' These,' he says, l contain more 
 grand and speculative views that might be brought to 
 bear upon this question than any found in the works 
 of modern electricians; but it is very unjust to the 
 experimentalists who by the laborious application of 
 new instruments, have discovered novel facts and 
 analogies, to refer them, to any such suppositions as 
 that all attractions, chemical, electrical, magnetical, 
 and gravitative, may depend upon the same cause.' It 
 is perfectly true, that such vague opinions, though 
 arising from that tendency to generalize which is the 
 essence of science, are of no value except so far as they 
 are both rendered intelligible, and confirmed by experi- 
 mental research. 
 
 The phenomena of chemical decomposition by means 
 of the voltaic pile, however, led other persons to views 
 very similar to those of Davy. Thus Grotthus in i8o5 5 
 published an hypothesis of the same kind. ' The pile 
 of Volta,' he says, ' is an electrical magnet, of which 
 each element, that is, each pair of plates, has a positive 
 and a negative pole. The consideration of this polarity 
 suggested to me the idea that a similar polarity may 
 come into play between the elementary particles of 
 water when acted upon by the same electrical agent; 
 and I avow that this thought was for me a flash of 
 light.' 
 
 6. The thought, however, though thus brought into 
 being, was very far from being as yet freed from vague- 
 ness, superfluities, and errours. I have elsewhere 
 noticed 6 Para day's remark on Davy's celebrated Me- 
 moir of 1806; that ' the mode of action by which the 
 effects take place is stated very generally, so generally, 
 indeed, that probably a dozen precise schemes of elec- 
 tro-chemical action might be drawn up, differing essen- 
 
 6 Ann. Chim. Ixviii. 54. 6 Hist. Ind. Sc. b. xiv. c. ix. sect. i. 
 
3/8 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 tially from each other, yet all agreeing with the state- 
 ment there given.' "When Davy and others proceeded 
 to give a little more definiteness and precision to the 
 statement of their views, they soon introduced into 
 the theory features which it was afterwards found 
 necessary to abandon. Thus 7 both Davy, Grottlras, 
 Riffault, and Chompre", ascribed electrical decompo- 
 sition to the action of the poles, and some of them 
 even pretended to assign the proportion in which the 
 force of the pole diminishes as the distance from it 
 increases. Faraday, as I have already stated, showed 
 that the polarity must be considered as residing not 
 only in what had till then been called the poles, but 
 at every point of the circuit. He ascribed 8 electro- 
 chemical decomposition to internal forces, residing in 
 the particles of the matter under decomposition, not 
 to external forces, exerted by the poles. Hence he 
 shortly afterwards 9 proposed to reject the word poles 
 altogether, and to employ instead, the term electrode, 
 meaning the doors or passages (of whatever surface 
 formed) by which the decomposed elements pass out. 
 What have been called the positive and negative poles 
 he further termed the Anode and Cathode; and he in- 
 troduced some other changes in nomenclature con- 
 nected with these. He then, as I have related in 
 the History 10 , invented the Volta-electrometer, which 
 enabled him to measure the quantity of voltaic action, 
 and this he found to be identical with the quantity of 
 chemical affinity; and he was thus led to the clearest 
 view of the truth towards which he and his predeces- 
 sors had so long been travelling, that electrical and 
 chemical forces are identical 11 . 
 
 7. It will, perhaps, be said that this beautiful train 
 of discovery was entirely due to experiment, and not 
 to any a priori conviction that co-existent polarities 
 
 7 See Faraday's Historical Sketch, Researches, 481492. 
 
 8 Art. 524. 
 
 9 In 1834. Eleventh Series of Researches. Art. 662. 
 
 1 Hist. Ind. Sc. b. xiv. c. ix. sect. 2. u Arts. 915, 916, 917. 
 
OF THE CONNEXION OF POLARITIES. 379 
 
 must be connected. I trust I have sufficiently stated 
 that such an a priori principle could not be proved, 
 nor even understood, without a most laborious and 
 enlightened use of experiment; but yet I think that 
 the doctrine, when once fully unfolded, exhibited 
 clearly, and established as true, takes possession of 
 the mind with a more entire conviction of its certainty 
 and universality, in virtue of the principle we are now 
 considering. When the theory has assumed so simple 
 a form, it appears to derive immense probability (to 
 say the least) from its simplicity. Like the laws 
 of motion, when stated in its most general form, it 
 appears to carry with it its own evidence. And thus 
 this great theory borrows something of its character 
 from the Ideas which it involves, as well as from the 
 Experiments by which it was established. 
 
 8. "We may find in many of Mr. Faraday's sub- 
 sequent reasonings, clear evidence that this idea of the 
 Connexion of Polarities, as now developed, is not limited 
 in its application to facts already known experiment- 
 ally, but, like other ideas, determines the philosopher's 
 researches into the unknown, and gives us the form of 
 knowledge even before we possess the matter. Thus, 
 he says, in his Thirteenth Series 12 , ' I have long sought, 
 and still seek, for an effect or condition which shall be 
 to statical electricity what magnetic force is to current 
 electricity; for as the lines of discharge are associated 
 with a certain transverse effect, so it appeared to me 
 impossible but that the lines of tension or of inductive 
 action, which of necessity precede the discharge, should 
 also have their correspondent transverse condition or 
 effect.' Other similar passages might be found. 
 
 I will now consider another case to which we may 
 apply the Principle of Connected Polarities. 
 
 9. Connexion of Chemical and Crystalline Polarities. 
 The close connexion between the Chemical Affinity 
 and the Crystalline Attraction of elements cannot be 
 overlooked. Bodies never crystallize but when their 
 elements combine chemically; and solid bodies which 
 
 i 2 Art. 1658. 
 
380 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 combine, when they do it most completely and exactly, 
 also crystallize. The forces which hold together the 
 elements of a crystal of alum are the same forces 
 which make it a crystal. There is no distinguishing 
 between the two sets of forces. 
 
 Both chemical and crystalline forces are polar, as we 
 stated in the last chapter; but the polarity in the two 
 cases is of a different kind. The polarity of chemical 
 forces is then put in the most distinct form, when it is 
 identified with electrical polarity ; the polarity of the 
 particles of crystals has reference to their geometrical 
 form. And it is clear that these two kinds of polarity 
 must be connected. Accordingly, Berzelius expressly 
 asserts 13 the necessary identity of these two polarities. 
 ' The regular forms of bodies suppose a polarity which 
 can be no other than an electric or magnetic polarity.' 
 This being so seemingly inevitable, we might expect to 
 find the electric forces manifesting some relation to the 
 definite directions of crystalline forms. Mr. Faraday 
 tried, but in vain, to detect some such relation. He 
 attempted to ascertain 14 whether a cube of rock crystal 
 transmitted the electrical force of tension with different 
 intensity along and across the axis of the crystal. In 
 the first specimen there seemed to be some difference ; 
 but in other experiments, made both with rock crystal 
 and with calc spar, this difference disappeared. Al- 
 though therefore we may venture to assert that there 
 must be some very close connexion between electrical 
 and crystalline forces, we are, as yet, quite ignorant 
 what the nature of the connexion is, and in what kind 
 of phenomena it will manifest itself. 
 
 10. Connexion of Crystalline and Optical Polari- 
 ties. Crystals present to us optical phenomena which 
 have a manifestly polar character. The double refrac- 
 tion, both of uniaxal and of biaxal crystals, is always 
 accompanied with opposite polarization of the two 
 rays ; and in this and in other ways light is polarized 
 in directions dependent upon the axes of the crystalline 
 form, that is, on the directions of the polarities of the 
 
 13 Essay on Chemical Prop. 113. 14 Researches. Art. 1689. 
 
OF THE CONNEXION OF POLARITIES. 381 
 
 crystalline particles. The identity of these two kinds of 
 polarity (crystalline and optical) is too obvious to need 
 insisting on; and it is not necessary for us here to 
 decide by what hypothesis this identity may most pro- 
 perly be represented. We may hereafter perhaps find 
 ourselves justified in considering the crystalline forces 
 as determining the elasticity of the luminiferous ether 
 to be different in different directions within the crystal, 
 and thus as determining the refraction and polarization 
 of the light which the crystal transmits. But at pre- 
 sent we merely note this case as an additional example 
 of the manifest connexion and fundamental identity of 
 two co-existent polarities. 
 
 n. Connexion of Polarities in general. Thus we 
 find that the Connexion of different kinds of Polarities, 
 magnetic, electric, chemical, crystalline, and optical, is 
 certain as a truth of experimental science. We have 
 attempted to show further that in the minds of several 
 of the most eminent discoverers and philosophers, such 
 a conviction is something more than a mere empirical 
 result: it is a principle which has regulated their 
 researches while it was still but obscurely seen and 
 imperfectly unfolded, and has given to their theories 
 a character of generality and self-evidence which expe- 
 rience alone cannot bestow. 
 
 It will, perhaps, be said that these doctrines, that 
 scientific researches may usefully be directed by prin- 
 ciples in themselves vague and obscure ; that theories 
 may have an evidence superior to and anterior to 
 experience; are doctrines in the highest degree dan- 
 gerous, and utterly at variance with the soundest 
 maxims of modern times respecting the cultivation of 
 science. 
 
 In the justice and wisdom of this caution I entirely 
 agree : and although I have shown that this principle 
 of the Connexion of Polarities, rightly interpreted and 
 established in each case by experiment, involves pro- 
 found and comprehensive truths; I think it no less 
 important to remark that, at least in the present stage 
 of our knowledge, we can make no use of this princi- 
 ple without taking care, at every step, to determine by 
 
382 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 clear and decisive experiments, its proper meaning and 
 application. All endeavours to proceed otherwise have 
 led, and must lead, to ignorance and confusion. At- 
 tempts to deduce from our bare Idea of Polarity, and 
 our fundamental convictions respecting the connexion 
 of polarities, theories concerning the forces which 
 really exist in nature, can hardly have any other result 
 than to bewilder men's minds, and to misdirect their 
 efforts. 
 
 So far, indeed, as this persuasion of a connexion 
 among apparently different kinds of agencies, impels 
 men, engaged in the pursuit of knowledge, to collect 
 observations, to multiply, repeat, and vary experiments, 
 and to contemplate the result of these in all aspects 
 and relations, it may be an occasion of the most import- 
 ant discoveries. Accordingly we find that the great 
 laws of phenomena which govern the motions of the 
 planets about the sun, were first discovered by Kepler, 
 in consequence of his scrutinizing the recorded obser- 
 vations with an intense conviction of the existence of 
 geometrical and arithmetical harmonies in the solar 
 system. Perhaps we may consider the discovery of 
 the connexion of magnetism and electricity by Profes- 
 sor CErsted in 1820, as an example somewhat of the 
 same kind ; for he also was a believer in certain com- 
 prehensive but undefined relations among the proper- 
 ties of bodies ; and in consequence of such views enter- 
 tained great admiration for the Prologue to the Che- 
 mistry of the Nineteenth Century, of Winterl, already 
 mentioned. M. CErsted, in 1803, published a summary 
 of this work; and in so doing, praised the views of 
 Winterl as far more profound and comprehensive than 
 those of Lavoisier. Soon afterwards a Review of this 
 publication appeared in France 15 , in which it was 
 spoken of as a work only fit for the dark ages, and as 
 the indication of a sect which had for some time 
 ' ravaged Germany,' and inundated that country with 
 extravagant and unintelligible mysticism. It was, 
 therefore, a kind of triumph to M. CErsted to be, after 
 
 15 Ann. Chim., Tom. L (1804), p. igi. 
 
OF THE CONNEXION OF POLARITIES. 383 
 
 some years' labour, the author of one of the most 
 remarkable and fertile physical discoveries of his time. 
 
 12. It was not indeed without some reason that 
 certain of the German philosophers were accused of 
 dealing in doctrines vast and profound in their aspect, 
 but, in reality, indefinite, ambiguous, and inapplicable. 
 And the most prominent of such doctrines had refer- 
 ence to the principle now under our consideration; 
 they represented the properties of bodies as consisting 
 in certain polarities, and professed to deduce, from the 
 very nature of things, with little or no reference to 
 experiment, the existence and connexion of these pola- 
 rities. Thus Schelling, in his Ideas towards a Philoso- 
 phy of Nature, published in 1803, says 10 , 'Magnetism 
 is the universal act of investing Multiplicity with 
 Unity; but the universal form of the reduction of 
 Multiplicity to Unity is the Line, pure Longitudinal 
 Extension : hence Magnetism is determination of pure 
 Longitudinal Extension; and as this manifests itself 
 by absolute Cohesion, Magnetism is the determination 
 of absolute Cohesion.' And as Magnetism was, by 
 such reasoning, conceived to be proved as a universal 
 property of matter, Schelling asserted it to be a con- 
 firmation of his views when it was discovered that 
 other bodies besides iron are magnetic. In like man- 
 ner he used such expressions as the following 17 : 'The 
 threefold character of the Universal, the Particular, 
 and the Indifference of the two, as expressed in their 
 Identity, is Magnetism, as expressed in their Differ- 
 ence, is Electricity, and as expressed in the Totality, 
 is Chemical Process. Thus these forms are only one 
 form ; and the Chemical Process is a mere transfer of 
 the three Points of Magnetism into the Triangle of 
 Chemistry.' 
 
 It was very natural that the chemists should refuse 
 to acknowledge, in this fanciful and vague language, 
 (delivered, however, it is to be recollected, in 1803,) 
 an anticipation of Davy's doctrine of the identity of 
 electrical and chemical forces, or of (Ersted's electro- 
 
 16 P. 22 3 . 17 P. 486. 
 
384 OF THE MECHANICO-CHEMICAL SCIENCES. 
 
 magnetic agency. Yet it was perhaps no less natural 
 that the author of such assertions should look upon every 
 great step in the electro-chemical theory as an illustra- 
 tion of his own doctrines. Accordingly we find Schel- 
 ling welcoming, with a due sense of their importance, 
 the discoveries of Faraday. When he heard of the 
 experiment in which electricity was produced from 
 common magnetism, he fastened with enthusiasm upon 
 the discovery, even before he knew any of its details, 
 and proclaimed it at a public meeting of a scientific 
 body 18 as one of the most important advances of modern 
 science. We have (he thus reasoned) three effects of 
 polar forces; Electro-chemical Decomposition, Elec- 
 trical Action, Magnetism. Yolta and Davy had con- 
 firmed experimentally the identity of the two former 
 agencies : (Ersted sjiowed that a closed voltaic circuit 
 acquired magnetic properties : but in order to exhibit 
 the identity of electric and magnetic action it was 
 requisite that electric forces should be extricated from 
 magnetic. This great step Faraday, he remarked, 
 had made, in producing the electric spark by means of 
 magnets. 
 
 13. Although conjectures and assertions of the kind 
 thus put forth by Schelling involve a persuasion of the 
 pervading influence and connexion of polarities, which 
 persuasion has already been confirmed in many in- 
 stances, they involve this principle in a manner so 
 vague and ambiguous that it can rarely, in such a 
 form, be of any use or value. Such views of polarity 
 can never teach us in what cases we are and in what 
 we are not expected to find polar relations ; and indeed 
 tend rather to diffuse error and confusion, than to pro- 
 mote knowledge. Accordingly we cannot be surprized 
 to find such doctrines put forward by their authors as 
 an evidence of the small value and small necessity of 
 experimental science. This is done by the celebrated 
 metaphysician Hegel, in his Encyclopaedia. 'Since/ 
 
 is Ueber Faraday's Neueste Entdeckung. Mttnclien. 1832. 
 i 9 Sec. 278. 
 
OF THE CONNEXION OF POLARITIES. 385 
 
 says he, ' the plane of incidence and of reflection in 
 simple reflection is the same plane, when a second 
 reflector is introduced which further distributes the 
 illumination reflected from the first, the position of 
 the first plane with respect to the second plane, con- 
 taining the direction of the first reflection and of the 
 second, has its influence upon the position, illumina- 
 tion or darkening of the object as it appears by the 
 second reflection. This influence must be the strongest 
 when the two planes are wjiat we must call negatively 
 related to each other : that is, when they are at right 
 angles.' l But,' he adds, ' when men infer (as Malus 
 has done) from the modification which is produced by 
 this situation, in the illumination of the reflection, that 
 the molecules of light in themselves, that is, on their 
 different sides, possess different physical energies; and 
 when on this foundation, along with the phenomena of 
 entoptical colours therewith connected, a wide laby- 
 rinth of the most complex theory is erected; we have 
 then one of the most remarkable examples of the infer- 
 ences of physics from experiment.' If Hegel's reason- 
 ing prove anything, it must prove that polarization 
 always accompanies reflection under such circum- 
 stances as he describes : yet all physical philosophers 
 know that in the case of metals, in whioh the reflection 
 is most complete, light is not completely polarized at 
 any angle; and that in other substances the polariza- 
 tion depends upon various circumstances which show 
 how idle and inapplicable is the account which he thus 
 gives of the property. His self-complacent remark 
 about the inferences of physics from experiment, is in- 
 tended to recommend by comparison his own method 
 of considering the nature of ' things in themselves ;' a 
 mode of obtaining physical truth which had been more 
 than exhausted by Aristotle, and out of which no new 
 attempts have extracted anything of value since his 
 time. 
 
 14. Thus the general conclusion to which we are 
 led on this subject, is, that the persuasion of the exist- 
 ence and Connexion or Identity of various Polarities in 
 nature, although very naturally admitted, and in many 
 VOL. i. cc 
 
386 OF THE MECHANICOCHEMICAL SCIENCES. 
 
 cases interpreted and confirmed by observed facts, is of 
 itself, so far as we at present possess it, a very insecure 
 guide to scientific doctrines. When it is allowed to 
 dictate our theories, instead of animating and extend- 
 ing our experimental researches, it leads only to errour, 
 confusion, obscurity, and mysticism. 
 
 This Fifth Book, on the subject of Polarities, is a 
 short one compared with most of the others. This 
 arises in a great measure from the circumstance that 
 the Idea of Polarity has only recently been appre- 
 hended and applied, with any great degree of clearness, 
 among physical philosophers ; and is even yet probably 
 entertained in an obscure and ambiguous manner by 
 most experimental inquirers. I have been desirous of 
 not attempting to bring forward any doctrines upon 
 the subject, except such as have been fully illustrated 
 and exemplified by the acknowledged progress of the 
 physical sciences. If I had been willing to discuss the 
 various speculations which have been published respect- 
 ing the universal prevalence of Polarities in the uni- 
 verse, and their results in every province of nature, I 
 might easily have presented this subject in a more 
 extended form j but this would not have been consist- 
 ent with my plan of tracing the influence of scientific 
 Ideas only so far as they have really aided in disclosing 
 and developing scientific truths. And as the influence 
 of this Idea is clearly distinguishable both from those 
 which precede and those which follow, in the character 
 of the sciences to which it gives rise, and as it appears 
 likely to be hereafter of great extent and consequence, it 
 seemed better to treat of it in a separate Book, although 
 of a brevity disproportioned to the rest. 
 
 END OF VOL. I. 
 
 Cambridge : Printed at the University Press. 
 
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