THE LIBRARY 
 
 OF 
 
 THE UNIVERSITY 
 
 OF CALIFORNIA 
 LOS ANGELES
 
 THE 
 
 POSITIVE PHILOSOPHY 
 
 AUGUSTE COMTE.
 
 THE 
 
 POSITIVE PHILOSOPHY 
 
 OF 
 
 AUGUSTE COMTE. 
 
 FREELY TRANSLATED AND CON DENSED 
 
 ET 
 
 HARRIET MARTINEAU. 
 
 IN TWO VOLUMES. 
 VOL. I. 
 
 THIRD EDITION. 
 
 LONDON: 
 KEGAN TAUL, TEENCH, TEUBNER, & CO. I/P 
 
 PATERNOSTER HOUSE, CHARING CROSS ROAD. 
 1893.
 
 TliC riijlds (ij IrunslutkiH and of rcjir<idiictii,n arc reserved
 
 B 
 
 THE PUBLISHERS' ANNOUNCEMENT, \^^i 
 
 V, 1 
 
 T^HE increasing desire on the part of the public to become 
 more thoroughly acquainted with Comte's Philosophy is 
 shown by the fact of two editions having been sold since his 
 death, while not one edition was exhausted during his lifetime. 
 The great extent of the work has been, and will always be, a 
 serious obstacle to its general acceptance ; but this obstacle is 
 removed by the condensation of the six volumes into two, which 
 Miss Martineau has effected in the work we now re-issue. It 
 is not for us to speak of the execution of this work; but we 
 may fitly mention, that it was so highly approved by the author 
 himself, that, in his annual issue of his catalogue of works 
 sanctioned by him, he substituted Miss Martineau's version 
 for the original. In consequence of this her version has been, 
 since his death, retranslated into French for the sake of its 
 diifasion among the author's own countrymen.* 
 
 TRUBNEr^ & CO. ■ 
 
 57 & 59 LuDGATE Hill, London, 
 March 1875. 
 
 * La Philosophie Positive d'Auguste Comte, condensde par Miss Harriet Mar- 
 tineau. Traduction frangaise de M. Ch. Avezac Lavigne. 2 octavo vols, of 700 pp. 
 each. Bordeaux, Feret & Fils. Paris, Dunod. 1871. 
 
 VOL. L
 
 PREFACE. 
 
 It may appear strange that, in these days, when tlie French lan- 
 guage is ahiiost as familiar to English readers as their own, I should 
 have spent many months in rendering into English a work which 
 presents no diflficulties of language, and which is undoubtedly 
 known to all philosophical students. Seldom as Comte's name is 
 mentioned in England, there is no doubt in the minds of students 
 of his great work that most or all of those who have added sub- 
 stantially 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 ofifending the prejudices 
 of the society in which they live. AVhichever way we look over the 
 whole field of science, we see the truths and ideas presented by 
 Comte cropping out from the surface, and tacitly recognized as 
 the foundation of all that is systematic in our knowledge. This 
 being the case, it may appear to be a needless labour 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 labour, while so much 
 Avork 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 honour, sooner or later. 
 Before the end of the century, society at large will have become 
 aware that this work is one of the chief honours of the century, and 
 that its author's name will rank with those of the worthies who 
 liave illustrated former ages : but it does not seem to me right to 
 assist in delaying the recognition till the author of so uoble a
 
 iv PRE FA CE. 
 
 service is beyond the reacli of our gratilude and honour : and that 
 it is demorah'zing to ourselves to accept and use such a boon as he 
 has given us in a silence which is in fact ingratitude. His honours 
 we cannot share : they are his own and incommunicable. His 
 trials we may share, and, by sharing, lighten; and he has 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 undertaking to reproduce his work in England in a form as 
 popular as its nature admits. 
 
 A stronger reason was that M. Comte's work, in its original form, 
 does no justice to its importance, even in France ; and much less 
 in 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 
 necessitated 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 scrui)ulous 
 honesty leads him to guard his enunciations with epithets so con- 
 stantly repeated, that though, to his own mind, they are necessary 
 in each individual instance, they become wearisome, especially 
 towards 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 condense 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 these two 
 volumes occupies in the original 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 conviction 
 of our need of this book in my own country, in a form which renders 
 it accessible to the lar<jcest number of iiilelliirent readers. We are
 
 PRE FACE. V 
 
 liviug in a remarkable lime, when tlie conflict of opinions renders a 
 firm foundation of knowledge indispensable, not only to our intel- 
 lectual, 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 science are 
 confounded together, and even mixed up with their application to 
 the arts,. and with natural history; and while the researches 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 students whose business 
 it is, not to explore, but to receive. . The growth of a scientific 
 taste among the working classes of this country 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 scientific 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 fbi- the good of nation or 
 race is that men should be adrift for want of an anchorasre for 
 their convictions. I believe that no one questions that a very large 
 proportion of our people are now so adrift. With pain and fear, 
 we see that a multitude, who might and should be among the wisest 
 and best of our citizens, are alienated for ever from the kindof failh 
 which sufficed for all in an organic period which has passed away, 
 while no one has presented to them, and they cannot obtain for 
 themselves, any ground of conviction as firm and clear as that 
 which sufficed for our fathers in their day. The moral dangers of 
 sueli a. state of fluctuation as has thus arisen are fearful in the ex- 
 treme, whether the transition stage from one order of convictions 
 to another be long or short. The work of M. Comte is unquestion- 
 ably the greatest single effort that has been made to obviate this 
 kind of danger ; and my deep persuasion is that it will be found 
 to retrieve a vast amount of wandering, of unsound speculation, of 
 listless or reckless doubt, and of moral uncertainty and depression.
 
 vi PREFACE. 
 
 Whatever else may be thought of the work, it will not be denied 
 that it ascertains with singular sagacity and soundness the founda- 
 tions of human knowledge, and its true object and scope ; and that 
 it establislies the true filiation of the sciences within the boundaries 
 of its own principle. Some may wish to interpolate this or that; 
 some to am])lify, 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 its 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 Philosophy 
 unfolds itself in order before their eyes, they will, I am persuaded, 
 find there at least a resting-place for their thought, — a rallying- 
 point of their scattered speculations, — and possibly an immovable 
 basis for their intellectual and moral convictions. The time will 
 come when the book itself will, for a while, be most discussed on 
 account of the deficiencies which M. Comte himself presses on our 
 notice ; and when his philosophy will sustain amplifications 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 it is only 
 a true book that could survive to be so treated : but, in the mean- 
 time, it gives us the basis that we demand, and the principle of 
 action that we want, and as much instruction in the procedure, and 
 information as to what has 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 mathematical 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 conditions 
 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
 
 PREFACE. vii 
 
 would avail more to extinguish the anarchy of popular and secilonal 
 opinion in this country than any other influence that has yet heeu 
 exerted, 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. 
 A few months afterwards, an 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, iu a 
 subsequent letter, that he had wished, for many years, to do wliat 
 I was then attempting, and had been prevented only by ill health. 
 My estimate of M. Comte's work, and my expectations from its 
 introduction into England in the form of a condensed translation, 
 were fully sliared by him ; and, to my utter amazement, he sent 
 me, as the first act of our correspondence, an order on his bankers 
 for £500. There was time, before his lamented death, for me to 
 communicate to him my views as to the disposal of this money, and 
 to obtain the assurance of his approbation. We planned that the 
 larger proportion of it should be expended 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 promulga- 
 tion 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 witli 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 ])hilosophy is, 
 accompanied by illustrations of my own devising. A wider expec- 
 tation 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 temptation of entering my protest, here and there, against a
 
 viii PREFA CE. 
 
 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 contioversy. 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, 
 and out of place to insert those of others. Those others can speak 
 for themselves, and tiie readers of the book can criticize 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 endeavoured 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 convey 
 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 labour 
 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 
 sections, except the omission of Comte's speculation on the possible 
 mathematical verification of Laplace's Cosmogony. But more had 
 1() be done with regard to the treatment of Piiysics. 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 fiivst, the author explains the fact, from the nature of the 
 case, — that Physics is rather a repository of somewhat fragmen- 
 tary portions of physical science, the correlation of which is not yet
 
 PREFACE. ix 
 
 cleai", than a single ciicumscribed science. And we must say for 
 liim, in regard to the other kind of imperfection, that such advances 
 liave been made in ahnost every department of Physics since his 
 second vohnue was pnbhslied, that it would be unfair to present 
 what he wrote under that head in 1835 as what he would have to 
 say now. Tiie 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 princi})les should remain. But as 
 the system of Positive Philosophy is much less an Expository than 
 a Critical work, the latter alternative alone seemed open, under due 
 consideration of jnstice 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 Philosoj)hy, or assents to 
 other conclusions held of main importance by the disciples of the 
 Positive Philosophy. The contrar}', 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 bomid to take the entiie resj)onsi- 
 bility, — 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 
 Ei)glisli 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 various 
 reasons, I thiidv it best to stop here, feeling assured that if this 
 AVork 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- 
 opiiy 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 
 version. It cannot be otherwise. The theological world cannot 
 but hate a book which treats of theological belief as a transient
 
 X PREFACE. 
 
 state of the human Qiind. 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 
 inwards, and not from within outwards, must necessarily think ill 
 of a work which exposes the futility of their method, and the worth- 
 lessness of the results to which it leads. As M. Comte treats of 
 theology and metaphysics as destined to pass away, theologians and 
 metaphysicians must necessarily abhor, dread, and despise his work. 
 They merely express their own natural feelings on behalf of the 
 objects 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 
 theology and metaphysics, and, finding what they are now worth, 
 have risen above them — will pronounce a very different judgment 
 on the contents of this book, though no appeal for such a judgment 
 is made in it, and this kind of discussion is nowhere expressly pro- 
 vided for. To those who have learned the difficult task of postpon- 
 ing dreams to realities till the beauty of reality is seen in its full 
 disclosure, while that of dreams melts into darkness, the moral 
 charm of this work will be as impressive as its intellectual satis- 
 factions. The aspect in which it presents Man is as favourable 
 to his moral discipline, as it is fresh and stimulating to his 
 intellectual taste. We find ourselves suddenly living and mov- 
 ing in the midst of the universe, — as a part of it, and not as its 
 aim and object. We find ourselves living, not under capricious and 
 arbitrary conditions, unconnected with the constitution and move- 
 ments of the whole, but under great, general, invariable laws, which 
 operate on us as a part of the whole. Certainly, I can conceive of 
 no instruction so favourable to aspiration as that which shows us 
 how great are our faculties, 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 indi- 
 cations 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 resig-
 
 PREFACE. XI 
 
 nation tluit are tlie natural consequence of pursuits so pure, anil 
 aims so true, as those of Positive Philosophy. Pride of intel- 
 lect 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 imaginations, to that which 
 is, and may be, referred to other judgments. If it be desired 
 to extinguif^h 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 Positive Phi- 
 losophy, with its train of noble truths and irresistible inducements. 
 The prospects it opens are boundless ; for among the laws it estab- 
 lishes that of human progress is conspicuous. The virtues it 
 fosters are all those of which Man is capable ; and the noblest are 
 those which are more eminently fostered. The habit of truth-seelc- 
 ing and truth-speaking, and of true dealing with self and with all 
 things, is evidently a primary requisite ; and tiiis habit once per- 
 fected, 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 w-hat it is that they are call- 
 ing in question. My hope is that this book may achieve, besides 
 the purposes entertained by its Author, the one more that he did 
 not intend, of conveying a sufiEicient rebuke to those who, in theo- 
 logical selfishness or metaphysical pride, speak evil of a philosophy 
 which is too lofty and too simple, too humble and too generous, 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 favourable to those virtues whose 
 repression would be incompatible with progress. 
 
 Autumn, 1853.
 
 CONTENTS. 
 
 INTRODUCTION. 
 
 CHAPTER I. 
 
 ACCOUNT OF TBE AIM OF THIS WORK. VIEW OF THE NATURE AND 
 
 IMPORTANCE OF THE POSITIVE PHILOSOPHY. 
 
 Preliminary survey . 
 
 Law of hum.TU development 
 
 First stage 
 
 Second stage . 
 
 Third stage 
 
 Ultimate point of each 
 
 Evidences of the Law 
 
 Actual .... 
 
 Theoretical 
 
 Character of the Positive Philosojihy . 
 
 History of the Positive Philosophy 
 
 New department of Positive Philosophy 
 
 Social Physics ..... 
 
 Secondary aim of this work 
 
 To review the philosophy of the sciences 
 
 Glance at specialitj'^ .... 
 
 Proposed new class of students . 
 
 Advantages of the Positive Philosophy 
 
 1. Llustrates Litellectual function 
 
 2. Must regenerate Education . 
 
 3. Advances Sciences by combining them 
 
 4. Must reorganize Society 
 
 No hope of reduction to a single Law 
 
 PAGE 
 1 
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 2 
 2 
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 2 
 3 
 3 
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 5 
 5 
 6 
 6 
 7 
 7 
 7 
 8 
 9 
 
 11 
 11 
 12 
 13
 
 CONTENTS. 
 
 CHAPTER II. 
 
 VIEW OF THE HIERARCHY OF THE POSITIVE SCIENCES. 
 
 Failure of proposed classifications 
 True principle of classification . 
 Boundaries of our field 
 Theoretical Inquiry . 
 Abstract science 
 Concrete science 
 Difficulty of classification . 
 Historical and Dogmatic Methods 
 True principle of classification . 
 Characters .... 
 
 1. Generality .... 
 
 2. Independence 
 
 Inorganic and Organic phenomena 
 I. Inorganic .... 
 
 1. Astronomy .... 
 
 2. Physics .... 
 
 3. Chemistry .... 
 II. Organic .... 
 
 1. Physiology .... 
 
 2. Sociology .... 
 Five Natural Sciences : their filiation 
 Filiation of their parts 
 Corroborations .... 
 
 1. This classification follows the order of di 
 
 2. Solves heterogeneousness 
 
 3. Marks relative perfection of sciences 
 
 4. Effect on Education 
 Effect on Method . 
 Orderly study of sciences 
 Mathematics . 
 A department . 
 A basis . 
 An instrument . 
 A double science 
 
 Abstract Mathematics, an instrument 
 Concrete Mathematics, a science 
 Mathematics pre-eminent in the scale 
 
 closure of
 
 CONTENTS. 
 
 BOOK I. 
 MATHEMATICS. 
 
 CHAPTER I. 
 
 
 
 
 
 MATHEMATICS, ABSTRACT AND CONCUETE. 
 
 
 PACK 
 
 Description of Mathematics .... 
 
 28 
 
 Object of Mathematics 
 
 
 
 
 
 
 29 
 
 General ^Method 
 
 . 
 
 
 
 
 
 29 
 
 Examples 
 
 
 
 
 
 
 30 
 
 True Definition of Mathematics 
 
 
 
 
 
 
 31 
 
 Its two Pakts 
 
 
 
 
 
 
 32 
 
 Their different objects 
 
 
 
 
 
 
 32 
 
 Their different natures 
 
 
 
 
 
 
 33 
 
 Concrete Mathematics 
 
 
 
 
 
 
 33 
 
 Abstract Mathematics 
 
 
 
 
 
 
 34 
 
 Extent of its domain 
 
 
 
 
 
 
 35 
 
 Its Universality- 
 
 
 
 
 
 
 35 
 
 Its limitations . 
 
 
 
 
 
 35 
 
 CHAPTER 11. 
 
 GENERAL VIEW OF MATHEMATICAL ANALYSIS. 
 
 Analysts 39 
 
 True idea of an equation . 
 
 
 
 . 
 
 39 
 
 Abstract functions . 
 
 
 
 
 40 
 
 Concrete functions . 
 
 
 
 . 
 
 40 
 
 Two parts of the Calculus . 
 
 
 
 , 
 
 41 
 
 Algebra .... 
 
 
 
 
 41 
 
 Arithmetic 
 
 
 
 . 
 
 41 
 
 Its extent 
 
 
 
 , 
 
 41 
 
 Its nature 
 
 
 
 , 
 
 42 
 
 Algebra .... 
 
 
 
 , , 
 
 43 
 
 Creation of new functions . 
 
 
 
 , 
 
 43 
 
 Finding equations between auxiliary quantities . 
 
 
 43 
 
 Division of the Calculus of functions 
 
 
 44 
 
 Section 1. Ordinary Analysis, or Calculus of Direct 
 
 Functions 
 
 45 
 
 Its object ....... 
 
 
 45 
 
 Classification of Equations 
 
 . 
 
 . 
 
 45 
 
 Algebraic equations . 
 
 
 . 
 
 45 
 
 Algebraic resolution of equations 
 
 . 
 
 . 
 
 4G 
 
 Our existing knowledge 
 
 
 
 
 
 
 47
 
 CONTENTS. 
 
 ISTuraerical 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 fluents . 
 Lagrange's Method 
 Identity of the three methods 
 Their comparative value . 
 The Differential and Integral Calculus 
 Its two parts . 
 Their mutual relations 
 Cases of union of the two 
 Cases of the Differential calculus alone 
 Cases of the Integral calculus alone 
 llie Differential Calculus . 
 Two portions .... 
 Subdivisions .... 
 Reduction to the elements 
 Transformation of derived functions for new variables 
 Analytical applications 
 ^'he Integral Calculus 
 Its divisions 
 Subdivisions . 
 One variable or several 
 Orders of differentiation 
 Quadratures 
 Algebraic functions . 
 Transcendental functions 
 Singular Solutions 
 Definite integrals 
 Prospects of the Integral Calculus 
 Calculus of Variations 
 I'roblems giving rise to this calculus 
 Other applications 
 liclation to the ordinary calculus 
 
 of In 
 
 direct
 
 CONTENTS. 
 
 CHAPTER III. 
 
 GKNERAL 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 modern geometry 
 
 Geometry of the ancients . 
 
 Geometry of the right line 
 
 Graphical solutions . 
 
 Descriptive Geometry 
 
 Algebraic solutions . 
 
 Trigonometry .... 
 
 Modern, or Analytical Geometry 
 
 Analytical representation of figures 
 
 Position ..... 
 
 Position of a point . 
 
 Plane curves .... 
 
 Expression 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, and of Equations by 
 
 Surfaces .... 
 Curves of double curvature 
 Imperfections of Analytical Geometry 
 Imperfections of Analysis . 
 
 PAGE 
 
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 84 
 85 
 
 86 
 87 
 87 
 87 
 
 CHAPTER IV. 
 
 RATIONAL MECHANICS. 
 
 Its nature ...... 
 
 Its characters ...... 
 
 Its object ...... 
 
 Matter not inert in Physics 
 VOL. I. 
 
 89 
 90 
 90 
 91
 
 CONTENTS. 
 
 Supposed inert in Mechanics 
 
 Field of Kational Mechanics 
 
 Three Laws of Motion 
 
 Law of inertia . . 
 
 Law of equality of action and re-action 
 
 Law of co-existence of motions 
 
 Two Primary divisions 
 
 Statics and Dynamics 
 
 Secondary divisions . 
 
 Solids and Fluids 
 
 Section 1. Statics . 
 
 Converse methods of treatment 
 
 First method 
 
 Statics by itself 
 
 Second method 
 
 Statics through Dynamics . 
 
 Moments 
 
 Want of unity in the method 
 
 Virtual Velocities 
 
 Theory of Couples . 
 
 Share of equations in producing equilibrium 
 
 Connection of the concrete with the abstract question 
 
 Equilibrium of fluids 
 
 Hydrostatics ..... 
 
 Liquids ...... 
 
 Gases ...... 
 
 Section 2. Dynamics 
 
 Object ...... 
 
 Theory of rectilinear motion 
 
 Motion of a point .... 
 
 Motion of a system .... 
 
 D'Alembert's principle 
 
 Results ...... 
 
 Statical theorems .... 
 
 Law of repose ..... 
 
 Stability and instability of equilibrium 
 Dynamical theorems .... 
 
 Conservation of the motion of the centre of gravity 
 
 Principle of areas 
 
 The invariable Plane 
 
 Moment of inertia 
 
 Principle axes . 
 
 Ccmclusion
 
 CONTENTS. 
 
 XIX 
 
 BOOK 11. 
 
 ASTKONOMY. 
 
 CHAPTER I. 
 
 GENERAL VIEW. 
 
 Its Nature 
 
 Definition 
 
 "Restriction 
 
 Means of exploration 
 
 Its rank . 
 
 When it became a science 
 
 Reduction to a single law 
 
 Relation to other sciences 
 
 Divisions of the science 
 
 Celestial Geometry . 
 
 Celestial Mechanics . 
 
 CHAPTER II. 
 
 METHODS OF STUDY OF ASTRONOMY, 
 
 114 
 115 
 
 115 
 116 
 116 
 117 
 117 
 118 
 119 
 119 
 120 
 
 Section 1. Instruments . . . . . . , ' . 121 
 
 Observation . . . . . . . . . .121 
 
 Shadows 121 
 
 Artificial methods . . . . . . . . .122 
 
 The pendulum 122 
 
 Measurement of angles . . . . . . . .123 
 
 Requisite corrections . . . . . . . .124 
 
 vSection 2. Refraction . . . . .' . . .124 
 
 Section 3. Parallax . . . . . . . .125 
 
 Section 4. Catalogue of stars ....... 127 
 
 CHAPTER III. 
 
 GEOMETRICAL PHENOMENA OF THE HEAVENLY BODIES. 
 
 Section 1. Statical Phenomena . . . . . .129 
 
 Two classes of phenomena . . . . . . .129 
 
 Planetary distances . . . . . . . . .129 
 
 Form and size .......... 131 
 
 Planetary atmospheres . . . . . . . .132 
 
 Earth's form and size . . . . . . . .132 
 
 Means of discovery . . . . . . . . .132
 
 CONTENTS. 
 
 Planetary motions . 
 
 Rotation ..... 
 
 Translation .... 
 
 Sidereal revolution . 
 
 Motion of the Earth . 
 
 Evidences of the Earth's motion 
 
 Ancient conceptions . 
 
 How they gave way . 
 
 Earth's rotation 
 
 Influence of centrifugal force upon gravity 
 
 Earth's translation . 
 
 Precession of the equinoxes 
 
 Retrogradations and stations of the planets 
 
 Aberration of light . 
 
 Influence of scientific fact upon opinion 
 
 Kepler s Laws .... 
 
 Annual parallax 
 
 Circles ..... 
 
 Kepler ..... 
 
 His three laws .... 
 
 First law .... 
 
 Second law .... 
 
 Third law .... 
 
 Three problems 
 
 Prediction of Eclipses 
 
 Transit of Venus 
 
 Foundation of Celestial Mechanics 
 
 Section 2. Dynamical Phenomena 
 
 Gravitation 
 
 Character of laws of Motion 
 
 Their history . 
 
 JSTewton's demonstration . 
 
 Old difficulty explained 
 
 Term Attraction inadmissible 
 
 Extent of the demonstration 
 
 Term Gravitation unobjectionable 
 
 Gravitation is that of Molecules 
 
 Secondary gravitation 
 
 Domain of the law .
 
 CONTENTS. 
 
 CHAPTER IV. 
 
 CELESTIAL STATICS. 
 
 Consummation by Newton 
 
 Statical considerations 
 
 First method of inquiry into masses 
 
 Second method 
 
 Third method .... 
 
 Si-CTioN 1. Weight of the earth 
 
 Section 2. Form of the planets 
 
 difficulty of the inquiry . 
 
 Geometrical 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 
 
 Requisites for exactitude 
 
 CHAPTER V. 
 
 CELESTIAL DYNAMICS. 
 
 Perturbations . . 
 
 Instantaneous . 
 
 Gradual perturbations 
 
 Perturbations of translation 
 
 Problem of three bodies 
 
 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 
 
 PAGE 
 
 154 
 154 
 154 
 
 155 
 155 
 156 
 157 
 
 157 
 157 
 158 
 158 
 158 
 158 
 158 
 159 
 160 
 160 
 160 
 161 
 
 162 
 
 162 
 163 
 164 
 164 
 165 
 165 
 165 
 166 
 167 
 167 
 168 
 168 
 169 
 169 
 170 
 171
 
 XXll 
 
 CONTENTS. 
 
 CHAPTER VI. 
 
 SIDEREAL ASTRONOMY AND COSMOGONY. 
 
 Multiple Stars .... 
 
 
 
 
 172 
 
 Our Cosmogony .... 
 
 
 
 . 174 
 
 Origin of Positive Cosmogony . 
 
 
 
 . 174 
 
 Cosmogony of Laplace .... 
 
 
 
 . 174 
 
 Recapitulation ..... 
 
 
 
 . 176 
 
 BOOK III. 
 
 PHYSICS. 
 
 CHAPTER I. 
 
 GENERAL VIEW. 
 
 Imperfect condition of the Science 178 
 
 Its domain ..... 
 
 
 
 
 178 
 
 Compared with Chemistry 
 
 
 
 
 179 
 
 Its generality ..... 
 
 
 
 
 179 
 
 Dealing with masses or molecules 
 
 
 
 
 179 
 
 Changes of arrangement or composition of i 
 
 molecules 
 
 
 
 . 180 
 
 Description of Physics 
 
 
 
 
 . 180 
 
 Instruments ..... 
 
 
 
 
 . 181 
 
 Methods of inquiry .... 
 
 
 
 
 181 
 
 Observation ..... 
 
 
 
 
 181 
 
 Experiment ..... 
 
 
 
 
 181 
 
 Application of Mathematical analysis . 
 
 
 
 
 181 
 
 Encyclopaedic rank of Physics . 
 
 
 
 
 182 
 
 Relation to Astronomy 
 
 
 
 
 182 
 
 To Mathematics ..... 
 
 
 
 
 183 
 
 To the other sciences 
 
 
 
 
 183 
 
 To human progress .... 
 
 
 
 
 184 
 
 Human power of modifying phenomena 
 
 
 
 
 184 
 
 Prevision imperfect .... 
 
 
 
 
 185 
 
 Characteristics of each science . 
 
 
 
 
 185 
 
 Philosophy of hypothesis . 
 
 
 
 
 185 
 
 Necessary condition .... 
 
 
 
 
 186 
 
 Two classes of hypothesis . 
 
 
 
 
 186 
 
 First class indispensable . 
 
 
 
 
 187 
 
 Second class chimerical 
 
 
 
 
 187
 
 
 CONTENTS. 
 
 
 
 
 xxiii 
 
 PAOK 
 
 History of the second class ....... 188 
 
 In Astronomy ....... 
 
 
 
 189 
 
 In Physics ....... 
 
 
 
 190 
 
 Rule of Arrangement in Physics 
 
 
 
 190 
 
 Order 
 
 
 
 . 191 
 
 CHAPTER IL 
 
 B.-^EOLOGY. 
 
 Divisions .,.......• 1*^^ 
 
 Section 1. Statics . 
 
 
 
 
 
 192 
 
 History . 
 
 
 
 
 
 . 192 
 
 Cases of liquids 
 
 
 
 
 
 193 
 
 First case 
 
 
 
 
 
 193 
 
 Second case 
 
 
 
 
 
 194 
 
 Case of gases . 
 
 
 
 
 
 195 
 
 History .... 
 
 
 
 
 
 . 195 
 
 Condition of the problem 
 
 
 
 
 
 196 
 
 Section 2. Dynamics 
 
 
 
 
 
 196 
 
 History . 
 
 
 
 
 
 197 
 
 Fluids .... 
 
 
 
 
 
 198 
 
 Case of liquids 
 
 
 
 
 
 198 
 
 Existing state of Barology 
 
 
 
 
 199 
 
 CHAPTER III. 
 
 THERMOLOGY. 
 
 Its Nature 200 
 
 History . . . . . 
 
 
 
 . 200 
 
 Relation to Mathematics .... 
 
 
 
 
 . 201 
 
 Section 1. Mutual thermological influence . 
 
 
 
 
 201 
 
 Two parts 
 
 
 
 
 201 
 
 Mutual influence ..... 
 
 
 
 
 201 
 
 Radiation of Heat ..... 
 
 
 
 
 202 
 
 Propagation by contact .... 
 
 
 
 
 202 
 
 Conductibility ...... 
 
 
 
 
 203 
 
 Permeability ...... 
 
 
 
 
 203 
 
 Penetrability ...... 
 
 
 
 
 203 
 
 Specific Heat ...... 
 
 
 
 
 204 
 
 Section 2. Constituent changes by heat 
 
 
 
 
 205 
 
 Latent heat ...... 
 
 
 
 
 205 
 
 Change of volume 
 
 
 
 
 205 
 
 Change in state of aggrega 
 
 tion . 
 
 
 
 
 20 ()
 
 xxiv CONTENTS. 
 
 Law of engagement and disengagement of heat 
 Vapours .... . . 
 
 Temperatures of ebullition 
 
 Hygrometrical equilibrium 
 
 Section 3. Thermology connected with analysis 
 
 Section 4. Terrestrial temperattLres . 
 
 Interior heat ...... 
 
 Temperature of the planetary intervals 
 Conditions of the problem .... 
 
 CHAPTER IV. 
 
 ACOUSTICS. 
 
 Its Nature ..... 
 Relation to the study of inorganic bodies 
 Relation to Physiology 
 To Mathematics 
 
 Divisions .... 
 
 Section 1. Propagation of sound 
 Effect of atmospheric agitation 
 Section 2. Intensity of sounds . 
 Section 3. Theory of tones 
 Composition of sounds 
 Recapitulation .... 
 
 CHAPTER V. 
 
 OPTICS. 
 
 Hypotheses on the nature of Light 
 
 Excessive tendency to systematize 
 
 Divisions of Optics . 
 
 Irrevelant matters 
 
 Theory of Vision 
 
 Specific Colour of bodies . 
 
 Section 1. Study of direct light 
 
 Optics proper . 
 
 Imperfections . 
 
 Photometry 
 
 Section 2. Catoptrics 
 
 Great law of reflection 
 
 Law of absorption not found 
 
 Section 3. Dioptrics
 
 CONTENTS. 
 
 Great law of refraction . . . . 
 
 Newton's discoveries on elementary colours . 
 Section 4. Diffraction .... 
 
 CHAPTER VI. 
 
 ELECTROLOGY, 
 
 History . 
 
 Condition 
 
 Arbitrary hypotheses 
 
 Relation to Mathematics 
 
 Unsound application 
 
 Sound application 
 
 Limits 
 
 Divisions . 
 
 Section 1. Electric production 
 
 Causes of electrization 
 
 Chemical action 
 
 Thermological action 
 
 Friction . 
 
 Pressure . 
 
 Contact . 
 
 Other causes 
 
 Instruments 
 
 Skction 2. Electric statics 
 
 Great law of distribution 
 
 Electric equilibrium . 
 
 Section 3. Electric dynamics 
 
 Ampere's experiments 
 
 Conclusion of Physics 
 
 227 
 
 228 
 229 
 
 230 
 230 
 23U 
 231 
 231 
 231 
 232 
 232 
 232 
 232 
 232 
 232 
 233 
 233 
 233 
 233 
 233 
 234 
 234 
 235 
 235 
 235 
 237 
 
 BOOK IV. 
 
 CHEMISTRY. 
 
 CHAPTER I. 
 
 Its nature 
 
 Great imperfection 
 
 Capacities 
 
 Object of Chemistry . 
 
 Specific character of its action 
 
 Condition of action . 
 
 239 
 240 
 240 
 240 
 241 
 242
 
 CONTENTS. 
 
 Definition , 
 
 Elements .... 
 
 Combination 
 
 Rational definition 
 
 Means of investigation 
 
 Observation 
 
 Experiment 
 
 Comparison 
 
 Chemical analysis and synthesis 
 
 Rank of the science . 
 
 Relation to Mathematics . 
 
 To Astronomy . 
 
 To Physiology . 
 
 To Sociology 
 
 Degree of possible perfection 
 
 Intrusion of hypotheses 
 
 Actual imperfection . 
 
 Comparative imperfection 
 
 Relation to human progress 
 
 Art of Nomenclature 
 
 State of chemical doctrine 
 
 Divisions of the science 
 
 No organic chemistry 
 
 Principles of composition and decomposition 
 
 CHAPTER 11. 
 
 INORGANIC CHEMISTRY. 
 
 Mode of beginning the study 
 
 Plurality of elements 
 
 Classification of elements 
 
 Classification of Berzelius 
 
 Premature efi'ort 
 
 Requisite preparation as to method 
 
 As to doctrine . 
 
 First condition 
 
 Second condition 
 
 Method of analysis . 
 
 Chemical dualism 
 
 Law of double saline decompositions 
 
 Chemical theory of air and water 
 
 Air ..... 
 
 Water
 
 CONTENTS. 
 
 CHAPTER III. 
 
 DOCTRINE OF DEFINITE PROPORTIONS, 
 
 
 
 
 
 
 
 
 PAGE 
 
 Scope of the doctrine ........ 267 
 
 As to Doctrine ..... 
 
 
 
 
 . 268 
 
 As to Method ...... 
 
 
 
 
 . 268 
 
 Its history ...... 
 
 
 
 
 . 268 
 
 Richter's law ...... 
 
 
 
 
 . 268 
 
 Berthollet's extension .... 
 
 
 
 
 . 269 
 
 Dalton's further extension 
 
 
 
 
 269 
 
 Atomic theory ..... 
 
 
 
 
 269 
 
 Extension by Berzelius .... 
 
 
 
 
 . 270 
 
 Extension by Gay-Lussac .... 
 
 
 
 
 . 271 
 
 Wollaston's verification .... 
 
 
 
 
 271 
 
 Scope of application of Numerical Chemistry 
 
 
 
 
 272 
 
 Objection of dissolution ..... 
 
 
 
 
 273 
 
 Of metallic alloys ..... 
 
 
 
 
 274 
 
 Of organic substances .... 
 
 
 
 
 274 
 
 Application of the principle of dualism 
 
 
 
 
 274 
 
 CHAPTER IV. 
 
 THE ELECTRO-CHEMICAL THEORY. 
 
 Relation of Electricity to Chemistry 278 
 
 History of the case . 
 
 
 
 
 
 
 278 
 
 Nicholson's discovery 
 
 
 
 
 
 
 
 278 
 
 Davy's discovery 
 
 
 
 
 
 
 
 279 
 
 Berzelius's extension 
 
 
 
 
 
 
 
 279 
 
 Synthetical process . 
 
 
 
 
 
 
 
 279 
 
 Becquerel's 
 
 
 
 
 
 
 
 280 
 
 Study of Combustion 
 
 
 
 
 
 
 
 280 
 
 Lavoisier's theory 
 
 
 
 
 
 
 
 280 
 
 First division 
 
 
 
 
 
 
 
 281 
 
 Berthollet's limitation 
 
 
 
 
 
 
 
 281 
 
 Second division 
 
 
 
 
 
 
 
 282 
 
 Difficulties of the theory 
 
 
 
 
 
 
 
 284 
 
 Its position and powers 
 
 
 
 
 
 
 
 28G 
 
 CHAPTER V. 
 
 ORGANIC CHEMISTRY. 
 
 Confusion of two kinds of fact . 
 Relation of Chemistry to Anatomy 
 To Physiology .... 
 
 287 
 288 
 289
 
 CONTENTS. 
 
 Partition of Organic Chemistry . 
 Ap23lication of dualism to organic compounds 
 Summary of the Chapter .... 
 Summary of the Book .... 
 
 291 
 293 
 294 
 294 
 
 BOOK V. 
 BIOLOGY. 
 
 CHAPTER I. 
 
 GENERAL VIEW OF BIOLOGY. 
 
 Opposite starting-points in philosophy 
 
 Starting-point of Physiology 
 
 Its present imperfection 
 
 Its field . 
 
 Relation to Medicine . 
 
 Its object . 
 
 Idea of Life 
 
 L)e Blainville's definition 
 
 Definition of Biology 
 
 Means of investigation 
 
 Observation . 
 
 Artificial apparatus . 
 
 Chemical exploration. 
 
 Experiment 
 
 By affecting the organism 
 
 By afi:ecting the medium 
 
 Comparative experiment 
 
 Pathological investigation 
 
 Comparison 
 
 Five kinds of comparison 
 
 Comparison of parts of the same organism 
 
 Of phases of the same organism 
 
 Of different organisms 
 
 Relation of Biology to other sciences 
 
 To Chemistry . 
 
 To Physics 
 
 To Astronomy . 
 
 To Mathematics 
 
 295 
 296 
 296 
 297 
 297 
 298 
 298 
 300 
 301 
 303 
 303 
 303 
 304 
 304 
 305 
 305 
 306 
 306 
 308 
 310 
 311 
 311 
 312 
 314 
 314 
 315 
 316 
 320
 
 CONTENTS. 
 
 XXIX 
 
 Use of scientific fictions .... 
 Condition and prospects of the science 
 Requisite Education ..... 
 Art of Classification ..... 
 Influence of Biology upon the Positive Spirit 
 Distribution of the science .... 
 Where to beufin ..... 
 
 PAOK 
 
 323 
 324 
 324 
 325 
 328 
 329 
 
 CHAPTER n. 
 
 ANATOMICAL PHILOSOPHY. 
 
 Development of Statical biology- 
 Process of discovery of the tissues 
 Combination with Comparative anatomy 
 Elements and Products 
 Vitality of the organic fluids 
 Classification of the tissues 
 Organic : Cellular tissue . 
 Dermous tissue 
 Sclerous tissue . 
 Kystous tissue . 
 Animal : Muscular and nervous tissues 
 Limitation of the inquiry . 
 
 331 
 331 
 332 
 333 
 335 
 337 
 337 
 337 
 338 
 338 
 338 
 339 
 
 CHAPTER in. 
 
 BIOTAXIC PHILOSOPHY. 
 
 Comparative anatomy of vegetables and animals 
 
 Animal anatomy our subject 
 
 Division of the Natural Method 
 
 Natural groups . 
 
 Co-ordination of them 
 
 Three laws of co-ordination 
 
 Question of permanence of organic species 
 
 Two logical conditions of the study 
 
 Subordination of characteristics . 
 
 Procedure, in use of the Natural Method 
 
 Division of animal and vegetable kingdom 
 
 Hierarchy of the animal kingdom 
 
 Attribute of symmetry 
 
 Division of Duplicate animals . 
 
 340 
 341 
 341 
 341 
 342 
 342 
 343 
 346 
 346 
 347 
 348 
 348 
 349 
 349
 
 CONTENTS. 
 
 Division of Articulated animals .... 
 Consideration of the envelope .... 
 Natural Method applied to the Vegetable Kingdom 
 Difficulty of co-ordination ..... 
 
 CHAPTER IV. 
 
 ORGANIC OR VEGETATIVE LIFE. 
 
 Condition of Dynamical Biology 
 Vital phenomena : their division 
 Theory of Organic Media . 
 Exterior conditions of Life 
 Mechanical conditions : Weight 
 Pressure ...... 
 
 Motion and rest .... 
 
 Thermological action 
 
 Light, Electricity, etc. 
 
 Molecular conditions 
 
 Air and water ..... 
 
 Study of Specifics .... 
 
 History of Physiology 
 
 Philosophical character of Physiology 
 
 Division of the study 
 
 Two functions of the organic life 
 
 Results of organic action . 
 
 State of composition and decomposition 
 
 Vital heat ..... 
 
 Electrical state .... 
 
 Production and development of living bodies 
 Decline of the organism .... 
 
 CHAPTER V. 
 
 THE ANIMAL LIFE. 
 
 Transition from the inorganic to the organic 
 
 Primitive nervous properties 
 
 Relation of the animal to the organic life 
 
 To inorganic philosophy . 
 
 Properties of tissue . 
 
 Sensibility .... 
 
 Irritability .... 
 
 Present state of analysis of animality 
 
 Movement .... 
 
 PAOK 
 
 350 
 350 
 351 
 351
 
 CONTENTS. 
 
 x.xxi 
 
 Exterior Sensation ..... ... 
 
 PAGE 
 
 373 
 
 Interior Sensation ........ 
 
 . 374 
 
 Mode of action of animality ...... 
 
 . 375 
 
 Intermittence ......... 
 
 . 375 
 
 Sleep ..... ..... 
 
 . 375 
 
 Habit 
 
 . 376 
 
 Need of activity 
 
 . 377 
 
 Association of the animal functions ..... 
 
 . 378 
 
 CHAPTER VI. 
 
 
 INTELLECTUAL AND MOEAL OR CEREBRAL FUNCTIONS. 
 
 
 Shortcoming of Descartes ....... 
 
 379 
 
 History till Gall's time ....... 
 
 379 
 
 Positive theory of Cerebral functions ..... 
 
 380 
 
 Its proper place ......... 
 
 380 
 
 Vices of Psychological systems ....... 
 
 381 
 
 Method ; Interior Observation ....... 
 
 381 
 
 Doctrine : Relation between the Affective and Intellectual facultie 
 
 s 383 
 
 Brutes and Man ......... 
 
 383 
 
 Theory of the 7 ........ . 
 
 383 
 
 Reason and Instinct ......... 
 
 384 
 
 Basis of Gall's doctrine ........ 
 
 386 
 
 Divisions of the brain ........ 
 
 387 
 
 Subdivision .......... 
 
 388 
 
 Objections .......... 
 
 389 
 
 Necessity of human actions ....... 
 
 389 
 
 Answered .......... 
 
 389 
 
 Hypothetical distribution of faculties ...... 
 
 390 
 
 Needed improvements ........ 
 
 391 
 
 Anatomical basis ...... ... 
 
 391 
 
 Physiological analysis of faculties ...... 
 
 392 
 
 Examination of historical cases ...... 
 
 393 
 
 Pathological and Comparative analysis ..... 
 
 393 
 
 Laws of action .......... 
 
 395 
 
 Intermittence and continuity ....... 
 
 395 
 
 Association .......... 
 
 395 
 
 Unity of the brain and nervous system ..... 
 
 396 
 
 Imperfect state of Phrenology ....... 
 
 396 
 
 Present state of Biology ........ 
 
 397 
 
 Retrospect of Natural Philosophy ...... 
 
 397
 
 THE 
 
 POSITIVE PHILOSOPHY 
 
 OP 
 
 AUGUSTE COMTE. 
 
 IN TROD UCTION. 
 
 CHAPTEB L 
 
 ACCOUNT OF THE AIM OF THIS WORK VIEW OF THE NATURE AND 
 
 IMPORTANCE OF THE POSITIVE PHILOSOPHY. 
 
 A GENERAL Statement of any system of pliilosophy may be either a 
 sketch of a doctrine to be estabhshed, or a summary of a doctrine 
 ab-eady estabhshed. If greater vaUie 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 especially 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 through all times, the discovery arises Law of human 
 of a great fundamental law, to which it is necessarily progress. 
 subject, and wdiicli has a solid foundation of proof, both in the facts 
 of our organization and in our historical experience. The law is 
 
 VOL. I. A
 
 2 V POSITIVE PHILOSOriiY. 
 
 tliis : — that each of our leading conceptions, — each branch of onr 
 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 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, 
 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 transi- 
 tion. 
 
 In the theological state, the human mind, seeking the essential 
 nature of beings, the first and final causes (the origin 
 sage. ^^^ purpose) of all effects, — in short, Absolute know- 
 ledge, — supposes all phenomena to be produced by the immediate 
 action of supernatural beings. 
 
 In the metaphysical state, which is only a modification of the 
 first, the mind supposes, instead of supernatural 
 s age. jj^j^gg^ 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 destina- 
 
 Third stage. . . ^ , , . , , , c i 
 
 tion of the universe, and the causes or 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 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 highest perfection of which 
 Ultimate point it is Capable when it substituted the providential 
 of each. action of a single Being for the varied operations of 
 
 the numerous divinities which had been before imngined. 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 ultimale perfection of the Posil^ive system would be 
 (if such })erfection could be hoped for) to represent all ])henomena 
 as particular aspects of a single general fact ; — such as Gravitation, 
 for instance. 
 
 The importance of the working of this general law will be estab-
 
 GROUNDS OF THE LA W OF PROGRESS. 3 
 
 Hslied hereafter. At present, it must suffice to point out some of 
 the (^^rounds of it. 
 
 Tiiere is no science which, having attained the positive stage, 
 does not hear marks of having passed through the Evidences 
 others. Some time since it was (wliatever it might ofthelaw. 
 he) composed, as we can now perceive, of metaphysical ahstrac- 
 tions; and, further hack 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 (nily an illusti'ation, 
 hut 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 coi-respond 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 meta- 
 })]iysician in his youth, and a natural philosopher in his manhood. 
 All inen 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 ^, . , 
 
 • • • ••■ ]£ licovct iccil 
 
 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 })resent advanced stage ; but, if we look back to the primi- 
 tive 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 cannot be observed 
 without the guidance of some theory. Without such guidance, our 
 facts would be desultoiy and fruitless ; we could not retain them : 
 for tlie 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 l)y 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 phe- 
 nomena, — should be the first to occur in a primitive state, while 
 tho.se which are really within our reach are regarded as almost 
 unworthy of serious study. The reason is evident enough : — that
 
 4 POSITIVE PHILOSOPHY. 
 
 experience alone can teach us the measure of oin- powei-s ; and if 
 men had not begun by an exaggerated estimate of wliat they can 
 do, they would never have done all that they are capable of. Oiu- 
 organization requires tiiis. At such a period there could have been 
 no reception of a positive philosophy, whose fnnction 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 reseai-ches with wliich men first occupied themselves. Such 
 inquiries offered the powerful charm of milimited 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 tlie 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 imaginations of astrologers and 
 alchemists. We have motive enough in the hope of discovering 
 the laws of j)henomena, with a view to the confirmation 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 spontaneous })hilosophy, the theo- 
 logical, the 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 philosopliy. The two 
 are so radically opposed, that an intermediate system of conceptions 
 has been necessaiy to render the transition possible. It is only in 
 doing this, that Metaphysical conceptions have any utility whatever. 
 In contemplating phenomena, men substitute for supei'uatiu'al direc- 
 tion 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 iiicts themselves, till, at 
 length, metaphysical agents have ceased to be anything more than 
 the abstract names of ])henomena. It is not easy to say by what 
 other ])rocess than this our minds could have passed from sujjer- 
 natural considerations to natural ; from the theological system to 
 the ])0sitive. 
 
 The Law of human development being thus established, let us 
 consider what is the proper nature of the Positive Pliilosophy.
 
 METAFTIYSICAL AMD POSITIVE PIIILOSOniY. 5 
 
 As we liave seen, tlie first cliaracteristic of the Positive Pliilos- 
 ojtliy is lliat it re<;avcls all phenomena as snLjected Character of 
 I0 invariable natni'al Zc^rs. Our bnsiness is, — seeing thePoaitive 
 liow vain is any i-esearch into what Jire called Causes, ^''''^losophi/. 
 whether first or final, — to })nrsiie an accurate discovery of these 
 Laws, wilh a view to reducing them to the smallest possible number. 
 By speculating upon causes, we could solve no difficulty about 
 oi-igin and purpose. Oiu- real business is to analyse accurately the 
 circumstances of phenomena, and to connect them by the natural 
 I'elations 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 tlie universe ai'e explained by it, because 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 jiroportioii 
 to the squares of their distances ; whilst the general fact itself is a 
 mere extension of one which is perfectly familiar to us, and whicli 
 we therefore say that we know ; — the weight of" bodies on the sur- 
 face 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 phenomena are identical; 
 which is the point from whicli 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 an universal ether. In treating his sub- 
 ject in the Positive method, he finds inexhaustible matei'ial for all his 
 activity of research, without betaking himself to insoluble questions. 
 
 P>efor6 ascertaining the stage which the Positive Philosophy has 
 reached, we must bear in mind that the different kinds History of the 
 of our knowledge have passed through the three Positive Pki- 
 stages of progress at different rates, and have not ^''^''My- 
 therefore arrived at the same time. The rate of advance dej)ends 
 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 
 ])ositive 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, simi)le, and inde])en- 
 dent 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
 
 6 POSITIVE PHI 10 SO PHY. 
 
 science. It mny be said, like everything else, to have been always 
 going on ; and especially since the labours of Aristotle and the 
 school of Alexandria; and then from the introduction of natural 
 science into the West of Europe by tlie Arabs. But, if we nuist fix 
 upon some marked period, to serve as a rallying point, it nuist be 
 that, — about two centuries ago, — when the human mind was astir 
 under the precepts of Bacon, the conceptions of Descartes, and the 
 discovei'ies of Galileo. Then it was that the spirit of the Positive 
 philosophy rose up in opposition to that of the superstitious and 
 scholastic systems which had hitherto obscuied the true character 
 of all science. Since that date, the progress of the Positive philos- 
 ophy, and the decline of the other two, have been so marked that 
 no rational mind now doubts that the revolution is destined to go 
 on to its completion, — every branch of knowledge being, sooner or 
 later, brought within the operation of Positive philosopliy. This is 
 not yet the case. Some are still lying outside : and not till they 
 are brought in will the Positive philosophy possess that character of 
 imiversality which is necessary to its definitive constitution. 
 
 In mentioning just now the four })rincipal categories of phenomena, 
 — astronomical, physiciil, chemical, and physiological, — there was 
 New departmmt ^" omission which will have been noticed. Nothing 
 of Positive was said of Social phenomena. Though involved 
 
 Philosophy. y][\\x the ])hysiological, Social phenomena demand a 
 distinct classification, 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 
 bi-anch of science has not hitherto entered into the domain of Positive 
 philosophy. Theological and metaphysical methods, exploded in 
 other departments, are 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 disputes about divine 
 i-ight and the sovereignty 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 animal, — 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 ])i-incipal aim of the jn-esent work to establish. 
 
 It would be absurd to pretend to offer this new science at once 
 Social Phusics ^" ^ Complete state. Others, less new, are in very 
 unequal conditions of forwardness. But the same 
 character of ])ositivity whicli is impressed on all the othei's will be 
 shown to belong to ihis. This once done, the philosophical system 
 of the moderns will be in fact complete, as there will then be no 
 phenomenon which does not naturally enter into some one of the
 
 NE W PROVINCE OF POSITIVE PIIIIOSOPH Y. 7 
 
 five great categories. All our fuiulauiental conceptions having 
 become homogeneous, the Positive state will bo 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 advantage vesting with the two preceding systems, it will 
 supersede them by its natural superiority, and leave to them only 
 an historical existence. 
 
 We have stated the special aim of this woi-k. Its secondary and 
 general aim is this : — to review what has been effected Secondaru aim, 
 m the Sciences, in oi'der to show that they are not of this work. 
 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 
 philoso[)hy, passing in review all the positive sciences already 
 formed. 
 
 The purpose of this work is not to give an account of the Natural 
 Sciences. Besides that it would be endless, and that Toreviciothe 
 it would require a scientific preparation such as no p/uiosuphi/ of 
 one man possesses, it would be apart from our object, i/^e Sciences. 
 which is to go throuuh a course of not Positive Science, but 
 Positive 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 j)hilosophy 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 anterior 
 facts which have a bearing upon social science. Though the funda- 
 mental sciences are not all equally interesting to ordinary minds, 
 there is no one of them that can be neglected in an inquiry like the 
 ])resent; and, in the eye of philosophy, all are of equal value to 
 human welfare. Even those which appear the least interesting have 
 their own value, either onaccoimtof 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 present . 
 
 the bane of a true positive philosophy, we will briefly ' '' ' 
 
 advert to the existing prevalence of such special ])ursuit. In the 
 primitive state of human knowledge there is no regular division of 
 intellectual labour. Every student cultivates all the sciences. As 
 knowledge accrues, the sciences part off ; and students devote them- 
 selves each to some one branch. It is owing to this division of
 
 8 POSITIVE PHIIOSOPHY. 
 
 employment, and concentration of whole minds upon a sini^le 
 department, that science has made so prodigjious 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 cannot 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 ui-gent. We all 
 acknowledge that the divisions established for the convenience of 
 scientific pursuit are radically artificial ; and yet there are very few 
 ■who can embrace in idea the whole of any one science : 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 know- 
 ledge. We must not be blind to the evil, nor slow in seeking a 
 remedy. AVe must not forget that this is the weak side of the 
 positive philosojihy, by which it may yet be attacked, with some 
 hope of success, by the adherents of the theological and metaphysi- 
 cal systems. As to the remedy, it 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 
 jierfecting 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, 
 Proposed veio Suitably prepared, whose business it shall be to take 
 class of stu- the respective sciences as they are, determine the 
 ^^"'^*- spirit of each, ascertain their relations and mutual 
 
 connection, and reduce their respective princi])les to the smallest 
 number of general principles, in conformity with the fundamental 
 j-ules of the Positive Method. At the same time, let other students 
 be prepared for their special pursuit by an education which recog- 
 nizes the whole scope of positive science, so as to profit by the 
 labours of the students of generalities, and so as to correct 
 j'eciprocally, under that guidance, the results obtained by each. We 
 see some ap{)roach already to this arrangement. Once established, 
 there would be nothing to a})prehend from any extent of division 
 of employments. Wlien we once have a clnss 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 being lost sight of in the pursuit of the details of 
 knowkulge. The organization of scientific research will then be 
 com])lete; and it will henceforth have occasion only to extend 
 its development, and not to chynge its charactei'. After all the
 
 FIRST BENEFIT. 9 
 
 f<ii"ination of such a new class as is proposed would be merely an 
 extension of tlie princii)le which lias created all the classes we have. 
 While science was nariow, there was only one class: as it expanded, 
 more were insliluted. With a further advance a fresh need arises, 
 and this new class will be the result. 
 
 The genei-al spirit of a course of Positive Philosophy having been 
 thus set forth, we must now glance at the chief ad- Advantcms of 
 vantages which may be derived, on behalf of" human the Pod'uve 
 ])i-ogression, from the study of it. Of these advan- Pi>->'iosoph!/. 
 tages, four may be especially pointed out. 
 
 I. The study of the Positive Philosophy affords the only rational 
 means of exhibiting the logical laws of the human illustrates the 
 mind, which have hitherto been sought by unfit intellectual 
 methods. To explain what is meant by this, we may fi^nctwn. 
 refer to a saying of M. de Blainville, in his work on Comparative 
 Anatomy, that every active, and especially every living being, may 
 be regarded 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 tliese heads. 
 Let us apply this classification to the intellectual functions. 
 
 If we regard these functions under their Statical aspect —that is, 
 if we consider the conditions under which they exist — we must de- 
 termine the organic circumstances of the case, which inquiry involves 
 it with anatomy and pliysiology. If we look at the Dynamic aspect, 
 we have to study simply the exercise and results of the intellectual 
 ])0wers of the human race, which is neither more nor less than the 
 general object of the Positive Philoso[)hy. 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 know- 
 ledge of logical laws. These being the only means of knowledge of 
 intellectual phenomena, the illusory psychology, which is the last 
 })hase of theology, is excluded. It jiretends to accomplish the dis- 
 coveiT of the laws of the human mind b}^ contemplating it in itself; 
 that is, by separating it from causes and effects. Such an attempt^ 
 n)ade in defiance of the physiological study of our intellectual organs, 
 and of the observation of rational methods of pioccdure, cannot 
 succeed at this time of day. 
 
 The Positive Philosoj)hy, which has been rising since the time 
 of Bacon, has now secured such a preponderance, that the meta- 
 physicians themselves profess to ground their pi-etended science on an 
 observation of facts. They talk of external and internal facts, and 
 say that their business is with the lattej-. This is much like saying 
 that vision is explained by luminous objects painting their images 
 npon the retina. To this the physiologists reply that another eye 
 would be needed to see the image. In the same mannei', the nn'nd 
 may observe all phenomena but its own. It \m\y ha said that a 
 man's intellect njay observe his passions, the scat of the reason
 
 lo POSITIVE PHIIOSOPIIY. 
 
 l>eing somewlial Jipnvt from that of the emotions in the brain; but 
 there can be nothin<^ like scientific observation of the passions, except 
 from without, as the stir of the emotions (hsturbsthe observing faculties 
 more or less. It is yet more out of the question to make an intel- 
 lectual observation of intellectual processes. The observing and 
 observed organ are here the same, and its action cannot be pure 
 and natural. In order to observe, your intellect must pause from 
 activity ; yet it is this very activity that you want to observe. If 
 you cannot effect the pause, you cannot 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 
 ])ursuit, no one proposition is established to the satisfaction of its 
 followers. They are divided, to this day, into a multitude of schools, 
 still disputing aljout the very elements of their doctrine. This in- 
 terior observation gives birth to almost 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 
 done some good in keeping up the activity of our understandings, 
 when there w\as 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 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 cannot 
 be 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 whicli loses time upon it. 
 We may talk for ever about the metliod, 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 psychol- 
 ogists, by dint of reading 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 a 
 'priori a true method of investigation, independent of a philosophical 
 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 cannot as 
 yet explain the great logical procedures apart from their application. 
 If we ever do, it will remain as necessary then as now to form good 
 intellectual habits by studying the regular application of the scientific 
 methods which we shall have attained. 
 
 This, then, is the first great result of the Positive Philosophy — 
 the manifestation by experiment of the laws which rule the Intel- 
 lect in the investigation of truth ; and, as a consequence, the know- 
 ledge of the general rules suitable for that object.
 
 SECOi\n AND TJinUJ BENEFITS. n 
 
 II. The socoml offoct of the Positive Philosopliy, an effect not 
 less important and far more urgently wanted, will be Must ra/merate 
 to regenerate Eilucation. Education. 
 
 The best minds are agreed that our European education, still 
 t'ssentially theological, metaphysical, and literary, must be su})er- 
 seded by a Positive ti'ainiug, 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 endeavours to the utmost, we 
 must not however conceal from ourselves that everything yet done 
 is inadequate to the object. The ])resent exclusive speciality of 
 our ])ursuits, and the consequent isolation of the sciences, spoil our 
 teaching. If any student desires to form an idea of nattu-al philos- 
 0})h3'' as a whole, he is com})elled to go through each department 
 as it is now taught, as if he 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 phenomena. It is such an aggregate of con- 
 ceptions, whether on a great or on a small scale, which must hence- 
 ibrth be the permanent basis of all human combinations. It will 
 constitute the mind of future generations. In order to this i-egen- 
 eration 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 important lesults. The speci- 
 alities of science can be pursued by those whose vocation lies in that 
 direction. They are indispensable ; 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 must also 
 aid the progress of the respective positive sciences : Ajranccs 
 and this constitutes our third head of advantages. acUurc,^ i,y 
 
 The divisions which we establish between the combmingthcm. 
 sciences are, though not arbitrary, essentially artificial. The sub- 
 ject of our researches is one: we divide it for our convenience, in 
 order to deal the more easily with its difficulties. But it sometimes 
 liappens — and especially with the most important doctrines of each 
 science — that we need what we cannot obtain under the present 
 isolation of the sciences, — a combination of several special points of 
 view ; and for want of this, veiy important problems wait for their 
 solution much longer than they otherwise need do. To go back 
 into the past for an example: Descartes' grand conception with 
 regard to analytical geometiy is a discovery which has changed the 
 whole aspect of mathematical science, and yielded the germ of all
 
 12 POSITIVE PHILOSOPHY. 
 
 future progress; and it issued from the union of two sciences which 
 liad always before been separately regarded and pursued. The case 
 of pending questions is yet more impi-essive ; as, for instance, in 
 Chemistry, the doctrine of Definite Proportions. Without entering 
 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 sliould necessarily 
 combine in fixed numbers, — it will be indispensable that the chemical 
 point of view should be united with the pliysiological. 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. 
 imdecided whether azote is a simple or a compound body. It was 
 concluded by almost all chemists that azote is a simple body ; the 
 iHustrious Berzelius hesitated, on purely chemical considerations ; 
 but he was also influenced by the physiological observation that 
 animals which receive no azote in their food have as much of it in 
 their tissues as carnivorous animals. Fi'om this we see how physi- 
 ology must unite with chemistry to inform us whether azote is 
 simple or compound, and to institute a new series of researches upon 
 the relation between the composition of living bodies and their mode 
 of alimentation. 
 
 Such is the advantage which, in the third place, we shall owe to 
 Positive pliilosophy — the elucidation of the respective sciences by 
 their combination. In the fourth place 
 
 IV. The Positive Philosophy offers the only solid basis for that 
 Must reorrjan- Social Reorganization which must succeed the 
 ize. society. critical condition in which the most civilized nations 
 
 are now living. 
 
 It cannot 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 genuine social ordei', 
 we ai'e suffering under an utter disagreement which may be called 
 universal. Till a certain number of general ideas can be acknow- 
 ledged as a rallying-point of social doctrine, the nations will remain 
 in a revolutionary state, whatever palliatives maybe devised; and 
 their institutions can he only provisional. But whenever the 
 iiecessaiy agreement on first principles can be obtained, appropriate 
 institutions will issue fi'om them, without shock or resistance; for 
 the causes of disorder will have been arrested by the mere fact of 
 the agreement. It is in this direction that those must look who 
 desiie 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
 
 FO UR TH BENEFIT. 1 3 
 
 tlieological, the metapliysicul, and the positive. Any one of these 
 might alone secnre some sort of social order ; but while the three 
 co-exist, it is impossible for ns to understand one another upon any 
 essential point Avhatever. 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 former 
 views, cannot but concr.r in its triumph. The problem once 
 recognized cannot I'emaiu 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, 
 thi'oughout which the others have been declining. The fact is 
 incontestable. Some may deplore it, but none can destro}' it, nor 
 therefore neglect it but under penalty of being betrayed by illusory 
 speculations. This general i-evolution of the human mind is neaily 
 accomplished. We have only to complete the Positive Philosoi)hy 
 by bringing Social phenomena within its comprehension, and aftei- 
 wards consolidating the whole into one body of homogeneous 
 doctrine. 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 reception 
 of tliis philosophy will be when it has acquired tlie only quality that 
 it now wants — a character of due generality. When it has becomi! 
 complete, its supremacy will take ])lace spontaneously, and will 
 le-establish order throughout society. There is, at present, n(» 
 conflict but between the theological and the metaphysical philoso- 
 ])hies. They are contending for the task of reorganizing society ; 
 but it is a woi-k too mighty for either of them. The })ositive 
 ])hilosophy has hitherto intervened oidy 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 
 controvensy. 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 propo.scd to consolidate the whole of our acquired 
 knowledge into one body of homogeneous doctrine, it j^^^ hor,eofre- 
 must not be supj)osed that we are going to study this auction to a 
 vast variety as proceeding from a single principle, smgielaw. 
 and as subiected to a sinjii-le law. There is something so chimerical 
 in attempts at universal explanation by a single law, that it niay 
 be as well to secure this Work at once from any imputation of the 
 kind, though its development will show how undeserved such an 
 imputation would be. Our intellectual resources are too narrow, 
 and the imiverse is too complex, to leave any liope tluit it will ever
 
 14 POSITIVE PHILOSOPHY. 
 
 be widiin our power to cany scientific perfection to its last degree 
 of simplicity. Moreover, it appears as if the value of such an 
 attainment, supposing it possible, were greatly overrated. The only 
 way, for instance, in which Ave could achieve the business, would 
 be by connecting all natural phenomena with the most general law 
 we know, — which is that of Gravitation, by which astronomical 
 phenomena are already connected with a poi-tion of terrestrial 
 })hysics. Laplace 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 cannot be while we are 
 Avithout data about the constitution of bodies), the difficulty of its 
 application would doubtless be found so great that we must still 
 maintain the existing division between astronomy and chemistry, 
 Avith the difference that Ave now regard as natural tliat division 
 which Ave should then call artificial. Laplace himself presented 
 his idea only as a philosophic device, incapable of exercising any 
 useful influence over the progress of chemical science. Moreover, 
 supposing this insuperable difficulty overcome, Ave should be no 
 nearer to scientific unity, since Ave then should still have to connect 
 the Avhole of physiological phenomena Avith the same law, Avhich 
 certainly would not be the least difficult part of the enterprise. 
 Yet, all things considered, the hypothesis Ave have glanced at Avould 
 be the most favourable to the desired unity. 
 
 The consideration of all phenomena as referable to a single origin 
 is by no means necessary to the systematic formation of science, 
 any more than to the realization of the great and happy consequences 
 that Ave anticipate from the positiv'C philosophy. The only neces- 
 sary unity is that of Method, Avhich is already in great part estab- 
 lished. As for the doctrine, it need not be one; it is enough that 
 it should be homogeneous. It is, then, under the double aspect of 
 unity of method and homogeneousness of doctrine that we shall 
 consider the different classe;? 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 natu- 
 ral phenomena, Ave shall regard as presumptuous every attempt, in 
 all future time, to reduce them rigorously to one. 
 
 Having thus endeavoured to determine the spirit and influence 
 of the Positive Philosophy, and to mark the goal of our labours, we 
 haA^e noAv to proceed to the exposition of the system ; that is, to 
 the determination of the universal, or encyclopaedic order, Avhich 
 must regulate the different classes of natural phenomena, and con- 
 sequently the corresponding positive sciences. 
 
 )
 
 IS 
 
 CHAPTEE II. 
 
 VIEW OF THE HIERARCHY OF THE POSITIVE SCIENCES. 
 
 Im proceedinp^ to offer a Classification of the Sciences, we must 
 leave on one side all others that have as yet been attempted. Such 
 scales as those of Bacon and D'Alembert are con- paihirc ofpro- 
 structed upon an arbitrary division of the faculties of posed dassiji- 
 the mind ; whereas, our principal faculties are often <^<^twns. 
 enga<;ed 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 
 cons])icuous, that the best minds feel a prejudice against this kind 
 of enterprise, in any shape. 
 
 Now, what is the reason of this ? — For one reason, the distribution 
 of the sciences, having become a somewh.at 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 personal reason, 
 however, is the want of homogeneousness in the different parts of the 
 intellectual system, — some having successively become positive, while 
 others remain theological or metaphysical. Among such incoherent 
 materials, classification is of course impossible. Every attempt at a 
 distribution has failed from this cause, without the distributor being 
 able to see why; — without his discovering that a radical contrariety 
 existed between the materials he was endeavouring to combine. 
 The fact was clear enough, if it had but been understood, that the 
 enterprise was prematui-e ; and that it was useless to undertake 
 it till our princi{)al scientific conceptions should all have become 
 positive. The ])receding chapter seems to show that this in- 
 dispensable condition may now be considered 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 i^ecent 
 botanists and zoologists, whose philosophical labours have exhibited 
 the true principle of classification; viz., that the classification must 
 proceed from the study of the things to be classified, and must by no 
 means be determined l3y dpriori considerations. The real affinities 
 and natural connections presented by objects being allowed to deter- 
 mine their order, the classification itself becomes the expression 
 of the most general fact. And thus docs the positive method apply
 
 i6 POSITIVE PHILOSOPHY, 
 
 to the question of classification itself, as well as to the objects in- 
 T rut 'principle cluded Under it. It follows that the mutual depend- 
 of classification, g^ce of the sciences, — a dependence resulting from 
 that of the corresponding phenomena, — must determine the arrange- 
 ment of the s3-stem of human knowledge. Before proceeding to 
 investigate this mutual dependence, we have only to ascertain the 
 real bounds of the classification proposed : in other words, to settle 
 Avhat we mean b}' human knowledge, as the subject of this work. 
 The field of human labour is either speculation or action : and 
 Boundaries thus, we are accustomed to divide our knowledge 
 of our field. j^to 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 whatevei", but of those 
 fundamental conceptions of the diflferent 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 appli- 
 cation may happen to throw back light on its speculative origin. 
 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 meta- 
 physical commentators. 
 
 There can be no doubt that Man's study of nature must furnish 
 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 power 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 science to art may be summed 
 up in a brief expression : 
 
 From Science comes Prevision : from Prevision comes Action, 
 We must not, however, fall 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 physiological 
 effects of consternatiQii, 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 
 the ])roper object of all scientific theories) is so inherent in our 
 
 1
 
 ABSTRACT AND CONCRETE SCIENCE. 17 
 
 organization, that if wo couUl not satisfy it by ]iositi\'e concepl-ions, 
 we must inevitably return to those theological and metaphysical 
 explanations which hiul their origin in this very fact of human 
 nature. — It is this original tendency which acts as a preservative, 
 in the minds of men of science, against the narrowness and incom- 
 ])]eteness whicli the practical habils of our age are apt to produce. 
 it, is through this that we are able to maintain just and noble ideas 
 of the importance and destination of the sciences; and if it were 
 not thus, the human untlerstanding would soon, as Condorcet has 
 observed, come to a stand, even as to the practical a{)i)lications 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, tSome of the most important arts are derived from speculations 
 ))m'sued during long ages with a purely scientific intention. For 
 instance, the ancient Greek geometers delighted themselves with 
 beautiful specidations on Conic Sections; those speculations wrought, 
 after a long series of generations, the renovation of astronumy ; and 
 out of this h;is the art of navigation attained a perfection which it 
 never could have reached otherwise than through the speculative 
 labours of Archimedes and Apollonius : so th;it, to use Condorcet's 
 illustration, "the sailor who is pi-eserved from shipwreck by the 
 exact observation of the longitude, owes his life to a the((fry conceived 
 two thousand yenrs before by men of genius who had in view simply 
 geometrical speculations." 
 
 Oui" business, it is clear, is with theoretical researches, letting 
 alone their })ractical application altogether. Though we may con- 
 ceive of a course of study wliich should unite the generalities of 
 speculation and ajiplication, the time is not come for it. To say 
 nothing of its vast extent, it would require preliminary achievements 
 which have not yet been attempted. We must first be in possession 
 of appropriate Special conceptions, formed according to scientific 
 theories; and for these we have yet to wait. Meantime, an inter- 
 mediate class is rising up, whose particidar destination is to oi-ganize 
 the relations of theory and practice; such as the engineei-s, who do 
 not labour in the advancement of science, but wdio study it in its 
 existing state, to apply it to practical purposes. Such classes are 
 furnishing us with the elements of a futiu-e body of doctrine on 
 the theories of the different arts. Already, Monge, in bis view of 
 descriptive geometry, has given us a general theory of the arts of 
 construction. But we have as yet only a few scattered instances 
 of this nature. The time Avill come when out of such i-esults, 
 a de[)artment of Positive ])hilosophy may arise : but it will be in 
 a distant future. If we remember that several sciences are im- 
 plicated in every important art, — that, for instance, a true theory 
 of Agricidture requires a combination of physiological, chemical, 
 mechanical, and even astronomical and mathematical science, 
 — it will be evident that true theories of tiie arts must wait 
 
 Vol. I. B
 
 1 8 POSITIVE PHIIOSOPHY. 
 
 for a large and equable development of these constituent 
 sciences. 
 
 One more preliminaiy remark occurs, before we finish the pre- 
 scription of our limits, — the ascertainment of our ticld of iuquiry. 
 We must distinguish between the two classes of 
 
 Abstract science. -kt i. i • ^ ii i j l i i • i i 
 
 JNatural science ; — the abstract or general, winch nave 
 for their object the discovery of the laws which regulate })henomena 
 „ . in all conceivable cases: and the concrete, particulai", 
 
 Concrete science. , . ^. , . , ^. n i tvt ^ i 
 
 or descriptive, which ai-e sometimes called JNatural 
 sciences in a restricted sense, whose function it is to apply these 
 laws to the actual history of existing beings. The*first are funda- 
 mental ; and our business is with them alone, as the second 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 miuerfdogy, which is secondary to it. — We 
 may say of Concrete Physics, as these secondary sciences are called, 
 the same thing that we said of theories of the arts, — that they 
 require a preliminary knowledge of several sciences, 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 of less in- 
 coherent materials than those which it now presents. At present, 
 too few of the students of these secondary sciences 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 ])ractical 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 j)roceed to the ascertainment of the true order of the funda- 
 mental sciences. 
 
 This classification of the sciences is not so easy a matter as it 
 
 Difficulty of may appear. However natural it may be, it will 
 
 classification. always involve something, if not arbitrary, at least 
 
 artificial ; and in so far, it will always involve inqiei'fection. It is 
 
 im[)ossil)le lo fulfil, quite rigorously, the object of presenting the 
 
 sciences in their natural connect ion, and according to their mutual 
 
 de})endence, so as to avoid the smallest danger ot being involved in 
 
 a vicious circle. It is eas}" to show why. 
 
 Historical and Every scieuce may be exhibited under two methods 
 
 do<jmatic Or procedures, the Historical and the Dogmatic. 
 
 vuthods. The.se are wholly distinct from each other, and any
 
 ME TIIODS OF S 1 UD Y. 1 9 
 
 other nietliod can Le noiliiiig Lut some combiiiaiion of these two. By 
 ihe iirst method knowledjj^e is ])reseiited in the same order in wliich 
 it. was actually obtained by the human mind, tog^ether with the way 
 in which it was obtaiired. By the second, the system of ideas is 
 ]>resented as it mi,<;ht be conceived of at this day, by a mind which, 
 duly jirepared and })Iaced at the ri,i2,ht point of view, should begin 
 to reconstitute the science as a whole. A new science must be 
 pursued historically, the only thing to be done being to study in 
 chronological oider the ditierent works which have contributed to 
 the ])rogress of the science. But wdien such materials have become 
 lecast to form a genei'al system, to meet the demand ibr a more 
 natuial logical order, it is because the science is too far advanced 
 i'or the liistorical order to be practicable or suitable. The more 
 discoveries are made, the gi-eater becomes the labour of the historical 
 method of study, and the more effectual the dogmatic, because the 
 new concej)tions 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 
 wiiiingsof Ai'chimedes and ApoUonius were, in fact, about all. On 
 the contrary, a modern geometer cojumonly finishes his education 
 without having read a single original work dating further back than 
 the most recent discoveries, which cannot be known by any other 
 means. Thus the Dogmatic Method is for ever sui)erseding the 
 liistorical, as we advance to a higher position in science. If every 
 mind had to jiass through all the stages that every predecessor in 
 the study had gone through, it is clear that, however easy it is to 
 learn rather than ins'ent, it would be impossible to effect the pur- 
 ])ose of education, — to place the student on the vantage-ground 
 gained by the labouis of all the men who have gone before. By 
 ihe dogmatic method this is done, even though the living student 
 may have only an ordinary intellect, and the dead may have been 
 men of lofty genius. By tlie dogmatic method therefore must every 
 advanced science be attained, with so nmcli of the historical com- 
 bined with it as is rendered necessary by discoveries too recent to 
 be studied elsewhere than in their own records. The only objection 
 to the preference of the Dogmatic method is that it does not show 
 how the science was atbuned ; but a moment's reflection will show 
 that this is the case also with the Historical method. To pursue 
 a science historically is quite a different thing from learning tlie 
 history of its ])rogress. This last ])ertains to the study of huujan 
 history, as we shall see when we reach the final division of this 
 work. It is trtie that a science cannot be completely understood 
 without a knowledge of how it arose ; and again, a dogmatic know- 
 ledge of any science is necessary to an understanding of its history ; 
 and therefore we shall notice, in treating of the fundamental 
 sciences, the incidents of their origin, when distinct and illustra-
 
 20 POSITIVE nilLOSOPIIY. 
 
 five ; nnd we slmll use their history, in a Rcientifio sense, in onr 
 treatment ot" Social Phj'sics; but the historical sl-udy, imjjortant, 
 even essential, as it is, remains entirely distinct from tlie proper 
 dogmatic study of science. These considerations in this place, tend 
 to define more precisely the spirit of onr course of inquiry, while 
 they more exactly detei-mine the conditions under which we may 
 hope to succeed in the construction of a ti'ue scale of the aggre- 
 gate fundamental sciences. Great confusion would arise from any 
 ;ittem])t to adhei'e strictly to historical oi'der 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 exposition of astronomy. 
 Minor defects, if inevitable, catmot invalidate a classification which, 
 on the whole, fulfils the princijial conditions of the case. They 
 belong to what is essentially artificial in our division of intellectual 
 laljour. In the main, however, our classification agrees with the 
 liistory of science; the more general and simple sciences actually 
 occurring first and advancing best in human histor}'', and being 
 followed by the more complex and i-estricted, though all were, since 
 the earliest times, enlarging simultaneously. 
 
 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 fimdamental sciences. They are six, 
 as we shall soon see. We cannot make them less; and most 
 scientific 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 
 proposed ; yet we might venture to say that there is probably net 
 one in favour of which some plausible reason might not be assigned ; 
 for we see the wildest divergences among the schemes whicii 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 
 possible systems. 
 
 Now we must remember that we have to look for the principle 
 True principle of classification in the comparison of the different 
 of classification, 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, Ibis dependence 
 can result only from that of the corresponding phenomena. All 
 observable phenomena may be included within a very few natural 
 categories, so arranged as that the study of each category may be
 
 PRINCIPLE OF classification: 21 
 
 t]^ronnded on tlio ])rincipal Inws of tlio precedini^, iuid serve as llie 
 basis of llie next ensning. This oi'der is determined by the degree 
 of sinii)licitv, or, what comes to tlie same thinjj^, of ^ 
 generality ot tlieu- phenomena. Jlence resnlts their 
 successive dependence, and tlie greater or lesser ^ 
 
 ...,., P , f . 1 • T Dependence. 
 
 hicility tor bemg studied. 
 
 It is clear, d 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 
 ])articular cases. We nuist 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, 
 ibr this order of generality or simplicity fixes the degree of facility 
 in the study of phenomena, while it determines the necessary con- 
 nection of the sciences hy the successive dependence of their pheno- 
 mena. 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 
 iMtional frame of mind than those in which he is more nearly 
 implicated ; and tliis constitutes a new ground for the correspond- 
 ing sciences being developed more rapidly. 
 
 We have now obtained our rule. Next we proceed to our 
 classiti cation. 
 
 We ai'e first struck by the clear division of all natural [)henomena 
 into two classes— of inorganic and of organic bodies, j^organicand 
 The organized are evidently, in fact, more complex Organic phe- 
 and less general than the inorganic, and depend upon ^omena. 
 them, instead of being de{)eiided on by them. Therefore it is that 
 ])hysiological study should begin with inorganic phenomena ; since 
 the organic include all the qualities belonging to them, with a 
 special oi'der added, viz., the vital {)hen()mena, which belong to 
 organization. We have not to investigate the nature of either ; for 
 the positive philosophy docs not inquire into natures. Whether 
 their nature 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 througli 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 nuist 
 be established before we can proceed with success to the examina- 
 tion of a de{)endent class of phenomena. 
 
 Each of these great halves of natural philosojiliy has subdivisions. 
 
 Inorganic ])hvsics must, in accordance with our rule ^ ^ 
 
 f 1-. " 1 *i 1 i- 1 1 c 1 I- Inorganic. 
 
 ot generality and tlie order ot dependence or pheno- 
 mena, be divided into two sections — of celestial and terrestrial 
 phenomena. Tims we have Astronomy, geometrical 
 and mechanical, and Terrestrial Physics. The ne- " " '"^' 
 cessity of this division is exactly the same as in the former case.
 
 22 POSITIVE PHILOSOPHY. 
 
 Astronomical plienomena are the most general, sim])le, and 
 abstract of all ; and therefore the study of natural |)liiloso{)liy 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 ])reponderate, in all terrestrial 
 ])henon]ena, over all effects which may be peculiar to them, and 
 modify the original ones. It follows that the analysis of the 
 simjilest teri-estrial phenomenon, not only chemical, but even purely 
 mechanical, pi'esents a greater complication than the most com- 
 jiound astronomical phenomenon. Tiie 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 tiiese 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 
 Physics into two, according as we regard bodies in their mechanical 
 
 or their chemical charactei-. Hence we have Physics, 
 
 properly so called, and Chemistry, Again, the second 
 class must be studied through the first. Chemical phenomena are 
 
 more complicated than mechanical, and depend upon 
 o. itmi>> iy. ^i^gj^-,^ without influencing them in return. Every 
 one knows tb.at all chemical action is first sid)mitted to the influence 
 of weight, heat, electricity, etc., and presents moreover something 
 which modifies all these. Thus, while it follows Physics, it presents 
 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 
 II. Okganic. jsfatural Philosophy — the science of organized bodies. 
 Here we find ourselves presented with two orders of phenomena ; 
 those which relate to the individual, and those which relate to the 
 species, especially when it is gregnrious. With regard to Man, 
 especially, this distinction is {'undan)ental. The last order of pheno- 
 mena is evidently dependent on the first, and is more complex. 
 
 Hence we have two great sections in organic physics 
 
 u/iioornj. — Physiology, pi'operly SO called, and Social Physics, 
 
 which is dependent on it In all Social phenomena we perceive 
 
 the working of the ])hysiological laws of the indivi- 
 ocioogy. ^\yy^\ -^ and moreover something which modifies their 
 effects, atid which belongs to the influence of individuals over each 
 other — singularly complicated in the case of the human race by the 
 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 suppose, 
 as some eminent ])hysiol()gists have done, that Social Physics is 
 only an aj)pendage to })hysiology. The phenomena of the two are 
 not identical, though they are homogeneous ; and it is of high
 
 ORGANIC PJIYSICS. 23 
 
 inipoilaiioe lo liold tlie two sciences sep;n-nto. As social cnndltioMs 
 modify the 0]ici'atioii of })liysiol()<;ical laws, {Social IMiysics must 
 liiive a set of observations of its own. 
 
 It would be easy to make the divisions of the Or<j:nnic lialf of 
 Science correspond with those of the ]uor<j;anic, b}' dividin<^ physi- 
 ology into vegetable and animal, according to popular custom. But 
 this (lislinction, however impoilant in Concrete Physics (in that 
 secontlary and special class of studies before declared to be inap[)ro- 
 ])riate to this Avork), hardly extends into those Abstract Physics 
 with which we have to do. Vegetables and animals come alike 
 undei' 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 Ihe 
 distinclion grows ever fainter and more dubious with new discoveries, 
 it bears no relation to our plan of research ; and we shall therefore 
 consider that there is only one division in the science of organized 
 bodies. 
 
 Thus we have before us Five fimdamental Sciences in successive 
 dependence, — Astronomy, Physics, Chemistry, Phy- Five Natural 
 siology, and finally Social Physics. The first considers Sciences. 
 the most general, simple, al)stract, and remote phenomena known 
 to us, ancl those which alfect all othei's without being affected by 
 them. The last considers the most j)articular, compound, concrete 
 jihenomena, and those which are the most interesting to Man. 
 Between these two, the degrees of speciality, of com[)lexity, and 
 individuality, are in regular proportion to the place of llie respective 
 sciences in the scale exhibited. This — castinii: out everything: 
 arbitrary — we must rerard as the true filiation of the ^, . ^,. . 
 
 •^ T ■ -. n ^ .-I 1 ,-.1 • 1 Their filiation. 
 
 sciences ; and in it we find tlie plan 01 tins work. 
 
 As we {)roceed, we siiall find that the same ])rinci[)le which gives 
 this order to the W'hole body of science arranges the Filiation of 
 ])arts of each science ; and its soundness will there- their vans. 
 fore be freshly attested as often as it presents itself afresh. There 
 is no refusing a ])rinci})le which distributes the interior of each 
 science after the same method witii 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 
 i-eview some of the leading properties of the hierarchy of science that 
 has been disclosed. 
 
 This gradation is in essential conformity with the order which 
 has spontaneouslv taken place anion<r the branches ., , 
 
 c"^,,.,-, ,1 ,0 , , Corroborations. 
 
 01 natural piiilosophy, when ])nrsue(l separat(My, and 
 without any purpose of establishing such order. Such an iiccordance 
 is a strong jaesumption that the arrangement is 1. This dwisi- 
 natural. Airain, it coincides with the actual develot)- fi^'-ation follows 
 
 tftC OTCiCV Of 
 
 inent of natural philosophy. If no leading science disclosure of 
 can be eftectually pursued otherwise than through sciences. 
 those which precede it in the scale, it is evident that no vast devel-
 
 24 POSITIVE PHILOSOPHY. 
 
 opment of any science could take place prior to the great astrono- 
 mical discoveries to which we owe the impulse given to the whole. 
 The progression mny since have been simultaneous ; but it has taken 
 phice in the order we have i-ecognized. 
 
 This consideration is so important that it is difficult to under- 
 
 2. Solves hete- stand without it the history of the human mind. 
 
 rogeneousncis. '\^\ie general law which governs this histor}', as we 
 have already seen, cannot be verified, unless we combine it with the 
 scientific gi-adation just laid down : for it is according to this grada- 
 tion that the diiferent human theories have attained in succession 
 the theological state, the metaj)hysical, and finally the positive. 
 ]f we do not bear in mind tlie law which governs progression, we 
 shall encounter insurmountable difficulties: for it is clear that the 
 theological or metaphysical state of some fundamental theories 
 must have temporarily coincided with the positive state of others 
 Avhich precede them in our established gradation, and actually have 
 at times coincided with them ; and this nuist involve the law itself 
 in an obscurity which can be cleared up only by the classification 
 we have proposed. 
 
 Again, this classification marks, with pi-ecision, the relative per- 
 „ ,, , , fection of the ditl'erent sciences, which consists in the 
 
 o. Marks rela- . . . ' i • i i ,• 
 
 live perfection degree ot precision or knowledge, and m the relation 
 in sciences. ot" 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 tliCvse 
 again terrestrial are less exact and systematic than those of iistronomy. 
 This fact is completely accounted for by the gradation we have laid 
 down ; and we shall see as we proceed, that the possibility of 
 applying mathematical analysis to the study of ])henomena is 
 exactly in proportion to the rank which they hold in the scale of 
 the whole. 
 
 There is one liability to be guai-ded ngainst, which we may men- 
 
 Defectsare ^'^'^ here. We must beware of conlbimding the 
 
 inns not degree of precision which we are able to attain in 
 
 m science. j-egard to any science, with the certainty of the science 
 
 itself. The certainty of science, and our precision in the knowledge 
 
 of it, are two very difiei-ent things, which have been too often con- 
 
 Ibunded ; and are so still, though less than formerly. Avery absurd 
 
 pro[)osition may l)e very j)reci.se ; 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 statement of it; as, for instance, when we assert that every 
 
 man will die. If the difi'erent sciences offer to us a v;irying degree 
 
 of precision, it is from no want of certainty in themselves, but of 
 
 our mastery of their phenomena.
 
 EFI'ECl' ON EDUCATION AND METHOD. 25 
 
 The most interest injjj property of our foiinula of gradation is its 
 effect on education, both general and scientific. Tiiis 4. Effect on 
 is its direct and unquestionable residt. It will be Education. 
 more and more evident as we ])roceed, tliat no science can be effect- 
 ually pursuetl witiiout the ])reparaliou of a competent knowledge of 
 the anterior sciences on wiiicli it depends. Physical philoso[)hers 
 cannot understand Pliysics without at least a general knowledge of 
 Astronomy ; nor Chemists, without Pliysics and Astronomy ; nor 
 Physiologists, without Chemistiy, Physics, and Astronomy ; nor, 
 above all, the students of Social ])hilosophy, without a general 
 knowledge of all the anterior sciences. As such conditions are, as 
 yet, rarely fulfilled, and as no organization exists for their fulfil- 
 ment, there is amongst us, in fact, no I'alional scientific education. 
 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 
 i-egai(l to scientific education, it is no less striking in regard to 
 geiieial education. Our intellectual system cannot be renovated 
 till the natural sciences are studied in their proper order. Even 
 the highest undeistandings are apt to associate their ideas accord- 
 ing to the order in which they were I'eceived : and it is only an 
 intellect hei'e and there, in any age, which in its utmost vigour can, 
 like Bacon, Descartes, and Leibnitz, make a clearance in their field 
 of knowledge, so as to reconstruct from the foundatiou their system 
 of ideas. 
 
 iSucli is the operation of our gieat law upon scientific education 
 through its effect on Doctrine. We camiot appre- Effect on 
 eiate it duly without seeing how it affects Method. Method. 
 
 As the phenomena which are homogeneous have been classed 
 under one science, while those which belong to other sciences are 
 lieteiogeneous, it follows that the Positive Method must be con- 
 stantly modified in an unifoi-m manner in the range of the same 
 fundamental science, and will undergo modifications, different and 
 more and more compotuid, in passing from one science to another. 
 Thus, under the scale laitl 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 alieady seen, we cannot 
 understand the positive method in the absfiact, but only by its 
 a[)plication, it is clear that we can have no adequate concepticai of 
 it but by studying it in its varieties of application. No one science, 
 however well chosen, could exhibit it. Though the ]\lethod is 
 always the same, its procedure is varied. For instance, it should 
 be Observation with regard to one kind of phenomena, and Experi- 
 ment with regard to another; and different kinds of exi)eriment, 
 according to the case. In the same way, a general precept, derived 
 ffom one fundamental science, however a})plicablc to another, nuist 
 have its spirit preserved by a reference to its origin ; as in the case
 
 26 POSITIVE Fin LO SOPHY. 
 
 «>r the theory of Classifications. The best idea of the Positive 
 Method would, of course, be obtained by tlie study of the most 
 })rimitive and exalted of the sciences, if we were confined to one; 
 but this isolated view would i^ive no idea of its capacity of a]){)lica- 
 lion to others in a modified form. Each science has its own proper 
 advantages ; and without some knowled<i^e of them all, no conception 
 can be formed of the ])ower of the Method. 
 
 One more consideration must be briefly adverted to. It is 
 Orderly study necessaiy not only to liave a <jjeneral knowledge of all 
 of sciences. w^q scieuccs, but to study them in their order. What 
 can come of a study of com})licated phenomena, if the student have 
 not learned, by the contemplation of the sinifjler, what a Law is, 
 what it is to Observe ; what a Positive conception is ; and even what 
 a chain of reasoning is ? Yet tliis is the way our young physiologists 
 proceed eveiy day, — plunging into the study of living bodies, with- 
 out 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 ])hilosophical 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 intellectual 
 instrument, through an adequate study of the range of antei-ior phe- 
 nomena ? 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 preparatoi-y study ; and thus, the admission remains barren, and 
 st)cial theories abide in the theological or metaphysical state, in spite 
 of the efforts of those who believe themselves positive reformers. 
 
 These, then, are the four points of view under which we have re- 
 cognized the importance of a Rational and Positive Classification. 
 
 It cannot but have been observed that in our enumeration of the 
 
 sciences there is a prodigious omission. We have 
 Mathematics. ., ., . . -.r .1 .• 1 • rm 
 
 said notlung 01 Mathematical science. I he omis- 
 sion was intentional ; and the reason is no other than the vast im- 
 ])ortance of mathematics. This science will be the first of wliich 
 we shall treat. IMeantime, in order not to omit from onr sketch a 
 department 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 regard Mathe- 
 matics less as a constituent ])art of natural philosophy 
 than as having been, since the time of Descartes and 
 Newton, the true basis of the whole of natural philosophy ; though 
 , , . it is, exactly s])eaking, both the one and the olher. 
 
 To us it is of less value for the knowledge of which 
 it consists, substantial and valuable as that knowledge is, than as 
 , . . Imuix the most ])owerful instrnment that the human 
 
 mind can employ in the investigation 01 tlie laws or 
 iiatural phenomeiia.
 
 MAI HEMATICS A DOUBLE SCIENCE. 27 
 
 In due piocisiou, Matheniaiics must Le divided into two great 
 sciences, quite distinct from each otlier — Abstract a douUe 
 Mathematics, or tlie Calcidus (taking the word in its science. 
 most extended sense), and Concrete Mathematics, wliicli is composed 
 of General Geometry and of llational Mechanics. The Concrete 
 ])art is necessarily founded on the Abstract, and it becomes in its 
 turn the basis of all natural philosophy ; all the phenomena of the 
 universe being I'egarded, as far as possible, as geometrical or 
 mechanical. 
 
 The Abstract portion is the only one which is purely instrumental, 
 it being simply an immense extension of natural logic Abstract ma- 
 to a certain order of deductions. Geometry and me- tkematics an 
 chanics must, on the contraiy, be regarded as true ^n^trwimit. 
 natural sciences, founded, like nil others, on observation, though, 
 by the extreme simplicity of their phenomena, they Concrete ma- 
 can be systematized to much greater perfection. It thenuuks a 
 is this capacity which has caused the experimental science. 
 character of their first pi-inci{)les to be too much lost sight of. But 
 these two ])hysical sciences have this peculiarity, that they are now, 
 .and will be more and more, employed rather as method than as 
 doctrine. 
 
 It needs scarcely be ]iointed out that in placing Mathematics at 
 the head of Positive Philosophy, we aie only extending the appli- 
 cation of the princi{)le which has governed om- whole Classification. 
 AVe are simply carrying back our pi-inciple to its first manifesta- 
 tion. Geometrical and Mechanical phenomena aie the most general, 
 the most simple, the most abstract of all, — the most irreducible to 
 others, the most inde{)endent 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 sciences, and be the 
 })oint of departure of all Education, whether general Mathematics 
 or special. In an empirical way, this has hitheito pre-eiaiveut 
 Ijeen the custom, — a custom which arose from the ■^^ the scaie. 
 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 ])roblem, 
 the rational plan of the study of the Positive Philosophy. The 
 oi'der that results is this ; an order which of all })ossible ai-range- 
 ments is the only one that accords with the natur;d manifestation 
 of all phenomena. jMathkmatics, A«tkokomy, Physics, Cuemisthy, 
 Physiology, Social Physics.
 
 2S 
 
 BOOK I. 
 
 MA THEM A TICS. 
 
 CHAPTEE I. 
 
 MATHEMATICS, ABSTRACT AND CONCRETK. 
 
 Wi'^ are now to enter npou the study of the first of the Six great 
 ISciences : and we begin by estabhshing the importance of the Posi- 
 tive Philosophy in {jeriecting the cl)aracter of eacli science in itself. 
 Tiiougli Matliematics is the most ancient and the most perfect 
 science of all, the general idea of it is far from being clearly deter- 
 mined. The definition of the science, and its chief divisions, have 
 I'emained np to this time vague and uncertain. The plural form of 
 the name (gi'ammatically 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 con- 
 ceived of as a whole ; and since that time geometers have been too 
 much engaged on its different branches, and in applying it to the 
 most important laws of the universe, to have much attention left 
 for the general system of the science. Now however 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 acliieve- 
 ments of mathematicians have prepared the way for this by evidenc- 
 ing 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 Fimctions and of Analytical Mechanics. 
 
 The common description of Mathematics, as the science of 
 Description of Magnitudes, or somewhat more positively, the science 
 maihcmatics. which rclatcs to the Measurcmeii t of 31c(g7iitudes, in 
 too vague and unmeaning to have been used but lor want of a 
 better. Yet tlie idea contained in it is just at bottom, and is even 
 sufficiently extensive, if properly understood; but it needs precision 
 and depth. It is important in such matters not to depart unneces-
 
 NA TURE AND OBJE CT OF MA THE MA TICS. 2 9 
 
 parily from notions generally admitted ; and we will therefore see 
 liow, from this point of view, we can rise to such a definition of 
 JMatheniatics as will be adequate to the importance, extent, and 
 diificulty of the science. 
 
 Our first idea of measuring a magnitude is simply that of com- 
 ]mring the magnitude in question with another sup- Object of 
 ]iosed to be known, which is taken for the unit of mathematics. 
 comparison among all others of the same kind. Thus, when we 
 define mathematics 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. Neverthe- 
 less, this definition has 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 m.easure- 
 ment of a magnitude is often an 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 be able to traverse 
 tlie line from one end to the other ; and this cannot be done with 
 the greater number of the distances which interest us the most. 
 We cannot 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, 
 and 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 measure- 
 ment 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 measure- 
 ment so often impossible, we are compelled to devise means of 
 doing it indirectly. Hence arose Mathematics. 
 
 The general method employed, and the only conceivable one, is 
 to connect the magnitudes in question with some General 
 that can be directly determined, and thus to ascertain viethod. 
 the former, through their relations with the latter. Such is the
 
 30 POSITIVE PHILOSOPHY. 
 
 ])recise object of Mathematics regarded as a wliole. To form 
 anything like a worthy idea of it, we must remember that the 
 indirect determination of magnitudes may have many degrees of 
 indirectness. It often happens that the magnitudes to whicli 
 undetermined magnitudes are to be referred cannot themselves be 
 measured directly, and must themselves be made the subject of a 
 ])rior process, and so on thi'ough a whole series ; and thus, the 
 inind is often obliged to establish a long course of intermediaries 
 between the one and the other point of the inquiry — points which 
 may appear at the outset to have no connection whatever. 
 
 If this a})pears too abstract, it may become plain by a few 
 exam[)les. In observing a falling body, we ai'e 
 ' aware that two quantities are involved : the height 
 from which the body falls, and" the time occupied in its descent. 
 These two quantities are connected, as they vary together, antl 
 together remain fixed. In the language of mathematicians, they 
 are functions of each other. The measurement of one being 
 impracticable, it is supplied by that of the other. By observing the 
 time occupied by a stone in falling down a precipice, we can 
 ascertain the height of the precipice as accurately as if we could 
 measure it with a horiztintal line. In another case, we may be 
 able to know the height whence a body has fallen, and unable to 
 observe the time with precision, and then we must have recourse 
 to the inverse question, — to determine the time by the distance; 
 as, for instance, if we were to inquii'e how long it would take for a 
 body to fall from the moon. In these cases, the question is very 
 simple, supposing we do not complicate it with considerations of 
 intensity of giavity, resistance of a fluid medium, etc. But, to 
 enlaige the question, we must contemplate the ])henomenon in its 
 greatest generality by supposing the fall to be oblique, and taking 
 into account all the principal circumstances. Then, instead of two 
 variable quantities, simply connected, the phenomenon will present 
 a considerable number, — the space traversed, whether in a vertical 
 or horizontal direction ; the time employed in traversing it : the 
 velocity of the body at each point of its course ; and even the 
 intensity and direction of the im[)ulse which sent it forth ; and 
 finally, in some cases, the resistance of the medium, and the 
 intensity of gravity. All these quantities are so connected that 
 each in its turn may be determined indirectly by means of the 
 others, and thus we shall have as many mathematical inquiries as 
 there are magnitudes coexisting in the phenomenon considered. 
 Such a very simi)le change as this iu the physical conditions of 
 a problem may {)lace a mathematical question, originally quite 
 elementary, in the raidc of those difficult questions whose com- 
 plete and I'igorous solution transcends the power of the human 
 understanding. 
 
 Again, — we may take a geometrical example. We want to
 
 METHOD OF ArPLICATWy. 3r 
 
 di'((M-niIno a dlistance not directly iiicnsnr;il>le. We sliall cniicolve 
 of it as niakiiii; a p:irt of some //V/^^re, or system of lines of some 
 sort, of which the oilier paits are directly mensurable ; let iis say ;i 
 trijin,<:jle (for this is the simplest, and to it all others are reducible). 
 The distance in question is supposed to form a portion of a triangle, 
 in which we are able to determine directly, either another side and 
 two angles, or two sides and one angle. The knowledge required 
 is obtained by the mathematical labour of deducing the uidcnown 
 distance from the observed elements, by means of the relation be- 
 tween them. The process may, and commonly does, become highly 
 complicated by the elements supposed to be known being themselves 
 determinable otdy in an indirect manner, by the aid of fresh aux- 
 iliary systems, the number of which may be veiy considerable. The 
 distance, once ascertained, will often enable us to obtain new quan- 
 tities, which will oiler occasion for new mathematical questions. 
 Thus, when we once know the distance of any object, the observa- 
 tion, sim[)le and always possible, of its apparent diameter, may dis- 
 close to us, with certainty, however indirectly, its real dimensions ; 
 and at length, by a series of analogous inquiries, its surface, its 
 volume, even its weight, and a niultitude of other qualities which 
 might have seemed out of the reach of our knowledge for ever. It 
 is by such labours that Man has learned to know, not only the dis- 
 tances of the planets from the earth and from each other, but their 
 actual magnitude, — their true form, even to the inequalities on 
 theii- surface, and (what seems much more out of his reach) their 
 respective masses, their mean densities, and the leading circumstances 
 of the fall of heavy bodies on their respeclive surfaces, etc. Through 
 the power of mathematical theories, all this and very much more 
 has been obtained by means of a veiy small number of straight lines, 
 properly chosen, and a larger number of angles. We might even 
 say, to describe the general beai'ing of the science in a sentence, 
 that, but for the fear of multiplying mathematical operations 
 unnecessarily, and for the consequent necessity of reserving them 
 for the determination of quantities which could not be measured 
 directly, the knowledge of all magnitudes susce[)tible of precise 
 estimate which can be offered by the vai'ious orders of phenomena, 
 woidd be finally reducible to the immediate measurement of a single 
 sti'aight line, and of a suitable munber of angles. 
 
 We can now define Mathematical science with precision. It has 
 for its object the indirect measurement of magni- Truedefini- 
 tudes, and it proposes to determine magnitudes hy Hon of ma- 
 each other, according to the inecise relations luhich <^"^"'«^"^-'*- 
 exist between them. Preceding definitions have given to Mathe- 
 matics the character of an Art ; this raises it at once to the I'ank of 
 a. true Science. According to this defim'tion, the sj)ii-it of Mathe- 
 matics consists in regaj'ding as nmtually connected ail the quantities 
 which can be presented by any phenomenon whatsoevei', in order to
 
 32 POSITIVE PHILOSOPHY. 
 
 deduce all from each other. Now, there is evidently no phenomenon 
 which may not be reo;arded as affording such considerations. Hence 
 I'esults the naturally indefinite extent, and the rigorous logical 
 universalit}^ of Mathematical science. As for its actual practical 
 extent, we shall see what that is hereafter. 
 
 These explanations justify the name of Mathematics, applied to 
 the science we are considering. By itself it signifies Scirnce. The 
 Greeks had no other, and we may call it tlie science; for its defini- 
 tion is neither more nor less (if we omit the specific notion of 
 magnitudes) than the definition of all science whatsoever. All 
 science consists in the co-ordination of facts ; and no science could 
 exist among isolated observations. It might even be said that 
 Mathematics might enable us to dispense with all direct observation, 
 by empowering us to deduce from the smallest possible number of 
 imme(h"ate data the largest possible amount of results. Is not this 
 the real use, both in speculation and in action, of the laws which 
 we discover .among natural phenomena ? If so, Mathematics 
 merely urges to the ultimate degree, in its own way, researches 
 which every real science ])ursues, in various inferioi- degrees, in its 
 own sphere. Tims it is only through Mathematics tiiat we can 
 ihorouglily understand what true science is. Here alone can we find 
 in the highest degree simplicity and severity of scientific law, and 
 such abstraction as the human mind can attain. Any scientific 
 education setting forth from any other point, is faulty in its basis. 
 
 Tiuis far, we have viewed the science as a whole. We must now 
 consider its primary division. Tiie secondar}' divisions will be laid 
 down afterwai'ds. 
 
 Every mathematical solution spontaneously separates into two 
 parts. The inquiry being, as we have seen, the 
 
 J TS TWO PARTS •. • • 
 
 determuiation of utdvuown magnitudes, thi'ough their 
 relation to the known, the student must, in the first place, ascertain 
 what these relations are, in the case under his notice. This first 
 Their differ- IS the Concrete pai't of the inquiry. When it is 
 ent objects. accomplished, what remains is a pure question of 
 numbers, consisting simply in the determination of unknown 
 numbers, when we know by what relation they are cmmected with 
 known numbers. This second opeivation is the Abstract part of 
 the inquiry. The primary division of Mathematics is therefore into 
 two great sciences: — Abstract Mathematics, and Concrete 
 Mathematics. This division exists in all complete mathematical 
 questions whatever, whether more oi- less simple. 
 
 Recurring to the simplest case of a falling body, we must begin 
 by learning the relation between the lieight from which it falls and 
 the time occupied in falling. As Geometers say, we must find the 
 equation which exists between them. Till this is done, there is no 
 basis for a computation. This ascertainment may be extremely 
 difficult, and it is incomparably tlie superior part of the problem.
 
 CHARACTER OF THE TWO DIVISIONS. 33 
 
 The true scientific spirit is so modern, llint :is fnr :is we know, no 
 one before Galileo had remarked the acceleration of velocity in a 
 falling body, the natural supposition having been that the lieight 
 Avas iu unitorm proportion to the time. This first inquiry issued 
 in the discovery of the law of Gnlileo. The Concrete part being 
 ;iCcom{)lished, the Abstract remains. We have ascertained that 
 the spaces traversed in eacii second inci'ease as the series of odd 
 numbers, and we now have only the tnsk of the computation of the 
 height from the time, or of the time from the lieight ; 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 ; 
 ■wlience we may finally detei-mine 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. If the same ]>henomenon were taken 
 in its greatest generality, the reverse would be 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 be 
 equivalent. 
 
 We have seen the difference in their objects. Tkcir differ- 
 They ai-e no less different in their nature. entnatm-es. 
 
 The Concrete must depend on the character of the objects 
 examined, and must vary when new })henomena present them- 
 selves: whereas, the Abstract is wholly independent of the nature 
 of the objects, and is concerned only with their numerical relations. 
 Thus, a great variety of phenomena may be brought nnder one 
 geometrical solution. Cases which appear as unlike each other as 
 possible may stand foi- one another under the Abstiact process, 
 which thus serves for all, while the Concrete process must be new 
 in each case. Thus the Concrete ])rocess is Special, and the 
 Abstract is General. The character of the Concrete is experimental, 
 ])hysical, 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 external 
 aid, to educe the results which these equations contain. 
 
 We see how natural and complete this main division is. We 
 will briefly prescribe the limits of each section. 
 
 As it is the business of Concrete Mathematics to discover the 
 equations of phenomena, we might suppose that it Concrete 
 must comprehend as many distinct sciences as there Mathematics. 
 are distinct categories of phenomena ; but we are very far indeed 
 from liaving discovered mathematical laws in all orders of pheno- 
 mena. In fact, there are as yet only two great categories of pheno- 
 
 VOL. I. * C
 
 34 POSITIVE PHILOSCniY. 
 
 inena whose equations are constantly known : — Geometrical and 
 Meclianical plienomena. Tims, the Concrete part of Mathematics 
 consists of GrEOMKTKY and Kational Mechanics. 
 
 There is a point of view from which all phenonuvna mi<i^ht he 
 included under these two divisions. All natural eifects, considered 
 statically or dynamically, miglit be referred to laws of extension 
 or laws of motion. But this point of view is too high for us at 
 present ; and it is only in the regions of Astronomy, and, partially, 
 of terrestrial Physics, that this vast transformation has taken place. 
 We will then proceed on the supposition that Geometry and 
 Mechanics are the constituents of Concrete Mathematics. 
 
 The nature of Absti'act Mathematics is jirecisely determined. It 
 Abstract ' is comjwsed of what is called the Calculus, taking 
 Mathematics. \\^\^ 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 Con- 
 crete Mathematics, — the knowledge of the precise relations — that is, 
 the equations — between different mngnitudes which are considei'ed 
 sinudtaneously. The object of the Calculus, however indirect or 
 complicated tlie relations may be, is to discover unknown quantities 
 by the known. This science, though more advanced than any other, 
 is, in reality, only at its begiimingyet ; 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 ajipears 
 to have arisen out of the contem})lation of geometrical and mechani- 
 cal facts; but it is not the less independent of these sciences, 
 logically speaking. Analytical ideas ai-e, 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 
 precision to our actual knowledge, but establishes a far moi'e 
 perfect co-oi-dination in the study of ])henomena which allow of 
 such an application. If a single analytical question, brought to an 
 abstract sohition, involves the implicit solution of a multitude of 
 physical questions, the mind is enabled to perceive relations between 
 phenomena apparently isolated, and to extract from them the 
 quality which they have in common. To the wonder of the 
 student, unsuspected relations arise between problems which, instead 
 of being, as they appeared before, wholly unconnected, turn out to 
 be identical. There appears to be no coimection between the deter- 
 mination of the direction of a curve at each of its points and that of 
 the velocity of a body at each moment of its variable motion ; yet, 
 in the eyes of the geoinetei', these questions are but one. 
 
 When we have seized the true <reneral character of Mathematical
 
 MATHEMATICAL ANALYSIS. 35 
 
 Analysis, wc easily see how perfect it is, in comparison wilh all 
 other branches of onr positive science. The perfection consists in 
 tiie simplicity of the ideas contemplated ; and not, as Condillac and 
 others have snpposed, to the conciseness and generality of the si^ns 
 nsed as instrnments of reasoning. The signs are of admirable nse 
 to work out the ideas, when once obtained ; but, in fact, all the 
 great analytical conceptions were formed without any essential aid 
 I'rom the signs. Subjects which are by tiieir nature inferior in 
 simplicity and generality cannot be raised to logical perfection by 
 any artitice of scientific language. 
 
 We liave now seen what is the object and what is the chai'acter 
 of ]\rathematical Science. It remains for us to con- Extent of its 
 sider the extent of its domain. domain. 
 
 We must first admit that, in a logical view, this science is 
 necessarily and rigorously universal. There is no 
 inquiry which is not finally I'educible to a question suniieisau;/. 
 of Numbers ; for there is none which may not be conceived of as 
 consisting in the determination of quantities by each other, accord- 
 ing to certain relations. Tlie fact is, we are always endeavouring 
 to arrive at numbers, at fixed quantities, whatever may be our 
 subject, however uncertain our methods, and however rough our 
 results. Nothing can appear less like a mathematical inquiry than 
 the study of living bodies in a state of disease ; yet, in studying the 
 cure of disease, we are endeavouring to ascertain the quantities of 
 the ditferent agents which are to modify the organism, in order to 
 bring it to its natural state, atlmitting, as geometers do, for some 
 of these quantities, 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 pheno- 
 mena ; but the logical extension of the science, which is what we 
 are now considering, comprehends such instances as this. 
 
 Kant has divided human ideas into the two categories of quantity 
 and quality, which, if true, would destroy the universality of Mathe- 
 matics ; but Descartes' fundamental concejition of the relation of 
 the concrete to the abstract in Mathematics abolishes this division, 
 and ])i-oves that all ideas of quality are reducible to ideas of 
 (piantity. He 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 
 admit that every phenomenon may be as logically ca[)able of being 
 represented by an equation as a curve or a motion, if only we were 
 always caj^able (which we are very far from being) of first discover- 
 ing, and then resolving it. 
 
 The limitations of Mathematical science aie not, then, in its 
 nature. The limitations are in our intelligence: , ,. . 
 
 „]i ii n ^ l^ ^ • c ^^ ■ Its limitations. 
 
 and by tliese we tind the domain of the science
 
 36 POSITIVE PHILOSOPHY. 
 
 remarlcably restricted, in proportion as phenomena, in becoming 
 special, become complex. 
 
 Though, as we have seen, every question may be conceived of as 
 reducible to numbers, the reduction cannot be made by us except 
 in the case of the simplest and most general phenomena. The 
 difficulty 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 Piiysics, — that we can even hope to subject 
 to the process. The proi)erties of inorganic bodies are nearly 
 invariable; and therefore, with regard to them, the first condition 
 of mathematical inquiry can be fulfilled : the different quantities 
 which they present may be resolved into fixed numbers; but the 
 variableness of the propei'ties of organic bodies is beyond our man- 
 agement. An inorganic body, possessing solidity, foi'm, consistency, 
 s})ecific gravity, elasticity, etc., presents qualities which are within 
 our estimate, and can be ti'eated mathematically; but the case is 
 altered when Chemical action is added to these. Com])lications 
 and variations then enter into the question Avhich at present bafHe 
 mathematical analysis. Hereafter, it may be discovered what 
 fixed numbers exist in chemical combinations : but we ai-e as yet 
 very far from having any practical knowledge of them. Still further 
 are we from being able to form such computations amidst the con- 
 tinual 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 theii* 
 incessant munerical variations, vital phenomena are, practically, 
 placed in opposition to mathematical processes. If we should desire 
 to com})ute, in a single case, the most simple facts of a living body, 
 — such as its mean densit}^ its temperature, the velocity of its cir- 
 culation, 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 exhalations, — and, 
 yet more, the energy of its muscular force, the intensity of its 
 impressions, etc., we must make as many observations 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 con- 
 dition, suiTOunding circumstances perpetually varying, such as the 
 constitution of the atmosphere, etc. It is clear that no mathe- 
 matical 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 matliematical basis does not exist in these cases, as truly as 
 in phenomena which exhibit, in all clearness, the law of gravitation : 
 but that our faculties are too limited for the working of problems 
 BO intricate. We are baffled by various phenomena of inorganic
 
 LIMITS OF ABSTRA CT MA THEM A TICS. 3 7 
 
 l)0(Hes, wlicii tlioy arc very complex. For instance, no one doubts 
 tliat meteorological plietioniena, are subject to luatlieuiatical laws, 
 liowev^er liUle 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 it" we were sure of the mathematical law which 
 governs each agent. Onr feeble faculties could not grasp and wield 
 such an aggregate of conditions, however certain might be our know- 
 ledge of each. In the simplest cases in which we desire to npproxi- 
 mate the abstract to the conci-ete 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 difSculty is such that we are, as 
 yet, without any raathematicnl solution of this very problem. The 
 same is the case with the yet more simple instance of the move- 
 ment of a solid projectile through a resisting medium. 
 
 To the popular mind it may appear strange, considering t.bese 
 facts, that we know so mucb as we do about the planets. But in 
 reality, that chiss of phenomena is the most simple of all within our 
 cogniznnce. The most complex problem which they present is 
 the influence of a third body acting in the same wny on two which 
 are tending towards each other in virtue of gravitation ; and this 
 is a more sim{)le question than any terrestrial problem whatever. 
 We have, however, attained only approximate solutions in this 
 case. And the high perfection to which solar astronomy has been 
 brought by the use of mathematical science is owing to our having 
 })rofited by those facilities that we may call accidental, which the 
 favourable constitution of our planetary system presents. The 
 jdanets which compose it are few; their masses are very unequal, 
 and much less than that of the sun ; they are far distant from each 
 other ; their foi-ms are nearly spherical ; their orbits ai'e nearly 
 circidar, and oidv 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 nearness, 
 of occasioning pci'ceptible derangements. If any of the conditions 
 mentioned above had been different, though the law of gravitation 
 had existed as it is, we might not at this day have discovered it. And 
 if we were now to try to investigate Chemical phenomena 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 MatheTuatical analysis cnn be a[)i)lied only to 
 Inorganic Physics, we are not restricting its domain. Its rigorous 
 universality, in a logical view, has been established. To pi-etend 
 tiiat it is practically a[)plicable to the same extent would be merely
 
 38 POSITIVE PHILOSOPHY. 
 
 lo 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 preliminary 
 state which prepares for the others the time when they too may 
 hecome capable of mathematical ti'eatment. Our business is to 
 study phenomena, in the characters and relations in which they 
 ])resent themselves to us, abstaining from introducing considerations 
 of quantities, and mathematical laws, whicli it is beyond our power 
 to apply. 
 
 We owe to Mathematics both the origin of Positive Philosophy 
 and its Method. When this method was intioduced into the other 
 sciences, it was natural that it should be urged too far. But ench 
 science modified the method by the operation of its own peculiar 
 })lienomena. TIius only could that true definitive charncter be 
 bi'ought out, which must [irevent its being ever confounded with 
 that of any otlier fundamental science. 
 
 The aim, character, and general relations of Mathematical Science 
 have now been exhibited as fully as they could be in such a sketch 
 {is this. We must next pass in review the three great sciences of 
 wiiich it is composed, — the Calculus, Geometry, and Rational 
 Mechanics.
 
 39 
 
 CHAPTER II. 
 
 GENERAL YIEW OF MATHEMATICAL ANALYSIS. 
 
 The historical development of the Abstract portion of Matlieraatical 
 science lias, since the time of Descartes, been for the 
 most jiart determined by that of the Concrete. Yet 
 the Calcnlns in all ils principal branches must be understood before 
 ]>assing on to Geometry and Mechanics. The Concrete ]iortions of 
 ihe science depend on the Abstract, which are wholly independent 
 of them. We will now therefore proceed to a rapid review of the 
 leading conceptions of the Analysis. 
 
 First, however, we must take some nolice of the general idea of 
 an equation, and sec how far it is from being the True idea of 
 true one on which geometers proceed in practice; for an equation. 
 without settling this point we cannot determine, with any precision, 
 tiie real aim and extent of abstract mathematics. 
 
 The business of concrete mathematics is to discover the equations 
 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 oidy by forming a true idea of an equation that we 
 can lay down the real line of separation between the concrete 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 
 belween 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, eveiy 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 we find in establishing the 
 relation of the concrete to the abstract which meets us in every 
 great mathematical question, taken by itself. If the word equation 
 meant what we ai'e apt to suppose, it is not easy to see what diffi- 
 cultv there could be, in jrcneral, in establishing the equations of any 
 
 . •••11 
 
 jiioblem whatever. This ordinary notion of an equation is widely 
 iiidike what geometers understand in the actual working of the 
 hcience. 
 
 According to my view, functions must themselves be divided into
 
 40 POSITIVE PHILOSOPHY. 
 
 Abstract and Concrete ; the first of wliicli alone can enter into trne 
 equations. Every equation is a relation of eqnality between two 
 abstract functions of the magnitudes in question, including with the 
 primary magnitudes all the auxiliary magnitudes which may be 
 connected with the problem, and the introduction of which may 
 facilitate the discovery of tlie equations sought. 
 
 This distinction may be established by both the d 'priori and 
 d posttriori methods; by characterizing each kind of function, and 
 by enn.merating all the abstract functions yet known, — at least with 
 regai'd to their elements. 
 
 A priori ; Abstract functions express a mode of dependence 
 between magnitudes which may be conceived between 
 
 Abstract numbers alone, without the need of pointins: out anv 
 
 fwn.cttoTis «•• r *^ 
 
 phenomena m which it may be found realized ; 
 Concrete while Concrete functions are those whose expression 
 
 functions. ■ -n i x \ c i • > 
 
 requu-es a speciiied actual case oi physics, geometry, 
 mechanics, etc. 
 
 Most functions were concrete in their origin, — even those which 
 are at present the most purely abstract ; and the ancients discovered 
 only through geometrical definitions elementary algebraic properties 
 of functions, to which a numerical value was not attached till long 
 afterwards, 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, whicli are still sometimes concrete, sometimes abstract, 
 according to the point of view from which they are regarded, 
 
 A 'poHtei'ioi'i ; the distinguishing character, abstract or concrete, 
 of a function having been established, the question of any deter- 
 minate 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 analy- 
 tical functions are infinite in number, because we are here speaking, 
 it must be remembered, of the elements — of the sim})le, not of the 
 compound. We have ten elementary foi-nuilas; and, few as they 
 are, they may give rise to an infinite number of analytical combina- 
 tions. 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 inli'oduce additions, 
 though there is so much difficulty attending their augmentation, 
 that we cannot hope tiiat it will proceed very lar. 
 
 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 pheno- 
 mena, 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
 
 ALGEBRA AND ARITHMETIC. 41 
 
 (lomaiii of tlie calciihis!. The dilliculiy of the pnssnf)^e from the 
 coiici'ete to tlie abstract in general consists in our having only these 
 i'ew analytical elements with which to represent all the precise rehi- 
 tions which the whole range of natural phenomena afford to us. 
 Amidst their infinite variety, our conceptions must he far below 
 the real difficnlty ; and especially because these elements of our 
 analysis have been snppliedtous by the mathematical consideration 
 of the simplest phenomena of a geometrical origin, which can afford 
 us LL priori no rational guarantee 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 munber of analytical elements. 
 
 Thus far we have considered the Calculus as a whole. We 
 must now consider its divisions. These divisions we Twopart^of 
 nnist call the Algebraic Calculus, or Algebra, and theCaiaUus. 
 the Arithmetical Calculus, or Aritliinetic, 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 the 
 task of finding the values of the formulas thus ^'^ '"" 
 
 ol)tained. The values of the numbers sought are already repre- 
 sented by certain ex{)licit functions of given numbers : these values 
 must be determined; and this is the arithmetical 
 question. Thus the algebraic and the arithmetical 
 calculus differ in their object. They differ also in their view of 
 quantities, — Algebra considei'ing quantities in regard to their ?'c'/a- 
 tions, and Ai'ithmetic 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 pi'ocesses 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 Functions, and Arith- 
 metic the Ccilculus of Valves. We have seen that the division of the 
 Calculus is iuto two branches. It remains for us to compare the two, 
 in oi'der to learn their respective extent, importance, and difficulty. 
 
 The Calculus of A^alues, Arithmetic, appears at first to have 
 as wide a field as Aljjrebra, since as many questions . .,, ,. 
 
 . , ^ . ..''./ "^ . 1 ,.^ Arithmetic. 
 
 nuglit seem to arise irom it as we can conceive uitier- 
 
 ent algebraic 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 sim[)le functions, there 
 
 will be no difficulty with the compound. In the algebraic relation,
 
 42 POSITIVE PHILOSOPHY. 
 
 a coinponnd function plnys a very different part from that of the 
 elementary functions which constitute it ; and this is the source of 
 oiu" chief analytical dififlcuh.ies. But it is quite otherwise with the 
 Arithmetical Calculus. Thus, the number of distinct arithmetical 
 operations is indicated by that of the abstract elementary functions, 
 which we have seen to be very few. The detei-mination of the 
 values of these ten functions necessarily affords that of all tlie infinite 
 number comprehended in the whole of mathematical analysis : and 
 there 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 questions 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 logarithms, and the 
 calculation of trigonometrical tables, and some distinct and higher 
 ])rocednres ; in short, every operation which has for its object the 
 determination of the values of functions. And we nnist 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 
 irdierent 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 counnonly 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 mathematical science essentialh'' 
 consists. This is evident, when we look into the real nature of 
 arithmetical questions. 
 
 Determinations of vahies are, in fact, nothing else than real 
 -., , ^ra?2s/br7?ia^iow6' of the functions to be valued. These 
 
 J.tiS 7lQ/tlI/7*C 
 
 transformations have a special end ; but they are 
 essentially of the same nature as all taught by analysis. In this 
 view, the Calculus of values may be regarded as a sup[)lement, and 
 a ])articular application of the Calculus of functions, so that 
 arithmetic disaj)pears, as it were, as a distinct section in the body 
 of abstract mathematics. To make this evident, we must obsei've 
 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 deter- 
 mining its value, we merely express it under another detei-minate 
 form, to which we are in the habit of referring the idea of each 
 ])articulai- number by making it re-enter into the regular system of 
 numci-ation. This is made clear by what happens wiien the mode 
 of numeration is such that the question is its own answer ; as, for
 
 ALGEBRA, 43 
 
 instance, wlien we want to add toi^eilier seven and tliiity, 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 
 togetlier twenty- three and fourteen : but not the less is the opera- 
 tion merely one of transformation of a question already defined and 
 expressed. In this view, the calculus of values might be regarded 
 as a jiarticular application of the calculus of functions, arithmetic 
 thereby disappearing, as a distinct section, from the domain of 
 abstract matiiematics. — And here we have done with the Calculus 
 of values, and pass to the Calculus of functions, of which abstract 
 mathematics is essentially com[)osed. 
 
 We have seen that the difficulty of establishing the relation of 
 ihe concrete to the abstract is owing to the insuffici- ,, , 
 cncy of the very small number of analytical elements 
 ihat we are in possession of. The obstacle has been surmounted 
 in a great number of important cases : and w^e will now see 
 how the establishment of the equations of ])henomena has beeri 
 achieved. 
 
 The first means of remedying the difficulty of the small number 
 of analytical elements seems to be to create new Creation of 
 ones. i3ut a little considei'ation will show that this neiv functions. 
 I'esource 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 
 appears almost impossible : but the introduction of another ele- 
 mentary abstract function into analysis sup})oses the simultaneous 
 creation of a new arithmetical operation ; which is cei-tainly ex- 
 tremely difficult. If we tiy to proceed according to the method 
 w^hich procured us the elements w^e 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 nmst not therefore conclude that we 
 have reached the limit appointed by the powers of our understand- 
 ing. Special impiovements in mathematical analysis have yieldeil 
 us some ])artial substitutes, which have increased oiu" resources: 
 but it is clear that the augmentation of these elements cannot pro- 
 ceed but with extreme slowness. It is not in this direction, then, 
 that the human mind has found its means of facilitating the estab- 
 lishment of equations. 
 
 This first method being discarded, thei'o I'emains only one other. 
 
 As it is im])Ossible to find the equations directly, we „. ,. 
 
 , Z ,. ,. J , ^ ''' Finding equa- 
 
 must .seek itir corresponcnng ones l)etween other tions between 
 
 auxihary quantities, connected with the first accoi'd- auxiliary 
 
 A.-1. -ii ir .1 1 quantities. 
 
 ing to a certam deternunate law, and from the rela- 
 tion between which we may ascend to that of the primitive magni- 
 tudes. This is the fertile conce[)tion whicli we term the (ransceu-
 
 44 POSITIVE PHILOSOPHY. 
 
 denial analysis, and use as our finest instrument for the matlie- 
 in;itical exploration of natural ])henoiuena. 
 
 This conce])tion lias a much larger scope than even profound 
 geometers have hitherto supposed; for the auxiliary quantities 
 resorted to might he 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 ])er- 
 haps he found in a new mode of derivation. But, at })res- 
 enr, the only auxiliary quantities habitually substituted for the 
 primitive quantities in transcendental analysis are what are 
 called— 
 
 1st, infinitely small elements, the differentials of different orders 
 of those qnauLities, if we conceive of this analysis in the manner of 
 Leibnilz : or 
 
 2nd, the fi^uxions, the limits of the ratios of the simultaneous 
 increments of the primitive quantities, compared with one another ; 
 01", more briefly, the prime and ultimate ratios of these increments, 
 if we adopt the conception of Newton : or 
 
 3rd, the derivatives, properly so called, of these quantities ; that 
 is, the coefficients of the different teitus 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 eacli 
 of them will be exhibited hereafter. 
 
 We now see that the Calcidus of functions, or Algebra, must 
 
 Division of consist of two distinct branches. The one has for 
 
 the Calculus its object the resolution ol equations when they are 
 
 of functions. Jii-ectly established between the magnitudes in ques- 
 tion : 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 corres})onding equations between the direct magnitudes in 
 question ; — bringing the problem within the domain of the preced- 
 ing calculus. — It might seem 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 fii-st ; for, the proposed ques- 
 tions always requiring to be completed by ordinary analysis, they 
 must be left in sus[)ense if the instrument of resolution had not 
 l)een studied beforehand. 
 
 To ordinary analysis I propose to give the name of Calculus of 
 DiRKCT Functions. To transcendental analysis (which is known 
 by the names 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) 1 sliall give the title of Calculus of Indiukct Fung-
 
 AL GEBRAIC E Q UA TJONS. 45 
 
 TiONS. I obtain these fenns by generabzin<^ niul .giving ]irecisi()ii 
 to the ideas of Lagrange, and employ them to indicate the exact 
 character of the two forms of analysis. 
 
 SECTION L 
 
 ORDINARY ANALYSIS, OR CALCULUS OF DIRT:CT FUNCTIONS. 
 
 Algebra is adeqnate to the solution of mathematical quest ions 
 \vhicii are so sim[)le that we can form directly the equations 
 between the magnitudes considered, without its being necessary to 
 bring into the problem, either in substitution or alliance, any system 
 of auxiliary quantities derived from the primary. It is line, in the 
 majority of important cases, its use requires to be preceded and 
 })repai'ed for by that of the calculus of indirect functions, by which 
 the establishment of equations is facilitated: but though algebra 
 then takes the second })Iace, it is not the less a necessaiy agent iu 
 1 he solution of the question; so that the Calculus of direct func- 
 tions 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 develo[)ment it has attained. 
 
 Its object being the resolution of equations (that is, the discovery 
 of the mode of formation of unknown quantities by r, r.- 
 the known, according to the equations which exist 
 between tliem), it presents as many parts as we can imagine distinct 
 classes of equations; and its extent is therefore rigorously indefinite, 
 because the number of analytical functions susceptible of entering 
 into equations is illin}itable, though, as we have seen, composed of 
 a very small number of j)rimitive elements. 
 
 Tlie rational classification of equations must evidently be deter- 
 mined by the nature of the analytical elements of Classification 
 which their members are composed. Accordingly, of Equations. 
 analysts first divide equations with one or more variables into two 
 ])rincipal classes, according as they contain functions of only the 
 first three of the ten couples, or as they include also either exponential 
 or circular functions. Though the names of algebraic and tran- 
 scendental functions given to these principal groups are inapt, the 
 division between the corresponding equations is real 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 only. In the 
 first place, we must observe that, though they may Ahjehraic 
 often contain irrational functions of the unknown equations. 
 quantities, as well as rational functions, the first case can always 
 be brought under the second, by transformations more or less easy ;
 
 46 POSITIVE PHILOSOPHY. 
 
 po tliat it is onl}' with the latter that analysts liave had to occupy 
 themselves, to resolve all tlie algehraic equations. As to their 
 classification, tlie early method of classing them according to the 
 numher of their terms has heeu retained only for equations with two 
 terms, which are, in fact, susce{)tihle of a resolution proper to them- 
 selves. The classification hy their degrees, long universally estah- 
 lished, is eminently natui'al; tor this distinction rigorously determines 
 the greater or less difficulty of their resolution. The gradation can 
 be independently, as well as practically exhibited : for the most 
 general equation of each degree necessarily comprehends all those of 
 the different inferior degrees, as must also the formula which deter- 
 mines tlie unknown quantity: and therefore, however slight we may, 
 d j)riori, suppose the difficulty to be of the degree under notice, 
 it must offer more and more obstacles, in proportion to the rank of 
 the degree, because it is complicated in the execution with those of 
 all the preceding degrees. 
 
 This increase of difficulty is so great, that the resolution of 
 Algebraic re- f^'gebraic equations is as yet known to us oidy in the 
 solution of four first degrees. In this respect, algebra has ad- 
 equations. vauced but little since the labours of Descartes and 
 the Italian analysts of the sixteenth century; though there has pro- 
 bably not been a single geometer for two centuries past who has not 
 striven to advance the resolution of equations. The general equa- 
 tion of the fifth degree has itself, thus far, resisted all attempts. 
 The formula of the fourth degree is so difficult as to be almost 
 inapplicable ; and analysts, while by no means despairing of the 
 I'esolution of equations of the fifth, and even higher degrees, being 
 obtained, have tacitly agreed to give u[) such researches. 
 
 The only question of this kind which would be of eminent impor- 
 tance, at least in its logical relations, would be the genei-al resolution 
 of algebraic equations of any degree whatever. But the more we 
 ponder this subject, the moi'e we are led to suppose, with Lagrange, 
 that it exceeds the scope of our understandings. Even if the 
 requisite formula could be obtained, it cotdd not be usefully applied, 
 unless we could simplify it, without im})airing its generality, 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, properly so called ; that 
 is, of the calculus of direct functions, comprehending the resolution 
 of all the equations that can be formed by the analytical functions 
 known to us at this day. Again, we must rememlier that by a law 
 of our nature, we shall always remain below the difficulty of science, 
 our means of conceiving of new questions being always more power- 
 ful than our i-esotuces for resolving them; in other words, the human 
 mind being more a[)t at imagining than at reasoning. Thus, if we 
 hud resolved all the analytical equations now known, and if, to do
 
 NUMERICAL RESOL UTION OF E QUA TJONS. 4 7 
 
 this, wo liad found new analytical elements, lliese again would intro- 
 duce classes of equations of which we now know nothing: and so, 
 however great might be the increase of our knowledge, the imper- 
 fection of our algebraic science woidd be perpetually reproduced. 
 
 Tiie methods that we have are, the coniplete resolution of the 
 equations of the first ibur degrees; of any binomial Ourcxisiivy 
 equations; of certain special equations of the superior knowicdae. 
 degrees ; and of a very small number of ex})onential, logarithmic, 
 and circular equations. These elements are very limited ; but geo- 
 meters have succeeded in treating with them a great number of 
 important questions in an admirable manner. The im{)rovements 
 introduced within a cenluiy into mathematical analysis have con- 
 tributed moi-e to render the little knowledge tliat we have immea- 
 surably useful, than to increase it. 
 
 To fill up the vast gap in the resolution of algebraic equations of 
 the higher degrees, analysts have had recourse to a .r . , 
 
 * . ^ .' •' Numert'-.al re- 
 
 new order of questions, — to Avtiat they call the numer- solution of 
 ical resolution of equations. Not being able to obtain equations. 
 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 opera- 
 tion 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 
 tins are now sufficiently general ; and what remains is to simplify 
 them so as to fit them for regular application. While such is the 
 state of algebra, we have to endeavour so to dispose tlie questions 
 to be worked as to require finally only this 7iumerical resolution 
 of the equations. We must not forget however that this is veiy 
 imperfect algebra ; and it is only isolated, or truly final questions 
 (wiiich are very few), that can be brought finally to depend upon 
 only the numerical resolution of equations. Most questions ai-e only 
 l)reparatoiy, — 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 \\\e formula which exhibits its deriva- 
 tion. Even in the most simple questions, when this luimerical 
 resolution is strictly sufficient, it is not the less a very imperfect 
 method. Because we cannot abstract and treat separately the alge- 
 braic ]iart of the question, which is coujmon to all the cases which 
 result from the mere variation of the given numbers, we are obliged 
 to go over again the whole series ol operations for the slightest 
 change that may take place in any one of the quantities concerned. 
 Tims is the calculus of direct functions at piesent divided into 
 two parts, as it is employed for the algebraic or tlie numei-ical 
 lesolution of equations. The first, the only satisfactory one, is 
 unfortunately very restricted, and there is little hope that it will 
 ever be otherwi«e : the second, usuallv insufficient, has at least the
 
 48 POSITIVE PHILOSOPHY. 
 
 n(lvanln<;e 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 considered by 
 them. Moreover, there is, in regard to their methods, an entirely 
 different ])rocedure in their rational distribution. In the first part, 
 Ave have nothing to do with the values of the unknown quantities, 
 and tlie division must take place according to the nature of the 
 equations which we are able to resolve ; whereas in the second, we 
 liave nothing to do with the degrees of the equations, as the methods 
 are applicable to equations of any degree whatever ; but the con- 
 cern is with the numerical character of the values of the unknown 
 quantities. 
 
 These two parts, which constitute the immediate object of the 
 Tiie Theory of Calculus of direct functions, are subordinated to a 
 equations. third, purely speculative, from which both derive 
 their most effectual resources, and which has 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 equations has, on account of its generality, special need of this 
 rational foundation. 
 
 Two orders of questions divide this important department of 
 algebra between them ; first, those which relate to the composition 
 of equations, and then those that relate to their transformation ; the 
 business of these last being to modify the roots of an equation with- 
 out knowing them, according to any given law, provided this law is 
 uniform in relation to all these roots. 
 
 One more theory remains to be noticed, to com])lete our rapid 
 exhibition of the different essential parts of the calculus of direct 
 Method of fuuctious. This theory, which relates to the trans- 
 indeifminate formation of functions into series by the aid of what 
 Coefficients. jg called the Method of indeterminate Coefficients, is 
 one of the most fertile and important in algebra. This eminently 
 analytical method is one of the most remarkable discoveries of 
 Desca.rtes. The invention and development of the infinitesimal 
 calculus, for 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 its applications and enlarged its 
 lesources ; so tliat, by the useful combination of the two theories, 
 the employment of the method of indeterminate coefficients has 
 become nnich more extensive than it was even before the formation 
 of the calculus of indirect functions. 
 
 I have now completed my sketch of the Calculus of Direct Func- 
 tions. AVe must next pass on to the moi'e important and extensive 
 branch of our science, the Calculus of Indirect Functions.
 
 TRANSCENDENTAL ANALYSIS. 49 
 
 SECTION II. 
 
 TRANSCENDENTAL ANALYSIS, OR CALCULUS OF INDIRECT FUNCTIONS. 
 
 We referred (p. 44) in a former sectioti to the views of tlie tran- 
 scendental analysis presented by Leibnitz, Newton, Tkreeprinci- 
 and Lno-i-ans^e. We shall see that each conception purviews. 
 has advantages of its own, that all are finally eqnivalent, and that no 
 method has yet been fonnd 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 oidy an historical interest ; and meanwhile, the 
 science must be regarded as in a merely provisional state, which 
 requires the use of all the three conce})tions at the same time; for 
 it is only by the use of them all that an adequate idea of the analy- 
 sis and its api)lications can be formed. The vast extent and diffi- 
 culty of this part of mathematics, and its recent formation, should 
 ])revent our being at all surprised at the existing want of system. 
 The concei)tion which will doubtless give a fixed and uniform 
 character to the science has come into the hands of only one new 
 generation of geometers since its creation ; and the intellectual 
 habits requisite to perfect it have not been sufficiently formed. 
 
 Tiie first germ of the intiin"tesimal method (which can be con- 
 ceived of independently of the Calculus) may be re- 
 cognized in the old Greek MetJiod 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 considei-ation of a polygon, inscribed or circumscribed, by 
 means of which the curve itself was reached, the limits of the 
 ]Mimitive ratios being suitably taken. Tliere is no doubt of the 
 tiliation 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 geome- 
 ters was to generalize the concei)tion of the ancients, and, consider- 
 ing it in an abstract manner, to reduce it to a system of calculation, 
 which was impossible to theni. 
 
 Liigrange justly ascribes to the great geometer Fermat the first 
 idea in this new direction. Fermat 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 tatigents, in which piocess he introdnced anxiliaries 
 which he afterwards suppressed as null when the ecpiations obtained 
 had undergone certain suitable transformations. After some modi- 
 fications of the ideas of Fermat in the intermediate time, Leibnitz 
 stripped the process of some complications, and formed the analysis 
 
 VOL. I. 1)
 
 50 POSITIVE PHILOSOPHY. 
 
 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 discoveiy was so ri[)e, as all great 
 conceptions are at the hour of their ndvent, 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 atid facility to the fundamental method. 
 Lagrange afterwards, discarding the lielerogeneous considerations 
 which had gnided Leibnitz and Newton, reduced the analysis to a 
 purely algebraic system, which only wants more aptitude for appli- 
 cation. 
 
 We will notice the three methods in their order. 
 
 The method of Leibnitz consists in introducing into the calcidus. 
 Method of in order to facilitate the establishment of equations, 
 Leibnitz. the infinitely small elements or dijftrentials whicli 
 are supposed to constitute the quantities whose relations we are 
 seeking. There are relations between these difierentials which are 
 simpler and more discoverable than those of the pi-imitive quantities ; 
 and by these we may afterwards (through a special calculus employed 
 to eliminate these auxiliary infinitesimals) recur to the equations 
 sought, which it would usually have beenimpossible to obtain dii-ectl v. 
 This indirect analysis may have various degiees of indirectness ; for, 
 when there is too much difficulty in forming the equation between tlie 
 differentials of the magnitudes under notice, a second application 
 of the method is required, the differentials being now treated as 
 new ])rimitive quantities, and a relation being sought between 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 be finally eliminated. 
 
 It may be asked by novices in these studies, how these aux- 
 iliary quantities can be of use while they are of the same species 
 with the magnitudes to be treated, seeing that the greater or less 
 value of any quantity cannot afiect any inquiry which has nothing 
 to do with value at all. The explanation is this. We must begin 
 by distinguishing the different orders of infinitely small quantities, 
 ol)taining a precise idea of this by considering them as being 
 either the successive powers of the same primitive infinitely small 
 quantity, or as being quantities which may be regarded as having 
 finite ratios with these powers ; so that, for instance, the second or 
 third or other differentials of the same variable ai'e classed as 
 infinitely small quantities of the second, third, or other order, 
 because it is easy to exhibit in them finite nniltiples of the second, 
 third, or other powers of a certain first differential. These ])i-e- 
 liminary ideas being laid down, the spirit of the infinitesimal 
 analysis consists in constantly neglecting the infinitely small 
 quantities in comparison with finite quantities ; and genei'ally, the
 
 METHOD OF LEIBNITZ. 51 
 
 infiintely small quantities of any order whatever in comparison with 
 all those of an inferior order. We see at once liow such a power must 
 facilitate the formation of equations between the difFerentials of 
 quantities, since \ve can substitute ior these difFerentials such other 
 elements as we may choose, and as will be more simple to treat, 
 only obscrvini:^ the condition that the new elements shall diller 
 iVom the prece^ling 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 elements, and 
 curved surfaces as formed of ])lane 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 vai'ied aj)[)licatiou ot" this method may give some idea 
 of the vast scope of the con("e})tion 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 conce[)ti()n w\as 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 
 abstract. In this lespect, we must regard it as the necessary com- 
 plement of" the great fundamental idea of Descartes on the general 
 analytical rejiresentation of natural j)henomena ; an idea which 
 could not be didy estimated or put to use till after the formation of 
 the infinitesimal analysis. 
 
 This analysis has another pi'operty, besides that of facilitating 
 the study of the mathematical laws of all phenomena, and perhaps 
 not less important than that. The differential formulas exhibit an 
 extreme generality, expressing in a single equation GeneraiiUjof 
 each deternn'nate phenomenon, however varied may the formulas. 
 be the subjects to wdiich 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 invaiiable formula ex- 
 presses the mathematical law of all variable motion; and one single 
 equation rej)resents the distribution of heat in any body, and for 
 any case. This remarkable genei'ality 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 perce[)tion of positive 
 approximations between classes of wholly different phenomena, 
 through the analogies presented by the differential expressions of 
 llieir mathematical laws. In virtue of this second property of the 
 analysis, the entire system of an immense science, like geometiy or 
 mechanics, has submitted to a condensation into a small number 
 of analytical formulas, from which the solution of all particular 
 problems can be deduced, by invariable rules.
 
 52 POSITIVE PIIILOSOFHY. 
 
 Tills beautiful meiliod is, liowever, imperfect in its lo<^icjil basis. 
 Justification of At first, j^^eoiiieters were naturally nioi-e intent upon 
 the Method. extending llie discovery and nniltiplying its applica- 
 tions, tlian 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 pi'oblems. 
 It became necessary, however, to recur to the basis of the new 
 analysis, to establish the rigorous exactness ot" the pi-ocesses 
 employed, notwithstanding their appai'ent 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 
 approximative calculus, the successive operations of which might, 
 it is evident, admit an augmenting amount of error. Some of his 
 successois were satisfied with showing that its results accorded 
 with those obtained by oi'dinary algebra, or the geometry of the 
 ancients, i-eproducing by these last some solutions which could be 
 at first obtained only by the new method. Some, again, demon- 
 strated the conformity of the new conception with others ; that of 
 Newton especially, which was unquestionably exact. This afforded 
 a practical justitication : but, in a case of such unequalled inipor- 
 tance, a logical justification is also required, — a direct proof of the 
 iiecessaiy rationality ot" 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 en-oi-s. We can- 
 not say that all the logical pcaffoldingof the infinitesimal method may 
 not have a merely provisional existence, vicious as it is in its nature: 
 but, in the present state of our knowledge, Oarnot's principle of the 
 necessary compensation of errors is of more importance, in legitimat- 
 ing the analysis of Leibnitz, than is even yet commonly suj)p()sed. 
 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 aux- 
 iliaries canuftt have occasioned other than infinitely small errors in 
 all the equations ; and when the lelations of finite quantities are 
 reached, these relations must be I'igorously exact, since the only 
 errors then possible nuist be finite ones, which cannot have entered: 
 and thus the final equations become perfect. Carnot's theory is 
 doubtless moi-e subtle than solid; but it has no other radical logical 
 vice than that of the infinitesimal method itself, of which it is, as 
 it seems to me, the natuial develo])ment: and general explanation ; 
 so that it nuist be adopted as long as that method is directly 
 em})loyed. 
 
 The philosophical character of the transcendental analysis has 
 now been sufficiently exhibited to allow of my giving only the 
 principal idea of the other two methods. 
 
 Newton's Newtou offered his conception under several dif- 
 
 MhTHOD. ferent forms in succession. That which is now
 
 NEWTON'S METHOD. 53 
 
 most commonly adopted, at least on the Continent, was called by 
 liinisell", sonu'tinies the Method of iw'imc and ullimale liatios, sonu'- 
 tiuies the Method of Limits, by which last term it is now usually 
 known. 
 
 Under this Method, the auxiliaries introduced are the limits of 
 the ratios of the simultaneous increments of the Method of 
 primitive quantities ; or, in other words, the final Limits. 
 ratios of these increments; limits or final ratios which we can 
 easily show to have a determinate and finite value. A special 
 calculus, wliich is the equivalent of the infinitesimal calculus, is 
 afterwards employed, to rise from the equations between these limits 
 to the coi-responding equations between the primitive quantities 
 themselves. 
 
 Tlie power of easy ex))ressijn of the mathematical laws of pheno- 
 mena <;iven by this analysis arises from the calculus api)lyin<;, not 
 to the increments themselves of the jjroposed quantities, but to the 
 limits ot the lalios of those increments ; and from our being 
 therefore able always to substitute for each increment any other 
 magnitude more eaty 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-, that the calculus of limits can be in no 
 way afifected by this substitution. Starting from this princi[)le, we 
 find neaily 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 jjolygons, and variable motions as the limits of an aggre- 
 gate of uniform motions of continually nearer a})[)roximation, etc. 
 etc. Such is, in Kubslance, Newton's conception ; or ratlier, that 
 which Maclauiin and D'AIembert have offered as the most j-ational 
 basis of the transcendental analysis, in the endeavour to fix and 
 arrange Newton's ideas on the subject. 
 
 Newton had another view, however, which ought to be presented 
 here, because it is still the special form of the cal- Fiuxiunsund 
 cuius of indirect functions commonly adopted by fluc7Us. 
 English geometers; and also on account of its ingenious clearness 
 in some cases, and of its having furnished the notation best adai)te(l 
 to this manner of legarding the transcendental analysis. I mean 
 the Calculus 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 ditferent quantities 
 presented by the curve, the abscissa, the ordinate, the arc, the 
 area, etc., will be I'egai-detl as simultaneously produced by succes- 
 sive degrees during this motion. The velocit// w'llh which each one 
 will have been described will be called the fluxion of that quantity, 
 which inversely would have been called {influent. Henceforth, the
 
 54 POSITIVE PHILOSOPHY. 
 
 transcendei)inl nnnlysis will, accordingly to tliis coDcepiion. consist in 
 ibrminii; directly tlie equations between the fluxions of the ])roposcd 
 quantities, to deduce from them afterwards, by a special Calculus, 
 the equations between the fluents themselves. What has just been 
 stated respecting curves may evidently be transferred to any magni- 
 tudes whatever, regarded, by the help of a suitable image, as some 
 being produced by the motion of others. This method is evidently 
 the same witb that of limits complicated wilh 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 ultimate 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 re- 
 quisite for us to offer special proofs of this. 
 
 Lagrange's conception consists, in its admirable simplicity, in 
 Lagrange's considering the transcendental analysis to be a great 
 Method. algebraic artifice, by which, to facilitate the establish- 
 
 ment of equations, we must introduce, in the place of or with the 
 ]trimitive functions, their derived functions; that is, according to 
 the definition of Lagrange the coefficient of the first term of the 
 increment of each function, arranged according to the ascending 
 ])owers of the increment of its variable. The Calculus of indiiect 
 fimctions, pi-o))erly 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 corre- 
 sponding equations between the primitive magnitudes. The tran- 
 scendental analysis is then only a simple, but very cons^iderable 
 extension of ordinary analysis. It has long been a common practice 
 with geometers to intioduce, in analytical investigations, in the 
 place of the magnitudes in question, their different powers, or tlieir 
 logarithms, or their sines, etc., in order to simplify the equations, 
 and even to obtain them more easily. Successive derivation is a 
 general artifice of the same nature, only of greater extent, and 
 eommanding, in consequence, much more important resources for 
 this connnon object. 
 
 But, though we may easily conceive, a priori, that the auxiliary 
 use of these derivatives may facilitate the study of equations, it is 
 not easy to ex})lain why it tiiust 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 j)iecise advantages, in an absti'act manner, and without 
 rocun-ence to the other conceptions of the transcendental analysis. 
 These advantages can be established only in the separate consider- 
 {ition of each j)rincipal question ; and this vei-ification becomes 
 laborious in the treatment of a complex problem. 
 
 Other theories have been proposed, such as Euler's Calculus of 
 vanisldvg quantities: but they are merely modifications of the three
 
 IDENTITY OF THE THREE METHODS. 55 
 
 just cxliihik'd. AVe must next compaie niul csliinate these 
 methods ; and in the liivst place observe their perfect and necessary 
 conformity. 
 
 Considering tlie lliree methods in re^^ard to their destination, in- 
 de[»endently of [)reliminary ideas, it is clear that they Jdcntity of the 
 all consist in the same j^eneral logical artifice; that three methods. 
 is, the introduction of a certain system of auxiliai'y magnitudes 
 uniformly 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 the}^ 
 must be finally eliminated by the help of a special calculus. It was 
 this which ck'tei'mined me to define the transcendental analysis as 
 the Calculus of indirect functions, in order to mark its true })hilo- 
 soi)hical character, while excluding all discussion about the best 
 manner of conceiving and applying it. Whatever may be the 
 Tiiethod 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 cannot hope that 
 the Calculus will ever lay hold of all questions of natural philosophy 
 — geometrical, mechanical, thermokigical, 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 establishment 
 of equations, which are the indispensable point of de])artuie for all 
 analytical investigations. But this preliminary elaboration has 
 been remaikably simplified by the creation of the transcendental 
 analy^sis, which lias thus hastened the moment at which general and 
 abstract processes may be uniforndy and exactly applied to the 
 solution, by reducing the operation to finding the equations between 
 auxiliaiy magnitudes, whence the Calculus leads to equations 
 directly ixdatitig to the proposed magnitudes, which had formerly 
 to be established directly. Whether these indirect equations 
 are differential equations, according to Leibnitz, or equations 
 of limits, according to Newton, or derived equations, accord- 
 ing to Lagrange, the general procedure is evidently always the 
 same. The coincidence is not only in the result but in the pro- 
 cess ; for the auxiliaries introduced are really identical, being only 
 legarded from different points of view. The conceptions of Leibnitz 
 and of Newton consist in making known in any case two general 
 necessary properties of the derived function of Lagrange. The 
 ti'anscendental analysi.s, then, examined abstractly and in its prin- 
 ciple, is always the same, whatever conception is adopted; and the 
 jirocessesof the Calculus of indirect functions are necessarily identical 
 ii' the.se diffeient methods, which must therefore, under any applicii- 
 liun whatever, lead to ligorously unilorm results.
 
 56 POSITIVE PHILOSOPHY. 
 
 If we endeavour to estimate their comparative value, we shall find 
 
 Their compara- in eacli of the three conceptions advantat^jes and 
 
 live value. inconveniences which are peculiar to it, and which 
 
 prevent geometers from adhering to any one of them, as exclusive 
 
 and final. 
 
 The method of Leihnitz has eminently the advantage in the 
 rapidity and ease with which it effects the formation of equations 
 hetween auxiliary magnitudes. We owe to its use the high per- 
 fection attained by all the general theories of geometry and mechanics. 
 Whatever may be the s{)eculative 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 
 un a new basis, rendered a candid and decisive homage to the con- 
 ception of Leibnitz, by employing it exclusively in the whole 
 system of his 'Analytical Mechanics.' Such a fact needs no com- 
 ment. Yet are we obliged to admit, with Lagrange, that the con- 
 ception of Leibnitz is radically vicious in its logical relations. Ho 
 himself declared the notion of infinitely small quantities 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 metaphysical age of its 
 birth and tendencies of its originator. By the ingenious principhj 
 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 com- 
 mit errors which have to be afterwards compensated. There is 
 nothing very logical in this ; nor is anything 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 upon princii)les widely 
 different from those which serve as a basis to ordinary analysis. 
 This division of analysis into two systems, almost wholly inde- 
 ])endent, tends to prevent the formation of general analytical con- 
 ceptions. To estimate the consequences duly, we must i-ecur in 
 thought to the state of the science before Lngrange had estab- 
 lished a genei'al and complete harmony between these two great 
 sections. 
 
 Newton's conception is free from the logical objections imputable 
 to that of Leibnitz. The notion of limits is in fact remarkable for 
 its distinctness and precision. The equations are, in this case, re- 
 ijrarded as exact from their oii<rin ; and the jieneral rules of reason-
 
 EMPLOYMENT OF ALL TLIE THREE. 57 
 
 ing .are as constantly observed as in ordinary analysis. I3iil, it is 
 weak in resources, and enibai-rassiu<2^ in operation, conipaied with 
 the infinitesimal metiiod. In its applications, the relative inferiority 
 of this theory is very strongly marked. It also separates tlie 
 ordinaiy and transcendental analysis, though not so C()ns[)icnonsly 
 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 
 the fundamental ideas, are found in the conception of Lagrange, 
 and thei-e alone. It is therefore the most {)hilosophical of all. 
 Discarding every heterogeneous consideration, Lagrange reduced 
 the transcendental analysis to its proper character, — that of present- 
 ing 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 sin)ple 
 extension of ordinary analysis. It is a superior algebi'a. All tiie 
 different parts of abstract mathematics, till then so incoherent, might 
 be from that moment conceived of as forming a single system. 
 This pliil(»soi)hical superiority marks it for adoption as the linal 
 theory of transcendental analysis ; but it presents too many diffi- 
 culties in its application, in comparison with the others, to admit 
 of its exclusive preference at ])resent. Lagrange himself had great 
 difficulty in rediscovering, by his own method, the principal results 
 already obtained by the infinitesimal method, on general questions 
 in geometry and mechanics ; and we may judge by that what 
 obstacles would occur in treating in the same way questions really 
 new and im})()rtant. Though Lagrange, stimulated by difficulty, 
 obtained results in some cases which otlier men would have 
 despaired of, it is not the less true that his conception has thus 
 far i-emained, as a whole, essentially unsuited to applications. 
 
 The result of such a com[)arison of these three methotls 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 enqjloy 
 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 other 
 departments of mathematical science, the consideration of difterent 
 methods for a single class of questions may be useful, apart from 
 the iiistorical interest which it presents; but it is not indispensable. 
 Here, on the contrary, it is strictly indispensable. Without it there 
 can be no j)iiilosophical judgment of this adn)irable creation of the 
 human mind; nor any success and facility in the use of tiiis powei- 
 ful instrument.
 
 58 POSITIVE PHILOSOPHY, 
 
 THE DIFFERENTIAL AND INTEGRAL CALCULUS. 
 
 The Calculus of Indirect functions is necessarily divided into two 
 iiarts ; or rather, it is composed of two distinct cal- 
 culi, 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 magni- 
 tudes ; 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. 
 
 Diffei'ent names have been given to the two systems, according 
 to the point of view from which the entire analysis has been re- 
 garded. Tiie infinitesimal metliod, properly so called, being most 
 in use; almost all geometers employ the terms Differential Calculus 
 and Integral Calculus established by Leibnitz. Newton, in accord- 
 ance with his method, called tlie first the Calculus of Fluxions, and 
 the second tlie Calculus of Fluents, terms which were till lately 
 commonly adopted in England. According to the theory of 
 Lngrange, the one would be called the Calculus of Derived Func- 
 tions, and the other the Calculus of Primitive Functions. I shall 
 make use of the terms of Leibnitz, as the fittest for the formation 
 of secondary expressions, thougli we must, as has been shown, 
 employ all tiie conceptions concurrently, approaching as nearly as 
 mny be to that of Lngiange. 
 
 The dififei'ential calculus is obviously the rational basis of the 
 Their mutual iutegj-al. We have seen that ten simple functions 
 relations. constitute the elements of our analysis. We cannot 
 
 Icnow how to integrate directly any other differential expressions 
 than those produced by the differentiation of those ten functions. 
 T^he 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 nil minds what can be the proper 
 utility of the differential calculus, independently of this necessary 
 connection with the integi-al calculus, wliich seems as if it must be 
 in itself the only dii-ectly indispensable one ; in fact, the elimination 
 of the infinitesimals or the derivatives, introduced as auxiliaries, 
 l)eing the final object of the calculus of indirect functions, it is 
 natural to tliink that the calculus wliich teaches us to deduce the 
 equations between the ]ii-imitive magiiiludes fVoni those between the 
 auxiliary magm'tndes must meet all the general needs of the tran- 
 scendental analysis, without oni- seeing at first whaf special and con- 
 stant part tlie solution of the inverse question can have in such an 
 analysis. A common answei- is assigning to the differential calculus 
 tl Hi office of forming the ditfei-ent.ial equations; but this is clearly 
 an error ; for the primitive formation of differential equations is not
 
 THE DIFFERENTIAL AND INTEGRAL CALCULUS. 59 
 
 llie business of any calculus, for ii is, on tlie contraiy, the point of 
 (le{)arture of any calculus whatever. The very use of the (liferent ial 
 calculus is enablin<j^ us to differentiate the various equations; and 
 it cannot therefoie he the process for eslahlishini; them. This 
 c'oininou ei'ior ai'ises fi-oui con found in 12^ the infiuitesiuial calcuhis 
 with the infinitesimal method, which last facilitates the fornjatioii 
 of equations, in eveiT application of the transcendental analysis. 
 The calculus is the indispensable complement of the method ; but 
 it is perfectly distinct from it. But a^ain, we should much miscon- 
 ceive the jK'culiar importance of this first branch of the calculus of 
 indirect functions if we saw in it only a pi-eliminary pi'ocess, 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 forming difierential 
 i'quations, we rarely restrict oui'selves to introducing Cases of union 
 difterentially only those magnitudes whose relations of the two. 
 are sought. It would often bo impossible to establish equatlon.s 
 without introducing olhei' magnitudes whose relations are, or are 
 .sup{K)sed to be, known. Now in such cases it is necessaiy that the 
 dill'erentials of these intermediaries should be eliminated before the 
 equations ai'e fit for integration. This elimination belongs to the 
 difierential calculus; for it must be done by determining, by means 
 of the equations between the intermediary functions, the relations 
 (if their differentials; and this is merely a question of differentia- 
 lion. This is tlie way in which the differential calculus not only 
 pre{)ares a basis for the integral, but makes it available in a multi- 
 lude of cases which could not otheiwise be treated. Cases of the 
 There are some questions, few, but highly important, i>ifferentiai 
 which admit of the employment of the difierential Calculus alone. 
 calculus alone. They are those in which the magnitudes sought 
 enter directly, and not by their differentials, into the jirimitive dif- 
 ferential equations, which then contain differentially only the various 
 known functions employed, as we saw just now., as intermediaries. 
 This calculus is here entirely sufficient for tlie elimination of the 
 infinitesimals, without the question giving rise to any integration. 
 There are also questions, few, l)ut highly im])ortant, wdiich are the 
 converse of the last, requiring the emj)loyment of the Cases of the in- 
 integral calculus alone. In these, the difierential tegrai Calculus 
 equations are found to be inuuedialely I'cady for «''<'"''• 
 intrgiation, because they contain, at their first formation, only t.he 
 infinitesimals which relate to the functions sought, or to the really 
 independent variables, without th(; inti-oduction, diffei-entially, of any 
 intermediaries being required. If intermediaiy functions are intro- 
 duced, they will, by the hypothesis, enter directly, and not by their 
 <lifferentials ; and then, ordinary algebra will serve lor their elimi- 
 nation, and to bring ihe question to depend on the integral calculu.« 
 only. The dillVientiiii calculus is, in such cases, not essential to the
 
 6o POSITIVE PIIILOSOrilY. 
 
 solution of llie prol)lein, wliicli will (le{)en(l entirely on the integral 
 calculus. Thus, all questions to whicii the analysis is ap|)licable 
 are contained in three classes. The first class comprehends the 
 ]iroblenis which may be resolved by the differential calculus alone. 
 The second, those which may be resolved by the inte<;i'al calculus 
 alone. These are only exceptional ; the third constituting the nor- 
 mal case; tliat in which the difierential and integral calcuhis have 
 each a distinct and necessary part in the solution of problems. 
 
 The Differential Calculus. 
 
 The entire system of the differential calculus is simple and 
 perfect, while the integral calculus remains extremely imperfect. 
 We have nothing to do here with tlie applications of either cal- 
 The Difftren- culus, whicli are quite a different study from that 
 Hal Calculus. q{- |^]jg abstract principles of differentiation and inte- 
 gration. The consequence of the common practice of confounding 
 these principles with their a{)plication, es[)ecially in geometry, is 
 that it becomes difficult to conceive of either analysis or geometry. 
 It is in the de[)artment of Concrete Mathematics that the applica- 
 tions should be studied. 
 
 The first division of the differential calculus is grounded on the 
 ^ . condition whether the functions to be differentiated 
 
 Txoo portions. t •, ■ t -l .1 • • • . ii 
 
 are explicit or implicit ; the one giving rise to the 
 differentiation oi' formulas, and the other to the differentiation of 
 equations. This classification is rendered necessary by the imper- 
 fection of ordinary analysis ; for if we knew how to resolve all 
 equations algebraically, it would be possible to render every implicit 
 function ex[)licit ; 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 algebraic 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 fimction without knowing it, though it is 
 determinate. Thus we have two classes of questions, the differen- 
 tiation 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 view 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 thedilferentiation 
 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 piepcaideiant importance.
 
 SUBDIVISIONS OF DIFFERENTIAL CALCULUS. 6i 
 
 E;icli of iliese parts of Uie (li(Fereiiti;il calculus is again divided : 
 ami this subdivision exhibits two vei y distinct theories, <:.,,■•• 
 
 ,. , .TO- J- i r i* c k>u.bdiVisions. 
 
 aecorduig as we have to dillerentiate functions ot a 
 .single variable, or functions of several independent variables, — the 
 second bi'anch being of far gieater 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 pai-is 
 of the dilferential calculus. The case in which it is i-equired to dif- 
 ferentiate at once dilferent im|)licit functions combined in certain 
 ])iimitive equations must be distinguished from that in which all 
 these functions are separate. The same imperfection of ordinary 
 analysis which ])revents our converting every implicit function into 
 an equivalent ex[)licit 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. 
 
 We have now seen the different ]iarts of this calculus in their 
 natural connection a!id rational distribution. The Reduction to 
 whole calculus is linally found to rest upon the dif- the elements. 
 ferentiation of explicit functions with a single variable, — the only 
 one which is ever executed directly. Now, it is easy to understand 
 ihat this first theory, this necessary basis of the whole system, simply 
 consists of the diffeientiation of the elementary functions, ten in 
 number, which compose all our analytical combinations; for the 
 differentiation of compound functions is evidently deduced, inmie- 
 diately and necessarily, from that of their constituent simple func- 
 lit)ns. We find, then, the whole system of differentiation i-educed 
 to the knowledge of the ten fundamental differentials, and to that 
 of the two general principles, by one of which the differentiation of 
 implicit functions is deduced from that of explicit, and by the other, 
 the difTei'entiation of functions of several variables is reduced to that 
 of functions of a single variable. Such is the simplicity and per- 
 fection of the system of the differential calculus. 
 
 The transfurmaiion of derived Functions for neio variables is a 
 theory which must be iust mentioned, to avoid the „ , 
 omission 01 an mdispensable complement of the sys- tion of derived 
 lem of differentiation. It is as finished and perfect f u-nct ions for 
 
 , , . , . , , , . h new rartaOles. 
 
 as tlie other parts or tins calculus; and its great impor- 
 tance is in its increa.sing our resources by permitting us to choose, 
 to facilitate the formation of differential equations, that system 
 of independnet variables which may appear to be most advan- 
 tageous, though it may afterwards be relinquished, as an inter- 
 mediate step, by which, through this theory, we may pass to 
 the final system, which sometimes could not have been considered 
 directly.
 
 62 POSITIVE PHILOSOPHY. 
 
 Thougli we cannot here consider tlie concrete applicalion.s of tliis 
 Analytical calculus, we innst glance at tliose which are analy- 
 appiications. tical, bccause they are of the same nature with the 
 theory, and slioukl be looked at in connection wiih it. These ques- 
 tions are reducible to three essential ones. First, tiie development 
 inlo series of functions of one or more variables ; or, more generally, 
 the transformation of functions, wbich constitutes the most beauliful 
 and the most important application of tlie differential calculus to 
 general analysis, and which comprises, besides the fundamental series 
 discovered by Taylor, the remarkable series discovered by Maclanrin, 
 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 become. The third is the 
 least im})ortant of tlie three : — it is the determination of the true 
 value of functions which present themselves under an indeterminate 
 appearance, for certain hypotheses made on the values of the corre- 
 sponding variables. In every view, the first question is the most 
 eminent ; it is also the most susceptible of future extension, especi- 
 ally by conceiving, in a larger manner than hitherto, of the employ- 
 ment of the dilFerential calculus for the transformation of functions, 
 about which Lagrange left some valuable suggestions which have 
 been neither generalized nor followed up. 
 
 It is with regret that I confine myself to the generalities which 
 are the })roper subjects of this work ; so extensive and so interesting 
 are the developments which might otherwise be offered. Insufficient 
 and summary as ai'e the views of the Differential Calculus just 
 offered, we must be no less ra])id in our survey of the Integral 
 Calculus, properly so called ; that is, the abstract subject of inte- 
 gration. 
 
 The Integral Calculus. 
 
 The division of the Integral Calculus, like that of the Differential, 
 The Integral proceeds ou the principle of distingin'shing the inte- 
 Caicuius. gration of explicit differential fornndas from the in- 
 
 tegration of implicit differentials, or of differential equations. The 
 r/. ^,-. -o^v,,,. sei)aration of these two cases is even more radical in 
 
 jI to (Xvv ISvOltSt *■ c * •1*1 1 Tl 
 
 the case of uitegration than m the otlier. In tiie 
 differential calculus this distinction rests, as we liave seen, oidy on 
 the extreme impeil'ection of ordinary analysis. But, on the other 
 iiand, it is clear that even if all equations could be algebraically 
 resolved, differential equations would nevertheless constitute a case 
 of integration altogether distinct from that presented by explicit 
 differential formulas. Their integration is necessarily more com- 
 l»licated than that of explicit differentials, by the elaboration of 
 which the integial calculus was oiiginated, and on which the others
 
 THE INTEGRAL CALCULUS. d^ 
 
 Imve l)eon made to dopcnd, as far as possible. All ilic vaiioiis 
 a/tialytical processes hitherto l)rop()se(l for the integration of did'eren- 
 tial eqnalions, 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 fbi-mulas, tiie only object which 
 can be directly undertaken. Unhappily, im{)erfect as is this neces- 
 sary basis of tlie whole integral calculus, tiie art of" reducing to 
 it the integration of ditFerential equations is even nnich less 
 advanced. 
 
 As in the case of the differential calculus, and for analogous 
 reasons, each of tiiese two branches of the integral „,,... 
 
 ,,'....-, . ,. V, iimdivisioiis. 
 
 calcuUis IS divided agam, according as we consider 
 functions with a single variable or functions with several indepen- 
 dent variables. Tiiis distinction is, like the preced- One variable, 
 ing, even more important for integration than for or several. 
 dilferentiation. This is especially remarkable with respect to differ- 
 ential equations. In fact, those which relate to several independent 
 variables may evidently present this characteristic and higher difii- 
 culty — that the function sought may be differentially defined by a 
 simple relation between its various 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 commoidy called the Integral Calculus of partial 
 difterences, created by D'Alembert, in which, as Lagrange truly 
 perceived, geometers shoukl have recognized 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 cannot completely reduce 
 the integration to that of the ordinaiy differential equations. 
 
 A new distinction, highly important here, though not in the 
 differential calculus, where it is a mistake to insist Orders of 
 upon it, is drawn from the higher or lower order of differentiation. 
 the differentials. We may regard this distinction as a subdivision 
 in the integration of explicit or implicit differentials. With regai'd 
 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 ordei-s is more 
 important still. In the first case, we know so little of integration 
 of even the first order of differential formulas, that differential 
 formulas of a high order produce new difficulties in arriving at the 
 primitive function wdiich is our object. And in the second case, 
 there is the additional difficulty that the higher order of the differ- 
 ential equations necessarily gives rise to questions of a new kind. 
 The higlier the order of differential equations, the more implicit are 
 the cases which they present; and they can be made to dejjend on 
 each other only by special methods, the investigation of which, in
 
 64 POSITIVE PBIIOSOPHY. 
 
 consequence, forms a new class of questions, with regard to the 
 simplest cases of which we as yet know next to nothing. 
 
 The necessary hasis of all other integrations is, as we see from 
 the foregoing considerations, that of explicit differential formulas of 
 1-he first order and of a single variable ; and we cannot succeed in 
 effecting other integrations hut by reducing them to this elementary 
 _ case, which is the only one capable of being treated 
 "' '" '"'^^' directly. This simple fundamental integi'ation, often 
 conveniently called quadratures, corresponds in the differential 
 calculus to tlie 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 extensive than 
 the ditfeiential question. We have seen that the latter 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 functions, 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 algebraic functions 
 Ahjehraic <iud transcendental fimctions. The algebraic class is 
 functions. the more advanced of the two. In relation to ii'ra- 
 tional functions, it is true, we know scarcely anything, the integrals 
 of them having been obtained only in very restricted cases, and 
 jiarticularly by rendering them rational. The integration of 
 rational functions is thus far the only theoiy of this calculus which 
 has admitted of coniplete treatment ; and thus it forms, in a logical 
 point of view, its most satisfactory part, though it is jierhaps the 
 least important. Even here, the imperfection of ordinaiy 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 conceins 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 calcu- 
 Transcendentai lus. The case of tiie integration of ti'anscendental 
 functions. functions is quite in its infancy as yet, as regards 
 
 either exponential, logarithmic, oi- circular functions. Very few 
 cases of these kinds have been treated; and though the simplest 
 have been chosen, the necessaiy calculations are extremely labo- 
 rious. 
 
 The theory of Singidcir Sohitions (sometimes called Particular 
 
 Sinf/ninr Solutions), fully developed by Lagrange in his Cal- 
 
 Soiutiom. cuius of Fuuctiotis, but Tiot yet duly appreciated by 
 
 geometei'S, must be noticed here, on account of its logical perfection 
 
 and the extent of its applications. This theory foi-ms im[)licitly 
 
 a portion of the general theory of the integi-ation of differential
 
 DEFINITE INTEGRALS. 65 
 
 cqnatJ(ins ; but I luivo left it till now, becuuso it is, as it were, oiit- 
 siiie of the integral calculus, and I wished to preserve the sequence 
 of its parts. Clairaut first observed tiie existence of these solutions, 
 and he saw in them a paradox of the integral calculus, since they 
 have the property of satisfying the dilFerential equations without 
 being conipreliended in the corresponding general iutcgi-als. La- 
 giange explainetl this paradox by showing how sucli solutions are 
 always derived from the general integral by the variation of the 
 arbitrary constants. This theory has a character of perfect gener- 
 ality ; for Lagrange has given invariable and veiy sim[)le processes 
 for finding the singular solution of any differential equation wliicli 
 admits of it ; and, what is very remarkable, these processes require 
 no integration, consisting only of differentiations, and being there- 
 lore always applicable. Thus has differentiation become, by a 
 happy artifice, a means of compensating, in certain circumstances, 
 for the imperfection of the integral calculus. 
 
 One more theory remains to be noticed, to complete our review 
 of that collection of analytical researches which con- Definite 
 
 stitutes the integral calculus. It takes its place out- inteurais. 
 side of the system, because, instead of being destined for true 
 integration, it proposes to supply the defect of our ignorance of 
 really analytical integrals. I refer to the determination of defi- 
 nife integrals. These definite integi-als are the values of the 
 required finictions for certain determinate values of the correspond- 
 ing variables. The use of these in transcendental analysis corre- 
 sponds to the numerical resolution of equations in ordinary analysis. 
 Analysts being usually unable to obtain the real integral (called in 
 opposition the general or indefinite integral), that is, the fimctioa 
 which, differentiated, has produced the proposed differential for- 
 nnda, 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 resolving the arithmetical question without having first 
 resolved the corresponding algebraic one, which is generally the 
 most important ; and such an analysis is, by its nature, as imperfect 
 as tliat of the numerical resolution of equations. Inconveniences, 
 logical and practical, result from such a confusion of arithmetical 
 and algebraic considerations. But, under our inability to obtain 
 the true integrals, it is of the utnjost 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 entii-ely 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 s[)ecial transformations. This kind of transcendental 
 arithmetic being considered perfect, the difficulty in its applications 
 is reduced to making the proposed inquiry finally depend only on a 
 
 VOL. I. E
 
 66 POSITIVE PHIIOSOPIIY. 
 
 simple determination of definite integrals ; a tiling which evidently 
 cannot be always possible, whatever analytical skill may be em- 
 ployed in effecting so forced a transformation. 
 
 We have now seen that while the differential calculns constitutes 
 Frospects of ^3' ^^^ nature a limited and perfect system, the integi'al 
 the Integral calculus, or the simple subject of integration, offers 
 Calculus. inexhaustible scope for the activity of the human 
 
 mind, independently of the indefinite applications of which tran- 
 scendental analysis is evidently capable. The reasons which con- 
 vince us of the impossibility of ever achieving the general resolution 
 of algebraic 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 cannot 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 labours of geometers must certainly add in time to our 
 knowledge of integration, and create procedures of a wider generality. 
 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 cannot 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 
 ])resent calculus of indirect functions ; and I am persuaded that 
 when geometers have exhausted the most important applications of 
 our present transcendental analysis, they will turn their attention in 
 this direction, instead of straining after perfection where it cannot 
 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 solution 
 of a multitude of questions of the highest importance in geometry, 
 mechanics, thermology, etc. The philosophical explanation of this
 
 CALCULUS OF VARIATIONS. 67 
 
 double general fact Is found in llio preponderating- iniporlancc and 
 scope of abstract science, the smallest portion of which naturally 
 corresponds to a multitude of concrete researches, Man haviug no 
 other resource for the successive extension of his intellectual means 
 than in the contemplation of ideas more and more abstract, and 
 nevertheless positive. 
 
 Calculus of Variations. 
 
 By his Calculus or Ilethod of Variations, Lngrange 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 ditferential equations pi-operly so 
 called. It is still too near its origin, and its applications have been 
 too few, to admit of its being understood by a purely abstract 
 account of its tlieory ; and it is therefore necessary to indicate briefly 
 the special nature of the problems which have given rise to this 
 liyper-transcendental analysis. 
 
 These problems are those which were long known by the name of 
 Isoperimcfrical Problems ; a name which is truly Prohiems giving 
 applicable to only a very small number of them, nsetotids 
 They consist in the investigation of the maxima and Calculus. 
 minima of certain indeterminate integral formulas which express the 
 analytical law of such or such a geometrical or mechanical pheno- 
 menon, 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 })articular 
 values must be assigned to these variables, in order that the 
 corresponding value of the proposed function may be a maximum 
 or a minimum with respect to those values which immediately pre- 
 cede and follow it : — that is, we inquire, 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 thnt 
 the values of the different variables which suit either the maximum 
 or minimum must always render null the different derivatives of the 
 first order of the given function, taken sepai-ately with relation to 
 each independent variable ; and by indicating moreover a character 
 suitable for distinjruishino: the maximum from the minimum, which 
 consists, in the case of a function of a single variable, for example, 
 in the derived function of the second order taking a negative value 
 for the maxinuim and a positive for the minimum. Such are the 
 fundamental conditions belonging to the mnjority 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
 
 68 POSITIVE PHILOSOPHY. 
 
 cliief interest of geometers in this kind of questions, they rose 
 ahnost immediately to the consideration of a new order of problems, 
 at once more important and more difficnlt, — those of isoperimeters. 
 It was then no longer the valnes of the variables proper to the 
 maximum or the minimum of a given function that liad to be 
 determined. It was the foim of the function itself that had to be 
 discovered, according to the condition of the maximnm or minimum 
 of a certain definite integral, merely indicated, which depended on 
 that function. We cannot here follow the history of these problems, 
 tlie oldest of which is that of the solid of least resistance, treated by 
 Newton in the second book of the ' Principia,' in which he deter- 
 mines what nmst bo the meridian curve of a solid of revolution, in 
 order that the resistance experienced by that body in the direction 
 of its axis njay be the least possible. Mechanics first furnished 
 this new class of problems ; but it was from geometiy that the 
 subjects of the princi[)al investigations were afterwards derived. 
 They were varied and com{)licated almost infinitely by the labours 
 of the best geometers, when Lngrange reduced their solution to 
 an abstract and entirely general method, the discovery of which 
 has checked the eagerness of geometers about such an 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 maxinuun or a minimum, with 
 regard to all those which it would take if the required function had 
 any other form whatever. In treating these pi-oblems, the predeces- 
 sors of Lagrange proposed, in substance, to reduce them to the ordi- 
 nary theory of maxima and minima. But they proceeded by ap})ly- 
 ing special simple artifices to each case, not reducible to certain 
 rules ; so that every new question reproduced analogous difficulties, 
 without the solutions ])reviously 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 pro- 
 vided. In his endeavours to bring all isoperimetrical problems to 
 depend on a common analysis, Lagrange was led to the conception 
 of a new kind of differentiation ; and to these new Diffei'entials he 
 gave the name of Variations. They consist of the infinitely small 
 increments wliich the integrals receive, not in virtue of analogous 
 increments on the part of the corresponding variables, as in the 
 common transcendental analysis, but by sui)posing 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 conmiou 
 theory of maxima and minima.
 
 SOME OF ITS AI PLICATIONS. 69 
 
 Tinporlant as is this great and liappy transformation, and though 
 tho Method of Variations luul at first no other ohject other apph- 
 ihan the rational and general resolution of isoperi- cations. 
 metrical piohlems, we should form a veiy inadequate estimate of 
 tliis beautiful analysis if we supposed it restricted to this application. 
 In fact, the abstract conception of two distinct natuies of differen- 
 tiation is evidently applical)le, not only to the cases for which it was 
 created, but for all wiiich present, for any reason whatever, two 
 ditferent ways of making the same magnitudes vary. Lagrange 
 liimself made an immense and all-important application of his Cal- 
 (idiis of Variations, in his 'Analytical Mechanics,' by employing 
 it to distinguish the two sorts of changes, naturally presented by 
 questions of rational Mechanics for the ditferent points we have to 
 consider, according as we compare the successive positions occupied, 
 in virtue of its motion, by the same point of each body in two con- 
 secutive instants, or as we pass from one point of the body to another 
 in the same instant. One of these compai-isons produces the common 
 differentials ; the other occasions variations which are, there as else- 
 where, only difllerentials taken from a new point of view. It is in 
 such a general acceptation as this that we must conceive of the 
 Calculus of Variations, to appreciate fitly the importance of this 
 admirable logical instrument ; the most powerful as yet constructed 
 by the human mind. 
 
 This ]\Iethod being only an immense extension of the general 
 transcendental analysis, there is no need of proof that it admits of 
 being considered under the difi"erent primary points of view allowed 
 by the calculus of indirect functions, as a whole. Lagrange invented 
 the calculus of variations in accordance with the infinitesimal con- 
 ception, 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 appli- 
 cable it was to the calculus of variations, which he exhibited with all 
 suitable development, according to his theory of derived functions. 
 liut the more difficult in the use the method of variations is found to 
 be, on account of the higher degree of abstraction of the ideas con- 
 sidered, the more important it is to husband the powers of our minds 
 in its application, by adopting the most direct and rapid analytical 
 conce[)tion, which is, as we know, that of Leibnitz. Lagrange himself 
 tlierefore 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 Calculus, I noticed D'Alembert's 
 creation of the Calculus of partial differences., in Relation to 
 which Lagrange recognized a new calculus. This the ordinary 
 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 i-eceive in viitue of the
 
 70 POSITIVE PHILOSOPHY. 
 
 change of each variable se{)arately, — seems to me to establisli 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 general basis for the Method of Variations. This last has, 
 in fact, done nothing more tlian transfer to the independent variables 
 tliemselves the view already adoj)ted for the functions of those vari- 
 ables ; a process which has remarkably extended its use. A recogni- 
 tion of such a derivation as this for the method of variations may 
 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 yet 
 attained. We had before, in that analysis, a powerful insti'ument 
 ibr the mathematical study of natural })henomena, 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 not any general 
 and abstract rules for the formation of these differential equations ; 
 nor were such supposed to be ])ossible. Now% the Analysis of 
 Variations brings the actual estabHshment of the differential equa- 
 tions within the reach of the Calculus ; for such is the general 
 eflfect, in a great number of important and difficult questions, of the 
 varied equations, which, still moie indirect than the simple differ- 
 ential equations, as regards the special objects of the inquiry, are 
 more easy to form : and, by invariable and complete analytical : 
 methods, employed to eliminate the new order of auxiliary infini- 
 tesimals introduced, we may deduce those ordinary diiferential 
 equations which we might not have been able to establish 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 incomparably the 
 noblest and most unquestionable testimony to the scope of the human 
 intellect. If, at the same time, we bear in mind that the employ- 
 ment of this metiiod 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-])laces, 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 ])hilosophers but Lagrange. 
 
 We have now reviewed Mathematical analysis, in its bases and 
 in its divisions, very briefly, but from a philosophical })oint of view, 
 neglecting those conceptions only which are not organized with tht; 
 great whole, or which, if ui'ged to their limit, would be found to 
 merge in some whi/";!! have been examined. I must next offer a
 
 END OF ABSTRACT MATHEMATICS. 71 
 
 similar outline of Concrete Mathematics. My particular task will 
 be to show how, — sui)[)o.sing the general science of the Calculus 
 to he in a ])ei-lect state, — it has been ])ossihle to reduce, by invaii- 
 able })rocetlures, to pure questions of analysis, all the problems of 
 Gkometuy and JMicciiaxics; and thus to invest these two great 
 bases of natuial philosophy with that precision and unity which can 
 only thus be attained, and which constitute high perfection.
 
 72 
 
 CHAPTER III. 
 
 GENERAL VIEW OF GEOMETRY. 
 
 We have seen tliat Geometry is a true natural science ; — on!}' more 
 simple, and therefore more perfect than any other. We must not 
 suppose that, because it admits tlie application of 
 mathematical analysis, it is therefore a purely logical 
 science, independent of observation. Every body studied by geo- 
 meters presents some primitive phenomena wliich, not being dis- 
 coverable by reasoning, must be due to observation alone. 
 
 The scienlific 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 i-equired. Thus 
 Geometry is the more genei'al 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 geo- 
 metry. The same is true in the comparison of abstract thermology 
 with geometry. For these I'easons, geometry holds the first place 
 inider 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 measurevient of extension. Even this does not include 
 all the opei-ations of geometry, for there are many investigations 
 which do not appear to have for their object the measurement of 
 extension. But regarding the science in its leading questions as 
 a whole, we mny accurately say that the measurement of lines, of 
 .surfaces, and of volumes, is the invariable aim, — son)etinies direct, 
 though oftener indirect, — of geometrical labours. 
 
 The rational study of geometry could never have begun if we 
 must have regarded at once and together all the phy- 
 
 Idcci of tSx')ciC(' • • • • • 
 
 sical properties of bodies, together with their magni- 
 tude aTid 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 im{)ression left by a body 
 on a fluid in which it has been ])laced, we arc able to retain an 
 image of the impression, which becomes a ground of geometrical 
 reasoning. We thus obtain, — apart from all metaphysical fancies,
 
 IDEA OF SPACE. 
 
 73 
 
 — nn idea of Spaco. This abstrncfion, now so fnmilinr to ns that 
 we cannot conceive the state we slioukl be in without it, is perhaps 
 ihe earliest pbilosopliical creation of the lunnan mind. 
 
 There is another abstraction whicb must be made before we can 
 enter on geometrical science. We must conceive of Kinds of 
 ihree kinds of extension, and learn to conceive of extension. 
 lheni se]):ira1ely. We cannot 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 
 1o one. Even when all thi-ee are to be finally considered, it is often 
 necessary, in order to avoid complication, to take only one at a time. 
 This is tbe second abstraction wliicli it is indispensable for us to 
 practise, — to think of surface and line apart from volume; and 
 again, of line apnrt 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 surface 
 becoming narrower and narrower, till it is reduced to the finest 
 imaginable thread ; and then we have the idea of a line. Though 
 we cannot speak of a ])oint as a dimension, we must have the abstract 
 idea of that too ; and it is obtained by reducing the line from one 
 end or both, till the smallest conceivable ])oitiou of it is left. This 
 ])oint indicates, not extension, of course, but position, or the place 
 of extension. Surfaces have clearly the property of circumscribing 
 volumes ; lines, again, circumscribe surfaces ; and lines, once more, 
 aie linn'ted by points. 
 
 The Mathematical meaning of measurement is simply the finding 
 the value of tiie ratios between any homogeneous Geometrical 
 magnitudes : but geometrically, the measurement is measurement. 
 always indirect. The comparison of two lines is direct ; that of two 
 surfaces or two volumes can never be direct. One lino may be con- 
 ceived to be laid upon another: but one volume cannot be conceived 
 of as laid upon another, nor one surface upon another, with any 
 convenience or exactness. The question is, then, how to measure 
 surfaces and volumes. 
 
 Whatever be the foi-m of a body, there must always be lines, the 
 length of which will define tlie magnitude of the sur- Measurement 
 face or volume. It is the business of geometry to use of surfaces arid 
 these lines, directly measuiable as they are, for the ■''"'"'"'^s- 
 ascertainment of the ra//o of the surface to the unity of surface, or of 
 the volume to the unity of volume, as either may be souglit. In brief, 
 the object is to leduce all comparisons of surfaces or of volumes to 
 simple com}iarisons of lines. Extending the ])rocess, 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 
 tlifficult, diiect comparisons of surfaces and volumes are employed : 
 bnt the procedure is hot geometrical. In the same way, the con-
 
 74 POSITIVE FIIIIOSOFIIY. 
 
 sideration of vveiglit is sometimes brought in, to determine volume, 
 or even surface ; but this device is derived from mechanics, and has 
 iiotliing to do with rational geometry. 
 
 In speaking of the direct measurement of hues, it is clear that right 
 Of curved lines are meant. When we consider curved lines, it 
 lines. js evident that their measurement must be indirect, 
 
 since we cannot conceive of curved lines being laid upon each other 
 with ail}' precision or certainty. Tiie procedure is first to reduce 
 the measurement of curved to that of right lines ; and consequently 
 to reduce to simi)le questions of right lines all questions relating 1o 
 the magnitude of any curves whatever. In every curve, there always 
 exist certain right lines, the length of which must determine that 
 of the curve ; as the length of the radius of a circle gives us that 
 of the circumference ; and again, as the length of an ellipse depends 
 on that of its two axes. 
 
 Thus, the science of Geometry has for its object the final reduc- 
 tion of the comparisons of all kinds of extent to comparisons of right 
 lines, which alone are capable of direct comparison, and are, more- 
 over, eminently easy to manage. 
 
 I must just notice that there is a primary distinct branch of 
 Geometr}^ exclusively devoted to the right line, on account of 
 occasionable insurmountable difficulties in making the direct com- 
 parison ; its object is to determine certain right lines from others 
 by means of the relations pi'oper to the figures resulting from 
 their assemblage. The importance of this is clear, as no ques- 
 tion could be solved if the measurement of right lines, on which 
 every other depends, 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 sur- 
 faces ; and, finally, that of volumes. 
 
 The field of geometrical science is absolutely unbounded. There 
 JtsiUimU- may be as many questions as there are conceivable 
 ablefieid. figures ; and the variety of conceivable figures is in- 
 finite. As to curved Lines, if we regard them as generated by the 
 motion of a point governed by a certain law, we cannot limit their 
 number, as the variety of distinct conditions is nothing short of 
 infinite; each genei'ating new ones, and those again others. Sur- 
 faces, again, are conceived of as motions of lines; and they not only 
 ])artake of the variety of hues, but have another of their own, arising 
 from the possible change of nature in the line. There can be 
 nothing like this in lines, as points cannot describe a figure. Thus, 
 there is a double set of conditions under which the figures of sur- 
 faces 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 
 S[)ecial consideration under this head. If we add the one further
 
 DOMAIN OF GEOMETRICAL SCIEXCE. 75 
 
 remark, that surfaces themselves furnish a new means of conceivin<jj 
 of new curves, as every curve may be regarded as produced by the 
 intersection of two surfaces, we shall perceive that, startin<^ from a 
 narrow oiound of observation, we can obtain an absolutely infinile 
 variety of forms, and therefore an illimitable field for geometrical 
 science. 
 
 The connection between abstract and concrete geometry is estab- 
 lished by the study of the properties of lines and sur- properties of 
 faces. Without multiplying in this way our means of lines and 
 recognition, we should not know, except by accident, surfaces. 
 bow to f]nd in nature the figure we desire to verify. Astronomy was 
 lecreated by Kepler's discovery tliat the ellipse was the curve wliicb 
 ihe planets describe about the sun, and the satellites about their 
 planet. This discovery could never have been made if geometers 
 bad known no more of the ellipse than as the oblique section of a 
 circular cone by a plane. All the properties of the conic sections 
 brought out by the speculative labours of the Greek geometei's, were 
 needed as prejtaration for this discovery, that Ke])ler might select 
 from them the characteristic wbicli was the true key to the planetary 
 orbit. In the same way, the spherical figure of the eaith could not 
 have been discovered if the j)rimitive cliaracter of the sphere had 
 been the only one known ; — viz., the equidistance of all its points 
 from an interior point. Certain properties of sui-faces were the 
 means used for connecting tbe abstract reasoning with the concrete 
 fact. And others, again, were required to prove that the earth is 
 not absolutely s))herical, and how much otherwise. The j)ursuit of 
 these labours does not intei-fere with tbe definition of Geometiy 
 given above, as they tend indirectly to the measurement of exten- 
 sion. The great body of geometrical researches relates to the pro- 
 })erties of lines and surfaces ; and the study of the properties of the 
 same figui'C is so extensive, that the labours 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 
 fai- as ever from being fiiu'shed. And here opens another infinity. 
 AVe 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 figiue may be regarded. 
 
 There are two general Methods of treating geometrical questions. 
 These are commonly called Synthetical Geometiy Tiooyenerai 
 and Analytical Geometry. I shall prefer tbe histori- ^icthods. 
 cal titles of Geometiy 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 the ancients, 
 as we perceive by their applications of the theory of proportions.
 
 76 POSITIVE PHILOSOPHY. 
 
 The diflference between tliem and us is not so mucli in the instru- 
 ment of dednclion as in the nature of the questions 
 anlknt^and Considered. The ancients studied geometry witli 
 general or reference to the bodies under notice, or specially: tlie 
 met^-y? ^^°' luoderns study it willi reference to the phenomena to 
 he considered, or generally. The ancients extracted 
 all the)' 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 abstract, to treat by itself, 
 every question relating to the same geometrical phenomenon, in 
 whatever bodies it may be considered. Geometers can thus rise 
 to the study of new geometrical conceptions, which, applied to the 
 curves investigated by the ancients, have brought out new properties 
 never suspected by them. The supei-iority of the modern method 
 is obvious at a glance. The time formerly spent, and the sagacity 
 and effort employed, in the path of detail, are inconceivably econo- 
 mized by the genei'al method used since the great revolution under 
 Descartes, The benefit to Concrete Geometry is no less than to 
 the Abstract ; for the recognition of geometricnl 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 
 onl}' accidental, and within a limited or doubtful range : whereas, 
 by the geneial method, no existing figure can escape application to 
 lis 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 modern. The experience of the ancients, and 
 the materials they accumulated 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 Calcnlus 
 cannot oi'iginate 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 
 discoveiy of precise relations. 
 
 We must briefly survey the geometry of the ancients, in its 
 
 Geomctrij of character of an indispensable introduction to that of 
 
 the ancients, w^q moderns. The one, special and preliminary, 
 
 must have its relation made 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 foimded upon observa- 
 tion, though the materials furnished by observation are few and 
 simple, and the structure of reasoning erected upon them vast and 
 coujplex. The only elementaiy materials, obtainable by direct 
 sludy alone, ai'e those which relate to the right line for the geometry 
 of lines ; to the quadrature of rectilinear })lane areas ; and to the 
 cuhaturc of bodies terminated by plane faces. The beginning of
 
 ANCIENT AND MODERN GEOMETRY. 77 
 
 geoniotiy must be fioin ilic observation of lines, of ilat suifaces 
 anguUuly boinided, and of bodies which have more or less bulk, 
 also angulaily bounded. Tiiese are all ; for all other figures, even 
 the circle, and the figures belonging to it, now come under the 
 head of analytical geometry. Tlie three elements just mentioned 
 allow a sufliciency of equations for the calculus to proceed upon. 
 More are not needed ; and we cannot do with less. Some have 
 endeavoured to extend analysis so as to dispense with a portion 
 of these facts : but to do so is merely to return to metaphysical 
 ])ractices, in presenting actual facts as logical abstraclions. The 
 more we perceive Geometry to be, in our day, essentially analytical, 
 the more careful we umst be not to lose sight of the basis of obser- 
 vation on wliich all geometrical science is founded. "When we 
 observe people attempting to demonstrate axioms and the like, we 
 may avow that it is better to admit more than may be quite necessary 
 of materials derived from observation, than to cairy logical demon- 
 stration into a region where direct observation will serve us better. 
 
 There ai-e two ways of studying tlie riglit line — the graphic and 
 the algebraic. The thing to be done is to ascertain. Geometry of the, 
 by means of one another, the different elements of right line. 
 any right line whatever, so as to understand, indirectly, a light line, 
 under any circumstances whatevei-. The way to do this is, first, to 
 study the figure, by constructing it, or otherwise Graphical 
 directly investigating it ; and then, to reason from sohuwns. 
 tliat observation. Tlie ancients, in the early days of the science, 
 made great use of the graphic method, even in the form of Con- 
 struction ; 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 right-angled 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 
 ihe triangle. Archimedes himself, though he was the first to intro- 
 duce calculated determinations into geometry, frequently used the 
 same means. The introduction of ti-igonometry lessened the prac- 
 tice ; but did not abolish it. The Greeks and Arabians employed 
 it still for a great number of investigations for which we now con- 
 sider 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 wliose parts lie in 
 dift'erent planes. Hence arises a new series of considerations, and 
 different systems of Projections. Whei-e we now employ s[)herical 
 trigonometry, especially for problems relating to the celestial si)here, 
 the ancients had to consiiler how they could replace constructions in 
 relief by plane constructions. This was the object of their analemmas, 
 and of the other plane figures which long supplied the place of the 
 Calculus. They were acquainted with the elements of what we
 
 78 POSITIVE PHILOSOPHY. 
 
 call Descriptive GeoraetiT, tliougli they did not conceive of it in a 
 distinct and general manner. 
 
 Digressing here for a moment into the region of application, I 
 Descriptive may obscrve that Descriptive Geometry, formed into 
 Geometry. ^ distinct systcm by Monge, practically meets the 
 difficulty 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 essentially 
 to the contacts and intei'sections of surfaces ; so that all the geo- 
 metrical questions which may arise in any of the various arts 
 of construction, — as stone-cutting, carpentry, perspective, dialing, 
 fortification, etc., — can always be treated as simple individual cases 
 of a single theory, the solution being certainly obtainable throu,2;h 
 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 towards a philosophical 
 renovation of the labours of mankind ; towards that precision and 
 logical character which can alone ensure the future progression of 
 all arts. Such a revolution must inevitably begin with that class of 
 arts which bears a relation to the simplest, the uiost perfect, and 
 the most ancient of the sciences. It must extend, in time, though 
 less readily, to all other industrial operations. Monge, who under- 
 stood the philosophy of the arts better than any one else, himself 
 indeed endeavoured to sketch out a philosophical system of mechan- 
 ical 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 students' faculty 
 of Imagination, — of conceiving of complicated geometrical com- 
 binations in space ; and that, while it belongs to the geometry of 
 the ancients by the character of its solutions, it approaches 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 gvaphic solution, we have to 
 notice the other branch, — the alrjehraic. 
 
 Some may wonder that this branch is not treated as belonging to 
 
 Algebraic General Geometry. But, not only were the ancients, 
 
 solutions. in fjict, 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 deconiposed into triangles, all
 
 TRIGONOMETRY. 79 
 
 that we want is to be .able to detorraiiie the difTercnt elements of a 
 triangle by means of one another. This rednces polygonometry 
 to simple irig-onometry. 
 
 The difticnlty lies in forming three distinct equations between 
 the angles and the sides of a triangle. These Trigonomttvy. 
 equations being obtained, all trigonometrical pro- 
 blems are reduced to mere questions of analysis. — There are two 
 methods of introducing the angles into the calculation. They are 
 either introduced diredhj, by themselves or by the circular arcs 
 which are jiroportional to them : or they are introduced indirecWj, 
 by the chords of these arcs, which are hence called their trigono- 
 metrical lines. Tiie second of these methods was the first adopted, 
 because the early state of knowledge admitted of its working, while 
 it did not admit the establishment of equations between the sides 
 of the triangles and the angles themselves, but only between the 
 sides and the trigonometrical lines. — The method which employs 
 the trigonometrical lines is still 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 intixxluce these lines, which have at last to 
 be eliminated, we nuist 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 logar- 
 ithms, 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 remaikable fact that the determination of angles by 
 their trigonometrical lines, and the converse, admits of an ai-ith- 
 metical solution, without the previous resolution of the corresi)onding 
 algebraic question. But for this, the ancients could not have 
 obtained trigonometry. Wiien Arcliimedes was at work upon the 
 rectification of the circle, tables of chords were prepared : from his 
 labours resulted the determination of a certain series of chords: 
 and, when Hipparchus afterwards invented trigonon)etry, he had 
 only to corn[)lete that operation by suitable intercalations. The 
 connection of ideas is here easily recognized. 
 
 For the same reasons which lead us to the employment of these 
 lines, we must employ several at once, instead of confining ourselves
 
 8o FOSITIVE PHILOSOPHY. 
 
 to one, as the ancients did. The Arahians, and after them tlie 
 moderns, attained to only four or five direct trigonometrical lines 
 altogether ; whereas it is clear that the numher 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 
 i-elating to an arc which is a very simple fnnction of the first. 
 Thus, we may say that the numher of trigonometrical lines actually 
 employed by modern geometers is unlimited through the augmen- 
 tations we may obtain b}' analysis. Special names have, however, 
 been given to those indirect lines only which refer to the comple- 
 ment of the primitive arc, — others being in much less frequent 
 use. 
 
 Out of this device arises a third section of trigonometrical 
 knowledge. Having introduced a new set of lines, — of auxiliary 
 magnitudes — we have to determine their relation to the first. And 
 this study, though preparatoiy, 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 nuist 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 trigonometiy has been given 
 chiefly to show how the most simple questions of elementary 
 geometry exhibit a close dependence and regular ramification. 
 
 Thus have we seen what is the peculiar character of Special 
 Geometry, strictly considered. We see that it constitutes an 
 indispensable basis to General Geometry. Next, we have to study 
 the philosophical character of the true science of Geometry, begin- 
 ning 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 considerations into equivalent analytical con- 
 siderations. Descartes established the constant possibility of doing 
 this in a uniform manner: and his beautiful conception is inter- 
 esting, not only from its carrying on geometrical science to a 
 logical perfection, but from its showing us how to organize the
 
 CO-ORDINATES. 8i 
 
 relations of the abstract to the concrete in Mathematics by the 
 analytical representation of natural ])henoniena. 
 
 The first thing to be done is evidently to find and fix a method 
 for expressing analytically the subjects which afford Analytical re- 
 the phenomena. If we can regard lines and sur- presentation 
 faces analytically, we can so regard, henceforth, the of figures, 
 accidents of these suhjecfs. 
 
 Here occurs the (lifflculty 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 
 mognitude, the Jigure, and the position of the exten- 
 sious in question. The relation of the first, magni- 
 tude, to numbers is immediate and evident : and the other two are 
 easily brought into one ; for tlie figure of a body is nothing else than 
 the natural position of the points of which it is composed: and its 
 ])osition cannot 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 established the system of General Greometry. 
 
 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 cannot point out, we say how it is related to objects 
 that ai-e known, by assigning the magnitude of the different geo- 
 metrical 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 i)lane the point is situated, the co-ordinates are two. 
 If the point may be anywhere in space, the co-ordinates cannot be 
 less than three. They may be multiplied without limit : but whether 
 few or many, the ideas of position will have been reduced to those 
 of magnitude, so that we shall represent the displacement of a point 
 as produced by ])ure 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 t\vo fixed right lines, supposed to be known, and 
 generally concluded to be i)erpendicular to each other. These are 
 called axes. Next, there may be the less sim})le process of deter- 
 mining the position by tiie distances from two fixed points ; and so 
 on to greater and greater complications. But, from some system or 
 other of co-ordinates being always employed, the question of position 
 is always reduced to that of magnitude. 
 
 It is clear that our only way of marking the position of a point is 
 by the intersection of two lines. AVhen the point is Position of 
 determined by the intersection of two right lines, each apomt. 
 parallel to a fixed axis, that is the system of rectilinear co-ordinates, 
 — the most common of all. The i^olar system of co-ordinates ex- 
 
 VOL. I. f
 
 82 POSITIVE PHiiosonn^. 
 
 hibits the point by the travelling of a right line round a fixed centre 
 of a circle of variable radius. Again, two circles may intei'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 endeavouring to form the 
 ])rocess 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 ti'eated only geometry of two dimensions in his analy- 
 tical method : and we will at first consider only this 
 
 Plane Curves. ,.,,.. -.1-01 ri t • i. 1 
 
 kmd, begnmmg witli Jrlane (Jurves. Lmes must be 
 expressed by equations : and again, equations must be expressed by 
 Exnression l'"*^'''. wheu the relation of geometrical conceptions to 
 of lines by numbers is established. — It comes to the same thing 
 Equations. whether we define a line by any one of its properties, 
 or supply tlie corresponding equation between the two variable co- 
 ordinates of the point which describes the line. If a point describes 
 a certain line on a plane, we know that its co-ordinates bear a fixed 
 relation to each 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 case, 
 a single co-ordinate suffices to fix its position. The second co- 
 ordinate is then a determinate function of the first; — that is, there 
 exists between them a certain equation of a nature corresponding to 
 that of the line on which the point is to be found. The co-ordinates 
 of the point each i-equire it to be on a certain line: and again, its 
 being on a certain line is the same thing as assigning the value of 
 one of the two co-ordinates ; which is then found to be entirely 
 dependent on the other. Thus are lines analytically expressed by 
 equations. 
 
 By a converse argument may be seen the geometrical necessity 
 
 Expression ^^ representing _ by a Certain line every equation of 
 
 of equations two variables, in a determinate system of co-ordi- 
 
 hy lines. nates. In the absence of any other known property, 
 
 such a relation would be a very characteristic definition ; and its 
 
 scientific eflect would be to fix llie attention immediately upon the 
 
 general course of the solutions of tlie 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
 
 GEOMETRY OF TWO DIMENSIONS. 83 
 
 ensy our conception of now g;onoral an.'ilytlcal views. This nielliod 
 lias become entirely elementary in our day ; and it is employed 
 when we want to get a clear idea of the general character of the 
 law which runs through a series of particular observations of any 
 kind whatever. 
 
 Kecurring to the representation of lines by equations, which is our 
 chief object, we see that this representation is, by its charujeintkc 
 nature, so faithful, that the line could not undergo line chamjcs 
 any modification, even the slightest, without causing '''^ cgwa^ton. 
 a corres})onding 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 ex[)ressly intended to distinguish 
 them always. It must be observed that general inconveniences of 
 this nature a[)pear to be strictly inevitable in analytical geometry; 
 since, ideas of position being the only geometrical ideas immediately 
 ]-educible to numerical considerations, and conceptions of form not 
 being referrible to them but by seeing in them relations of situation, 
 it is impossible that analysis siiould not at first confound phenomena 
 of form with simple phenomena of position ; which are the only 
 ones that equati')ns express directly. 
 
 To complete our description of the basis of analytical geometry, 
 it is necessary to point out that not only must every Every defini- 
 defined line give rise to a certain equation between UonofaUne 
 the two co-ordinates of any one of its points, but ^s an equation. 
 every 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 i)roperty 
 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 genera- 
 tion, and in that case the definition is ex'planatory . For instance, if 
 we say that tlie circle is the line wliicli in the same form contains the 
 largest area, we offer a characteristic definition ; whereas if we choose 
 its property of having all its ])oints equally distant from a fixed 
 ]ioint, we have an explanatory definition. It is clear moreover that 
 the characteristic definition always leaves room for an explanatory 
 one, which further study nuist 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 ap[)ly. 
 AVe cannot 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
 
 84 POSITIVE PHILOSOPHY, 
 
 general conception of wliat a system of co-ordinates is, and if we 
 admit all possible systems, it is clear that sncli 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 genera- 
 tion considered ; as in the case of the circle, the common definition 
 of which may be regarded as being the polar equation of that curve, 
 taking the centre of the circle for the pole. 
 
 This view not only exhibits the necessaiy representation of every 
 line by an equation, but it indicates the general difficulty which 
 occurs 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 definition 
 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 charactei-istic properties, the difficulty must proceed 
 from our taking up a designated system of co-ordinates, instead of 
 admitting indiftei-ently all possible systems. These systems are not 
 all equally suitable ; and, in regard 1o curves, geometers think 
 that they should almost always be referred, as far as possible, to 
 rectilinear co-ordinates. Now, these particular co-ordinates are 
 often not those with reference to which the equation of the curve 
 will be found to be established by the proposed definition. It is in 
 a certain transformation of co-ordinates then that the chief difficulty 
 in the formation of the equation of a line really consists. The view 
 I have given does not furnish us with a complete and certain general 
 method for the establishment of these equations ; but it may cast a 
 useful light on the course which it is best to pursue to attain the 
 end proposed. 
 
 The choice of co-ordinates — the preference of that system which 
 
 Choice of may be most suitable to the case — is the remaining 
 
 co-ordinates, poj^t which we have to notice. 
 
 First, we must distinguish very carefully the two views, the 
 converse 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, founded on the picturing of equations by lines. Though 
 the two are necessarily combined in every investigation of general 
 geometry, and we have to pass from the one to the other alter- 
 nately, and almost insensibly, we must be able to separate them 
 here, for the answer to the question of method which we are con- 
 sidering is far from being the same under the two I'elations : so 
 that without this distinction we could not form any clear idea of it. 
 
 In the case of the representation of linos by equations, the first 
 object is to choose those co-ordinates which afford the greatest 
 simplicity in the equation of each line, and the greatest facility in 
 arriving at it. There can be no constant preference here of one
 
 CHOICE OF CO-ORDINATES. 85 
 
 pystom of co-ordinatos. The rectilinear system itself, though often 
 :i(lvaiitaLreons, cannot bo always so, and inny be, in turn, less so 
 than any other. But it is far ()thcr\vis(; in the conveise case of the 
 representation of equations by lines. Here the rectilinear system is 
 always to be ])i-eferred, as the most simple and trustworthy. If we 
 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 oiu* choice to the rectilinear system. In constructin<^ 
 •geometrical Joel, that system of co-ordinates must be the best in 
 which it is easiest to conceive the chani^e of phice of a point result- 
 in;^ from the chanij^e 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 
 represent equations in a more simple, and even in a more faithful 
 )nanner. 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 tigure 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 
 analytical geometry, the ordinaiy 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 
 theories in modern geometry are generally expressed by the recti- 
 linear system. The polar system is preferred next to it, both because 
 its opposite character enables it to solve in the simplest way the 
 equations which are too complicated for management under the 
 iirst; 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 labours of Descartes, his conce])tion of 
 analytical geometry being designed only for the study of Plane 
 Curves. It was Clairaut who, about a century later, extended it to 
 liie study of Surfaces and Curves of double curvature. The concep- 
 liou having been explained, a very brief notice will suffice for the rest. 
 
 With regard to Surfaces the determination of a point in space 
 requires that the values of three co-ordinates should Dctcrviination 
 be assigned. The system generally adopted, which ofapuintiu 
 corresponds with the rectilinear system of plane *^"^^* 
 geometiy, is tliat of distances from the [xant to three fixed planes,
 
 86 POSITIVE PHILOSOPHY. 
 
 usually perpendicular to each other, whereby the pohit is pi'esented 
 as the intersection of three planes whose direction is invarial)le. 
 Beyond this, there is tlie same infinite variety araonp^ possible 
 sj^stems 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 sur- 
 faces 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 suffice for determining its situation at any 
 moment; as the proposed surface will take the place of" the con- 
 dition imposed by the third co-ordinate. This last co-ordinate 
 then becomes a determinate function of the two others, they re- 
 Uiaining 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 corre- 
 spond to the precise degree of indetermination in the position of 
 the point. 
 
 In the expression of Surfaces by Equations, and again in the 
 expi-ession of Equations by Surfaces, the same con- 
 ^^TuT/aces'by ce[)tion is pursucd as in the analytical geometry of 
 Equations, and two dimeusious. Ill the first case, the equation will 
 fc^^wrface"^ ^® ^^^^ analytical definition of the proposed surface, 
 since it must bo 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 phenomena relating to surfaces may 
 be translated by certain equivalent 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 con- 
 verse case, every equation of three variables may, in general, 
 be represented geometrically by a determinate surface, defined 
 by the characteristic property that the co-oi'dinates 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 every one can 
 extend to sui'faces the other considerations which have been indicated 
 with regard to lines. I will only add that the superiority of the 
 rectilinear system 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.
 
 IMPERFECTIONS OF ANALYTICAL GEOMETRY. 87 
 
 In deterniiuini; Curves of double curvature, — which is the Uust 
 elenientnry poiut of view of analytical <^eonietry of Cm-ves of 
 three dinieusious, — the 8ame principle is employed. double 
 Accordiuj^ to it, it is clear that when a ])oint is curvature. 
 required to be situated npon some certain curve, a single co-ordinate 
 is enough to determine its position completely, by the intersection 
 of this curve with the sm-face resulting from this co-ordinate. The 
 two other co-ordinates of the point nnist thus be regarded as func- 
 tions necessarily determinate, and distinct from the first. Con- 
 sequently, every line, considered in space, is represented analytically 
 no longer by a single equation, but by a system 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 
 Avhich, considered separately, express a certain surface, must in 
 combination pi'esent the line sought as the intersection of two 
 determinate 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-ordin- 
 ates, and by which the line sought may be hidden under endless 
 algebraical disguises, it must be got rid of by giving up the facilities 
 resulting from such a variety of geometrical constructions. It is 
 sufficient in fact, to obtain from the analytical system established 
 for a certain line, the system corresponding to a single couple of 
 surfaces 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 
 recognizing the identity of lines in spite of the extensive diversity 
 of their equations. 
 
 Analytical Geometry still presents some im- imperfections 
 iierfections on the side both of geometry and of of Analytical 
 
 '■ -, ■ o J Geometry. 
 
 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 necesvsary, occasionally, to be able to express analytically 
 a part of a line or surface, or even a discontinuous line or surface, 
 composed of a series of sections b(>longing to distinct geometrical 
 figures. Some })rogress has been made in supplying means for 
 this purpose, to which our analytical geometry is inap))licable ; but 
 the method introduced by M. Fourier, in his labours on discon- 
 tinuous functions, is too complicated to be at present introducKl 
 into our established system. 
 
 In regard to analysis, we are so far from having a complete 
 command of analytical geometr}', that we cannot imperfections 
 fuinish anything like an adequate geometrical of Analysis. 
 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
 
 88 POSITIVE FIIIIO SOPHY. 
 
 impossible to find among geometrical phenomena a concrete repre- 
 sentation of all the laws expressed by analysis : but there is another 
 evil which is due to our own imperfect conceptions ; that, in our 
 I'epresentatious of equations of two or of three variables liy lines or 
 surfaces, we regard oidy 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 representation of the equation is always imperfect ; and 
 it fails altogetlier when the equation admits ot" only imaginary 
 solutions. This brings after it, in analytical geometry of two or 
 three dimensions, many inconveniences of less consequence, arising 
 from the want of correspondence between various analytical modi- 
 fications and any geometrical phenomena. 
 
 We have now seen what Analytical Geometry is. By this science 
 we determine what is the analytical expression of such or such a 
 geometrical phenomenon belonging to lines or surfaces: and, 
 leciprocally, 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 
 exemplify the manner in which geometers have actually established 
 this beautiful hai'mony : but such a review is not necessary to the 
 purpose of this Work, and would occupy too much space. We 
 have seen wiiat 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.
 
 89 
 
 CHAPTEE IV. 
 
 PiATlONAL MECHANICS. 
 
 IMEcnANiCAL plionomcna are by their nature more pnrticular, more 
 com{)licatecl, and more concrete than geometi'ical plienomena. 
 Therefore tlicy come after geomel ry in our survey ; and 
 therefore must they he pronounced to be more difficult 
 to studj', and, as yet, more imj)erfect. Geometrical questions are 
 always completely independent of Mechanics, while mechanical ques- 
 tions are closely involved witii geometrical considerations, — the form 
 (if bodies necessarily influencing tlie])hen()mena of motion and equili- 
 brium. 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 gi'avitation 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 
 j)art 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 ajipears iu an 
 nggravated form in ]\Iechanics, from the greater complexity of the 
 science. We encounter a perpetual confusion between the abstract 
 and the concrete points of view ; between the logical and the 
 j)hysical ; between the aitificial conceptions necessary to help us 
 lo general 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 ]\Iathematical analysis to Mechanics, 
 it lias set us back in some respects. The tendency to a 'priori sup- 
 })ositions, 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 mathematical 
 analysis is for canying the science on and upwards, 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 constiucted 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
 
 90 POSITIVE PHILOSOPHY. 
 
 of Meclianics is that the science is founded on some general facts, 
 furnished by observation, of which we can give no exphmation 
 whatever. Our business now is to point out exactly tiie philo- 
 sophical character of the science, distinguishing the abstract from 
 the concrete point of view, and separating the experimental depart- 
 ment from the logical. 
 
 We have nothing to do here with the causes or modes of produc- 
 tion of motion, but only with the motion itself. Thus, 
 Its characters. ^^ ^^^ ^^,^ ^^^ treating of Physics, but of Mechanics, 
 
 forces are only motions produced or tending to be produced ; and 
 two forces which move a body with the same velocity in the same 
 direction are regarded ns identical, whether they proceed from mus- 
 cular contractions in an animal, or fi'om a gravitation towards 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 philoso})by. 
 
 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 bo 
 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 precisely what 
 are the data and what the unknown parts of every mechanical 
 question. The science has nothing to do with the action of a single 
 force ; for this is, by the terms of the statement, supposed to be known. 
 It is concerned solely with the combination of forces, whether there 
 results from that combination a motion to be studied, or a state of 
 equilibrium, whose conditions have to be described. 
 
 Tiie two general questions, the one direct, the other inverse, 
 which constitute the science, are equivalent in importance, as regards 
 their application. Simple motions ai'e a matter of observation, and 
 their combined operation can be understood only through a theory : 
 and again, the compound result being a matter of observation, 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 forces 
 to which it is subject, — the direction and uniform velocity which 
 would be caused by the impulsion alone ; and again, the acceleia- 
 t ion of the vertical motion by its weight alone. The problem is to 
 discover thence the dilierent circumstances of the compound move- 
 ment produced by the combination of the two, — to determine the 
 path of the body, its velocity at each moment, 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 example of the 
 inverse problem is found in celestial mechanics, where we have to 
 determine tiie forces which cany the planets round the sun, and the 

 
 DOMAIN OF RATIONAL MECHANICS. 91 
 
 Kiilellites ronrul the ])lancls. We Iciiow iminediutely only the coiii- 
 ])onnd nioveiuent : Kepler's laws give us the characteristics of the 
 iiiovenient ; and then we have to go hack to the elementary forces 
 hy which tiie heavenly bodies are supposed to he impelled, to cor- 
 lespond with the observed result : and these forces 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. 
 
 In ancient times, men conceived of matter as being passive or 
 inert, — all activity being produced by some external Matter not in- 
 ageucy, — either of supernatural beings or some meta- ^fi in Physics. 
 l)hysical 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 (dl bodies whatever. The difference is merely of degree 
 between what men call hrutc matter and aniioated beings. More- 
 <n'er, science shows us that there are not different kinds of matter, 
 but that the eleujents 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 par- 
 ticles 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 spontaneous 
 activity, very various, and at most, in some cases, peculiar ; though 
 ])hysiologists are more and more disposed to regard the most pecu- 
 liar as a modification of antecedent kinds. However this may be, 
 it would be purely absurd now to regard any })ortion of matter 
 whatever as inert, as a matter of fact, or under the head of Physics. 
 But in Mechanics it must be so regarded, because we cannot estab- 
 lish any general proposition upon the abstract laws Supposed inert 
 of equilibrium or motion without putting out of the in Mechanics. 
 question all interference with them by other and inherent forces. 
 Wliat we have to beware of is mixing up this logical supposition 
 with the old notion of actual inertia. 
 
 As for how this is to be done, — we must remember v;hat has been 
 just said, — that in Mechanics, we have nothing to do Field of 
 with the origin or different nature of forces ; and Rational 
 they are all one while their mechanical oi)eration is Mechanics. 
 uniform. It is impossible to conceive of any substance as devoid of 
 weight, for instance; yet geometerfi 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
 
 92 POSITIVE PHILOSOPHY. 
 
 of properties of bodies. When we have so abstracted natural pro- 
 jierties, in our lo<2:;ical view, as to have before us an unmixed case of 
 the action of certain forces, and have ascertained their hiws, — 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 
 restoration is so difficult to effect, — the transition from the abstract 
 to the concrete in Mechanics is so difficult, — that, while its theo- 
 I'etical domain is unbounded, its practical application is singularly 
 limited. In fact, tlie a])plication of rational mechanics is limited 
 (accurately speaking) not only to celestial phenomena, but to those 
 of our own solar system. One would suppose that tlie single pro- 
 perty 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 cannot be estimated with any accuracy. And when we 
 l)roceed to electrical and chemical, and especially to physiological 
 l)henomena, we are yet more baffled. General gravitation affords 
 lis the only simple and determinate law ; and even there we are 
 perplexed, when we come to regard certain secondary actions. It 
 may be doubted whetlier questions of terrestrial mechanics will ever 
 admit — restricted as our means are — of a study at once purely 
 rational and pi-ecisely accordant with the general laws of abstract 
 mechanics, — though the knowledge of these laws, primarily indis- 
 ])ensable, may often lead us to frequent and valuable indications 
 and suggestions. 
 
 Bodies being supposed inert, the general facts, or laws of motion 
 Three laws to whicli they are subject, are three; all results of 
 of motion. observation. 
 
 The first is that law discovered by Ke[)ler, which is inaptly 
 called the laio of inertia. According to it, all motion 
 
 Law of inertia. • ,••,■ ' -, • r ^i j. • i i • n i 
 
 IS rectumear aiui iiniiorm ; tliat is, any body impelled 
 by a single force will move in a right line, and with an invariable 
 velocity. Instead of resorting to the old ways of ])ronouncing or 
 imagining ivhy it must be so, the Positive Philosophy instructs us 
 lo recognize the simple fact that it is so; that, through the whole 
 rnnge of nature, bodies move in a right line, and with a uniform 
 velocity, when impelled by a single force. 
 
 Tiie second law we owe to Newton. It is that of the constant 
 La of c ual- ^(luality of actlon and reaction ; that is, whenever one 
 ity of action body is moved by another, tlie reaction is such that 
 and reaction, the secoud loses precisely as much motion, in propor- 
 tion to its masses, as the first gains. Whether the movement pro- 
 ceeds from impulsion or attraction, is, of course, of no consequence. 
 Newton treated this general fact as a nnitter of observation, and 
 most geometers have done the same ; so that there has been less
 
 LA U'S OF MOTION. 93 
 
 fruitless search into the ivlnj witli reg-ard to this second Uxw than to 
 the first. 
 
 Tlie third fundamental Law of motion involves the principle of 
 the independence, or coexistence of motions, wliich r v 
 
 1 1 • T 1 1 • 1 1111 -Law of co- 
 
 leads immediately to wliat is commonly called the existence 
 composition o/forces. Galileo is, strictly speakiiij]:;, of motions. 
 the true discoverer of tliis law, though he did not regard it precisely 
 under the form in which it is presented here : — that any motion 
 common to all the hodies of any system whatever does not affect the 
 particular motions of these hodies with 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 
 general motion, whatever may he its velocity and direction, will not 
 affect relative motions. No a priori considerations can enter here. 
 There is no seeing why the fact should be so, and therefore no an- 
 ticipating that it would be so. On the conti'aiy, when Galileo stated 
 this law, he was assailed by a host of objections that his fact was 
 logically impossible. Philosophers were ready with plenty of d 
 priori reasons that it could not be true: and the fact was not unani- 
 mously admitted till men had quitted the logical for the physical 
 ])oint of view. We now find, however, that no proposition iu 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 he overthrown, from enil 
 to end, but for this law. A ship impelled smoothly, without rolling 
 and 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 motion 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 ohstacle to the establishment of the Copernican theory. The 
 Copernicans struggled to get rid of the insurmountahle ohjections to 
 which their doctrine was liahle by vain metaphysical subtleties, till 
 Galileo cleared up the difficulty. Since his time, the movement of 
 the globe has been considered an all-sufficient confirmation 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, but 
 must vary with the diversities of their direction, and of the angle 
 that each direction Avould 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 comparison 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 instance.
 
 94 POSITIVE PHILOSOPHY. 
 
 all the parts are not at an equal distance from the centre of rotation. 
 When the interior of a sliip is affected, it is by the rollinf^ and 
 pitching, whicli 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. 
 
 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 scientifiG term is, the 
 effect of a second will be to describe the diagonal of the paral- 
 lelogram. 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 immediately to 
 the geometrical construction expressed by the rule of the parallelo- 
 gram of forces. And thus it appears that this fundamental theoi-em 
 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 a 
 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 con:position 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 pro{)ortion of forces to velocities. The rela- 
 tions of forces may be determined either by a statical or dynamical 
 l)rocedure. No purpose is answered by the transposition of the 
 general fict 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 assign- 
 ing any a priori reasons why more laws may not be hereafter dis-
 
 DIVISIONS OF RATIONAL MECHANICS. 95 
 
 oovorod, if these tliree should ])fove to he incomplete. There cannot, 
 in the nature of thino;s, be many more ; and 1 would ralhor incur 
 the inconvenience of the introduction of one or two, than run any risk 
 of surrenderini^ the j)ositive chaiacter of the science and overstrain- 
 ing its logical considerations. We caimot 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 analytical equations easily 
 obtained. As for the most extensive, important, and difficult part 
 of the science, the mechanics of varied motion or continuous forces, 
 we can perceive the possibility of reducing it to elementary 
 Mechanics by tlie ajjplication of the infinitesimal method. For 
 each intinitely small point of time, we must substitute a uniform 
 motion in the place of a varied one, whence will immediately result 
 the differential equations relative to these varied motions. AVe 
 may hereafter see what results have been obtained in regard to the 
 abstract laws of equilibrium and motion. Meantime, we see that 
 the whole science is founded on the combination of the three 
 ])hysical laws just established ; and here lies the distinct boundary 
 between the physical and the logical parts of the science. 
 
 As for its divisions, the first and most important is into Statics 
 and Dvnamics ; that is, into questions relating to ^ „ . 
 
 .,.,'. ' . 1 i- X ..• cu i- Two Primary 
 
 oquilibruim and questions relatmg to motion, fetatics divisions. 
 are the easiest to treat, because we abstract from them Statics and 
 the element of time, which must enter into Dynamical 
 questions, and complicate them. The whole of Statics corresponds 
 to the very small portion of Dynamics which relates to the theory 
 of imiform motions. This division corresponds well with the facts 
 of human education in this science. The fine researches of Archi- 
 medes 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 without the most rudimentary knowledge of Dynamics. 
 Galileo, in fact, created that department of the science. 
 
 The next division is that of solids from fluids-. This division is 
 generally placed first, but it is unquestionable that 
 
 111 e i I- ^ 1 • , , -iii Secondary 
 
 the laws of statics and dynamics must enter into the divisions. 
 study of solids and fluids, that of fluids requiring the Soiuisand 
 addition of one more consideration, — variability of 
 form. This liowever is a consideration which introduces the 
 necessity of treating 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 importanco 
 and difficulty of the researches under this division cannot be exag- 
 gerated. Their complication places even the easiest cases beyond
 
 95 POSITIVE PHILOSOPHY. 
 
 our reach, except Ly the aid of extremely pi-ecarioiis hypolheses. 
 We must admit tlie vast necessary difficulty of hydrostatics, and 
 yet more of hydrodynamics, in comparison with statics and dy- 
 namics, 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 
 result 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 
 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 department 
 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. AVe 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 " impeifect 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 
 Kational 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, nccording as 
 Converse Statics are 1-egarded directly, or as a particular case 
 
 methods of of Dynamics. By the first method we have to dis- 
 treatment. cover a principle of equilibrium so general as to be 
 applicable to the conditions of equilibrium of all systems of possible
 
 STATICS. 97 
 
 forces. By the second method, we reverse the process, — ascertaining 
 what motion wonld result from the simnUaneous action of any pro- 
 ])Osed differing forces, and tlien determining what reUitions of these 
 forces wouhl render motion null. 
 
 The first method was the only one possihle in the early days of 
 science; for, as I have said befoi-e, Galileo was the First method. 
 creator of the science of Dynamics. Archimedes, the statics by Ustif. 
 founder of Statics, estahlishod the condition of equilibrium of two 
 weights suspended at the end of a straight lever; that is, he showed 
 that the weights must he in an inverse ratio to their distances from 
 the fulcrum of the lever. He endeavoured to refer to this })iinci[)le 
 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 deficiencies, were 
 relinquished when the establishment of Dynamics Second mtiwd 
 ])ermilte(l the use of the second method, — of seeking statics throwjii 
 the conditions of equilibrium through the laws of the dynamics. 
 composition of forces. It is by this last method that Varignon dis- 
 covered the theory of the equilibrium of a system of forces applied 
 upon a single point; and that D'Alembeit afterwards 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 the first 
 glance, it does not ai)pear the most rational, — Dynamics being more 
 complicated than {Statics, and precedence being natural to the 
 simi)ler. It would, in fact, be more philosophical to refer dynamics 
 to statics, as has since been done ; but we may observe that it is 
 only the most elementary part of dynamics, the theory of uniform 
 motions,, that we are concei-ned with in treating statics as a pai'ti- 
 cular case of dynamics. The complicated considerations of varied 
 jnotions do not enter into the process at all. 
 
 The easiest method of aj)plying the theory of uniform motions to 
 statical questions is through the view that, when forces are in equili- 
 hrio, each of them, taken singly, may be regarded as destroying the 
 eifect of all the others together. Thus, the thing to be done is to show 
 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 ha.ving compounded the two first forces, we can deduce 
 the composition of any number of forces. 
 
 After having established the elementary laws of the composition 
 of forces, geometers, before a})[)lying them to the investigation of the 
 conditions of equilibrium, usually subject them to an important 
 transformation, wliich, without beirsg indis})ensab]e, is of enn"nent 
 utility, in an analytical view, from the extreme simplification which 
 
 VOL. I, G
 
 98 POSITIVE PHILOSOPHY. 
 
 it inti-oduces into tlie 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 elementai-y 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 
 ])ractical advantage gained through the treatment of statics as a 
 particular case of elementary Dynamics, it would be satisfactory to 
 
 Want of unity relurn, if we could, to the method of the ancients, — 
 
 in the method, to leave Dynamics on one side, and proceed directly 
 to the investigation of the laws of equilibrium regarded by itself, by 
 means of a direct general princi{)le of equilibrium. Geometers strove 
 after this as soon as the general equations of equilibrium were dis- 
 covered by the dynamic method. But a higher motive than even 
 the desire to place statics in a more philosophical position 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'Alerabert 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 funda- 
 mental 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 nuist 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 necessaiy 
 thus to pass backwards and forwards 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 
 admirable treatise on ' Analytical Mechanics,' the leading conception 
 of which must be the basis of all future labours 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 Lagrange 
 
 Virtual sclectcd from among the properties of equilibrium, 
 
 Velocities. — j^r,,-] \)Qqxi 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 afterwards expressly pointed out
 
 VIRTUAL VELOCITIES. 
 
 99 
 
 IliG universal use tliat inl<;lit 1)0 made of it in Statics. The com- 
 bination of it with D'AU'nibert's principle led Lagrange to conceive 
 of the whole of Rational Mechanics as deduced fi-oni a single funda- 
 mental theorem, and to give it that rigorous unity which is the 
 highest philosopiiical perfection of a science. 
 
 The clearest idea of the system of virtual velocities may Ite 
 obtained 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 ibrces 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 
 vek)cities which would take place if the equilibrium did not exist. In 
 this primitive state, the principle, easily verified with regard to all 
 Icnown machines, oftei's great ])ractical utility ; for it permits us to 
 obtain with ease the mathematical condition of equilibrium of any 
 machine whatever, whether its constitution is known or not. If we 
 give the name of virtual momentum (or simply of momentum in its 
 })rimitive sense) to the product of each force by its virtual velocity, 
 — a product which in fact then measures the effort of the foi'ce to 
 move the machine, — we may greatly simplify the statement of the 
 jirinciple in merely saying that, in this case, the momentum of the two 
 forces must be equal and of opposite signs, that there may be equili- 
 brium, and that the positive or negative sign of each momentum is 
 determined according to that of the virtual velocity, which will be 
 considered ])0sitive or negative according as, by the supposed motion, 
 the projection of the point of application would be found to fall 
 upon the direction of the force or upon its prolongation. This 
 abridged expression of the principle of virtual velocities is especially 
 useful for the staten]ent of this principle in a general manner, with 
 regard to any system of forces whatever. It is simply this : that 
 the algebraic sum of the virtual moments of all forces, estimated 
 according to the preceding rule, must be null to cause equilibrium: 
 and this condition must exist distinctly with regard to all the 
 elementary motions which the system might assume in virtue of the 
 forces by which it is animated. In the equation, containing this 
 ])riiiciple, 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 simimary, useful by its sim- 
 plicity and uniformity. But, if it was to be a fundaniental princi[)le, 
 a basis of the whole science, it must be underived, or at least cap- 
 able of being presented in its preliminary propositions as a mattci- 
 of observation. Tliis was done by Lagrange, by his ingenious 
 demonstration through a system of j)ulleys. He exhibited the
 
 loo POSITIVE PHILOSOPHY. 
 
 theorem of viilual velocities very easily by iuiaj^Ining a single 
 weight which, by means of pulleys suitably constructed, replaces 
 simultaneously all the forces of the system. Many other demonstra- 
 tions have been furnished ; but, while more complicated, they are 
 not logically superior. 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 
 Mechanics. A perfect unity having been established by this prin- 
 ciple, 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 admii-able for this purpose than Lagrange's 
 adaptation of the principle of virlual velocities to the uniform 
 ajiplication of mathematical analysis. 
 
 Striking as is the philosoj^hical eminence of this princi[)le, there 
 are difficulties enough in its use to prevent its being considered 
 elementary, so far as to preclude the consideration of any othei' in 
 a course of dogmatic teaching. It is for this reason that I have 
 referred to the dynamic metliod, 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 
 Descartes : and this general progress would be almost accomplished 
 if the fundamental ideas of transcendental analysis were as widely 
 s})read as they ought to be. 
 
 The greatest acquisition, since the regeneration of the science by 
 Theory of Lagrange, is the conception of M. Poinsot, — the 
 Couples. theory of Couples, which appeal's 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 paradox in Statics. He seized upon this idea, and made it the 
 subject of an extended and original theory relating to the trans- 
 form;ition, coui})Osition, and use of these singular groups, which 
 he has shown to be endowed with pi'operties remarkable for their 
 generality and sim[)licity. lie 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 conception will do more for dynamics than for statics; and it 
 has hardly yet entered u))on its chief office. Its value will bo 
 appreciated when it is found to render the notion of the movements 
 of rotation as natuial, as familiar, and almost as simple as that of 
 forward movement or translation.
 
 EQUATIONS OF EQUILIBRIUM. loi 
 
 One moi'e coiisiilci'ation sliould, I tliiiik, be adveited to before we 
 quit the subieet of statics as a whole. When we „, . 
 
 1 .1 J i- ^1 J- 1-1 ii Share of equa- 
 
 Kliuly tlie nature of the equations wliicli express tlie tionsinpro- 
 conditions of equilibrium of any system of forces, it ducing eqmli- 
 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 eacli 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 equations nuist produce 
 equilibiium by leaving an impossibility for the body to move in 
 any way whatever. Now the natural state of things is for move- 
 ment to consist of I'otation and translation. Either of these may 
 exist without the other ; but the cases are so extremely i-are 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 rotation of the sun upon its axis was established, 
 every geometer concluded that it had also a progressive motion, 
 carrying all its planets with it, before astronomers had produced 
 any evidence that such was actually the case. In the same way we 
 conclude that certain planets, travelling in their oibits, rotate round 
 their axes, though the fact has not yet been verified. Some equa- 
 tions must therefore tend to destroy all progressive motion, and 
 others all motion of rotation. How many equations of each kind 
 must there be ? 
 
 It is clear that, to keep a body motionless, it must be hindered 
 from moving according to three axes in different planes, — commonly 
 supposed to be perpendicular to each other. If a body cannot 
 move from north to south, nor the reverse ; nor from east to west, 
 nor tlie reverse ; nor up, nor down, it is clear that it cannot move 
 at all. Movement in any intermediate direction might be conceived 
 of as partial progression in one of these, and is therefore impossible. 
 On the other hand, we cannot reckon fewer than three independent 
 elementary motions ; for the body might move in the direction of 
 one of the axes, without having any translation in the direction of 
 either of the others. Thus we see that, in a general way, three 
 equations are necessary, and three are sufficient to establish the 
 absence of translation ; each being specially adapted to destroy one 
 of the three progressive motions of which the body is capable. 
 The same view presents itself with regard to the other motion, — of 
 rotation. The mechanical conception is more complicated ; but it 
 is true, as in the sim[)ler case, 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 sufiicient to stop all motion 
 whatever. 
 
 When, instead of supposing any system of forces whatever as the
 
 I02 POSITIVE FHILOSOPHY. 
 
 sul)ject of the question, we paiticnlarize any, we get rid of more or 
 fewer possible movements. Having excluded these, we may exclude 
 also their corresponding equations, retaining only those which relate 
 to the possible motions that remain. Thus, instead of having to 
 deal with six equations necessary to equilibrium, there may be only 
 three, or two, or even one, which it will be easy enough to obtain in each 
 case. These remarks may be extended to any restrictions npon 
 motion, whether resulting from the special constitution of the system 
 of forces, or from any other kind of control, affecting the body under 
 notice. 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 establish 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 equilibrium will have been obtained : but the different 
 general methods ai-e far from being equally suitable to the applica- 
 tion of this rule. The one which is best adapted to it is, undoubtedly, 
 the Statical one, properly so called, founded, as has been shown, on 
 the principle of virtual velocities. In fact, one of the characteristic 
 properties of this principle 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 immediately an equation of equilibrium 
 specially relating to this motion. 
 
 When we come to the inquiry how geometers apply the principles 
 ^ of abstract Mechanics to the properties of real bodies, 
 
 the. concrete. ' we must State that the ordy complete application yet 
 
 with the ah- accomplished is in the question of terrestrial gravity. 
 stract question. -»t -i • • i • i. i • i t i • n i 
 
 JNow, this IS a subject winch cannot, Jogicaliy, be 
 
 treated under the head of Mechanics, as it belongs to Piiysics. It 
 is sufficient to explain that the statical study of terrestrial gravity 
 becomes convertible 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 })ure 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 pro- 
 blem of general geometry : — Given, any system of points disposed 
 in a determinate way with regard to each other, to find a jjoint 
 whose distance to any plane shall be a mean between the distances 
 of all the given j)oints to the same plane. — The abstraction of all 
 consideration of gravity is an assistance in every way. The simple
 
 CONCRETE AND ABSTRACT QUESTION. 103 
 
 geometrical idea is precisely what we want in most of tlie piinoipal 
 theories of Rational Mechanics, and especially when we conteniplulc 
 the great dynamic properties of the centre of mean distances ; in 
 which stndy the idea of gravity becomes a mere encnmhrance and 
 perplexity. It is trne that, by proceeding tlms, we exclude tiic 
 question from the domain of Mechanics, to place it in that of Geo- 
 metry. 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 misap- 
 prehend the true nature of the question. — The integral calculus 
 oilers the means of surmounting those difficulties in determining 
 the centre of gravity which are imposed by the conditions of the 
 question. But, the integrations in this case being more complicated 
 than those to which they are analogous, — those of quadratures and 
 cubatures, — their precise solution is, owing to the extreme imper- 
 fection of the integral Calculus, much more rarely obtained. It 
 is a matter of high importance, 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 com))lete 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 com})uto 
 the mutual attraction of all known bodies, if the conditions of each 
 body were understood by us : but this is not the case. For 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 pi-actically concerned 
 with in Rational Statics: and we see, by this, how backward this 
 science is in regard to universal gravitation. As for the exterior 
 general circumstances, such as friction, resistance of media, and the 
 like, which are altogether excluded in the establishment of the 
 rational laws of Mechanics, we can only say that we are absolutely 
 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 hypotheses, 
 unfit for scientific use. 
 
 As for the theory of equilibrium in regard to fluid bodies, — the 
 application which it remains for us to notice, — those FquiUhrium 
 bodies must be regarded as either liquid or gaseous. ofjiuids. 
 
 Hydrostatics niay be treated in two ways. We may seek the 
 laws of the equilibrium of fluids, accordin<j: to statical „ , 
 
 • 1 , • -^ > ,1 , 1 r 1 T Hydrostaticn. 
 
 considerations ])roper to that class 01 bodies : or we 
 
 may look for them among the laws whicli lelate to solids, allowing 
 
 for the new characteristics resulting from fluidity.
 
 104 POSITIVE PHILOSOPHY. 
 
 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 gi'and question of the figure of the 
 earth, on the supposition that it was once fluid. Huyghens first 
 endeavoured 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 incontest- 
 able, 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 sufiicient for equilibi-ium. 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 isolated equilibrium of any infinitely small canal; and, tried 
 by this criterion, he showed that the combination required by 
 Bouguer might take place without equilibrium happening. Several 
 great geometers, proceeding on Claii-aut'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 Enler 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 constitution of fluids indicates this as a general law ; 
 and it furnishes the requisite equations with extreme facility. 
 
 It was inevitable that the mathematical theory of the equilibrium 
 of fluids should, in the fiist place, be founded, as we have seen that 
 it was, on statical principles peculiar to this kind of bodies : for, in 
 . . eai-ly dnys, the characteristic differences betw^een solids 
 
 and fluids must have appeared too great for any geo- 
 meter to think of applying to the one the general principles appro- 
 priated 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 endeavour to 
 attach them to the same general principles, and perceive the neces- 
 sary applif-'ibility of the fundamental rules of Statics to the equili- 
 brium of fluids, making allowance for the attendant variability of 
 form. But, before hydrostatics could be comprehended under 
 Statics, it was necessary that the abstract theory of equilibrium 
 sliould be made so general as to apply directly to fluids as well as 
 solids. This was accomplished when L;igrango supplied, as the 
 l)asis of the whole of Eational Mechanics, the single principle of 
 Vii'tual Velocities. One of its most valuable properties is its being 
 as directly applicable to fluids as to solids. From that time, Hydro-
 
 DYNAMICS. 105 
 
 statics, ccasinf; to Lc a natural bi'anch of science, has taken its {)lace 
 as a secondary division of Statics. Tins 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 funda- 
 mental equations of the equilibrium of fluids, we have to consider 
 that all that such an application requires, is to introduce anwng the 
 forces of the system under notice one new force, — the pressure exerted 
 upon each molecule, which will introduce one term more into the 
 jreneral equation. Proceeding thus, the three general equations of 
 the equilibrium of fluids, employed when hydrostatics was treated 
 as a separate branch, will be immediately I'eached. If the fluid bo 
 a liquid, we must have regard to the condition of incompressibility, — 
 of change of form without change of volume. If the fluid be gaseous, 
 we must substitute for the incompressibility that condition which 
 subjects the volume of the fluid to vary according to a determinate 
 function of the pressure ; for instance, in the inverse ratio of the 
 pressuie, according to the physical 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 cnn at present 
 be regarded only as an a})proximation, — 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 
 that high degree of speculative perfection which transforms its 
 questions into simple problems of Mathematical Analysis. We 
 nuist 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. 
 
 The object of Dynamics is the study of the varied motions \ixo- 
 ^ucedhy continxous forces. The Dynamics of varied 
 motions or continuous forces includes two depart- 
 ments, — the motion of a 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 determina- 
 tion of one particle serves for the whole; while in the more general case, 
 each particle of the body, or each body of the system, assuming a
 
 io6 POSITIVE PHILOSOPHY. 
 
 distinct motion, it is necessary to examine these different effects, 
 and the action u|)on them of the rehitions belonging to the system 
 under notice. The second tlieory being more comphcated tlian 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 determine 
 the circumstances of the compound curvilinear motion, resulting 
 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 problem admits 
 of a converse solution. 
 
 But here intervenes a preliminary theoiy, which must be noticed 
 Theoni of recti- before either of the two de])artments can be entered 
 linear motion, npon. Tliis theory is popularly called the theory of 
 rectilinear motion, produced by a single continuous force acting inde- 
 finitely 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 continu- 
 ous 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 necessity 
 of being able to pass from any one of them to all the rest. The 
 })reliminary theory of vaiied motion relates to these transformations, 
 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 correlated 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, — time, space, 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 for- 
 midas which express, as a function of time, velocity and foi-ce, when 
 the law of space is supposed to be known. The infinitesiinal method 
 leads to these formulas with the utmost ease, the motion being con- 
 sideied uniform during an infinitely small interval of time, and as 
 uniforndy accelerated during two consecutive intervals. Thence the 
 velocity, supposed to be constant at the instant, according to the 
 first consideration, will be naturally expressed by the differential of 
 the space, divided by that of the lime ; and, in the same way, the 
 continuous force, according to the second consideration, will evi- 
 dently be measured by the relation between the infinitely small
 
 THE OR V OF RE CTIEINEAR MO TION. i o 7 
 
 increment of the velocity, and the time employed in producing; this 
 increment. 
 
 Jjii^rani^o's conception of transcendental analysis excluding him 
 trom tliis use of the infinitesimal metliod for the establishment of 
 the two foregoing dynamic formulas, he was led to present this 
 theory under another point of view, more impoi'tant than seems to be 
 generally supposed. In his Theory of Analytical Functions, he has 
 shown that this dynamic consideration really consists in conceiving 
 any varied motion as compounded, each moment, of a certain 
 uniform motion and another motion uniformly varied, — likening it 
 to the veilical motion of a lieavy body under a first im{)ulsion. 
 Lagrange has not given its due advantage to this conception, by 
 developing it as he might have done! In fact, it supplies a com- 
 l)lete theory of the assimilation of motions, exactly like the theory 
 of the contacts of curves and surfaces, in the department of 
 geometry. Ijike 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 pei'fect measure of all varied motion than we 
 obtain by the ordinary theory, though reasons of convenience 
 compel us to abide by the method originally adopted. 
 
 The fiist case or department of rational dynamics, — that of the 
 motion of a jwint, or of a body which has all its points Motion of 
 or j)ortions affected by the same force, — relates to the * pomt. 
 study of the ciuvilinear motion ])roduced 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 
 compelled to move in a single curve, or on a given surface. The 
 fundamental theory of curvilinear motion may be established in 
 either case, in a difTerent way ; each being susceptible of direct 
 treatment, and of being connected with the olher. 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 
 the curve which it must traverse ; and this is enough to afford the 
 fundamental equations, though this curve may then be primitively 
 unknown. 
 
 Tiie other, more real and more difficult case, is that of the 
 motion of a system of bodies in any way connected, Motion of 
 whose proper motions are altered by the conditions aai/stem. 
 of their connection. There is a new elementary conception about 
 the measurement of forces which some geometers declare to be 
 logically deducible from antecedent considerations, and to which 
 tliey would assign the j)lace and title of a fburih law of motion. 
 For the sake of convenience we may make it into a fouith law of 
 motion ; but such is not its philosophical character. The idea is, 
 that forces which impress the same velocity on different masses are
 
 io8 POSITIVE PHlLOSOrHY. 
 
 to each other exactly as those masses ; or, in other words, that the 
 forces are proportional to the masses, as we have seen them, under 
 the third law of motion, to be proportional to the velocities. All 
 ])henomena, 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 un- 
 equal 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 quantUy of motion ; and it deter- 
 mines 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 
 characteristic 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 d 
 priori knowledge of what the alterations will be. In the case of 
 the pendulum, for instance, the particles nearest the point of 
 suspension, 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 fundamental 
 dynamic equations. Lagrange has given us, in his ' Analytical 
 Mechanics,' the general history of this series of labours : and very 
 interesting it is, as a study of the progressive march of the luunan 
 D'AiemhcH's intellect. This method of proceeding continued till 
 lirincipie. 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 principle 
 is simply this. In the case supposed, tlie natural motion clearly 
 divides itself into two, — the one wliich 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, necessaiily balance 
 each other, under the conditions of the connection which chai'ac- 
 terizes the proposed system. James Bcrnouilli saw this with regard 
 to the particular case of the })enduhini ; 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 resoiu'ce 
 lio further ; and what he did detracts nothing from the credit of
 
 UALEMBERTS PRINCIPLE. 109 
 
 D'Alembevt's conce})Hon, the excellence of wliich consi.sls in its 
 entire ii;enerality. 
 
 In D'AlemLert'.s Imnds tlie principle seemed to have a purely 
 logical character. But its germ mny be recognized in the second 
 law of motion, established by Newton, under the nnnie of llie 
 equality of reaction and action, "^i'liey are, in fact, the same, witli 
 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'Alemhert did not 
 impress ujion it. Henceforth thei'efore we i-ecognize in it the second 
 law of motion, extended to any number of bodies connected in any 
 manner. 
 
 We see how every dynamical question is thus convertible into 
 one of Statics, by foiming, in each case, equations of equilibrium 
 between the destroyed motions. But then comes the difficulty of 
 making out what the destroyed motions are. In endeavouring to 
 get rid of the embarrassing consideration of tiie 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 contraiy motion could be imposed 
 upon it, equilibrium would be the result. This method contem})lates 
 only the })rimitive and the actual motions which are the true elements 
 of the dynamic pi-oblem, — the given and the unknown ; and it is 
 under this method that D'Alembert's principle is habitually con- 
 ceived of. Questions of motion being thus reduced to questions of 
 equilibrium, the next step is to combine D'Alembert's principle 
 with that of virtual velocities. This is the combination proposed 
 by Lagrange and developed in his ' Analytical Mechanics,' 
 wliich has carried uj> the science of abstract Mechanics to the high- 
 est degree of logical perfection, — that is, to a rigorous unity. All 
 questions that it can comprehend are brought under a single prin- 
 ciple, through which the solution of any problem whatever offers 
 only analytical difficulties. 
 
 D'Alemhert immediately apj)lied his principle to the case of fluids 
 — liquid and gaseous, whicli evidently admit of its use as well as 
 solids, their peculiar conditions being considered. The i-esult was 
 our obtaining general equations of the motion of fluids, wholly 
 unknown before. The principle of virtual velocities I'endered this 
 perfectly easy, and again left nothing to be desired, in regard to 
 concrete considei'ations, and presented none but analytical diffi- 
 culties. We must admit however that our actual knowledge 
 obtained under this theory is extremely imperfect, owing to insur- 
 mountable difficulties in the integrations required. If it was so in 
 questions of pure Statics, much more nmst it be so in the more 
 complex dynamical questions. The problem of the flow of a gravi-
 
 no POSITIVE FHILOSOPHY. 
 
 tating liquid through a given oi-ifice, simple as it appears, has never 
 yet been resolved. To siinplify as far as they could, geonietei-s 
 have had recourse to Daniel Bernouiili's hypothesis of the parallelism 
 of sections, which admits of our considering motion in regard to 
 horizontal laminae instead of particle by [)article. But this method 
 of consideringeach horizontal lamina ota liquid as moving altogether, 
 and taking tlie place of the following, is evidently contrary to the 
 fact in almost all cases. Tiie lateral motions are wholly abstracted, 
 and their sensible existence imposes on ns 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 the 
 fundamental equations of the motions of fluids are irreversibly 
 established, it is clear that what remains to be accomplished is 
 in the direction of mathematical analysis alone. 
 
 Such is the Method of Eational Mechanics. As for the great 
 
 theoretical results of the science, — the principal 
 
 general properties of equilibrium and motion thus 
 
 far discovered, — they were at first taken for real principles, each 
 
 being destined to furnish the solution of a certain order of new 
 
 ])roblems in Mechanics. As the systematic character of the science 
 
 has come out, however, these supposed princi})les have shown 
 
 Statical themsclves to be mere theorems, — necessary results 
 
 theorems. q[ \\^q 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 
 ])0ssible point, in comparison with all the positions it might take 
 under any other situation of the system. — Maupertuis afterwards, 
 by his working out of his Law of repose, gave a large generalization 
 to this theorem of Torricelli's, which at once became a mere parti- 
 cular case under that law ; Torricelli's applying merely to cases of 
 terrestrial gravitation ; while that of Maupertuis extends through- 
 out the whole sphere of the great natural attractive forces. 
 
 The other general pioperty relating to equilibrium may be 
 
 stabiiitv and I'^g'^rded as a necessary complement of the former. 
 
 ivstabiiiti/ of It consists iu the fundamental distinction between 
 
 equilibrium. ^-[^q cases of stability and instability of equilibrium. 
 
 Th(n"e being no such thing in nature as abstract repose, the term is 
 
 a])[)lied here to that state of stable equilibrium which exists whei'e 
 
 the centre of gravity is placed as low as ])()ssible ; while unstable 
 
 equilibrium is that which is popularly called equilibrium ; and it 
 
 exists when the centre of gravity is placed as high as possible.
 
 DYNAMICAL THEOREMS. in 
 
 Manpertuis's ilicorom consisted in this, — tliat the silnation of 
 oqnilibriuni of any system is always tliat in wliicli tlic snni o'^ vires 
 viva (active forces) is a maxinnnn or a minininm ; and the one under 
 notice, developed by Lagrange, consists in this, — tliat in any system 
 equilibrium 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 lai-ge 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 dynamics, 
 deduced from the combination of D'Alembert's prin- Dynamical 
 ciple with the principle of virtual velocities. The theorems. 
 first theorem is that of the conservation of the motion of the centre 
 of gravity, discovered by Newton. Newton showed that the mutual 
 action of the bodies of any system, whether of attrac- ^ 
 
 ,. "li !!• iii Conservation 
 
 tion, mipulsion, or any other, — regard bemg had to ofthe^notion 
 the constant equality between action and reaction, of the centre 
 — cannot in any way affect the state of the centre of ^j^^^'^^v- 
 gravity ; so that if there were tio accelerating forces besides, and if 
 the exterior forces of the system were reduced to instantaneous 
 forces, the centre of gravity would remain immovable, or would 
 move uniformly in a right line. D'Alembert generalized this 
 property, and exhibited it in such a form that every case in which 
 the motion of the centre of gravity has to be considered may be 
 treated as that of a single molecule. It is seldom that we form an 
 idea of the entire theoretical generality of such great results as 
 those of rational Mechanics. We think of them as relating to 
 inorganic bodies, or as otherwise circumscribed; but we cannot too 
 carefully remember that they apply to all phenomena whatever; 
 and in virtue of this universality alone are the basis of all real 
 science. 
 
 The second general theorem of dynamics is the ]irlnciple of areas, 
 the first perception of which is attributable to Kepler. Principle 
 Li its simplest form it is this ; that if the accelerat- of areas. 
 ing force of any molecule tends constantly towards a fixed point, 
 the vector radius of the moving body describes equal areas in equal 
 times round the fixed point ; so that the area described at the end 
 of any time increases in proportion to the time : and the lecijirocal 
 fact is clear, — that the evidence of the areas and the times ])roves 
 the action upon the body of a force dii'ected towards the fixed point. 
 This discovery of Kepler's is the more remarkable for having been 
 made before dynamics had been really created by Galileo. Its 
 importance in astronomy we shall see hereafter. But though, in 
 its simplest form, it is one of the bases of celestial Mechanics, it 
 is, in fact, only the simplest particular case of the great general
 
 IT2 POSITIVE PHILOSOPHY. 
 
 tlieorcra of arefis, exhibited in the mi(klle of the last centuty hy 
 D'Arcy, Daniel Bernouilli, and Euler. Kepler's discovery related only 
 to the motion of a point, while the later one refers to the motion of 
 any system of bodies, acting on each other in any manner whatever ; 
 which constitutes a case, not only more comjjlex, but different, on 
 account of the mutual actions involved. It yields proof, however, 
 that though the area described by the vector radius of each mole- 
 cule may be altered by reci{>rocal actions, the sum of the areas 
 described will remain invariable in a given time, and will incretise 
 therefore in proportion to the time. As the theorem of the centre 
 of gravity determines all that relates to motions of translation, this 
 determines all that relates to motions of rotntion : and the two 
 together are sufficient for the complete study of the motion of any 
 sj'stera of bodies, in either direction. And here comes in the 
 facility afforded by M. Poinsot's conception — referred to under the 
 liead of Statics. By substituting for the areas or momentum of 
 the geometers, the couples engendered by the proposed forces, a 
 philosophical completeness is given to the theory, and a concrete 
 value, and proper dynamic direction, to what was before a simple 
 geometrical expression of a part of the fundamental equations of 
 motion. 
 
 Laplace elicited from the theory of areas that dynamic property 
 Theinvari- whicli he Called the invariable plane, the consider- 
 abie plane. ation of wliicli is highly important in celestial 
 mechanics. It is in the study of astronomy that the importance 
 fully appears of the determination of a plane, whose direction is 
 unaffected by the mutual action of different bodies in our own solar 
 system ; for we thus obtain a })oint of reference, a necessarily fixed 
 term of comparison, by which to eslimate the variations of the 
 heavenly bodies. We are far from having yet attained precision 
 in the determination of the situation of this plane ; but this does 
 not impair the character of the theorem in its relation to rational 
 Mechanics. Ag;iin, we are indebted to Poinsot, who, by simplify- 
 ing, has once more extended the process to which his method is 
 applied : and he has repaired an important omission made by Lap- 
 lace, in taking into the account the smaller areas described by 
 satellites, and their rotation, and that of the sun itself ; whereas 
 Laplace has attended only to the larger areas described by the 
 planets in their course round the sun. 
 
 Finally, there are Euler's theorems of the moment of inertia, 
 Moment of J^nd the principal axes, which are among the most 
 inertia. important general results of rational Mechanics. By 
 
 Principal axes. means of these we are able to arrive at a complete 
 analysis of the motion of rotation. — By means of all the theorems 
 just touched upon, we are put in the direct way to determine 
 the entire motion of any body, or system of bodies whatever. — 
 Besides them, geometers have discovered some which are less
 
 CONCL USION OF MA THEM A TICS. 1 1 3 
 
 p^ciieral, but, ihou^h by no means indispensable, yet very ini[)orLaiit 
 iVoni the simplification they introduce into special i-esearches. 
 Students will recognize their functions, when their mere names are 
 ])resented ; which is all that our space allows : — I refer to the 
 theorem of the conservation of active forces, — singularly important 
 in its applications to industrial Mechanics: the theorem, in){)roperIy 
 called the principle of the least action, as old as Ptolemy, who 
 observed that reflected light takes the shortest way from one ])oint 
 to another, — an observation which was the basis of Maupertuis's 
 discovery of this property : and lastly, a theorem not usually classed 
 with the foregoing, yet worthy of no less esteem, — the theorem of 
 i\\e coexistence of small oscillations, ol Daniel Bernouilli. This 
 discovery is as important in its physical as its logical bearings; and 
 it explains a multitude of iacls which, clearly known, could not be 
 referred to their principles. It consists in showing that the infinitely 
 small oscillations caused by the return of any system of forces to a 
 state of stable equilibrium coexist without interference, and can be 
 treated separately. 
 
 This reference to the principal general theorems hitherto dis- 
 covered in Rational Mechanics concludes our review of the second 
 branch of Concrete Mathematics. 
 
 As for our review of the whole science, I wish I could better have 
 communicated my own profound sense of the nature _ . 
 of this immense and admirable science, which, the 
 necessary basis of the whole of Positive Philosophy, constitutes the 
 most unquestionable proof of the com])ass of the human intellect. 
 But I hope that those who have not the misfortune to be wholly 
 ignorant of this fundamental science may, according to the process 
 of thought which I have indicated, attain some clear idea of its 
 philosophical chaiacter. 
 
 To preserve complete the philosophical arrangement of Mathe- 
 matics in its present state, I ought to consider here a third branch 
 of Concrete Mathematics ; — the application of analysis to thermo- 
 logical phenomena, according to the discoveries of Fom-ier. But, 
 to avoid too great a breach of customai-y arrangement, I have 
 reserved the subject, and shall place Thermology among the 
 dcjiartments of Physics. 
 
 Mathematical philosophy being now completely characterized, we 
 shall proceed to examine its application to the study of Natural 
 Phenomena, in iheir various orders, ranked according to their 
 degree of simplicity. By this character alone can lliey cast light 
 back again upon the science which explains them ; and under this 
 character alone can they be suitably estimated. According to the 
 natural order laid down at the beginning, we now proceed to that 
 class of phenomena with which Mathematics is most concerned, — 
 the phenomena of Astronomy. 
 
 VOL. I. H
 
 114 
 
 BOOK 11. 
 ASTRONOMY, 
 
 CHAPTER I. 
 
 GENERAL VIEW. 
 
 It is easy to describe clearly the character of astronomical science, 
 from its being thoroughly separated, in our time, from 
 all theological and metaphysical influence. Looking 
 at the simple facts of the case, it is evident that tiiough three of our 
 censes take cognizance of distant objects, only one of the three per- 
 seives the stars. The blind could know nothing of them ; and we 
 who see, after all our preparation, know nothing of stars hidden by 
 distance, except by induction. Of all objects, the planets are those 
 which appear to us under the least varied aspect. We see how we 
 may determine their forms, their distances, their bulk, and their 
 motions, but we can never know anything of their chemical or 
 mineralogical structure ; and, much less, that of organized beings 
 living on their surface. We may obtain positive knowledge of 
 their geometrical and mechanical phenomena; but all physical, 
 chemical, physiological, and social researches, for which our powers 
 fit us on our own earth, are out of the question in regard to the 
 planets. AVhatever knowledge is obtainable by means ot the sense 
 of Sight, we may hope to attain with regard to the stars, whether 
 we at present see the method or not ; and whatever knowledge 
 requires the aid of other senses, we must at once exclude from our 
 ex[)cctations, in spite of any appeai'ances to the contrary. As to 
 questions about which we are uncertain whether they finally depend 
 on Sight or not, — we must patiently wait, for an ascertainment of 
 their character, before we can settle whetber they are applicable to 
 the stai'S or not. The only case in which this rule will be pronounced 
 too severe is that of questions of temperatures. The mathematical 
 Ihermology created by Fourier may tempt us to hope that, as he 
 has estimated the temperature of the space in which we move, we
 
 ASTRONOMY, 115 
 
 niny in time ascertain the mean temperature of the heavenly 
 bodies: but I regard this onler of facts as for ever excluded from 
 our recognition. We can never learn their internal constitution, 
 nor, in regard to some of them, how heat is absorbed by tlieir atinos- 
 })liere. Newton's attempt to estimate tJie temperature of the comet 
 of 1G80 at its perihelion could accomplish nothing more, even with 
 the science of our day, than show what would be the temperature 
 of our globe in the circumstances of that comet. We T^ ^ . . 
 may therefore define Astronomy as the science by " **"' 
 
 which we discover the laws of the geometrical and mechanical 
 phenomena presented by the heavenly bodies. 
 
 It is desirable to add a limitation which is important, though not 
 of primary necessity. The part of the science which 
 
 ^ z ii iii-ioi 'i. Restriction. 
 
 we command from what we may call the feolar pomt 
 of view is distinct, and evidently capable of being made complete 
 and satisfactory ; while that which is regarded from the Universal 
 ])oint of view is in its infancy to us now, and must ever be illimit- 
 able to our successors of the remotest generations. Men will never 
 compass in their conceptions the whole of the stars. The difference 
 is very striking now to us who find a perfect knowledge of the solar 
 system at our command, wliile we have not oljtained the first and 
 most simple element in sidereal astronomy — the determination of 
 the stellar intervals. Whatever may be the ultimate progress of 
 our knowledge in certain portions of the larger field, it v/ill leave 
 us always at an immeasurable distance from understanding the 
 universe. 
 
 Throughout the whole range of science there exists a constant and 
 necessary harmony between our needs and our knowledge. We 
 shall find this to be true everywhere. The fact is, we need to know 
 oidy what, in some way or other, acts upon us ; and the influence 
 Avhich acts upon us becomes, in turn, our means of knowledge. This 
 is evidently and remarkably ti'ue in regard to Astronomy. It is of 
 the highest importance to us to know the laws of the solar system : 
 and we have attained great precision with regard to them ; but, if 
 the knov.'ledge of the starry universe is forbidden to us, it is clear 
 that it is of no real consequence to us, except as a gratification of 
 our curiosity. The interior mechanism of each solar system is 
 essentially independent of the mutual action of distant suns ; as it 
 may well be, considering the distance of these suns from each othei", 
 ill comparison with the distance of planets from their suns. Our 
 tables of astronomical events, constructed in advance, proceed on 
 the supposition of there being no other system than our own ; and 
 they agree with our direct observations, precisely and necessarily. 
 This is our proper field ; and we must remember that it is so. We 
 must keep carefully apart the idea of the solar system and that of 
 the universe, and be always assured that our only true interest is in 
 the former. Within this boiuidary alone is astronomy the supreme
 
 ii6 POSITIVE PHILOSOPHY. 
 
 and ])ositive science tliat we have determined it to be ; and, in 
 fact, the innumerable stars that are scattered tlirough space serve 
 us scientifically only as providing positions which may be called 
 fixed, with which we may compare the interior movements of our 
 system. 
 
 We shall find, as we proceed through the whole gradation of the 
 Means of ex- scicnce, that the more complex the science, the more 
 pioration. various are the means of exploration ; whereas, it 
 does not at all follow, as we shall see, that the completeness of the 
 knowledge obtained is in any })roportion to the abundance of our 
 means. Our knowledge of astronomy is more perfect than that of 
 any of the sciences which follow it ; yet in none are our means of 
 exploration so few. 
 
 The means of exploration are three : — direct observation ; obser- 
 vation by experiment ; and observation by comparison. In the 
 first case, we look at the phenomenon before our eyes ; in the 
 second, we see how it is modified by artificial circumstances to 
 which we have subjected it ; and in the third, we contemplate a 
 series of analogous cases, in which the })henomenon is more and 
 more simplified. It is only in the case of organized bodies, whose 
 phenomena are extremely difficult of access, that all the three 
 methods can be employed ; and it is evident that in astronomy we 
 can use only the first. Experiment is, of course, impossible ; and 
 comparison could take place only if we were familiar with abun- 
 dance of solar systems, which is equally out of the question. Even 
 simple observation is reduced to the use of one sense, — that of sight 
 alone. And again, even this sense is very little used. Keasoning 
 bears a greater })roportion to observation here, than in any science 
 that follows it ; and hence its high intellectual dignity. To meas- 
 ure angles and compute times are the only methods by which we 
 can discover the laws of the heavenly bodies ; and they are enough. 
 The few incoherent sensations concerned would be, of themselves, 
 very insignificant ; they could not teach us the figui-e of the earth, 
 nor the path of a planet. They are combined and rendered service- 
 able by long-drawn and complex reasonings ; so 
 that we might truly say that the phenomena, how- 
 ever real, are constructed by our understanding. The simplicity of 
 the phenomena to be studied, and the difficulty of getting at them, 
 constitute, by their combination, the eminently mathematical charac- 
 ter of the science of astronomy. On the one hand, the perpetual 
 necessity of deducing from a small number of direct measures, 
 whether angular or horary, quantities which are not themselves 
 immediately observable, renders the use of abstract mathematics 
 indispensable ; while, on the other hand, astronomical questions 
 being always, in themselves, pi-oblems of geometry, or else of 
 mechanics, must fall into the depart ment of concrete mathematics. 
 Again, the regularity of astronomical forms admits of geometrical
 
 ORIGIN AND POSITION AS A SCIENCE. 117 
 
 treatment ; and tlio simplicity of astronomical movements admits of 
 mechanical tiealment with a very hi<]^h degree of precision. There 
 is perhaps no analytical process, no geometrical or mechanical doc- 
 tiine which is not employed in astronomical researches, and many of 
 ihem have as yet had no other aim. Considering the simple natnre 
 of astronomical investigations, and the easy application to them 
 of mathematical means, it is e\"ident why astronomy is, hy common 
 consent, placed at the head of the natnral sciences. It deserves 
 this place, first, hy the perfection of its scientific character; and, 
 next, hy the preponderant importance of the laws which it discloses. 
 
 Passing over, for the present, its utility in the raeasnrement of 
 time, the exact description of the glohe, and the perfecting of navi- 
 gation, which are not circumstances that could determine its rank, 
 we may just ohserve that it affords an instance of the necessity of 
 the loftiest scientific speculations to the satisfaction of the most 
 oidinary wants. Hipparchus began to apply astronomical theory 
 to the finding the longitude at sea. A prodigious amount of 
 geometrical science has gone to improve our tables of longitude up 
 to their present point; and if we cannot now get within half-n- 
 dozen miles of a true estimate in the seas under the line, it is for 
 M'ant of more science still. 
 
 Those who say tliat science consists in an accumulation of 
 observed facts may here see how imperfect is their account of the 
 matter. Tiie Chaldeans and Egyptians collected facts from obser- 
 vation of the heavens; but there was no astronomical when it became 
 science till the early Greek philosophers referred the a science. 
 diurnal movement to geometrical laws. The aim of astronomical 
 researches was to establish what would be the state of the sky at 
 some future time ; and no accumulation of facts could eflect this, 
 till the facts were made the basis of reasonings. Till the rising of 
 the sun, or of some star, could be accurately predicted, as to time 
 and place, there was no astronomical science. Its whole progress 
 since has been l)y introducing more and more certainty and preci- 
 sion into its predictions, and in using smaller and smaller data from 
 direct observation for a moie and more distant prevision. No part 
 of natural philosophy manifests more strikingly the truth of the 
 axiom that all science has iwevision for its end: an axiom which 
 separates science from erudition, which relates the events of the 
 past, without any regard to the future. 
 
 However impossible may be the aim to reduce the phenomena of 
 the respective sciences to a single law, supreme in Reduction to 
 each, this should be the aim of philosophers, as it is a single laio. 
 only the impoifection of our knowledge which prevents its accom- 
 plishment. The perfection of a science is in exnct proportion to 
 its approach to this consummation ; and, accoiding to this test, 
 astronomy distances all other sciences. Supposing it to relate to 
 our solar system alone, the point is attained ; tor the single general
 
 1 1 8 POSITl VE PHIL OSOFIl Y, 
 
 l.iw of f;^ravItatioii comprehends tlie whole of its plicDomena. It in 
 1o this that wc must recur wlien we wisli to show what we mean hy 
 the explanation of a plienomenon, without any inquiry into its first 
 or final cause ; and it is here that we learn the true character and 
 conditions of scientific hypothesis,- — no other science having applied 
 this powerful insti'ument so extensively or so usefully. After 
 having exhihited these great general pi'operties of astronomical 
 philosophy, I shall apply them to perfect the philosophical character 
 of the other principal sciences. 
 
 Regarding astronomical science, apart from its method, and with 
 a view to the natural laws which it discloses, its pre-eminence is no 
 less incontestable. I have always admired, as a stroke of philo- 
 sophic genius, Newton's title of his treatise on Celestial Mechanics, 
 — ' The Mathematical Principles of Natural Philoso[)hy ; ' for it 
 would be impossible to indicate with a moi'e energetic conciseness 
 that the general laws of astronomical phenomena are the basis of 
 all our real knowledge. 
 
 We may see at a glance that astronomy is independent of all the 
 Relation to natural sciences, depending on Mathematics alone : 
 other sciences. f^„(j thougli, philosophically speaking, we put Mathe- 
 matics at the head of the whole series, we practically regard it less 
 as a natuial science of itself (from the paucity of phenomena which 
 it presents to observation) than as the repository of piinciples by 
 which the natural sciences aie interpreted and investigated. Philo- 
 sophically speaking, astronomy depends on Mathematics alone, 
 owing nothing to Physics, Chemistry, or Physiology, which were 
 either imdiscovered, or lost in theological and metaphysical con- 
 fusion, while astronomy was a true science in the hands of the 
 ancient geometers. But the phenomena of the other sciences are 
 dependent, naturally as well as systematically, on asti'onomical facts, 
 and can be perfectly studied only through astronomy. We cannot 
 thoroughly understand any terrestrial phenomenon without consider- 
 ing what our globe is, and what part it bears in the solar system, 
 as its situation and motions affect the conditions of everything upon 
 it ; and what would become of our physical, chemical, and physio- 
 logical ideas, without consideration of the law of gravitation ? In 
 the remotest case of all, that of Social phenomena, it is certain that 
 changes in the distance of the earth from the sun, and consequently 
 in the duration of the year, in the obliquity of the ecliptic, etc., 
 which in astronomy would merely modify some coefficients, would 
 largely affect or completely destroy our social development. It is 
 no exaggeration to say that Social physics would be an impossible 
 science, if geometers had not shown us that the perturbations of our 
 solar system can never be more than gradual and restricted oscilla- 
 tions round a mean condition which is invariable. If astronomical 
 conditions were liable to indefinite variations, the human existence 
 which depends upon them could never be reduced to laws. 
 
 I
 
 RELATION OF ASTRONOMY TO THE HUMAN MIND. 1 19 
 
 Not less iinportaiit is the influence of astronomical science onoui- 
 own intellii;-ence. It lias done much more than relieve us (Voiii 
 .superstitious terrors and ahsurd notions about comets and echpses, 
 — notions which, as La{)lace observed, would spring up again 
 immediately if our astronomy were forgotten. This science has 
 done much more for our underslandings than that. It has done 
 more than any other pursuit — simply because it is the most scien- 
 tific of all — to expose and destroy the doctrine of final causes, which 
 is generally regarded by the moderns as the basis of every religious 
 system, though it is in fact a consequence and not a cause. The 
 knowledge of the motion of tlu; earth has overthrown the very 
 foundation of the doctrine, which supposed the universe to be 
 subordinated to our globe, and therefore to Man. Since Newton's 
 time, the dev^elopment of Celestial Mechanics has deprived theolo- 
 gical philosophy of its principal intellectual office, by proving that 
 the order maintained throughout our system and the whole universe 
 is by the simple gravitation of its parts. If we look an a lyriori 
 view, we should say that, as we exist, our system must be such as 
 to admit of our existence ; and one necessaiy condition of this is 
 such a degree of stability in our system as we actually find. This 
 stability we scientifically perceive to be a simple consequence of 
 mechanical laws working among the incidents of our system, — the 
 extremely small planetary bodies in their relation to the larger sun; 
 the small eccentricity of their orbits, and moderate inclination of 
 their planes ; which incidents, again, are necessary consequences of 
 the mode of formation of the entire system. The stability by virtue 
 of which we hold our existence is not foimd in the case of comets, 
 whose perturbations are not only great, but liable to indefinite 
 increase ; and their being inhabited is inconceivable. Thus, tlu; 
 doctrine of final causes would be reduced to the truism that there 
 are no inhabited bodies in oiu" system but those wliich are habitable. 
 This brings us back to the principle of the conditions of existence, 
 which is the true positive ti-ansformation of the doctrine of final 
 causes, and of far superior scope and profit in every way. 
 
 We have next to consider the divisions of the science. These 
 arise immediately out of the fact, now familiar to us, Divisions of 
 that astronomical phenomena are either geometrical the science. 
 or mechanical. Tliey are Celestial Geomeiry, which is still called 
 Astronomy, from its having possessed a scientific character befbi-(! 
 the other ; and Celestial Mechanics, of which Newton was tlu; 
 immortal founder. Though our business is with our own system, 
 the same division extends to Sidereal astronomy, CciesHid 
 supposing that kind of exploration to be within our Geomctri/. 
 power. As before, we see geometry to be more simple in its })heno- 
 niena than mechanics, and that mechanics is dependent on geometry, 
 without reciprocity. In fact, men were successfully inquiring into 
 the forms and sizes of the heavenly bodies, and studying their
 
 120 POSITIVE PHILOSOPHY, 
 
 geometrical laws, befoi-e anythin2: was known of the forces whicli 
 Celestial changed their positions. Whereas, the province of 
 Mechanics. Celestial Mechanics is to analyse the motions of the 
 stars, in order to refer them, by the rules of Kational IMechanics, to 
 the elementary motions regulated by a universal and invariable 
 mathematical law ; — thence, again, departing to perfect the know- 
 ledge of real motions by scientifically determining them a priori, 
 taking from observation the necessary data — the fewest possible — • 
 for tiie calculations of general mechanics. This is the link by 
 which astronomy and physics are connected, and connected so 
 closely that some great phenomena render the ti-ansition almost 
 insensible ; as in the theory of the Tides. But it is evident that 
 the whole reality of celestial mechanics consists in its having issued 
 from the exact knowledge of true movements, furnished by celestial 
 geometiy. It was for want of this point of depai'ture that all 
 attempts before the time of Newton, even Descartes', however valu- 
 able in other ways, failed to establish systems of celestial mechanics. 
 This division of the science into two parts has therefore nothing 
 arbitrary in it, nor even scholastic : it is derived from the nature 
 of the science, and is at once historical and dogmatic. As for the 
 subdivisions, we need not trouble ourselves with them now. 
 
 In regard to the point of view from which the science shotild be 
 regarded, Lacaille thought it would sim[)lify matters extremely to 
 place his observer on the surface of the sun. And so it would, if 
 the thing could be done in accordance with positive knowledge ; 
 but undoubtedly the solar station should be the ultimate and not 
 the original one, under a rational system of astronomical study. 
 And when, as in the case of this work, the object is the analysis of 
 the scientific method, and the observation of the logical filiation of 
 the leading scientific ideas, it matters less to obtain a clearer expo- 
 sition of general results than to adliere to the positive method. 
 
 I suppose my readers to be well acquainted with the two funda- 
 mental facts of the diurnal and aimual rotation of our globe, as data 
 without which nothing could be clearly understood of the essential 
 methods and general results of astronomical science. I am not 
 giving a treatise on asti'onomy, nor even a summary ; but a series 
 of philosophical considerations upon the different parts of the 
 science, in which any extended special exposition would be mis- 
 placed. 
 
 We must first see what methods of observation astronomers need, 
 and are possessed of.
 
 CHAPTER 11. 
 
 METHODS OF STUDY OF ASTKONOMT. 
 SECTION I. 
 
 INSTRUMENTS. 
 
 All astronomical observation is, as we have seen, comprehended In 
 tlie measurement of limes and of an^^les. The two ^, 
 considerations concerned in attaining the gi-eat per- 
 fection we have reached are the perfecting the instruments, and tlie 
 application by tlieory of certain corrections, without wliich their 
 precision would be misleading. 
 
 The observation of shadows was the first resource of astronomers, 
 when the rectilinear propagation of light was estab- sfruimc; 
 lished. Solar shadows, and also lunar, were very 
 valuable in the beginning ; and much was obtained from the sim])le 
 device of a style, so fixed as to cast a shadow corresponding with 
 the diurnal rotation to be observed : but the alterations rendered 
 necessary by the annual motion, and impossible to make on that 
 apparatus, rendered the instrument unfit for precise observations. 
 Again, by comparing the length of the shadow cast by a vertical 
 style with the height of the style, the corresponding angular 
 distance of the sun from the zenith was computed : and a valuable 
 method this was : but the penumbra rendered the accurate measure- 
 ment of the shadow impossible. The difficulty, aggravated by its 
 unequal amount at different distances from the zenith, was partly 
 removed by the use of very large gnomons ; but not completely. 
 These imperfections determined astronomers to get rid as soon as 
 ])ossible of the process of gnomonic measurement. Shadows will 
 always be at hand to measure by when better means are wanting : 
 and one application of this instrument remains in our observatories, 
 — as the basis of the meridian line, regarded as dividing into two 
 equal parts the angle formed ])y the horizontal shadows of the same 
 length which correspond to the two equivalent parts of the same 
 day. In this case, the penumbra is harmless, as it affects the 
 two parts equally ; and as for the obliquity of the sun's motion, 
 that may be mainly got rid of by choosing the period of either 
 solstice,— especially the summer one. It is easy, too, to rectify the 
 observation by the stars.
 
 122 rOSITIVE PIULOSOFHY. 
 
 Proceeding to more exact methods, and, first, witli regard to 
 measurement of time, it is clear that tlie most perfect of all chrono- 
 meters is the sky. It seems as it it would be enough, after knowing 
 precisely the latitude of one's observatory, to measure the distance 
 of any star from the zenith, and learn its hoi-ary angle, and, as an 
 immediate consequence, the tiiue that has elapsed, by resolving the 
 spherical triangle formed by the pole, the zenith, and the star. If 
 a sufficiently wide observation of this kind liad been made, and 
 numerical tables formed for certain selected stars, great results 
 might have been obtained from this natural method ; but it is 
 insufficient ; and it has the defect of making the measure of time 
 depend on that of angles, which is the least perfect of the two, in 
 our day. This method is therefore used only in the absence of a 
 better, as in nautical astronomy ; and its commonest service is in 
 regulating other chronometers, by a comparison with that of the 
 heavens themselves. Artificial methods of measuring time are 
 therefoie indispensable in astronomy. 
 
 Every phenomenon which exhibits continuous change might 
 
 Artificial scrvc, in a rough way, to mark time : various chemi- 
 
 niethods. ^al processes, or even the beating of our own pulses, 
 
 might afford a measure, more or less inaccurate : astronomical 
 
 ])henomena are excluded, because they are what we want to 
 
 measure : and we therefore have recourse to physical means, and 
 
 find weight the best. The ancients tried it in the form of the flow 
 
 of liquids ; and to water clocks succeeded the hour-glass ; but the 
 
 uncertainty of these led to solids being prefei'red ; and in the form 
 
 of weight having a vertical descent. By no care, however, could the 
 
 disturbances caused by natural forces be remedied, till Galileo, by his 
 
 creation of rational dynamics, suggested the pendulum. Whether 
 
 ^, , , it is or is not correct to assign to Galileo the idea of 
 The pendulum. • ,i i i ,• . • -^ • 
 
 usmg the pendulum as a measurer oi tune, it is cer- 
 tain that his discoveries suggested it, and that Huyghens enabled 
 us to use it. He had recourse to the highest principles of science 
 to render this service, and discovered the princi))le of vires vivae, 
 which, besides being scientifically indispensable, afforded to art new 
 means of modifying oscillations without changing the dimensions of 
 the apparatus. Considered as a collection of discoveries for a single 
 aim, Huyghens' treatise I)e Horologio oscilhiiorio, is perhaps the 
 most remarkable examj)le of special researches that the histoiy of 
 the human mind has yet exhibited. From that time, the perfecting 
 of astionomical clocks became merely a matter of art. In regard to 
 fixed clocks, two things have to be attended to ; — the diminution of 
 fi'iction, by improved methods of sus{)ension, and the correction by 
 a compensating ap})aratus of irregularities caused by variations of 
 tem[)erature. As for ])ortal)le chronometers, woiked by a spiral 
 s])ririg, they are a marvellous invention ; but they belong to the 
 province of arl, and not science.
 
 MEASUREMEN2' OF ANGLES. \ 23 
 
 In ro<:!jard to llie inensureineiit of angles, il is clear that an inwtru- 
 Tiient which would admit of an allowance for minutes Measurement 
 and seconds, must be of a size incompatible with min- of angles. 
 ute ])rccision. It must always be that large apparatus must be so 
 adecled in its weight and temperature as to be impaired in its 
 accm-acy. The lai-ge telescopes of modern times are intended to 
 !-how us stars otherwise invisible : and no one thinks of using them 
 for ])iu'poses of precise measurement. It is generally agreed now 
 that instruments for measuring angles should not be more than ten 
 feet in diameter when we are dealing with an entire circle ; and 
 they are usually not more than six or seven. The wonder then is 
 how we are to estimate angles to a seco.nd, as we do every day, with 
 ciicles whose size would scarcely indicate minutes. It is done by 
 the concui-rent use of three methods, — the eye piece, the use of the 
 vernier (so called after its inventor), and the repetition of angles. 
 
 It was long before it occurred to astronomers to use their lenses 
 for any other purpose than the discovery of new objects : but at last 
 it occurred to them to replace the ancient transoms and modern 
 sights by an eye-piece which should secure the advantages without 
 the inaccuracies of a large instrument. Morin first made this use 
 of a lens. Auzont followed with his invention of the reticle ; and, 
 a century after, Dollond gave us a power of absolute precision by 
 his invention of the achromatic object-glass. Vernier proposed in 
 1631 to divide intervals into parts much more minute than could be 
 marked. He enabled us to ascertain angles, within half a minute, 
 or circles divided only into sixth-parts of a degree. The precision 
 obtainable by his simple apparatus is indefinite, being limited only 
 l)y our difficully in detecting the coincidence of the line of the 
 vernier with that of the limb. The union of the third method with 
 these two gives us the perfection we have attained. It is strange 
 that we should have been so long in perceiving that, the imperfec- 
 tion of angular instruments having nothing to do with the dimen- 
 sions of the angle to be measured, we should gain much by increasing, 
 in fixed proportions, the magnitude of the angles, which is equivalent 
 to diminishing the impeifection of the instrument. The repetition 
 of angles served every purpose immediately, with regard to terres- 
 trial objects, on account of the steadiness of the point of view ; but 
 there was the difficulty, with regard to the heavenly bodies, of their 
 perpetual change of place. Borda a})plied himself to measure the 
 distance from tlie zenith of the stars when they crossed the meridian ; 
 and the- star tiien remains sensibly at the same distance from the 
 zenith long enough to allow the operation of the nuilti})lication of 
 tlie angle. By these means, angular instruments are matched with 
 horary in regard to pi-ecision. They require from the observer a 
 diligent patience in ap{)]ying all the minute precautions and recti- 
 fications which expeiience has proved to be indispensable to the 
 fullest use of these instruments.
 
 124 POSITIVE PHILOSOPHY. 
 
 Then, we have Roeraer's meridional eyeglass, whicli fixes tlie 
 instant of the passage of a star over the meridian. The plnne of 
 the meridian is made in this case purely geometrical, by being 
 described by the optic axis of a simple eyeglass, properly disposed ; 
 Avhich is enough when all we want to know is the precise moment 
 of the star's passage. Then there are the micrometrical instruments, 
 by which we measure the diameters of stars, and, generally all small 
 angular intervals. These are the material instruments of observa- 
 tion, — horary and angular. We must now advert to the intellectual 
 means, — that is, to the corrections which astronomers must apply 
 lo the results exliibited by their instruments. There would be little 
 use in perfecting our instruments, if refraction and parallax intro- 
 duced as much error into our observations as we had got rid of by 
 the improvement of our apparatus. 
 
 The corrections required are of two kinds. The first relate to 
 Requisite cw- the crrors caused by the position of the observer, — • 
 rections. the Ordinary refraction and parallax. No deep 
 
 astronomical knowledge is reqiiired for the correction of these. 
 The second class, arising from the snme cause, since they proceed 
 from the observer being on a moving planet, are founded on primary 
 astronomical theories : they are the annual parallax, the precession 
 of the equinoxes, aberration, and nutation. Our business now is 
 with the first and most important class. 
 
 SECTION II. 
 
 REFRACTION. 
 
 The light which comes to us from any star must be more or less 
 turned aside by the action of the terrestrial atmos- 
 phere. We must estimate the amount of this devi- 
 ation befoi-e our observations can answer any theoretical purpose. 
 The star is, by this refraction, made to appear too near the zenith, 
 while left in the same vertical plane. Only at the zenith is the 
 error absent, while it increases as the star descends to the horizon. 
 This error, pi'imaiily affecting distances from the zenith, must affect, 
 indirectly, ail other astronomical measurements, except azimuths : 
 but it would be easy to calculate them, if we once knew the law of 
 diminution and increase of refraction at different distances from the 
 zenith. Philosophers have tried the logical way and the empirical, 
 and have ended by combining the two. 
 
 If our atmosphere wei'e homogeneous, the refraction of light 
 would be uniform and calculable. But our atmosphere is composed 
 of strata ; and the consequent refractions are excessively unequal, 
 a!id increasing as the light pcnetiates a denser stratum, so that 
 its passage constitutes a curve of the last degree of complication. 
 Even this would be calculable, with more or less pains, if we knew 
 
 I 
 
 I
 
 DIFFIC UL TIES FR OM REEK A CTION. 1 2 5 
 
 Hie law of variation of lliese ainiosplicric deiisiiies : l)ut we do not 
 and cannot know that law. We have no exact knowledge of the 
 laws of teni})eiatnre, and cannot estimate atmospheric changes, 
 either as to nnmber or degree: and all mathematical processes 
 founded on laws of pressure, etc., may be good as exercises, but are 
 of no value in estimating refi-action. As to the em[)iric;d method, 
 if the refraction remained always constant at the same height, wo 
 might construct tables ; and, by extending our observations, and 
 instituting various comparisons, we might hope to obtain such 
 a mass of materials as would afford us some certain results. This 
 is what astronomers have, in fact, patiently and laboriously done, 
 by the help of the improved instruments, we have spoken of. They 
 have used whatever geometrical help they could make aj)plicable : 
 but the results are discouraging enough. Thei-e is nothing like 
 uniformity in the results: for the changes in the atmosphere are 
 beyond our calculation and measurement. We study the barometer, 
 the thermometer, and the hygrometer, at the right moment ; we can 
 learn from them only the changes taking place on the spot in which 
 we are ; and our tables of i-efraction vary as our observatories, and 
 even in one observatory at different times. Delambre found differ- 
 ences of four or five minutes between one day and another, after 
 taking all imaginable pains. All that we can do is to confine our 
 observations to the nearest possible approach to the zenith, and to 
 ])lace no reliance on what we attempt near the horizon. By doing 
 this, we shall find our astronomical observations less affected by the 
 unmanageable difficulties of refraction than might be anticipated. 
 
 SECTION III. 
 
 PARALLAX. 
 
 The difficulty of the parallaxes can be dealt with much more 
 easily and satisfactorily than that of the refiactions. 
 Observations of the heavenly bodies made in ditFerent 
 ])laces could not be exactly compared without a reference, in idea, 
 to those which would be made from an imaginary observatory, 
 situated in the middle of the earth, which is besides the true centre 
 of apparent diurnal motions. This correction, which is called the 
 paiallax, is analogous to that which is constanlly made in measure- 
 ments of the earth's surface, under the more logical name of 
 reduction to the centre of the station. 
 
 The effect of the parallax, like that of refraction, is upon the 
 distance of stars from the zenith alone, leaving the star in the same 
 vertical plane, and placing it too far from the zenith, instead of too 
 near, as in the case of refraction. In this instance too, as in the 
 other, though not according to the same law, the deviation increases 
 as the star descends to the horizon. In like maimer, loo, there
 
 126 POSITIVE PHILOSOPHY. 
 
 must be socondary raodlficalions for all tlie other astronomical 
 quantities, except with regard to the azimuths. The rectification 
 is easy in comparison with tlie other case, from the absence of the 
 hopeless difficulties caused by our ill-understood atmosphere. The 
 similar course of the two difficulties, producing counteracting 
 effects, has, we may observe, relaxed the attention of astronomers 
 to the facts of refraction and parallax, by partly concealing their 
 influence on actual observations. 
 
 The parallax does not, like refraction, affect all the stars alike, 
 but, on the contrary, affects all unequally, and each according to its 
 position. It is insensible with regard to all which lie outside the 
 limits of our system, on account of their immense distance ; and it 
 varies extremely within our system, from the horizontal parallax of 
 Uranus, wliich can never reach a half-second, to tiiat of the moon, 
 which may at times exceed a degree. Here lies the radical distinc- 
 tion, in astronomical calculations, between the theory of parallaxes 
 and that of refractions. The determination of questions of paralhixes 
 does not wholly depend, like that of refraction, on methods of obser- 
 vation in astronomy, but is truly a portion of science. Depending 
 as it does, ultimately, on the estimate of the distances of the stars 
 from the earth, it pertains to celestial geometry, through the 
 necessity of knowing the law of motion of each star. Thus, it 
 constitutes a part of the science itself ; though, in the absence of 
 direct knowledge of the distances of stars, an empirical method of 
 determining the coefficients, analogous to that employed in the case 
 of refraction, may be adopted. The method wliich will suffice is to 
 choose a place and time which will show the proposed star passing 
 the meridian very near the zenith : then to measure, for several 
 consecutive days, its polar distance, so as to know pretty nearly the 
 amount of this distance at any moment of the process : and this 
 being laid down, then to calculate, for this instant, according to the 
 horary angle antl its two sides, the true distance from the star to 
 the zenith, when it is considerably remote from it, without being 
 too near the horizon (say from 75° to 80°) : and then, the compari- 
 son of this distance with that which is actually observed at the 
 moment, will evidently disclose the corresponding parallax, and 
 therefore the horizontal i)arallax, provided the due correction for 
 the refraction has been made. This is the method by which it is 
 most easily established that the parallax of all tl\f stars is absolutely 
 insensible. 
 
 It is a serious inconvenience in this method, that all the uncer- 
 tainty of the case of refraction is introduced into that of parallax. 
 In regard to a body whose parallax is very great, as the moon, the 
 uncertainty is of small consequence ; but in regard to the sun, or 
 other distant body, an error of one-third, or even one-hnlf, in the 
 value of its horizontal parallax, might be occasioned. The method 
 is absolutely inapplicable to the remotest of our planets; and not
 
 CATALOGUES OF STARS. 127 
 
 only to Uranus, but to Saturn and Jupiter. Tlie rational metliod 
 must be lesorted to, in the case of tliese. The empirical method 
 has been mentioned here from the philosophical inteiest which 
 attaches to the fact that, up to a cerliiin point, the true distances of 
 stars from the earth, at least in proportion to its radius, may be 
 ascertained by observations made in one place ; a thing which 
 appears, at first sight, geometrically impossible. 
 
 SECTION IV. 
 
 CATALOGUE OF STARS. 
 
 I am disposed to give a place hero, contrary to custom, to the 
 Catalogue of stai'S, which I think should be reckoned Catalogue 
 among our necessary means of observation in astro- of stars. 
 noniy. This catalogue is a mathematical table of direction.^ by 
 which we find the different stars. Such a detei'mination is a basis 
 of direct knowledge in regard to Sidei'eal astronomy: while, in 
 regard to our own system, it is simply a valuable means of observa- 
 tion, which suiiplies us w'ith terms of comparison indispensable for 
 the study of the interior movements of the system. Such has been 
 the essential use of catalogues of stars, from Hipparchus, who 
 began them, to this day. — In order to fulfil their purposes, these 
 catalogues should contain the greatest possible mnnber of stars, 
 spread over every region of the sky. Astronomers have done their 
 duty well ; for it is a settled habit with them to determine, as far 
 as they can, the co-ordinates of every new star which they observe ; 
 and thus our catalogues are very voluminous, and for ever augment- 
 ing. Our business here is not with the system of classitication 
 and nomenclature adopted in these catalogues. The nomenclature, 
 bearing as it does the marks of the primitive theological state of 
 astronomy, might be easily replaced by one of a methodical character, 
 — the objects to be classified being of the simplest nature, and the 
 distinctions being, in fact, only those of position. But it is this 
 very simplicity which prevents the need from being felt as it would 
 among more com])lex elements, — useful as a rational system would 
 no doubt be in finding and assigning the places of stars. The 
 change will be made in time, no doubt, and the need is not urgent. 
 Stars are not known by their names, for astronomical pui-poses, but 
 by their descriptions ; and the classitication and nomenclature 
 in the catalogue, resulting from the fundamental division of the 
 circle, are as perfect as possible; and all else is of little importance. 
 I would only ask that we should cease to speak of the magnitudes of 
 stars, as marking their lank, and substitute the word brightness, in 
 order to avoid all risk of supposing stars to be large or small in 
 proportion to their brightness or dimness. The word hrighfncss
 
 128 POSITIVE PHILOSOPHY. 
 
 would be a simple declaration of the fact, without judgiug the 
 causes, which we are far froui understandiug. 
 
 By viewing these methods as I have brought thera together, we ■: 
 may trace the progress of the science from its earliest days. With f j 
 regard to angular uieasurement, for inslauce, tlie ancients observed 
 with exactness a degi-ee at the utmost; Tyclio Brahe carried up the 
 precision to a minute, and the moderns to a second ; — a perfection 
 so recent, that observations which lie more than a century behind 
 our time are considered, from their want of precision, inadmissible 
 in the formation of astronomical theories. 
 
 My object has been, chiefly, to show the harmony which exists 
 among these difTerent methods of observation ; a harmony which, 
 while it tends to perfect them all, up to a certain point, still re- 
 stricts them all, by making each a limit to the rest. No improve- 
 ment in horary or angular instruments, for instance, could carry us 
 far, while our knowledge of refraction remains as imperfect as it is. 
 But there is no reason to sui)pose that we have approached the 
 limits imposed by the conditions of the subject.
 
 129 
 
 CHAPTER III. 
 
 GEOMETRICAL PHENOMENA OF THE HEAVENLY BODIES. 
 SECTION I; 
 
 STATICAL PHENOMENA. 
 
 The phenomena of our solar system divide themselves into two 
 classes, — the Statical and the Dynamical. The first Tioo classes of 
 class comprehends the circumstances of the star itself, phenomena. 
 independent of its motions ; as its distance, magnitude, form, atmos- 
 phere, etc. : the other comprehends the facts of its dis[)lacemeiits, 
 and the mathematical considerations belonging to its different 
 positions. According to the usual analogy, the first is independent 
 of the second ; while the second could have no existence without 
 the first. The Statical phenomena would exist if the system was 
 immovable: while the dynamical are wholly determined by the 
 statical conditions. 
 
 The first thing necessary to be known about any heavenly body 
 is its distance from the earth : and the difficulty of Planetary 
 obtaining this ground for further observations is distances. 
 extremely great, — the smallness of the base of our triangle, and 
 the immensity of the distance of the planet, rendering all accuracy 
 hopeless in very many cases. Towards the middle of the last 
 century, when it was desired to determine the horizontal parallax of 
 the moon, — the most manngeable of the heavenly bodies, — Lacaille 
 went to the Cape of Good Hope, and Lalande to Berlin, to observe 
 its distance at the same moment from the zenith, — that moment 
 being appointed, — as the middle of an anticipated eclipse. The 
 stations were so chosen as to afford a pretty accurate knowledge of 
 the extent of the line of the base, — which was about as long a one as 
 our globe could afford. The observations of the two distances of 
 the moon from the zenith must thus afford the necessary data for 
 the resolution of the triangle which must give the distance sought: 
 and thus we have obtained a very exact knowledge of the moon's 
 distance, which, at its mean, is about sixty terrestrial diameters, and 
 about wliich we are sure that we cannot be mistaken to the extent 
 of more than twelve miles. The same method might serve to 
 give us, though with much less precision, the distance of Venus and 
 
 VOL. L I
 
 I30 POSITIVE PHILOSOPHY. 
 
 even M;iis, if the observation was made wlien they were nearest to 
 the earth ; but it becomes too uncertain with regard to the sun. It 
 would leave an uncertainty of at least an eiglith, or about twelve 
 millions of miles. Of course, it is of no avail with regard to yet 
 more distant bodies. 
 
 The method used by astronomers under this diflflculty is to measure, 
 first, distances for which our small terrestrial bases will serve ; and 
 on these, according to their related phenomena, to erect other cal- 
 culations ; thus making of the first a basis for the support of new 
 estimates. Aristarchus of Samos conceived of an ingenious method 
 of discovering the distance of the sun through that of the moon ; 
 but the uncertainty about seizing the exact moment of the quadrature 
 of the moon introduced fatal inaccuracy into the calculation. 
 Halley's method, by means of the passage of Mercury and Venus 
 over the sun, is more circuitous, and suitable only to an advanced 
 state of geometrical science ; but it is far more accurate, and the 
 only one now admissible, for determining the parallax of those 
 planets and of the sun, and therefore the distance of the sun from 
 the earth, through the differences in the transit observable at two 
 very distant stations. By this method, we can estimate, within a 
 hundredth part, the distance of the sun from the earth. This dis- 
 tance being ascertained, we have it for a basis for other calculations. 
 We have only to observe the angular distance from the sun to the 
 proposed body, at two periods separated by six months, — that is, 
 from opposite points of the earth's orbit. This gives us an immense 
 triangle, the base of which is twice the length of the distance of 
 the earth from the sun : and thus it is that our knowledge of the 
 earth's motion has helped us to a base twenty-four thousand times 
 longer than the longest that can be conceived on our own globe. It 
 is true, the planet observed will have changed its place in the interval ; 
 but the remoter planets,— which alone are in question here, — move 
 very slowly ; — Saturn's circuit, for instance, occupying thirty years; 
 and our times of observation being practically reducible to a shorter 
 time than six months, — even to two or one, with regard to those 
 planets of our system wiiich move more ra})idly ; while the slower 
 ones may be considered almost stationary, during such short periods 
 of time ; and again, allowance can be made for this small change of 
 place, according to the geometrical theoiy of its proper motion. It 
 is in this way that astronomers have attained to their knowledge of 
 the positions of the remotest bodies of our system. The numbers 
 by which we express their relations to the distance of the earth 
 from the sun, are now certain to the third decimal at least. 
 
 The vast increase of the basis of observation afforded us by our 
 knowledge of the earth's movement is clearly the greatest that we 
 can attain. If we have cleared the bounds of our globe, we certainly 
 cannot go beyond its orbit. Great as tiiis distance appears to us, 
 it vanishes when we want to ascertain the distances of stars outside
 
 PLANETAR Y DISTANCES. 1 3 1 
 
 onr system. All measurement is here so out of llic question that 
 llie most we can do is to fix a limit within which they certainly are 
 not, — sayiti<;^, for instance, that the nearest star is at least two 
 hundred thousand times more remote than the sun, or ten thcusand 
 limes further olT than the remotest planet of our system ; which is 
 quite sufficient to estahlish the independence of our system. 
 
 When we have ascertained the distance of the planets from the 
 earth, it is easy to understand how we may find their distances 
 from each othei-, since, in the triangle in which each is contained, 
 two sides are already given, and the angle to the eai-th can always 
 he measured. It is only with regard to the sun and the moon that 
 the distances to the earth are of im[)ortaiice. It is enougli to know 
 the distances of the planets from the sun, and of the satellites from 
 their planets, which involve little variation. These ai-e our means 
 for ascertaining astronomical distances. As we might anticipate, 
 our assurance is in proportion to the nearness ; and great remoteness 
 haffles us entirely. We see here again, as everywhere, that the 
 most simple and elementary determination depends on the most 
 delicate and comple.x scientific theories. This first case exhihits 
 so much of the s})irit of astronomical procedure, that we may go 
 more rapidly through the other statical heads of celestial geometry. 
 
 The distances of the stars from our glohe being once ascertained, 
 
 we can learn whatever we desire about their form _ , . 
 
 , . , , , . • r • i 1 1 i • 1 Form and size. 
 
 and size by observation, ir it be but precise enougli. 
 
 Their very distance is favourable to this ; for, while their motion or 
 ours displays in turn all their possible aspects, ©ur distance enables 
 US to see at once the whole of each aspect. With regard to the most 
 distant and the smallest, however, — to the stars outside our system, 
 and the satellites of Uranus, and the small planets between Mars 
 and Jupiter, — they can appear to us only as points of vivid light, 
 and their sphericity is concluded upon only through a bold induc- 
 tion. But, in observing the larger [)lanets of our system, vve have 
 only to measure their apparent diameter in all directions, after 
 allowing for refraction and parallax. It is much easier to us to 
 learn the form and size of sun and moon than of our own globe, 
 since we have had the aid of glasses. The only case of difficulty 
 is that of Saturn's rings ; as it once was with the moon, whose 
 changing aspects greatly puzzled the ancients. The most simple 
 geometry now solves the last difficidiy, and Huyghens has helped 
 us over the first. With these exceptions, direct observation assures 
 us that the plnnets are all round, with more or less flattening at 
 the poles and bulging at tlie equator, in proportion to the rapidity 
 of their rotation. 
 
 As for the size of the heavenly bodies, it is easily calculated from 
 the measurement of the apparent diameter combined with the 
 cetermination of the distance; and the only reason why men were 
 BO long and so widely mistaken about tlie dimensions of the planets
 
 132 POSITIVE PHILOSOPHY. 
 
 was tliat their real distances were unknown. No rule as yet appears 
 Vv'hich connects these results with the order of tlie distance of the 
 planets from the sun. All we know is that the sun is larger than 
 all the other bodies of the system put together, and in general that 
 the satellites are much smaller than their planets, as the laws of 
 celestial mechanics require. With regard to the bodies outside of 
 our S3'stem, as we have no knowledge of their distances, we are, of 
 course, ignoi-ant of their dimensions. 
 
 It is by the occultation of stars, as starry eclipses ai-e called, that 
 Planetary we make observations on the atmospheres of the 
 atmospheres, planets, by seeing what deviation their atmospheres 
 cause in the light of the remote stars which they eclipse. As the 
 sun's light is prolonged to us by the refraction of our atmosphere, 
 the atmosphere of a planet defers (only in a much greater degree) 
 the occultation of the star, and also shortens it ; and the comparison 
 of the apparent duration of the eclipse with that which it would 
 otherwise be, gives us data for the calculation of the atmosphere 
 which causes the deviation. It is thus that we learn that the moon 
 has no appreciable atmosphere. The horizontal refraction which, 
 on our globe, would reach ihirty-fonr minutes, does not in the moon 
 amount to a single second. And the inference that an atmosphere 
 is wanting there is confirmed by M. Arago, who in a different path of 
 inquir}', about the polarization of light reflected from liquid surfaces, 
 lias eslabliishetl the fact that there are not, on the surface of the 
 moon, any great liquid masses, fitted to form an atmosphere. The 
 next best-known case is that of Venus, which exhibits a horizontal 
 refraction of thirty minutes, twenty-four seconds. As for the extent 
 of the atmospheres, it may be roughly conjectured from the cessation 
 of the refracting power ; but sucli conjectures must be veiy loose, 
 as the refracting power may become imperceptible to us, f;ir within 
 the limits of an atmosphere becoming attenuated towards its verge. 
 The strangest phenomenon is that of the telescopic planets, with 
 the exception of Vesta ; the atmosphere of Pallas, for instance, being 
 more than twelve times the diameter of the planet. The usual 
 condition, however, appears to be that shared by our globe, — of an 
 atmosphere whicii is very shallow in pioportion to the dimensions 
 of the planet : and this is nearly all we know. 
 
 The remaining statical topic is that of the form and size of the 
 Earth's form earth, which has been left to the last, on account of 
 and size. \[^ special iiat uie. 
 
 No glance of the eye will aid us here, nor any direct observation 
 Means of whatever. A long accumulation of indirect observa- 
 discovcry. tions, serving as a basis for comi)lex mathematical 
 reasonings, are our only means. The geometrical aspect of the 
 question must be taken first, though it depends on the highest 
 mechanical theories, and arises from a mechanical beginning. In 
 the infancy of mathematical astronomy, the variations exhibited in
 
 EARTITS FORM AND SIZE. 133 
 
 (linercnt jilaces by the diuniiil jnoveinciit furiiislied \\w first 
 i^oometrical proof of the earUi being loiiiid. It was enough lo 
 t'stablish its evidently and exclusively spherical charactei-, that the 
 change exhibited by the heiglit of the jkiIc on each horizon was 
 always in exact ])rojK)ition to the length traversed according to 
 any meridian whatever : and this remains the source of all our 
 geometrical knowledge of the form and dimensions of our planet. 
 Astronouiers reached their knowledge of its precise foi-m through 
 that of its size ; for it was long before its deviation from the perfect 
 spherical form was understood. In this, as in every case, the form 
 of any body is appreciable oidy by measuring its diiuensions in 
 various directions; and here the only difficulty is in the measuring. 
 The first piinciples of the discovery were given by Eratosthenes, in 
 tiie early days of the school of Alexandria; but his meth.od was 
 never effectually emp)loyed till the middle of the seventeenth 
 century, when Picard undertook to measure the degree between 
 Paris and Amiens. This was the great starting-point of the 
 measuring operations, which nuist have revealed, as they became 
 more perfect, the truth that the earth is not a perfect sphere; but 
 Newton, by his theory of gravitation, and with his one fact of the 
 shortening of the seconds-pendulimi at Cayenne, settled the 
 matter, by deciding that our globe nuist necessarily be flattened at 
 liie poles, and bulge at the equator, in the relation of 229 to 230. 
 The astronomers could not at once pronounce against the evidence 
 of direct measurement, while the geometers saw the fact to be 
 certain ; and the controversy between these two orders of philos- 
 ophers, for half a centuiy, led to those scientific operations which 
 have brought us all to one mind. The question was settled by the 
 great expedition sent forth, above a century ago, by the French 
 Academy, to measure, at the equator and the pole, the two extreme 
 degi-ees of latitude which must exhibit the widest variation from 
 each other : and the comparison of these with each other, and with 
 Picard's degree, terminated the controversy, and established, not 
 Old}' the truth of Newton's discovery, but the very near accuracy 
 of his calculation. All the experiments made since, in various 
 countries, have united in confirming the fact of the continual 
 lengthening of degrees in approaching the pole. It does not follow 
 that the figure of the eailh has been ascertained with absolute 
 ])recision. There are slight discrepancies which may either be from 
 imperfection in our estimates, or from the earth not being precisely 
 an ellipsoid of revolution; but whatever may be the result of 
 future labours, we know that we are near enough to the truth for 
 all practical purposes, uidess in questions of extreme delicacy. 
 We have no absolute knowledge here, any more than in any other 
 department; and we must be content to make our approximations 
 more ccnnplex as new ])henomena arise to demand it. Such is the 
 true character of the advances that have been made in this science
 
 134 POSITIVE PHILOSOniY. 
 
 from the beginning. Superficial observers may call its theories 
 arbitrary, from the incessant clumges of view that have arisen : but 
 the knowledge gained has always been positive ; every scientific 
 opinion has corresponded with the focts which gave rise to it ; and 
 such opinions remain therefore useful and sound at this day, within 
 their own range. The science has thus always exhibited a character 
 of stability, through all incidents of progression, from the earliest 
 days of the Alexandrian school till now. 
 
 Such are the statical aspects of the planets of oiir system. We 
 have now to look at the geometrical theory of their motions. 
 
 Like all other bodies, the planets have a motion composed of 
 
 Planetary translation and rotation. The connection of these 
 
 motions. f,^vo motious is SO natural, that when we know of the 
 
 one we look for the other. Yet they present very different degrees 
 
 of difficulty, and require separate consideration. 
 
 The progression of the stars was observed long before their rota- 
 tion, — the unassisted eye being enough for the first; yet the geo- 
 metrical study of their rotations is easier, because the motions of 
 the observer have no effect upon them ; whereas they hu-gely afTect 
 questions of translation. And again, the question of orbits is the 
 chief difficulty of the study of translations ; and it does not enter 
 into that of rotations. The hitter nearly approaches to the character 
 of statical questions ; and therefore it ought to be taken first in the 
 exposition of celestial geometry. 
 
 Galileo introduced the study of rotations by discovering that of 
 tlie sun, which was sure to follow closely on the in- 
 vention of the telescope. The method used is obvious 
 enough, and the same in all cases ; — to observe any marks that may 
 exist on the surface of the body, their displacement and return. 
 The more such points of observation are multiplied, the more accu- 
 rate and com[)lete will be the calculations of time, magnitude, uni- 
 formity of movement, etc., deducible from them. There is no more 
 delicate task than this, except with regard to the sun and moon ; 
 and none that more absolutely requires a special training of the eye. 
 It is a proof of this, that a careful and honest observer, Bianchini, 
 supposed the rotation of Venus to be twenty-four times slower than 
 it is. Some bodies, as Uranus, are too remote, and others, as the 
 satellites and new planets, too small, to have their rotation estab- 
 lished at all, though it is concluded from analogy and induction. 
 We, as yet, know of no law determining the time of these rotations : 
 they are not connected with distances, nor with magnitudes; and 
 they seem only to have some general, but not invariable, connection 
 with the degree of flattening at the poles.* But if the duration is, 
 though regular in each case, altogether irregular as regards the dif- 
 ferent bodies, the case is much otherwise with the direction ; for it 
 
 * The rotations of some of the satellites are known. They all follow the law of 
 the moon'd rotation, namely, the time corresponds with the orbital periods. — J. P. N.
 
 PROOFS OF THE PLANETARY ORBITS. 135 
 
 is always, tlirou<;]i()iit our system, from west to east, and on planes 
 slightly inclined to that of the solar equator : and this constitutes 
 an important general datum in the study of our globe. 
 
 The study of translations, much more complex, is also mucli more 
 important, if we consider the great end of astrono- 
 mical pursuit — the exact prevision of the state of the 
 heavens at some future time. Besides that the movement of the 
 earth constitutes an important element in such a study, it must 
 make a did'erence with regard to other stars, whether the observer 
 is fixed or moving, as his own movement must affect his observation 
 of other motions. We might indeed decide with certainty, without 
 this introductory knowledge, that the sun and not the earth is the 
 true centre of the motions of all the planets, as Tycho Brahe did 
 when he denied our own motion ; for it is enough, with this view, 
 to establish that the distances from the planets to the sun scarcely 
 vary at all, while their distance from us varies excessively ; and 
 again, that the solar distance between each inferior planet and the 
 sun is less, and between each superior {)lanet and the sun is greater, 
 than our distance from the sun. But we cannot go further than 
 this, — we cannot determine the form of the ])lanetary orbits, or the 
 mode in which they are traversed, without making a careful and 
 exact allowance for the displacement of the observer. Deferring 
 for the present the subject of the earth's motion, we will briefly 
 notice some important data connected with the planetary motions, 
 which may be obtained without reference to our own movement, 
 and which are so simple as to rank among statical researches. 
 I mean jiarticularly the knowledge of the planes of orbits, and of 
 the duration of the sidereal revolutions, which has nothing to do 
 with the form of the orbits or the variable velocity of the planets. 
 A plane being determined by three points, it is enough to observe 
 three positions of a star to draw a geometrical conclusion about the 
 situation of the plane of its orbit. Astronomers do not now use, in 
 these operations, the declinations and right ascensions, which are 
 the only co-ordinates directly observed, but, for the sake of conveni- 
 ence, two other spherical co-ordinates, improperly called astrono- 
 mical latitude and longitude, which are analogous, with regard to 
 the ecliptic, to the others with regard to the equator. After having 
 detei-mined the latitude and longitude of the planet in the three 
 positions, its nodes ai-e found ; that is, the points at which its orbit 
 meets the jjlaiie of the ecliptic, and the inclination of the orbit to 
 this plane. It is evident that confirmation may be obtained by 
 observing other positions of the body, if they are chosen sufficiently 
 remote from each other ; and thus we may obtain a far gi-ea1er pre- 
 cision than in the case of rotations. It is thus that we have learned 
 that the planes of all the planetary orbits pass through the sim ; 
 and the same with regard to the satellites of any planet; and that 
 these planes are in general slightly inclined to the ecliptic, and more
 
 136 POSITIVE PHIIOSOPHY. 
 
 slightly still to the plane of the solar eqnntor, except the newly-dis- 
 covered planets, in whose case we find the inclination much more 
 considerable. 
 
 The duration of the sidereal revolutions may, of course, he 
 Sidereal directly observed, in the first instance, by looking for 
 revolution, the return of the star to the same spot in relation to 
 the centre of its motion. If we suppose its motion to be uniform, 
 which we may for a first approximation, we can estimate its course 
 by observing the time required between any of the three positions, 
 without waiting for the total revolution, which is sometimes very 
 slow. The geometrical law of this motion permits us to determine, 
 i'rom this kind of observation, the exact time of the planetary revo- 
 lution. The values of these periodic times are not irregularly 
 divided atnong the bodies of our system, like the other data that 
 we have noticed. The shorter the course, the more rapid the 
 motion ; and the duration increases more rapidly than the corre- 
 sponding distance ; so that the mean velocity diminishes in pro- 
 portion as the distance increases. We owe to Kepler the discovery 
 of the harmoii}^ between these two essential elements, and it is one 
 of the most indispensable hases of celestial mechanics. 
 
 Such is the spirit of the methodshy which celestial geometry is made 
 to yield us the elementary data which characterize the bodies of the 
 solar system. We have still to consider those of our own planet, he- 
 fore we proceed to the geometrical laws of the planetary motions. 
 
 Motion of the Earth. 
 
 We are accustomed to think of the motions of translation and 
 rotation as inseparable ; but, in the transition from supposing the 
 earth to be motionless, to the present state of our knowledge, 
 Evidences of the ^ theory existed that it whirled roimd its axis, but 
 Earth's Motion, was stationary in space. We now pei'ceive that, in 
 addition to the general evidence of the double motion of the 
 planetary bodies, we have special evidence about our own globe, — 
 that the animal motion could not exist without the diurnal, though 
 we might logically suppose beforehand that it could. 
 
 As the rotfition of the earth cannot be absolutely uniform in all 
 ])arts of its surface, some indications of its course must exist among 
 terrestrial phenomena. We must therefore distinguish between 
 the celestial and tei'restrial proofs of our diurnal motion, while the 
 annual motion admits only of the former. 
 
 Immediate appearances go for nothing in this case; for it is clear 
 
 Aiidentcon- that, to our cycs (as we do not feel the rotation), 
 
 cepiwus. it must be exactly the same thing whether we 
 
 move round among the heavenly bodies, or whetiier they, fixed in 
 
 a system, move round us in a contrary direction. There was 
 
 nothing absurd in the latter supposition, in the old days when men
 
 r ROOFS OF THE EARTH'S ROTATION. 137 
 
 liad no doubt of llie stars being very near, and not niucli larger 
 than they appear I0 the eye, while they exaggerated the size of the 
 earth. They could not avoid supposing that such a mass must be 
 immovable, while the small stars, with their little intervals, were 
 seen moving every day. Even when the Greek astronomers bad 
 sketched out the true geometrical theory of the movements of the 
 ])lanets, they treated only of the directions, and bad no idea of 
 measnring distances ; and it required the whole strength of positive 
 evidence of dimensions and distances to uproot men's strong and 
 natural persuasion of the stability of tbeir globe. From the 
 moment of our obtaining an idea of the proportions of tbe universe, 
 the old conception became too revolting to reason to be sustained. 
 When it was understood that the eartli is a mere point in the 
 midst of prodigious intervals, and that its dimensions are extremely 
 small in comparison with that of the sun, and even of other bodies 
 of our own system, it was absurd to suppose that such a universe 
 could travel round us every day. What velocities would be required 
 to enable the outlying stars to complete such a daily circuit, — mak- 
 ing allowance fur their being twenty-four thousand times nearer 
 the earth, if the earth describes no orbit, — and how small the move- 
 ment of the earth, while those prodigious masses were travelling at 
 such speed ! On mechanical grounds, the centrifugal force would 
 be seen to be unmanageable. In every way, the supposition was 
 ])erceived to be monstrous. iVgain, the passage of stars before each 
 other, and in a contrary direction to that of the general movement 
 of the sky, showed that they were at different distances from each 
 other, and not bound into an unvarying fabric. Hence arose the 
 notion of Aristotle and Ptolemy, of a system of solid and transpar- 
 ent firmaments. But the existence of comets alone was enough to 
 confute this, appearing as they do in all regions of the sky in turn. 
 As Fontenelle said, this theory put the universe in Hoiothey 
 <langer of being fractured. It was, curiously enough, ^"'''^ ^'^y- 
 Tycho Brahe, the most illustrious opponent of the Copernican 
 system, who provided for the overthrow of his own arguments by 
 first presenting the true geometrical theory of comets. Long betore 
 modern precision was attained, men had been pre- Earth's 
 jiared by such consideiations as the above to conclude rotation. 
 upon the rotation of the earth. Long before Copernicus, a rough 
 conception of the truth existed. Even Tycho Brahe felt the astro- 
 nomical superiority of the true theory; but it seemed to be con- 
 tradicted by what is before our eyes, — the fall of heavy bodies, etc. 
 Copernicus himself could not remove the objections which arose 
 out of men's ignorance of the laws of Mechanics. These objections 
 held their ground for a century, till Galileo established the great 
 law which we have recognized as one of the three on which Eational 
 Mechanics is based, — that the relative motions of dilferent bodies 
 are independent of the common motion of the whole. Till this was
 
 133 POSITIVE PHILOSOPHY. 
 
 established, the supposition of the rotation of the earth was inad- 
 missible. It is a curious fact, casting nuich h'ght upon the action 
 of the human mind, that the opponents of Galileo taunted him with 
 the so-called fact that a ball let down from the top of the mast of a 
 ship in motion would not fall at the foot of the mast, but some way 
 behind, — neither they nor anybody else having tried the experiment, 
 which would have shown them that their supposed fact was a 
 mistake. The followers of Copernicus did worse, — they admitted 
 tiie so-called fact, but tried to reason away its bearings with 
 fantastic subtleties. The matter was not settled even by the 
 demonstrations of Galileo, nor till Gassendi compelled observation 
 by a public experiment in the port of Marseilles. 
 
 That order of experiments has been carried on, and would be of 
 high value if we could obtain perpendicular stations of sujfficient 
 height ibr the purpose. It is clear that a lofty tower must describe 
 a larger circle in the same time at the top than at the base ; and 
 that any body dropped from it must share the higher rate of 
 velocity, having a slight horizontal velocity in the direction of the 
 earth's rotation, — falling therefore a little to the east of the base of 
 the tower. Omitting the consideration of the resistance of the air, 
 this amount is calculable in the function of the height of the tower 
 and of its latitude ; but experiment would also be valuable ; and it 
 is to be hoped that it will be tried at the equator, where the devia- 
 tion must be greater than anywhere else. 
 
 The most certain terrestrial })roof of the earth's rotation is found 
 Influence of ^^ tracing the influence of the centrifugal force upon 
 centrifugal force the direction and intensity of weight. This has been 
 upon gravity. done by that observation of Richer, on the shortening 
 of the seconds-pendulum at Cayenne, which has been mentioned as 
 having emboldened Newton to declare the true figure of the earth. 
 The deviation from the spherical form is too small to account for 
 more than one-third of the effect observed ; and the other two- 
 thirds are precisely what would be required, at the equator, where 
 the centrifugal force is greatest, on the supposition of the earth's 
 rotation. Wherever the delicate observation can be made with 
 sufficient precision on other points of the globe's surface, the result 
 answers to the theory. Thus, we should have sufficient assurance, 
 in the absence of the abundant astronomical proofs that we possess 
 of the rotation of the earth. Probably no one fact has ever, in the 
 history of our race, produced such consequences as that observation 
 of Richer's, — two-thirds of the estimated effect having completely 
 established the rotation of our globe, and the other third having 
 led Newton to the ascertainment of its form. 
 
 The movement of translation is ascertainable only by astrono- 
 
 Earth's trans- mical proofs, for the difference in velocity of the 
 
 lation. various pai-ts of the globe, in virtue of this motion, 
 
 is too slight to be sensible to us, or to produce any effect oa
 
 PROOFS OF THE EARTFI'S TRANSLATION. 139 
 
 terrestrial ]ilienonicna. When the circuits of otlier planets were 
 known, men's minds were prepared for that of the earth, — tiie 
 question then being whether the earth was in analogy with Venus, 
 Mai's, Jnpiter, etc., or whether, while they continued their courses 
 round the sun, the sun made a yearly circuit round the earth. 
 Reason must declare, in such a case, that any uncertainty must 
 arise simply from the position of the observer, who, placed on any 
 other planet, would have doubted whether he was not the centre of 
 the heavenly motions. Any observation of mere appearances must 
 evidently go for nothing in this case ; as appearances must be exactly 
 the same, — the parallelism of its Jixis of rotation being unaltered, — 
 whether the earth or the sun is in the Ecliptic, and the other in the 
 centre. The proofs must be derived fi-oni better testimony than 
 mere appearances ; and they so abound that we have only to choose 
 among those which are piesented by the whole range of the heavens. 
 The phenomenon called the precession of the equinoxes was 
 observed by Hipparchus, who was struck by the Precession of 
 difference of two degrees which he observed between the equinoxes. 
 the longitudes of stars in his time and those which had been recorded 
 a century and a half before. To accoimt for such a phenomenon, 
 successive astronomers imagined other heavens ; a process that 
 they repeated with regard to nutations, which was a phenomenon 
 too mimite for their observation. To account for it, on the sup[)Osi- 
 tion of the earth being stable, a third general movement of the 
 whole heavens must be supposed. Newton indicated and Bradley 
 afterwards proved, that very slight alterations in the parallelism 
 of the earth's axis, — such alterations as must residt from the 
 influence of the sun, and yet more of the moon, upon the equa- 
 torial bulge, — precisely account for the perturbations which create 
 such confusion under the ancient view of the earth's stability. The 
 most unquestionable proof of all, however, is in that class of pheno- 
 mena called the retrogradations and stations of the Rctronradatiov^ 
 planets, which are perfectly explained by the annual and stations of 
 circuit of our globe, and are otherwi.se quite incom- n^e iMnets. 
 prehensible. If two boats are gliding down a river, at different 
 rates of speed, the one must appear to the other advancing, 
 stationary, or retrograde, according as its own speed is smaller, 
 equal, or greater. With regard to the heavenly bodies, their 
 velocities and other circumstances are known to us, so that we 
 can calculate what their courses ought to be to our eyes, on the 
 supposition of our own annual movement. The appearances answer- 
 ing to our scientific expectation, the proof is practically complete. 
 If the earth moves, the retrogradation of the larger planets ought 
 to happen, as it does, when they are in opposition, and that of Venue 
 and Mercury when they are in inferior conjunction. The regular 
 occurrence of this coincidence was not even attempted to be ex- 
 plained by the ancients.
 
 I40 POSITIVE PHILOSOPHY. 
 
 We have called these proofs practically complete ; and thev were 
 held to he so by Copernican philosophers before the time of Kepler 
 and Galileo : but our age is not satisfied without a more strict 
 mathematical evidence, amounting to demonstration. 
 
 The one demonstration on which modern science rests is that 
 Aherraiion derived from the various phenomena of the aberration 
 of light. of light, which are quite incompatible with the stabil- 
 
 ity of the globe. Roemer's observations of the satellites of Jupiter 
 suggested to him the use of light as a measure of distance. Know- 
 ing what changes must be taking place at various distances from 
 us in the heavens, and knowing the velocity of light, the variations 
 in time at which the changes become visible to us will be a measure 
 of our change of place and distance. For instance, the first 
 satellite of Jupiter is eclipsed every forty-two hours and a half. 
 The eclipse will take place in a shorter or a longer time than this 
 to our eyes, in proportion as we are removed to the one side or the 
 other of our mean distance from Jupiter, on account of the smaller 
 or greater space that the light will have to travel through. By 
 extending our observation, not only to the other satellites of Jupiter, 
 but to those of Saturn and Uranus, we have obtained further 
 verifications of the relation of our orbit to theirs, and also, proof of 
 the uniformity in the passage of light, — at least within our own 
 system. If the earth wei'e immovable, we might have an error of 
 time, with regard to distant stars, but not of place: but, by cora- 
 ])0undiiig the velocity of the earth in its orbit with that of light, 
 which is about ten thousand times greater, we can calculate how far 
 any star ought to appear to deviate from its position. This devia- 
 tion is found not to exceed, at its maximum, twenty seconds in any 
 direction ; and therefore forty seconds is the greatest deviation 
 which can appear in the position of any star in the course of the 
 year. It was the striking periodicity of these deviations which led 
 Bradley to seek for the true theory in the combination of the motion 
 of the earth with that of light, and to work it out with the mathe- 
 matical exactness permitted by modern science: and there is 
 nothing in the case to prevent the direct application of the mathe- 
 matical process to the visible phenomena. The result is an unques- 
 tionable demonstration of the annual movement of the earth, with 
 which all tiie phenomena of the case precisely agree, and without 
 which they coukl not exist. 
 
 It is evident that this knowledge of aberrations compels us to add 
 another correction to those of refraction and parallax, and the same 
 is the case with regard to the precession of the equinoxes and nuta- 
 tion. Thus, as science advances, the preparation of a phenomenon, 
 observed with the best instruments, for scientific use, becomes a 
 delicate and laborious operation. 
 
 'J'hese aie the considerations which have led men to the know- 
 ledge of the double motion of the planet we inhabit. No other in-
 
 KNOWLEDGE OF THE EARTH'S MOTION. 141 
 
 tellectual revolution has ever so llioron<;lily asserted the luitunil recti- 
 tude of the human mind, or so well shown the action influeweof 
 of positive demonstration upon definitive opinions ; for scientific fact 
 no other has had such ohstacles to surmount. A very '^p°^ Opinion. 
 small number of ])hil()sophei'S, workin<i; apart, without any other 
 social superiority than that which attends positive i;enius and real 
 science, have overthrown, within two centuries, a doctrine as old as 
 our intelli<;ence, directly established upon the plainest and com- 
 monest appearances, intimately connected with the whole system of 
 existing: opinions, general interests, and dominant authorities, and 
 supported moreover by luunau pride, powerfid in the recesses of 
 each individual mind. The whole system of tlieolofj^ical belief 
 rested on the notion that the entire universe was ordained for Man, 
 a notion which ap})ears truly absurd the moment it is seen that our 
 globe is only a subaltern star, — not any centre whatever, but circu- 
 lating in its place and season, among others, round the sun, whose 
 inhabitants might, with more reason, claim the monopoly of a system 
 which is itself scarcely j)erceptible in the universe. The notion of 
 final causes and providential laws undei-goes dissolution at the same 
 time ; for, the once clear and reasonable idea of the subordination 
 of all things to the advantage of Man being exploded, no assignable 
 purpose remains for such pi'ovidential action. As the admission of 
 the motion of the earth overthrows the whole theory founded on the 
 human destination of the universe, it is no wonder that religious 
 minds revolted from the great disclosure, and that the sacerdotal 
 power maintained a bitter rage against its illustrious discoverer. 
 
 The Positive philosophy never destroys a doctrine without 
 instantly substituting a conviction, adequate to the needs of our 
 human nature. If the vanity of Man was grievously humbled when 
 science disabused him of his notion of his supreme importance in 
 the universe, to this vanity at once succeeded a lofty sentiment of 
 his true intellectual dignity, when he saw what means were in his 
 ))ower, under such difficulties as his position imposed upon him, for 
 the discovery of such a truth as he had attained. Laplace has 
 pointed this out, showing how to the fantastic and enervating notion 
 of a universe arranged for Man has succeeded the sound and vivify- 
 ing conception of Man discovering, by a positive exercise of his 
 intelligence, the general laws of the world, so as to be able to modify 
 them, for his own good, within certain limits. AVhich is the nobler 
 lot? Which is most in harmou}^ with our highest instincts? 
 Which is the most stimulating to our faculties? And which is the 
 most animating to our feelings ? 
 
 One more remark suggested by these discoveries is that a clear 
 distinction is for ever established between our system and the uni- 
 verse at large. The old notion of the univeise as a single system 
 was founded on the error of the stability of the earth as its centre. 
 The discovei-y of the earth's revolution at once transported all the
 
 142 rOSITIVE PHILOSOPHY. 
 
 exteiDal stars to distances infinitely more considerable than the 
 greatest planetary intervals, and has left no place for the idea of 
 system at all, beyond the limits of our sun's influence. We do not 
 know, more or less, and men will probably never know, whether the 
 innumerable suns that we see compose a general system, or any 
 number, lai-ge or small, of partial systems entirely independent of 
 each other. The idea of the universe therefore is excluded from 
 positive j)hilosophy ; and that philosophy is, strictly speaking, 
 bounded by the limits of the solar system, in regard to definite 
 results ; and this circumscription is, as elsewhere, to be regarded as 
 real progress. This restriction is further justified by tiie knowledge 
 we have obtained of all really universal phenomena being essentially 
 independent of the interior phenomena of our system, since the 
 astronomical tables of the state of our system, prepared without ref- 
 erence to any other sun than our own, invariably coincide with the 
 minutest direct observations. Tiie theory of the earth's revolution 
 has not as yet exerted its due influence on our views, and especially 
 in regard to this last consideration. This is doubtless owing to the 
 imperfections of our education, which keep back these high philo- 
 sophical truths till even the best minds have been possessed with 
 an op{)Osite doctrine : so that the positive knowledge which they 
 afterwards attain commonly does little more than modify and 
 restrain the bad tendencies of their education, instead of ruling and 
 guiding their highest faculties. 
 
 Kepler s Laios. 
 
 The first idea that occurs to us when we are once satisfied of the 
 revolution of the earth is that our point of view ought henceforth to be 
 
 Annual the Centre of the sun. This transformation of our ob- 
 
 paraiiax. servatiou is called the annual parallax, and follows the 
 same rules as the diurnal parallax, allowance being made for the 
 much greater distance. Whether our observations of the sidereal 
 heavens are geocentric or heliocentric, — from the middle of the 
 earth or of the sun, — is of no appreciable consequence ; but within 
 our system the annual parallax is of sensible importance. When, 
 from the central point of view, the orbits of the planets are deter- 
 mined, we can proceed to that great aim and end of the science, — 
 the prevision of future conditions of the heavens at appointed times. 
 
 The earliest supposition was lliat the motions of the planets were 
 „. unifoiin and ciicular. The ancients had a, supersti- 
 
 tion, as their writings abundantly show, that tiie circle 
 was the most perfect of all forms, and therefore the most suitable 
 for the motion of such divine existences as the stars. Their choice 
 of the form was wise : they had to suppose some form, while that of 
 the circle answered best to what they saw ; and we ourselves now
 
 KEPLER S LAWS, 143 
 
 tnlce it provisionally in Ibrniini^ the theory of a new s(ar. But the 
 superstitious attachment of the ancients to this form was a serious 
 impediment to the advance of astronomy. For every deviation and 
 new appearance a new circle was supfjosed, till all the simplicity 
 of the original hypothesis was lost in a complication of epicycles. 
 By the end of the sixteenth century the numher of circles supposed 
 necessary for the seven stars then known amounted to seventy-four, 
 while Tyclio Brahe was discovering^ more and more planetary 
 movements for which these circles could not account. Tims it is 
 that men cleave to old ideas and methods till they are utterly worn 
 out, and proved beyond recal to be ineffectual, under all additions 
 that can be made to them. 
 
 Then came Kepler, the first man for twenty centuries who had 
 the couraire to jro back to the beG^inninc^, as if nothinjj^ 
 
 ^ . K.KPLER- 
 
 had been done in the way of theory. He took for his 
 materials the complete system of exact observations which weie the 
 result of the life of his illusti'ious precursor, Tyclio Brahe. Not- 
 withstanding the natural hardihood of his genius, his works reveal 
 to us how strenuously he had to maintain his enthusiasm, in order 
 to support the toils of so bold and difficult an enterprise — rational 
 as it was. He chose the planet Mars for study ; and it was a 
 happy choice ; because the marked eccentricity of that planet was 
 most apt to suggest the true law of irregularity. Mercury is more 
 eccentric still ; but it does not admit of continuous observation. 
 He discovered three great laws, which, extended from the case of 
 Mars to that of all the other planets in our system, constitute the 
 foundation of Celestial Mechanics. Tlie first law „. ^ 
 
 ,, ., ■!•■ ,1 11. • ji His three laws. 
 
 regulates the velocity ; the second determines the 
 
 figure of the crbit: the third establishes harmony among all the 
 
 planetary motions. 
 
 It had long been remarked that the angular velocity (that is, the 
 larger or smaller angle described, in a given time, by p-. ti 
 its vector radius) of each planet increases constantly 
 in proportion as the body approaches the centre of its motion ; but 
 the relation between the distance and the velocity remained wholly 
 unknown. Kepler discovered it by comparing the maximum and 
 minimum of these quantities, by which their relation became more 
 sensible. He found that the angular velocities of Mars at its nearest 
 and furthest distance from the sun were in inverse proportion to tlie 
 squares of the corresponding distances. Another way of expressing 
 this law is used by himself; that the area described in a given time 
 by the vector radius of the planet is of a constant magnitude, though 
 its form is variable: or, again, in other words, that the areas described 
 increase in proportion to the times. Thus he destroyed the old 
 notion of the uniformity of the planetary motions, and showed that 
 the uniformity was not in the arcs described, but in the areas. 
 
 The second law was less difficult to discover, when once Kepler
 
 144 POSITIVE PHILOSOPHY. 
 
 had surrendered his attachment to the circle. The next figure tliat 
 ]M-esented itself must nnturally be the ellipse, which 
 
 Second law. • ,^ • ^ l c c ^ i cl \^ • ^ 
 
 IS the snnplest lorm or closed curve, alter the circle. 
 The Greek geometers had advanced the abstract theory of this 
 cm've some way. Kepler could not long hesitate whei'c to place the 
 sun in it : it must be either in the centre or in one of the two foci. 
 No mathematical labour was needed to show him that it could not 
 be in the centre : and thus, in constructing elliptic orbits, Kepler 
 was necessarily led to place the sun in the focus for all the planets 
 at once. His hypothesis once formed, it was easy to verify it by 
 comparison with observations, the first principles of the required 
 calculations being laid down beforehand. The second law of 
 Kepler then is that the planetary orbits are elliptical, having the 
 sun for their common focus. 
 
 These two laws determined the course of each planet ; but the 
 movements of all round their common focus seemed 
 to be purely arbitrary, till Kepler discovered his 
 third law. Being distinguished by the most remarkable genius for 
 analogy ever seen in man, Kepler sought, and successfully, to 
 establish some kind of harmony among all these various move- 
 ments. He spent much time in pursuing the old metaphysical 
 ideas of certain mystic harmonies which must exist in the universe : 
 but, beyond the general conception of harmony, he obtained no 
 assistance from these vague notions. The ground on which he pro- 
 ceeded was, in fact, the observation of astronomers that the plane- 
 tary revolutions are always slow in proportion to the extent of their 
 orbits. If he had confined himself to this ground, this discovery 
 would certainly not have occupied seventeen years of assiduous toil. 
 At last his labour issued in the discov^ery that the squares of the 
 times of the planetary revolutions are proportional to the cubes of 
 their mean distances from the sun : a law which all subsequent 
 observations have verified. Out. important result of this law is that 
 we may determine the periodic times and mean distances of all the 
 planets by any one. By it, for instance, we have determined the 
 duration of the year of Uranus, when once we knew its distance 
 from the sun : and, conversely, if we discovered a new planet very 
 near the sun, we need only observe its short revolution, to be able 
 to calculate its distance, which, in that position, we could not effect 
 by other means. Asti'onomers are every day using this double 
 facility, afibrded them by Kepler's third law. 
 
 These are the three laws which will for ever constitute the basis of 
 celestial geometry, in regard to planetary motions. They answer 
 for all the bodies in our system, regulating the satellites, by placing 
 the origin of areas and the focus of tiie ellipse in the centre of the 
 respective planets. Since Kepler's time, the number of bodies in 
 our system has more than trebled ; and all have in turn verified 
 these laws. By them, motions of translation require for their 
 
 I
 
 ECLIPSES— THEIR USES. 145 
 
 delerrnination notliing more tlian a simple geometrical problem, 
 vhich clemaiids from direct observation only a certain number 
 of data, — six for each planet. And thus is a perfectly logical 
 character given to astronomy. 
 
 The application of these laws, restricted to oni- own system, is 
 naturally divided into three problems: the problem 
 of the planets ; that of the satellites; and that of the 
 comets. These are tiie three general cases of our system ; and, by 
 the ajiplication to them of Kepler's laws, we niay assign to every 
 body within the system, its ])recise position, in all time past and all 
 lime to come: and thence again, we can exhibit all the secondary 
 phenomena, past and future, which must result from such relative 
 positions. The next striking fact of this kind to the general mind 
 is the prediction of eclipses, absolutely conclusive as Prediction 
 it is, with regard to the accuracy of our geometrical ^/ Eclipses. 
 knowledge. This kind of prediction, quite apart from the vague 
 prophesying of ancient times, when eclipses occiu'red, as tliey do 
 now, necessarily from the planetary orbits being all closed curves, 
 and which men's experience told them must return, — began in the 
 immortal school of Alexandria ; and its degree of precision, to the 
 hour, then to the minute, then to the second, faithfully represents 
 the great historical phases of the gradual perfecting of celestial 
 geometry. It is this which will, apart from all other considerations, 
 for ever make the observation of eclipses a spectacle as interesting 
 for philosophers as for the public, and on grounds which the spread 
 of the positive spirit will render, we may hope, more and more 
 analogous, though unequally energetic. 
 
 We are learning to make more use of this class of ])henoraena, 
 and to make out new uses from them, as time goes on. Independently 
 of their practical utility in regard to the great problem of the 
 longitudes, they have been found, within a century, very important 
 in determining with more exactitude the distance of the sun from 
 our earth. AVhether it be an eclii)se by the moon, or the transit of 
 Venus or Mercury, the difference in duration of the phenomenon, 
 observed in different parts of the earth, will furnish Transit of 
 the relative parallax of that body and the sun, and Venus. 
 consequently the distance of the sun itself. Some bodies are more 
 fit than others for this experiment, certain conditions being neces- 
 sary, which are not common to all. Of the three known bodies 
 which can pass between us and the sun, two — the Moon and Mercury 
 — are excluded by these conditions ; and there remains only Venus. 
 Halley taught us how to conduct and u.se the observation. The 
 parallax, in such a position, offers suitable proportions, being nearly 
 three times that of the sun ; and the angular velocity is smaU 
 enough to allow the phenomenon (lasting from six to eight hour.s) 
 to present differences of at least twenty miimtes between well-chosen 
 observatories. I have specified this ca.-5e, on account of its extreinii 
 
 VuL I. K
 
 146 POSITIVE PinLOSOPHY. 
 
 importance to the whole S3'stem of astronomical science; but it 
 would be quitting our object and plan to notice any other secondary 
 cases. 
 
 I must remark upon one very striking^ truth which becomes ap- 
 parent during the pursuit of astronomical science ; — its distinct and 
 ever-increasing opposition as it attains a higher perfection to the 
 theological and meta{)hysical spirit. Theological philosophy sup- 
 poses every thing to be governed by will ; and that phenomena 
 are therefoi'e eminently variable and irregular, — at least virtually. 
 The Positive philosophy, on the contrary, conceives of them as 
 subjected to invariable laws, which permit us to predict with 
 absolute precision. The radical incompatibility of these two views 
 is nowhere more marked than in regard to the phenomena of the 
 heavens ; since, in that direction, our prevision is proved to be 
 perfect. Tiie punctual arrival of comets and eclipses, with all their 
 train of minute incidents, exactly foretold, long before, by the aid 
 of ascertained laws, must lead the common mind to feel that such 
 events must be free from the control of any will, which could not 
 be will if it was thus subordinated to our astronomical decisions. 
 The three laws of Kepler form tiie foundation of the higher 
 Foundation Conception to which we are next to pass on ; the 
 of Celestial mecliauical theory of astronomical phenomena. By 
 Mechanics. ^|jjg ^ilferior study, we obtain new determinations; 
 but a more important office of the Mechanical theory is to perfect 
 celestial geometry itself, by giving more precision to its theories, and 
 establishing a sublime connection among all the parts of our solar 
 system, without exception. The laws of Kepler, inestimable as 
 they are, have come to be regarded as a sort of approximation, — 
 supposing, as they do, various elements to be constant, while they 
 are subject to more or less alteration. The exact knowledge of the 
 laws of these variations constitutes the principal astronomical result 
 of celestial mechanics, independently of its own high philosophical 
 importance. 
 
 SECTION II. 
 
 DYNAMICAL PHENOMENA. 
 
 Gravitation. 
 
 The laws of Motion, more difficult to discover than those of ex- 
 Charactcr of tension, and later in being discovered, are quite as 
 lavs of Motion, certain, Universal, and positive in character: and of 
 course it is the same with their application. Every curvilinear dis- 
 placement of any kind of body, — of a star as well as a cannon-ball, 
 — may be studied under the two points of view which are equally 
 mathematical : geometrically, in determining by direct observation 
 the form of the trajectoiy and the law by which its velocity varies, 
 
 I
 
 HISTORY OF THE LAWS OF MOTIOiV. 147 
 
 as Kepler did witli llio lioaveidy bodies ; and ineclianically, by 
 seeking (lie law of molioii whicb ])i-eveiits the body fi-oni piirsuinjj; 
 its Tiutural straight course, and which, combined with its actual 
 velocity, makes it describe its trajectory, which may henceforib be 
 known d priori. These inquiries are evidently equally positive, 
 and in like manner Ibunded upon phenomena. If we find still in 
 use some terms which seem to relate to tiie nature and cause of 
 motion, they ai-e only vestiges of a mode of thinking long gone by; 
 and they do not ailect the positive character of the research. 
 
 The two motions whicli constitute the coui'se of tlie cannon-ball 
 are perfectly known to us bel'orehand ; but we bave not the geo- 
 metrical knowledge of its trajectory. With regard to the star, oui- 
 knowledge of its trajectory compensates exactly for the difficulty 
 of our preliminary ignorance about its elementary motions. If the 
 law of the fall of weights had not been directly established, we 
 should have le.irned it, indirectly, but no less surely, from the 
 observation of the curvilinear motions produced by weight. 
 
 Celestial Mechanics was then founded on a firm basis, when 
 through Kepler's laws, and by the rules of rational 
 ynamics, discovery was made or the law of duection 
 and intensity of the foi'ce wliich must act uikju the ])lnnet to divert 
 it from the tangent whicli it woidd naturally describe. This 
 fundamental law once discovered, all astronomical researches enter 
 into the domain of Mechanics, in which the motions of bodies are 
 calculated fiom the forces whicli impel them. This was the course 
 philosophically and perseveringly pursued by Newton. 
 
 It does not detract from Newton's merits that Kepler had some 
 foresight of the results of his great laws. He carried their dynamic 
 interpretation as far as the science of his day permitted ; and, seeking 
 for what could not yet be found, he wandered off among fantasies. 
 The true precursoi-s of Newton, as founders of dynamics, were 
 Huyghens and Galileo, — especially the last : yet history tells of no 
 such succession of philosophical efforts as in the case of Kepler, 
 who, after constituting celestial geometry, strove to pursue that 
 science of celestial mechanics which was, by its nature, reserved for 
 a future generation. As the means were wanting, he failed ; but 
 the example is not the less remarkable. 
 
 The first of Kepler's laws proves that the accelerating force of 
 onch planet is constantly directed towards the sun. The accelerat- 
 ing force, however great it may be supposed, does not at all affect 
 the magnitude of the area which would be described in a given time 
 by the vector radius of the })lanet, in virtue of its velocity, if its 
 direction passes exactly thi'ough the sun, while it would inevitably 
 change it on any other supposition. Thus, the ])ermanence of this 
 area, — the first general datum of observation, — discloses the law of 
 direction. The great difficulty of the problem, gloriously solved by 
 Newton, lies in the discovery, by means of Kepler's other two theorems
 
 148 POSITIVE PHILOSOPHY. 
 
 of the law of the intensity of this action, which we speak of as 
 exercised by the sun on the phmets. 
 
 When Newton began to work on this conception, he toi)k 
 Kepler's third law as his basis, supposing the orbits, as he might do 
 for such a purpose, to be circular and uniform. The solar action, 
 equal, and opposed to the centrifugal force of the planet, thus 
 became necessarily constant at the different points of the orbit, and 
 could not vary but in passing from one planet to another. Tliis 
 variation between one planet and another was provided for by the 
 theorems of Huyghens relating to the centrifugal force in the 
 circle. This force being in proportion to the relation between the 
 radius of the orbit and the square of the periodic time, must vary 
 from one star to another inversely to the square of its distance 
 from the sun, in virtue of the permanence which Kepler showed 
 to exist of the relation between the cube of this distance and 
 this same squai-e of the periodic time, for all the planets. It was 
 this mathematical consideration which put Newton in the way of 
 his great discovery, and not any metaphysical reasonings, such as 
 prevailed before it, and which probably never entered his niind, 
 one way or another. 
 
 There remained the difficulty of explaining how this law of the 
 variation of the solar action agreed with the geometrical nature of 
 the orbits, as exhibited by Ke[)ler. The elliptical orbit presented 
 two remarkable points, — the aphelion and the perihelion, in which 
 the centrifugal force was directly opposed to the action of the sun, 
 and consequently equal to it ; and the change in this action there 
 must be at the same time more marked. The curve of the orbit 
 was evidently identical at these two points; the action then had 
 simply to be measiu^ed, according to Huyghens' theorems, by the 
 square of the corresponding velocity. Thence, it was easily deduced, 
 from Kepler's first law, that the decrease of the solar action, from 
 the perihelion to the aphelion, must be inversely to the square of 
 the distance. Here was a full confirmation of the law which 
 related to the different planets by an exact coujparison between the 
 two principal positions of each of them. Still, however, the elliptical 
 motion had not been considered. Any other curve would, thus far, 
 have served as well as the ellipse, provided its two extremities had 
 shown an equal curvature. The remaining portion of the demonstra- 
 tion, — the measurement of the solar action throughout the extent 
 of the orbit, — is to be obtained only by transcendental analysis. The 
 process is necessary for carrying on the conqjarison of the solar 
 action and tiie centrifugal force ; and the theoi'y of the curvature of 
 the ellipse is required. Huyghens made a near approach to the 
 principle of this great process ; but it could not be completed without 
 the aid of the differential analysis, of which Newton was the 
 inventor, as well as Leibnitz. By the aid of this analysis, the force 
 of the solar action in all [larts of the orbit is easily estimated, in
 
 NF.IVTON'S DEMONSTRATION. 149 
 
 vnvious wnys ; mid it is foiiiKl lo vary inversely to tlie square of the 
 distance, and tliat it is independent of tlie direolion. Ncu-tun's 
 Furlliermore, tlie same method shows, in accordance demonstration. 
 with Kepler's tliird law, that the action varies in proportion lo 
 distance alone ; so that the sun acts upon all the pUmets alike, 
 whatever may be their dimensions, their distance only being the 
 (;iicurastance to be considered. Thus Newton completed his 
 demonstration of the fundamenlfd law that the solar action is, in 
 every case, ]iroportionale, at the same distance, to the mass of the 
 ]ilanet; in the same way that, by the identity of the fall of all ter- 
 restrial bodies in a vacuum, or by the, {precise coincidence of their 
 oscillations, jiroof had already been obtained of the jiroportion 
 between their weight and their masses. We thus see how the 
 three laws of Kejiler have concnned in establishing, according to 
 the rules of rational mechanics, this fundamental law of nature. 
 The fiist shows the tendency of all the planets towards the sun ; 
 the second shows that this tendency, the same in every direction, 
 changes with the distance from the sun, inversely to its square; 
 and the third teaches that this action is always simply pro])orlionate, 
 the distance being equal, to the mass of each planet. In accord- 
 ance with the laws of Kepler, which relate to the whole interior of 
 our system, the same theory applies to the connection between 
 the satellites and their planets. 
 
 Newton thouglit it necessary to complete his demonstration by 
 presenting it in an inverse manner ; that is, by determining a priori 
 the planetaiy motions which must result from such a dynamic law. 
 The process brought him back, as it must do, to Kepler's laws. 
 Besides fuinishing some means of simplifying the study of these 
 motions, this labour ])roved that, whereas, by Kepler's laws, the 
 oibit might have had more figures than one, the ellipse was the only 
 one possible under the Newtonian law. 
 
 It was once a great perplexity to some people, which others could 
 not satisfactorily ex[)lain, that wdien the planet is Old difficulty 
 li'avelling towards its aphelion we cannot say that it explained. 
 tends towaids the sun. But the difHculty arose out of the use of 
 inappropi-iate language. The question is, not whether the planet is 
 iieaier to tlie sun than it lately was, but whether it is nearer than it 
 would have been without the force that sends it forward. It is 
 always tending towards the sun to the utmost that is allowed by the 
 other force to which it is subjected. The orbit is always concave 
 towards the sun ; and it would evidently have been insurmountable 
 if the trajectory could have been convex. In the same w'ay, when a 
 bomb ascends, its weight is not suspended or reversed : it always 
 lends towards the earth, and is, in fact, falling towards it more rapidly 
 (-very moment, even if ascending, because it is every moment fmlher 
 below the point at which it would have been but for the action of the 
 earth upon it; and its trajectory is always concave to the ground.
 
 156 POSITIVE PHILOSOPHY. 
 
 I have tlius far carefully avoided givins^ any naine to ilie tendency 
 
 rr^^^ A+j- n of tlie planets towards the sun, and of the satellites 
 
 Won. inadmU- towards the planets, io call it atlr action \vo\\\ii ho 
 
 ^*^''^- misleading; and we, in truth, can know nothing of 
 
 its nature. All that we know is that these hodies are connected, 
 
 and that their effect upon each other is mathematically calculable. 
 
 It is by quite another j)roperty of Newton's great discovei-y that this 
 
 effect is explained, in the true sense of the word, — that is, compre- 
 
 liended from its conformity with the ordinary phenomena which 
 
 gravity continually produces on the surface of our globe. Let us 
 
 now see what this property of the discovery is. 
 
 "We owe a great deal to the moon. If the earth had no satellite, 
 we might calculate the celestial motions by the rules of dynamics, 
 but we could not connect them with those which are under our 
 immediate observation. It is the moon which affords this connec- 
 tion by enabling us to establish the identity of its tendency towards 
 the earth with weight, properly so called : and from this knowledge, 
 we have risen to the view that the mutual action of the heavenly 
 bodies is nothing else than weight properly generalized ; or, putting 
 it the other way, that weight is only a particular case of the general 
 action. The case of the moon is susce[)tible of the most precise 
 testing. The data are known ; and by dynamical analysis, the 
 intensity of the action of the earth upon the moon is exactly ascer- 
 tainable. We liave only to suppose the moon close to the earth, 
 with the due increase of this intensit3% inversely to the square of the 
 distance, and compare it with the intensity of weight on the earth, 
 as manifest to us by the fall of bodies, or by the pendulum. A 
 coincidence between the two amounts to proof; and we have, in fact, 
 mathematical demonstration of it. It was in pursuing this method 
 of proof that Newton evinced that philosophical severity which we 
 find so interesting in the anecdote of his long delay, because he 
 could not establish the coincidence, while confident that he had dis- 
 covered the fact. He failed for want of an accurate measurement 
 of a degree on the earth's surface ; and he put aside this important 
 part of his great conception till Picard's measurement of the earth 
 enabled him to establish his demonstiation. 
 
 The identity of weight and the moon's tendency towards the 
 Extent of the earth places the whole of celestial mechanics in a 
 demonstration, jjew light. It shows US the motions of the stars as 
 exactly like that of projectiles which we have under our immediate 
 observation. If we could stai't our projectiles with a sufficient and 
 continuous ibrce, we should, except for the resistance of the air, finil 
 them the models of the planetary system: or, in other words, 
 astronomy has become to us an artillery ])roblem, simi)litied by the 
 absence of a resisting medium, but com{)licated by the variety and 
 plurality of weights. — If our observation of weight on our globe has 
 helped us to a knowledge of planetary relations, our celestial obser-
 
 EXTENT OF THE DEMONSTRATION. 151 
 
 vatlons have in turn lauf:jlit us the hiw of the variation of \vei<jjht, 
 imperceptible in terrestrial plienoiuena. Men had always conceived 
 weight to he an inalterahle property of hodies, iindinr:^ that no 
 metanioi-phosis, — not even from life to death, — made any change 
 in the weight of a hody, while it remained entire. This was the 
 one particnlar in whicii men might suppose they had found the 
 Ahsolute. In a moment, the Newtonian demonstration overthrew 
 tiiis fast-rooted notion, and sliowed that weigiit was a relative 
 quality, — not under the circumstances in which it had hitherto heen 
 observed, hut under the new one, — the position of the observed 
 hody in the system, — its distance from the centre of the earth. The 
 human mind could hardly have sought out this fact directly : hut, 
 once revealed in the coiu'se of astronomical study, the verification 
 easily followed ; and experiments on our own globe, in the vertical 
 direction, and yet more in the horizontal, have established the 
 reality of the law, by experiments too delicate, from the necessity 
 of the case, to be appreciable, if we had not known beforehand what 
 differences must be found to exist. 
 
 It is to express briefly the identity between weight and the 
 accelerating force of the planets that the happy term TermGxKwx- 
 Graviiation has been devised. This term has every tationMMo6- 
 merit. It expresses a simple fact, without any refer- J<^ctionabie. 
 ence to the nature or cause of this universal action. It affords the 
 only ex[)lanation which positive science admits ; that is, the connec- 
 tion between certain less known facts and other better known facts. 
 Since the creation of this term, there has been no excuse for the 
 continued use of the word attraclion. It is desirable to avoid ped- 
 antry in language; but it is of high importance to ])reserve pure the 
 })Ositive character of so fundamental a conception as this, by using 
 Ji term which expresses exactly what we know, and dismissing one 
 whicli assumes what is purely fanciful, and wholly incorrect. 
 Attraction is a draicing towards. Now, when we draw anything 
 towards us, the distance is of no importance : tlie same force draws 
 the same body with equal ease three feet or thirty feet, whicli is 
 directly contradictory to the facts of gravitation. Our business is 
 with the fact of the action, and not at all with its nature. It was 
 the use of this metaj)hysical term, it now appears, which occasioned 
 the opposition that the Newtonian theory encountered so long, and 
 especially in France. Descartes had. by laborious efforts, banished 
 the notions of occidt qualities, which he perceived to be so fatal to 
 science ; and in this theory of attraction, his followers saw a falling 
 back into the old metaphysical delusions. We perceive this in the 
 writings of John Bernouilli and Fontenelle : and it appears that the 
 clear and positive scientific intellect of France did good service in 
 stripping off from the sublime discovery of Newton the ujetaphysical 
 appearance which obscured its reality for a time. 
 
 One more consideration remains to be adverted to. AVe have
 
 152 rOSlTIVE PHILOSOPHY. 
 
 regarded t]ie heavenly bodies tliiis far as points, witliont reference to 
 Gravitation ^^1^"" forms and dimensions. But as it is proved tliat 
 is that (if the intensity of the action of the snn on the planets, 
 molecules. j^^j ^^^f |||y pl^nets on their satellites, is proportioned 
 to the mnss of the hody acted upon, it is clear that the force operates 
 directly only on molecules, which ai-e all independently affected by 
 it ; and eqnally, their distance being the same. The gravitatioti 
 of molecules is therefore the only real one; and that of masses 
 is simply its mathematical result. In the mathematical study of 
 motions bowevei- it is necessary to have a conception of a single force, 
 instead of such an infinity of elementary actions : and hence arises 
 that preliminary part of celestial mechanics which consists in com- 
 pounding in one result all the mutual gravitation of the molecule.*^ 
 of two stiu-s. Newton founded this portion, with all the rest ; and 
 the two theorems which he established for the purpose still remain 
 the commonest expression of this important theory. He proved 
 that if the stars were truly spherical, and their strata were homo- 
 geneous, the gravitation of their particles would be so balanced that 
 the bodies might be treated as points, in the study of their motions 
 of translation. But the irregularity of their forms, however slight, 
 must be considered in tlie theory of their rotations, to which these 
 theorems cease to be applicable. For any other form thnn the 
 sphere, the problem becomes \evy complicated ; and the analytical 
 difficulties can be surmounted only by approximation, notwith- 
 standing all the perfections introduced into the theory in recent 
 times. And unless we could also lenrn what is the law of density 
 in the interior of the stars, — a kind of knowledge which seems 
 to be for ever beyond our reach,— we cannot attain a perfect 
 solution. 
 
 The fundamental law of Rational Mechanics, which declares the 
 Secondary ncccssary equality of action and renction, shows that 
 gravitation. gravitation must be mutual, — that the sun must tend 
 towards the planets, and the planets towards their satellites. The 
 extreme inequality of the masses renders the ascei-tainment of the 
 inverse gravitation extremely difficult ; yet its reality is established 
 by various secondary phenomena. The gravitation of the planets 
 towards each otJier is :\ necessnry part of the whole conception ; but 
 it was not mnthematicnily demonstrated till Newton's successors 
 deduced from it an exact explanntion of the pei'turbations ob- 
 served in the principal moticMis of the planets. Their labours 
 have established secondary gravitation as positively as the 
 I)riraary. 
 
 Thus has every kind of proof concurred to establish that great 
 fundamental law which is the noblest result of our aggregate studies 
 of nature. All the molecules of our system gravitate towards each 
 other, in proportion to their masses, and invei'sely to the squares of 
 their distances.
 
 OrRRA TION OF NE WTON 'S DISCO VER Y. 153 
 
 I (Inro not, ns mnny do, confidently exicri(] tlie npplication of this 
 Inw to llic eiiliro universe. There can be no objec- Domain of 
 lion to enl,eit;iinin<j^ it annlo^^ically till we obtain some theiaw. 
 laiowle(l<2;e of the mechanism of the sidereal heavens ; bnt we must 
 remember that we have not yet that knowledge, and that we cannot 
 ])romise oni'selves tliat we ever shall. Without the phenomena of 
 our own system, the theoiy of its motions would be only an intel- 
 lectual exei'cise and sport: there can be no positive science apart 
 from phenomena, and of the phenomena of the universe beyond our 
 own system we are not in scientific possession.* It must be under- 
 stood that I advocate simply a suspension of judgment where there 
 is no ground for either affirmation or denial. I merely desire to 
 keep in view that all our positive knowledge is relative; and, in ray 
 dread of our resting in notions of anything absolute, I would venture 
 to say that I can conceive of such a thing as even our theory of 
 gravitation being hereafter supei'seded. I do not think it probable ; 
 and the fact will ever remain that it answers completely to our 
 present needs. It sustains us, up to the last point of precision that 
 we can attain. If a future generation should reacli a greater, and 
 feel, in consequence, a need to construct a new law of gravitation, it 
 will be as true as it now is that the Newtonian theory is, in the 
 midst of inevitable variations, stable enough to give steadiness and 
 confidence to our understandings. It will appear hereafter how 
 inestimable this theoiy is in the interpretation of the phenomena of 
 the interior of our system. We already see how much Ave owe to 
 it, apart from all si)ecific knowledge whicli it has given ns, in the 
 advancement of our philosophical progress, and of the general edu- 
 cation of human reason. Descartes could not rise to a mechanical 
 conception of general phenomena without occupying himself witb 
 a baseless hypotliesis about their mode of producti(jn. This was, 
 doubtless, a necessary pi'ocess of transition from the old notions of 
 the absolute to the positive view ; but too long a continuance in 
 this stage would have seriously impeded human progress. The 
 Newtonian discovery set us forward in the true positive direction. 
 It retains Descartes' fundamental idea of a Meclianism, but casts 
 aside all inquiry into its origin and mode of production. It shows 
 ])ractically how, without attempting to penetrate into the essence of 
 ]thenomena, we may comiect and assimilate them, so as to attain, 
 with precision and certainty, the ti'ue end of oui' studies, — that 
 exact prevision of events which d 'priori conceptions are necessarily 
 unable to supply. 
 
 * M. Comte omits liere all notice of such positive applications .as we are able to 
 make in Sidereal astronomy. He takes no notice of the fact that the motion of 
 the multiple stars in elliptical orbits, and in accordance with Kepler's law of the 
 velocities, demonstrates the existence of a law of force, according to the inverse 
 Bcjuare of the distance. — J. P. N.
 
 ^54 
 
 CHAPTER IV, 
 
 CELESTIAL STATICS. 
 
 Kepler's k.ws connected celestial plienouieria to a certain degree, 
 Consnynmation before Newton's theory was propounded : but they left 
 by Newton. \\^{q imperfection, — that phenomena which ranked 
 Tinder two of these laws had no necessary connection with each 
 other. Newton brought under one head all the three classes of 
 general facts, uniting them in one moi'C general still ; and since 
 that time we have been able to perceive exactly the relation between 
 any two of the phenomena which are all connected with the common 
 theory. As far as we can see, there is nothing more to gain in this 
 direction. 
 
 We have seen what this great conception is in itself. We have 
 now to observe its a])plication to the mathematical explanation of 
 
 Statical con- celestial phenomena, and the perfecting of their study. 
 
 siderations. j^or this jmrpose, we will recur to our former division 
 of subjects, and contemplate the j)henomena of i)lanets as immov- 
 able first, and of j)lanets in motion afterwards ; the statical 
 phenomena first, and the dynamical afterwards. 
 
 To know the mutual gravitation of the heavenly bodies, we must 
 know their masses. Such knowledge once appeared inaccessible 
 from its very nature ; but the Newtonian theory has put it within 
 our power, and furnished us with a wholly new set of ideas about 
 these bodies. There are three ways in which the inquiry has been 
 prosecuted, all differing from each other, both in generality and in 
 
 First method simplicity. The first method, the most general, the 
 
 of inquiry only onc iu fact which is applicable to all cases, is 
 
 into masses. ^jj^ „^ost difficult. It consists in analyzing the special 
 share of each body in the perturbations obsei'ved in the principal 
 motions of another, — both of translation and rotation. Here two 
 elements are concerned, — the distance, and the mass of the star iii 
 question. The first is well known, the other is not ; and only an 
 approximate determination is possible. It is difficult to apportion 
 the shares in the action ; and geometers place little dependence on 
 the computation of masses obtained by this method, in comparison 
 with that obtained by either of the (;thcrs. 
 
 Nearest in generality to this first method is that which Newton 
 employed with regard to planets that had a satellite ; that of com-
 
 STATICAL INQUIRIES. 155 
 
 pnriiig tlie motion of tlie satellite rouiul the pliiiiet wiih that of tiio 
 
 planet round the sun. The law which determines the „ , ., , 
 ' , . 1 , 1 T i 1 • 1 • • / 1 , Second method. 
 
 Mction by tlie distance being comparetl, in its resulls, 
 
 in the two cases, ijjives the relation of the masses of the sun and the 
 planet. The mass of Jnpiler, delermined by Newton in this way, 
 lias under^'one little chanj^e of statement by methods since em- 
 ployed ; and what difference there is is almost wholly owing to the 
 data of the process l)eing now better known. 
 
 The third method is the most direct and simple of all ; hut it is 
 
 the most restricted, as it is necessarily confined to ^, . , 
 
 ,, , 4 • 1 1 •( 1 1 ji 1 ji. ■ ^ • Tlurd method. 
 
 the planet inhal)ile(l l)y tlie observer. ■ it consists in 
 
 estiinatiiig the relative masses by the com])ari.son of the weiglits 
 which they produce. If we knew the mass of any planet, we should 
 know what would be the weight of things on its surface, or at a 
 given distance; and reciprocally, the weight being known, we are 
 able to estimate the mass. With the pendulum, we have measured 
 terrestrial weight with absolute precision ; and, diminishing it, 
 inversely to the square of the distance, we shall know its value at 
 the distance of the sun. We have then only to com{)ai-e it with 
 the amount, before well known, which expresses the sun's action 
 upon the earth, to find immediately the relation of the mass of the 
 earth to that of the sun. With regard to every other planet, on 
 the contrary, it must be the estimate of its mass which would yield 
 that of its corresponding gravity. All these methods being practi- 
 cable in the case of the earth, its mass, in comparison with that of 
 the sun, must be considered the best known of all within our system. 
 The mass of the moon, and that of Jupiter, are now estimated 
 almost as perfectly ; and those of Saturn and Uranus come next. 
 We are less sure about the other three which have been calculated, 
 — Mercury, Venus, and Mars ; though the uncertainty about them 
 cannot be very great. Of the telescopic planets and the comets wi; 
 know scarcely anything, owing to their extreme smallness, which 
 })recludes their exerting any sensible influence on perturbations. 
 Comets pass, during their prodigious course, near very small stars, 
 such as the satellites of Jupiter and Saturn, without producing any 
 perceptible derangement. As for the satellites, we have no know- 
 ledge except of the moon, and approximately, of those of Jupiter. 
 No comparison of results has as yet exhibited any harmony what- 
 ever between them. The only essential circumstance which they 
 present is the vast superiority of the size of the sun to the whole 
 contents of the system. Those entire contents, if" thrcnvn together, 
 would scarcely amount to a thousandth part of the mass of the sun. 
 Looking abroad from the sun, we see alternating, without any 
 visible order, here decreasing, there increasing masses. We might 
 have supposed, a 2y^'io7'i, as Kepler did, that the masses were 
 regularly connected with the volumes (which are themselves 
 irregular however), so that the mean densities should be contiu-
 
 156 POSITIVE PHILOSOrHY. 
 
 tially less in mafliematical j)voportion to tJieir distances from tlia 
 sun. But, independently of this numerical law, which is never 
 exactly ohserved, the simple fact of the decrease of density presents 
 some exceptions, in reji^ard to Uranus, among others. No rational 
 ground can be assigned for this. 
 
 SECTION I. 
 
 "WKIGHT OF THE EARTH, 
 
 These are the means by which the masses of the bodies of our 
 WeioMof system are ascertained. The remaining process is to 
 the earth. bring them into relation with our estimates of weight, 
 by ascertaining the total weight of the earth. Bouguer was the 
 first who distinctly perceived the jiossibility of such an estimate, 
 during his scientific expedition lo Pei'U, when he found that the 
 neighbourhood of vast mountains slightly affected the direction of 
 -weight. We see how, in accordance with the law of gravitation, a 
 considerable mass, regarded as condensed in its centre of gravity, 
 may affect the plumbline, however slightly, if it be brought close 
 enough, subjecting it to a secondary gravitation, which aftbrds data 
 for a comparison between the action of the earth and that of the 
 mountain. By this, sou)e estimate may be formed of the propor- 
 tion of the mountain to the globe. In the time of Bouguer science 
 was not advanced enough to admit of more than the conception of 
 how the thing could be done. Half a century later, Maskelyne 
 observed the mountain Schehallion in Scotland, and fomid that it 
 occasioned an alteration in the natural direction of weight of from 
 five to six seconds ; and Hutton deduced from this that the weight 
 of the earth is equal to four and a half times that of a similar 
 volume of distilled water at its maxinumi of density. Anything 
 like exactness, however, is out of the question while there nnist be 
 so much uncertainty about the weight of the mountain, which can 
 be calcidated only from its volume. 
 
 Wlien Coulomb had invented his Torsion Balance, intended to 
 measure the smallest forces, Cavendish saw how the earth might be 
 weighed by comparing it, by means of this balance, with artificial 
 masses which might be computed. By his immortal experiments, 
 he discovered the mean density of our globe to be five and a half 
 times equal to that of water; whence we can, if we thiidc proj)er, 
 deduce tiie weight of the earth in cwts. and tons. — We thus obtain, 
 among other advantages, some insight into the constitution of our 
 globe, which by its positivily, puts to flight many fanciful notions. 
 The density of the parts near the surface is so far below the average, 
 — water occupying much spjice, for instance, — that the density 
 nearer the centre must be much above the aveiage. This is iu
 
 FORM OF THE PLANETS. 157 
 
 accordance with the indications of Celestial Mechanics; iuul it, im- 
 nislies us with one comHtion ot" tlie interior of the glohe. Tlierc 
 can be no void there. What there is we know not, further than 
 that it must be something consistent with the condition of superior 
 density. 
 
 SECTION 11. 
 
 FORM OF THE PLANKT3. 
 
 The next great statical inquiry relates lo the foini of the heavenly 
 bodies, as deduced from the theory of their equilibrium, pomx of the 
 Geometers sujipose the phitietary bodies to have been planets. 
 originally fluid, because their equilibrium can thus consist with 
 only one form ; whereas, if they had been always solid, as our earth 
 is now, their equilibrium might have been compatible with any 
 form whatever. Several phenomena indicate this supposition, and 
 it agrees lemarkably with the whole of our direct observations. 
 
 If the planets had no motion of rotation, their being perfectly 
 spherical would accord with the equilibrium of their molecules : 
 but the centrifugal force engeiuk'red by the rot;ition must necessarily 
 modify the primitive foruj, by altering, more or le.-^s, the direction, 
 or the intensity of weight, properly so called, lluyghens established 
 this with regard to the direction, and Newton with niijicuity of 
 regard to the intensity. We thus become easily assured ^''^ inquiri/. 
 of the general fact of the nearly spherical form of all the planetx, and 
 of their being slightly flattened at the poles : but, when we go 
 further, and attem{)t to estimate their forms mathematically, and 
 learn the precise degree of the flattening at the poles, the question 
 becomes one of transcendental analysis, and is involved in difflculty 
 which can never be entirely surmounted. The inquiry involves a 
 sort of vicious circle, which does not admit of a logical issue, in 
 order to form an equation of the surface, we ought, by the law of 
 equilibrium of fluids, to know the weight of the molecules con- 
 cerned ; whereas, by the law of gravitation, this can Geometrkil 
 be ascertained only through the knowledge of the estimate. 
 form of the planet, and even of the mode of variation of its interior 
 density. All that can be done is to discover whether the proposetl 
 form fulfils such and such conditions. Maclaiu'in discovered a 
 theorem, highly valued by geometers, which has become the basis 
 of all our inquiries on this subject, and which shows that the 
 ellipsoid of revolution precisely fulfils- the conditions of equilibrium. 
 But this supposes the strueture of the body lo be houjogeneous ; 
 which it is not, in any case. The labours of geometers have however 
 brought within very narrow limits the possible variations of the 
 polar flattening. The result with regard to the earth is that the 
 mathematical rule perfectly agrees with direct observation.
 
 t;SS POSITIVE PHILOSOPHY, 
 
 In the case of the planets we have another resource. Tlieir 
 Estimate from flattenin<^ atfects certain plienoraena of pertnrhation, 
 ■ perturbations. \^y tlie study of which we ohtain materials for an 
 estimate. Altogether, the calculations and measurements agree 
 more closely than we could have ventured to hope. The only case 
 which seems to present a real exception is that of Mars, which, hy 
 its magnitude, its mass, and the time of its rotation, should he little 
 more flattened than the earth ; whereas, if the ohservations of 
 Herschell are exact, it is almost as much so as Jupiter. — We 
 must observe, moreover, that tliough, as Maclaurin has shown, 
 equilibrium is compatible with the ellipsoid form, this form is not 
 to be supposed the only one : — witness, in our own system, the rings 
 of Saturn, which are a remarkable exani{)le to the contrary: and 
 Laplace has demonstrated how these rings could, even in a fluid 
 state, be in equilibriimi. 
 
 The most useful consequence of the mathematical theory of the 
 Indirect esti- P^ii-Hetary forms is that it has established an important 
 mate of the relation between the value of the different degrees on 
 earths form. ^|jg g,qi-t]|'t5 sui'facc and the intensity of the correspond- 
 ing gravity, measured by the length of the seconds-pendulum in 
 different latitudes. We can thus, with great ease, multiply our 
 indirect observations about the form of our globe ; whereas the 
 geometrical estimate of degrees is a long and laborious operation, 
 which cannot be often repeated with due care. But, generally 
 speaking, the more indirect a measurement is, cceteris paribus, the 
 more uncertain it is : and there remains the uncertainty arising 
 from our ignorance of the law of interior density in our earth ; so 
 that our chief reliance should still be on mathematical measurement, 
 conducted with due care. 
 
 An interesting question belonging to the hydrostatic theory of 
 
 Hiidrosiatic ^''® planetary forms is of the conditions of stability of 
 
 theorij ofp^a- equilibrium of the fluids which are collected on a part 
 
 netary forms, ^j. ^jj^ whole of the suvfacc of the })lanets. Laplace 
 
 shows this stability to depend, under all circumstances, on the 
 
 density of the fluid being less than the mean density of the planet; 
 
 a view established with regard to the earth by Cavendish's fine 
 
 experiment. 
 
 SECTION III. 
 
 THE TIDES. 
 
 There remains the question- of the tides, — the last important in- 
 
 Quesiionof quiry mider the head of celestial statics. Under the 
 
 the tides. aslj-ouomical point of view, this is evidently a statical 
 
 question, — the earth being, in that view, regarded as motionless : 
 
 and it is not less a statical question in a mathematical view, because 
 
 wliat we are looking at is the figure of the ocean during periods
 
 THE TIDES. 
 
 fS9 
 
 ^f cqnilibiinm, witliotit tliiiikiiij^ of the motions wliicli proJuocd 
 that equilibrium. Moreover, this inquiiy natuniliy belongs to the 
 study of the planetary forms. 
 
 A particular interest attaches to this qiiostion, from its being tlio 
 link between celestial and terrestrial physics, — tlie celestial expla- 
 nation of a great terrestrial })henonienon. — Descartes did uuicb for 
 \is in establishing this. He failed to explain tbe phenomenon ; but 
 lie cast aside the metaphysical conceptions whicli bad prevailed 
 before, and showed that there was a connection between the change 
 of the tides and the motions of tbe moon ; and this certainly helped 
 to put Newton in the way of tbe true, theory. As soon as it was 
 known that tbe cause of tbe tides Wiis to be looked for in the sky, 
 the theory of gravitation was certain to afford its true explanation. 
 Newton therefore gave out the simple principle that the unequal 
 gravitation of tbe different parts of the ocean towards any one of tbe 
 bodies of our system, and particularly towards tbe sun and moon, 
 was tbe cause of tbe tides: and Daniel Bernouilli afterwards per- 
 fected tbe theory. Tbe same theory answers for the atmosphere : 
 but we bad better study it in tbe case of the seas alone ; on account 
 of the uncertainty of our knowledge of tbe vast gaseous covering of 
 our globe, wbose diffused mass almost defies precise observation. 
 
 Suppose tbe earth joined to any lieavenly body by a line passing 
 tbrough the earth's centre. It is clear that the point Theory of 
 of tbe earth's surface whicb is nearest tbe other body *'*c '»'^e«- 
 will gravitate towards it more, and tbe remoter point less, tban tbe 
 centre, inversely to the squares of tbeir respective distances. Tbe 
 first point tends away from the centre : and tbe centre tends away 
 from tbe second point ; and in each case tbe fluid surface must 
 rise ; and in nearly the same degree in both cases. The effect must 
 diminisb in proportion to tbe distance from tbese points in any 
 direction : and at a distance of ninety degrees it ceases. But there 
 tbe level of tbe waters must be lowered because of tbe exbaustion in 
 that place caused by tbe overflow elsewliere. And here enters a new 
 consideration, difflcult to manage : — tbe changes in the terrestrial 
 gravity of the waters, occasioned by tbeir changes of level. — Thus 
 tiie action of any beaverdy body causes tbe ocean to assume the 
 form of a spheroid, elongated in the direction of that body. Newton 
 calculated tbe chief part of the phenomenon of tbe tides on tbe 
 supposition of an ellipsoid of homogeneous structure, as he bad done 
 in estimating the effect of the centrifugal force on the earth's 
 figure, substituting for tbe centrifugal force the difference between 
 the gravitation of the centre of the globe and that of its surface 
 next the proposed body. After that, Maclaurin's theorem served 
 Daniel Bernouilli for a basis of an exact theory of tbe tides. 
 
 Thus far, we have regarded tbe tides oidy as if tbey were a fixed 
 accumulation of waters under tbe proposed star. This is the 
 mathematical basis of tbe whole question ; but tbe most striking
 
 i6o POSITIVE PHILOSOPHY. 
 
 part has yet to be considered, — tlie periodical rise and Call. It is 
 the diurnal motion of our globe which causes this rise and fall, by- 
 carrying the waters successively into all the positions in which the 
 other body can raise or depress them. Hence arise the four nearly 
 equal periodical alternations, when the two greatest elevations take 
 place during the two passages of the heavenly body over the 
 meridian of the jilace, and the lower levels at its rising and setting ; 
 the total period being precisely fixed by combining the terrestrial 
 rotation with the pro[)er daily movement of the heavenly body. The 
 last indispensable element of the question is the valuation of the 
 powers of the different heavenly bodies. This calculation is easily 
 made from the difference between the gravitation of the centi'e of 
 our globe and that of the extreme points of its surface next the ob- 
 served body. Guided by the law of gravitation, we can determine 
 which, among all the bodies of our system, are those which can par- 
 ticipate in the phenomenon, and what is the share taken by each. 
 
 Influence of We thus find that the sun by its immense mass, and 
 
 the sun. i\^q moou by its proximity, are the only ones which 
 
 produce any appreciable tides : that the action of the moon is from 
 two and a half to three times more powerful than 
 lemoon. ^-^^^^ ^^^ ^^^ suu ; and that, consequently, when they 
 act in opposite directions, that of the moon prevails ; which explains 
 the primary observation of Descartes about the coincidence of the 
 tidal period with the lunar day. 
 
 Thus far, we have considered only the effect of a single heavenly 
 Composite body upou the tides ; that is, the case of a simi)le and 
 influence. abstract tide. The complication is very great, when 
 the action of two such bodies has to be considered. But the resources 
 of science are sufficient to meet tiiis case, — even deriving from it 
 new means of estimating the mass of tiie suii and moon ; — and also 
 of calculating the modifications arising out of the various distances ot 
 the earth, from either body ; and again, of tracing the changes of 
 direction caused by the diurnal movement of tiie proposed body, — 
 whether in accordance with the earth's axis of rotation, or parallel 
 with the equator, which makes the difference between the tides of 
 our equinoctial and solstitial lunar months. As for the difference 
 of the phenomenon in various climates, the con.sideration of lati- 
 tude is the only one which affords much result. At the poles, there 
 can of course be no other tides than such as are caused by the flux 
 and reflux of waters elsewhere, as the earth has no rotation there. 
 The equiitor must exhibit the tides at their extremes, not only on 
 account of the diminished gravitation there, but yet more on account 
 of the more com{)lete diversity of the successive positions occupied 
 by the waters during the daily rotation. Elsewhere the greatness 
 of the tide must vaiy in proportion to the force of the rotation. 
 
 Tiie mathematical theory of the tides accords with direct obser- 
 vation to a degiee which is really wonderful, considering how many
 
 PROOF OF THE THE ORY. 1 6 1 
 
 hypotheses geometers must liave recourse to, to make the questions 
 calcnhible at all, and how many inaccessible data Requisites for 
 would he required to uiake an estimate thoroughly exactitude. 
 logical. It would not even be enough to know the extent and form 
 of the bed of the ocean. Somelhing beyond that in difficulty is le- 
 quired, — the true law of density, in the interior of the earth, as with 
 regard io the figure of the planets. AVe ought to know too wlielher 
 the interior strata are solid or fluid, in order to know whether they 
 ]>articipate in tidal phenomena, and whether they therefore modify 
 lliose at the surface or not. These considerations show the sound- 
 ness of the advice given by one who was full of the true mathematical 
 spirit, consisting above all in the relation of the concrete to the 
 abstract, Daniel Bernouilli, who recommended geometers " not to 
 urge too far the results of formulas, for fear of drawing conclusions 
 contrary to truth." 
 
 The comparison between mathematical theory and direct observa- 
 tion has never been carried out to any advantage, — all the measure- 
 ments having been taken in the ports, or near the shore. The tides 
 in such ])laces are very indirect ; and they cannot properly represent 
 the regular tides from which they issue, their force being chiefly 
 determined by the form of the soil, — at the bottom as well as on the 
 surface, — and even perhaps affected by its structure. These are 
 incidents which cannot enter into mathematical estimates ; and to 
 them we must doubtless refer the vast differences in the height of 
 the tides at the same time, and in nearly the same place, — as, for 
 instance, the tides of Bristol and Liverpool, of Granville and Diep))e. 
 The only way of making an efTectnal direct observation would be 
 to note the phenomena of the tides in a very small island, at the 
 equator, and thirty degrees at least from any continent, for such a 
 course of years as would allow of repeated record of variations as 
 repeatedly foreseen. In this way, and in no othei-, might the mathe- 
 matical theory of the tides be verified and perfected. 
 
 Whatever may be the uncertainty with regard to some of the 
 data of this great theory, it has that conclusive sanction, — the ful- 
 filment of its previsions ; — a fulfilment so exact as to guide our 
 conduct ; and this, as we know, is the true end of all science. The 
 principal local circumstances, except the winds, being calculable, it 
 lias been found practicable to assign for each port the mean height 
 of the tides and their times ; and thus have mathematical deter- 
 minations been proved to be sufficiently conformable to reality, 
 and a class of phenomena which, a century ago, were regarded as 
 inexplicable, have been referred to invariable law.s, and shown to 
 be as little arbitrary as anything else. 
 
 Such are the philosophical characteristics of the three great 
 questions which coujpose the statical department of Celestial 
 Mechanics. We must next look into the dynamical department, 
 as represented by the phenomena of our system. 
 
 VOL. I. L
 
 l62 
 
 CHAPTER V. 
 
 CELESTIAL DYNAMICS. 
 
 The principal motion of the planets is, as wo have seen, deter- 
 mined by the gravitation of each of them towaids the 
 focus of its orbit. The regularity of this movement 
 must be impaired by the mutual gravitation of the bodies of the 
 system. The most striking of these derangements were observed by 
 the School of Alexandria, in the first days of Mathematical Astron- 
 omy ; others have been observed, in proportion as our knowledge 
 became more precise ; and now all are explained with such com- 
 pleteness by the theory of gravitation, that the smallest perturbations 
 are known before they are observed. This is the last possible test 
 and triumph of the Newtonian system. 
 
 There are, as Lagrange pointed out, two principal kinds of per- 
 turbations which differ as much in their mathematical theory as in 
 the circumstances which constitute them ; instantaneous changes, 
 from shocks or explosions, and gradual changes or perturbations, 
 l)roperly so called, caused by secondary gravitation, requiring time. 
 , ^ The first kind may never have taken ])lace in our 
 
 Instantaneous. , i , •, • , • ^ -i i_ i 
 
 system ; but it is necessary to consider it, not only 
 because it is of possible occurrence, but because it is a necessary 
 [ireliminary to the study of the other kind, — the gradual perturba- 
 tions being treated theoretically as a series of little shocks. 
 
 The first case is easy of treatment. No collision or explosion 
 would affect Kepler's laws: and, if the form of the orbit was 
 altered, the accelerating forces would remain the same ; and thus, 
 the new variation once understood, our calculations might proceed 
 as before. Supposing a collision between two planets, or the 
 breakage of one planet into several fragments by an internal ex- 
 plosion ; there might be any variations whatever in the astronomical 
 elements of their elliptical movement; but there are two relations 
 which are absolutely unalterable, and which might, in my opinion, 
 generally enable us to establish the leality of such an event at any 
 period whatever : these are the essential properties of the continuous 
 motion of the centre of gravity, and the invariableness of the sum 
 of the areas, — both resting on that great law of the equality of 
 action and reaction to which all changes must conform. From 
 these must result two impoitant equations between the masses, the
 
 PLANETARY PERTURBATIONS. 163 
 
 velocities, and the positions of the two bodies, or the two frn2;nionts of 
 the same body, considered before and after the event. No indica- 
 tion at present leads us to suppose that the case of collision has 
 ever occurred in our system ; and it is evident that such an 
 enooiuiter, though not mathematically impossible, would be very 
 (lirticult. But it is far otherwise with regard to explosions. 
 
 The little planets discovered between Mars and Jupiter have mean 
 distances and periodic times so nearly identical, that Mr Olbers 
 lias conjectured that they once formed a single planet, which had 
 exploded into fi-jiginents. Lagi'ange added a supposition, from the 
 irregulnrily of their form, th;it the event must hnve happened after 
 the consolidation of the primitive planet. When their masses 
 become known, I thiidc this conjecture may be subjected to mathe- 
 matical proof, — in this way. By calculating the positions and 
 successive velocities of the centre of gravity of the system of these 
 four planets, we might, if they had sucli an origin, retrace the 
 pi-incipal motion of the ])rimiLive plnnet. If we should then find 
 this centre of gravity describing an ellipse round the sun as a focus 
 and its vector I'ndius tracing arens proportioned to the times, this 
 event would be as completely established as any fact that we have 
 Tiot witnessed. We have not \'et the materials for such a test ; but 
 it is interesting to see how celestial mechanics may establish, in a 
 ])Ositive manner, events like tliese which appear to have left no 
 evidence behind them. It is obvious that the instantaneous 
 character of such a change must preclude our fixing any date for 
 it, since the phenomena would be precisely the same, whether the 
 explosion were recent or long ago. It is otherwise with regard to 
 perturbations, properly so called. 
 
 Lagrange believed that these explosions had been frequent in 
 our system, and that this was the true explanation of comets, judg- 
 ing from the greatness of their eccentricity and inclination, and 
 the smallness of their mas.ses. We have only to conceive that a 
 ]»lanet may have burst into two very unequal fragments, the larger 
 of which would proceed pretty nearly as before, while the smaller 
 must describe a very long ellipse, much inclined to the ecliptic. 
 Lagrange showed that the amount of impulsion necessary for this 
 change is not great ; and that it is less in proportion as the primi- 
 tive planet is remote from the sun. This opinion is far from having 
 been demonstrated ; but it appears to me moi-e satisfactory than 
 any other that has been proposed on the subject of comets. 
 
 The important and difficult subject of perturbations is tiie 
 ])rincipal object of celestial mechanics, for the per- Gradual po-- 
 fecting of astionomical tables. They are of two turbatwng. 
 classes ; the one relating to motions of translation, the other of 
 rotation. The latter are, as before, the most difficult: but the 
 motions of rotation are less altered than the other class, within our 
 own system : and tliey are less important to be known.
 
 i64 POSITIVE PHILOSOPHY. 
 
 In the study of motions of translation, the ])hine(s must be 
 Perturbations treated as if they were condensed in their centres 
 
 of translation, q^ o'ravilv. 
 
 The direst method, the only rational one, of calculatini; 
 the difFerential equations of the motion of any one planet, 
 under the influences of all the rest, is impracLicahle, from the 
 umnanageable complication of the problem. It would make an 
 inextricable analytical enigma. Geometers have therefore been 
 obliged to analyze directly the motion of each planet round that 
 which is its focus, taking for modification only one at a time. Tiiis 
 is what constitutes in general the celebrated problem of three 
 
 Problem of bodics, tliough this denomination was at first 
 
 three bodies, employed only for the theory of the moon. It is 
 easy to see what circumvolutions are involved in this method, since 
 the modifying body, being in its turn modified by others, compels 
 a return to the study of the primitive body, to understand its 
 perturbations. The determination of the motions of the whole of 
 our system must, by its very nature, be a single problem. It is 
 the imperfection of our analysis which obliges us to divide it into 
 detached problems, and to overload our formulas with multiplied 
 modifications. The elementary problem of two bodies, — one of 
 these even being regarded as fixed, — is the only one that we are 
 capable of bringing to a solution ; tlie problem of the elliptical motion, 
 represented by Kepler's laws ; and here the calculations are extremely 
 laborious. It is to this type that geometers have to refer the motions 
 of the planets, by extremely complicated approximations, accumu- 
 lating the perturbations separately produced by every body that 
 can be supposed to exert any influence ; and these perturbations 
 prescribe the series required for the integration of the equations 
 belonging to the case of the three bodies. 
 
 Then follows the task of choosing the perturbations which have 
 to enter into the estimate. The law of gravitation enables us to 
 compare the secondary influences involved in each case, — the masses 
 of all within our own system being supposed to be known. It is a 
 favourable circumstance to mathematical research that our system 
 is constituted of bodies of veiy small mass in comparison with the 
 sun (making the perturbations extremely small) ; moreover, very 
 few, very far from each other, and very unequal in mass; the result 
 of all which is that, in almost every case, the principal motion is 
 modified by only one body. If the contrary had been the case, the 
 perturbations must have been very great, and extremely varied, 
 since a great number of bodies must have powerfully acted in each 
 disturbance. Celestial Mechanics nmst then, we should think, 
 have presented an inextricable complication, being incapable of 
 reduction to the problem of three bodies. 
 
 This study of modified motions divides itself into three parts, 
 answering, as in a former case, to the planets,, satellites and comets.
 
 THE THREE PROBLEMS. 1O5 
 
 Rijjjorously speaklii<jj, wo onj^lit to make a fourlli case of the 8un, 
 which cantiot liero be regarded as niotioidess, because the ])lanets 
 react upon it. In fact, we cannot allow ourselves I0 consider any 
 ])oint within the system as motionless, except the centre of gravity 
 of the system itself, which is the true focus of plane- Centre of the 
 lary motion, and round which the sun ilself must Solar Sii stem. 
 oscillate, in directions which vary according to the positions of the 
 j)lanets. This point is always between the centre and the surface 
 of the sun. But we cannot approach nearer to the fact than this: 
 we shall })robal)ly never be able to indicate this centre precisely ; 
 and it is enough lor practical purposes, and necessary to them, to 
 consider the sun as fixed, excei)t as to its rotary motion. The same 
 conclusion nuist be come to with regard to the planets and their 
 satellites, — even in the case of the earth and moon, where the varia- 
 tions of the primary body are greatest. The centi'e of gravity falling 
 within the mass of the primary body, its variations from that centre 
 may be neglected as having no appreciable influence on the motion 
 of translation ; and thus, celestial mechanics presents, in this branch, 
 no other problems than those treated, under another point of view, 
 by celestial geometiy. 
 
 The simplest problem is here, as before, that of the planets, and 
 for the same reasons, — the smallness of their eccen- Problem of 
 Iricities, and of the inclinations of their orbits. There '^^ Planets. 
 is also a considerable uniformity of perturbations, since each planet 
 remaining in the same regions of the sky, contimies in the same 
 mechanical relations, though their intensity varies within certain 
 limits. The least privileged of these bodies in these matters is 
 uidiappily our own planet, on account of the heavy satellite wdiich 
 escorts it so closely, and to which its chief perturbations are due ; 
 though this does not save it from being sensibly troubled by others, 
 at the period of opposition, and es{)ecially by such a mass as that of 
 Jupiter. No other planet with satellites, not even Jupiter, is in so 
 unfavourable a case ; for Ju[)iter's motion could not be veiy much 
 deranged by the action of his satellites, however near in position, 
 since the mass of the largest is less than a ten-thousandth part of 
 his, while the mass of our moon is a sixty-eighth part of that of the 
 earth. Jupiter's circulation is sensibly afTected by Satui'ii alone. 
 The simplest case of all seems to be that of Uranus, from its being 
 the last planet, and very remote from the next ; and its six satellites 
 do not appear to trouble its motion. 
 
 The problem of the satellites is necessarily more complicated than 
 that of the planets, on account of the instability of ProMem of 
 the focus of the principal motion, as in celestial tite Satellites. 
 geometry. Besides theii- own perturbations, the satellites have 
 lefiected upon them all those to which their planet is liable. The 
 founders of Celestial Mechanics were long {)erplexed, for instance, 
 by the peipetual acceleration of the mean motion of the moon ; it
 
 1 66 POSITIVE FHILOSOrnV. 
 
 was considered inexplicable, till Laplace discovered its cause in tliG 
 slight variation to wliicli the eccentricity of the earth's orbit is sub- 
 ject. In regard to the direct perturbations of the satellites, there is 
 an essential distinction between the case of one, and that of several 
 satellites. In the first, — the single case of our moon, — the disturb- 
 ing body is the sun, on account of its unequal action on the planet 
 and the satellite. If the difficulties arising out of this position are 
 greater than in the case of any other satellite, it is partly because 
 the case more immediately concerns us, and because our oppor- 
 tunities of observation disclose more fully the imperfection of our 
 means. For, in the mathematical point of view, there must be moi-o 
 complexity in the case of several satellites ; all that is true in regard 
 to one being true in regard to each one, with the addition of" the 
 mutual action of the members of the group. Their perturbations 
 are reduced by the preponderating size of their planet ; but from 
 there being so many of them, of such nearly equal sizes and direction, 
 and all so close together, the difficulty of calculating their motions 
 is so great that the only theory as yet established is that of the 
 satellites of Jupiter. For the motions of three of them, Laplace 
 found means completely to account. Those of Saturn and Uranus 
 are known only geometrically, we having not even an approximate 
 estimate of their masses. It is to be remembered, however, that we 
 do not need so perfect a knowledge of them as of the moon ; and 
 that a much less exact theory will suffice for them than for the 
 moon, whose slightest irregularity is very evident to us. 
 
 The comets intervene to increase our difficulties about the satellites. 
 Problem of From the extreme prolongation of their orbits, and 
 the Comets. their inclination in all directions, comets are in a state 
 of ever variable mechanical relations, from tlie number of bodies that 
 they approach in their course; whilst the planets, and even the satel- 
 lites, have always the same relations, the variation being only in the 
 intensity. The perturbation which, in every other case, bears a 
 very small proportion to the gravitation, may, in the case of con^ets, 
 exceed it ; so that it is conceivable that a comet might be diverted 
 from its orbit, and become a satellite, when it passes near so con- 
 siderable a body as Jupiter, Saturn, or even Uranus. Besides the 
 eccentricities of comets, there are other circumstances, such as their 
 small weight, and their possible loss of weight by parting with some 
 of their atmosphere to the bodies they approach, which tend to per- 
 })lex the study of their j)erturbations. These are the incidents 
 which make it so difficult to foresee exactly the return of these little 
 bodies. When we liave studied them so long and so laboriously as 
 to liave, to the best of oiu' belief, mastered their case, we find that 
 their periods are entii^ely changed through one omitted circumstance. 
 A memorable example of this was the comet of 1770, calculated by 
 Lexell. This comet had then a i-evolution of less than six years: 
 but it has never appeared since, having been entirely deranged by
 
 PER TURBA TIONS OF ROTATION: 167 
 
 passing too near Jupiter. The imperfection of our knowledge about 
 these small bodies is from the same cause that renders them of very 
 little consequence to us. From their vast distances, their action 
 upon an}' one body of the system is little more than momentary ; 
 and their lightness prevents even the satellites from being affected 
 by their passage. The passage of the comet of 1770 among the 
 satellites of Jui)iter proved this, in a striking manner. Theii- tables, 
 constructed beforehand, without any idea of such an incident, per- 
 fectly agreed with direct observations ; a proof that the intrusion 
 of the comet did not sensibly affect their motions. There is, there- 
 ibre, no more occasion for the puerile iears of our day than for the 
 i-eligious tei'rors of former times, in regard to the passage of comets. 
 Their collision with the earth is all but impossiide ; and they could 
 not otherwise be felt at all. Their mere approach, however near, 
 could have no other effect than to raise somewhat the corresponding 
 tide. If a comet could pass two or three times nearer to us than 
 the moon (which no known comet could do) its very small mass 
 could produce no other effect than an imperceptible rise of the tides. 
 We have therefore no immediate and practical reason to regret the 
 imperfection of our cometary theories. 
 
 Passing from the perturbations proper to motions Pertarhations 
 of translation, we must notice those belonging to f>f rotation. 
 rotation. 
 
 The ellipsoid bodies of our system must, whether they began or 
 not, have ended, sooner or later, with turning round one of their 
 axes, — and that one the most stable, — that of their smallest dia- 
 meter : for, as we have seen, it is their rotation that has produced 
 their deviation iVom a perfectly splierical form, and determined the 
 direction favourable to stability. The regularity of this rotation is 
 evidently so indispensable to the existence of living 
 bodies on the surface of a planet, that we might ' vanes. 
 d 'priori assert this stability wherever life is possible, from the time 
 when it became possible. But, stable as each planet is in itself, its 
 mutual gravitation with others must introduce certain secondary 
 modifications, the bearing of which must be upon the direction of 
 its axis in space. It is only with regard to the earth that these 
 modifications concern us ; for however great they might be in any 
 other body, they could in no way affect us. 
 
 If the planets were perfect spheres, the total gravitation of their 
 particles must pass through their centres of gravity ; and tlius, it 
 is only through their sligiit fiiilure in sphericity that they can act 
 at all upon one another's rotation ; that failure being caused by the 
 rotation itself. We see here how the same necessity which secures 
 the stability of the rotations, with regard to their duration and their 
 poles, determines, from another point of view, the inevitable altera- 
 tion of the parallelism of their axes. — In our own planet the preces- 
 sion of the equino.Kes, modified by the nutation, results from the
 
 i68 POSITIVE PIIIIOSOPHY. 
 
 action of the otlier bodies of our system, — especially of the snn and 
 moon, — upon our equatorial pi'otuherance. The power of each body 
 is, as in the case of the tides, in the direct ratio of its mass, and 
 inversely to the cube of its distance ; so that the sun and moon are 
 the only bodies whose influence need be considered. Further, the 
 extent of the deviation depends on the mass and magnitude of the 
 earth, on the time of its rotation, on its degree of flattening, and on 
 the obliquity of the ecliptic. Tlie intensity of the influence must 
 vary, as in the case of the tides, with the variable distance of the 
 sun from the earth, and yet more of the moon ; but the want of 
 uniformity is too slight to be pei-ceptible to direct observation. — 
 These are the general causes which deteimine the small changes 
 which the rotation of our globe undergoes, in regard to the direction 
 of its axis in space. — The case of the other ])lanets bears a general 
 likeness to that of the earth, varied according to the different inclina- 
 tions of their axes to their orbits, their position, their mass, their 
 size, the duration of their rotation, and the degree of their flattening 
 at the poles. On all these grounds, the perturbations of Mars are 
 the most remarkable. 
 
 The rotation of the satellites presents one consideration of the 
 highest interest, — that remarkable equality between 
 the duration of this rotation and that of their circuit 
 round their planet, by which they present always the same hemi- 
 sphere, except from those veiy small oscillations called librations, 
 whose law is well understood. The fact is absolutely certain only 
 with regard to the moon ; but our mechanical principles justify our 
 erecting it into a general law of all the satellites. Lagrange has 
 shown that it results from the preponderance that, by the action of 
 the planet, the nearer hemisphere must acquire at the outset, whence 
 arises a natural tendency in the satellite to return perpetually to 
 the same position. If it is thus with the moon, there is every 
 rciison to suppose the same fact with regard to satellites belonging 
 to heavier planets, to which they ai'e proportionally nearer. 
 
 Such are the various kinds of perturbations produced in the 
 
 movements of the bodies of our system, by their mutual action. 
 
 This study may be simplified and rendered much more exact, by 
 
 the device of referring all these movements to a plane whose posi- 
 
 Devicevfan ^'*^" must ncccssarily be. independent of all their 
 
 invariable Variations. — Among several planes which have been 
 
 plane. proposed, diftering in their degrees of variableness, 
 
 M. Poinsot has discovered one which is the only truly invariable 
 
 one, but which is extremely difficult to determine, since it requires 
 
 not only an estimate of the planetary masses, but data dependent 
 
 on the mathematical law of the interior density of the heavenly 
 
 bodies, — a law which is still very hypothetical. The theory is 
 
 com})lete; but its precise ap])licati()n is at present impossible. 
 
 Whatever may be the practical difficulties, we cannot but feel a deep
 
 STABILITY OF THE SOLAR SYSTEM. 169 
 
 interest in seoing how Celestial Meclianics lias nccomplisliod the 
 iixin;j^ of an invariablo plane in the midst of all the interior per- 
 turbations of our system, as Newton h;ul first recoj^nized an inalter- 
 able velocity, — that of the centre of f^eneral gravity. These are the 
 only two elements in our system which are rigorously independent of 
 all the events that can occui- in its interior ; — of even the vastest 
 commotions that onr imagination can suggest. Such variations as 
 they can be conceived to have could relate only to the most general 
 ])henomena of the universe, produced by the mutual action of 
 different suns, of which they would afford us the clearest manifes- 
 tation, if such knowledge were within our reach. 
 
 We end this study of perturbations with a recognition of the 
 stability of our own system, in regard to all its most stahuity of 
 important constituent bodies. Setting aside the our si/stem. 
 comets, all the variations whatever of any perceptible value are 
 periodical ; and their period is usually very long, while their extent 
 is very small ; so that the whole of our planetary system can only 
 oscillate with extreme slowness round a mean state, from which it 
 deviates very little. Through all starry changes the translations of 
 our planets present the almost rigorous invariableness of the great 
 axes of their elliptical orbits, and of the duration of their sidereal 
 revolutions : and their rotation shows a regidarity even more perfect, 
 in its duration, in its poles, and even, though in a somewhat smaller 
 degree, in the inclination of its axis to the corresponding orbit. We 
 know, for instance, that from the time of Hipparchus, the length 
 of the day has not varied the hundredth part of a second. Amidst 
 all this general regularity, we perceive a special and most marked 
 stability with regard to the elements which are concerned in the 
 continued existence of living beings. — Such are the sublime theorems 
 of natural philosophy for which humanity is indebted to the sum 
 of the great works executed in the last century by the successors of 
 Newton. 
 
 Tiie general cause of these important results lies in the small 
 eccentricity of all the principal orbits, and the small divei-gence of 
 their planes. If the planets had had cometary orbits and planes, 
 there would have been no regularity — no periodicity, — and, we may 
 add, no life upon their surface. No })lanets can be habitable but 
 such as have their oscillations restricted within very narrow limits. 
 
 The Mathematical theory of celestial mechanics has taken no 
 notice, thus far, of the resistance of any general Resistance of 
 medium, in which these motions are proceeding, a Medium. 
 The conformity of our mathematical tables with observed facts 
 shows that the resistance is imperceptible in degree ; yet, as it is 
 manifestly impossible that it should be null, the geometers have 
 endeavoured to prepare befoiehand a genei-al analysis of it. Con- 
 sidered apart fi'om its intensity, this action is of a totally different 
 nature from that of peiluibations, though gradual like them: for
 
 I70 POSITIVE FJIILOSOPHY. 
 
 it cannot be periodical, and ninst always be exercised in the same 
 direction, so as continually to diminish all velocities, and the more 
 the greater they are. It cannot alter the positions of the orbits, 
 but can by possibility affect only tlieir dimensions, and periodic 
 times, and the duration of rotations : that is, it aflfects the elements 
 which are spared by the perturbations. Thus, the rotations must 
 become slower, the orbits must grow smaller and rounder, and their 
 periodic times shorter ; because, as velocity diminishes, the solar 
 action must become more powerful, and these effects are not only 
 continuous, but always increasing in rapidity. So, in a future too 
 remote to be assigned, all the bodies of our system must be united 
 to the solar mass, from which it is probable that they proceeded : 
 and thus the stability of the system is simply in relation to the 
 perturbations properly so called. These are among the incontest- 
 able indications of Celestial Mechanics. 
 
 As yet, we practically fail to recognize the effect of a resisting 
 medium. We neither trace its operations, nor should know how to 
 calculate it if we could trace it. Whenever we do, it will be by 
 the study of comets ; for their small mass, and the great surface 
 which they present to the action of the medium when their atmos- 
 pheres are widely diffused, must render its resistance much more 
 appreciable than in the case of planets, — their velocity being besides 
 naturally at its maximum at the moment of this expansion. Some 
 contemporary astronomers believe that they have established the 
 effect of this resistance in regard to one or two comets. Hitherto 
 the study of these bodies seems to be only negatively useful, to 
 prevent the return of the absurd terrors which they formerly occa- 
 sioned. We now see that there is no body in our system, however 
 insignificant, whose theory may not offer to us a direct and positive 
 intei'est, since we may owe to comets the knowledge of one of the 
 most important general laws of the system to which we belong, and 
 that which, in a remote future, must chiefly rule its destinies.* 
 In our geometrical review we saw, by the agreement of astro- 
 independence nomical tables with direct observation, that our system 
 of the solar is independent of all that lies outside. This incon- 
 sijstem. testable truth is confirmed by the mechanical view. 
 
 If our system gravitated towards any of the suns outside, the action 
 of other suns would nearly neutralize the tendency. Again, it 
 would be only by an unequal action of those suns upon our planets 
 that any change could be occasioned. Again, the vast distances 
 would, according to om* law of gravitation, make the action of 
 remote suns imperceptible. The nearest body, if a million times 
 heavier than our system, would produce an effect incalculably 
 smaller than the action which occasions our tides. We may there- 
 fore pronounce the independence of our system to be perfectly 
 
 * M. Comte estimates too lightly the indications of a medium i:iven by Encke's 
 comet. — J. r. N.
 
 INDEPENDENCE OE THE SOLAR SYSTEM. i-ji 
 
 certain. I notice this becanse we seem to tind liere tlie only excep- 
 tion to tlie <;reat encyclopedical law which is the basis of this 
 woik, — that the most j^eneral phenomena rule the most particular, 
 without being in any degree reciprocally influenced. Thus our 
 astronomical phenomena regulate those of our own globe, — whether 
 physical, chemical, physiological, or social. Yet here we find thiit 
 the phenomena of the universe have no influence over those of 
 the solar system. There is no difficulty about this to persons who, 
 like myself, admit that oui- researches are limited by the boundaries 
 of otn- own system, and that positive knowledge cannot go beyond 
 it. The study of the universe forms no part of natural pliilosopliy : 
 a truth which will become more apparent, and be seen to be mure 
 im})oi-tant the further our studies extend. 
 
 At the close of this brief review of celestial dynamics, we see that, 
 great as are the achievements since Newton's time, we are reminded 
 in many directions of the imperfection which results from the 
 insufficiency of our mathematical analysis. In the execution of 
 astronomical tables it has to borrow from celestial geometry other 
 aid than the estimate of indispensable data, derived from direct 
 observation; and this in regard not only to bodies whose mechanical 
 theory is but just initiated, but with regard to some with which we 
 are best acquainted. 
 
 We see however that besides the sublime direct knowledge 
 afforded to us, celestial dynamics have powerfully con- Achievements 
 ti'ibuted to perfect the whole body of astronomical of Celestial 
 theories in regard to theii- definitive aim, — the exact Dynamics. 
 ])revision of tlie state of the heavens at any period whatever, pastor 
 futiu'e. Kei)ler's laws might suffice to determine tlie state of our 
 system for a short time, ])roper data being chosen ; but if we wish 
 to extend the inquiry, back or forwards, to any considerable period, 
 we find the most perfect theory of ])erturbations absolutely necessary. 
 It is to celestial dynamics that we owe our power of ranging up and 
 down the centuries, to fix the precise moments of various celestial 
 ])henomena, such as eclipses, with certainty, and with a minuteness 
 only inferior to that which is possible in the case of present events. 
 
 Though we have, according to my view, completed our considera- 
 tion of astronomical science, it would be felt to be a great omission 
 if we passed over altogether what is now called Sidereal Asti-onomy. 
 We will therefore see how much there is that we can conceive to be 
 positive in regard to cosmogony.
 
 1]2 
 
 CFIAPTER VI. 
 
 SIDEREAL ASTUONOMY AND COSMOGONY. 
 
 The only bnincli of Sidereal Astronomy wliich appears to admit of 
 _ exact study is that of the relative motions of the 
 ip e ais. ]y[^||^Jp|Q gtars, first discovered by Herschell. By 
 multiple stars astronomers understand stai-s very near each other, 
 whose angular distance never exceeds a half minute, and which, for 
 this reason, appear to be one, not oidy to the naked eye, but to ordi- 
 nary telescopes, only the most powerful lenses being able to separate 
 tliem. The relative movements of these stars tend to deceive us 
 as to their precise multiple character, as, for instance, by mutual 
 occultations, which do not permit us to separate them. Among 
 some thousands of midti])le stars registered in the catalogues, before 
 the southern heavens had been really explored, almost all were only 
 double, and we have found none which are more than triple, — a 
 circumstance which may be owing solely to the imperfection of our 
 telescopes, as we knew of none but single stars before Herschell's 
 time. However interesting the study of them is, Ihey constitute 
 only a particular case in the universe, as the intervals of the stars 
 which compose them are probably much smaller than those which 
 divide the stms of the universe, so that the study of their relative 
 motions does not lead us up to any of the great general phenomena 
 of the heavens, and the speciality would be more conspicuous if as- 
 tronomers did what I think they ought, — form their catalogues of 
 those double stars only whose motions they have fully established. 
 With regard to others, we cannot be sure whether their duality is a 
 I'eal relation or an accident. Knowing nothing whatevei' of their 
 interval, or of the distance of either of them from us, we cannot be 
 sure whether they form a system any more than any other two stars 
 combined by chance in the heavens. Because a few incontestable 
 examples are before us of a binary system, in which the smaller 
 circulates round the larger, it is anything but philosophical to con- 
 clude the same to be the case with the whole uudtitude of double 
 stars, some of which nuiy appear so merely through an accident of 
 position, apparent only to our own system. Analogy is not a})pli- 
 cablc here ; as what looks like analogy is merely the imperfection of 
 our investigation.s. No astronomer would venture to assert that if 
 our telescopes were what they may one day become, we might not
 
 MULTIPLE STARS. 173 
 
 fiiul bchveeii slnrs now appnroiilly iiKlf'jXMidcnt a muUlludc of 
 clustered interinediate stars which shoukl rciuler tlie case of duality 
 ahnost general. The apparent nearness would not then be a sufll- 
 cient ground for ])iesuniing then- mutual revolutions, because it is 
 in virtue of their very small numbei- that analogy now suggests that 
 presmnption. The only ])ositive study in sidereal astronomy is that 
 ot the known relative motions of certain double stars, at present not 
 more than seven or eight in number. We could never hope to 
 assign with accuracy their oi-bits, or their periodic times, or any 
 solid basis for dynamical conclusions.*. The importance of such 
 inquiries is much diminished by the consideration that our system, 
 which, in such a case, means our sun, belongs to no groups of the 
 kind,— either investigated or merely pointed out. This circumstance 
 seems to me not at all accidental ; for, if our system made a part of 
 a double star, which it is not difficult to imagine, it would probably 
 be impossible for us ever to be aware of the other part of such a 
 duality, because in the direction of the sun it would be so near that 
 its light would be lost to us in that of our sun. Such a case might, 
 however, have a scientific interest for us, not only as elucidating 
 the displacements of our system, but as allowing such great precision 
 as might arise from the position of the inqm'rer on one of the stars 
 of the coujjle. 
 
 The first of the few orbits of double stars known to us was 
 investigated by Savary. They all present a veiy considerable 
 eccentricity, the smallest of which is double, and the greatest four 
 times greater than that of the most eccentric in our system. Of 
 their periodic times, the shortest slightly exceeds forty years, and 
 the longest six hundred. We cannot perceive that the eccentricity 
 and the duration bear any fixed relation to each other, and neither 
 seems to depend at all on the angular distance of tlie respective 
 ])airs of stars. This is the sum of what we know about the 
 double stars; and unless we could learn something of their 
 linear distance from our system and from each other, our con- 
 ceptions can neither be accurate nor of great imi)ortance. M. 
 Savary has })roposed a method, foimded on the known velocity 
 of light, by which these distances will, if ever, be estimated: f but 
 the uncertainty of some of the elements which must enter into the 
 question is so great that the most that can be hoped for is the fixing 
 of certain limits within which the real distance may be sui){)osed to 
 lie : and this is all that M. Savary himself proposed. At present, 
 we know only the nearer limit, beyond which, not only the double 
 stai's, but the whole starry host, are known to lie. 
 
 * M. Comte quite underrates the importance of the phenomena of the multiple 
 stars. The orbits of a very considerable number are now distinctly ascertained, and 
 the laws of motion in their orbits. The existence of a motion of revolution is fi.\ed, 
 with regard to the far greater number. — J. P. N. 
 
 t They are calculated ; but by strictly geometrical methods.
 
 174 rOSITIVE PHILOSOPHY. 
 
 Proceeding now to ascertain wliaf. we may rationally conceive of 
 Our Cos- our own cosmogony, I need hardly say that we must 
 mof/onii. p^t aside altogether any notion of creation, as 
 unintelligible, — all that we are able to conceive of being successive 
 transformations in the sky ; and of these, oidy such as have pro- 
 duced its present state. Here, again, we find our own system to be 
 1he only subject of knowledge. We are in possession of some facts 
 in regard to it which may bear testimony to its inimediate oi'igin ; 
 but we can form no rensonable conjectures about the formation of 
 the suns themselves. The phenomena necessary for such a pm-pose 
 ju'e not only not explored but not explorable. Whatever may be 
 tlie interest of Herschell's curious observations on the progressive 
 condensation of the nebula3, they do not warrant his conclusion of 
 tlieir transformation into stars;* for from such a conclusion must 
 flow consequences about form and motion which must be in har- 
 mony with established phenomena; and of these we have absolutely 
 none. 
 
 The beginning of positive cosmogony was when geometers, pur- 
 Oriqin of suing the mathematical theory of the figures of the 
 Positive planets, showed that they were originally in a state of 
 
 Cosmogony, fluijity. We cnnnot go furtlier back than this : and 
 we must set out with an existing sun, turning on its axis with an 
 indeterminate velocity, admitting, for the formation of the planetary 
 system, no agencies which we do not now see at work, in the pheno- 
 mena which we habitually witness, though they may have wrought 
 formerly on a larger scale. These restrictions are indispensable to 
 the scientific character of the inquiry; and, after all, our cosmogonic 
 theories, however guaixled, must remain essentially conjectural, if 
 ever so plausible. No mathematical principles can enter here as 
 into celestial mechanics, leading us up to a definite theory and 
 excluding every other. No abstract theory of formations is possible ; 
 and the utmost we can do is to collect such information as can be 
 had, construct hypotheses from it, and compnre them carefully and 
 continuously with the whole of the phenomena that we explore. 
 Such hypotheses, whatever degree of consistency they may attain, 
 can never, like the law of gravitation, take rank among general 
 facts : for we can never be sure that some other hypothesis may not 
 turn up which would equally well answer the present purpose, and 
 some others besides. 
 
 The cosmogony of Laplace seems to me to ])resent the most 
 
 Cosmogony plausiblc theoi-y of any yet proposed. It has the 
 
 of Laplace. eminent merit of requiring, for the formation of our 
 
 system, only the simple agents, weight and heat, which meet us 
 
 everywhere, and which are the only two principles of action which 
 
 * This portion of Herscbell's speculation must be abandoned. What he fancied 
 to be instances of nebulous matter turn out to be galaxies, or vast groups of stars.^ 
 J. P. N.
 
 COSMOGONY OF LAFLACE. 175 
 
 are absolutely general. The point in wliicli I differ from La[)lace 
 is with regard to comets, which he regai'ds as strangers in our sys- 
 tem ; whereas Lagrange's view of them, before cited, appears to be 
 preferable, as being consistent with the independence of our solar 
 group. 
 
 The hypothesis of Laplace tends to explain the general circum- 
 stances of our system, viz., the common direction of all the planets 
 from west to east ; timt of their rotations ; and tliat of all the 
 satellites : also, the small eccentricity of all the orbits ; and finally, 
 the small inclination of their planes, especially in comparison with 
 that of the solar equator. 
 
 It is supposed by this tlieory that the solar atmosphere was 
 originally extended to the limits of our system, in virtue of its 
 extreme heat; that it was successively contracted by cooling ; and 
 that the planets were formed by this condensation. Tlie theory 
 rests on two mathematical considerations. The first involves the 
 necessary relation between the successive expansions or contractions 
 of any body whatever and the duration of its rotation; by which the 
 rotation should be quickened as the dimensions lessen and becomes 
 slower as they increase, so that the angular and linear variations 
 sustained by the sum of the areas become exactly compensated. 
 The other consideration relates to the connection between the angu- 
 lar velocity of the sun's rotation and the possible extension of its 
 atmosphere, the mathematical limit of wiiich is at the distance at 
 which the centrifugal force, due to this rotation, becomes equal to 
 the corresponding gravity: so that if any portion of the atmosphere 
 should be outside of this limit, it would cease to belong to the sun, 
 though it must continue to revolve with the velocity it had at the 
 moment of separation. From that moment, it ceases to be involved 
 in any further consequences from the cooling of the solar atmos- 
 phere. It is evident, from this, how the solar atmosphere nnist 
 have diminished, as to its mathematical limit, without intermission, 
 in regard to the parts situated at the solar equator, as the cooling 
 was for ever accelerating the rotation. Portions of the atmosphere, 
 thus parted with, must form gaseous zones, situated just beyond the 
 respective limits; and this constituted the first condition of our 
 planets. By the same process the satellites were formed out of the 
 atmospheres of their respective planets. Once detached from the 
 sun, our planets must become first liquid, and then solid, in the 
 course of their own cooling, without being further affected by solar 
 changes : but the irregularity of tiie cooling, and the unequal den- 
 sity of parts of the same body must change, in almost every case, 
 the primitive annular form, which remains in the rings of Saturn 
 alone. In most cases, the whole gaseous zone has gatheied, in the 
 way of absorption, round the preponderating portion of the zone as 
 a nucleus : thence the body assumed its spheroidal form, with a 
 revolving motion in the same direction as its movement of transla-
 
 176 POSITIVE riilLOSOPIlY. 
 
 lion, on account of the excess of the velocity of the upper molecules 
 in comparison with that of the lower. 
 
 This theory answers to all the appearances of our system, and 
 explains the difficulty of the primitive impulsion of the planets. It 
 shows, also, that the formation of the system has been successive, 
 the remotest planets being the most ancient, and the satellites the 
 most modern.* 
 
 If from points of view like these the stability of our system can 
 scarcely be regarded as absolute, what it may lead us to suspect is 
 that, by the continuous resistance of the general medium, our system 
 must at length be re-united to the solar mass from which it came 
 forth, till a new dilatation of this mass shall occur in the immensity 
 of a future time, and organize in the same way a new system, to 
 follow an analogous career. All these prodigious alternations of 
 destruction and renewal must take place without affecting the most 
 general phenomena, occasioned by the mutual action of the suns ; 
 so that these revolutions of our system, too vast to be more than 
 barely conceived of by our minds, can be only secondary, even local 
 events, in relation to really universal transformations. It is not less 
 remarkable that the natural history of our system should be, in its 
 turn, as certainly independent of the most prodigious changes that 
 the rest of the universe can undergo : so that whole systems are, 
 perhaps frequently, developed or condensed in other regions of space, 
 without our attention being in any way drawn towards these 
 immense events. 
 
 The end I had in view in this exposition of astronomical philos- 
 . ophy will be attained if I have clearly exhibited, in 
 regard both to method and to doctrine, the true 
 general chaincter of this admirable science, which is the immediate 
 foundation of the whole of Natural Philosophy. We have seen the 
 human mind, by means of geometrical and mechanical researches, 
 and with the help of constantly improving mathematical aids, 
 attaining to a precision of logical excellence superior to any that 
 other branches of knowledge admit of. We see the various pheno- 
 mena of our system numerically estimated, as the different aspects 
 of the same general fact, rigorously defined, and continually repro- 
 duced before our eyes in the commonest terrestrial phenomena; so 
 that the great end of all our positive studies, the exact prevision of 
 events, has been attained as completely as could be desired, in regard 
 alike to the certainty and extent of the prevision. We have seen 
 how this science must operate in liberating the human intellect for 
 
 * The author subjoins a jiruposed tnatlieinatical wrification of Laplace's cosmogoin-, 
 wliicb is not given in the text, as it does not seem to rest on adequate foundations. 
 If an arithmetical verification be ever obtained, it will probably be in connection with 
 the periods of the rotations of tlie different planets ; — periods already in so far con- 
 nected with the nebular hypothesis by the investigatious of an American inquirer — 
 Mr Kirkwood. — J. V. N.
 
 CONCLUSION. 177 
 
 ever from all tlieologic.il and metaphysical thraldom by sho\vin<; 
 that the most gi^eneral phenomena are snbjected to invariable rela- 
 tions, and that the order of the heavens is necessary and spontaneous. 
 This last consideration belongs more particularly to a subsequent 
 part of this work ; but it has been our business to point out as we 
 went aloni; how the development of astronomical science has shown 
 us that the universe is not destined for the passive satisfaction of 
 Man ; but that Man, superior in intelligence to whatever else he 
 sees, can modify for his good, within certain deteiininate limits, the 
 system of phenomena of which he forms a part, — being enabled 
 to do this by a wise exercise of his activity, disengaged from all 
 oppressive terror, and directed by an accurate knowledge of natural 
 laws. Lastly, we have seen that the field of positive philosophy lies 
 wholly wilhin the limits of our solar system, the study of the uni- 
 verse being inaccessible in any positive sense.* 
 
 • As before remarked, M. Comte speaks much too absolutely here, iu oversight of 
 what modern astronomical researches have really accomplished. — J. P. N. 
 
 VOL. I. M
 
 178 
 
 BOOK IIT. 
 PHYSICS. 
 
 CHAPTEE I. 
 
 GENERAL VIEW. 
 
 Astronomy was a positive science, in its geometrical aspect, from tlie 
 Imperfect earliest days of the School of Alexandria ; but Physics, 
 condition of which WO are now to consider, had no positive char- 
 the science. acter at all till Galileo made his great discoveries on 
 the fall of heavy bodies. We shall find the state of Physics far less 
 satisfactory than that of Astronomy, not only on account of the 
 greater complexity of its phenomena, but under its speculative aspect, 
 from its theories being less pure and systematized, and, under its 
 ])ractical aspect, from its previsions being less extended and exact. 
 The precepts of B.icorj and the conceptions of Descartes have advanced 
 it considerably in the last two centuries, in its character of a positive 
 science ; but the empire of the primitive metaphysical habits is not to 
 be at once overthrown ; and Physics could not be immediately imbued 
 with the positive spirit, which Astronomy itself, our only completely 
 positive science, did not assume in its mechanical aspect till the 
 middle of that period. The further we go among the sciences, the 
 more we shall find of the old unscientific spirit, and not only in their 
 details, but impairing their fundamental conceptions. If we now 
 compare the philosophy of Physics with the perfect model offered to 
 us by astronomical Philosophy, I hope we shall perceive the posi- 
 bility of giving to it, and afterwards to the other sciences in their 
 turn, the same positivity as the first, though their phenomena are far 
 from admitting of an equal ])erfection of simplicity and generality. 
 
 First we must see what is the domain of Physics, properly so 
 called. 
 
 Taken together with Ciiemistry (for the present), the object of the 
 
 two is the knowledge of the general laws of the In- 
 
 omam. o,-g.|nic woi'ld. This study has marked characters, 
 
 to be analysed hereafter, distinguishing it from the science of Life, 
 
 Avhich follows it in our encyclopedic scale, as well as from that of
 
 PBYSICS AND CHEMISTRY. 170 
 
 astronomy wliicli })roce(los it. The distinction between Physics 
 
 and Chemistry is ninch less easy to establish ; and it 
 
 is one more difficult to pronounce upon from day to c/!aa^strt ^^'^^ 
 
 day, as new discoveries l)rin<;^ to liiijht closer relations 
 
 between them. Though the divisions between these sciences is less 
 
 obvious tiian between any other two in the scale, it is not the less 
 
 real and indispensable, ns we shall see by three considei'ations which, 
 
 })ei-haps, might be insufficient apart, but wliich, when united, leave no 
 
 uncertainty. 
 
 First, the generality which characterises physical researclies con- 
 trasts with the speciality inherent in the chemical. 
 Every physical consideration is applicable to all bodies * ff<^nera i y. 
 whatever, while chemistry studies the action appropriate to a par- 
 ticular substance. It' we look attlieir classes of phenomena, we find 
 that gravity manifests itself in the same way in all bodies ; and tUe 
 same with phenomena of heat, of sound, of light, and even electrical 
 effects. The difference is only in degree. But, in the compositions 
 and decompositions of Cliemistry, we have to deal with specific jn-o- 
 j)erties, which vary not only in elementary substances, but in their 
 most analogous combinations. The only exception which can be 
 alleged, in the whole domain of Piiysics, is that of magnetic pheno- 
 mena ; but modern researches tend to prove that they are a mere 
 modification of electrical phenomena, w-hich are unquestionably 
 general. The general pi-operties of Physics were, in the metaphysical 
 days of the science, regarded as consisting of two classes ; those which 
 were necessarily, and those which were contingently universal. But 
 the false distinction arose from the notion of that age, that the busi- 
 ness of science was to inquire into the nature of bodies, — the study 
 of their properties being a mere secondary affair. Now that we know 
 our business to be with the properties alone, we see the error, and 
 need only ask whether we can conceive of any body absolutely devoid 
 of weight, or of temperature. 
 
 In the second place. Physics relates to masses, and Chemistry to 
 molecules; insomuch that chemistry was formerly ncaiina loith 
 called Molecular Physics. But, real as this distinction vmsscs or mole- 
 Is we must not carry it too far, but I'emember that *^"'^** 
 purely physical action is often as molecularaschemical action; as in the 
 case of gravity. Physical phenomena observed in masses are usually 
 only the sensible results of those which are going on among their ])ar- 
 ticles: and, at most, we can expect from this only phenoruena of sound, 
 and perhaps of electricity. As for the necessity of a certain mass, to 
 Tuanifest piiysical action, that is equally indispensable in chemistry. 
 Tlie best way of expressing the general fact which lies at the bottom 
 of this distinction is, perhaps, that in chemistry, one at least of the 
 bodies concerned must be in a state of extreme division ; while this is 
 so far from being a necessary condition of physical action that it is 
 rather an impediment to it. This is a proof of a I'eal distinction 
 between the two sciences, though it may not be a very nuu'ked one.
 
 i8o ♦ POSITIVE PHILOSOPHY. 
 
 In the third place, the constitution of bodies, — ilie arrangement 
 „, . of their molecules, — may be changed, in exhibitin<!: 
 
 xJildTlOCS of CIV' ' */ <r> ' o 
 
 raiu/ement or physical phenomena ; but the composition of their 
 composition moleculcs remains unchangeable : whereas, in Chem- 
 istry, not only is there always a change of state in one 
 of the bodies concerned, but the mutual action of the bodies alters 
 their nature ; and it is this alteration which constitutes the pheno- 
 menon. Many physical agents can, no doubt, work changes of com- 
 position and decomposition, if their operation be very energetic and 
 prolonged ; and it is this which forms such connection as there is 
 between the two sciences : but, at that point of activity, physical 
 agencies pass the boundary, and become chemical. 
 
 Positive philosophy requires that we should draw off altogether 
 from the study of agents, to which it may be imagined that pheno- 
 mena are to be referred. Any number of persons may discover a 
 supposed agent, — as, for instance, the universal ether of modern 
 philosopliers, by wdiich a variety of phenomena may be supposed to 
 be explained ; and we may not be able to disprove such an agency. 
 But we have no more to do with modes of operation than with the 
 nature of the bodies acted upon. We are concerned with phenomena 
 alone; and what we have to ascertain is their laws. In departing 
 from this rule, we leave behind us all the certainty and consistency 
 of leal vscience. 
 
 Keeping within our true limits, then, we see that if chemical 
 phenomena should be rednced by analysis into the form of purely 
 physical actions, — an achievement very possible to the present 
 generation of scientific men, — our fundamental distinction between 
 the two sciences will not be shaken. It will still be true that in a 
 chemical fact something more is involved than in a simply physical 
 one : namely, tlie characteristic alteration undergone by the mole- 
 cular composition of the bodies, and therefore by the whole of their 
 })roperties. Such a distinction is secure amidst any scientific revolu- 
 tion that can ever happen. 
 
 From these three considerations, taken together, we derive our 
 Description description of Physics. Tiiis science consists in study- 
 of Physics. Jng the laws which regulate the general properties 
 of bodies, commonly regarded in the mass, and always placed in cir- 
 cumstances wiiich admit of their molecules remaining unaltered, anil 
 generally in their state of aggregation. With a view to the great entl 
 of all science, we nmst add that the aim of physical theories is to 
 foresee, as exactly as possible, all the ])henomena that will be exhibited 
 by a body placed inanyset of given circumstances, excluding, of course, 
 such as could alter its nature. This is not the less true because we 
 can rarely attain the prescribed aim. The imperfection is in our 
 knowledge alone. In estimating the true character of any science, 
 the only way is, first, to suppose the science perfect, and then to 
 study the fundamental difficulties presented by this ideal perfection.
 
 METHODS OF PHYSICAL INQUIRY. ,8i 
 
 OiirdoRcriplion shows ns how iinicli more complexity we shnll find 
 in })liysical than in astronomical inquiries. In astronomy we study 
 bodies, known to us only by sight, imder two aspects 
 only, their I'orms and motions. All considerations but '""'cm .■?. 
 
 these are excluded. But in Physics, on the contrary, the bodies we 
 have to study are recognized by all our senses, and are regarded 
 imder an aggregate of general conditions, and therefore amidst u 
 complication of i-elations. It is clear, not only that this science is 
 inferior to asti'onomy, but that it would be impracticable if the group 
 of fundamental obstacles was not compensated for, up to a certain 
 ])oint, by the extension of oiu* means of exploration. We meet here 
 the law, before laid down, that in proportion as phenomena become 
 complicated they thereby become exploi-able under a propiortionate 
 variety of rehitions. 
 
 Of the three procedures which constitute our ai't of observing, the 
 last, Comparison, is scarcely more applicable here than Methods of 
 with regard to astronomicsd phenomena. Its proper inquiry. 
 application is, in fact, to the phenomena of organized bodies, as we 
 shall see hei'eafter. But the other two methods are entirely suitable to 
 Physics. Observation was, in astronomy, restricted to the use of a 
 single sense; but in Physics, all our senses find occupa- . 
 
 lion. Yet would Observation effect little without 
 the aid of Ex{)eriment, the regulated use of which is the great 
 icsource of physicists in all questions that involve any complexity. 
 This procedure consists in observing beyond the range 
 
 . ... J^JCt)€Tt'l)i€tlt 
 
 of natural circumstances; — in placingbodiesin artificial 
 conditions, expressly instituted to enable us to examine the action of 
 the phenomena we wish to study under a particular point of view. 
 We can see at once how eminently this art is adapted to physical 
 researches ; and how it must there find its trinmj)hs : since there are 
 hardly any bounds to our jjower of modifying bodies, for the purpose 
 of studying their phenomena. In chemistry, experiment is commonly 
 supposed to be more complete than in any other depai'tment : but I 
 think it is of a highei- order in physics, for the reason that in chemis- 
 try the circumstances are always artificially arranged, while in physics 
 we have the choice of natural or artificial circumstances ; and the 
 philosophical character of experimentation consists in choosing the 
 freest possible case that will show us what we want. We have a 
 wider range, and a choice of simpler cases, in physics than in chem- 
 istry ; and in physics, therefore, is experiment supreme. 
 
 The next great virtue of ])hysics is its allowing the application of 
 mathematical analysis, which enters into this science. Application of 
 and at present goes no further ; — not yet, with real mathematical 
 ( fficacy, into chemistry. It is less perfect in physics <^naiysis. 
 than in astronomy ; but there is still enough of simplicity and fixed- 
 ness in physical phenomena to allow of its extended use. Its employ- 
 ment ma v be dii-ect or indirect : — direct when we can seize the funda-
 
 1 82 POSITIVE PHILOSOrilY. 
 
 mental miniGiical law of plienomena, so as to malce it the basis of a 
 series of analytical deductions; as when Fourier founded his tlieory of 
 the distiibution of heat on the principle of thermological action be- 
 tween two bodies being proportionate to tlie difference of their tem- 
 peratures : and indirect, when the phenomena have been referred 
 to some geometrical or mechanical laws; when, however, it is not 
 pi-operly to physics that the analysis is applied, but to geometry 
 or mechanics. Such are the cases of reflection or refraction in the 
 geometrical i-elations ; and in the mechanical, the investigation of 
 weiglit, or of a pai't of acoustics. In either case extreme care is 
 i-equisite in the first application, and the further development should 
 1)0 vigilantly regulated by the spirit of physical research. The 
 domain of physics is no proper field for mathematical pastimes. 
 The best security would be in giving a geometrical training to phys- 
 icists, who need not then have recourse to mathematicians, whose 
 tendency it is to despise experimental science. By this method will 
 that union between the abstract and the concrete be eff'ected which 
 will perfect the uses of mathematical, while extending the positive 
 value of physical science. Meantime, the uses of analysis in physics 
 are clear enough. Without it we should have no precision, and no 
 co-ordinntion ; and what account could we give of our study of heat, 
 weight, light, etc. ? We should have merely series of unconnected 
 facts, in which we could foresee nothing but by constant recourse 
 to experiment ; whereas, they now have a character of rationality 
 which fits them for purposes of prevision. From the complexity of 
 physical phenomena, however, the difficulty of the mathematical 
 application is great. In some, a part of the essential conditions of 
 the problem nuist be thrown out, to admit of its transformation into 
 a mathematical question ; and hence the necessity for reserve in the 
 enployment of analysis. The art of combining analysis and experi- 
 ment, without subordinating the one to the other, is still almost 
 unknown. It constitutes the last advance of the true method of 
 physical study; and it will be developed when physicists, and not 
 geometers, conduct the analytical process, and not till then. 
 
 Having seen what is the object of Physics, and what the means 
 
 Encvdopedi- ^^ investigation, we have next to fix its position in 
 
 cai Rank of the Scientific hierarchy. 
 
 Physics. rpj^^ phenomena of Physics are more complicated 
 
 than those of Astronomy; and Astronomy is the scientific basis 
 and model of Physics, which cannot be effectually studied otherwise 
 than through the study of the n)ore simple and general science. In 
 
 Relation to this, we individually follow the coui'se of our race. 
 
 Astronomy. Jt was by Astronouiy that the positive spirit was 
 introduced into natural philosophy, after it had been sufiiciently 
 developed by purely mathematical investigations. Our individual 
 education is in analogy with this: tor we have learned from astron- 
 on)y what is the real meaning of the cociolanation of a phenomenon,
 
 RELATIONS TO OTHER SCIENCES. 183 
 
 without any impracticable inquiry about its cause, first or final, 
 or its mode of proilucliDii. IMiysics should, more than the other 
 natural sciences, follow closely upon astronomy, because, after 
 astronomy, its phenomena are less complex than any. 
 
 Besides these reasons belonging to Method, there is the grand 
 consideration that tiie theories of astronomy afford the only data 
 for the study of terrestrial physics. Our position in the solar 
 system, and the motions, form, size, and equilibrium of the mass of 
 our world among the other planets, must be known before we can 
 understand the })heuomena going on at its surface. What couKl 
 we make of weight, for instance, or of the tides, without the data 
 afforded by astronomical science ? These })henomena indeed make 
 the transition from astronomy to physics almost in- Relation lu 
 sensible. In this way Physics is indirectly connected Mathematics. 
 with ]\Iathematics. There is also a direct connection, as some 
 ])hysical phenomena have a geometrical and mechanical character, — 
 as much as those of astronomy, though under a great complication 
 of the circumstances. Tiie abstract laws of space and motion must 
 })revail as much in the one science as in the other. If tlie relation 
 is thus unquestionable in the doctrine, it is not less so in the spirit 
 and method which we must bring to the study of physics. It must 
 be ever remembered that the true positive spirit first came forth from 
 the pure sources of mathematical science ; and it is only the mind 
 that has imbibed it there, and which has been face to face with the 
 lucid truths of geometry and mechanics, that can bring into full 
 action its natural positivity, and apply it in bringing the most 
 complex studies into the reality of demonstration. No other disci- 
 pline can fitly j)repare the intellectual organ. We might further 
 say that, as geometrical ideas are more clear and fundamental than 
 mechanical ideas, the former are more necessary, in an educational 
 sense, to physicists than the latter, though the use of mechanical 
 ideas is the more immediate and extended in physical science. 
 Thus we see how we must conclude that the education of physicists 
 must be more complicated than that of astronomers. Both have 
 need of the same mathematical basis, and physicists must also 
 have studied astronom}', at least in a general way. And this, 
 again, assigns the position of their science. 
 
 Its rank is equally clear, it" we look in the opposite direction, — 
 at the sciences which come after it. 
 
 It can hardly be by accident that, in all languages of thinking 
 peoples, the word which originally indicated the Relation to 
 study of the whole of natiu'e sliould have become the the rest. 
 name of the particular science we are now considering. Astronomy 
 is, in fact, an emanation from mathematics. Every other natural 
 science was once comprehended under the term Physics ; and that 
 to which it is now restricted must be supreme over the rest. Its 
 relation to the rest is just this : that it investigates the general
 
 1 84 POSITIVE PHILOSOPHY. 
 
 ])roperties common to all bodies ; that is, the fundamental constitu- 
 tion of matter ; while the other sciences exhibit the modifications 
 of those pi'operties peculiar to each : and the study of those proper- 
 ties in the general must, of course, precede that of their particular 
 cases. In regard to Physiology, for instance, it is clear that 
 organized bodies are subject to the general laws of matter, those 
 laws being modified in their manifestations by the characteristic 
 circumstances of the state of Life. The same is the case with 
 Chemistry. Without admitting the questionable hypothesis under 
 which some eminent men of our time refer all chemical phenomena 
 to purely physical action, it is yet evident that the concurrence of 
 physical influences is indispensable to every chemical act. What 
 could we make of any phenomenon of composition or decomposition, 
 if we left out all data of weight, heat, electricity, etc.? And how 
 could we estimate the chemical power of these various agents with- 
 out first knowing the laws of the general influence proper to each? 
 Chemistry is closely dependent on Physics ; while Physics is wholly 
 independent of Chemistry. 
 
 As for the direct operation of this science on the human intellect, 
 p , ,- ,^ it is less marked than that of the two natural sciences 
 
 Relation to . . ,. , , 
 
 human pro- which occupy the extremities oi the scale, — astronomy 
 yress. rj^^^j physiology, — which immediately contemplate 
 
 the two great objects of human interest, — the Universe and Man : 
 but one striking fact with regard to Physics is that it has been the 
 great battle-ground between the old theological and metaphysical 
 spirit and the positive philosophy. In Astronomy the positive 
 philosophy took possession, and triumphed almost without opposi- 
 tion, except about the earth's motion ; while in the domain of 
 Physics the conflict has gone on for centuries ; a circumstance 
 attributable to the imperfection of physical, in comparison with 
 astronomical science. 
 
 With this science begins the exhibition of human power in 
 Humanpoicer modifying phenomena. In Astronomy, human inter- 
 of modifyiiKj vcution was out of the question : in Physics it begins ; 
 phenomena. ^^^^ ^g ^\va\\ scc how it bccomes more powerful as we 
 descend the scale. This power counterbalances that of exact pre- 
 vision which we have in astronomy, through its extreme simplicity. 
 The one power or the other, — the power of foreseeing or of modi- 
 fying, — is necessary to our out-growth of theological philosophy. 
 Our prevision disproves the notion that ])henomena proceed from a 
 supernatural will, which is the same thing as calling them variable : 
 and our ability to modify them shows that the powers under which 
 they proceed are subordinated to our own. The first is the higher 
 order of proof ; but both are complete in their way, and certain 
 to command, sooner or later, universal assent. The proof which 
 FiankUn afibrded of human control over the lighlning destroyed 
 the religious terror of thunder as eflectually as the superstition
 
 HUMAN INTER I 'ENTION BEGINS 185 
 
 al)Oiit comets was destroyed by the prevision of their return ; tlioii<r]i 
 ihe experiments by which Frniddin established tlie identity of the 
 li<i;litnin,i5 wiih tlie common electric dischar<;e conld be decisive only 
 with physicists, while the j^enerality of men coidd understand liow 
 the return of comets was foi-eseen. As the opposition between tlie 
 theological and the positive philosophies becomes less simply 
 evident, our power of intervention becomes more varied and ex- 
 tended ; the amount of proof yielded in the two cases being equal 
 in the eyes of men in general, though not strictly equivalent. 
 
 In regard to the speculative rank of Physics, it is clear that it 
 does not admit of prevision to any extent at all com- Prevision 
 parable to that of astronomy, because it consists of imperfect. 
 numerous branches, scarcely at all connected with each other, and 
 concurring only in a feeble and doubtful way in its chief phenomena. 
 We can therefore see only a little way forward ; often scarcely 
 beyond the experiment in hand : but we shall see its speculative 
 superiority to the sciences wlu'ch come after it, when, in studying 
 chemistry and j)hysiology, we find another kind of incoherence 
 existing among their phenomena, making prevision more imperfect 
 still. The great distinction of Physics is that which has been 
 referred to before, — that it instructs us in the art of Experiment. 
 Philosophers must ascend to this source of experimentation, what- 
 ever their special objects may be, to learn what are the spirit and 
 conditions «)f true exi)erimentation, and what the necessary pi-e- 
 cautions. — Each of the sciences in the scale presents, besides the 
 characters of the positive method which are common to them all, 
 some indication appropriate to itself, which ought to ckaracteris- 
 be studied at its source, to be duly appreciated. Uvsofeach 
 IMathematical science exhibits the elementary con- *'^'«"^^- 
 dilions of positivity : astronomy deteimines the true study of Nature : 
 ])hysics teaches us the theory of experimentation : chemistry offers 
 us the art of nomenclature : and physiology discloses the true 
 theory of classification. 
 
 I have deferred till now what I have to offer on the important 
 subject of the rational construction and scientific use of Jnjpotheses, 
 regarded as a powerful and indispensable auxiliary rhUosophy of 
 to our study of nature. It is in the i-egion of astrono- hijpothe&is. 
 my that I uuist take my stand in discussing this subject, though it 
 was not necessary to advert to it while we were surveying that region. 
 Hypothesis is abundantly emjiloyed in asti'onomy; but there it may 
 be said to prescribe the conditions of its own use, — so simple are 
 the phenomena in question there. From thence do I think it 
 necessary to derive, theretbie, our conceptions of the character and 
 rules of this valuable resource, in order to its employment in the 
 olhei- departments of natural ]»hiloso])liy. 
 
 There are only two genoral methods by which we can get at the 
 law of any phenomenon, — the immediate analysis of the course of
 
 i86 POSITIVE PHILOSOFHY. 
 
 tlie phenomenon, or its relation to some more extended law already 
 established ; in other words, by induction or deduction. Neither of 
 these methods would help us, even in regard to the simplest pheno- 
 mena, if we did not begin by anticipating the results, by making a 
 provisional supposition, altogether conjectui'al in the first instance, 
 with regard to some of the very notions which are the object of the 
 inquiry. Hence the necessary introduction of hypotheses into 
 natural philosophy. The method of approximation employed by 
 geometers first suggested the idea ; ami without it all discovery of 
 natural laws would be impossible in cases of any degree of com- 
 plexity ; and in all, very slow. But the employment of this instru- 
 ment must always be subjected to one condition, the neglect of 
 which would impede the development of real knowledge. This 
 Necessary couditiou is to imagine such hypotheses only as admit, 
 coyidition. \,y their nature, of a positive and inevitable verifica- 
 tion at some future time, — the precision of this verification being 
 proportioned to what we can learn of the corresponding phenomena. 
 In other words, philosophical hypotheses must always have the 
 character of simple anticipations of what we might know at once, 
 by experiment and reasoning, if the circumstances of tiie problem 
 had been more favourable than they are. Provided this rule be 
 scrupulously observed, hypotheses may evidently be employed 
 without danger, as often as they are needed, or rationally desired. 
 It is only substituting an indirect for a direct investigation, when 
 the latter is impossible or too difficult. But if the two are not 
 employed on the same general subject, and if we tiy to reach by 
 hypothesis what is inaccessible to observation and reasoning, the 
 fundamental condition is violated, and hypothesis, wandering out of 
 the field of science, merely leads us astray. Our study of nature is 
 restricted to the analysis of phenomena in order to discover their 
 laws ; that is, their constant relations of succession or similitude; 
 and can have nothing to do with their nature, or their cause, first 
 or final, or the mode of their production. Every hypothesis which 
 strays beyond the domain of the positive can merely occasion 
 interminable discussions, by pretending to pronounce on questions 
 wliich our understandings are incomjietent to decide. — Every man 
 of science admits this lule, in its simple statement ; but it cannot 
 be practically understood, — so often as it is violated, and to such 
 a degree as to alter the whole character of Physics. The use of 
 conjecture is to fill up provisionally the intervals left here and there 
 by reality; but practically we find the two materials entirely 
 separated, and the real subordinated to the conjectural. It is 
 necessary, therefore, to ascertain and explain the actual state of the 
 question with regaid to Physics. 
 
 Two classes of The hypotheses employed by ])hysical inquirers in 
 
 hypothesis. qx^v (lay are of two classes : the first, a very small class, 
 
 relate simi)ly to the laws of phenomena : the other, and larger class,
 
 ABUSE OF IlYrOTHESlS. 187 
 
 nim at (letcrmliiing the general aireiits to wliicli dlfTerent kinds of 
 iiiitural effects may be referred. Now, according to tiie rule just 
 laid down, the first kind alone are admissible : the second have an 
 anti-scientific character, are chimerical, and can do nothing but 
 hinder the progress of science. 
 
 In Astronomy, the first class only is in use, because the science 
 has a wholly positive character. K fact is obscure; First class 
 or a law is utdaiown : weproceed to form a hypothesis, indispensable. 
 in agreement, as far as possible, with the whole of the data we are 
 in possession of; and the science, thus left free to develop itself, 
 always ends by disclosing new observable consequences, tending to 
 confirm or invalidate, indisputably, the primitive supposition. We 
 have before noticed frequent and happy examples of this method of 
 discovering the primary truths of astronomy. But, since the estab- 
 lishment of the law of gravitation, geometers and astronomers have 
 put away all their fancies of chimerical fluids causing planetary 
 motions ; or have, at least, indulged in them merely as a matter of 
 personal taste and not of scientific investigation. It would be well 
 if, in a study so much more difficult as Physics, phil- second class 
 osophers would imitate the astronomers. It would chimericaL 
 be well if they would confine their hypotheses to the yet unknown 
 circumstances of phenomena, or their yet hidden laws, and would 
 entirely let alone their mode of jtroduction, which is altogether beyond 
 the limit of our faculties. Wliat scientific use can there be in 
 fantastic notions about fluids and imaginary ethers, which are to 
 account for phenomena of heat, light, electricity and magnetism? 
 Such a mixtm-e of facts and dreams can only vitiate the essential 
 ideas of physics, cause endless controversy, and involve the science 
 itself in the disgust which the wise must feel at such proceedings. 
 These fluids are supposed to be invisible, intangible, even imponder- 
 able, and to be inseparable from the substances which they actuate ; 
 Their very definition shows thera to have no place in real science, 
 for the question of their existence is not a subject for judgment : it 
 can no more be denied than affirmed : our reason has no grasp of 
 them at all. Those who, in our day, believe in caloric, in a lumin- 
 ous ether, or electric fluids, have no right to despise the elementary 
 spirits of Paracelsus, or to refuse to admit angels and genii. We 
 find them spurning Lamarck's notion of a resonant fluid ; but the 
 misfortune of this hypothesis was that it came too late, — long after 
 the establishment of acoustics. If it had been put forth as early in 
 the days of science as the hypotheses about heat, light, and electricity, 
 this resonant fluid would no doubt have prospered as well as therest. 
 Without going into the history of more of these baseless inquiries, 
 it is enough to point out that they are irreconcilable with each other ; 
 and when sui)erficial minds witness the ease with wliich they destroy 
 each other, they naturally conclude the whole science to be arbitrary, 
 consisting more in futile discussion than in anything else. Each sect
 
 1 88 POSITIVE PIIILOSOPIIY. 
 
 or pliilosoplier can show liow untenable is the hypothesis of anotlier, 
 hut cannot establish his own ; and it would <2;enerally be easy to devise 
 a third which might agree with botli. It is true, physicists are 
 now eager to declaie that they do not attribute any intrinsic reality 
 to these hypotheses ; and that they countenance thera merely as indis- 
 ])ensable means for facilitating the cnncejition and combination of 
 phenomena. But we see here the working of an incomplete positivity, 
 which feels the inanity of such systems, and yet dares not surrender 
 them. But besides that it is scarcely ])ossible to employ a fictitious 
 instrument as a reality without at times falling into tiie delusion of 
 its reality, what rational ground is there for })roceeding in such a 
 way, when we have befoie us the procedure and achievements of 
 astronomical science, for a pattern and a promise ? These hypotheses 
 explain nothing. For instance, the expansion of bodies by heat is 
 not explained, — that is, cleared up, — by the notion of an imaginary 
 iiuid interposed between the molecules, which tends constantly to 
 enlarge their intervals ; for we still have to learn how this supposed 
 fluid came by its spontaneous elasticit}', which is, if anything, more 
 unintelligible than the primitive fact. And so on, through the 
 whole range. These hy[)otheses clear away no difficulties, but only 
 make new ones, while they divert our attention from the true object 
 of our inquiries. As for the plea that habit has so taken hold of the 
 minds of inquirers, that they would be adrift if deprived of all their 
 moorings at once, and that their language must be superseded by 
 a wholly new one, I think this kind of difficulty is very much exag- 
 gerated. We have seen, within half a centmy, how often men have 
 contrived to pass from some physical systems to their opposites 
 without being much hindered by obstacles of language. There would 
 be scarcely more difficulty in casting aside futile hypotheses ; and 
 we might, as we see by existing examples, gradually substitute the 
 real and permanent meaning of scientific terras for the fanciful and 
 variable interpretation. 
 
 These fluids are nothing more than the old entities materialized. 
 Histonj of the Whichever way we look at it, what is heat apart from 
 second class. f\^Q warm body, light apart from the hmiinous body, 
 — electricity apart from the electric body? Are they not pure 
 entities, like thought apart from the thinking body, and digestion 
 apart from the digesting body ? Here we have, instead of abstract 
 beings, imaginary fluids deprived, by their very definition, of all 
 material qualities, so that we cannot even suppose in them the limit 
 of the most larefied gas. If the descent of these from the old 
 entities be not recognized, what filiation of ideas can ever be 
 admitted ? Tiie essential character of metaphysical conceptions 
 is to attrihute to properties an existence separate from the sub- 
 stance which manifests them. What does it matter whether we 
 call these abstractions souls or fluids? The origin is always the 
 Barae; and it is connected with that inquisition into the essence of
 
 APPLICA TION IN ASTR ONOM V. 1 89 
 
 tilings which always characterizes (he infancy of the hinnun mind, 
 occnsioning, first, the conception of gods, wliich grew into that of 
 souls, which hecanie in time imaginary fluids. In all positive 
 science, our understandings, unable to pass abrnptly fi-om the meta- 
 physical to the positive stage, have travelled throngh this transition 
 state of development. Melai)hysics itself is the transition stage 
 from theology to positive science ; but a secondary transilion is also 
 necessary, as we see by the fact; a transition from metaphysical to 
 positive conceptions. The mathematicians and astronomers have 
 attained the positive basis. The physicists, the chemists, the 
 physiologists, and the social philosophers, are now in the last period 
 of transition ; the physical inquirers, ready to pass uj) to the level 
 of the astronomers and geometers, and all the others held back foi- 
 a while by the complexity of their respective subjects ; as we shall 
 see hereafter. This bastai-d positivism was the way out of ihe old 
 metaphysical condition, in which men would, but for it, have beeu 
 im{)risoned to this day. Nascent science first humoured the con- 
 stitutional need, and then led us on by offering to our minds, in the 
 place of the old scholastic entities, new entities, more tangible, which 
 must by their natni'e introduce into our studies the contemplation 
 of phenomena and their laws, restricting us to these more and more. 
 This seems to have been the important temporary use of this system 
 of hypotheses ; to enable us to pass from the meta^jhysical to the 
 positive stage. 
 
 Astronomy has not been exempted froiu this transition state, any 
 more than the other sciences ; but it was over so long 
 ago that it is forgotten, — so few are those who are in- " * rommy. 
 terested in the history of philosophy ! If we look back to the action 
 of the human mind in the seventeenth century, we shall see how 
 geometers and astronomers were preoccupied with hypotheses of 
 the kind we are considering. There is no better example of them 
 than that famous conce[)tion, the Vortices of Descartes ; for it pi-e- 
 sents clearly the three stages of existence connnon to them all ; th(! 
 creation of the hypothesis, its temporary use, and its rejection when 
 its purpose is answered. These vortices, so ridiculed by men who 
 believe in caloric, ethei-, and electric fluids, helped us to a sound 
 philosophy by introducing the idea of mechanism, where even Kepler 
 had imagined only the incom})rehensible action of souls and 
 genii. When the discussion had attained the firm ground of 
 Celestial Mechanics, founded upon the Newtonian theory, the in- 
 fluence of the Cartesian hypothesis ceased to be progressive, and 
 became retrograde. To the last, the Cartesian philosophers insisted, 
 in arguments as plausible as those of our existing ])hysicists, that 
 it was impossible to philosophize without such a hypothesis. They 
 were answered in tlie only efiectual way, by philosophizing in an- 
 other mode : and the vortices were heard of no more when geometers 
 and astronomers apprehended the true object of scientific studies.
 
 iQo POSITIVE PHILOSOPHY. 
 
 The Cartesian hypothesis contributed to the education of the human 
 mind b}^ leading it to see that we have nothing to do with the 
 primitive agents, or mode of production of phenomena, but only 
 with their laws. If, in the other sciences, we have, as their pro- 
 fessors assert, reached the stage of positivity, hypotheses like that 
 of the vortices may be dismissed, as no longer needed to bring us 
 out of the metaphysical state. As soon as they are needless, they 
 become pernicious. 
 
 The transition has been obvious elsewhei'e than in astronomy. 
 It has taken place in the most advanced departments 
 lysics. ^j. pii^.gjgg. ^^^^ especially with regard to Weight. 
 There was scarcely a philosopher, even long after Galileo's time, 
 who had not some system to offer about the causes of the fall of 
 bodies. At tiiat time, such hypotheses appeai-ed the only method of 
 studying weight; but who hears of them now? Acoustics was 
 emancipated about the same time. The labours of Fourier will 
 evidently release thermology; and then there will remain only the 
 study of light and of electricity ;* and no reason can be assigned 
 for tiieir exclusion from the general rule. The question will be 
 regarded as settled henceforward by all who believe that the 
 historical dev^elopment of the human mind is subject to natural 
 laws, determinate and uniform ; and such will admit, as the principle 
 of the true theory of hypotheses, that every scientific hypothesis, to 
 be a matter of judgment, must relate exclusively to the laws of 
 phenomena, and never to their mode of production. 
 
 Here we find, as in every analogous case of difficulty, the use of 
 the comparative historical method which I have just employed. 
 "We shall enlai-ge on this hereafter: meantime, I nnist offer the 
 observation that the philosophy of the sciences cannot be pi'operly 
 studied apart from their history ; and, conversely, that the history 
 apart from the pliiloso])hy, would be idle and unintelligible. 
 
 In reviewing the different departments of Physics, I shall follow 
 
 p,,;. ^/„^ the rule whicli determines the order of the sciences 
 
 raiifiement themselves I that IS, 1 shall take them m the order 
 
 %n Physics. q^' ^|^^J generality of their phenomena, their simplicity, 
 
 the relative pei-fection of our knowledge of them, and their mutual 
 
 dependence. Under this rule, we shall find that the departments 
 
 that offer themselves first border upon Astronomy, and those that 
 
 come last upon Ciiemistry. 
 
 First will come Weight, in solids and fluids, regarded statically 
 and dynamically. About this assignment, there are no two opinions; 
 weight being absolutely universal. Its phenomena are simple, in- 
 dependent of others, and so exactly understood that science is here 
 almost as positive as in astronomy, to which it is very nearly allied. 
 Electric phenomena being the most opposite of all to those of weight, 
 
 * In Electricity, the hypotliesis of fluids ia nij-.idly yielding before the rational 
 idea of Pularitv.— J. P. N.
 
 DEPARTMENTS OF PHYSICS. 1 9 1 
 
 in all these particulars, will C(mie last; atul tlioy arc closcl}^ allied 
 to chemistry. Between them will come theiin()l()f::;y, acoustics, and 
 optics. Fourier put away, in his stutly of heat, all fantastic notions 
 about imaginary fluids, and brought his subject up to such a point 
 of positivity as to place it next to the study of gravity. Acoustics 
 might, perhaps, contest its place with thermology, but for the gener- 
 ality of the phenomena of the latter. In regard to })ositivity, there 
 is little to choose between them : but there are gaps in our know- 
 ledge of acoustics which also indicate^ the lower ])lace for it. Our 
 order theti is, — Barology, or the science of weight : ^, , 
 thermology, or the science of heat : acoustics, optics, 
 and electricity. But we must beware of attaching too much impor- 
 tance to this arrangement, which is really little better than arbitrary, 
 though as good as our ])resent knowledge admits. We shall now 
 proceed to a philosophical review of them, exempt from details ; 
 having, in this chapter, analysed the proper object of Physics ; the 
 modes of investigation appropriate to it ; its position in regard to 
 the other sciences ; its influence upon the education of human 
 reason ; its degree of scientific perfection ; its incomplete positivity 
 at present; the means of remedying this by a sound institution of 
 hypotheses ; and finally, the rational distribution of its different 
 departments.
 
 li)Z 
 
 CHAPTER II. 
 
 BAUOLOGY. 
 
 Notwithstanding the advanced state of our means for the study 
 of Barology, we have no complete theoiy of weight, but only frag- 
 mentary portions of a theory, dispersed through treatises on rational 
 mechanics or physics. It will be of great advantage to bring them 
 together. 
 
 The division of tiie subject is into two principal sections, sub- 
 . divided into three ; the Statical and Dynamical con- 
 
 sideration of weight, in its application to solids, 
 liquids, and gases. Both })hilosophically and historically, this 
 division is indicated. 
 
 SECTION I. 
 
 Taking the statics of gravity first, we must point out that we owe 
 statics of our elementary notions of positive barology to Archi- 
 Oraviti,. medes. He first clearly established that" the statical 
 effort produced in a body by gravity, — that is, its iveigltt, — is en- 
 tirely independent of the form of the surface, and depends only on 
 . the volume, as long as the nature and constitution of 
 
 the body lemain unchanged. This may appear to us 
 very simple ; but it is not the less the true germ of a leading pro- 
 position in natural philosophy, which was perfected only at the end 
 of the last century ; namely, that not only is the weight of a body 
 independent of its form, and even of its dimensions, but of the mode 
 of aggregation of its pailicles, and of all variatit)ns which can occur 
 in tiieir composition, even by the different vital oi)erations ; in a 
 word, that this quality is absolutely unalterable, except by the cir- 
 cumstance of its distance from the centre of the earth. Archimedes 
 could take none but geometrical circuiustances into the account ; 
 but, in this elementary relation, his work was complete. He not 
 only discovered that, in homogeneous masses, the weight is always 
 in proportion to the volume ; but he disclosed the best means 
 of measuring, in solid bodies, by his famous hydrostatic principle, 
 the specific coefficient which enables us to estimate, according to
 
 CA SE OF LIQ UIDS. 1 93 
 
 this law, the weight and vohime of the body, by means of each other. 
 We owe to him too the idea of the centre of gravity, together with 
 the first development of the corresponding geometrical theory. 
 Under this view, all problems respecting the equilibrinnx of solids 
 are included in the domain of rational mechanics : so that, excej)t 
 the important relation of weight to masses, which could be fully 
 known only to the moderns, Archimedes ought to be regarded as 
 the true founder of statical barology, in relation to solids. There 
 is, however, another leading idea which was not clear in the time 
 of Archimedes, though it became so soon afterwards; that of the 
 law of the direction of gravity, which men spontaneously considered 
 to be constant, and which the school of Alexandria ascertained to vary 
 from place to place, always being perpendicular to the surface of the 
 terrestrial globe ; a discovery which is evidently due to astronomy, 
 by which alone the means are offered of manifesting and measuring, 
 by comparison, the divergence of verticals. 
 
 The ancients had no accurate ideas about the equilibrium of liquids; 
 for Archimedes contemplated only the equilibrium Cases of 
 of solids sustained by liquids. His principle did not liquids. 
 result, as with us, from an analysis of the pressure of the liquid 
 against the vessel containing it, thus disclosing the total force 
 exercised by the liquid in sustaining the weight. The theory of 
 gravitating liquids is to be ascribed to tho moderns. 
 
 The mathematical character of fluids is that their molecules are 
 absolutely inde[)endent : and the geometrical character of liquids is 
 that they are absolutely incompressible. Neither of these is strictly 
 true. The mutual adherence of fluid molecules now forms a-n 
 interesting section of physics ; and, as for the compressibleness of 
 liquids, it was indicated by several phenomena, — especially the trans- 
 mission of sound by water, — and it is now proved by unquestionable 
 experiments. The contraction as yet producible is very small ; and 
 we do not know what law the phenomenon follows, in its relation to 
 variation of pressure. But this uncertainty does not affect the theory 
 of the equilibrium of natural liquids, owing to the extreme smallness 
 of the condensation. In the same way, imperfect fluidity is no 
 hindrance, provided the mass has a certain extension. We may 
 therefore put aside these exceptions, and proceed to consider the 
 equilibrium of gravitating liquids, in the two cases in which they 
 are studied : in a mass so limited that the verticals are })arallel, 
 which is tlie ordinary case ; or in a great mass, as that of the sea, in 
 which we have to allow foi- the variable direction of gravity. 
 
 In the first case, there is clearly no difficulty about the surface : 
 and the whole question is of the pressure against the p.^_^^ ^^^^_ 
 enclosing vessel. Stevinus, following the principle of 
 Archimedes, showed that the pressure u{)Oii a horizontal boundary, or 
 floor, is always equal to the weight of the liquid column of the same 
 base which should issue at the surface of equilibrium : and he after- 
 
 VOL. I. N
 
 194 POSITIVE PIIIIO SOPHY. 
 
 wards resolved into this the case of an incUned boundary, by decom- 
 posing it into horizontal elements, as we now do. From this it 
 appeared that the pressure is always equal to the weight of a vertical 
 column which should have the proposed boundary for its base; and 
 for its height that of the surface of equilibrium above the centre 
 of gravity of this boundary. According to that, the infinitesimal 
 analysis enables us easily to calculate the pressure ngainst any definite 
 portion of any curved surface. The most interesting physical result 
 is the estimate of the total pressure supported by the whole of the 
 vessel, which is necessarily equivalent to the weight of the liquid it 
 contains. The equilibrium of floating bodies is only a simple appli- 
 cation of this rule of measurement of jn-essure. The immersed part 
 of the solid is a boundary ; and it is clear that the pressure of the 
 liquid to sustain the body is equivalent to a vertical force equal to 
 the weight of the displaced fluid, and applied to the centre of gravity 
 of the immersed portion. This rule is precisely the principle of 
 Archimedes. The main problem was geometrically treated by him. 
 The only really difficult research in this matter relates to the con- 
 ditions of stability of this equilibrium, and the analysis of the oscil- 
 lations of the body floating round its stable position ; and this is one 
 of the most complex applications of the dynamics of solids. If the 
 question was of the vertical oscillations of the centre of gravity, the 
 study would be easy, because we can estimate at once the way in 
 Avhich the pressure increases as the body is further immersed, and 
 diminishes as it rises, tending always to a return to the primitive 
 state. But it is otherwise with oscillations from rotation, whether of 
 rolling or pitching, the theory of which is a matter of much interest 
 in naval art. Here, the mathematical difficulties of the problem can 
 be met only by abstracting the resistance and agitation of the liquid ; 
 .and the labours of geometers become merely mathematical exercises, 
 of no practical use. 
 
 The question of the equilibrium of the vast liquid masses which 
 compose the greaterpart of the earth's surfaceis clearly 
 connected with the genei'al theory of the form of the 
 planets : but difficulties, unconnected with the figure of the planets, 
 intervene, and cannot be entirely surmounted. Rational hydrostatics 
 shows us that equilibrium is possible when there is the same density 
 at all points equally distant from the earth's centre ; a condition 
 which is impossible under our variety of temperatures in different 
 positions. There is no rational result from any practicable study of 
 currents, of varying temperatures, of the com{)ressibleness of liquids, 
 all of which, though following utdcnown laws, are necessary to the 
 solution of the problem. We iiave no better resource, at present, than 
 in empirical studies : and these, which belong more to the natural 
 history of the globe than to physics, are very imjierfect. 
 
 The theory of tides will hereafter, when sciences and their 
 arrangements are more peifected, take its place in the department
 
 CASE OF GASES. 
 
 J95 
 
 of barology. The periodical disturbances of the equilibrium of tbe 
 ocean are a proper subject for study in connection witb terrestrial 
 gravity ; and it can make no difference tbat the cause of those per- 
 turbations is found in tiie planetary system. 
 
 In studying the third question of equilibrium, that of gases, we 
 Tueet with a difficulty which does not ])ertain to that 
 of solids and liquids ; — we have to discover the grav- ' ° ^"^^^' 
 ity of the general medium in which we live. In the case of liquids, 
 we have only to weigh an empty vessel, and then the same vessel 
 filled ; whereas, in the case of the atmosphere, a vacuum cannot be 
 created but by artificial means, which nnist themselves be fomided 
 on a knowledge of the weight of the medium to be weighed. The 
 fact can be ascertained only by indirect means ; and those are 
 derived from the theories of pressure which we have been noticing. 
 Stevinus was not thinking of the atmosphere in elaborating his 
 theory ; but, as it answered for heterogeneous liquids, it nmst 
 answer for the atmosphere. From that date, the 
 means for ascertaining the equilibrium of the atmos- 
 l)here, in a positive manner, were provided. Gralileo projected the 
 work, in his last years ; and it was well executed by his illustrious 
 disci[)le, Torricelli. He proved the existence and measurement of 
 atmospheric pressure by showing that this pressiu'e sustained dif- 
 ferent liquids at heights inversely proportioned to their densities. 
 Next, Pascal established the necessary diminution of this pressure 
 at increasing heights in the atmosphere ; and Guericke's invention 
 of the air-pump, an inevitable result of Torricelli's discovery, gave 
 us direct demonstration in the power of making a vacuum, and 
 consequently of estimating the specific gravity of the air which 
 surrounds us, Avhich had hitherto been only vaguely computed. 
 The ci'eation and improvement of instruments of observation is an 
 invariable consequence of scientific discovery; and, in this case, 
 the fruits are the barometer and the air-pump. 
 
 One condition remained, before we could ai)ply the laws of hydro- 
 statics to atmospheric equilibiium. We had to learn the relation 
 between the density of an elastic fluid and the pressure which it 
 supports. In liquids (supposed incompressible) the two pheno- 
 mena are mutually independent ; whereas, in gases they are inevit- 
 ably connected : and herein lies the essential dilference between 
 the mechanical theories of the two fluids. The discovery of this 
 elementary relation was made about the same time by ]Mariotte in 
 France and Boyle in England. These illustrious ])hil()sophers 
 proved by their experiments that the ditferent volumes successively 
 occupied by tlie same gaseous mass are in an inverse ratio to the 
 different pressures it receives. This law has since been verified by 
 increasing the pressure to nearly that of thirty atmospheres ; and 
 it has been ad()i)ted as the basis of the whole Mechanics of gas and 
 vapours. But we must beware of accepting it as the mathematical
 
 196 POSITIVE PHILOSOPHY. 
 
 expression of the reality ; for it is evidently the same thing as 
 regarding elastic fluids as always equally compressible, however 
 compressed they may already be ; or, conversely, as always equally 
 dilatable, however dilated they may already be : — suppositions 
 which cannot he indefinitely extended. But thus it is, more or less, 
 with all the laws we have ascertained in our study of nature. They 
 approximate, within narrow limits, we must suppose, to the mathe- 
 matical reality : but they are not that reality itself, even in the 
 grand instance of gravitation. These laws are sufficient for our use 
 and guidance ; and that is all the result that positive science pre- 
 tends to. 
 
 Under the law of Mariotte and Boyle, the theory of atmospheric 
 Condition of equilibrium falls into the department of Rational 
 thejprobiem. Mechanics. We see at once that the air can no more 
 be in a state of real equilibrium than the ocean ; and so much the 
 further from it as heat expands air more than water. Yet we must 
 conceive of the partial equilibrium of a very narrow atmospheric 
 column, to form a just general idea of the mode of diminution in 
 regard to the density and pressure of the different sti-ata. Putting 
 aside considerations of heat, and the small effects of gravitation in 
 such a case, we see that density and pressure must diminish in a 
 geometrical progression for altitudes increasing in arithmetical pro- 
 gression: but this abstract variation is retarded by the diminishing 
 heat of the loftier atmospheric strata, which makes each stratum 
 more dense than it would be from its position. Here, therefore, 
 we are stopped by the intervention of a new element which we do 
 not understand, — the law of the vertical variation of atmospheric 
 temperatures, — our ignorance of which can be supplied only by 
 inexact and uncertain expedients. Great caution is necessary iu 
 using Bouguer's method of measuring altitudes by the barometer ; 
 a method very ingenious, but depending on such complex and un- 
 certain conditions, and requiring sometimes so much delay, that it 
 is even preferable, when circumstances permit, to enter upon a 
 geometrical measurement, which has so greatly the advantage in 
 certainty. Yet, considered by itself, the method of measurement 
 by the barometer is valuable for its contributions to our knowledge 
 of the surface reliefs of our globe. 
 
 SECTION 11. 
 
 DYNAMICS. 
 
 We have now to consider the laws of notion of gravitating bodies ; 
 and of Solids, in the first place. 
 
 The last elementary notion about gravity, — that of the necessary 
 proportion between weight and mass, — which was still wanting to 
 statical barology, was established by the admirable observation that
 
 D YNAMICS OF BAR OLOGY. 197 
 
 all bodies in a vnciuim fall tlirou<;h the same s])nce in the same time. 
 Proceeding from this, we must examine the discoveiy of the laws of 
 motion produced by gravity. We shall find herein, 
 not only the historical origin of physics, but the most 
 perfect method of philosophizing of which the science admits. 
 Aristotle observed the natural acceleration of bodies in their fall ; 
 but he could not discover the law of the case, for want of the 
 elementary principles of rational dynamics. His hypothesis, that 
 the velocity increases in proportion to the space traversed, was 
 ])lausible, till Galileo found the true theory of varied motions. 
 When Galileo had discovered the law that the velocity and the 
 space travei'scd were necessarily proportioned, the one to the time, 
 and the other to its square, he showed how it could be verified in 
 two ways ; by the immediate observation of the fall, or by retarding 
 the descent at will by the aid of a sufficiently inclined plane, — 
 allowance being made for the friction. An ingenious instrument, 
 which affords a convenient verification, was afterwards offered by 
 Atwood : it retards the descent, while leaving it vertical, by com- 
 pelling a smnll mass to move a very large one in that dii-ection. 
 
 By this one law of Galileo, all the problems relating to the 
 motions of falling bodies resolve themselves into questions of rational 
 dynamics. They, indeed, compelled its formation, in the seventeenth 
 century ; as, in the eighteenth, questions of celestial mechanics 
 thoroughly developed it. In all that relates to the motion of trans- 
 lation of a body in space, this study is due to Galileo, who established 
 the theory of the curvilinear motion of projectiles— allowance being 
 made for the resistance of tlie air. All attempts, however, to ascer- 
 tain the effect of this resistance have hitherto been in vain ; and 
 therefore the study of the real motion of projectiles is still extremely 
 imperfect. 
 
 As for the motions that gravity occasions in bodies that are not 
 free in space, — the only important case is that of a body confined to 
 a given curve. It constitutes the problem of the pendulimi, which 
 we have already considered, as the immortal achievement ot 
 Iluyghens. Its practical interest, as the basis of the most perfect 
 chronometry, presented itself first : but it has besides furnished two 
 general consequences, very essential to the progress of barology. 
 First, it enabled Newton to verify the proportion of weight to mass 
 with much more exactness than would have been possible by the 
 experiment of the fall of bodies in a vacuum ; and, in the second 
 }ilnce, the pendulum has enabled us, as we saw before, to observe the 
 differences of the universal gravity at different distances from the 
 earth's centre. It is by the use of this method that we are con- 
 1 inually adding to our knowledge of the measure of gravity at various 
 ])oints on the globe, and, therefoie, of the figure of the earth. 
 
 In these different sections of dynamical barology, solid bodies are 
 regarded as points, — considerations of dimensions being discarded :
 
 198 POSITIVE PHILOSOPHY. 
 
 but a new order of diflQculties arises when we have to consider the 
 particles ot which the body is really formed. With regard to cases 
 of restricted motion, as the pendulum, the thing to be done is to 
 find out under what laws the different points of the body modify 
 the unequal times of their respective oscillations, so that the whole 
 may oscillate as one point, real or ideal. This law, discovered by 
 Hu3'gliens, and afterwards obtained by James Bernouilli in a more 
 scientific manner, easily transforms the compound into the simple 
 pendulum, when the moment ofinei'tia of the body is known. The 
 study of the pendulum is involved in all the questions of the 
 dynamics of solids. To give it the last degree of precision, it is 
 necessary to consider the resistance of the air, though that resistance 
 is small in compai'ison with the case of projectiles. This is done, 
 with ease and certainty, by comparing theoretical oscillations with 
 real ones, exposed to the resistance of the air ; when, of course, the 
 difference between the two gives the amount of that resistance. 
 We have seen enough of the difficulties of hydrodynamics to 
 understand that the part of dynamical barology which 
 relates to fluids must still be very imperfect. In the 
 case of the gases, and especially of the atmosphere, next to nothing 
 has been attempted, from the sense of the impracticable nature of 
 the inquiry. The only analysis which hns been proceeded with, 
 in regard to liquids, is that of their flow by very 
 iquits. gj^-j^^n Qi-jfjQgg ifj \\-^Q bottom or sides of a vessel : that 
 is, the purely linear motion, mathematically presented by Daniel 
 Bernouilli, in his celebrated hypothesis of the parallelism of laminae. 
 Its principal result has been to demonstrate the rule, empirically 
 proposed by Torricelli, as to the estimate of the velocity of the 
 liquid at the orifice, as equal to that of a weight falling from the 
 entire height of the liquid in the vessel, Now, this rule has been 
 reconciled with observation, even on the supposition of an invariable 
 level, only by an ingenious fiction, suggested by the singular pheno- 
 menon of the contraction of the liquid filament. The case of a 
 variable level is scarcely entered upon yet; or any which involves 
 the form and size of the orifice. As for more complex cases, — their 
 theory is yet entirely in its infancy.* 
 
 From this cursory review of Barology we may carry away some 
 idea of its spirit, and of the progress of which it admits. Imperfect 
 as our survey has been, we may perceive that this first province of 
 Physics is not only the purest, but the richest. We m;iy observe 
 in it a character of rationality, and a degree of co-ordination which 
 we shall not meet with in other parts of the science. It is because 
 
 * The subjects here spoken of by M. Comte have recently received remarkable 
 elucidation throngli experiments entered upon chiefly by the instrumentality of the. 
 " British Association for the Advancement of Science." There cannot be much ho]>e, 
 in the present state of our knowledge, of ascending otherwise than empirically 
 towards any general law or rules which will either comprehend phenomena, or he of 
 use in practice. — J. P. N.
 
 EXISTING STA TE OF BAROLOG Y. 199 
 
 we look for a consistency and ])reci.sion almost like tliose thai we 
 find in astronomy tliat we consider barok\2;y so imjierfect as we do. 
 It lias long attained its j)osilion of jwsitivity : there is no one of its 
 snbdivisions which is not at least sketched out : all Exhtinci state 
 the general means of investigation, Observation, ofBaroiogy. 
 Experiment, and Comparison, have been successively applied to it; 
 and thus its future progress depends only on a more complete 
 harmony between these three methods, and on a more uniform and 
 close combination between the mathematical spirit and the physical.
 
 200 
 
 CHAPTER III. 
 
 THERMOLOGY. 
 
 Next to the phenomena of gravity, those of heat are, unquestion- 
 ably, the most universal in the province of Physics. Throughout 
 the economy of terrestrial nature, dead or living, the function of 
 
 heat is as important as tliat of gravity, of which it is 
 
 the chief antagonist. The consideration of gravity 
 presides over the geometrical and mechanical study of bodies : 
 Avliile that of heat prevails in its turn, when we investigate deeper 
 modifications, relating to either the state of aggregation or the com- 
 position of molecules : and finally, vitality is subordinated to it. 
 The intelligent application of heat constitutes the cliief action of 
 man upon nature. Thus, after barology, therraology is, of all the 
 parts of physics, the one wliich most deserves our study. 
 
 The earliest scientific observations in tliermology are almost as 
 old as the discoveries of Stevinus and Galileo on gravity, the inven- 
 tion of the thermometer having taken place at the beginning of the 
 seventeenth century : but, owing to the complication of its pheno- 
 
 mena, it has always been far distanced by barology. 
 
 At the end of the seventeenth centur}^ the indications 
 of the thermometer could not be compared, for want of two fixed 
 points, the necessity for which was, at that time, pointed out by 
 Newton. The greatest difference between the two studies, however, 
 was in their spirit. While philosophers were already inquiring, 
 with regard to weight, what were its phenomena and their laws, 
 those who were studying heat were looking for the nature of fire, 
 and reducing the facts of the case to something merely episodical. 
 It was only at the end of the last centui-y, that the great discovery 
 of Black imparted anything of a scientific character to thermology, 
 while barology was almost as much advanced as it is now. Our 
 ]ihilosophers still entertain some of the old chimeras ; but now very 
 loosely, and, as they say, to iacilitate the study of the phenomena. 
 The labours of Fourier, however, must soon establish a thoroughly 
 scientific method ; and this result cannot but be aided by the fact 
 ihat the two great modern hypotheses about the nature of heat are 
 in direct collision. It is certain that, of all the provinces of physics, 
 thermology is the nearest to a complete emancipation from the 
 anti-scientific spirit.
 
 PR OPA GA TION OF HE A T. 2 & i 
 
 Of all the branches of Physics which admit Mathematical 
 Analj'sis, this is the one which exhibits the most neiation to 
 special application of it. Barology enters into the Mathematics. 
 province of rational mechanics; and so, in a less degree, does 
 acoustics. The analytical theory of heat now offers a scientific 
 character as satisfactory as that of gravity and sound ; and it may 
 be treated as a dependency of abstract Mechanics, without any 
 resort to chimerical hypotheses. Our present business is, however, 
 with the purely physical study of Hejat. 
 
 SECTION 1. 
 
 MUTUAL THERMOLOaiCAL INFLUENCE. 
 
 Physical thermology consists of two parts, distinct, but nearly 
 connected. The first relates to the mutual influence 
 of two bodies in altering their respective tempera- ^"°'pa^ 
 tnres. The second consists in the study of those alterations : that 
 is, of the modifications or entire change which the physical consti- 
 tution of bodies may undergo in consequence of their variations of 
 temperature, stopping at the point at which chemistry intervenes, 
 — that is, where the molecular composition becomes affected. The 
 first of these consists in the theory of warming and cooling. 
 
 No thermological effect is produced between bodies of precisely 
 the same tempeiature. The action begins when the First .- Mutual 
 temperatures become unequal. The warmer body influence. 
 then raises the temperature of the cooler ; and the cooler depresses 
 that of the warmer, till, sooner or later, they reach a common tem- 
 j)erature. Though the final state may usually be at an unequal 
 distance from the two extremes, action is not, properly speaking, 
 the less truly equivalent to reaction in a contrary direction. This 
 case again divides itself into two. The bodies may act at a distance, 
 greater or smaller ; or they may be in contact. The first case con- 
 Ntitutes what is called the radiation of heat. 
 
 The direct communication of heat between two isolated bodies 
 was long denied by philosophers, who regarded the air, or some 
 other medium, as indispensable to the effect. But there is now no 
 doubt about it ; as thermological action takes place in a vacuum : 
 and the small density and conducting power of the air could not 
 account for the effects observed in the majority of ordinary cases. 
 This action, like that of gravity, extends, no doubt, to all distances, 
 in conformity to the fundamental approximation between these two 
 great phenomena, pointed out by Fourier : for we can conceive of 
 the planets of our system as exerting an appreciable mutual influ- 
 ence in this respect: and it seems as if the temperature of the wliole 
 solar system were attributable to the thermometrical equilibrium 
 to which all the parts of the universe are for ever tending.
 
 '202 POSITIVE FIIIIO SOPHY. 
 
 The first law relating? to such an action consisLs in its rectilinear 
 Badiation propagation. Thongh the term radiation has heen 
 of Heat. connected with nntenable h3'potheses, we may retain 
 it, provided we carefully restrict it to the meaning that it is in a 
 right line that two points act thermologically on each other. It 
 thus implies that in placing bodies to prevent this mutual action 
 between two others, the absorbing body must ])e placed in a right 
 line. — Tills radiating heat can be reflected like light, and in con- 
 formity with the same rule : and it undergoes the same refractions 
 as light, with some modifications. — Another question about this 
 action relates to the influence of the direction of radiation considered 
 in regard to the surface of either the warming or the warmed body. 
 The experiments of Leslie, confirmed by the mathematical results 
 of the case, have established that, in either case, the intensity of the 
 action is greater as the rays approach the perpendicular, and that 
 it varies in proportion to the sine of the angle that they form with 
 each surface. — The last consideration, and the most important, is 
 the difference of temperatures between the two bodies. When this 
 difference is not very great, the intensity of the action is in precise 
 proportion to it : but this relation appears to cease when the tem- 
 peratures become very unequal. 
 
 When the radiation is not direct, but transmitted through an 
 intermediate body, the conditions just noticed become complicated 
 with new circumstances relating to the action of the interposed body. 
 We owe to Saussure a fine series of experiments — hardly varied 
 enough, however — upon the effect of a set of transparent coverings 
 in changing remarkably the natural mode of accumulation or dis- 
 persion of heat whether radiant or obscure. More recently, M. 
 Melloni has pointed out an essential distinction between the trans- 
 mission of heat and that of light, in proving that the most translucent 
 bodies are not always those through which heat passes most easily ; 
 as was previously supposed. 
 
 It is well for physicists to separate the radiation of heat from its 
 Propagation propagation by contact, for analytical purposes ; but 
 hy Contact. ^^ is evident that the separation cannot be found in 
 nature. They are always found in connection, however unequall}'. 
 Besides that the atmosphere can hardly be absent, and is always 
 establishing an equilibrium of heat between bodies that are apart, 
 it is clear that it is only the state of the surface that can be deter- 
 mined by radiation. The interior parts, which have as much to do 
 with the final condition as the surface, can grow warmer or cooler 
 only by contiguous and gradual ])ropagation. Thus, no real case 
 can be analysed by the study of its radiation alone. And again, 
 the action by contact of two bodies can take place only in those 
 small portions which are in contact, while the bodies are acting 
 upon each otlier by the radiation of all the rest of their surfaces. 
 Tluis, though the two modes of action are really distinct, the
 
 CONDUCTIBILITY. 203 
 
 analysis of either is rendered extremely difficult by their perpetual 
 combination. 
 
 Of the three conditions noticed above, relating to the action 
 exerted at a distance, the only one which applies equally to the pro- 
 pagation of heat by contact is the difference of temperatures. The 
 temperature of the parts in question can have so little inequality, 
 that the law which makes the action increase in proportion to the 
 difference may be regarded almost as an exact expression of the 
 fact. As for the law relating to dii-eotion, it probably subsists here 
 too, though we cannot be perfectly assured of it. But that law 
 which relates to the distance must be totally changed ; for, on the 
 one hand, the action of the almost contiguous particles cannot be 
 nearly so great as the variations which we are able to estimate 
 would lead us to suppose ; and, on the other hand, when we com- 
 pare the various small intervals between them, the diminution is 
 certainly much more rapid than in the case of distant bodies. 
 
 Whatever maybe the mode of thewarming and cooling of respective 
 bodies, the final state which is established under the laws just 
 noticed is numerically determined by three essential coefficients, 
 projier to every natural body, as its sj)ecific gravity is in barology. 
 
 Under the old term conductihility , two properties were con- 
 founded, which Fourier separated, giving them the ^ , ,.,.,., 
 
 ' . 1 -T, 1 1 -Ti -iX -G *. CohductlhlllltJ. 
 
 names 01 'penetraoiLity and permeability ; the iirst 
 signifying that by which heat is admitted at the surface, or dis{)erse(l 
 from it, and the other that by which the changes at the surface are 
 propagated through the interior. Permeability de- 
 pends altogether on the nature of the body and its state " '""'^' ' ''' 
 of aggregation. The differences of bodies in this respect have 
 always been open to observation ; for instance, the rapid propagation 
 of heat in metallic bodies in comparison with coal, which may be 
 burning at one point and scarcely warm a few inches off, while the 
 heat rapidly pervades the whole body of metal. It varies with the 
 })hysical constitution of bodies, so diminishing in fluids that even 
 Rum ford went so far as to deny permeability in them altogether, 
 ascribing the propagation of lieat in them to interior agitation. 
 This was a mistake ; but permeability is very weak in liquids, and 
 weaker still in gases. As to penetrability, while pcneirahUitu 
 partly depending on the state of aggregation of bodies, 
 it depends much more on the state of their surfaces, — on colour, 
 polish, and the legularity in which radiation in various directions, 
 can take place, and divers other modifications : and it changes in the 
 same surface as it is exposed to the action of different media. 
 
 Strictly speaking, the different degrees of these two kinds of 
 conductihility cannot alfect the final tliermological state of the two 
 bodies, but only the time at which it is reached. Yet, as real 
 questions often become mere questions of time, it is clear that it the 
 inequalities are very marked, they must affect the intensity of the
 
 204 POSITIVE PHILOSOPHY. 
 
 ])henomena under shidy ; for instance, where the permeability is so 
 ieeble that the requisite interior heat cannot be obtained in time, 
 but by applyinj]^ such heat to the surface as will break or burn it ; 
 in which case, the phenomenon cannot take place ; or not within any 
 ])racticable time. In general, the more perfect both kinds of 
 conductibility, the better the bodies will obey the laws of thermologi- 
 cal action, at a distance or in contact. It would therefore be very 
 important to measure the value of these two coefficients for all 
 bodies under study : but unhappily such estimates are at present 
 extremely imperfect. The business was vague enough, of course, 
 when the two kinds of conductibility were confounded ; but Fourier 
 lias taught us how to estimate permeability directly, and, of course, 
 penetrability indirectly, by subtracting the permeability from the 
 total of conductibility. But the application of his methods is as 
 yet hardly initiated. 
 
 One consideration, that of specific heat, remains to be noticed, as 
 s e m h at concurring to regulate the results of thermological 
 pea c ea . ^^^^^^^ Whether under conditions of equal weight, or 
 of equal volume, the different substances consimie distinct quantities 
 of heat to raise their temperature equally. This jiroperty, of which 
 little was known till the latter half of the last century, depends, 
 like permeability, only in a less degree, on the physical constitution 
 of bodies, Avhile it is independent of the state of their surfaces. It 
 must considerably affect the equalized temperature common to two 
 bodies, which cannot be equally different from the primitive tem- 
 perature of each, if they differ from each other in the point of their 
 specific heat. Physicists have achieved a good deal in the estimate 
 of specific heat. The best method is that of experiment with the 
 calorimeter, invented by Lavoisier and Laplace, for its measure- 
 ment. The quantity of heat consumed by any body at a deter- 
 minate elevation of temperature, is estimated by the quantity of ice 
 melted by the heat it gives out in its passage from the highest to the 
 lowest temperature. The apparatus is so contrived as to isolate the ex- 
 periment from all thermological action of the vessel and of the medium ; 
 and thus the results obtained are as precise as can be desired. 
 
 These are thethi-ee coefficients which serve to fix the final tempera- 
 tures which result from the thermological equilibrium of bodies. 
 Till we know more of the laws of their variations, it is natural to 
 suppose them essentially uniform and constant : but it would not 
 be rational to conceive of conductibility as identical in all directions, 
 in all bodies, however their structure may vary in diffeient directions ; 
 and specific heat probably undergoes changes at extreme tempera- 
 tures, and especially in the neighbourhood of those which determine 
 a new state of aggregation, as some experiments already seem to 
 indicate. However, these modifications are still so uncertain and 
 obscure, that physicists cannot be blamed for not keeping them, at 
 this day, perpetually in view.
 
 CONSTITUENT CHANGES B V HE A T. 205 
 
 SECTION II. 
 
 CONSTITUENT CHANGES BY HEAT. 
 
 The second part of tliorraology is that wliich relates to the altera- 
 tions caused by heat in the physical constitution of 
 bodies. These alterations are of two kinds; changes constituent' 
 of volume, and the production of a new state of aggre- duuiucs by 
 gation ; and this is the part of thermology of which *'" ' 
 we are least ignorant. 
 
 These phenomena are independent of those of warming and cool- 
 ing, though they are found together. When we heat any substance, 
 the elevation of temperature is determined solely by the j)ortion of 
 heat consumed, the rest of which (often the greater part), insensible 
 to the thermometer, is absorbed to modify the physical constitution. 
 This is what we mean when we say that a portion of 
 
 ijii lii i 1-1 L • Latent heat. 
 
 lieat lias become latent; a term winch we may retaui, 
 though it was originally used in connection with a theory about the 
 nature of heat. This is the fundamental law discovered by Black-, 
 by observation of the indisputable cases in which a physical modi- 
 fication takes place without any change of temperature in the modi- 
 fied body. When the two effects coexist, it is much more diflficult 
 to analyse and apportion them. 
 
 Considering first the laws of change of volume, it is a general 
 truth that every homogeneous body dilates with heat Change 0/ 
 and contracts with cold; and the fact holds good with volume. 
 heterogeneous bodies, such as organized tissues especially, in regard 
 to their constituent parts. There are very few exceptions to this 
 rule, and those few extend over a very small portion of the ther- 
 mometrical scale. The principal anomaly however being the case 
 of water, it has great importance in natural history, though not 
 much in abstract physics, except from the use that philosophers have 
 made of it to procure an invariable unity of density, always at com- 
 mand. Tliese anomalies, too rare and restricted to invalidate any 
 general law, are sufficient to discredit all a priori explanations of 
 expansions and contractions, according to which every increase of 
 temperature should cause an expansion, and every diminution a 
 contraction, contrary to the facts. 
 
 Solids dilate less than liquids, under the same elevation of tempera- 
 ture, and liquids than gases ; and not only when the same substance 
 passes through the tluee states, but also when different substances 
 are emi)loyed. The expansion of solids proceeds, as far as we know, 
 with perfect uniformit}'. We know more of the case of liquids, 
 which is rendered extremely important from its connection with the 
 true theory of the thermometer, without which all thermological 
 inquiries would be left in a very dubious state. Ex[)erinients,
 
 2o6 POSITIVE PHIIOSOPIIY. 
 
 devised hy Dulong and Petit, have shown that for ahove three hun- 
 dred centigrade degrees the expansion of the mercury follows an 
 exactly uniform course, — equal increase of volume being pi'oduced 
 by heat able to melt equal weights of ice at zero. This is the only 
 case fully established ; but we have reason to believe that the rule 
 extends to that of all liquids. The most marked case of such regu- 
 larity is that of gases. Not only does the expansion take place by 
 equal gradations, as usually in liquids and solids, but it affects all 
 gases alike. Gases differ from each other, like liquids and solids, 
 in their density, their specific heat, and their permeability; yet they 
 all dilate uniformly and equally, their volume increasing three- 
 eighths, from the temperature of melting ice to that of boiling 
 water. Vapours are like gases in this particular, as in so many 
 others. These are the sim])le general laws of the expansion of elastic 
 fluids, discovered at once by Gay-Lnssac at Paris, and Dalton at 
 Manchester. 
 Change in state Ncxt, we liavc to noticc the changes produced by 
 of aggregation. \}qqX in the state of aggregation of bodies. 
 Solidity and fluidity used to be regarded as absolute qualities of 
 bodies ; whereas, we now know them to be relative, and are even 
 certain that all solid bodies might be rendered fluid if we could 
 apply heat enough, avoiding chemical alteration. In the converse 
 way, we used to suppose that gases must preserve their elasticity, 
 through all degrees of cooling and of compression ; whereas Bussy 
 and Faraday have shown us that most of them easily become liquid, 
 when they are seized in their nascent state ; and there is every reason 
 to believe that by a due combination of cold and pressure, they may 
 be always liquefied, even in their developed state. Under this view, 
 therefore, different substances are distinguished only by the different 
 parts of the indefinite thermometrical scale to which their successive 
 stales, solid, liquid, and gaseous, correspond. But this simple 
 inequality is an all-important characteristic, which is not yet 
 thoroughly connected with any other fundamental quality of each 
 substance. Density is the relation which is the least obscure and 
 variable, — gases being in geneial less dense than liquids, and liquids 
 than solids. But there are striking exceptions in the second case, 
 and might be in the first, if we knew more of gases in regard to 
 compression, and in varied circumstances of other kinds. As for the 
 three states of the same substance, there is always, except in some 
 cases of scarce anomaly, rarefaction in the fusion of solids and in 
 the evaporation of liquids. All these changes have been brought 
 by the illustrious Black under one fundamental law, which is, both 
 from its importance and its universality, one of the 
 gagemcntand finest discoverics in uatui-al i)hilosophy. It is this: 
 disengage- that in tlic passage froui the solid to the liquid state, 
 mentof eat. ^^^^ {vova that to the gaseous, evciy substance always 
 absorbs a more or less distinguishable quantity of heat, without
 
 VAPOURS. 
 
 20J 
 
 raising its temperature ; wliile the inverse process occasions a 
 disengagement of lieat ]M'ecise]y correspondent to the ahsorption. 
 These disengagements and ahsorptions of lieat are evidently, after 
 chemical phenomena, the principal sources of heat and cold. In 
 an experiment of Leslie's, an evajioration, rendered extremely rapid 
 by artifici.'d means, has produced the lowest temperatures known. 
 Eminent natiu'al pliilosophers liave even believed that the heat which 
 is so abundantly disengaged in most great chemical combinations 
 could proceed only from the different changes of state which com- 
 monly result from them. But this opinion, though true in regard 
 to a great number of cases, has too many exceptions to deal with to 
 become a general principle. 
 
 We have now done with physical thermology. But the laws of 
 the formation and tension of vapours now form an 
 appendix to it; and also of course hygrometry. The "^^^^^ 
 
 theory is, in fact, the necessary complement of the doctrine of 
 changes of state ; and this is ils proper place. 
 
 Before Saussure's time, evaporation was regarded as a chemical 
 fact, occasioned by the dissolving action of air upon liquids. He 
 showed that the action of the air was adverse to evaporation, except 
 in the case of the renovation of the atmosphere. The test was 
 found in the formation of vapour in a restricted space. Saussure 
 found that, in such a case, with a given time, temperature, and 
 space, the quantity and elasticity of the vapour were always the 
 same, whether the space was a vacuum or tilled with gas. Tlie 
 mass and tension of the vapour increased steadily with the tem- 
 perature ; whereas it apjiears that no degree of cold suffices to stop 
 the process entirel}'', since ice itself produces a vapour appreciable 
 by very delicate means of observation. We do not know by what 
 law the increase of temperature accelerates the evaporation, at least 
 while the liquid remains below boiling point ; but the variations 
 in elasticity of the vapour produced have been successfully studied. 
 
 One term common to all liquids is the boiling point. At that 
 ])oint, the rising tension of the vapour formed has become equal to 
 the atmospheric pressure; a fact which can be ascertained by direct 
 experiment. Proceeding from this point, Dr Dalton Temperatures 
 discovered the important law that the vapours of of ebullition. 
 dififerent liquids have tensions always equal between themselves to 
 temperatures equidistant from the corresponding boiling points, 
 whatever may be otherwise the direction of the difference. Thus, 
 the boiling of water taking place at one hundred degrees, and that 
 of alcohorat eighty degrees, the two vapours, having at that ])oint 
 the same tension, equal to the atmospheric pressure, will then have 
 equal elasticities, superior or inferior to the pieceding, wiicn their 
 two temperatures are made to vai-y in the same number of degrees. 
 The many new liquids discovered by chemists since this law was 
 found have all tended to confirm it. It is very desirable that some
 
 2o8 POSITIVE PHILOSOPHY. 
 
 harmony slioiild be discovered between the boiling temperatures of 
 different liquids and their other properties ; but this remains to be 
 done, and these temperatures appear to us entirely incoherent, 
 though there is every reason to believe that they are not so. 
 
 It is evident that this law of Dalton's simplifies prodigiously the 
 inquiry into the variation of the tension of vapours, according to 
 their temperatures ; since the analysis of these variations in one 
 instance will serve for all the rest. The experiments undertaken 
 for this purpose by Dr Dalton and his successors have not fully 
 established the rule of proportion between the tension and the tem- 
 perature : but an empirical law proposed by Dulong has thus far 
 answered to the observed phenomena. All a friori determinations 
 of the law have utterly failed. 
 
 The study of hygrometrical equilibrium between moist bodies 
 Hygrometricai scems a natural adjunct to the theory of evaporation. 
 equilibrium. gaussuro and Deluc have given us a valuable instru- 
 ment for this inquiry ; but we know scarcely anything of the laws 
 which regulate the equilibrium of moisture. Prevision, which is 
 the exact measure of science of eveiy kind, is almost non-existent 
 in the case of hygromeiry. The small part that it plays in the 
 inorganic departments of nature is, no doubt, the reason of the little 
 attention that physicists have devoted to it : but we shall hereafter 
 find how important is its share in vital phenomena. According to 
 M. de Blaiaville, hygrometrical action constitutes the first degree 
 and elementary mode of the nutrition of living bodies, as capillarity 
 is the germ of the most simple organic motions. In this view, the 
 neglect is much to be regretted. It is one instance among a multi- 
 tude of the mischiefs arising from the restricted training of natural 
 philosophers. In this case, two important studies, which can be 
 accomplished only by physical inquirers, are neglected, merely 
 because their chief destination concerns another department of 
 science. 
 
 After this survey, we can form some idea of the charactei-istics of 
 this fine section of Physics. We see the rational connection of the 
 different questions conjprised in it ; the degree of perfection which 
 each of them has attained ; and the gaps which remain to be filled 
 up. A vast advance was made when, by the genius of Fourier, the 
 most simple and fundamental ])henomena of" heat were attached to 
 an admirable mathematical theory. 
 
 SECTION III. 
 
 CONNECTION WITH ANALYSIS. 
 
 This theory relates to the first class of cases, — those in which an 
 equalization of heat takes place between bodies at a distance or in 
 contact ; and not at all to those in which the physical constitution
 
 ANALYSIS, 209 
 
 is altered by heat. It is only by an indirect investigation that we 
 can learn liowlieat, once introduced into a body from Thermoiom, 
 the surface, extends thronirh its mass,assi<rnins to each connected with 
 point, at any fixed moment, a determuiate tem- -analysis. 
 perature ; or tlie converse — liow this interior heat is dissij[)ated, by 
 a gradual dispersion through the surface. As direct observation 
 could not liclp us here, we must remain in ignorance, if Fourier had 
 not brought mathematical analysis to the aid of observation, so 
 as to discover the laws by which these processes take place. The 
 perfection with which this has been dune opens so wide a held 
 of exploration and application, unites so strictly the abstract and 
 the concrete, and is so pure an example of the positive aim and 
 method, that future generations will probably assign to this achieve- 
 ment of Fourier's the next ])lace, as a mathematical creation, to the 
 theory of gravitation. Many contemi)oraries have hastened into the 
 new field thus opened ; but most of them liave used it only for 
 analytical exercises which add nothing to our permanent knowledge ; 
 and perhaps the labours of M. Duhamel are hitherto the only ones 
 which aflford really any extension of Fourier's theory, by perfecting 
 the analytical representation of thermological j)lienomena. 
 
 According to the plan of this work, we ought not to quit the 
 limits of natural philosophy to notice any concrete considerations 
 of natural history, — the secondary sciences being only derivatives 
 from the primaiy. It is departing from our rule, therefore, to bring 
 forward the important theory of terrestrial temperatures : yet this 
 most important and difficult ap))lication of mathematical thermology 
 forms so interesting a part of Fourier's doctrine, that I cannot 
 refrain from ofTerinir some notice of it. 
 
 SECTION IV. 
 
 TERRESTRIAL TEMPERATURES. 
 
 The temperature of each point of our globe is owing (putting aside 
 local or accidental influences) to the action of three Terrestrial 
 general and permanent causes variously combined : temperatures. 
 first, the solar heat, affecting different parts unequally, and sub- 
 jected to periodical variations : next, the interior heat proper to the 
 earth since its formation as a planet : and thirdly, the general ther- 
 mometrical state of the si)ace occupied by the solar system. The 
 second is the only one of the three which acts uj)on all the points of 
 the globe. The influence of the two others is confined to the sur- 
 face. The order in wliich they have become known to us is that 
 in which I have placed them. Before Fourier's time, the whole 
 subject had been so vaguely and carelessly regarded, that all the 
 })henomena were ascribed to solar heat alone. It is true, the notion 
 of a central heat was very ancient ; but this hypothesis, believed in 
 and rejected without sufficient reason, had no scientific consistency, 
 
 vor-. T,
 
 2IO POSITJVE PHILOSOPHY. 
 
 — the question having never heen raised of the effect of Mils original 
 
 heat on the thermological variations at the surface. The theoiy of 
 
 Fourier afforded him niatlieniatical evidence that at the surface the 
 
 temperatures woukl be widely different from wliat tiiey are, both in 
 
 degree and mutual })roporti()n, if the globe were not pervaded by a 
 
 . , heat of its own, independent of the action of the sun ; 
 Interior heat. i - i • i . i i. t • r ^i c i 
 
 a heat which tends to dispersion irom the surface, by 
 
 radiation towards the planets, thougli the atmosphere must con- 
 siderably retard this dispersion. This original heat contributes 
 very little, in a direct way, to the temperatures at the surface ; but 
 without it, the solar influence would be almost wholly lost, in the 
 total mass of the globe ; and it therefore prevents tlie periodical 
 variations of temperature from following other laws than those which 
 result from the solar influence. Immediately below the surface, the 
 central heat becomes preponderant, and soonest in the parts nearest 
 the equator ; and it becomes the sole regulator of temperatures, and 
 in a rigorously uniform manner, in proportion to the depth. — As to 
 Temperature ^^^® third cause, Fourier was the first to conceive of 
 of the planet- it. He was wout to give, in a simple and striking 
 ary intervals, form^ the results of iiis inquiries in the saying that if 
 the earth left a thermometer behind it in any part of its orbit, the 
 instrument (supposing it protected from solar influence) could not 
 fall indefinitely : the column would stop at some point or other, 
 which would indicate the temperature of the space in which we 
 revolve. This is one way of saying that the state of tlie tempera- 
 tures on the surface of the globe would be inexplicable, even con- 
 sidering the interior heat, if the surrounding space had not a 
 determinate temperature differing but little from that which we 
 should find at the poles, if we could precisely estimate it. It is 
 remarkable that, of the two new thermological causes discovered by 
 Fourier, one may be directly observed at the equator and the other 
 at the poles ; whilst, for all the intermediate jioints, our observation 
 must be guided and interpreted by mathematical analysis. 
 
 New as this difficult inquiry is, our progress in it depends only 
 on the perfecting of the observ\atioiis which Fourier's theory has 
 marked out for us. When the dala of the problem thus become 
 better known, this theory will enable ns to lay hold of some certain 
 evidences of the ancient thermological state of our globe, as well as 
 of its future modifications. We have already learned one fact of 
 high importance ; that the periodical state of the earth's surface 
 has become essentially fixed, and cannot undergo any but imper- 
 ceptible changes by the coulinuous cooling of the interior mass 
 through future ages. This rapid sketch will suffice to show what 
 a sudden scientific consistency has been given, by the labours of one 
 man of genius, to this fundamental ])ortion of natural history, which, 
 before Fourier's time, was made up of vague and arbitrary opinions, 
 mingled with incomplete and incoiierent observations, out of which 
 no exact general view could possibly arise.
 
 211 
 
 CHAPTER IV. 
 
 ACOUSTICS. 
 
 This science had to pass, like all the rest, through the theological 
 aiul metaphysical stages ; but it assumed its positive 
 
 • • 1-^ J.tiS TKttUVC 
 
 character about the same tiuie with Barology, and as 
 completely, though our knowledge of it is, as yet, very scanty, in 
 comparison with what we have learned of gravity. The exact 
 information wliicli was obtained in the middle of the seventeenth 
 century about the elementary mechanical pro[)erties of the atmos- 
 phere, opened upa clear conceptionof the production and transmission 
 of sonorous vibrations. The analysis of the phenomena of sound 
 shows us that the doctrine of vibrations offers the exact expression 
 of an incontestable reality. Besides its philosophical interest, and 
 the direct importance of the phenomena of Acoustics, this depart- 
 ment of Physics appeals to special attention in two principal 
 relations, arising from its use in ]ierfecting our fundamental ideas 
 regarding inorganic bodies, and Man himself. 
 
 By studying sonorous vibrations, we obtain some insight into the 
 interior mechanical constitution of natural bodies, p,; ,• ^^ 
 manifested by the modifications undergone by the the study of 
 vibratory motions of their molecules. Acoustics inorganic 
 affords the best, if not the only means lor this 
 inquiry ; and the small i)resent amount of our acquisitions seems to 
 me no reason why we should not obtain abundant I'esults when the 
 study of acoustics is more advanced. It has already revealed to us 
 .some delicate properties of natural bodies which could not have 
 })een perceived in any other way. For instance, the capacity to 
 contract habits, — a faculty which seemed to belong exclusively to 
 living beings (I mean the power of contracting fixed dispositions, 
 according to a prolonged series of luiiform im^jressions), — is clearly 
 .shown to exist, in a greater or smaller degree, in inorganic apparatus. 
 By vibratory motions, also, two mechanical structures, j)laced 
 apart, act remarkably upon each other, as in the case of two clocks 
 2)laced upon the same pedestal. 
 
 On the other hand, acoustics forms a basis to physiology for the 
 analysis of the two elementary functions which are jieiadonto 
 most important to the establishment of social relations, Physiology. 
 — hearing and the utterance of sound. Putting aside, in this place,
 
 2r2 POSITIVE PHIIOSOPHY. 
 
 all the nervous phenomena of the case, it is clear that the inquiry 
 rests on a knowledge of the general laws of acoustics, which regulate 
 the mode of vibration of all auditory apparatus. It is remarkably 
 so with regard to the production of the voice, — a phenomenon of 
 the same character with that of the action of any other sonorous 
 Instrument, except for its extreme complication, through the 
 organic variations which affect the vocal system. Yet, it is not to 
 physicists that the study of these two great phenomena belongs. 
 The anatomists and physiologists ought not to surrender it to them, 
 but to derive from physics all the ideas necessary for conducting the 
 research themselves: for physicists are not prepared with the 
 anatomical data of the problem, nor yet to supply a sound physiolo- 
 gical interpretation of the results obtained. Science has indeed 
 suffered from the prejudices which have grown out of the introduc- 
 tion into ])hysics of superficial theoi-ies of hearing and phonation, 
 from physical inquirers having intruded upon the province of the 
 physiologists. 
 
 After Barolog}', there is no science which admits of the application 
 Relation to of mathematical doctrines and methods so well as 
 Mathematics. Acoustics. In the most general view, the phenomena 
 of sound evidently belong to the theory of very minute oscillations 
 of any system of molecules round a situation of stable equilibrium ; 
 for, in order to the sound being produced, there must be an abrupt 
 perturbation in the molecular equilibrium ; and this transient 
 derangement must be followed by a quick return to the primitive 
 state. Once produced, in the body directly shaken, the vibrations 
 may be transmitted at considerable intervals, by means of an elastic 
 medium, by exciting a gradual succession of expansions and con- 
 tractions which ai-e in evident analogy with the waves formed on 
 the surface of a liquid, and have given occasion to the term sonorous 
 undulations. In the air, in particular, so elastic as it is, the 
 vibration must propagate itself, not only in the direction of the 
 primitive concussion, but in all directions, in the same degree. The 
 transmitted vibrations, we nnist obsci-ve, are always necessarily 
 isochronal with the primitive vibrations, though their amplitude 
 may be widely different. 
 
 It is clear from the outset that the science of acoustics becomes, 
 almost from its origin, subject to the laws of rational Mechanics. 
 Since the time of Newton, who was the first to attempt to determine 
 the rate of propagation of sound in the air, acou.stics has always 
 been moi-e or less mixed up wilh th(^ labour of geometers to develop 
 absti-act Meciianics. It was from simple considerations of acoustics 
 that Daniel Bernouilli derived the general principle relating to the 
 necessary and separate coexistence, or independency, of small and 
 various oscillations occasioned at the same time in any system, by 
 distinct concussions. The phenomena of sound afford the best 
 realization of that law, without which it would be impossible to
 
 SCIENTIFIC RELA TIONS OF A CO USTICS. 2 1 3 
 
 explain tlie commonest plienomenon of acoustics, — the simultaneous 
 existence of numerous and distinct sounds, such as we are every 
 moment hearini:^. 
 
 TIiou<;li the comiection of acoustics with rational Mechanics is 
 fdmost as direct and C()m[)lete as that of Barolog^y, this matheumtical 
 character is far less manageahlo in the one case than the other. 
 The most important questions in l)arolo<:i^y are immediately connected 
 with the clearest and most primitive mechanical theories ; whereas 
 the mathematical study of sonorous vihrations depends on that 
 dilficult and delicate dynamical theory, — the theory of the pertur- 
 halions of equilibrium, and the diherential equations which it 
 I'urnishes relative to the highest and most imperfect part of the 
 integral calculus. Vibratory motion of one dimension is the oidy 
 one, even in regard to solids, whose mathematical theory is complete. 
 Of such motion of three dimensions we are, as yet, wholly ignorant. 
 
 To foi-m any idea of the difficulties of the case, we nuist remember 
 that vibratory motions must occasion certain physical modifications 
 of another nature in the molecular constitution of bodies ; and that 
 these changes, though affecting the vibratory result, are too minute 
 and transient to be appreciable. The only attempt that has been 
 made to analyse such a complication is in the case of the thermo- 
 logical effects which result from the vibratory motion. Laplace 
 used this case to explain the difference between the velocity of 
 sound in the air as determined by experiment and that prescribed 
 by the dynamic formula, which indicated a variation of about one- 
 sixth. This difference is accounted for by the heat disengaged by 
 the compression of the atmospheric stiata, which must make their 
 elasticity vary in a greater proportion than their density, thereby 
 accelerating the pro{)agation of the vibratoiy motion. It is true, a 
 great gap is left here; since, as it is impossible to measure this disen- 
 gagement of heat, we must assign to it conjecturally tlie value which 
 compensates for the difference of the two velocities. But we learn 
 from this procedure of Laplace the necessity of combining thermo- 
 logical considerations with the dynamical theory of vibratory 
 motions. The modification is less marked in the case of liquids; 
 and still less in that of solids ; but we are too far behind witii our 
 comparative experiments to be able to judge whether the modification 
 is or is not too inconsiderable for notice. 
 
 Notwithstanding the eminent difficulties of the mathematical 
 theory of sonorous vibrations, we owe to it such ))rogress as has yet 
 been made in acoustics. The formation of the differential equations 
 proper to the phenomena is, independent of their integration, a very 
 important acquisition, on account of the approximations which 
 mathematical analysis allows between questions, otherwise hetero- 
 geneous, which lead to similar equations. This fundamental 
 pioperty, whose value we have so often to recognize, applies 
 remarkably in the [)resent case; and especially since the creation of
 
 214 POSITIVE PHIIOSOFIIY. 
 
 mathematical theniiology, whose principal equations are strongly 
 analogous to those of vibratory motion. — This means of investigation 
 is all the more valuable on account of the difficulties in the way of 
 direct inquiry into the phenomena of sound. We may decide upon 
 the necessity of the atmospheric medium for the transmission of 
 sonorous vibrations ; and we may conceive of the possibility of deter- 
 mining by experiment the duration of the propagation, in the air, 
 and then through other media ; but tlie general laws of the vibra- 
 tions of sonorous bodies escape immediate observation. We should 
 know almost nothing of the whole case if the mathematical theory 
 did not come in to connect the different phenomena of sound, 
 enabling us to substitute for direct observation an equivalent 
 examination of more favourable cases sul)jected to the same law. 
 For instance, when the analysis of the problem of vibrating chords 
 has shown us that, other things being equal, the number of oscilla- 
 tions is in inverse proportion to the length of the chord, we see 
 that the most rapid vibrations of a very short chord may be counted, 
 since the law enables us to direct our attention to very slow vibra- 
 tions. The same substitution is at our command in many cases in 
 which it is less direct. Still, it is to be regretted that the process 
 of experimentation has not been further improved. 
 
 Acoustics consists of three parts. We might perhaps say four, 
 _. . . includino; the timbre (ring or tone) arising from the 
 
 particular mode or vibration or each resonant body. 
 This quality is so real that we constantly speak of it, both in daily 
 life and in natural history: but it would be wandering out of the 
 department of general physics to inquire what constitutes the ring 
 or tone peculiar to different bodies, such as stones, wood, metals, 
 organized tissues, etc., whose properties lie within the scope of con- 
 crete physics. But, if we regard this quality as capable of modifi- 
 cation by changes of circumstances, then we bring it into the domain 
 of acoustics, and recognize its proper position, though we know 
 nothing else about it. That part of the science presents a mere 
 void. 
 
 The three parts referred to are, first, the mode of propagation of 
 ^sounds ; next, their degree of intensity ; and thirdly, their musical 
 tone. Of these departments, the second is that of which our know- 
 ledge is most imperfect. 
 
 SECTION I. 
 
 PROPAGATION OF SOUND. 
 
 As to the first, the propagation of sound, the simplest, most 
 
 Pnypmiation interesting, and best known question is the measure- 
 
 of sound. ment of the duration, especially when the atmosphere 
 
 is the medium. Newton enunciated it very simply, apart from all
 
 INTENSITY OF SOUNDS. 
 
 215 
 
 modifying causes : — tliat the velocity of sourul is thai acquired by 
 a gravitating body falling from a height equal to half the weight 
 of the atmosphere, — sui)posiiig the atmosphere homogeneous. In 
 an analogous way, we may calculate the velocity of sound in the 
 different gases, according to their respective densities and elastici- 
 ties. By this law the speed of sound in the air must be regarded 
 as independent of atmospheric vicissitudes, since, by Mariotte's 
 rules, the density and elasticity of the air always vary in proportion; 
 and their mutual relation alone influences the velocity in question. 
 Of Laplace's rectification of Newton's formula, we took notice just 
 now. — One important result of this law is the necessary identity 
 of the velocity of different sounds, notwithstanding their varying 
 degrees of intensity or of acuteness. If any inequality existed, we 
 should be able to establisli it, from the irregularity which must take 
 place in musical intervals at a certain distance. 
 
 All mathematical calculations about the velocity of sound suppose 
 the atmosphere to be motionless, except in regard to Effect of at- 
 the vibrations under notice, and it is one of the inter- vioxpheric 
 esting points of the case to ascertain what effect is cgitutwn. 
 produced by agitations of the air. The result of experiments for 
 tliis purpose is that, within the limits of the connuou winds, there 
 is no perceptible effect on the velocity of sound when the direction 
 of the atmospheric current is perpendicular to that in which the 
 sound is propagated ; and that when the two directions coincide, 
 the velocity is slightly accelerated if the directions agree, and retarded 
 if they are opposed : but the amount and, of course, the law of this 
 slight perturbation are unknown. — It is only in regard to the air 
 that the velocity of sound has been effectually studied, 
 
 SECTION II. 
 
 INTENSITY OF SOUNDS. 
 
 We cannot pretend to be any wiser about the intensity of sounds, 
 — which is the second pait of acoustics. Not oidy intensity of 
 have the phenomena never been analysed or estimated, sounds. 
 but the labours of the student have added nothing essential to the 
 results of popular experience about the influences which regulate the 
 intensity of sound ; such as the extent of vibrating surfaces, the dis- 
 tance of the resonant body, and so on. These subjects have therefore 
 no right to figure in our programmes of physical science ; and to ex- 
 patiate upon them is to misconceive the character of science, which 
 can never be anything else than a special carrying out of universal 
 reason and experience, and which therefore has for its starting-pouit 
 the aggregate of the ideas spontaneou.sly acquired by the generality 
 of men in regard to the subjects in question. If we did but attend 
 to this truth, we should simplify our scientific expositions not a
 
 2i6 POSITIVE FIIILOSOPHY. 
 
 little, by stripping tlism of a ranltitude of superfluons details which 
 only obscure the additions that science is able to make to the funda- 
 mental mass of human knowledge. 
 
 With regard to the intensity of sound, the only scientific inquiry, 
 — a very easy one, — wliich has been accomplished, relates to the 
 effect of tlie density of the atmospheric medium on the force of 
 sounds. Here acoustics confirms and explains the common obser- 
 vation on the attenuation of sound in proportion to tlie rarity of the 
 air, without informing us whetlier the weakening of the sound is in 
 exact proportion to the rarefaction of the medium, as it is natural 
 to suppose. In my opinion, we know nothing yet of a matter usually 
 understood to be settled, — the mode of decrease of sound, in propor- 
 tion to the distance of the sounding body ; as to which science has 
 added nothing to ordinary experience. It is commonly supposed 
 that the decrease is in an inverse ratio to the square of the distance. 
 This would be a very important law if we could establish it : but it 
 is at present only a conjecture; and I prefer admitting our ignorance 
 to attempting to conceal a scientific void, by arbitrarily extending 
 to this case the mathematical formula which belongs to gravitation. 
 A natural prejudice may dispose us to find it again here ; but we 
 have no proof of its presence. 
 
 It would be strange if we had any notion of the law of the case, 
 when we have not yet any fixed ideas as to the way in which inten- 
 sity of sound may be estimated ; nor even as to the exact meaning 
 of the term. We have no instrument which can fulfil, with regard 
 to the theory of sound, the same oflfice as the pendulum and the 
 barometer with regard to gravity, or the thermometer and electro- 
 meter with regard to iieat and electricity. W^e do not even discern 
 any clear princi[)le by which to conceive of a sonometer. While the 
 science is in this state, it is much too soon to hazard any numerical 
 law of the variations in intensity of sound. 
 
 SECTION III. 
 
 THEORY OF TONES. 
 
 The third department of acoustics, — the theory of tones, — is by 
 Theory of fai" the most interesting and satisfactory to us in its 
 Toves. existing state. 
 
 The laws which determine the musical nature of different sounds, 
 Ihat is, their ])iecise degree of acuteness or gravity, marked by the 
 number of vibrations executed in a given time, are accurately known 
 only in the elementary case of a series of linear, even rectilinear, 
 vibrations produced either in a metallic rod, fixed at one end and 
 free at the other, or in a column of air filling a very narrow cylin- 
 drical pipe. It is by a combination of expeiiment and of mathe- 
 matical tlieory that this case is understood. It is the most impor-
 
 THEORY OF TONES. 217 
 
 taut for tlie analysis of tlie connnonest inorganic instruments, but 
 not for the stnd^'of the mechanism of hearing and utterance. With 
 regard to stretcliod chords, the established mathematical theory is 
 that the number of vibrations in a given time is in the direct ratio 
 of the square root of the tension of the chord, and in the inverse 
 ratio of the product of its length by its thickness. In straight and 
 liomogeneous metallic rods this number is in proportion to the rela- 
 tion of their thickness to the square of their length. This essential 
 difference between the laws of these two kinds of vibrations is owing 
 to the flexibility of the one sounding body and the rigidity of the 
 other. Observation pointed it out first, and especially with regard 
 to the effect of thickness. These laws relate to ordinary vibrations, 
 which take place transversely ; but there ai'e vibrations in a longi- 
 tudinal direction much more acute, which are not affected by thick- 
 ness, and in which the difference between strings and rods disapj)ears, 
 the vibrations varying reciprocally to the length ; a result which 
 might be antici[)ated from the inextensibility of the string being 
 equivalent to the rigidity of the rod. A third order of vibrations 
 arises from the twisting of metallic rods, when the direction becomes 
 more or less oblique. It ought to be observed, however, that recent 
 experiments have shown that these three kinds are not radically 
 distinct, as they can be mutually transformed by varying the direc- 
 tion in which the sounds are propagated. As for the sounds yielded 
 by a slender column of air, the number of vibrations is in inverse 
 ])rojH)i-tion to the length of each column, if the mechanical state of 
 ihe air is undisturbed: otherwi.se, it viiries as the square root of the 
 lelation between the elasticity of the air and its density. Hence it 
 is that changes of temperature which alter this relation in the same 
 direction have here an action absolutely inverse to that which they 
 produce on strings or rods : and thus it is explained by acoustics 
 why it is impossible, as musicians have always found it, to maintain 
 through a changing temperature the harmony at first established 
 between stringed and wind instruments. 
 
 Thus far the resonant line has been supposed to vibrate through 
 its whole length. But if, as usually happens, the slightest obstacle 
 1o the vibrations occurs at any point, the sound undergoes a radical 
 inodification, the law of which could not have been mathematically 
 discovered, but has been cleai-ly apprehended by the great acoustic 
 experimentalist, Sauveur. He has established that the sound pro- 
 duced coincides witli that wliich woidd be yielded by a similar but 
 shorter chord, equal in length to that of the greatest common 
 measure between the two parts of the whole string. The same dis- 
 covery explains another fundamental law, which we owe to the same 
 philosopher, — that of the series of harmonic sounds which always 
 accompanies the principal sound of every resonant string, their 
 acuteness increasing with the natural series of whole numbers ; the 
 truth of which is easily tested by a delicate ear or by experiment.
 
 2i8 POSITIVE nillOSOPIIY. 
 
 Tlie phenomenon is, if not explained, exactly represented by refer- 
 ring it to the preceding case ; though we cannot conceive how the 
 spontaneous division of the string takes place, nor how so many 
 vibratory motions, so nearly simultaneous, agree as they do. 
 
 These are the laws of simple sounds. Of the important theory of 
 Composition the Composition of sounds we have yet very imperfect 
 of sounds. notions. It is supposed to be indicated by the experi- 
 ment of the musician Tartini, with regard to resulting sounds. He 
 showed that the precisely simultaneous production of any two sounds, 
 sufficiently marked and intense, occasions a single sound, graver 
 than the other two, according to an invariable and simple rule. 
 Interesting as this fact is, it relates to physiology, and not to acoustics. 
 It is a phenomenon of the nerves; a sort of normal hallucination of 
 the sense of hearing, analogous to optical illusions. 
 
 The vibrations of resonant surfaces have exhibited some curious 
 phenomena to observation, though the mathematical theory of the 
 case is still in its infancy : and M. Savart's observations on the 
 vibratory motions of stretched membranes must cast much light on 
 the auditory mechanism, in regard to the effects of degrees of tension, 
 the hygrometrical state, etc. 
 
 The study of the most general and most complicated case, that 
 of a mass which vibrates in three dimensions, is scarcely begun, 
 except with some hollow and regular solids. Yet this analysis is 
 above all important, as without it, it is clearly impossible to com- 
 ])lete the explanation of any real instrument ; even of those in which 
 the principal sound is produced by simple lines, the vibrations of 
 which must always be more or less modified by the masses which 
 are connected with them. We may say that the state of acoustics 
 is such that we cannot explain the fundamental properties of any 
 musical instrument whatever. Dam'el Bernouilli worked at the 
 theory of wind instruments ; a subject which may appear very 
 simple, but which really requires the highest perfection of the 
 science, even putting aside those extraordinary eftects, far transcend- 
 ing scientific analysis, which the art of a musician may obtain from 
 any instrument whatever, and restricting ourselves to influences 
 which may be clearly defined and durably characterized. 
 
 Imperfect as is our review of Acoustics, I hope we nosv understand 
 something of its general character, the importance of its laws, as far 
 as we know them, the connection of its parts, the development that 
 they have obtained, and the intervals which are left void, to be filled 
 up by future knowledge.
 
 219 
 
 CHAPTER V. 
 
 OPTICS. 
 
 Teie emancipation of natural pliilosopliy from theological and meta- 
 physical influence has thus far gone on by means of a succession of 
 partial efforts, each isolated in intention, though all converging to 
 a final end, amidst the entire unconsciousness of those Hypothesis on 
 who were bringing that result to pass. Such an the nature 
 incoherence is a valuable evidence of the foice of that ''-' '^''' 
 instinct which universally characterizes modern intelligence; but it 
 is an evil, in as far as it has retarded and embarrassed and even 
 introduced hesitation into the course of our liberation. No one 
 having hitherto conceived of the positive philosophy as a whole, and 
 the conditions of positivity not having been analysed, much less pre- 
 scribed, with the modifications appropriate to different orders of 
 researches, it has followed that the founders of natural philosophy 
 have remained under theological and metaphysical influences in 
 all departments but the one in which they were working, even while 
 their own labours were preparing the overthrow of those influences. 
 It is certain that no thinker has approached Descartes in the clear- 
 ness and completeness with which he a})prehended the true character 
 of modem philosophy ; no one exercised so intentionally an action 
 so direct, extensive, and effectual on this transformation, though the 
 action might be transitory ; and no one was so independent of the 
 spirit of his contemporaries; yet Descartes, who overtluew llie whole 
 ancient philosophy about inorganic phenomena, and the physical 
 phenomena of the organic, was led away by the tendency of his age 
 in a contrary direction, when he strove to put new life into the old 
 theological and metaphysical concei)t!ons of the moral^ nature of 
 man. If it was so witli Descartes, who is one of the chief types of 
 the progress of the general development of humanity, we cannot 
 be surprised that men of a more special genius, who have been 
 occupied rather with the development of science than of the human 
 mind, should have followed a metaphysical direction in some matters, 
 while in others not very remote they have manifested the true posi- 
 tive spirit. 
 
 These observations are particularly applicable to the philosophical 
 history of Oi)tics,— the department o"f I'hysics in which an imperfect 
 positivism maintains the strongest consistence, — chiefly through the
 
 2 20 POSITIVE PHILOSOPHY. 
 
 mathematical labours whicli are connected with it. The founders 
 of this science are those who have done most towards hiying the 
 foundations of the Positive Philosophy, — Descartes, Huyghens, and 
 Newton ; yet each one of them was led away by the old spirit of the 
 absolute to create a chimerical hypothesis on the nature of light. 
 That Newton should have done this is tlie most remarkable, con- 
 sidering how his doctrine of gravitation had raised the conception of 
 modern philosophy above the point at which Descartes had left it, 
 by establisliing the radical inanity of all research into the nature 
 and mode of production of phenomena, and by showing that the 
 great end of scientific effort is the reduction of a system of particular 
 facts to one singular and general fact. Newton himself, whose 
 favourite saying was, " ! Physics, beware of Metaphysics ! " 
 allowed himself to be seduced by old habits of philosophizing to 
 personify light as a substance distinct from and independent of the 
 luminous body: a conception as metaphysical as it would have been 
 to imagine gravity to have an existence separate from that of the 
 gravitating bod3^ 
 
 After what has been said about the ])hilosophical theory of hypo- 
 thesis, there can be no occasion to expose the fictitious character of 
 the respective doctrines of philosophers on the nature of light. Each 
 one has exposed the untenableness of those of others ; and each 
 explorer has confined himself to the evidence which favoured his 
 own conception. Euler brought fatal objections against the doctrine 
 of emission ; yet, at the present day, our instructors conceal the fact 
 that the advocates- of the emission doctrine have offered equally fatal 
 o])jections to that of undulations. To take the most simple instance 
 — Has the fact of propagation in all directions, charactei-istic of the 
 vibratory motion, ever been reconciled with the common phenomenon 
 of night ; that is, of darkness produced by the interposition of an 
 opaque body ? Does not tlie fundamental objection of the Newto- 
 nians about this matter hold its ground against the system of Des- 
 cartes and Huyghens, untouched at this hour as it was above a 
 century ago, after all the subterfuges that have been in use ever 
 since ? The case is made clearer by the fact that there are pheno- 
 mena which the two theories will suit equally well. If the laws of 
 jcflection and refraction issue with equal ease from the hypotheses 
 (if emission and undulation, it is pretty clear that our business is 
 with the laws, and not with the hypotheses. The mathematical 
 labours (expended on the opposite theories will not have been thrown 
 away ; they will show, in a very short time, that the analytical 
 ap[)aratus is no cei-tain instrument of truth, as it has served the 
 ])urpose of both hypotheses equally well ; as it would, quite as 
 easily, of many others, if the progress of positivity was not exclud- 
 ing, more and more, this vicious method of philosophizing. It is 
 true, the most enlightened advocates of both systems are ready to 
 give up the reality of emission and of undulation, and hold to them
 
 n 1 TOTHESES ABOUT LIGJl T. 221 
 
 only as a matter of logical convenience, — asa rallyiiig-pointof ideas. 
 But if we can pass from the one hypothesis to the other without 
 affecting the science at all, it is clear that such au artifice is need- 
 less. We must admit, as we before said, that the combination of 
 scientific ideas w'ould bo extremely diflicult to minds trained under 
 the prevalent habits of thought, if they were suddenly de[)rived of 
 such a mode of connection as they here contend for ; but it is not 
 the less true that the next generation of scientific thinkers would 
 combine their ideas more easily, and much more perfectly, it" they 
 were trained to regard directly the relations of phenomena, without 
 being troubled by artifices like these, which only obscure scientific 
 realities. 
 
 The history of Optics, regarded as a whole, seems to show that 
 these hypotheses have not sensibly aided the progress of the theory 
 of light, since all our important acquisitions have been entirely 
 indefiendent of them. Tiiis is true not only of the laws of reflection 
 and refraction, which were discovered before these hypotheses were 
 created, but with regard to all the other leading truths of Optics. 
 The hypothesis of emission no more suggested to Newlonthe notion 
 of the unequal refrangibility of the different colours, than that of 
 undulation disclosed to Huyghensthe law of double refraction proper 
 to certain substances. Great discoverers like these observe a con- 
 nection of facts, and then create a hypothesis to account for the con- 
 nection ; and then those who come after them conclude that the 
 chimerical conceptions nmst be inseparable from the immortal 
 discoveries. There is a use, as I have before asserted, in these 
 imaginary conceptions, which, in regard to their one function, arc 
 indispensable. Tiiey serve, transiently, to develop the scientific 
 spirit by carrying us over from the metaphysical to the positive 
 system. They can do this and nothing more, and they accomplished 
 their task some time ago. Their action can henceforth be only 
 injurious, and especially in the case of Optics, as any one may see 
 wlio will inquire into the state of this science, — particularly since 
 the almost universal adoption of the undulatory in the place of the 
 emissive system. 
 
 One more error must be noticed before we leave the subject of 
 the unscientific pui'suit of Optics. Some enlightened Excessive ten- 
 students imagine that the science acquires a satisfac- daiq/ to si/n- 
 tory rationality by being attached to the fundamental ^«'"«''-<^- 
 laws of universal mechanics. The emission doctrine, if it means 
 anything, must suppose luminous phenomena to be in analogy with 
 those of ordinary motion ; and if the doctrine of undulation means 
 anything, it means that the phenomena of light and sound are alike 
 in their vibratoiy agitation ; and thus the one party likens optics 
 to barology and the other to acoustics. But not only is nothing 
 gained by Ihe supposition, but if either was the case, there would 
 be no room for imajiinatiou or for argument. The connection would
 
 42 2 POSITIVE PHILOSOPHY. 
 
 be ut once apparent to all eyes on tlie simple view of the phenomena. 
 Such a reference of phenomena to those fl:;eneral laws has never been 
 a matter of question or of conjecture. The only difficulty has been 
 to know those laws well enough to admit of the npplication. No 
 one doubted the mechanical nature of the principal effects of gravity 
 and sound long before the progress of rational dynamics admitted 
 of their exact analysis. The application powerfully tended, as we 
 have seen, to the perfecting of barology and acoustics; but this 
 was precisely because there was nothing forced or hypothetical 
 about it. It is otherwise with Optics. Notwithstanding all 
 arbitrary suppositions, the phenomena of light will always constitute 
 a category sid generis, necessarily irreducible to any other : a light 
 will be for ever heterogeneous to a motion or a sound. 
 
 Again, physiological considerations discredit this confusion of 
 ideas, by the characteristics which distinguish the sense of sight 
 from those of hearing, and of touch or pressure. If we could abolish 
 such distinctions as these by gratuitous hypotheses, there is no saying 
 where we should stop in our wanderings. A chemical philosopher 
 might make a type of the senses of taste and smell, and proceed to 
 explain colours and tones by likening them to flavours and scents. 
 It does not require a wilder imagination to do this, than to issue as 
 a supposition, now become classical, that sounds and colours are 
 radically alike. It is much better to leave such a pursuit of scientific 
 unity, and to admit that the categories of heterogeneous phenomena 
 are more numei'ous than a vicious systematizing tendency would 
 suppose. Natural philosophy would no doubt be more perfect if it 
 were otherwise ; but co-ordination is of no use unless it rests on 
 real and fundamental assimilation. — Physicists must then abstain 
 from fancifully connecting the phenomena of light and those of 
 motion. All that Optics can admit of mathematical treatment is 
 with relation, not to mechanics, but to geometry, which is eminently 
 a})plicable to it, from the evidently geometrical character of the 
 piincipal laws of light. The only case in which we can conceive 
 of a direct ap[)lication of analysis is in certain optical researches in 
 which observation would immediately furnish some numerical rela- 
 tions : and in no case must the positive study of light give place to 
 a dynamical analysis. These are the two directions in which geome- 
 ters may aid the progress of Optical science, which they have only 
 too effectually im[)eded by prolonging the influence of anti- 
 scientific hypotheses through inappropriate and ill-conceived 
 analyses. 
 
 The genius of Fourier released us from the necessity of applying 
 the doctrine of hypotheses, as previously laid down, to the case of 
 thermology : and neither barology nor acoustics required it. As to 
 olectrology, there are abundance of chimerical conceptions prepon- 
 derant in that de])artment: but their absurdities are so obvious, 
 that almost all their advocates acknowledge them. It is in Optics
 
 DJVJ SIGNS OF OPTICS. 223 
 
 iliat tlie plausibility and consistence of sncli chimeras give tlieni 
 the most inipoitance ; and I have Iheielore chosen that dei)artiiient 
 as the <;round on whicli the_y should be judged. 
 
 We will now pass iVom these useless hypotheses to the real know- 
 ledge that we are in jiossession of about the theory of light. The 
 whole of 0[>tics is iiiiturally divided into four depnrt- Divisions 
 inents, as light, whether homogeneous or c()lnui-ed, is of Optics. 
 direct, reflected, refracted, or diffracted. These elementary efTects 
 usually coexist in ordinary phenomena; but they are distinct, and 
 must therefore be separately considered. These four parts compre- 
 hend all optical ])henomeria which are rigorously universal ; but we 
 must add, as an indispensable complement, two other sections, 
 lelating to double refraction and polarization. These orders of 
 ])henomena are proper to certain bodies ; but, besides that they are 
 a remarkable modification of fundamental phenomena, they appear 
 in more and more bodies, as the study proceeds, and their condi- 
 tions refer more to general circumstances of structure than to inci- 
 dents of substance. For these reasons they ought to be exactly 
 analysed. As for the rest, it is not our business to classify the 
 application of these six departments either to natural history, as in 
 the beautiful Newtonian theory of the rainbow, or to irrelevant 
 ihe arts, as in the analysis of optical instruments. matters. 
 These aj)plications serve as the best measure of the degree of perfec- 
 tion of the science ; but they do not enter into the field of optical 
 j>hilosophy, with which alone we are concerned. 
 
 For the same reasons which have led us to condemn theories of 
 hearing and utterance, in connection with Physics, we Theory of 
 must now refuse to include among optical phenomena vision. 
 ihe theory of vision, which certainly belongs to physiologj\ When 
 physicists undertake the study of it, they bring only one of the 
 special qualifications necessary, being otherwise on a level with the 
 multitude ; and, however im{)ortant their one qualification may be, 
 it cannot fulfil all the conditions. It is in consequence of so many 
 conditions being unfulfilled, that the explanations hitherto otTered 
 have been so incomplete, and therefoi'e illusory. There is scarcely 
 a single law of vision which can be regarded as established on a 
 sound basis, even where the simplest and commonest phenomena are 
 in question. The elementary faculty of seeing distinctly at unequal 
 distances remains without any satisfactory explanation, though 
 physicists have attempted to refer it to almost every part of the 
 ocular apparatus in succession. This humbling ignorance is no 
 doubt owing to scientific men, both physiologists and physicists, 
 having left the theory of sensations in the hands of the meta- 
 ]thysicians, who have got nothing out of it but some decei)tive 
 ideology : but before this time we should have approached to some- 
 thing like positive solutions, but for the bad oi-ganization of scientific 
 labour among us. If, from the time of these questions beginning
 
 2 24 POSITIVE PHILOSOFHY. 
 
 to assume a positive clinivnctei-, anatomists and pliysiologists bad 
 
 occupied themselves with a tlieory of vision grounded on the materials 
 
 furnished by Optical science, instead of looking to physicists for 
 
 solutions which they could not furnish, our condition in regard to 
 
 this important subject would be somewhat less deplorable than it is. 
 
 Another study which must be excluded from Optics, and from 
 
 e ■^i all natural philosophy, is the theory of the colour of 
 
 Specijic 1 V 1 • 4-1 4- T 4. (■ ■ 
 
 Colour of bodies. I need not explam that i am not reierrmg 
 
 bodies. |.^ |]^Q admirable Newtonian experiments on the 
 
 decomposition of light, which have supplied a fundamental idea, 
 
 common to all the departments of Optics. I refer to the attempts 
 
 made to ascertain, now through the theory of emission, and now 
 
 through that of undulation, the inexplicable [)rimitive phenomenon 
 
 of the" elementary colour proper to every substance. The so-called 
 
 explanations, about the supposed faculty of reflecting or transmitting 
 
 sucii and such a kind of rays, or of exciting such and such an order 
 
 of ethereal vibrations, in virtue of certain supposed arrangements of 
 
 the molecules, are more difficult to conceive than the fact itself, and 
 
 are, in truth, as absurd as the explanations that Moliere puts into 
 
 the mouth of his metaphysical doctors. It is lamentable that we 
 
 should have such comments to make in these days. Nobody now 
 
 tries to explain the specific gravity proper to any substance or 
 
 structure : and why should we attempt it with regard to specific 
 
 colour, which is quite as primitive an attribute ? — In physiology, 
 
 the consideration of colours is of high importance, in connection 
 
 with the theory of vision ; and in natural history, it mny prove a 
 
 useful means of classification : but, in optics, the object of the true 
 
 theory of colours is merely to perfect the analysis of light, so as to 
 
 estimate the influence of structure or other circumstance upon 
 
 transmitted or reflected colour, without entei-ing into the causes of 
 
 specific colouring. The field of inquiry is vast enough, without any 
 
 such illusory research as this. 
 
 SECTION L 
 
 STUDY OF DIRECT LIGHT. 
 
 The first department is that of Optics, properly so called, or the 
 
 study of direct light. This and catoptrics are the 
 
 Optics proper. ^^^^ ^.^^.j. ^^ ^j^^ science cultivated by the ancients ; 
 
 but this branch is as old as the knowledge of the law of the 
 rectilinear propagation of light in every homogeneous medium. 
 This primary law makes purely geometrical questions of problems 
 relating to the theory of shadows ; questions difficult to manage in 
 many cases, but not in the most important, — those of very distant 
 luminous bodies, or bodies of extremely small dimensions. The 
 theory depends, both for the shadow and the penumbra, on the
 
 OPTICS. 
 
 225 
 
 determination of an extensible surface, circumscribed at once by 
 the luminous and the illuminated body. — Wiiatevcr its real anliqiiily 
 maybe, this first part of 0})tics is still very imperfect, regarded Ironi 
 the second i)()int of view : that is, with regard to the , , . 
 
 1 t i\ • i -i I- T 1 i I t • 11 1 Imperfections. 
 
 laws 01 tlie inlensily ot light, or what is called 
 pJiotometry. Important as it is to have a clear knowledge, our 
 notions are as yet either vague or precarious as to how the inten- 
 sity of light is modified by such circumstances as its direction, 
 whether emei'gent or incident ; its distance ; its absorption by ihe 
 medium ; and, finally, its coluui'. 
 
 We are met by a grand difficulty at the outset. We have no 
 photometrical instruments that can be de|)ended on 
 for enablmg us to verity our conjectures ori the 
 different modes of gradation of light. All our photometers I'cst on 
 a sort of vicious circle, being devised in accordance with the laws 
 which they are destined to verify, and generally according to the 
 most doubtful of all, in virtue uf its metaphysical origin, — that 
 which relates to distance. We have called light an emanation ; 
 have calculated its intensity by the square of its distance; and then, 
 without confirming this conjecture by any experiment whatever, 
 we have proceeded to found the whole of photometry upon it. And 
 when this conjecture was replaced by that of imdulations, we accepted 
 the same photometry, neglecting the consideration that it must 
 require revision from its very basis. It is clear what our present 
 photometry must be, after such treatment as this. The law relating 
 to direction, in the ratio of the sine of the angle of emergence or of 
 incidence, is no better demonstrated than that of distance, though 
 it comes from a less suspicious source. It has nothing about it at 
 present like Fourier's labours on radiating heat ; and yet it seems 
 as if it would admit of an analogous mathematical elaboration. 
 The only part of photometry which has, as yet, any scientific 
 consistency is the mathematical theory of gradual absorption of 
 light by any medium. Bouguer and Lambert have given us some 
 interesting knowledge about this : but even here we are on unstable 
 ground, for want of precise and unquestionable experiments. 
 Again, the photometrical influence of colour has been the subject of 
 some exact observations ; but we are not yet in possession of general 
 and precise conclusions, unless it be the fixing of the maximum of 
 brightness in the middle of the solar spectrum. Tims, to siun up, 
 in this first, oldest, and simplest department of ojjtics, phiIost>pheis 
 have scarcely outstripped popular observation, — leaving out what 
 belongs to geometry, and the measurement of the velocity of the 
 propagation of light, which is furnished by astronomy. 
 
 VOL. I.
 
 226 POSITIVE PHILOSOPHY. 
 
 SECTION II. 
 
 CATOPTRICS. 
 
 It is otherwise with regard to catoptrics, and yet more, dioptrics, 
 if we discard questions about the first causes of reflection and refrac- 
 tion. Scientific studies have largely extended and perfected universal 
 ideas about those two orders of general phenomena; and the vai'ied 
 effects belonging to them are now referred with great precision to 
 a very small number of uniform laws, of remarkable simplicit}'. 
 The fundamental law of catoptrics, well known by the ancients. 
 Great laxo of and abundantly confirmed by experiment, is that 
 reflection. whatever may be the form and nature of the reflect- 
 
 ing body, and the colour and intensity of the light, the angle of 
 reflection is always equal to the angle of incidence, and in the same 
 normal plane. Under this law, the analysis of the efTects pro- 
 duced by all kinds of mirrors is reduced to simple geometrical 
 problems, which might, it is true, involve some long and difficult 
 calculations, according to the forms of some bodies, if it were not 
 usually sufficient to examine the simple forms of the plane, the 
 sphere, and, at most, the circular cylinder. If we pretended to ab- 
 solute precision in the analysis of images, we might encounter con- 
 siderable geometrical difficulties : but this is not necessary. This 
 analysis depends, in general, mathematically speaking, on the theory 
 (if caustic curves, created by Tschirnhausen. But even in the 
 application of this theory, some conjectures are hazarded ; and the 
 w^ant of direct and exact experiments, and the uncertainty which 
 attends almost all the parts of the theory of vision, pi'event our 
 depending too securely on the reality of the remote results of any 
 general principle that we can yet employ. 
 
 Every luminous reflection upon any body whatever is accompanied 
 Lawofahsorp- by an absorption of more or less, but always of a 
 tion not found, great part of the incident light ; and this gives rise to 
 a second interesting question in catoptrics. But our knowledge about 
 it amounts to very little, from our backwardness in photometry ; so 
 that we have not yet laid hold of any law. We do not know 
 whether the loss is the same in all cases of incidence : nor whether 
 it is connected with the degree of brightness : nor what is the 
 influence of colour upon it: nor whether its variations in diff'erent 
 reflecting bodies are in harmony with other specific, and especially 
 optical characters. These questions are not only untouched: they 
 have never been proposed. All that we know is simply that the 
 absorption of light appears to be always greater (but to what degree 
 we are ignorant) by reflection than by transmission. From this has 
 resulted, in recent times, the use of lenticular beacons, introduced 
 by Frcsnel.
 
 DIOPTRICS. 
 
 ■227 
 
 A move advanced kind of inquiry belongs to the study of trans- 
 parent substances ; but here, again, the laws are ill understood. In 
 these bodies, reflection accompanies refraction, and we have the 
 o))portunity of inquiring by what laws, general or special, the 
 division between transmitted and reflected light takes place. We 
 only know that the last is more abimdant in proportion as the inci- 
 dence is more oblique ; and tliat reflection begins to become total 
 from a certain inclination proper to each substance, and measured 
 exactly with regard to several bodies. The inclination a])pears 
 to be less in proportion as the substance is more I'efracting : but 
 the supposed law of the case is connected with chance conjectures 
 upon the nature of" light, and requires to be substantiated by direct 
 experiment. 
 
 SECTION III. 
 
 DIOPTRICS. 
 
 Of all the departments of Optics, dioptrics is at present the 
 richest in certain and exact knowledge, reduced to a few simple laws, 
 embracing a large variety of phenomena. The fundamental law of 
 refraction was wholly unknown to the ancients, and Great law of 
 was discovered at the same time, under two distinct refraction. 
 and equivalent forms, by Snellius and Descartes. It consists of the 
 constant proportion of the sines of the angles that the refracted ray 
 and the incident ray, always contained in the same normal plane, 
 foim with the perpendicular to the refracting surface, in whatever 
 direction the refraction may be. The fixed relation of these two 
 .sines, when the light passes from a vacuum into any medium what- 
 evei-, constitutes the most important optical coefficient of every 
 natural body, and holds a real rank in the aggregate of its physical 
 characteristics. The philosophers have laboured at its deter- 
 mination with much care and success, by ingenious and exact pro- 
 cesses : they have prepared very extensive tables, which may rival, 
 as to pi'ecision, our tables of specific gravity — the uncertainty not 
 exceeding a hundredth part of the numerical value of the refracting 
 power. If the light passes from one medium to another, the relation 
 of the refraction depends on the nature uf both : but in every case, 
 the inverse passage gives it always a precisely reciprocal value ; 
 as experiment has constantly tshown. Again, while a body under- 
 goes no chemical change, and becomes only more or less dense, the 
 relation of refraction which belongs to it varies in proportion to the 
 specific gravity ; as may be easily shown, especially with regard to 
 liquids, and yet more to gases, in which we can so extensively 
 modify density by temperature and pressure. This is why philos- 
 ophers have adopted, in preference to the proper relation of refrac-
 
 2 28 POSITIVE PHILOSOPHY. 
 
 lion, its quotient by tlie density, which they have named refractinf;^ 
 power ; in order to obtain more fixed and specific characters in the 
 dioptric comparison of different substances. There is substantial 
 ground for this distinction, though its origin was suspicious. But 
 it must be observed that the refracting power varies when the sub- 
 stance does not undergo any chemical change, but passes, as we 
 have seen in the case of water, through different states of aggregation. 
 These variations in the refracting power have given occasion to 
 conflicts between the advocates of the two hypothetical systems, — 
 each of which requires an invariability in the refracting power 
 which we do not know to exist : and the difficulty of separating 
 what is really established from what they require is one of the mis- 
 chievous consequences of anti-scientific hypotheses, and one which 
 may well render tlie actual character of" the science itself doubtful 
 to impartial minds. 
 
 Newton's discoveries of the unequal refrangibility of the different 
 elementary colours form an indispensable comple- 
 SS'o« ''" i"ent of the law of refraction. From the fact of the 
 demcntari/ decom[)Osition of light in a prism, it clearly follows 
 colours. ^i^j^j- ^ijg relation of the sine of incidence, though con- 
 
 stant for each colour, varies in the different portions of the solar 
 spectrum. The total increase which it undergoes from the red rays 
 to the violet measures the dispersum proper to each substance, and 
 must complete the determination of its refracting power in the 
 common tables, where only the mean refraction can be inserted. 
 This estimate constitutes, from its minuteness, one of the most 
 delicate operations of optics, and does not admit of so much exact- 
 ness as that of the refracting action properly so called, especially in 
 bodies which bend the light but little, as the gases ; but it is ascer- 
 tained for a considerable number of substances, solid or liquid. In 
 comparing the changes of the dispersive power as we pass from 
 one body to another, we discover that the variations are not, as 
 Newton supposed, in proportion to the refracting power : and indeed 
 we find, in more than one case, that the light is least dispersed 
 by substances which refract it most. Tiie discovery of this discre- 
 pancy between two qualities which appear to be analogous was 
 made by Dollond, about the middle of the last century. It is an 
 idea of high importance in 0})tics, as it indicates the possibility of 
 Mcluomatism by tiie compensation of the opposite action pertaining 
 to two different substances which, without that, could not cease to 
 dis{)erse the light but by ceasing to bend it. 
 
 The laws of refraction show us that there can be none but 
 purely geometrical difficulties in the analysis of the effects of 
 homogeneous media upon the light which traverses them. The 
 great complication which might arise from the form of the refract- 
 ing body is diminished in ordinary cases by our satisfying ourselves
 
 DIFFRACTION. 229 
 
 Avllli plane, s[)hcrical, or cyliuclrical surfaces : but we should yet 
 find the inquiry embarrassing, and especially in regard to the dis- 
 jiersion, if we did not confine it to an approximate estimate of the 
 few commonest circumstances. 
 
 SECTION IV. 
 
 DIFFRACTION- 
 
 The modification called diffraction has now become one of the 
 essential parts of Optics. It was entered upon by Grimaldi and 
 Newton, advanced by the researches of Dr Young, and completed 
 by those of Fresnel. It consists of the deviation, always accom- 
 panied by a more or less marked dispersion, that light undergoes, in 
 passing close by the edges of any body or opening. Its simplest 
 way of manifesting itself is by the unequal and variously-coloured 
 fringes, some exterior and some interior, which surround the 
 shadows produced in a darkened room. The famous general 
 principle of interferences, discovered by Dr Young, is the most 
 important idea connected with this theory. It was not appreciated, 
 remarkable as it is, till Frosnel made use of it to exjilain sevei-al 
 interesting phenomena, difficult to analyse; and, among others, the 
 celebrated phenomenon of the coloured rings, which were by no 
 means fully accounted for by Newton's admirable efforts. The law 
 of interferences is this : that when two luminous cones emanate 
 from the same point, and follow, for any reason, two distinct 
 courses, but little inclined towards each other, the intensities 
 jiropcr to the two lights neutralize and augment each other alter- 
 nately, increasing by equal and minute degrees, the value of which 
 is determined, the difference in length between the entire paths 
 traversed by the two cones. It is a pity that this important prin- 
 ciple should have suffered, like the rest, from being iiuj^licated wilh 
 chimerical conceptions on the nature of light. 
 
 We have done all that the nature of this Work admits, in regard 
 I0 Optics ; and we must pass over the subjects of the double re- 
 fraction proper to vai'ious crystals, the general law of which was 
 discovered by Huyghens. We must also omit the phenomena of 
 2')olarization, disclosed by Mains. In what I have brought foi-ward, 
 I hope that, while I have pointed out the gaps in this science, of 
 which we are too little conscious at present, I have also placed in 
 a clear light the great and numerous results obtained during the 
 last two centuries, notwithstanding the disastrous preponderance of 
 vain hypotheses about the nature of light over the spirit of rational 
 experimentation.
 
 230 
 
 1 
 
 CHAPTER VI. 
 
 ELECTRO LOGY. 
 
 This last branch of Physics, relatin*^ as it does to the most com- 
 plex and least manifest phenomena, could not be 
 ihstonj. developed till after the rest. The electrical machine 
 indeed is as old as the air-pump; but it was not till a century later 
 that the study assumed a scientific character, through the distinc- 
 tion of the two electricities, Muschenbroek's experiments with 
 the Leyden jar, and then through Franklin's great meteorologi- 
 cal discovery, which was the first manifestation of the influence 
 of electricity in the general system of nature. Up to that time, the 
 isolated observations of philosophers had only suggested the character 
 of generality inherent in this part of Physics, as in all others, by 
 continually adding to the number of substances susceptible of 
 electrical phenomena: and it was not till the end of the last century 
 that this department of Physics presented anything like the rational 
 character which belongs to the others. It is owing to the labours 
 of Coulomb that it takes its place, and still an inferior place, with 
 the rest. 
 
 No other science offers so great a variety of curious and impor- 
 tant phenomena ; but facts do not constitute science, 
 though they are its foundation and material. Science 
 consists in the systematizing of facts under established general 
 laws : and, regarded in this way, Electrology is the least advanced 
 of all the branches of Physics, imperfect as they all are. In the 
 absence of ascertained laws, arbitrary hypothesis has run riot. The 
 Arbitrary simple confidence with which students have explained 
 hypotheses. all phenomena by endowing imaginary fluids with 
 new properties for every fresh occurrence, reminds us of the old 
 metaphysical explanations, — the ancient entities being merely re- 
 placed by supposed fluids. But the delusion is less mischievous 
 liere than in Oj)tics, where the arbitrary conjectures are closely con- 
 nected with real laws, and share their imposing character. In 
 electrology the hypotheses, standing alone, exhibit their barrenness; 
 and everybody can see that they have borne no share in the great 
 discoveries of the last half-century, though the discoveries, once 
 made, have been afterwards attached to the hypotheses. Most 
 people regard them now as a sort of mnemonic apparatus, useful
 
 HYPOTHESES ABOUT ELECTRICITY. 231 
 
 for connect in<; facts in the memory, tliouf^li orIi;inally designed for 
 a very difTerent purpose. They are a bad a{)paratus ibr even this 
 object, which woukl be much better answered by a system of 
 scientific formuhis especially adapted to that use. And, though less 
 mischievous than in Optics, hypotlieses of this order do harm in 
 electrology, as evei-ywhere else, by concealing from most minds the 
 real needs of the science. It should be remembered, moreover, that 
 anti-scientific action like this extends its influence over the succeed- 
 ing and more complex sciences, which, on account of their greater 
 difficulty, require the severest method, the type of which will 
 naturally be looked for in the antecedent sciences. It is a serious 
 injury to transmit to them a radically vicious model. While 
 physicists ai'c using these hypotheses as having avowedly no intrinsic 
 reality, their very use leads students of the successive sciences, and 
 especially physiologists, to consider them the very sublimity of 
 physics, and to proceed to take them for the basis of their own 
 labours. We see how the notion oF magnetic and electric fluids 
 tends to confirm that of a nervous fluid, and to encourage wild 
 dreams about the nature of what is called animal magnetism, in 
 which even eminent physicists have shared. Such consequences 
 show how a study which is naturally favourable to the positive 
 develo})ment of human intelligence may, by vicious methods of 
 philosophizing, become fatal to our understandings. 
 
 From the complex nature of the phenomena, there can be but 
 little application of mathematics in electrology. It Reiationto 
 has as yet borne only a small share in the |)rogress of Mathematics. 
 the science : but it is as well to point out the two ways, — the one 
 illusory, the other real, — in which the application of mathematics 
 has been attempted. 
 
 Those who have occupied themselves with imaginary fluids as the 
 causes of electrical and magnetic phenomena, have Unsound 
 transferred the general laws of rational mechanics to application. 
 the mutual action of their molecules ; thus making the body under 
 notice a mere substratum, necessary for the manifestation of the 
 phenomenon, but unconcerned in its production ; with Mdiicli office 
 the fluid is charged. It is clear that mathematical labours so base- 
 less can serve no other purpose than that of analytical exercise, 
 without adding a particle to our knowledge. In the Sound aiq^U- 
 other case, — of a sound application, — the mathema- catwn. 
 tical process has been based on some general and elementary laws, 
 established by ex])eriment, according to which the study of pheno- 
 mena proper to the bodies themselves has been pursued, — all 
 chimerical hypotheses being discarded. Tiiis is the character of 
 the able researches of ]\1. Ampere and his successors, on the 
 mathematical investigation of electro-magnetic phenomena, in which 
 the laws of abstract dynamics have been efficaciously applied to cer- 
 tain cases of mutual action between electric conductors or magnets.
 
 232 POSITIVE PHILOSOPHY. 
 
 In exfiminiiig the principal parts of electrolop:y, we must exclude 
 all that belongs to the chemical or physiological influence of 
 electricity, and all connection of electricity with concrete physics ; 
 and especially with meteorology. 
 
 Thus limited to the physical and abstract, electrology at present 
 comprehends three orders of researches. The first 
 
 Divisions. 
 
 relates to the production, manifestation, and measure- 
 ment of electrical phenomena: the second, to the comparison of the 
 electric state proper to the diflferent parts of the same mass, or to 
 different contiguous bodies : the third, to the laws of the motions 
 which result from electrization : we may add, as a fourth head, the 
 application of the results under the other three to the special study 
 of magnetic phenomena, which can never henceforth be separated 
 from them. 
 
 SECTION I. 
 
 ELECTRIC PRODCCTIOX. 
 
 Tlie sum of our observations leads us to regard the electric 
 
 condition of bodies as being, more or less evidently, an invariable 
 
 consequence of almost all the modifications they can undergo : but 
 
 Causes of eke- the chief causes of electrization offer themselves, in 
 
 trization. i\iq order of their power and scientific importance, 
 
 thus: chemical compositions and decompositions: variations of 
 
 temperature : friction : pressure : and, finally, simple contact. 
 
 This distribution differs widely from that first indicated by inquiry, 
 
 — friction being long supposed the only, and then the most powerful 
 
 means of producing the electric condition. The comparison of 
 
 means is very far from being exhausted ; but we may be assured 
 
 that the order specified above will never be radically changed. 
 
 There is no doubt that chemical actions are the most general 
 
 Chemical sourccs of electricity, as well as the most abundant ; 
 
 action. as they are with regard to Heat. In the most 
 
 powerful electrical apparatus, and especially in the Voltaic pile, the 
 
 (;hemical action, which at first passed unnoticed, is now recognized, 
 
 thanks to the labours of Wollaston and others, as the principal 
 
 so]u'ce of electrization, which becomes indeed almost insensible 
 
 when care is taken to exclude chemical action. — After this, the 
 
 Thermoiofji- uext inost powerful cause is thcrmological action, 
 
 ml. action. though, till recently, it was recognized only in the 
 
 single case of heated tourmalin. We now know that marked 
 
 (lilierenccs of temperature between consecutive bars of different 
 
 kinds, whether homogeneous or otiierwise in the particular case, 
 
 suffice to induce a marked electrical condition, the more intense as 
 
 the elements are more numei'ous, — the therraometrical conditions 
 
 remaining the same. — These two causes are so powerful, and so
 
 CA USES OF EL E CTRIZA TJOiV. 2 3 3 
 
 (llfficult to exclude, that tlie estimate of tlie otliers becomes a very 
 delicate matter. It is difficult to determine how much influence 
 lo ascribe to any cause after these two, while yet they are ahnost 
 imavoidably present. Thus, even about friction, 
 which used to be regarded as so powerful a cause, it 
 is now doubtful whether the friction itself has any influence, and 
 whether the electrization is not due to the thermometrical, and 
 even the chemical effects which always accompany friction, but 
 whicb used to be altogether overlooked in this instance. 
 
 The case is nearly the same with Pressure, the electric influence 
 of which, however, is, if less marked, more unquestion- 
 able, from our being able to isolate it more. But the 
 remark is above all applicable to the production of the electric state 
 by the simple contact of heterogeneous bodies. It was by this con- 
 tact that Volta brought out the power of his won- 
 derful instrument, while it is well known now that 
 chemical action bears a chief part in it, and that contact contri- 
 l)utes to it in only a secondary manner, if even it be not altogether 
 doubtful. 
 
 Besides these leading causes of electrization, there are many less 
 important, — as changes in the mode of aggregation. 
 The fusion of solids, and the evaporation of liquids. 
 Even simple motion suffices, under special conditions, to induce an 
 electric state, as M. Arago has shown in the experiment of the 
 influence of the rotation of a metallic disc upon a magnetized needle, 
 near but not contiguous. Our philosophers, however, must beware 
 of passing into the other extreme from that with which they justly 
 reproach their predecessors. It is, no doubt, prejudicial to electro- 
 logy to neglect all sources of electrization but the most conspicuous: 
 but it may be not less so to carry analysis too far, and see causes of 
 electrization in all sorts of minute phenomena.* 
 
 A special instrument, or class of instruments, naturally corre- 
 sponds to each of the general modes of electrization, ,. ^ 
 
 • • J)\st/VWiTt€fltS 
 
 in order to realize the most favourable conditions for 
 the production and support of the electric state. However important 
 these may be, it is clear that we cannot here enter upon the con- 
 sideration of them. But we must not pass over the instruments 
 invented for the manifestation and measurement of the electric con- 
 dition, — the electroscope and the electrometer. The most eminent 
 })hilosopliers have always attached the highest importance to the 
 perfecting of these instruments, in the invention of which real genius 
 lias often been exhibited. Their perfection is of more consequence 
 than that of electric jiroducers ; because very weak electric powers 
 often answer best in delicate experiments, from their simplicity ; 
 
 * In this paragrajjh, M. Comte allude.s to the now most fertile, but wlien he wrote, 
 the comparatively unknown subject of the development of Electricity by Induction. 
 
 —J. r. N.
 
 234 POSITIVE FIIILOSOPIIY. 
 
 while tlie utmost ingenuity is required in instituting means of mani- 
 festing and measuring the minutest electric effects. — Though the 
 electric condition cannot be measured without being first manifested, 
 and the manifestation leads to some sort of estimate, there is a real 
 distinction between electroscopes and electrometers. Among simple 
 electroscopes, the most remarkable for use in very delicate researches, 
 is that kind called condensers, which render feeble electrical effects 
 sensible through their gradual accumulation : and all these instru- 
 ments are so arranged as to show, by the method of experimentation 
 itself, the positive or negative character of the electricity under 
 notice. — Coulomb's electrical balance is certainly the most perfect 
 of electrometers. It was by its means that he discovered, and that 
 we every day demonstrate, the fundamental law of the variation of 
 electric action, repulsive or attractive, inversely to the square of the 
 distance ; a law which could not be unquestionably obtained by any 
 other means. As we have advanced in the science of electro-mag- 
 netism, a new class of electrometei's has been introduced, ibr purposes 
 of measurement, for which Coulomb's bnlance would not answer. 
 These are the class of multipliers. Valuable and delicate as they 
 are, they have not yet been applied, with so much certainty as the 
 balance, to exact measurements, from the difficulty of proportioning 
 the graduation to the intensity of the observed phenomenon. 
 
 SECTION II. 
 
 ELECTRICAL STATICS. 
 
 The second part of electrology includes what is improperly called 
 electrical statics ; a term imputable to illusory hypotheses about 
 the nature of electricity : yet it is not a wholly absurd title, as it 
 relates, in fact, to the distribution of electricity in a mass, or in a 
 system of bodies, the electric state of which is regarded as invari- 
 able. We may therefore continue to use this abridged term, if we 
 carefully keep clear of all mechanical notions of the equilibrium of 
 any supposed electric fluid, and attach to it a sense analogous to 
 that of Fourier, when he spoke of an equilibrium of heat, and of 
 economists when they speak of an equilibrium of population. 
 
 Considering first the case of an isolated body, Coidomb has estnb- 
 Greatiawof lished a fundamental law which is (metaphorically 
 distribution, expressed) the constant tendency of electricity to the 
 .surface, or, in rational langunge, that after an inappreciable instant 
 of time electrization is always limited to the surface, however it may 
 have been in the first place produced. As for the distribution of 
 the electric state among the different parts of the surface, it depends 
 on the form of bodies, being uniform for the si)here alone, unequal 
 for all other foruLS, but always subject to regular laws. The ana- 
 lysis of these may be supposed to present insurmountable diffi-
 
 ELECTRICAL DYNAMICS. 235 
 
 culties ; nevertheless, Coulomb lias eslablislied a general fact of 
 g^reat importance, by compariiijjj the electric states proper to the 
 extremities of an ellipsoid <;-radually elongated : he has perceived 
 that their electrization increases rapidly as the figure is elongated, 
 diminishing in the rest of the body; whence he deduced an expla- 
 nation of that remarkable power of points, disclosed by Franklin.* 
 
 The laws of electric equilibrium between several contiguous 
 bodies afford a yet more difficult and extensive in- ejchhc 
 quiry. Coulomb studied them only in the limited equilibrium. 
 and insufficient single case of spherical masses. However, we learn 
 from his labours that the nature of substances exercises no influence 
 over the electric distribution established among them, the mode 
 depending merely on their form and their magnitude ; only, the 
 electric state assumed by each surface is more or less persistent, and 
 manifests itself with more or less rapidity, according to the degree 
 of conductibility in the body. Coulomb analysed completely the 
 mutual action of two equal spheres; discovering that the electric 
 condition is always null at the point of contact, scarcely sensible at 
 20 degrees from that point, fast increasing from 60 to 90 degrees, 
 and then more slowly increasing up to 180 degrees, which is its 
 maximum. If the globes are unequal, the smallest is the most 
 strongly affected : and it makes no difference whether they are 
 electrized together, or the one before the other. The question 
 becomes more complex when more than two bodies are concerned. 
 Coulomb examined only a series of globes ranged in a straight line ; 
 but if they had been so placed as that each should touch three or 
 four others, the mode of electric distribution would inevitably have 
 undei'gone great changes. The subject must be regarded as merely 
 initiated by this great philosopher ; and no one has added anything 
 to it since his time. It offers to electricians a subject of almost 
 inexhaustible research. f 
 
 SECTION III. 
 
 ELECTRICAL DYNAMICS. 
 
 The third part of electrology is very properly called Electrical 
 Dynamics, because it relates to the motions wliich Amjicrc'sex- 
 result from electrization. Recent as is its origin, it i^eriments. 
 is superior to the others in its scientific condition, through the 
 labours of M. Ampere; always supposing conjectures about the 
 nature of electric phenomena to be discarded. M. Ampere has 
 referred the analysis of the effects observed in this branch of elec- 
 trology to one great and general })lienomenon, the laws of which he 
 has fully ascertained ; the direct and mutual action of two threads, 
 
 * Much has since been added to this class of invest igationsi. — J. P. X. 
 
 f These specific facts are now comprehended within general laws. — J. P. X.
 
 2.^6 POSITIVE PHIIOSOPHY. 
 
 charged with electricity hy Voltaic piles, hahitually reduced to their 
 gi'eatest simplification ; that is, almost always composed of a single 
 element. 
 
 M. Ampere so arranged his experiment as to guard the conduct- 
 ing threads from the j^erturbing influence of the earth's electricity ; 
 and this done, he could easily seize the elementary laws of the 
 ])henomenon under his notice. He foiuul that when the two con- 
 ductors are sufficiently mobile, they tend to place themselves in 
 directions parallel to each other ; and that they then attract or repel 
 each other, according to the conformity or contrariety of the two 
 electric currents. In looking for the laws of the case, it is necessar}^ 
 for the sake of generality and simplicity, to keep in view only 
 infinitely small portions of the different conductors. These laws, 
 mathematically considered, relate either to the influence of the 
 direction, or to that of the distance. 
 
 As to the direction, there are the two cases to be considered of 
 the conducting elements being in the same plane, or in different 
 planes. In the first case, the intensity of the action depends only 
 on the angle formed by each of the two elements with the line which 
 joins their middle points : it is null at the same time with this angle, 
 and increases with it, attaining its maxiuuim when it becomes right. 
 All phenomena, direct or indirect, appear to be exactly represented 
 if this intensity is made to vary in proportion to the sine of the 
 inclination, according to the formula adopted by all the successors 
 of M. Ampere. In the other case, — of the conductors not being in 
 the same plane, — the action depends moreover on the mutual 
 inclination of the planes indicated by each of them, and by the 
 common line of their middle jioints ; and the result of this second 
 relation is wholly different. The perpendicularity of the two planes 
 determines the absence of all action : there is attraction while the 
 angle is acute, and it increases as the angle diminishes, its maximum 
 taking place at the moment of coincidence ; when the angle is obtuse, 
 the action becomes repellent, and increases as each plane approaches 
 towards the prolongation of the other, a situation which produces 
 the maximum of repulsion. The supposition which ai-ises in this 
 case is that the action is in proportion to the cosine of the angle of 
 the two planes ; but we have not yet attained such certainty as in 
 the former case. 
 
 As for the influence of distance, M. Ampere supposed that, in 
 analogy with Coulomb's law of common electric attraction and 
 )-ei)ulHion, the action of two conducting elements is always reciprocal 
 to the square of the distances of their middle points. ])ut analogy 
 is not sufficient to conclude upon ; and direct observation is out of 
 the question when the parts taken are infinitely small, and the result 
 sought must be aflected by the fwm and magnitude of the conduc- 
 tors. However, it may be mathematicnlly demonstrated that, in 
 the hypothesis adopted by M. Ampere, the action of a rectilinear
 
 CONCLUSION OF PIIYSJCS. 237 
 
 conductor, of an indefinite lenfjjtli, upon a magnetized needle, must 
 vary exactly in the inverse ratio of their shortest distance. This 
 consequence has been precisely verified by experiment ; and it places 
 beyond a doubt the reality of the proposed law. 
 
 Under this law, electric action would seem to be, mathematically, 
 in analogy with that of gravitation. But this case afl'ords a lesson 
 against iucaution in transferring to the study of these singular move- 
 ments the ordinary procedure of abstract dynamics. Gravitation is 
 independent of mutual direction, which is the determining influence 
 in electrical dynamics: and thus the parallel fails. We see, further, 
 liow many more difficulties are in the way of the analysis of electric 
 forces than in that of molecular gravitation. If this last is, from 
 its complexity, unmanageable except in the simplest cases, it is no 
 wonder that electrical dynamics has not been mathematically studied 
 further than in one dimension, and never at all in surface. Even 
 this much would be hardly effected but for a last fundamental idea, 
 established by M. Ampere ; that in an infinitely small extent, and 
 as long as the distance is not sensibly changed, the electric action is 
 identical for two conducting elements issuing at the same exti-emities, 
 whatever may be otherwise their difference of form. Such a pro- 
 perty must introduce valuable analytical simplifications, tending 
 to establish a remarkable analogy between electric and ordinary 
 dynamic decompositions. 
 
 These are the grounds on which the study of the various action 
 of electrized threads proceeds. Among the many dispositions of 
 these conductors, the most interesting case is that of the spiral form ; 
 and especially when the turns are very close together. M. Ampere 
 has shown the high importance of this form, in order to imitate, as 
 exactly as possible, the phenomena characteristic of magnetized, 
 bodies.* 
 
 We have now reviewed the philosophy of Physics, noticing in 
 turn the aspects presented by the study of the pro- Conclusion 
 ])erties common to all substances and all structures. of Physia. 
 These are not so much branches of a single study as distinct sciences. 
 Part of our business has been to carry on a philosophical operation, 
 hardly necessary in astronomy, but becoming more and more so as 
 we descend to the more complex sciences ; — that of disengaging 
 veal science from the influence of the old metaphysical i)hil()sophy, 
 under which it still suffers deplorably, and which manifests itself in 
 Physics through illusory and arbitrary conceptions about the primi- 
 tive agents of phenomena. I have been able oidy to indicate the 
 
 * M. Comte concludes the section on Electricity by a slight reference to the dis- 
 coveries of Oersted, Arago, and others, ref,'arding its virtual identity with all we term 
 the magnetic forces. But as the whole of this most interesting and inniortant part 
 of Physics has taken a new form since the date of his work, it has not, for reasons 
 assigned in the Preface, been thought necessary to reproduce bia remarks in thia 
 place. — J. P. Js'.
 
 238 POSITIVE PHIIOSOPHY. 
 
 mischief, and where it resides ; and I must leave the work of puri- 
 fication to rational philosophers, whose attention will, we must hojie, 
 be more and more drawn to this vital question. It is with the same 
 view that I have endeavoured to assign the true application of mathe- 
 matical theories to the principal branches of physics, pointing out 
 by the way the danger of the excessive systematization which is too 
 often sought by carrying the use of this powerful instrument further 
 than the complex nature of the corresponding phenomena would 
 fairly allow. While giving my chief attention throughout to the 
 method, I have pointed out, in brief, the principal natural laws 
 relating to each department of science, discovered by human effort 
 during the two centuries which have elapsed since the birth of 
 Physics, properly so called : and I liave shown what gaps are dis- 
 closed in the course of such a survey. 
 
 Our next study will be of the last science which belongs to the 
 class of general knowledge, or that of inoi'ganic nature. Chemistry 
 relates to the molecular and specific reactions which different sub- 
 stances exert upon each other. It is a more comple-c, and conse- 
 quently more imperfect science than those which we have reviewed : 
 but its general character may be perfected, through the means 
 afforded by its subordination to the anterior sciences. 
 
 f
 
 BOOK IV. 
 
 CHEMISTRY. 
 
 CHAPTER I. 
 
 We have now to review the last of the sciences which relate to the 
 inorganic world. Chemistry has for its object the 
 modifications that all substances may imdergo in their 
 composition in virtue of their molecular reactions. Without this 
 new order of phenomena, the most important operations of terrestrial 
 nature would be incomprehensible to us ; and there is no other class 
 of phenomena so intimate and so complex. Inert bodies can never 
 appear so nearly like vital ones as when they produce in each other 
 those rapid and profound ])ertur])ations which characterize chemical 
 effects. We shall see hereafter that the spirit of all theological and 
 metaphysical philosophy consists in conceiving of all phenomena as 
 analogous to the only one which is known by immediate conscious- 
 ness, — Life: and we can easily understand that the primitive 
 method of philosophizing must have exerted a more powerful and 
 obstinate dominion over chemical phenomena than any other, in the 
 inorganic world. — We must consider, too, that direct and spon- 
 taneous observation must have been applied in the first place only 
 to very complicated phenomena, such as vegetable combustions, 
 fermentations, etc., the analysis of which now requires all the 
 resources of our science : and that the most important chemical phe- 
 nomena are produced only in artificial circumstances, which were 
 long in being devised, and very difficult at first to institute. Easy 
 as it is now for even the most ordinary inquirers to use known 
 substances for the disclosure of new relations, we can hardly imagine 
 the difficulty there must have been, in the infancy of chemistry, in 
 creating suitable subjects for observation : and we caimot suppose 
 that the ancient investigators of nature could have had energy and 
 perseverance to discover the pi-incipal phenomena of the science if 
 they had not been constantly stimulated by the unbounded hopes 
 arisiner from their chimerical notions of the constitution of matter.
 
 240 POSITIVE PHILOSOFUY. 
 
 The complex and doubtful nature of the phenomena , in the first 
 
 Great imper- place, and next, the difficulty of <^ettin<^ at them, are 
 
 fection. quite euougli to account for the tardy and incomplete 
 
 positivity of chemical conceptions, in comparison with all others in 
 
 the inorganic region of nature. If, as we have seen, Physics is 
 
 defective in several respects, much more must that science be so 
 
 which, being at once more difficult and more recent, seeks the laws 
 
 of composition and decomposition. Whichever way we look at it, 
 
 whether speculatively, as to the value of its explanations, or actively, 
 
 as to the previsions which they admit of, this science is evidently 
 
 the least advanced of all the branches of inorganic philosophy. 
 
 Indeed, it is hardly possible to call chemistry a science at all while 
 
 it scarcely ever leads to that precise prevision which is the criterion 
 
 of perfection in speculative knowledge. We can rarely tell what 
 
 will be the result of the smallest and fewest modifications introduced 
 
 among the best explored chemical operations ; and while that is the 
 
 case, however important and numerous may be the facts collected, 
 
 we are in possession of only erudition, and not science. To suppose 
 
 otherwise is to mistake a quarry for an edifice, 
 
 Ik is not to be hoped that chemistry can ever attain a state of 
 
 _, . . rationality so satisfactory as that of the sciences which 
 
 Capacities. i^ii c ' -ii ^ i 
 
 relate to phenomena or a more snnple character ; and 
 
 especially that of the eternal type of natural philosophy, — Astron- 
 omy. But so much of its inferiority seems to be due to a vicious 
 philosophy, and to the defective education of philosophers, that I 
 cannot but hope that a judicious philosophical analysis may con- 
 tribute to a speedy perfecting of so important a science. This is 
 the conviction that I desire to awaken by the rapid sketch which I 
 propose to offer of chemical philosopliy, regarded in all its essential 
 aspects. Little as can be done within the bounds of this section, it 
 is possible that some one eminent inquirer may be impressed by the 
 necessity of submitting to a new and more rational elaboration the 
 fundamental conceptions which constitute the science. 
 
 First, — wiiat is the general object of Chemistry ? Vast and 
 Object of complex as is its subject, the definition of Chemistry 
 Chemistry. jg easier than that of Physics. We ai-e already pre- 
 ])ared for it, indeed, by having contrasted that of Physics with it. 
 It is easy to characterize the phenomena of chemistry, in a direct 
 and marked manner ; for all indicate an alteration, greater or 
 smaller, in the constitution of bodies: that is, a composition or 
 decomposition, and generally both, taking into the account the 
 whole of tlic substances which participate in the action. Thus, at 
 all epochs of scientific development, since chemistry first became an 
 object of speculative study, chemical researches have steadily mani- 
 fested a remarkable originality, which has prevented their being 
 confounded with other parts of natural philosophy ; even while 
 Physics itself was mixed up, as its title shows, with physiology ;
 
 OBJECT OF CHEMISTRY. 241 
 
 which was the case up to a very recent tuue. — It is by this general 
 cliaracter of its ])heiiouiena that Chemistry is (listinguished from 
 Physics which precedes it, and Phy!sioh)i^y which follows it. The 
 three sciences may be considered as having for their object the 
 molecular activity of matter, in all the different modes of which it 
 is susceptible. Each corresponds to one of three successive degrees 
 of activity, which are essentially and natiu'ally distinguished from 
 each other. The chemical action obviously presents something 
 more than the physical action, and something less than the vital. 
 The pliysical activity modifies the arrangement of particles in bodies ; 
 and these moditications are usually slight and transient, and never 
 alter the substance. The chemical activity, on the contrary, besides 
 these alterations in the structure and the state of aggregation, 
 occasions a profound and durable change in the very composition of 
 the particles : the bodies which occurred in the phenomenon are no 
 longer recognizable, — so much has the aggregate of their properties 
 been disturbed. — Again, physiological phenomena show us the 
 molecular activity in a much higher degree of energy ; for, as soon 
 as the chemical combination is effected, the bodies become, once 
 more, completely inert ; whilst the vital state is characterized, over 
 and above all physical and chemical effects, by a double continuous 
 motion of composition and decomposition, adapted to maintain, 
 within certain limits of variation and of time, the organization of 
 the body by incessantly renewing its substance. This is the grada- 
 tion, which no sound philosophy can ever confound, of the three 
 modes of molecular activity. 
 
 Two more characteristics of this science must be pointed out : one 
 relating to its nature, and the other to its general conditions. 
 
 Chemistry would not be classed among the inorganic sciences 
 unless its phenomena were general ; that is, unless sjKcOic ckar- 
 every substance were susceptible of chemical action, acterofUs 
 more or Ics.s. And it is because chennstry is thus "<'''^'^"- 
 radically different from })hysiology that it ranks as the last of the 
 inorganic sciences, — physiological phenomena being, by their nature, 
 })eculiar to certain substances, organized in certain modes. Never- 
 theless, it is incontestable that chemical phenomena present, in every 
 case, something specific, or, to use Bergmann's energetic expression, 
 elective. Not only does each material element ])roduce chemical 
 effects which are altogether peculiar to it, but it is the same with 
 their innumerable combinations of different orders, among the most 
 analogous of which certain fundamental differences are utseivable, 
 even so as to be adopted as their characteristics. While therefore 
 physical differences among difierent bodies are those of degree only, 
 chemical properties are specific. Physical properties affoid thc^ 
 common foundation of material existence ; and it is by chemical 
 pro])erties that individuality is manifested. 
 
 The other characteristic relates to the mode of chemical action. 
 
 Vol. I, Q
 
 242 .POSITIVE PHILOSOPHY, 
 
 The immediate contact of antagonistic particles is absolutely neces- 
 Condition ^'"T ^^ cliemical action ; and therefore one at least of 
 of action. the substances concerned must be fluid or gaseous. 
 When this condition does not alread}' exist, it must be artificially 
 procured by liquefying the substance. It is the earliest axiom in 
 the science, that combination cannot take place, except under this 
 condition ; and there is not an instance upon record of chemical 
 action between two solids, unless at a temperature which obscures 
 the true state of aggregation of substances ; and the action is never 
 so powerful as when both substances are liquid. These facts estab- 
 lish the eminently molecular character of chemical effects, and 
 especially in comparison with physical effects. The distinction from 
 physiological effects is, though less marked, as real, the latter 
 requiring, as we shall see hereafter, the junction of solids with 
 fluids. 
 
 The definition of Chemistry, then, is that it relates to the laws of 
 the phenomena of composition and decomposition, 
 "wliich result from the molecular and specific mutual 
 action of different substances, natural or artificial. 
 
 It will be long, we must fear, before a more precise definition 
 than this can be given. Meantime, however incomplete, the most 
 rational that can as yet be offered is of importance as far as it goes. 
 In this view, and connecting, as usual, the consideration of science 
 with that of prevision, the aim proposed should be this : — the 
 characteristic properties of substances, simple or compound, being 
 given, and those properties being placed in a chemical relation in 
 well-defined circumstances, to determine in what their action will 
 consist, and what will be the chief properties of the new products. 
 This problem is, at all events, determinate ; and nothing contained 
 in it could be omitted without its ceasing to be so ; and the formula 
 therefore contains nothing superfluous. On the other hand, if we 
 could obtain such solutions as are indicated, the application of 
 chemistry to the three great objects, vital phenomena, the natural 
 history of the globe, and industrial operations, would be rationally 
 organized, instead of being, as now, the almost accidental result of 
 the spontaneous development of science. Each question would at 
 once be referred to our fornnila, the data of which would be supplied 
 by the circumstances peculiar to tlie application. Far distant as 
 we are from being able thus to conduct our inquiries, this is the 
 end to be kept in view : and chemists all agree that the most 
 advanced portions of their science are those few and simple questions 
 in which this aim has been more or less completely attained. 
 
 By a continued application of this method, all the data must 
 finally be reducible to the knowledge of the essential properties of 
 simple substances, which would lead to that of the different 
 immediate ])rincii)les : and consequently, to the most complex and 
 remote combinations. As for the study of the elements, that must,
 
 DEFINITION OF CHEMISTR V. 243 
 
 of course, be a matter of direct, experimental elaboration, divided 
 into as many parts as there are undecomposed sub- 
 stances. Whether or not it may be possible to 
 discover, by rational methods, relations between the chemical 
 properties of each element and its ai^gregate physical projjcrties, we 
 mnst lay down as imlispensable a direct exploration of the chemical 
 characters of each element. This general basis once obtained from 
 experiment, all other chemical problems must be suscejjtible of a 
 rational solution, under a small number of invariable laws. 
 
 The classes of combinations naturally divide themselves into two, 
 according, first, to the simi)licity, or the greater or ^ ,. ,. 
 less degree m which the immediate principles are 
 com[)ounded: and, secondly, the number of elements combined. 
 Chemical action is observed to become more difficnlt the more 
 substances are compounded : the greater part of compound atoms 
 belong to the first two orders ; and beyond the third their composi- 
 tion seems almost impossible : and, in the same way, in regard to 
 the number of elements, combinations lose their stability in pro- 
 portion as the elements are multiplied: — there are usually only two ; 
 and scarcely any body involves more than four. Thus, the number 
 of chemical classes must always be very small in regard to the 
 distinction under notice: and each of them must have a correspond- 
 ing law of combination, according to which the result might be 
 certainly anticipated through a knowledge of the data. This would 
 be the scientific perfection of chemistry. Our prodigious remote- 
 ness from such a state is ascribable to the feebleness of our faculties, 
 and, in an accessory way, to their vicious direction. We must 
 remember that the great aim has begun to be fulfilled in one 
 secondary department of chemical research, — the study of propor- 
 tions, as we shall see hereafter. What has been done in that one 
 category makes us ask why an analogous perfection should not be 
 attained in other departments. We may sum up this account of 
 the requisites, with the fully rational definition of Chemistry, that 
 it has for its object, — the properties of all simple national 
 bodies being given, to find those of all the compound difiaitwu. 
 bodies which may be formed from them. Every science falls short 
 of its definition : but a real definition is the first evidence that a 
 science has attained some consistency : it then measures its own 
 advancement from one epoch to another ; and it always keeps 
 inquirers in a right direction, and supports them in a philosophic 
 progress. 
 
 Looking now at our means of investigation, we shall find that in 
 chemistry the law holds good that the complication Means of in- 
 of phenomena coincides with the extension of onr ■vesu-jaium. 
 means of inquiry. 
 
 Here Obsei-valion begins to find its fnll development. Up to 
 this time it has been more or less partial. In astronomy, it
 
 244 POSITIVE PHILOSOPHY. 
 
 is confined to the sense of sight : in physics we use hearing and 
 touch also ; and chemistry employs, hesides these, 
 
 serva ion. ^^g^g ^j^^j gniell. How much is tiuis gained we may 
 know by imagining what would become of chemistry, if we were 
 without taste and smell, which are often the only means by which 
 we can recognize effects produced. The important thing to observe 
 imder this head is that there is nothing accidental, nor even 
 empirical, in such a correspondence ; for, as we shall hereafter see, 
 the sound physiological theory of sensations shows that the apparatus 
 of taste and smell, unlike that of the other senses, operates in a 
 chemical manner, and thus shows these two senses to be specially 
 adapted for the perception of phenomena of composition and 
 decom})osition. 
 
 As for Experiment, it is enough to say that the greater number 
 of chemical phenomena, and especially the most 
 
 xpenmen . instructive, are of artificial production. Still, we 
 must remember that the essential character of experimentation 
 consists in the institution, or the choice of the circumstances of the 
 phenomenon, in order to a more evident and decisive investigation. 
 This process is more difficult in chemistry than in physics, because 
 it is more difficult to institute two parallel cases, undisturbed by 
 the intrusion of irrelevant influences ; and yet this is the fundamen- 
 tal condition of experimentation. On this account, I dissent from 
 the ordinary supposition that the experimental method is more 
 appropriate to chemical than to physical researches. Though this 
 is my view, and though the greater advancement of physics gives it 
 the advantage over chemistry in the use of experiment, I can have 
 no doubt of the powerful influence of experimentation in chemistry, 
 independently of its having supplied new subjects of observation. 
 From the early days of the science, the immortal series of Priestley's 
 experiments, and yet more, those of Lavoisier, have offered admir- 
 able models, almost comparable to the most perfect researches in 
 })hysics, and quite enough to prove that there is nothing in the 
 nature of chemical phenomena to prevent the extended and luminous 
 employment of the experimental method. 
 
 The third means, Comparison, which avc have before seen to be 
 . inapplicable in Astronomy, and of especial use in 
 
 ompaiibon. pj^ygjology, begins to have a real use in Chemistry. 
 The essential condition of this valuable method is that there shall 
 be an extended series of cases, analogous but distinct, in which a 
 phenomenon shall be modified more and more, whether by successive 
 simplifications or gradations. It is evident that this can take place 
 fully only with regard to vital phenomena; according!}', it is only 
 by ])hysiological analysis that a clear idea of its value can be 
 obtained. But chemical ])henomena appioach those of ])hysiology 
 nearly enough, not only to demand this method, but to indicate that 
 without it the science can never find the road to perfection. The 
 
 I 
 
 I
 
 ANALYSIS AND SYNTHESIS. 245 
 
 existence of natuial families in clicuiistry is now admitted by the best 
 inquirers : but the chissification remains to be made. The need of 
 llie classification must lead to the use of the comparative method, 
 both being based on the common consideration of the uniformity 
 of certain preponderant ])henomena in a long series of dilYerent bodies. 
 There is even such a ct)nnection between the two orders of ideas 
 that the construction of a natural cheniical classification is impos- 
 sible without a large application of the comparative art, as the 
 ))hysiologists understand it ; and conversely, comparative chemistry 
 cannot be regularly cultivated without the guidance of some sketch 
 of a natural classification. Chemistry is at present only a nascent 
 science; general methods are as yet scarcely recognized in connec- 
 tion with it ; and only a very few i-esearches afford an example of 
 the comparative method ; but I am persuaded, not only of the 
 fundamental suitability of that method in chemistry, but of its 
 ai»plication, before very long, to the perfecting of the science. Such 
 an anticipation, somewhat preceding the spontaneous development 
 of any science, may be a contribution to its actual progress. 
 
 All the means employed are subject, — especially, but not solely 
 in chemistry, — to a verification by the precise coUa- chemical ana- 
 tiou of the two procedures of analysis and synthesis ; lysis and syn- 
 — or (as these terms have been corrupted by meta- '''^^'^• 
 physical uses) composition and decomposition. — Every substance 
 which has been decomi)o.sed must evidently be capable of recom- 
 position, whether the process be otherwise practicable or not. If 
 I he inverse operation reproduces })recisely the primitive substance, 
 the chemical demonstration is complete. Unfortunately, the vast 
 extension of chemical resources in this century has had a much 
 stronger bearing on analytical powers than synthetical means ; so 
 that there is at present little proportion and harmony between the 
 two methods. — Such harmony is indispensable to the establishment 
 of certainty in some cases, as we see when we duly distinguish two 
 widely differing kinds of chemical analysis : the preliminary, 
 consisting of the simple separation of the immediate principles ; 
 and the final, leading to the determination of the elements, ])voi)ev]y 
 so called. Though both are essential to chemical research, the 
 first is of the most important and extensive use. The elementaiy 
 analysis might be spared a synthetical verification, — because the 
 composition of the reacting substances may be compared witii the 
 results obtained, thus indicating the composition of the proposed 
 substance, the different elements of wiiich will in this way have been 
 in some sort separated. The impossibility of recombining the ele- 
 ments, to reproduce the primitive body, ought not to throw any 
 doubt on the solution, unless there is some special reason for sus- 
 ])ecting the simplicity of any one of the elements. Synthesis can, 
 in this case, only add a valuable confiitnation to what was before 
 not doubtful. But the case is very different when we have to
 
 246 POSITIVE PHILOSOPHY. 
 
 determine only the immediate principles. As the elements con- 
 cerned can produce combinations of different orders, we can never 
 be sure that one or more of the supposed immediate principles 
 obtained does not result from the reactions caused by the analysis 
 itself. It is only synthesis which, by reconstructing; the proposed 
 substance with the materials concerned, can decide the question 
 conclusively, thouo;h in some cases of feeble agency in the reactives, 
 and strong analogical induction, there is no room left for reasonable 
 doubt. In immediate analyses of great complexity, when the agree- 
 ment of various analytical means strongly corroborates the con- 
 clusions obtained, we cannot rely on real chemical demonstration 
 without the synthetical confirmation. This maxim of chemical 
 philosophy is abundantly exemplified in the analysis of mineral 
 waters, and yet more of organic substances. — It is noticeable that 
 synthesis is easiest where it is most necessary, and would be most 
 difficult in the case of elementary analysis, where it can, as we have 
 seen, be dispensed with. This is owing to the combinations becom- 
 ing less tenacious as the order of composition of the constituent 
 particles is higher ; and if the decomposition is easy, so is the 
 I'ecomposition. Tiie cases of immediate analysis require only feeble 
 antagonisms, offering no great obstacles to the synthetical operations 
 indispensable for Iheir demonstration. 
 
 nank of the We have next to consider the encyclopedical posi- 
 
 science. tion of Chemistry, to justify the rank assigned to 
 
 it in our scale. 
 
 It is from no vain and arbitrary consideration that Chemistry is 
 })laced between Physics and Physiology in our scale. By the impor- 
 tant series of electro-chemical phenomena Chemistry becomes, as it 
 were, a prolongation of Physics : and at its other extremity, it lays 
 the foundations of physiology by its research into organic combina- 
 tions. These relations are so real that it has sometimes happened 
 that chemists, untrained in the philosophy of science, have been 
 uncertain whether a pailicular subject lay within their department, 
 or ought to be referred cither to physics or to physiologj'. 
 
 The phenomena of Chenu'stry are more complex than those of 
 Physics, and are certainly dependent on them. Their degree of 
 generality is inferior, — chemical effects requiring a much more ex- 
 tended concurrence of varied conditions. Physical properties belong 
 not ordy to all substances, but, with simj)le modifications, to all the 
 states of aggregation, and even of combination, of each of them : 
 whereas, it is only in a more or less determined and restricted con- 
 dition that each body manifests its chemical properties. In a word, 
 nature often shows us physical effects apart from the chemical, while 
 there can be no chemical effects apart from certain physical pheno- 
 mena. Thus Chemistry cannot be rationally studied without a pre- 
 vious knowledge of physics. Besides, the most powerful chemical 
 agents are derived from physics, which presents, in its different
 
 RELATIONS OF THE SCIENCE. 247 
 
 orJets of plionomena, the first distinctive characters of difTcnent 
 substances. It is impossible in our day to conceive of scientific 
 chemistry without givin<]f it the whole of physics for its basis : and 
 tluis is its first relation in the scale established. And, as physics 
 is dependent on astronomy and mathematics, so must its own de- 
 pendent be. But it uuist be owned that, with regard to doctrine, 
 the connection of Chemistry with the first two sciences is neither 
 extensive nor very important. 
 
 Every attempt to refer chemical questions to mathematical doc- 
 trines must be considered, now and always, profoundly irrational, 
 as being contrary to the nature of the phenomena. lu the case of 
 ] ihysics, the mischief would be, as we have seen, merely Eeiation to 
 from the misuse of an instrument which, properly Mathematics. 
 directed, may be of admirable efficacy : but if the employment 
 of mathematical analy.sis should ever become so preponderant in 
 chemistry (an aberration which is happily almost impossible) it 
 Avould occasion vast and rapid retrogradation, by substituting 
 vague conceptions for positive ideas, and an easy algebraic ver- 
 biage for a laborious investigation of facts. The direct subordina- 
 tion of chemistry to astronomy is also slight, but more m . ., 
 marked. It is almost insensible in regard to abstract ' ' ^ ' ' 
 chemistry, which alone is cultivated in our day. But, when the 
 time shall come for the develo{)ment of concrete chemistry, — that 
 is, the methodical application of chemical knowledge to the natural 
 history of the globe, — astronomical considerations will no doubt 
 enter in where now there seems no point of contact between the two 
 sciences. Geology, inunature as it is, hints to us such a future 
 necessity, some vague inxtinct of which was probably in the minds 
 of philosophers in the theological age, when they were fancifully 
 and 3'et obstinately bent on uniting astrology and alchemy. It is, 
 in fact, impossible to conceive of the great intestinal operations of 
 the globe as radically inde{)endent of its planetary conditions. — 
 Inconsiderable as are the relations of chemistry with mathematics 
 and astronomy, in regard to doctrine, it is far otherwise with regard 
 to method. It is easy to see how the perfection of chemistry might 
 be secured and hastened by the training of the minds of clieniists 
 in the mathematical spirit and astronomical philosophy. Besides 
 that mathematical study is the necessary foundation of all positive 
 science, it has a sjiecial use in chemistry in discij)lining the mind to 
 a wise severity in the conduct of analysis : and daily observation 
 shows the evil efi'ects of its absence. Yet, it can never be said that 
 chemists have so much need of a mathematical education as phy- 
 sicists, because they do not need it as an instrument in daily use, 
 but as an intellectual preparation for the rational study of nature. 
 As to astronomy, we have seen that it constitutes the most peifect 
 type of the stutly of nature ; and this at once establishes its relation 
 of superiority to chemistry. The more complex the phenomena, the
 
 248 POSITIVE PHILOSOPHY. 
 
 Tiiore in)])ortant is the influence of such a model ; and it is only hy 
 having always before their eyes such an exemplification of the true 
 Nl)irit of natural })liilosoph3% that chemists can rightly estimate the 
 inanity of the meta[)iiysical explanations which vitiate their doc- 
 trine, and can acquire an adequate sense of the true character, 
 conditions, and destiny of chemical science. Under this point of 
 view, astronomy is more useful to chemists than even physics, in 
 pi-oportion to the superiority of its method. 
 
 So much for the sciences which precede chemistry. As for those 
 that follow, physiology depends upon chemistry both 
 as a pomt oi departure and as a prmcipal means or 
 investigation. If we separate the j)henomena of life, properly so 
 called, from those of animality, it is clear that the first, in the double 
 intestinal movement which characterizes them, are essentially chem- 
 ical. The processes which result from organization have peculiar 
 characteristics ; but apart from such modifications, they are neces- 
 sarily subjected to the general laws of chemical effects. Even in 
 studying living bodies under a sim])ly statical point of view, chemis- 
 tiy is of indispensable use in enabling us to distinguish with pre- 
 cision the different anatomical elements of any organism. — We shall 
 see hereafter that the new science of Social Physics is subordinated 
 to chemical science. In the first place, it depends on 
 ocioogy. j^ ^^^ j^^ immediate and manifest connection with 
 ])hysiology : but, besides that, as social phenomena are the most 
 complex and particular of all, their laws must be subject to those 
 of all the preceding orders, e;ich of which manifests, in social science, 
 its own peculiar influence. In regard to Chemistry especially, it is 
 evident that among the conditions of man's social existence several 
 chemical harmonies between man and external circumstances are 
 involved. Even if individual existence could be sustained, society 
 could not, if these harmonies were destroyed, or even only somewhat 
 disturbed, — as by changes in the atmospheric medium, or in the 
 ■waters oi" the soil. 
 
 The position of Chemistry among the sciences being thus deter- 
 Degree of pos- mined, the next inquiry is about the degree of scien- 
 sible perfection, ^ific ])erfecti()n that its nature admits, in comparison 
 Avith othci-s. As for the ructhod, if physics sutlers from the intiusion 
 of hypotheses, we may say that chemistry has been their absolute 
 ])i-ey, through its more difficult and tardy development. The doctrine 
 of affinities appears to me more ontological than that of fluids and 
 imaginary ethers. If the electric fluid and the luminous ether are, 
 as I called them before, materialized entities, ajftni/ies are at bottom 
 ])ure entities, as vague and indeterminate as liiose of the scholastic 
 ])hilosopliy of the Middle Age. The pretended solutions that they 
 
 ivtr%ision<f yield are of the usual charact(M- of metaphysical ex- 
 
 hypothescx. plauatious, — a mere reproduction, in abstract terms, of 
 the statement of the phenomenon. The advance of chemical know-
 
 HYPOTHESES. 249 
 
 lodge which must at last discredit for ever sncli vjiin philosojihy 
 lins as yet only modified it, so far as to disclose its radical futility. 
 While affinities were regarded as absolute and invariable, there was 
 at least something imposing in them ; but since facts have com- 
 ])elled the belief of their being variable according to a multitude of 
 circumstances, their use has only tended to piove, more and more, 
 their utter inanity. Thus, for instance,, it is known that at a certain 
 temperature, iron decomposes watei-, or protoxide of hydrogen : and 
 yet, it has been since discovered that, under the influence of a higher 
 temperature, hydrogen in its turn decomposes oxide of iron. What 
 signifies, in this case, any order of affinity that we may ascribe to 
 iion and hydrogen with oxygen ? If we make the order vary with 
 the temperature, we have a merely verbal, and therefore pretended 
 explanation. Chemistry affords us now many such cases, apparently 
 contradictory, independently of the long series of decisive considera- 
 tions that have made us reject absolute affinities, — the only ones, 
 after all, that have any scientific consistency whatever. The old 
 habit is, however, so strong that even Berthollet, in the very work 
 in which he overthrows the old doctiine of iuvarial)]e or elective 
 affinities, proposes vague affinities under many modifications. The 
 strange doctrine of predisposing affinity is to be found in the work, 
 among others, of the most rational of recent chemists, the illustrious 
 Berzelius. When, for instance, water is decomposed by iron through 
 the action of sulphuric acid, so as to disengage the hydrogen, this 
 remarkable phenomenon is commonly attributed to the affinity of 
 the sulphuric acid for the oxide of iron which toMds to become 
 i'ormed. Now, can anything be imagined more metaphysical, or 
 more radically incomprehensible, than the sympathetic action of 
 one substance upon another which does not yet exist, and tlie for- 
 mation of the last by virtue of this mysterious affection ? The 
 strange fluids of physicists are rational and satisfactory in comparison 
 with such notions. These considerations justify the desire that 
 chemists should have a sufficient training in mathematical, astro- 
 nomical, and then in physical philosophy, which have already put 
 an end to such chimerical researches within their own domain, and 
 would discard them speedily from the more complex parts of natural 
 jihilosophy. It is only by having witnessed the purification in the 
 anterior sciences that chemists could realize it in their own : and 
 there could not be complete positivity in chemistry if metaphysics 
 lingered in asti'onomy or physics. This, again, justifies the place 
 assigned to chemistry among the sciences. The individual must 
 follow the genei-al course of his race in his passage to the positive 
 state. He must find that true science consists, everywhere, in exact 
 relations, established among observed facts, allowing the deduction 
 of the most extensive series of secondary phenomena from the small- 
 est possible number of oi-iginal phenomena, putting aside all vain 
 inquiiT into causes and essences. And this is the spirit which has
 
 2 50 POSITIVE PHILOSOPHY. 
 
 to be made })repon{lerniit in cliem'ustiy, — dissolving for ever tlie 
 metaphysical doctrine of affinities. 
 
 The inferiority of chemistiy to physics, in regard to method and 
 Actual im- doctrine, explains its relative imperfection with regard 
 perfection. \q actual scicncc. We have only to compare with 
 the formula which told us what chemistry ought to be what it 
 actually is, to see that it is at an immense distance — much further 
 than physics — from its true scientific aim. Chemical facts are at 
 this day essentially incoherent, or, at best, feebly co-ordinated by a 
 small number of partial and insufficient relations, instead of those 
 certain, extended, and uniform laws of which physics is so justly 
 proud. As for prevision, if it is imperfect in physics in comparison 
 with astronomy, it can hardly be said to exist in chemistry at all : 
 the issue of each chemical event being usually known only by 
 specially consulting the immediate experiment, when, as it w^ere, 
 the event is already accomplished. 
 
 Imperfect as chemistry is, in regard to method and doctrine, it is 
 Comparative JQt Superior to physiology, and still more, to social 
 imperfection, scieuce, not only because, from the comparative sim- 
 plicity of its phenomena, the facts and investigations are clearer 
 and more decisive, but because it has a few, though very few, real 
 theories, capable of affording complete previsions ; a thing as yet 
 impracticable, except in a general manner, with living bodies. We 
 siiall have occasion to notice the theory of proportions, the equivalent 
 of which is not, in any sense, to be looked for in physiology. Wo 
 must remember, while estimating the comparative imperfection of 
 the sciences, that the importance to us of their perfection is in pro- 
 portion to their simplicity; our available means being always found 
 to correspond with our reasonable w^ants. I hope, too, that this 
 severe estimate of the actual state of each science will stimulate 
 rather than discourage the student ; for it is more gratifying to our 
 human activity to conceive of the sciences as susceptible of vast, 
 varied, and indefinite progress, than to suppose them perfect, and 
 therefore stationary, except in tiielr secondary developments. 
 
 This leads us to consider the function of Chemistry in the educa- 
 tion of the human mind. 
 
 It may be said to train us in the great art of experimentation : 
 
 Relation to hu- uot as being our exclusive teacher, for, as we have seen, 
 
 man progress. physics is Superior to it in this: and it is more the 
 
 art of observing than of experimenting that Chemistry is chiefly 
 
 distinguished for. But there is an important part of the positive 
 
 metliod which chemistry seems destined to carry to the highest 
 
 perfection. I do not mean the theory of classifications, of which 
 
 Artofnomen- chcmists know too little at present; but the art of 
 
 ckuure. rational nomenclatures, which is quite unconnected 
 
 "with classifications. Since the reform in chemical language, attempts 
 
 have been incessantly made, to this hour, to form a systematic
 
 EDUCATIONAL FUNCTION. 251 
 
 Yiomenclaliire in anatomy, in ])aili(>l( )*;}', and especially in zo()lo<;y : 
 bnt these endeavonrs have not had, and never can have, any success 
 to compare with that of" the reformers of chemical language; for 
 the nature of the phenomena does not admit of it. It is not by 
 accident that the chemical nomenclature is alone in its perfection. 
 The more complex phenomena are, and the more varied and less 
 restricted tlie comparisons of objects, the more difficult it becomes 
 to subject them to a system of denominations, at once i-ational and 
 abridged, so as to facilitate the habitual combination of ideas. If 
 the organs and tissues of the living body differed only from one 
 ])oint of view; if maladies were sufficiently defined by their seat; 
 if, in zoology, genera, or at least families could be established by a 
 homogeneous consideration, the corresponding sciences might at 
 once admit of systematic nomenclatures as rational and as efficacious 
 as that of Chemistry. But the diversity of aspects, rarely reducible 
 to one head, renders such an arrangement extremely difficult and 
 not very advantageous. 
 
 The case of chemistry is the only one in which, by its nature, the 
 ])henomena are simple, uniform, and determinate enough to allow 
 of a rational nomenclature at once clear, rapid, and complete, so as 
 to contribute to the general progress of the science. The idea of 
 composition, the great end of the science, is always preponderant. 
 Thus, the systematic name of each body, expressing its composition, 
 indicates fii'st a correct general view, and then, the sum of its 
 chemical history ; and, by the nature of the science, the more it ad- 
 vances towards perfection, the more must this double property of the 
 nomenclature be developed. In another view, dualism being tlie 
 commonest constitution in chemistry, and the most essential, and 
 that to which all other modes of composition are more and moi-e 
 referred by science, we see that the conditions of the problem are as 
 fiivourable as possible to a rapid and expressive nomenclature. 
 Thus, there has always been some system of nomenclature, more or 
 less rough, though none to be compared to that so ha])pily founded 
 by Guyton-Morvcau. Though the art can manifest its excellence 
 only in pi-oportion to the advance of chemistry, it is in such har- 
 mony with the nature of the science that, in its present imperfect 
 state, it upholds it, by provisionally supplying, as it were, the almost 
 absolute deficiency of true rationality. Thus chemistry may be re- 
 garded as specially adapted to develop one of the few fundamental 
 means, the aggregate of which constitutes the general power of the 
 human m.ind. The formation of a similai- aid in the more com- 
 plex sciences offers a real and strong interest : and I iiave only 
 desired to show that we nuist resort to chemistry for the true 
 j)rinciples and general spirit of llie art of nomenclature, according 
 to the rules so olten set forth in this work, that each great logical 
 artifice should be directly studied in the department of natural 
 philosophy where it is found in the greatest perfection, that it may
 
 252 POSITIVE PHILOSOPHY. 
 
 Le afterwards applied in aid of the sciences to which it less specially 
 lielongs. 
 
 The high philosophical properties of Chemistry are more striking 
 state of chemi- in regard to doctrine than to method. However im- 
 cai doctrine. perfect our chemical science is, its development has 
 operated largely in the emancipation of the human mind. Its oppo- 
 sition to all theological philosophy is marked by the two general tacts 
 in which it has a share with all the rest of positive philosophy, — 
 first, the prevision of plienomena, and next, our voluntary modifica- 
 tion of them. We have already seen that the more the complexity 
 of phenomena baffles our prevision, the greater becomes our power 
 of modifying them, through the variety of resources afforded by the 
 complexity itself ; so that the anti-theological influence of science is 
 infallible, in the one way or the other. In chemistry, our modify- 
 ing power is so strong that the greater part of chemical plienomena 
 owe their existence to human intervention, by which alone circum- 
 stances could be suitably arranged for their production : and if the 
 phenomena of physiology and social science admit of modification 
 in a yet greater degree, chemistry will always, in this particular, 
 hold the first rank, since the higliest order of modifications is that 
 which we here find, — those wiiich are most important for the 
 amelioration of the condition of Man. In the system of the action 
 of man upon nature, chemistry must ever be regarded as the chief 
 source of power, though all the fundamental sciences participate in 
 it more or less. 
 
 In this way, chemistry effectually discredits the notion of the rule 
 of a providential will among its phenomena. But there is another 
 way in which it acts no less strongly ; by abolisliing the idea of 
 destruction and creation in nature. Before anything was known of 
 gaseous materials and products, many striking appearances must 
 inevitably have inspired the idea of the real annihilation or produc- 
 tion of matter in the general system of nature. These ideas could 
 not yield to the true conception of decomposition and composition 
 till we had decomposed air and wa(er, and then analysed vegetable 
 and animal substances, and then iinished with the analysis of alka- 
 lies and earths, thus exhibiting the fundamental principle of the 
 indefinite perpetuity of matter. In vital phenomena, the chemical 
 examination of not only the substances of living bodies, but their 
 functions, — imperfect as it yet is, — must cast a strong light upon 
 the economy of vital nature by showing that no organic matter 
 radically heterogeneous to iiioiganic matter can exist, and that 
 vital transformations are subject, like all others, to the universal 
 laws of chemical phenomena. Chemical analysis seems to have 
 fulfilled its function in this direction : henceforth it must be by the 
 more difficult, but more luminous method of synthesis that this 
 great philosoi)hical revolution uuist be completed : and attempts 
 enough have been successfully made to prove the possibility of it. 
 
 i
 
 PRINCIPLE OF COMPOSITION. 253 
 
 The divisions of tlie science have not been clearly and permanently 
 settled, partly because of its very recent origin, and Divhiomoj 
 partly on account of its nature. In the first place, the science. 
 students have been more occupied in multiplying observations than 
 in classifying them ; and in the next, the homogeneous character 
 of chemical phenomena causes essential differences to be less \no- 
 found, and therefore less marked, than in any other of the funda- 
 mental sciences. In astronomy, there can be no question of a 
 division into geometrical and mechanical phenomena. Physics is 
 less a unique science than a group of almost isolated sciences ; and 
 they indicate their own arrangement. We shall see hereafter that 
 nearly the same thing happens, though from a different cause, in 
 physiology. But in chemistry, the conditions are less favourable, 
 the distinctions being scarcely more marked than those which exist 
 in a single department of physics, — as thermology, and yet more, 
 electrology. The imperfection and small importance of its present 
 divisions are easily explained : and there are strong symptoms of 
 an approaching discussion of this great subject ; for the majority 
 of eminent chemists are more or less dissatisfied with the provisional 
 division which they have been hitherto obliged to accept as guidance 
 in their labouis. 
 
 The general division of organic and inorganic chemistry cannot 
 be sustained, on account of its evident irrationality. jvo ornanic 
 What is at present called organic chemistry has an chcmistrn. 
 essentially bastard character, half chemical, half physiological, and 
 not, in fact, either the one or the other, as we shall have occasion 
 to see. The division cannot even be sustained under another form, 
 as equivalent to the general distinction between cases of dualism 
 and of other composition. For if inorganic combinations are usually 
 binary, there are some which are composed of three elements and 
 even of four ; while, conversely, we very often meet with a true 
 dualism in bodies which are called organic. For a genuine division 
 we must look to general ideas relating to composition principles of 
 and decomposition; and in this form, attending to composition and 
 the rule of following the gradual complication of (frcompositwn. 
 phenomena: first, the growing plurality of constituent principles 
 (mediate or immediate), according as the coml}inations are binary, 
 ternary, etc. ; and secondly, the higher or lower degree of composi- 
 tion of the immediate principles, each of which may (as in the case 
 of a continual dualism) be decomposable into two others, for a 
 greater or smaller number of consecutive times. Though each of 
 these two points of view is of high importance, the preponderance 
 of the one or the other must be agreed u])on before the rational 
 division of chemistry can be organized. Though this is not the 
 place to discuss this new question of high chemical philosophy, it 
 may be well to state that 1 regard it as solved ; and that the con- 
 sideration of the degree of composition is, in my eyes, evidently
 
 254 • POSITIVE PHILOSOPHY, 
 
 superior to tliat of the number of elements, inasmuch as it affects 
 more profoundly the aim and spirit of chemical science, as tliey 
 have been characterized in this chapter. As for the rest, whatever 
 the decision may be, we may remark that the two classifications 
 differ from each other much less than we might at first be tempted 
 to suppose ; for they necessarily concur, whether in the preliminary 
 or in the final case, and diverge only in the intermediate parts. 
 
 We have now reviewed the nature and spirit of chemical science ; 
 the means of investigation proper to it ; its true encyclopedical posi- 
 tion ; the kind and degree of perfection of which it is susceptible ; 
 its philosophical properties in regard to method and to doctrine ; 
 and, finally, the mode of division which would be suitable to it. 
 We must complete the survey of the science by a special and direct 
 notice of the few essential doctrines which have been disclosed by 
 the spontaneous development of chemical philosophy. It must be 
 remembered that the object of this work is not to present a treatise 
 on each science, or to enlarge upon it in proportion to its proper 
 importance, or the multiplicity of its facts : but to ascertain its 
 relative importance, as one head of positive philosophy. No one 
 will expect that chemical philosophy, in its present state, can be 
 examined here as fully or satisfactorily as, for instance, astronomical 
 philosophy, the perfection of which admits of a methodical analysis, 
 clear and complete, though summary, such as befits that immutable 
 type of natural philosophy. 
 
 I
 
 255 
 
 CHAPTER II. 
 
 INORGANIC CHEMISTRY. 
 
 "Whatever may be the principles of division and classification pre- 
 ferred in the general system of cliemical studies, it is Modeofhegin- 
 agreed by almost all chemists that the preliminary ning the study. 
 and fundamental study should be the successive and continuous 
 history of all the simple bodies. The plan of M. Chevreul is an 
 exception to this, his method being to proceed at once from the study 
 of each element to all the combinations, binary, ternary, etc., that 
 it can form with those already examined ; confining himself, how- 
 ever, to compounds of the first order. This plan has the advantage 
 that simple bodies are more completely known from the beginning 
 than by the usual method, which scatters through the different parts 
 of the science the most important chemical properties of each of 
 them. But, on the other hand, tlie history of any element remains 
 incomplete; a factitious inequality is established among chemical 
 researches into different elementary substances ; and the didactic 
 inconvenience which M. Chevreul proposed to escape seems to be 
 unavoidable, under any method. On no plan can any chemical 
 history be completed by a first study. The provisional information 
 obtained by a first study must be followed by a revision which nllows 
 us to take into consideration the whole series of phenomena relative 
 to each substance. The question is merely a didactic one, only of 
 secondary importance in this work, though of great practical interest. 
 On any scheme, it remains certain that the preliminary study of 
 elementary substances is, by the nature of the science, the necessary 
 foundation of chemical knowledge. 
 
 On account of the considerable and always increasing number of 
 substances regarded as simple, some modern philosophers, possessed 
 with the notion of the simplicity and economy of Plurality of 
 natuie, have concluded h priot'i that most substances elements. 
 must be the various compounds of a much smaller number of others. 
 But, while endeavouring to conceive of nature under the simplest 
 aspect possible, we must do so under the teaching of her own 
 ])henomena, not substituting for that instruction any thoughtless 
 desires of our own. We have no right to presume beforehand that 
 the number of simple substances must be either very small or very 
 large. Chemical research alone should settle this ; and all that we
 
 256 POSITIVE PHILOSOPHY. 
 
 are eiiiltleLl to Siiy is tliat our minds sire disposed to prefer tlie 
 smaller number, even, if it were possible, so far as there being but 
 two. But not the less are we bound to sup[)ose all substances which 
 have never in any way been decomposed to be simple, though we 
 should not pronounce them to be for ever undecomposable. All 
 chemists now admit this rule as the first axiom of sound chemical 
 philosophy. 
 
 Aristotle first saw this rule, though he did not conceive of its 
 rational grounds. His doctrine of the four elements, popularly 
 cried down in our time, should be judged of as the first attem})t 
 of the true philosophical spirit to conceive of the composition of 
 natural bodies, amidst the then existing deficiency of all suitable 
 means of research. To appreciate it we must compare it with 
 anterior notions. Now, up to that time, all the schools, however 
 they might differ about other things, agreed that there was only 
 one elementary substance ; and their dispute was about the choice 
 of the principle. Aristotle, with his rational character of mind, put 
 sm end to all those barren controversies by establishing the plurality 
 of elements. This immense progress must be considered the true 
 origin of chemical science, which would be radically impossible ou 
 the supposition of a single element, excluding all idea of composi- 
 tion and decomposition. Whatever appearances may be, there is 
 no doubt that it must be much more difficult for the human mind 
 to pass from the absolute idea of unity of principle to the relative 
 idea of plurality, than to rise gradually, by means of research, from 
 the four elements of Aristotle to the fifty-six simple bodies of our 
 chemistry of this day. Our NaiiD'ists, who are all for simplicity 
 and economy without caring much for reality, have no right to ap- 
 peal to the authority of Aristotle, who had so much reverence for 
 reality as to infringe the notion of simplicity which he found pre- 
 vailing. They should go back further than Aristotle, — to Empe- 
 docles or Heraclitus, and attain the utmost simplicity at once, by 
 admitting only a single principle. 
 
 Other ])hilosophers, among whom was Cuvier, have objected to 
 the simplicity of most of the elements now admitted by chemists, 
 that some of them seem to be extremely abundant in nature, while 
 others are scantily and partially distributed : whereas, it seems 
 natural to presume that the dilTerent elements must be almost 
 equally diffused throughout the globe, and that therefore chemical 
 analysis will sooner or later prove the rare ones to be compound 
 substances, requiring peculiar and rare influences for their formation. 
 It would be enough to say that the presum[)tion, though plausible, 
 is nothing more than a presumption : but it may be added tliat we 
 know nothing of our planet beyond the upper strata; and we can 
 form no prejudgment of the coujposition of the whole. It would be 
 too much to s;iy that there should be an equality of elements on the 
 surface, even the probability being the otlier way ; for the heaviest
 
 CLASSIFICATION OF ELEMENTS. 257 
 
 elemeuts are the rarest at the surface, and the commonest are tliose 
 which '^Q to the composition of livin<^ bodies; and the probability is 
 strong that the preponderance is reversed in the interior of the 
 globe, to make up the mean density, which is not to be found 
 among the solids, liquids, and gases which are required for the 
 existence of life. Thus the objection seems to be converted by 
 chemical analysis into a sort of confirmation. 
 
 Since the time, — recent, it is true, — of the decomposition of the 
 elements of Aristotle, there has not been a single instance of a sub- 
 stance having passed from the class of simple to that of compound 
 bodies, while the inverse case has been frequent. Yet, no chemist 
 disputes the possibility of a reduction of the elements by a more 
 thorough analysis ; for chemical simplicity, as it is to us, is a purely 
 negative quality, not admitting of those irreversible demonstrations 
 proper to positive compositions and recompositions. The great 
 general example of substances called organic, the chemical theory of 
 which is so complex, notwithstanding the small number of their 
 elements, might lead us to suppose that such a reduction would not 
 be after all so very great an advantage : but in this case, the diffi- 
 culty seems to me to be referrible to the deficiency of duality. 
 iS[ot withstanding this example, we cannot but think that chemistry 
 would become more rational and more systematic, if the elements 
 were fewer, from the closer and more general relation which must 
 then subsist among the different classes of phenomena. But the 
 apparent perfection could be only barren and illusory if we were to 
 assume it by conjecture anticipating the real progress of chemical 
 analysis. 
 
 This profusion of elements has naturally led to endeavours to 
 classify them. The high importance of the question Classification 
 has become manifest through the deep persuasion of dements. 
 that the rational classification of simple bodies must determine 
 tliat of compound substances, and therefore that of the whole 
 chemical system. The first principle to be laid down is that 
 the hierarchy of elementary substances is not to be dcterminetl 
 only by their proper essential characters, but by the less direct 
 consideration of the princi[)al phenomena of the compounds which 
 they form. Without this requisition, the classification would 
 have little use or interest; for it would be of small consequence in 
 what conventional order we studied fifty-six bodies all independent 
 of each other: whereas, with its proper condition, this question is 
 as important as any that chemical philosophy can present. 
 
 The old division of the elements into the comburent and com- 
 bustible (those which burn in the active and in the neuter sensej, 
 and the subdivision of these into metallics and non-mctallics, are 
 evidently too artificial to be maintained, except provisionally. For 
 many years, endeavours have been made to supersede it ; but no 
 irreversible classification has been yet obtained. M. Ampere seems 
 to have been the first who pointed out the necessity ; and he pro- 
 
 VOL. I. K
 
 258 POSITIVE PHILOSOPHY. 
 
 posed a system in 1816; but it was not one which induced the 
 chemists to abandon their ancient distribution, the binary structure 
 of which made it easy of application, whatever might otherwise be 
 its defects. A few years after, BerzeHus offered, in a simple and 
 almost incidental form and manner, a far superior system of classi- 
 fication. He first understood the necessity of rising finally to a 
 unique series, constituting, by a uniform and preponderant character, 
 a true hierarchy ; whereas, M. Ampere saw only the importance of 
 natural groups, which might be arbitrarily co-ordinated. Both con- 
 ditions are imposed by the general theory of classifications ; but 
 that which BerzeHus had chiefly in view is unquestionably superior to 
 the other; and especially in the present case, when the small number 
 of objects to be classified renders the formation of groups a matter 
 of secondary importance, provided the series be naturally ordained. 
 M. Berzelius's conception is grounded on the consideration of 
 Classification electro-clicmical ])henomena. Its simple and lucid 
 of BerzeHus. principle is that the elements are to be so disposed as 
 that each shall be electro-negative to those which precede it, and elec- 
 tro-positive to those which follow it. The series thus derived, appears, 
 thus far, to be in conformity with the whole of the known properties 
 of both the elements themselves and their principal compounds. It 
 is too soon, however, to speak decisively of this : and, on the other 
 hand, the chemical preponderance of electric characters is by no 
 means so logically established as to compel us to seek the bases of 
 a natural classification in that order of phenomena. It must, it 
 seems to me, be clearly proved, at the outset, that the point of 
 departure is a real one, — that is, that a constant order of electriza- 
 tion exists among the different elements, which is maintained under 
 all conditions of exterior circumstances, of aggregation and decom- 
 position : but, not only has this never been adequately undertaken, 
 but there is some reason to apprehend that its. result would be 
 opposite to the proposed principle. Whatever may be the issue 
 of future labours, Berzelius has secured the eternal honour of 
 liaving first exhibited the true nature of the problem, and the 
 aggregate of its principal conditions, and perhaps the order of ideas 
 in which its solution is to be sought. Whenever this solution is 
 obtained, chemistry will have made a great stride towards a truly 
 rational state : for, under a hierarchy of the elements, the system- 
 atic nomenclature of compound substances will almost suffice to 
 give a first indication of the general issue proper to each chemical 
 event ; or, at least, to restrict the uncertainty within narrow limits. 
 Yet, through this very connection of such a research with the whole 
 of chemical studies, ] do not think it can be efficaciously pursued 
 while we sepai-ate it, as has hitherto been done, from the general 
 question about the establishment of a complete system of chemical 
 Premature classification for all bodies, simple and compound. 
 fjfort. Now, this great question seems to me at present pre- 
 
 mature. The pieliininaiy conditions, both of method and of doc-
 
 SYSTjEM of classification. 259 
 
 trino, are, as we have seen, far from being comi)iete(l. As sucli a 
 
 general system of classification must constitute both the sinnming 
 
 lip and the fundamental view of tlie wliole of chemical philoso[)hy, 
 
 I shall further expand my idea about it in this place. 
 
 As for the method, it requires perfecting in two ways, for which 
 
 chemists must resort to iihysiology. Tiiey must 7?„„„-<-,. 
 
 11 n t • liequisite pre- 
 
 understand tlie fundamental theory or natural classi- paratimiasto 
 
 fications, which can be obtained nowhere else : and '^^tiwd, 
 they must, for the same reason, study in the same school the general 
 spirit of the comparative method, of which chemists have very little 
 idea, and without which they can never proceed properly in search 
 of a rational classification. Tiiese two improvements must be de- 
 rived from biological philosoj)hy; the one to lay down the problem of 
 chemical classification, and the other to undertake its solution. It 
 will be by perceiving these harmonies and mutual applications 
 among the sciences commonly treated as isolated and independent, 
 that philosophers in all departments will at length become aware 
 of the reality and utility of the fundamental conception of this work ; 
 the cultivation of the different branches of natural philosophy under 
 the impulsion and dii'cction of a genei'al system of positive jihil- 
 <isophy, as a common basis and uniform connection of all scientific 
 labours. We have little iilea what we lose by the narrow and 
 irrational spirit in which the different sciences are cultivated, and 
 especially with regard to method. When the great scientific rela- 
 tions of the future shall be regularly organized, men will scarcely be 
 able to imagine, otherwise than historically, that the study of nature 
 could ever have been conceived and directed in any other way. 
 
 As to the doctrine, we have seen that the desired classification 
 cannot take place till we have settled the preponder- , , ^ . 
 
 c .^ ii ii • 1 i • ii -^s to doctrine. 
 
 ance of the one or the otiier consideration, — the 
 order of composition of the immediate principles, or their degree of 
 l)lurality. Now, such a problem has not yet been rationally pro- 
 posed. If we suppose it resolved, adojiting the rule which I think 
 almost incontestable, as I explained before, of treating the first 
 point of view as necessarily superior to the second, we must still 
 attend to two special conditions, before we can proceed to the 
 rational construction of the system of chemical substances. 
 
 By the first of these conditions, we must dismiss the irrational 
 distinction of substances into orffiinic and inorganic. „. , „„„ ,,vv>. 
 
 --_, , '^ . . ^ First condition. 
 
 We shall see hereafter that organic chemistry must 
 soon dissolve, parting with some of its questions to chemistry 
 proper, and others to physiology. When any combination is sus- 
 ceptible of a chemical examinalion, it must be subjected to a fixed 
 order of homogeneous considerations, whatever may have been its 
 origin and mode of concrete existence, with which chemistry has 
 nothing to do, unless as a source of information. As long as any 
 classification must be adapted to the strange conception of a sort
 
 2 6o POSITIVE PHILOSOPHY, 
 
 ot" double cliemibtiy, establislieJ upon a false division of suLstance.*?, 
 it must be precarious and artificial in its details, because it is viti- 
 ated in principle. The evil is felt, as is shown more and more by 
 the tendency to refer organic combinations to the general laws of 
 inorganic combinations : but it would not be enough, as might be 
 supposed, that a distinguished chemist should take the initiative, 
 in a large and direct manner, to accomplish this important reform. 
 Such a work demands a specinl and difficult operation, requiring a 
 delicate combination of the chemical and physiological ])oint of 
 view, in order to make a true division of what should remain with 
 chemistry, and what should return to physiology. 
 
 The second condition is closely connected with the first. It 
 Second, requires that all combinations should, if possible, be 
 condition. submitted to the law of dualism, erected into a con- 
 stant and necessary principle of chemical philosophy. Great as 
 would be such an improvement in the way of simplification of chemi- 
 cal conceptions, it nnist, however, be admitted that it is not so 
 indispensable to classification as the preceding. Without the first 
 condition, rational classification would be impossible: whereas, it 
 might take place, with imperfection and difficulty, without the 
 second. As ibr the prospects of the case, the tendency to improve- 
 ment is as real and marked in the one case as the other ; as any 
 one may observe for himself. 
 
 It is of the more importance to set the consideration of the order 
 Method of of compositiou of immediate principles above that 
 analysis. of their degree of plurality, as before proposed, because 
 the first is, by its nature, clear and incontestable, while the other 
 Chemical is always more or less obscure and dubious. The one 
 dualism. jg^ j[jj \.^^q^^ ^]^q simple appreciation of an analytical 
 or synthetical fact : the second has always a certain hypothetical 
 character, since we then pronounce upon the mode of agglomeration 
 of elementary particles ; which is a thing radically inaccessible to 
 us. Thus, for example, a chemist may establish with certainty that 
 such or such a salt is a compound of the second order, and that 
 certain acids and alkalies are, on the contraiy, of the first order ; 
 for analysis and synthesis can demonstrate that each of the last 
 bodies is composed of two elementary substances, and that, on the 
 contrary, the immediate principles of the salt are decomposable into 
 two elements. But, in another view, when the analysis of any 
 substance has established the existence in it of three or four 
 elements, as in the case of vegetable or animal matters, we cannot, 
 without resort to hypothesis, [)ronounce that this combination is 
 really ternary or quaternary, instead of being simply binary: for 
 we can never assert that we could not, by a preliminary analysis 
 less violent than this final one, resolve the proposed substance into 
 two immediate ])rincii)les of the first order, each of which should be 
 further susceptible of a new binary deconipositiou.
 
 CHEMICAL DUALISM. 261 
 
 If an unskilled clicinist should at this day apply unduly stronj^ 
 means to the analysis of saltpetre, the results might authorize him, 
 hdlowing our present erroneous ])rocednre, to conceive of this 
 suhstance as a ternary combination of oxygen, azole, and potasshim : 
 and yet wc know that such a conclusion Avould be false, as the 
 substance mny be easily reconstructed by a direct combination 
 between niti-ic acid and potash, which' might have been separated 
 by a less disturbing analysis, without occasioning their decomposi- 
 tion. How do we know that it may not be so with every combina- 
 tion habitually classed as ternary or quaternary ? Immediate 
 analysis being as yet so imperfect in comparison with elementary 
 analysis, especially with regard to these substances, would it bo 
 rational to proclaim, for the time to come, its necessary and eternal 
 impotence with regard to them ? Such judgments seem to be 
 founded on a confusion between these two kinds of analysis, so really 
 different in themselves, and so characterized in their operations by 
 delicacy in the one case and energy in the other. — One important 
 consideration, relating to the synthetical point of view, is evidence 
 of this confusion between the two analyses : and that is, the extreme 
 difUculty, if not impossibility, of verifying by synthesis the ana- 
 lytical results proper to these substances. We have seen that 
 immediate synthesis is usually very easy, while elementary synthesis 
 is scarcely practicable. Thus, reciprocally, it seems to me rational 
 to suppose that when the recomposition cannot be effected, the 
 analysis has not been immediate, — there being no other objection 
 to such a conclusion. For example, we exhibit the impossibility of 
 reproducing by synthesis vegetable and animal substances : and 
 this lias been even set up as a sort of empirical principle. But is 
 not this impossibility owing to our persisting in an elementary 
 synthesis when we ought to proceed by an immediate synthesis, the 
 materials of which might in many cases be discovered beforehand ? 
 This remark is true with regard to a multitude of combinations the 
 dualism of which is, however, very certain, with the sole difference 
 that the immediate principles are better known. If we tried to 
 recompose saltpetre by directly combining oxygen, azote, and 
 potassium, we should succeed no better than in j-eproducing organic 
 substances by throwing together their three or four elements : the 
 obstacles which we admit in the last case apply equally to the first. 
 The most striking achievement is that of M. Woehler, in_ producing 
 the animal substance urea, lie could not have done this if he had 
 tried, according to the common prejudice, to combine directly oxy- 
 gen, hydrogen, carbon, and azote, which concur in the elementary 
 constitution of this substance, instead of uniting only its two imme- 
 diate principles, till then unknown in this quality. Is there any 
 reason to suppose that it is otherwise in arty other case ? — It appeai-s 
 then that chemists will be safe in attributing an entire generality 
 to the fundnmental principle of the dualism of all combinations,
 
 262 rO SI JIVE PHILOSOPHY. 
 
 under the one easy condition of regarding as still very imperfect tlie 
 analysis of substances exceeding the binary composition ; and 
 especially the substances called organic, the true immediate prin- 
 ciples of which would thus remain to be discovered. These prin- 
 ciples can be conceived of only by imagining a considerable number 
 of new binary combinations, of the first and second orders, between 
 oxygen, hydrogen, carbon, and azote : and the realization of this 
 may seem, in the present .state of our knowledge, almost impossible. 
 But we have no right to conclude it to be so, while our analytical 
 procedures are what they are ; and there is no scientific objection 
 to our supposing that there may be many more direct and binary 
 combinations among the elements of ternary or quaternary sub- 
 stances than chemistry has yet established. 
 
 It must be observed, however, that universal and indefinite 
 dualism cannot be maintained unless chemists will scientifically 
 determine the sense of the word substance ; that is, restrict it to 
 mean real combination : for it would be easy to cite, and especially 
 in physiological chemistry, very marked cases of the defect of 
 dualism. But we cannot regard as a true chemical substance an 
 accidental assemblage of heterogeneous substances, whose agglomera- 
 tion is evidently mechanical, such as sap, blood, a biliary calculus, 
 etc., unless we confound the notion of dissolution, and even of 
 mixture, with that of combination. If we extend in this way the 
 use of the term substance, so valuable in chemistry, we might as 
 well treat, as so many chemical substances, the waters of different 
 seas, difterent mineral waters, soils, etc. : and even more, artificial 
 mixtures of a variety of salts dissolved together in water or alcohol. 
 We shall see hereafter that all difliculties in this subject may be 
 disposed of by our learning that they proceed from our not having 
 clearly and rigorously separated the chemical from the physiological 
 point of view. We may be assured that the most elementary 
 notions of chemical philosophy cannot be rationally established, in 
 their due clearness, generality, and stability, without being founded 
 on a full comparison with biology ; a comparison which can be 
 organized only under a complete system of positive philosophy. 
 
 Meantime, there is a marked tendency in the present movement 
 of chemical ideas, towards a complete dualism. Tlie increasing 
 assimilation attempted between organic and inorganic substances 
 is an indirect advance in that road : but much more striking, in 
 this view, are experiments like those of M, Washier, wliich refer the 
 most refractory compounds to dualism, either by analysis or synthe- 
 sis. A binary formula is adojjted, too, 1o represent the pro}x)rtion 
 of elements projjer to the most complex substances : and, though 
 this is not a true dualism, it helps to prepare minds for the estab- 
 lishment of a real and general one. The sum of wliat has been 
 said on this important subject of chemical dualism is this: — the 
 real mode of agglomeration of elementary particles is, and ever
 
 CHEMICAL D UALISM. 263 
 
 must be, unknown to us, and tlierefore no proper object of our 
 study :— our positive researches being thus circumscribed, we may 
 rationally conceive of the immediate composition of any substance 
 as binary ; but so as to represent all the phenomena that chemistry 
 can ofFer to us, in any future state of perfection. Tiuis, I do not 
 propose universal dualism as a law of nature ; for this we could 
 never establish: but I declare it to be a fundamental artifice of 
 true chemical philosophy, destined to simplify our elementary con- 
 ceptions, by using our optional intellectual liberty in accordance 
 with the true end and aim of positive chemistry. 
 
 These are the conditions necessary to the institution of a system 
 of natural classification, answering in chemistry to the universal 
 hierarchy of living bodies in biology, if the complication of pheno- 
 mena would admit of our obtaining such a system. Up to this time, 
 perhaps no one has formed an adequate idea of the nature and 
 spirit of such an operation : but, in my view, chemical classification, 
 thus conceived of, is the science itself, condensed into the most 
 substantial summary. All I claim to have done is to have intro- 
 duced into chemical science the special kind of philosophical s[)irit 
 which is naturally developed by biological science, as it has been 
 conceived of by all its great masters, from Aiistotle downwards. 
 
 It is because I have high expectations of what Chemistry will 
 become, that I attach so much importance to the preceding discus- 
 sion. The science is now weak and desultory, notwithstanding its 
 rich collection of facts : but, extended and complex as it is, there is 
 no fundamental science, except astronomy, whose phenomena are so 
 homogeneous, and therefore so fit for a true systematization, in the 
 positive spirit. Now, this future constitution of chemical science 
 must, it seems to me, consist in a complete system of natural classi- 
 fication, which cannot be obtained till all combinations, whatever 
 their origin, are subjected to a fixed order of homogeneous considera- 
 tions, and, on the other hand, constantly referred to a fundamental 
 dualism. 
 
 We cannot form any certain expectation of the future condition 
 of Chemistry from its present state : but, before proceeding to 
 examine the two doctrines which at this day approach nearest to 
 positive rationality, — that of definite proportions and the electro- 
 chemical theory, — I will indicate two points of doctrine which seem, 
 by their nature, to indicate with precision the true dogmatic forma- 
 tion towards which the science, as a whole, must tend. 
 
 First, there is the great law of double saline decompositions, 
 discovered by Berthollet, and completed by M. Law of double 
 Dulong's investigations on the reciprocal action of saline <iecom- 
 soluble and insoluble salts. The case of double solu- po'^^^^^'- 
 bility, considered by Berthollet, is this : two soluble salts, of any 
 kind, mutually decompose each other whenever their reaction may 
 produce an insoluble salt, or one less soluble than either of the two.
 
 264 POSITIVE PHILOSOPHY. 
 
 This theorem stands first among genei-al propositions in chemistry, 
 and is the only one wliich can as yet give an exact idea of what, in 
 chemistiy, constitutes a true laiu. It has all the characters of a law : 
 it relates to the proper subject of chemical science ; it establishes a 
 relation between two classes of phenomena before independent ; and, 
 above all, it admits of prevision of phenomena according to their 
 positive relations. In establishing this law, Berthollet escaped some 
 metaphysical snares, rejecting hypotheses of aftinities ; but he fell 
 into one when he attempted to explain the law which he had just 
 discovered. No law can be explained otherwise than by showing 
 that it enters into another, more general thnn itself: but this law of 
 Berthollet's is alone of its kind ; and it therefore admits of no 
 explanation. It may hereafter be attached to a fundamental theory 
 of the reciprocal action of all compounds of tbe second order ; and 
 such a relation will truly explain it : but at present it is simply a 
 general fact which, inexplicable itself, serves to explain each of the 
 jiarticular facts which it comprehends. 
 
 The influence of air and water in the production of chemical 
 
 Chemical phenomena is another of the most perfect doctrines of 
 
 theory of air chemistry as it stands. The importance of the action 
 
 and ivater. ^^ .^jj. r^j^^ water in the terrestrial economy has induced 
 
 some German philoso})hers irrationally to set up the system of these 
 
 two fluids into a sort of third reign, between the inorganic and the 
 
 oi-ganic : but abstract chemistry has nothing to do with natural 
 
 history, and regards the study of air and water from a different 
 
 point of view, wdiile aware of its fundamental importance. 
 
 All chemical phenomena take place in the presence of air ; and 
 they almost invariably require the intervention of water : it is clear 
 therefore that before we study any chemical reaction, we must be 
 able to analyse the participation of these two fluids. Thus the 
 chemical theory of air and water is a sort of necessary introduction 
 to the system of chemistry, properly so called, as belonging more 
 to method than to doctrine, and as immediately following the study 
 of simple bodies. It is an historical fact that the double analysis of 
 air and water marked the fii'st great advance in modern chemistry. 
 The influence of the air, not less important than that of water in 
 chemical phenomena, was less difficult to charncterize: 
 for the air is simply a mixtm-e, and its chemical 
 action is merely that of the gnscs which compose it, each of Avhich 
 acts as if it were isolated, allowing for the diminution of intensity 
 from its diffusion, and for the very few cases in which the accom- 
 ])lishment of the pi'oposed phenomenon determines the combination 
 of the gases in an accessory way. Chemistry has only to analyse it, 
 leaving all other study of it to the (lci)artment of natural histor3\ 
 This analysis was effected in the early days of modern chenu'stry, 
 except that there is still some uncertainty about the proportion of 
 carbonic acid gas, and perhaps of some other more considerable
 
 WAl'ER. 265 
 
 ]iiiiicip]es, as, for instance, liydroj:i;en, the existence of wliich begins 
 to be generally suspected. Though no appreciable change in the 
 composition of the atmosphere has taken place within half a century, 
 it is impossible to conceive that some alteration must not happen, 
 in some direction, in course of time, among the many pertui'bing 
 influences which act upon the mixture. Their antagonism, and 
 that of vegetable and animal action, partly neutralizes them: but 
 the equilibrium cannot be precise and continuous. Geological con- 
 siderations and botanical fossils lead us to suppose that at some 
 remote periods the composition of the air must have been sensibly 
 different: and chemists themselves have actually established some 
 slight periodical variations, dependent on the proportion of carbonic 
 acid at different seasons. Our analytical resources are, however, 
 very imperfect with regard to the accessory principles of the atmos- 
 phere; for chemists can ascertain nothing of the distinctions which 
 are proved to exist in the best-marked localities, by their influences 
 on living beings. The study of these vai'iations, all-important in 
 its way, — even as possibly indicating the limits of human life in a 
 remote future, — belongs to natural history ; and that is probably 
 the reason why chemists trouble themselves so little about it: and 
 if there is neglect, it should be charged upon the naturalists. It 
 is true that a preparation is required for their order of study, like 
 all others, — a provision of knowledge, rising from physiology to 
 astronomy itself : but the research is not especially a matter of 
 chemical duty. 
 
 The study of water requires much more extended and complex 
 I'esearches than that of the air ; and it is indispen- 
 sable to the general system of chemical science : for 
 water being a real combination, and perhaps the most perfect known 
 I0 us, may exercise chemical effects proper to itself, independently 
 of those attributable to its elements, and apart from its importance as 
 a solvent, — to say nothing of it as a simple mixture. Thus tliere are 
 three aspects under which water nuist be considered by chemist.s, 
 all distinct and all essential ; and the appreciation of them has been 
 slow and difficult, if even we may say that this fundamental examina- 
 tion is yet complete. 
 
 The analysis of water, represented by a quantity of hydrogen 
 double in volume that of oxygen, and unquestionably confirmed by 
 synthesis, is the finest of the early discoveries of modern chemi.stry, 
 not only from the light it ca.sts upon the whole of chemical pheno- 
 mena and the general economy of nature, but also from its conquest 
 of prodigious difficulties. In regard to the first view, chemical 
 science leaves nothing to desire. Yet, a notion has arisen, in recent 
 times, of the existence of a new and more higldy oxygenated com- 
 bination between the two elements of water, which may rai.-^e some 
 interesting questions, not about the irreversible composition of water, 
 but about the kind of chemical influence which is taken for granted
 
 266 POSITIVE nilLOSOPIIY. 
 
 in its decomposition and recoinposition in a multitude of plienomena ; 
 and especially, about the true mode of union of 0x3' gen and hydro- 
 gen in all substances, and above all in liquids, which cannot be 
 obtained without water. Some doubts have lately been proposed 
 about this, which seem to me to deserve mature examination. 
 
 The dissolving action of water has been the subject of a long 
 series of laborious researches, much less difficult, and not far from 
 complete. Yet more attention ought to be paid than is paid to the 
 fine experiment of Vauquelin, in which it is shown that water, satu- 
 rated with one salt, remains capable of receiving another, and even 
 acquires by that the siiigular i)roperty of dissolving a new quantity 
 of the first. This experiment, which has been in a manner despised, 
 seems to me of the first order in its way, and a fit basis of a series 
 of interesting researches about the apparently capricious laws of 
 solubility, the study of which is yet essentially empirical. 
 
 Chemists were long in conceiving that water, besides being a 
 solvent, might act in a really chemical manner, otherwise than by 
 its elements. It seemed as if a combination so eminently neuter 
 must be inoffensive, and inoperative, except by its decomposition. 
 It was Proust who thought that this neutrality itself afforded a 
 presumption of certain chemical affections, independently of its com- 
 position. This was the rational consideration which led him to 
 create the important study of the hydrates, regarded as a sort of new 
 salts, in which water plays the part, with regard to the alkalies, of a 
 kind of hydric acid. 
 
 The examination of these combinations, and of all others that 
 water can form with any substances without being decomposed, 
 constitutes the third and last part of the fundamental study of water, 
 regarded as an indispensable preliminary to the general system of 
 chemical studies.
 
 267 
 
 CHAPTER III. 
 
 DOCTUINE OF DEFINITE PROPORTIONS. 
 
 There are two general doctrines in chemistry, as it now exists, 
 which present a systematic appearance, and invest the science with 
 such rationality as it has attained. The first of these is the impor- 
 tant doctrine of Definite Proportions. 
 
 Even if this doctrine were complete, it could exert only 
 a secondary influence on the solution of the great scope of the. 
 ])i'oblem of the science, — the study of the haws of doctrine. 
 the phenomena of composition and decomposition. The essential 
 question is, what separations and new combinations must take 
 place under determinate circumstances ; and the theory of de- 
 finite proportions aff'ords no assistance to this kind of prevision. 
 It proceeds, indeed, on the supposition that the question is already 
 solved ; and that it is to be taken as the ])oint of departure for the 
 estimate of each of the new products, — of their quantity and the 
 proportion of their elements. Thus, the theory of definite pro})or- 
 tions presents the singular scientific character of rendering rational, 
 in its numerical details, a solution which usually remains empirical 
 in its most important aspect. 
 
 It was natural that the founders of modern chemistry should 
 have attended to the laws of coniposition and decomposition, in pre- 
 ference to a study which they regarded as subordinate ; and it was 
 natiu-al also that, as the advance of science disclosed to them the vast 
 difficulties of the main problem, they should attend more and more 
 to the secondary study, which promised an easier and more speedy 
 success. But the most important office of this subordinate theory, — 
 that of supplying the defect of immediate experiment, — can be but 
 very imperfectly fulfilled, while it is regarded apart from the 
 I)rincipal theory ; and thus, the doctrine of definite proportions 
 will never acquire its full scientific value till it is connected with 
 an unquestionable basis of chemical laws, of which it will be the 
 indispensable numerical complement. 
 
 Meanwhile, howevei-, it affords a real, though secondary assist- 
 ance to chemists, in rendering their analyses more easy and more 
 precise. Moreover, it restricts the number of cases of combination 
 logically possible, by exhibiting the very small number of distinct 
 proportions; and by thus diminishing the uncertainty in cases of
 
 268 POSITIVE PHIIOSOPIIY. 
 
 chemical action, it is, in fact, a natural preliminary to the establish- 
 ment of those chemical laws lo which it will he, under another view, 
 a necessary supplement. 
 
 In regard to doctrine, this theory offers a perfect type of the 
 precise kind of rationality which must hereafter 
 
 As to doctrine. k ■, j r^t • j i i t i j 
 
 belong to Chemistry as a whole. Jn regard to 
 method, the inquirers who have devoted themselves to establish the 
 theory have adv^anced chemical science while appear- 
 ing to diverge from it ; simplifying the vast problem 
 which their successors will solve, and preparing for the disclosure 
 of the great laws of composition and decomposition, which would 
 be undiscoverable amidst the infinity of products, if substances could 
 combine, within certain limits, in all imaginable proportions. Such 
 are the claims of this theory, as to doctrine and to method. 
 
 It assumed its existence and present form during the first quarter 
 ^, , . , of this century : and it arose from a phenomenon 
 
 J.ts tllStOW • • • • • 
 
 discovered by Eichter, and a speculative discussion 
 established by Berthollet. — During the latter half of the last century, 
 several chemists had observed that, in the mutual decomposition of 
 two neutral salts, the two new salts thus formed are always equally 
 neuter. Bergraann, among others, had steadily and specially dwelt 
 upon this. Yet the fact was neglected or underrated till Eichter, 
 at the end of the century, generalized the observation, saw what 
 it imported, and derived from it the fundamental law which bears 
 , his name. The law is this : that the ponderable 
 
 quantities of the diH'erent alkalies requisite to neutral- 
 ize a given weight of any acid are always proportionate to those 
 requii-ed for the neutralization of the same weight of every other 
 acid. This is, in fact, evidently the immediate consequence of the 
 maintenance of neutrality after the double decomposition. Such a 
 transformation would appear almost spontaneous if it related to a 
 simpler and more developed science than Chemistry ; but amidst 
 its complications and the imperfection of our intellectual habits, 
 the closest deductions are difficult if they have any character of 
 generality, and therefore of abstraction ; and this achievement of 
 Eichter's is, in consequence, eminently meritorious, on other 
 grounds than its high utility. — His law, witli the comi)lements it 
 lias since received, is the original basis of the general doctrine of 
 definite proportions. It exhibited, in the case of a considerable 
 number of compounds, the great end of this doctrine; viz., the 
 assignment to every substance of a certain chemical coefficient, 
 invariable and specific, indicating the proportions in which it can 
 combine with each of those that have been similarly characterized. 
 When it had been determined, by a double series of trials, what 
 was the numerical composition of all the salts that mny be formed 
 by any one acid with the difierent alkalies, and any one alkali with 
 the different acids, Eichter's law enabled us to deduce immedi-
 
 THE ATOMIC THEORY. ^69 
 
 ately the proportions relatiiij^ to all the compounds that can 
 result from the binary combination of these two orders of sub- 
 stances. Richter himself brought his discovery up to this result, 
 and prepared (l)ut on a basis of experiment too narrow and 
 imi)eriect) the first table of what were afterwards called chemical 
 equivalents. 
 
 These neutral salts constituted a particular case, which could 
 hardly have led on to a general theory of definite licrthoiicVs 
 proportions. The idea of perfect neutralization must extension. 
 probably, at all times, have suggested to chemists that of a single 
 proportion, on either side of which the neutrality must be destroyed ; 
 and thus the neutral salts were a natural first stage of the general 
 theory ; but they could not in themselves involve such a theoiy. 
 It was Berthollet who extended the consideration of ])roj)ortions to 
 the whole of chemical phenomena. Some years after Richter's dis- 
 covery, he established as a fundamental piinciple, in his ' Chemical 
 Statics,' the necessary existence of definite proportions for certain 
 compounds of all orders ; and he assigned the essential conditions 
 of this characteristic propeily, which he attiibuted to all causes 
 which can release the product of chemical reaction, as it forms, 
 from the ulterior influence of tiie primitive agents. He thus added 
 to Richter's restricted case the idea of a great number of cases sul>- 
 jected to the same jjrinciple, and able to lead on to its entire 
 generalization. It is assigning much too little honour to Berthollet 
 to recognize only the influence of his controversy with Proust, 
 eminent as was the service rendered by Proust in that conflict, 
 in establishing directly the general principle of determinate and 
 invariable proportions. 
 
 Such was the double origin, experimental and speculative, of 
 numerical chemistry. The next development had also a double 
 character, arising from the harmony between the con- Daiton's further 
 ception of Dr Dalton and the experimental researches extension. 
 of Berzelius, Gay-Lussac, and Wollaston. The inquiry was in a 
 nascent state when Daiton's philosophic mind discerned its possible 
 generality. He proposed the great Atomic theory, under which 
 tiie doctrine of definite proportions was developed to . 
 
 the whole extent that it has reached, and which 
 serves as the basis of its daily application. The general principle 
 of the theory is this: all elementary bodies are conceived of as 
 formed of individual atoms, the ditierent species of which unite, 
 generally by twos, in a small number of groups, constituting com- 
 ])Ound atoms of the first order, always mechanically indivisible, but 
 thenceforth chemically divisible, and, in their turn, constituting all 
 the other orders of composition by a series of analogous combina- 
 tions. The principle is in such harmony with scientific conceptions 
 in all departments, that it appeared like a happy generalization 
 of the most familiar ideas of scientific men in every province of
 
 2 70 POSITIVE PHILOSOPHY, 
 
 natural philosophy ; and its universal aud immediate admission 
 took place as a matter of course. 
 
 It was observed by Berzelius that the deduction of the existence 
 of definite proportions from this principle would be illusory if the 
 combinations were not restricted to a very small number of atoms : 
 for otherwise, — if the number vi^as, though limited, very great, — 
 the binary assemblages would be so multiplied that we might as 
 well have combinations in any proportions whatever ; and then the 
 atomic theory might almost equally well represent the opposite 
 doctrines of definite and indefinite proportions. Dalton was well 
 aware of this ; and the restrictions that he enunciated were presently 
 declared too narrow by his successors, who found that they would 
 not comprehend all existing combinations. His assertion was, that, 
 in every combination, one of the immediate principles always enters 
 for a single atom, and the other generally for a single atom also, 
 and always for a very small number, rarely exceeding six. Taken 
 with the expansion proposed by his successors, the atomic concep- 
 tion evidently represents the entire doctrine of definite proportions. 
 But it is the theory of successive multiples, derived from the 
 })rimary doctrine, which especially distinguishes Dr Dalton's in- 
 fluence upon numerical chemistry. From the ground of his doc- 
 trine he easily saw that if two substances can combine in various 
 distinct proportions, the ponderable quantities of the one which 
 correspond, in the different compounds, to the same weight in the 
 other, must naturally follow the series of whole numbers, since these 
 compounds will have resulted from the union of one atom of the 
 second substance with one, two, three, etc., of the first : and this 
 constitutes a principal element, then perceived for the first time, of 
 the theory of chemical proportions. 
 
 Berzelius followed, with his vast experimental study of the whole 
 Extmaioahy of the important points concerned in numerical 
 Berzelius. chemistry, the diffierent parts of which he has done 
 more than any other chemist to develop and systematize. He 
 first perfected Richter's law, so as to connect it closely with tiie 
 atomic theory ; by which it became susceptible of the extension 
 given to it by Berzelius himself, to all compounds of the second 
 order. But the most important new knowledge has arisen from his 
 numerical study of compounds of the first order. By comparing 
 the composition of the metallic sulphurets and that of the corre- 
 sponding oxides, he discovered a law, analogous to Richter's in 
 regard to the salts. This law, — that the quantity of sulphur of 
 the first is alw;iys proportionate to the quantity of oxygen com- 
 bined with a like weight of the base in the second, — is now 
 regarded, by induction, as applicable to all the compounds of the 
 first order to which the same degree of chemical neuti-ality is 
 jissignable. And again, the luminous series of the analyses of 
 Berzelius have precisely verified in another direction the law of
 
 WOLLASTON'S VERIFICATION. 271 
 
 successive miiUiples discovered by Dalton in pursuance ot his atomic 
 theor}'. 
 
 Gay-Luspac followed, with the valuable numerical analyses he 
 effected by having recourse to gaseous combinations, Extcvsionhy 
 considered, not as to weight, but to volume. He Gaii-Lusxac. 
 thus not only verified, in a special manner, the general principle of 
 definite proportions, but ]iresented it under a new aspect, which, by 
 a wise induction, comprehends all possible cases, — showing that all 
 bodies in a gaseous state combine in invariable and simple numeri- 
 cal relations of volume. An accessory advantage of this achieve- 
 ment was that the specific gravity of the gases might be obtained 
 with a precision often comparable to that of experimental estimate. 
 It is necessary, however, to warn inquirers not to be led away, in their 
 application of the theory of volumes to substances which have never 
 been vaporized, from the point of view which in Gay-Lussac's 
 application is equivalent to Dal ton's, as adopted by Berzelius. 
 
 The labours of Wollaston bore a great part in establishing the 
 doctrine of definite proportions. I do not refer WoUaston's 
 chiefly to his transformation of the atomic theoiy into verification. 
 that of chemical equivalents, though it has a more positive character, 
 and tends to restrain the student from wandering after inaccessible 
 objects, to which the first might tempt him, if not judiciously 
 directed. The substitution would be of high value, no doubt, if it 
 were not less a change of conception than an artifice of language. 
 Nor have I in view tlie ingenious expedients by which Wollaston 
 ]iopularized numerical chemistry by rendering its use more clear 
 and convenient. A greater service, in our present view, was his 
 furnishing us with the indispensable complement of Richter's dis- 
 covery, by establishing the theory in regard to the acid salts, since 
 extended by analogy to the alkaline salts. The case of the acid 
 salts was perhaps the most unfavourable possible for the ascertain- 
 ment of the principle of invariable proportions. Wollaston effected 
 the proof in the most satisfactory manner ; and this special con- 
 firmation of the princijile is considered, from its nature, the most 
 decisive of all. 
 
 Such has been the logical and historical progress of the researches 
 which have constituted numerical chemistry as it is now. We can 
 represent by an invariable number, appropriated to each of the 
 different elementary bodies, their fundamental relations of chemical 
 equivalence, whence, by very simple formulas, immediately express- 
 ing the laws just indicated, we easily pass to the uTUuerical composi- 
 tion proper to each combination. No further evidence of the truth 
 of the doctrine is needed, than the fact of so many illustrious 
 inquirers having attained the same view by ways which each one 
 opened for himself, and all agreeing as to its positive application to 
 all cases of importance, diffeiing only as to the mode of expression 
 of the results, in as far as the atomic tlieory left it indeterminate,
 
 272 rOSITIVE PHILOSOPHY. 
 
 and tlierefove optional. But we must glance at the difficulties 
 thrown in the way of its application by a consideration of the 
 aggregate of chemical phenomena, in order to form a clear idea of 
 the final improvement of which this doctrine yet stands iu need. 
 Among the points which are beyond dispute, it is, first, evident, 
 e / ;■ and no chemist has ever doubted it, that substances 
 
 Scope of appUca- i • ji j.- • ^i ^ c 
 
 tionof Numeri- diiier as much m the proportion as m the nature of 
 cai Chemistry, their Constituent principles. It is an axiom of 
 chemical philosophy that any change whatever in the numerical 
 composition causes a change in the whole of the specific properties, 
 in a more marked degree as the alteration is greater. Varied and 
 gradual above all others as are the proportions produced by the 
 chemical phenomena proper to living bodies, they afford a striking 
 confirmation to this universal maxim. Therefore, in the lowest 
 stages of chemical analysis, chemists have always endeavoured to 
 assign, as a characteristic property, the proportion of the elements 
 of each substance, as far as was possible : and when this was omitted, 
 it was on the understanding that the proposed combination admitted 
 of only a certain proportion ; as in the c;ise of the neutral salts. 
 
 Again, it has long been acknowledged that there always exists, 
 between any two substances, a certain minimum and maximum of 
 reciprocal saturation, beyond or short of which all combination 
 becomes impossible. At the utmost, certain variations, themselves 
 restricted, have been supposed procurable. BerthoUet established, 
 more directly than any one else, the general and necessary existence 
 of these limits of combination, — one of the principal characters 
 which distinguish it from simple mixture. It is clear that the two 
 extreme degrees of all combination must be subject to special and 
 invariable proportions : and, as all agree in this, all argument about 
 the opposite doctrines of indefinite and definite proportions is 
 reduced to the question whether the passage from the minimum to 
 the maximum of saturation can be effected gradually and almost 
 imperceptibly, or whether it takes place always abruptly, through a 
 small number of well-marked degrees. 
 
 Thirdly, the possibility and actual existence of intermediary 
 definite propoi'tions are admitted by all chemists, who can have no 
 other dispute than about the greater or smaller generality of such 
 a property. We have seen that the idea of neutrality must, sooner 
 or later, bring after it that of a determinate and unchangeable 
 ])roportion ; and the gradual development of chemical knowledge 
 lias extended this character to more and more varied cases. BerthoUet 
 disclosed several other causes of definite pro[)ortions, which were 
 entirely misconceived before his time, and which may meet in almost 
 all combinations, modifying certain circumstancesof the phenomenon. 
 The })recise question now is, therefore, whether, besides these deter- 
 minate compounds, subject to fixed proportions, within the tvyo 
 mits of possible combination, there does or does not exist, in
 
 VALUE OF THE THEORY. 273 
 
 t^eneml, a coiitinnons series of oMier iniennecliate compounds of a 
 less maikod character; in a word, wliether definite proportion con- 
 stitutes the rule, as is now generally supposed, or, as Berthollet 
 endeavoured to establish, the exception. This is now the oidy 
 dispute. It is no derogation from the interest of the doctrine of 
 definite proportions to say, as some preceding considerations compel 
 us to do, that the decision of this disputed point is not of the 
 importance commonly supposed. The doctrine lias tended to 
 simplify the general jiroblem of chemistry ; but it must not be sup- 
 ])Osed that the solution would have been impossible without this 
 aid : — it would have been simply more difficult and less precise. 
 The eminent chemists who concurred iu establishing the doctrine 
 were naturally engrossed by that labour ; but tlieir successors, who 
 find numerical chemistry constituted to their hand, must beware 
 of losing sight in it of the true scientific aim of chemistry. They 
 must not linger in this vestil)ule of the science, to the neglect of 
 the direct construction of Chemistry itself, — an enterprise scarcely 
 begvm, and to wliich it is high time that attention should be once 
 more fully dii-ected. 
 
 If we inquire, as we must do, how far the doctrine of definite 
 proportions is irrevocably established, we shall bear in mind that 
 the founders of numerical chemistry have accomplished that chief 
 ])art which depends on an investigation of all known compounds, 
 leaving only the question whether the doctrine is compatible with 
 certain chemical })henomena, neglected during its formation, and 
 remaining to be since referred to it. 
 
 The first general objection relates to the important phenomenon 
 of dissolution, evidently possible in an infinity of dif- objection of 
 fei-ent proportions. It must be acknowledged that the dissolution. 
 distinctions between the state of dissolution and that of combination, 
 by which the difficulty has been met, afford little satisfaction. In 
 my opinion the only effectual reply nmst consist in the extension of 
 ihc principle of definite [)rop()rtions to the phenomena of dissolution ; 
 and, difficult as it may be to do it, it does not seem to me impossible. 
 The way is by the use of an hypothesis already proposed for other 
 cases in which it might appear less admissible. All the successive 
 degrees of concentration of the liquid must be regarded as simple 
 mixtui'es of the small number of definite dissolutions which shall 
 have been established, either between themselves or with the dis- 
 solvent, in the manner of habitual mixtures of water with alcohol, 
 with sulphuric acid, etc. In any case, the positive verification of 
 Ibis hypothesis must be extremely delicate. Furthei'more, to render 
 the study of dissolutions fully rational, in this point of view, it is 
 necessary to combine with it that of other analogous chemical 
 phenomena, relating to the absorption of gases by liquids or by 
 l)orous solids. All these different modes of molecular union are 
 often energetic enough to resist influences able to destroy certaia 
 
 VOL. I. S
 
 2 74 POSIT J VE PHIL OSOPH Y. 
 
 combinations, properly so called : why should they not be, like them, 
 subject to the rule of definite proportions, if that rule is truly a 
 fundamental law of nature ? 
 
 Tlie next case, that of various metallic alloys, is very extensive, 
 Of metallic tliougli more particular. The difficulty lies in the 
 alloys. question whether these are cases of combination or of 
 
 mixture. The state of combination has been taken for granted in 
 the case of alloys ; whereas the general application of the principle 
 of numerical chemistry requires that they should be mixtures; 
 while, again, it is difficult to conceive of such a mixture of solids 
 as could resist perturbing influences which would appear to be 
 necessarily destructive ; as great changes of temperature, the in- 
 fluence of crystallization, etc. The question can be decided only by 
 a series of special experiments, devised to find the general limits 
 of the permanence of unquestionable mixtures ; and the results 
 might be extended to other questions of numerical chemistry, as of 
 certain oxides, on which explanations have been hazarded too freely. 
 When a true chemical theory of mixtures is established on a proper 
 basis of experiment, and we leave ofii" referring to an hypothesis of 
 mixture all cases in which combination seems susceptible of an 
 indeterminate proportion, in order to bring them under the law of 
 definite proportions, we shall get rid of a formidable objection to the 
 principle of numerical chemistry. 
 
 The remaining case constitutes the greatest obstacle of all in the 
 Of organic Way of the generalization of the law of definite pro- 
 substances. portions : and if it cannot be surmounted, the law 
 sinks to the rank of an empirical rule, fit for nothing more than 
 facilitating a certain order of chemical analyses. I refer to the class, 
 anomalous in this view, of substances called organic. And this is 
 the effect, in fact, of the declaration of the chemists of our time, 
 that organic substances do not come under the principle of definite 
 proportions. It amounts to saying tliat the law rules all the 
 elements, except oxygen, hydrogen, carbon, and azote. The division 
 between inorganic and organic chemistry is merely scholastic ; for 
 all chemistry is, by its nature, homogeneous, — that is, inorganic. 
 And thus, if we admit of the enormous exception of the numerical 
 composition of so-called organic substances, the doctrine of definite 
 proportions is overthrown as a rational theory. As it evidently can- 
 not be founded on any d priori considerations, it is only by a strict 
 generality that it can become a rational theory. 
 
 If we could not hold at once the grand princi[)le of the dualism 
 
 Application of wliicli pcrvadcs chemistry, and constitutes its homo- 
 
 the princijde geneous character, and the doctrine of definite pro- 
 
 of dualism. portions, I should not hesitate to sacrifice the latter : 
 
 for it is more important for chemical progress to grasp the great 
 
 principle of systematic dualism, than to advance our investigations 
 
 by the use of the numerical rule. But there is not, in fact, any
 
 PRINCIPLE OF DUALISM. 275 
 
 incompatibility between tliese two means of progress : and such a 
 brief sketcli of my conception on this subject as my limits allow 
 may show how the doctrine of definite proportions can be duly gen- 
 eralized only by discarding organic chemistry as a separate body 
 of doctrine, and extending the principle of dualism to all organic 
 compounds. 
 
 If we are to include all organic compounds under one uniform 
 system of chemistry, properly so called, we must refer to physi- 
 ology, vegetable and animal, the study of the numerous secondary 
 substances which owe their transient and variable existence to the 
 development of vital phenomena, and which have no scientific 
 interest except under the head of biology. We shall see, under 
 that head, hereafter, what the precise classification is ; and all that 
 we have to do with it now is to show that it proceeds from the 
 fundamental distinction between the state of death and that of life. 
 The second, and most extended class of organic substances is chiefly 
 composed of mixtures which, as such, admit of all imaginable pro- 
 l)ortions, within the limits of vital conditions. As for the substances 
 which exhibit real combinations, w^e must conceive of them as sub- 
 ject to the law of definite proportions ; but the complexity, and yet 
 more the instability of such compounds will probably for ever forbid 
 their being successfully studied under the numerical point of view, 
 which is indeed of very inferior interest in biology. — Even after 
 this clearing of the field, we could not accomplish the desired 
 geneialization if we had not taken a new stand with regard to the 
 ternary and quaternary substances contemplated by ordinary chem- 
 istry. The rigorous dualism which I have before, and in a higher 
 view, shown to be necessary, seems to supply, naturally and finally, 
 the needs of the doctrine of definite proportions. 
 
 As long as chemists persist in regarding organic combinations as 
 ternary and quaternary, — that is, in confounding their elementary 
 with an immediate analysis ; — while oxygen, hydrogen, carbon, and 
 azote are regarded as immediately united, the compounds from them 
 which must be recognized as distinct, after the severest sifting, will 
 be enough to constitute an invincible objection to the principle of 
 a numerical chemistry. But if they become binary comj)ounds of 
 the second, or at most the third order, whose principles are formed 
 by the direct and binary combination of those three or four elements, 
 we find ourselves able to represent all the actual numerical varieties 
 established by an elementary analysis, conceiving, for each degree 
 of combination, a very small number of distinct and entirely defi- 
 nite proportions. — In the ternary case, — appropriate to compounds 
 of a vegetable origin, — their three elements may be united in three 
 kinds of binary combinations. Combining these again, still em- 
 ploying at once oxygen, hydrogen, and carbon, we have three prin- 
 cipal classes of compounds of tiie second order. But then again, 
 each term of the new compounds really corresponds to two distinct
 
 2 76 POSITIVE PHILOSOPHY, 
 
 substances : and thus, while admitting only one proportion for the 
 binary composition of" these bodies, we have already provided for 
 the numerical composition of twelve substances at present called 
 ternary. But, further, we are compelled to suppose at least three 
 different proportions for each binary combination : one producing 
 perfect neutralization, and the other two the extreme limits of the 
 reciprocal saturation : and chemical analogies indicate a much 
 larger number of compounds. Putting those aside, we have thirty- 
 six compounds, without going beyond the second order, by the 
 known combination of three elements on the principle of dualism. 
 We are also entitled now to conceive of a third possible combina- 
 tion between oxygen and carbon, or between carbon and hydrogen, 
 etc., which already furnish two, after being long supposed to admit 
 of only one. Hence, and in view of all these considerations, we 
 may be assured that by dualism we might completely and naturally 
 subject to the law of definite proportions eighty-one compounds of 
 the second order, formed from oxygen, hydrogen, and carbon ; and 
 this would unquestionably more than suffice to represent the ele- 
 mentary analysis of all distinct substances in the range of vegetable 
 chemistry. 
 
 Passing on to the quaternary case, — characterizing what is called 
 Animal Chemistry, — it seems as if the principal class of compounds 
 of the second order must be more numerous than in the ternary 
 case : but the indispensable condition of employing all the four 
 elements at once restricts the classes to three. But when we ex- 
 amine the terms of the secondary compounds, we find that while 
 two represent only one compound each, a third represents five. 
 Thus, the three pairs of compounds yield fourteen of one proportion, 
 and forty-two of the three proportions indicated in the last case. 
 But applying, at each degree, the rational rule of a triple binary 
 combination, without stopping at the inevitable gaps of our existing 
 chemistry, we find ourselves in possession of ninty-nine compounds 
 of the second order, now regarded as quaternary. This is probably 
 a laiger number than a rational analysis of animal substances will 
 be found to require. Moreover, as animal substances have undergone 
 a greater degree of vital elaboration than vegetable matters, it would 
 be philosophical to admit, with respect to them, the possibility of a 
 higher order of composition, such as physiological combinations must 
 eminently tend to realize. On such an hypothesis, without going 
 beyond the third order, we might obtain ten thousand perfectly 
 distinct compounds from these four elements, all formed by an 
 invariable dualism, and strictly subject to the law of definite pro- 
 portions. It is true, nature would not permit the realization of 
 more than a small part of these speculative combinations ; but I 
 have pursued the consequences of my conception to this extreme 
 ideal limit, to show how abundant are the rational resources supplied 
 by this new theoiy for the generalization of the laws of numerical
 
 APPLICATION OF THE PRINCIPLE. 277 
 
 chemistry. If tliis view is not followed up, or somo equivalent one 
 proposed, it is evident that we must give up the doctrine of definite 
 j)roportions as a law of natural philoso})hy, and return to Berihollet's 
 theory, merel}^ enlarging the cases of fixed pro2)ortions which he 
 admitted. In the present state of the question there is no other 
 choice. But my theory having been, not instituted for this desti- 
 nation, but naturally arrived at by another way, and with higher 
 views, and proceeding from established })rinciples, to meet the needs 
 of chemical philosophy, seems to me to be presumptively entitled to 
 a future, and perhaps speedy realization. 
 
 This account of the present aspects of the doctrine of definite 
 proportions will enable any one to judge of its real progress from 
 its institution to this day ; of the conditions which must be fulfilled 
 before its i)rinciple can be converted into a great law of nature; and 
 of the rational course which alone can leap to such a final constitution 
 of numerical chemistry.
 
 278 
 
 CHAPTEE IV. 
 
 THE ELECTRO-CHEMICAL THEORY. 
 
 From the beginning of modern chemistry, the chemical influence 
 Relation of ^^ clectricity manifested itself unequivocally in many 
 Electricity to important phenomena, and above all, in the grand 
 Chemistry. experiment of the recomposition of water by the direct 
 combination of oxygen with hydrogen, effected by the aid of the 
 electric spark. But the special attention of chemists was not 
 strongly drawn towards this agency till Volta's immortal discovery 
 disclosed its princi])al energy, in rendering the electric action at 
 once more complete, more profound, and more continuous. Since 
 that time, various series of general phenomena have taught us that 
 electricity is a chemical agent more universal and irresistible than 
 heat itself, both for decomposition and combination. The danger 
 now is of exaggerating the relation it bears to the general system of 
 chemical science. Though chemistry is united to physics by this 
 agency more than by any other, it must yet be remembered that the 
 two sciences are distinct, and that there should be no confounding 
 of chemical with electrical properties. In order to ascertain with 
 precision what are the relations of chemistry with electrology, we 
 must briefly review the gradation of ideas which have led up to the 
 present electro-chemical theory, as systematized by Berzelius. 
 The first important chemical effect of the voltaic influence was 
 History of the decomposition of water, established by Nicholson 
 the case. Jq 1801. It was a nccessary result of examination 
 
 into the action of the pile, without any chemical intention. It 
 Nicholson's Confirmed a truth before well known : but it had a 
 discovery. high chemical value, as revealino: the chemical eners^y 
 of the instrument ; and it thus constituted the starting-point of 
 electro-chemical research. We may even refer to this origin the 
 first attempts at founding a general theory of electro-chemical 
 ])henomena ; for the conception offered by Grothuss, to explain 
 Nicholson's observation by the electric polarity of molecules, con- 
 tains the germ of all the essential ideas whicli have expanded, accord- 
 ing to the requisitions of the phenomena, into the present electro- 
 chemical theory. 
 
 The analytical power of the voltaic pile having been once dis- 
 covered, it was natural for the chemists to aj)ply tlie new agent to
 
 SYNTIIESJS. 
 
 279 
 
 the decomposition of substances which had liitlicrto resislcd all 
 known means. This first series of attempts ])roduced, after a few 
 years, the brilliant discovery by the illustrious Davy, Davrjsdis- 
 — of the analysis of the alkalies, properly so called, covery. 
 and the earths. Lavoisier's theory, which showed that every 
 salifiable base must be a result of the combination of oxygen with 
 some metal, had foreshown this analysis ; but no chemical means 
 had sufficed to effect it. It was believed in, in spite of Berthnllet's 
 discovery of the true composition of ammonia; and the brilliant 
 i-esult obtained by Davy was an easy consequence of a discovery 
 completely prepared for. M. Gay-Lussac soon followed, with a 
 more difficult but less striking achievement, — the confirmation, by 
 a jiurely chemical process, of the electrical analysis of potash. 
 
 Nicholson's observation having originated electi'O-chemistry, and 
 Davy's given it a great impulse, the next step was to investigate 
 the chemical influence of electricity, in a scientific view. This was 
 effectually, though indirectly, determined by Davy's great feat ; for 
 by it chemistry was proved to have achieved the most imi)ortant, 
 and hithei-to inaccessible analj'ses : and in fact, the science has not 
 since made any essential acquisition. Electro-chemical action was 
 presently and permanently subjected to direct and regular study ; 
 and it was irrevocably constituted a fundamental part of chemical 
 science when Berzelius accomplished his series of BerzeUus's 
 investigations on the voltaic decomposition of all the extension. 
 salts, and then of the principal oxides and acids. It was in conse- 
 quence of these researches that the habitual consideration of electric 
 ])roperties has assumed a growing inqwrtance in the chemical study 
 of all substances, which are now scientifically divided into the 
 classes of electro-negatives and electro-positives. It was thus the 
 ])rivilege of Berzelius to be the first to conceive of the electro- 
 chemical theory under an entirely systematic form. 
 
 One condition remained to be fulfilled, to give its due scientific 
 character to this new branch of chemistry. The Synthetical 
 voltaic action had thus far been regarded only ana- p»"occss. 
 lytically : — it must be regarded synthetically also. This was 
 (lone by the labours of i3ecquerel, who fully established the 
 synthetical influence of electricity, fitly applied ; and who, more- 
 over, employed it in effecting new and valuable combinations, 
 hitherto impracticable. From this procedure arose tlie necessity 
 of modifying the method of experimentation. The apparatus 
 which was powerful enough to deconq)ose was much too i)()werful 
 to combine, because it would probably decompose the immediate 
 ])rinciplcs which were intended to be combined ; and thence arose 
 the method of employing the protracted action of veiy feeble 
 electric powers, — every advantage being given to the disposition of 
 the substances to be acted upon. M. Becquerel did this very suc- 
 cessfully by operating with" a single voltaic element, and seizing
 
 2 So POSITIVE PHILOSOPHY. 
 
 each substance in the state called nascent, which is a<;reed upon 
 as most favourable to combination. This change of 
 ecquoe s. procedure is the distinctive honour of M. Becquerel. 
 He not only determined direct combinations, not before obtainable, 
 but exhibited in others, which were obtainable, the remarkable 
 property of clearly manifesting their geometrical structure, through 
 the slowness and regularity of their gradual formation ; — a character 
 especially marked in the case of certain metallic sulphurets, some 
 oxides, and several salts. It does not lie within our province to 
 point out the results of this method in regard to the natural history 
 of the globe, in explaining a great number of mineral origins, when 
 the time for such concrete questions shall have arrived. It is more 
 in our way to observe the importance of these labours in bringing 
 up chemical synthesis to something like harmony with the progress 
 of analysis, — favoured as the latter pursuit had been by the ease 
 with which we destroy, in comparison with the difficulty with 
 which we recreate. Once more, these researches of M. Becquerel 
 liave completed the general conslitutionof electro-chemistry, which, 
 being henceforth at once synthetical and analytical, can only expand 
 in one of these two directions, however great the improvements 
 remaining to be attained. 
 
 Such has been the filiation of electro-chemical discoveries, since 
 the beginning of our century. A little attention to the great pheno- 
 menon which was the original subject of the electro-chemical theory 
 will show how this study has gradually led to a new fundamental 
 conception for the whole of Chemistry. 
 
 It has been said, through all periods of chemical research, that 
 study of the study of Combustion must be the central point 
 combustion. q{ ^|jg science. So it was thought in the ancient 
 theological period of the science ; and also in the more recent 
 metaphysical stage, when combustibility was called phlogiston, and 
 regarded as an intangible materialized entity: and the advent of 
 the positive period of chemistry was marked by the establishment 
 of Lavoisier's new theory of combustion. In our day it is the 
 recognized necessity of modifying this theory that has especially 
 led to the electric conception of chemical ])henomena. 
 
 The pneumatic theory of combustion of Lavoisier had two entirely 
 Lavoisier's different objects in view ; objects which are too com- 
 theoru. monly confounded, but which we must be careful to 
 
 keep apart. First, the analysis of the general phenomenon of com- 
 bustion : and, secondly, the explanation of the ellccls of heat and 
 light, which is, in the eyes of the vulgar, the more important of 
 the two. Jjoth were treated in the most admirable manner possible 
 in the existing state of knowledge ; and in the characteristics of 
 positivity and rationality Lavoisier's theory has not since been sur- 
 passed. All combustion, abrupt or gradual, was regarded as con- 
 sisting in the combination of the combustible body with oxygen,
 
 THEOR V OF COMB USTION. 28 1 
 
 whence, wlien (lie body was sini])le, must result an oxide, f^cnerally 
 susceptible of becoming the basis of a salt, and, if the oxygen was 
 preponderant, a true acid, the principle of a certain kind of Siilts. 
 As for the disengagement of heat and light, it was attributed, in a 
 general way, to the condensation of the oxygen, and in an accessory 
 way, to that of the combustible body, in this combination. 
 
 The first part of this anti-{)hlogistic theory has a much more 
 philosophical character than the second. It was ^,. ,. . . 
 
 '■•ti j-ij 1 ^1 1 c i ir St division. 
 
 eminently rational to analyse the phenomenon 01 
 combustion, so as to seize whatever was common to all cases. The 
 conclusions drawn might be too general ; and were afterwards 
 proved to be so ; but whatever would stand the test of time must 
 form a body of indestructible truth, constituting an essential })art of 
 chemical science through all future revolutions. The case is dif- 
 ferent with the explanation of heat and light. The question is not, 
 like the first, of a chemical, but a physical nature ; and, whatever 
 may be its final solution, it cannot affect chemical conceptions. It 
 would have been wiser to abstain fiom offering any general explan- 
 ation of the effects of heat and light through such a supposition as 
 that of a condensation, which does not necessarily take place, and 
 which is, in fact, found to be often absent. Lavoisier hoped to 
 attach the thermological effect to the great law discovered by 
 Black, of the disengagement of heat proper to the passage of any 
 body from one state to another more dense ; but such a connection 
 could not be established on the ground of phenomena not invariably 
 present, or indisputably manifested. However, it would be too 
 much to expect a perfect scientific reserve in discoverers who bring 
 out scientific truths from a region of metaphysical fantasies. It is 
 from their followers that we have a right to demand it ; and we are 
 compelled to charge upon the chemists who have been eager to 
 substitute the electro-chemical theory for the anti-phlogistic theory, 
 properly so called, a want of care in constructing explanations ana- 
 logous to those which are dismissed as insufficient. To justify this 
 charge, we must i-eview the proofs of the imperfection of Lavoisier's 
 theory, — still regarding it under the two divisions just exhibited. 
 
 Berthollet saw presently that Lavoisier's method of analysis of 
 combustion must be modified. One of the chief con- BertiwUct's 
 sequences of this analysis was that every acid and Umihuion. 
 every salifiable base must be a residt of combustion ; that is, of the 
 combination of any element with oxygen ; whereas, Berthollet dis- 
 covered that one of the most marked of the alkalies, ammonia, is 
 formed of hydrogen and azote alone, without any oxygen ; and soon 
 after, he proved that sul[)hurette(l hydrogen gas, in which abso 
 there is no oxygen, nevertheless presents all the essential properties 
 of a real acid. These facts have since been confirmed in every 
 possible way, and especially by the electric method ; and the excep- 
 tions, both as to alkalies and acids, have become so nudtiplied, that
 
 282 POSITIVE PHILOSOPHY. 
 
 the investigation and comparison of them have given that high 
 character of generahty to the study of alkahes and acids which 
 belongs to it in our time. 
 
 Moreover, the primitive theory of combustion has been gradually 
 modified by the discovery that a rapid disengagement of heat and 
 light is not always an indication of a combination with oxygen. 
 Chlorine, sulphur, and several other bodies, even non-elementary, 
 have been found to occasion true combustion. And again, the 
 phenomenon of fire is no longer attributed exclusively to any special 
 combination, but, in general, to all chemical action at once very 
 intense and vivid. 
 
 It does not follow that because Lavoisier's discoveries have parted 
 with some of their character of generality, they have lost any of 
 their direct value : and such alteration of views as there is relates 
 chiefly to artificial phenomena, while the natural facts remain 
 securely established. Thus, though there are acids and alkalies 
 without oxygen, it is unquestionable that the greater number of 
 them, and especially the most powerful, are oxygenated : and again, 
 if oxygen be not indispensable to combustion, it remains the chief 
 agent, and especially in natural combustions. In natural history, 
 the theory is applicable, almost without reserve, though it is insuf- 
 ficient for the severe conditions of abstract science. If the universal 
 sovereignty of oxygen has been overthrown, it will yet be for ever 
 the chief element of the whole chemical system. 
 
 As for the second aspect of the discover}^ — the explanation of 
 fire, — it was destroyed by the first direct examination 
 
 Second division. c •- -vr r i • i r v ii 
 
 01 it. JNo new facts were requu-ed tor its overthrow, 
 
 but merely a more scientific appreciation of common phenomena. 
 It will not even serve naturalists for their concrete purposes in any 
 degree, having never really explained the most ordinary effects. 
 The required condensation is found to be only occasionally present, 
 and often absent in the most important cases ; so that if it were 
 not for the connection of this aspect of the theory with a sounder 
 one, it would be inconceivable how it could have held its ground 
 up to a recent period, — busy as the chemists were with other 
 theoretical speculations. After finding that in cases where conden- 
 sation was supposed, expansion exists instead ; and that where we 
 find vivid combustion, there should, by the theory, be a great cool- 
 ing, we are brought to the reflection that if the fiie on our hearths 
 was not a matter of daily fact to us, its existence must become 
 doubtful, or be disbelieved, through those very explanations by which 
 the phenomenon has been proposed to bo established. To my 
 mind, this is a clear indication that the chemical i)roduction of 
 fire does not admit, in a general way, of any lational explanation. 
 Otherwise it appears incomprehensible that men of such genius 
 and such science, at a time so near our own, should have been so 
 deluded. The electric fire, now proposed for an explanation, must
 
 THEORY OF COMBUSTION. 283 
 
 have been sufficiently known to Lavoisier, Cavendish, BerthoUet, 
 and others, to have served as a basis to their theory, if its prepon- 
 derance over the merits of their hypothesis had been so great as is 
 now commonly supposed. This consideration, however, strikin^^ 
 as it may be, is no dispensation from the duty of examining the 
 electro-chemical concej^tion, for which we have been prepared by 
 this sliort account of its antecedents. 
 
 According to this theory, the tire produced in the greater number 
 of strong chemical reactions must be attributed to a real electric 
 discharge which takes place at the moment of combination (by the 
 nuitual neutralization, more or less complete, of the opposite electric 
 conditions) of the two substances under consideration, — one of which 
 must be electro-positive and the other electro-negative. There is 
 every reason to fear, however, that when this theory has been effect- 
 ually examined, it will be found as defective in rationality as its 
 l)redecessor. If electric effects are concerned in all chemical pheno- 
 mena, as seems to be now agreed upon by most chemists and 
 })hysicists, they must oftener be supposed than found : and the 
 t'lectric symptoms are most impossible to detect in precisely those 
 chemical phenomena which have been most relied on for overthrow- 
 ing the old theory. And in the cases in which electrization is 
 evident, its chemical influence is so equivocal that some regard it 
 as the cause, and others as the effect, of the combination. The 
 explanation is not yet positively established for any phenomenon 
 whatever that has been duly analysed : while its vague nature 
 leaves room for fear that it will not be so radically or so speedily 
 destroyed as its predecessor. It could be plainly shown whether 
 the requisite condensation did or did not exist : but there is always 
 a resource, in the more recent case, in the faint or fugitive 
 character of the electric condition, which defies our means of posi- 
 tive exploration, — qualities which are far from being a ground of 
 recommendation of a theory which is to account for very striking 
 and intense effects. I do not desire to say that the disengagement 
 of light and heat can never have an electric origin, any more than 
 ihat it can never proceed from the condensation ])roposed : but I 
 lliiidcan impartial observation would decide that in most cases of 
 combustion there is neither condensation nor electrization. My own 
 view is that these vain attempts to explain the chemical i)roduction 
 of fire proceed from the lingering metaphysical tendency to penetrate 
 into the nature and mode of being of phenomena : and that chemical 
 action is one of the various primitive sources of heat and light, 
 which cannot, from their nature, usually admit of any positive 
 explanation ; that is, of being referred, in this relation, to any other 
 fundamental influence. 
 
 If our chemical science were more advanced than it is, we should 
 not have to point out that the consideration of fire, which, however 
 important, is only a physical accessory of chemical phenomena,
 
 284 rOSITlVE rillLOSOFHY. 
 
 cannot afford a rational ground for a radical change in our con- 
 ceptions of chemical action. When our predecessors regarded heat 
 Difficulties of as the chicf physical agent in composition and decom- 
 tke theory. position, they did not pervert such a consideration to 
 the point of assimilating chemical to thermological effects. We 
 are less cautions at the present day : we confound the auxiliary, or 
 the general i)hysical agent of the phenomenon, with the phenomenon 
 itself, and pervert chemistry hy confounding it with electrology, hy 
 irrationally assimilating chemical to electrical properties, as is seen 
 especially in the theory of M. Berzelius. How can there he any 
 scientific comparison between the tendency of two bodies to a 
 mechanical adhesion after a certain mode of electrization, and the 
 disposition to unite all their molecules, external and internal, by a 
 true chemical action ? M. Berzelius has frankly declared that 
 cohesion, properly so called, admits of no electric explanation. 
 Nothing is gained towards explaining the molecular connection, 
 indissoluble by any mechanical force, in contrast with the magnetic 
 imion so easily overcome, by talking of voltaic elements with their 
 positive or negative pole, and their connection by the electric anta- 
 gonism of the opposite poles. Such inventions give no idea what- 
 ever of molecular cohesion. Nor is affinity, or the tendency to 
 combination, any better explained by the electro-chemical theory. 
 Electrical phenomena, in physics, are eminently general, offering 
 only differences of intensity in different bodies ; whereas, chemical 
 phenomena are essentially special or elective : and therefore every 
 attempt to make chemistry, as a whole, enter into any branch of 
 l)hysics, is thoroughly anti-scientific. This would be enough : but 
 we see besides that the smallest changes in the mode of electriz- 
 ation reverse the electric antagonism, and destroy the proposed 
 electrical order of elementary bodies : we find ourselves unable to 
 deduce the new electric properties which the theory bids us look 
 for in the compounds of" different orders ; we do not know by what 
 laws they derive their positive or negative character from the 
 electric condition of each of the two elements ; nor are we able to 
 approach the great end of chemical science, — the prevision of the 
 qualities of compounds by those of their constituent elements. 
 Moreover, in any case, the great body of chemical phenomena 
 opposes insurmountable obstacles ; as when oxygen, the most nega- 
 tive element, when entering largely into certain oxides, finds them 
 positive towards certain acids, into which it enters nuich more 
 sparingly, — the radicals of the first being often as negative as those 
 of the last. According to M. Berzelius's own frank declaration, 
 organic compounds cast insuperable difficulties in the way of 
 his arjangement of electric relations ; and he alleges the transience 
 of the combinations in that class of cases as an explanation of the 
 anomaly : but chemical science would be imi)0ssible if compounds 
 were not throughout considered stable till causes of decomposition
 
 DIFFICULTIES OF THE THEORY. 285 
 
 arise : and if orgatiic compounds are guarded from these, tlicy 
 remain chemically stable, like inorganic substances. The funda- 
 mental obstacle of the whole case, — liie identity of the elements, in 
 opposition to the electric variety, — cannot by any means whatever 
 be got over. 
 
 Leaving all these difficulties on one side, we learn nothing about 
 chemical phenomena by likening them to electric action, for we 
 establish thus no harmony between the pretended causes and the 
 real effects. Every attempt seems to prove simply the auxiliary 
 influence of electricity on chemical effects, — acting as heat does, 
 only with a different intensity. In fact, there are scarcely any 
 electro-chemical combinations which cannot be effected by ordinary 
 chemical processes, without any electric indications; and the few 
 exceptional cases still existing may, by analogy, be expected to be 
 brought under the rule. If, in the face of all tliis, Ave were to per- 
 sist in investing the electrical influence with the specific and mole- 
 cular attributes of chemistry, we should merely be restoring the old 
 entity of affinity, decked out with some hypothetical material attri- 
 butes, which would be far from rendering it more positive: and such 
 a procedure would be as hurtful to physics as to chemistry, by 
 infusing new vagueness into our notions of electricity, which are at 
 present far from being sufficiently distinct. And then might follow, 
 as likely as not, the founding on electrology, not only the whole of 
 chemistry, but the theories of heat, of weight, and probably, as a 
 consequence, that of celestial mechanics. And then, if we added to 
 this heterogeneous assemblage a conlbunding of the supposed nervous 
 fluid with the pretended electric fluid, we should have attained to 
 the show of an universal system, devoid of all scientific use, which 
 would fall to pieces as soon as tested by real study, parting off' into 
 categories of independent doctrine, and encumbering natural phil- 
 osophy with insoluble questions, which must be discarded, to enable 
 us to begin afresh. 
 
 Thus, to sum up, the great chemical influence of electricity, like 
 that of weight, and yet more of heat, is unquestionable ; and 1 have 
 endeavoured to exhibit the high importance of electro-chemistry to 
 the improvement of chemical science, of which it is one of the essen- 
 tial elements. But I must, once for all, reject the conception 
 through which it has been attempted to transform all chemical into 
 electrical phenomena. In a ])hilosophical point of view, Lavoisier's 
 theory ap])ears to me, notwithstanding its serious imperfections, 
 very superior as a scientific composition to that to which it has given 
 place. It related directly to the aim of chemical science, — the 
 establishment of general laws of composition and decomposition ; 
 whereas tlie newer theory draws attention away from it, in a vain 
 inquiry into the intimate nature of chemical phenomena. Thus the 
 anti-i)hlogistic conception has suggested mmierous and important 
 chemical discoveries, while it is very doubtful whether as nuich will
 
 286 POSITIVE PHILOSOPHY. 
 
 ever be said for the electric theory. In an indirect way, however, it 
 Its position n^ay operate favourably for chemical science, by its 
 andpoicers. binary antagonism suggesting the extension of dualism 
 among compounds which are as yet supposed to be more than bin- 
 ary. Berzelius appears to have felt this connection ; and he would 
 probably have erected dualism into a fundamental principle but for 
 his subjection to the old division of chemistry into organic and 
 inorganic. Others, who are free from this entanglement, may be 
 prepared by the electro-chemical theory for general dualism ; though 
 it is not, in principle, desirable to recur to faulty means to attain 
 good results in an indirect way. In another view, this theory may 
 be useful, — in fixing the attention of chemists on the influence of 
 time in the production of chemical effects; — an influence remarkabl}'' 
 manifested by many phenomena, but not, as yet, directly analysed. 
 Not only does time naturally increase the mass of the products of 
 chemical re-action ; it also causes formations which would not 
 otherwise exist. The chemical theory of time is at present a blank 
 in science ; and tlie phenomena of electro-chemistry seem likely to 
 enlighten us on this head, — so all-important in its connection with 
 chemical geology, while constituting an indispensable element in 
 the conceptions of abstract chemistry. 
 
 To complete our survey of the philosophy of Chemistry, we must 
 turn to some considerations already suggested, on the subject of 
 what is commonly called Organic Chemistry.
 
 28; 
 
 CHAPTER Y. 
 
 ORGANIC CHEMISTRY. 
 
 Organic Chemistry coniprelieiKls, it cannot be denied, two kinds of 
 researches, chemical and physiological, which are perfectly distinct. 
 For instance, the study of organic acids, and especially the vegetable 
 acids, and alcohol, ethers, etc., has a character as Confusion of 
 purely chemical as that of any inorganic substances two kinds 
 whatever ; and, on the other hand, there is no doubt ^ff"''^^- 
 of the biological cliaracter of inquiries into the composition of sap 
 or blood, and of the analysis of the products of respiration, and 
 many other matters included in organic chemistry. The confusion 
 of the two orders is prejudicial to both sciences, and especially to 
 physiology. The division of Chemistry conceals or violates essential 
 analogies, and hinders the extension of dualism into the organic 
 region, where it is seldom found, though, as I have shown, it is 
 optional, in fact, throughout the whole range of chemistry ; and thus 
 the arbitrary arrangement is the chief obstacle in the way of the 
 entire generalization of the doctrine of definite proportions. And 
 whenever a true chemical theory shall fitly replace the anti-phlogistic 
 conception, it must necessarily comprehend all organic, as well as 
 inorganic compounds. The most philosophical chemists are tending 
 more and more to a recognition of the identity of the two depart- 
 ments ; and there can be no doubt that the establishment of that 
 identity will bean immediate consequence of a rational classification 
 of chemical knowledge. 
 
 The confusion is more mischievous, but less felt by chemists, 
 under the other view, — the comprehension of biological phenomena 
 among those of organic chemistry. The confusion arose from the 
 need of chemical researches in very many physiological questions ; 
 and these chemical researches, being usually extensive and difficult, 
 were out of the range of the physiologists, and were taken possession 
 of by the chemists, who annexed them to their own domain. Both 
 classes were to blame for the vicious arrangement, and both must 
 amend their scientific habits before a division can be elfected in 
 entire conformity to natural analogies. The physiologists have the 
 most difficult task before them, having to qualify themselves for 
 inquiries from which the chemists have only to abstain. 
 
 It is scarcely possible to characterize or to circumscribe tho
 
 288 POSITIVE PHIIOSOPHY. 
 
 physiological part of oi-ganic chemistry, formed as it is by successive 
 encroachments. It coiiipreliends tlie chemical analysis of all the 
 anatomical elements, solid or fluid, and that of tlie products of the 
 organism; and if its usurpations remained unchecked, it would 
 soon include the phenomena of what Bichat called the orgo.nic life; 
 that is, the functions of nutrition and secretion, the only ones 
 common to all living bodies, and in which the chemical point of 
 view might well appear the natural one. In such a state of things, 
 physiology would he reduced to the study of the functions of animal 
 life, and to that of the development of the living being. It is easy 
 to see what would become of biological science, if it were reduced 
 to this fragmentary state. Chemists cannot but be unfit for the 
 rational examination of the important questions of anatomy and 
 physiology, vegetable and animal, because the research requires that 
 comprehension of view which their studies, as chemists, preclude 
 them from obtaining. In the anatomical relation, they are per- 
 petually overlooking the fundamental division, established by M. de 
 Blainville, between the true elements oi the organism, and its simple 
 ]jroducts ; and they take for one another, almost indifferently, the 
 tissues, the parenchyma, and the organs. The spirit of biological 
 investigation being unknown to them, they can neither choose their 
 subject well, nor direct their analysis wisely. If these are grave in- 
 conveniences in anatomical questions, they are much more serious in 
 physiological problems, properly so called, the essential conditions 
 of which are not understood by chemists. The rational direction 
 of physiological analysis can take place only by the subordination 
 of the chemical to the physiological view ; and therefore by the 
 employment of chemistry by the physiologists themselves, as a 
 simple means of investigation. It is an analogous case to that 
 before exhibited, of the application of mathematical analysis to 
 physical questions. If it is important that physicists should employ 
 the instrument of analysis, instead of delivering over physical sub- 
 jects to the mathematicians, to be a mere theme for analytical 
 exercises, much more important is it that the greater diversity of 
 view in chemistry and physiology should not be lost sight of. The 
 irrational and incoherent studies comprised in the organic chemistry 
 of our day give us no idea of the true nature of the aids that biology 
 may derive from chemistry. A few instances will show the high 
 importance of an improved organization of scientific labour. 
 
 In the anatomical order, almost all the researches of chemists 
 
 Relation of ^^''^^^ ^^ undergo an entire revision by the physiologists, 
 
 Chemistnj to before they can be applied to the studies of the ele- 
 
 Anatomy. ments or the products of the organism. The fine 
 
 series of researches of M. Chevreul on fatty bodies are perhaps the 
 
 only important chemical study innnediately applicable to biology, 
 
 animal or even vegetable. In the chemical annlysis of blood or 
 
 sap, or almost any other element, a single case, taken at hnzard, is
 
 RELATION OF CHEMISTRY TO PHYSIOLOGY. 289 
 
 usually presented as a satisfactory type, without any comparative 
 investigation either of each species of organism in its normal state, 
 or of the degree of development of the living heing, — ils sex, its 
 temi)erament, its mode of alimentation, the system of its exterior 
 conditions of existence, etc., and other modificalions which physi- 
 ologists alone can duly estimate. Such analyses correspond to 
 nolhing in anatomy hut the single case ohserved ; and even that is 
 seldom sufficiently characterized. Hence inevitable divergences 
 among chemists, who choose different types, and discussions of no 
 scientific use, as the discordance is attributed to the diderent 
 analytical methods employed, instead of to the variations which 
 })hysiology would have led them to anticipate. The case is the 
 same with regard to products first secreted and then excreted, as 
 bile, saliva, etc., which offer a still greater complication. The 
 chemists make no inquiry about the parts from which these pro- 
 ducts are taken, or the uiodifications which may have been occasioned 
 by their remaining some time after their production, etc. ; and 
 therefore the analyses of these products, however often renewed, are 
 still incoherent and thoroughly defective. We owe to M. llaspail 
 a full exposure of the practice of the chemists of multiplying 
 organic principles almost without limit, from differences of char- 
 acter which imply no distinction of nature, but arc merely marks 
 of various degrees of elaboration of the same principle in different 
 developments of vegetation ; and even from confounding the 
 observed substances with their anatomical envelope. It is to be 
 regretted that M. Raspail did not complete his great service to 
 science by founding rationally the physiological portion of organic 
 chemistry, instead of vainly attempting to systematize organic 
 chemistiy, under the bias of our crude chemical education. 
 
 If we turn from the anatomical order of questions to the physio- 
 logical, we shall find yet stronger evidences of the ,p p, • , 
 inaptitude of chemists for biological inquiries. All ° yswogy. 
 endeavours have yet failed to establish any point of general doctrine 
 in biology ; and we find ourselves mei-ely with sim[)le materials, 
 which must be newly elaborated by physiologists, under the view of 
 vitality, before they can be put to use. To give an example or two : 
 — the experiments of Priestley, Sennebier, Saussm-e, and others, on 
 the mutual chemical action of vegetables and atmos^jheric air, were 
 of the highest value, as instituting positive knowledge of the vege- 
 table econoujy ; but the inquiiy is by no means so simple as its 
 founders naturally supposed, after having analysed one separate 
 ])ortion of the phenomenon of vegetation. The absorj)tion of car- 
 bonic acid, and the exhalation of oxygen, though very important in 
 relation to the action of leaves, are oidy one aspect of the double 
 vital motion, and cannot be understood but in the physiological view 
 of both. This action cannot explain the elementaiy couq)osition of 
 vegetable substances, or determine the kind of alteration sustained 
 
 VOL. I. . T
 
 2 90 POSITIVE PHILOSOPHY. 
 
 by tlie air tlirongli vegetation, because it is, in otlier ways, partially 
 compensated by the precisely inverse action produced by the ger- 
 mination of seeds, the ripening of fruits, etc., and even, as regards 
 the leaves, by the mere passage from light to darkness. It is much 
 1o have indicated the true nature of the requisite research, and to 
 have supplied some materials for it. The rest is the business of tlie 
 physiologists. If we turn to animal physiology for examples, the 
 case is yet more striking. 
 
 After all the inquiry that has been made into the chemical pheno- 
 mena of respiration, no fixed point is yet established. It was long 
 supposed that the absorption by the lungs of atmospheric oxygen, 
 and its transformation into carbonic acid, would explain the gi-eat 
 phenomenon of the conversion of venous into arterial blood. But 
 the problem is much more complicated than was supposed by the 
 chemists who established this essential part of the phenomenon, and 
 whose labours present the most contradictory conclusions in regard 
 to the facts under their notice. We do not know, for instance, 
 whether the quantity of carbonic acid formed really corresponds to 
 the quantity of oxygen absorbed ; and even the simple general 
 difference between the inhaled and exhaled air, which is the first 
 point to be ascertained, is far from being positively established. 
 The atmospheric azote appears to some to be increased after respira- 
 tion, while others say it is diminished, and others ngain that it 
 remains the same. The disagreements about the changes in the 
 composition of the blood are yet more marked. Perhaps the inapti- 
 tude of chemists and physicists for physiological researches is more 
 striking still in the case of animal heat. In the early days of 
 modern chemistry, this phenomenon seemed to be sufficiently 
 accounted for by the disengagement of heat corresponding to the 
 decarbonizing of the blood in the lungs, which the chemists regarded 
 as the focus of a real combustion ; but this explanation was soon 
 found to be inadequate, even in a normal condition ; and much 
 more in various pathological cases. Uncertain as we still are as to 
 the pulmonary influence in the process, we know that all the vital 
 functions must concur in it, in a greater or smaller degree. There 
 is even some reason to suppose, in direct opposition to the opinion 
 of the chemists, that respiration, far from aiding to produce animal 
 heat, constantly tends to diminish it. No doubt, the chemical 
 eft'ects occasioned by vital action must always be taken into the 
 account in the study of animal heat ; but it is only the physiolo<;ists 
 who can deal with them in the light of the whole subject. When 
 we have learned to combine the chemical and the physiological view, 
 we may proceed to the formation of jwsitive doctrine, without having 
 to deal with preliminary obstacles ; as, in regard to such questions 
 as the general agreement between the chemical composition of living 
 bodies, and that of the whole of their aliments ; a case which con- 
 stitutes one of the principal aspects of the vital state.
 
 DIFFICULTIES. 291 
 
 Tt is evident, at tlio outset, tliat every livin^i; body, wliatcver its 
 origin, must be, in tlie long run, composed of the dillerent cliemical 
 elements concerned in the substances, solid, liquid, and gaseous, by 
 which it is habitually nourished ; since, on the one hand, the vital 
 motion subjects its parts to a continual renovation, and, on the other, 
 Ave cannot without absurdity suppose it capable of spontaneously 
 producing any real element. This consideration is so far from 
 involving any difliculty, that it might lead us to divine tlie general 
 nature of the principal elements of living bodies ; for animals feed 
 in the first place on vegetables, or on other animals which have 
 eaten vegetables ; and in the second place, on air and water, which 
 are the basis of the mitrition of plants : and thus, the organic world 
 evidently admits of those chemical elements only which are furnished 
 by the decomposition of air and water. When these two fluids 
 liave been duly analysed, physiologists can, in a manner, foresee 
 that animal and vegetable substances must be composed of oxygen, 
 liydrogen, azote, and carbon, as chemistry taught them. It is true, 
 such a prevision must be very imperfect, as it indicates nothing of 
 the difference between animal and vegetable substances, nor why 
 the latter usually contain so much carbon and so little azote. But 
 this first glimpse, though it suggests some of the difficulty of the 
 problem, yet indicates the possibility of establishing such a general 
 harmony. But. when we proceed with the comparison, we encounter 
 imjwrtant objections, which are at present insoluble. The chief 
 difficulty is that azote appears to be as abundant in the tissues of 
 herbivorous as of carnivorous animals, though the solid aliments of 
 the former contain scarcely any azote. The opinions of chemists, iis 
 of Berzelius and Kaspail, as to the nature of azote, do not solve the 
 difficulty, as they cast no light upon its origin. This is one of a 
 nudtitude of cases in which we cannot at all explain the chemical 
 composition of anatomical elements by that of the exterior substances 
 from which they are unquestionably derived. Another striking case 
 is that of the constant presence of carbonate, and, yet more, phos- 
 phate of lime in the bony tissue, though the nature of the aggregate 
 of aliments appears to afford no room for the formation of those 
 salts. This system of investigations, considered in its whole range, 
 constitutes one of the most important genei-al questions that can 
 arise from the chemical study of life ; and, if it is at present hardly)' 
 initiated, the l)ackwardness is owing, not only to its eminent diffi- 
 culty, but to the biologists having abandoned to the chemists a task 
 which, under a wise organization of labour, would have belonged to 
 themselves alone. 
 
 In order to eflfect a rational division of organic chemistry, and to 
 assign its portions to chemistry on tlie one hand and partition of 
 physiology on the other, we must take our stand on ^''''""'''^ 
 the separation between the state of life and that of "^"*''* '-'' 
 death ; or, what comes to nearly the same thing, between the
 
 292 POSITIVE PHILOSOPHY, 
 
 stability and instability of the proposed combinations, submitted tc 
 the influence of common agents. Among the compounds indis- 
 criminately called organic, some owe their existence only to vital 
 motion ; they exhibit perpetual variations, and usually constitute 
 mere mixtures: and these cannot belong to chemistry, but nuist 
 enter into the domain of biology,' — statical or dynamical, according 
 as we study their fixed condition, or the vital succession of their 
 7'egular changes. Such, for instance, are blood, lymph, fat, etc. 
 The others, which form the more immediate principles of these, are 
 substances essentially dead, admitting of a remarkable permanence, 
 and presenting all the characters of true combinations, independent 
 of life : their natural place is evidently, in the general system of 
 chemical science, among the substances of inorganic origin, from 
 which they differ in no important respect. Of these, the organic 
 acids, alcohol, albumen, etc., are examples. These are the substances 
 which truly belong to organic chemistiy ; and no reason exists for 
 Iheir separation i'rom analogous inorganic substances, even if no 
 injury was done by such an arbitrary division ; and there is more 
 reason for giving the title of organic to them than to the theory 
 of oxygen, hydrogen, carbon, and azote, or to the study of many 
 other substances, acid, alkaline, saline, etc., without which chemical 
 anatomy and })hysiology would be unintelligible. As for chemical 
 phenomena truly common to all compounds of this class, in conse- 
 quence of the necessary identity of their chief elements, it is cer- 
 tainly important to assign to them their precise relations. The 
 most important of these phenomena relate, at present, to the inter- 
 esting and very imperfect theory of the different kinds of fermenta- 
 tion. But the consideration of these properties does not constitute 
 a different order from that which results from the same ground in 
 the case of many other compounds, purely inorganic. The property 
 of fei-mentation, however important, has not a higher scientific 
 value than that of burnins:, and has no more rioht to an exclusive 
 classification. It is admitted that too much was attributed to com- 
 bustion formerly, in regard to inoiganic substances ; and we may 
 be attributing too much now to fermentation, or any other common 
 ]u-operty, among so-called organic bodies. We cannot yet assign 
 their pi'oper place to these compounds in the rational system of 
 chemical science ; but we are able to affirm that, in that systeu), 
 they must be more or less separated from each other, and interposed 
 among substances called inorganic. Nothing more than this is 
 needed to settle the question of the maintenance or the suppression 
 of organic chemistry as a distinct body of doctrine. In applying 
 the principle which I have proposed, to ascertain to which science 
 any question belongs, it is enough to inquire whether chemical 
 knowledge will serve the purposes of the research, or whether 
 any biological considerations enter into it. The proposing such an 
 alternative is, in fact, making the classification.
 
 niP'JSIONS OF ORGANIC CHEMISTRY. 293 
 
 It is not our business to treat of any special api)lication of this 
 })rinciple ; but it is desirable to point out that in tiiis partition of 
 organic chemistry, its two portions are very unequally divided — 
 the study of vegetable substances contributing most to chemistry, 
 and that of animal substances to biology. At the first ghmce, we 
 miglit suppose tlie difference to be the other way ; for the impor- 
 tance of chemical considerations is really much greater with regard 
 to living vegetables than animals, whose chemical functions are, 
 except among the lowest orders of the zoological hierarchy, subor- 
 dinated to a superior order of new vital actions. Yet, in virtue of 
 the higher degree of vital elaboration that matter undergoes in the 
 animal than iu the vegetable organism, the chemical part of animal 
 ])hysiology presents a much greater extent and complexity than the 
 vegetable, in which, for instance, the whole series of the phenomena 
 of digestion is absent, and in which assimilation and secretion are 
 much simplified. But, on account of the superior elaboration, and 
 of the greater number of elements, animal substances are much less 
 stable than vegetable : they rarely remain separate from tlie oigan- 
 ism ; and, at the same time, the new immediate ])rinci])les proper 
 I0 them are so few that their very existence h;is been questioned. 
 Vegetation is evidently the chief source of true organic comj)ounds, 
 ■which are derived thence by the animal organism, and modified by 
 it, either through their mutual combinations or new external influ- 
 ences. Thus, the true domain of chemical science must necessarily 
 be more extended by the study of vegetable than by that of animal 
 substances. 
 
 Enough has been said about the necessity of subjecting organic 
 compounds to the law of dualism; but there is a particular aspect, 
 nnder which the importance of this conception in improving 
 chemical theories is worth a brief notice. 
 
 In consideririg substances as ternary or quaternary, their raulti- 
 T)licity is accounted for only by the difference in the . ,. ,. 
 ])roportions of their constituent elements, — their of dualism 
 component princit)les being identical. Very great to organic 
 (liilerences are sometnnes explanied by mequauties 
 of ])roportion so small as to shock the s})irit of chemical analysis 
 and in other cases, the projwrtions being the same, the differences 
 remain unaccounted for; — as, for instance, in the cases of sugar and 
 gum, in which we find the same elements, combined in the same 
 })roportions. If we extend dualism to organic compoimds, this class 
 of anomalies disappeai's ; for the distinction between immediate 
 and elementary analysis enables us to resolve by dualism, in the 
 most natural manner, the general paradox of the real diversity of 
 two substances composed of the same elements, in the same propor- 
 tions. In fact, these substances, identical in their elementary, 
 would differ in the immediate analysis, as we may understand fiom 
 ■what was offered in my chapter on the law of definite proportions.
 
 294 POSITIVE PHILOSOPHY. 
 
 In another connection, chemists have remarked the possihiliiy of 
 exactly representing the nnnierical composition of alcohol or ether, 
 etc., according to several binary formnlas, radically distinct from 
 each other, and yet finally equivalent with regard to the elementary 
 analysis. Now, if such fictitious combinations should ever be 
 lealized, they would produce highly distinct substances, which 
 might differ much in the aggregate of their chemical properties, 
 and yet coincide by their elementary composition. It is only 
 necessary to transfer the same spirit into the study of organic com- 
 binations, by the establishment of a universal dualism, to dissipate 
 all these anomalies: and the resource may thus be happily prepared, 
 before the cases of isomerism (as Berzelius calls this fact) have 
 become very numerous. 
 
 We have now seen how heterogeneous is the body of doctrine 
 
 Summary of included Under the name of organic chemistry ; how 
 
 the chaiiter. j^ gliould be divided ; what is the duty of physiologists 
 
 with regard to their share of it; and how the extension of dualism 
 
 will establish a natural agreement between the composition of 
 
 substances and their collective characters. 
 
 With regard to Chemistry at large, I have pointed out the true 
 Summai-y of Spirit of the scicncc, under a philosophical view of its 
 the Book. present aspects, and of the indispensable conditions 
 
 of its advancement. We do not want new materials so much as 
 the rational disposition of the details which already abound : and I 
 have offered two prominent ideas, in my survey of chemical phi- 
 losophy ; the fusion of all genuinely chemical studies into one body 
 of homogeneous doctrine ; and the reduction of all combinations to 
 the indispensable conception of a dualism always optional. These 
 two conditions, distinct but connected, have been presented as 
 necessary to the definite constitution of chemical science. The 
 application of such a conception to the only part of chemical 
 research which yet exhibits anything of a positive rationality has 
 removed all doubt about its general fitness, by showing its spon- 
 taneous aptitude to resolve the anomalies of numerical chemistry. 
 
 With this division closes our survey of the whole of natural 
 philosophy that relates to universal or inorganic phenomena. In 
 the order of phenomena to which we next proceed, there is at once 
 much more complexity, and much less established order. The 
 study of them is scarcely yet organized ; and yet, out of the 
 speciality of the phenomena arises the most indispensable part of 
 natural philosophy, — that of which Man is first the chief object, 
 and then Society.
 
 295 
 
 BOOK V. 
 
 BIOLOGY. 
 
 CHAPTER I. 
 
 GENERAL VIEW OF BIOLOGT. 
 
 The study of the external world and of Man is the eternal business 
 of philosophy ; and there are two methods of pro- opposite 
 ceeding ; by passing from the study of Man to that starUnr/- 
 of external nature, or from the study of external nature ^I^Uosophv 
 to that of Man. Whenever philosopliy shall be per- 
 fect, the two methods will be reconciled : meantime, the contrast of 
 the two distinguishes the opposite philosophies, — the theological and 
 tiie positive. We shall see hereafter that all theological and meta 
 pliysical philosophy proceeds to explain the phenomena of the 
 external world from the starting-point of our consciousness of human 
 phenomena ; whereas, the positive philosophy subordinates the con- 
 ception of Man to that of the external world. All the multitude of 
 incompatibilities between the two philosophies proceed from this 
 radical 0{)position. If tlie consideration of Man is to prevail over 
 that of the universe, all phenomena are inevita])ly attributed to 
 tviU, — first natural, and then outside of nature; and this constitutes 
 the theological system. On the contrary, the direct study of the 
 universe suggests and develops the great idea of the laivs of nature, 
 which is the basis of all positive philosophy, and capable of exten- 
 sion to the whole of phenomena, including at last those of Man and 
 Society. The one point of agreement among all schools of theology 
 and metaphysics, which otlierwiso dilfer without limit, is tiiat they 
 regard the study of Man as primary, and that of the universe as 
 secondary,— usually neglecting the latter entirely. Whereas, the
 
 296 POSITIVE PHILOSOPHY. 
 
 most marked cliaracteristic of the positive scliool is that it founds 
 tlie study of Man on the prior knowledge of the external world. 
 This consideration affects physiology further than by its bearing 
 starting-point on its encyclopedical rank. In this one case the 
 of Physioiofiy. character of the science is affected by it. The basis 
 of its positivity is its subordination to the knowledge of the external 
 world. Any multitude of facts, however well analysed, is useless as 
 long as the old method of j)hilosophizing is persisted in, and physi- 
 ology is conceived of as a direct study, isolated from that of inert 
 nature. The study has assumed a scientific character only since 
 the recent period when vital phenomena began to be regarded as 
 subject to general laws, of which they exhibit only simple modifica- 
 tions. This revolution is now irreversible, however incomplete and 
 liowever imperfect have been the attempts to establish the positive 
 character of our knowledge of the most complex and individual of 
 ])hysiologicaI phenomena ; especially that of the nerves and brain. 
 Yet, unquestionable as is the basis of the science, its culture is at 
 jiresent too like that to which men have been always accustomed, 
 l)ursued independently of mathematical and inorganic philoso])hy, 
 which are the only solid foundations of the positivity of vital 
 .studies. 
 
 There is no science with regard to which it is so necessary to 
 ascertain its true nature and scope ; because we have not only to 
 assign its place in the scale, but to assert its originality. On the 
 one hand, metaphysics strives to retain it ; and on the other, the 
 inorganic philosophy lays hold of it, to make it a mere outlying 
 ])ortion of its scientific domain. For more than a century, during 
 which biology has endeavoured to take its place in the hierarchy of 
 fundamental sciences, it has been bandied between metaphysics and 
 physics ; and the strife can be ended only by the decision of positive 
 ])hilosophy, as to what position shall be occupied by the study of 
 living bodies. 
 
 The present backwardness of the science is explained by the 
 Its present extreme complexity of its phenomena, and its recent 
 imperfection, d^te. That Complexity forbids the hope that biologi- 
 cal science can ever attain a perfection comparable to that of the 
 more simple and general parts of natural philosophy : and, from its 
 recent date, minds which see in every other province the folly of 
 looking for first causes and modes of production of phenomena, 
 still carry these notions into the study of living bodies. For more 
 than a centuiy intelligent students have in physics put aside the 
 search after the mystery of weight, and have looked only for its 
 laws; 3'et they reproach physiology with teaching us nothing of the 
 nature of life, consciousness, and thought. It is easy to see how 
 ])liysiology may thus be supposed to be far more imperfect than it 
 is ; and if it be, from una.voidable circumstances, more backward 
 than the oilier fundamental sciences, it yet includes some infinitely
 
 RELA TION OF BIOL C V TO MEDICINE. 2 9 7 
 
 valuable conceptions, and its scitMitific cliavacter is far less inferior 
 than is comniouly supposed. 
 
 We must first describe its domain. 
 
 There is no doubt that the gradual develo]iment of human intelli- 
 «;ence would, in course of time, lead us over from the 
 theological and metaphysical state to the positive by a 
 series of logical conceptions. But such an advance would be extremely 
 slow ; and we see, in fact, that the process is much quickened by a 
 special stimulus of one sort or another. Oiu- historical experience, 
 Avhich testifies to every great advance having been made in this way, 
 shows that the most common auxiliary influence is the need of a))j)lica- 
 tion of the science in question. Most philosophers have said that 
 every science springs from a corresponding art ; — a maxim wliich, 
 amidst much exaggeration, contains solid truth, if we restrict it, as we 
 ought, to the separation of each science from the theological and meta- 
 physical philosophy which was the natural product of Relation to 
 early human intelligence. In this view it is true that a Medicine. 
 double action has led to the institution of science, the arts furnishing 
 positive data, and then leading speculative researches in the dii'ec- 
 tion of real and accessible questions. But thei'C is another side to 
 this view. When the science has once reached a certain degree of 
 extension, the progress of speculative knowledge is checked by a too 
 close connection of theory with practice. Oin- power of speculation, 
 limited as it is, still far surpasses our capacity for action : so that 
 it would be radically absurd to restrict the progress of the one to 
 tliat of the other. The rational domains of science and art are, in 
 general, ])erfectly distinct, though philosophically connected : in the 
 one we learn to know, and therefore to foreknow ; in the other to 
 become capable, and ihei'efore to act. If science springs from art, 
 it can be matured only when it has left art behind. This is pal- 
 pably true with legard to the sciences whose character is clearly 
 recognized. Archimedes was, no doubt, deeply aware of it wlien 
 he apologized to posterity for having for the moment applied iiis 
 genius to practical inventions. In the case of mathematics and 
 astronomy we have almost lost sight of this truth, from the remote- 
 ness of their formation ; but, in the case of physics and of 
 chemistry, at whose scientific birth we may almost be said to have 
 been present, we can ourselves testify to their dependence on the 
 arts at the outset, and to the rapidity of their progress after their 
 separation fiom tliem. The first series of chemical facts were fur- 
 nished by the labours of ai't ; but the prodigious recent develop- 
 ment of the science is certainly due, for the most part, to the 
 speculative chaiacter that it lias assumed. 
 
 These considerations are eminently apj)licable to physiology. 
 No other science has been closely connected with a corresponding 
 art, as biology has with medical art, — a fact accounted for by the 
 high importance of the art and the complexify of the science. But
 
 298 POSITIVE PHILOSOPHY. 
 
 for the growing needs of practical medicine, and tlie indications it 
 affords about the chief vital phenomena, physioh:)gy would have 
 probably stopped short at those academical dissertations, half 
 literary, half metaphysical, studded with episodical adornments, 
 which constituted what was called the science little more than a 
 century ago. The time however has arrived when biology must, 
 like the other sciences, make a fresh start in a purely speculative 
 direction, free from all entanglement with medical or any other art. 
 And when this science and the others shall have attained an 
 abstract completeness, then will arise the further duty, as I have 
 indicated before, of connecting the system of the arts with that of 
 the sciences by an intermediate order of rational conceptions. 
 Meanwhile such an operation would be premature, because the 
 system of the sciences is not completely formed ; and, with regard 
 to physiology especially, the first necessity is to separate it from 
 medicine, in order to secure the originality of its scientific character, 
 by constituting organic science as a consequence of inorganic. 
 Since the time of Haller this process has gone on ; but with ex- 
 treme uncertainty and imperfection ; so that even now the science 
 is, with a few valuable exceptions, committed to physicians, who 
 are rendered unfit for such a charge both by the eminent impor- 
 tance of their proper business, and by the profound imperfection 
 of their existing education. Physiology is the only science which 
 is not taken possession of by minds exclusively devoted to it. It 
 has not even a regularly assigned place in the best instituted scien- 
 tific corporations. This state of things cannot last, the importance 
 and difficulty of the science being considered. If we would not 
 confide the study of astronomy to navigators, we shall not leave 
 physiology to the leisure of physicians. Such an organization as 
 this is a sufficient evidence of the prevalent confusion of ideas about 
 physiological science ; and when its pursuit has been duly pro- 
 vided for, that reaction for the benefit of art wmII ensue which 
 should put to flight all the fears of the timid about the separ- 
 ation of theory from practice. We have seen before how the 
 loftiest truths of science concur to put us in possession of an art ; 
 and the verification of this truth, which physics and chemistry 
 have afforded before our eyes, will be repeated in the case of physi- 
 ology when the science has advanced as far. 
 
 Having provided for a speculative view of physiology, we must 
 inquire into its object ; and, as the vital laws consti- 
 tute the essential subject of biology, we must begin 
 by analysing the fundamental idea of life. 
 
 Before the time of Bichat, this idea was wrapped in a mist of 
 
 Id f L'f n'ietai)hysical abstractions ; and Bichat himself, after 
 
 havitig jierceived that a definition of life could be 
 
 founded on nothing else tiian a general view of phenomena proper 
 
 to living bodies, so far fell under the influence of the old philosophy
 
 IDEA OF LIFE. 
 
 299 
 
 ns to call life a struggle between dead nature and living nature. 
 The irrationality of this conception consists especially in its sup- 
 ])ressing one of the two elements whose concurrence is necessary to 
 tlie general idea of life. This idea supposes, not only a being so 
 organized as to admit of the vital state, but sucii an arrangement of 
 external influences as will also admit of it. The harmony between 
 the living being and the corresponding medium (as I shall call its 
 environment) evidently characterizes the fundamental condition of 
 life; whereas, on Bicliat's supposition, the whole environment of 
 living beings tends to destroy them. If certain perturbations of 
 tlie medium occasionally destroy life, its influence is, on the wliole, 
 jireservative ; and the causes of injury and death proceed at least as 
 often from necessary and spontaneous modifications of the organism 
 as from external influences. Moreover, one of the main distinctions 
 between the organic and the inorganic regions is that inorganic 
 phenomena, from their greater simplicity and generality are pro- 
 duced under almost any external influences which admit of their 
 existence at all ; while organic bodies are, from their complexity, 
 and the variety of actions always proceeding, very closely dependent 
 on the influences around them. And the higher we ascend in the 
 ranks of oiganic bodies, the closer is this dependence, in proportion 
 to the diversity of functions ; though, as we must bear in mind, the 
 power of the organism in modifying the influences of the medium 
 rises in proportion. The existence of the being then requires a 
 more complex aggregate of exterior circumstances ; but it is com- 
 patible with wider limits of variation in each influence taken by 
 itself. In the lowest rank of the organic hierarchy, for instance, 
 we find vegetables and fixed aiu'mals which have no effect on the 
 medium in which they exist, and which would therefore perish by 
 the slightest changes in it, but for the very small number of 
 distinct exterior actions required by their life. At the other extremity 
 we find Man, who can live only by the conciu-rence of the most 
 complex exterior conditions, atmosphei'ical and terrestrial, under 
 various physical and chemical aspects ; but, by an indispensable 
 compensation, he can endure, in all these conditions, nuich wider 
 differences tlian inferior orgam'sms could support because he has 
 a superior power of reacting on the surrounding system. However 
 gieat this power, it is as contnidictoiy to Bicliat's view as hisde])end- 
 ence on the exterior world. But this notion of Man's independence 
 of exterior nature, and antagonism to it, was natural in Bicliat's 
 case, when ])hysiological considerations bore no relation to any 
 hierarchy of organisms, and when Man was studied as an isolated 
 existence. However, the radical vice of such a starting-point for 
 such a study could not but impair the whole system of Bicliat's 
 physiological conceptions; and we shall see how seriously the effects 
 have made themselves felt. 
 
 Next ensued the abuse by philoso])hers, and especially in Germany,
 
 300 POSITIVE PHILOSOPHY. 
 
 of the benefits disclosed by comparative anatomy. They j^eneral- 
 ized extravagantly the abstract notion of life yielded by the study of 
 the aggregate of organized beings, making the idea of life exactly 
 equivalent to that of spontaneous activity. As all natural bodies 
 are active, in some manner and degree, no distinct notion could be 
 attached to the term ; and this abuse must evidently lead us back 
 to the ancient confusion, which arose from attributing life to all 
 bodies. The inconvenience of having two tei'ms to indicate a single 
 general idea should teach us that, to prevent scientific questions 
 from degenerating into a contest of words, we must cai'efully restrict 
 the term life to the only really living beings, — that is, those which 
 are organized, — and not give it a meaning which would include all 
 })ossible organisms, and all their modes of vitality. In this case, as 
 in all primitive questions, the philosophers would have done wisely 
 to respect the rough but judicious indications of popular good sense, 
 which will ever be the true starting-point of all wise scientific 
 s2)eculation. 
 
 I know of no other successful attempt to define life than that of 
 DcBiainviUes M, de Blainvillc, proposed in the inti'oduction to his 
 dejinition. treatise on Comparative Anatomy. He characterizes 
 life as the double interior motion, general and continuous, of composi- 
 tion and decomposition, which in fact constitutes its true universal 
 nature. I do not see that this leaves anything to be desired, unless 
 it be a more direct and explicit indication of the two correlative 
 conditions of a determinate organism and a suitable medium. This 
 criticism however applies rather to the formula than to the conception ; 
 and the conditions are implied in the conception, — the conditions 
 of an organism to sustain the renovation, and a medium to minister 
 to the absorption and exhalation ; yet it might have been better to 
 express them. With this modification, the definition is unexcep- 
 tionable, — enunciating the one phenomenon which is common to all 
 living beings, and excluding all inert bodies. Here we have, in my 
 view, the first elementary basis of true biological philosojihy. 
 
 It is true, this definition neglects the eminent distinction between 
 the organic and the cmimal life, and relates solely to the vegetative 
 life; and it appears to violate the general principle of definitions, — 
 that they should exhibit a phenomenon in the case in which it is 
 most, and not in that in which it is least developed. But the pro- 
 ])Osed definition is shown by these very objections, to rest upon a 
 due estimate of the whole biological hierarchy : for the animal life 
 is sim])ly a complementary advancement upon the organic or funda- 
 mental life, adapted to procure matei-ials for it by reaction upon the 
 external world, and to prepare or facilitate its acts by sensations, 
 locomotion, etc., and to preserve it fi'om unfavourable influences. 
 The higher animals, and Man especially, are the only ones in which 
 this relation is totally subverted, — the vegetative life being destined 
 to support the animal, which is erected into the chief end and pre-
 
 VEGETATIVE AND ANIMAL LIFE. 301 
 
 ponderant chninctev of orj2;aiiic existence. But in Man Iiirnself, 
 this admiiablo inversion of tlie usual order becomes comprehensible 
 only by the aid of a remarkable development of inte]li<i;enco and 
 sociality, which tends more and more to transform the species 
 artificially into a single individual, immense and eternal, endowed 
 with a constantly progressive action upon external nature. This is 
 the only just view to take of this subordination of the vegetative to 
 the animal life, as the ideal type towards which civilized humanity 
 incessantly tends, though it can never be fully realized. We shall 
 hereafter show bow this conception is related to tlie new funda- 
 mental science which I propose to constitute : but in pure biology. 
 the view is unscientific, and can only lead us astra}'. It is not with 
 the essential ])roperties of humanity that biology is concerned; but 
 with the individual in his relation to other oi-gnnic beings ; and it 
 uuist therefore rigorously maintain the conception of animal life 
 being subordinated to the vegetative, as a geneial law of the organic 
 realm and the only apparent exception to which forms the special 
 object of a wholly different fundamental science. It should be 
 added that, even wdiei-e the animal life is the most developed, the 
 organic life, besides being the basis and the end, remains common 
 to all the tissues, while, at the same time, it alone proceeds in a 
 uecessaiily continuous manner, — the animal life, on the contrary, 
 being intermittent. These are the grounds on which M. de 
 Blainville's definition of life must be confirmed, while, nevertheless, 
 we may regard the consideration of animality, and even of humanity, 
 as the most important object of biology. 
 
 This analysis of the phenomenon of life will help us to a clear 
 definition of the science which relates to it. We Definition of 
 have seen that the idea of life supposes t'.ie uuitual Binioyy. 
 relation of two indispensable elements, — an oiganisra, and a suitable 
 medium or environment. It is from the reciprocal action of these 
 two elements that all the vital phenomena proceed ; — not only the 
 animal, but also the organic. It immediately follows that the 
 great problem of positive biology consists in establishing, in the 
 most general and simple manner, a scientific harmony between 
 these two inseparable powers of the vital conflict, and the act which 
 constitutes that conflict : in a word, in connecting, in both a 
 general and special maimer, the double idea of organ and medium 
 with that of function. The idea of function is, in fact, as double as 
 the other; and, if we were treating of the natural history of vital 
 beings, we must expiessly consider it so : for, by the law of the 
 equivalence of action and reaction, the oi-gani.'-m must act on the 
 medium as much as the medium on the organism. In treating of the 
 human being, and especially in the social state, it would be necessary 
 to use the term function in this largei- sense : but at present there 
 will be little inconvenience in adopting it in its ordinary sense, 
 signifying organic acts, independently of their exterior consequences.
 
 302 POSITIVE PHILOSOPHY. 
 
 Bioloo;y, then, may be regarded as having for its object tlie con-- 
 necting,in each determinate case, the anatomical and the physiological 
 point of view; or, in other words, the statical and dynamical. This 
 perpetnal relation constitutes its true philosophical character. Placed 
 in a given system of exterior circumstances, a definite organism must 
 always act in a necessarily determinate manner; and, inversely, the 
 same action could not be precisely produced by really distinct 
 organisms. We may then conclude interchangeably, the act from 
 the subject, or the agent from the act. The surrounding system 
 being always supposed to be known, according to the other fimda- 
 mental sciences, the double biological problem may be laid down 
 thus, in the most mathematical form, and in general terms: Given, 
 the organ or organic modification, to find the function or the act ; 
 and reciprocally. This definition seems to me to fulfil the chief 
 philosophical conditions of the science ; and especially it provides 
 for that rational prevision, which, as has been so often said, is the 
 end of all true science ; an end which abides through all the degrees 
 of imperfection which, in any science, at present prevents its attain- 
 ment. It is eminently important to keep this end in view in a 
 science so intricate as this, in which the multitude of details tempts 
 to a fatal dispersion of efforts upon desultory researches. No one 
 disputes that the most perfect portions of the science are those in 
 which prevision has been best realized ; and this is a sufficient 
 justification of the proposal of this aim, whether or not it shall ever 
 be fully attained. My definition excludes the old division between 
 anatomy and physiology, because I believe that division to have 
 marked a very early stage of the science, and to be no longer 
 sustainable. It was by the simple and easy considerations of 
 anatomy that the old metaphysical view was discredited, and posi- 
 tivity first introduced into biology: but that service once accom- 
 plished, no reason remains for the separation ; and the division, in 
 fact, is growing fainter every day. 
 
 Not only does my definition abstain from separating anatomy 
 from physiology ; it joins to it another essential part, the nature of 
 which is little known. If the idea of life is really inseparable from 
 that of oi'ganization, neither can be severed, as we have seen, from 
 that of a medium or environment, in a determinate relation with 
 them. Hence arises a third elementary aspect ; viz., the general 
 tlieory of organic media, and of their action upon the organism, 
 abstractedly regarded. Tliis is what the German philosophers of 
 our day confusedly asserted in their notion of an intermediate realm, 
 — of air and water,— uniting the inorganic and organic worlds : and 
 this is what M. de Blainville had in view in what he called the 
 study of exterior modifiers, general and special. Unhappily, this 
 ])ortion which, after anatomy proper, is the most indispensable 
 preliminary of biology, is still so obscuie and imperfect that few 
 physiologists even suspect its existence.
 
 DEFINITION OF BIOLOGY. 303 
 
 The definition that I have proposed aids ns in describinj^ not 
 only the object or nalnre of the science, bnt its snbject, or domain : 
 for, according to this fornmhi, it is not in a single organism, bnt in 
 all known, and even possible organisms, that biology must endea- 
 vour to establish a constant and necessary harmony between the 
 anatomical point of view and the })hysiological. This unity of 
 snbject is one of the chief" philosophical beauties of biology ; and, 
 in order to maintain it, we must here avow that, in the midst of an 
 almost infinite diversity, the study of man must always prevail, 
 and rule all the rest, whether as starting-point or aim. Our hope, 
 in studying other organisms, is to arrive at a more exact knowledge 
 of Man : and again, the idea of Man is the only possible standard 
 to which we can refer other organic systems. In this sense, and in 
 this only, can the point of view of the antiquated philosophy be 
 sustained by the deeper philosophy which is taking its place. Such 
 is, then, the necessary consolidation of all the parts of biological 
 science, notwithstanding the imposing vastness of its rational 
 domain. 
 
 As for the means of investigation in this science, — the first 
 observation that occurs is that it affords a striking Means ofin- 
 confirmation of the philosophical law before laid vestujatwn. 
 down, of the inevitable increase of our scientific resources in pro- 
 jiortion to the complication of the phenomena in question. If 
 biological phenomena are incomparably more complex than those 
 of any preceding science, the study of them admits of the most 
 extensive assemblage of intellectual means (many of them new) 
 and develops human faculties hitherto inactive, or known only 
 in a rudimentary state. The logical resources which are tlius 
 obtained w'ill be exhibited hereafter. At present, we must notice 
 the means of direct exploration and analysis of phenomena in this 
 science. 
 
 First, Observation acquires a new extension. Chemistry ad- 
 mitted the use of all the five senses ; but biology is, ^ 
 in this respect, an advance upon chemistry. We 
 can here employ an artificial a{)paratus to perfect the natural sen- 
 sations, and especially in the case of sight. Much needing precau- 
 tion in the use, and very subject to abuse, as is this resource, it 
 ■will always be eagerly employed. In a statical view. Artificial 
 such an apparatus helps us to a much better estimate apparatus. 
 of a structure whose least perceptible details may acquire a primary 
 importance, in various relations : and, even in the dynamical view, 
 though much less favourable, we are sometimes enabled by these arti- 
 ficial means to observe directly the elementary play of the smallest 
 organic parts, which are the ordinary basis of the principal vital 
 phenomena. Till recently, these aids were limited to the sense of 
 sight, which here, as everywhere else, is the chief agent of scientific 
 observation. But some instruments have been devised in our day
 
 304 POSITIVE PHILOSOPHY. 
 
 to assist the lieariug ; and, though invented for pathological inves- 
 tigations, they are equally fit for the study of the healthy organism. 
 Though rough at present, and not to be compared to microscopic 
 apparatus, these instruments indicate the improvements that may 
 be made hereafter in artificial hearing. Moreover, they suggest, 
 by analogy, that the other senses, not excepting even touch, may 
 admit of such assistance, hinted to the restless sagacity of explorers 
 by a better theory of the corresponding sensations. 
 
 Next, the biologist has an advantage over the chemist in being 
 Chemical able to employ the whole of chemical procedures, as 
 exploration. ^ sort of new power, to perfect the preliminary ex- 
 ploration of the subject of his researches, according to the evident 
 rule of philosophy that each doctrine may be converted into a 
 method with regard to those that follow it in the scientific hierarchy ; 
 but never with regaid to those which precede it. In anatomical 
 observations, especially, as might be foreseen, a happy use is made 
 of chemical procedure, to characterize with precision the differ- 
 ent elementary tissues, and the chief products of the organism. In 
 physiological observations also, though they are less favourable 
 to the use of such means, they are of real and notable efficac}', 
 — always supposing, in both cases, that they are used under the 
 guidance of sound philosophy, and not overcharged with the 
 minute numerical details which too often burden the chemical 
 analyses of the organic tissues. One more resource may be men- 
 tioned, which was often employed by Bichat to make up for the 
 absence or imperfection of chemical tests ; the examination of 
 alimentary effects, — the substances which immediately compose 
 organized bodies being, usually, by their nature, moi-e or less fit 
 for nutrition. In an anatomical view, this study may become a 
 useful complement of the other means of investigation. 
 Proceeding to the second class of means, — Experiment cannot 
 but be less and less decisive, in proportion to the com- 
 plexity of the phenomena to be explored : and there- 
 fore we saw this resource to be less effectual in chemistry than in 
 ])hysics ; and we now find that it is eminently useful in chemistry 
 in comparison with physiology. In fact, the nature of the pheno- 
 mena seems to offer almost insurmountable impediments to any 
 extensive and prolific application of such a procedure in biology. 
 These phenomena require the concurrence of so large a number of 
 distinct influences, external and internal, which, however diverse, 
 are closely connected with each other, and yet within narrow limits, 
 that, however easy it may be to disturb or suspend the process under 
 notice, it is beyond measure difficult to effect a determinate pertur- 
 bation. If too powerful, it would obviate the phenomenon : if too 
 feeble, it would not sufficiently mark the artificial case. And, on 
 the other hand, though intended and directed to nK)dify one only 
 of the phenomena, it nuist presently affect several others, in virtue
 
 EXPERIMENT. 305 
 
 of their mutual sympathy. Thus, it requires ahi<^hly philosophical 
 spirit, acting with extreme circumspection, to conduct physiological 
 experiments at all ; and it is no wonder if such experiments have, 
 with a few happy exceptions, raised scientific difficulties greater 
 tiian those proposed to he solved, — to sny nothing of those innumer- 
 ahle experiments which, having no definite aim, have merely encum- 
 bered the science with idle and unconnected details. 
 
 In accordance with what has been said of the mutual relations of 
 the organism and its environment, we must bear in mind that ex- 
 ])eriments in physiology must be of two kinds. We must introduce 
 determinate perturbations into the medimu as well as the organism ; 
 whereas the latter process has alone been commonly attempted. If 
 it is objected that the organism must itself be dis- Bij affectmy 
 lurbed by such affection of the medium, the answer the oiv/antsm. 
 is that the study of this reaction is itself a part of the ex[)eriinent. 
 It should be remarked that experimentation on the organism is 
 much the less rational of the two methods, because the conditions 
 of experiment are much less easily fulfilled. The first rule, that the 
 change introduced shall be fully compatible with the existence of the 
 phenomenon to be observed, is rendered often impracticable by the 
 incompatibility of life with much alteration of the organs : and the 
 second rule, — that the two compared cases shall ditier under only 
 one point of view, — is baffled by the mutual sympathy of the organs, 
 which is very diflierent from their harmony with their environment. 
 In both lights, nothing can be imagined more futile in the way of 
 experiment than the practice of vivisection, which is the commonest 
 of all. Setting aside the consideration of the cruelty, the levity, 
 and the bad moral stimulus involved in the case, it must be pro- 
 nounced absurd ; for any positive solution is rentlered impossible 
 by the induced death of a system eminently indivisible, and the 
 imiversal disturbance of the organism under its approach. 
 
 The second class of physiological experiments appears to me 
 much more promising ;— that in which the system of Bu affcdimj 
 exterior circumstances is modified for a determinate themedi,um. 
 purpose. Scarcely anything has been done in this direction beyond 
 some incomplete researches into the action of artificial atmospheres, 
 and the comparative influence of ditierent kinds of alimentation. 
 We are here better able to circumscribe, with scientific precision, 
 the artificial perturbation we produce ; we can control the action 
 upon the organism, so that the general disturbance of the system 
 may affect the observation very slightly ; and we can suspend the 
 process at pleasure, so as to allow the restoration of the normal 
 state before the organism has undergone any irreparable change. 
 It is easy to see how favourable, in comparison, these conditions are 
 to rational induction. And to these considerations may be added 
 the one more, that under this method we can observe varying states 
 in one individual ; whereas, under the practice of vivisection, we 
 
 VOL. I. u
 
 3oa POSITIVE PHILOSOPHY. 
 
 liave to observe tlie norm.al state in one individual, and the artificial 
 in another. Thus we are justified In our satisfaction that the least 
 violent method of experimentation is the most instructive. 
 
 As to the application of experiment in the various degrees of 
 Comparative tiie blologlcal scale ; — it is easiest in the lower order 
 experiment. of Organisms, because their organs are simpler and 
 fewer, their mutual sympathy is less, and their environment is 
 more definite and less complex ; and these advantages, in my 
 opinion, more than compensate for the restriction of the field of ex- 
 periment. It is true, we are remote from the human type, which is 
 the fundamental unity of biology; and our judgment is thus impaired, 
 especially with regard to the phenomena of animal life : but, on the 
 other hand, we are all the nearer to the scientific constitution proper 
 to inorganic physics, which I consider to be the ultimate destination 
 of the art of experiment. The advantages at the other end of the 
 scale are that the higher the organism, the more is it susceptible of 
 modification, both from its own complexity and from the greater 
 variety of external influences involved ; — every advantage bringing 
 with it, as we have seen, an increase of difficulty. 
 
 No one will suppose, I trust, that from anything I have said I 
 have the slightest desire to undervalue the use of experiment in 
 biology, or to slight such achievements as Harvey's experiments 
 on circulation ; Haller's on irritability ; part of Spallanzani's 
 on digestion and generation ; Bichat's on the triple harmony 
 between the heart, the brain, and the lungs in the superior 
 animals ; those of Legallois on animal heat ; and many analogous 
 efforts which, seeing the vast difficulty of the subject, may rival the 
 most perfect investigations in ])hyslcs. My object is simply to 
 rectify the false or exaggerated notions of the capacity of the experi- 
 mental method, misled by its apparent facility to suppose it the best 
 method of physiological research ; which it is not. One considera- 
 tion remains in this connection; the consideration of the high 
 scientific destination of pathological investigation, regarded as 
 offering, in biology, the real equivalent of experimentation, properly 
 so called. 
 
 Precisely in the case in which artificial experimentation is the 
 Pathological most difficult, nature fulfils tiie conditions for us ; 
 Investigation, and It would surely be mistaking the means for the 
 end to insist on introducing into the organism perturbations of our 
 own devising, when we may find them taking place without that 
 additional confusion which is caused by the use of artificial methods. 
 Physiological phenomena lend themselves remarkably to that spon- 
 taneous experimentation which results from a comparison of the 
 normal and abnormal states of the organism. The state of disease 
 is not a radically difterent condition from that of health. The 
 pathological condition is to the physiological simply a prolongation 
 of the limits of variation, higher or lower, proper to each phenomenon
 
 PAXHOLOGICAL INVESTIGATION. 507 
 
 of the normal organism ; and it can never produce any entirely new 
 ])henomenon. Tlierefore, tlie accurate idea of the physiolof^ical state 
 is the indispensable ground of any sound pathological theory ; and 
 therefore, again, must the scientific stud3^of pathological phenomena 
 be the best way to perfect our investigations into the normal state. 
 The gradual invasion of a malady, and the slow passage from an 
 almost natural condition to one of fully marked disease, are ftir from 
 being useless preliminaries, got rid of by the abrupt introduction of 
 what may be called the violent malady of direct experiment : they 
 offer, on the contrary, inestimable materials to the biologist able to 
 ])ut them to use. And so it is also in the happy converse case, of 
 the return, spontaneous or contrived, to health, which presents a sort 
 of verification of the primitive analysis. Moreover, the direct exami- 
 nation of the chief phenomenon is not obscured, but much elucidated, 
 by this natural process. And again, it may be applied directly 
 to Man himself, without prejudice to the pathology of animals, and 
 even of vegetables. We may enjoy our power of turning our 
 disasters to the profit of our race : and we cannot but deplore the 
 misfortune that our great medical establishments are so constituted 
 as that little rational instruction is obtained from them, for want of 
 complete observations and duly prepared observers. 
 
 Here, as elsewhere, the distinction holds of the phenomena be- 
 longing to the organism or to the medium ; and here, as before, we 
 find the maladies produced from without the most accessible to in- 
 quiry. Pathological inquiry is also more suitable than experimental, 
 to the whole biological series : and thus it answers well to extend 
 our observations through the entire hierarchy, though our object 
 may be the study of Man ; for his maladies may receive much light 
 from a sound analysis of the derangements of other organisms, — even 
 the vegetable, as we shall see when we treat of the comparative 
 process. 
 
 Again, pathological analysis is applicable, not only to all organ- 
 isms, but to all phenomena of the same organism ; whereas direct 
 experimentation is too disturbing and too abrupt to be ever applied 
 with success to certain phenomena which require the most delicate 
 harmony of a varied system of conditions. For instance, the obser- 
 vation of the numerous maladies of the nervous system offers us a 
 special and inestimable means of improving our knowledge of the 
 laws of intellectual and moral phenomena, imperfect as are yet our 
 qualifications for using them. There remains one other means of 
 knowledge under this head ; the examination of exceptional organi- 
 zations, or cases of monstrosity. As might be anticipated, these 
 organic anomalies were the last to pass over from the gaze of a 
 barren curiosity to the investigation of science ; but we are now 
 learning to refer them to the laws of the regular organism, and to 
 subject them to pathological procedures, regarding such exceptions 
 as true maladies, of a deeper and more obscure origin than others,
 
 30$ POSITIVE PHILOSOPHY. 
 
 and of a more Incurable nature; — considerations wlilcli, of course, 
 reduce their scientific value. This resource shares with patholo;;y 
 the advantage of being applicable through the whole range of the 
 biological system. 
 
 It is still necessary to insist that, in either method of experimen- 
 tation, direct or indirect, artificial or natural, the elementary rules 
 should be kept in view ; first, to have a determinate aim ; that is, 
 to seek to illustrate an organic phenomenon, under a special aspect ; 
 and, secondly, to understand beforehand the normal state, and its 
 limits of variation. In regard to the more advanced sciences, it 
 would seem puerile to recommend such maxims as these ; but we 
 must still insist on them in biology. It is through neglect of them 
 that all the observations yet collected on the derangements of the 
 intellectual and moral phenomena have yielded scarcely any know- 
 ledge of their laws. Thus, whatever may be the value of the most 
 suitable method of experimentation, we must ever remember that 
 here, as elsewhere, and more than elsewhere, pure observation must 
 always hold the first rank, as casting light, primarily, on the whole 
 subject, which it is proposed to examine afterwards, as a special 
 study, with a determinate view, by the method of experimentation. 
 
 In the third place, we have to review the method of Comparison, 
 which is so specially adapted to the study of living 
 bodies, and by which, above all others, that study 
 must be advanced. In Astronomy, this method is necessarily 
 inapplicable : and it is not till we arrive at Chemistry that this 
 third means of investigation can be used ; and then, only in subor- 
 dination to the two others. It is in the study, both statical and 
 dynamical, of living bodies, that it first acquires its full develop- 
 ment ; and its use elsewhere can be only through its application 
 here. 
 
 The fundamental condition of its use is the unity of the principal 
 subject, in combination with a great diversity of actual modifications. 
 According to the definition of life, this combination is eminently 
 realized in the study of biological phenomena, however regarded. 
 The whole system of biological science is derived, as we have seen, 
 from one great })hilosophical conception ; the necessary correspon- 
 dence between the ideas of organization and those of life. There 
 cannot be a more perfect fundamental unity of subject than this ; 
 and it is unnecessary to insist upon the almost indefinite variety of 
 its modifications, — statical and dynamical. In a purely anatomical 
 view, all possible organisms, all the piu'ts of each organism, and all 
 the difierent states of each, necessarily present a common basis 
 of structure and of composition, whence proceed successively the 
 different secondary organizations which constitute tissues, organs, 
 and systems of organs, more or less complicated. In the same way, 
 in a physiological view, all living beings, from vegetable to man, 
 , considered in all the acts and periods of their existence, are endowed
 
 COMPARISON. 309 
 
 \vllli a corlain common vitality, which is a noccssavy basis of the 
 inimmcrable phenomena which characterize them in their degrees. 
 lioth these aspects present as most important, and really fnnda- 
 niental, what there is in common among all the cases ; and their 
 ])articnlarities as of less conseqnence ; which is in accordance with 
 ihe great prevalent law, tliat the more general })lienomena overrule 
 the less. Thus broad and sound is the basis of the comparative 
 method, in regard to biology. 
 
 At the first glance the immensity of the science is overwhelming 
 to the understanding, embracing as it, does all organic and vital 
 cases, which it appears imjiossible ever to reduce within the com- 
 pass of our knowledge : and no doubt, the discouragement hence 
 :i rising is one cause of the backwardness of biological philosophy. 
 Yet tlie truth is that this very magnitude affords, not an obstacle, 
 but a facility to the perfecting of the science, by means of the lumi- 
 nous comparison which results from it, when once the human 
 mind becomes familiar enough with the conditions of the study 
 I0 dispose its materials so as to illustrate each other. The science 
 could make no real progress while Man was studied as an isolated 
 subject. Man must necessarily be the type ; because he is the 
 most complete epitome of the whole range of cases : Man, in his 
 adult and normal state, is the representative of the great scientific 
 unity, whence the successive terms of the great biological series 
 recede, till they terminate in the simplest oiganizations, and the 
 most imperfect modes of existence. But the science would remain 
 in the most defective state in regard to Man himself, if it were not 
 ])ursued through a peipetual comparison, imder all possible aspects, 
 of the first tei'm with all inferior ones, till the simplest was reached : 
 and then, back again, through the successive complications which 
 occur between the lowest type and the highest. This is the most 
 general, the most certain, tlie most effectual method of studying 
 ])hysiological as well as anatomical phenomena. Not only is there 
 ihus a greater number of cases known, but each case is much 
 lictter undeistood by their approximation. This would not be the 
 case, and the problem would be embarrassed instead of simplified, 
 if there were not a fundamental resemblance among the whole series, 
 accompanied by gradual modifications, always regulated in their 
 course : and this is the reason why the comparative method is 
 appropriate to biology alone, of all the sciences, except, as we shall 
 see hereafter, in social physics. 
 
 Complete and si)()ntaneous as this harmony really is, no philos- 
 opher can contemplate without admiration the eminent art by 
 which the human mind has been aided to convert into a potent 
 means what appeared at first to be a formidable difficulty. I know 
 no stronger evidence of the force of human reason than such a 
 transformation affords. And in this case, as in every other in 
 which primordial scientific powers are concerned, it is the work
 
 310 POSITIVE PHILOSOPHY. 
 
 of the whole race, j^jvadiially developed in the course of ages, and 
 not the original product of any isolated mind, — however some 
 moderns may he asserted to he the creators of comparative hiology. 
 Between the primitive use that Aristotle made of this method in 
 the easiest cases, — as in comparing the structure of man's upper 
 and lower limbs, — to the most profound and abstract approxima- 
 tions of existing biology, we find a very extensive series of inter- 
 mediate states, constantly progressive, among which history can 
 point out individually only labours which prove what had been the 
 advance in the spirit of the comparative art at the corresponding 
 period, as manifested by its larger and more effectual application. 
 It is evident that the comparative method of biologists was no 
 more the invention of an individual than the experimental method 
 of the physicists. 
 
 Five kinds of There are five principal heads under which bio- 
 comparison. logical Comparisons are to be classed. 
 
 1. Comparison between the different parts of the same organism. 
 
 2. Comparison between the sexes. 
 
 3. Comparison between the various phases presented by the 
 whole of the development. 
 
 4. Comparison between the different races or varieties of each 
 species. 
 
 5. Lastly, and pre-eminently, Comparison between all the 
 organisms of the biological hierarchy. 
 
 It must be understood that the organism is always to be sup- 
 posed in a normal state. When the laws of that state are 
 fully established, we may pass on to pathological comparison, 
 which will extend the scope of those laws : but we are not yet 
 advanced enough in our knowledge of normal conditions to under- 
 take anything beyond. Moreover, though comparative pathology 
 would be a necessary application of biological science, it cannot 
 form a part of that science, but rather belongs to the future medical 
 science, of which it must form the basis. — Again, biological com- 
 parison can take place only between the organisms, and not between 
 them and their medium. When such comparison comes to be 
 instituted, it will be, not as biological science, but as a matter 
 of natural history. 
 
 The spirit of biological comparison is the same under all forms. 
 It consists in regarding all cases as radically analogous in respect 
 to the proposed investigation, and in representing their differences 
 as simple modifications of an abstract type ; so that secondary 
 differences may be connected with the i)rimary according to uniform 
 laws ; these laws constituting the biological philosophy by whicli 
 each determinate case is to be explained. If the question is anato- 
 mical, Man, in his adult and normal state, is taken for the funda- 
 mental unity, and all other organizations as successive simplifica- 
 tions, descending from the primitive type, whose essential features
 
 KINDS OF COMPARISON, 311 
 
 will be found in tlic remotest cases, stripped of all complication. 
 If the question is physiological, we seek the fundamental identity 
 of the chief })henonienon which characterizes the function proposed, 
 amidst the graduated modifications of the series of comparative 
 cases, till we find it isolated, or nearly so, in the simplest case of 
 all ; and thence we may trace it back again, clothed in successive 
 complications of secondary qualities. Thus, the theory of analo- 
 gous existences, which has been offered as a recent innovation, is 
 only the necessary principle of the comjiarative method, under 
 a new name. It is evident that this methotl nnist be of surpassing 
 value when philosophically applied : ai'id also that, delicate as it is, 
 and requiring extreme discrimination and care in its estimate and 
 use, it may be easily converted into a hindrance and embarrassment, 
 by giving occasion to vicious speculations on analogies which arc 
 only apparent. 
 
 Of the five classes specified above, three only are so marked as ta 
 require a notice here : the comparison between the different parts 
 of the same organism ; between the different phases of each 
 development; and between the distinct terms of the gi-eat hierarchy 
 of living bodies. 
 
 The method of comparison began with the first of these. Look- 
 ing no further than Man, no philosoidu'cal mind can » 
 
 1 7 1 • I ^ -i A 111 11 Comparison 
 
 lielp bemg struck by tlie remarkable resemblance of parts of 
 that his different chief parts bear to each other in the same 
 many respects, — both as to structure and function. °^ o"'^"''"'- 
 First, all the tissues, all the apparatus, in as far as they are oi-gan- 
 ized and living, offer those fundamental characteristics which are 
 inherent in the very ideas of organization and life, and to which the 
 lowest organisms are reduced. But, in a more special view, the 
 analogy of the organs becomes more and more marked as that of 
 the functions is so ; and the converse ; and this often leads to 
 luminous comparisons, anatomical and physiological, passing from 
 the one to the other, alternately. This original and simple method 
 of comparison is by no means driven out by newer processes. It 
 was thus, for instance, that Bicl)at, whose subject was Man only, 
 and adult Man, discovered the fundamental analogy between the 
 mucous and the cutaneous systems, which has yielded so much 
 advantage to both biology and pathology. And again, with all M. 
 de Blainville's mastery of the principle of the comparative method, 
 we cannot doubt the sufficiency of the analysis of the human organ- 
 ism to establish the resemblance he exhibited between the skull and 
 the other elements of the vertebral column. 
 
 A new order of resources presents itself when we compare the 
 different ])hases of the same oi-ganism. Its chief of phases of 
 value is in its offering, on a small scale, and, as it were, the same or- 
 under one aspect, the whole series of the most marked oanism. 
 organisms of the biological hierarchy ; for it is obvious that the
 
 512 POSITIVE PHILOSOPHY. 
 
 primitive state of the highest organism mnst present the esseniia! 
 characters of tlie complete state of the lowest ; and thus successively, 
 — without, however, compelling us to tind the counterpart of every 
 inferior term in the superior organism. Such an analysis of ages 
 unquestionahly offers the property of realizing in an individual 
 that successive complication of" organs and functions which charac- 
 terizes the biological hierarchy, and which, in this homogeneous 
 and compact form, constitutes a special and singular order of 
 luminous comparisons. Useful through all degrees of the scale, it 
 is evidently most so in the case of the highest type, the adult Man, 
 as the interval from the origin to the utmost complexity is in that 
 case the greatest. It is valuable chiefly in the visible ascendant 
 ])eriod of life ; for we know very little of the foetal period ; and the 
 declining stage, which is in fact onl}' a gradual death, presents 
 little scientific interest: for, if there are many ways of living, there 
 is only one natural w^ay of dying. The rational analysis of death, 
 however, has its own importance, constituting a sort of general 
 corollary, convenient for the verification of the whole body of 
 biological laws. 
 
 The popular notion of comparative biology is that it consists 
 Ofdifferent wholly of the last of the methods I have pointed out: 
 orcjaiiisms. ^•^^^[ w^\^ sliows how pre-eminent it is over the others ; 
 the popular exaggeration however being mischievous by concealing 
 the origin of the art. The peculiarity of this largest application 
 consists in its being founded on a very protracted comparison of a 
 very extensive series of analogous cases, in which the modification 
 proceeds by almost insensible graduated declension. The two 
 )nore restricted methods could not otier a series of cases extensive 
 enough to establish, without confirmation, the nature and value of 
 the comparative method, though, that point once fixed, they may 
 then come into unquestionable use. As for the value of the largest 
 application, it demonstrates itself There is clearly no structure or 
 function whose analysis may not be perfected b}' an examination of 
 what all oi-ganisms offer in common with regard to that structure 
 and function, and by the simplification effected by the stripping 
 away of all accessory characteristics, till the quality sought is found 
 alone, from whence the process of reconstruction can begin. It 
 may even be fairly said that no anatomical arrangement, and no 
 ])hysiological phenomenon, can be really understood till the abstract 
 notion of its principal element is thus reached, by successively 
 attaching to it all secondaiy ideas, in the rational order prescribed 
 by their greater or less persistence in the oi'ganic series. Such a 
 method seems to me to offer, in biology, a philosophical character 
 very like mathematical analysis genuinely applied ; when it presents, 
 as we have seen, in every indefinite series of analogous cases, the 
 essential part which is common to all, and which was before hidden 
 under the secondary specialities of each sej^arate case. It cannot
 
 COMFAJilSON OF ORGAXJSMS. ^it, 
 
 1)C doubted tliat tlie comparntive art of biulonists will pvodnce an 
 tqnivalent result, up to a certain point ; and especially, by the 
 jational consideration of the organic hierarchy. 
 
 This great consideration was at first established only in regard 
 1o anatomy ; but it is yet more necessary in physiology, and not 
 less apphcable, except from the difficulty of that kind of observa- 
 tion. In I'cgard to physiological })roblems particularly, it should 
 l)e remarked that not only all animal organisms, but the vegetable 
 also, should be included in the comparison. Many important 
 jjhenomena, and among others those, of organic life, properly so 
 called, cannot be analysed without an inclusion of the vegetable 
 form of them. There we see them in their simplest and most 
 marked condition, for it is by the great act of vegetable assimila- 
 tion that brute matter passes really into the organized state, all 
 ulterior transformations by means of the animal organization being 
 much less marked. And thus, the laws of nutrition, which are of 
 the highest importance, are best disclosed by the vegetable organ- 
 ism. The method is nnquestionably applicable to all organs and 
 all acts, without any exception ; but its scientific value diminishes 
 as it is api)lied to the higher apparatus and functions of the supe- 
 rior organisms, because these are restricted in proportion to their 
 comjilexity and superiority. This is eminently the case with 
 the highest intellectual and moral functions which below Man dis- 
 jippear almost entirely ; or, at least, almost cease to be recognizable 
 below the fii'st classes of the mammifers. We cannot but feel it to 
 be an imperfection in the comparative method that it serves us 
 least where we are most in need of all our resources ; but it would 
 be nnphilosophical to dej)rive ourselves, even in this case, of the 
 light which is cast upon the analysis of Man as moi-al, by the study 
 of the intellectual and affective qualities of the superior animals, and. 
 of all others which present such attributes, however imperfect our 
 management of the comparison may yet be. And we may observe 
 that the comparative method finds a partial equivalent in the 
 lational analysis of ages. — thus rendered more clear, extensive, and 
 (■om[ilete, — for the disadvantages which belong to the same stage 
 of the biological hierarchy. 
 
 Thus I have presented the principal philosophical characters of 
 the comparative method. It being the aim of biological study to 
 ascertain the general laws of organic existence, it is })lain that no 
 course of inquiry could be more favourable than that which exhibits 
 organic cases as ]-adically analogous, and deducible from each 
 otiier. 
 
 This study of our means of exploration has shown that our re- 
 sources do indeed increase with the complexity of our subject. The 
 two first methods — of Observation and Experiment — we have seen 
 lo acquire a large extension in the case of this science : while the 
 third, befoie almost imperceptible, becomes, by the nature of the
 
 314 POSITIVE PHILOSOPHY. 
 
 plienoniGiia, wellnigli unbounded in its scoj>e. We have next to 
 examine tlie tine rational position of Biology in the hierarchy of 
 Ihe fundamental sciences ; that is, its relation to those that precede 
 it, and to the one which follows it, in order to ascertain what kind and 
 degree of speculative perfection it admits of, and what preliminary 
 training is best adapted to its systematic cultivation. By this 
 inquiry we shall see why we are justified in assigning to it a place 
 between chemistry and social science. 
 
 Of the relation of Biology to social science, I need say little here, 
 
 Relation of Bi- ^^ ^ shall have to speak ot" it at length in the next 
 
 oiornj to other volume. My taslv will then be to separate them, 
 
 sciences. rather than to establish their connection, which it is 
 
 ihe tendency of our time to exaggerate, through the spontaneous 
 
 development of natural philosophy. None but purely metaphysical 
 
 philosophers would at this day persist in classing the theory of the 
 
 human mind and of society as anterior to the anatomical and 
 
 physiological study of individual man. We may therefore regard 
 
 this point as sufficiently settled for the present, and pass on to the 
 
 relation of Biology to inorganic philosophj'. 
 
 It is to chemistry that Biology is, by its nature, most directly and 
 completely subordinated. In analysing the pheno- 
 menon ot lire, we saw that theiiindamental acts which, 
 by their perpetuity, characterize that state, consist of a series of 
 compositions and decompositions ; and they are therefore of a 
 chemical nature. Though in the most imperfect organisms, vital 
 reactions are widely separated from common chemical effects, it is 
 not the less true that all the functions of the proper organic life are 
 necessarily controlled by those fundamental laws of composition 
 and decomposition which constitute the subject of chemical science. 
 If we could conceive throughout the whole scale the same separa- 
 tion of the oi-ganic from the animal life that we see in vegetables 
 alone, the vital motion would offer only chemical conceptions, 
 except the essential circumstances which distinguish such an order 
 of molecular reactions. Tiie general source of these important 
 differences is, in my opinion, to be looked for in the result of each 
 chemical conflict not depending only on the simple composition of 
 the bodies between which it takes place, but being modified by 
 their proper oiganization ; that is, by their anatomical structure. 
 Chemistry must clearly furnish the starting-point of every rational 
 theory of nutrition, secretion, and, in short, all the functions of the 
 vegetative life, considered separately ; each of which is controlled 
 by the influence of chemical laws, except for the special modifica- 
 tions belonging to organic conditions. If we now bring in agnin 
 the consideration, discarded for the moment, of the animal life, we 
 see that it could in no way alter this fundamental subordination, 
 4,hough it must greatly complicate its actual application : for we
 
 RELA TION OF BIOL OGY TO CHEMISTR Y. 315 
 
 have seen that the animal life, notwithstanding its vast importance, 
 can never be regarded in biology otherwise than as destined to 
 extend and perfect the organic life, whose general natnre it cannot 
 change. Such an intervention modifies, anew and largely, the 
 chemical laws of the purely organic functions, so as to render the 
 effect very difficult to foresee; but not the less do these laws 
 continue to control the aggregate of the plienomena. If, for instance, 
 a change in the nervous condition of a superior organism disturbs 
 a given secretion, as to its energy or even its nature, we cannot 
 conceive that such an alteration can, be of a random kind: such 
 modifications, irregular as they may appear, are still submitted to 
 the chemical laws of the fundamental organic phenomenon, which 
 permit certain variations, but interdict many more. Thus, no com- 
 plication produced by animal life can withdraw the organic functions 
 I'rom their subordination to the laws of composition and decompo- 
 sition. This relation is so important, that no scientific theory 
 could be conceived of in biology without it ; since, in its absence, 
 the most fundamental phenomena might be conceived of as 
 susceptible of arbitrary variations, which would not admit of any 
 true law. When we hear, at this day, on the subject of azote, such 
 a doctrine as that the organism has the power of spontaneously 
 creating certain elementary substances, we perceive how indispen- 
 sable it still is to insist directly on those principles which alone can 
 restrain the spirit of aberration. 
 
 Besides tliis direct subordination of biology to chemistry, there 
 are relations of method between them. Observation and experi- 
 mentation being much more perfect in chemistry, they serve as an 
 admirable training for biological inquiry. Again, a special property 
 of chemistry is its developing the art of scientific nomenclature ; 
 and it is in chemistry that biologists must study this important 
 part of the positive method, though it cannot, from the complexity 
 of their science, be of so much scientific value as in chemistry. It 
 is on the model of the chemical nomenclature that those systematic 
 denominations have been laid down by which biologists have 
 classified the most simjjle anatomical arrangements, certain well- 
 defined patliological states, and the most general degrees of the 
 animal hierarchy : and it is by a continued pursuit of the same 
 method that further improvements will be effected. 
 
 We thus see why biology takes its place next after chemistry, 
 and why chemical inquiries constitute a natural transition from the 
 inorganic to the organic philosophy. 
 
 The subordination of biology to Physics follows from its relation 
 to Chemistry : but theie are also direct reasons, . 
 
 relating both to doctrine and method, why it should 
 be so. 
 
 As to doctrine, — it is clear that the general laws of one or more 
 ^branches of Physics must be applied in the analysis of any physio-
 
 3i6 POSITIVE PHILOSOPHY. 
 
 logical phenomenon. This npplication is necessary in the examination 
 of the medium, in the first place ; and the analysis of the medium 
 is required to be very exact, on account of the strong effect of its 
 variations on phenomena so easily modified as those of the organism. 
 And next, the organism itself is no less dependent on those laws, 
 relating as they do to weight, heat, electricity, etc. It is obvious 
 that if biology is related to chemistry through the organic life, it is 
 related to physics by the animal life, — the most special and noble 
 of the sensations, those of sight and hearing, requiring for the 
 starting-point of their investigation an a[)plication of optics and 
 acoustics. The same remark holds good in regard to the theory of 
 utterance, and the study of animal heat and the electric properties 
 of the organism. It remains to be wished that the biologists 
 would study and api)ly these laws themselves, instead of committing 
 the task to physicists: but they have hitherto followed too much 
 the example of the physicists, who, as w^e have seen, have committed 
 the application of mathematical analysis in their own science to the 
 geometers; whereas, it cannot be too carefully remembered that if 
 the mora general sciences are independent of the less general, which, 
 on the other hand, must be dependent on them, the students of 
 the higher must be unfit, in virtue of that very independence, to 
 apply them to a more complex science, whose conditions they 
 cannot sufficiently understand. If the case was clear in regard to 
 the intrusion of the geometers into physics, it is yet more so with 
 regard to the intrusion of the physicists into biology ; on account of 
 the more essential difference in the nature of the two sciences. The 
 biologists should qualify tliemselves for the application of the 
 preceding sciences to their own, instead of looking to the physicists 
 for guidance which can only lead them astray. 
 
 In regard to Method, biology is indebted to physics for the most 
 ])erfect models of observation and experimentation. Observations 
 in physics are of a sufficient complexity to serve as a type for the 
 same method in biolog}^, if divested of their numerical considerations, 
 which is easily done. Chemistry however can furnish an almost 
 equally good model in simple observation. It is in experimentation 
 that biologists may find in physics a special ti'aining for their work. 
 As the most perfect models are found in the study of physics, and 
 the method is singularly difficult in physiolog}^ we see how impor- 
 tant the contemplation of the best type nmst be to biologists. 
 Such is the nature of the dependence, as to doctrine and method, 
 of biology on ])hysics. We turn next to its relations with Astro- 
 nomy ; and first, with regard to doctrine. 
 
 The relation of physiology to astronomy is more important than 
 is usually sujiposed. I mean something more than 
 s ronomy. ^|^^ impossibility of understanding the theory of 
 weight, and its effects upon the organism, apart from the considera- 
 tion of general gravitation. I mean, besides, and more specially,
 
 TO ASTRONOMY. 317 
 
 tliat it is impossible to form a scientific conception of the condiliuns 
 of vital existence witliont taking into the acconnt the ai^gregate 
 astronomical elements that characterize tlie planet which is the home 
 of that vital existence. We shall see more fully, in the next volume, 
 how humanity is allected by these astronomical conditions ; but we 
 must cursorily review these relations in the present connection. 
 
 The astronomical data })roper to our planet are, of course, stati- 
 cal and dynamical. The biological importance of the statical con- 
 ditions is immediately obvious. No one questions the importance 
 to vital existence of the mass of our planet in comparison with that 
 of the sun, which determines the intensity of gravity ; or of its 
 form, which regulates the direction of the force ; or of the funda- 
 mental equilibrium and the reguhir oscillations of the fluids which 
 cover the greater part of its surtace, and with which the existence 
 of living beings is closely implicated; or of its dimensions, which 
 limit the indefinite multiplication of races, and especially the human ; 
 or of its distance from the centre of our system, which chiefly detei- 
 mines its temperature. Any sudden change in one or more of these 
 conditions would largely modify the phenomena of life. But the 
 influence of the dynamical conditions of astronomy on biological 
 study is yet more important. Without the two conditions of the 
 fixity of the poles as a centre of rotation, and the uniformity of the 
 angular velocity of the earth, there would be a continual perturba- 
 tion of the oi-ganic media which would be incompatible with life. 
 Bichat pointed out that the intermittence of the proper animal life 
 is subordinate in its periods to the diurnal rotation of our planet ; 
 and we may extend the observation to all the periodical phenomena 
 of any organism, in both the normal and pathological states, allow- 
 ance being made for secondary and transient influences. Moreover, 
 there is every reason to believe that, in every organism, the total 
 duration of life and of its chief natural phases depends on the angu- 
 lar velocity proper to our planet ; for we are authorized to admit 
 that, other things being equal, the duration of life must be shorter, 
 especially in the animal organism, in proportion as the vital pheno- 
 mena succeed each other more rapidly. If the earth were to rotate 
 luuch faster, the course of physiological phenomena would be 
 accelerated in proportion ; and thence life would be shorter ; so that 
 the duration of life may be regarded as dependent on the duration 
 of the day. If the duration of the year were changed, the life of 
 the organism would again be affected : but a yet more striking 
 consideration is that vital existence is absolutely implicated with 
 the form of the earth's orbit, as has been observed before. If that 
 ellipse were to become, instead of nearly circular, as eccentric as 
 the orbit of a comet, both the medium and the organism would 
 undergo a change fatal to vital existence. Thus the small eccen- 
 tricity of the earth's orbit is one of the main conditions of biological 
 phenomena, almost as necessary as the stability of the earth's rota-
 
 3i8 POSITIVE PHILOSOPHY. 
 
 tion ; and every other element of the annual motion exercises an 
 influence, more or less marked, on biolo^s^ical conditions, though not 
 so great as the one we have adduced. The inclination of the plane 
 of the orbit, for instance, determines the div^ision of the earth into 
 climates, and, consequently, the geographical distribution of living 
 species, animal and vegetable. And again, through the alternation 
 of seasons, it influences the phases of individual existence in all 
 organisms ; and there is no doubt that life would be affected if the 
 revolution of the line of the nodes were accelerated; so that its 
 being nearly immovable has some biological value. These con- 
 siderations indicate how necessary it is for biologists to inform 
 themselves accurately, and without any intervention, of the real 
 elements proper to the astronomical constitution of our planet. An 
 inexact knowledge will not suffice. The laws of the limits of vari- 
 ation of the different elements, or, at least, a scientific analysis of 
 the chief grounds of their permanence, are essential to biological 
 investigation ; and these can be obtained only through an acquaint- 
 ance with astronomical conceptions, both geometrical and me- 
 chanical. 
 
 It may at first appear anomalous, and a breach of the ency- 
 clopedical arrangement of the sciences, that astronomy and biology 
 should be thus immediately and eminently connected, while two other 
 sciences lie between. But, indispensable as are physics and chemi- 
 stry, astronomy and biology are, by their nature, the two principal 
 branches of natural philosophy. They, the complements of each 
 other, include in their rational harmony the general system of our 
 fundamental conceptions. The solar system and Man are the 
 extreme terms within which our ideas will for ever be included. 
 The system first, and then Man, according to the positive course of 
 our speculative reason : and the reverse in the active process : the 
 laws of the system determining those of Man, and remaining un- 
 affected by them. Between these two poles of natural philosophy the 
 laws of physics interpose, as a kind of complement of the astronom- 
 ical laws ; and again, those of chemistry, as an immediate preliminary 
 of the biological. Such being the rational and indissoluble constitu- 
 tion of these sciences, it becomes apparent why I insisted on the 
 subordination of the study of Man to that of the system, as the 
 primary ])hilosophical characteristic of positive biology. 
 
 Though in the infancy of the human mind, when it was in its 
 theological state, and in its youthful metaphysical stage, the order 
 of these sciences was reversed, there was a preparation for the true 
 view. Through all the fanciful notions of the ancient philosophy 
 about the physiological influence of the stars, we discern a strong 
 though vague perception of some connection between vital and 
 celestial phenomena. Like all primitive intuitions of the under- 
 standing, this one needed only rectification by the positive philos- 
 ophy ; under the usual condition, however, of being partially
 
 BIOL OGY AND ASTR ONOM V. 319 
 
 overthrown in order to be reorganized. But niodorn students, 
 finding no astronomical conditions in tlie course of their anatomical 
 and physiological observations, have discarded the idea of them 
 altogether, — as if it were ever possible for facts to bear immediate 
 testimony to the conditions without which they could not exist, and 
 ■which do not admit of a moment's suspension ! Such an order 
 of primitive conditions is however now established beyond dispute. 
 In order to prevent any return to vicious or exaggerated notions 
 about the physiological influence of the stars, it is enough to bear 
 in mind two considerations : first, that the astronomical conditions 
 of vital existence are comprised within our own planetary system ; and 
 secondly, that they relate, not directly to the organism, but to its 
 environment, affecting as they do the constitution of our globe. 
 
 In regard to method, — the importance of astronomical study 
 to biologists consists, as in other cases, in its offering the most 
 perfect model of philosophizing on any phenomena whatever ; the 
 importance of this example becoming greater in proportion to the 
 complexity of the subordinate science, on account of the stronger 
 temptation to discursive and idle inquiries offered by the latter. 
 The more difficult their researches become, the more sedulous 
 should physiologists be to refresh their positive forces at the source of 
 positive knowledge ; and, in the contemi)lation of the few general 
 and indisputable conceptions wdiich constitute this lofty science, to 
 be on their guard against the baseless notions of a vital principle, 
 vital forces, and entities of that character. Hitherto, all advance 
 in positivity in biology has been obtained at the expense of its dignity, 
 which has always been implicated with an imaginary origin of life, 
 of sensibility, etc.: but when physiologists have learned from their 
 study of gravitation and other primary laws how to confine them- 
 selves to true science, their subject will rise to the highest elevation 
 that positivity admits of, — that rational prevision of events which 
 is, as I have so often said, the end of true science: — an end to bo 
 aimed at in biology, as it is j)erfectly fulfilled in astronomy. 
 
 Here, too, must biologists learn the character of sound scientific 
 hypothesis. This method is eminently wanted in so complex a 
 study as physiology ; but it has been as yet used with very little 
 effect. The way is, undoubtedly, to determine the organ from the 
 function or the function from the organ. It is peimissible to 
 form tlie most plausible hypothesis as to the unknown function of 
 a given organ, or the concealed organ of a manifest function. If 
 the supposition be in harmony with existing knowledge, if it be 
 held provisionally, and if it be capable of a positive verification, it 
 may contribute to the progress of discovery, and is simply a use of 
 a right of the human mind, exercised as in astronomy. The only 
 eminent example known to me of sound hypothesis in biology is 
 that of M. Broussais, in proposing the mucous membrane of the 
 alimentary canal as the seat of so called essential fevers. Whether
 
 320 POSITIVE PHILOSOPHY. 
 
 lie was mistaken or not, is not the question. His hypothesis beln^ 
 open to unquestionable confiiniation or subversion, it gave a great 
 impulse to the study of pathology in a positive manner: and it 
 Avill stand in the history of the human mind, as the first example 
 of the spontaneous introduction of a sound hypothetical method 
 into the positive study of living beings: a method derived from the 
 region of" astronomy. 
 
 It remains to consider the relation of biology to mathematics. 
 The encroachments of the pure geometers upon the domain of 
 To Maihe- blology have been attended with the same mischief, 
 matics. |)iit in {^n aggravated form, that we have witnessed 
 
 in the case of other sciences. This mischief has led physiologists 
 to repudiate mathematics altogether, and open an impassable gulf 
 between themselves and the geometers. This is a mistake ; inas- 
 much as their science cannot be severed from that which is the 
 basis of the whole of natural philosophy ; and it is only through 
 the admission of this that they can maintain the originality and 
 independence of their scientific labours. The rational study of 
 nature proceeds on the ground that all phenomena are subject to 
 invariable laws, which it is the business of philosophical specula- 
 tion to discover. It Is needless to prov(i that on any other supposi- 
 tion, science could not exist, and our collections of facts could yield 
 no result. In the phenomena of living bodies, as in all othei-s, 
 every action proceeds according to precise, that is, mathematical 
 laws, which we should ascertain if we could study each phenomenon 
 by itself. The phenomena of the inorganic world are, for the most 
 })art, simple enough to be calculable : those of the organic world are 
 too complex for our management : but this has nothing to do witli 
 any diil'erence in their nature. And this is the view which both 
 geometers and biologists should bear in mind. 
 
 If in astronomy our calculations are baffled when we pass beyond 
 two or three essential conditions, it is evident how impracticable 
 they must be amidst the inextricable complications of physiology. 
 And again, this complexity prevents our ever effecting a mathema- 
 tical disclosure of the elementary laws of the science. Tins 
 excludes all idea of this method of philosophizing in biology ; for 
 these laws are no otherwise accessible than by the immediate 
 analysis of their numerical effects. Now, whichever way vital pheno- 
 mena are looked at, they present such endless and incessant variations 
 in their numbers, that geometers are baffled as completely as if 
 those degrees were entirely arbitrary. Even numerical chemistry is 
 inapplicable to bodies whose molecular composition varies Incessantly ; 
 and this is })recisely the distinguishing character of living organisms. 
 However hurtful may have been the incursions of the geometers, 
 direct and indirect, into a domain which it is not for them to 
 cultivate, the physiologists are not the less wrong in turning 
 away from matiiematlcs altogether. It is not only that without
 
 BIOL OGY AND MA THEM A TICS. 3 2 1 
 
 mathematics they could not receive their duo preliiniiiary training 
 in the interveniii*]^ sciences : it is further necessary lor them to have 
 geometrical and mechanical knowledge, to understand the strucliu-e 
 and the play of the complex apparatus of the living, and especially 
 the animal organism. Animal mechanics, statical and dynamical, 
 must be unintelligible to those who are ignorant of the general laws 
 of rational mechanics. The laws of equilibrium and motion are, as 
 we saw when treating of them, absolutely universal in their action, 
 de[)ending wholly on the energy, and not at all on the nature of the 
 forces considered : and the oidy difficulty is in their numerical appli- 
 cation in cases of complexity. Thus, discarding all idea of a numer- 
 ical a})plication in biology, we perceive that the general theorems of 
 statics and dynamics must be steadily verified in the mechanism 
 of living bodies, on the rational study of which they cast an indis- 
 ])ensable light. The highest orders of animals act, in re})ose and 
 motion, like any other mechanical apparatus of a simiUir complexity, 
 with the one difference of the mover, wdiicli has no power to alter 
 the laws of motion and equilibrium. The participation of rational 
 mechanics in positive biology is thus evident. Mechanics cannot 
 dispense with geometry ; and besides, we see how anatomical and 
 physiological speculations involve considerations of form and posi- 
 tion, and require a familiar knowledge of the principal geometrical 
 laws which may cast light U{)on those complex relations. 
 
 In regard to Method, the necessity of reciuring to a perfect model 
 of reasoning, the moi-e earnestly in proportion to the complexity of 
 the science concerned, is applicable in regard to Mathematics, as to 
 Astronomy ; only with still greater urgency. In mathematics we 
 iind the primitive source of rationality ; and to mathematics must 
 the biologists resort for means to carry on their researches. If 
 biologists have hitherto not done this, but contented themselves 
 with what is called logic, apart from all determinate reasoning, 
 much of the fault is chargeable upon the indifference of geometers 
 about duly organizing the whole of mathematical knowledge. The 
 imperfect and inadequate character of the elementary treatises on 
 mathematics that have hitherto been given to the world quite 
 accounts for the neglect of the fundamental logical properties of 
 mathematical science by even intelligent minds. It accounts also 
 for the exaggerations of some philosophers, who maintain that, far 
 from preparing the intellectual organ for the rational interpretation 
 of nature, a mathematical education rather tends to develop a spirit 
 of sophistical argumentation and illusory speculation. Such au 
 abuse, however, cannot affect the real value of mathematics as a 
 means of positive education ; but rather exhibits the necessity of 
 a philosophical renovation of the whole system of mathematical 
 instruction. Whatever advantage can bo attributed to logic in 
 directing and strengthening the action of the understanding is 
 found in a higher degree in mathematical study, with the immense 
 
 VOL. I. X
 
 32 2 POSITIVE PHILOSOPHY. 
 
 added advantage of a determinate subject, distinctly clrcnmsciibed, 
 admitting of the utmost precision, and free fi'om the danger which 
 is inherent in all abstract logic, — of leading to useless and puerile 
 rules, or to vain ontological speculations. The positive method, 
 being everywhere identical, is as much at home in the art of 
 reasoning as anywhere else : and this is why no science, whether 
 biology or any other, can offer any hind of reasoning, of which 
 mathematics does not supply a simpler and purer counterpart. 
 Thus, we are enabled to eliminate the only remaining portion of 
 the old i^hilosophy which could even appear to offer any real utility; 
 the logical part, the value of which is irrevocably absorbed by 
 mathematical science. Hither, then, must biologists come, to study 
 the logical art so as to apply it to the advancement of their difficult 
 researches. In this school must they learn familiarly the real 
 characters and conditions of scientific evidence, in order to transfer 
 it afterwards to the province of their own theories. The study of it 
 here, in the most simple and ])erfect cases, is the only sound pre- 
 paration for its recognition in the most complex. 
 
 Tlie study is equally necessary for the formation of intellectual 
 habits; for obtaining an aptitude in forming and sustaining positive 
 abstractions, without which the comparative method cannot be used 
 in either anatomy or physiology. The abstraction which is to be 
 the standard of comparison must be first clearly formed, and then 
 steadily maintained in its integrity, or the analysis becomes abortive: 
 and this is so completely in the spirit of mathematical combinations, 
 that practice in them is the best preparation for it. A student who 
 cannot accomplish the process in the more simple case may be assured 
 that he is not qualified for the higher order of biological researches, 
 and must be satisfied with the humbler office of collecting materials 
 for the use of minds of another order. Hence arises another use of 
 mathematical training; — that of testing and classifying minds, as 
 well as preparing and guiding them. Probably as much good would 
 be done by excluding the students who only encumber the science 
 by aimless and desultory inquiries, as by fitly instituting those who 
 can better fulfil its conditions. 
 
 There seems no sufficient reason why the use of scientific fictions, 
 UseofscAen- SO common in the hands of geometers, should not be 
 tijic fictions, introduced into biology, if systematically employed, 
 and adopted with sufficient sobriety. In mathematical studies, 
 great advantages have arisen from imagining a series of hypothetical 
 cases, the consideration of which, though artificial, may aid the 
 clearing up of the real subject, or its fundamental elaboration. This 
 art is usually confounded with that of hypotheses ; but it is entirely 
 different ; inasmuch as in the latter case the solution alone is 
 imnginaiy ; whereas in the former, the problem itself is radically 
 ideal. Its use can never be in biology comparable to what it is in 
 mathematics : but it seems to rae that the abstract character of the
 
 CONDITION AND PROSPECTS OF BIOIOGY. 323 
 
 lili;lier conceptions of coraparcative biology renders tlieni siisccj^tible 
 of sucli treatment. Tlie process would be to intei'calute, amon<r 
 iliflerent known organisms, certani purely fictitious orgainsms, so 
 imagined as to facilitate their comparison, by rendering the biolo- 
 gical series more homogeneous and continuous : and it miglit be 
 that several miglit bereafter meet with more or less of a realization 
 among organisms bitherto unexplored. It may be possible, in the 
 present state of our knowledge of living bodies, to conceive of a new 
 organism capable of fulfilling certain given conditions of existence. 
 However that may be, the collocation of real cases with well- 
 imagined ones, after the manner of geometers, will doubtless be 
 practised hereafter, to complete the general laws of comparative 
 anatomy and physiology, and possibly to anticipate occasionally the 
 dii'ect exploration. Even now, the rational use of such an artifice 
 might greatly simplify and clear up the ordinary S3'stem of pure 
 biological instruction. But it is only the highest order of investigators 
 who can be intrusted willi it,. Wlienever it is adopted, it will con- 
 stitute another ground of relation between biology and mathematics. 
 We have now gone over all those grounds, — both of doctrine and 
 of method. Of the three parts of niatliemalics. Mechanics is con- 
 nected with biology in the scientific point of view ; and geometry 
 in the logical : while both rest upon the analytical theories which 
 are indispensable to their systematic development. 
 
 This specification of the relations of biology determines its rank 
 in the hierarchy of sciences. From this again we learn the kind 
 and degree of perfection of which biology is susceptible ; and, more 
 directly, the rational plan of preliminary education which it 
 indicates. 
 
 If the perfection of a science were to be estimated by the means 
 of its pursuit, biology would evidently excel all Condition and 
 others; for we can concentrate upon it the wliole of prospects of 
 the resources of observation and of reasoning offered the science. 
 by all the others, together witb some of high importance appropriate 
 to itself. Yet, all this wealth of resources is an insufficient com- 
 ])ensation for the accumulated obstacles whicb beset the science. The 
 difficulty is not so much in its recent passage into the positive state 
 as in the high complexity of its phenomena. After the wisest use 
 of all our resources, this study must ever remain inferior to all the 
 departments of inorganic philosophy, not excepting chemistiy itself. 
 Still, its speculative improvement will be greater than might be 
 supposed by those who are unawai'e bow incomjdete and barien is 
 the accumulation of observations and heterogeneous conce{)tions 
 %vhich now goes by the name of the science. All that has yet 
 been done should be regarded as a preliminary operation, — an 
 ascertainment and trial of means, hitherto provisional, but hence- 
 forth to be organized. Such an organization having reallv taken
 
 •324 POSITIVE PHILOSOPHY, 
 
 place among a few qualified investigators, tlie state of the science 
 may be regarded as very satisfactory. As for the direct establisli- 
 ment of biological laws, the few positive ideas that we have ob- 
 tained justify the expectation that the science of living bodies may 
 attain to a real co-ordination of phenomena, and therefore to their 
 prevision, to a greater or smaller extent. 
 
 As for the requisite education, — as it comprehends the study of 
 Requisite the preceding sciences, from mathematics downwards, 
 Education, jj; jg clearly of a more extensive and difficult order 
 than any hitherto prescribed. But the time saved from the useless 
 study of words, and from futile metaphysical speculations, would 
 suffice for all the purposes of the regenerated science, which dis- 
 cards these encumbrances. 
 
 If, next, we look at the reaction of the science on the education 
 of the general mind, the first thing that strikes us is that the positive 
 study of ]\Ian affords to observers the best test and measure of the 
 mental power of those who pursue the study. In other sciences, 
 the real power of the inquirer and the value of his acquisitions are 
 concealed from popular estimate by the scientific artifices which 
 are requisite fur the pursuit ; as in the case of mathematics, whose 
 hieroglyphic language is very imposing to the uninitiated : so that 
 men of extremely small ability, rendering very doubtful services, 
 have obtained a high reputation for themselves and their achieve- 
 ments. But tliis can hardly take place in biology ; and the prefer- 
 ence which popular good sense has accorded to the study of Man 
 as a test of scientific intelligence is therefore well-grounded. Here 
 the most important phenomena are common to all ; and the race 
 may be said to concur in the study of Man : and, the more difficult 
 and doubtful the ascertainment of general laws in so complex a 
 science, the higher is the value of individual and original meditation. 
 When these laws become better known, this originality will yield 
 some of its value to the ability which will then be requisite for 
 their application. The moral world will, under all future, as under 
 all past circumstances, regard the knowledge of human nature as 
 the most indubitable sign, and the commonest measure, of true 
 intellectual superiority. 
 
 The first intellectual influence of the science is in perfecting, or 
 rather developing, two of the most important of our elementary 
 powers, which are little required by the ])receding sciences ; — the 
 arts of comparison and of classification, which, however necessary 
 to each other, are perfectly distinct. Of the first, I have said enough ; 
 and of the second I shall speok hereafter ; so that I have now only 
 to indicate its function in biology. 
 
 The universal theory of philosophical classifications, necessary not 
 
 Art of ciassi- Only to aid the memory but to perfect scientific com- 
 
 fication. binatious, cannot be absent from any branch of 
 
 •natural philosophy : but it is incontestable that the full develop-
 
 I 
 
 BIOLOGY AND POSITIVISM. 335 
 
 ment of the art of classification was reserved for biological science. 
 As we have seen before, each of our elenientaiy powers must be 
 s[)ecially developed by that one of our positive studies which re- 
 quires its most urgent application, and which, at the same time, 
 olFers it the most extended iield. Under both aspects, biology tends, 
 more than any other science, to favour the spontaneous rise of the 
 general theory of classifications. First, no otlier so urgently claims 
 a series of rational classifications, on account both of the multi- 
 plicity of distinct but analogous beings, and of the necessity of 
 organizing a systematic comparison of them in the form of a bio- 
 logical hierarchy ;• and next, the same characteristics which demand 
 these classifications facilitate their spontaneous establishment. The 
 multiplicity and complexity are not, as might at first appear, 
 obstacles to the systematic arrangement of subjects : on the con- 
 trary, they are aids, as the diversity of their relations offers a greater 
 ntunber of analogies, more extensive and easy to lay hold of. This 
 is the reason why the classification of animals is superior to that of 
 vegetables ; the greater variety and complexity of animal organisms 
 affording a better hold for the art of classifying. And thus we see 
 that the very difficulties of the science are of a nature at once to 
 require and permit the most marked and spontaneous development 
 of the general art of classification ; and hither must the student in 
 every other department of science resort, to form his conceptions of 
 this all-important method. Here alone can geometers, astronomers, 
 ])hysicists, and even chemists learn the formation of natural groups, 
 and their rational co-ordination ; and, yet more, the general prin- 
 ci[)le of the subordination of characteristics, which constitutes the 
 chief artifice of the method. The biologists alone, at this day, can 
 be in habitual possession of clear and positive ideas in these three 
 relations. — Each of the fundamental sciences has, as we have already 
 so often seen, the exclusive property of specially developing some 
 one of the great logical procedures of which the whole positive 
 method is composed ; and it is thus that the more complex, while 
 dependent on the simpler, react on their superiors by affording 
 them new rational powers and instruments. In this view of the 
 hierarchical character and unity of the system of human knowledge, 
 it becomes clear that the isolation still practised in the organization 
 of our positive studies is as hurtful to their special progress as to 
 Iheir collective action upon the intellectual government of the 
 human race. 
 
 Looking now to the higher function of this science, — its influence 
 upon the positive spirit, as well as method, — we have 
 only to try it by the test proposed before; — its power Bioioyi] upon 
 of destroying theological conceptions in two ways: — the Positive 
 by the rational prevision of phenomena, and by the *''"*• 
 voluntary modification of them which it enables Man to exercise. 
 As the phenomena of any science become more complex, the first
 
 326 FOSITIVE PHILOSOPHY. 
 
 power decreases, and tlie other increases, so that the one or the 
 other is always present to show, unquestionably, that the events of 
 the worhl are not ruled by supernatural will, but by natural laws. 
 Biological science eminently answers to this test. While its com- 
 plexity allows little prevision, at present, in regard to its phenomena, 
 it supplies us with a full equivalent, in regard to theological 
 conceptions, in the testimony afforded by the analysis of the con- 
 ditions of action of living bodies. The natural opposition of this 
 species of investigation to every kind of theological and metaphysical 
 conception is particularly remarkable in the case of intellectual and 
 affective phenomena, — the positivism of which is very recent, and 
 which, with the social phenomena that are derived from them, are 
 the last battle-ground, in the popular view, between the positive 
 philosophy and the ancient. In virtue of their complexity, these 
 phenomena are precisely those which I'equire the most determinate 
 and extensive concurrence of various conditions, exterior and 
 interior; so that the positive study of them is eminently fitted to 
 expose the futility of the abstract explanations derived from the 
 theological or metaphysical philosophy. Hence, we easily under- 
 stand the marked aversion which this study is privileged to arouse 
 among different sects of theologians and metaphysicians. As the 
 labours of anatomists and physiologists disclose the intimate depen- 
 dence of moral phenomena on the oi"ganism and its environment, 
 there is something very striking in the vain efforts of followers of 
 the old philosopliies to harmonize with these facts the illusory play 
 of supernatui'al influences or psychological entities. Thus has the 
 development of biological science put the positive philosophy in 
 possession of the very stronghold of the ancient philosophy. The 
 same effect becomes even more striking in the other direction, from 
 biological phenomena being, beyond all others, susceptible of 
 modification from human intervention. We have a lai'ge power of 
 affecting both the organism and its environment, from the very 
 considerable number of the conditions which concur in their exist- 
 ence: and our voluntary power of disturbing phenomena, of 
 suspending, and even destroying them, is so striking as to com])el 
 us to reject all idea of a theological or metaphysical direction. As 
 in the other case, of which indeed it is a mere extension, this effect 
 is most particularly marked in regard to moral phenomena, properly 
 so called, Avhich are more susceptible of modification than any 
 others. The most obstinate ])sych(>logist could not well persist in 
 maintaining the sovereign independence of his intellectual entities, 
 if he would consider that the mere standing on his head for a 
 moment would [)ut a complete stop to the course of his own specu- 
 lations. Much as we may wish that, in addition to these evidences, 
 we had that of an extensive power of scientific })revision in biology, 
 such a power is not needed for the conclusions of popular good 
 sense. This prevision is not always baffled : and its success in a
 
 POSITIVE BIOLOGY. 327 
 
 few marked capcs is sufficient to satisfy tlie general mind tlmt tlie 
 phenomena of living bodies are subject, like all others, to invariable 
 natural laws, which we are prevented from interpreting in all cases 
 only by their extreme complexity. 
 
 But, moreover, positive biology has a special conquest of its own 
 ever the theological and metaphysical systems, by which it has 
 converted an ancient dogma into a new ])rinciple. In chemistry the 
 same thing occurred when the primitive notion of absolute creation 
 and destruction was converted into the precise conception of 
 perpetual decomposition and recomposition. In Astronomy, the 
 same thing occurred when the liypothesis of final causes and provi- 
 dential rule gave place to the view of the solar system as the 
 necessary and spontaneous result of the mutual action of the 
 principal masses which compose it. Biology, in its close connection 
 with astronomy, has completed this demonstration. Attacking, in 
 its own way, the elementary dogma of final causes, it has gradually 
 transformed it into the fundamental princijjle of the conditions of 
 existence, which it is the particular aptitude of biology to develop 
 and systematize. It is a great error in anatomists and physiologists, 
 — an error fatal both to science and theology, — to endeavour to 
 imite the two views. Science compels us to conclude that there is 
 no organ without a function, and no function without an organ. 
 Under the old theological influences, students are apt to fall into a 
 state of anti-scientific admiration when they find the conditions and 
 the fulfilment coincide, — when, having observed a function, ana- 
 tomical analysis discloses a statical position in the organism which 
 allows the fulfilment of the function. This irrational and barren 
 admiration is hurtful to science, by habituating us to su})pose that 
 all organic acts are effected as perfectly as we can imagine, thus 
 repressing the expansion of our biological speculations, and inducing 
 ns to admire complexities which are evidently injurious : and it is 
 in direct opposition to religious aims, as it assigns human wisdom 
 as the rule and even the limit of the divine, which, if such a i)arallel 
 is to be established, must often appear to be the inferior of the two. 
 Though we cannot imagine radically new organisms, we can, as I 
 showed in my suggestion about the use of scientific fictions, conceive 
 of oi-ganizations which should differ distinctly from any that are known 
 to us, and which should be incontestably superior to them in certain 
 determinate respects. The philosophical principle of the conditions 
 of existence is in fact simply the direct conception of the necessaiy 
 harmony of the statical and the dynamical analyses of the subject 
 ])roposed. This principle is eminently adapted to the science of 
 biology, which is continually engaged in establishing a harmony 
 between the means and the end ; and nowhere else, therefore, is 
 seen in such perfection, that double analysis, statical and dynamical, 
 which is found everywhere. 
 
 These, then, are the philosophical properties of positive biology.
 
 328 POSITIVE rillLOSOPHY. 
 
 To complete our review of tlie science as a whole, we have only to 
 
 note briefly the division and rational co-ordination of its parts. 
 
 It does not fall within the scope of this work to notice several 
 
 JDistribution branches of positive biological knowledge, which are 
 
 of the science, of extreme importance in their own place, but 
 
 secondary in regard to the principles of positive philosophy. We 
 
 have no concern here with pathology, and the corresponding medical 
 
 art ; nor with natural history, and the corresponding art of the 
 
 education of organisms. These are naturally, and not untruly, 
 
 called biological studies : but we must here coqtine the term 
 
 strictly to the speculative and abstract researches which are the 
 
 foundation of the science. Tlie interior distribution of the science, 
 
 thus regarded, is this. 
 
 The speculative and abstract study of the organism must be 
 divided, first, into statics and dynamics ; according as we are seek- 
 ing the laws of organization or those of life: and again, statical 
 biology must be divided into two parts, to which M. de Blainville 
 has given the name, in regard to animals, of zootomy and zoofax//, 
 according as we study the structure and composition of individual 
 organisms, or construct the great biological hierarchy which results 
 from the comparison of all known organisms. It would be easy to 
 modify M. de Blainville's terms so as to make them common to 
 animals and vegetables. Dynamical biology, to which we may give 
 the name hionomy, as the end and aim of the whole set of studies, 
 evidently admits of no analogous subdivision. The general name 
 of Biology thus includes the three divisions, biotomy, biotaxy, and 
 pure bionomy, or physiology properly so called. 
 
 Their definition exhibits their necessary dependence ; and there- 
 by determines also their philosophical co-ordination. While it is 
 universally allowed that anatomical ideas are indispensable to 
 ])hysiological studies, because the structure must be known before 
 its action can be judged of, the subordination of bionomy to biotaxy 
 is not so well understood. Yet it is easy to see that the place of 
 any organism in the scale must be known before its aggregate 
 phenomena can be effectually studied : and again, the consideration 
 of this hierarchy is indispensable to the use of that grandest instru- 
 ment of all, — the comparative method. Thus, from every point of 
 view, the double relation of dynamical to statical biology is unques- 
 tionable. 
 
 The two divisions of statical biology are less clearly marked; and 
 it even appears as if, in regard to them, we were involved in a 
 vicious circle : for if, on the one hand, the rational classification of 
 living beings requires the antecedent knowledge of their organization, 
 it is certain, on the other hand, that anatomy itself, like physiology, 
 cannot be studied, in regard to all organisms, without an antece- 
 dent formation of the biological hierarchy. Thus we must admit 
 a consolidation of the respective advancements of biotomic and
 
 I 
 
 DJVJSIONS OF 7 BE SCIENCE. 329 
 
 biotaxic studies through tlieir intimate connection. In such a case, 
 as a separation and deterininato co-ordination are required hy our 
 luiderstandings, it appears to me that we cannot hesitate to make 
 a dogmatic arrangement, — placing tlie theory of organization before 
 that of classification, — for the last is absolutely dependent on the 
 first ; while the first could meet some wants, though in a restricted 
 way, without the second. In a word, none but known organisms 
 can be classified ; whereas they all can and must be studied, to a 
 certain extent, without being mutually compared. And again, 
 there is no reason why, in a systematic exposition of anatomical 
 j)hilosophy, we should not borrow directly from biotaxy its con- 
 struction of the organic hieraichy ; an anticipation which involves 
 much less inconvenience than severing the complete study of 
 structure. — However, it must be always borne in mind that any 
 system will have to nndergo a general revision, with a view to 
 bringing out the essential relations of its parts : the relations, not 
 only of the two sections of statical biology, but of both to the 
 dynamical. This consideration goes far to diminish the importance 
 assigned to these questions of priority : and the only reason why 
 such a revision appears more necessary in biology than in the other 
 sciences is, that there is a profounder accordance between its 
 departments than we find in theirs. 
 
 The interior distribution of these three departments is determined, 
 as usual, by the order of dependence of phenomena, on the ground 
 of their relative generality. Thus, the theory of the organic life 
 precedes that of the animal : and the theory of the highest functions 
 and oi'gans of Man terminates the biological system. 
 
 It has often been a question whether, in studying each organ 
 and function in the whole scale, it is best to begin at Whereto 
 the one end or the other; — to begin with Man or the hefjin. 
 simplest known organism. I do not consider this question so all- 
 important as it is often supposed, as all qualified inquirers admit the 
 necessity of using the two methods alternately, whichever is taken 
 first : but I think that a distinction should be made between the 
 study of the organic and that of the animal life. The functions of 
 the first being chemical, it is less necessary to begin with Man ; 
 and I think there may be a scientific advantage in studying the 
 vegetable organism first, in which that kind of functions is the 
 more pure and more marked, and therefore the more easily and 
 completely studied : but every investigation, anatomical or physio- 
 logical, relating to animal life, must be ohscure if it began elsewhere 
 than with Man, who is the only being in which such an order of phe- 
 nomena is immediately intelligible. It is evidently the obvious state 
 of Man, more and more degraded, and not the indecisive state of the 
 sponge, more and more improved, that we shoidd pursue, through 
 the animal series, when we are analysing any of the constituent 
 characters of animality. If we seem to be by tliis procedure desert-
 
 33 o POSITIVE nilLOSOPHY. 
 
 \x\^ ilie ordinary course of passing; from tlio most <]jeneral and simple 
 subject to the most particular and complex, it is only to conform 
 the better to the pliilosophical principle which prescribes that very 
 course, and which leads us from the most known subjects to the 
 least known. In all cases but this, the usual course is the fittest, 
 in biological studies. 
 
 Here we conclude our review of biological science as a whole. 
 The extent to which I have carried out the survey will allow us to 
 consider its separate portions very briefly. In doing so, I shall 
 follow the order just laid down, passing from the simj)le consider- 
 ations of pure anatomy to that positive study of the phenomena of 
 the intellect and the affections, as the highest part of human nature, 
 which will carry us over from biology to Social Physics, — the final 
 object of this work.
 
 33^ 
 
 I 
 
 CHAPTER ir. 
 
 ANATOMICAL PIIlEOSOPHY. 
 
 It was during tlie second half of tlie last century that Dauhcnfon 
 and Vicq-d'Azyr achieved the extension of the Development 
 statical study of living bodies to the whole of known of statical 
 organisms; and the lectures and writings of Cuvier ^"'%y- 
 carried on, and spread abroad, the regenerating influence of this 
 great view. But, indispensable as was this conception to the 
 development of anatomical science, it couhl not complete the 
 character of statical biology without the aid and addition of Bichat's 
 grand idea of the general decomposition of the organism into its 
 various elementary tissues ; the high pliilosophical importance of 
 which appears to me not yet to be worthily appreciated. 
 
 The natural development of comparative anatomy would, no doubt, 
 liave disclosed this analysis to us sooner or later : but how slow the 
 process would have been we may judge by what we see of the 
 reluctance of comparative anatomists to abandon the exclusive study 
 of systems of oigans while unable to deny the preponderant impor- 
 tance of the study of the tissues. Of all changes, those which relate 
 to method are the most difficult of accomplishment; and perhaps 
 there is no example of their resulting spontaneously from a regular 
 advance under the old methods, without a direct impulsion from a 
 new original conception, energetic enough to work a revolution in 
 the system of study. Biology must, from its great complexity, be 
 more dependent on such a necessity than any other science. 
 
 Though zoological analysis furnishes the best means of separating 
 the various oi'ganic tissues, and especially of giving precision to the 
 true philosophical sense of this great notion, patho- Process of 
 logical aTialysis offered a more direct and rapid way discover)/ of 
 to suggest the first idea of such a decomposition, the tissues. 
 even regarding the human orgam'sm alone. When pathological 
 anatomy had been once founded by Morgagni, it was evident that 
 in the best-marked maladies no organ is ever entirely diseased, and 
 that the alterations are usually contnied to some of its constituent 
 parts, while the others preserve their normal condition. In no other 
 way could tlie distinction of the elementary tissues have been so 
 clearly established. By the coexistence in one oigan of sound and
 
 332 2'OSITIVE PHILOSOFIIY. 
 
 impaired tissues, and, ngain, by different organs being affected by 
 similar maladies, in virtue of the disease of a common tissue, tlie 
 analysis of the chief anatomical elements was spontaneously in- 
 dicated, at the same time that the study of" the tissues was shown to 
 be more important than that of the organs. It is not consistent 
 with ray objects to go further into this : but it was necessary to 
 show that we owe to pathological analysis the perception of this 
 essential truth. It was Pinel who suggested it to Bichat, by his 
 happy innovation of studying at once all the diseases proper to the 
 different mucous membranes. Bichat then, while knowing nothing 
 of the study of the organic hierarchy, cariied off from the students 
 of comparative anatomy the honour of discovering the primitive idea 
 which is most indispensable of all to the general advancement of 
 anatomical philosophy. His achievement consisted in rationally 
 connecting with the normal condition a notion derived from the 
 ])athological condition, in virtue probably of the natural reflection 
 that if the different tissues of the same organ could each be separ- 
 ately diseased in its own way, they must have, in their healthy con- 
 <lition, distinct modes of existence, of which the life of the organ is 
 really composed. This principle was entirely overlooked before 
 Bichat published the treatise in which he established the most sat- 
 isfactor}^ a posteriori development of it : and it is now placed beyond 
 all question. The only matter of regret is that, in creating a 
 wholly new aspect of anatomical science, Bichat did not better mark 
 its spirit by the title he gave it. If he had called it abstract or 
 elementary instead of general anatomy, he would have indicated its 
 philosophical function, and its relation to other anatomical points of 
 view. 
 
 The anatomical philosophy began to assume its definitive 
 ^ ,. ,. character from the verv recent time when the human 
 
 Combination - i i ^ , ", • ji ^ , • •^• 
 
 with Com- mmd learned to combme the two great prumtive 
 parative idcas of the Organic hierarchy, and of Bichat's dis- 
 
 covery, which applies the universal conception to the 
 statical study of living bodies. These combined ideas are necessarily 
 the subject of our present examination. Putting aside the irrational 
 distinctions, still too common among biologists, of many different 
 kinds of anatomy, we must here lecognize only one scientific 
 anatomy, chiefly characterized by the philosophical combination of 
 the comparative method with the fundamental notion of the decom- 
 position of the organs into tissues. It is apparently strange that, 
 after Bichat's discovery, comparative anatomists, with Cuvier at 
 their head, should have persisted in studying organic apparatus in 
 its complex state, instead of beginning with the investigation of the 
 tissues, pursuing the analysis of the laws of their combinations into 
 organs, and ending with the grouping of those organs into apparatus, 
 })roperly so called : but not even Cuvier's great name can now 
 prevent the application of the comparative method to the analysis
 
 ANATOMICAL FIIILOSOFIIY. 7^7,^ 
 
 of tissues, tlirougliout the whole biological hiernrchy. The worlc, 
 though at present neither energetically nor profoundly pursued, is 
 begun, and will reform the habitual direction of anatomical spec- 
 ulation. 
 
 Bichat's studies related to ]\Ian alone ; and his method of com- 
 })arison bore only upon the simplest and most restricted cases of 
 all ; the comparison of parts and that of ages. His principle must, 
 therefore, necessarily undergo some transformations, to fit it for a 
 more extensive application. The most important of these improve- 
 ments, especially in a logical view, appears to me Elements and 
 to consist in the great distinction introduced by M. de Products. 
 Blainville between the true anatomical elements and the rjimple 
 'products of the organism, which Bichat had confounded. We saw 
 l)efore the importance of this distinction in the chemical study of 
 organic substances : and we meet it again now, face to face, as an 
 anatomical conception. 
 
 We have seen that life, reduced to the simplest and most general 
 notion of it, is characterized by the double continuous motion of 
 absorption and exhalation, owing to the reciprocal action of the 
 organism and its environment, and adapted to sustain during a 
 certain time, and within certain limits of variation, the integrity of 
 the organization. It results from this that, at every instant of its 
 existence, every living body unist present, in its structure and com- 
 position, two very different orders of principles: absorbed matters 
 in a state of assimilation, and exhaled matters in a state of separa- 
 tion. This is the ground of the great anatomical distinction between 
 organic elements and organic j)roducts. The absorbed matters, 
 once completely assimilated, constitute the whole of the real materials 
 of the organism. The exhaled substances, whether solid or fluid, 
 become, from the time of their separation, foreign to the organism, 
 in which they cannot generally remain long without danger. Ee- 
 garded in a solid state, the true anatomical elements are always 
 necessarily continuous in tissue with the whole of the organism : 
 and again, the fluid elements, whether stagnant or circulating, 
 remain in the depths of the general tissue, from which they are 
 equally insej)arable : whereas the products are only deposited, for a 
 longer or shorter time, on the exterior or interior surface of tin; 
 organism. The differences are not less characteristic, in a dynamical 
 view. The true elements alone must be regarded as really living : 
 they alone participate in the double vital motion : and they alone 
 grow or decrease by absorption or exhalation. Even before they 
 are finally excreted, the products are already essentially dead sub- 
 stances, exhibiting the same conditions that they would manifest 
 anywhere else, under similar molecular influences. 
 
 The separation of the elements from the products is not always 
 easy to effect when, as frequently happens, they combine in the same 
 anatomical anan":ement to concur in the same function. All
 
 334 POSITIVE PHIIOSOPIIY. 
 
 l)iodticts are not, like sweat, urine, etc., destined to be expelled 
 witliout furtlier use in the organic economy. Several others, as 
 saliva, the gastric fluid, bile, etc., act as exterior substances, and in 
 virtue of their chemical conaposition in ]ireparing for the assimi- 
 lation of the organic materials. It is difficult to fix the precise 
 moment when these bodies cease to be products and become 
 elements ; — the moment, that is, when they pass from the inorganic 
 to the organic state, — from death to life. But these difficulties 
 arise from the imperfection of our analysis, and not from any 
 uncertainty in the principle of sepai-ation. It luay be observetl 
 however that there are circumstances in which products, and par- 
 ticularly among the solids, are closely united to true anatomical 
 elements in the structure of certain apparatus, to which they supply 
 essential means of improvement. Such are, for instance, the greater 
 number of epidermic productions, the hair, and eminently the teeth. 
 But even in this case, a sufficiently delicate dissection, and a careful 
 analysis of the whole of the function will enable us to ascertain, with 
 entire precision, how much is organic and how much inorganic in 
 the proposed structure. Such an investigation was not prepared for 
 when Bichat confounded the teeth with the bones, and concluded 
 the epidermis and the hair to be tissues, of a piece with the cutane- 
 ous tissue ; but the rectification which ensued was all-important, as 
 enabling us to define the idea of tissue or anatomical element, which 
 is the preferable terra. It was through comparative anatomy that 
 the rectification took place ; for the study of the biological series 
 showed that the inorganic parts which in Man appear inseparable 
 from the essential apparatus are in fact only simple means of 
 advancement, gradually introduced at assignable stages of the 
 ascending biological series. 
 
 If we assert that in the order of purely anatomical speculations, 
 the study of products must be secondary to that of elements, it will 
 not be supposed that we undervalue the study of products. This 
 study is of extreme importance in physiology, whose principal 
 phenomena would be radically unintelligible without it ; and without 
 it pathological knowledge must come to a stand. As results, they 
 indicate organic alterations ; and as modifiers, they exhibit the 
 origin of a great number of tho.se alterations. In fact, the know- 
 ledge of them is much promoted by their separation from the 
 anatomical elements, which withdraw the attention of biologists 
 from the leal claims of the whole class of products. 
 
 The consideration of products being once dismissed to its proper 
 place, anatomical analysis has assumed its true character of com- 
 pletene.ss and clearness. Thus we may undertake now what was 
 before impossible — an exact enumeration of anatomical elements. 
 And again, these tissues can be classified according to their true 
 general relations; and may even be reduced to a single tissue, 
 modified by detei-minate laws. These two are the otiier chief trans-
 
 VITAL FART OF FLUIDS. 335 
 
 formations undergone by the great anatomical theory of Bicliat, 
 tlirongli the np{)lica(ion of the comparative metliod: and these two 
 we now ])rocced to review. 
 
 The first is connected with the gi-eat question of the vitality of 
 the organic fluids, about which our ideas are far from vitaiUi/ of the 
 being, I think, sufficiently settled. Every living organic fluids. 
 body consists of a combination of solids and fluids, the respective 
 proportions of which vary, according to the species, within very 
 wide limits. The very definition of the vital state supposes this 
 conjunction ; for the double motion of composition and decompo- 
 sition which characterizes life could not take place among solids 
 alone ; and, on the other hand, a liquid or gaseous mass not only 
 requires a solid envelope, but could admit of no real organization. 
 If the two great primitive ideas of Life and Organization were not 
 inseparable, we might imagine the first to belong to fluids, because 
 they are so readily modified ; and the other to solids, as alone 
 capable of structural formation : and here, under another view, we 
 should find the necessary harmony of the two elements. Thecomjiarison 
 of types in the biological series confirms, in fact, the general rule that 
 the vital activity increases with the preponderance of fluid elements 
 in the organism ; while a greater persistence of the vital state 
 attends the preponderance of solids. This has long been regarded 
 as a settled law by philosophical biologists, in studying the series 
 of ages alone. These considerations seem to show that the contro- 
 versy about the vitality of fluids rests, like many other famous 
 controversies, on a vicious proposal of the problem, since such a 
 mutual relation of the solids and the fluids excludes at once both 
 humourism and solidism. Discarding, of course, the products from 
 the question, there can be no doubt that the fluid elements of the 
 organism manifest a life as real as that of the solids. The founders 
 of modern pathology, in their reaction against the old humourism, 
 have not paid sufficient attention, in the theory of diseases, to the 
 direct and spontaneous alterations of which the organic fluids, 
 especially the blood, are remarkably susceptible, in virtue of the 
 complexity of their composition. It would ap[)ear, from a philo- 
 sophical point of view, very strange if the most active and susceptible 
 of the anatomical elements did not participate, primarily or consec- 
 utively, in the perturbations of the organism. But, on the other 
 hand, it is not less certain that the fluids, animal and vegetable, 
 cease to live as soon as they have quitted the organism ; as, for 
 instance, the blood after venesection. They then lose all organiza- 
 tion, and are in the condition of products. The vitality of the 
 fluids, considered separately, constitutes them an ill-defined, and 
 therefore interminable, question. 
 
 A truly positive inquiry however arises out of the question — the 
 inquiry as to which of the immediate principles of a fluid are vital ; 
 for it cannot be admitted that all are so indiscriminately. Thus,
 
 336 POSITIVE PHILOSOPHY. 
 
 the blood being chiefly composed of water, it would be absurd to 
 suppose such an inert vehicle to partici{)ate in the life of the fluid ; 
 but then, which of the other constituents is the seat of lifeP 
 Microscopic anatomy gives us the answer, — that it resides in the 
 globules, properly so called, which are at once organized and living. 
 However valuable such a solution would be, it can be regarded at 
 })resent only as an attempt ; for it is admitted that these globules, 
 though determinate in form, shrink moie and more as the arterial 
 blood passes through an inferior order of vessels, — that is, as it 
 approaches its incorporation with the tissues ; and that, at the 
 precise moment of assimilation, there is a complete liquefaction of 
 the globules. It would thus appear that we must cease to regard 
 the blood as living at the very moment when it accomplishes its 
 chief act of vitality. Before any decision can be made, we must 
 have the counter proof, — the acknowledgment that true globules 
 are exclusively characteristic of living fluids, in opposition to those 
 which, as simple products, are essentially inert, and hold in suspen- 
 sion various solids, which make them difflcult to be distinguished 
 from true globules, notwithstanding the determinate form of the 
 latter. Microscopical observations are too delicate, and sometimes 
 deceptive, to admit at present of the irreversible establishment of 
 this essential point of anatomical doctrine. 
 
 The statical study of living bodies would form but a very incom- 
 plete introduction to the dynamical, if the fluids were left out of 
 ihe investigation of the organic elements, however much remains to 
 be desired in our knowledge of them. The omission of them in 
 Bichat's treatise leaves a great gap. Still, as the anatomy of solids 
 must always take precedence of that of fluids, Bichat chose the 
 true point of departure, though he did not undertake the whole 
 subject. It must be added that the examination of the fluids is so 
 much the more difficult of the two as to be wellnigh impracticable. 
 In an anatomical sense, it is impracticable : and the two only 
 methods, — microscopical and chemical examination, — are impaired 
 by the ra})id disorganization which ensues when the fluids quit the 
 organism. The chemical method is in itself the more valuable of 
 tile two : but, besides that the chemists habitually confound the 
 elements and the products, they have always examined the former 
 in a more or less advanced slate of decomposition : and, being 
 unaware of this, they have off'ered only the most false and incoherent 
 notions of the molecular constitution of the organized fluids. In 
 such a state of things, it is only by a full preparation, from the 
 study of the solid elements, that anything can be done in the study 
 of the fluids. It is almost needless to say that by the same rule 
 which })rescribes this order, we should study fluids in the order of 
 their increasing liquefaction, — taking the fatly substances first, then 
 the blood and other liquids, and lastly the vaporous and gaseous 
 elements, which will always be the least understood.
 
 ORGANIC TISSUES. 337 
 
 The oi'der of inquiiy beinj:: thus settled, the next subject is the 
 rational classification of the tissncs, according to their Classification 
 anatomical filiation. It was not by such a stnd}^ as of the tissues. 
 ]>ichat's, — of Man alone, — that anything certain conld become 
 Icnown of such obscure differences as those of the fnndamental 
 tissues. In order to obtain such knowledge the study of the whole 
 biological series is indispensable. 
 
 The first piece of knowledge thus obtained is that the cellular 
 tissue is the primitive and essential web of every organic. 
 organism ; it being the only one that is present throngh Cellular tissue. 
 the whole range of the scale. The tissiies which appear in Man so 
 multiplied and distinct lose all their characteristic attributes as we 
 descend the series, and tend to merge entirely in the general cellnlar 
 tissue, which remains the sole basis of vegetable, and perhaps of the 
 lowest animal organization. This fact harmonizes well with the 
 j)hilosophical account of the basis of life, in its last degree of sim- 
 ])licity ; for the cellnlar tissne is eminently fitted, by its structure, 
 for absorption and exhalation. At the lower end of the series, the 
 living organism, placed in an unvarying medium, does nothing but 
 absorb and exhale by its two surfaces, between which are ever 
 oscillating the fluids destined for assimilation, and those which 
 result from the contrary process. For so simple a function as this 
 the cellnlar tissue snfiices. It remained to be ascertained under 
 what laws the original tissue becomes gradually modified so as to 
 engender all the others with those attributes which at first disguise 
 their common derivation : and this is what Comparative anatomy 
 has begnn to establish, with some distinctness. 
 
 The characteristic modifications of the tissue are of two prominent 
 classes : the first, more common and less profound, are limited to 
 the simple structure ; the other class, more special, and more 
 lirofound, affect the composition itself. 
 
 Of the first order the prominent case is that of the dermous 
 tissue, properly so called, which is the basis of the Dermous 
 general organic envelope, exterior and interioi'. The tissue. 
 modification here is mere condensation, differently marked, in 
 regard to animal organisms, according as the surface is, as in 
 exterior surfaces, more exhalant than absorbent, or, as in interior 
 snrfaces, more absorbent than exhalant. Even this first transfor- 
 7nation is not rigorously universal; and we must ascend the scale a 
 little way to find it clearly characterized. Not only in some of the 
 lowest of the animal organisms, are the exterior and interior essen- 
 tially alike, so that the two surfaces maybe interchanged, but if 
 we go a little lower, we find no anatomical distinction between the 
 envelope and the whole of the organism, which is uniformly cellnlar. 
 
 By an increasing condensation of the parent tissue, three distinct 
 but inseparable tissues proceed from the derma, all of which are 
 destined to an important, though passive office in the animal 
 
 VOL. I. Y
 
 S3S POSITIVE PHILOSOPHY. 
 
 economy, either as envelopes protecting the nervous organs, or as 
 
 auxiliaries of the locomotive apparatus. These are the fihrous, 
 
 cartilaginous, and bony tissues, ranged by Bichat in their rational 
 
 order, and named by M. Laurent, in their corabina- 
 
 iiCierous tissue. .. ,, 7 , • n^^ tcc l 1 c 
 
 tion, the sclerous tissue, ihe dirierent degrees or 
 consolidation here ai'ise from the deposition in the cellular network 
 of a heterogeneous substance, organic or inorganic, the extraction of 
 which leaves no doubt as to the nature of the tissue. When, on the 
 other hand, by a last direct condensation, the original tissue becomes 
 itself more compact, without being incrusted by a foreign substance, 
 we recognize a new modification, in which impermeability becomes 
 compatible with suppleness, which is the characteristic of the serous, 
 
 or (as M. Laurent calls it) the h/stous tissue, the 
 
 RystOUS tissue. nn r l'1'j.'i. 1j il 
 
 oiiice of wlucli IS to mterpose between the various 
 mobile organs, and to contain liquids, both circulating and stag- 
 nant. 
 
 The second order of transformations exhibits two secondary kinds 
 Animal ^^ tissue which distinguish the animal organism, and 
 
 Muscular and which appear at about the same degree of the scale — 
 nervous tissues, ^jj^ nuiscular and the nervous tissues. In each there 
 is an anatomical combination of the fundamental tissue with 
 a special organic element, semi-solid and eminently vital, which, 
 having long gone by the name of fihrine in the first case, has sug- 
 gested the corresponding name of 7ieurme (given us by M. do Blain- 
 ville) for the other. Here the transformation of the parent tissue 
 is so complete, that it would be difficult to establish, and yet more 
 to detect it in the higher organisms ; but the analogies of compara- 
 tive anatomy leave no doubt, and only make us wish that we could 
 understand with more precision the mode of anatomical union of 
 the muscular and nervous substances with the cellular tissue. 
 
 Passing on to the chief subdivision of each of the secondary 
 tissues, the first consideration is of the general position, which is 
 always related to a modification, greater or smaller, of the structure 
 itself. Comparative analysis shows us that in the case of both the 
 muscular and the nervous system, the organization of the tissue 
 becomes more special and elevated, exactly in proportion to its 
 deeper position between the exterior and interior surfaces of the 
 animal envelope. Thence arises the rational division of each of 
 tliese systems into superficial and profound. This distinction is 
 more especially remarkable with regard to the nervous system, 
 arranged, first, in the form of filaments, and afterwards that of 
 ganglions, with or without external apparatus. 
 
 This is the lamily of tissues, the study of which forms the basis 
 of anatomical analysis. It would be departing from my object to 
 inquire into the laws of composition under which the ascent is 
 made from this ])rimary study to that of porous substances, and 
 thence on to the theory of the organs, and then to that of systems
 
 STATE OF ANATOMICAL SCIENCE. ^y-^ 
 
 of organs, whidi \v()ul(l Iciul lis on to pliysiologlcal analysis. I 
 liave I'ulfilleil the aim ot" this seclion in exhibitini^ the nielliodical 
 connection oi the four deg-rees of aualoniical .speculation, about 
 whicli no real uncertainly exists. 
 
 Deeper than this we cannot go. The last term in our abstract, 
 intellectual decomposition of the organism is the idea Limitation of 
 of tissue. To attempt the passage from this idea to the inqidr>i.' 
 that of molecule, whicli is appropriate to inorganic philosophy, is to 
 quit the ])ositive method altogether : and those who do so, under 
 tlie fancy that they may possibly establish a notion of organic mole- 
 cules, and who give that vain search the nanae of transcendental 
 anatomy, are in fact imitating the chemists in a region into which 
 Chemistry must enter in its own shape where admissible at all, and 
 are asserting in other words that, as bodies are formed of indivisible 
 molecules, animals are formed of animalcules. This is simply an 
 attempt, in the old spirit, to peneti'ate into the nature of existences, 
 and to establish an imaginary analogy between orders of phenomena 
 which are essentially heterogeneous. It is little creditable to the 
 scientific spirit ot" our time that this aberration should call for 
 ex})osure and rebuke, and that it should need to be asserted that 
 the idea of tissue is, in organic s[)eculation, the logical equivalent 
 of the idea of molecule in inorganic speculation. 
 
 We heie find ourselves in possession of a sufficient basis of 
 anatomical science, while we need yet a more complete and pro- 
 found combination of the ideas of comparative and textural anatomy. 
 This want will be supplied when we become universally familiarized 
 Avith the four analytical degrees, complementary to each other, 
 which must henceforth be recognized and treated as the basis of 
 anatomical speculation.
 
 340 
 
 CHAPTER III. 
 
 BIOTA XIC PHILOSOPHY. 
 
 After tlie statical analv.sis of living bodies, there must be a hierar- 
 chical co-ordination of all known, or even possible organisms, in a 
 single sei-ies, which must serve as a basis for the whole of biological 
 speculations. The essential principles of this philosophical opera- 
 tion ai-e what I have now to point out. 
 
 We have already seen that it is the distinction of biological 
 science to have developed the theory of classifica- 
 anatomyor ^'^^^^^' which, existing in all sciences, attains its per- 
 vegctahies and fectiou when applied to the complex attributes of the 
 animals. animal organisms. In all ages, the vegetable organism 
 
 was the direct subject of biological classification ; but it was pursued 
 on the principles furnished by the considei'ation of animals, whence 
 the type was derived which guided philosophical speculation in the 
 case. It could not be otherwise, so marked and incontestable as 
 are the distinctions among animal organisms : and even the zoo- 
 logical classification of Aristotle, imperfect as it is, is infinitely 
 superior to anything which could then have been attempted with 
 regard to vegetables. This natural original classification has been 
 rather rectified than changed by the labours of modern times ; 
 while that of vegetables has met with an opposite fate. As a fact, 
 the first successful atten)pts in the animal region long preceded 
 the establishment of the true principles of classification ; whereas, 
 it was only by a laboi-ious systematic application of these principles 
 that it has been possible, even within a century, to effect any rational 
 co-ordination in the vegetable region, so lit,tle marked, in comparison, 
 are the distinctions in the latter case. The natural resuU was that 
 the animal realm, used as a type, became more and more attended 
 to, till the imi)rovements in zoological classification have gone so 
 far as perhaps to lead us to fear that the vegetable organism, owing 
 1o its great simplicity, can never become subject to a much better 
 classification than that in wliich it was left in the last century. The 
 labours of the reformers of lliat time are very far indeed from having 
 been useless ; only, what they undci'took for the vegetable kingdom 
 has turned rather to the profit of the animal ;— an inevitable cii-- 
 cnmstance, since the propeiiy which rendered the animal kingd(»ni 
 the natural type of the taxonomical series must adapt it to receive
 
 VEGETABLE AND ANIMAL ANATOMY. 34i 
 
 all the improvements arising; from (lie gencial priiici])les of the 
 theory. The character oCthe theorj' coiihliiothuL remain incomplete, 
 however, as lonijj as the vegetable classification continued to be re- 
 garded as the chief end of the research ; and the classification became 
 rational only wlien it was seen that the vegetable region was the 
 further end of the series, in which the most complex animal organism 
 must hold the first place ; an order of arrangement under which the 
 vegetable organism will be more effectually studied than it ever was 
 while made an t)bject of exclusive investigation. All that is needed 
 is that naturalists should extend to the whole series the anatomical 
 and physiological considerations which' have been attached too ex- 
 clusively to animal organisms ; and this will certainly be done now 
 that the human mind is fairly established at the true point of view, 
 commanding the fundamental theory of natural classification. 
 
 These prefatory remarks indicate our theme. We must have 
 the whole series in view ; but the animal region Animal 
 must be our immediate and explicit subject, — both as anatomy our 
 furnishing the rational bases of the general theory ^^^^'jcct. 
 of classification, and as exhibiting its most eminent and perfect 
 application. 
 
 The subject divides itself into two parts: the formation of 
 natural groups, and their hieiarchicul succession ; — Division of 
 a division necessary for pui-{)oses of study, though the natural 
 ihe two parts ultimately and logically coalesce. method. 
 
 In contemplating the groups, the process is to class together tho.s(^ 
 species which present, amidst a variety of differences, 
 
 1 , i- 1 1 • 1 xi Ti 1 Natural groups. 
 
 such essential analogies as malce tliem more hlveeacli 
 other than like any others, — without attending, for the present, to 
 tiie gradation of the groups, or to their interior distribution. If this 
 were all, the classification must remain either doubtful or arbitrary, 
 as the circumscription of each group could seldom be done so 
 certainly as inevitably to include or exclude nothing that might not 
 belong to another group: and great discordance was therefbi'o 
 observed in the early division into orders, families, and even genera. 
 But the difficulty disappears on the foundation of the fundamental 
 hierarchy, which rigorously assigns its place to each sjiecies, and 
 clearly defines the ideas of genera, families, and classes, which 
 hencelbrlh indicate different kinds of decomposition, effected through 
 certain modifications of the principle which graduates the whole 
 series. The animal realm, especially in its higher parts, is as yet 
 the only one in which the successive degrees have admitted a fully 
 scientific description. The rough clas.sification into natural groups 
 was an indispensable preparation for the mar.shalling into a series 
 of the immeasurable mass of materials presented by nature. The 
 groups being thus separated, and the study of their interior distri- 
 bution postponed, the innumerable throng of organic existences 
 became manageable. This irreat benefit has misled botanists into
 
 342 POSITIVE PHILOSOPHY. 
 
 Ibe supposition that the formation of these groups is tlie most 
 scientific part of the natural method, — otherwise than as a prelimi- 
 nary process. Tlie regular establishment of natural families offers, 
 no doubt, great facilities to scientific study, by enabling a single case 
 to serve for a whole group : but this is a wholly different matter 
 from the value of the natural method, regarded as it must hence- 
 forth be, as the highest rational means of the whole study, statical 
 and dynamical, of tiie system of living bodies ; and the great condi- 
 tion of which is that the mere position assigned to each body makes 
 manifest its whole anatomical and physiological nature, in its rela- 
 tion to the bodies which rank before or after it. These properties 
 Co-ordination could never belong to any mere establishment of 
 of them. natural families, if they could be grouped with a 
 
 perfection which is far fiom being possible ; for the arbitrary 
 arrangement of the families, and the indeterminate decomposition 
 of each of them into species, would destroy all aptitude for compre- 
 hensive anatomical or physiological comparison, and open the way 
 for that search after partial and secondary analogies which we see 
 to be so mischievous in the study of the vegetable kingdom at this 
 tlay. 
 
 The Natural Method, then, is philosophically characterized by the 
 general establishment of the organic hierarchy, reduced, if desired, 
 to the rational co-ordination of genera and even of families, the 
 realization of which is found only in the animal region ; and there 
 only in an initiatory state. And the co-ordination proceeds under 
 Three laivs of ihvee great laws, which are these: first, that the 
 co-ordination, animal sjiccies present a perpetually increasing com- 
 plexity, both as to the diversity, the multiplicity, and the speciality 
 of their organic elements, and as to the composition and augment- 
 ing variety of their organs and systems of organs. Secondly : that 
 this order corresponds precisely, in a dynamical view, with a life 
 more complex and more active, composed of functions more 
 numerous, more varied, and better defined. Thirdly : that the 
 living being thus becomes, as a necessary consequence, more and 
 more susceptible of modification, at the same time that he exercises 
 an action on the external world, continually more profound and 
 more extensive. It is the iniion of these three laws which 
 rigorously fixes the philosophical direction of the biological hierarch}', 
 each one dissipating any uncertainty v^'hich might hang about the 
 other two. Hence results the possibility of conceiving of a final 
 arrangement of all living species in such an order as that each 
 shall be always inferior to all that precede it, and superior to all 
 lliat follow it, whatever might otherwise, from its nature, be the 
 difficulty of ever realizing the hierarchical type to such a degree of 
 precision as this. 
 
 All adequate inquirers are now agreed upon this conception as the 
 starting-point of biological speculation ; and I need not therefore
 
 LAMARCK'S HYPOTHESIS. 343 
 
 stop to take notice of any prior controversies, excei)t one, wliicli is 
 noticeable from its liavini^ tended to iilnstrale and advance the prin- 
 ciple of the natural mctiiod. I refer to the discussion raised by 
 
 Lamarck, and maintained, thout'h in an imperfect n^.^.a^^ ^f 
 \ r\ • -11 1 r 41 1 Question of 
 
 manner, by Cuvier, witb regard to the general per- permanence of 
 
 nianence of organic species. The first consideration organic species. 
 in this matter is, that whatever may be the final decision of this 
 great biological question, it can in no way affect the fundamental 
 existence of the organic hierarchy. Instead of there being, as 
 Lamarck conjectured, no real zoological series, all animal organisms 
 being identical, and their characteristics due to external circum- 
 stances, we shall see, by a closer examination, that the hypothesis 
 merely presents the series under a new aspect, which itself renders 
 the existence of the scale more clear and unquestionable than before ; 
 for the whole zoological series would then become, in fact and in 
 speculation, ])erfectly analogous to the development of the individual ; 
 at least, in its ascending period. There would be simply a long 
 determinate succession of organic states, gradually deduced from 
 each other in the course of ages by transformations of growing com- 
 plexity, the order of which, necessarily linear, would be precisely 
 comparable to that of the consecutive metamorphoses of hexapod 
 insects, only much more extended. In brief, the progressive course 
 of the animal oi-ganism, which is now only a convenient abstraction, 
 adapted to facilitate thought by abridging discourse, would thus be 
 converted into a real natural law. This controversy, then, in which 
 Lamarck showed by far the clearer and profounder conception of the 
 organic hierarchy, while Cuvier, without denying, often misconceived 
 it, leaves, in fact, wholly untouched the theory of the biological 
 series, which is quite independent of all opinion about the perma- 
 nence or variation of living species. 
 
 The only attribute of this series which could be affected by this 
 controversy is the continuity or discontinuity of the organic progres- 
 sion : for, if we admit Lamarck's hypothesis, in which the different 
 organic states succeed each other slowly by imperceptible transitions, 
 the ascending series must evidently be conceived of as rigorously 
 continuous; whereas, if we admit the stability of living species, we 
 nuist lay down as a fundamental princi{)le the discontinuousness of 
 the series, without pretending, either, to limit, a i^riori, in any way 
 the small elementary intervals. Tiiis is the question to be con- 
 sidered; and thus restricted, the discussion is of extreme importance 
 I0 the general advancement of the Natural Method, which will be, 
 in fact, much more clearly described if we are able to regard the 
 species as essentially stable, and the organic series therefore as com- 
 })osed of distinctly se[)arate terms, even at its highest stages of devel- 
 opment ; for the idea of species, which is the principal biotaxic unity, 
 would no longer allow any scientific definition, if we must admit 
 the indefinite transformation of different species into each other,
 
 344 POSITIVE FHIIOSOrilY. 
 
 under the sufficiently-prolonged influence of chcumslances suffi- 
 ciently intense. However certain might be the existence ot" tlie bio- 
 logical hierarchy, we should have almost insurmountable difficulty 
 in realizing it ; and this proves to us the high philosophical interest 
 which belongs to this great question. 
 
 Lamarck's reasoning rested on the combination of these two 
 incontestable but ill-described principles : first, the aptitude of any 
 organism (and especially an animal organi.sm) to be modified to a 
 conformity to the exterior circumstances in which it is placed, and 
 which solicit the predominant exercise of some special organ, corre- 
 sponding to some faculty become requisite ; and secondly, the ten- 
 dency of direct and individual modifications to become fixed in races 
 by hereditary transmission, so that they may increase in each new 
 generation, if the environment remains unaltered. It is evident that 
 it" this double property is admitted without restriction, all organisms 
 may be regarded as having been produced by each other, if we only 
 dispose the environment with that freedom and prodigality so easy 
 to the artless invagination of Lamarck, The falseness of this hypo- 
 thesis is now so fully admitted by naturalists that I need only briefly 
 indicate where its vice resides. 
 
 We need not stop to object to the immeasurable time required for 
 each system of circumstances to effect such an organic transforma- 
 tion ; nor yet to expose the futility of imagining organic environ- 
 ments, purely ideal, which are out of all analogy with existing media. 
 We may pass on to the consideration that the conjecture rests on a 
 deeply erroneous notion of the nature of the living organism. The 
 organism and the medium must doubtless be mutually related; but it 
 does not follow that either of them produces the other. The question 
 is simply of an equilibrium between two heterogeneous and indepen- 
 dent powers. If all possible organisms had been placed in all possible 
 media, for a suitable time, the greater number of them would neces- 
 sarily disappear, leaving those only which were accordant with the 
 laws of the fundamental equilibrium ; and it is probably by a series 
 of eliminations like this that a biological harmony has become gra- 
 dually established on our globe, where we see such a process now 
 for ever going on. But the whole conception would be overthrown 
 at once if the organism could be supposed capable of modification, 
 ad infinitum, by the influence of the medium, without having any 
 proper and indestructible energy of its own. 
 
 Though the solicitation of external circumstances certainly does 
 change the primitive organization by developing it in some particu- 
 lar direction, the limits of the alteration are very narrow : so that, 
 instead of wants creating faculties, as Lamarck would have us 
 believe, those wants merely develop the powers to a very inconsider- 
 able <legree, and could have no influence at all without a primitive 
 tendency to act upon. The disappearance of the superior races of 
 animals bei'ore the encroachments of Man shows how limited is the
 
 CUVIER'S EXrOSlTION. 345 
 
 power of the orj^^anisni to ndnpt itself to nii altered erivii'onnieni ; 
 even the human baibarian gives way to civilized Man : and yet the 
 power of adaj)(ation is known to be greatest in the liighest organisms, 
 whereas the hypothesis of Lamarck woidd require the fact to be 
 the other way. In a statical view, too, this conception would cum- 
 pel ns to regard tlie introductory animal as containing, at least in 
 a rudimentary state, not only all the tissues (which might be admis- 
 sible, reducible as they are to the cellular tissue), but all the oigans 
 and systems of organs ; which is incompatible with anatomical 
 comparison. Thus, in every view is Lamarck's conception 
 condemned : and it even tends to' destroy the philosophical 
 balance between the two fundamental ideas of organization and life, 
 by leading us to suppose most life where there is least organization. 
 The l(>sson that we may learn from it is to study more efFectually 
 tlie limits within which, in each case, the medium may modify the 
 organism, about which a very great deal remains to be learned : 
 and meantime, there can scarcely be a doubt, especially after the 
 luminous exposition of Cuvier, that species remain essentially fixed 
 through all exterior variations compatible with their existence. 
 
 Cuvier's argument rests upon two chief considerations, comple- 
 mentary to each other ; — the permanence of the most ancient 
 known species ; and the resistance of existing species to the most 
 powerful modifying forces : so that, first, the number of species 
 does not diminish ; and next, it does not increase. We go back for 
 evidence to the descriptions of Aristotle, twenty centuries ago : we 
 find fossil species, identical with those before our eyes : and we 
 observe in the oldest mummies even the simple secondary differences 
 which now distinguish the races of men. And, as to the second 
 view, we derive evidence from an exact analysis of the effects of 
 domestication on races of animals and vegetables. Human inter- 
 vention, affording, as it does, the most favourable case for altera- 
 tion of the organism, has done nothing more, even when combined 
 with change of locality, than alter some of the qualities, without 
 touching any of the essential characters of any species ; no one of 
 which has ever been transformed into any other. No modification 
 of race, nor any influences of the social state, have ever varied the 
 fundamental and strongly marked nature of the human species. 
 Thus, without straying into any useless speculations about the 
 origin of the diflfereiit organisms, we rest upon the great natural 
 law that living species tend to perpetuate themselves indefinitely, 
 with the same chief characteristics, through any exterior changes 
 com])atible with their existence. In non-essentials the species is 
 modified within certain limits, beyond which it is not modified 
 but destroyed. To know thus much is good : but we must re- 
 member that it teaches us nothing, with any completeness, of the 
 kind of influence exercised by the medium on the organism. The 
 rational theorv of this action remains to be formed : and the
 
 346 POSITIVE PHILOSOPHY. 
 
 laying down the question was the great result of the Lamai-ck con- 
 tioversy, which thus rendered an eminent service to the progress of 
 sound biological philosophy. 
 
 We may now proceed on the authoiized conception that the 
 great biological series is necessarily discontinuous. Tiie transitions 
 may ultimately become more gradual, by the discovery of interme- 
 diate organisms, and by a better directed study of those already 
 known : but the stability of species makes it certain that the series 
 will always be composed of clearly distinct terms separated by im- 
 practicable intervals. It now appears that the preceding exami- 
 nation was no needless digression, but an inquiry necessary to 
 establish, in the hierarchy of living bodies, this characteristic 
 property, so directly involved in the rational establishment of the 
 liierarcliy itself. 
 
 Having surveyed the two great conceptions of Natural Groups, 
 
 Twologicai ^^^^ ^^^® biological Series, which together constitute 
 
 conditions of what is Called the Natural Method, we must now 
 
 the study. notice two great logical conditions of the study. The 
 
 first, or primordial, is the principle of the subordination of 
 
 characters : the other, the final, prescribes the translation of the 
 
 interior characters into exterior, which, in fact, results from a radical 
 
 investigation of the same principle. 
 
 From the earliest use of the natural method, even before the 
 Subordination investigation had passed on from the natural groups 
 of character- to the scrics of them, it was seen that the taxonomic 
 tstics. characters must be not only numbered but weighed, 
 
 according to the rules of a certain fundamental subordination which 
 must exist among them. The only subordination which is strictly 
 scientific, and free from all arbitrary intermixture, is that which 
 results from a comparative analysis of the different organisms: this 
 analysis is of recent date, and even yet is adequately applied only 
 in the animal region ; and thus the subordination was no more than 
 liai'ely conceived of before the institution of comparative anatomy, 
 and the weighing of attributes is closely connected with the con- 
 ception of the organic hierarchy. The subordination of taxonomic 
 characters is effected by measuring their respective importance 
 according to the relation of the corresjionding organs to the pheno- 
 mena which distinguish the species under study, — the ])henomena 
 becoming more special as we descend to smaller subdivisions. In 
 short, here as elsewhere, the philosophical task is to establish a true 
 harmony between statical conditions and dynamical properties ; 
 between ideas of life and ideas of organization, which should never 
 be separated in our scientific studies but in order to their ulterior 
 combination. Thus our aim, sometimes baffled but always hopeful, 
 is to subordinate the taxonomic characters to each other, without 
 the admission of anything arbitrary into any arrangement of 
 importance. We thus meet with gaps in our schedules which we
 
 APPLICATION OF THE NATURAL AIETHOD. 347 
 
 sliould avoid, or be insensible to, under an arbiirnry system ; but we 
 limy subdue our natural impatience under this imperfoetiou by 
 .•iccustoming our minds to regard rational classiiicatiou as a true 
 science, continually progressive, always perfectible, ami therefore 
 always more or less imperfect, like all positive science. 
 
 In conducting the process of comparison, the characters must be 
 admitted without restriction, in virtue of their posi- procedure, in. 
 live rationalitv, however inconvenient to manage and use of thcNa- 
 difficult to verify. This is the foundation of the I'^rai Method. 
 proposed classification. The next step is to discard from the 
 collection those whose verification would be too difficult, substitut- 
 ing for them some customary equivalents. Without this second 
 })rocess, which is as yet inadequately appreciated, the passage Ironi 
 Ihe abstract to the concrete would be inextricably embarrassed. 
 The anatomist and physiologist may be satisfied with a definition 
 of groups which will not suit the zoologist, and still less the natural- 
 ist. The kind of transformations required is easily specified. First, 
 it is clear how important it is to discard the characters which are 
 not permanent, and those Avhicb do not belong to the various natural 
 modifications of the species under study. They can be admitted 
 only as provisional attributes till true equivalents, permanent and 
 common, have been discovered. But the very nature of the problem 
 indicates that the aim of the chief substitution should be to leplace 
 all the interior characters by exterior : and it is this which con- 
 stitutes the main difficulty, and, at the same time, the highest per- 
 fection of this final operation. When such a condition is fulfilled, 
 on the basis of a rational primitive classification, the natural method 
 is irrevocably constituted, in the plenitude of its various essential 
 properties, as we now find it in the case of the animal kingdom. 
 
 This transformation appears to be necessarily possible ; for, as a 
 chief characteristic of animality is action upon the external world and 
 corresponding reaction, the most important primitive phenomena 
 of animal life must take place at the surface of separation between 
 the organism and its environment ; and considerations with regard 
 to this envelope, its form, consistence, etc., naturally furnish the 
 principal distinctions of the different animal organizations. The 
 interior oi-gans, which have no direct and continuous relation to the 
 medium, will always be of the highest importance among vegetative 
 phenomena, the primitive and uniform basis of all life : but they 
 are of secondary consequence in considering tlie degrees of animality ; 
 so that the interior part of the animal envelope, by which various 
 materials for use are elaboi-ated, is less important, in a taxonomic 
 j)()int of view, than the exterior part, which is the seat of the most 
 characteristic phenomena. Accordingly, the transformation of in- 
 terior into exterior zoological characters is not merely an ingenious 
 and indispensable artifice, but a simple return from the distraction 
 of an overwhelming mass of facts to a direct philosophical course of
 
 34S rosirivE philosophy. 
 
 investigation. When tlierefore we see a recourse to interior charac- 
 teristics in the study of the animal series, we must recognize the 
 truth that not only is the classification as yet unfinished, but that the 
 operation is imperfectly conceived of ; that the inquirer has not 
 ascended, through sound biological analysis, to the original source 
 of analogies empirically discovered. 
 
 After thus ascertaining the nature of the Natural Method, we 
 must, before quitting the subject, glance at the mode of its appUca- 
 tion in the co-ordination of the biological series, condensed into its 
 principal masses. 
 
 The most general division of the organic world is into the animal 
 .. and vegetable kingdoms; a division which i-emains 
 
 animal arid i^s an instance of thorough discontinuity, in spite of 
 vegetable ^\\ efforts to represent it as an ai'tificial arrangement. 
 
 '"^ ' ■ The deeper we go in the study of the inferior animals, 
 the more plainly we perceive that locomotion, partial at least, and 
 a corresponding degree of general sensibility, are the predominant 
 and uniform characters of the entire animal series. These two 
 attributes are even more universal in the animal kingdom than the 
 existence of a digestive canal, which is commonly regarded as its 
 chief exclusive characteristic: a predominance which would not 
 have been assigned to this attribute of the organic life but for its 
 being an inevitable consequence, and therefore an unquestionable 
 test, of the double property of locomotion and sensibility; to which 
 we must, in consequence, assign the first place. Such a transfor- 
 mation, however, i-elales only to movable animals; so that for the 
 rest, we should have still to seek some other yet more general 
 indication of universal animalit}', if we must despair of finally 
 discovering in it every direct anatomical condition of these two 
 animal properties. As for the case of certain gyrating plants which 
 appear to manifest some signs of these properties, our imperfect 
 analysis of their motions discloses no true character of animality, 
 since we can discover no constant and immediate relation with 
 either exterior impressions or the mode of alimentation. 
 
 Next to the division of the two organic kingdoms comes the 
 
 Hierarchy of qi^<^stion of the rational hierarchy of the animal 
 
 the animal kingdom, by itself. In the place of the irrational 
 
 kuifjduin. considerations, so much relied on formerly, of abode, 
 
 mode of nutrition, etc., we now rest upon tiie supreme consideration 
 
 of the greater or less complexity of the organism, of its relative 
 
 perfection, speciality, elevation ; in short, of the degree of animal 
 
 dignitij, as M. Jussieu lias well expressed it. The next preparatory 
 
 step was in the anatomical field, to determine the successive degrees 
 
 of animality proper to the different organs. The combination of 
 
 the two great inquiries, — into the bases of the zoological hierarchy 
 
 as residing in tlie organizaticni, and into the rank of the organs in 
 
 their relation to life, — has furnished, since the beginning of this
 
 ANIMAL HIERARCHY. 349 
 
 century, ilio first, direct nnd ixeneral slcetch of a definitive 2;raduniion 
 <if tlie animal kingdom. Henceforth, it became admitted that, as 
 the nervous system constitutes the most animal of the anatomical 
 elements, the classification must be directed by it; other organs, 
 and, yet more, inorganic conditions, being recurred to only on the 
 i'ailure of the chief in the most special subdivisions; and the 
 substitutes being employed according to their decreasing animality. 
 Whatever share other zoologists may have contributed by their 
 labours to the formation of this theory, it is to M. de Blainville that 
 the credit of it especially belongs : and it is by his classification that 
 Ave must proceed in estimating the application of the Natural Method 
 to tlie direct construction of the true animal hierarchy. 
 
 The happiest innovation which distinguishes this zoological sj^stem 
 is that it attributes a high taxonomic importance to AUrihuteo/ 
 the general form of the animal envelope, which had symmetry. 
 before been neglected by naturalists, and which offers the most 
 siriking feature, in regard to description, in the symmetry, which is 
 the prevailing character of the animal organism. We must here 
 reserve the case of the non-symmetrical animals ; and this shows 
 that the idea is insufficiently analysed as yet. The principle is 
 perhajis saved, or the difficulty distanced, by the fact that in these 
 animals no ti'ace can be discovered of a nervous system ; but there 
 is a sufficient want of precision and clearness to mark this as a case 
 reserved for further analysis. We shall not wonder at this imper- 
 fection if we remember how erroneous were the notions, no further 
 back than two generations ago, about very superior orders of 
 animals, — the whole of the radiated, a part of the mollusks, and 
 even of the lower articulated animals. Among the orders thus 
 restricted, there are two kinds of symmetiy, the most perfect of 
 which relates to a plane, and the otlier to a. })()int, or rather to an 
 axis : hence the further classification of animals into the duplicate 
 and the radiated. It is impossible to admii'e too much the exact- 
 ness with which an attribute, apparently so unimportant, corresponds 
 with the aggregate of the highest biological comparisons, which are 
 all found siiontaneously converging towards this simple and limiin- 
 ous distinction. Still it i-emains em])irical in its prepondei-ance ; 
 and we yet need a clear and rational explanation, both physiological 
 and anatomical, of the extreme necessary inferioi-ity of the radiated 
 to the duplicate animals, which, by their nature, must be nearer to 
 Man, the fimdamental unity in zoology. 
 
 Taking the duplicate animals, ox artiozoaries, their oi-der is again 
 divided according to the consistence of the envelope, nirinonof 
 — whether it is hard or soft, — and therefore more or duplicate 
 less fit for locomotion. This is, in fact, a proti'action 'Animals. 
 of the last consideration, as sxinmetiy must be moi'c mai-ked in 
 the case of a hard than of a soft covering. The two gi-eat attributes 
 of animality, — hjcoraotion and sensatinn,- -establish profound and.
 
 350 POSITIVE PHILOSOPHY. 
 
 unquGstionaLle differences, anatomical and pliysiologlcal, between 
 these two cases; and we may easily connect tiiem rationally willi 
 this primitive distinction, and perceive how they exhibit inarticulated 
 animals as necessarily inferior to the articnlated. 
 
 The articulated animals must next be distinguished into two 
 
 Division of gi'^at classes, according to the mode of articulation ; 
 
 articulated whether uuder the envelope, by a bony skeleton or a 
 
 animals. cartilaginous one, in the lowest degrees ; or whether 
 
 the articulation is external by the consolidation of certain horny 
 parts of the envelope, alternating with the soft parts. The inferi- 
 ority of this latter organization, especially with regard to the high 
 functions of the nervous system, must be seen at a glance. It is 
 observable that the more imperfect development of this eminently 
 animal system always coincides with a fundamental difference in 
 the position of its central part, which is always above the digestive 
 canal in vertebrated animals, and below it in those which have an 
 external articulation. 
 
 The rational hieraichy of the chief oi'ganisms in the upper part 
 of the animal series is, then, composed of the three great classes ; — 
 the vertebrated animals, those which are articulated externally, and 
 the moUusks ; or, in scientific language, the osteozoaries, the ento- 
 mozoaries, and the malacozoaries. 
 
 Glancing, finally, at the division of the first of these classes, I 
 Consideration may remark that all former descriptions and defini- 
 of the envelope, tious may merge in the consideration of the envelope. 
 It will be enough here to refer merely to the secondary view of the 
 envelope, — that of the inorganic ])roductions which separate it from 
 its environment. M. de Blainville has shown us how the descent 
 Irom Man, through all the mammifera, the birds, reptiles, amphibi- 
 ous animals, and fishes, is faithfully represented by the consideration 
 of a cutaneous surface furnished with hair, feathers, scales, or left 
 bare. The same determining importance of the envelope is per- 
 ceived in the next order of animals, in which the descent is measured 
 by the increasing number of pairs of locomotive appendages, from 
 the hexapods to the myriapods, and even to the apodes, which are 
 at the lower extremity. 
 
 It is not consistent with the object of this Work to go further 
 into a description of the animal hieraichy. My aim in giving the 
 above details has been to fix the reader's attention on my prelimin- 
 ary recommendation to study the present co-ordination of the animal 
 kingdom as an indispensable concrete ex})lanation of the abstract 
 conceptions which I had offered, in illustration of the natural method. 
 We must pass by, thcMefore, all speculations and studies which 
 belong to zoological philosophy, and mei'ely observe that there is 
 one portion of the fundamental system which remains to be consti- 
 tuted, and the general principles of which are as yet only vaguely 
 jicrceived ; — I mean the rational distribution of the species of each 
 
 «
 
 NO VEGETABLE HIERARCHY. 351 
 
 natural genus. This extreme and delicate application of llic iaxo- 
 nomic theory wouUl have been inopportune at an earlier stai^o of 
 the development of the science : but the time has arrived for it to 
 be undertaken now. 
 
 We cannot but see that the natural method does not admit of 
 anything like the perfection in the vegetable king- j^j- . , , 
 dom that it exhibits ni even the lower stages ot the applied to the 
 animal. Tiie iamilies may be regarded as estab- venetahie kiwi- 
 lished, though in an empirical way; but their 
 natural co-ordination remains almost entirely arbitrary, for 
 want of a hierarchical principle by which to subordinate them 
 rationally. The idea of animality yields a succession Difficulty of co- 
 ol degrees, deeply marked, so as to supply the basis ordination. 
 of a true animal hierarchy : but there is nothing of the kind in our 
 conception of vegetable existence. The intensity in this region is 
 not always equal ; but the character of vegetable life is homogeneous; 
 — -it is always assimilation and the contrary, continuous, and issuing 
 in a necessar}'^ reproduction. The mere differences of intensity in 
 such phenomena cannot constitute a true vegetable scale, analogous 
 to the animal ; and the less because the graduation is owing at 
 least as much to the preponderant influence of external circumstances 
 as to the characteristic organization of each vegetable. Thus, we 
 liave here no sufficient rational basis for a hierarchical comparison. 
 A second obstacle ought to be noticed, — serious enough, though of 
 less importance than the first ; — that each vegetable is usually an 
 agglomeration of distinct and independent beings. The case does 
 not resemble that of the polypus formation. In the compound 
 structure of the lowest animal orders a scientific definition is still 
 possible. There is a vital basis common to all the animal structures 
 which are otherwise independent of each other : but in the vegetable 
 case, it is a mere agglomeration, such as we can often produce by 
 grafting, and where the only common elements are inorganic parts, 
 aiding a mechanical consolidation. There is no saying, in the 
 present state of our knowledge, how far such a system may extend, 
 without being limited by any organic condition, as it seems to 
 depend on purely physical and chemical conditions, in combination 
 witii exterior circumstances. Faintly marked as the original or- 
 ganic diversity is by nature, it is evident how all rational subordina- 
 tion of the vegetable families in a common hierarchy is impeded by 
 the coalescent tendency just noticed. 
 
 The principal division which is the starting-point of M. Jussieu's 
 classification is the only beginning of a true co-ordination in the 
 vegetable kingdom. It consists in distinguishing the vegetables by 
 the presence or the absence of seminal leaves ; and when they are 
 present, by their having several or only one. For the successive 
 passage from those which have several to those that have none may 
 be regarded as a continuous descent, like that of the biological
 
 352 POSITIVE PHILOSOPHY. 
 
 series, tliongli less marked. Such a view has been verified by the 
 investigation of the organs of nutrition, according to the discovery 
 of Desfbntaines, — as yet the only eminent example of a large and 
 happy application of comparative anatomy to the vegetable organism. 
 By this concurrence of the two modes of comparison, — of the 
 reproductive and nutritive arrangements, — this proposition has 
 taken its rank among the most eminent theorems of natural phi- 
 losoph}'. But this beginning of a hierai'chy remains obviously 
 insufficient, — the numerous families in each of the three divisions 
 remaining under a purely arbitrary arrangement, which we can 
 hardly hope to convert into a rational one. The interior distribution 
 of species, and even of genera, within each family must be radically 
 imperfect, as the requisite taxonomical principles cannot be applied 
 to their arrangement till the difficulty of the co-ordination of the 
 families, — a difficulty much less, but as yet insurmountable, — has 
 been overcome. The Natural Method has, therefore, as yet 
 yielded no other result, as to the vegetable kingdom, than the 
 more or less empirical establishment of families and genera. We 
 cannot be surprised that it has not yet excluded the use of artificial 
 methods, and above all that of Linnteus, — true as it is that, up to 
 our time, the co-ordination of the vegetable kingdom was the field 
 for the application of the Natural Method. It should ever be 
 remembered, however, that the Natural Method is not merely a 
 means of classification, but an important system of real knowledge 
 as to the true relations of existing beings : so that even if it should 
 be disused for the purposes of descriptive botau}', it would not the 
 less be of high value for the study of plants, the comparative results 
 of which would be fixed and combined by it. In the present con- 
 dition and prospects of the science, as to the establishment of 
 a vegetable series, we must take the whole vegetable kingdom 
 together as the last term of the great biological series, — as 
 the last of the small number of essential modes of organization 
 which (when the subdivisions are disregarded) are markedly sepa- 
 rated from eacli other in the classification that gives us the 
 logical command of the study of living beings. Applied first to the 
 vegetable kingdom, the natui-al method is now seen to be the means 
 by which the animal realm, the type of all our knowledge of organic 
 life, is to be perfected. To whatever orders of phenomena natural 
 classification is to be applied, here its theory must first be studied ; 
 and hence it is that biological science bears so important a part in 
 the advancement of the whole of the positive method. It is much 
 that a considerable progress has been made in langing, in a due 
 order of dignify, the immense series of living beings, from Man to 
 the simplest plant : but, moreover, this theory of classification is an 
 indis{)ensable element of the whole positive method ; an element 
 which could not have been developed in any other way, nor even 
 ctherwise appreciated. 
 
 I
 
 353 
 
 CHAPTER IV. 
 
 OUGANIC OR VEGETATIVE LIFE, 
 
 We have to pass on to dynamical biology, which is very far indeed 
 from having attained the clearness and certainty condUionof 
 of the statical department of the science. Important Di/namicai 
 as are the physiological researches of recent times, ^>-o'orjii. 
 they are only preliminary attempts, which must be soundly systema- 
 tized before they can constitute a true dynamical biology. The 
 n)inds which are devoted to mathematical, astronomical, and 
 ])hysical studies are not of a difTerent make from those of ])hysiolo- 
 gists ; and the sobriety of the former classes, and the extravagance 
 of the latter, must be ascribed to the definite constitution of the 
 simpler sciences and the chaotic state of physiology. The melan- 
 choly condition of this last is doubtless owing in part to the vicious 
 education of those who cultivate it, and who go straight to the 
 study of the most com{)lex phenomena without having prepared 
 their understandings by tlie practice of the most simple and positive 
 speculation : but I consider the prevalent licence as due yet more 
 to the indeterminate condition of the spirit of physiological science. 
 In fact, the two disadvantages are one; for if the true character 
 of the science were established, the preparatory education would 
 immediately be rectified. 
 
 This infantine state of physiology prescribes the method of treat- 
 ing it here. I cannot proceed, as in statical biology, to an analytical 
 estimate of established conceptions. I can only examine, in pure 
 ])hysiology, the notions of method ; that is, the mode of organization 
 of the researches necessary to the ascertainment of the laws of vital 
 ])henomena. The progress of biological philosophy depends on the 
 distinct and rational institution of physiological questions, and not 
 on attem])ts, which must be premature, to resolve them. Concep- 
 tions relating to method are always important in i)roportion to the 
 complexity of the phenomena in view : therefore are they especially 
 valuable in the case of vital plienomena ; and above all, while the 
 science is in a nascent state. 
 
 Though all vital phenomena are truly interconnected, we must, 
 as usual, decompose them, for purposes of speculative vital pheno- 
 study, into those of greater and those of less generality, vuna .■ their 
 This distinction answers to Bichat's division into the '^'>'^sion. 
 organic or vegetative life, which is the common basis of existence of 
 
 VOL, I. z
 
 354 POSITIVE PHILOSOPHY. 
 
 all living bodies; and animal life, pi-oper to animals, but tlie chief 
 characters of which are clearly marked only in tiie higher part of 
 the zoological scale. But since Gall's time, it has become necessary 
 to add a third division, — the positive study of the intellectual and 
 moral phenomena which are distinguished from the preceding by a 
 yet more marked speciality, as the organisms which rank nearest 
 to Man are the only ones which admit of their direct exploration. 
 Though, under a rigorous definition, this last class of functions may 
 doubtless be implicitly included in what we call the animal life, yet 
 its restricted generality, the dawning posilivity of its systematic 
 study, and the pectdiar nature of the higher difficulties that it offers, 
 all indicate that we ought, at least for the present, to regard this 
 new scientific theory as a last fundamental branch of physiology, 
 in order that an unseasonable fusion should not disguise its high 
 importance, and alter its true character. These, then, are the 
 three divisions which remain for us to study, in our survey of 
 biological science. 
 
 Before proceeding to the analysis of organic or vegetative life, I 
 Theory of Or- must Say a fcw words on the theory of organic media, 
 (janic Media. without Considering which, there can be no true 
 analysis of vital phenomena. 
 
 This new element may be said to have been practically introduced 
 into the science by that controversy of Lamarck, already treated of, 
 about the variation of animal species through the prolonged influ- 
 ence of external circumstances. It is our business here to exclude 
 from the researches thus introduced, everything but what concerns 
 physiology properly so called, reduced to the abstract theory of the 
 living organism. We have seen that the vital state supposes the 
 necessary and permanent concurrence of a certain aggregate of 
 external actions with the action of the organism itself : it is the 
 exact analysis of these conditions of existence which is the object of 
 the preliminary theory of organic media ; and I think it should be 
 effected by considering separately each of the fundamental influences 
 under which the general phenomenon of life occurs. It cannot be 
 necessary to point out the importance of the study of this half of 
 the dualism which is the condition of life : but I may just remark 
 on the evidence it afibrds of the subordination of the organic to the 
 inorganic philosophy ; the influence of the medium on the organism 
 being an impracticable study as long as the constitution of the 
 medium is not exactly known. 
 
 The exterior conditions of the life of the organism are of two 
 Exterior con- classes, — physical and chemical ; or, in other words, 
 ditions of Life, mechanical and molecular. Both are indispensable ; 
 but the first may be considered, from their more rigorous and 
 sensible permanence, the most general, — if not as to the different 
 organisms, at least as to the continued duiation of each of 
 them.
 
 EXTERNAL CONDITIONS OF LIFE. 355 
 
 First in generality we must rank tlie action of Weiglit. There 
 is no (lenyint; that Man himself must obe}', whether Mechanical 
 as weight or projectile, the same mechanical laws conditions. 
 that govern every other equivalent mass : and by '^^'^'i'^''^- 
 reason of the universality of these laws, weight participates largely 
 in the production of vital phenomena, to which it is sometimes 
 favourable, sometimes opposed, and scarcely ever indifferent. There 
 is great difficulty in the analysis of its effects, because its influence 
 cannot be suspended or nuich modified for the purpose ; but we 
 have ascertained somethiug of them, both in the normal and the 
 pathological states of the organism. In the lower, the vegetable 
 ])ortion of the scale, the physiological action of weight is less varied 
 but more preponderant, the vital state being there extremely simple 
 and least removed from the inorganic condition. The laws and 
 limits of the growth of vegetables appear to depend essentially on 
 this influence, as is proved by Mr Knight's experiments on germina- 
 tion, as modified by a quicker or slower motion of rotation. Much 
 liigher organisms are subject to analogous conditions, without which 
 we could not explain, for instance, why the largest animal masses 
 live constantly in a fluid sufficiently dense to support almost their 
 whole w^eight, and often to raise it spontaneously. However, the 
 superior part of the animal series is least fit for the ascertainment 
 of the physiological influence of weight, from its concurrence with 
 a great number of heterogeneous actions: but this again enables 
 us to study it in a variety of vital operations : for there is scarcely 
 a function, organic, animal, or even intellectual, in which we may 
 not point out the indispensable intervention of weight, which 
 specially manifests itself in all that relates to the stagnation or 
 movement of fluids. It is therefore much to be regretted that a 
 subject so extended and important has not been studied in a rational 
 spirit and method. 
 
 The next mechanical condition, — pressure, liquid or gaseous, — 
 is an indirect consequence of weight. Some few 
 scientific results have here been obtained, from the 
 facility with which pressure may be modified by artificial or natm-al 
 circumstances. There are limits in the barometrical scale, outside 
 of which no atmospherical animal, — Man or any other, — can exist. 
 We cannot so directly verify such a law in the case of aquatic 
 animals : but it would seem that in proportion to the density of the 
 medium must be the narrowness of the vertical limits assignable to 
 the abode of each species. Of the relation between these intervals 
 and the degree of organization, it nnist be owned, however, that we 
 have no scientific knowledge, our ideas being, in fact, wholly con- 
 fused as to the inferior organisms, and especially in the vegetable 
 kingdom. Though, through many difficulties and complexities, 
 the science is in a merely nascent state, some inquiries, such as 
 those relating to the influence of atmospheric pressure on the venous
 
 356 POSITIVE PHILOSOPHY. 
 
 circulation, and recent observations on its co-operation in tlie 
 mechanism of standing and moving, etc., sliow that biologists are 
 disposed to study this order of questions in a rational manner. 
 
 Among the physical conditions, and perhaps first among them, 
 Motion the physiological influence of motion and rest should 
 and rest. ]yQ investigated. Amidst the confusion and obscurity 
 which exist on this subject, I think we may conclude that no organism, 
 even the very simplest, could live in a state of complete immobility. 
 The double movement of the earth, and especially its rotation, may 
 probably be as necessary to the development of life as to the peri- 
 odical distribution of heat and light. Too much care, however, 
 cannot be taken to avoid confounding the motion produced by the 
 organism itself with that by which it is affected from witliout ; 
 and the analysis had therefore better be applied to communicated 
 than spontaneous motion. And as rotary motion tends, by the laws 
 of mechanics, to disorganize any system, and therefore, eminently, 
 to trouble its interior phenomena, it is this kind of motion which 
 may be studied with the best result ; for which object we should 
 do well to investigate, in a comparative way, the modifications 
 undergone by the principal functions from the organism being 
 made to rotate in such a gradual variety as is compatible with a 
 normal state. The attempt has as yet been made only with plants, 
 and for another purpose ; while, in the case of the su})erior animals, 
 including Man, we have only incomplete and disjointed observalions, 
 scarcely transcending mere popular notions. 
 
 After the mechanical influences, we reach one which affects 
 
 Thermoiorjicai structure — the tlicrmological action of the medium. 
 
 action. Jt ig the best known of all ; for nothing is plainer 
 
 than that life can exist only within certain limits of the thermo- 
 metrical scale, and that there are limits affecting every family, and 
 even every living race ; and again, that the distribution of organisms 
 over our globe takes place in zones sufficiently marked, as to differ- 
 ences of heat, to furnish thermometrical materials to the physicists, 
 in a general way. But, amidst the multitude of facts in our posses- 
 sion, all the essential points of rational doctrine are still obscure 
 and uncertain. We have not even any satisfactory series of obser- 
 vations about the thermometrical intervals corresponding to the 
 diflerent organic conditions; — much less any law relating to such a 
 harmony, which has never, in fact, been connected with any other 
 essential biological character. This great gap exists as much with 
 regard to the successive states of the same organism as to the scale 
 of organisms. The necessary revision might be best applied first 
 to the lowest states ; as the iig^^ and the lowest organisms appear 
 able to sustain wider difierences of temperature than those of a 
 higher order : and several philosophical biologists have even believed 
 that life may have been always possible on our planet, notwith- 
 standing the different systems of temperature through which its
 
 EXTERNAL CONDITIONS OF LIFE. 357 
 
 suiTacG has successively passed. On the whole, the sum of our 
 analyses may seem to disclose, amidst many anomalies, a general 
 law : that the vital state is so subordinated to a determinate ther- 
 raometrical interval, as that this interval perpetually diminishes as 
 life becomes more marked, — in the case both of the individual and 
 of the series. If even tliis general law is not yet scientifically 
 ostahlishcd, it may be su{)posed how ignorant we are of the modi- 
 fications produced in the organism by variations of external temper- 
 ature, within the limits compatible with life. There has even been 
 much confusion between the results of abrupt and gradual changes 
 of temperature, though experiment has shown that graduation 
 vastly expands the limits within which the human organism can 
 exist ; and again, between the influence of external, and the organic 
 ])roduction of vital heat. This last great error shows that even 
 the hiying down the question remains to be done. The same thing 
 may be said of the other exterior conditions, such as light and 
 electricity, of which all that we know in this connec- Light, euc- 
 lion is that they exert a permanent influence needful tridti/,etc. 
 for the production and support of life. Besides the confusion and 
 uncertainly of our observations, we have to contend with the in- 
 feriority of otu- knowledge of those branches of physics, and with the 
 mischief of the baseless hypotheses which we before saw to infest the 
 study of them. While physicists talk of fluids and ethers, avowing 
 that they do so in an artificial sense, for purposes of convenience, 
 l)hysiologists speak of them as the real principles of two orders of ex- 
 terior actions indispensable to the vital state. Till reform becomes 
 substantial and complete in the study of light and electricity, these 
 Avill remain the exterior conditions of vitality of which our know- 
 ledge is the most imperfect. 
 
 Passing on from the physical to the chemical conditions of the 
 medium, we find our amount of knowledge scarcely Molecular 
 more satisfactory. Instrict generality, thisstudy relates conditions. 
 I0 the physiological influence of air and water, the mingling of which, 
 in various degrees, constitutes the common medium 
 
 . V ,. A T\r 1 T->i • -ii 11 Air a7id Water. 
 
 necessary to vitahty. As M. de Ijlamvdle remarked, 
 they must not be considered separately, as in a physical or chemical 
 inquiry, but in that mixture which varies only in the proportions of 
 its elements. This might be anticipated from our knowledge of the 
 ciiemical constitution of living bodies, the essential elements of which 
 are found only in the combination of air and water : but we have 
 physiological evidence also, which shows that air deprived of mois- 
 ture, and water not aerated, are fatal to vital existence. In this 
 view there is no difference between atmospheric and aquatic being.s, 
 animal and vegetable, but the unequal proportion of the tw^o fluids ; 
 the air, in the one case, serving as a vehicle for vaporized water ; 
 and the water, in the other case, conveying liquefied air. In both 
 cases, water furnishes the indispensable basis of all the organic
 
 3SS POSITIVE PHILOSOPHY. 
 
 liquids : and the iiir the essential elements of nutrition. We know 
 that the higher mammifera, including Man, perish when the air 
 reaches a certain degi-ee of dryness, as fishes do in water which has 
 been sufficiently deprived of air by distillation. Between these 
 extreme terms there exists a multitude of intermediaries in which 
 moister conditions of air and more aerated states of water correspond 
 with determinate organisms ; and the observation of Man iu the 
 different hygrometrical states of atmospheres shows how, in the 
 individual case, physiological phenomena are modified within the 
 limits compatible with the vital state. If we may say that the 
 question has been laid down in this inquiry, it is only in a vague 
 and obscure way. Besides our ignorance of the varying proportions, 
 we have none but the most confused notions about the way in which 
 each fluid participates in the support of life. Oxygen is the only 
 element of the air about which we have made any intelligent inquir}', 
 while physiologists entertain the most contradictory notions about 
 azote: and the uncertainty and obscurity are still greater with regard 
 to water. In this state of infancy, it can be no wonder that the 
 science offers as yet no law as to the influence of the medium on the 
 organism — even in regard to the question whether a certain condi- 
 tion of existence becomes more or less inevitable as the organism 
 rises in the scale. 
 
 The study of the influence of specifics does not enter here, on 
 study^ of account, of course, of the absence of generality ; but 
 specifics. it should be just pointed out that, reduced as is the 
 number of substances called specifics, there are still enough, — as 
 aliments, medicines, and poisons, — to afford a hint of what might 
 be learned by an exploration of them, in regard to the harmony 
 between the organic world and the inorganic. The very quality of 
 their operation, that it is s])ecial and discontinuous, and therefore 
 not indispensable, indicates the experimental method in this case, 
 as being certain, well circumscribed, and very various. This study 
 may then be regarded as a needful appendix, completing the pre- 
 liminary biological doctrine which I have called that of organic 
 media, and offering resources which are proper to it, and cannot be 
 otherwise obtained. Unhappily, this complement is in even a 
 more backward state than tlie more essential portions, notwith- 
 standing the multitude of observations, unconnected and unfinished, 
 already assembled in this path of research. 
 
 If such is the state of preliminary knowledge, it is clear how little 
 
 History of luis yet bccu learned of the laws of life themselves. 
 
 Physiology. ']^\^q inquiry has gone through revolutions, as other 
 questions have, before reaching the threshold of positivity, in our 
 day. From the im[)ulse given by Descartes, the illustrious school 
 of Boerhaave arose in physiology, which exaggerated the subordina- 
 tion of biology to the simj)ler parts of natural philosopliy so far as 
 to assign to the study of life the place of appendix to the general
 
 HISTOR Y OF niYSIOLOG Y. 359 
 
 system of inorganic physics. From tlie consequent reaction against 
 this absurdity arose tlie theory of Slahl, whicli may be [considered 
 the most scientific Ibrmula of the metaphysical state of physiolog}^ 
 Tlie struggle has since lain between these two schools, — the strength 
 of the metaphysical one residing in its recognition of physiology as 
 a distinct science, and that of the physico-chemical in its princi[)le 
 of the dependence of the organic on the inorganic laws, as daily dis- 
 closed more fully by the progress of science. The eifect of this 
 improved knowledge has been to modify the conceptions of meta- 
 physical })hysiology : the formula of Barthez, for instance, repre- 
 senting a further departure from the theological state than that of 
 Stahl ; as Stahl's already did than that of Van Helmont, though 
 the same metaphysical entity might be in view when Van Helmont 
 called it the urclieus, and Stahl the soul, and Barthez the vital 
 principle. Stahl instituted a reaction against the })hysico-chemical 
 exaggerations of Boerhaave; but Barthez established, in his pre- 
 liminary discourse, the characteristics of sound philosophizing, and 
 exposed the necessary futility of all inquisition into causes and 
 modes of production of phenomena, reducing all real science to the 
 discovery of their laws. For want of the requisite practice in the 
 ])Ositive method, the scheme of Barthez proved abortive ; and, after 
 having proposed his conception of a vital principle as a mere term 
 to denote the unknown cause of vital phenomena, he was drawn 
 away by the prevalent spirit of his time to regard the assumed 
 ])rinciple as a real and complex existence, though profoundly unin- 
 telh'gible. Ineffectual as his enterprise proved, its design with 
 regard to the advancement of positive science cannot be mistaken. 
 The progressive spirit is still more marked in the physiological 
 theory of Bichat, though we find entities there too. These entities 
 however show^ a great advance, as a determinate and visible seat is 
 assigned to them. The vital forces of Bichat however still intervene 
 in phenomena, like the old specific entities introduced into physics 
 and chemistry, in their metaphysical period, under the name of 
 faculties or occult virtues, which Descartes so vigorously Inmted 
 down, and Moliere so happily ridiculed. Such is the character of 
 tlie su))posed organic sensihiliiy, by which, though a mere terra, 
 Bichat endeavoured to explain physiological phenomena, which he 
 thus merely reproduced under another name : as when, for instance, 
 he thought he had accounted for the successive flow of different 
 liquids in one canal by saying that the organic sensibility of the 
 canal was successively in harmony with each fluid, and in antipathy 
 to the rest. But lor his untimely death, however, there can be no 
 doubt that he would have issued into an entire positivity. His 
 treatise on General Anatomy, though appearing a very few years 
 after his treatise on Life and Death, is a great advance upon it ; 
 and even in the construction of his metaphysical theory of vital 
 forces he certainly first introduced, under the title properties of
 
 36o POSITIVE PHILOSOPHY. 
 
 tissue, a conception of the highest vahie, destined to absorb all 
 ontological conceptions, and to prepare for the entire positivity of 
 tlie elementary notions of physiology. The thing required is to 
 substitute properties iox forces ; and Bichat's treatment of tissue 
 fulfilled this condition with regard, to a very extensive class of 
 eflfects : and thus his theor}', while it amended the metaphysical 
 doctrine of Stahl and Barthez, opened the way to its entire reforma- 
 tion by presenting at once the germ and the example of purely posi- 
 tive conceptions. This is now the state of physiological philosophy 
 in the minds of the majority of students ; and the conflict between 
 the schools of Stahl and Boerhaave, — between the metaphysical and 
 the physico-chemical tendency, — remains at the point to which it 
 was brought up by the impulse communicated by Bichat. It would 
 be hopeless to look to the oscillations of this antagonism for an 
 advance in science. If the one doctrine pievailed, science would 
 be in a state of retrogi'cssion ; if the other, in a state of dissolution ; 
 as in our social condition, in the conflict of the two political ten- 
 dencies, the retrograde and the revolutionary. The progress of 
 ])hysiology depends on the growth of positive elementary concep- 
 tions, such as will remand to the domain of history the controversy 
 from which nothing more is to expected. Abundant promise of 
 such an issue now appears: the two schools have annulled each 
 other ; and the natural development of the science has furnished 
 means for its complete institution to be begun. This I look upon 
 as the proper task of the existing generation of scientific men, who 
 need only a better training to make them adequate to it. If, from 
 its complexity, physiology has been later than other sciences in its 
 rational formation, it may reach its maturity more rapidly from the 
 ground having been cleared by the ])ursuit of the anterior sciences. 
 Many delays were occasioned in their case by transitory phases 
 which were not understood in the earlier days of positivity, and 
 which need never again arrest experienced investigation. It may 
 be hoped that physiologists will spare their science the useless and 
 humbling delay in the region of metaphysical hypothesis which long 
 embarrassed the progress of physics. 
 
 The true philosophical character of physiology consists, as we 
 Philosophical ^'^^^'^ seen, in establishing an exact and constant 
 character of haiinouy bctweeu the statical and the dynamical 
 Physiology. pojuts of vicw, — between the ideas of orgam'zation 
 and of life, — between the notion of the agent and that of the act ; 
 and hence arises the obligation to reduce all abstract conceptions 
 of physiological properties to the consideration of elementary 
 and general [)henomena, each of which conveys the idea of a deter- 
 minate seat. In other words, the reduction of /?«?c^/ons to corre- 
 sponding p)yoperties must be regarded as the simple consequence 
 of decompounding the general life into the different functions, — 
 discarding all notions about causes, and inquiring only into laivs.
 
 CHARACTER OF FJIYSIOLOGY. 361 
 
 13icliat's coiicc[)li(ni of the propeities of tissue contains the first 
 ,i;erm of tliis renovated view ; but it only indicates the nature of the 
 )»hilosophical operation, and contains no solution of the problem. 
 Not only is there a secondary confusion between tlie properties of 
 tissue and sinii)le physical properties, but the principle of the con- 
 ce[)tion is vitiated by the irrational distinction between the proper- 
 lies of tissue and vital })roperties ; for 110 property can be admitted 
 in physiology without its being at once vital and belonging to 
 tissue. In endeavouring to harmonize the difTerent degrees of 
 physiological and of anatomical analysis, we may lay down the 
 ])hilosophical princij)le that the idea of property which indicates 
 tbe last term of the one must correspond witli tissue, which is the 
 extreme term of the other ; whilst the idea o'i function, on the other 
 hand, corresponds to that of organ : so that the successive ideas of 
 iunclii)u and of property present a gradation of thoughts similar to 
 that which exists between the ideas of organ and of tissue, except 
 that the one relates to the act and the other to the agent. This 
 relation appears to me to constitute an incontestable and important 
 rule in biological })hilosophy ; and on it we may establish the first 
 great division among physiological properties. We Division of 
 have seen how in anatomy there is a division between the study. 
 the fundamental, generating tissue, the cellular, and the secondary 
 tissues which result from the combination of certain substances 
 with this original web ; and in the sauie way must physiological 
 properties be divided into two groups,— the one com})rising the 
 general properties which belong to all the tissues, and which consti- 
 tute the proper life of the cellular tissue ; and the other, the special 
 properties which characterize its most marked modifications, — that 
 is, the muscular and nervous tissues. This division, indicated by 
 anatomy, strikingly agrees with the great physiological distinction 
 between the organic or vegetative and the animal life; as the first 
 order of properties must afford the basis of that general life, com- 
 mon to all organized beings, to which vegetable existence is reduced ; 
 while the second i-elales exclusively to the special life of animated 
 beings. Such a conespondence at once make's the principle more 
 unquestionable, and facilitates the application of the rule. 
 
 If we look at what has been done towards the construction 
 of this fundamental theory, we shall find that it is fairly accom- 
 j)lished with regard to the secondary, or animal tissues, — all the 
 general phenomena of animal life being unanimously connected 
 with irritability and sensibility, — these being considered as attributes 
 each of a definite tissue : and thus, tlie most marked case is the best 
 understood. But the other division, — the properties which are wholly 
 general, belonging to the universal life, are i'ar niore important, as 
 underlying the others ; and an extreme confusion and diveigence 
 exist with regard to them. No clear and satisfactory conception 
 of the second class can be formed while the first is left in obscurity;
 
 362 POSITIVE PHILOSOPHY. 
 
 and tlius, tlie science remains in a purely ])rc)visional state, — lis 
 development having taken place in an order inverse to that which 
 its nature requires. 
 
 The functions which belong to the vegetative life are two, — the 
 Two functions ''^"tagonism of which corresponds to the definition of 
 of the or- life itself: first, the mteviov absorption of nutritive 
 
 game Uje. materials from the surrounding medium ; whence 
 results, after their assimilation, final nutrition; and secondly, the 
 exhalation of molecules, which then become foreign bodies, to be 
 })arted with, or disassimilated, as nutrition proceeds. It appears 
 to be an error to make digestion and circulation characteristics of 
 animality ; as we certainly find them here in the fundamental sense 
 of both. Digestion is properly a pre])aration of aliment for assimila- 
 tion ; and tliis takes place in a simple and almost unvaried manner 
 in vegetable organisms: and circulalion, though nothing like what 
 it is in animals, where there is a central organ to effect it, is not 
 less essential in vegetative life, — the lowest organism showing the 
 continual motion of a fluid holding in suspension or dissolved, 
 matters absorbed or thrown out ; and this perpetual oscillation, 
 which does not require a system of vessels to itself, but may take 
 place through the cellular tissue, is equally indispensable to animal 
 and vegetable existence. These, then, are the two great vegetative 
 ])rocesses, performed by properties which are provisionally supposed 
 (alter the analysis of M. de Blainville, which is open to some objec- 
 tions) to be three, — hygrometricity, capillarity, and retractility. 
 This analysis shows clearly that the actions which constitute vege- 
 table life are simple physico-chemical phenomena; physical as to 
 the motion of the molecules inwards and outwards ; and chemical 
 in what relates to the successive modifications of these different 
 substances. Under the first aspect, they depend on the properties, 
 hygrometrical, capillary, and retractile, of the cellular tissue : under 
 the second, and much more obscure at pi'esent, they lelate to the 
 molecular action which its composition admits. This is the spirit 
 in which the analyses of organic phenomena should be instituted ; 
 whereas that of animal phenomena should be regarded from a 
 wholly different point of view, as we shall see hereafter. 
 
 The study of this vegetative life is not even yet rationally organ- 
 ized. We have seen that, in the anatomical view, the vegetable 
 kingdom is regarded as the last term of an unique series, — the 
 various degrees of which differ, for the most })art, more widely from 
 each other than any one of them from this exti-eme term. The 
 same conce[)ti()n should direct physiological speculations on the 
 organic life, analysed imiformly for all living beings : but this has 
 not hitherto been even attempted. Till it is accomplished, no 
 essential ])oint of physiological doctrine can be established, however 
 able may be the investigations carried on, and however valuable the 
 siijaterials su])plicd. It may be alleged that the phenomena relating
 
 RESULTS OF ORGANIC ACTION. ^G^ 
 
 to pjeneral life may be studied in the broadest simplicily in vegetable 
 organisms: but it is no more possible in physiology than in anatomy 
 to interpret the extreme cases in the scale by each other without 
 having passed tlirongh the intermediate degrees : and the dynamical 
 case is the moi-e difficult of the two : so that the isolated study of 
 the organic life in vegetables cannot illustrate that of the higher 
 order of animals. And one natural consequence of this irrational 
 isolation of the vegetable case is that chemists and physicists have 
 engrossed researches which properly belong to biologists alone. 
 The comparative method, Avhicli we have seen to be the character- 
 istic resource of biological philosophy, has not as yet been duly in- 
 troduced into the general study of oi'ganic life, though it is at once 
 more indispensable, and more completely a])plicable, than in the 
 case of animal life. If it were consistent with the character of this 
 Work, we could j^oiut out gaps at almost every step, and about the 
 simplest phenomena, which must shock any inquiring mind : — the 
 darkness, doubts, and difTerences about digestion ; and again about 
 gaseous digestion, or respiration; — in regard to which the most 
 contradictory opinions are held : — divergences about the simplest 
 preliminary phenomena of vegetative life, which show how much 
 has to be done before we can undertake any direct investigation into 
 the ]ihenomena of assimilation and the converse process. 
 
 We shall find ourselves even further from satisfaction if we turn 
 from the consideration of the functions oi organic life Results of 
 to those more compound phenomena which are usually organic action. 
 confounded with them, but which M. de Blainville has taught us to 
 distinguish as results from the action of, not one organ or set of 
 organs, as in the case of function, but of the simultaneous action of 
 all the principal organs. Of these results, the most 
 immediate and necessary is the continuous state of composition 
 composition and decomposition which characterizes anddecom- 
 the vegetative life. Ignorant as we are of assimilation ■^°*' *''*'■ 
 and secretion, the very questions cannot have been as yet suitably 
 laid down. No one has thought, for instance, of instituting an 
 exact chemical comparison between the total comi)Osition of each 
 organism and the corresponding system of alimentation : nor, con- 
 versely, between the exhaled products and the whole of the agents 
 which had supplied or modified them ; so that we can give no pre- 
 cise scientific account of the genei-al phenomenon of the composition 
 and decomposition of every organism as a necessary consequence of 
 the concurrence of the different functions. We have at present 
 only incomplete and disjointed materials, which have never been 
 referred to any general fact. 
 
 It is acknowledged now that all organisms have, more or less, 
 the character which used to be ascribed to only the 
 liighest, of sustaining a determinate temperature, 
 notwithstanding variations of heat in their environment ; and thid
 
 364 POSITIVE PHILOSOPHY. 
 
 is a second result of the whole of the vegetative functions, which 
 almost always coexists with the first. But tliis important study is 
 not only in a backward state, hut ill-conceived. Besides the error 
 before noticed, of confounding vital heat with the temperature of 
 the medium, the fundamental character of the phenomenon appears 
 to me to have been misconceived. Its modification by the animal 
 i'unctions can never be understood till it has been studied in its 
 primitive universal manifestation in all living bodies, each of which 
 represents a chemical centre, able to maintain its temperature 
 against external influences, within certain limits, as a necessary 
 consequence of the plienomena of composition and decomposition. 
 This is doubtless the point of view from which the positive study of 
 vital heat must be regarded; and to consider it under the modifica- 
 tions of animal life, is to place the accessory before the principal, 
 and to propose views which are merely provisional, if not erroneous. 
 In the most I'ecent works upon this leading subject, the organic 
 foundations are, it is true, more carefully considered : but the 
 investigation cannot be said to be duly instituted as long as the 
 vegetable oi-ganism is not regularly introduced into it. 
 
 These remarks are even more applicable to the electrical study of 
 Electrical living bodies. Here we find again, and with aggra- 
 state. vation, the confusion between organic action and 
 
 external influence, as well as the aberrations remarked on in physics 
 about ethers and electric fluids. Here, too, we meet with the error 
 observed upon in the last case, about the physiological origin of 
 the plienomenon. And here, again, we are bound to conclude 
 that a permanent electrization is ascribable to acts of composition 
 and decomposition, notwithstanding the electrical variations of the 
 medium. And again we find that the animal functions can only 
 modif}', by accelerating or augmenting, more or less, the funda- 
 mental phenomenon. But the electrical analysis of the organism 
 is yet further than the thermological from being conceived of and 
 pursued in a rational view. 
 
 Next follow the general phenomena which result in a less direct 
 and necessary manner from the whole of the vegetative functions : — 
 the production and development of living bodies. 
 
 Notwithstanding the original investigations of Harvey and of 
 
 _, , Haller, with regard to the superior animals, this 
 
 and develop- investigation may be considei-ed, owing to its com- 
 
 mcat of HviiHj plexity, to be more in the rear of a positive institution 
 
 than any of the preceding. The tendency to search 
 
 for causes and modes of production of phenomena, instead of for 
 
 their laws, has acted with fatal effect hei-e ; and, amidst every kind 
 
 of deficiency, the main cause of the obscurity of the case is, 
 
 undoubtedly, that students have occupied themselves in looking for 
 
 what cannot be found. However, the labours of anatomists and 
 
 zoologists liave evidently prepared the way for a more rational
 
 VITAL DE VEL OPMENT AND DE CLINE. 365 
 
 study. It is even wortliy of remark tliat some students who were 
 most bent on the search into causes have been led on by Ihe s[)rea(l 
 of the positive spirit, to spend their efforts on inquiries into ovoloi^y 
 and embryology, which are assuming a more scientific character 
 every day. Still, the preliminary requisite (or the formation of 
 doctrine,— a fundamental analysis, — remains unfulfilled ; and the 
 ascertainment of the laws of })roducti()n and development is not, 
 therefore, to be attempted at present. In the lowest departments 
 of the scale, the multiplication of organisms takes place by a simple 
 prolongation of any part of the parent mass, which is almost homo- 
 geneous ; and in this extreme case, we understand the phenomenon 
 to be analogous to every other kind of reproduction of the primitive 
 cellular tissue. In the higher degrees of the scale, we are in the 
 dark from the moment we depart from immediate observation ; and 
 wlien the simplest previsions are so radically uncertain and even 
 erroneous as in this case, the science may be pronounced to be in a 
 state of infancy, notwithstanding the imposing a})pearance of the 
 mass of works accumulated for its illustration. 
 
 The comparative method has been applied in a yet more incom- 
 plete way to the phenomena of organic development. The question 
 has never yet been laid down under a form common to all organisms, 
 including the vegetable. The grave error is still committed of 
 studying the development in the animal cases alone; so that the 
 most eminently animal of the systems, the nervous, is represented 
 as the first to appear in the embr}'© of the higher orders, — a 
 supposition adverse to the institution of any really general concep- 
 tion of the theory of development, and in direct opposition to one 
 of the most constant laws of biological philosophy, — the perpetual 
 accordance between the chief phases of the individual evolution and 
 tiie most marked successive degrees of the organic hierarchy ; for 
 in this last view the nervous tissue is seen to be the latest and most 
 special transformation of the primitive tissue. The preliminary 
 analysis of organic development is, then, still far from being 
 conceived of in a rational spirit, governed by the high philosophical 
 intention of reconciling, as much as possible, the various essential 
 aspects of the science of living bodies. 
 
 To be complete, this analysis should evidently be followed by the 
 inverse, and yet correlative study of the decline of Decline of the 
 the organism, from its maturity to its death. The organism. 
 general theory of death is certainly in a very backward state, since 
 the ablest physiological researches on this subject have usually 
 related to violent or accidental death ; considered, too, in the highest 
 organisms exclusively, and affecting functions and systems of organs 
 of an essentially animal nature. As for the deterioration of the 
 organic life, we have yet attained to only one initiatoiy philosophical 
 glimpse, which exhibits it as a necessary consequence of life itself, 
 by the growing predominance of the movement of exhalation over
 
 366 POSITIVE PHILOSOPHY. 
 
 that of absorption, whence results gradually an exaggerated consol- 
 idation of the organism which was originally almost fluid, a process 
 which, in the absence of more rapid influences, tends to produce 
 a state of desiccation incompatible with all vital phenomena. 
 Valuable, however, as is such a glimpse, it serves only to charac- 
 terize the true nature of the question, by indicating the general 
 direction of the researches which it requires. The important 
 considerations relative to animal life could not be rationally intro- 
 duced into such a subject till this preliminary doctrine shall have 
 been established ; as in regard to all the other points of view before 
 examined. 
 
 Summary as this review has been, we have seen enough to be 
 authorized to conclude that the backward state of physiological 
 .science is owing mainly to the vicious training of physiologists, 
 and the irrational institution of their habitual labours. The 
 circulation of the blood, the first general foct which gave birth to 
 positive physiology ; and the laws of the fall of bodies, the first 
 acquisition of sound physics, are discoveries almost absolutely 
 contemporaneous ; and yet, what an immense inequality there is 
 now in the progress of two sciences setting out from so similar a 
 disclosure ! Such a difference cannot be attributed wholly to the 
 greater complexity of physiological phenomena, and must have 
 depended much also on the scientific spirit which directed their 
 general study, to the level of which the greater number of those 
 who cultivate it have been unable to rise. The phenomena of the 
 vegetative life obviously require, both for their analysis and their 
 explanation, an intimate combination of the leading notions of 
 inorganic philosophy with physiological considerations, obtained 
 through a thorough familiarity with the preliminary laws relating 
 to the structure and classification of living bodies. Now, each of 
 these inseparable conditions is, in our day, the separate property of 
 a particular order of positive investigators. Hence we have, on the 
 one hand, the supposed organic chemistry, a bastard study, which 
 is only a rough first sketch of vegetable physiology, undertaken by 
 inquirers who know nothing of the true subject of their labours: 
 and, on the other hand, vague, incoherent, and partly metaphysical 
 doctrines, of which physiology has been chiefly constituted by minds 
 almost entirely destitute of the most indispensable preliminary ideas. 
 The barren anarchy which has resulted from so vicious an organi- 
 zation of scientific labour would be enough of itself to testify to the 
 direct utility of the general, and yet positive point of view which 
 characterizes the foregoing survey.
 
 367 
 
 CHAPTER V. 
 
 TllK ANIMAL LIFE. 
 
 It was only by a late and long-prepared etlort that the human mind 
 could attain that state of abstraction and physiological generality 
 necessary for the comprehension of all vital beings, — from Man to 
 the vegetable, — as one series. It is only in our own day that a point 
 of view so new and so difiticult has been established ; and as yet, 
 among only the most advanced minds, even as regards the simplest 
 general aspects of biology, — in the statical study of the oi-ganisni. 
 It is not at all surprising that physiological comparison should 
 have been first applied to the animal functions, because they first 
 suggest its importance and possibility, however clearly it may after- 
 Avards appear that the organic life at once requires and admits a 
 larger and more indispensable application of the comparative method. 
 Looking more closely, however, into this evident existing superiority 
 of animal aver organic physiology, we must bear in mind the dis- 
 tinction between the two elementary aspects of every positive study, 
 — the analysis of phenomena and their explanation. It is only with 
 regard to the first that the animal lite has been in reality better ex- 
 plored than the organic. It is not possible that the explanation of 
 the most special and complex phenomena shotdd be more advanced 
 thnn that of the most simple and general, wliich serve as a basis to 
 the others. Such a state of the science would be in opposition to all 
 the established laws of the human mind. 
 
 However imperfect the theory of organic phenomena still is, it is 
 unquestionably conceived in a more scientific spirit than we find in 
 any explanations of animal physiology. We have seen that the 
 vegetative phenomena approach most nearly to the inorganic ; and 
 that the school of Boerhaave sinned only in exaggeration, pro- 
 ceeding from insufficient knowledge : and it must _, 
 be by this time evident that this is the link between from thein- 
 the inorganic and the biological philosophy, by which organic to the 
 we are enabled to regard the whole of natural philos- 
 ophy as forming a homogeneous and continuous body of doctrine. 
 By a natural consequence, a wholly difFeient view must be taken of 
 ihe rational theories of animal life : tliat is, of the ])henomena of 
 irritability and sensibility, which offer no basis of analogy with in- 
 organic phenomena. AVith regard to sensibility, no one will ques-
 
 368 POSITIVE PHILOSOPHY. 
 
 tion this : and as to irritaLIlity, — tliougli contraction may be seen 
 as a movement occasioned by beat, and yet more, by electricity, 
 tbese pbenomena must be carefully separated from the contractile 
 effect of the irritable fibre wbicb is a product of tbe nervous action ; 
 and especially wben it is voluntary. Irritability is as radically 
 foreign to tbe inorganic world as sensibility ; witb wbicb, too, it is 
 Primitive inseparably connected. This double property is, tben, 
 nervouspro- strictl}" primitive in tbe secondary tissues, and tbere- 
 periies. f^^.g j-,q niore a subject of explanation tban weigbt, 
 
 heat, or any other fundamental physical property. Whenever we 
 have a true theory of animal life, it will be by comparing all the 
 general phenomena which are connected with this double property, 
 according to their preparatory analysis, in order to discover their 
 laws ; that is, as in all other cases, their constant relations, both of 
 succession and similitude. This will be done in order to the usunl 
 end of obtaining a rational prevision ; the subject here being the 
 mode of action of a given animal organism, placed in determinate 
 circumstances ; or, reciprocally, the animal an-angement that may 
 be induced by any given act of" animality. All attempts to explore 
 the nature of sensibility and irritability are mere hindrances in the 
 way of this final aim, by drawing off our attention from the laws of 
 animality in a vain search after what can never be found. 
 
 The true relation of the animal to the organic life must throughout 
 Relation of the be Carefully kept in mind. This relation is double. 
 animal to the The orgauic life first serves as the basis of the animal ; 
 onjamciije. ^^^^ ^\^q^ gg jjg general end and object. We have 
 dwelt enough on the first, if even any one would think of con- 
 testing that, in order to move and feel, the animal must first live; 
 and that the fundamental vegetative life could not cease without 
 extinguishing the other. As for the second relation, it is evident 
 that the phenomena of irritability and sensibility are directed by 
 the general needs of the organic life, which they serve by procuring 
 better materials, and by guarding against unfavouiable influences. 
 Even tbe intellectual and moral functions have usually no other 
 primitive office. Without such a destination, these properties would 
 either destroy the organism or themselves perish. It is oidy in the 
 human species, and even there only under a high degree of civiliza- 
 tion, that any kind of inversion of this order can be conceived of. 
 In that case, the vegetative life is essentially subordinated to the 
 animal, the development of which it is alone destined to aid ; and 
 this, it seems to me, is the noblest scientific notion that we can 
 form of humanity, distinct from animality: — a transformation which 
 can be safely considered as possible only by transferring to the 
 whole species, or at least to society, the primitive end which, in 
 the case of animals, is limited to the individual, or, at the utmost, 
 to the family, as we shall see hereafter. It is only among a small 
 number of men, and it is very far indeed from being a just matter
 
 THE ANIMAL LIFE. 369 
 
 of expoctation from the whole species, that the intellect can acqniie 
 Knch a preponderance in the whole of the organism as to become liie 
 end and object of luiman existence. An exception so special, and 
 so easy to explain in the case of Man, cannot alter the universulity 
 of a consideration verified by the whole animal kingdom, wherein 
 the animal life is seen to be always destined to perfect the organic. 
 It is oidy by a scientific abstraction, necessary for pnrposes of pro- 
 gress, that we can provisionally conceive of the first as isolated from 
 the second, which is, strictly speahing, inseparable from it under the 
 double aspect just exhibited. Thus, as the positive theory of ani- 
 malily must contimially rest on that of general vitality, it is indis- 
 solubly combined with the whole of inorganic philos- To inorganic 
 ophy, which furnishes the basis of organic physiology. vii'<'<^osop]aj. 
 In a secondary sense, the same dependence exists. We admitted, 
 while reviewing mathematical philoso[)hy, that the laws of equili- 
 bi-ium and motion operate among all orders of ])henomena, being 
 absolutely universal. Among physiological phenomena w^e find 
 them accordingly ; and, when contraction is produced by the irrita- 
 bility of the muscular fibre, all the phenomena of animal mechanics 
 which result, whether for rest or locomotion, are dependent on the 
 general laws of mechanics. In an inverse way the same Ihing 
 takes place with regard to the functions of sensibility, in which the 
 inorganic philosophy must intervene in connection with the primitive 
 impression on the sentient extremities, carefully distinguished from 
 its transmission by the nervous filament, and its perception by the 
 cerebral organ. This impression acts through an intermediate 
 ])hysical apparatus, optical, acoustic, or other, the study of which, 
 according to appropriate })hysical laws, constitutes a chief element 
 of the positive analysis of the phenomenon. Not only must we use 
 the knowledge already established, but we want, for our analysis, 
 further progress in it, and even the creation of new doctrines, as 
 the theory of flavours, and yet more of odours, in regard to the mode 
 of ])ropagation of which there are doubtless several general laws, of 
 a purely inorganic character, remaining to be established. In in- 
 vestigating these connections between biology and inorganic science, 
 we find again wdiat we saw before, that chemistry is spontaneously 
 related tovegetable])hysiology,and physics especially to animal physi- 
 ology ; though neither could be altogether dispensed with in either 
 department, where they are lequired more or less in combination. 
 
 Our ideas of the double ])roperty of irritability and sensibility 
 cannot be truly scientific till each is irreversibly Properties 
 assigned to a corresponding tissue. Bichat conceived of tiamie. 
 of all tissues as necessarily sensitive and irritable, but in dilferent 
 degrees ; — an error which was natural or inevitable at a time when 
 so little was known of tissue in the way of anatomical analysis, but 
 one which, if maintained now, W(»uld hand over the whole science 
 to the physico-cliemical school, and efface all real distinction be- 
 
 YOL. I. 2 a
 
 .370 POSITIVE PHILOSOPHY. 
 
 t.ween the inorganic and organic departments of natural pliilosophy. 
 Rational biology requires that the two properties should he inherent 
 in determinate tissues, — themselves modifications, profound and 
 distinctly marked, of the primitive cellular tissue, — that our ana- 
 tomical data may be in harmony with the physiological ; in other 
 words, that the elementary ideas of tissue and of property should 
 be in perfect correspondence. The scientific character of physiology 
 in this direction is essentially defective among biologists in general. 
 But the new explorations continually made show us 
 
 ensitiy. how it was that Bichat was misled. He considered 
 it ]>roved that sensibility existed where there were no nerves: but 
 further investigation proves that the symptoms of sensibility were 
 erroneously attributed to an organ deprived of nerves, instead of 
 being referred to the simultaneous injury of neighbouring nerves ; 
 or that the nervous tissue existed, though it was difficult to find. 
 If cases apparently contradictory still remain, it would be obviously 
 absurd to reject on their account a conception required by the 
 principles of rational physiology, and founded on unquestionable 
 cases, by far more numerous and decisive than those which still 
 seem to be exceptional. This considei'ation should be applied to 
 different organisms, as well as to the different tissues of the human 
 organism. The animals supposed to be without nerves, on which 
 the metaphysical school has insisted so much, disappear as com- 
 parative anatomy enables us to generalize more and more the idea 
 of nervous tissue, and to detect it in the inferior organisms. It is 
 thus, for instance, that it has been recently found in several radiated 
 animals. The time has come for its being established as a philoso- 
 ])hical axiom that nerves are necessary for any degree of sensibility, 
 the apparent exceptions being left as so many anomalies to be re- 
 solved by the future progress of anatomical analj'sis. 
 
 The same process must be instituted with the common notions of 
 ,. . irritability, which are still ruled by Bichat's theory. 
 
 ma til/. jj^ supposed, for instance, that the contractions of 
 the heart were determined, independently of all nervous action, by 
 the immediate stimulus of the flow of the blood towards it; whereas, 
 it is now established that a provision of nerves is as indispensable 
 to the irritability of this muscle as of any other; and generally, 
 that the great distinction laid down by Bichat, between organic and 
 animal contractility, must be abandoned. All irritability is then 
 necessarily animal; that is, it requires a corresponding nervous 
 provision, whatever may be the immediate centre from which the 
 nervous action proceeds. Much illustration of this subject is needed 
 for its scientific use, thou<2;h not for the logical sanction of a principle 
 already })laced beyond dispute. We need this further enlighten- 
 ment, not only in regard to the use of the modcM-u distinction made 
 by many physiologists between the sensory and the motory nerves, 
 though such a question has considerable philosophical importance ; 
 
 I 
 I
 
 IRRITABILITY. 371 
 
 but niiicli more in regard to another consideration, more direct and 
 more eminent, in regard to which we are in a state of most incon- 
 venient uncertainty and obscurity: I mean the scientific distinction 
 which must be maintained, sooner or later, between the vohmtary 
 and in vohmtary motions. Tiie doctrine of Bichat had the advan- 
 tage of representing this difference, which we see, in fact, to have 
 i'urnished him with liis cliief arguments; whereas, now that we 
 insist on irritability being of one kind only, and dependent on a 
 nervous jtiovision, we find ourselves involved in a very delicate 
 fundamental difficulty, the solution of which is however indispen- 
 sable, to enable us to understand how all motions must not be 
 indistinctly voluntary. For this solution we must obtain — what 
 we certainly have not as yet — an exact co-ordination of anatomical 
 (lifTerenccs with incontestable physiological differences. There can 
 be no question that such a phenomenon as the voluntary movements 
 of the locomotive muscles while that of the cardiac muscle remains 
 absolutely involuntary, must admit of analysis, however difficult it 
 may be. Here then we find a chasm among the very principles 
 of the science, by which the positive theory of irritability is much 
 perplexed, certain as is its princi})le. In almost all cases the ablest 
 anatomist is unable to decide otherwise than by the fact itself, if 
 any definite motion is necessarily voluntary or involuntary ; which 
 affords sufficient proof of the absence of any real law in the case. 
 The solution will probably be obtained by an analysis of the inter- 
 mediate motions, as we may call them, — those which, involuntary 
 at first, end in becoming voluntary ; or the reverse. These cases, 
 which are very common, appear to me eminently fit to prove that 
 the distinction between voluntary and involuntary motions arises 
 from no radical difference of muscular irritability, but only from the 
 mode, and perhaps the degree of innervation, modified by long 
 liabit. If this be as generally true as it seems to be in some cases 
 of acquired control, it must be supposed that the most involuntary 
 motions, which are those most indispensable to life, would have 
 been susceptible of voluntar}'' suspension (not even excepting the 
 motions of the heart) if their incessant rigorous necessity had not 
 hindered the contraction of suitable habits in their case. Wliile we 
 conclude it to be probable that the difference between the two kinds 
 of motion proceeds indirectly from the action of the entire nervous 
 system upon the muscular system, we cannot help perceiving how 
 greatly science stands in need of a thorough new examination into 
 this obscure fact. 
 
 This brief survey shows us the general imi)erfection of the study 
 of animality. We shall find that even in the depart- presmt state 
 ment of the ])rimltive analysis of its general pheno- of anai/jsis of 
 mena, in which it appears so superior to tliat of the <^nunahty. 
 organic life, it is veiy far indeed from being yet fit for exploration 
 by positive laws.
 
 372 POSITIVE PHILOSOPHY. 
 
 la regard to irritability, first, — the mechanism of no animal 
 movement lias 3'et been satisfactorily analysed, — all the 
 chief cases being still the subject of radical controversy 
 among equally qualified pliysiologists. We retain a vicious dis- 
 tinction among movements, contrary to all mechanical judgment, — 
 a distinction between the general motion which disjilaces the whole 
 mass, and the partial motions which subserve the organic life, — as 
 for the reception of aliment, or the expulsion of any residuum, or 
 the circulation of fluids; yet the first order are partial, though their 
 object is unlike that of the second ; for, in a mechanical view, the 
 organism allows of no others. By the great laws of motion, the 
 animal can never displace its centre of gravity by interior motion, 
 without co-operation from its environment, any more than a steam- 
 carriage which should work without friction on a horizontal plane, 
 turning its wheels without result. The movements wliich produce 
 locomotion are not mechanically different from those, for instance, 
 which carry food along the alimentary canal ; — the difference is in 
 the apparatus, which, for locomotion, consists of exterior appendages, 
 so disposed as to cause a reaction in the medium, which produces 
 the displacement of the whole body. Certain moUusks furnish an 
 illustration of this, when they change their place by means of con- 
 tractions of the cardiac muscle, or of intestinal muscles. The sim- 
 plest notions of animal mechanics being thus obscured and corrupted 
 in their origin, it is no wonder that the physiologists still dispute 
 about the mechanism of the circulation, and most of the means of 
 locomotion, as leaping, flying, swimming, etc. In the way in which 
 they proceed, they are remote i'rom any mutual understanding, and 
 the most opposite ojiinions may be maintained with equal plausi- 
 bility. It needs but a word to suggest to those who have attended 
 to what has gone before, that this extreme imperfection results from 
 the inadequate and faulty education of physiologists, who are too 
 often ignorant of the inorganic science which is here directly in- 
 volv^ed. The comj)lexity of the animal apparatus, and the impos- 
 sibility of bringing the primitive moving powers under any mathe- 
 matical theory, will for ever forbid the application of numerical 
 methods : but the great laws of equilibrium and motion are 
 applicable, through all varieties of apparatus, and are the same in 
 animal mechanics, or celestial, or industrial, or any other mechanics 
 whatever. Some physiologists, finding their difiicully, have handed 
 over their study to the geometers and physicisls : and these, with 
 their habits of numerical precision, and their ignorance of anatomy, 
 have brought out only absurd results. The remedy is, as we know, 
 in the work being consigned to physiologists, duly prepared by a 
 sufficient training in inorganic science. The study of animal sounds, 
 or phonation, for instance, cannot be carried on to any purpose 
 without such knowledge as physicists have of the theory of sound ; 
 and the general production of the voice, and the dill'erences of utter-
 
 SENSATIO.y. 373 
 
 nnce anionu^ niiliiials, require for tlieir ex])lanation a Icnowlodge at 
 once of acoustics and anatomy ; and s))eech itself requires lliis pre- 
 ])aration no less, while demanding; other requisites with it. It is 
 1o be hoped that all experience, in each department of scientific 
 inqiiir}', will convince students more and more of the folly and mis- 
 chief of the anarchical parcelling; out of natural jihilosophy; but the 
 ])hysiolog;ists are those who, above all, must see the need of a better 
 ()i"ganization of scientific labour, — so remarkable as is the suboidina- 
 tion of their particular science to all that have gone before. 
 
 The analysis of the phenomena of sensibility is not more satis- 
 factoiy than that of irritability: and even less so, if we leave out of 
 the account the great knowledge that we liave obtained, by anatom- 
 ical study, of the corresponding organs; a knowledge which, how- 
 ever, must here be connected with physiology. The least imperfect 
 ])art of this study relates to the simple exterior sensations. The 
 ])henomenon of sensation is composed of three ele- Exterior 
 inents, as we bave seen : the impression of the external sensation. 
 agent or the nervous extremities, by the aid of some physical appa- 
 latus ; the transmission by the nervous fibre ; and the reception by 
 Ihe cei'cbral organ. The first of these suggests, like the mechanical 
 facts we have been considering, the immediate dependence of the 
 ])henomenon on the laws of the inorganic world ; as the relation of 
 ihe theory of vision to optics : of the theory of hearing to acoustics, 
 in all that concerns the mode of action proper to the apparatus of 
 sight and hearing. And yet, more expressly than even in the case 
 of mechanics, have these theories been delivered into the hands of 
 the physicists, who, again, bring out results from their treatment of 
 them which are manifestly absurd. The only difference between 
 Ibis case and the preceding is that the metaphysicians have kept a 
 longer hold upon this part of animal physiolog}', — the theory of 
 sensations having been abandoned to them till a veiy recent time. 
 It was not, indeed, till Gall imparted his ever-memorable impulse 
 to the investigation, that ph3'siologists claimed this department at 
 all. It is no wonder, therefore, that the positive theory of sensa- 
 tions is less well conceived, and more recent, than that of motion ; 
 and naturally more imperfect, independently of its superior difHculty, 
 and the backwardness of those branches of ])hysics to which it 
 relates. The simplest modifications of the phenomenon of vision 
 and of hearing cannot as yet be referred with certainty to deter- 
 minate organic conditions; as, for instance, the adjustment of the 
 eye to see distinctly at very various distances; a faculty which the 
 ])hysiologists have allowed the physicists to attribute to various 
 circumstances of structure, always illusory or inadequate, the 
 ]>hysiologists the while playing the part of critics, instead of appro- 
 ])riating a study which belongs exclusively to them. Even the 
 limits of the fimction are usually very vaguely defined: that is, the 
 kind of exterior notions furnished by each sense, abstracted from
 
 374 POSITIVE PHILOSOPHY. 
 
 all intellectual reflection, is rarely circumscribed with any distinct- 
 ness. Thus it is no wonder if we are still ignorant of almost all 
 positive laws of sight and hearing, and even of smell and taste. — 
 The only point of doctrine, or rather of method, that we may con- 
 sider to have attained any scientific stability, is the fundamental 
 order in which the different kinds of sensations should be studied : 
 and this notion has heen supplied by comparative anatomy rather 
 than by physiology. It consists in classifying the senses by their 
 increasing speciality, — beginning with the universal sense of contact, 
 or touch, and proceeding by degrees to the four special senses, taste, 
 smell, sight, and finally, hearing. This order is rationally deter- 
 mined by the analysis of tlie animal series, as the senses must be 
 considered more special and of a higher kind in proportion as they 
 disappear from the lower degrees of the zoological scale. It is 
 remarkable that this gradation coincides with the degree of impor- 
 tance of the sensation in regard to sociality, if not to intelligence. 
 Unhappily, it measures yet moie evidently the increasing imper- 
 fection of the theory. — We ought not to pass over the luminous 
 distinction introduced by Gall between the passive and the active 
 state of each special sense. An analogous consideration to this, 
 but more fundamental, would consist, it seems to me, in distinguish- 
 ing the senses themselves as active and passive, according as their 
 action is, from their nature, voluntary or involuntary. This dis- 
 tinction seems very marked in the case of sight and hearing ; the 
 one requiring our free participation, to a certain extent, while the 
 other affects us without our will, or even our consciousness. The 
 more vague, but more profound influence that music influences over 
 us than we receive from painting, seems to be chiefly attributable 
 to such a diversity. An analogous difference, but less marked, 
 exists between taste and smell. 
 
 There is a second class of sensations, forming the natural tran- 
 inierior sition from the study of the sensations to that of the 
 sensation. affective and intellectual functions, which all physio- 
 logists, since the time of Cahanis, and yet more, of Gall, have 
 found it necessary to admit, to complete the study of the sensa- 
 tions. They are the interior sensations which relate to the satis- 
 faction of the natural wants ; and, in a pathological state, the t)ains 
 produced hy bodily alteration. These are still more indispensable 
 than the first to the perfection of the organic life ; and, though they 
 procure no direct notions of the external world, they radically 
 modify, hy their intense and continuous action, the general course 
 of intellectual operations, which are, in most animal species, en- 
 tirely subordinated to them. This great department of the theory 
 of sensations is even more obscure and unadvanced than the fore- 
 going. The ordy positive notion which is fairly established in 
 regard to it is that the nervous system is indispensable to both 
 kinds of sensibility.
 
 A CTION OF ANIMALITY. 3 7 5 
 
 Once agnin we see liow llie extreme iinperfectioii of doctrine here 
 is owing lo llie imperCection of method ; and that again, as before, 
 to the inadequate preparation of the inquirers to whom the study 
 belongs. It is a great thing, however, to have withdrawn the 
 subject from the control of the meta])hysicians ; and some labours 
 of contemporary physiologists authorize us to hope that the true 
 spirit of the inquiry is at length entered into; and that the study 
 of the sensations will be directed, as it ought to be, to develop the 
 radical accordance between anatomical and physiological analysis. 
 
 Having reviewed the two orders of animal functions, we must 
 consider the coniplementary part of the theory of Mode of 
 animality ; — the ideas about the mode of action action of 
 which are common to the phenomena of irritability animaUty. 
 and sensibility. It is true, these ideas belong also to intellectual 
 and moral phenomena ; but we must review them here, to com- 
 ])lete our delineation of the chief aspects of the study of animal 
 life. 
 
 The considerations about such mode of action naturally divide 
 themselves into two classes: the one relating to the function of 
 either motion or sensation, separately; and the other to the associa- 
 tion of the two functions. The first may relate to either the mode 
 or the degree of the animal phenomenon. Following this order, 
 the first theory that i)resents itself is that of the ,. , 
 intermiltence of action, and consequently, that of 
 habit, which results from it. Bichat was the first who pointed out 
 the intermittent character of every animal faculty, in contrast with 
 the continuousness of vegetative phenomena. The double move- 
 ment of absorption and exhalation which constitutes life could not 
 be suspended for a moment without determining the tendency to 
 disorganization : whereas, every act of irritability or sensibility is 
 necessarily intermittent, as no contraction or sensation can be con- 
 ceived of as indefinitely prolonged ; so that continuity would imply 
 as great a contradiction in animal life as interruption in the organic. 
 All tlie progress made, during the present century, in physiological 
 anatomy, has contributed to the perfecting of this theory of inler- 
 mittence. Rationally understood, it applies immediately to a very 
 extensive and im})ortant class of animal phenomena; that is, to 
 those which belong to the different degrees of bleep. The state of 
 sleep thus consists of the simultaneous susj)ension, ^ 
 for a certain time, of the principal actions of irrita- 
 bility and sensibility. It is as complete as the organization of the 
 superior animals admits when it suspends all motions and sensations 
 but such as are indispensable to the oiganic life, — their activity 
 being also remarkably diujinished. The phenomenon admits of 
 great vaiiety of degrees, from simple sonmolence to tiie torpor of 
 hibernating animals. But this theory of sleep, so well instituted 
 by Bichat, is still merely initiated, and presents many fundamental
 
 376 POSITIVE PIIIIO SOPHY. 
 
 difficulties, when we consider thecliief modifications of such a state, 
 even the organic conditions of wliich are very imperfectly known, 
 except the stagnation of the venous blood in the brain, which 
 nppears to be generally an indispensable preliminary to all extended 
 and durable letliargy. It is easy to conceive how the prolonged 
 activity of the animal functions in a waking state may, by the law 
 of intermittence, occasion a proportional suspension : but it is not 
 so easy to see why the suspension should be total when the activity 
 luis been only partial. Yet we see how profound is the sleep, 
 intellectual and muscular, induced by fatigue of the muscles alone 
 in men who, while awake, have given very little exercise to their 
 sensibility, interior, or even exterior. We know still less of incom- 
 ])lete sleep ; especially when only a part of the intellectual or affec- 
 tive organs, or of the locomotive apparatus is torpid ; whence arise 
 dreams and various kinds of somnauibulism. Yet such a state has 
 certainly its own general laws, as well as the waking state. Some 
 experiments, not duly attended to, perhaps justify the idea that, in 
 animals, in which the cerebral life is much less varied, the nature 
 of dreams becomes, to a certain point, susceptible of being directed 
 at the pleasure of the observer, by the aid of external impressions 
 produced, during sleep, upon the senses whose action is involuntary; 
 and especially smell. And in the case of Man, there is no thought- 
 ful physician who, in certain diseases, does not take into the account 
 the habitual chaiacter of the patient's dreams, in order to perfect 
 the diagnosis of maladies in which the nervous system is especially 
 implicated : and this supposes that the state is subject to determi- 
 nate laws, though they may be unknown. But, however imperfect 
 the theory of sleep may still be, in these essential respects, it is 
 fairly constituted upon a positive basis of its own ; for, looked at 
 as a whole, it is explained, according to the scientific acceptation 
 of the term, by its radical identity with the phenomena of partial 
 repose offered by all the elementary acts of the animal life. When 
 the theory of intermittence is perfected, we shall, I imagine, adopt 
 Gall's view of connecting it with the symmetry which characterizes 
 all the organs of animal life, by regarding the two parts of the 
 syinmetrical apparatus as alternately active and passive, so that 
 their function is never simultaneous : and this, as much in regard 
 to the external senses as the intellectual organs. All this, however, 
 deserves a fresh and thorough investigation. 
 
 The theory of Habit is a sort of necessary appendix to that of 
 intermittence; and, like it, due to Bichat. A con- 
 tiniious phenomenon would be, in fact, capable of 
 persistence, in virtue of the law of inertia ; but intermittent })heno- 
 raena alone can give rise to habits, properly so called : that is, can 
 tend to reproduce themselves spontaneously through the influence 
 of a preliminaiy repetition, sufficiently prolonged at suitable inter- 
 vals. The importance of this animal property is now universally
 
 ANIMAL HABIT. 377 
 
 acknowledged among aLlo inquirers, wlio see in it one of llie chief 
 bases of tlie gradual jierfectibility oC animals, and especially of Man. 
 Through this it is that vital phenomena may, in some sort, partici- 
 pate in the admirable regularity of those of the inorganic world, by 
 liecoming, like them, periodical, notwithstanding their greater com- 
 plexity. Thence also results the transformation, — optional uj) to a 
 certain point of inveteracy of habit, and inevitable beyond that 
 jioint, — of voluntary acts into involuntary tendencies. But tlie 
 study of habit is no further advanced than that of intermittence, in 
 regard to its analysis: for we have paid more attention hitherto to 
 the influence of habits once contracted than to their origin, with 
 regard to which scarcely any scientific doctrine exists. What is 
 known lies in the department of natural history, and not in that of 
 biology. Perhaps it may be found, in the course of scientific study, 
 that we have been too hasty in calling this an animal property, 
 though the animal structure may be more susceptible of it. In 
 fact, there is no doubt that inorganic apparatus admits of a more 
 easy reproduction of the same acts after a sufficient regular and 
 prolonged reiteration, as I had occasion to observe in regard to the 
 phenomenon of sound : and this is essentially the character of animal 
 liabit. According to this view, which I commend to the attention 
 of biologists, and which, if true, would constitute the most general 
 I)oint of view on this subject, the law of habit may be scientifically 
 attached to the law of inei-tia, as geometers understand it in the 
 positive theory of motion and equilibrium. 
 
 In examining the phenomena common to irritability and sensi- 
 bility under the aspect of their activity, physiologists have to 
 examine the two extreme terms, — exaggerated action, and insuffi- 
 cient action, in order to determine the intermediate noimal degree: 
 tor the study of intermediate cases can never be successfully undei- 
 laken till the extreme cases which comprehend them have been 
 first examined. 
 
 The need of exercising the faculties is certainly the most general 
 and important of all those that belong to the animal Need of 
 life : we may even say that it comprehends them all, activity. 
 if w^e exclude what relates merely to the organic life. Tlie existence 
 of an animal organ is enough to awaken the need immediately. 
 We shall see, in the next volume, that this considei-ation is one of 
 thechief bases that social physics derives from individual physiology. 
 Uidiappily this study is still very impei-fect with regard to most of 
 the animal functions, and to all the three degrees of their activity. 
 To it we must refer the analysis of all the varied phenomena of 
 pleasure and ])ain, physical and moi'al. The case of defect has been 
 even less studied than that of excess ; and yet its scientific examina- 
 tion is ceitainly not less important, on account of the theory of 
 annul, the consideration of wliich is so prominent in social physics, 
 — not only in connection with an advanced state of civilization, but
 
 378 POSITIVE PHILOSOPHY. 
 
 even in the roughest periods, in which, as we shall see liereafler, 
 ennui is one of the chief moving springs of social evolution. As 
 for the intermediate degree, wiiich characterizes health, welfare, 
 and finally happiness, it cannot he well treated till the extremes are 
 hetter understood. The only positive principle yet estahlished in 
 this part of physiology is that which prescrihes that we should not 
 contemplate this normal degree in an absolute manner, but in sub- 
 ordination to the intrinsic energy of the corresponding faculties ; as 
 popular good sense has already admitted, however difficult it may 
 be practically to conform to the precept in social matters, from the 
 unreflecting tendency of every man to erect himself into a necessary 
 type of the whole species. 
 
 We have now only to notice, further, the third order of considera- 
 tions ; the study of the association of the animal functions. 
 
 This great subject should be divided into two ])arts, relating to 
 As&odaiion of ^'^^ sympatliies, to which Bichat has sufficiently drawn 
 the animal the attention of physiologists, and the synergies, as 
 functions. Barthez has called them, which are at present too 
 much neglected. The difference between these two sorts of vital 
 association corresponds to that between the normal and the patho- 
 logical states : for there is synergy whenever two organs concur 
 simultaneously in the regular accomplishment of any function ; 
 whereas sympathy supposes a certain j)erturbation, momentary or 
 permanent, partial or general, which has to be stopped by the 
 intervention of an organ not primarily affected. These two modes of 
 physiological association are proper to the animal life, any appear- 
 ance to the contrary being due to the influence of animal over 
 organic action. The study is fairly established on a rational basis ; 
 the physiologists of our time seeming to be all agreed as to the 
 nervous system being the necessary agent of all sympathy ; and 
 this is enough for the foundation of a positive theory. Beyond this, 
 we have only disjointed though numerous facts. The study of the 
 synergies, though more simple and better circumscribed, does not 
 ])resent, as yet, a more satisfactory scientific character, either as to 
 the mutual association of the different motions, or as to the differ- 
 ent modes of sensibility ; or as to the more geneial and complex 
 association between the phenomena of sensibility and those of irri- 
 tability. And yet this great subject leads directly to the most 
 important theory that physiology can finally present, — that of the 
 fundamental unity of the animal organism, as a necessary result of 
 a harmony between its various chief functions. Here alone it is 
 that, taking each elementaiy faculty in its normal state, we can find 
 the sound theoiy of the Ego, so absurdly perverted at present by 
 the vain dreams of the meta[)hysicians : for the general sense of the 
 /is certainly determined by the equilibrium of the faculties, the 
 disturbance of which impairs that consciousness so profoundly iu 
 many diseases.
 
 379 
 
 CHAPTER VI. 
 
 INTELLECTUAL AND MORAL, OU CEREBRAL FUNCTIONS. 
 
 The remaining portion of biolopfical philosophy is that which re- 
 hites to the study of the atFective and intellectual iaculties, Avhicli 
 leads us over from individual pliysiology to Social Physics, as vege- 
 tative physiology does from tlie inorganic to the organic philosophy. 
 While Descartes was rendering to the world the glorious service 
 of instituting a complete system of positive philo- shortcominri 
 sophy, the reformer, with all his bold energy, was of Descartes. 
 imable to raise himselt' so far above his age as to give its complete 
 logical extension to his own theory by comprehending in it the part 
 of physiology that relates to intellectual and moral phenomena. 
 After having instituted a vast mechanical hypothesis upon the 
 fundamental theory of the most simple and universal phenomena, 
 he extended in succession tlie same philosophical spirit to the 
 different elementaiy notions relating to the inorganic world ; and 
 finally subordinated to it the study of the chief physical functions 
 of the animal organism. But, when he arrived at the functions of 
 the aflfections and the intellect, he stopped abruptly, and expressly 
 constituted from them a special study, as an appurtenance of the 
 raetaphysico-theological philosophy, 1o which he thus endeavoured 
 to give a kind of uew life, after having wrought far more success- 
 fully in sapping its scientific foundations. We have an unquestion- 
 able evidence of the state of his mind in his celebrated paradox 
 about the intelligence and instincts of animals. He called brutes 
 automata, rather than allow the application of the old philosophy 
 to them. Being unable to pursue this method with Man, he 
 delivered him over expressly to the domain of metaphysics and 
 theology. It is difficult to see how he could have done otherwise, 
 in the then existing stale of knowledge : and we owe to his strange 
 hypothesis, which the physiologists went to work to confute, the 
 clearing away of the ]»artition which he set uf> between the study 
 of animals and that of Man, and consequently, the entire elimination 
 among the higher order of investigators, of theological and meta- 
 physical philosophy. What the first contradictory constitution of 
 the modern philosophy was, we may ^ee in the great nisionitui 
 work of Malebranche, Avho was the chief interpreter Gail's time. 
 of Descartes, and who shows how his philosophy continued to apply
 
 3 So POSITIVE PHILOSOPHY. 
 
 to the most com]jlex parts of the intellectual system the same 
 methods which had heen shown to be necessarily futile with regard 
 to the simplest subjects. It is necessary to indicate this state of 
 things because it has remained essentially unaltered during the last 
 two centuries, notwithstanding the vast progress of positive science, 
 which has all the while been gradually preparing ibr its inevitable 
 transformation. The school of Boerhaave left Descartes's division 
 of subjects as they found it : and if they, the successors of Descartes 
 in physiology, abandoned this department of it to the metaphysical 
 method, it can be no wonder that intellectual and moral phenomena 
 remained, till this century, entirely excluded from the great scien- 
 tific movement originated and guided by the impulse of Descartes. 
 The growing action of the positive spirit has been, during the 
 Avhole succeeding interval, merely critical, — attacking the inefiicacy 
 of metaphysical studies, — exhibiting the perpetual reconciliation of 
 Ihe naturalists on points of genuine doctrine in contrast to the in- 
 cessant disputes of various metaphysicians, arguing still, as from 
 Plato downwards, about the very elements of their pretended 
 science : this criticism itself relating only to results, and still offer- 
 ing no objection to the supremacy of metaphysical philosophy, in 
 the study of Man, in his intellectual and moral aspects. It was 
 not till our own time that modern science, with the illustrious 
 Gall for its organ, drove the old philosophy from this last portion 
 of its domain, and passed on in the inevitable course from the 
 critical to the organic state, striving in its turn to treat in its own 
 way the general theory of the highest vital functions. However 
 imperfect the first attempts, the thing is done. Subjected for half 
 a century to the most decisive tests, this nev/ doctrine has clearly 
 manifested all the indications which can guarantee the indestruct- 
 ible vitality of scientific conceptions. Neither enmity nor irrational 
 advocacy has hindered the continuous spread, in all parts of the 
 scientific world, of the new system of investigation of intellectual 
 and moral man. All the signs of the progressive success of a 
 happy philosophical revolution are present in this case. 
 
 The positive theory of the affective and intellectual functions is 
 p therefore settled, irreversibly, to be this : — it consists 
 
 theory of i'l the exjierimcntal and rational study of the pheno- 
 Cerebrcd nicua of interior sensibility proper to the cerebral 
 June ions, ganglions, apart from all immediate external appar- 
 atus. These phenomena are the most complex and the most special 
 of all belonging to physiology ; and therefore they have naturally 
 been the last to attain to a positive analysis ; to say nothing of their 
 relation to social consideiations, which must be an impediment in 
 the way of their study. This study could not precede the principal 
 lis proper scientific conceptions of the organic life, or the first 
 place. notions of the animal life; so that Gall nnist follow 
 
 Jjichat : and our surprise would be that he followed him so soon,
 
 HISTOR V OF MENTAL PHIL OSOPJI Y. 3 S i 
 
 if the maturity of liis task did not explain it sufiiciently. The 
 j;ronnds of my provisional separation of this part of physiology 
 from the province of animal life generally are — the eminent differ- 
 ences between this order of phenomena and those that have gone 
 before, — their more direct and striking importance, — and, above 
 all, the greater imperfection of our present study of them. This 
 new body of doctrine, thus erected into a third section of physiology, 
 will assume its true place within the boundaries of the second when 
 we obtain a distincter knowledge of oi"ganic, and a more phih»- 
 sophical conception of animal physiology. We nuist bear in mind 
 what the proper arrangement should be, — this third department 
 differing much less from the second than the second differs from 
 the iirst. 
 
 We need not stop to draw out any parallel or contrast between 
 phrenology and psychology. Gall has fully and vices of Psy 
 clearly exposed the powerlessness of metaphysical choioi/icai 
 methods for the study of intellectual and moral s^/^'^em. 
 phenomena : and in the present state of the human mind, all 
 discussion on this subject is superfluous. The great philosophical 
 cause is tried and judged ; and the metaphysicians have passed from 
 a state of domination to one of protestation, — in the learned world 
 at least, where their opposition would obtain no attention but for 
 the inconvenience of tlieir still impeding the ])rogress of popular 
 reason. The triumph of the positive method is so decided that it 
 is needless to devote time and effort to any demonstration, exce[)t 
 in the way of instruction : but, in order to characterize, by a strik- 
 ing contrast, the true general spirit of phrenological physiology, it 
 may be useful here to analyse very briefly the radical vices of the 
 pretended psychological method, considered merely in regard to 
 what it has in common in the principal existing schools ; — in those 
 called the French, the German, and (the least consistent and also 
 the least absurd of the three) the Scotch school : — that is, as far 
 as we can talk of schools in a philosophy which, by its nature, must 
 engender as many incompatible opinions as it has adepts gifted with 
 any degree of imagination. We may, moreover, refer confidently 
 to these sects for the mutual refutation of their most essential 
 points of difference. 
 
 As for their fundamental principle of interior observation, it 
 would certainly be superfluous to add anything to Method 
 what I have already said about the absurdity of the interior 
 supposition of a man seeing himself think. It was Obsen-atwn. 
 well remarked by M. Broussais, on this point, that such a method, 
 if possible, would extremely restrict the .study of the underslamling, 
 by necessarily limitiug it to the case of adult and healthy Man, 
 without any hope of illustrating this difficult doctrine by any com- 
 parison of different ages, or consideration of pathological states, 
 which yet are unanimously recognized as indispensable auxiliaries
 
 382 POSITIVE PHILOSOPHY. 
 
 in the simplest researches about Man. But, fuither, we must be 
 also struck by the absolute interdict which is laid upon all intel- 
 lectual and moral study of animals, from whom the psychologista 
 can hardly be expecting any interior observation. It seems rather 
 strange that the philosophers who have so attenuated this immense 
 subject should be those who are for ever reproaching their adver- 
 saries with a want of comprehensiveness and elevation. The case 
 of animals is the rock on which all psychological theories have 
 split, since the naturalists have compelled the metaphysicians to 
 part with the singular expedient imagined by Descartes, and to 
 admit that animals, in the higher parts of the scale at least, mani- 
 fest most of our affective, and even intellectual faculties, with mere 
 differences of degree ; a fact which no one at this day ventures to 
 deny, and which is enough of itself to demonstrate the absurdity of 
 these idle conceptions. 
 
 Eecurring to the first ideas of philosophical common sense, it is 
 at once evident that no function can be studied but with relation 
 to the organ that fulfils it, or to the phenomena of its fulfilment : 
 and, in the second place, that the affective functions, and yet more 
 the intellectual, exhibit in the latter respect this particular charac- 
 teristic, — that they cannot be observed during their operation, but 
 only in their results, — more or less immediate, and more or less 
 durable. There are then only two ways of studying such an order 
 of functions ; either determining, with all attainable precision, the 
 various organic conditions on which they depend, — which is the 
 chief object of phrenological physiology; or in directly observing 
 the series of intellectual and moral acts, — which belongs rather to 
 natural history, properly so called : these two inseparable aspects of 
 one subject being always so conceived as to throw light on each 
 other. Thus regarded, this great study is seen to be indissolubly 
 connected on the one hand with the whole of the foregoing parts of 
 natural philosophy, and especially with the fundamental doctrines 
 of biology ; and, on the other hand, with the whole of history, — of 
 animals as well as of man and of humanity. But when, by the 
 pretended psychological method, the consideration of both the agent 
 and the act is discarded altogether, what material can remain but 
 an unintelligible conflict of words, in which merely nominal entities 
 are substituted for real phenomena? The most difficult study of all 
 is thus set up in a state of isolation, without any one point of sup- 
 j)ort in the most simple and perfect sciences, over which it is yet 
 proposed to give it a majestic sovereignty : and in this all psycholo- 
 gists agree, however extreme mny be their dilferences on other 
 points. 
 
 About the method of psychology or ideology, enough has been 
 said. As to the doctrine, the first glance shows a radical fault in 
 it, common to all sects, — a false estimate of the general relations 
 between the affective and the intellectual faculties. However vari-
 
 FORMER DOCTRINE. 383 
 
 (^ns may lie the theories about the preponderance of the latter, all 
 metaphysicians assert that pre])ond('rance by tnak- Doctrine 
 iiig these faculties their start iiii:;-p()int. The intel- Relation be- 
 lect is almost exclusively the subject of their specn- jfccUveayid 
 lations, and the affections liave been almost entirely intellectual 
 neixlected ; and, moreover, always suboidinated to the Muities. 
 understandin<2^. Now, such a conception represents precisely the 
 reverse of the reality, not only for animals, but also for Man: for 
 daily experience shows that the atfections, the pi'opensities, the 
 passions, are the <z;reat springs of human life ; and that, so far from 
 resulting from intelligence, their spontaneous and independent 
 hnpulse is indispensable to the first awakening and continuous 
 development of the various intellectual faculties, by assigning to 
 them a permanent end, without which — to say nothing of the vague- 
 ness of their general direction — they would remain dormant in the 
 in;ijority of men. It is even but too certain that the least noble 
 and most animal propensities are habitually the most energetic, and 
 therefore the most influential. The whole of human nature is thus 
 very unfaithfully represented by these futile systems, which, if 
 noticing the affective faculties at all, have vaguely connected them 
 Avith one single principle, sympathy, and, above all, self-conscious- 
 ness, always supposed to be directed by the intellect. Thus it is 
 that, contrary to evidence, Man has been represented as essentially 
 a reasoning being, continually carrying on, unconsciously, a multi- 
 tude of imperceptible calculations, with scarcely any spontaneity of 
 action, from infancy upwards. This false conception has doubtless 
 been supported by a consideration worthy of all respect, — that it is 
 by the intellect that Man is modified and improved ; but science 
 requires, before all things, the reality of any views, independently 
 of their desirableness; and it is always this reality which is the 
 basis of genuine utility. Without denying the secondary influence 
 of such a view, we can show that two purely philosophical causes, 
 quite unconnected with any idea of application, and inherent in the 
 nature of the method, have led the metaphysicians of all sects to 
 this hypothesis of the supremacy of the intellect. The first is the 
 radical separation which it was thought necessary to make between 
 brutes and man, and which would have been effaced Brutes and 
 at once by the admission of the preponderance of the ■''^«"- 
 affective over the intellectual faculties ; and the second was the 
 necessity that the metaphysicians found themselves under, of pre- 
 serving the unity of what they called the /, that it might corre- 
 spond with the unity of the soul, in obedience to the requisitions of 
 the theological philosoph3% of which metaphysics is, 
 as we must ever bear in mind, the final traTisformation. *'^'"^ 
 But the positive philosophers, who approach the question with the 
 simple aim of ascertaining the true state of things, and reproducing 
 it with all possible accuracy in their theories, have perceived that,
 
 384 POSITIVE PHILOSOPHY. 
 
 according to universal experience, human natiiie is so far from 
 being single that it is eminently multiple; tiiat is, usually induced 
 in various directions by distinct and independent powders, among 
 which equilibrium is established with extreme difficulty when, as 
 usually happens in civilized life, no one of them is, in itself, suffici- 
 ently marked to acquire spontaneously any considerable })reponder- 
 ance over the rest. Thus, tlie famous theory of the I is essentially 
 without a scientific object, since it is destined to represent a purely 
 fictitious state. Theie is, in this direction, as I have ah-eady pointed 
 out, no other real subject of positive investigation than the study 
 of the equilibrium of the various animal functions, — both of irrita- 
 bility and of sensibility, — which marks the normal state, in which 
 each of them, duly moderated, is regularly and permanently 
 associated with the whole of the others, according to the laws of 
 sympathy, and yet more of synergy. The very abstract and indirect 
 notion of the / proceeds from the continuous sense of such a 
 harmony ; that is, from the universal accordance of the entire 
 organism. Psychologists have attempted in vain to make out of 
 this idea, or rather sense, an attribute of humanity exclusively. It 
 is evidently a necessary result of all animal life ; and therefore it 
 must belong to all animals, whether they are able to discourse upon 
 it or not. No doubt a cat, or any other vertebrated animal, with- 
 out knowing how to say " I," is not in the habit of taking itself for 
 another. Moreover, it is probable that among the superior animals 
 the sense of personality is still more marked than in Man, on account 
 of their more isolated life; though if we descended too far in the 
 zoological scale we should reach or<ranisms in which the continuous 
 degradation of the nervous system attenuates this compound sense, 
 together with the various simple feelings on which it depends. 
 It must not be overlooked that though the psychologists have 
 Reason and agreed in neglectingthe intellectual and moral faculties 
 Instinct. of brutes, which have been happily left to the natu- 
 
 ralists, they have occasioned great mischief by their obscure and 
 indefinite distinction between intelligence and instinct, thus setting 
 up a division between human and animal nature which has had too 
 nuich eftect even upon zoologists to this day. The only meaning 
 tliat can be attributed to the word instinct, is any spontaneous 
 impulse in a determinate direction, inde[)endently of any foreign 
 influence. In this primitive sense, the term evidently apj)lies to 
 the pro})er and direct activity of any faculty whatever, intellectual 
 as well as affective; and it therefore does not conflict with the term 
 intelligence in any way, as we so often see when we speak of those 
 Avho, without any education, manifest a marked talent for music, 
 painting, mathematics, etc. In this way there is instinct, or 
 rather, there are instincts in Man, as nuich or more than in brutes. 
 If, on the other hand, we describe intelligence as the aptitude to 
 modify conduct in conformity to the circumstances of each case,—
 
 J^EASON AND INSTINCT. 385 
 
 wliicli, in fact, is the main practical athibiite of reason, in its 
 proper sense, — it is more evident than before that there is no other 
 essential dilFerence between huiuanit}' and aniniality tlian that of 
 the degree of development admitted by a facnlty Avhich is, by its 
 nature, common to all animal life, and without which it could not 
 even be conceived to exist. Thus the famous scholastic definitiou 
 of" Man as a reasonahle animal offers a real no-meanin<^, since no 
 animal, especially in the higher parts of the zoological scale, could 
 live without being to a certain extent reasonable, in proportion 
 to the complexity of its organism. . Tliough the moral nature of 
 animals has been but little and very imperfectly explored, we can 
 yet perceive, without [)Ossibility of mistake, among those that live 
 with us and that are familiar with us, — ^judging of them by the 
 same means of observation that we should employ about men whose 
 language and ways were previously unknown to us, — that they not 
 only apply their intelligence to the satisfaction of their organic 
 wants, much as men do, aiding themselves also with some sort of 
 language ; but that they are, in like manner, susceptible of a kind 
 of wants more disinterested, inasmuch as they consist in a need to 
 exercise their faculties for the mere pleasure of the exercise. It is 
 the same thing that leads children or savages to invent new sports, 
 and that renders them, at the same time, liable to ennui. That 
 state, erroneously set up as a special privilege of human nature, is 
 sometimes sufficiently marked, in the case of certain animals, to 
 urge them to suicide, when captivity has become intolerable. An 
 attentive examination of the facts therefore discredits the perversion 
 of the word instinct when it is used to signify the fatality under 
 Avhich animals are impelled to the mechanical performance of acts 
 uniformly determinate, without any possible modification from cor- 
 responding circumstances, and neither requiring nor allowing any 
 education, properly so called. This gratuitous supposition is evi- 
 dently a remnant of the automatic hypothesis of Descartes. Leroy 
 has demonstrated that among mammifers and birds this ideal fixity 
 in the construction of habitations, in the seeking of food by hunting, 
 in the mode of migration, etc., exists only in the eyes of closet- 
 naturalists or inattentive observers. 
 
 After thus much notice of the radical vice of all psychological 
 systems, it would be departing from the object of this work to show 
 how the intellectual faculties themselves have been misconceived. 
 It is enough to refer to the refutation by which Gall and Si)urzheini 
 have introduced their labours : and I would particularly point out 
 the philosophical demonstration by which tiiey have exhibited the 
 conclusion that sensation, memory, imagination, and even judgment, 
 — all the scholastic faculties, in sliort, — are not, in fact, fundamental 
 and abstract faculties, but only different degrees or consecutive 
 modes of the same phenomenon, })roper to each of the true elementary 
 phrenological functions, and necessarily variable in different cases, 
 
 VOL. 1. 2 B
 
 386 POSITIVE PHILOSOPHY. 
 
 with a proportionate activity. One virtue of this admirable analyila 
 is that it depi'ives the various metaphysical theories of their one 
 remaining credit, — their mutual criticism, which is here effected, 
 once for all, with more efficacy than by any one of the mutually 
 opposing schools. 
 
 Again, it would be departing from the object of this portion of 
 our work to judge of the doctrines of the schools by their results. 
 What these have been we shall see in the next volume; the deplor- 
 able influence on the political and social condition of two generations 
 of the doctrines of the French school, as presented by Helvetius, and 
 of the German psychology, with the ungovernable I for its subject ; 
 and the impotence of the Scotch school, through the vagueness of 
 Avhat it called its doctrines, and their want of mutual connection. 
 Dismissing all these for the present, we must examine the great 
 attempt of Grail, in order to see what is wanting in phrenological 
 philosophy to form it into the scientific constitution which is proper 
 to it, and from which it is necessarily still more remote than 
 organic, and even animal physiology. 
 
 Two philosophical principles, now admitted to be indisputable, 
 Basis of G all's Serve as the immovable basis of Gall's doctrine as a 
 doctrine. whole : viz., the innateness of the fundamental disposi- 
 
 tions, affective and intellectual, and tlie plurality of the distinct and. 
 independent faculties, though real acts usually require their more 
 or less complex concurrence. Within the limits of the human race, 
 all cases of marked talents or character prove the first ; and the 
 second is proved by the diversity of such marked cases, and by most 
 pathological states, — especially by those in which the nervous system 
 is directly affected. A comparative observation of the higher 
 animals would dispel all doubt, if any existed in either case. These 
 two principles, — aspects of a single fundamental conception, — are 
 but the scientific expression of the results of experience, in all times 
 and places, as to the intellectual and moral nature of Man, — an 
 indispensable symptom of truth, with regard to all parent ideas, 
 which must always be connected with the spontaneous indications 
 of popular reason, as we have seen in preceding cases in natural 
 philosophy. Thus, besides all guidance from analogy, after the 
 study of the animal life, we derive confirmation from all the methods 
 of investigation that physiology admits ; from direct observation, 
 experiment, pathological analysis, the comparative method and 
 popular good sense, — all of which converge towards the establish- 
 ment of this double principle. Such a collection of proofs secures 
 the stability of this much of phrenological doctrine, whatever trans- 
 formations other parts may have to undeigo. In the anatomical 
 view, this physiological conception corresponds with the division of 
 the brain into a certain number of partial organs, symmetrical like 
 those of the animal life, and, though more contiguous and mutually 
 resembling than in any other system, and therefore more adapted
 
 DISTRIBUTIOi^/ OF THE BRAIN. 38 7 
 
 hoth for sympathy and syiier<;y, still distinct and mutually inde- 
 pendent, as we were already aware was the case with the ganglions 
 appropriate to the external senses. In brief, the bi'ain is no longer 
 an organ, but an apparatus of organs, more coiuplex in proj)ortion 
 to the degree of animality. The proper object of phrenological 
 physiology thence consists in determining the cerebral organ appro- 
 priate to each clearly marked, simple disposition, affective or intel- 
 lectual ; or, reciprocally, which is more difficult, what function is 
 ftdfilled by any portion of the mass of the brain which exhibits 
 the anatomical conditions of a distinct organ. The two processes 
 are directed to develop the agreement between physiological and 
 anatomical analysis which constitutes the true science of living 
 beings. Unfortunately, our means are yet further from answering 
 our aims than in the two preceding divisions of the science. 
 
 The scientific principle involved in the phrenological view is that 
 the functions, affective and intellectual, are more Divisions of 
 elevated, more human, if you will, and at the same thehrain. 
 time less energetic, in proportion to the exclusiveness with which 
 they belong to the higher part of the zoological series, their positions 
 being in portions of the brain more and more restricted in extent, 
 and further removed from its immediate origin, — according to the 
 anatomical decision that the skull is simply a prolongation of the 
 vertebral column, which is the primitive centre of the entire nervous 
 system. Thus, the least developed and anterior j)art of the brain 
 is appropriated to the characteristic faculties of humanity ; and the 
 most voluminous and hindmost part to those which constitute the 
 basis of the whole of the animal kingdom. Here we have a new 
 and confirmatory instance of the rule which we have had to follow 
 in every science ; that it is necessary to proceed from the most 
 general to the more special attributes, in the order of their dimin- 
 ishing generality. We shall meet with it again in the one science 
 which remains for us to review ; and its constant presence, through 
 the whole range, points it out as the first law of the dogmatic pro- 
 cedure of the positive spirit. 
 
 A full contemplation of Gall's doctrine convinces us of its faith- 
 ful representation of the intellectual and moral nature of Man and 
 animals. All the psychological sects have misconceived or ignored 
 the pre-eminence of the affective faculties, plainly manifest as it is 
 in all the moral phenomena of brutes, and even of Man ; but we 
 find this fact placed on a scientific basis by the discovery that the 
 affective organs occupy all the liinder and middle portion of the 
 cerebral apparatus, while the intellectual occupy only the front 
 portion, whicii, in extreme cases, is not more than a fourth, or even 
 a sixth part of the whole. The difference between Gall and his 
 predecessors was not in the separation of tlie two kinds of faculties, 
 but that they assigned the brain to the intellectual faculties alone, 
 regarding it as a single organ, and distributing the passions among
 
 3SS POSITIVE PHILOSOriiY. 
 
 the or<;;ins pertaining to the vegetative life, — the heart, the liver, 
 etc. Bichat supported this view by the argument of the sympathies 
 of these organs, under the excitement of the respective passions ; 
 but the variableness of the seat of sympathy, according to native 
 susceptibility or to accident, is a sufficient answer to such a plea, 
 and teaches us simply the importance of considering the influence 
 exercised by the state of the brain upon the nerves which supply the 
 apparatus of the organic life. 
 
 Next comes the subdivision established by Gall and Spm-zheim 
 . in each of these two orders. The affective faculties 
 are divided into the ])ropensities, and the affections 
 or sentiments : the first residing in the hindmost and lowest part 
 of the brain ; and the other class in the middle ])ortion. The in- 
 tellectual faculties are divided into the various perceptive faculties, 
 which together constitute the range of observation : and the small 
 number of reflective faculties, the highest of all, constituting the 
 power of combination, by comparison and co-ordination. The upper 
 part of the frontal region is the seat of these last, which are the 
 chief characteristic attribute of human nature. There is a certain 
 deficiency of precision in this description ; but, besides that we 
 may expect improving knowledge to clear it up, we shall find, on 
 close examination, that the inconvenience lies more in the language 
 than in the ideas. The only language we have is derived from a 
 philosophical period when all moral and even intellectual ideas 
 were shrouded in a mysterious metaphysical unity, which allows us 
 now no adequate choice of terms. 
 
 Taking the ordinary terms in their literal sense, we should mis- 
 conceive the fundamental distinction between the intellectual 
 faculties and the others. When the former are very marked, they 
 unquestionably produce real inclinations or propensities, which are 
 distinguished from the inferior passions only by their smaller energy. 
 Nor can we deny that their action occasions true emotions or 
 sentiments, more rare, more pure, more sublime than any other, 
 and, though less vivid than others, ca[)able of moving to tears ; as 
 is testified by so many instances of the rapture excited by the dis- 
 covery of truth, in the most eminent thinkers that have done honour 
 to their race — as Archimedes, Descartes, Kepler, Newton, etc. 
 Would any thoughtful student take occasion, by such approxima- 
 tions, to deny all real distinction between the intellectual and 
 affective faculties? The wiser conclusion to be drawn from the 
 case is that we nnist reform oin- philoso[)hical langunge, to raise it, 
 by rigorous precision, to the dignity of scientific language. We 
 may say as much about the subdivision of the aHective faculties 
 into propensities and sentiments, the distinction being, though less 
 marked, by no means less real. Apart from all useless discussion 
 of nomenclature, we may say that the real difference has not been 
 clearly seized. In a scientific view, it would suffice to say that the
 
 OBJECTIONS. 389 
 
 first and fundamental class relates to tlie individual alone, or, at. 
 most, to the family, regarded successively in its principal needs of 
 preservation, — such as rejiroduction, the rearing of young, the mode 
 of alimentation, of habitation, etc. Whereas, the second more 
 special class supposes the existence of some social relations, either 
 among individuals of a different species, or especially between 
 individuals of the same species, apart from sex, and determines the 
 character whicii the tendencies of the animal must impress on 
 each of these relations, whether transient or permanent. If we 
 keep this distinctive character of .the two classes in view, it will 
 matter little what terms we use to indicate them, when once they 
 shall have acquired a sufficient fixedness, through rational use. 
 
 These are ihii great philosophical results of Gall's doctrine, 
 regarded, as I hav(; now pi-esented it, apart from all vain attempts 
 to localize in a special manner the cerebral or phrenological func- 
 tions. I shall have to show how such an attempt was imi)0sed 
 upon Gall by the necessities of his glorious mission : but, notwith- 
 standing this unfortunate necessity, the doctrine embodies already 
 a real knowledge of human and brute nature very far superior to 
 all that had ever been offeied before. 
 
 Among the innumerable objections which have been aimed at 
 Ibis line doctrine, — considered always as a whole, — • ohjecUons. 
 the only one which merits discussion here is the Necessity of 
 supposed necessity of human actions. This objection '""«a« «c/u,«.?. 
 is not only of high importance in itself, but it casts new light back 
 upon the sjiirit of the theory ; and we must briefly examine it from 
 tiie point of view of positive philosophy. 
 
 When objectors confound the subjection of events to invariable 
 laws with tiieir necessary exemption from modification, they lose 
 sight of the fact that phenomena become susceptible of modification 
 in j)roportion to their complexity. The only irresistible action that 
 we know of is that of weight, which takes place under ^ , 
 
 the most general and simple of all natural laws. But 
 the phenomena of life and acts of the mind are so highly complex 
 as to admit of modification beyond all estimate ; and, in the inter- 
 mediate regions, phenomena are under control precisely in the order 
 of their complexity. Gall and Spurzheim have shown how human 
 action depends on the combined operation of several faculties ; how 
 exercise develops them ; how inactivity wastes them ; and how the 
 intellectual faculties, adapted to modify the general conduct of the 
 animal according to the variable exigencies of his situation, may 
 overrule the practical influence of all his other faculties. It is only 
 in mania, when disease interferes with the natural action of the 
 faculties, that fatality, or what is popularly called irresponsibility, 
 exists. It is therefore a great mistake to accuse cerebral physiology 
 of disowning the influence of education or legislation, because it 
 fixes the limits of their power. It denies the possibility^ asserted
 
 390 POSITIVE PHIIOSOPIIY. 
 
 Ly the ideology of the French school, of converting by suitahia 
 arrangements, all men into so many Socrates, Homci-s, or Archi- 
 medes ; and it denies the ungovernable energy of the /, asserted by 
 the German school ; but it does not therefore affect IMan's reason- 
 able liberty, or interfere with his improvement by the aid of a wise 
 education. It is evident indeed that improvement by education 
 supposes the existence of requisite predispositions: and that each 
 of them is subject to determinate laws, without which they could 
 not be systematically influenced ; so that it is, after all, cerebral 
 physiology that is in possession of the philosophical problem of 
 education. Furthermore, this physiology shows us that men arc 
 commonly of an average constitution ; that is, that, apart from a 
 very few exceptional organizations, every one possesses in a moderate 
 degree all the propensities, all the sentiments, and all the elementary 
 aptitudes, without any one faculty being remarkably preponderant. 
 The widest field is thus open for education, in modifying in almost 
 any direction organisms so flexible, though the degree of their 
 development may remain of that average amount which consists 
 very well with social harmony ; as we shall have occasion to sec 
 hereaftei-. 
 
 A much more serious objection to Gall's doctrine arises out of 
 Hvpotheticai ^^^^ veuturesouie and largely erroneous localization 
 diittrihution of the faculties which he thought proper to propose. 
 of faculties. jf ^,g ]qq|, ^^ ]-^j^ position, we shall see that he merely 
 used the right, common to all natural philosophers, of instituting a 
 scientific hypothesis, in accordance with the theory on that subject 
 which we examined in connection with Physics. He fulfilled the 
 conditions of this theory ; his subject being, not any imaginary 
 fluids, ethers or the like, but tangible organs, whose hypothetical 
 attributes admit of positive verifications. Moreover, none of those 
 who have criticized his localization could have proposed any less 
 imperfect, or, probably, so well indicated. The advice of prudent 
 mediocrity, to abstain from hy[)othesis, is very easy to offer ; but 
 if the advice was followed, nothing would ever be done in the way 
 of scientific discovery. It is doubtless inconvenient to have to 
 withdraw or remake, at a subsequent period, the hypotheses to 
 which a science owes its existence, and which, by that time, have 
 been adopted by inferior inquirers with a blinder and stronger faith 
 than that of the original proposers: but there is no use in dwelling 
 upon ;i liability which arises from the infirmity of our intelligence. 
 The practical point for the future is that strong minds, prepared by 
 a suitable scientitic education, should plant themselves on the two 
 great principles which have been laid down as the foundation of 
 the science, and thence explore the principal needs of cerebral 
 physiology, and the character of the means by which it may be 
 cariied forwards. Nor need tliei'e be any fear that the science 
 will be held back by such a metluxl. Nothing prevents us, when
 
 REQUISITE IMPROVEMENTS. 391 
 
 reasoning, as geometers do, upon indeterruinale seals, or positions 
 Kupj)Osed to be indeterminate, from arriving at real conclusions, 
 involving actual utilit}^ as I hope to show, from my own experience, 
 in the next volume ; though it is evident that it will be a great 
 advantage to the exactness and efficacy of our conclusions, wlien- 
 ever the time arrives for the positive determination of the cerebral 
 organs. Meantime it is clear that we owe to Gall's hypothetical 
 localization our view of the necessity of such a course; and that if 
 he had confined himself to the high philosophical generalities with 
 which he has furnished us, he would never have constituted a 
 science, nor formed a school ; and the truths which we see to be 
 inestiu:iable would have been strangled in their birth by a coalition 
 of hostile influences. 
 
 We see what is the philosophical character of cerebral physiology. 
 We must next inquire what are the indispensable Needed im- 
 improvements that it demands. provements. 
 
 First, we want a fundamental rectification of all the organs and 
 fjiculties, as a necessary basis for all further progress. Taking an 
 anatomical view of this matter, Ave see that the distribution of 
 organs has been directed by physiological analyses alone, — usually 
 imperfect and superficial enough, — instead of being Anatomical 
 subjected to anatomical determinations. This has ^«^*^- 
 entitled all anatomists to treat such a distribution as arbitrary and 
 loose, because, being subject to no anatomical consideration about 
 the difference between an organ and a part of an organ, it admits of 
 indefinite subdivisions, which each phrenologist seems to be able 
 to multiply at will. Though the analysis of functions no doubt 
 casts much light on that of organs, the original decomposition of 
 the whole organism into systems of organs, and those again into 
 single organs, is not the less independent of physiological analysis, 
 to which, on the contrary, it must furnish a basis. This is estab- 
 lished in regard to all other biological studies ; and there is no reason 
 why cerebral inquiries should be an exception. We do not need 
 to see the digestive, or llie respiratory ap})aratus in action, before 
 anatomy can distinguish them from each other : and why should it 
 be otherwise with the cerebral apparatus ? The anatomical diffi- 
 culties are no doubt much greater, on account of the resemblance 
 and proximity of the organs in the cerebral case : but we must not 
 give up this indispensable analysis for such a reason as that. If it 
 were so, we must despair of conferring a special scientific character 
 on phrenological doctrine at all ; and we must abide by those 
 generalities alone which I have just laid down. When we propose 
 to develop the harmony between the anatomical and the physio- 
 logical analysis of any case, it is supposed that each has been 
 neparately established, and not that the one can be copied from the 
 other. Nothing therefore can absolve the phrenologists from the 
 obligation to pursue the analysis of the cerebral system by a
 
 392 . POSITIVE PHILOSOPIIY. 
 
 series of vigorous anatomical labours, diseaiding for tlie time all 
 ideas of function, or, at most, employing them only as auxiliary to 
 anatomical exploration. Such a consideration will be most ear- 
 nestly supported by those phrenologists who perceive that, in 
 determining the relative preponderance of each cerebral organ in 
 difierent subjects, it is not only the bulk and weight of the organ 
 that has to be taken into the account, but also its degree of activity, 
 anatomically estimated by, for instance, the energy of its partial 
 circulation. 
 
 Next, following a distinct but parallel order of ideas, there must 
 Phonological ^^ a purely physiological analysis of the various 
 analysis of elementary faculties ; and m this analysis, which has to 
 faculties. |jg harmonized with the other, every anatomical idea 
 
 must be, in its turn discarded. The position of phrenology is 
 scarcely more satisfactory in this view than any other, for the dis- 
 tinction between the dififerent faculties, intellectual and even affective, 
 and their enumeration are conceived of in a very superficial way 
 though incomparably more in the positive spirit than any meta- 
 jihysical analyses. If metaphysicians have confounded all their 
 psychological notions in an absurd unity, it is probable that the 
 phrenologists have gone to the other extreme in multiplying ele- 
 mentary functions. Gall set up twenty-seven ; which was, no 
 doubt, an exaggeration to begin with. Spurzheim raised the 
 number to thirty-five ; and it is liable to daily increase for want of 
 a rational })rinciple of circumscription for the regulation of the easy 
 enthusiasm of popular explorers. Unless a sound philosophy inter- 
 poses, to establish some order, we may have as many faculties and 
 organs as the psychologists of old made entities. However great 
 may be the diversity of animal natures, or even of human types, it 
 is yet to be conceived (as real acts usually suppose the concurrence 
 of several fundamental faculties) that even a greater multiplicity 
 might be represented by a very small number of elementary func- 
 tions of the two orders. If, for instance, the whole number were 
 reduced to twelve or fifteen well-marked faculties, their combinations, 
 binary, ternary, quaternary, etc., would doubtless correspond to 
 many more types than can exist, even if we restricted ourselves to 
 distinguishing, in relation to the normal degree of activity of each 
 function, two other degrees, — one higher and the other lower. But 
 tiie exorbitant multiplication of faculties is not in itself so shocking 
 as the levity of most of the pretended analyses which have regulated 
 their distribution. In the intellectual order, especially, the apti- 
 tudes have been usually ill-described, apart from the organs : as 
 when a mathematical aptitude is assigned on grounds which would 
 justify our assigning a chemical aptitude, or an anatomical aptitude, 
 if the whole bony casket had not been previously ])arcelled off into 
 irremovable compartments. If a man could do sums according to 
 rules quickly and easily, he had the mathematical aptitude, accord-
 
 REQUISITE IMPROVEMENTS. 393 
 
 iiig to those who do not suspect that roalheinatical speculations 
 require any superiority of intellect. Thougli the analysis of the 
 adective faculties, which are so much better marked, is less iraper- 
 lect, there are several instances of needless multiplication in that 
 department. 
 
 To rectify or improve this analysis of the cerebral faculties, it 
 would be useful to add to the observation of Man and Examination 
 society a physiological estimate of the most marked ofkisioncai 
 individual cases, — especially in past times. The '^"^'^^• 
 intellectual order, wliich most needs revision, is that which best 
 admit of this procedure. If, for instance, it had been applied to 
 the cases of the chief geometers, the absurd mistake that I have 
 just pointed out could not have been committed ; for it would have 
 been seen what compass and variety of faculties are lequired to 
 constitute mathematical genius, and how various are tlie forms in 
 which that genius manifests itself. One great geometer has shone 
 by the sagacity of his inventions ; another by the strength and 
 extent of his combinations ; a third by the luippy choice of his 
 notations, and the perfection of his algebraic style, etc. We might 
 discover, or at least verify, all the real fundamental intellectual 
 faculties by the scientific class alone. In an inferior degree it would 
 be the same with an analogous study of the most eminent artists. 
 This consideration, in its utmost extent, is connected with the 
 utility of the philosophical study of the sciences, under the histori- 
 cal as well as the dogmatical point of view, for the discovery of the 
 logical laws concerned : the difference being that in this last case, 
 we have first to determine the elementary faculties, and not the 
 laws of their action : but the grounds must be essentially ana- 
 logous. 
 
 Phrenological analysis has, then, to be reconstituted ; first in the 
 anatomical, and then in the physiological order ; and finally, the 
 two must be harmonized ; and not till then can phrenological 
 ])hysiology be established upon its true scientific basis. 8uch a 
 procedure is fairly begim, as we have seen, with regard to the two 
 ])ieceding divisions of our science ; but it is not yet even conceived 
 of in relation to cerebral physiology, from its greater complexity 
 and more recent positivity. 
 
 The phrenologists must make a much more extensive use than 
 hitherto of the means fuinished by biological ])hilos- pathological 
 ophy for the advancement of all studies relating to and Cvmpara- 
 living bodies : that is, of pathological, and yet more t^'^^e analysis. 
 of com])arative analysis. The luminous maxim of M. Broussais, 
 which lies at the foundation of medical philosophy, — that the 
 ])henomena of the pathological state are a simple prolongation of 
 the phenomena of the normal state, beyond the ordinary limits of 
 variation, — has never been duly apj)lied to intellectual and moial 
 phenomena: yet it is impossible to understand anything of the
 
 394 POSITIVE rniLOSOPIlY. 
 
 differenl, kinds of madness, if they are not examined on tliis prin- 
 ciple. Here, as in a former division of the science, we see that the 
 study of malady is the way to understand the healthy state. No- 
 thing can aid us so well in the discovery of the fundamental faculties 
 as a judicious study of the state of madness, when each faculty 
 manifests itself in a degree of exaltation which separates it dis- 
 tinctly from others. There has heen plentiful study of monomania ; 
 but it has been of little use, for want of a due connection and com- 
 parison with the normal state. The works that have appeared on 
 the subject have been more literary than scientific; those who have 
 had the best opportunity for observation have been more engaged in 
 governing their patients than in analysing their cases ; and the 
 successors of Pinel hav^e added nothing essential to the ameliorations 
 introduced by him, half a century ago, in regard to the theory and 
 treatment of mental alienation. As for the study of animals, its 
 use has been vitiated by the old notions of the difference between 
 instinct and intelligence. Humanity and animality ought recipro- 
 cally to cast light upon each other. If the whole set of faculties 
 constitutes the complement of animal life, it must surely be that 
 all that are fundamental must be common to all the superior 
 animals, in some degree or other : and differences of intensity are 
 enough to account for the existing diversities, — the association of 
 the laculties being taken into the account on the one hand, and on 
 the other the improvement of Man in society being set aside. If 
 there are any faculties which belong to Man exclusively, it can only 
 be such as correspond to the highest intellectual aptitudes: and 
 this much may appear doubtful if we compare, in an unprejudiced 
 way, the actions of the highest maramifers with those of the least 
 developed savages. It seems to me more rational to suppose that 
 power of observation and even of combination exists in animals, 
 though in an immeasurably inferior degree ; — the want of exercise, 
 resulting chiefly from tiieir state of isolation, tending to benumb 
 and even starve the organs. Much might be learned from a 
 study of domestic animals, though they are far from being the 
 most intelligent. Much might be learned by comparing their 
 moral nature now with what it was at periods nearer to their 
 first domestication ; for it would be strange if the changes that they 
 liave undergone in so many physical respects had been unaccom- 
 panied by variations in the functions which more easily than any 
 others admit of modificaiion. The extreme imperfection of phreno- 
 logical science is manifest in the pride with which Man, from the 
 height of his supremacy, judges of animals as a despot judges of his 
 subjects ; that is, in the mass, without perceiving any inequality 
 in them worth noticing. It is not the less certain that, surveying 
 the whole animal hierarchy, the principal orders of this hierarchy 
 sometimes differ more from each other, in intellectual and moral 
 respects, than the highest of them vaiy from the human type. The
 
 SUBJECTS FOR INQUIRY. 395 
 
 rational study of the niiiid and tlie ways of animals has still to be 
 instituted, — nothing- having yet been done but in the way of pre- 
 paration. It promises an ainple harvest of important discovery 
 directly apjilicable to the advancement of the study of Man, if only 
 the naturalists will disregard the declamation of theologians and 
 metaphysicians about their pretended degradation of human nature, 
 while they are, on the contrary, rectifying the fundamental notion 
 of it by establishing, rigorously and finally, the profound differences 
 which positively separate us from the animals nearest to us in the 
 scale. 
 
 The two laws of action, — intermission and association, — require 
 much more attention than they have yet received in 
 
 , • -.1 1 1 1 '• 1 n^\ 1 c Laxijs of action. 
 
 connection with cerebral physiology. The law of 
 intermittence is eminently applicable to the functions of the brain, 
 — the symmetry of the organs being borne in mind. But this 
 ^reat subject requires a new examination, seeing that it is requisite 
 for science to reconcile their evident intermittence 
 with the perfect continuity that seems to be involved r,£i!Sv. 
 in the connection which mutually unites all our 
 intellectual operations, from earliest infancy to extreme decrepitude, 
 and which cannot be interrupted by the deepest cerebral perturba- 
 tions, provided they are transient. This question, for which meta- 
 physical theories allowed no place, certainly offers serious diffi- 
 culties : but its positive solution must throw great light upon the 
 general course of intellectual acts. As for the association of the 
 faculties, in sympathy or synergy, the physiologists . . 
 
 begin to understand its high importance, though its 
 general laws have not yet been scientifically studied. Without this 
 consideration, the number of propensities, sentiments, or aptitudes 
 Avould seem to be susceptible of any degree of multiplication. For 
 one instance, investigators of human nature have been wont to 
 distinguish various kinds of courage, under the names of civil, 
 military, etc., though the original disposition to brave any kind of 
 danger must always be uniform, but more or less directed by the 
 understanding. No doubt, the martyr who endures the most 
 horrible tortures witli unshaken fortitude rather than deny his con- 
 victions, and the man of science who undertakes a perilous experi- 
 ment after having calculated the chances, might fly in the field of 
 battle if compelled to fight for a cause in which they felt no inter- 
 est ; but not the less is their kind of courage the same as that of 
 the brave soldier. Apart from inequalities of degree, there is no 
 other difference than the superior influence of the intellectual 
 faculties. Without the diverse cerebral synergies, either between 
 the two great orders of faculties, or between the ditferent functions 
 of each order, it would be impossible to analyse the greater pro- 
 portion of mental actions ; and it is in the positive interpretation 
 of each of them by such association that the ap[)]icatiou of phreno-
 
 396 rosnivE niiLOSoriiY. 
 
 logical doctrine will cliiefly consist, wlien such doctrine shall have 
 
 been scientifically erected. When tlie elementary analysis shall 
 
 have been instituted, allowing us to pass on to the study of these 
 
 compound phenomena, we may think of proceeding to the more 
 
 delicate inquiry whether, in each cerebral organ, a distinct part is 
 
 not especially appropriate to the establishment of these synergies 
 
 and sympathies. Some pathological observations have given rise 
 
 to this suspicion, — the grey substance of the brain appearing more 
 
 inflamed in those perturbations which affect the phenomena of the 
 
 will, and the white in those which relate to intellectual operations. 
 
 If our existing phrenology isolates the cerebral functions too 
 
 Unity of the niuch, it is yet more open to reproach for separating 
 
 hrain and ncr- the brain from the whole of the nervous system. 
 
 voussysicra. i^-^^^^^ ^^^^^^^^ ^^ y,^^|. ||^(. intellectual and affective 
 
 phenomena, all-important as they are, constitute, in tlie whole 
 system of the animal economy, only an intermediate agency between 
 the action of the external world upon the animal through sensorial 
 impressions, and the final reaction of the animal by muscular con- 
 tractions. Now, in the present state of phrenological physiology, 
 no positive conception exists with regard to the relation of the 
 series of cerebral acts to this last necessary reaction. We merely 
 suspect that the S})inal marrow is its immediate organ. Even if 
 cerebral physiology carefully comprehended the whole of the 
 nervous system, it would still, at present, separate it too much from 
 the rest of the economy. While rightly discarding the ancient 
 error about the seat of the passions being in the organs of the 
 vegetative life, it has too much neglected the great influence to 
 Avhich the chief intellectual and moral functions are subject from 
 other physiological phenomena; as Cabanis pointed out so emphati- 
 cally, wliile preparing the way for the philosophical revolution 
 ■which we owe to Gall. 
 
 We have now seen how irrational and narrow is the way in 
 which intellectual and moral physiology is conceived of and studied : 
 and that till this is rectified, the science, which really appeals not 
 to have advanced a single step since its institution, cannot make 
 any true progress. We see how it requires, above even the other 
 
 branches of physiology, the preparation of scientific 
 ijFiienoio^^. l^^^ifs, and familiarity with the foregoing departments 
 
 of natural philosopliy ; and how, from its vicious 
 isolation, it tends to sink to the level of the most superficial and 
 ill-])repared minds, which will make it the groundwork of a gross 
 and mischievous quackery, if the true scientific inquirers do not 
 talce it out of their hands. No inconveniences of this kind, how- 
 ever, should blind us to the eminent merits of a conception which 
 will ever be one of the principal grounds of distinction of the })hilos- 
 ophy of the nineteenth century, in comparison with the one which 
 preceded it. 
 
 I
 
 RE TR OSPE CT OF NA TURA L PHIL OS OP 1 1 Y. 3 '; 7 
 
 Looklnp: back, 011 tlio comi)letIon of tliis survey of tlie positive 
 study of livinjjj bodies, we see that, imperfect as it is, Present state 
 and unsatisfactory as are the parts which relate to of nioiogn. 
 life, compared with those whicli relate to organization, still the 
 most imperfect have bei^un to assume a scientific clraracter, more 
 or less clearly indicated, in proportion to tlie comi)lexity of the 
 phenomena. 
 
 We have now surveyed the whole system of natural philosopliy, 
 from its basis in malhemntical, to its termination Retrospect of 
 in biological philoso})hy. Notwithstanding the vast Natural Pi'a- 
 interval embraced by these two extremities, we ^osophij. 
 have passed through the whole by an almost insensible gradation, 
 finding nothing hypothetical in the transition, through chemistry, 
 from inorganic to organic philosophy, and verifying as we proceeded 
 the rigorous continuity of the system of the natural sciences. That 
 system, though comprehending all existing knowledge, is, however, 
 still incomplete, leaving a wide area to the retrograde influence of 
 the theologico-metaphysical ])hilosophy, to which it abandons a 
 whole oi'der of ideas, the most inunediately ap{)licable of all. There 
 is yet wanting, to complete the body of positive philosophy, and to 
 organize its universal preponderance, the subjection to it of the 
 most complex and special phenomena of all, — those of humanity in 
 a state of association. I shall therefore venture to propose the 
 new science o^ Social Physics, which I have found myself compelled 
 to create, as the necessary complement of the system. This new 
 science is rooted in biology, as every science is in the one which 
 precedes it ; and it will render the body of doctrine complete and 
 indivisible, enabling the human mind to proceed on positive prin- 
 ciples in all directions whatever, to which its activity may be in- 
 cited. Imperfect as the preceding sciences are, they have enough 
 of the positive character to render this last transformation possible : 
 and when it is effected, the way will be open for their future 
 advancement, through such an organization of scientific labour as 
 must put an end to the intellectual anarchy of our present con- 
 dition. 
 
 END OF VOLUME I. 
 
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