mathkmatics, abstract and concrete.
Description of Mathematics Object of Mathematics General Method E.xamples...True Definition of Mathematics Its two parts C0^• TENTS.
Their difterent oLjects Their different natures Concrete Mathematics Abstract Mathematics Extent of its domain Its Universality Its limitations.
XXXlll PAGE CHAPTER II.
general view of mathematical analysis.
Analysis.
True idea of an equation Abstract functions.
Concrete functions.
Two parts of the Calculus Algebra.
Arithmetic Its extent.
Its nature.
Algebra.
Creation of new functions.. • • Finding equations between auxiliary quantities Division of the Calculus of functions..
Section 1. Ordinary Analysis, or Calculus of Direct tions...•••• Its object...••••• Classihcation of Equations...Algebraic equations Algebraic resolution of equations.
Numerical resolution of equations.
The Theory of equations. _. _.
Method of indeterminate coefficients •,, • Section 2. Transcendental Analysis, or Calculus Functions.. Three principal views History '.
Method of Leibnitz. Generality of the formulas Justification of the Method Newton's Method. Method of Limits Fluxions and Ihients. Lagkange's Method. Identity of the three methods Their comparative value. - The Differenti.al and Integral Calculus Its two parts I. C of Indirect CONTENTS.
Tlieir mutual relations Cases of union of the two.
Cases of the Differential calculus alone Cases of the Integral calculus alone.
The Differential Cdlculus.
Two ])ortions...Subdivisions...Reduction to the elements Transformation of derived functions for new variables Analytical applications.
The Integral Caleulus Its divisions...Subdivisions...One variable or several Orders of differentiation.
Quadratures...Algebraic functions.
Transcendental functions.
Singular Solutions.
Definite integrals Prospects of the Integral Calculus Caleitlus of Variations Problems giving rise to this calculu> Other applications.
Relation to the ordinary calculus CHAPTER III.
GENERAL VIEW OF GEOMETRY.
Its Nature Definition.
Idea of Space.
Kinds of extension.
Geometrical measurement Measurement of surfaces and volumes Of curved lines.
Its illimitable field.
Properties of lines and surfaces Two general Methods Special or ancient, and general or moder (ieometry of the ancients.
Geometry of tiie right line Graphical solutions.
Descrii)tive Geometry Algebraic solutions.
Trigonometry.
Modern, or Ana.lytiral Geometry Analytical representation of figures CONTENTS.
XXXV Position of a jioint...Plane curves...Exjjression of lines by equations Expression of equations by lines Change in the line changes the equation. Every definition of a line is an equation Choice of co-ordinates Determination of a point in space. Determination of Surfaces by Equations, Surfaces...Curves of double curvature Imperfections of Analytical Geometry Imperfections of Analysis and of Equations by CHAPTER IV.
KATIONAL MECHANICS.
Its nature.
Its characters.
Its object...Matter not inert in Physics Supposed inert in Mechanics Field of Rational Mechanics Three Laws of IVIotion Law of inertia.
Law of equality of action and i Law of co-existence of motions Two Primary divisions Statics and Dynamics.
Secondary divisions.
Solids and Fluids Section L Statics.
Converse methods of treatment First' method.
Statics by itself Second method Statics through Djaiamics Moments...Want of unity in the method Virtual Velocities Theory of Couples.
Sh.are of equations in producin Connection of the concrete witl Equilibrium of tluids Hydrostatics Object...e-action e([uilibrium 1 the abstract quesi tion XXXVl CONTENTS.
PAGE Tlieory of rectilinear motion 137 Motion of a point...139 Motion of a system...139 Results 143 Statical theorems...143 Law of repose...143 Stability and instability of equilibrium...143 Dynamical theorems...144 Conservation of the motion of the centre of gravity.. 144 Principle of areas...144 The invariable plane...145 Moment of inertia...146 Principal axes...146 Conclusion...147 BOOK 11.
ASTRONOMY.
CHAPTER I.
GENERAL V I E AV.
Its Nature 148 Definition...149 Restriction...149 Means of exploration...150 Its rank...151 When it became a science...152 Reduction to a single law...152 Relation to other sciences...153 Divisions of the science...155 Celestial Geometry...155 Celestial Mechanics 155 CHAPTER II.
METHODS OF STUDV OF ASTRONOMY.
Section 1. Instmments 157 01)servation...157 Shadows...157 Artificial methods...158 The pendulum...159 Measurement of angles...159 Requisite corrections...161 CONTENTS. XXXvii PAGE Section 2. Refraction 161 Section 3. Parallax 163 Section 4. Catalogue of stars 165 CHAPTER III. geometrical phenomena of the heavenly bodies.
Section 1. Statical Phenomena 167 Two classes of phenomena...167 Planetary distances...167 Form and size...170 Planetary atmospheres...171 Earth's form and size...171 Means of discovery...172 Planetary motions..173 Rotation...174 Translation...174 Sidereal revolution...176 Motion of the Earth 176 Evidences of the Earth's motion...177 Ancient conceptions...177 How they gave way...178 Influence of centrifugal force upon gravity...179 Precession of the equinoxes...180 Retrogradations and stations of the planets...181 Aberration of light...181 Influence of scientific fact upon opinion...182 Kepler's Laws...184 Annual parallax...184 Circles I85 Kepler." 185 His three laws 186 First law 186 Second law...186 Third law I87 Three problems 188 Prediction of Eclipses...188 Transit of Venus...189 Foundation of Celestial Mechanics 189 Character of laws of Motion 190 Their history 191 Old difficulty explained...194 Term Attraction inadmissible 194 Extent of the demonstration.195 xxxviir CONTENTS.
Term Gravitation unobjectionable Gravitation is that of Molecules Secondary gravitation Domain of the laAV.
CELESTIAL STATICS.
Consummation by NeAvton Statical considerations First method of inquiry into masses Second method...Third method Section 1. Weight of tlic earth Section 2. Form of the i^lanets Difficulty of the inquiry. (xeometrical estimate Estimate from perturbations. Indirect estimate of the earth's form Hydrostatic theory of planetary forms Section 3. The Tides Question of the tides Theory of the tides. Influence of the Sun. Of the moon Composite influence. Kequisites for exactitude CHAPTER V.
celestial dynamics.
Perturbations.
Instantaneous.
Gradual perturbations Perturbations of translation Problem of three l)odies.
Centre of the Solar System Problem of the Planets.
Of the Satellites Of the Comets.
Perturbations of rotation.
The Planets.
The Satellites..
Device of an Invariable Plane Stability of our system.
Resistance of a Medium.
Independence of the solar system Achievements of celestial dynamics CONTENTS.
CHAPTER VI.
SIDEREAL ASTRONOMY AND COSMOGONY.
I'A(iE Multiple Stars "224 Our Cosmogony...226 Origin of Positive Cosmogony.
227 Cosmogony of Laplace 227 Recapitulation 230 BOOK III.
PHYSICS.
CHAPTER I.
GENERAL VIEW.
Imperfect condition of the Science 2.31 Its domain 232 Compared with Chemistry 232 Its generality 232 Dealing with masses or molecules.
233 Changes of arrangement or composition ( )f molecules 233 Description of Physics 234 Instruments...235 Methods of inquiry...235 Observation 235 Experiment...235 Application of INIathematical analysis 236 Encyclopiedic rank of Physics.
237 Relation to Astronomy.
237 To jNIathematics...238 To the other sciences 238 To human j^rogress...239 Human power of modifying jihenomena 239 Prevision imperfect...240 Characteristics of each science.
241 Philosophy of hypothesis.
241 Necessary condition...241 Two classes of hypothesis 242 First class indispensable.
243 Second class chimerical.
243 History of the second class 245 In Astronomy...246 In Physics 247 Rule of arrangement in Physics 247 Order 248 xl CONTENTS.
CHAPTER II.
BAROLOGY.
PAGE Divisions...249 Section 1. Statics 249 History 249 Cases of liquids...250 First case...251 Second case...252 Case of g.ises 253 History.253 Condition of the problem...254 Section 2. Dynamics 255 History 255 Fluids 257 Case of liquids 257 Existing state of Barology 258 CHAPTER III.
THERMOLOGY.
Its nature 259 History 259 Relation to Mathematics 260 Section 1. Mutual tlmruwlogical itiflncncc...260 Two parts...' 260 Mutual influence 260 Radiation of Heat 261 Propagation by contact...262 Condiictibility 263 Permeability 263 Penetrability 263 Specific Heat 264 Section 2. Constituent changes hy heat 265 Latent heat 266 Change of volume...266 Change in state of aggregation...267 Law of engagement and disengagement of heat...268 Vapours 268 Temperatures of ebullition...269 Hygrometrical equilil)riuni...269 Sections. The7'niolugy connected tvith analysis...270 Section 4. Terrestrial temperatures...271 Interior heat...272 Temperature of the planetary intervals...272 Conditions of the problem...273 CONTENTS.
xli CHAPTER IV.
ACOUSTICS.
Its nature Relation to the study of Inorganic bodies Relation to Physiology. To Mathematics Divisions...Section 1. Propagation of sound Eftect of atmospheric agitation Section 2. Intensity of sounds Section 3. Theory of tones. Composition of sounds Recapitulation...PAGE 274 274 275 CHAPTER V.
optics.
Hypotheses on the nature of Light Excessive tendency to systematize Divisions of Optics.
Irrelevant matters.
Theory of Vision Specitic Colour of bodies.
Section 1. Study of direct light Optics proj^er...Imperfections...Photometry...Section 2. Catoptrics.
Great law of reflection Law of absorption not found.
Section 3. Dioptrics Creat law of refraction Newton's discoveries on elementary colours CHAPTER VL electrology.
History 300 Condition 300 Arbitrary hypotheses 300 Relation to Mathematics 301 Unsound application...302 Sound application...302 Limits 302 Divisions 302 xlii CONTENTS.
Section 1. Electric production Causes of electrization Chemical action Theiniological action Friction Pressure.
Contact (Jther causes Instruments Section 2. Electric stedics Great law of Distril)ution Electric equilibrium.
Section 3. Electric dynamics Amjjere's experiments Conclusion of Physics PAGE BOOK IV. CHEMISTRY.
CHAPTER I.
Its nature...Great imperfection.
Capacities.
Object of Chemistry.
Specific character of its action Condition of action.
Rational definition.
Means of investigation Observation Experiment Comparison Chemical analysis and synthesis Rank of the science.
Relation to Mathematics.
To Astronomy.
To Physiolo_<,'y.
To Sociology.
Degree of possible perfection Intrusion of hypotheses.
Actual imperfection.
Comparative imperfection Relation to human progress Art of Nomenclature CONTENTS.
xliii State of chemical doctrine Divisions of the science...No organic chemistry...Princiijles of comp(jsition and decomposition PAGE 329 330 331 331 CHAPTER II.
INORGANIC CHEMISTRY.
Mode of beginning the study. Phirality of elements Classification of elements. Classification of Berzelius. Premature effort Requisite preparation as to method As to doctrine...First condition...Second condition...Method of analysis. Chemical dualism Law of double saline decomposition Chemical theory of air and water Air AVater CHAPTER III.
DOCTRINE OF DEFINITE PROPORTIONS.
Scope of the doctrine As to Doctrine.
As to Method.
Its history Richter's law.
Berth ollet's extension Dalton's further extension Atomic theory.
Extension by Berzelius.
Extension by Gay-Lussac.
WoUaston's verification.
Scope of application of Numerical Chemistry Objection of dissolution, Of metallic alloys Of organic substances Application of the principle of dualism xliv CONTENTS.
CHAPTER IV.
THE ELECTRO-CHEMICAL THEORY.
Relation of Electricity to Chemistry History of the case.
Nicholson's discovery Davy's discovery Berzelius's extension Synthetical process.
Recquerel's Study of Combustion Lavoisier's theory First division.
Berthollet's limitation Second division.
Difficulties of the theory Its position and powers I'AliK CHAPTER V.
ORGANIC CHEMISTRY.
Confusion of two kinds of fact.
Relation of Chemistry to Anatomy.
To Physiolof,''y...Partition of Organic Chemistry A])plication of dualism to organic compoundi- Summary of the Chapter...Summary of the Rook...375 377 37S 3S1 383 384 384 THE POSITIVE PHILOSOPHY OF AUGUSTE COMTE.
INTEODUCTION. CHAPTER I.
ACCOUNT OF THE AIM OF THIS WOKK. 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 doctrine 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 proposed study is vast and hitherto indeterminate, it is esjjecially important that the field of research should be marked out with all possible accuracy. For this purpose, I will glance at the considerations 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 progressive 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 intelligence, in all directions, and progress"™*" through all times, the discovery arises of a ° * great fundamental law, to which it is necessarily subject, and which has a solid foundation of pi'oof, both in the facts of our organization and in our historical experience. The law is this: — that each of our leading conceptions, — I. B 2 POSITIVE PHILOSOPHY.
each braucli of our knowledge, — passes successively through three different theoretical conditions: the Theological, or fictitious; the Metaphysical, or abstract; and the Scientific, or positive. In other words, the human mind, by its nature, employs in its progress three methods of philoso- phizing, the character of which is essentially different, and even radically oj^posed: viz., the theological method, the metaphysical, and the positive. Hence arise three philoso- phies, or general systems of conceptions on the aggregate of phenomena, each of which excludes the others. The first is the necessary point of departure of the human understanding; and the third is its fixed and definitive state. The second is merely a state of transition.
In the theological state, the human mind, ^ ' ^ • seeking the essential nature of beings, the first and final causes (the origin and i^urpose) of all effects, — in short, Absolute knowledge, — supposes all phenomena to be produced by the immediate action of suj)ernatural beings.
, „ In the metaphysical state, which is only a econ e. j^Q(j^f^(,g^^|Qjj^ Qf the first, the mind supposes, instead of supernatural beings, abstract forces, veritable entities (that is, personified 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. ™,., ^^ In 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 observation, duly com- bined, are the means of this knowledge. What is now understood when we speak 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.
The Theological system arrived at the of each.*'' ^''''"*' highest perfection 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 GROUNDS OF THE LAW OF PROGRESS. 3 way, in the last stage of the Metaphysical system, meu substitute 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 j^articular aspects of a single general fact; — such as Gravitation, for instance.
The importance of the working of this general law will be established hereafter. At j^resent, it must suffice to point out some of the grounds of it.
There is no science which, having attained the positive stage, does not bear marks of the Kw^^*^ *^ having passed through the others. Some time since it was (whatever it might be) composed, as we can now perceive, of metaphysical abstractions; 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 illus- tration, but an indirect evidence of that of the general mind. The j^oiut 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 metaphysician 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 rpi..g+jcni 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 rej^eated, since Bacon's time, that there can be no real knowledgs but that which is based on observed facts. This is incontestible, in our present advanced stage; but, if we look back to the primitive stage of human knowledge, we shall see that it 4 POSITIVE PHILOSOPHY, must have been otherwise then. If it is true that every theory must be based upon observed facts, it is equally true that facts cannot 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 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 human mind. It is remarkable that the most inaccessible questions, — those of the nature of beings, and the origin and purpose of phenomena, — 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 coxild have been no reception of a positive philosophy, whose function is to discover the laws of phenomena, and whose leading characteristic it is to 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 the external world, — a world destined wholly for our use, and involved in every way with our existence. The theological philosophy^ presenting this view, administered exactly the stimulus necessary to incite the human mind to the irksome labour 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 needing any such stimulus as wrought upon the imagina- METAPHYSICAL AND POSITIVE PHILOSOPHY. 5 tions of astrologers and alchemists. We have motive enough in the hope of discovering the laws of phenomena, with a view to the confii-mation or rejection of a theory. But 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 Berthollet on behalf of chemistry. Thus was a spon- taneous philosophy, the theological, the only possible be- ginning, 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 conceptions 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 supernatural 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 for 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 theological 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 Character of the Positive Philosophy is that it regards all the Positive phenomena as subjected to invariable natural Philosophy. 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 number. By speculating upon causes, we could solve no difficulty about 6 POSITIVE PHILOSOPHY.
oi'igin and purpose. Our real business is to analyse accu- rately 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, becaiise it connects under one head the whole immense variety of astronomical facts; exhibiting the constant tendency of atoms towards each other in direct proportion to their masses, and in inverse proportion to the squares of their distances; whilst the general fact itself is a mere extension of one which is per- fectly familiar to us, and which we therefore say 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. Theologians and metaphysicians may imagine and refine about such questions; 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 attraction: that is, that the two orders of j^henomena are identical; which is the point fi'om 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 pheno- mena 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 an universal ether. In treating his subject in the Positive method, he finds inexhaustible material for all his activity of research, without betaking himself to insoluble questions. History of the Before ascertaining the stage which the Positive Phi- Positive Philosophy has reached, we must losophy. bear in mind that the different kinds of our