THE POSITIVE PHILOSOPHY OP AUGUSTE COMTE. TRANSLATED BT HARRIET MARTINEAU PUBLISHERS, BELFORD, CLARKE & CO., CHICAGO, NEW YORK, AND SAN FRANCISCO. a 1 PREFACE It may appear strange that, in these days, when the French lan- guage is almost 3 3 iamiliar to English readers as their own, I should have spent many months in rendering into English a work which presents no difficulties of language, and which is undoubtedly known to all philosophical students. Seldom as Comte's name is men- tioned in England, there is no doubt in the minds of students of his great work that most or all of those who have added substan- tially to our knowledge for many years past are fully acquainted with it, and are under obligations to it which they would have thankfully acknowledged, but for the fear of offending the preju- dices of ths- society in which they live. Whichever way we look over the whole held of science, we see the truths and ideas presented by Comte cropping out from the surface, and tacitly recognised as the foundation of all that is systematic in our knowledge. This being the case, it may appear to be a needless labor to render into our own tongue what is clearly existing in so many of the minds which are guiding and forming popular views. But it was not without reason that I undertook so serious a labor, while so much work was waiting to be done which might seem to be more urgent. One reason, though not the chief, was that it seems to me unfair, through fear or indolence, to use the benefits conferred on us by M. Comte without acknowledgment. His fame is no doubt safe. Such a work as this is sure of receiving due honor, sooner or later. Before the end of the century, society at large will have become aware that this work is one of the chief honors of *he century, and that its authors name will rank with those of the worthies who have illustrated former ages: but it does not seem to me right to 4 PREFACE. assist in delaying the recognition till the author of so noble a ser- vice is beyond the reach of our gratitude and honor: and that it is demoralizing to ourselves to accept and use such a boon as ho has given us in a silence which is in fact ingratitude. His honors we can not share : they are his own and incommunicable. His trials we may share, and, by sharing, lighten ; and lie lias the strongest claim upon us for sympathy and fellowship in any popular disrepute which, in this case, as in all cases of signal social service, attends upon a first movement. Such sympathy and fellowship will, I trust, be awakened and extended in proportion to the spread among us of a popular knowledge of what M. Comte has done: and this hope was one reason, though, as I have said, not the chief, for my under- taking to reproduce his work in England in a form as popular as its nature admits. A stronger reason was that M. Comtc's work, in its original form, does no justice to its importance, even in France ; and much less m England. It is in the form of lectures, the delivery of which was spread over a long course of years ; and this extension of time ne- cessitated an amount of recapitulation very injurious to its interest and philosophical aspect. M. Comte's style is singular. It is at the same time rich and diffuse. Every sentence is full fraught with meaning ; yet it is overloaded with words. His scrupulous hon- esty leads him to guard his enunciations with epithets so constantly repeated, that though, to his own mind, they are necessary in each individual instance, they become wearisome, especially toward the end of his work, and lose their effect by constant repetition. This practice, which might be strength in a series of instructions spread over twenty years, becomes weakness when those instructions are presented as a whole ; and it appeared* to me worth while to con- dense his work, if I undertook nothing more, in order to divest it of the disadvantages arising from redundancy alone. My belief is that thus, if nothing more were done, it might be brought before the minds of many who would be deterred from the study of it by its bulk. What I have given in this volume occupies in the origi- nal six volumes averaging nearly eight hundred pages : and yet I believe it will be found that nothing essential to either statement or illustration is omitted. My strongest inducement to this enterprise was my deep convie tion of our need of this book in my own country, in a form which renders it accessible to the largest number of intelligent rollers. We are living in a remarkable time, when the conflict of opinions PREFACE. £ renders a firm foundation of knowledge indispensable, not only to our intellectual, moral, and social progress, but to our holding such ground as we have gained from former ages. While our science is split up into arbitrary divisions ; while abstract and concrete sci- ence are confounded together, and even mixed up with their appli- cation to the arts, and with natural history ; and while the research- es of the scientific world are presented as mere accretions to a heterogeneous mass of facts, there can be no hope of a scientific progress which shall satisfy and benefit those large classes of stu dents whose business it is, not to explore, but to receive. The growth of a scientific taste among the working classes of this coun- try is one of the most striking of the signs of the times. I believe no one can inquire into the mode of life of young men of the middle and operative classes without being struck with the desire that is shown, and the sacrifices that are made, to obtain the means of sci- entific study. That such a disposition should be baffled, and such study rendered almost ineffectual, by the desultory character of scientific exposition in England, while such a work as Comte's was in existence, was not to be borne, if a year or two of humble toil could help, more or less, to supply the need. In close connection with this was another of my reasons. The supreme dread of every one who cares for the good of nation or race is that men should be adrift for want of an- anchorage for their convictions. I believe that no one questions that a very large pro- portion of our people are now so adrift. With pain and fear, we see that a multitude, who might and should be among the widest and best of our citizens, are alienated for ever from the kind of faith which sufficed for all in an organic period which has passed away, while no one has presented to them, and they can not obtain for themselves, any ground of conviction as firm and clear as that which sufficed for our fathers in their day. The moral dan- gers of such a state of fluctuation as has thus arisen are fearful in o the extreme, whether the transition stage from one order of convic- tions to another be long or short. The work of M. Comte is un- questionably the greatest single effort that has been made to obvi- ate this kind of danger ; and my deep persuasion is that it will be found to retrieve a vast amount of wandering, of unsound specula- tion, of listless or reckless doubt, and of moral uncertainty and de- pression. Whatever else may be thought of the work, it will not be denied that it ascertains with singular sagacity and soundness the foundations of numan knowledge, and its true object and scope ; 6 PREFACE. and that it establishes the true filiation of the sciences within the boundaries of its own principle. Some may wish to interpolate thip or that ; some to amplify, and perhaps, here and there, in the most obscure recesses of the great edifice, to transpose, more or less : but any who question the general soundness of the exposition, or of the relations of iis parts, are of another school, and will simply neglect the book, and occupy themselves as if it had never existed. It is not for such that I have been working, but for students who are not schoolmen ; who need conviction, and must best know when their need is satisfied. When this exposition of Positive Philoso- phy unfolds itself in order before their eyes, they will, I am per- suaded, find there at least a resting-place for their thought — a rallying-point of their scattered speculations — and possibly an im- moveable basis for their intellectual and moral convictions. The time will come when the book itself will, for a while, be most dis- cussed on account of the deficiencies which M. Comte himself pres- ses on our notice ; and when his philosophy will sustain amplifica- tions of which he himself does not dream. It must be so, in the inevitable growth of knowledge and evolution of philosophy; and it is the fate which the philosopher himself should covet, because k is only a true book that could survive to be so treated : but, in the meantime, it gives us the basis that we demand, and the princi- ple of action that we want, and as much instruction in the proce- dure, and information as to what ha3 been already achieved, as could be given in our time ; perhaps more than could have been given by any other mind of our time. Even Mathematics is here first constituted a science, venerable and unquestionable as mathe- matical truths have been for ages past : and we are led on, tracing as we go the clear genealogy of the sciences, till we find ourselves among the elements of Social science, as yet too crude and confused to be established, like the others, by a review of what had before been achieved ; but now, by the hand of our master, discriminated, arranged, and consolidated, so as to be ready to fulfil the condi- tions of true science as future generations bring their contributions of knowledge and experience to build upon the foundation here laid. A thorough familiarity with the work in which all this is done would avail more to extinguish the anarchy of popular and sectional opin- ion in this country than any other influence that has yet been exert- ed, or, I believe, proposed. It was under such convictions as these that I began, in the spring of 1851, the analysis of this work, in preparation for a translation. PREFACE. 7 A few months afterward, au unexpected aid presented itself. My purpose was related to the late Mr. Lombe, who was then residing at Florence. He was a perfect stranger to me. He told me, in a subsequent letter, that he had wished, for many years, to do what I was then attempting, and had been prevented only by ill health. My estimate of M. Comte's work, and my expectations from its in- troduction into England in the form of a condensed translation, were fully shared by him ; and, to my utter amazement, he sent me, as the first act of our correspondence, an order on his bankers for five hundred pounds sterling. There was time, before his la- mented death, for me to communicate to him my views as to the disposal of this money, and to obtain the assurance of his approba- tion. We planned that the larger proportion of it should be ex- pended in getting out the work, and promoting its circulation. The last words of his last letter were an entreaty that I would let him know if more money would, in any way, improve the quality of my version, or aid the promulgation of the book. It was a matter of deep concern to me that he died before I could obtain his opinion as to the manner in which I was doing my work. All that remained was to carry out his wishes as far as possible ; and to do this, no pains have been spared by myself, or by Mr. Chapman, who gave him the information that called forth his bounty. As to the method I have pursued with my work — there will be different opinions about it, of course. Some will wish that there had been no omissions, while others would have complained of length and heaviness, if I had offered a complete translation. Some will ask why it is not a close version as far as it goes ; and others, I have reason to believe, would have preferred a brief account, out of my own mind, of what Comte's philosophy is, accompanied by illustrations of my own devising. A wider expectation seems to be that I should record my own dissent, and that of some critics of much more weight, from certain of M. Comte's views. I thought long and anxiously of this ; and I was not insensible to the tempta- tion of entering my protest, here and there, against a statement, a conclusion, or a method of treatment. I should have been better satisfied still to have adduced some critical opinions of much higher value than any of mine can be. But my deliberate conclusion was that this was not the place nor the occasion for any such contro- versy. What I engaged to do was to present M. Comte's first great work in a useful form for English study ; and it appears to me that it would be presumptuous to thrust in my own criticisms. 8 PREFACE. and out of place to insert those of others. Those others can speak for themselves, and the readers of the book can criticise it for them- selves. No doubt, they may be trusted not to mistake my silence for assent, nor to charge me with neglect of such criticism as the work has already evoked in this country. While I have omitted some pages of the Author's comments on French affairs, I have not attempted to alter his French view of European politics. In short, I have endeavored to bring M. Comte and his English readers face to face, with as little drawback as possible from intervention. This by no means implies that the translation is a close one. It is a very free translation. It is more a condensation than an abridgment : but it is an abridgment too. My object was to con- vey the meaning of the original in the clearest way I could ; and to this all other considerations were made to yield. The serious view that I have taken of my enterprise is proved by the amount of labor and of pecuniary sacrifice that I have devoted to my task. Where I have erred, it is from want of ability ; for I have taken all the pains I could. One suggestion that I made to Mr. Lombe, and that he approved, was that the three sections — Mathematics, Astronomy, and Physics — should be revised by a qualified man of science. My personal friend Professor Nichol, of Glasgow, was kind enough to undertake this service. After two careful readings, he suggested nothing material in the way of alteration, in the case of the first two sec- tions, except the omission of Comte's speculation on the possible mathematical verification of Laplace's Cosmogony. But more had to be done with regard to the treatment of Physics. Every reader will see that that section is the weakest part of the book, in regard both to the organization and the details of the subject. In regard to the first, the author explains the fact, from the nature of the case, that Physics is rather a repository of somewhat fragmentary por- tions of physical science, the correlation of which is not yet clear, than a single circumscribed science. And we must say for him, in regard to the other kind of imperfection, that such advances have been made in almost every department of Physics since his second volume was published, that it would be unfair to present what he wrote under that head in 1835 as what he would have to say now. The choice lay therefore between almost rewriting this portion of M. Comte's work, or so largely abridging it that only a skeleton presentment of general principles should remain. But as the sys- tem of Positive Philosophy is much less an Expository than a Crifc- PREFACE. ï* ical work, the latter alternative alone seemed open, under due consid- eration of justice to the Author. I have adopted therefore the plan of extensive omissions, and have retained the few short memoranda in which Professor Nichol suggested these, as notes. Although this gentleman has sanctioned my presentment of Comte's chapters on Mathematics and Physics, it must not be inferred that he agrees with his Method in Mental Philosophy, or assents to other conclu- sions held of main importance by the disciples of the Positive Phi- losophy. The contrary, indeed, is so apparent in the tenor of his own writings, that so far as his numerous readers are concerned, this remark need not have been offered. With the reservation I have made, I am bound to take the entire responsibility — the Work being absolutely and wholly my own. It will be observed that M. Comte's later works are not referred to in any part of this book. It appears to me that they, like our English criticisms on the present Work, had better be treated of separately. Here his analytical genius has full scope ; and what there is of synthesis is, in regard to social science, merely what is necessary to render his analysis possible and available. For vari- ous reasons, I think it best to stop here, feeling assured that if this Work fulfils its function, all else with which M. Comte has thought fit to follow it up will be obtained as it is demanded. During the whole course of my long task, it has appeared to me that Comte's work is the strongest embodied rebuke ever given to that form of theological intolerance which censures Positive Philos- ophy for pride of reason and lowness of morals. The imputation will not be dropped, and the enmity of the religious world to the book will not slacken for its appearing among us in an English ver- sion. It can not be otherwise. The theological world can not but hate a book which treats of theological belief as a transient state of the human mind. And again, the preachers and teachers, of all sects and schools, who keep to the ancient practice, once inevitable, of contemplating and judging of the universe from the point of view of their own minds, instead of having learned to take their stand out of themselves, investigating from the universe inward, and not from within outward, must necessarily think ill of a work which exposes the futility of their method, and the worthlessness of the results to which it leads. As M. Comte treats of theology and metaphysics as destined to pass away, theologians and metaphysi- cians must necessarily abhor, dread, and despise his work. They uieivly express their own natural feelings on behalf of the objects 10 PREFACE. of their reverence and the purpose of their lives, when they charge Positive Philosophy with irreverence, lack of aspiration, hardness, deficiency of grace and beauty, and so on. They are no judges of the case. Those who are — those who have passed through theol- ogy and metaphysics, and. finding what they are now worth, have risen above them — will pronounce a very dill» rent judgment on the contents of this hook, though no appeal for such a judgment is made in it. and this kind of discussion is nowhere expressly provided for. To those who have learned the difficult task of postponing dreams to realities till the beauty of reality is Been in its full disclosure, while that of dreams melts into darkness, the moral charm of this work will he as impressive as its intellectual satisfactions. The aspect in which it presents .Man is as favorable to his moral disci- pline, as it Is fresh and stimulating to his intellectual taste. We find ourselves suddenly living and moving j n the midst of the uni- verse. — ;iv ;l pari of it. and not as its aim and object. We find ourselves living, not under capricious and arbitrary conditions, un- connected with the constitution and movements of the whole, hut under great, general, invariable laws, which operate on us as a part of the whole. Certainly, 1 can conceive of no instruction so favor- able to aspiration as that which -hows us how great are our facul- ties, how small our knowledge, how sublime the heights which we may hope to attain, and how boundless an infinity may be assumed to spread out beyond. We find here indications in passing of the evils we suffer from our low aims, our selfish passions, and our proud ignorance ; and in contrast with them, animating displays of the beauty and glory of the everlasting laws, and of the sweet serenity, lofty courage, and noble resignation, that are the natural consequence of pursuits so pure, and aims so true, as those of Pos- itive Philosophy. Pride of intellect surely abides with those who insist on belief without evidence and on a philosophy derived from their own intellectual action, without material and corroboration from without, and not with those who are too scrupulous and too humble to transcend evidence, and to add, out of their own imagi nations, to that which is, and may be, referred to other judgments. If it be desired to extinguish presumption, to draw away from low aims, to fill life with worthy occupations and elevating pleasures, and to raise human hope and human effort to the highest attainable point, it seems to me that the best resource is the pursuit of Posi- tive Philosophy, with its train of noble truths and irresistible in- ducements. The prospects it opens are boundless ; for among the PREFACE. 11 laws it establishes that of human progress is conspicuous. The vir- tues it fosters are all those of which Man is capable ; and the no- blest are those which are more eminently fostered. The habit of truth-seeking and truth-speaking, and of true dealing with self and with all things, is evidently a primary requisite ; and this habit once perfected, the natural conscience, thus disciplined, will train up all other moral attributes to some equality with it. To all who know what the study of philosophy really is — which means the study of Positive Philosophy — its effect on human aspiration and human discipline is so plain that any doubt can be explained only on the supposition that accusers do not know what it is that they are calling in question. My hope is that this book may achieve, be- sides the purposes entertained by its author, the one more that he did not intend, of conveying a sufficient rebuke to those who, in theological selfishness or metaphysical pride, speak evil of a phi- losophy which is too lofty and too simple, too humble and too gen- erous, for the habit of their minds. The case is clear. The law of progress is conspicuously at work throughout human history. The only field of progress is now that of Positive Philosophy, under whatever name it may be known to the real students of every sect ; and therefore must that philosophy be favorable to those virtues whose repression would be incompatible with progress. CONTENTS. INTRODUCTION. CHAPTER I. Account of the Aim of this Work. View of the Nature and Importance of the Positive Philosophy. Preliminary Survey page 25 Law of Human Development 25 First Stage 26 Second Stage 26 Third Stage 26 Ultimate Point of each 26 Evidences of the Law 26 Actual 26 Theoretical 2*7 Character of the Positive Philosophy. 28 History of the Positive Philosophy. . . 29 New Department of Positive Philoso- phy 30 Social Physics 30 Secondary Aim of this Work 30 To Review the Philosophy of the Sciences 81 Glance at Speciality 31 Proposed new Class of Students 82 Advantages of the Posiïve Philosophy 32 1 Illustrates Intellectual Function. .. 32 2. Must regenerate Education 84 S. Advances Sciences by combining them 85 4. Must reorganize Society 36 No hope of Reduction to a single Law 37 CHAPTER IL View of the Hierarchy of the Positive Sciences. Failure of Proposed Classifications... 38 True Principle of Classification 39 Boundaries of our Field 39 Theoretical Inquiry 89 Abstract Science 41 Concrete Science 41 Difficulty of Classification 42 Historical and Dogmatic Methods. ... 42 True Principle of Classification 44 Characters 44 1. Generality 44 2. Independence 44 Inorganic and Organic Phenomena. . . 44 I. Inorganic 45 1 . Astronomy 45 2. Physics 45 3. Chemistry paok 45 II. Organic 45 1. Physiology 45 2. Sociology. . . 4. 45 Five Natural Sciences: their Filiation. 46 * Filiation of their Parts 46 Corroborations 46 1. This Classification follows the Order of Disclosure of Sciences 46 2. Solves Heterogeneousness 47 3. Marks Relative Perfection of Sciences 47 4. Effect on Education 48 Effect on Method 48 Orderly Study of Sciences 49 Mathematics 49 A Department 49 A Basis 49 An Instrument 49 A Double Science 50 Abstract Mathematics, an Instrument. 50 Concrete Mathematics, a Science 50 Mathematics Pre-eminent in the Scale 50 BOOK I.— MATHEMATICS. CHAPTER L Mathematics, Abstract and Concrete. Description of Mathematics 51 Object of Mathematics 52 General Metnod , 52 Examples 53 True Definition of Mathematics 54 Its Two Parts 55 Their Different Objects 55 Their Different Natures 56 Concrete Mathematics 5« Abstract Mathematics 57 Extent of its Domain 58 Its Universality 58 Its Limitations 58 CHAPTER IL General View of Mathematical Analysis. Analysis 61 True Idea of an Equation 61 Abstract Functions K'l Concrete Functions 69 Two parts of the Calculus 68 Algebra 6S Arithmetic 6? 14 CONTENTS. Its Extent PAGE 63 Its Nature 64 Algebra 65 Creation uf New Functions 65 Finding Equations between Auxiliary Quantities 65 Division of the Calculus of Functions. 66 Section 1. Ordinary Analysis, or Cal- 86 cuius of Direct Functions 66 Its Object 67 Classification of Equation/ 67 Algebraic Equations 67 Algebraic Resolution of Equations. . . 68 Our Existing Knowledge 68 Numerical Résolution of Equations. . . 69 The Theory of Equations 69 Method of Indeterminate Coefficients. 70 SkOTXON 2. Transcendental Analysis, or Calculus of Indirect Functions .... 70 Three Principal Views 7o History 71 Method of Leiunitz 71 Generality of the Formulas 73 Justification of the Method 73 Newton's Mktiiod 74 Method of Limits 74 Fluxions and Fluents. 75 Lagkanuk's Mktiiod 75 blent i t y of the Three Methods 76 Their Comparative Value 77 The Differential and Integral Cal- culus 79 Its Two Pnrts 79 Their Mutual Relations 79 Cases of Union of the Two 80 Casesof the Differential Calculus Alone 80 Cases of the Integral Calculus Alone. . 80 The Differential Calculus 81 Two Portions 81 Subdivisions 82 Reduction to the Elements 82 Transformation of Derived Functions for New Variables 82 Analytical Applications 82 The Integral Calculus 83 Its Divisions 83 Subdivisions 84 One Variable or Several 84 Orders of Differentiation 84 Quadratures 85 Algebraic Functions 85 Transcendental Functions 85 Singular Solutions 85 Definite Integrals 86 Prospects of the Integral Calculus. ... 86 Calculus of Variations 87 Problems Giving Rise to this Calculus. 88 Other Applications 89 Relation to the Ordinary Calculus. ... 90 CHAPTER III. General View of Geometry. Its Nature 92 Definition 92 Idea of Space page 92 Kinds of Extension 93 Geometrical Measurement 93 Measurement of Surfaces and Volumes 93 Of Curved Lines 93 Its Illimitable Field 94 Properties of Lines and Surfaces 95 Two General Methods 95 Special or Ancient, and General or Modern Geometry 95 Geometry of the Ancients 96 Geometry of the Right Line 97 Graphical Solutions 97 Descriptive Geometry 97 Algebraical Solutions 9* Trigonometry 99 Modern, or Analytical Geometry 10O Analytical Representation of Figures. 100 Position 1 00 Position of a Point 101 Plane Curves 101 Expression of Lines by Equations. . . .101 Expression of Equations by Lines. . . .102 Change in the Line Changes the Equa- tion 102 Every Definition of a Line is an Equa- tion 103 Choice of Co-ordinates 104 Determination of a Point in Space. . . 1<)5 Determination of Surfaces by Equa- tions, and of Equations by Surfaces. 105 Curves of Double Curvature 106 Imperfections of Analytical Geometry. 106 Imperfections of Analysis .107 CHAPTER IV. Rational Mechanics. Its Nature 1 08 Its Characters 109 Its Object 109 Matter not Inert in Physics 110 Supposed Inert in Mechanics 110 Field of Rational Mechanics 110 Three Laws of Motion Ill Law of Inertia Ill Law of Equality of Action and Re- Action Ill Law of Co-Existence of Motions Ill Two Primary Divisions 114 Statics and Dynamics 114 Secondary Divisions. 114 Solids and Fluids 114 Section 1. Statics 115 Converse Methods of Treatment 115 First Method 115- Statics by Itself 1 1& Second Method 115 Statics through Dynamics 116 Moments 116 Want of Unity in the Method 116 Virtual Velocities 1 17 Theory of Couples 11» Share of Equations in Producing Equi- librium 11?" CONTENTS 15 Connection of the Concrete with the Abstract Question pagk 1 21 Equilibrium of Fluids 122 Hydrostatics 122 Liquids 123 Gases 123 Suction 2. Dynamics 124 Object 124 Theory of Rectilinear Motion 124 Motion of a Point 125 Motion of a System 126 D'Alembert's Principle 126 Results 128 Statical Theorems 128 Law of Repose 128 Stability and Instability of Equilib- rium 128 Dynamical Theorems 129 Conservation of the Motion of the Centre of Gravity 129 Principle of Areas 129 The Invariable Plain 130 Moment of Inertia 130 Principal Axes 130 Conclusion 131 BOOK IT.— ASTRONOMY. CHAPTER I. General View. Its Nature 132 Definition 133 Restriction 133 Means of Exploration 134 Its Rank 134 When it became a Science 135 Reduction to a Single Law 135 Relation to other Sciences 136 Divisions of the Science 137 Celestial Geometry 137 Celestial Mechanics .137 CHAPTER II. Methods of Study of Astronomy. Section 1. Instruments 138 Observation 138 Shadows 139 Artificial Methods 139 The Pendulum 140 Measurement of Angles 140 Requisite Corrections 141 Section 2. Refraction 142 Section 3. Parallax 143 Section 4. Catalogue of Stars 144 CHAPTER III. Geometrical Phenomena of the Heavenly Bodies. Section 1. Statical Phenomena 145 Two Classes of Phenomena 145 Planetary Distances page 146 Form and Size 147 Planetary Atmospheres 148 Earth's Form and Size 149 Means of Discovery 1 4y Planetary Motions 150 Rotation 150 Translation 151 Sidereal Revolution 1 52 Motion of the Earth 152 Evidences of the Earth's Motion 152 Ancient Conceptions 153 How they Gave Way 153 Earth's Rotation 154 Influence of Centrifugal Force upon Gravity 1 54 Earth's translation 1 55 Precession of the Equinoxes 155 Rétrogradations and Stations of the Planets 155 Aberration of Light 156 Influence of Scientific Fact upon Opinion 157 Kepler's Laws 158 Annual Parallax 158 Circles 158 Kepler 159 His Three Laws 159 First Law 159 Second Law 159 Third Law ..160 Three Problems , -161 Predictions of Eclipses .161 Transit of Venus 161 Foundation of Celestial Mechanics.. .162 Section 2. Dynamical Phenomena., 162 Gravitation 162 Character of Laws of Motion 162 Their History 163 Newton's Demonstration 164 Old Difficulty Explained 165 Term Attraction Inadmissible 1C5 Extent of the Demonstration 166 Term Gravitation Unobjectionable. . .167 Gravitation is that of Molecules 167 Secondary Gravitation 1 68 Domain of the Law 168 CHAPTER IV. Celestial Statics. Consummation by Newton 169 Statical Considerations 169 First Method of Inquiry into Masses. .170 Second Method 170 Third Method 170 Section 1. Weight of the Earth 171 Section 2. Form of the Planets 172 Difficulty of the Inquiry 172 Geometrical Estimate 172 Estimate from Perturbations 173 Indirect Estimate of the Earth's Form .173 Hydrostatic Theory of Planetary Forms 173 Section 3. The Tides 174 16 CONTENTS. Question of the Tides page 174 Theory of the Tides 174 Influence of the Sun 175 Of the Moon 17.~> Composite Influence 175 Requisites for Exactitude 176 CHAPTER V. Celestial Dynamic*. Perturbations 177 Instantaneous 177 Gradual Perturbations 17s Perturbations of Translation 179 Problem of Tine.- Bodies 179 Centre of the Solar System lso Problem of the Planets 160 Of the Satellites 180 Of the Com. Ms 1S1 Perturbations of Rotation 182 The Planets 188 The Satellites 188 Device of an iu variable Plane 188 Stability of our System 181 Résistance of s Medium 184 Independence of the Solar System. . . . 1*5 Achievements of Celestial Dynamics. . 186 CHAPTER VI. Sidereal Astronomy and Cosmogony. Multiple Stars 186 Our Cosmogony 188 Origin of Positive- Cosmogony 188 Cosmogony of Laplace 189 Recapitulation 191 BOOK III.— PHYSICS. CHAPTER I. General View, Imperfect Condition of the Science. . .192 Its Domain 192 Compared with Chemistry 193 Its Generality 193 Dealing with Masses or Molecules. .. .193 Changes of Arrangement or Composi- tion of Molecules 194 Description of Physics 194 Instruments 195 Methods of Inquiry 195 Observation 195 Experiment 195 Application of Mathematical Analysis. 195 Encyclopaedic Rank of Physics 196 Relation to Ast ronomy 196 To Mathematics 197 To the other Sciences 197 To Human Progress 198 Human Power of Modifying Phenom- ena 198 Prevision Imperfect 199 Characteristics of each Science 199 Philosophy of Hypothesis' 199 N ssary Condition page 20C Two Classes of Hypothesis 200 First Class Indispensable .....200 Second (.'lass Chimerical 201 History of the Second Class 202 I ii Astronomy 203 In l'hvsics '. . . .203 Rule of Arrangement in Physics 204 Order 204 CHAPTER II. Barology. Divisions 205 Skction 1. Sialics 205 Historv 2(>5 Cases of Liquids 2<>6 First Case 2<>6 Second Case 207 ( lase of Gtases 208 History 209 Condition oï the Problem 209 Sbotiom 2. Dynamics 209 History 206 Fluids. 211 Case of Liquids 211 Exist ing Mate of Barology 211 CHAPTER I'll. Thermology. Its Nature 212 History 212 Relation to Math. -ma tics 212 >i.'ii"N 1. Mutual Thermological In- fluence 218 Two l'art s 218 Mutual Influence 213 Radiation of Heat 213 Propagation !>y Contact 214 Conductibility 215 Permeability 215 Penetrability 215 Specific Heat 216 Skction 2. Constituent Changes by Heat.216 Latent Heat 217 Change of Volume 217 Change in State of Aggregation 218 Law of Engagement and Disengage- ment of Heat 218 Vapors 21t Temperatures of Ebullition 219 Hygrometrical Equilibrium 219 Section 3. Thermology Connected with Analysis 220 Section 4. Terrestrial Temperatures. .221 Interior Heat 221 Temperature of Planetary Intervals. .22 J Conditions of the Problem . . 222 CHAPTER IV. Acoustics. Its Nature 222 Relation to the Study of Inorganic Bodies 228 CONTENTS. 17 Relation to Physiology page 223 To Mathematics 223 Divisions 225 Section 1. Propagation of Sound. . . . .226 Effect of Atmospheric Agitation 226 Section 2. Intensity of Sounds 226 Section 3. Theory of Tones 227 Composition of Sounds 229 Recapitulation 229 CHAPTER Y. Optics. Hypotheses on the Nature of Light.. 230 Excessive Tendency to Systemize 232 Divisions of Optics 233 Irrelevant Matters 234 Theory of Vision 234 Specific Color of Bodies 234 Section 1. Study of Direct Light 235 Optics Proper 235 Imperfections .235 Photometry 235 Section 2. Catoptrics 236 Great Law of Reflection 236 Law of Absorption not Found 237 Section 3. Dioptrics 237 Great Law of Refraction 237 Newton's Discoveries on Elementary Colors 238 Section 4. Diffraction 239 CHAPTER VI. Electrology History 240 Condition 240 Arbitrary Hypotheses 240 Relation to Mathematics 241 Unsound Application 241 Sound Application 241 Limits 242 Divisions 242 Section 1. Electric Production 242 Causes of Electrization 242 Chemical Action 242 Thermological Action 242 Friction 243 Pressure 243 Contact 243 Other Causes 243 Instruments 243 Section 2. Electric Statics 244 Great Law of Distribution 244 Electric Equilibrium 245 Section 3. Electric Dynamics 245 Ampere's Experiments 245 Conclusion of Physics 247 BOOK IV.— CHEMISTRY. CHAPTER I. Its Nature 249 Great Imperfection 250 Capacities page 250 Object of Chemistry 250 Specific Character of its Action 251 Condition of Action 251 Definition 252 Elements 252 Combination 253 Rational Definition 253 Means of Investigation 253 Observation „ 253 Experiment 254 Comparison 254 Chemical Analysis and Synthesis 255 Rank of the Science 256 Relation to Mathematics 25C To Astronomy 257 To Physiology 257 y To Sociology ." 258 Degree of Possible Perfection 258 Intrusion of Hypotheses 25S Actual Imperfection 259 Comparative Imperfection 259 Relation to Human Progress 260 Art of Nomenclature 260 State of Chemical Doctrine 261 Divisions of the Science 262 No Organic Chemistry 262 Principles of Composition and Decom- position 263 CHAPTER II. Inorganic Chemistry. Mode of Beginning the Study 264 Plurality of Elements 264 Classification of Elements 266 Classification of Berzelius 267 Premature Effort 267 Requisite Preparation as to Method. .267 As to Doctrine 268 First Condition 268 Second Condition 269 Method of Analysis 269 Chemical Dualism 269 Law of Double Saline Decompositions. 272 Chemical Theory of Air and Water. .278 Air 273 Water 274 CHAPTER III. Doctrine of Definite Proportions. Scope of the Doctrine. . 275 As to Doctrine 276 As to Method 276 Its History 276 Richter'g Law 276 Berthollet's Extension 277 Dalton's Further Extension '277 Atomic Theory 277 Extension by Berzelius 278 Extension by Gay-Lussac 27fc Wollaston's Verification 279 Scope of Application of Numerical Chemistry '280 18 CONTENTS. Oljection of Dissolution FAGK281 Of Metallic Alloys 281 Of Organic Substances 289 Application of the Principle of Dualism. 282 CHAPTER IV. The filectro-Chemical Theory. Relation of Electricity to Chemistry. .285 History of the Case 285 Nicholson's Discovery 285 Davy's Discovery 286 Berzelius's Extension 286 Synthetical Process 286 Becquerel's 286 Study of Combustion 28Ï Lavoisier's Theory 2x~ First Division 288 Berth ol let's Limitation 288 Second Dii tsion 289 Difficulties of the Theory 290 Its Position and Powers 292 CHAPTER V. Orgû 7i ic Ch em 1st ry. Confusion of two Kinds of Fact 2'.'.°> Relation of Chemistry to Anatomy. . .295 To Physiology .296 Partition of Organic Chemistry 297 Application of Dualism to Organic Compounds 2'.» 1 .' Summary of the Chapter BOO Summary of the Book 3n4 De Blainville's Definition 3u6 Definition of Biology 307 Means of Investigation 308 Observation 3<»8 Artificial Apparatus 393 Pathological and Comparative Analynis.89^ Laws of Action 395 Intermittence and Continuity 390 Association 396 Unity of the Brain and Nervous S\ s- tem .396 Imperfect State of Phrenology 397 Present State of Biology 397 Retrospect of Natural Philosophy. . . .397 BOOK VI.— SOCIAL PHYSIOS. CHAPTER I. Necessity of an Opportuneness of this New Science. Proposal of the Subject 399 Conditions of Order and Progress. . . .401 The Theological Polity 402 Criterion of Social Doctrine 40b Failure of the Theological Polity 403 The Metaphysical Polity 406 Becomes Obstructive 408 Dogma of Liberty of Conscience 408 Dogma of Equality 411 Dogma of the Sovereignty of the Peo- ple 412 Dogma of National Independence. . . .412 Inconsistency pf the Metaphysical Doctrine 413 Notion of a State of Nature 413 Adhesion to the Worn-out 414 Recurrence to War 415 Principle of Political Centralization . .415 The Stationary Doctrine 418 Dangers of the Critical Period 420 Intellectual Anarchy 420 Destruction of Public Morality 422 Private Morality 423 Political Corruption 425 Low Aims of Political Questions 427 Fatal to Progress 427 Fatal to Order 428 Incompetence of Political Leaders. . . .429 Advent of the Positive Philosophy. . .480 Logical Coherence of the Doctrine. . .431 Its Effect on Order 432 Its Effect on Progress 484 Anarchical Tendencies of the Scientific Class 437 Conclusion 438 CHAPTER II. Principal Philosophical Attempts io Constitute a Social Science. History of Social Science 489 Aristotle's "Politics" 442 20 CONTKNTS. Montesquieu pagk 442 Condoreet 444 Political Economy 446 Growth of Historical Study 449 CHAPTER III. Characteristic* of the Positive Method in its Application to Social Phenomena. Infantile State of Social Science 452 . The Relative Superseding the Absolute. 4 53 Presumptuous Character of the Exist* ing Polities] Spirit 454 Prevision of Social Phenomena 456 Spirit of Social Science 457 Statical Study 457 Social Organization 458 Political and Social Concurrence 459 Interconnection of the Social Organ- ism 461 Order of8tatical Study 462 Dynamical Study 468 Social Continuity 164 Produced l>v Natural Laws 4ti4 Notion of Human Perfectibility 4f,7 Limits of Political Action * 469 Social Phenomena Modifiable 469 Order of Modifying Influences -17 1 Means of Investigation in Social Science.478 Direct Meaii9 474 Observation 474 Experiment 477 Comparison 478 Comparison with Interior Animals. . .478 Comparison of Co-Existing States of Society 479 Comparison o\' Consecutive States. . . .4SI Promise of B Fourth Method of Inves- tigation 484 CHAPTER IV. Relation of Sociology to the other Depart- ments of Positive Philosophy. Relation to Biology 486 Relation to Inorganic Philosophy. .. .489 Man's action on the External World. .491 Necessary Education 492 Mathematical Preparation 492 Pretended Theory of Chances 492 Reaction of Sociology 494 As to Doctrine 494 As to Method 495 Speculative Rank of Sociology 497 CHAPTER V. Social Statics, or Theory of the Spon- taneous Order of Human Society. Three Aspects 498 1. The Individual 498 2. TheFamilv 502 The Sexual Relation 504 The Parental Relation 506 ». Societ v 508 Distribution of Employments. . .page 51C Inconveniences 511 Basis of the True Theory of Govern- ment 512 Elementary Subordination 512 Tendency of Society to Government. .514 CHAPTER VI. Social Dynamics; or Theory of the Nat- ural Progress of Human Society. Scientific View of Human Progression .515 Course of Man's Social Development. 51 6 Rate of Progress 517 En n ui 517 Duration of Human Life 518 Increase of Population 619 The Order of Evolution 521 Law of the Three Periods 522 The Theological Period 523 Intellectual Influence of the Theologi- cal Philosophy 526 Social influences of the Theological Philosophy . . . .528 Institution of a Speculative ('lass. .. .529 The Positive Stage 530 Attempted Union of the Two Philoso- phies 581 The Metaphysical Period 533 Co existence of the Three Periods in the Same Mind 584 Corresponding Material Development. 535 Primitive Military Life 535 Primitive Slavery 536 The Military Regime Provisional 536 Affinity between the Theological and Military Régime 537 Affinity between the Positive and In- dustrial Spirit 539 Intermediate Regime 540 CHAPTER VII. Preparation of the Historical Question. — Fîrit Theological J'Ikisp : petichixm. — Beginning of the Theological and Mil- itary System. Limitations of the Analysis 541 Ahstract Treatment of History 542 Ahsfract Inquiry into Laws 543 Co-existence of Successive States 544 Fetichism 545 Starting-point of the Human Race. . . .545 Relation of Fetichism to Morals 548 To Language 548 To Intellect 549 To Society 549 Astrolatry 550 Relation of Fetichism to Human Knowl- edge 551 To the Fine Arts 552 To Industry 552 Political Influence 558 Institut ion of Agriculture 555 Protection to Products 556 CONTENTS. 21 Transition to Polytheism page 557 The Metaphysical Spirit Tvaceable. . .560 CHAPTER VIII. Second Phase : Polytheism. — Development of the Theological and Military System. True Sense of Polytheism 562 Its Operation on the Human Mind... 562 Polytheistic Science 564 Polytheistic Art 566 Polytheistic Industry 571 Social Attributes of Polytheism 572 Polity of Polytheism 573 Worship 574 Civilization by War 574 Sacerdotal Sanctions 575 Two Characteristics of the Polity. . . .576 Slavery 576 Concentration of Spiritual and Tem- poral Power 578 Morality of Polytheism 580 Moral Effects of Slavery 580 Subordination of Morality to Polity.. 581 Personal Morality 682 Social Morality 583 Domestic Morality 583 Three Phases of Polytheism 584 The Egyptian, or Theocratic 584 Caste 7 585 The Greek, or Intellectual 588 Science 589 Philosophy ! 591 The Roman, or Military 592 Conquest 593 Morality 594 Intellectual Development 594 Preparation for Monotheism 595 The Jews 598 CHAPTER IX. Age of Monotheism. — Modification of the Theological and Military System. Catholicism, the Form 599 Principle of Political Rule 600 The Great Problem 603 Separation of Spiritual and Temporal Power 603 Transposition of Morals and Politics. .604 Function of Each 604 The Speculative Class 605 The Catholic System 606 Ecclesiastical Organization 607 Elective Principle 607 Monastic Institutions 608 Special Education of the Clergy 608 Restriction of Inspiration 609 Ecclesiastical Celibacy 610 Temporal Sovereignty of the Popes. .610 Educational Function 612 Dogmatic Conditions T. . . .614 Dogma of Exclusive Salvation 614 Of the Fall of Man 815 I Of Purgat.-v 615 Of Christ's Divinity pagf.615 Of the Real Presence 615 Worship 616 Significance of Controversies 616 Temporal Organization of the Régime. 6\7 The Germanic Invasions 617 Rise of Defensive System 618 Of Territorial Independence 618 Slavery Converted into Serfage 619 Intervention of the Church Through- out 619 Institution of Chivalry 621 Operation of the Feudal System 621 Moral Aspect of the Regime 622 Rise of Morality over Polity 622 Source of Moral Influence of Cathol- icism 623 Moral Types 626 Personal Morality under Catholicism. 626 Domestic 627 Social 628 Intellectual Aspect of the Régime. . . .629 Philosophy 630 Science 631 Art 632 Industry 633 Provisional Nature of the Regime. . . .633 Division between Natural and Moral Philosophy 634 The Metaphysical Spirit 634 Temporal Decline 635 Conclusion 636 CHAPTER X. Metaphysical State, and Critical Period of Modern Society. Conduct of the Inquiry 637 Necessity of a Transitional State 638 Its Commencement 638 Division of the Critical Period 640 Causes of Spontaneous Decline 640 Decline under Negative Doctrine. . . .642 Character of the Provisional Philoso- phy 612 Christian Period of the Doctrine 643 Deistical Period 643 Organs of the Doctrine 645 Scholasticism 645 The Legists 646 Period of Spontaneous Spiritual De- cline 647 Spontaneous Temporal Decline 649 True Character of the Reformation. . .650 The Jesuits 652 Final Decay of Catholicism 652 Vices of Protestantism 653 Temporal Dictatorship 654 Royal and Aristocratic 655 Rise of Ministerial Function 657 Military Decline 657 Rise of Diplomatic Function 659 Intellectual Influence of Protestant- ism 660 Cntlioli*! Share in Protestant Result-. .662 22 CONTENTS. Jansenism pa(;k G t> -J Quiei ism 662 Moral Influence of Protestantism 688 Three Si ages of Dissolution titi4 Lutheran ism 664 Calvinism 664 Soeiiiiiinism 66a Quakerism 666 Political Revolutions of Protestantism . 665 Holland 666 England 666 America 666 Attendant Errors «'>e>7 Subjection of Spiritual Power 661 Moral Changes under Protestantism. .668 Stage of Full Development of the Critical Doctrine 669 Protestantism Opposed to Progress. ..67 U The Negative Philosophy 671 Three Periods of the Negative Philos- ophy ' 678 itemised 673 liol.be* 673 Its Intellectual Character 67:^ Ik Moral Character 674 Its Political Character 675 lis Propagation 676 School of Voltaire 678 Its Political Action 679 School of Rouss.au 680 The Economists 681 Attendant Evils 681 CHAPTER XI. Rise of (he Element* of the Positive Slate. — Prejxirai^-'i for Social Reorganiza- tion. Date of Modern History 683 Rise of New Elements f>84 Philosophical Order of Employments. 685 Classifications 685 Order of Succession 686 The Industrial Movkmknt 6S8 Birth of Political Liberty 691 Characteristics of the Industrial Move- ment 692 Personal Effect 692 Domestic Effect 693 Social Effect 693 Industrial Policy 695 Relation to Catholicism 695 Relation to the Temporal Authority. .696 Administration 696 Three Periods 697 Paid Armies 697 Rise of Public Credit 698 Political Alliances 698 Mechanical Inventions 698 The Compass 698 Fire-Arms 699 Printing 699 Maritime Discovery 700 Second Period 701 Colonial System 702 Slavery page 703 Third 'Period 704 Final Subordination of the Military Spirit '.704 Spread of Industry 704 Tiik Intkli-kcttal Movkmknt 706 The yEsthetic Development 7i>6 Intellectual Originality 7<>8 Relation of Art to In.luslrv 7<»9 Critical Character of Art.'. 711 Retrograde Character 712 Relation of Art to Polities 713 Spread of Art 715 The Scient i tic Development 716 New Birth of Science 7 17 Relation to Monotheism 718 Ast rology 720 Alchemy 720 Fust Modern Phase of Progress 721 Second Phase 723 Filiation of Discoveries 724 Relation of Science to Old Philosophy .724 Galileo ". .725 Social Relations of Science 725 Third Phase 726 Relations of I discoveries 727 Stage of Speciality 728 The Philosophical Development 729 Reason and Faith 729 Bacon ami Descartes 731 Political Philosophy 733 The Scotch School/ 734 Political Philosophy 7:!4 Idea of Progression 735 (Japs to he Supplied 788 In Industry 736 In Art 738 In Philosophy 738 In Science 738 Existing Needs 738 CIIAPTFR XII. Review of the Revolutionary Crisis. — Ascertainment of the Final Tendency of Modern Society. France first Revolutionized 739 Precursory Events 740 First Stage. — The Constituent Assem- bly .. . 740 Second Stage. — The National Conven- tion 742 Alliance of Foes 742 Constitutional Attempt 744 Military Ascendency 745 Napoleon Bonaparte 745 Restoration of the Bourbons 747 Fall of the Bourbons 748 The Next Reign 748 Extension of the Movement 749 Completion of the Theological Decay. 750 Decay of the Military System 752 Recent Industrial Progress 754 Recent ./Esthetic Progress 755 Recent Scientific Progress 758 CONTENTS. 23 Abuses page 757 Récent Philosophical Progress 761 The Law of Evolution 703 Speculative Preparation 764 The Spiritual Authority 765 Its Educational Function 769 Regeneration of Morality 771 International Duty 772 Basis of Assent , 773 The Temporal Authority 774 Public and Private Function 774 Principle of Co-Ordination 775 Speculative Classes Highest 776 The Practical Classes 776 Privileges and Compensations 777 Practical Privacy 778 Practical Freedom 779 Popular Claims 779 Reciprocal Effects 781 Preparatory Stage 781 Promotion of Order and Progress. . . .782 National Participation 784 France 784 Italy 784 Germany 785 England 785 Spain 786 Co-operation of Thinkers 787 Summary of Results under the Socio- logical Theory 787 CHAPTER XIIL Final Estimate of the Positive Method. Principle of Unity 788 Which Element shall Prevail 788 First General Conclusion 789 The Mathematical Element 790 The Sociological Element 793 Solves AntHgonisms 796 Spirit of the Method 798 Nature of the Method 799 Inquiry into Laws 799 Accordance with Common Sense 799 Conception of Natural Laws 801 Logical Method page 808 Scientific Method 803 Stability of Opinions 805 Destination of the Method 806 The Individual 806 The Race 807 Speculative Life 808 Practical Life 808 Liberty of Method 809 Extension of the Positive Method. . . .810 Abstract of Concrete Science 811 Relations of Phases 813 Mathematics 813 Astronomy 818 Physics and Chemistry 814 Biology 815 Sociology 816 CHAPTER XIV. Estimate of the Results of Positive Doc- trine in its Preparatory Stage. The Mathematical Element 818 Application to Sociology 819 The Astronomical Element 821 The Physical 822 The Chemical 823 The Biological 824 The Sociological...- .825 CHAPTER XV. Estimate of the Final Action of the Posi- tive Philosophy. The Scientific Action 828 Abstract Speculation 829 Concrete Research 830 The Moral Action 831 Personal Morality 832 Domestic Morality 832 Social Morality 838 Political Action 834 Double Government 834 The esthetic Action 836 The Five Nations 838 POSITIVE PHILOSOPHY. INTRODUCTION. CHAPTER I. ACCOUNT OF THE AIM OF THIS WORK.— VIEW OF THE NATURE AND IMPORTANCE OF THE POSITIVE PHILOSOPHY. A general statement of any system of philosophy may be either a sketch of a doctrine to be established, or a summary of a doc- trine already established. If greater value belongs to the last, the first is still important, as characterizing from its origin the subject to be treated. In a case like the present, where the pro- posed study is vast and hitherto indeterminate, it is especially important that the field of research should be marked out with all possible accuracy. For this purpose, I will glance at the conside- rations which have originated this work, and which will be fully elaborated in the course of it. In order to understand the true value and character of the Positive Philosophy, we must take a brief general view of the pro- gressive course of the human mind, regarded as a whole ; for no conception can be understood otherwise than through its history. From the study of the development of human intelli- Litw of human , gence, in all directions, and through all times, the dis- i ,ro s«*B. covery arises of a great fundamental law, to which it is necessarily subject, and which has a solid foundation of proof, both in the facts of our organization and in our historical experience. The law is this : — that each of our leading conceptions — each branch of our knowledge — passes successively through three different theoretical condititions : the Theological, or fictitious ; the Metaphysical, or abstract ; and the Scientific, or positive. In other words, the hu- man mind, by its nature, employs in its progress three methods of philosophizing, the character of which is essentially different, and even radically opposed: viz., the theological method, the meta- physical, and the positive. Hence arise three philosophies, or general systems of conceptions on the aggregate of phenomena, 26 POSITIVE PHILOSOPHY. eacli of which excludes the others. The first is the necessary point of departure of the human understanding; and the third is its fixed and definite state. The second is merely a state of tran- sition. e In the theological state, the human mind, seeking the essential nature of beings, the first and final causes (the origin and purpose) of all effects — in short, Absolute knowl- edge — supposes all phenomena to be produced by the immediate action of supernatural beings. In the metaphysical state, which is only a modifica- tion of the first, the mind supposes, instead of super- natural beings, abstract force 8, veritable entities (that is, personi- fied abstractions) inherent in all beings, and capable of producing all phenomena. What is called the explanation of phenomena is, in this stage, a mere reference of each to its proper entity. \n the final, the positive state, the mind has given over the vain search after Absolute notions, the origin and destination of the universe, and the causes of phenomena, and applies itself to the study of their laws — that is, their invariable relations of succession and resemblance. Reasoning and observa- tion, duly combined, are the means of this knowledge. What is now understood when we Bpeak of an explanation of facts is simply the establishment of a connection between single phenomena and some general facts, the number of which continually diminishes with the progress of science. Ufa»*» point of The Theological system arrived at the highest per- ,,h fection of which it is capable when it substituted the providential action of a single Being for the varied operations of the numerous divinities which had been before imagined. In the same way, in the last stage of the Metaphysical system, men sub- stitute one great entity (Nature) as the cause of all phenomena, instead of the multitude of entities at first supposed. In the same way, again, the ultimate perfection of the Positive system would be (if such perfection could be hoped for) to represent all phenomena as particular aspects of a single general fact — such as Gravitation, for instance. The importance of the working of this general law will be es- tablished hereafter. At present, it must suffice to point out some of the grounds of it. Evidences of the There is no science which, having attained the posi- Iavv - tive stage, does not bear marks of having passed through the others. Some time since it was (whatever it might be) composed, as we can now perceive, of metaphysical abstrac- tions ; and, further back in the course of time, it took its form from theological conceptions. We shall have only too much occasion to see, as we proceed, that our most advanced sciences still bear very evident marks of the two earlier periods through which they have passed. The progress of the individual mind is not only an illustration, GROUNDS OF THE LAW OF PROGRESS. 27 but an indirect evidence of that of the general mind. The point of departure of the individual and of the race being the same, the phases of the mind of a man correspond to the epochs of the mind of the race. Now, each of us is aware, if he looks back upon his own history, that he was a theologian in his childhood, a metaphy- sician in his youth, and a natural philosopher in his manhood. All men who are up to their age can verify this for themselves. Besides the observation of facts, we have theoretical reasons in support of this law. The most important of these reasons arises from the necessity that always exists for some theory to which to refer our facts, combined with the clear impossibility that, at the outset of human knowledge, men could have formed theories out of the observation of facts. All good intellects have repeated, since Bacon's time, that there can be no real knowledge but that which is based on observed facts. This is incontestable, in our present advanced stage ; but, if we look back to the primitive stage of human knowledge, we'shall see that it must have been otherwise then. If it is true that every theory must be based upon observed facts, it is equally true that facts can not be observed without the guidance of some theory. Without such guidance, our facts would be desultory and fruitless ; we could not retain them : for the most part we could not even perceive them. Thus, between the necessity of observing facts in order to form V/ a theory, and having a theory in order to observe facts, the human /\ mind would have been entangled in a vicious circle, but for the natural opening afforded by Theological conceptions. This is the fundamental reason for the theological character of the primitive philosophy. This necessity is confirmed by the perfect suitability of the theological philosophy to the earliest researches of the hu- man mind. It is remarkable that the most inaccessible questions — those of the nature of beings, and the origin and purpose of phe- nomena — should be the first to occur in a primitive state, while those which are really within our reach are regarded as almost unworthy of serious study. The reason is evident enough : — that experience alone can teach us the measure of our powers ; and if men had not begun by an exaggerated estimate of what they can do, they would never have done all that they are capable of. Our organization requires this. At such a period there could have been no reception of a positive philosophy, whose function is to discover v , the laws of phenomena, and whose leading characteristic it is to yC regard as interdicted to human reason those sublime mysteries which theology explains, even to their minutest details, with the most attractive facility. It is just so under a practical view of the nature of the researches with which men first occupied themselves. Such inquiries offered the powerful charm of unlimited empire over x the external world — a world destined wholly for our use, and in- volved in every way with our existence. The theological phi- losophy, presenting this view, administered exactly the stimulus 28 positive philosophy". necessary to incite the human mind to the irksome labor without which it could make no progress. We can now scarcely conceive of such a state of things, our reason having become sufficiently mature to enter upon laborious scientific researches, without need- ing any such stimulus as wrought upon the imaginations of astrolo- gers and alchemists. We have motive enough in the hopo of discovering the laws of phenomena, with a view to the confirmation or rejection of a theory. Hut it could not be so in the earliest days; and it is to the chimeras of astrology and alchemy that we owe the long series of observations and experiments on which our positive science is based. Kepler felt this on behalf of astronomy, and Berthollel on behalf of chemistry. Thus was a spontaneous philosophy, the theological, ihe only possible beginning, method, and provisional system, out of which the Positive philosophy could grow. It is easy, after this, to perceive how Metaphysical methods and doctrines must have afforded the means of transition from the one to the other. The human understanding, slow in its advance, could not step at once from the theological into the positive philosophy. The two are so radically opposed, that an intermediate system of concep- tions has been necessary to render the transition possible. It is only in doing this, that metaphysical conceptions have any utility whatever. In contemplating phenomena, men substitute for super- natural direction a corresponding entity. This entity may have been supposed to be derived from the supernatural action: but it is more easily lost sight of, leaving attention free from the facts themselves, till, at length, metaphysical agents have ceased to be anything more than the abstract names of phenomena. It is not easy to say by what other process than this our minds could have passed from supernatural considerations to natural ; from the theo- logical system to the positive. The law of human development being thus established, let us consider what is the proper nature of the Positive Philosophy. , ... As we have seen, the first characteristic of the Posi- Charneffr of the . ' Positive ihiioso- tive Philosophy is that it regards all phenomena as subjected to invariable natural Laws. Our business is, — seeing how vain is any research into what are called Causes, whether first or final. — to pursue an accurate discovery of these Laws, with a view to reducing them to the smallest possible num- ber. By speculating upon causes, we could solve no difficulty about origin and purpose. Our real business is to analyse accurately the circumstances of phenomena, and to connect them by the natural relations of succession and resemblance. The best illustration of this is in the case of the doctrine of Gravitation. We say that the general phenomena of the universe are explained by it, because it connects under one head the whole immense variety of astronomical facts ; ex- hibiting the constant tendency of atoms toward each other in direct proportion to their masses, and in inverse proportion to the squares of their distance ; while the general fact itself is a mere extension HISTORY OF POSITIVE PHILOSOPHY. 29 ot one which is perfectly familiar to us, and which we therefore gay that we know ; — the weight of bodies on the surface of the earth. As to what weight and attraction are, we have nothing to do with that, for it is not a matter of knowledge at all. Theolo- gians and metaphysicians may imagine and refine about such ques- tions ; but positive philosophy rejects them. When any attempt has been made to explain them, it has ended only in saying that attraction is universal weight, and that weight is terrestrial attrac- tion : that is, that the two orders of phenomena are identical ; which is the point from which the question set out. Again, M. Fourier, in his fine series of researches on Heat, has given us all the most important and precise laws of the phenomena of heat, and many large and new truths, without once inquiring into its nature, as his predecessors had done when they disputed about calorific matter and the action of a universal ether. In treating his subject in the Positive method, he finds inexhaustible material for all his activity of research, without betaking himself to insol- uble questions. Before ascertaining the stage which the Positive . Philosophy has reached, we must bear in mind that the romtiv- Phi- different kinds of our knowledge have passed through l0 ° hy the three stages of progress at different rates, and have not there- fore arrived at the same time. The rate of advance depends on the nature of the knowledge in question, so distinctly that, as we shall see hereafter, this consideration constitutes an accessary to the fundamental law of progress. Any kind of knowledge reaches the positive stage early in proportion to its generality, simplicity, and independence of other departments. Astronomical science, which is above all made up of facts that are general, simple, and independent of other sciences, arrived first ; then terrestrial Physics then Chemistry ; and, at length, Physiology. It is difficult to assign any precise date to this revolution in science. It may be said, like everything else, to have been al ways going on ; and especially since the labors of Aristotle and the school of Alexandria ; and then from the introduction of natu- ral science into the West of Europe by the Arabs. But, if we m list fix upon some marked period, to serve as a rallying point, it must be that, — about two centuries ago, — when the human mind was astir under the precepts of Bacon, the conceptions of Descartes, and the discoveries of Galileo. Then it was that the spirit of the Positive philosophy rose up in opposition to that of the supersti- tious and scholastic systems which had hitherto obscured the true character of all science. Since that date, the progress of the Posi- tive philosophy, and the decline of the other two, have been so marked that no rational mind now doubts that the revolution is des- tined to go on to its completion, — every branch of knowledge be- ing, sooner or later, brought within the operation of Positive philoso- phy. This is* not yet the case. Some are still lying outside : and not till they are brought in will the Positive philosophy 30 POSITIVE PHILOSOPHY. that character of universality which is necessary to its definitive constitution. In mentioning just now the four principal categories of phenom- ena, — astronomical, physical, chemical, and physiological, — there was an omission which will have been noticed. Nothing was said of Social phenomena,. Though involved with the phy- o\ Pontîve pu- Biological, Social phenomena demand a distinct classi- fication, both on account of their importance and of their difficulty. They are the most individual, the most complicated, the most dependent on all others ; and therefore they must be the latest, — even if they had no special obstacle to encounter. This branch of science lias not hitherto entered into the domain of Positive philosophy. Theological and metaphysical methods, ex- ploded in other departments, aie as yet exclusively applied, both in the way of inquiry and discussion, in all treatment of Social subjects, though the best minds are heartily weary of eternal dis- putes about divine right and the sovereignity of the people. This is the great, while it is evidently the only gap which has to be filled, to constitute, solid and entire, the Positive Philosophy. Now that the human mind has grasped celestial and terrestrial physics, — mechanical and chemical : organic physics, both vegetable and ani- mal, — there remains one science, to fill up the series of sciences of observation. — Social physics. This is what men have now most need of: and this it is the principal aim of the present work to establish. It would be absurd to pretend to offer this new science at once in a complete state. Others, less new, are in rery unequal conditions of forwardness. But the same character of positivitv which is impressed on all the others will be shown to belong to this. This once done, the philosophical system of the moderns will be in fact complete, as there will then be no phenom- enon which does not naturally enter into some one of the five great categories. All our fundamental conceptions having become homo- geneous, the Positive state will be fully established. It can never again change its character, though it will be for ever in course of development by additions of new knowledge. Having acquired the character of universality which has hitherto been the only ad- vantage resting with the two preceding systems, it will supersede them by its natural superiority, and leave to them only an histor- ical existence. We have stated the special aim of this work. Its ?f c tSwJrk. aim secondary and general aim is this : — to review what has been effected in the Sciences, in order to show that they are not radically separate, but all branches from the same trunk. If we had confined ourselves to the first and special object of the work, we should have produced merely a study of Social physics : whereas, in introducing the second and general, we offer a study of Positive philosophy, passing in review alT the positive Fciences already formed. DESULTORY DIVISION OF RESEARCH. 31 The purpose of this work is not to give an account of the Natural Sciences. Besides that it would be end- philosophy of less, and that it would require a scientific preparation tbe Sciencefl such as no one man possesses, it would be apart from our object, which is to go through a course of not Positive Science, but Posi- tive Philosophy. We have only to consider each fundamental science in its relation to the whole positive system, and to the spirit which characterizes it ; that is, with regard to its methods and its chief results. The two aims, though distinct, are inseparable ; for, on the one hand, there can be no positive philosophy without a basis of social science, without which it could not be all-comprehensive ; and, on the other hand, we could not pursue Social science without having been prepared by the study of phenomena less complicated than those of society, and furnished with a knowledge of laws and an- terior facts which have a bearing upon social science. Though the fundamental sciences are not all equally interesting to ordinary minds, there is no one of them that can be neglected in an inquiry like the present ; and, in the eye of philosophy, all are of equal value to human welfare. Even those which appear the least inter- esting have their own value, either on account of the perfection of their methods, or as being the necessary basis of all the others. Lest it should be supposed that our course will lead . . us into a wilderness of such special studies as are at " pec present the bane of a true positive philosophy, we will briefly ad- rert to the existing prevalence of such special pursuit. In the primitive state of human knowledge there is no, regular division of intellectual labor. Every student cultivates all the sciences. As knowledge accrues, the sciences part off; and students devote themselves each to some one branch. It is owing to this division of employment, and concentration of whole minds upon a single department, that science has made so prodigious an advance in modern times ; and the perfection of this division is one of the most important characteristics of the Positive philosophy. But, while admitting all the merits of this change, we can not be blind to the eminent disadvantages which arise from the limitation of minds to a particular study. It is inevitable that each should be possessed with exclusive notions, and be therefore incapable of the general superiority of ancient students, who actually owed that general superiority to the inferiority of their knowledge. We must consider whether the evil can be avoided without losing the good of the modern arrangement ; for the evil is becoming urgent. We all acknowledge that the divisions established for the conve- nience of scientific pursuit are radically artificial ; and yet there are very few who can embrace in idea the whole of any one sci- ence : each science moreover being itself only a part of a great whole. Almost every one is busy about his own particular section, without much thought about its relation to the general system of positive knowledge. We must not be blind to the evil, nor slow 32 POSITIVE PHILOSOPHY. in Becking a remedy. We must not forget that this is the weak side of the positive philosophy, by which it may yet be attacked, with some hope of success, by the adherents of the theological and metaphysical systems. As to the remedy, ; t certainly does not lie in a return to the ancient confusion of pursuits, which would be mere retrogression, if it were possible, which it is not. It lies in perfecting the division of employments itself, — in carrying it one degree higher, — in constituting one more speciality from the study of scientific generalities. Let us have a new class of students, Propped new suitably prepared, whose business it shall be to take ci.iBflofstudenu. the respective sciences as they are, determine the spirit of each, ascertain their relations and mutual connection, and reduce their respective principles to the smallest number of general prin- ciples, in conformity with the fundamental rules of the Positive Method. At the same time, let other students be prepared for their special pursuit by an education which recognises the whole scope of positive science, so as to profit by the labors of the stu- dents of generalities, and so as to correct reciprocally, under that guidance, the results obtained by each. We see some approach already to this arrangement. Once established, there would be nothing to apprehend from any extent of division of employments. When we once have a class of learned men, at the disposal of all others, whose business it shall be to connect each new discovery with the general system, we may dismiss all fear of the great whole bung lost Bight of in the pursuit of the details of knowledge. The organization of scientific research will then be complete; and it will henceforth have occasion only to extend its development, and not to change its character. After all, the formation of such a new class as is proposed would be merely an extension of the principle which has created all the classes we have. While science was narrow, there was only one class: as it expanded, more were instituted. With a further advance a fresh need arises, and this new class will be the result. Kthnntt c -, The general spirit of a course of Positive Philosophy ih- roitm- pi.- having been thus set forth, we must now glance at the chief advantages which may be derived, on behalf of human progression, from the study of it. Of these advantages, four may be especially pointed out. n^. . ,. . I. The study of the Positive Philosophy affords the Illustrates tliMn- .. . J ,., .. i i • i « /» teiiectuai fane- only rational means of exhibiting the logical laws of the human mind, which have hitherto been sought by unfit methods. To explain what is meant by this, we may refer to a saying of M. de Blainville, in his work on Comparative Anatomy, that every active, and especially every living being, may be re- garded under two relations — the Statical and the Dynamical ; that is, under conditions or in action. It is clear that all considera- tions range themselves under the one or the other of these heads. Let us apply this classification to the intellectual functions. If we regard these functions under their Statical aspect — that is, FIBST BENEFIT. 33 if we consider the conditions under which they exist — we must determine the organic circumstances of the case, which inquiry involves it with anatomy and physiology. If we look at the Dy- namic aspect, we have to study simply the exercise and results of the intellectual powers of the human race, which is neither more nor less than the general object of the Positive Philosophy. In short, looking at all scientific theories as so many great logical facts, it is only by the thorough observation of these facts that we can arrive at the knowledge of logical laws. These being the only means of knowledge of intellectual phenomena, the illusory psy- chology, which is the last phase of theology, is excluded. It pre- tends to accomplish the discovery of the laws of the human mind \J by contemplating it in itself ; that is, by separating it from causes and effects. Such an attempt, made in defiance of the physio- logical study of our intellectual organs, and of the observation of rational methods of procedure, can not succeed at this time of day. The Positive Philosophy, which has been rising since the time of Bacon, ha'i now secured such a preponderance, that the metaphysi- cians themselves profess to ground their pretended science on an observation of facts. They talk of external and internal facts, and say that their business is with the latter. This is much like saying that vision is explained by luminous objects painting their images upon the retina. To this the physiologists reply that another eye would be needed to see the image. In the same manner, the mind may observe all phenomena but its own. It may be said that a man's intellect may observe his passions, the seat of the reason being somewhat apart from that of the emotions in the brain ; but there can be nothing like scientific observation of the passions, \/ except from without, as the stir of the emotions disturbs the /\ observing faculties more or less. It is yet more out of the question to make an intellectual observation of intellectual processes. The observing and observed organ are here the same, and its action can not be pure and natural. In order to observe, your intellect must pause from activity ; yet it is this very activity that you want tu observe. If you can not effect the pause, you can not observe : if you do effect it, there is nothing to observe. The results of such a method are in proportion to its absurdity. After two thousand years of psychological pursuit, no one proposition is established to the satisfaction of its followers. They are divided, to this day, into a multitude of schools, still disputing about the very elements, of their doctrine. This interior observation gives birth to almost "V as many theories as there are observers. We ask in vain for any ^ one discovery, great or small, which has been made under this method. The psychologists have do.ie some good in keeping up the activity of our understandings, when there was no better work for our faculties to do ; and they may have added something to our stock of knowledge. If they have done so, it is by practising the Positive method — by observing the progress of the human mind " 3 X 34 POSITIVE PHILOSOPHY. in the light of science ; that is, by ceasing, for the moment, to be psychologists. The view just given in relation to logical Science becomes yet more striking when we consider the logical Art. The Positive Method can be judged of only in action. It can not he looked at by itself, apart from the work on which it is employed. At all events, such a contemplation would be only a dead study, which could produce nothing in the mind which loses time upon it. We may talk for ever about the method, and state it in terms very wisely, without knowing half so much about it as the man who has once put it in practice upon a single particular of actual research, even without any philosophical intention. Thus it is that psychologists, by dint of leading the precepts of Bacon and the discourses of Descartes, have mistaken their own dreams for science. Without saying whether it will ever be possible to establish, à prion ', a true method of investigation, independent of a philo- sophical study of the sciences, it is clear that the thing has never been done yet, and that we are not capable of doing it now. We can not, as vet, explain the great logical procedures, apart from their applications. \i we ever do, it will remain as necessary then as now to form good intellectual habits by studying the reg- ular application of the scientific methods which we shall have attained. Tins, then, is the first great result of the Positive Philosophy — the manifestation by experiment of the laws which rule the Intellect in tiie investigation of truth ; and, as a consequence, the knowledge of the general rides suitable for that object, mut regenerate H* r l" m ' second effect of the Positive Philosophy, an Education. effect not less important and far more urgently wanted, will be to regenerate Education. The best minds are agreed that our European education, still essentially theological, metaphysical, and literary, must be super- seded by a Positive training, conformable to our time and needs. Even the governments of our day have shared, where they have not originated, the attempts to establish positive instruction ; and this is a striking indication of the prevalent sense of what is wanted. While encouraging such endeavors to the utmost, we must not however, conceal from ourselves that everything yet done is inade- quate to the object. The present exclusive specialty of our pur- suits, and the consequent isolation of the sciences, spoil our teach- ing. If any student desires to form an idea of natural philosophy as a whole, he is compelled to go through each department as it is now taught, as if lie were to be only an astronomer, or only a chemist; so that, be his intellect what it may, his training must remain very imperfect. And yet his object requires that he should obtain general positive conceptions of all the classes of natural phe- nomena. It is such an aggregate of conceptions.whether on a great or on a small scale, which must henceforth be the permanent basis of THIRD BENEFIT. 35 all human combinations. It will constitute the mincl of future genera tions. In order to this regeneration of our intellectual system, it is necessary that the sciences, considered as branches from one trunk, should yield us, as a whole, their chief methods and their most im- portant results. The specialities of science can be pursued by those whose vocation lies in that direction. They are indispensa- ble ; and they are not likely to be neglected ; but they can never of themselves renovate our system of Education ; and, to be of their full use, they must rest upon the basis of that general instruction which is a direct result of the Positive Philosophy. III. The same special study of scientific generalities Advance8 ecÎPn . must also aid the progress of the respective positive ces by combining sciences : and this constitutes our third head of advan- tages. The divisions which we establish between the sciences are, though not arbitrary, essentially artificial. The subject of our researches is one : we divide it for our convenience, in order to deal the more easily with its difficulties. But it sometimes hap- pens — and especially with the most important doctrines of each science — that we need what we can not obtain under the present isolation of the sciences — a combination of several special points of view ; and for want of this, very important problems wait for their solution much longer than they otherwise need do. To go back into the past for example : Descartes' grand conception with re- gard to analytical geometry is a discovery which has changed the whole aspect of mathematical science, and yielded the germ of all future progress ; and it issued from the union of two sciences which had always before been separately regarded and pursued. The case of pending questions is yet more impressive ; as, for instance, in Chemistry, the doctrine of Definite Proportions. Without enter- ing upon the discussion of the fundamental principle of this theory, we may say with assurance that, in order to determine it — in order to determine whether it is a law of nature that atoms should neces- sarily combine in fixed numbers — it will be indispensable that the chemical point of view should be united with the physiological. The failure of the theory with regard to organic bodies indicates that the cause of this immense exception must be investigated ; and such an inquiry belongs as much to physiology as to chemistry. Again, it is as yet undecided whether azote is a simple or a com- pound body. It was concluded by almost all chemists that azote is a simple body ; the illustrious Berzilius hesitated, on purely chemical considerations ; but he was also influenced by the physio- logical observation that animals which receive no azote in their food have as much of it in their tissues as carnivorous animals. From this we see how physiology must unite with chemistry to in- form us whether azote is simple or compound, and to institute a new series of researches upon the relation between the compositior of living bodies and their mode of alimentation. such is the advantage which, in the third place, we shall owe to 36 POSITIVE PHI )SOPHY. Positive philosophy — the elucidation of the respective sciences by their combination. In the fourth place Must reorganise IV. The Positive Philosophy offers the only solid society basis for that Social Reorganization which must suc- ceed the critical condition in which the most civilized nations are now living. It can not be necessary to prove to anybody who reads this work that Ideas govern the world, or throw it into chaos ; in other words, that all social mechanism rests upon Opinions. The great political and moral crisis that societies are now undergoing is shown by a rigid analysis to arise out of intellectual anarchy. While stability in fundamental maxims is the first condition of gen- uine social order, we are suffering under an utter disagreement which may be called universal. Till a certain number of general ideas can be acknowledged as a rallying-point of social doctrine, the nations will remain in a revolutionary state, whatever palliatives may be devised ; and their institutions can be only provisional. But whenever the necessary agreement on first principles can be obtained, appropriate institutions will issue from them, without shock or resistance ; for the causes of disorder will have been ar- rested by the mere fact of the agreement. It is in this direction that those must look who desire a natural and regular, a normal state of society. Now, the existing disorder is abundantly accounted for by the existence, all at once, of three incompatible philosophies — the the- ological, the metaphysical, and the positive. Any one of these might alone secure some sort of social order ; but while the three co-exist, it is impossible for us to understand one another upon any essential point whatever. If this is true, we have only to ascertain which of the philosophies must, in the nature of things, prevail ; and, this ascertained, every man, whatever may have been his for- mer views, can not but concur in its triumph. The problem once recognised, can not remain long unsolved ; for all considerations whatever point to the Positive Philosophy as the one destined to prevail. It alone has been advancing during a course of centuries, throughout which the others have been declining. The fact is in- contestable. Some may deplore it, but none can destroy it, nor therefore neglect it but under penalty of being betrayed by illusory speculations. This general revolution of the human mind is nearly accomplished. We have only to complete the Positive Philosophy by bringing Social phenomena within its comprehension, and after- ward consolidating the whole into one body of homogeneous doc- trine. The marked preference which almost all minds, from the highest to the commonest, accord to positive knowledge over vague and mystical conceptions, is a pledge of what the recep- tion of this philosophy will be when it has acquired the only quality that it now wants — a character of due generality. When it has become complete, its supremacy will take place spontaneously, and will re-establish order throughout society. There is, at près- PRECAUTIONARY OBSERVATION. 37 ent, no conflict but between the theological and the metaphysical philosophies They are contending for the task of reorganizing society ; bu j it is a work too mighty for either of them. The positive philosophy has hitherto intervened only to examine both, and both are abundantly discredited by the process. It is time now to be doing something more effective, without wasting our forces in needless controversy. It is time to complete the vast intellectual operation begun by Bacon, Descartes, and Galileo, by constructing the system of general ideas which must henceforth prevail among the human race. This is the way to put an end to the revolutionary crisis which is tormenting the civilized nations of the world. Leaving these four points of advantage, we must attend to one precautionary reflection. Because it is proposed to consolidate the whole of our acquired knowledge into one body of homogeneous auction to a It doctrine, it must not be supposed that we are going to gle law * study this vast variety as proceeding^ from a single principle, and as subjected to a single law. There is something so chimerical in attempts at universal explanation by a single law, that it may be as well to secure this Work at once from any imputation of the kind, though its development will show how undeserved such an imputa- tion would be. Our intellectual resources are too narrow, and the universe is too complex, to leave any hope that it will ever be within our power to carry scientific perfection to its last degree of simpli- city. Moreover, it appears as if the value of such an attainment, supposing it possible, were greatly overrated. The only way, for instance, in which we could achieve the business, would be by con- necting all natural phenomena with the most general law we know — which is that of gravitation, by which astronomical phenomena are already connected with a portion of terrestrial physics. La- place has indicated that chemical phenomena may be regarded as simple atomic effects of the Newtonian attraction, modified by the form and mutual position of the atoms. But supposing this view proveable (which it can not be while we are without data about the constitution of bodies), the difficulty of its application would doubt- less be found so great that we must still maintain the existing division between astronomy and chemistry, with the difference that we now regard as natural that division which we should then call artificial. Laplace himself presented his idea only as a philosophic device, incapable of exercising any useful influence over the prog- ress of chemical science. Moreover, supposing this insuperable difficulty overcome, we should be no nearer to scientific unity, since we then should still have to connect the whole of physiological phe- nomena with the same law, which certainly would not be the least difficult part of the enterprise. Yet, all things considered, the hypothesis we have glanced at would be the most favorable to the desired unity. The consideration of all phenomena as referable to a single ori* X 88 POSITIVE IHTLOSOPHY. gin is by no means necessary to the systematic formation of science, any more than to the realization of the great and happy conse- quences that we anticipate from the positive philosophy. The only necessary unity is that of Method, which is already in great part established. As for the doctrine, it need not be one ; it is enough that it be homogeneous. It is, then, under the double aspect of unity of method and homogeneousness of doctrine that we shall consider the different classes of positive theories in this work. While pursuing the philosophical aim of all science, the lessening of the number of general laws requisite for the explanation of natural phenomena, we shall regard as presumptuous every attempt, in all future time, to reduce them rigorously to one. Having thus endeavored to determine the spirit and influence of the Positive Philosophy, and to mark the goal of our labors, we have now to proceed t<> the exposition of the system; that is, to the determination of the universal, or encyclopaedic order, which must regulate the different classes of natural phenomena, and con- sequently the corresponding positive sciences. CHAPTER II. VIEW OK THE HIERARCHY OF THE POSITIVE SCIENCES. In proceeding to oiler a Classification of the Sciences, we must leave on one side all others that have as yet been at- posed ciaaeiflc*- tempted. Such scales as those of Bacon and D'Alem- t1on8 " bert are constructed upon an arbitrary division of the faculties of the mind ; whereas, our principal faculties are often engaged at the same time in any scientific pursuit. As for other classifications, they have failed, through one fault or another, to command assent: so that there are almost as many schemes as there are individuals to propose them. The failure has been so conspicuous, that the beat minds feel a prejudice against this kind of enterprise, in any shape. Now, what is the reason of this ? — For one reason, the distribu- tion of the sciences, having become a somewhat discredited task, has of late been undertaken chiefly by persons who have no sound knowledge of any science at all. A more important and less per- sonal reason, however, is the want of homogeneousness in the dif- ferent parts of the intellectual system, — some having successively become positive, while others remain theological or metaphysical. Among such incoherent materials, classification is of course impos- sible. Every attempt at a distribution has failed from this cause, without the distributor being able to see why: — without his dis- covering that a radical contrariety existed between the materials PRINCIPLE OF CLASSIFICATION OF THE SCIENCES. 39 he was endeavoring to combine. The fact was clear enough, if it had but been understood, that the enterprise was premature ; and that it was useless to undertake it till our principal scientific con- ceptions should all have become positive. The preceding chapter seems to show that this indispensable condition may now be con- sidered fulfilled : and thus the time has arrived for laying down a sound and durable system of scientific order. We may derive encouragement from the example set by recent botanists and zoologists, whose philosophical labors have exhibited the true principle of classification ; viz., that the classification must x proceed -from the study of the things to be classified, and must by \ no means be determined by à priori considerations. The real affinities and natural connections presented by objects being allowed to determine their order, the classification itself becomes the ex- pression of the most general fact. And thus does the positive method apply to the question of classification itself, as well as to the objects included under it. It follows that the mu- T ^ io princi|)le of tual dependence of the sciences, — a dependence result- c° Nation. ing from that of the corresponding phenomena, — must determine the arrangement of the system of human knowledge. Before pro- ceeding to investigate this mutual dependence, we have only to ascertain the real bounds of the classification proposed : in other words, to settle what we mean by human knowledge, as the subject of this work. The field of human labor is either speculation or Boundaries of action : and thus, we are accustomed to divide our our fidd - knowledge into the theoretical and the practical. It is obvious that, in this inquiry, we have to do only with the theoretical. We are not going to treat of all human notions whatever, but of those fundamental conceptions of the different orders of phenomena which furnish a solid basis to all combinations, and are not founded on any antecedent intellectual system. In such a study, speculation is our material, and not the application of it, — except where the application may happen to throw back light on its speculative ori- gin. This is probably what Bacon meant by that First Philosophy which he declared to be an extract from the whole of Science, and which has been so differently and so strangely interpreted by his metaphysical commentators. There can be no doubt that Man's study of nature must furnish V the only basis of his action upon nature ; for it is only by knowing /\ the laws of phenomena, and thus being able to foresee them, that we can, in active life, set them to modify one another for our advantage. Our direct natural x »)wcr over everything about us is extremely weak, and altogether disproportioned to our needs. Whenever we effect anything great, it is through a knowledge of natural laws, by which we can set one agent to work upon another, — even very weak modifying elements producing a change in the results of a large aggregate of causes. The relation of scicucn tc art mav be summed up in a brief expression : 40 POSITIVE PHILOSOPHY. From Science comes Prevision : from Prevision comes Action. We must not, however, tail into the error of our time, of regard- ing Science chiefly as a basis of Art. However great may be the services rendered to Industry by science, however true may be the saying that Knowledge is Power, we must never forget that the sciences have a higher destination still ; and not only higher, but more direct — that of satisfying the craving of our understanding to know the laws of phenomena. To feel how deep and urgent this need is, we have only to consider for a moment the physiologieal effects of consternation, and to remember that the most terrible sensation we are capable of, is that which we experience when any phenomenon seems to arise in violation of the familiar laws of nature. This need of disposing facts in a comprehensible order (which is tin 1 proper object of all scientific theories) is so inherent in our organization, that if we could not satisfy it by positive con- ceptions, we must Inevitably return to those theological and meta- physical explanations which had their origin in this very fact of human nature. It is this original tendency which acts as a pre- servative, in the minds of men of science, against the narrowness and incompleteness which the practical habits of our age are apt to produce. \i is through this that we are able to maintain just and noble ideas of the importance and destination of the sciences ; and it* it were not thus, the human understanding would soon, as Con- dorcet has observed, come to a stand, even as to the practical applications for the sake of which higher things had been sacrificed ; for, if the arts flow from science, the neglect of science must destroy the consequent arts. Some of the most important arts are derived from speculations pursued during long ages with a purely scientific intention. For instance, the ancient Greek geometers delighted themselves with beautiful speculations on Conic Sections ; those speculations wrought, after a long series of generations, the renova- tion of astronomy ; and out of this has the art of navigation attained a perfection which it never could have reached otherwise than through the speculative labors of Archimedes and Apollonius : so that, to use Condorcet's illustration, " the sailor who is preserved from shipwreck by the exact observation of the longitude, owes his life to a theory conceived two thousand years before by men of genius who had in view simply geometrical speculations." Our business, it is clear, is with theoretical researches, letting- alone their practical application altogether. Though we may con- ceive of a course of study which should unite the generalities of speculation and application, the time is not come for it. To say nothing of its vast extent, it would require preliminary achieve- ments which have not yet been attempted. We must first be iL possession of appropriate Special conceptions, formed according to scientific theories ; and for these we have yet to wait. Meantime, an intermediate class is rising up, whose particular destination is- to organize the relation of theory and practice ; such as the engi- neers, who do not laboi in the advancement of science, but who ABSTRACT AND CONCRETE SCIENCE. 4l study it in its existing state, to apply it to practical purposes. Such classes are furnishing us with the elements of a future body of doctrine on the theories of the different arts. Already, Monge, in his view of descriptive geometry, has given us a general theory of the arts of construction. But we have as yet only a few scat- tered instances of this nature. The time will come when out of such results, a department of Positive philosophy may arise ; but it will be in a distant future. If we remember that several sciences are implicated in every important art, — that, for instance, a true theory of Agriculture requires a combination of physiological, chemical, mechanical, and even astronomical and mathematical sci- ence, — it will be evident that true theories of the arts must wait for a large and equable development of these constituent sciences. One more preliminary remark occurs, before we finish the pre- scription of our limits, — the ascertainment of our field of inquiry. We must distinguish between the two classes of Natural science ^ — the abstract or general, which have for their object the discovery of the laws which regulate phenomena in e.S™ 01 8C1 " all conceivable cases : and the concrete, particular, or descriptive, which are sometimes called Natural sci- Sj "" 616 8ci " ences in a restricted sense, whose function it is to apply these laws to the actual history of existing beings. The first are fundamental ; and our business is with them alone, as the sec- ond are derived, and however important, not rising into the rank of our subjects of contemplation. We shall treat of physiology, but not of botany and zoology, which are derived from it. We shall treat of chemistry, but not of mineralogy, which is secondary to it. — We may say of Concrete Physics, as these secondary sci- ences are called, the same thing that we said of theories of the arts, — that they require a preliminary knowledge of several sci- ences, and an advance of those sciences not yet achieved ; so that, if there were no other reason, we must leave these secondary classes alone. At a future time Concrete Physics will have made progress, according to the development of Abstract Physics, and will afford a mass o? less incoherent materials than those which it now pre- sents. At present, too few of the students of these secondary sci- ences appear to be even aware that a due acquaintance with the primary sciences is requisite to all successful prosecution' of their own. We have now considered, First, that science being composed of speculative knowledge and of practical knowledge, we have to deal only with the first ; and Second, that theoretical knowledge, or science properly so called, being divided into general and particular, or abstract and concrete science, we have again to deal only with the first. Being thus in possession of our proper subject, duly prescribed, we may proceed to the ascertainment of the true order of the fun darwmtal sciences. 42 POSITIVE PHILOSOPHY. Difficulty of cil» Tliis classification of the sciences is not so easy a Bitication. matter as it may appear. However natural it maybe, it will always involve something, if not arbitrary, at least artificial; and in so far, it will always involve imperfection. It is impossible to fulfil, quite rigorously, the object of presenting the sciences in their natural connection, and according to their mutual dependence, so as to avoid the smallest danger of being involved in a vicious circle. It is easy to show why. Hietoricni m,i E y ery science may be exhibited under two methods dojrmntic or procedures, the Historical and the Dogmatic. These are wholly distinct from each other, and any other method can be nothing but some combination of these two. By the first method knowledge is presented in the same order in which it was actually obtained by the human mind, together with the way in which it was obtained. By the Becond, the system of ideas is presented as it [night be conceived of at this day, by a mind which, duly prepared and placed at the right point of view, should begin to reconstitute the science as a whole. A DOW science must be pursued historically, the only thing to be done being to study in chronological order the different works which have contributed to the progress of the science. Bnt when such materials have become recast to form a general system, to meet the demand for a more natural logical order, it i> because the Bcience is too far advanced for the historical order to be practicable or suitable! The more discoveries are made, the greater becomes the labor of tin; histor- ical method of study, and the more effectual the dogmatic, because the new conceptions bring forward the earlier ones in a fresh light. Thus, the education of an ancient geometer consisted simply in the study, in their due order, of the very small number of original treatises then existing on the different parts of geometry. The writings of Archimedes and Apollonius were, in fact, about all. On the contrary, a modern geometer commonly finishes his educa- tion without having read a single original work dating further back than the most recent discoveries, which can not be known by any other means. Thus the Dogmatic Method is for ever superseding the Historical, as we advance to a higher position in science. If every mind had to pass through all the stages that every prede- cessor in the study had gone through, it is clear that, however easy it is to learn rather than invent, it would be impossible to effect the purpose of education, — to place the student on the vantage- ground gained by the labors of all the men who have gone before. By the dogmatic method this is done, even though the living stu- dent may have only an ordinary intellect, and the dead may have been men of lofty genius. By the dogmatic method therefore must every advanced science be attained, w^ith so much of the historical combined with it as is rendered necessary by discoveries too recent to be studied elsewhere than in their own records. The only ob- jection to the preference of the Dogmatic method is that it doefr not show how the science was attained ; but a moment's reflection HISTORICAL AND DOGMATIC METHODS OF STUDY. 43 will show that this is the case also with the Historical method. To pursue a science historically is quite a different thing from learning the history of its progress. This last pertains to the study of human history, as we shall see when we reach the final division of this work. It is true that a science can not be completely under- stood without a knowledge of how it arose ; and again, a dogmatic .knowledge of any science is necessary to an understanding of its history ; and therefore we shall notice, in treating of the funda- mental sciences, the incidents of their origin, when distinct and illustrative ; and we shall use their history, in a scientific sense, in our treatment of Social Physics ; but the historical study, import- ant, even essential, as it is, remains entirely distinct from the proper dogmatic study of science. These considerations, in this place, tend to define more precisely the spirit of our course of in- quiry, while they more exactly determine the conditions under which we may hope to succeed in the construction of a true scale of the aggregate fundamental sciences. Great confusion would arise from any attempt to adhere strictly to historical order in our exposition of the sciences, for they have not all advanced at the same rate ; and we must be for ever borrowing from each some fact to illustrate another, without regard to priority of origin. Thus, it is clear that, in the system of the sciences, astronomy must come before physics, properly so called : and yet, several branches of physics, above all, optics, are indispensable to the complete exposi- tion of astronomy. Minor defects, if inevitable, can not invalidate & classification which, on the whole, fulfils the principal conditions of the case. They belong to what is essentially artificial in our •division of intellectual labor. In the main, however, our classifica- tion agrees with the history of science ; the more general and simple sciences actually occurring first and advancing best in hu- man history, and being followed by the more complex and restrict- ed, though all were, since the earliest times, enlarging simultane- ously. A simple mathematical illustration will precisely represent the difficulty of the question we have to resolve, while it will sum up the preliminary considerations we have just concluded. We propose to classify the fundamental sciences. They are six, as we shall soon see. We can not make them less ; and most sci- entific men would reckon them as more. Six objects admit of 720 different dispositions, or, in popular language, changes. Thus we have to choose the one right order (and there can be but one right) out of 720 possible ones. Very few of these have ever been pro- posed ; yet we might venture to say that there is probably not one in favor of which some plausible reason might not be assigned ; for we see the wildest divergences among the schemes which have been proposed, — the sciences which are placed by some at the head of the scale being sent by others to the further extremity. Our problem is, then, to find the one rational order, among a host of posvoble systems. 44 POSITIVE PHILOSOPHY. True principle Now we must remember that we have to look for the t.i clarification, principle of classification in the comparison of the dif- ferent orders of phenomena, through which Science discovers the laws which are her object. What we have to determine is the real dependence of scientific studies. Now, thi& dependence can result only from that of the corresponding phenomena. All ob- servable phenomena may be included within a very few natural categories, so arranged as that the study of each category may be grounded on the principal laws of the preceding, and serve as the basis of the next ensuing. This order is determined by the degree Générant- °^ simplicity, or, what comes to the same thing, of generality of their phenomena. Hence results their Dependence. successive dependence, and the greater or lesser facility for being studied. It is clear, à priori, that the most simple phenomena must be the most general ; for whatever is observed in the greatest number of cases is of course the most disengaged from the incidents of particular cases. We must begin then with the study of the most general or simple phenomena, going on successively to the more particular or complex. This must be the most methodical way, for this order of generality or simplicity fixes the degree of facility in the study of phenomena, while it determines the necessary connec- tion of the sciences by the successive dependence of their phenom- ena. It is worthy of remark in this place that the most general and simple phenomena are the furthest removed from Man's ordi- nary sphere, and must thereby be studied in a calmer and more rational frame of mind than those in which he is more nearly im- plicated ; and this constitutes a new ground for the corresponding sciences being developed more rapidly. We have now obtained our rule. Next we proceed to our clas- sification. We are first struck by the clear division of all natu- Inorgramc and . J ., . . . , „ organic phe- ral phenomena into two classes — of inorganic and of organic bodies. The organized are evidently, in fact, more complex and less general than the inorganic, and depend upon them, instead of being depended on by them. Therefore it is that physiological study should begin with inorganic phenomena ; since the organic include all the qualities belonging to them, with a special order added, viz., the vital phenomena, which belong to organization. We have not to investigate the nature of either; for the positive philosophy does not inquire into natures. Whether their natures be supposed different or the same, it is evidently necessary to separate the two studies of inorganic matter and of living bodies. Our classification will stand through any future decision as to the way in which living bodies are to be regarded ; for, on any supposition, the general laws of inorganic physics must be established before we can proceed with success to the examina- tion of a dependent class of phenomena. INORGANIC AND ORGANIC PHYSICS. 45 Each of these great halves of natural philosophy has subdivisions. Inorganic physics must, in accordance ' lN0BOANia with our rule of generality and the order of dependence of phenom- ena, be divided into two sections — of celestial and ter- restrial phenomena. Thus we have Astronomy, geo- ' 8tronomy - metrical and mechanical, and Terrestrial Physics. The necessity of this division is exactly the same as in the former case. Astronomical phenomena are the most general, simple, and abstract of all ; and therefore the study of natural philosophy must clearly begin with them. They are themselves independent, while the laws to which they are subject influence all others whatsoever. The general effects of gravitation preponderate, in all terrestrial phenomena, over all effects which may be peculiar to them, and modify the original ones. It follows that the analysis of the sim- plest terrestrial phenomenon, not only chemical, but even purely mechanical, presents a greater complication than the most com- pound astronomical phenomenon. The most difficult astronomical question involves less intricacy than the simple movement of even a solid body, when the determining circumstances are to be com- puted. Thus we see that we must separate these two studies, and proceed to the second only through the first, from which it is derived. In the same manner, we find a natural division of Terrestrial Fhysics into two, according as we regard bodies in their mechani- cal or their chemical character. Hence we have . Physics, properly so called, and Chemistry. Again, ' ya the second class must be studied through, the first. Chemical phenomena are more complicated than mechanical, and depend upon them, without influencing them in return. Every one knows that all chemical action is first submitted to the influence of weight, heat, electricity, etc., and presents moreover something which modifies all these. Thus, while it follows Physics, it pre- sents itself as a distinct science. Such are the divisions of the sciences relating to inorganic matter. An analogous division arises in the other half of Natural Philosophy — the science of organized bodies. Here we find ourselves presented with two orders of phenom ena ; those which relate to the individual, and those which relate to the species, especially when it is gregarious. With regard to Man, especially, this distinction is fundamental. The last order of phenomena is evidently dependent on the first, and is more com- plex. Hence we have two great sections in organic physics — Physiology, properly so called, and Social Physics, ' < ' . which is dependent on it. In all Social phenomena we perceive the working of the physiological laws of the individual ; and moreover something which modifies their effects, and which belongs to the influence of individuals over each other — singularly complicated in the case of the human race by the X Xi 46 POSITIVE PHILOSOPHY. influence of generations on their successors. Thus it is clear that our social science must issue from that which relates to the life of the individual. On the other hand, there is no occasion to sup- pose, as some eminent physiologists have done, that Social Physics is only an appendage to physiology. The phenomena of the two are not identical, though they are homogeneous ; and it is of high importance to hold the two sciences separate. As social conditions modify the operation of physiological laws, Social Physics must have a set of observations of its own. It would be easy to make the divisions of the Organic half of Science correspond witli those of the Inorganic, by dividing phys- ology into vegetable and animal, according to popular custom. But this distinction, however important in Concrete Physics (in that secondary and special class of studies before declared to be inappropriate to this work), hardly extends into those Abstract Physics with which we have to do. Vegetables and animals come alike under our notice, when our object is to learn the general laws of life — that is. to study physiology. To say nothing of the fact that the distinction grows ever fainter and more dubious with new discoveries, it bean no relation to our plan of research ; and we shall therefore consider that there is only one division in the sci- ence of organized bodies. Fiv ,. N tm . 1Sli . Thus we have before us Five fundamental Sciences in successive dependence — Astronomy, Physics, Chem- istry. Physiology, and finally Social Physics. The first considers the most general, simple, abstract, and remote phenomena known to us, and those which affect all others without being affected by them. The last considers the most particular, compound, concrete phenomena, and those which are the most interesting to Man. Between these two, the degrees of speciality, of complexity, and individuality, are in regular proportion to the place of the respec- tive sciences in the scale exhibited. This — casting out everything _ . __. . arbitrary — we must regard as the true filiation of the Their filta ion. . J , . 011 i pi- i sciences ; and in it we find the plan ot this work. As we proceed, we shall find that the same principle Fii^onoftimr wn i cn gi ves ^^g order to the whole body ot science arranges the parts of each science ; and its soundness will therefore be freshly attested as often as it presents itself afresh. There is no refusing a principle which distributes the interior of each science after the same method with the aggregate sciences. But this is not the place in which to do more than indicate what we shall contemplate more closely hereafter. We must now rapidly review some of the leading properties of the hierarchy of science that has been disclosed. corroborations. This gradation is in essential conformity with the order which has spontaneously taken place among the tion^oii d .w* lfi th a R branches of natural philosophy, when pursued sepa- nreorMirac 10 ' ra t ei 7> an d without any purpose of establishing such order. Such an accordance is a strong presumption CLASSIFICATION CORROBORATED. 47 that the arrangement is natural. Again, it coincides with the actual development of natural philosophy. If no leading science can be effectually pursued otherwise than through those which pre- cede it in the scale, it is evident that no vast development of any science could take place prior to the great astronomical discove- ries to which we owe the impulse given to the whole. The pro- gression may since have been simultaneous ; but it has taken place in the order we have recognised. This consideration is so important that it is difficult 2 SolvP8 hetero . to understand without it the history of the human mind. g^< ousmss. The general law which governs this history, as we have already seen, can not be verified, unless we combine it with the scientific gradation just laid down : for it is according to this gradation that the different human theories have attained in succession the theo- logical state, the metaphysical, and finally the positive. If we do not bear in mind the law which governs progression, we shall en- counter insurmountable difficulties ; for it is clear that the theologi- cal or metaphysical state of some fundamental theories must have temporarily coincided with the positive state of others which pre- cede them in our established gradation, and actually have at times coincided with them ; and this must involve the law itself in an ob- scurity which can be cleared up only by the classification we have proposed. Again, this classification marks, with precision, the relative perfection of the different sciences, which con- perfection m sci- sists in the degree of precision of knowledge, and in ences ' the relation of its different branches. It is easy to see that the more general, simple, and abstract any phenomena are, the less they depend on others, and the more precise they are in themselves, and the more clear in their relations with each other. Thus, organic phenomena are less exact and systematic than inorganic ; and of these again terrestrial are less exact and systematic than those of astronomy. This fact is completely accounted for by the gradation we have laid down ; and we shall see as we proceed, that the pos- sibility of applying mathematical analysis to the study of phenom- ena is exactly in proportion to the rank which they hold in the scale of the whole. There is one liability to be guarded against, which D fr Ct8nrP inu8, we may mention here. We must beware of confound- not iu science, ing the degree of precision which we are able to attain in regard to any science, with the certainty of the science itself. The cer- tainty of science, and our precision in the knowledge of it, are two very different things, which have been too often confounded ; and ore so still, though less than formerly. A very absurd proposition may be very precise ; as if we should say, for instance, that the sum of the angles of a triangle is equal to three right angles ; and a very certain proposition may be wanting in precision in our state- ment of it; as, for instance, when we assert that every man will die. If the different sciences offer to us a varying degree of pre 48 POSITIVE PHILOSOPHY. ciêion, it is from no want of certainty in themselves, but of our mastery of their phenomena. 4. Effect on Ed- The most interesting property of our formula of gra- uction. dation is its effect on education, both general and sci- entific. This is its direct and unquestionable result. It will be more and more evident as we proceed, that no science can be effec- tually pursued without the preparation of a competent knowledge of the anterior sciences on which it depends. Physical philoso- phers can not understand Physics without at least a general knowl- edge of Astronomy ; nor Chemists, without Physics and Astrono- my ; nor Physiologists, without Chemistry, Physics, and Astrono- my ; nor, above all, the students of Social philosophy, without a general knowledge of all the anterior sciences. As such conditions axe, as yet, rarely fulfilled, and as no organization exists for their fulfilment, there is among us, in fact, no rational scientific educa- tion. To this may be attributed, in great part, the imperfection of even the most important sciences at this day. If the fact is so in regard to scientific education, it is no less striking in regard to general education. Our intellectual system can not be renovated till the natural sciences are studied in their proper order. Even the highest understandings are apt to associate their ideas accord- ing to the order in which they were received : and it is only an in- tellect here and there, in any age, which in its utmost vigor can, like Bacon, Descartes, and Leibnitz, make a clearance in their field of knowledge, so as to reconstruct from the foundation their system of ideas. tiffeat „n Men.- Such is the operation of our great law upon scientific 0,1 education through its effect on Doctrine. We can not appreciate it duly without seeing how it affects Method. As the phenomena which are homogeneous have been classed under one science, while those which belong to other sciences are heterogeneous, it follows that the Positive Method must be con- stantly modified in a uniform manner in the r?uge of the same fundamental science, and will undergo modifications, different and more and more compound, in passing from one science to another. Thus, under the scale laid down, we shall meet with it in all its varieties ; which could not happen if we were to adopt a scale which should not fulfil the conditions we have admitted. This is an all- important consideration ; for if, as we have already seen, we can not understand the positive method in the abstract, but only by its application, it is clear that we can have no adequate conception of it but by studying it in it3 varieties of application. No one science, however well chosen, could exhibit it. Though the Method is al- ways the same, its procedure is varied. For instance, it should be Observation with regard to one kind of phenomena, and Experiment with regard to another ; and different kinds of experiment, accord- ing to the case. In the same way, a general precept, derived from one fundamental science, however applicable to another, must have its spirit preserved by a reference to its origin ; as in the case of ORDERLY STUDY OF SCIENCES. 49 the theory of Classifications. The best idea of the Positive Method would, of course, be obtained by the study of the most primitive and exalted of the sciences, if we were confined to one; but this isolated view would give no idea of its capacity of application to others in a modified form. Each science has its own proper advan- tages ; and without some knowledge of them all, no conception can be formed of the power of the Method. One more consideration must be briefly adverted to. orderly study of It is necessary, not only to have a general knowledge sciences - of all the sciences, but to study them in their order. What can come of a study of complicated phenomena, if the student have not learned, by the contemplation of the simpler, what a Law is, what it is to Observe ; what a Positive conception is; and even what a chain of reasoning is? Yet this is the way our young physiolo- gists proceed every day — plunging into the study of living bodies, without any other preparation than a knowledge of a dead language or two, or at most a superficial acquaintance with Physics and Chemistry, acquired without any philosophical method, or reference to any true point of departure in Natural philosophy. In the same way, with regard to Social phenomena, which are yet more compli- cated, what can be effected but by the rectification of the intellec- tual instrument, through an adequate study of the range of anterior phenomena? There are many who admit this: but they do not see how to set about the work, nor understand the Method itself, for want of the preparatory stud} 7 ; and thus, the admission remains barren, and social theories abide in the theological or metaphysical state, in spite of the efforts of those who believe themselves posi- tive reformers. These, then, are the four points of view under which we have recognised the importance of a Rational and Positive Classifi- cation. It can not but have been observed, that in our enu- ,. , , . , , . t • • Mathematics. meration of the sciences there is a prodigious omission. We have said nothing of Mathematical science. The omission was intentional ; and the reason is no other than the vast importance of mathematics. This science will be the first of which we shall treat. Meantime, in order not to omit from our sketch a depart- ment so prominent, we may indicate here the general results of the study we are about to enter upon. In the present state of our knowledge, we must re- gard Mathematics less as a constituent part of natural epar philosophy than as having been, since the time of Descartes and Newton, the true basis of the whole of natural philos- ophy; though it is. exactly speaking, both the one and the other. To us it is of less value for the knowledge of which it consists, substantial and valuable as that knowledge is, than as being the most powerful instrument that the human • j | * . ,% • ,. r ±i i /• An instrument- mind can employ in the investigation of the laws ot natural phenomena. 4 50 POSITIVE PHILOSOPHY. a double ed- I n due precision, Mathematics must be divided into ,>,1,V two great sciences, quite distinct from each other — Ab- stract Mathematics, or the Calculus (taking the word in its most extended sense), and Concrete Mathematics, which is composed of General Geometry and of Rational Mechanics. The Concrete part is necessarily founded on the Abstract, and it becomes in its turn the basis of all natural philosophy ; all the phenomena ot the universe being regarded, as far as possible, as geometrical or mechanical. Ab-tract nmthe- The Abstract portion is the only one which is purely maticflaniiutra. instrumental, it being simply an immense extension of natural logic to a certain order of deductions. Geom- Ooncrrta math etl T an( * mechanics must, on the contrary, be regarded ematica a sd- as true natural sciences, founded, like all others, on observation, though, by the extreme simplicity of their phenomena, they can be systematized to much greater perfection. It i- tins capacity which has caused the experimental character of their first principles to be too much lost sight of. But these two physical sciences have this peculiarity, that they are now, and will be more and more, employed rather as method than as doctrine. It needs scarcely to be pointed out that, in placing Mathematics at the head of Positive Philosophy, we are only extending the application of the principle which has governed our whole Classi- fication. We are simply carrying back our principle to its first manifestation. Geometrical and Mechanical phenomena are the most general, the most simple, the most abstract of all,— the most irreducible to others, the most independent of them ; serving, in fact, as a basis to all others. It follows that the study of them is an indispensable preliminary to that of all others. Therefore must Mathematics hold the first place in the hierarchy of the eminent in the sciences, and be the point of departure of all Educa- tion, whether general or special. In an empirical way, this has hitherto been the custom, — a custom which arose from the great antiquity of mathematical science. We now see why it must be renewed on a rational foundation. We have now considered, in the form of a philosophical problem, the rational plan of the study of the Positive Philosophy. The order that results is this ; an order which of all possible arrange- ments is the only one that accords with the natural manifestation of all phenomena. Mathematics, Astronomy, Physics, Chemistry, Physiology, Social Physics. BOOK I. MATHEMATICS. CHAPTER I. MATHEMATICS, ABSTRACT AND CONCRETE. We are now to enter upon the study of the first of the Six great Sciences : and we begin by establishing the importance of the Positive Philosophy in perfecting the character of each science in itself. Though Mathematics is the most ancient and the most perfect science of all, the general idea of it is far from being clearly de- termined. The definition of the science, and its chief divisions, have remained up to this time vague and uncertain. The plural form of the name (grammatically used as singular) indicates the want of unity in its philosophical character, as commonly conceived. In fact, it is only since the beginning of the last century that it could be conceived of as a whole ; and since that time geometers have been too much engaged on its different branches, and in ap- plying it to the most important laws of the universe, to have much attention left for the general system of the science. Now, how- ever, the pursuit of its specialities is no longer so engrossing as to exclude us from the study of Mathematics in its unity. It has now reached a degree of consistency which admits of the effort to reduce its parts into a system, in preparation for further advance. The latest achievements of mathematicians have prepared the way for this by evidencing a character of unity in its principal parts which was not before known to exist. Such is eminently the spirit of the great author of the Theory of Functions and of Analytical Mechanics. The common description of Mathematics, as the set- D( ,. arriptl0n ^ ence of Magnitudes, or somewhat more positively, the Mathematics. science which relates to the Measurement of Magnitudes, is too vague and unmeaning to have been used but for want of a better. Yet the idea contained in it is just at bottom, and is even suffi- 52 POSITIVE PHILOSOPHY. ciently extensive, if properly understood ; but it needs precision and depth. It is important in such matters not to depart unneces- sarily from notions generally admitted ; and we will therefore see how, from this point of view, we can rise to such a definition of Mathematics as will be adequate to the importance, extent, and difficulty of the science. object of M„th Our first idea of measuring a magnitude is simply that ematui. f comparing the magnitude in question with another supposed to be known, which is taken for the unit of comparison among all others of the same kind. Thus, when we define mathe- matics as being the measurement of magnitudes, we give a very imperfect idea of it, and one which seems to bear no relation, in this respect, to any science whatever. We seem to speak only of a series of mechanical procedures, like a superposition of lines, for obtaining the comparison of magnitudes, instead of a vast chain of reasonings, inexhaustible by the intellect. Nevertheless, this defi- nition bas no other fault than not being deep enough. It does not mistake the real aim of mathematics, but it presents as direct an object which is usually indirect : and thus it misleads us as to the nature of the science. To rectify this, we must attend to a general fact, which is easily established — that the direct measurement of a magnitude is often au impossible operation ; so that if we had no other means Of doing what we want, we must often forego the knowledge we desire. We can rarely even measure a right line by another right line: and this is the simplest measurement of all. The very first condition of this is. that we should lie able to traverse the line from one end to the other: and this can not be done with the greater number of the distances which interest us the most. AVe can not do it with the heavenly bodies, nor with the earth and any heavenly body, nor even with many distances on the earth; and again, the length must be neither too great nor too small, a'nd it must be conveniently situated : and a line which could be easily measured if it were horizontal, becomes impracticable if vertical. There are so few lines capable of being directly measured with precision, that we are compelled to resort to artificial lines, created to admit of a direct determination, and to be the point of reference for all others. If there is difficulty about the measurement of lines, the embarrassment is much greater when we have to deal with surfaces, volumes, velocities, times, forces-, etc., and in general with all other magnitudes susceptible of estimate, and, by their nature, difficult of direct measurement. It is the general fact of this difficulty, inherent in almost every case, which necessitates the formation of mathematical science ; for, finding direct measurement so often impossible, we are compelled to devise means of doing it indirectly. Hence arose Mathematics. The general method employed, and the only conceiv- discovery of the law of Galileo. The Concrete part being ac- complished, the Abstract remains. We have ascertained that thu spaces traversed in each second increase as the series of odd nun» 56 POSITIVE PHILOSOPHY. hers, and we now have only the task of the computation of the height from the time, or of the time from the height; and this con- sists in finding that, by the established law, the first of these two quantities is a known multiple of the second power of the other; whence we may finally determine the value of the one when that of the other is given. In this instance the concrete question is the more difficult of the two. \ï the same phenomenon were taken in its greatest generality, the reverse would lie the case. Take the two together, and they may be regarded as exactly equivalent in difficulty. The mathematical law may be easy to ascertain, and difficult to work : or it may be difficult to ascertain, and easy to work. In importance, in extent, and in difficulty, these two great sections of .Mathematical Science will be seen hereafter to he equivalent. Ti,.ir different ^ e Mî - Vr Been tm ' difference in their objects. They are m> less different in their nature. The Concret» 1 must dépend on the character of tin 1 objects exam- ined, and must Miry when new phenomena present themsel\ whereas, the Attract is wholly independent of the nature of the objects, and is concerned only with their numerical relations. Thus, a great variety of phenomena may he brought under one geo- metrical solution. Cases which appeal- as unlike each other as possible may stand for one another under the Abstract proc which thus serves for all. while the Concrete process must be new in each case. Thus the Concrete process is Special, and the Ab- stract is General. The character of the Concrete is experimental, physical, phenomenal: while the Abstract is purely Logical, rational. The Concrete part of every mathematical question is necessarily founded on consideration of the external world : while the Abstract part consists of a series of logical deductions. The equations being once found, in any case, it is for the understanding, without exter- nal aid, to educe the results which these equations contain. AVe see how natural and complete this main division is. We will briefly prescribe the limits of each section. emerge Mathe- ^ s ^ * s the business of Concrete Mathematics to dis- matics. cover the equations of phenomena, we might suppose that it must comprehend as many distinct sciences as there are dis- tinct categories of phenomena ; but we are very far indeed from having discovered mathematical laws in all orders of phenomena. In fact, there are as yet only two great categories of phenomena whose equations are constantly known — Geometrical and Mechani- cal phenomena. Thus, the Concrete part of Mathematics consists of Geometry and Rational Mechanics. There is a point of view from which all phenomena might be included under these two divisions. All natural effects, considered statically or dynamically, might be referred to laws of extension or laws of motion. But this point of view is too high for us at pres- ent ; and it is only in the regions of Astronomy, and, partially, of terrestrial Physics, that this vast transformation has taken place. CHARACTER OF THE TWO DIVISIONS. 57 We will then proceed on the supposition that Geometry and Me- chanics are the constituents of Concrete Mathematics. The nature of Abstract Mathematics is precisely A!)gtract Math(> determined. It is composed of what is called the matics Calculus, taking this word in its widest extension, which reaches from the simplest numerical operations to the highest combinations of transcendental analysis. Its proper object is to resolve all questions of numbers. Its starting-point is that which is the limit of Concrete Mathematics — the knowledge of the precise relations — that is, the equations — between different magnitudes which are considered simultaneously. The object of the Calculus, however indirect or complicated the relations may be, is to discover un- known quantities by the known. This science, though more advanced than any other, is, in reality, only at its beginning yet ; but it is necessary, in order to define the nature of any science, to suppose it perfect. And the true character of the Calculus is what we have said. From an historical point of view, Mathematical Analysis appears to have arisen out of the contemplation of geometrical and mechani- cal facts ; but it is not the less independent of these sciences, logi- cally speaking. Analytical ideas are, above all others, universal, abstract, and simple ; and geometrical and mechanical conceptions are necessarily founded on them. ' Mathematical Analysis is there- fore the true rational basis of the whole system of our positive knowledge. We can now also explain why it not only gives pre- cision to our actual knowledge, but establishes a far more perfect co-ordination in the study of phenomena which allow of such an application. If a single analytical question, brought to an abstract solution, involves the implicit solution of a multitude of physical questions, the mind is enabled to perceive relations between phe- nomena apparently isolated, and to extract from them the quality which they have in common. To the wonder of the student, unsus- pected relations arise between problems which, instead of being, as they appeared before, wholly unconnected, turn out to be identical. There appears to be no connection between the determination of the direction of a curve at each of its points and that of the velo- city of a body at each moment of its variable motion ; yet, in tho eyes of the geometer, these questions are but one. When we have seized the true general character of Mathematical Analysis, we easily see how perfect it is, in comparison with all other branches of our positive science. The perfection consists in the simplicity of the ideas contemplated ; and not, as Condillac and others have supposed, in the conciseness and generality of the signs used as instruments of reasoning. The signs are of admirable use to work out the ideas, when once obtained ; but, in fact, all the great analytical conceptions were formed without any essential aid from the signs. Subjects which are by their nature inferior in sim- plicity and generality can not be raised to logical perfection by any nrtiiice of scientific language. 68 PosrnvB philosophy. Extent of ii« do- ^ e M;,Vt> nmv StM ' n what is the object and what ia umiu - the character of Mathematical Science. It remains for us to consider the extent of its domain. We musl first admit that, in a logical view, this sci- mice is necessarily and rigorously universal. There is no inquiry which is not finally reducible to a question of Numbers; for there is none which may not be conceived of as consisting in the determination of quantities by each other, according to certain relations. The fact is, we are always endeavoring to arrive at numbers, at fixed quantities, whatever may be our subject, however uncertain our method-, and however rough our results. Nothing can appear Less like a mathematical inquiry than the study of liv- ing bodies in a state of disease ; yet. in studying the cure of dis- ease, we are endeavoring to ascertain the quantities of the différent agents which are to modify the organism, in order to bring it to its natural state, admitting, as geometers do, for some of these quan- tities, in certain cases, values which are equal to zero, negative, or even contradictory. It is not meant that such a method can be actually followed in the case of complicated phenomena; but the logical extension of the science, which is what we are now con- sidering, comprehends such instances as this. Kant has divided human idea- into the two categories of quantity and quality, which, if true, would destroy the universality of Math- ematics; but Descartes'ti fundamental conception of the relation of tin; concrete to the abstract in Mathematics abolishes this divis- ion, and proves that all ideas of quality are reducible to ideas of quantity, lie had in view geometrical phenomena only; but his successors have included in this generalization, first, mechanical phenomena, and, more recently, those of heat. There are now no geometers who do not consider it of universal application, and ad- mit that every phenomenon may be as logically capable of being represented by an equation as a curve or a motion, if only we were always capable (which we are very far from being) of first discover- ing, and then resolving it. The limitations of Mathematical science are not, then, in its nature. The limitations are in our intelligence: and by these we find the domain of the science remarkably re- stricted, in proportion as phenomena, in becoming special, become complex. Though, as we have seen, every question may be conceived of a;* reducible to numbers, the reduction can not be made by us except in the case of the simplest and most general phenomena. The dif- ficulty of finding the equation in the case of special, and therefore complex phenomena, soon becomes insurmountable, so that, at the utmost, it is only the phenomena of the first three classes, — that is, only those of Inorganic Physics, — that we can even hope to sub- ject to the process. The properties of inorganic bodies are nearly invariable ; and therefore, with regard to them, the first condition of mathematical inquiry can be fulfilled: the different quantities LIMITATIONS OF MATHEMATICAL SCIENCE. 59 which they present may be resolved into fixed numbers ; but the variableness of the properties of organic bodies is beyond our man- agement. An inorganic body, possessing solidity, form, consistency, specific gravity, elasticity, etc., presents qualities which are within our estimate, and can be treated mathematically ; but the case is altered when Chemical action is added to these. Complications and variations then enter into the question which at present baffle mathematical analysis. Hereafter, it may be discovered what fixed numbers exist in chemical combinations : but we are as yet very far from having any practical knowledge of them. Still further are we from being able to form such computations amidst the continual agitation of atoms which constitutes what we call life, and therefore from being able to carry mathematical analysis into the study of Physiology. By the rapidity of their changes, and their incessant numerical variations, vital phenomena are, practi- cally, placed in opposition to mathematical processes. If we should desire to compute, in a single case, the most simple facts of a liv- ing body, — such as its mean density, its temperature, the velocity of its circulation, the proportion of elements which at any moment compose its solids or its fluids, the quantity of oxygen which it consumes in a given time, the amount of its absorptions or its exhalation, — and, yet more, the energy of its muscular force, the intensity of its impressions, etc., we must make as many observa- tions as there are species or races, and varieties in each ; we must measure the changes which take place in passing from one individual to another, and in the same individual, according to age, health, interior condition, surrounding circumstances perpetually varying, such as the constitution of the atmosphere, etc. It is clear that no mathematical precision can be attained amidst a complexity like this. Social phenomena, being more complicated still, are even more out of the question, as subjects for mathematical analysis. It is not that a mathematical basis does not exist in these cases, as truly as in phenomena which exhibit, in all clearness, the law of gravita- tion : but that our faculties are too limited for the working of prob- lems so intricate. We are baffled by various phenomena of in- organic bodies, when they are very complex. For instance, no one doubts that meteorological phenomena are subject to mathematical laws, however little we yet know about them ; but their multiplicity renders their observed results as variable and irregular as if each cause were free of all such conditions. We find a second limitation in the number of conditions to be studied, even if we were sure of the mathematical law which governs each agent. Our feeble faculties could not grasp and wield such an aggregate of conditions, however certain might be our knowledge of each. In the simplest cases in which we desire to approximate the abstract to the concrete conditions, with any completeness, — as in the phenomenon of the flow of a fluid from a given orifice, by virtue of its gravity alone, — the difficulty is such that we are, as yet, without any mathematical solution of this very 60 POSITIVE PHILOSOPHY. problem. The same is the case with the yet more simple instance of the movement of a solid projectile through a resisting medium. To the popular mind it may appear strange, considering these facts, that we know so much as wc do about the planets. But in reality, that class of phenomena is the most simple of all within oui* cognizance. The most complex problem which they present is the influence of a third body acting in the same way on two which are tending toward each other in virtue of gravitation ; and this is a more simple question than any terrestrial problem what- ever. We have, however, attained only approximate solutions in this case. And the high perfections to which solar astronomy has been brought by the use of mathematical science is owing to our having profited by those facilities thai we may call accidental, which the favorable constitution of our planetary system presents. The planets which compose it arc few; their masses are very un- equal, and much less than that of the sun : they are far distant from each other ; their forms arc nearly spherical ; their orbits are nearly circular, ami only slightly inclined in relation to each other; and so on. Their perturbations are, in consequence, inconsiderable, for the most part : and all we have to do is usually to take into the account, together with the influence of the sun on each planet, the influence of one other planet, capable, by its size and its near- ness, of occasioning perceptible derangements. If any of the con- ditions mentioned above had been different, though the law of grav- itation had existed as it is, we might not at this day have discov- ered it. And if we were now to try to investigate Chemical phenom- ena by the same law, we should find a solution as impossible as it would be in astronomy, if the conditions of the heavenly bodies were such as we could not reduce to an analysis. In showing that Mathematical analysis can be applied only to Inorganic Physics, we are not restricting its domain. Its rigor- ous universality, in a logical view, has been established. To pre- tend that it is practically applicable to the same extent would be merely to lead away the human mind from the true direction of scientific study, in pursuit of an impossible perfection. The most difficult sciences must remain, for an indefinite time, in that pre- liminary state which prepares for the others the time when they too may become capable of mathematical treatment. Our business is to study phenomena, in the characters and relations in which they present themselves to us, abstaining from introducing considera- tions of quantities, and mathematical laws, which it is beyond our power to apply. We owe to Mathematics both the origin of Positive Philosophy and its Method. When this Method was introduced into the other sciences, it was natural that it should be urged too far. But each science modified the method by the operation of its own peculiar phenomena. Thus only could that true definitive character be brought out, which must prevent its being ever confounded with that of anv other fundamental science. GENERAL VIEW OF MATHEMATICAL ANALYSIS. 61 The aim, character, and general relations of Mathematical Sci- ence have now been exhibited as fully as they could be in such a sketch as this. We must next pass in review the three great sci- ences of which it is composed, — the Calculus, Geometry, and Ra- tional Mechanics. CHAPTER II. GENERAL VIEW OF MATHEMATICAL ANALYSIS. The historical development of the Abstract portion of Mathematical science has, since the time of Descartes, been for the most part determined by that of the Concrete. Yet the Calculus in all its principal branches must be understood before passing on to Geometry and Mechanics. The Concrete portions of the science depend on the Abstract, which are wholly indepen- dent of them. We will now, therefore, proceed to a rapid review of the leading conceptions of the Analysis. First, however, we must take some notice of the gen- True idea of an eral idea of an equation, and see how far it is from equation. being the true one on which geometers proceed in practice ; for without settling this point we can not determine, with any pre- cision, the real aim and extent of abstract mathematics. The business of concrete mathematics is to, discover the equa- tions which express the mathematical laws of the phenomenon under consideration ; and these equations are the starting-point of the calculus, which must obtain from them certain quantities by means of others. It is only by forming a true idea of an equation that we can lay down the real line of separation between the con crete and the abstract part of mathematics. It is giving much too extended a sense to the notion of an equa tion to suppose that it means every kind of relation of equality between any two functions of the magnitudes under consideration ; for, if every equation is a relation of equality, it is far from being the case that, reciprocally, every relation of equality must be an equation of the kind to which analysis is, by the nature of the case, applicable. It is evident that this confusion must render it almost impossible to explain the difficulty wo find in establishing the rela- tion of the concrete to the abstract which meets us in every great mathematical question, taken by itself. If the word equation meant what we are apt to suppose, it is not easy to see what difficulty there could be, in general, in establishing the equations of any problem whatever. This ordinary notion of an equation is widely unlike what geometers understand in the actual working of the science. According to my view, functions must themselves be divided into Ï>Z POSITIVE PHILOSOPHY. Abstract and Concrete : the first of which alone can enter into truo equations. Every equation is a relation of equality between two abstract functions of the magnitudes in question, including with die primary magnitudes all the auxiliary magnitudes which may be connected with the problem, and the introduction of which may facilitate the discovery of the equations Bought. This distinction may l»e established by both the à priori and à posteriori methods : by characterizing each kind of function, and by enumerating all the abstract functions yet known, — at least with regard to their elements. ,., lutu .. A priori ; Abstract functions express a mode of de- pendence between magnitudes which may be conceived between numbers alone, without the need of pointing out any phe- ConcratB fen» nomena in which it may be found realized ; while Con- crete functions are those whose expression requires a specified actual case of physics, geometry, mechanics, etc. Most functions were concrete in their origin, — even those which are at present the most purely abstract ; and the ancients discov- ered only through geometrical definitions elementary algebraic prop- erties of functions, to which a numerical value was not attached till long afterward, rendering abstract to us what was concrete to the old geometers. There is another example which well exhibits the distinction just made — that of circular functions, both direct and inverse, which are still sometimes concrete, sometimes abstract, Recording to the point of view from which they are regarded. A posteriori; the distinguishing character, abstract or concrete, of a function having been established, the question of any determi- nate function being abstract, and therefore able to enter into true analytical equations, becomes a simple question of fact, as we are acquainted with the elements which compose all the abstract func- tions at present known. We say we know them all, though ana- lytical functions are infinite in number, because we are here speak- ing, it must b.> remembered, of the elements — of the simple, not of the compound. We have ten elementary formulas ; and, few as tli jy are, they may give rise to an infinite number of analytical combinations. There is no reason for supposing that there can never be more. We have more than Descartes had, and even Newton and Leibnitz ; and our successors will doubtless introduce additions, though there is so much difficulty attending their aug- mentation, that we can not hope that it will proceed very far. It is the insufficiency of this very small number of analytical elements which constitutes our difficulty in passing from the con- crete to the abstract. In order to establish the equations of phe- nomena, we must conceive of their mathematical laws by the aid of functions composed of these few elements. Up to this point the question has been essentially concrete, not coming within the domain of the calculus. The difficulty of the passage from the concrete to the abstract in general consists in our having only theso few analytical ebments with which to represent all the precise GENERAL VIEW OF MATHEMATICAL ANALYSIS. 63 relations which the whole range of natural phenomena afford to us. Amid their infinite variety, our conceptions must be far below the real difficulty; and especially because these elements of our analysis have been supplied to us by the mathematical considera- tion of the simplest phenomena of a geometrical origin, which can afford us, à priori, no rational guaranty of their fitness to represent the mathematical laws of all other classes of phenomena. We shall hereafter see how this difficulty of the relation of the concrete to the abstract has been diminished, without its being necessary to multiply the number of analytical elements. Thus far we have considered the Calculus as a whole. "We must now consider its divisions. These divisions we must Two part8 of the call the Algebraic Calculus, or Algebra, and the Arith- cakuius. metical Calculus, or Arithmetic, taking care to give them the most extended logical sense, and not the restricted one in which the terms are usually received. It is clear that every question of Mathematical Analysis presents two successive parts, perfectly distinct in their nature. The first stage is the transformation of the proposed equations, so as to exhibit the mode of formation of unknown quantities by the known. This constitutes the algebraic question. Then ensues bra the task of finding the values of the formulas thus obtained. The values of the numbers sought are already repre- sented by certain explicit functions of given numbers : these values must be determined ; and this is the arithmetical ques- . tion. Thus the algebraic and the arithmetical calcu- lus differ in their object. They differ also in their view of quanti- ties, — Algebra considering quantities in regard to their relations y and Arithmetic in regard to their values. In practice, it is not always possible, owing to the imperfection of the science of the cal- culus, to separate the processes entirely in obtaining a solution ; but the radical difference of the two operations should never be lost sight of. Algebra, then, is the Calculus of Function, and Arithmetic the Calculus of Values. We have seen that the division of the Calculus is into two branches. It remains for us to compare the two, in order to learn their respective extent, importance, and difficulty. The Calculus of Values, Arithmetic, appears at first -, .-i /?-ii A-ii • Arithmetic. to have as wide a field as Algebra, since as many ques- tions might seem to arise from it as we can conceive different alge- braic formulas to be valued. But a very simple reflection will show that it is not so. Functions being divided into simple and compound, it is evident that when we become able to determine the value of simple functions, there will be no difficulty with the compound. In the algebraic relation, a com- pound function plays a very different part from that of the elemen- tary functions which constitute it ; and this is the source of our chief analytical difficulties. But it is quite otherwise with the Arithmetical Calculus. Thus, the number of distinct arithmot ; cal <>l POSITIVE PHILOSOPHY. operations is indicated by that of the abstract elementary functions, which we have seen to be very few. The determination of the values of these ten functions necessarily affords that of all the infinite number comprehended in the whole of mathematical anal- ysis : and I here can be no new arithmetical operations otherwise than by the creation of new analytical elements, which must, in any case, for ever be extremely small. The domain of arithmetic then is, by its nature, narrowly restricted, while that of algebra is rigorously indefinite. Still, the domain of arithmetic is more extensive than is commonly represented ; for there are many ques- tions treated as incidental in the midst of a body of analytical researches, which, consisting of determinations of values, are truly arithmetical. Of this kind are the construction of a table of log- arithm.-, and the calculation of trigonometrical tables, and some distinct and higher procédures : in short, every operation which has l'or its object the determination of the values of functions. And we must also include that part of the science of the Calculus which we call the Theory of Numbers, the object of which is to discover the properties inherent in different numbers, in virtue of their values, independent of any particular system of numeration. It constitutes a sort of transcendental arithmetic. Though the domain of arithmetic IS thus larger than is commonly supposed, this Calculus of values will yet never be more than a point, as it were, in comparison with the calculus of functions, of which math- ematical science essentially consists. This is evident, when we look into the real nature of arithmetical questions. Determinations of ruines are. in fact, nothing else than real transformations of the functions to be valued. These transformations have a special end ; but they are essen- tially of the same nature as all taught by analysis. In this view, the Calculus of values may be regarded as a supplement, and a par ticular application of the Calculus of functions, so that arithmetic disappears, at it were, as a distinct section in the body of abstract mathematics. To make this evident, we must observe that when we desire to determine the value of an unknown number whose mode of formation is given, we define and express that value in merely announcing the arithmetical question, already defined and expressed under a certain form ; and that, in determining its value, we merely express it under another determinate form to which we are in the habit of referring the idea of each particular number by making it re-enter into the regular system of numeration. This is made clear by what happens when the mode of numeration is such that the question is its own answer ; as, for instance, when we want to add together seven and thirty, and call the result seven-and- thirty. In adding other numbers, the terms are not so ready, and we transform the question ; as when we add together twenty-three and fourteen : but not the less is the operation merely one of trans formation of a question already defined and expressed. In this view, the calculus of values might be regarded as a particular ap- ALGEBRA. 65 plication of the calculus of functions, arithmetic thereby disappear- ing, as a distinct section, from the domain of abstract mathematics. — And here we have done with the Calculus of values, and pass to the Calculus of functions, of which abstract mathematics is essen- tially composed. We have seen that the difficulty of establishing the relation of the concrete to the abstract is owing to the ge ra * insufficiency of the very small number of analytical elements that we are in possession of. The obstacle has been surmounted in a great number of important cases : and we will now see how the establish- ment of the equations of phenomena has been achieved. The first means of remedying the difficulty of the creation of new small number of analytical elements seems to be to ere- fonctions, ate new ones. But a little consideration will show that this re- source is illusory. A new analytical element would not serve unless we could immediately determine its value : but how can we determine the value of a function which is simple ; that is, which is not formed by a combination of those already known ? This ap- pears almost impossible : but the introduction of another elemen- tary abstract function into analysis supposes the simultaneous cre- ation of a new arithmetical operation ; which is certainly extremely difficult. If we try to proceed according to the method which pro- cured us the elements we possess, we are left in entire uncertainty ; for the artifices thus employed are evidently exhausted. We have thus no idea how to proceed to create new elementary abstract functions. Yet, we must not therefore conclude that we have reached the limit appointed by the powers of our understanding. Special improvements in mathematical analysis have yielded us some partial substitutes, which have increased our resources : but it is clear that the augmentation of these elements can not proceed but with extreme slowness. It is not in this direction, then, that the human mind has found its means of facilitating the establish- ment of equations. This first method being discarded, there remains Fading equa. only one other. As it is impossible to find the equa- SSiJ^ES tions directly, we must seek for corresponding ones be- ties - tween other auxiliary quantities, connected with the first accord- ing to a certain determinate law, and from the relation between which we may ascend to that of the primitive magnitudes. This is the fertile conception which we term the transcendental analysis y and use as our finest instrument for the mathematical exploration of natural phenomena. This conception has a much larger scope than even profound geometers have hitherto supposed ; for the auxiliary quantities re- sorted to might be derived, according to any law whatever, from the immediate elements of the question. It is well to notice this ; because our future improved analytical resources may perhaps be found in a new mode of derivation. But, at present, the only aux- fi 66 POSITIVE PHILOSOPHY. • iliary quantities habitually substituted for the primitive quanti tie» in transcendental analysis are what are called — 1st, infinitely small elements, the differentials of dilTerent orders of those quantities, if we conceive of this analysis in the manner of Leibnitz: or 2d, the fluxions, the limits of the ratios of the simultaneous in- crements of the primitive quantities, compared with one another; or, more briefly, the prime and ultimate ratios of these increments, if we adopt the conception of Newton: or 8d, the derivatives, properly so called, of these quantities; that is, the coefficients of the different terms of their respective incre- ments, according to the conception of Lagrange. These conceptions, and all others that have been proposed, are by their nature identical. The various grounds of preference of each of them will be exhibited hereafter. Divi ,„„ of fh . We now see that the Calculus of functions, or Alge- tfloffunc- bra, must consist of two distinct branches. The one has for its object the resolution of equations when they ore directly established between the magnitudes in question : the other, setting out from equations (generally much more easy to form) between quantities indirectly connected with those of the problem, has to deduce, by invariable analytical procedures, the correspond- ing equations between the direct magnitudes in question ; — bringing the problem within the domain of the preceding calculus. — It might Beem that the transcendental analysis ought to be studied before the ordinary, as it provides the equations which the other has to resolve. But, though the transcendental is logically independent of the ordinary, it is best to follow the usual method of study, taking the ordinary first; for, the proposed questions always re- quiring to be completed by ordinary analysis, they must be left in suspense if the instrument of resolution had not been studied before- hand. To ordinary analysis I propose to give the name of Calculus op Direct Functions. To transcendental analysis (which is known by the name of Infinitesimal Calculus, Calculus of fluxions and of fluents, Calculus of Vanishing quantities, the Differential and Integral Calculus, etc., according to the view in which it has been conceived) I shall give the title of Calculus of Indirect Functions. I obtain these terms by generalizing and giving pre- cision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis. section i. ordinary analysis, or calculus of direct functions. Algebra is adequate to the solution of mathematical questions which are so simple that we can form directly the equations be- tween the magnitudes considered, without its being necessary to bring into the problem, either in substitution or alliance, any sys- ALGEBRAIC EQUATIONS. 67 tern of auxiliary quantities derived from the primary. It is true, in the majority of important cases, its use requires to be preceded and prepared for by that of the calculas of indirect functions, by which the establishment of equations is facilitated : but though algebra then takes the second place, it is not the less a neces- sary agent in the solution of the question ; so that the Calculus of direct functions must continue to be, by its nature, the basis of mathematical analysis. We must now, then, notice the rational composition of this calculus, and the degree of development it has attained. Its object being the resolution of equations (that is, the discovery of the mode of formation of unknown t8 ° Ject quantities by the known, according to the equations which exist be- tween them), it presents as many parts as we can imagine distinct classes of equations ; and its extent is therefore rigorously indefi- nite, because the number of analytical functions susceptible of en- tering into equations is illimitable, though, as we have seen, com- posed of a very small number of primitive elements. The rational classification of equations must evi- classification of dently be determined by the nature of the analytical Kquniiona. elements of which their numbers are composed. Accordingly, ana- lysts first divide equations with one or more variables into two principal classes, according as they contain functions of only the first three of the ten couples, or as they include also either expo- nental or circular functions. Though the names of algebraic and transcendental functions given 1o these principal groups are inapt, the division between the corresponding equations is really enough, in so far as that the resolution of equations containing the transcendental functions is more difficult than that of algebraic equations. Hence the study of the first is extremely imperfect, and our analytical methods relate almost exclusively to the elaboration of the second. Our business now is with these Algebraic equations A1 , pl raic only. In the first place, we must observe that, though equations, they may often contain irrational functions of the unknown quan- tities, as well as rational functions, the first case can always be brought under the second, by transformation more or less easy ; so that it is only with the latter that analysts have had to occupy themselves, to resolve all the algebraic equations. As to their clas- sification, the early method of classing them according to the num- ber of their terms has been retained only for equations with two terms, which are, in fact, susceptible of a resolution proper to them- selves. The classification by their degrees, long universally estab- lished, is eminently natural ; for this distinction rigorously deter- mines the greater or less difficulty of their resolution. The gra- dation can be independently, as well as practically exhibited : for the most general equation of each degree necessarily comprehend? all those of the different inferior degrees, as must also the formula which determines the unknown quantity : and therefore, however slight we may, à priori, suppose the difficulty to be of the degree 68 POSITIVE PHILOSOPHY, under notice, it must offer more and more obstacles, in proportion to the rank of the degree, because it ta complicated in the execution with those of all the preceding degrees. This increase of difficulty is so great, that the résolu- lution of cqaa. tioii of algel Tair equations is as yet known to us only in the first four degrees. In this respect, algebra has advanced but little since the Labors of Descartes and the Italian analysts of the sixteenth century ; though there has probably not been a Bingle geometer for two centuries past who has not striven to advance the resolution <>!' equations. The general equation of the fifth degree has itself, thus far, resisted all attempts. The for- mula of the fourth degree is so difficult as to be almost inapplicable; and analysts, while by no means despairing of the resolution of equations of the fifth, and even higher degrees, being obtained, have tacitly agreed to give ap Buch researches. The only question of this kind which would be of eminent im- portance, at Least in its Logical relations, would be the general olution of algebraic equations of any degree whatever. But the morr we ponder this Bubject, the more we are Led to suppose, with Lagrange, that it exceeds the scope of our understandings. Even if the requisite formula could be obtained, it could not be usefully applied, uide-- we could simplify it. without impairing its general- ity, by the introduction of a new class of analytical elements, of which we have as yet no idea. And, besides, if we had obtained the resolution of algebraic equations of any degree whatever, we should still have treated only a very small part of algebra, prop- erly so called: that is, of the calculus of direct functions, com- prehending the resolution of all the equations that can be formed by the analytical functions known to US at this day. Again, we must remember that by a law of our nature, we Bhall always remain below the difficulty of science, (.ur means of conceiving of new questions being always more powerful than our resources lor resol- ving them; in other words, the human mind being more apt at imagining than at reasoning. Thus, if we had resolved all tin; ana- lytical equations now known, and if to do this, we had found new analytical elements, these again would introduce classes of equa- tions of which we now know nothing : and so, however great might be the increase of our knowledge, the imperfection of our algebraic science would be perpetually reproduced. our exisrin- The methods that we have are, the complete resolu- knowiedge. tion of the equations of the first four degrees ; of any binomial equations ; of certain special equations of the superior degrees ; and of a very small number of exponential, logarithmic, and circular equations. These elements are very limited ; but geom- eters have succeeded in treating with them a great number of im- portant questions in an admirable manner. The improvements introduced within a century into mathematical analysis have contrib- uted more to render the little knowledge that we have immeasur- ably useful, than to increase it. NUMERICAL RESOLUTION OF EQUATIONS. 69 To fill up the vast gap in the resolution of algebraic equations of the higher degrees, analysts have had re- oiutionofeqS course to a new order of questions, — to what they call tlons - the numerical resolution of equations. Not being able to obtain the real algebraic formula, they have sought to determine at least the value of each unknown quantity for such or such a designated system of particular values attributed to the given quantities. This operation is a mixture of algebraic with arithmetical questions ; and it has been so cultivated as to be rendered possible in all cases, for equations of any degree and even of any form. The methods for this are now sufficiently general ; and what remains is to sim- plify them so as to fit them for regular application. While such is the state of algebra, we have to endeavor so to dispose the ques- tions to be worked as require finally only this numerical resolution of the equations. We must not forget however that this is very imperfect algebra ; and it is only isolated, or truly final questions (which are very few), that can be brought finally to depend upon only the numerical resolution of equations. Most questions are only preparatory, — a first stage of the solution of other questions ; and in these cases it is evidently not the value of the unknown quantity that we want to discover, but the formula which exhibits its derivation. Even in the most simple questions, when this nu- merical resolution is strictly sufficient, it is not the less a very im- perfect method. Because we cannot abstract and treat separately the algebraic part of the question, which is common to all the cases which result from the mere variation of the given numbers, we are obliged to go over again the whole series of operations for the slightest change that may take place in any one of the quantities concerned. Thus is the calculus of direct functions at present divided into two parts, as it is employed for the algebraic or the numerical res- olution of equations. The first, the only satisfactory one, is un- fortunately very restricted, and there is little hope that it will ever be otherwise : the second, usually insufficient, has at least the ad- vantage of a much greater generality. They must be carefully distinguished in our minds, on account of their different objects, and therefore of the different ways in which quantities are consid- ered by them. Moreover, there is, in regard to their methods, an entirely different procedure in their rational distribution. In the first part, we have nothing to do with the values of the unknown quantities, and the division must take place according to the nature of the equations which we are able to resolve ; whereas in the second, we have nothing to do with the degrees of the equations, as the methods are applicable to equations of any degree whatever ; but the concern is with the numerical character of the values of the unknown quantities. These two parts, which constitute the immediate ob- Thp Theoiy oJ ject of the Calculus of direct functions, are subordina- wiwtoona ted to a third, purely speculative, from which both derive their most 70 POSITIVE PHILOSOPHY. effectual resources, and which haa been very exactly designated by the general name of Theory of Equations, though it relates, as yet ; only to algebraic equations. The numerical resolution of equa- tions has, on account of its generality, special need of this rational foundation. Two orders of question divide this important department of al- gebra between them; first, those which relate to the composition of equations, and then those that relate to their transformation: tlif business of these Last being to modify the roots of an equation without knowing them, according to any given law, provided this law is uniform in relation t<> all these roots. On» 1 more theory remains to he noticed, to complete our rapid exhibition of the different essential parts of the calculus of direct functions. This theory, which relates to the trausfor- tenninwecJeffli niation «>f functions into series by the aid of what is called the Method of indeterminate Coefficients, is one of the most fertile and important in algebra. This eminently ana- lytical method is one of the most remarkable discoveries of Des- cartes. The invention and development of the infinitesimal calcu- lus, tor which it might be very happily substituted in some respects, has undoubtedly deprived it of some of its importance; but the growing extension of the transcendental analysis has. while Lessening its necessity, multiplied it- applications and enlarged it- resources; so that, by the useful combination of the two theories, the employ- ment of the method of indeterminate coefficients has become much more extensive than it was even before the formation of the cal- culus of indirect functions. 1 have now completed my sketch of the Calculus of Direct Func- tions. We must next pass on to the more important and extensive branch of our science, the Calculus of Indirect Functions. SECTION II. TRANSCENDENTAL ANALYSIS, OR CALCULUS OF INDIRECT FUNCTIONS. Three pm.cipai ^ e referred (p. 65^) in a former section to the views riew of the transcendental analysis presented by Leibnitz, Newton, and Lagrange. We shall see that each conception has advantages of its own, that all are finally equivalent, and that no method has yet been found which unites their respective character- istics. Whenever the combination takes place, it will probably be by some method founded on the conception of Lagrange. The other two will then offer only an historical interest ; and meanwhile, the science must be regarded as in a merely provisional state, which requires the use of all the three conceptions at the same time ; for it is only by the use of them all that an adequate idea of the anal- ysis and its applications can be formed. The vast extent and dif- ficulty of this part of mathematics, and its recent formation, should prevent our being at all surprised at the existing want of system. The conception which will doubtless give a fixed and uniform char- METHOD OF LEIBNITZ. 71 acter to the science has come into the hands of only one new gen- eration of geometers since its creation ; and the intellectual habits requisite to perfect it have not been sufficiently formed. The first germ of the infinitesimal method (which can be conceived of independently of the Calculus) may bo recognised in the old Greek Method of Exhaustions, employed to pass from the properties of straight lines to those of curves. The method consisted in substituting for the curve the auxiliary consideration of a polygon, inscribed or circumscribed, by means of which the curve itself was reached, the limits of the primitive ratios being suitably taken. There is no doubt of the filiation of ideas in this case ; but there was in it no equivalent for our modern methods ; for the ancients had no logical and general means for the determination of these limits, which was the chief difficulty of the question. The task remaining for modern geometers was to gen- eralize the conception of the ancients, and, considering it in an abstract manner, to reduce it to a system of calculation, which was impossible to them. Lagrange justly ascribes to the great geometer Fermât the first idea in this new direction. Fermât may be regarded as having initiated the direct formation of transcendental analysis by his method for the determination of maxima and minima, and for the finding of tangents, in which process he introduced auxiliaries which he afterward suppressed as null when the equations obtained had undergone certain suitable transformations. After some modi- fications of the ideas of Fermât in the intermediate time, Leibnitz stripped the process of some complications, and formed the analysis into a general and distinct calculus, having its own notation : and Leibnitz is thus the creator of transcendental analysis, as we employ it now. This pre-eminent discovery was so ripe, as all great con- ceptions are at the hour of their advent, that Newton had at the same time, or rather earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much more logical in itself, but less adapted than that of Leibnitz to give all practicable extent and facility to the fundamental method. La- grange afterward, discarding the heterogeneous considerations which had guided Leibnitz and Newton, reduced the analysis to a purely algebraic system, which only wants more aptitude for ap- plication. We will notice the three methods in their order. The method of Leibnitz consists in introducing into Method of the calculus, in order to facilitate the establishment of Leibnitz. equations, the infinitely small elements or differentials which are supposed to constitute the quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the primitive quantities: and by these we may afterward (through a special calculus employed to eliminate these auxiliary infinitesimals) recur to the equations sought, which it would usually have been impossible to obtain directly. 72 POSITIVE PHILOSOPHY. This indirect analysis may have varions degrees of indirectness; for, when there is too much difficulty in forming the equation be* tween the differentials of the magnitudes under notice a second ap- plication of the method is required, the differentials being now treated as new primitive quantities, and a relation being Bought be- tween their infinitely small elements, or second differentials, and so on : the same transformation being repeated any number of times, provided the whole number of auxiliaries l>e finally eliminated. It may be asked by novices in these studies, how these auxiliary quantities Can I f OSe while they are of the s;nne species with the magnitudes to be treated, seeing that the greater or Less value of any quantity can n«'t affect any inquiry which has aothing to do with value at all. The explanation is this. We must begin by distinguishing the different orders of infinitely small quantities, obtaining a precise idea <.f this l'y considering them as being either the successive powers <»t' the same primitive infinitely small quan- tity, or as being quantities which may he regarded as having finite ratios with these powers; bo that, for instance, the second or third or other differentials of tla- Bame variable are classed a- infinitely small quantities of the Becond, third or other order, because it is easy t" exhibit in them finite multiples of the second, third, or other powers of a certain first differential. These pre- liminary ideas being laid down, the spirit of the infinitesimal analy- sis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities'; ami generally the infinitely small quantities of any older whatever in comparison with all those of an inferior order. We gee at once how such a power must facil- itate the formation of equations between the differentials of quanti- ties, since we can Substitute for these differentials Such other ele- ments as we may choose, and as will he more simple to treat, only observing the condition that the new elements shall differ from the preceding only by quantities infinitely small in relation to them. It is thus that it becomes possible in geometry to treat curved lines as composed of an infinity of rectilinear (dements, and curved surfaces as formed of plane elements : and. in mechanics, varied motions as an infinite series of uniform motions, succeeding each other at infinitely small intervals of time. Such a mere hint as this of the varied application of this method may give some idea of the vast scope of the conception of transcendental analysis, as formed by Leibnitz. It is, beyond all question, the loftiest idea ever yet attained by the human mind. It is clear that this conception was necessary to complete the basis of mathematical science, by enabling us to establish, in a broad and practical manner, the relation of the concrete to the ab- stract. In this respect, we must regard it as the necessary com- plement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena ; an idea which, could not be duly estimated or put to use till after the formation of the infinitesimal analysis. METHOD OF LEIBNITZ. 73 This analysis has another property besides that of facilitating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential formulas Generality of th» exhibit an extreme generality, expressing in a single lormulas - equation each determinate phenomenon, however varied may be the subjects to which it belongs. Thus, one such equation gives the tangents of all curves, another their rectifications, a third their quadratures ; and, in the same way, one invariable formula ex- presses the mathematical law of all variable motion ; and one sin- gle equation represents the distribution of heat in any body, and for any case. This remarkable generality is the basis of the loftiest views of the geometers. Thus, this analysis has not only furnished a general method for forming equations indirectly which could not have been directly discovered, but it has introduced a new order of more natural laws for our use in the mathematical study of natural phenomena, enabling us to rise at times to a perception of positive approximations between classes of wholly different phenomena,, through the analogies presented by the differential expressions of their mathematical laws. In virtue of this second property of the analysis, the entire system of an immense science, like geometry or mechanics, has submitted to a condensation into a small number of analytical formulas, from which the solution of all particular prob- lems can be deduced, by invariable rules. This beautiful method is, however, imperfect in its . InRtific . ltion of logical basis. At first, geometers were naturally more the Method, intent upon extending the discovery and multiplying its applica- tions than upon establishing the logical foundation of its processes. It was enough, for some time, to be able to produce, in answer to objections, unhoped-for solutions of the most difficult problems. It became necessary, however, to recur to the basis of the new analy- sis, to establish the rigorous exactness of the processes employed, notwithstanding their apparent breaches of the ordinary laws of reasoning. Leibnitz himself failed to justify his conception, giving, when urged, an answer which represented it as a mere approxima- tive calculus, the successive operations of which might, it is evident, admit an augmenting amount of error. Some of his successors were satisfied with showing that its results accorded with those obtained by ordinary algebra, or the geometry of the ancients, reproducing by these last some solutions which could be at first obtained only by the new method. Some, again*, demonstrated the conformity of the new conception with others ; that of Newton especially, which was unquestionably exact. This afforded a prac- tical justification: but, in a case of such unequalled importance T a logical justification is also required, — a direct proof of the neces- sary rationality of the infinitesimal method. It was Carnot who furnished this at last, by showing that the method was founded on the principle of the necessary compensation of errors. We can not say that all the logical scaffolding of the infinitesimal method may not have a merely provisional existence, vicious as it is in its nature : 74 POSITIVE PHILOSOPHY. but, in the present state of our knowledge, Carnot's principle of the necessary compensation of errors is of more importance, in legiti- mating llif analysis of Leibnitz, than is even yet commonly sup- posed. His reasoning is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries ean not have occasioned other than infinitely small errors in all the equations ; and when the relations of finite quantities are reached, these relations must be rigorously exact, since the only errors then possible must be finite ones, which can not have en- tered : and thus the finite equations become perfect. Carnot's the- ory is doubtless more subtile than solid : but it lias no other radical logical vice than that of the infinitesimal method itself, of which it is, as it seems to me, the natural development and general explana- tion : so that it must be adopted as long as that method is directly employed. The philosophical character <>f the transcendental analysis has now been sufficiently exhibited to allow of my giving only the prin- cipal idea of the other two methods. Nkwton .. s Newton offered his conception under several differ- mkihod. en t forms in succession. That which is now most com- monly adopted, at leasl on the continent, was called by himself, sometimes the Method of prime r. in other words, the final ratios of these increments : limits or final ratios which we can easily show to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is afterward em- ployed, to rise from the equations between these limits to the corresponding equations between the primitive quantities them- selves. The power of easy expression of the mathematical laws of phe- nomena given by this analysis arises from the calculus applying, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those increments ; and from our being therefore able always to substitute for each increment any other magnitude more easy to treat, provided their final ratio is the ratio of equality ; or, in other words, that the limit of their ratio is unity. It is clear, in fact, thai; the calculus of limits can be in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offered by the analysis of Leibnitz, which are merely considered from another point of view. Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uni- form motions of continually nearer approximation, etc., etc. Such is, in substance, Newton's conception ; or rather, that which Mac- laurin and D'Alembert have offered as the most rational basis of Lagrange's method. 75 the transcendental analysis, in the endeavor to fix and arrange Newton's ideas on the subject. Newton had another view, however, which ought to Fluxiond and be presented here, because it is still the special form of flulirs - the calculus of indirect functions commonly adopted by English geometers ; and also, on account of its ingenious clearness in some cases, and of its having furnished the notation best adapted to this manner of regarding the transcendental analysis. I mean the Cal- culus of fluxions and of fluents , founded on the general notion of velocities. To facilitate the conception of the fundamental idea, let us con- ceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the curve, the abscissa, the ordinate, the arc, the area, etc., will be regarded as simultaneously produced by successive de- grees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluent. Henceforth, the transcendental analysis will according to this conception, consist in forming directly the equations between the fluxions of the proposed, quantities, to deduce from them afterward, by a special Calculus, the equations between the fluents themselves. What has just been stated respecting curves may evidently be transferred to any mag- nitudes whatever, regarded, by the help of a suitable image, as some being produced by the motion of others. This method is evi- dently the same with that of limits complicated with the foreign idea of motion. It is, in fact, only a way of representing, by a comparison derived from mechanics, the method of prime and ulti- mate ratios, which alone is reducible to a calculus. It therefore necessarily admits of the same general advantages in the various principal applications of the transcendental analysis, without its being requisite for us to offer special proofs of this. Lagrange's conception consists in its admirable sim- Lagrange . 3 plicity, in considering the transcendental analysis to be method. a great algebraic artifice, by which, to facilitate the establishment of equations, we must introduce, in the place of or with the primi- tive functions, their derived functions ; that is, according to the definition of Lagrange, the coefficient of the first term of the incre- ment of each function, arranged according to the ascending powers of the increment of its variable. The Calculus of indirect functions, properly so called, is destined here, as well as in the conceptions of Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equa- tions between the primitive magnitudes. The transcendental anal- ysis is then only a simple, but very considerable extension of ordi- nary analysis. It has long been a common practice with geometers to introduce, in analytical investigations, in the place of the magni- tudes in question, their different powers, or their logarithms, 01 their sines, etc., in order to simplify the equations, and even to ob 76 POSITIVE PHILOSOPHY. tain them more easily. Successive derivation is a general artifice of the same nature, only of greater extent, and commanding in con- sequence, much more important resources for this common object. But, though we may easily conceive, à priori, that the auxiliary use of these derivatives may facilitate the study of equations, it is not easy to explain why it must be so under this method of deriva- tion, rather than any other transformation. This is the weak side of Lagrange's great idea. We have not yet become able to lay hold of its precise advantages, in an abstract manner, and without recurrence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate considera- tion of each principal question : and this verification becomes labo- rious, in the treatment of a complex problem. Other theories have been proposed, such as Eider's Calculus of vanishing quantities: but they are merely modifications of the three just exhibited. We must next compare and estimate these methods; and in the first place observe their perfect and necessary conformity. identity of the Considering the three methods in regard to their des- •ree method* tinatioii, independently of preliminary ideas, it is clear that they all consist in the same general logical artifice ; that is, the introduction of a certain system of auxiliary magnitudes uni- formly correlative with those under investigation; the auxiliaries being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated by the help of a special calculus. It was this which determined me to define the transcendental analysis as the Calculus of indirect functions, in order to mark its true philo- sophical character, while excluding all discussion about the best manner of conceiving and applying it. Whatever may be the method employed, the general effect of this analysis is to bring every mathematical question more speedily into the domain of the calculus, and thus to lessen considerably the grand difficulty of the passage from the concrete to the abstract. We can not hope that the Calculus will ever lay hold of all questions of natural philoso- phy — geometrical, mechanical, thermological, etc. — from their birth. That would be a contradiction. In every problem there must be a certain preliminary operation before the calculus can be of any use, and one which could not by its nature be subjected to abstract and invariable rules : — it is that which has for its object the establish- ment of equations, which are the indispensable point of departure for all analytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcend- ental analysis, which lias thus hastened the moment at which gen- eral and abstract processes may be uniformly and exactly applied to the solution, by reducing the operation to finding the equations between auxiliary magnitudes, whence the Calculus leads to equa- tions directly relating to the proposed magnitudes, which had for- merly to be established directlv. Whether those indirect equation? VALUE OF THE THREE METHODS. 77 are differential equations, according to Leibnitz, or equations of limits, according to Newton, or derived equations, according to Lagrange, the general procedure is evidently always the same. The coincidence is not only in the result but in the process ; for the auxiliaries introduced are really identical, being only regarded from different points of view. The conceptions of Leibnitz and Newton consist in making known in any case two general necessary prop- erties of the derived function of Lagrange. The transcendental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted ; and the processes of the Calculus of indirect functions are necessarily identical in these different methods, which must therefore, under any application whatever, lead to rigorously uniform results. If we endeavor to estimate their comparative value, Their compara . we shall find in each of the three conceptions advan- tive van- tages and inconveniencies which are peculiar to it, and which prevent geometers from adhering to any one of them, as exclusive and final. The method of Leibnitz has eminently the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitudes. We owe to its use the high perfec- tion attained by all the general theories of geometry and mechan- ics. Whatever may be the speculative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed the analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his 'Analytical Mechanics.' Such a fact needs no comment. Yet we are obliged to admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical rela- tions. He himself declared the notion of infinitely small quanti- ties to be a false idea : and it is in fact impossible to conceive of them clearly, though we may sometimes fancy that we do. This false idea bears, to my mind, the characteristic impress of the met- aphysical age of its birth and tendencies of its originator. By the ingenious principle of the compensation of errors, we may, as we have already seen, explain the necessary exactness of the processes which compose the method ; but it is a radical inconvenience to be obliged to indicate, in Mathematics, two classes of reasonings so unlike, as that the one order are perfectly rigorous, while by the others we designedly commit errors which have to be afterward compensated. There is nothing very logical in this ; nor is any- thing obtained by pleading, as some do, that this method can be made to enter into that of limits, which is logically irreproachable. This is eluding the difficulty, and not resolving it ; and besides, the advantages of this method, its ease and rapidity, are almost entirely lost under such a transformation. Finally, the infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by creating a transcendental analysis founded 78 POSITIVE PHILOSOPHY. upon principles widely different from those which serve as a basis to ordinary analysis. This division of analysis into two systems, almost wholly independent, tends to prevent the formation of gen- eral analytical conceptions. To estimate the consequences duly, we must recur, in thought, to the state of the science before Lagrange had established a general and complete harmony be- tween these two great sections. Newton's conception is free from the logical objections imputable to that ol* Leibnitz. The notion of limits is in fact remarkable for its distinctness and precision. The équations are, in this case, regarded as exact from their origin ; and the general rules of rea- soning are as constantly observed as in ordinary analysis. But it is weak in resources, and embarrassing in operation, compared with the infinitesimal method. In its applications, the relative inferior- ity of this theory is very Strongly marked. It also separates the ordinary and transcendental analysis, though not so conspicuously as the theory of Leibnitz. As Lagrange remarked, the idea of limits, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent. This perfect unity of analysis, and a purely abstract character in tin 1 fundamental ideas, are found in the conception of Lagrange, and there alone. It is, therefore, the most philosophical of all. Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis to its proper character, — that of pre- senting a very extensive class of analytical transformations, which facilitate, in a remarkable degree, the expression of the conditions of the various problems. This exhibits the conception as a simple extension of ordinary analysis. It is a superior algebra. All the different parts of abstract mathematics, till then so incoherent, might be from that moment conceived of as forming a single system. This philosophical superiority marks it for adoption as the final theory of transcendental analysis; but it presents too many difficulties, in its application, in comparison with the others, to admit of its exclu- sive preference at present. Lagrange himself had great difficulty in rediscovering, by his own method, the principal results already obtained by the infinitesimal method, on general questions in geom- etry and mechanics ; and we may judge by that what obstacles would occur in treating in the same way questions really new and important. Though Lagrange, stimulated by difficulty, obtained results in some cases which other men would have despaired of, it is not the less true that his conception has thus far remained, as a whole, essentially unsuited to applications. The result of such a comparison of these three methods is the conviction that, in order to understand the transcendental analysis thoroughly, we should not only study it in its principles according to all these conceptions, but should accustom ourselves to employ them all (and especially the first and last) almost indifferently, in the solution of all important questions, whether of the calculus of indirect functions in itself, or of its applications. In all the DIFFERENTIAL AND INTEGRAL CALCULUS. 79 other departments of mathematical science, the consideration of different methods for a single class of questions may be useful, apart from the historical interest which it presents ; but it is not indis- pensable. Here, on the contrary, it is strictly indispensable. Without it there can be no philosophical judgment of this admirable creation of the human mind ; nor any success and facility in the use of this powerful instrument. THE DIFFERENTIAL AND INTEGRAL CALCULUS. Its Two Parts. The Calculus of Indirect Functions is necessarily divided into two parts ; or rather, it is composed of two distinct calculi, having the relation of converse action. By the one we seek the relations between the auxiliary magnitudes, by means of the relations between the corresponding primitive magnitudes ; by the other we seek, conversely, these direct equations by means of the indirect equations first established. This is the double object of the transcendental analysis. Different names have been given to the two systems, according to the point of view from which the entire analysis has been regarded. The infinitesimal method, properly so called, being most in use, almost all geometers employ the terms Differential Calculus and Integral Calculus established by Leibnitz. Newton, in accordance with his method, called the first the Calculus of Fluxions, and the second the Calculus of Fluents, terms which were till lately com- monly adopted in England. According to the theory of Lagrange, the one would be called the Calculus of Derived Functions, and the other the Calculus of Primitive Functions. I shall make use of the terms of Leibnitz, as the fittest for the formation of secon- dary expressions, though we must, as has been shown, employ all the conceptions concurrently, approaching as nearly as may be to that of Lagrange. The differential calculus is obviously the rational Thpir mutua] basis of the integral. We have seen that ten simple relations. functions constitute the elements of our analysis. We cannot know how to integrate directly any other differential expressions than those produced by the differentiation of those ten functions. The art of integration consists therefore in bringing all the other cases, as far as possible, to depend wholly on this small number of simple functions. It may not be apparent to all minds what can be the proper util- ity of the differential calculus, independently of this necessary con- nection with the integral calculus, which seems as if it must be in itself the only directly indispensable one ; in fact, the elimination of the infinitesimals or the derivatives, introduced as auxiliaries, being the final object of the calculus of indirect functions, it is nat- ural to think that the calculus which teaches us to deduce the equa- tions between the primitive magnitudes from those between the auxiliary magnitudes must meet all the general needs of the trans 80 POSITIVE PHILOSOPHY. cendental analysis, without our seeing at first what special and constant part the solution of the inverse question can have in such an analysis. A common answer is assigning to die differential cal- culus the office of forming the differential equations; but this is clearly an error ; for the primitive formation of differential equa- tions is not the business of any calculus, for it is, on the contrary, the point of departure of any calculus whatever. The very use of the differential calculus is enabling us to differentiate the various equations; and it cannot therefore be the process for establishing them. This common error arises from confounding the infinitesimal calculus with the infinitesimal t/irt/torf, which last facilitates the formation of equations, in every application of the transcendental analysis. The calculus is the indispensable complement of the method : but it is perfectly distinct from it. But again, we should much misconceive the peculiar importance of this first branch of the calculus of indirect functions if we saw in it only a preliminary process, designed merely to prepare an indispensable basis for the integral calculus. A few words will show that a primary direct and necessary office is always assigned to the differential calculus. In of anion fonning differential equations, we rarely restrict our- etwo. selves to introducing differentially only those magni- tudes whose relation- are Bought. It would often be impossible to establish equations without introducing other magnitudes whose relations arc, or are supposed to be, known. Now in such cases it i< accessary thai the differentials of these intermediaries should be eliminated before the (Minutions arc lit for integration. This elimi- nation belongs to the differential calculus ; for it must be done by determining, by means of the equations between the intermediary functions, the relations of their differentials; and this is merely a question of differentiation. This is the way in which the differential calculus not only prepares a basis for the integral, but makes it available in a multitude of cases which could not otherwise be treated. There are some questions, few, but highly ferïS caicni'ua important, which admit of the employment of the dif- ferential calculus alone. They are those in which the magnitudes sought enter directly, and not by their differentials, into the primitive differential equations, which then contain differ- entially only the various known functions employed, as we saw just now, as intermediaries. This calculus is here entirely sufficient for the elimination of the infinitesimals, without the question giving rise to any integration. There are also questions, few, but highly important, which are the converse of the last, requiring the em ployment of the integral calculus alone. In these, the te$S ° calculus* differential equations are found to be immediately ready aione. f or integration, because they contain, at their first for- mation, only the infinitesimals which relate to the functions sought, or to the really independent variables, without the introduction, differentially, of any intermediaries being required. If intermediary functions are introduced, they will, by the hypothesis, enter directly, THE DIFFERENTIAL CALCULUS. 81 and not ny their differentials ; and then, ordinary algebra will serve for their elimination, and to bring the question to depend on the integral calculus only. The differential calculus is, in such cases, not essential to the solution of the problem, which will depend en- tirely on the integral calculus. Thus, all questions to which the analysis is applicable are contained in three classes. The first class comprehends the problems which may be resolved by the differential calculus alone. The second, those which may be resolved by the integral calculus alone. These are only exceptional ; the third constituting the normal case ; that in which the differential and in- tegral calculus have each a distinct and necessary part in the solu- tion of problems. The Differential Calculus. The entire system of the differential calculus is simple and per- fect, while the integral calculus remains extremely imperfect. We have nothing to do here with the applications of The Differentia | either calculus, which are quite a different study from caieuius. that of the abstract principles of differentiation and integration. The consequence of the common practice of confounding these principles with their application, especially in geometry, is that it becomes difficult to conceive of either analysis or geometry. It is in the department of Concrete Mathematics that the application should be studied. The first division of the differential calculus is grounded on the condition whether the functions to be differentiated are explicit or implicit ; the one giving rise to the differentiation of formulas, and the other to the differentiation, of equa- wo portl ^ na * tions. This classification is rendered necessary by the imperfection of ordinary analysis ; for if we knew how to resolve all equations algebraically, it would be possible to render every implicit function explicit ; and, by differentiating it only in that state, the second part of the differential calculus would be immediately included in the first, without giving rise to any new difficulty. But the alge- braic resolution of equations is, as we know, still scarcely past its infancy, and unknown for the greater number of cases ; and we have to differentiate a function without knowing it, though it is de- terminate. Thus we have /two classes of questions, the differentia- tion of implicit functions being a distinct case from that of explicit functions, and much more complicated. We have to begin by the differentiation of formulas, and we may then refer to this first case the differentiation of equations, by certain analytical considerations which we are not concerned with here. There is another new in which the two general cases of differentiation are distinct. The relation obtained between the differentials is always more indirect, in comparison with that of the finite quantities, in the differentia- tion of implicit, than in that of explicit functions. We shall meet with this consideration in the case of the integral calculus, where it acquires a preponderant importance. 6 82 POSITIVE PHILOSOPHY. Each of these parts of the differential calculus is again divided : and this subdivision exhibits two very distinct theories, according as we have to differentiate functions of a single variable, or functions of several independent variables, — the second branch being of far greater complexity than the first, in the case of explicit functions, and much more in that of implicit. One more distinction remains, to complete this brief sketch of the parts of the differential calculus. The case in which it is required to differentiate at once different implicit functions combined in certain primitive equations must be distinguished from that in which all these functions are separate. The same imperfection of ordinary analysis which prevents our converting every implicit function into an equivalent explicit one, renders us unable to separate the func- tions which enter simultaneously into any system of equations ; and the functions are evidently still more implicit in the case of com- bined than of separate functions : and in differentiating, we are not only unable to resolve the primitive equations, but even to effect the proper elimination among them. i; t un t., dw We have now seen the different parts of this calculus in their natural connection and rational distribution. The whole calculus is finally found to rest upon the differentiation of explicit functions with a single variable, — the only one which is ever executed directly. Now, it is easy to understand that this first theory, this necessary basis of the whole system, simply con- sists of the differentiation of the elementary functions, ten in num- ber, which compose all our analytical combinations; for the differ- entiation of compound functions is evidently deduced, immediately and necessarily, from that of their constituent simple functions. We lind, then, the whole system of differentiation reduced to the knowledge of the ten fundamental differentials, and to that of the two general principles, by one of which the differentiation of im- plicit functions is deduced from that of explicit, and by the other, the differentiation of functions of several variables is reduced to that of functions of a single variable. Such is the simplicity and perfection of the system of the differential calculus. The transformations of derived Functions for new Trans rn a ion of r t Oil- L rial les. 8 r vë'i a f Inc- variables is a theory which must be just mentioned, to ,r new va- avo iol t| ie omission of an indispensable complement of the system of differentiation. It is as finished and perfect as the other parts of this calculus ; and its great import- ance is in its increasing our resources by permitting us to choose, to facilitate the formation of differential equations, that system of independent variables which may appear to be most advantageous, though it may afterward be relinquished, as an intermediate step, by which, through this theory, we may pass to the final system, which sometimes could not have been considered directly. Analytical appH- Though we can not here consider the concrete appli- cations, cations of this calculus, we must glance at those which are analytical, because they are of the same nature with the theory, THE INTEGRAL CALCULUS. 83 ana should be looked at in connection with it. These questions are reducible to three essential ones. First, the development into series of functions of one or more variables ; or, more generally, the transformation of functions, which constitutes the most beauti- ful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental series discovered by Taylor, the remarkable series discovered by Maclaurin, John Bernouilli, Lagrange and others. Secondly, the general theory of maxima and minima values for any functions whatever of one or more variables ; one of the most interesting problems that analysis can present, however elementary it has be- come. The third is the least important of the three : — it is the determination of the true value of functions which present them- selves under an indeterminate appearance, for certain hypotheses made on the values of the corresponding variables. In every view, the first question is the most eminent ; it is also the most suscepti- ble of future extension, especially by conceiving, in a larger man- ner than hitherto, of the employment of the differential calculus for the transformation of functions, about which Lagrange left some valuable suggestions which have been neither generalized nor fol- lowed up. It is with regret that I confine myself to the generalities which are the proper subjects of this work ; so extensive and so interest- ing are the developments which might otherwise be offered. In- sufficient and summary as are the views of the Differential Calculus just offered, we must be no less rapid in our survey of the Integral Calculus, properly so called ; that is, the abstract subject of integ- ration. Tlie Integral Calculus, The division of the Integral Calculus, like that of the Thp [l]tegral Cal . Differential, proceeds on the principle of distinguishing culus - the integration of explicit differential formulas from the integration of implicit differentials, or of differential equations. The separation of these two cases is even more radical in the case of integration than in the other. In the differential calculus this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is clear that even if all equations could be algebraically resolved, differential equations would nevertheless constitute a case of inte- gration altogether distinct from that presented by explicit differen- tial formulas. Their integration is necessarily more complicated than that of explicit differentials, by the elaboration of which the integral calculus was originated, and on which the others have been made to depend, as far as possible. All the various analytical processes hitherto proposed for the integration of differential equa- tions, whether by the separation of variables, or the method of multipliers, or other means, have been designed to reduce these integrations to those of differential formulas, the only object which M POSITIVE PHILOSOPHY. can be directly undertaken. Unhappily, imperfect as is this neces- sary basis of the whole integral calculus, the art of reducing to it the integration of differential equations is even much less advanced. . A6 in the case of the differential calculus, and for analogous reasons, each of these two branches of the integral calculus is divided again, according as we consider func- one variable, or tions with a single variable or functions with several ■everai. independent variables. This distinction is, like the preceding, even more important for integration than for differenti- ation. This is especially remarkable with respect to differential equations. In feet, those which relate to several independent variables may evidently present this characteristic and higher diffi- culty — that the function sought may be differentially denned by a simple relation between its varions special derivatives with regard to the different variables taken separately. Thence results the most difficult, and also the most extended branch of the integral calculus, which is commonly called the Integral Calculus of partial differences, created by D'Alembert, in which, as Lagrange truly perceived, geometers should have recognised a new calculus, the philosophical character of which has not yet been precisely decided. This higher branch of transcendental analysis is still entirely in its infancy. In the very simplest case, we can not completely reduce the integration to that of the ordinary differential equations. Orders of dis* ^ new distinction, highly important here, though not •oiiat-oa \ n the differential calculus, when; it is a mistake to insist upon it, is drawn from the higher or lower order of the differ- entials. We may regard this distinction as a subdivision in the integration of explicit or implicit differentials. With regard to explicit differentials, whether of one variable or of several, the necessity of distinguishing their different orders is occasioned merely by the extreme imperfection of the integral calculus; and. with reference to implicit differentials, the distinction of orders is more important still. In the first case, we know so little of inte- gration of even the first order of differential formulas, that differ- ential formulas of a high order produce new difficulties in arriving at the primitive function which is our object. And in the second case, there is the additional difficulty that the higher order of the differential equations necessarily gives rise to questions of a new kind. The higher the order of differential equations, the more implicit are the cases which they present ; and they can be made to depend on each other only by special methods, the investigation of which, in consequence, forms a new class of questions with regard to the simplest cases of which we as yet know next to nothing. The necessary basis of all other integrations is, as we see from the foregoing considerations, that of explicit differential formulas of the first order and of a single variable ; and we can not succeed in effecting other integrations but by reducing them to this ele- mentary case, which is the only one capable of being treated QUADRATURES. 85 directly. This simple fundamental integration, often conveniently called quadratures, corresponds in the differential calculus to the elementary case of the differentiation of explicit functions of a single variable. But the integral question is, by its nature, quite otherwise complicated, and much more ex- tensive than the differential question. We have seen that the lat- ter is reduced to the differentiation of ten simple functions, which furnish the elements of analysis ; but the integration of compound functions does not necessarily follow from that of the simple func- tions, each combination of which may present special difficulties with respect to the integral calculus. Hence the indefinite extent and varied complication of the question of quadratures, of which we know scarcely anything completely, after all the efforts of analysts. The question is divided into the two cases of alge- A] „ e i,raié func- braic functions and transcendental functions. The t[o ° n *- algebraic class is the more advanced of the two. In relation to irrational functions, it is true, we know scarcely anything, the integrals of them having been obtained only in very restricted cases, and particularly by rendering them rational. The integra- tion of rational functions is thus far the only theory of this calculus which has admitted of complete treatment ; and thus it forms, in a logical point of view, its most satisfactory part, though it is per- haps the least important. Even here, the imperfection of ordinary analysis usually comes in to stop the working of the theory, by which the integration finally depends on the algebraic solution of equations ; and thus it is only in what concerns integration viewed in an abstract manner that even this limited case is resolved. And this gives us an idea of the extreme imperfection of the integral calculus. The case of the integration of transcenden- Tran=ceT1(1 ..„,.,, tal functions is quite in its infancy as yet, as regards fonction*, either exponential, logarithmic, or circular functions. Very few cases of these kinds have been treated ; and though the simplest have been chosen, the necessary calculations are extremely labo- rious. The theory of Singular Solutions (sometimes called Hn „ ll]ar Sllu . Particular Solutions), fully developed by Lagrange in tlul ^ his Calculus of Functions, but not yet duly appreciated by geom- eters, must be noticed here, on account of its logical perfection and the extent of its applications. This theory forms implicitly a por- tion of the general theory of the integration of differential equa- tions ; but I have left it till now, because it is, as it were, outside of the integral calculus, and I wished to preserve the sequence of its parts. Clairaut first observed the existence of these solutions, and he saw in them a paradox of the integral calculus, since they have the property of satisfying the differential equations without being comprehended in the corresponding general integrals. La- grange explained this paradox by showing how such solutions are always derived from the general integral by the variation of the 86 POSITIVE PHILOSOPHY. arbitrary constants. This theory has a character of perfect gene- rality ; for Lagrange has given invariable and very simple pro- cesses for finding the singular solution of any differential equation which admits of it ; and, what is very remarkable, these processes require no integration, consisting only of differentiations, and being therefore always applicable. Thus lias differentiation become, by a happy artifice, a means of compensating, in certain circumstances, for the imperfection of the integral calculus. d finite int.- ® ne niore thorny remains to be noticed, to complete B rMla our review of that collection of analytical researches which constitutes the integral calculus. It takes its place outside of the system, because, instead of being destined for true intégra- tion, it proposes to supply the defect of our ignorance of really analytical integrals. I refer to the determination of definite inte- grals. These definite integrals are the values of the required func- tions for certain determinate values of the corresponding variables. The use of these in transcendental analysis corresponds to the numerical resolution of equations in ordinary analysis. Analysts being usually unable to obtain the real integral (called in oppo- sition the general or indefinite integral), that is, the function which, differentiated, has produced the proposed differential formula, have been driven to determining, at least, without knowing this function, the particular numerical values which it would take on assigning certain declared values to the variables. This is evidently resolv- ing the arithmetical question without having first resolved the cor- responding algebraic one. which is generally the most important; and such an analysis is. by its nature, as imperfect as that of the numerical resolution of équations. Inconveniences, logical and practical, result from such a confusion of arithmetical and algebraic considerations. But, under our inability to obtain the true inte- grals, it is of the utmost importance to have been able to obtain this solution, incomplete and insufficient as it is. This has now been attained for all cases, the determination of the value of definite integrals having been reduced to entirely general methods, which leave nothing to be desired, in many cases, but less complexity in the calculations ; an object to which analysts are now directing all their special transformations. This kind of transcendental arith- metic being considered perfect, the difficulty in its applications is reduced to making the proposed inquiry finally depend only on a simple determination of definite integrals ; a thing which evidently can not be always possible, whatever analytical skill may be em- ployed in effecting so forced a transformation. We have now seen that while the differential calculus intejra?" calcu. constitutes by its nature a limited and perfect system, tas the integral calculus, or the simple subject of integra- tion, offers inexhaustible scope for the activity of the human mind, independently of the indefinite applications of which transcendental analysis is evidently capable. The reasons which convince us of the impossibility of ever achieving the general resolution of alge- CALCULUS OF VARIATIONS. 87 feraic equations of any degree whatever, are yet more decisive against our attainment of a single method of integration applicable to all cases. " It is," said Lagrange, " one of those problems whose general solution we can not hope for." The more we meditate on the subject, the more convinced we shall be that such a research is wholly chimerical, as transcending the scope of our understanding, though the labors of geometers must certainly add in time to our knowledge of integration, and create procedures of a wider gener- ality. The transcendental analysis is yet too near its origin, it has too recently been regarded in a truly rational manner, for us to have any idea what it may hereafter become. But, whatever may be our legitimate hopes, we must ever, in the first place, consider the limits imposed by our intellectual constitution, which are not the less real because we can not precisely assign them. I have hinted that a future augmentation of our resources may probably arise from a change in the mode of derivation of the auxiliary quantities introduced to facilitate the establishment of equations. Their formation might follow a multitude of other laws besides the very simple relation which has been selected. I discern here far greater resources than in urging further our present cal- culus of indirect functions ; and I am persuaded that when geom- eters have exhausted the most important applications of our present transcendental analysis, they will turn their attention in this direc- tion, instead of straining after perfection where it can not be found. I submit this view to geometers whose meditations are fixed on the general philosophy of analysis. As for the rest, though I was bound to, exhibit in my summary exposition the state of extreme imperfection in which the integral calculus still remains, it would be entertaining a false idea of the general resources of the transcendental analysis to attach too much importance to this consideration. As in ordinary analysis, we find here that a very small amount of fundamental knowledge respecting the resolution of equations is of inestimable use. However little advanced geometers are as yet in the science of integrations, they have nevertheless derived from their few abstract notions the solu- tion of a multitude of questions of the highest importance in geom- etry, mechanics, thermology, etc. The philosophical explanation of this double general fact is found in the preponderating import- ance and scope of abstract science, the smallest portion of which naturally corresponds to a multitude of concrete researches, Mar» having no other resource for the successive extension of his intel lectual means than in the contemplation of ideas more and more abstract, and nevertheless positive. Calculus of Variations. By his Calculus or Method of Variations, Lagrange improved the capacity of the transcendental analysis for the establishment of equations in the most difficult problems, by considering a class of -equations still more indirect than differential equations properly so 88 POSITIVE PHILOSOPHY. called. It is si ill too near its origin, and its applications have been too few, to admit of its being understood by a purely abstract ac- count of its theory; and it is therefore necessary to indicate briefly the special nature of the problems which have given rise to this hyper-transcendental analysis. These problems are those which were long known 'hs V!!i~ by the name of Isoperimetrical Problems; a name which is truly applicable to only a very small number of them. They consist in the investigation of the maxima and minima of certain indeterminate integral formulas which express the analytical law of such or such a geometrical or mechanical phenomenon, considered independently of any particular subject. In the ordinary theory of maxima and minima, we seek, with regard to a given function of one or more variables, what particular values must be assigned to these variables, in order thai the cor- responding value of the proposed function may be a maximum or a minimum with respect to those values which immediately precede and follow it : — that is. we impure, properly speaking, at what instant the function ceases to increase in order to begin to decrease, or the reverse. The differential calculus fully suffices, as we know, for the general resolution of this class of questions, by showing that the values of the different variables which suit either the max- imum or minimum must always render null the different deriv- atives Of the first order of the given function, taken separately with relation to each independent variable ; and by indicating more- oxer a character suitable lor distinguishing the maximum from the minimum, which consists, in the case of a function of a single variable, for example, in the derived function of tin.' second order taking ;i negative value for the maximum and a positive for the minimum. Such are the fundamental conditions belonging to the majority of cases : and where, modifications take place, they are equally subject to invariable, though more complicated abstract rules. The construction of this general theory having destroyed the chief interest of geometers in this kind of questions, they rose al- most immediately to the consideration of a new order of problems, at once more important and more difficult, — those of isoperimeters. It was then no longer the values of the variables proper to the max- imum or the minimum of a given function that had to be deter- mined. It was the form of the function itself that had to be dis- covered, according to the condition of the maximum or minimum of a certain definite integral, merely indicated, which depended on that function. We can not here follow the history of these problems, the oldest of which is that of the solid of least resistance, treated by Newton in the second book of the " Principia," in which he- determines what must be the meridian curve of a solid of revolu- tion, in order that the resistance experienced by that body in the* direction of its axis may be the least possible. Mechanics first furnished this new class of problems ; but it was from geometry SOME OP ITS APPLICATIONS. 89 that the subjects of the principal investigations were afterward derived. They were varied and complicated almost infinitely by the labors of the best geometers, when Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has checked the eagerness of geometers about such at order of researches. It is evident that these problems, considered analytically, consist in determining what ought to be the form of a certain unknown function of one or more variables, in order that such or such an integral, dependent on that function, may have, within assigned limits, a value which may be a maximum or a minimum, with regard to all those which it would take if the required function had any other form whatever. In treating these problems, the predecessors of Lagrange proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But they proceeded by applying special simple artifices to each case, not reducible to certain rules T so that every new question reproduced analogous difficulties, with- out the solutions previously obtained being of any essential aid. The part common to all questions of this class had not been dis- covered ; and no abstract and general treatment was therefore provided. In his endeavors to bring all isoperimetrical problems to depend on a common analysis, Lagrange was led to the concep- tion of a new kind of differentiation ; and to these new Differentials he gave the name of Variations. They consist of the infinitely small increments which the integrals receive, not in virtue of anal- ogous increments on the part of the corresponding variables, as in the common transcendental analysis, but by supposing that the form of the function placed under the sign of integration undergoes an infinitely small change. This abstract conception once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems of isoperimeters to the simple common theory of maxima and minima. Important as is this great and happy transformation, 0her npplica . and though the Method of Variations had at first no ,ioi!H other object than the rational and general resolution of isoperi- metrical problems, we should form a very inadequate estimate of this beautiful analysis if we supposed it restricted to this applica- tion. In fact, the abstract conception of two distinct natures of differentiation is evidently applicable, not only to the cases for which it was created, but for all which present, for any reason whatever, two different ways of making the same magnitudes vary. Lagrange himself made an immense and all-important application of his Calculus of Variations, in his ' Analytical Mechanics,' by employing it to distinguish the two sorts of changes, naturally pre- sented by questions of rational Mechanics for the different point» we have to consider, according as we compare the successive posi- tions occupied, in virtue of its motion, by the same point of each body in two consecutive instants, or as we pass from one point of the body to another in the same instant. One of these compari- 90 POSITIVE PHILOSOPHY. sons produces the common differentials ; the other occasions varia- tions which are. there as elsewhere, only differentials taken from a new point of view. It is in such a general acceptation as this that we must conceive of the Calculas of Variations, to appreciate fitly the importance of this admirable Logical instrument; the most pow- erful as yet constructed by the human mind. Tins Method being only an immense extension of the general transcendental analysis, there is no need of proof that it admits of being considered under the different primary points of view allowed by the calculus of indirect functions, as a whole. Lagrange in- vented the calculus of variations in accordance with the infinitesi- mal conception, properly so called, and even some time before he undertook the general reconstruction of the transcendental analysis. When he had effected that important reform, he easily showed how applicable it was to the calculas of variations, which he exhibited with all suitable development, according to his theory of derived functions. Hut the more difficult in the use the method of varia- tions is found to be, on account of tic higher degree of abstraction of the ideas considered, the more important it is to husband the powers of our minds in its application, by adopting the most direct and rapid analytical conception, which is. as we know, that of Leibnitz. Lagrange himself, therefore, constantly preferred it in the important use which he made of the calculus of variations in his ''Analytical Mechanics." There is not, in fact, the slightest hesitation about this among geometers. . .. In the section on the Integral Calcul us, I noticed ordinary Calcu- J) Alenihert 8 creation 01 the Calculus of partial differ- ences^ in which Lagrange recognised a Dew calculus. This new elementary idea in transcendental analysis. — the notion of two kinds of increments, distinct and independent of each other, which a function of two variables may receive in virtue of the change of each variable separately, — seems to me to establish a natural and necessary transition between the common infinitesimal calculus and the calculus of variations. D'Alembert's view appears to me to approximate, by its nature, very nearly to that which serves -as a gênerai basis for the Method of Variations. This last has, in fact, done nothing more than transfer to the independent variables themselves the view already adopted for the functions of those variables ; a process which has remarkably extended its use. A recognition of such a derivation as this for the method of variations maj exhibit its philosophical character more clearly and simply ; and this is my reason for the reference. The Method of Variations presents itself to us as the highest degree of perfection which the analysis of indirect functions has jet attained. We had before, in that analysis, a powerful instru- ment for the mathematical study of natural phenomena, inasmuch as it introduced the consideration of auxiliary magnitudes, so ■chosen as that their relations were necessarily more simple and easy to obtain than those of the direct magnitudes. But we had END OP ABSTRACT MATHEMATICS. 91 not any general and abstract rules for the formation of these dif- ferential equations ; nor were such supposed to be possible. Now, the Analysis of Variations brings the actual establishment of the differential equations within the reach of the Calculus ; for such is the general effect, in a great number of important and difficult questions, of the varied equations, which, still more indirect than the simple differential equations, as regards the special objects of the inquiry, are more easy to form : and, by invariable and com- plete analytical methods, employed to eliminate the new order of auxiliary infinitesimals introduced, we may deduce those ordinary differential equations which we might not have been able to estab- lish directly. The Method of Variations forms, then, the most sublime part of that vast system of mathematical analysis, which, setting out from the simplest elements of algebra, organizes, by an uninterrupted succession of ideas, general methods more and more potent for the investigation of natural philosophy. This is incom- parably the noblest and most unquestionable testimony to the scope of the human intellect. If, at the same time, we bear in mind that the employment of this method exacts the highest known degree of intellectual exertion, in order never to lose sight of the precise object of the investigation in following reasonings which offer to the mind such uncertain resting-places, and in which signs are of scarcely any assistance, we shall understand how it may be that so little use has been made of such a conception by any philos- ophers but Lagrange. We have now reviewed Mathematical analysis, in its bases and in its divisions, very briefly, but from a philosophical point of view, neglecting those conceptions only which are not organized with the great whole, or which, if urged to their limit, would be found to merge in some which have been examined. I must next offer a similar outline of Concrete Mathematics. My particular task will be to show how, supposing the general science of the Calculus to be in a perfect state, — it has been possible to reduce, by invariable procedures, to pure questions of analysis, all the problems of Geometry and Mechanics ; and thus to invest these two great bases of natural philosophy with that precision and unity which can only thus be attained, and which constitute high perfection 92 POSITIVE PHILOSOPHY. CHAPTER III. GENERAL VIEW OF GEOMETRY. We have seen that Geometry is a true natural sci- ence ; — only more simple, and therefore more perfect, than any other. We must not suppose that, because it admits the application of mathematical analysis, it is therefore a purely logi- cal science, independent of observation. Every body studied by geometers presents some primitive phenomena which, not being discoverable by reasonim:. must be due to observation alone. The scientific eminence of Geometry arises from the extreme generality and simplicity of its phenomena. If all the parts of the universe were regarded as immovable, geometry would still exist ; whereas, for the phenomena of Mechanics, motion is required. Thus Geometry is the more general of the two. It is also the more simple, for its phenomena are independent of those of Mechanics, while mechanical phenomena are always complicated with those of geometry. The same is true in the comparison of abstract ther- mology with geometry. For these reasons, geometry holds the first place under the head of Concrete Mathematics. Instead of adopting the inadequate ordinary account of Geometry, that it is the science of extension, I am disposed to give as a general description of it, that it is the science of the measurement of extension. Even this does not include all the operations of geometry, for there are many investigations which do not appear to have for their object the measurement of exten- sion. But regarding the science in its leading questions as a whole, we may accurately say that the measurement of lines, of surfaces, and of volumes, is the invariable aim — sometimes direct, though oftener indirect — of geometrical labors. The rational study of geometry could never have begun if we must have regarded at once and together all the physical properties of bodies, together with their magnitude and form. By the character of our minds we are able to think of the dimensions and figure of a body in an abstract way. After observation has shown us, for instance, the impression left by a body on a fluid in which it has been placed, we are able to retain an image of the impression, which becomes a ground of geometrical reasoning. We thus obtain, apart from all metaphysical fancies, an idea of Space. This abstraction, now so familiar to us that we can not perceive the state we should be in without it, is perhaps the earliest philosophical creation of the human mind. GEOMETRICAL MEASUREMENT. 93 There is another abstraction which must be made be- Kind3 cf extell fore we can enter on geometrical science. We must 8ion - conceive of three kinds of extension, and learn to conceive of them separately. We can not conceive of any space, filled by any object, which has not at once volume, surface, and line. Yet geo- metrical questions often relate to only two of these ; frequently only to one. Even when all three are to be finally considered, it is often necessary, in order to avoid complication, to take only one at a time. This is the second abstraction which it is indispensable for us to practise — to think of surface and line apart from volume ; and again, of line apart from surface. We effect this by thinking of volume as becoming thinner and thinner, till surface appears as the thinnest possible layer or film : and again, we think of this sur- face becoming narrower and narrower till it is reduced to the finest imaginable thread ; and then we have the idea of a line. Though we can not speak of a point as a dimension, we must have the ab- stract idea of that too ; and it is obtained by reducing + he line from one end or both, till the smallest conceivable portion of it is left. This point indicates, not extension of course, but position, or the place of extension. Surfaces have clearly the property of circum- scribing volumes ; lines, again, circumscribe surfaces ; and lines, once more, are limited by points. The Mathematical meaning of measurement is sim- Geometrical ply the finding the value of the ratios between any ho- me ^ ur, ' melit - mogeneous magnitudes : but geometrically, the measurement is always indirect. The comparison of two lines is direct ; that of two surfaces or two volumes can never be direct. One line may be conceived to be laid upon another : but one volume can not be con • ceived of as laid upon another, nor one surface upon another, with any convenience or exactness. The question is, then, how to meas- ure surfaces and volumes. Whatever be the form of a body, there must always be lines, the length of which will define the magnitude surfaces and vol- of the surface or volume. It is the business of geome- umes try to use these lines, directly measurable as they are, for the ascer- tainment of the ratio of the surface to the unity of surface, or of the volume to the unity of volume, as either may be sought. In brief, the object is to reduce all comparisons of surfaces or of volumes to simple comparison of lines. Extending the process, we find the possibility of reducing to questions of lines all questions relating to surfaces and volumes, regarded in relation to their magnitude. It is true that when the rational method becomes too complicated and difficult, direct comparisons of surfaces and volumes are employed : but the procedure is not geometrical. In the same way, the con- sideration of weight is sometimes brought in, to determine volume, or even surface ; but this device is derived from mechanics, and has nothing to do with rational geometry. In speaking of the direct measurement of lines, it is clear that right lines are meant. When we consider 94 POSITIVE PHILOSOPHY. curve lines, it is evident that their measurement must be indirect, since we can not conceive of curved lines being laid upon each other with any precision or certainty. The procedure is first to reduce the measurement of curved to that of right lines ; and consequently to reduce to simple questions of right lines all questions relating to the magnitude of any curves whatever. In every curve, there al- ways exist certain right lines, the length of which must determine that of the curve; as the length of the radius of a circle gives ns that of the circumference; and again, as the length of an ellipse dépends on that of its two axes. Thus, the Bcience of Geometry has for its object the final reduc- tion of the comparisons of all kinds of extent to comparisons of right lines, wh'u-h alone are capable of direct comparison, and are, moreover, eminently easy to manage. 1 must just notice that there is a primary distinct branch of Ge- ometry, exclusively devoted to the right line, on account of occa- sional insurmountable difficulties in making the direct compari- son ; its object is to determine certain right lines from others by means of the relations proper to the ligures resulting from their as- semblage. The importance of this is clear, as no question could be solved it" the measurement of right lines, on which every other de- pends, were Left, in any case, uncertain. The natural order of the parts of rational geometry IS therefore, first the geometry of line, beginning with the right line; then the geometry of surfaces; and, finally, that of volumes. i im1i1i . The field of geometrical science is absolutely un- ri ' bounded. There may he as many questions as there are conceivable ligures: and the variety of conceivable figures is infinite. As to curved Lines, if we regard them as generated by the motion of a point governed by a certain law, we can not limit their number, as the variety of distinct conditions is nothing short of infinité ; each generating new ones, and those again others. Sur- faces again, are conceived of as motions of lines ; and they not only partake of the variety of lines, but have another of their own, arising from the possible change of nature in the line. There can I-e nothing like this in lines, as points can not describe a figure. Thus, there is a double set of conditions under which the figures of surfaces may vary : and we may say that if lines have one infinity of possible change, surfaces have two. As for volumes, they are distinguished from one another only by the surfaces which bound them ; so that they partake of the variety of surfaces, and need no special consideration under this head. If we add the one fur- ther remark, that surfaces themselves furnish a new means of con- ceiving of new curves, as every curve may be regarded as produced by the intersection of two surfaces, we shall perceive that, starting from a narrow ground of observation, we can obtain an absolutely infinite variety of forms, and therefore an illimitable field for geo- metrical science. The connection between abs + ract and concrete geometry is estab- ANCIENT AND MODERN GEOMETRY. 95 lished by the study of the properties of lines and sur- pro ^ . faces. Without multiplying in this way our means of iin°sand%r. recognition, we should not know, except by accident, ace8 " how to find in nature the figure we desire to verify. Astronomy was recreated by Kepler's discovery that the ellipse was the curve which the planets describe about the sun, and the satellites about their planet. This discovery could never have been made if geome- ters had known no more of the ellipse than as the oblique section of a circular cone by a plane. All the properties of the conic sec- tions brought out by the speculative labors of the Greek geometers, were needed as preparation for this discovery, that Kepler might select from them the characteristic which was the true key to the planetary orbit. In the same way, the spherical figure of the earth could not have been discovered if the primitive character of the sphere had been the only one known ; — viz., the equidistance of all its points from an interior point. Certain properties of surfaces were the means used for connecting the abstract reasoning with the concrete fact. And others, again, were required to prove that the earth is not absolutely spherical, and how much otherwise. The pursuit of these labors does not interfere with the definition of Ge- ometry given above, as they tend indirectly to the measurement of extension. The great body of geometrical researches relates to the properties of lines and surfaces ; and the study of the properties of the same figure is so extensive, that the labors of geometers for twenty centuries have not exhausted the study of conic sections. Since the time of Descartes, it has become less important ; but it appears as far as ever from being finished. And here opens another infinity. Wa had before the infinite scope of lines, and the double infinity of surfaces : and now we see that not only is the variety of figures inexhaustible, but also the diversity of the points of view from which each figure may be regarded. There are two general Methods of treating geometri- Two grneral cal questions. These are commonly called Synthetical Me&oiis. Geometry and Analytical Geometry. I shall prefer the historical titles of Geometry of the Ancients and Geometry of the Moderns. But it is, in my view, better still to call them Special Geometry and General Geometry, by which their nature is most accurately conveyed. The Calculus was not, as some suppose, unknown to s P ec ; ai or a», the ancients, as we perceive by their applications of the ^f ' or a n^E theory of proportions. The difference between them Geometry. and us is not so much in the instrument of deduction as in the na- ture of the questions considered. The ancients studied geometry with reference to the bodies under notice, or specially ; the mod- erns study it with reference to the phenomena to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another ; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They 96 POSITIVE PHILOSOPHY. abstract, to treat by itself, even' question relating to the same geo metrical phenomenon, in whatever bodies it may be considered Geometers can thus rise to the study of new geometrical concep- tions, which, applied to the curves investigated by the ancients, have brought out new properties never suspected by them. The superiority of the modern method is obvions at a glance. The time formerly spent, and the sagacity and effort employed, in the path of detail, are inconceivably economized by the general method used since the great revolution under Descartes. The benefit to Con- crete Geometry is no less than to the Abstract : for the recognition of geometrical figures in nature was merely embarrassed by the study of lines in detail ; and the application of the contemplated figure to the existing body could be only accidental, and within a limited or doubtful range: whereas, by the general method, no existing ligure can cscnpc application to its true theory, as soon as its geometrical features are ascertained. Still, the ancient method was natural ; and it was necessary that it should precede the mod- ern. The experience of the ancients, and the materials they ac- cumulated by their special method, were indispensable to suggest the conception of Descartes, and to furnish a basis for the general procedure. It is evident that the Calculus can not originate any science. Equations must exist as a starting-point for analytical operations. No other beginning can be made than the direct study of the object, pursued up to the point of the discovery of precise relations. (; , )(11 .. IV ot -,,„. We must briefly survey the geometry of the ancients, H " ••■ I|N in its character of an indispensable introduction to that of the moderns. The one, special and preliminary, must have its relation mad» 4 clear to the other, — the general and definitive geometry, which now constitutes the science that goes by that name. We have seen that Geometry is a science founded upon observa- tion, though the materials furnished by observation are few and simple, and the structure of reasoning erected upon them vast and complex. The only elementary materials, obtainable by direct study alone, are those which relate to the right line for the geom- etry of lines ; to the quadrature of rectilinear plane areas ; and to the cubature of bodies terminated by plane faces. The beginning of geometry must be from the observation of lines, of flat surfaces angularly bounded, and of bodies which have more or less bulk, also angularly bounded. These are all ; for all other figures, even the circle, and the figures belonging to it, now come under the head of analytical geometry. The three elements just mentioned allow a sufficiency of equations for the calculus to proceed upon. More are not needed ; and we can not do with less. Some have endeavored to extend analysis so as to dispense with a portion of these facts : but to do so is merely to return to metaphysical prac- tices, in presenting actual facts as logical abstractions. The more we perceive Geometry to be, in our day, essentially analytical, the DESCRIPTIVE GEOMETRY. 97 more careful we must be not to lose sight of the basis of observa- tion on which all geometrical science is founded. When we ob- serve people attempting to demonstrate axioms and the like, we may avow that it is better to admit more than may be quite neces- sary of materials derived from observation, than to carry logical demonstration into a region where direct observation will serve us better. There are two ways of studying the right line — the Geometry of the graphic and the algebraic. The thing to be done is to ri s ht line ascertain, by means of one another, the different elements of any right line whatever, so as to understand, indirectly, a right line, under any circumstances whatever. The way to do this Graphical 80lu . is, first, to study the figure, by constructing it, or other- nons - wise directly investigating it ; and then, to reason from that obser- vation. The ancients, in the early days of the science, made great use of the graphic method, even in the form of Construction ; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the rightangled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observation of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice ; but did not abolish it. The Greeks and Arabians employed it still for a great number of investigations for which we now consider the use of the Calculus indispensable. While the graphic or constructive method answers well when all the parts of the proposed figure lie in the same plane, it must receive additions before it can be applied to figures whose parts lie in different planes. Hence arises a new series of considerations, and different systems of Projections. Where we now employ sphe- rical trigonometry, especially for problems relating to the celestial sphere, the ancients had to consider how they could replace con- structions in relief by plane constructions. This was the object of their analFmmas, and of the other plane figures which long sup- plied the place of the Calculus. They were acquainted with the elements of what we call Descriptive Geometry, though they did not conceive of it in a distinct and general manner. Digressing here for a moment into the region of ap- Descriptive Go- plication, I may observe that Descriptive Geometry, emetry. formed into a distinct system by Monge, practically meets the dif- ficulty just stated, but does not warrant the expectations of its first admirers, that it would enlarge the domain of rational geometry. Its grand use is in its application to the industrial arts ; — its few abstract problems, capable of invariable solution, relating essen- tially to the contacts and intersections of surfaces ; so that all the geometrical questions which may arise in any of the various arts of construction, — as stone-cutting, carpentry, perspective, dialling 7 98 POSITIVE PHILOSOPHY. fortification, etc., — can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circumstances of each case. This creation must be very important in the eyes of philosophers who think that all human achievement, thus far, is only a first step toward a phil- osophical renovation of the labors of mankind ; toward that pre- cision and logical character which can alone insure the future pro- gression of all arts. Such a revolution must inevitably begin with that class of arts which bears a relation to the simplest, the most perfect, and the most ancient of the sciences. It must extend, in time, though less readily, to all other industrial operations. Mongc, who understood the philosophy of the arts better than any one else, himself indeed endeavored to sketch out a philosophical system of mechanical arts, and at least succeeded in pointing out the direction in which the object must be pursued. Of Descriptive Geometry, it may further be said that it usefully exercises the stu- dents' faculty of Imagination. — of conceiving of complicated ge- ometrical combinations in space; and that, while it belongs to the geometry of the ancients by the character of its solutions, it ap- proaches to the geometry of the moderns by the nature of the questions which compose it. Consisting, as we have said, of a few abstract problems, obtained through Projections, and relating to the contacts and intersections of surfaces, the invariable solutions of these problems are at once graphical, like those of the ancients, and general, like those of the moderns. Yet, as destined to an industrial application. Descriptive Geometry has here been treated of only in the way of digression. Leaving the subject of graphic solution, we have to notice the other branch, — the algebraic. Aicehrau- eoiu- Some may wonder that this branch is not treated as belonging to General Geometry. But not only were the ancients, in fact, the inventors of trigonometry, — spherical as well as rectilinear, — though it necessarily remained imperfect in their hands ; but algebraic solutions are also no part of analytical geometry, but only a complement of elementary geometry. Since all right-lined figures can be decomposed into triangles, all that we want is to be able to determine the different elements of a triangle by means of one another. This reduces polygonometry to simple trigonometry. The difficulty lies in forming three distinct equations between the angles and the sides of a triangle. These equations being obtained, all trigonometrical problems are reduced to mere questions of analysis. — There are two methods of introdu- cing the angles into the calculation. They are either introduced directly, by themselves or by the circular arcs which are propor- tional to them : or they are introduced indirectly, by the chords of these arcs, which are hence called their trigonometrical lines. The second of these methods was the first adopted, because the early state of knowledge admitted of its working, while it did not admit ihe establishment of equations between the sides of the triangles TRIGONOMETRY. 99 and the angles themselves, but only between the sides and the trig- onometrical lines. — The method which employs the trigonometrical lines is till preferred, as the more simple, the equations existing only between right lines, instead of between right lines and arcs of circles. To meet the probable objection that it is rather a complication than a simplification to introduce these lines, which have at least to be eliminated, we must explain a little. Their introduction divides trigonometry into two parts. In one, we pass from the angles to their trigonometrical lines, or the con verse : in the other we have to determine the sides of the triangles by the trigonometrical lines of their angles, or the converse. Now, the first process is done for us, once for all, by the formation of numerical tables, capable of use in all conceivable questions. It is only the second, which is by far the least laborious, that has to be undertaken in each individual case. The first is always done in advance. The process may be compared with the theory of loga- rithms, by which all imaginable arithmetical operations are decom- posed into two parts — the first and most difficult of which is done in advance. We must remember, too, in considering the position of the ancients, the remarkable fact that the determination of angles by their trigonometrical lines, and the converse, admits of an arith- metical solution, without the previous resolution of the correspond- ing algebraic question. But for this, the ancients could not have obtained trigonometry. When Archimedes was at work upon the rectification of the circle, tables of chords were prepared : from his labors resulted the determination of a certain series of chords : and, when Hipparchus afterward invented trigonometry, he had only to complete that operation by suitable intercalations. The connection of ideas is here easily recognised. For the same reasons which lead us to the employment of these lines, we must employ several at once, instead of confining our- selves to one, as the ancients did. The Arabians, and after them the moderns, attained to only four or five direct trigonometri- cal lines altogether ; whereas it is clear that the number is not limited. Instead, however, of plunging into deep complications, in obtaining new direct lines, we create indirect ones. Instead, for instance, of directly and necessarily determining the sine of an angle, we may determine the sine of its half, or of its double, — taking any line relating to an arc which is a very simple function of the first. Thus, we may say that the number of trigonometrical lines actually employed by modern geometers is unlimited through the augmentations we may obtain by analysis. Special names have, however, been given to those indirect lines only which refer to the complement of the primitive arc — others being in much less fre- quent use. ( hit of this device arises a third section of trigonometrical knowl- edge. Having introduced a new set of lines — of auxiliary magni 100 POSITIVE PHILOSOPHY. tudes — we have to determine their relation to the first. And this study, though preparatory, is indefinite in its scope, while the two other departments are strictly limited. The three must, of course, be studied in just the reverse order to that in which it has been necessary to exhibit them. First, the student must know the relations between the indirect and direct trigonometrical lines : and the resolution of triangles, properly so called, is the last process. Spherical trigonometry requires no special notice here (all-im- portant as it is by its uses) — since it is, in our day, simply an application of rectilinear trigonometry, through the substitution of the corresponding trihedral angle for the spherical triangle. This view of the philosophy of trigonometry has been given chiefly to show how the most simple questions of elementary geom- etry exhibit a close dependence and regular ramification. Thus have we seen what is the peculiar character of Special Ge- ometry, strictly considered. We see that it constitutes an indis- pensable basis to General Gh letry. Next, we have to study the philosophical character of the true science of Geometry, beginning with the great original idea of Descartes, on which it is wholly founded. Modern, or Analytical Geometry. General or Analytical Geometry is founded upon the transfor- mation of geometrical considérations into equivalent analytical considerations. Descartes established the constant possibility of doing this in a uniform manner : and his beautiful conception is interesting, not only from its carrying on geometrical science to a logical perfection, but from its showing us how to organize the relations of the abstract to the concrete in Mathematics by the analytical representation of natural phenomena. . ^ The first thing to be done is evidently to find and fix Bentadon of fig- a method for expressing analytically the subjects which afford the phenomena. If we can regard lines and surfaces analytically, we can so regard, henceforth, the accidents of these subjects. Here occurs the difficulty of reducing all geometrical ideas to those of number : of substituting considerations of quantity for all considerations of quality. In dealing with this difficulty, we must observe that all geometrical ideas come under three heads : — the magnitude, the figure, and the position of the extensions in question. The relation of the first, magnitude, to numbers is immediate and evident : and the other two are easily brought into one ; for the figure of a body is nothing else than the natural position of the points of which it is composed : and its po- sition can not be conceived of irrespective of its figure. We have therefore only to establish the one relation between ideas of position and ideas of magnitude. It is upon this that Descartes has estab- lished the system of General Geometry. CO-ORDINATES. 101 The method is simply a carrying out of an operation which is natural to all minds. If we wish to indicate the situation of an object which we can not point out, we say how it is related to objects that are known, by assigning the magnitude of the different geometrical elements which connect it with known objects. Those elements are what Descartes, and all other geometers after him, have called the co-ordinates of the point considered. If we know in what plane the point is situated, the co-ordinates are two. If the point may be anywhere in space, the co-ordinates can not be less than three. They may be multiplied without limit : but whether few or many, the ideas of position will have been re- duced to those of magnitude, so that we shall represent the dis- placement of a point as produced by pure numerical variations in the values of its co-ordinates. The simplest case of all, that of plane geometry, is when we determine the position of a point on a plane by considering its distances from two fixed right lines, sup- posed to be known, and generally concluded to be perpendicular to each other. These are called axes. Next, there may be the less simple process of determining the position by the distances from two fixed points ; and so on to greater and greater complications. But, from some system or other of co-ordinates being always em- ployed, the question of position is always reduced to that of mag- nitude. It is clear that our only way of marking the position Position of a of a point is by the intersection of two lines. When p«nt the point is determined by the intersection of two right lines, each parallel to a fixed axis, that is the system of rectilinear co-ordi- nates — the most common of all. The polar system of co-ordinates exhibits the point by the travelling of a right line round a fixed centre of a circle of variable radius. Again, two circles may inter- sect, or any other two lines : so that to assign the value of a co-ordinate is the same thing as to determine the line on which the point must be situated. The ancient geometers, of course, were like ourselves in this necessary method of conceiving of position : and their geometrical loci were founded upon it. It was in endeav- oring to form the process into a general system that Descartes created Analytical Geometry. Seeing, as we now do, how ideas of position — and, through them, all elementary geometrical ideas — can be reduced to ideas of number, we learn what it was that he effected. Descartes treated only geometry of two dimensions in his ana- lytical method : and we will at first consider only this Plane CurveB kind, beginning with Plane Curves. Lines must be expressed by equations : and again, equations must be expressed by lines, when the rela- Hnos'by Eqoa- tion of geometrical conceptions to numbers is estab- tlona ' lished. It comes to the same thing whether we define a line by any one of its properties, or supply the corresponding equation between the two variable co-ordinates of the point which describes 102 POSITIVE PHILOSOPHY. the line. If a point describes a certain line on a plane, we know that its co-ordinates bear a fixed relation to eacli other, which may be expressed by an appropriate equation. If the point describes no certain line, its co-ordinates must be two variables independent of each other. Its situation in the latter case can be determined only by giving at once its two co-ordinates, independently of each other : whereas, in the former ease, a single co-ordinate suffices to fix its position. The second co-ordinate is then a determinate function of the first: — that is, there e\i>ts between them a certain equation of a nature corresponding to that of the line on which the point is to lie found. The co-ordinates of the point eacfi require it ,o be <>n a certain line : and airain, its being on a certain line is the tame thing as assigning the ralue of one of the two co-ordinates; which is then found t<» be entirely dependent on the other. Thus are lines analytically expressed by equations. By a converse argument may be seen the geometrical ^quat:oM 0n by necessity of represent ing by a certain line every equa- ,ines tion of two variables, in a determinate system of co- ordinates. In the absence of any other known property, such a relation would he a very characteristic definition ; and its scientific effect would be to fix the attention immediately upon the general course of the solutions of the equation, which will thus be noted in the most Striking and simple manner. There is an evident and vast advantage in this picturing of equations, which reacts strongly upon the perfecting of analysis itself. The geometrical locus stands before our minds as the representation of all the details that have gone to its preparation, and thus renders comparatively easy our conception of new general analytical views. This method has become entirely elementary in our day: and it is employed when we want to net a clear idea iA' the general character of the law which runs through a series of particular observations of any kind whatever. Recurring to the representation of lines by equations, ime'lfhang"s tile which is our chief object, we see that this representa- equ mon ^ on j^ | )y -^ narure? so faithful, that the line could not undergo any modification, even the slightest, without causing a corresponding change in the equation. Some special difficulties arise out of this perfect exactness ; for since, in our system of analytical geometry, mere displacements of lines affect equations as much as real variations of magnitude or form, we might be in danger of confounding the one with the other, if geometers had not discovered an ingenious method expressly intended to distinguish them always. It must be observed that general inconveniences of this nature appear to be strictly inevitable in analytical geometry ; since, ideas of position being the only geometrical ideas immedi- ately reducible to numerical considerations, and conceptions of form nor being referrible to them but by seeing in them relations of sit- uation, it is impossible that analysis should not at first confound phenomena of form with simple phenomena of position ; which aro the only ones that equations express directly. GEOMETRY OF TWO DIMENSIONS. lOo To complete our description of the basis of anal- ytical geometry, it is necessary to point out that not o/^iufe^ïï only must every denned line give rise to a certain equa- p< î uatIon - tion between the two co-ordinates of any one of its points, but overy definition of a line is itself an equation of that line in a suit- able system of co-ordinates. Considering, first, what a definition is, we say it must distinguish the defined object from all others, by assigning to it a property which belongs to it alone. But this property may not disclose the mode of generation of the object, in which case the definition is merely characteristic; or it may express one of its modes of gen- eration, and in that case the definition is explanatory. For instance, if we say that the circle is the line which in the same form contains the largest area, we offer a characteristic definition ; whereas if we ohoose its property of having all its points equally distant from a fixed point, we have an explanatory definition. It is clear more- over that the characteristic definition always leaves room for an explanatory one, which further study must disclose. It is to explanatory definitions only that what has been said of the definition of a line being an equation of that line can apply. We can not define the generation of a line without specifying a certain relation between the two simple motions, of translation or of rotation, into which the motion of the point which describes it will be decomposed at each moment. Now, if we form the most general conception of what a system of co-ordinates is, and if we admit all possible systems, it is clear that such a relation can be nothing else than the equation of the proposed line, in a system of co-ordinates of a corresponding nature to that of the mode of gen- eration considered ; as in the case of the circle, the common defini- tion of which may be regarded as being the polar equation of that ourve, taking the centre of the circle for the pole. This view not only exhibits the necessary representation of every line by an equation, but it indicates the general difficulty which oc- curs in the establishment of these equations, and therefore shows us how to proceed in inquiries of this kind which, by their nature, do not admit of invariable rules. Since every explanatory defini- tion of a line constitutes the equation of that line, it is clear that when we find difficulty in discovering the equation of a curve by means of some of its characteristic properties, the difficulty must proceed from our taking up a designated system of co-ordinates, instead of admitting indifferently all possible systems. These .systems are not all equally suitable ; and, in regard to curves, geometers think that they should almost always be referred, as far as possible, to rectilinear co-ordinates. Now, these pafticular co- ordinates are often not those with reference to which the equation of the curve will be found to be established by the proposed defini- tion. It is in a certain transformation of co-ordinates then that the chief difficulty in the formation of the equation of aline really con- sists. The view I have given does not furnish us with a complete 104 POSITIVE PHILOSOPHY. and certain general method for the establishment of these equa- tions ; but it may cast a useful light on the course which it is best to pursue to attain the end proposed. choice of oo. The choice of co-ordinates — the preference of that ° ,lm r 8 system which may be most suitable to the case — is the remaining point which we have to notice. First, we must distinguish very carefully the two views, the con- verse of each other, which belong to analytical geometry, viz. the relation of algebra to geometry, founded on the representation of lines by equations, and, reciprocally, the relation of geometry to algebra, rounded ou the picturing of equations by lines. Though the two are necessarily combined in every investigation of general geometry, and we have to pass from tin 4 one to the other alternately, and almost insensibly, we musl be able to separate them here, for the answer to the question of method which we are considering is far from being the same under the two relations: so that without this distinction we could not form any clear idea of it. In the case <>f the representation of lines by equations, the first object is t«» choose those co-ordinates which afford the greatest simplicity in the equation of each line, and the greatest facility in ar- riving at it. Thera can lie no constant preference here of one sys- tem of co-ordinates. The rectilinear Bystem itself, though often ad- vantageous, can not be always so. and may he. in turn, Less so than any other. Bui ii is far otherwise in the converse case of the rep- utation of equations by line-. Here the rectilinear system is always to he preferred, a- tin 1 mOSl Minple and t rus1 worthy. If wc seek to determine a point by the intersection of two lines, it must be best that those lines should be the simplest possible; and this confines our choice to the rectilinear Bystem. In constructing geom- etrical loci, that system of co-ordinates must be the best in which it is easiest to conceive the change of place of a point resulting from the change in the value of its co-ordinates ; and this is the case with the rectilinear system. Again, there is great advantage in the common usage of taking the two axes perpendicular to each other, when possible, rather than with any other inclination. In representing lines by equations, we must take any inclination of the axes which may best suit the particular question ; but, in the con- verse case, it is easy to see that rectangular axes permit us to rep- resent equations in a more simple, and even in a more faithful manner. For if we extend the geometrical locus of the equation into the several unequal regions marked out by oblique axes, we shall have differences of figure which do not correspond to any analytical diversity ; and the accuracy of the representation will be lost. On the whole then, taking together the two points of view of an- alytical geometry, the ordinary system of rectilinear co-ordinates is superior to any other. Its high aptitude for the representation of equations must make it generally preferred, though a less perfect system may answer better in particular cases. The most essential GEOMETRY OF THREE DIMENSIONS. 105 theories in modern geometry are generally expressed by the recti- linear system. The polar system is preferred next to it, both be- cause its opposite character enables it to solve in the simplest way the equations which are too complicated for management under the first ; and because polar-co-ordinates have often the advantage of admitting of a more direct and natural concrete signification. This is the case in Mechanics, in the geometrical questions arising out of the theory of circular movement, and in almost all questions of celestial geometry. Such was the field of the labors of Descartes, his conception of analytical geometry being designed only for the study of Plane Curves. It was Clairaut who, about a century later, extended it to the study of Surfaces and Curves of double curvature. The con- ception having been explained, a very brief notice will suffice for the rest. With regard to Surfaces, the determination of a Determination of point in space requires that the values of three co- a p° int in *p Rce - ordinates should be assigned. The system generally adopted, which corresponds with the rectilinear system of plane geometry, is that of distances from the point to three fixed planes, usually perpendicular to each other, whereby the point is presented as the intersection of three planes whose direction is invariable. Beyond this, there is the same infinite variety among possible systems of co-ordinates, that there is in geometry of two dimensions. Instead of the intersection of two lines, it must be that of three surfaces which determines the point ; and each of the three surfaces has, in the same way, all its conditions constant, except one, which gives rise to the corresponding co-ordinates, whose peculiar geometrical effect is thus to compel the point to be situated upon that surface. Again, if the three co-ordinates of a point are mutually independent, that point can take successively all possible positions in space ; but, if its position on any surface is defined, two co-ordinates suf- fice for determining its situation at any moment ; as the proposed surface will take the place of the condition imposed by the third co-ordinate. This last co-ordinate then becomes a determinate function of the two others, they remaining independent of each other. Thus, there will be a certain equation between the three variable co-ordinates which will be permanent, and which will be the only one, in order to correspond to the precise degree of indé- termination in the position of the point. In the expression of Surfaces by Equations, and again DetPrm in n tion in the expression of Equations by Surfaces, the same of surfaces by ... T . ,-,•■ i j» i , /» Equations, and conception is pursued as in the analytical geometry of of Equations two dimensions. In the first case, the equation will by Surf,,ce8 - be the analytical definition of the proposed surface, since it must be verified for all the points of this surface, and for them only. If the surface undergoes any change, the equation must, as in the case of changing lines, be modified accordingly. All geometrical phe- nomena relating to surfaces may be translated by certain equivalent 106 POSITIVE PHILOSOPHY. analytical conditions, proper to equations of three variables: and it is in the establishment and interpretation of this harmony that the science of analytical geometry of three dimensions essentially consists. In the second and converse case, every equation of three variables may, in general, be represented geometrically by a deter- minate surface, defined by the characteristic property that the co- ordinates of all its points always preserve the mutual relation exhibited in this equation. Thus we see in this application the complement of the original idea of Descartes: and it is enough to say this, as everyone can extend to surfaces the other considerations which have been indi- cated with regard to lines. 1 will only add that the superiority of the rectilinear Bystem of co-ordinates becomes more evident in analytical geometry of three dimensions than in that of two, on account of the geometrical complication which would follow the choice of any other. caTYMofdoutto 1" determining Curves of double curvature, — which ,un " is the Last elementary point of view of analytical geometry of three dimensions. — the Mime principle is employed. According to it, it is clear that when a point is required to be situ- ated upon some certain curve, a single co-ordinate is enough to determine its position completely, by the intersection of this curve with the Burface resulting from this co-ordinate. The two other co-ordinates of the point must thus be regarded as functions neces- sarily determinate, and distinct from the first. Consequently, every line, considered in space, is represented analytically no longer by a single equation, bul by a Bystem of two equations between the three co-ordinates of any one of its points. It is evident, indeed, from another point of view, that the equations which, considered separately, express a certain surface, must in combination present the line sought as the intersection of two de- terminate surfaces. As for the difficulty occasioned by the infinity of the number of couples of equations, through the infinity of couples of surfaces which can enter the same system of co-ordinates, and by which the line sought may be hidden under endless algebra- ical disguises, it must be got rid of by giving up the facilities result- ing from such a variety of geometrical constructions. It is suffi- cient, in fact, to obtain from the analytical system established for a certain line, the system corresponding to a single couple of sur- faces uniformly generated, and which will not vary except when the line itself shall change. Such is a natural use of this kind of geometrical combination, which thus affords us a certain means of recognising the identity of lines in spite of the extensive diversity of their equations. imp recti. n - Analytical Geometry still presents some imperfec- oi An i y t cai tions on the side both of geometry and analysis. In regard to Geometry, the equations can as yet represent only entire geometrical loci, and not determinate portions of those loci. Yet it is necessary, occasionally, to be able to IMPERFECTIONS OF ANALYTICAL GEOMETRY. 107 express analytically a part of a line or surface, or even a discon- tinuous line or surface, composed of a series of sections belonging to distinct geometrical figures. Some progress has been made in supplying means for this purpose, to which our analytical geometry is inapplicable ; but the method introduced by M. Fourier, in his labors on discontinuous functions, is too complicated to be at present introduced into our established system. In regard to analysis, we are so far from having a ^perfections of -complete command of analytical geometry, that we can Analysis. not furnish anything like an adequate geometrical representation of analytical processes. This is not an imperfection in science, but inherent in the very nature of the subject. As Analysis is much more general than geometry, it is of course impossible to find among geometrical phenomena a concrete representation of all the laws expressed by analysis : but there is another evil which is due to our own imperfect conceptions ; that, in our representa- tions of equations of two or of three variables by lines or surfaces, we regard only the real solutions of equations, without noticing any imaginary ones. Yet these last should, in their general course, be as capable of representation as the first. Hence the graphic repre- sentation of the equation is always imperfect ; and it fails alto- gether when the equation admits of only imaginary solutions. This brings after it, in analytical geometry of two or three dimen- sions, many inconveniences of less consequence, arising from the want of correspondence between various analytical modifications and any geometrical phenomena. We have now seen what Analytical Geometry is. By this sci- ence we determine what is the analytical expression of such or such a geometrical phenomenon belonging to lines or surfaces : and, reciprocally, we ascertain the geometrical interpretation of such or such an analytical consideration. It would be interesting now to consider the most important general questions which would exem- plify the manner in which geometers have actually established this beautiful harmony : but such a review is not necessary to the pur- pose of this Work, and would occupy too much space. We have seen what is the character of generality and simplicity inherent in the science of Geometry. We must now proceed to ascertain what is the true philosophical character of the immense and more complex science of Rational Mechanics. 108 POSITIVE PHILOSOPHY. CHAPTER IV. RATIONAL MECHANICS. Mechanical phenomena are by their nature more particular, more complicated, and more concrete, than geometrical phenomena. Therefore they come after geometry in our survey; and therefore must they be pronounced to be more difficult to study, and, as yet, more imperfect. Geometrical ques- tions are always completely independent of Mechanics, while me- chanical questions are closely involved with geometrical considera- tions, — the form of bodies necessarily influencing the phenomena of motion and equilibrium. The simplest change in the form of a body may enhance immeasurably the difficulties of the mechanical problem relating to it, as we see in the question of the mutual gravitation of two bodies, as a result of that of all their molecules ; a question which can be completely resolved only by supposing the bodies to be spherical ; and thus, the chief difficulty arises out of the geometrical part of the circumstances. Our tendency to look for the essences of things, instead of study- ing concrete facts, enters disastrously into the study of Mechanics. We found something of it in geometry ; but it appears in an aggra- vated form in Mechanics, from the greater complexity of the sci- ence. We encounter a perpetual confusion between the abstract and the concrete points of view ; between the logical and the phys- ical ; between the artificial conception necessary to help us to gen- eral laws of equilibrium and motion, and the natural facts furnished by observation, which must form the basis of the science. Great as is the gain of applying Mathematical analysis to Mechanics, it has set us back in some respects. The tendency to à priori suppo- sitions, drawn by us from analysis where Newton wisely had recourse to observation, has made our expositions of the science less clear than those of Newton's days. Inestimable as mathe- matical analysis is for carrying the science oh and upward, there must first be a basis of facts to employ it upon ; and Laplace and others were therefore wrong in attempting to prove the elementary law of the composition of forces by analytical demonstration. Even if the science of Mechanics could be constructed on an analytical basis, it is not easy to see how such a science could ever be applied to the actual study of nature. In fact, that which constitutes the reality of Mechanics is that the science is founded on some general facts, furnished by observation, of which we can give no explanation whatever. Our business now is to point out exactly the philosoph- CHARACTER AND OBJECT OF RATIONAL MECHANICS. 109 ical character of the science, distinguishing the abstract from the concrete point of view, and separating the experimental department from the logical. We have nothing to do here with the causes or modes Ita character of production of motion, but only with the motion it- self. Thus, as we are not treating of Physics, but of Mechanics, forces are only motions produced or tending to be produced ; and two forces which move a body wi th the same velocity in the same direction are regarded as identical, whether they proceed from muscular contractions in an animal, or from a gravitation toward a centre, or from collision with another body, or from the elasticity of a fluid. This is now practically understood ; but we hear too much still of the old metaphysical language about forces, and the like ; and it would be wise to suit our terms to our positive philosophy. The business of Rational Mechanics is to determine how a given body will be affected by any different forces whatever, acting together, when we know what motion would be produced by any one of them acting alone : or, taking it the other way, what are the simple motions whose combination would occasion a known compound motion. This statement shows pre- cisely what are the data and what the unknown parts of every mechanical question. The science has nothing to do with the ac- tion of a single force ; for this is, by the terms of the statement, supposed to be known. It is concerned solely with the combina- tion of forces, whether there results from that combination a mo- tion to be studied, or a state of equilibrium, whose conditions have to be described. The two general questions, the one direct, the other inverse, which constitute the science, are equivalent in importance, as re- gards their application. Simple motions are a matter of observa- tion, and their combined operation can be understood only through a theory : and again, the compound result being a matter of obser- vation, the simple constituent motions can be ascertained only by reasoning. When we see a heavy body falling obliquely, we know what would be its two simple movements if acted upon separately by the force to which it is subject, — the direction and uniform velo- city which would be caused by the impulsion alone ; and again, the acceleration of the vertical motion by its weight alone. The prob- lem is to discover thence the different circumstances of the com- pound movement produced by the combination of the two, — to de- termine the path of the body, its velocity at each movement, the time occupied in falling ; and we might add to the two given forces the resistance of the medium, if its law was known. The best ex- ample of the inverse problem is found in celestial mechanics, where we have to determine the forces which carry the planets round the sun, and the satellites round the planets. We know immediately only the compound movement : Kepler's laws give us the character- istics of the movement ; and then we have to go back to the ele- mentary forces by which the heavenly bodies are supposed to be 110 POSITIVE PHILOSOPHY. impelled, to correspond with the observed result : and these force» once understood, the converse of the question can be managed by geometers, who could never have mastered it in any other way. Such being the destination of Mechanics, we must now notice its fundamental principles, after clearing the ground by a preparatory observation. not in rt In ancient times, men conceived of matter as being in il. y ks. passive or inert, — all activity being produced by sonic external agency, — either of supernatural beings or some metaphys- ical entities. Now that science enables us to view things more truly, we are aware that there is some movement or activity, more or less, in all bodies whatever. The difference is merely of degree between what men call brute matter and animated beings. More- over. Bcience shows us that there arc not different kinds of matter, but that the elements are the same in the most primitive and the most highly organized. If we knew of any substance which had nothing but weight, we could not deny activity even to that; for in falling it is as active as the globe itself, — attracting the earth's particles precisely as much as its own particles are attracted by the earth. Looking through the whole range of substances, up to those of the highest organization, we find everywhere a spontane- ous activity, very various, and at most, in some cases, peculiar ; though physiologists are more and more disposed to regard the most peculiar as a modification of antecedent kinds. However this may be, it would be purely absurd now to regard any portion of matter whatever as inert, as a matter of lact, or under the head of , t . Phvsics. But in Mechanics it must be so regarded, be- cause we can not establish any general proposition upon tli«' abstract laws of equilibrium or motion without putting out of the question all interference with them by other and inherent forces. What we have to beware of is mixing up this logical supposition with the old notion of actual inertia. , iOBd As for how this is to be done — we must remember what has been just said — that in Mechanics, we have nothing to do with the origin or different nature of forces; and they are all one while their mechanical operation is uniform. It is impossible to conceive of any substance as devoid of weight, for in- stance ; yet geometers have logically to treat of bodies as without an inherent power of attraction. They treat of this power as an external force ; that is, it is to them simply a force ; and it does- not matter to them whether it is inherent or external — whether it is attraction or impulsion — while it is the fall of the body that they have to study. And so on, through the whole range of properties of bodies. When we have so abstracted natural properties, in our logical view, as to have before us an unmixed case of the action of certain forces, and have ascertained their laws — then we can pass from abstract to concrete Mechanics, and restore to bodies their natural active properties, and interpret their action by what we have learned of the laws of motion and equilibrium. This rostora- THREE LAWS OF MOTION. Ill tion is so difficult to effect — the transition from the abstract to the concrete in Mechanics is so difficult — that, while its theoretical domain is unbounded, its practical application is singularly lim- ited. In fact, the application of rational mechanics is limited (accurately speaking) not only to celestial phenomena, but to those of our own solar system. One would suppose that the single prop- erty of weight was manageable enough ; and that of a given form intelligible enough : but there are such complications of physical circumstances — as the resistance of media, friction, etc. — even if bodies are conceived of as in a fluid state, that their mechanical phenomena can not be estimated with any accuracy. And when we proceed to electrical and chemical, and especially to physiological phenomena, we are yet more baffled. General gravitation affords us the only simple and determinate law ; and even there we are perplexed, when we come to regard certain secondary actions. It may be doubted whether questions of terrestrial mechanics will ever admit — restricted as our means are — of a study at once purely rational and precisely accordant with the general laws of abstract mechanics — though the knowledge of these laws, primarily indispen- sable, may often lead us to frequent and valuable indications and Suggestions. Bodies being supposed inert, the general facts, or Three lawg of ]aws of motion to which they are subject, are three ; all motion ' results of observations. The first is that law discovered by Kepler, which is . inaptly called the law of inertia. According to it, all motion is rectilinear and uniform ; that is, any body impelled by a single force will move in a right line, and with an invariable ve- locity. Instead of resorting to the old ways of pronouncing or im- agining why it must be so, the Positive Philosophy instructs us to recognise the simple fact that it is so ; that, through the whole range of nature, bodies move in a right line, and with a uniform velocity, when impelled by a single force. The second law we owe to Newton. It is that of the T . 7 . , , . Law of cqnnlitv constant equatity of action ana reaction; that is, when- ofaetumandre. ever one body is moved by another, the reaction is such RCtl0D, that the second loses precisely as much motion, in proportion to its masses, as the first gains. Whether the movement proceeds from impulsion or attraction, is, of course, of no consequence. Newton treated this general fact as a matter of observation, and most geom- eters have done the same ; so that there has been less fruitless search into the why with regard to this second law than to the first. The third fundamental law of motion involves the prin- Law of co . (>xist . ciple of the independence or co-existence of motions, «we of motion* which leads immediately to what is commonly called the composi- tion of force. Galileo is, strictly speaking, the true discoverer of this law, though he did not regard it precisely under the form in which it is presented here : — that any motion common to all the bodies of any system whatever does not affect the particular mo- 112 POSITIVE PHILOSOPHY. tions of these bodies v^ith regard to each other ; which motions proceed as if the system were motionless. Speaking- strictly, we must conceive that all the points of the system describe at the same time parallel and equal straight lines, and consider that this gene- ral motion, whatever may be its velocity and direction, will not affect relative motions. No d-priori considerations can enter here. There is no seeing why the fact should be BO, and therefore no anticipating that it would be so. On the contrary, when Galileo stated this law, lie was assailed by a host of objections that his fact was Logically impossible. Philosophers were ready with plenty of i-priori reasons that it could not be true : and the fact was not unanimously admitted till men had quitted the logical for the physi- cal point of view. We now find, however, that no proposition in the whole range of natural philosophy is founded on observations so simple, so various, so multiplied, so easy of verification. In fact, the whole economy of the universe would bo overthrown, from end to end, but for this law. A ship impelled smoothly, without roll- ing ;ind pitching, has everything going on within it just-the same as if it were at rest : and, in the same way, but on the grandest scale, the great globe itself rushes through space, without its mo- tion at all affecting the movements going on on its surface. As we all know, it was ignorance of this third law of motion which was the main obstacle to the establishment of the Copernican theory. The Copernicans struggled to get rid of the insurmountable objec- tions to which their doctrine was liable by vain metaphysical subtle ties, till Galileo cleared up the difficulty. Since his time, the movement of the globe has been considered an all-sufficient confir- mation of the law. Laplace points out to us that if the motion of the globe affected the movements on it, the effect could not be uniform, bnt must vary with the diversities of their direction, and of the angle that each direction would make with that of the earth : whereas, we know how invariable is, for instance, the oscillating movement of the pendulum, whatever may be its direction in com- parison with that of the travelling globe. It may be as well to point out that rotary motion does not enter into this^case at all, but only translation, because the latter is the only motion which can be, in degree and direction, absolutely com- mon to all the parts of a system. In a rotating system, for in- stance, all the parts are not at an equal distance from the centre of rotation. When the interior of a ship is affected, it is by the rolling and pitching, which are rotary movements. We may carry a watch any distance without affecting its interior movements ; but it will not bear whirling. And, again, the forward motion of the globe could be discovered by no other means than astronomical observation ; whereas, the changes which occur on the surface of the earth, produced by the inequality of the centrifugal force at its different points, are suffi- cient evidence of its rotation, independently of all astronomical considerations whatever. DIVISIONS OF RATIONAL MECHANICS. 113 The law or rule of the composition of forces, which is involved in the general fact just stated, is, in fact, identical with it. It is only another way of expressing the same law. If a single impulsion describes a parallelogram of forces, as the scientific term is, the effect of a second will be to describe the diagonal of the parallelo- gram. This is nothing more than an application of the law of the independence of forces ; since the motion of any body along a straight line is in no way disturbed by a general motion which car- ries away, parallel with itself, the whole of this right line along any other right line whatever. This consideration leads immedi- ately to the geometrical construction expressed by the rule of the parallelogram of forces. And thus it appears that this funda- mental theorem of Rational Mechanics is a true natural law ; or, at least, a direct application of one of the greatest natural laws. And this is the best account to give of it, instead of looking to logic for a fallacious à priori deduction of it. Any analytical demonstration, too, must suppose certain portions of the case to be evident ; and to talk of a thing being evident is to refer back to nature, and to depend on observation of nature. It is worthy of remark that those who wish to make a separate law of the composition of forces, in order to avoid introducing the third law into the prolegomena of Mechanics, and to dispense with it in the exposition of Statics, are brought back to it when entering upon the study of Dynamics. Upon this alone can be based the important law of the proportion of forces to velocities. The rela- tions of forces may be determined either by a statical or dynamical procedure. No purpose is answered by the, transposition of the general fact of the independence of forces, to the dynamical depart- ment of the science : it is equally necessary for the statical ; and a world of metaphysical confusion is saved by laying it down as the broad basis that in fact it is. These three laws are the experimental basis of the science of Mechanics. From them the mind may proceed to the logical con- struction of the science, without further reference to the external world. At least, so it appears to me ; though I am far from as- signing any à-priori reasons why more laws may not be hereafter discovered, if these three should prove to be incomplete. There can not, in the nature of things, be many more ; and I would rather incur the inconvenience of the introduction of one or two, than run any risk of surrendering the positive character of the sci- ence and overstraining its logical considerations. We can not however conceive of any case which is not met by these three laws of Kepler, of Newton, and of Galileo ; and their expression is so precise, that they can be immediately treated in the form of ana- lytical equations easily obtained. As for the most extensive, im- portant, and difficult part of the science, the mechanics of varied motion or continuous forces, we can perceive the possibility of re- ducing it to elementary Mechanics by the application of the infini- tesimal method. For each infinitely small point of time, we must 8 114 POSITIVE PHILOSOPHY. substitute a uniform motion in the place of a varied one, whence will immediately result the differential equations relative to these varied motions. We may hereafter see what results have been ob- tained in regard to the abstract laws of equilibrium and motion. Meantime, we see that the whole science is founded on the com- bination of the three physical laws just established ; and here lies the distinct boundary between the physical and the logical parts of the science. Tw.prim T.v.u. As for its divisions, the first and most important is tea «Id Dm» i nto Statics and Dynamics ; that is, into questions re- »o«- biting to equilibrium and questions relating to motion. Statics are the easiest to treat, because we abstract from them the element of time, which must enter into Dynamical questions, and complicate them. The whole of Statics corresponds to the Yery small portion of Dynamics which relates to the theory of uniform motions. This division corresponds well with the facts of human education in this science. The fine researches of Archimedes show us that the ancients, though far from having obtained any complete system of rational Statics, had acquired much essential knowledge of equilibrium — both of solids and fluids — while as yet wholly with- out the most rudimentary knowledge of Dynamics. Galileo, in fact, created that department of the science. d The next division is that of Solids from Fluids, ion* soiid$ and This division is generally placed first, but it is unques- tionable that the laws of statics and dynamics must enter into the study of solids and fluids, that of fluids requiring the addition of one more consideration, — variability of form. This however is a consideration which introduces the necessity of treat- ing separately the molecules of which fluids are composed, and fluids as systems composed of an infinity of distinct forces. A new order of researches is introduced into Statics, relative to the form of the system in a state of equilibrium ; but in Dynamics the questions are still more difficult to deal with. The importance and difficulty of the researches under this division can not be exag- gerated. Their complication places even the easiest cases beyond our reach, except by the aid of extremely precarious hypotheses. We must admit the vast necessary difficulty of hydrostatics, and yet more of hydrodynamics, in comparison with statics and dynam- ics, properly so called, which are in fact far more advanced. Much of the difficulty arises from the mathematical statement of the question differing from the natural facts. Mathematical fluids have no adhesion between their particles ; whereas natural fluids have, more or less ; and many natural phenomena are due to this adherence, small though it be in comparison with that of solids. Thus, the result of an observation of the quantity of a given fluid which will run out of a given orifice will differ widely from the re- sult of the mathematical calculation of what it should be. Though the case of solids is easier, yet there perplexity may be introduced by the disrupting action of forces, of which abstraction must be STATICS. 115 made in the mathematical question. The theory of the rupture of solids, initiated by Galileo, Huyghens, and Leibnitz, is still in a very imperfect and precarious state, great as are the pains which have been taken with it, and much needed as it is. Not so much needed however as the mechanics of fluids, because it does not affect questions of celestial mechanics ; and in this highest depart- ment alone can we, as I said before, see the complete application of rational mechanics. There is a gap left between these two studies, which should be pointed out, though it is of secondary importance. We want a Mechanics of semi-fluids, or semi-solids, — as of sand, in relation to solids, and gelatinous conditions of fluids. Some considerations have been offered with regard to these " imperfect fluids," as they are called ; but their true theory has never been established in any direct and general manner. Such is the general view of the philosophical character of Ra- tional Mechanics. We must now take a philosophical view of the composition of the science, in order to see how this great second department of Concrete Mathematics has attained the theoretical perfection in which it appears in the works of Lagrange, who has rendered all its possible abstract questions capable of an analytical solution, like those of geometry. We must first take a view of Statics, and then proceed to Dynamics. SECTION I. STATICS. There are two ways of treating Rational Mechanics, according as Statics are regarded directly, or as a par- oda of treat. ticular case of Dynamics. By the first method we mnt have to discover a principle of equilibrium so general as to be ap- plicable to the conditions of equilibrium of all systems of possible forces. By the second method, we reverse the process, — ascertain- ing what motion would result from the simultaneous action of any proposed differing forces, and then determining what relations of these forces would render motion null. The first method was the only one possible in the First method, early days of science ; for, as I have said before, ,statics by ifcelf - Galileo was the creator of the science of Dynamics. Archimedes, the founder of Statics, established the condition of equilibrium of two weights suspended at the ends of a straight lever ; that is, he showed that the weights must be in an inverse ratio to their dis- tances from the fulcrum of the lever. He endeavored to refer to this principle the relations of equilibrium proper to other systems of forces ; but the principle of the lever is not in itself general enough for such application. The various devices by which it was attempted to extend the process, and to supply the remaining defi- ciencies, were relinquished when the establishment of Dynamics permitted the use of the second method, — of 116 POSITIVE PHILOSOPHY. statics through seeking the conditions of equilibrium through the laws Dynamics, f the composition of forces. Jt is by this last method that Varignon discovered the theory of the equilibrium of a system of forées applied upon a single point ; and that D'Alembert after- ward established, for the first time, the equations of equilibrium of any system of forces applied to the different points of a solid body of an invariable form. At this day, this is the method universally employed. At t he first glance, it does not appear the most rational, — Dynamics being more complicated than Statics, and precedence being natural to the simpler. It would, in fact, be more philosoph- ical to refer dynamics to statics, as has niice been done ; but we may observe that it is only the most elementary part of dynamics, the theory of uniform motions, that we art 1 concerned with in treat- ing Btatics as a particular case of dynamics. The complicated considerations of varied motions do not enter into the proa at all. The easiest method of applying the theory of uniform motions to statical questions is through the view that, when forces art; in equitibrio, each of them, taken Bingly, may be regarded as destroy- ing the effect of all tl there together. Thus, the thing to be done is to >liii\v that any one of the forces of the system is equal, and directly opposed, to the resulting force of all the rest. The only difficulty here is in determining the resulting force; that is, in mutually compounding the given forces. Here comes in the aid of the third great law of motion, and having compounded the two firsl forces, we can deduce the composition of any number of forces. After having established the elementary laws of the composition of forces, geometer.-, before applying them to the Investigation of the conditions of equilibrium, usually subject them to an important transformation, which, without being indispensable, is of eminent utility, in an analytical view, from the extreme simplification which it introduces into the algebraical expression of the conditions of equilibrium. The transformation consists in what is called the theory of Moments, the essential property of which is to reduce, analytically, all the laws of the composition of forces to simple additions and subtractions. Without going into an examination of this theory, it is necessary simply to say that it considers statics as a particular case of elementary dynamics, and that its value is in the simplicity which it gives to the analytical part of the process of investigation into the conditions of equilibrium. Simple, however, as may be the operation, and great as may be the practical advantage gained through the treatment of statics as a particular case of elementary Dynamics, it would be satisfactory to return, if we could, to the method of the ancients, — to leave Dy- wam of unity in namics on one side, and proceed directly to the inves- * method. tigation of the laws of equilibrium regarded by itself, by means of a direct general principle of equilibrium. Geometers strove after this as soon as the general equations of equilibrium METHODS OF STATICS. 117 were discovered by the dynamic method. But a higher motive than even the desire to place statics in a more philosophical posi- tion impelled them to establish a direct Statical method : and this it was which caused Lagrange to carry up the whole science of Rational Mechanics to the philosophical perfection which it now enjoys. D'Alembert made a discovery (to be treated of hereafter), by the help of which all investigation of the motion of any body or system might be converted at once into a question of equilibrium. This amounts, in fact, to a vast generalization of the second fun- damental law of motion ; and it has served for a century past as a permanent basis for the solution of all great dynamical questions ; and it must be so applied more and more, from its high merits of simplification in the most difficult investigations. Still, it is clear that this method compels a return into statics ; and Statics as in- dependent of Dynamics, which are altogether derived from Statics. A science must be imperfectly laid down, as long as it is necessary thus to pass backward and forward between its two departments. In order to establish the necessary unity, and to provide scope for D'Alembert's principle, a complete reconstitution of Rational Mechanics was indispensable. Lagrange effected this in his admi- rable treatise on " Analytical Mechanics," the leading conception of which must be the basis of all future labors of geometers upon the laws of equilibrium and motion, as we have seen that the great idea of Descartes is with regard to geometrical speculations. The principle of Virtual Velocities, — the one which VirtUfl] Veloci . Lagrange selected from among the properties of equi- ties - librium, — had been discovered by Galileo in the case of two forces, as a general property manifested by the equilibrium of all machines. John Bernouilli extended it to any number of forces, composing any system. Varignon afterward expressly pointed out the universal use that might be made of it in Statics. The combination of it with D'Alembert's principle led Lagrange to conceive of the whole of Rational Mechanics as deduced from a single fundamental theorem, and to give it that rigorous unity which is the highest philosophical perfection of a science. The clearest idea of the system of virtual velocities may be ob- tained by considering the simple case of two forces, which was that presented by Galileo. We suppose two forces balancing each other by the aid of any instrument whatever. If we suppose that the system should assume an infinitely small motion, the forces are, with regard to each other, in an inverse ratio to the spaces traversed by their points of application in the path of their directions. These spaces are called virtual velocities, in distinction from the real velocities which would take place if the equilibrium did not exist. In this primitive state, the principle, easily verified with regard to all known machines, offers great practical utility ; for it permits us to obtain with ease the mathematical condition of equilibrium of any machine whatever, whether its constitution is 118 POSITIVE PHILOSOPHY. known or not. If we give the name of virtual momentum (or simply of momentum in its primitive sense") to the product of each force by its virtual velocity. — a product which in fact then measures the effort of the force to move the machine, — we may greatly simplify the statement of the principle in merely saying that, in this case, the momentum of the two forces must be equal and of opposite signs, that there maybe equilibrium, and that the positive or negative BigD Of each momentum is determined ac- cording to that of the virtual velocity, which will be considered positive or negative according as, by the supposed motion, the pro- jection of the point of application would be found to fall upon the direction of the force or upon its prolongation. This abridged ex- pression of the principle of virtual velocities is especially useful for the statement of this principle in a general manner, with regard to any system of forces whatever. It is simply this: that the al- gebraic sum of the virtual moments of all forces, estimated accord- ing to the preceding rule, must be null to cause equilibrium s and this condition must exist distinctly with regard to all the element- ary motions which the Bystem might assume in virtue of the forces by which it is animated. In the equation, containing this principle, furnished by Lagrange, the whole of Rational Mechanics may be considered to be implicitly comprehended. While the theorem of virtual velocities was conceived of only as a general property of equilibrium, it could be verified by observing its constant conformity with the ordinary laws of equilibrium, otherwise obtained, of which it was a summary, useful by its sim- plicity and uniformity. But, if it was to be a fundamental principle, a basis of the whole science, it must DC underived. or at least capable of being presented in its preliminary propositions as a matter of ob- servation. This was done by Lagrange, by his ingenious demon- stration through a system of pulleys. He exhibited the theorem of virtual velocities very easily by imagining a Bingle weight which, by means of pulleys suitably constructed, replaces simultaneously all the forces of the system. Many other demonstrations have been furnished ; but, while more complicated, they are not logically su- perior. From the philosophical point of view it is clear that this general theorem, being a necessary consequence of the fundamental laws of motion, can be deduced in various ways, and becomes practically the point of departure of the whole of Rational Me- chanics. A perfect unity having been established by this principle, we need not look for any others ; and we may rest assured that Lagrange has carried the co-ordination of the science as far as it can go. The only possible object would be to simplify the ana- lytical researches to which the science is now reduced ; and nothing can be conceived more admirable for this purpose than Lagrange's adaptation of the principle of virtual velocities to the uniform ap- plication of mathematical analysis. Striking as is the philosophical eminence of this principle, there are difficulties enough in its use to prevent its being considered THEORY OF COUPLES. 119 elementary, so far as to preclude the consideration of any other in a course of dogmatic teaching. It is for this reason that I have referred to the dynamic method, properly so called, which is the only one in general use at present. All other considerations must however' be only provisional. Lagrange's method is at present too new ; but it is impossible that it should for ever remain in the hands of a small number of geometers, who alone shall be able to make use of its admirable properties. It must become as popular in the mathematical world as the great geometrical conception of Des- cartes : and. this general progress would be almost accomplished if the fundamental ideas of transcendental analysis were as widely spread as they ought to be. The greatest acquisition, since the regeneration of Theory of the science by Lagrange, is the conception of M. Poin- coupi.s. sot, — the theory of Couples, which appears to me to be far from being sufficiently valued by the greater number of geometers. These Couples, or systems of parallel forces, equal and contrary, had been merely remarked before the time of M. Poinsot, as a sort of par- adox in Statics. He seized upon this idea, and made it the sub- ject of an extended and original theory relating to the transfor- mation, composition, and use of these singular groups, which he has shown to be endowed with properties remarkable for their general- ity and simplicity. He used the dynamic method in his study of the conditions of equilibrium : but he presented it, by the aid of his theory of couples, in a new and simplified aspect. But his con- ception will do more for dynamics than for statics ; and it has hardly yet entered upon its chief office. Its value will be appreciated when it is found to render the notion of the movements of rotation as natural, as familiar, and almost as simple, as that of forward movement or translation. One more consideration should, I think, be adverted to before we quit the subject of statics as a whole, fions 6 in° P Sd£ When we study the nature of the equations which ex- tin ?"q uilibriui ». press the conditions of equilibrium of any system of forces, it seems to me not enough to establish that the sum of these equations is indispensable for equilibrium. I think the further statement is necessary, — in what degree each contributes to the result. It is clear that each equation must destroy some one of the possible motions that the body would make in virtue of existing forces ; so that the whole of the equation must produce equilibrium by leaving an impossibility for the body to move in any way whatever. Now the natural state of things is for movement to consist of rotation and translation. Either of these may exist without the other ; but the cases are so extremely rare of their being found apart, that the verification of either is regarded by geometers as the strongest presumption of the existence of the other. Thus, when the rota- tion of the sun upon its axis was established, every geometer con- cluded that it had also a progressive motion, carrying all its planets with it, before astronomers had produced any evidence that such 120 POSITIVE PHILOSOPHY. was actually the case. In the same way we conclude that certain planets, travelling in their orbits, rotate round their axes, though the fact has not yet been verified. Sonic equations must therefore tend to destroy all progressive motion, and others all motion of ro- tation. How many equations of each kind must there he': It is (dear that, to keep a body niotionlos. it must be hindered from moving according to three axes in different planes — commonly supposed to he perpendicular to each other. If a body can not move from north to south, nor the reverse; nor from east to west, nor the reverse : nor up. nor down, it is clear that it can not move at all. Movement in any intermediate direction mighl be conceived of as partial progression in one of these, and is therefore impossi- ble. On the other hand, we can not reckon fewer than three in- dependent elementary motions; for the body might move in the direction of one of the axe8, without having any translation in tli«' direction of either of tic others. Thus we Bee that, in a general w ay, three equations are necessary, and three arc sufficient to estab- lish the absence of translation; each being specially adapted to destroy our of the three progressive motions of which the body is capable. The same \ ievi presents itself with regard to the other motion — of rotation. The mechanical conception is more compli- cated : hut it is true, as in the simpler Ca8e, that motion is possible in only three directions — in three co-ordinated planes, or round three axes. Three equations are necessary and sufficient here also; and thus we have six which are indispensable and sufficient to stop all motion whatever. W hen. instead of supposing any Bystem of forces whatever as the subject of the question, we particularize any, we get rid of more or fewer possible movements. Having excluded these, we may ex- clude also their corresponding equations, retaining only those which relate to the possible motions that remain. Thus, instead of hav- ing to deal with six equations necessary to equilibrium, there may be only three, or two, or even one, winch it will be easy enough to obtain in each case. These remarks may be extended to any re- strictions upon motion, whether resulting- from the special constitu- tion of the system of forces, or from any other kind of control, affecting the body under nolice. If, for instance, the body were fastened to a point, so that it could freely rotate but not advance, three equations would suffice : and again, if it is fastened to two fixed points, two equations are enough ; and even one, if these two fixed points are so placed as to prevent the body from moving on the axis between them. Finally, its being attached to three fixed points, not in a right line, will prevent its moving at all, and estab- lish equilibrium without any condition, whatever may be the forces of the system. The spirit of this analysis is entirely independent of any method by which the equations of equilibriums will have been obtained : but the different general methods are far from being equally suitable to the application of this rule. The one which is- best adapted to it is, undoubtedly, the Statical one, properly so* th>- concrete with the an- nexion. TERRESTRIAL GRAVITATION. 121 called, founded, as has been shown, on the principle of virtual ve- locities. In fact, one of the characteristic properties of this princi- ple is the perfect precision with which it analyses the phenomena of equilibrium, by distinctly considering each of the elementary motions permitted by the forces of the system, and furnishing imme- diately an equation of equilibrium specially relating to this motion. When we come to the inquiry how geometers apply connection of the principles of abstract Mechanics to the properties of real bodies, we must state that the only complete ap- «tract qi plication yet accomplished is in the question of terrestrial gravity. Now, this is a subject which can not, logically, be treated under the head of Mechanics, as it belongs to Physics. It is sufficient to explain that the statical study of terrestrial gravity becomes con vertible into that of centres of gravity ; and that all confusion between the two departments of research would be avoided if we accustomed ourselves to class the theory of centres of gravity among the questions of pure geometry. In seeking the centre of gravity as (according to the logical denomination of the ancient geometers) the centre of mean distances, we remove all traces of the mechanical origin of the question, and convert it into this prob- lem of general geometry : — Given, any system of points disposed in a determinate way with regard to each other, to find a point whose distance to any plane shall be a mean between the distances of all the given points to the same plane. — The abstraction of all consid- eration of gravity is an assistance in every way. The simple geo- metrical idea is precisely what we want in most of the principal theories of Rational Mechanics, and especially when we contem- plate the great dynamic properties of the centre of mean distances ; in which study the idea of gravity becomes a mere encumbrance and perplexity. It is true that, by proceeding thus, we exclude the ques- tion from the domain of Mechanics, to place it in that of Geometry. I should have so classed it but for an unwillingness to break in upon established customs. However it may be as to the matter of arrangement, it is highly important for us not to misapprehend the true nature of the question. — The integral calculus offers the means of surmounting those difficulties in determining the centre of gravity which are imposed by the conditions of the question. But, the in- tegrations in this case being more complicated than those to which they are analogous — those of quadratures and cubatures — their pre- cise solution is, owing to the extreme imperfection of the integral Calculus, much more rarely obtained. It is a matter of high im- portance, however, to be able to introduce the consideration of the centre of gravity into general theories of analytical mechanics. Such is, then, the relation of terrestrial gravitation to the science of abstract Statics. As for universal gravitation, no complete study has yet been made of it, except in regard to spherical bodies. What we know of the law of gravitation would easily enable us to compute the mutual attraction of all known bodies, if the conditions of each body were understood by us : but this is not the case. For 122 POSITIVE PHILOSOPHY. instance, we know nothing of the law of density in the interior of the heavenly bodies. It is still true that the primitive theorems of Newton on the attraction of spherical bodies are the most useful part of our knowledge in this direction. Gravity is the only natural force that we are practically con- cerned with in Rational Statics : and we see, by this, how back- ward this science is in regard to universal gravitation. As for the exterior general circumstances, such as friction, resistance of media, and the like, which arc altogether excluded in the establishment of the rational laws of Mechanics, we can only say that we are abso- lutely ignorant of the way to introduce them into the fundamental relations afforded by analytical Mechanics, because we have nothing to rely on, in working them, but precarious and inaccurate hypothe- ses, unfit for scientific use. Equilibrium «.f ^ s *' or tno theory of equilibrium in regard to fluid fluid* bodies — the application which it remains for us to no- tice — those bodies must be regarded as cither liquid or gaseous. Hydrostatics may be treated in two ways. We may seek the laws of the equilibrium of fluids, according to statical considerations proper to that class of bodies: or we may look for them among the laws which relate to solids, allowing for the new characteristics resulting from fluidity. The first method, being the easiest, was in early times the only one employed. Till a rather recent time, all geometers employed themselves in proposing statical principles peculiar to fluids ; and especially with regard to the grand question of the figure of the earth, on the supposition that it was once fluid. Huyghens first endeavored to resolve it, taking for his principle of equilibrium the necessary perpendicularity of weight at the free surface of the fluid. Newton's principle was the necessary equality of weight between the two fluid columns going from the centre — the one to the pole, the other to some point of the equator. Bouguer showed that both methods were bad, because, though each was incontestable, the two failed, in many cases, to give the same form to the fluid mass in equilibrium. But he, in his turn, was wrong in believing that the union of the two principles, when they agreed in indicating the same form, was sufficient for equilibrium. It was Clairaut who, in his treatise on the form of the earth, first discovered the true laws of the case, setting out from the evident consideration of the isola- ted equilibrium of any infinitely small canal ; and, tried by this cri- terion, he showed that the combination required by Bouguer might take place without equilibrium happening. Several great geome- ters, proceeding on Clairaut's foundation, have carried on the theory of the equilibrium of fluids a great way. Maclaurin was one of those to whom we owe much ; but it was Euler who brought up the subject to its present point, by founding the theory on the principle of equal pressure in all directions. Observation of the statical con- stitution of fluids indicates this as a general law ; and it furnishes the requisite equations with extreme facility. EQUILIBRIUM OF FLUIDS. 123 It was inevitable that the mathematical theory of the equilibrium of fluids should, in the first place, be founded, as we have seen that it was, on statical principles peculiar to this kind of bodies : for, in early days, the characteristic differences between solids and fluids must have appeared too great for any geometer L,qmd8 - to think of applying to the one the general principles appropriated to the other. But, when the fundamental laws of hydrostatics were at length obtained, and men's minds were at leisure to estimate the real diversity between the theories of fluids and of solids, they could not but endeavor to attach them to the same general principles, and perceive the necessary applicability of the fundamental rules of Statics to the equilibrium of fluids, making allowance for the attendant variability of form. But, before hydro- statics could be comprehended under Statics, it was necessary that the abstract theory of equilibrium should be made so general as to apply directly to fluids as well as solids. This was accomplished when Lagrange supplied, as the basis of the whole of Rational Me- chanics, the single principle of Virtual Velocities. One of its most valuable properties is its being as directly applicable to fluids as to solids. From that time, Hydrostatics, ceasing to be a natural branch of science, has taken its place as a secondary division of Statics. This arrangement has not yet been familiarly admitted ; but it must soon become so. To see how the principle of Virtual Velocities may lead to the fundamental equations of the equilibrium of fluids, we have to con- sider that all that such an application requires, is to introduce among the forces of the system under notice one new force, — the pressure exerted upon each molecule, which will introduce one term more into the general equation. Proceeding thus, the three general equations of the equilibrium of fluids, employed when hydrostatics was treated as a separate branch, will be immediately reached. If the fluid be a liquid, we must have regard to the con- dition of incompressibility, — of change of form without change of volume. If the fluid be gaseous, we must substitute for the incom- pressibility that condition which subjects the volume of the fluid to vary according to a determinate function of the pressure ; for in- stance, in the inverse ratio of the pressure, according to the phys- ical law on which Mariotte has founded the whole Mechanics of the gases. We know but too little yet of these gaseous conditions ; for Mariotte's law can at present be regarded only as an approxi- mation, — sufficiently exact for average circumstances, but not to be rigorously applied in any case whatever. Some confirmation of the philosophical character of this method of treating hydrostatics arises from its enabling us to pass, almost insensibly, from the order of bodies of invariable form to that of the most variable of all, through intermediate classes, — as flexible and elastic bodies, — whereby we obtain, in an analytical view, a natural filiation of subjects. We have seen how the department of Statics has been raised to 1-4 POSITIVE PHILOSOPHY. that high degree of speculative perfection which transforms its questions into simple problems of Mathematical Analysis. We must now take a similar review of the other department of general Mechanics, — that more extended and more complicated study which relates to the laws of Motion. SECTION II. DYNAMICS. ob . The object of Dynamics is the study of the varied motions produced by continuous forces. The Dynam- ics of varied motions or continuous forces includes two depart- ments, — the motion of B point, and that of a body. From the positive point of view, this means that, in certain cases, all the parts of the body in question have the same motion, so that the de- termination of one particle Berves for the whole ; while in the more general ease, each particle of the body, or each body of the system, assuming a distinct motion, it is necessary to examine these dif- ferent effects, and the action upon them of the relations belonging to the system under notice. The second theory being more com- plicated than the first, the first is the one to begin with, even if both are deduced from the same principles. With regard to the motion of a point, the question is to deter- mine the circumstances of the compound curvilinear motion, result- ing from the simultaneous action of different continuous forces, it being known what would be the rectilinear motion of the body if influenced by any one of these forces. Like every other, this prob- lem admits of a converse solution. Theory of mcti- But nere intervenes a preliminary theory, which must linear motion, be noticed before either of the two departments can be entered upon. This theory is popularly called the theory of recti- linear motion, produced by a single continuous force acting indefi- nitely in the same direction. It may be asked why we want this, after having said that the effect of each separate force is supposed to be known, and the effect of their union the thing to be- sought. The answer to this is, that the varied motion produced by each continuous force may be defined in several ways, which depend on each other, and which could never be given simultaneously, though each may be separately the most suitable ; whence results the neces- sity of being able to pass from any one of them to all the rest. The preliminary theory of varied motion relates to these transfor mations, and is therefore inaptly termed the study of the action of a single force. These different equivalent definitions of the same varied motions result from the simultaneous consideration of the three distinct but co-related functions which are presented by it, — space, velocity, and force, conceived as dependent on time elapsed. Taking the most extended view, we may say that the definition of a varied motion may be given by any equation containing at once these four variables, of which only one is independent, THEORY OF RECTILINEAR MOTION. 120 velocity, and force. The problem will consist in deducing from this equation the distinct determination of the three characteristic laws relating to space, velocity, and force, as a function of time, and, consequently, in mutual correlation. This general problem is always reducible to a purely analytical research, by the help of the two dynamical formulas which express, as a function of time, velocity, and force, when the law of space is supposed to be known. The infinitesimal method leads to these formulas with the utmost ease, the motion being considered uniform during an infinitely small in- terval oftime, and as uniformly accelerated during two consecutive intervals. Thence the velocity, supposed to be constant at the instant, according to the first consideration, will be naturally ex- pressed by the differential of the space, divided by that of the time ; and, in the same way, the continuous force, according to the second consideration, will evidently be measured by the relation between the infinitely small increment of the velocity, and the time employed in producing this increment. Lagrange's conception of transcendental analysis excluding him from this use of the infinitesimal method for the establishment of the two foregoing dynamic formulas, he was led to present this the- ory under another point of view, more important than seems to be generally supposed. In his Theory of Analytical Functions, he has shown that this dynamic consideration really consists in con- ceiving any varied motion as compounded, each moment, of a cer- tain uniform motion and another motion uniformly varied, — liken- ing it to the vertical motion of a heavy body under a first impulsion. Lagrange has not given its due advantage to this conception, by developing it as he might have done. In fact, it supplies a com- plete theory of the assimilation of motions, exactly like the theory of the contracts of curves and surfaces, in the department of geom- etry. Like that theory, it removes the limits- within which we supposed ourselves to be confined, by disclosing to us, in an abstract way, a much more perfect measure of all varied motion tli an we obtain by the ordinary theory, though reasons of conve- nience compel us to abide by the method originally adopted. The first case or department of rational dynamics, — Motion of a that of the motion of a point, or of a body which has i K,int all its points or portions affected by the same force, — relates to the study of the curvilinear motion produced by the simultaneous action of any different continuous forces. This case divides itself again into two, — according as the mobile point is free, or as it is com- pelled to move in a single curve, or on a given surface. The fun- damental theory of curvilinear motion may be established in either case, in a different way ; each being susceptible of direct treat- ment, and of being connected with the other. In the first case, in order to deduce the second, we have only to regard the active or passive resistance of the prescribed curve or surface as a new force to be added to the others proposed. In the other way, we have only to consider the moving point as compelled to describe 126 POSITIVE PHILOSOPHY. the curve which it must traverse ; and this is enough to afford the fundamental equations, though this curve may then be primi- tively unknown. Motion of a ay». Tne other, more real and more difficult case, is that ,,>m of the motion of a system of bodies in any way con- nected, whose proper motions are altered by the conditions of their connection. There is a new elementary conception about the measurement of forces which some geometers declare to be logi- cally deducible from antecedent considerations, and to which they would assign the place and title of a fourth law of motion. For the sake of convenience, we may make it into a fourth law of mo- tion ; but such is not its philosophical character. The idea is, that forces which impress the Barae velocity on different masses are to each other exactly as those masses ; or, in other words, that the forces are proportional to tin* masses, as we have seen them, under the third law of motion, to be proportional to the velocities. All phenomena, such as the communication of motion by collision, or in any other way. have tended to confirm the supposition of this new kind of proportion. It evidently results from this, that when we have to compare forces which impress different velocities on unequal masses, each must be measured according to the product of the mass upon which it acts by the corresponding velocity. This product is called by geometers quantity of motion; and it de- termines the percussion of a body, and also the pressure that a body may exercise against any fixed obstacle to its motion. Proceeding to the second dynamical case, we see that the char- acteristic difficulty of this order of questions consists in the way of estimating the connection of the different bodies of the system, in virtue of which their mutual reactions will necessarily affect the motions which each would take if alone ; and we can have no à-priori knowledge of what the alterations will be. In the case of the pendulum, for instance, the particles nearest the point of sus- pension, and those furthest from it, must react on each other by their connection — the one moving faster and the other slower than if they had been free ; and no established dynamic principle exists revealing the law which determines these reactions. Geometers naturally began by laying down a principle for each particular case ; and many were the principles thus offered, which turned out to be only remarkable theorems furnished simultaneously by funda- mental dynamic equations. Lagrange has given us, in his "Ana- lytical Mechanics," the general history of this series of labors ; and very interesting it is, as a study of the progressive march of the D-Aipmherfs human intellect. This method of proceeding continued p.incipie. till the time of D'Alembert, who put an end to all these isolated researches by seeing how to compute the reactions of the bodies of a system in virtue of their connection, and establishing the fundamental equations of the motion of any system. By the aid of the great principle which bears his name, he made questions of motion merge in simple questions of equilibrium. The princi] le •d'alembert's principle. 127 is simply this. In the case supposed, the natural motion clearly divides itself into two — the one which subsists and the one which has been destroyed. By D'Alembert's view, all these last, or in other words, all the motions that have been lost or gained by the different bodies of the system by their reaction, necessarily balance each other, under the conditions of the connection which character- izes the proposed system. James Bernouilli saw this with regard to the particular case of the pendulum ; and he was led by it to form an equation adapted to determine the centre of oscillation of the most simple system of weight. But he extended the resource no further ; and what he did detracts nothing from the credit of D'Alembert's conception, the excellence of which consists in its entire generality. In D'Alembert's hands the principle seemed to have a purely logical character. But its germ may be recognised in the second law of motion, established by Newton, under the name of the equality of reaction and action. They are, in fact, the same, with regard to two bodies only acting upon each other in the line which connects them. The one is the greatest possible generalization of the other ; and this way of regarding it brings out its true nature, by giving it the physical character which. D'Alembert did not im- press upon it. Henceforth therefore we recognise in it the second law of motion, extended to any number of bodies connected in auy manner. We see how every dynamical question is thus convertible into one of Statics, by forming, in each case, equations of equilibrium between the destroyed motions. But then comes the difficulty of making out what the destroyed motions are. In endeavoring to get rid of the embarrassing consideration of the quantities of motion lost or gained, Euler, above others, has supplied us with the method most suitable for use — that of attributing to each body a quantity of motion equal and contrary to that which it exhibits, it being evident that if such equal and contrary motion could be imposed upon it, equilibrium would be the result. This method contemplates only the primitive and the actual motions which are the true elements of the dynamic problem — the given and the un- known ; and it is under this method that D'Alembert's principle is habitually conceived of. Questions of motion being thus reduced to questions of equilibrium, the next step is to combine D'Alem- bert's principle with that of virtual velocities. This is the combi- nation proposed by Lagrange, and developed in his "Analytical Mechanics," which has carried up the science of abstract Mechanics to the highest degree of logical perfection — that is, to a rigorous unity. All questions that it can comprehend are brought under a single principle, through which the solution of any problem what- ever offers only analytical difficulties. D'Alembert immediately applied his principle to the case of fluids — liquid and gaseous, which evidently admit of its use as well as solids, their peculiar conditions being considered. The result was our obtaining general equations of the motion of fluids, wholly 128 POSITIVE PHILOSOPHY. unknown before. The principle of virtual velocities rendered thin perfectly easy, and again left nothing to be desired, in regard to concrete considerations, and presented none but analytical diffi- culties. We must admit, however, that our actual knowledge ob- tained under this theory is extremely imperfect owing to insur- mountable difficulties in the integrations required. If it vas so in questions of pure Statics, much more must it be bo in the more complex dynamical questions. The problem of the flow of a gravir tating Liquid through a given orifice, simple as it appears, has never vet been resolved. To simplify as far as they could, geometers have had recourse t<> Daniel Bernouilli's hypothesis of the parallel- ism of sections, which admits of our considering motion in regard to horizontal lamime instead of particle by particle. Bui this method of considering each horizontal lamina of a liquid as moving altogether, and taking the place of the following, is evidently con- trary to tin- fact in almost all cases. The Lateral motions are wholly abstracted, and their sensible existence imposes on us the necessity of studying the motion of each particle. We must then consider the science of hydrodynamics as being still in its infancy, even with regard to Liquids, and much more with regard to gases. Yet, as tie fundamental equations of the motions of fluids arc irre- versibly established, it is clear thai what remains to be accomplished is in the direction of mathematical analysis alone. Such is tic Method of Rational Mechanics. As for the great theoretical results of the science — the princi- pal general properties of equilibrium and motion thus far discovered — they were at tii'M taken for real principles, each being destined to furnish the solution of a certain order of new problems in Me- chanics. As the systematic character of the science has come out, th.-,,. however, these supposed principles have shown them- ******* selves to be mere theorems — necessary résulta of the fundamental theories of abstract Statics and Dynamics. Of these theorems, two belong to Statics. The most remarkable is that discovered by Torricelli with regard to the equilibrium of heavy bodies. It consists in this ; that when any system of heavy bodies is in a situation of equilibrium, its centre of gravity is necessarily placed at the lowest or highest pos- sible point, in comparison with all the positions it might take under any other situation of the system. Maupertuis afterward, by his working out of his Laiv of repose, gave a large generalization to this theorem of Torricelli' s , which at once became a mere particu- lar case under that law ; Torricelli' s applying merely to cases of terrestrial gravitation, while that of Maupertuis extends throughout the whole sphere of the great natural attractive forces. The other general property relating to equilibrium Stability and in- . & , , f f J S> "1 stability ot equi- may be regarded as a necessary complement oi the iibnum. former. It consists in the fundamental distinction between the cases of stability and instability of equilibrium. There being no such thing in nature as abstract repose, the term is applied DYNAMICAL THEOREMS. 129 here to that state of stable equilibrium which exists where the cen- tre of gravity is placed as low as possible ; while unstable equilib- rium is that which is popularly called equilibrium ; and it exists when the centre of gravity is placed as high as possible. Mauper- tuis's theorem consisted in this — that the situation of equilibrium of any system is always that in which the sum of vires vivce (active forces) is a maximum or a minimum ; and the one under notice, developed by Lagrange, consists in this — that in any system equi- librium is stable or unstable according as the sum of vires vivce is a maximum or a minimum. Observation teaches the facts in the most simple cases ; but it requires a large theory to exhibit to geometers that the distinction is equally applicable to the most compound systems. Proceeding to the theorems relative to dynamics, the most direct way of establishing them is that used by Lagrange — exhibiting them as immediate consequences of the general equation of dynam- ics, deduced from the combination of D'Alembert's Dynamical theo . principle with the principle of virtual velocities. The r