THE LIBRARY OF THE UNIVERSITY OF CALIFORNIA LOS ANGELES FROM THE LIBRARY OF ERNEST CARROLL MOORE BOHN'S PHILOSOPHICAL LIBRARY THE POSITIVE PHILOSOPHY OF AUGUSTS COMTE VOL. I GEORGE BELL & SONS LONDON : YORK STREET, COVENT GARDEN AND NEW YORK, 66, FIFTH AVENUE CAMBRIDGE : DEIGHTON, BELL & CO. THE POSITIVE PHILOSOPHY OF AUGUSTE COMTE FREELY TRANSLATED AND CONDENSED BY > HARRIET MARTINEAU WITH AN INTRODUCTION BY FREDERIC HARRISON IN THREE VOLUMES VOL. I. LONDON GEORGE BELL & SONS T896 CHISWICK PRESS : — CHARLES WHITTlNliHAM AND CO. TOOKS COURT, CHANCERY LANE, LONDON. Oonege Library B V.I INTRODUCTION. "If it cann(»t be said of Comte that lie has created a science, it may l)e said truly that he has, for the first time, made the creation possible. This is a great achievement, and, with the extraordinary merit of his historical analysis, and of his philosophy of the phy- sical sciences, is enough to immortalize his name." — John Stuart Mill. "Comte is now geiieially admitted to have been the most emi- nent and impf)rtant of tiiat iiiteresting groujt of thinkers whom the overthrow of old institutions in France turned towards social speculations." — John Morley. THE foregoing- quotations f]-om the two English authorities who have most severely criticized the " Positive Polity " of Auguste Comte, bear witness to the profound imj^ulse given to modern thought by the publication of the " Positive Philosophy," more than half a century ago. Miss Martineau's condensation appeared eleven years later, during the lifetime of Comte and before the completion of his later works. It was warmly welcomed by the philosopher himself, and adopted by him as the popular form of his own voluminous treatise. Since that time an immense amount of discussion has arisen about the philosophy itself, about the subsequent development of Comte's own career and sj^eculations, and on the incidents of his strenuous life. In placing before the public Miss Martineau's version of the " Philosophie Positive " in a new form, it seems a fitting occasion to in- troduce it by some notice of Comte's own life and labours, 1569S98 VI INTRODUCTION. as well as by some account of that which he called his "fundamental work," and of the very remarkable version by which Harriet Martineau gave it a new literary form. Auguste Comte Avas born at Montpellier, in the south of France. 19th Jan., 1798, the eldest son of Louis Comte, treasurer of taxes for the department of Hcrault, and of Rosalie Boyer, whose family produced some eminent l)hysicians. Both father and mother were sincere Catholics and ardent royalists. Their son was christened Isidore Auguste Marie Francois Xavier. The house in which he was born is still to be seen opposite the church of Sainte Eulalie. At the age of nine, a small and delicate child, he was placed as a boarder in the Lycee of his native city. He soon showed extraordinary intelligence and industry, a character of singular coui*age and resolution, and a spirit of defiance towards religious and civil authority. He re- fused to conform to any worship, and avowed an open hatred of Napoleon and his schemes of conquest. Anec- dotes are still told of his prodigious memory ; he could repeat a hundred verses after a single recital, and could recite backwards the words of a page that he had once read. He carried off all prizes, and at the age of fourteen and a half lie had passed through the entire course of the Lycee. He then studied mathematics under Daniel Encontre, a teacher of great ability, whose place he was able to take in his fifteenth year. At the age of sixteen he passed in the Ecole Pojyfech/ilque, the first on the list of candidates for the south and centime of France. In October, 1814, the young Comte, then in his seven- teenth year, entered the great college at Paris, and there applied himself with his usual energy to mathematics and physics under the illustrious Poinsot. He was called " the philosopher," and took the lead amongst his fellow pupils by his energy as well as his abilities. He was known as an ardent republican, a fierce opponent of tyranny, whether INTRODUCTION. VU theological, political, or academic. In 1816, one of the tutors having given offence to the younger pupils, Comte took the lead in demanding his resignation, and drew up a curt memorial to this effect. The college was sent down, and Comte, who was only in his second year of residence, as the author of the insurrection, was sent home to his despairing parents and placed under the surveillance of the police, with his hopes of a future career entirely destroyed. For some time he studied biology in the medical school of Montpellier, but in September, 1816, being then eighteen, he returned to Paris with the brave intention of suj)port- ing himself l\v lessons. He now dropped the mediaeval name of Isidore by which he had been known from infancy, and took his Roman baptismal name of Auguste. In the following year he was introduced to Saiut-Simou, with whom he remained in relations for four or five years. The vague, optimistic, aud humanitarian dreams of this singular reformer did undoubtedly exercise a certain fascination over the youthful mind oi Comte, and gave his genius aud character an iuHexible bent towards a scheme of social reorganization. But the shallowness of Saint-Simon's acquirements could not impart anything of a solid kind to such a mind as Comte's ; and the vanity and charlatanry of the famous socialist alienated his young follower. They soon came into direct opposition on Saint-Simon's con- tention that intellectual aud moral re-organization could only proceed from the authority of government. Saint- Simon claimed as liis own the work of his youfig colleague, and when he fell back on a mystical theologism, the rupture became final. Auguste Comte wrote a few pieces for various periodicals in Paris, to which he attached but little importance. His first great philosophical woi'k was a pamphlet in 191 pages, published in May. 1822, with an introduction by Vin INTRODUCTION. Saint-Simon. It was entitled a " Prospectus of the scientific works required for the reorganization of Society, by Auguste Comte, former pnpil of the Ecole Polytechniqne." He republished his pamphlet witli some small modifica- tions and additions in 1824, under the title " System of Positive Polity," and this is reprinted in vol. iv. of the " Politique Positive," 1854. A full account and the text of both editions is given in the " Revue Oceidentale " (1895, vol. xi. p. 1). This essay of 1822 contains a statement of the classification of the sciences, of the law of the three states, and the suggestion of a science of sociology. It is in truth the prospectus of "^that Avhich for thirty years Comte continued to elaborate. It has not the smallest connection with Saint-Simon, nor with contemporary socialism or mysticism, and has always l>een treated by Comte and by his adherents as the the first sketch of the " Positive Philosophy." Between 1816 and 1826 (jetat. 18 to 28) Comte laboured and read with extraordinary energy, frequently absorbed for twenty -four hours at a stretch, and writing all through the night. By his essay of 1822 and one or two other pieces in the " Producteur." 1825-26. he had won the favourable opinion of many eminent men of science and literature. Amongst these are mentioned Delambre, Fourier, Blainville, Bonniu, Poinsot. Carnot. Guizot. J. B. Say, Dnnoyer, Professor Buchholtz of Berlin, de VillMe. Lamennais. For a few weeks he was private secretary to Casimir-Perier, but his independent spirit declined to accept the duties required. In April, 1826 (fetat. 28). he opened iu his own rooms a course of jtublic leetiu'es on the Positive Philosophy, which was to extend to seventy-two lectures, from 1st April, 1826. to 1st April, 1827. Amongst his audience were such men as Broussais, Blainville, Poinsot, J. Fourier, Alexander von Humboldt, D'Eichthal, Montebello, Carnot, son of the famous yeneral, Cerclet, INTRODUCTION. IX Montgerv, and other young students. The series was in fact that which was subsequently published. At the fourth lecture the course was abrviptly broken off. Intense mental strain, together with domestic misery, brought on an attack of insanity. He left his home in a state of dis- traction, and was placed in an asylum by his friend Broussais. There he remained for seven months. The devotion of his mother and his wife, who took him from the care of Dr. Esquii'ol whilst still suffering from the disease, succeeded in gradually restoring his reason. An epoch of profound despair followed, during which he threw himself into the Seine, but was rescued ; and thence- forth he resolved to devote himself with patience and resignation to the work of his life, supporting himself with private lessons. In January, 1829, he resumed his course of lectures on the Positive Philosojjhy, and he had the satisfaction of seeing the same eminent men amongst his audience, with the excepti(m of Humboldt, who Avas no longer in France. On this occasion he completed the whole series of lectures, and in December, 1829, he re- peated them in a public coi;rse at the Athetiee. He also gave other gratuitous public lectures, including the series on Popular Astronomy whicii he repeated during eighteen years, from 1830 to 1848. In 1832, Comte was apj^ointed repetiteur of analytic mathematics at the Ecole Poly- technique, at the instance of M. Navier, then professor there ; and in 1837 he was named examiner of the > andi- dates for admission. For a short time he filled the place of the Professor. The work of which these three volumes are a condensa- tion was published at intervals from 1830 to 1842. The first volume, containing the Introduction and the philo- sophy of Mathematics, was published sejmrately, with a dedication to Baron Fourier and M. de Blainville. A brief note described it as the result of the author's labours X INTRODUCTION, from the year 1816, and as a development of the new ideas put forth in his early essay of 1822, entitled a " System of Positive Polity." The second volume, comprising Astronomy and Physics, did not appear until 1835, owing to the commercial disasters of the Revolution of July. The third volume, comprising Chemistry and Biology, appeared in 1838. The new science of Sociology, which was intended to he comprised in a single volume, ulti- mately extended to three volumes, published in 1839, 1841, and 1842. The last volume, containing nearly a thousand pages, was introduced by a personal preface to explain the prolongation of the work over twelve years, and the grounds for devoting one half of the entire work to the new Social Science. And it contained in notes Comte's vehement repvidiation of Saint-Simon, and his do less vehement condemnation of M. Arago and the official directors of the Ecole Poly technique. M. Littrc has described, with the knowledge of intimacy and the warmth of a disciple, the colossal task which Comte had now brought to a conclusion. " Twelve years had passed," he says, ^ during which his life had been closed against any kind of disti'action. No wish for pre- mature publication was suffered to lead his mind oft' the conscientious completion of his task. No ambition of gaining popularity was allowed to modify a single line in conformity with the opinions of the time. With stern resolution, and deaf to all external distractions, he con- centrated his whole soul upon his Avork. In the history of men who have devoted their lives to great thoughts, I know nothing nobler than that of these twelve years." ' There was indeed nothing exceptional in these twelve years. Precisely the same may be said of the whole forty years of Comte's life from the time of his leaving the college, at the age of eighteen, until his death in 1857. ' " Auguste Conite, et la Philosopliie Positive," 1863, p. 188. INTRODUCTION. XI His method, of coraposition was unique and has been dwelt upon by all his biographers. His marvellous memory and power of mental concentration enabled him to think out an entire volume in all its parts, plan, sub- divisions, ideas, arguments, and details, without putting a word to paper. When this was completed, he regarded the work as ready. His courses of lectures were all delivered without writing. When he commenced to pre- pare them for the press, he simply wrote them down from memory with great rapidity, composing the matter as fast as the sheets were printed, and without altering the proofs. For example, the first chapter of the sixth volume consists of 343 octavo j>ages and was written in twenty-eight days, althougli its mental elaboration was the outcome of years of meditation. As M. Littrc remarks, this method of composition was only possible to abnormal powers, and it secured an extraordinary unity of conception and organic symmetry of plan. But it had the obvious disadvantages of a certain multiplicity of phrase, a monotony, and that repetition which is only proper to oral exposition. These defects have been universally imputed to the written style of Comte, who has shown that on occasions he could rise into dignity and 23athos, or illumine his discourse with a profound epigram or even a brilliant sally. But habitually and on system, he suppressed any such gifts, and uniformly cast his philosophic thoughts into a very formal, artificial, and undoubtedly cumbrous style which he elaborated for himself and which gi-adually became a confirmed man- nerism. Tedious and even repidsive as it is to the average reader, to the serious student of Positivism this method of exposition has rare and paramount advantages. It is unerringly precise, lucid, qualified, and suggestive. Comte certainly had nothing of the literary genius of Bossuet and Voltaire, Hume and Berkeley. But his long-drawn and XU INTRODUCTION. over-elaborated sentences never leave the student in doubt for a moment as to his meaning, as to his whole meaning, as to all that he wishes to express, and all that he means to disclaim or exclude. The result is, that the genei'al reader can hardly follow these crowded and closely welded paragraphs without the assistance of an expert, whilst the serious student of the Positive Philosophy finds some new light or some needful warning in everyone of these pregnant epithets and precise limitations. Comte saw this clearly himself ; and hence, in his " Popular Library," embodied in his later works, he inserts — not his own " Positive Philosophy " in six volumes — but Miss Marti- neau's condensed English version. Unfortunately not only the general reader, but the professed critics of Positivism have too ofteu adopted his generous sug- gestion. " The Philosojihie Positive " as a whole received an earlier and more open welcome in England than in France. Sir David Brewster, the eminent physicist, a strong oppo- nent of Positivism as a religious and social philosophy, reviewed the first two volumes in the " Edinburgh Review," (No. 136, 1838, vol. Ixvii., p. 271). In this essay, which is far from being the work of a partisan or even a friend, Brewster pays homage to the depth and sagacity of Comte's mind, and he accepts in principle the law of the Three States, the Classification of the Sciences, and the ultimate extension of the methods of Science to Sociology. Mr. Mill followed in his " System of Logic," 1843, in which he spoke of Auguste Comte as amongst the first of European thinkers, and by his institution of a new social science, as in some respects, the first. In 1845-6, George Henry Lewes published his " Biographical History of Philosophy," enlarged in 1857, 1867, 1870. and 1880, in which he treated of Auguste Comte as " the greatest of modern thinkers," and as crowning the general history INTRODUCTION. XIU of ijbilosophical evolution. lu 1853, Lewes published Comte's " Philosophy of the Sciences," a volume in Bohn's Philosophical Library. And in the same year Miss Martineau published the condensed translation which at once made Comte familiar to all English students. This has been translated into French by M. Avezac-Lavigne, and has passed through more than one edition. It is a singular fact in literary history, and a striking testimony to the merit of Miss Martineau, that the work of a French philosopher should be studied in France in a French re- translation from his English translator — and that at his own formal desire and by his own special followers. An interesting account of Miss Martineau's own labours on the translation may be found in her " Autobiography and Letters" (2nd edition, 1877, vol. ii., p. 385, etc., etc.). The work appeared fiually, after some interruptions, in the beginning of November, 1853, and it was received with a chorus of approval by the French philosopher and by bis English readers. Comte's own opinion is set forth in the letters of his printed by M. Littrc in his biographical work, to which we shall presently return. George Grrote, the historian, wrote to Miss Martineau : " Not only is it ex- tremely well done, but it could not be better done." The French translation of Miss Martineau's condensation by M. .\vezac-Lavigne, a Bordeaux disciple of Comte, appeared in May, 1871. The correspondence between him and Miss Martineau is set out in the " Autobiographv " (vol. iii., p. 310;. The outspoken language of the " personal preface " to the sixth volume of the " Philosoijhie " brought down upon Comte even severer sufferings than either he or his friends had anticipated. He was deprived first of one, then of both his official posts, was treated as an outcast from the academic world, and was reduced to absolute penury. But in August, 1842, just before the actual publication of the XIV INTRODUCTION. sixth volume, liis wife carried out the intention which she had long meditated and announced, and insisted on a separation. The story of Comte's married life is full of interest and of tragedy, but it is too intricate, and still too much disputed, to be here fully told. The case of Madame Comte has been presented by M. Littre in the work cited above, and the case of Auguste Comte has been recently set forth by M. Lonchampt, one of his executors, in the "Revue Occidentale" (vol xxii., p. 271 ; and vol. xxiii., pp. 1, 135). As a young man of twenty-three, Comte casually fell in with a certain Caroline Massin, a young Parisian, of a degraded past life, of singular intelligence, with gi'eat ambition, and many fascinating gifts. He felt for her affection and pity, took her under his protection, and ultimately married her. In spite of real affection on his side, real admiration on hers, long-suffering self- control on his ])art, and some fitful acts of self-devotion on her part, their union became unhappy, and at last intoler- able. She never learned either to love her husband or to respect her own position as a wife. His entire absorption in his work, and his defiance of the academic and literary world, and all that it had to offer, alienated her selfish nature; she left him more than once, and, on the com- pletion of the polemical preface to vol. vi., she left him for ever, after seventeen years of married life. They continued to correspond for some years ; but separation ultimately passed into mutual estrangement and bitter feeling. In his last will he spoke of her with poignant reproaches, the pround of which has now been divulged, and he described his marriage as the one great error of his life. It is not proposed in this brief introduction to Miss Martineau's work to enlarge on the subsequent life and the later works of Auguste Comte. By the intervention of Mr. Mill, three Englishmen, Mr. Grote, Mr. Raikes Currie, and Sir W. Molesworth, provided, in 1844, the salary of INTRODUCTION. XV d£200 of which he was dejjrived ; but to the surprise, and even the indignation, of Conite, they declined to make this permanent. Mr. Grote and other friends made some further contributions ; and ultimately, by the help of M. Littrc, Dr. Charles Robin, Dr. Seg-onc], and others, a regular subsidy was established in 1849. It began with 8,000 francs (^120), idtimately rose to 8,000 francs (^6320), and it has been continued until the pi-esent time, in order to carry out the purposes of the last will. On this pittance Comte lived until his death, absorbed in his philosophic work, and continuing the allowance to his wife. He adopted an almost ascetic life, avoiding the use of alcohol, coffee, tobacco, and all stimulants, limiting his food bv weight to the minimum of two meals per diem, one of these being of bread and milk only. During a few years his income had been d8400, Init for the greater part of his life it had fallen much below this amount. There can be no question that his whole career was one of the niost intense concentration of mind, gigantic industry, rigid economy, and singular punctuality and exactness in all his habits. Though far from conforming to any saintly ideal, it was a life of devotion to philosophy, as all his biographers agree to describe it. John Morley truly says, "Neither Franklin nor any man that ever lived, could surpass him in the heroic tenacity with which, in the face of a thousand obstacles, he pursued his own ideal of a vocation." In 1844, two years after the desertion of his wife, Comte saw Madame Clotilde de Vaux, the sister of one of his dis- ciples, the wife of a man of good family, condemned for life to penal servitude. In the course of the next year, he fell in love with her, entered into the closest intimacy with her, which she succeeded in maintaining quite irreproach- able, whilst he insisted on claiming her as his spiritual wife. After one year of devoted friendship, she died in his arms, leaving him inconsolable in what he called his veuvage XVI INTRODUCTION. i'ternel. From this point began the second period of his life, and of his philosophic career. He gave public lectures again in 1848-1850, until the hall was closed by the Empire, and he published his second great work, the "Positive Polity," in four vols., 1851-1854. The "Cate- chism " was published in 1852, the " Appeal to Conserva- tives " in 1855, and the " Subjective Synthesis " in 1856. In the year following his health, perhaps affected by his rigid austerity of life, began to give way, and lie died of cancer on September 5th, 1857. He was buried in Pcre la Chaise ; the day of his death has since been commemorated yearly by his followers, who now for thirty-eight years have maintained his rooms, books, and effects intact, and have carried out the dii'ections of his last will. This is not the place to enter on the comjjlex question whether the subsequent works of Comte were a normal and legitimate development of his fundamental " Philo- sophy." Gr. H. Lewes and John Morley have amply shown that it was, though both of them refuse their assent to the teaching of the " Polity." But, as Mr. Morley says, for the purposes of Comte' s career the two " ought to be regarded as an integral whole." And he also remarks, " A great analysis was to precede a great synthesis, but it was the synthesis on which Comte' s vision was centred from the first." This is now so clear from the mass of correspondence and biogra2:)hy which recent years have produced, that it would no doubt modify the contrary opinion expressed by Mr. Mill, thirty years ago. When Miss Martineau translated the " Philosopliy," more than forty years ago, the later works of Comte were not before her ; and, as she frankly states in her preface, the later works of Comte are not referred to in her book at all. She carried this decision to the very extreme point of suppress- ing, without any mention, the last ten pages of the sixth and concluding volume of the " Philosophy." Now, from the INTRODUCTION. XVll pointof view of the unity of Comte's career these ten pages are crucial, for they contain the entire scheme of Comte's future philosojjhical labours as he designed them in 1842. ainl as they were ultimately carried out in the "Polity," "Cate- chism," " Synthesis," etc., etc. These important pages have been added by the present writer, in the condensed form adopted by Miss Martineau. A few words only are needed as to her very remarkable work. It has been already shown that the singularly arti- ficial style in which Comte chose to express his ideas, with elaborate qualifications, provisoes, suggestions, and con- notations crowded into every sentence, made his " Philo- sophy " very irksome reading to any but a patient student. The language is not at all verbose, nor are the qualifying words useless, for every one of them adds some new idea or guards against some misconception. But the mass of these reiterated adjectives and adverbs certainly wearies the average reader. Miss Martineau seized the dominant idea of each sentence or rather paragraph— not without much sacrifice of the continuity of thought, no little loss in precision and accuracy of definition, sometimes a serious omission of important matter — but on the whole with an extraordinary gain to the freshness of impi-ession on the general reader. In the work already cited. M. Littre has printed three letters of Comte to Miss Martineau on receipt of her trans- lation of his work. He welcomes it with gratitude and enthusiasm. He says, " I have already read the noble preface and the excellent table of contents, as well as some decisive chapters. And I am convinced that you have displayed clearness of thought, truth, and sagacity in your long and diflicult task." " The important undertaking that you so happily conceived and have so worthily accom- ])lislied will give my ' Positive Philosophy ' a competent audience greater than I could have hoped to find in mj I. b xvm l^JTRODUCTION. own lifetime." " It is due to you, that the arduous study of my fundamental treatise is now indispensable only for the small number of those who purpose to become sys- tematic students of philosophy. But the majority of readers, with whom theoretic training is only intended to provide them with practical good sense, may now prefer, and even ought to prefer for ordinary use, your admirable condensa- tion [sic in orig.]. It realises a wish of mine that I formed ten years ago. And looking at it from the point of view of future generations, I feel sure that your name will be linked with mine, for you have executed the only one of those works that will survive amongst all those which my fundamental treatise has called forth." With great generosity Miss Martiueau offered to Comte a considerable share in the profits of her book. He declined on the ground that he had made it a rule that all his literary work should be gratuitous. On her still pressing on him this offer, he consented so far as to accept her gift towards meeting the cost of printing his " Polity." It is right to point out that the systematic students of Comte' s works, whilst fully accepting the condensation of Miss Martineau for a poj^ular exposition, and admiring the energy and skill Avith which Miss Martineau performed a most difficult literary feat, do not admit that for purposes of serious study, much less of hostile criticism, the English condensation can ever dispense with knowledge of the French original. As Comte justly said, that original remains indispensable to students of philosophy who look for more than a popular exposition. It could not be other- wise. Miss Martineau reduced more than four thousand jmges to somethmg over one thousand. And as no one of these four thousand pages was without its careful limita- tions cf the author's meaning, it follows that much of his thought has Iteen presented in outline and not in detail. Nor can it be denied that there are points, and even points INTRODUCTION. xix of great importance, iu which the translator failed to grasp the author's meaning. In a treatise of a scope so vast, ranging over the whole held of knowledge, some such slips were quite inevitable. It is an extraordinary fact that they were not more numerous. Whatever they were, the present writer has made no attemj^t to modify or even to indicate them. It has been no jiart of his task to edit Miss Martineau's version, which will long remain for ordinary use, as Comte himself said, the popular form of his great fundamental treatise of " that great analysis," to use the words of John Morley, *' which was to precede the great synthesis." Frederic Harrison. 189."). For biographies and criticisms of Comte the following woi'ks may be consulted : G. H. Lewes. "History of PIuloso]ihy." 5th eilition, 1S8U. vol. 11. J. Stuart Mill. "System of Logic," vol. ii ., and " Auouste Comte anil Po.sltlvism,"' I860. John Morley. CoMTE, " EncyclopcTtUa Brit.," vol. vl. ; " Critical Miscellanies," vol. Hi. Littre. " Aujjnste Comte et la Phllosophle Positive," 1863. Herbert Spencer. " Essays," vol. 111. T.Huxley. "Essays.' "Revue Occldentale," jxtssiiu, 1878-1S95, and especially three articles hy Lon('lianii)t, vol. xxii. \>. 271, vol. xxlll. pp. 1, 135 (1889). Koblnet. " Vie d'Auguste Comte," 1860. Dr. Bridges. " LTnity of Comte's Life and Doctrine," 1866. Professor E. S. Beesly. " Comte as a Moral Type," 1885. "The Positlvlst Review,"' 1893-1895 (monthly), W. Reeves, 185, Fleet Street. Dr. Congreve. "Collected Works." "Lettres d'.\uguste Comte," a Valat (1870), a John Stuart Mill (1877). "Testament ' (1884). PREFACE BY HAERIET MARTINEAU.' IT may appear strange that, in these days, when the French language 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 diffi- culties 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 substantially to our knowledge for many years past are fully acquainted with it, and are under obligations to it which they would have thankfully acknowledged, bul for the fear of offending the prejudices of the society in which they live. Whichever 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 recog- nized as the fuumhition 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 work was waiting to be done which might seem to be more urgent. One reason, though not the chief, was that it seems to me unfair, through fear or indolence, to use the benefits conferred on us by M. Comte without acknowledgment. His fame is no doubt safe. Such a work as this is sure ' To the first edition pviljlisihed by John Chapman in 1853, in two vohunes 8vo, of wliicli tlic present edition is a rejuint. XXH PREFACE. of receiving due honour, sooner or later. Befon^ the end of the century, society at large will have becume aw^are that this work is one of the chief honour? of the century, and that its author's name vpill rank with those of the worthies who have illustrated former ages : hut it does not seem to me right to assist in delaying the recognition till the author of so noble a service is beyond the reach of our gratitude and honour : and that it is demoralizing 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 awakeaed 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 iu 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 m 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. Evcny sentence is full fraught with meaning ; yet it is overloaded with words. His scrupulous honesty leads him to guard his enuncia- tions with epithets so constantly repeated, that though, to his own mind, they are necessary in each individual in- stance, 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 in- structions spread over twenty yeai's. 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 dis- advantages arising from redundancy alone. My belief is PREFACE. XXlll 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 illus- tration 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 largest number of intelligent readers. We are living in a remarkable time, when the conflict of opinions renders a firm foundation of knowledge indispensable, not only to our intellectual, moral, and social progress, but to our holding such ground as we have gained from former ages. While our science is split up into arbitrary divisions ; while abstract and concrete science are confounded together, and even mixed up with their application to the arts, and with natural history ; and while the reseai-ches of the scientific world are pre- sented 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 ono can inquire into the mode of life of young men of the middle an 1 operative classes without being struck with the desire that is shown, and the sacri- fices 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 boi'ne, 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 eveiy one who cares for the good of nation or race is that men should be adrift for want of an anchorage for their convictions. I believe that no one questions that a very large proportion of our people are now so adrift. With pain and fear, we see that a multi- XXIV PREFACE. tude, who might and should he among the wisest and best of our citizens, are alienated for ever from the kind of faith which sufficed for all in an organic period which has passed away, while no one has presented to them, and they cannot obtain for themselves, any ground of con- viction as firm and clear as that which sufficed for our fathers in their day. The moral dangers of such 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 unquestionably the greatest single eft'oi-t that has been made to obviate this kind of danger ; and my dee]) 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. Whatever else may be thought of the work, it will not be denied that it ascertains with singular sagacity and sound- ness the foundations of human knowledge, and its true object and scope ; and that it establishes the true tiliation of the sciences within the bcnindaries of its own principle. Some may wish to interpolate this or that ; some to amplify, and perhaps, here and there, in the most obscure recesses of the great edifice, to transpose, more or less : but any who cpiestion the general soundness of the exposi- tion, or of the relations of its j^arts, are of another school, and will simply neglect the book, and occupy themselves as if it had never existed. It is not for such that I have been working, but for students who are not schoolmen ; who need conviction, and must best know when their need is satisfied. When this exposition of Positive Philoso]>liy 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 immoveable 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 bo so, in the inevitable growth of knowledge and evolution of philosophy ; and it is the fate which the philosopher him- PREFACE. XXV self should covet, because it is only a true book that could survive to be so treated : but, in the meantime, it gives us the basis that we demand, and the 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 unquestion- able 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 estab- lished, like the others, by a review of what had before been achieved ; but now, by the band of our master, dis- criminated, 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 liuild upon the foundation here laid. A thorough fami- liarity with the work in which all this is done would avail more to extinguish the anarchy of popular and sectional opinion in this country than any other influence that has yet been 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, in a subsequent letter, that he had wished, for many years, to do what I was then attempting, and had been prevented only by ill health. My estimate of M. Comte's work, and my expecta- tions from its introduction into England in the form of a condensed translation, were fully shared by him ; and, to my utter amazement, he sent me, as the first act of our correspondence, an order on his bankers for .£500. There was time, before his lamented death, for me to communi- cate 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. Tlie last words of his last letter were an entreaty that I would ;XXV1 PREFACE. let him know if more money would, in any way, improve the quality of my version, or aid the pi'oniulgation 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 oi;t his wishes as far as possible ; and to do this, no pains have been spared by myself, or by Mr. Chapman, who gave him the information that called forth his bounty. As to the method I have pursued with my worlv, — thei'e 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 ottered a complete translation. Some will ask why it is not a close version as far as it goes ; and others, I have reason to believe, would have preferred a brief account, out of my own mind, of what Comte's philosophy is, accompanied by illustrations of my own devising. A wider ex]!ectation 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 ])rotest, here and there, against a statement, a conclusion, or a method of treatment. I should have been better satisfied still to have adduced some critical opinions of much higher value than any of mine can be. But my deliberate conclusion was that this was not the place nor the occasion for any such controversy. What I engaged to eak for themselves, and the readers of the book can criticize it for themselves. 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. PREFACE. xxvn This by no means implies tliat the translation is a close one. It is a very free translation. It is more a condensa- tion 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, Astro- nomy, and Physics — should be revised by a qualified man of science. My personal friend, Professor Nichol, of Grlasgow, 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 Conite's speculation on the possible mathematical verification of Laplace's Cosmogony. But more had to be done with regard to the treatment of Physics. Every reader will see that that section is the weakest part of the book, in regard both to the organization and the details of the subject. In regard to the first, the author explains the fact, from the nature of the case, — that Physics is rather a repository of somewhat fragmentary portions of physical science, the correlation of which is not yet clear, than a single circumscribed science. And we must say for him, in regard to the other kind of imperfec- tion, that such advances have been made in almost every department of Physics since his second volume was pub- lished, that it would Vie unfair to ]:)resent what he wrote under that head in 1835 as what he would have to say now. The choice lay therefore between almost re-writing this portion of M. Comte's work, or so Jargelv abridging it that only a skeleton presentment of general principles 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 justice to the Author. I have adopted therefore the plan of extensive omissions, and have retained the few short memoranda in which Professor Nichol sutrgested these, as XXVni PlIEFACE. uotes. Although this gentleman has sanctioned my pre- sentment of Comte's chapters on Mathematics and Physics, it must not be inferred that he agrees with his Method iu Mental Philosophy, or assents to other conclusions held of main importance by the disciples of the Positive Philo- sophy. The conti'ary, indeed, is so apparent in the tenour of his own writings, that so far as his numerous readers are concerned, this remark need not have l>een offered. With the reservation I have made, I am bound to take the entire responsibility, — the Work being absolutely and wholly my own. It will be observed that M. Comte's later works are not referred to in any part of this book. It appears to me that they, like our English criticisms on the present Work, had better be treated of sepai-ately. 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 possilile and available. For various reasons, I think it best to stop here, feeling assured that if this Work fulfils its function, all else with which M. Comte lias 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 Philosophy for pride of reason and low- ness of morals. The imputation will not be dropped, and the enmity of the religious world to the book wall not slacken for its appearing among us in an English version. It cannot lie otherwise. The theological world cannot but hate a book which ti'eats of theological belief as a transient state of the human mind. And again, the preachers and teachers, of all sects and schools, who keep to the ancient practice, once inevitable, of contemplating and judging of the universe from the point of view of their own minds, instead of having learned to take their stand out of them- selves, investigating from the universe inwards, and not from wdthin 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 PREFACE. XXIX away, theologians and metapliysiciaus 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 metapthysics. and. finding what they are now worth, have risen above them — will pronounce a very ditt'erent 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 expiessly pro- vided for. To those who have learned the difficult task of postponing 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 im- pressive as its intellectual satisfactions. The aspect in which it presents Man is as favourable to his moral discip- line, as it is fresh and stinndating to his intellectual taste. We find ourselves suddenly living and moving in the midst of the univei'se, — as a part of it. and not as its aim and object. We find ourselves living, not under capricious and arbi- trary conditions, unconnected with the constitution and movements of the whole, but under great, general, in- variable laws, which operate on us as a part of the whole. Certainly, I can conceive oi no instruction so favourable to aspii'ation 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. Wc find here indications in passing of the evils we suffer from our low aims, our selfish passions, and our proud ignor- ance ; and in contrast with them, animating displays of the beauty and glory of the everlasting laws, and of the sweet serenity, lofty courage, and noble resignation that are the natural consequence of pursuits so pure, and aims so true, as those of Positive Philosophy. Pride of intellect surely abides with those Avho insist on belief without evidence and on a pliilost)phy derived from their own in- tellectual action, without material and corroboration from without, and not with those Avho are too scrupulous and XXX PREFACE. too humble to transcend evidence, and to add, ovit of their own imaginations, to that which is, and may be, referred to other judgments. If it be desired to extinguish ]»re- sumption, to diuw 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 Philosophy, with its train of noble truths and irresistible inducements. The prospects it opens are boundless ; for among the laws it establishes that of human progress is conspicuous. The virtues it fosters are all those of which Man is capable ; and the noblest ai"e those which are more eminoDtly fostered. The habit of truth-seeking and truth-speaking, and of true dealing with self and with all things, is evidently a primary requisite ; and this habit once perfected, the natural conscience, thus disciplined, will train up all other moral attributes to some equality with it. To all who know what the stndy of philosophy really is, — which means the study of Positive Philosophy, — its effect on human aspiration and human discij)line is so plain that any doubt can be explained only on the supposition that accusers do not know what it is that they are calling in question. My hoj^e is that this book may achieve, besides the purposes entertained by its author, the one more that he did not intend, of conveying a sufficient rebuke to those who, m theological 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 ])rogress is conspicuously at work throughout human his- tory. The only field of progress is now that of Positive Philosophy, under whatever name it may be known to the real students of eveiy sect ; and therefore must that philo- sophy be favourable to those virtues whose repression would be incompatible with progress. CONTEXTS. INTKODUCTION. CHAPTER I. ACCOUNT OF THE AIM OF THIS WORK.— VIEW OF THE NATUUE AND IMPORTANCE OK THE POSITIVE PHILOSOPH^i . PAGE Preliiiiiuaiy survey ......... 1 Law uf human develojimt 111 ....... 1 First stage ... ...... '2 Second stage ........... Third stage -' Ultimate point of eacii .... ... 2 Evidences of the Law ........•'{ Actual ........... 3 Theoretical .......... 3 Character of the Positive Pliilosopliy ..... o History of the Positive Philoscjphy ...... 6 New department of Positive Pliilosophy ..... 7 Social l^hysics .......... S ^ Secondary aim of this work ....... ^S To review the jiliilosophy of the sciences .... 8 J Glance at speciality .........!) Prcjposed new class of students . . . . . .10 Advantages of the Positive Philosophy . . . . .11 1. Illustrates Intellectual function . ..... 11 2. ]\Iust regenerate Education ...... 13 3. Advances Sciences by combining them .... 14 4. Must reorganize Society . . . . . . .15 No hope of reduction to a .-• ingle Law 17 CHAPTER IL VIEW OF THE HlEItARCHV OF THE POSITIVE .SCIENCES. Failure of proposed classifications ...... True principle of classification ....... Boundaries of our held ........ 19 •20 20 CONTENTS. Theoretical Inquiry .... Abstract science .... Concrete science .... Difficulty of classification Historical and Dogmatic Methods . True principle of classification. Characters ..... 1. Generality ..... 2. Independence .... 1 norganic and Organic phenomena . 1. Inokganic 1. Astronomy ..... 2. Physics ..... 3. Chemistry ..... II. Organic 1. Physiology ..... 2. Sociology ..... Five Natural Sciences : their filiation Palliation of their parts Corroborations ..... 1 . This classification follows the order of 2. Solves heterogeneousness . 3. Marks relative perfection of sciences 4. Effect on Education Eftect on Method .... ( )rderly study of sciences. Mathematics A department ..... .\ basis ...... An instrument A double science .... Abstract Mathematics, an instrument Concrete Mathematics, a science Mathematics pre-eminent in the scale disclosure of sciences pa(;k 22 23 23 23 24 26 26 26 26 27 27 28 28 28 28 29 29 29 30 30 30 30 31 32 32 33 34 34 34 34 34 35 35 35 BOOK I. MATHEMATICS. CHAPTER I. mathkmatics, abstract and concrete. Description of Mathematics Object of Mathematics General Method E.xamples ..... True Definition of Mathematics Its two parts 37 37 38 38 40 41 C0^• TENTS. Their difterent oLjects Their different natures Concrete Mathematics Abstract Mathematics Extent of its domain Its Universality Its limitations . XXXlll PAGE 42 43 43 44 45 45 46 CHAPTER II. general view of mathematical analysis. Analysis . True idea of an equation Abstract functions . Concrete functions . Two parts of the Calculus Algebra . Arithmetic Its extent . Its nature . Algebra . Creation of new functions . . • • Finding equations between auxiliary quantities Division of the Calculus of functions . . Section 1. Ordinary Analysis, or Calculus of Direct tions ....•••• Its object ...••••• Classihcation of Equations .... Algebraic equations Algebraic resolution of equations . Om- existing knowledge . . . . • Numerical resolution of equations . The Theory of equations . _. _ . Method of indeterminate coefficients • , , • Section 2. Transcendental Analysis, or Calculus Functions . . Three principal views History ' . Method of Leibnitz . Generality of the formulas Justification of the Method Newton's Method. Method of Limits Fluxions and Ihients. Lagkange's Method . Identity of the three methods Their comparative value . - The Differenti.al and Integral Calculus Its two parts I. C of Indirect 50 50 51 51 52 53 53 53 54 55 55 56 57 58 58 58 59 59 60 61 62 62 63 63 63 64 66 67 68 68 69 70 71 CONTENTS. Tlieir mutual relations Cases of union of the two. Cases of the Differential calculus alone Cases of the Integral calculus alone. The Differential Cdlculus. Two ])ortions .... Subdivisions .... Reduction to the elements Transformation of derived functions for new variables Analytical applications . The Integral Caleulus Its divisions .... Subdivisions .... One variable or several Orders of differentiation . Quadratures .... Algebraic functions . Transcendental functions . Singular Solutions . Definite integrals Prospects of the Integral Calculus Caleitlus of Variations Problems giving rise to this calculu> Other applications . Relation to the ordinary calculus CHAPTER III. GENERAL VIEW OF GEOMETRY. Its Nature Definition. Idea of Space . Kinds of extension . Geometrical measurement Measurement of surfaces and volumes Of curved lines. Its illimitable field . Properties of lines and surfaces Two general Methods Special or ancient, and general or moder (ieometry of the ancients. Geometry of tiie right line Graphical solutions . Descrii)tive Geometry Algebraic solutions . Trigonometry . Modern, or Ana.lytiral Geometry Analytical representation of figures Position ..... n ii-eometr CONTENTS. XXXV Position of a jioint .... Plane curves ..... Exjjression of lines by equations Expression of equations by lines Change in the line changes the equation. Every definition of a line is an equation Choice of co-ordinates Determination of a point in space . Determination of Surfaces by Equations, Surfaces ..... Curves of double curvature Imperfections of Analytical Geometry Imperfections of Analysis and of Equations by CHAPTER IV. KATIONAL MECHANICS. Its nature. Its characters . Its object .... Matter not inert in Physics Supposed inert in Mechanics Field of Rational Mechanics Three Laws of IVIotion Law of inertia . Law of equality of action and i Law of co-existence of motions Two Primary divisions Statics and Dynamics . Secondary divisions . Solids and Fluids Section L Statics . Converse methods of treatment First' method . Statics by itself Second method Statics through Djaiamics Moments .... Want of unity in the method Virtual Velocities Theory of Couples . Sh.are of equations in producin Connection of the concrete witl Equilibrium of tluids Hydrostatics Liquids .... Gases .... Section 2. Dynamics . Object .... e-action e([uilibrium 1 the abstract quesi tion XXXVl CONTENTS. PAGE Tlieory of rectilinear motion 137 Motion of a point ......... 139 Motion of a system . . . . . . . . .139 D'Alembert's principle ........ 140 Results 143 Statical theorems ......... 143 Law of repose . . . . . . . . . .143 Stability and instability of equilibrium ..... 143 Dynamical theorems ........ 144 Conservation of the motion of the centre of gravity . . 144 Principle of areas ......... 144 The invariable plane . . ...... 145 Moment of inertia ......... 146 Principal axes .......... 146 Conclusion .......... 147 BOOK 11. ASTRONOMY. CHAPTER I. GENERAL V I E AV. Its Nature 148 Definition ........... 149 Restriction .......... 149 Means of exploration . . . . . . . .150 Its rank ........... 151 When it became a science ....... 152 Reduction to a single law ....... 152 Relation to other sciences . . . . . . .153 Divisions of the science ........ 155 Celestial Geometry . . . . . . . . .155 Celestial Mechanics 155 CHAPTER II. METHODS OF STUDV OF ASTRONOMY. Section 1. Instmments 157 01)servation . . . . . . . . . .157 Shadows ........... 157 Artificial methods ......... 158 The pendulum . . . . . . . . . .159 Measurement of angles ........ 159 Requisite corrections . . . . . . . .161 CONTENTS. XXXvii PAGE Section 2. Refraction 161 Section 3. Parallax 163 Section 4. Catalogue of stars 165 CHAPTER III. geometrical phenomena of the heavenly bodies. Section 1. Statical Phenomena 167 Two classes of phenomena . . . . . . .167 Planetary distances . . . . . . . . .167 Form and size .......... 170 Planetary atmospheres ........ 171 Earth's form and size ........ 171 Means of discovery . . . . . . . . .172 Planetary motions . .173 Rotation ........... 174 Translation . . . . . . . . . .174 Sidereal revolution ......... 176 Motion of the Earth 176 Evidences of the Earth's motion ...... 177 Ancient conceptions. ........ 177 How they gave way. . . . . . . . .178 Earth's rotation . . . . . . . . .178 Influence of centrifugal force upon gravity .... 179 Earth's translation . . . . . . . . .ISO Precession of the equinoxes ....... 180 Retrogradations and stations of the planets .... 181 Aberration of light ......... 181 Influence of scientific fact upon opinion ..... 182 Kepler's Laws .......... 184 Annual parallax ......... 184 Circles I85 Kepler ." 185 His three laws 186 First law 186 Second law .......... 186 Third law I87 Three problems 188 Prediction of Eclipses ........ 188 Transit of Venus . . . . . . . . .189 Foundation of Celestial Mechanics 189 Section 2. Dynamical Phenomeiui ..... 190 Gravitation . . . . . . . . . .190 Character of laws of Motion 190 Their history 191 Newton's demonstration . I93 Old difficulty explained . . . . . . . .194 Term Attraction inadmissible 194 Extent of the demonstration .195 xxxviir CONTENTS. Term Gravitation unobjectionable Gravitation is that of Molecules Secondary gravitation Domain of the laAV . PAGE 196 197 198 198 CHAPTER IV. CELESTIAL STATICS. Consummation by NeAvton Statical considerations First method of inquiry into masses Second method ..... Third method Section 1. Weight of tlic earth Section 2. Form of the i^lanets Difficulty of the inquiry . (xeometrical estimate Estimate from perturbations . Indirect estimate of the earth's form Hydrostatic theory of planetary forms Section 3. The Tides Question of the tides Theory of the tides . Influence of the Sun . Of the moon Composite influence . Kequisites for exactitude CHAPTER V. celestial dynamics. Perturbations . Instantaneous . Gradual perturbations Perturbations of translation Problem of three l)odies . Centre of the Solar System Problem of the Planets . Of the Satellites Of the Comets . Perturbations of rotation . The Planets . The Satellites . . Device of an Invariable Plane Stability of our system . Resistance of a Medium . Independence of the solar system Achievements of celestial dynamics CONTENTS. CHAPTER VI. SIDEREAL ASTRONOMY AND COSMOGONY. I'A(iE Multiple Stars "224 Our Cosmogony .... 226 Origin of Positive Cosmogony . 227 Cosmogony of Laplace 227 Recapitulation 230 BOOK III. PHYSICS. CHAPTER I. GENERAL VIEW. Imperfect condition of the Science 2.31 Its domain 232 Compared with Chemistry 232 Its generality 232 Dealing with masses or molecules . 233 Changes of arrangement or composition ( )f molecules 233 Description of Physics 234 Instruments ..... 235 Methods of inquiry .... 235 Observation 235 Experiment ..... 235 Application of INIathematical analysis 236 Encyclopiedic rank of Physics . 237 Relation to Astronomy . 237 To jNIathematics .... 238 To the other sciences 238 To human j^rogress .... 239 Human power of modifying jihenomena 239 Prevision imperfect .... 240 Characteristics of each science. 241 Philosophy of hypothesis . 241 Necessary condition .... 241 Two classes of hypothesis 242 First class indispensable . 243 Second class chimerical . 243 History of the second class 245 In Astronomy ..... 246 In Physics 247 Rule of arrangement in Physics 247 Order 248 xl CONTENTS. CHAPTER II. BAROLOGY. PAGE Divisions ........... 249 Section 1. Statics 249 History 249 Cases of liquids ......... 250 First case ........... 251 Second case .......... 252 Case of g.ises 253 History .253 Condition of the problem ........ 254 Section 2. Dynamics 255 History 255 Fluids 257 Case of liquids 257 Existing state of Barology 258 CHAPTER III. THERMOLOGY. Its nature 259 History 259 Relation to Mathematics 260 Section 1. Mutual tlmruwlogical itiflncncc .... 260 Two parts. . . . ' 260 Mutual influence 260 Radiation of Heat 261 Propagation by contact ........ 262 Condiictibility 263 Permeability 263 Penetrability 263 Specific Heat 264 Section 2. Constituent changes hy heat 265 Latent heat 266 Change of volume ......... 266 Change in state of aggregation ...... 267 Law of engagement and disengagement of heat . . . 268 Vapours 268 Temperatures of ebullition ....... 269 Hygrometrical equilil)riuni ....... 269 Sections. The7'niolugy connected tvith analysis . . . 270 Section 4. Terrestrial temperatures . . . . .271 Interior heat .......... 272 Temperature of the planetary intervals ..... 272 Conditions of the problem ....... 273 CONTENTS. xli CHAPTER IV. ACOUSTICS. Its nature Relation to the study of Inorganic bodies Relation to Physiology . To Mathematics Divisions ..... Section 1. Propagation of sound Eftect of atmospheric agitation Section 2. Intensity of sounds Section 3. Theory of tones . Composition of sounds Recapitulation .... PAGE 274 274 275 275 278 279 279 280 281 2S.S 2S3 CHAPTER V. optics. Hypotheses on the nature of Light Excessive tendency to systematize Divisions of Optics . Irrelevant matters . Theory of Vision Specitic Colour of bodies . Section 1. Study of direct light Optics proj^er .... Imperfections .... Photometry .... Section 2. Catoptrics . Great law of reflection Law of absorption not found . Section 3. Dioptrics Creat law of refraction Newton's discoveries on elementary colours Section 4. Diffraction .... 285 288 289 289 289 291 292 292 292 293 294 294 294 295 295 297 298 CHAPTER VL electrology. History 300 Condition 300 Arbitrary hypotheses 300 Relation to Mathematics 301 Unsound application ........ 302 Sound application ......... 302 Limits 302 Divisions 302 xlii CONTENTS. Section 1. Electric production Causes of electrization Chemical action Theiniological action Friction Pressure . Contact (Jther causes Instruments Section 2. Electric stedics Great law of Distril)ution Electric equilibrium . Section 3. Electric dynamics Amjjere's experiments Conclusion of Physics PAGE 303 303 303 303 304 304 304 304 304 306 306 306 307 307 310 BOOK IV. CHEMISTRY. CHAPTER I. Its nature .... Great imperfection . Capacities. Object of Chemistry. Specific character of its action Condition of action . 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 Physiolo_<,'y . To Sociology . Degree of possible perfection Intrusion of hypotheses . Actual imperfection . Comparative imperfection Relation to human progress Art of Nomenclature 312 313 313 314 315 315 316 317 317 318 318 318 318 319 320 321 322 322 323 324 324 325 326 326 327 327 CONTENTS. xliii State of chemical doctrine Divisions of the science .... No organic chemistry .... Princiijles of comp(jsition and decomposition PAGE 329 330 331 331 CHAPTER II. INORGANIC CHEMISTRY. Mode of beginning the study . Phirality of elements Classification of elements. Classification of Berzelius. Premature effort Requisite preparation as to method As to doctrine .... First condition .... Second condition ... Method of analysis . Chemical dualism Law of double saline decomposition Chemical theory of air and water Air AVater 333 334 336 337 338 338 339 339 340 340 340 344 345 346 .347 CHAPTER III. DOCTRINE OF DEFINITE PROPORTIONS. Scope of the doctrine As to Doctrine . As to Method . Its history Richter's law . Berth ollet's extension Dalton's further extension Atomic theory . Extension by Berzelius . Extension by Gay-Lussac. WoUaston's verification . Scope of application of Numerical Chemistry Objection of dissolution , Of metallic alloys Of organic substances Application of the principle of dualism 349 350 350 350 350 .351 352 352 3,53 354 3.54 355 357 358 358 359 xliv CONTENTS. CHAPTER IV. THE ELECTRO-CHEMICAL THEORY. Relation of Electricity to Chemistry History of the case . Nicholson's discovery Davy's discovery Berzelius's extension Synthetical process . Recquerel's Study of Combustion Lavoisier's theory First division . Berthollet's limitation Second division. Difficulties of the theory Its position and powers I'AliK 363 363 363 364 364 365 365 366 366 367 367 368 370 373 CHAPTER V. ORGANIC CHEMISTRY. Confusion of two kinds of fact. Relation of Chemistry to Anatomy . To Physiolof,''y . ..... Partition of Organic Chemistry A])plication of dualism to organic compoundi- Summary of the Chapter .... Summary of the Rook .... 375 377 37S 3S1 383 384 384 THE POSITIVE PHILOSOPHY OF AUGUSTE COMTE. INTEODUCTION. CHAPTER I. ACCOUNT OF THE AIM OF THIS WOKK. VIEW OF THE NATURE AND IMPORTANCE OF THE POSITIVE PHILOSOPHY. A GENERAL statement of any system of philosophy may be either a sketch of a doctrine to be established, or a summary of a doctrine already established. If greater value belongs to the last, the first is still important, as characterizing from its origin the subject to be treated. In a case like the present, where the proposed study is vast and hitherto indeterminate, it is esjjecially important that the field of research should be marked out with all possible accuracy. For this purpose, I will glance at the considerations which have originated this work, and which will be fully elaborated in the course of it. In order to understand the true value and character of the Positive Philosophy, we must take a brief general view of the progressive course of the human mind, regarded as a whole ; for no conception can be understood otherwise than through its history. From the study of the development of human intelligence, in all directions, and progress"™*" through all times, the discovery arises of a ° * great fundamental law, to which it is necessarily subject, and which has a solid foundation of pi'oof, both in the facts of our organization and in our historical experience. The law is this : — that each of our leading conceptions, — I. B 2 POSITIVE PHILOSOPHY. each braucli of our knowledge, — passes successively through three different theoretical conditions : the Theological, or fictitious ; the Metaphysical, or abstract ; and the Scientific, or positive. In other words, the human mind, by its nature, employs in its progress three methods of philoso- phizing, the character of which is essentially different, and even radically oj^posed : viz., the theological method, the metaphysical, and the positive. Hence arise three philoso- phies, or general systems of conceptions on the aggregate of phenomena, each of which excludes the others. The first is the necessary point of departure of the human understanding ; and the third is its fixed and definitive state. The second is merely a state of transition. In the theological state, the human mind, ^ ' ^ • seeking the essential nature of beings, the first and final causes (the origin and i^urpose) of all effects, — in short, Absolute knowledge, — supposes all phenomena to be produced by the immediate action of suj)ernatural beings. , „ In the metaphysical state, which is only a econ e . j^Q(j^f^(,g^^|Qjj^ Qf the first, the mind supposes, instead of supernatural beings, abstract forces, veritable entities (that is, personified abstractions) inherent in all beings, and capable of producing all phenomena. What is called the explanation of phenomena is, in this stage, a mere reference of each to its proper entity. ™, . , ^^ In the final, the positive state, the mind " ' has given over the vain search after Absolute notions, the origin and destination of the universe, and the causes of phenomena, and applies itself to the study of their laws, — that is, their invariable relations of succession and resemblance. Reasoning and observation, duly com- bined, are the means of this knowledge. What is now understood when we speak of an explanation of facts is simply the establishment of a connection between single phenomena and some general facts, the number of which continually diminishes with the progress of science. The Theological system arrived at the of each.*'' ^''''"*' highest perfection of which it is capable when it substituted the providential action of a single Being for the varied operations of the numerous divinities which had been before imagined. In the same GROUNDS OF THE LAW OF PROGRESS. 3 way, in the last stage of the Metaphysical system, meu substitute one great entity (Nature) as the cause of all phenomena, instead of the multitude of entities at first supposed. In the same way, again, the ultimate perfection of the Positive system would be (if such perfection could be hoped for) to represent all phenomena as j^articular aspects of a single general fact ; — such as Gravitation, for instance. The importance of the working of this general law will be established hereafter. At j^resent, it must suffice to point out some of the grounds of it. There is no science which, having attained the positive stage, does not bear marks of the Kw^^*^ *^ having passed through the others. Some time since it was (whatever it might be) composed, as we can now perceive, of metaphysical abstractions ; and, further back in the course of time, it took its form from theological conceptions. We shall have only too much . , , occasion to see, as we proceed, that our most advanced sciences still bear very evident marks of the two earlier periods through which they have passed. The progress of the individual mind is not only an illus- tration, but an indirect evidence of that of the general mind. The j^oiut of departure of the individual and of the race being the same, the phases of the mind of a man correspond to the epochs of the mind of the race. Now, each of us is aware, if he looks back upon his own history, that he was a theologian in his childhood, a metaphysician in his youth, and a natural philosopher in his manhood. All men who are up to their age can verify this for themselves. Besides the observation of facts, we have theoretical reasons in support of this law. The most important of these reasons arises rpi ..g+jcni from the necessity that always exists for some theory to which to refer our facts, combined with the clear impossibility that, at the outset of human knowledge, men could have formed theories out of the observation of facts. All good intellects have rej^eated, since Bacon's time, that there can be no real knowledgs but that which is based on observed facts. This is incontestible, in our present advanced stage ; but, if we look back to the primitive stage of human knowledge, we shall see that it 4 POSITIVE PHILOSOPHY, must have been otherwise then. If it is true that every theory must be based upon observed facts, it is equally true that facts cannot be observed without the guidance of some theory. Without such guidance, our facts would be desultory and fruitless ; we could not retain them : for the most part we could not even perceive them. Thus, between the necessity of observing facts in order to form a theory, and having a theory in order to observe facts, the human mind would have been entangled in a vicious circle, but for the natural opening afforded by Theological conceptions. This is the fundamental reason for the theological character of the primitive philosophy. This necessity is confirmed by the perfect suitability of the theological philosophy to the earliest researches of the human mind. It is remarkable that the most inaccessible questions, — those of the nature of beings, and the origin and purpose of phenomena, — should be the first to occur in a primitive state, while those which are really within our reach are regarded as almost unworthy of serious study. The reason is evident enough : — that experience alone can teach us the measure of our powers ; and if men had not begun by an exaggerated estimate of what they can do, they would never have done all that they are capable of. Our organization requires this. At such a period there coxild have been no reception of a positive philosophy, whose function is to discover the laws of phenomena, and whose leading characteristic it is to regard as interdicted to human reason those sublime mysteries which theology explains, even to their minutest details, with the most attractive facility. It is just so under a practical view of the nature of the researches with which men first occupied themselves. Such inquiries offered the powerful charm of unlimited empire over the external world, — a world destined wholly for our use, and involved in every way with our existence. The theological philosophy^ presenting this view, administered exactly the stimulus necessary to incite the human mind to the irksome labour without which it could make no progress. We can now scarcely conceive of such a state of things, our reason having become sufficiently mature to enter upon laborious scientific researches, without needing any such stimulus as wrought upon the imagina- METAPHYSICAL AND POSITIVE PHILOSOPHY. 5 tions of astrologers and alchemists. We have motive enough in the hope of discovering the laws of phenomena, with a view to the confii-mation or rejection of a theory. But it could not be so in the earliest days ; and it is to the chimeras of astrology and alchemy that we owe the long series of observations and experiments on which our positive science is based. Kepler felt this on behalf of astronomy, and Berthollet on behalf of chemistry. Thus was a spon- taneous philosophy, the theological, the only possible be- ginning, method, and provisional system, out of which the Positive philosophy could grow. It is easy, after this, to perceive how Metaphysical methods and doctrines must have afforded the means of transition from the one to the other. The human understanding, slow in its advance, could not step at once from the theological into the positive philosophy. The two are so radically opposed, that an intermediate system of conceptions has been necessary to render the transition possible. It is only in doing this, that Metaphysical conceptions have any utility whatever. In contemplating phenomena, men substitute for supernatural direction a corresponding entity. This entity may have been supposed to be derived from the supernatural action : but it is more easily lost sight of, leaving attention free for the facts themselves, till, at length, metaphysical agents have ceased to be anything more than the abstract names of phenomena. It is not easy to say by what other process than this our minds could have passed from supernatural considerations to natural ; from the theological system to the positive. The Law of human development being thus established, let us consider what is the proper nature of the Positive Philosophy. As we have seen, the first characteristic of Character of the Positive Philosophy is that it regards all the Positive phenomena as subjected to invariable natural Philosophy. Laws. Our business is, — seeing how vain is any research into what are called Causes, whether first or final, — to pursue an accurate discovery of these Laws, with a view to reducing them to the smallest possible number. By speculating upon causes, we could solve no difficulty about 6 POSITIVE PHILOSOPHY. oi'igin and purpose. Our real business is to analyse accu- rately the circumstances of phenomena, and to connect them by the natural relations of succession and resemblance. The best illustration of this is in the case of the doctrine of Gravitation. We say that the general phenomena of the universe are explained by it, becaiise it connects under one head the whole immense variety of astronomical facts ; exhibiting the constant tendency of atoms towards each other in direct proportion to their masses, and in inverse proportion to the squares of their distances ; whilst the general fact itself is a mere extension of one which is per- fectly familiar to us, and which we therefore say that we know ; — the weight of bodies on the surface of the earth. As to what weight and attraction are, we have nothing to do with that, for it is not a matter of knowledge at all. Theologians and metaphysicians may imagine and refine about such questions ; but positive philosophy rejects them. When any attempt has been made to explain them, it has ended only in saying that attraction is universal weight, and that weight is terrestrial attraction: that is, that the two orders of j^henomena are identical ; which is the point fi'om which the question set out. Again, M. Fourier, in his fine series of researches on Heat, has given us all the most important and precise laws of the pheno- mena of heat, and many large and new truths, without once inquiring into its nature, as his predecessors had done when they disputed about calorific matter and the action of an universal ether. In treating his subject in the Positive method, he finds inexhaustible material for all his activity of research, without betaking himself to insoluble questions. History of the Before ascertaining the stage which the Positive Phi- Positive Philosophy has reached, we must losophy. bear in mind that the different kinds of our knowledge have passed through the three stages of progress at different rates, and have not therefore arrived at the same time. The rate of advance depends on the nature of the knowledge in question, so distinctly that, as we shall see hereafter, this consideration constitutes an accessory to the fundamental law of progress. Any kind of knowledge reaches the positive stage early in proportion to its gene- rality, simplicity, and independence of other departments. NEW PROVINCE OF POSITIVE PHILOSOPHY. 7 Astronomical science, wliicli is above all made np of facts tliat are general, simple, and independent of other sciences, arrived first ; then terrestrial Physics ; then Chemistry ; and, at length, Physiology. It is difficult to assign any precise date to this revolution in science. It may be said, like everything else, to have been alvrays going on ; and especially since the labours of Aristotle and the school of Alexandria ; and then from the introdiiction of natural science into the West of Europe by the Arabs. But, if we must fix upon some marked period, to serve as a rallying point, it must be that, — about tvro centuries ago, — when the human mind was astir under the precepts of Bacon, the conceptions of Descartes, and the discoveries of Galileo. Then it was that the spirit of the Positive philosophy rose \vp in opposition to that of the superstitious and scholastic systems which had hitherto obscured the true character of all science. Since that date, the progress of the Positive philosophy, 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 philosophy. 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 universality which is necessary to its definitive constitution. In mentioning just now the four principal categories of phenomena, — astronomical, physical, chemical, and jjhysio- logical, — there was an omission which will have been noticed. Nothing was said of Social pheno- ]s[ew depart- mena. Though involved with the physio- ment of Posi- logical. Social phenomena demand a distinct ^ive philoso- elassification, both on account of their im- 1' ^^' portance and of tbeir difficulty. They are the most indi- vidual, the most complicated, the most dependent on all others ; and therefore they must be the latest, — even if they had no special obstacle to encounter. This branch of ^ science has not hitherto entered into the domain of Positive , philosophy. Theological and metaphysical methods, ex- ■ ploded in other departments, are as yet exclusively applied, both in the way of inquiry and discussion, in all treatment 8 POSITIVE PHILOSOPHY. of Social subjects, though the best minds are heartily weary of eternal disputes about divine right and the sove- reignty 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 vege- table 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 principal aim of the jDresent work to establish. r, • 1 Til • It would be absurd to pretend to offer this Social Physics. . , . ^ i , , , new science at once m a complete state. Others, less new, are in very unequal conditions of forward- ness. But the same character of positivity which is im- pressed on all the others will be shown to belong to this. This once done, the philosophical system of the moderns will be in fact complete, as there will then be no pheno- menon which does not naturally enter into some one of the five great categories. All our fundamental conceptions having become homogeneous, the Positive state will be fully established. It can never again change its character, though it will be for ever in course of development by additions of new knowledge. Having acquired the character of universality which has hitherto been the only advantage resting with the two preceding systems, it will supersede them by its natural supeinority, and leave to them only an historical existence. „ , . We have stated the special aim of this becondary aim i tx t i i • ■ of this work "work. Its secondary and general aim is this : — to review what has been effected in the Sciences, in order to show that they are not radically separate, but all branches from the same trunk. If we had confined ourselves to the first and special object of the work, we should have produced merely a study of Social physics : whereas, in introducing the second and general, we offer a study of Positive philosophy, passing in review all the positive sciences already formed. To review the ^^^ jDurpose of this work is not to give au Ijhik)sophy of account of the Natural Sciences. Besides tlie Sciences, that it would be endless, and that it would DESULTORY DIVISION OF RESEARCH. 9 require a scientific preparation sucli as no one man possesses, it would be ajiart from our object, which is to go through 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 jjliilosophy 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 fundamental sciences are not all equally interesting to ordinary minds, there is no one of them that can be neglected in an inquiry like the pi'esent ; 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 on account of the perfection of their methods, or as being the necessary basis of all the others. Lest it should be suY>posed that our course ^ p„j„i ;*. , will lead us into a wilderness of such special studies as are at present the bane of a true positive philo- sophy, we will briefly advert to the existing prevalence of such special pursuit. 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 de- vote themselves each to some one branch. It is owing to this division of employment, and concentration of whole minds upon a single department, that science has made so pro- digious 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 particidar study. It is inevitable that each should be possessed with exclusive notions, and be thei'efore in- capable of the general superiority of ancient students. 10 POSITIVE PHILOSOPHY. who actually owed that general superiority to the in- feriority of their knowledge. We must consider whether the evil can be avoided without losing the good of the modern arrangement ; for the evil is becoming urgent. We all acknowledge that the divisions established for the 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 knowledge. We must not be blind to the evil, nor slow in seeking a remedy. We must not forget that this is the weak side of the posi- tive philosophy, by which it may yet be attacked, with some hope of success, by the adherents of the theological and metaphysical systems. As to the remedy, 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 perfecting the division of em- ployments 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, suitably Proposed new prepared, whose business it shall be to take class of stu- the respective sciences as they are, determine dents. i\^Q sjjirit of each, ascertain their relations and mutual connection, and reduce their respective princi- ples to the smallest number of general principles, in con- formity with the fundamental rules of the Positive Method. At the same time, let other students be j)rei3ared for their special pursuit by an education which recognizes the whole scope of positive science, so as to profit by the labours of the students of generalities, and so as to correct recipro- cally, under that guidance, the residts obtained by each. We see some aj^proach already to this arrangement. Once established, there would be nothing to apprehend from any extent of division of employments. When we once have a class of learned men, at the disposal of all others, whose business it shall be to connect each new discovery with the general system, we may dismiss all fear of the great whole being lost sight of in the pursuit of the details of know- ledge. The organization of scientific research will then be FIRST BENEFIT. 11 complete ; and it will henceforth have occasion only to ex- tend its development, and not to change its character. After all, the formation of such a new class as is proposed would be merely an extension of the principle which has created all the classes we have. While science was narrow, there was only one class : as it expanded, more were insti- tuted. With a further advance a fresh need arises, and this new class will be the result. The general spirit of a course of Positive Advantages of Philosophy having been thus set forth, we the Po.sitive must now glance at the chief advantages t*liilosopliy. which may be derived, on behalf of human progression, from the study of it. Of these advantages, four may be especially pointed out. I. The study of the Positive Philosophy lllnstratesthe affords the only rational means of exhibiting Intellectual thelogical laws of the human mind, which have function, hitherto been sought by unfit methods. To explain what is meant by this, we may refer to a saying of M. de Blain- ville, 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 con- siderations range themselves under the one or the other of these heads. Let us apply this classification to the intel- lectual functions. If we regard these functions under their Statical aspect — that is, if we consider the conditions under which they ■exist — we must determine the organic circumstances of the case, which inquiry involves it with anatomy and physiology. If we look at the Dynamic aspect, we have to study simply the exercise and results of the intellectual powers of the human race, which is neither more nor less than the general object of the Positive Philosophy. In short, looking at all scientific theories as so many great logical facts, it is only by the thorough observation of these facts that we can arrive at the knowledge of logical laws. These being the only means of knowledge of intellectual phenomena, the illusory psychology, which is the last phase of theology, is excluded. It pretends to accomplish the discovery of the laws of the human mind by contemplating it in itself ; 12 POSITIVE PHILOSOPHY. that is, by separating it from causes and effects. Such an attempt, made in defiance of the physiological study of our intellectual organs, and of the observation of rational methods of procedure, cannot succeed at this time of day. The Positive Philosophy, which has been rising since the time of Bacon, has now secured such a preponderance, that the metaphysicians themselves profess to ground their pre- tended science on an observation of facts. They talk of ex- ternal and internal facts, and say that their business is with the latter. This is much like saying that vision is explained by luminous objects painting their images upon the retina. To this the physiologists reply that another eye woiild be needed to see the image. In the same manner, the mind may observe all phenomena but its own. It may be said that a man's intellect may observe his passions, the seat of the reason being somewhat apart from that of the emotions in the brain ; but there can be nothing like scientific observation of the passions, except from, without, as the stir of the emotions disturbs the observ- ing faculties more or less. It is yet more out of the ques- tion to make an intellectual 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 j^ause from activity ; yet it is this very activity that you want to observe. If you can- not 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 jjursuit, no one proposition is established to the satisfaction of its followers. They are divided, to this day, into a multitude of schools, still dis- jjuting about 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 was no better work for our faculties to do ; and they may have added something to our stock of knowledge. If they have done so, it is by practising the Positive method — by observing SECOND BENEFIT. 13 the progress of tlie liuman 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 contempla- tion would be only a dead study, which could produce nothing in the mind which loses time upon it. We may talk for ever about the method, and state it in terms very wisely, withoi:t knowing half so much about it as the man who has once j^ut it in practice ujDon a single particular of actual research, even without any philosophical intention. Thus it is that psychologists, by dint of reading the pre- cepts of Bacon and the discourses of Descartes, have mis- taken their own dreams for science. Without saying whether it will ever be possible to estab- lish 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 ajiplications. 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 Philo- sophy — the mai^ifestation by experiment of the laws which rule the Intellect in the investigation of truth ; and, as a consequence the knowledge of the general rules suitable for that object. II. The second effect of the Positive Must re^ene- Philosophy, an effect not less important and late Educa- far more urgently wanted, will be to regene- *^'^"- rate Education. The best minds are agreed that our European education, still essentially theological, metaphysical, and literary, must be superseded by a Positive training, conformable to our time and needs. Even the governments of our day have shared, where they have not originated, the attempts to establish positive instruction ; and this is a striking in- 14 POSITIVE PHILOSOPHY. dication 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 j^resent exclusive speciality of our pursuits, and the consequent isolation of the sciences, spoil our teaching. If any student desires to form an idea of natural philosophy as a whole, he is compelled to go through each department as it is now taught, as if 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 concej^tious of all the classes of natural phenomena. It is such an aggregate of concep- tions, whether on a great or on a small scale, which must henceforth be the permanent basis of all human combina- tions. It will constitute the mind of future generations. In order to this regeneration of our intellectual system, it is necesary that the sciences, considered as branches from one trunk, should yield us, as a whole, their chief methods and their most important results. The specialities 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. Advances HI- The same special study of scientific sciences by generalities must also aid the progress of the combining respective positive sciences : and this consti- them. tutes our third head of advantages. The divisions which we establish between the sciences are, though not arbitrary, essentially artificial. The sub- ject of our researches is one : we divide it for our con- venience, in order to deal the more easily with its difficul- ties. But it sometimes happens — -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 sevei'al special points of view ; and for want of this, very important problems wait for their solution much longer than they otherwise need do. To go back into the past for an examj)le : Descartes' grand THIRD AND FOURTH BENEFITS. 15 conception with regard to analytical geometry is a dis- covery which has changed the whole aspect of mathe- matical science, and yielded the germ of all future pro- gress ; aud it issued from the union of two sciences which had always before been separately regai'ded and pursued. The case of pending questions is yet more impressive ; as, for instance, in Chemistry, the doctrine of Definite Propor- tions. Without entering upon the discussion of the funda- mental principle of this theoxy, we may say with assurance that, in order to determine it — in order to determine whether it is a law of nature that atoms should necessarily combine in fixed numbers, — it will be indispensable that the chemical point of view should be united with the physiological. The failure of the theory with regard to organic bodies indicates that the cause of this immense exception must be investigated ; and such an inquiry be- longs as much to physiology as to chemistry. Again, it is as yet undecided whether azote is a simj^le or a compound body. It was concluded by almost all chemists that azote is a simple body ; the illustrious Berzelius liesitated, 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. From this we see how physiology must unite with chemistry to inform us whether azote is simple or compound, and to institute a new series of re- searches 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 philosophy — the elucidation of the respec- tive sciences by their combination. In the fourth place IV. The Positive Philosophy offers the only solid basis for that Social Reorganiza- „^,j.\.^„;„(!;' ' tion which must succeed the 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 funda- 16 - POSITIVE PHILOSOPHY. mental maxims is the first condition of genuine social order, we are suffering under an utter disagreement which may be called universal. Till a certain number of general ideas can be acknowledged as a rallying-point of social doctrine, the nations will remain in a revolutionary state, whatever palliatives may be devised; and their institutions can be only provisional. But whenever the necessary agreement on first principles can be obtained, appropriate institutions will issue from them, without shock or resis- tance ; 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 desire a natural and regular, a normal state of society. Now, the existing disorder is abundantly accounted for by the existence, all at once, of three incompatible philo- sophies, — the theological, the metaphysical, and the posi- tive. Any one of these might alone secure some sort of social order ; but while the three co-exist, it is impossible for us to understand one another upon any essential point whatever. If this is true, we have only to ascertain which of the philosophies must, in the nature of things, prevail ; aud, this ascertained, every man, whatever may have been his former views, cannot but concur in its triumph. The problem once recognized cannot remain long unsolved ; for all considerations whatever point to the Positive Philosophy as the one destined to prevail. It alone has been advancing during a course of centuries, throughout which the others have been declining. The fact is incontestable. Some may deplore it, but none can destroy it, nor therefore neglect it but under penalty of being betrayed by illusory speculations. This genei'al revolution of the human mind is nearly accomplished. We have only to complete the Positive Philosophy by bringing Social phenomena within its comprehension, and afterwards consolidating the whole into one body of homogeneous doctrine. The marked pre- ference 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 this philosophy will be when it has acquired the only quality that it now wants — a character of due generality. When it has become complete, its supremacy will take PRECAUTIONARY OBSERVATION. 17 place spontaneously, and will re-establisli order throughout society. There is, at present, no conflict but between the theological and the metaphysical philosophies. They are contending for the task of reorganizing society ; but it is a work too mighty for either of them. The positive philo- sophy has hitherto intervened only to examine both, and both are abuudaiitly discredited by the process. It is time now to be doing something more effective, without wasting our forces in needless controversy. It is time to complete the vast intellectual operation begun by Bacon, Descartes, and Galileo, by constructing the system of general ideas which must henceforth j^revail among the human race. This is the way to put an end to the revolu- tionary crisis which is tormenting the civilized nations of the world. Leaving these four points of advantage, we must attend to one precautionary reflection. Because it is proposed to consolidate the No hope of re- whole of our acquired knowledge into one chiction to a body of homogeneous doctrine, it must not single law. be supposed that we are going to study this vast variety as proceeding from a single principle, and as subjected to a single law. There is something so chimerical in attempts at universal explanation by a single law, that it may be as Avell to secure this Work at once from any imputation of the kind, though its development will show how lui deserved such an imputation would be. Our intellectual resources are too narrow, and the universe is too complex, to leave any hope that it will ever be within our power to carry scientific perfection to its last degree of simplicity. More- over, it appears as if the value of such an attainment, sup- posing it possible, were greatly overrated. The only way, for instance, in which we 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 witli a portion of terrestrial physics. 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 sup- posing this view jjroveable (which it cannot be wliile we are I. c 18 POSITIVE PHILOSOPHY. without 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, with the difference that we now regard as natural that division which we should then call artificial. Laplace himself presented his idea only as a philosophic device, incapable of exercising any useful in- fluence over the progress of chemical science. Moreover, supposing this insuperable difficulty overcome, we should be no nearer to scientific unity, since we then should still have to connect the whole of physiological phenomena with the same law, which certainly would not be the least diffi- cult part of the enterprise. Yet, all things considered, the hypothesis we have glanced at would be the most favour- able 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 we anticipate from the positive philosophy. The only necessary unity is that of Method, which is already in great part established. As for the doctrine, it need not be one ; it is enough that it should be homogeneous. It is, then, under the double aspect of unity of method and homogeneousness of doctrine that we shall consider the different classes of positive theories in this work. While pursuing the philosophical aim of all science, the lessening of the number of general laws requisite for the explanation of natural phenomena, we shall regard as presumptuous every attemj^t, in all future time, to reduce them rigorously to one. Having thus endeavoured to determine the spii'it and influence of the Positive Philosophy, and to mark the goal of our labours, we have now to proceed to the exposition of the system ; that is, to the determination of the universal, or encyclopaedic order, which must regulate the different classes of natural j^henomena, and consequently the corre- sjionding jDositive sciences. 19 CHAPTER II. VIEW OF THE HIERARCHY OF THE POSITIVE SCIENCES. IN proceeding to offer a Classificatiou of the Sciences, we must leave on one side all others that have as yet been attempted. Snch scales are those of Bacon and D' Alembert are constructed upon an arbitrary division of Faihireof pro- the faculties of the mind ; whereas, our posed classiti- principal faculties are often engaged at the cations, same time in any scientific pursuit. As for other classifica- tions, 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 failui-e has been so conspicuous, 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 somewhat discredited task, has of late been undertaken chiefly by persons who have no sound knowledge of any science at all. A more important and less personal reason, however, is, the want of homogeueousness in the different parts of the intellectual system, — some having successively become posi- tive, while others remain theological or metai^hysical. 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 con- trariety existed between the materials he was endeavouring to combine. The fact was clear enough, if it had but been understood, that the enterprise was premature ; and that it was useless to undertake it till our principal scientific conceptions should all have become positive. The preceding chapter seems to show that this indispensable condition may now be considered fulfilled : and thus the time has 20 POSITIVE PHILOSOPHY. arrived for laying down a sound and durable system of scientific order. We may derive encouragement from tbe example set by recent 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 deter- mined by a priori considerations. The real afiinities and natural connections presented by objects being allowed to determine their order, the classification itself becomes the expression of the most general fact. And thus does the positive method apply to the question of classification itself, as well as to the objects included under it. It follows True principle that the mutual dependence of the sciences, — of classifica- a dependence resulting from that of the corre- tion. sponding phenomena, — must determine the arrangement of the system of human knowledge. Before proceeding to investigate this mutual dependence, we have only to ascertain the real bounds of the classification pro- posed : in other words, to settle what we mean by human knowledge, as the subject of this work. „ , . . The field of human labour is either specula- lioundaries or , ■ , . , ,i ^ , -v our field ""^^^ ^■'' ^^ction : and thus, we are accustomed to divide our knowledge into the theoretical and the practical. It is obvious that, in this inquiry, we have to do only with the theoretical. We are not going to treat of all human notions whatever, but of those funda- mental conceptions of the different orders of phenomena which furnish a solid basis to all combinations, and are not founded on any antecedent intellectual system. In such a study, speculation is our material, and not the application of it, — except where the application may happen to throw back light on its speculative origin. This is probably what Bacon meant by that First Philosophy which lie declared to be an extract from the whole of Science, and which has been so differently and so strangely interpreted by his metaphysical commentators. There can be no doubt that Man's study of nature must furnish 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 THE VALUE OF SCIENTIFIC THEORIES. 21 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 regarding Science chiefly as a basis of Art. However great may be the services rendered to Industry by science, how- ever true may be the saying that Knowledge is Power, we must never forget that the sciences have a higher destina- tion 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 physio- logical effects of consternation, and to remember that the most terrible sensation we are capable of, is that which we experience when aiiy phenomenon seems to arise in violation of the familiar laws of nature. This need of disposing facts in a comprehensible order (which is the proper object of all scientific theories) is so inherent in our organization, that if we could not satisfy it by positive conceptions, we must inevitably return to those theological and metaphy- sical explanations which had 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 incompleteness which the practical habits 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 understanding would soon, as Con- dorcet has observed, come to a stand, even as to the practical applications for the sake of which higher things had been sacrificed ; for, if the arts flow from science, the neglect of science must destroy the consequent arts. Some of the most important arts are derived from speculations pvu'sued 22 POSITIVE PHILOSOPHY, duriBg long ages with a purely scientific intention. For instance, the ancient Greek geometers delighted themselves with beautiful sj^eculations on Conic Sections ; those specu- lations wrought, after a long series of generations, the renovation of astronomy ; and out of this has the art of navigation attained a perfection which it never could have reached otherwise than through the speculative labours of Archimedes and Apollonius : so that, to iise Condorcet's illustration, " the sailor who is preserved from shipwreck by the exact observation of the longitude, owes his life to a theory conceived two thousand years before by men of genius who had in view simply geometrical speculations." Our business, it is clear, is with theoretical researches, letting alone their practical application altogether. Though we may conceive of a course of study which should unite the generalities of speculation and application, the time is not come for it. To say nothing of its vast extent, it would require preliminary achievements which have not yet been attempted. We must first be in possession of appropriate Special concejjtions, formed according to scien- tific theories ; and for these we have yet to wait. Mean- time, an intermediate class is rising up, whose particular destination is to organize the relations of theory and prac- tice ; such as the engineers, who do not labour in the advancement of science, but who study it in its existing state, to apply it to practical purposes. Such classes are furnishing us with the elements of a future body of doc- trine on the theories of the different arts. Already, Monge, in his view of descriptive geometry, has given us a general theory of the arts of construction. But we have as yet only a few scattered instances of this nature. The time will come when out of such results, a dejjartment of Positive philosophy may arise : but it will be in a distant future. If we remember that several sciences are implicated in every important art, — that, for instance, a true theory of Agriculture requires a combination of physiological, chemi- cal, mechanical, and even astronomical and mathematical science, — it will be evident that true theories of the arts must wait for a large and equable develo])ment of these constituent sciences. One more preliminary remark occurs, before we finish the THE PRIMARY AND SECONDARY SCIENCES. 23 prescription of our limits, — the ascertainment of our field of inquiry. We must distinguish between the two classes of Natural science ; — the abstract or general, Abstract which have for their object the discovery of science, the laws which regulate phenomena in all conceivable cases : and the concrete, particu- Concrete lar, or descriptive, which are sometimes called science. Natural sciences in a restricted sense, whose function it is to apply these laws to the actual history of existing beings. The first are fundamental ; and oiir 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 chemisti-y, but not of mineralogy, 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 preliminai'y 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 incoherent 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 know- ledge and of practical knowledge, we have to deal only with the first ; and Second, that theoretical knowledge, or science properly so called, being divided into general and particular, or abstract and concrete science, we have again to deal only with the first. Being thus in possession of our proper subject, duly prescribed, we may proceed to the ascertainment of the true order of the fundamental sciences. The classification of the sciences is not so easy a matter as it may appear. However elLificSion. natural it may be, it will always involve 24 POSITIVE PHILOSOPHY. something, if not arbitrary, at least artificial ; and in so far, it will always involve imperfection. It is impossible to fulfil, quite rigorously, the object of presenting the sciences iu their natural connection, and according to their mutual dependence, so as to avoid the smallest danger of being in- volved in a vicious circle. It is easy to show why. Historical and Every science may be exhibited under two dogmatic methods or procedures, the Historical and methods. ^i^q Dogmatic. These are wholly distinct from each other, and any other method can be nothing but some combination of these two. By the first method know- ledge is presented in the same order in which it was actu- ally obtained by the human mind, together with the way iu which it was obtained. By the second, the system of ideas is presented as it might be conceived of at this day, by a mind which, duly prepared and placed at the right l>oint 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 order the different works which have contributed to the progress of the science. But when such materials have become re- cast to form a general system, to meet the demand for a moi*e natural logical order, it is because the science is too far advanced for the historical order to be practicable or suitable. The more discoveries are made, the greater be- comes the labour of the historical method of study, and the more effectual the dogmatic, because the new conceptions bring forward the earlier ones in a fresh light. Thus, the education of an ancient geometer consisted simply in the study, in their due order, of the very small number of original treatises then existing on the dilferent parts of geometry. The writings of Archimedes and Apollonius were, in fact, about all. On the contrary, a modern geometer comnronly 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 superseding the Historical, as we advance to a higher j^osition in science. If every mind had to pass through all the stages that every predecessor iu the study had gone through, it is clear that, however easy it is to learn rather than invent, it would be impossible HISTORICAL AND DOGMA'lIC METHODS OF STUDY. 25 to effect the purpose of education, — to place the student on the vantage-ground gained by the labours of all the men who have gone before. By the 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 the dogmatic method, therefore, must every advanced science be attained, with so much of the historical combined with it as is rendered necessary by discoveries too recent to be studied elsewhere than in their own records. The only objection to the preference of the Dogmatic method is that it does not show how the science was attained; but a moment's reflection will show that this is the case also with the Historical method. To pursue a science histori- cally is quite a different thing from learning the history of its progress. This last pertains to the study of human history, as we shall see when we reach the final division of this work. It is true that a science cannot l)e completely understood without a knowledge of how it arose ; and again, a dogmatic knowledge of any science is necessary to an un- derstanding of its history ; and therefore we shall notice, in treating of the fundamental sciences, the incidents of their origin, when distinct and illustrative ; and we shall use their history, in a scientific sense, in our treatment of Social Physics ; but the historical study, important, even essential, as it is, remains entirely distinct from the 2)roper dogmatic study of science. These considerations, in this place, tend to define more precisely the spirit of our course of inquiry, while they more exactly determine the conditions imder which we may hope to succeed in the construction of a true scale of the aggregate fundamental sciences. Great con- fusion would arise from any attempt to adhere strictly to historical order in our exposition of the sciences, for they have not all advanced at the same rate ; and we must be for ever borrowing from each some fact to illustrate another, without regard to priority of origin. Thus, it is clear that, in the system of the sciences, astronomy must come before physics, properly so called : and yet, several branches of physics, above all, optics, are indispensable to the complete exposition of astronomy. Minor defects, if inevitable, can- not invalidate a classification which, on the whole, fulfils the principal conditions of the case. They belong to what 26 POSITIVE PHILOSOPHY. is essentially artificial in our division of intellectual labour. In the main, however, our classification agrees with the history of science ; the more general and simple sciences actually occurrring first and advancing best in human his- tory, and being followed by the more complex and restricted, though all wei'e, since the earliest times, enlarging simul- taneously. A simple mathematical illustration will precisely represent the difficulty of the question we have to resolve, while it will sum up the preliminary considerations we have just concluded. We propose to classify the fundamental sciences. They are six, as we shall soon see. We cannot make them less ; and most scientific men would reckon them as more. Six objects admit of 720 different dispositions, or, in populai" 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 not one in favour of which some plausible reason might not be assigned ; for Ave see the wildest divergences among the schemes which have been proposed, — the sciences which are placed by some at the head of the scale being sent by others to the further extremity. Our problem is, then, to find the one rational order, among a host of possible systems. True principle Now we must remember that we have to of classifica- look for the principle of classification in the t'ion. comparison of the different orders of pheno- mena, through which science discovers the laws which are her object. What we have to determine is the real de- pendence of scientific studies. Now, this dependence can result only from that of the corresponding phenomena. All observable ]>henomena may be included within a very few natural categories, so arranged as that the study of each category may be grounded on the principal laws of the preceding, and serve as the basis of the next ensuing. This order is determined by the degree of simplicity, GeneraUty. <^>r. what comes to the same thing, of gene- rality of their phenomena. Hence results Dependence. their successive dependence, and the greater or lesser facility for being studied. FU>DAMENTAL PRINCIPLES OF CLASSIFICATION. 2^ It is clear, a priori, that the most simple phenomena must be the most general; for whatever is observed in the greatest number of cases is of course the most disengaged from the incidents of particular cases. We must begin then with the study of the most general or simple pheno- mena, going on successively to the more particular or com- plex. This must be the most methodical way, for this order of generality or simplicity fixes the degree of facility in the study of phenomena, while it detennines the neces- sary connection of the sciences by the successive depen- dence of their phenomena. It is worthy of remark in this j^lace that the most general and simple phenomena are the furthest removed from Man's ordinary sphere, and must thereby be studied in a calmer and more rational frame of mind than those in which he is more nearly implicated ; and this constitutes a new ground for the corresponding sciences being developed more rapidly. We have now obtained our rule. Next we j^roceed to our classification. We ai'e first struck by the clear division of Inorranches. The one has for its object the of functions. resolution of equations when they are directly established between the magnitudes in question : the other, setting out from equations (generally much more easy to form) between quantities indirectly connected with those of the problem, has to deduce, by invariable analytical procedures, the corre- sponding equations between the direct magnitudes in ques- tion ; — bringing the problem within the domain of the pre- ceding calculus. — It might seem that the transcendental analysis ought to be studied before the ordinary, as it pro- vides the equations which the other has to resolve. But, though the transcendental is logically indej^endent of the ordinary, it is best to follow the usual method of study, taking the ordinary first ; for, the proposed questions always requiring to be completed by ordinary analysis, they must be left in suspense if the instrument of resolution had not been studied beforehand. To ordinary analysis I propose to give the name of Cal- CTTLIJS OF DiKECT 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., ac- cording to the view in which it has been conceived,) I shall give the title of Calculus of Indikect Functions. I obtain these terms by generalizing and giving precision to the ideas of Lagrange, and employ them to indicate the exact character of the two forms of analysis. 58 POSITIVE PHILOSOPHY. SECTION I. ORDINARY ANALYSIS, OR CALCULUS OF DIRECT FUNCTIONS. Algebra is adequate to the solution of mathematical ques- tions which are so simple that we can form directly the equations between the magnitudes considered, without its being necessary to bring into the problem, either in substi- tution or alliance, any system of auxiliary quantities de- rived from the primary. It is true, in the majority of im- portant cases, its use requires to be preceded and prepared for by that of the calculus of indirect functions, by which the establishment of equations is facilitated : but though algebra then takes the second place, it is not the less a necessary agent in the solution of the question ; so that the Calculus of direct functions must continue to be, by its nature, the basis of mathematical analysis. We must now, then, notice the rational composition of this calculus, and the degree of development it has attained. J. -I ■ f Its object being the resolution of equations (that is, the discovery of the mode of forma- tion of unknown quantities by the known, according to the equations which exist between them), 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 illimitable, though, as we have seen, composed of a very small number of primitive elements. „, .„ ,. The rational classification of equations Classiiic.ation , •liii i. ■ ni .1 j. of Eouations ^m^^t evidently be determiued by the nature of the analytical elements of which their members are comjiosed. Accordingly, analysts first divide equations Avith one or more variables into two principal classes, according as they contain functions of only the first three of the ten couples, or as they include also either ex- ponential or circular functions. Though the names of algebraic and transcendental functions given to these prin- cipal groups are inapt, the division between the corre- sponding equations is real enough, insofar as that the re- solution of equations containing the transcendental func- RESOLUTION OF ALGEBRAIC EQUATIONS. 59 tions 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 AUr i " equations only. In the first place, we must equations, observe that, though they may often contain irrational functions of the unknown quantities, as well as rational functions, the first case can always be brought under the second, by transformations more or less easy ; so that it is only with the latter that analysts have had to occupy themselves, to resolve all the algebraic equations. As to their classification, the early method of classing them according to the number of their terms has been retained only for equations with two terms, which are, in fact, sus- ceptible of a resolution proper to themselves. The classi- fication by their degrees, long universally established, is eminently natural ; for this distinction rigorously deter- mines the greater or less difficulty of their resolution. The gradation can be independently, as well as practically ex- hibited : for the most general equation of each degree necessarily comprehends all those of the diifereut inferior degrees, as miist also the formula which determines the unknown quantity : and therefore, however slight we may, d priori, 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 com- plicated in the execution with those of all the preceding degrees. This increase of difficulty is so great, that Algebraic re- the resolution of algebraic equations is as yet solution of known to us only in the four first degrees, equations. In this respect, algebra has advanced but little since the labours of Descartes and the Italian analysts of the six- teenth century ; though there has probably not been a single geometer for two centuries past who has not sti'iven 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 resolution of equations of the fifth, 60 POSITIVE PHILOSOPHY. and even higher degrees, being obtained, have tacitly agreed to give up such researches. The only question of this kind which would be of eminent importance, at least in its logical relations, would be the general resolution of algebraic equations of any degree whatever. But the more we ponder this subject, the more we are led to suppose, with Lagrange, that it exceeds the scope of our understandings. Even if the requisite formula could be obtained, it could not lie usefully applied, unless we could simplify it, without impairing 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 ti'eated only a very small part of algebra, properly so called ; that is, of the calculus of dii'ect 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 remember that by a law of our nature, we shall always remain below the difficulty of science, our means of conceiving of new ques- tions being always more powerful than our resources for resolving them ; in other words, the human mind being more apt at imagining than at reasoning. Thus, if we had resolved all the analytical equations now knowu, and if, to do this, we had found new analytical elements, these again would introduce classes of equations of which we now know nothing : and so, however great might be the increase of our knowledge, the imperfection of our algebraic science would be perpetually reproduced. . The methods that we have are, the com- knowlecb'-e " plete resolution of the equations of the first four degrees ; of any binomial equations ; of certain sjiecial equations of the superior degrees ; and of a very small number of exponential, logarithmic, and circular equations. These elements are very limited ; but geometers have succeeded in treating with them a great number of important questions in an admirable manner. The im- provements introduced within a century into mathema- tical analysis have contributed more to render the little knowledge that we have immeasurably useful, than to in- crease it. NUMERICAL RESOLUTION OF EQUATIONS. 61 To fill up the vast gap in the resolution of Numerical re- algebraic equations of the higher degrees, solutions of analysts have had recourse to a new order of equations, questions, — to what they call the numerical resolution of equations. Not being able to obtain the real algebraic formula, they have sought to determine at least the value of each unknown quantity for such or such a designated system of particular values attributed to the given quantities. This operation is a mixture of algebraic with arithmetical questions ; and it has been so cultivated as to be rendered possible in all cases, for equations of any degree and even of any form. The methods for this are now sufficiently general ; and what remains is to simplify them so as to fit them for regular application. While such is the state of algebra, we have to endeavour so to dispose the questions to be worked as to require finally only this numerical re- solution of the equations. We must not forget however that this is very imperfect algebi'a ; and it is only iso- lated, or truly final questions (which are very few), that can be brought finally to depend upon only the numerical resolution of equations. Most questions are only prepara- tory, — a first stage of the solution of other questions ; and in these cases it is evidently not the value of the unknown quantity that we want to discover, but the formula which exhibits its derivation. Even in the most simple questions, when this numei'ical resolution is strictly sufficient, it is not the less a veiw imperfect method. Because we cannot abstract and treat separately the algebraic part of the question, which is common to all the cases which result from the mere variation of the given numbers, we are obliged to go over again the whole series of operations for the slightest change that may take place in any one of the quantities concerned. Thus is the calculus of direct functions at present divided into two parts, as it is employed for the algebraic or the numerical resolution of equations. The first, the only satisfactory one, is unfortunately very restricted, and there is little hope that it will ever be otherwise: the second, usually insufficient, has at least the advantage of a much greater generality. They mvist be carefully distinguislied in our minds, on account of their different objects, and 62 POSIllYK PHILOSOPHY. therefore of the different ways in which quantities are con- sidered by them. Moreover, there is, in i-egard to their methods, an entirely different procedure in their rational distribution. In the first part, we have nothing to do with the values of the unknown quantities, and the division must take place according to the nature of the equations which we are able to resolve ; whereas in the second, we have nothing to do with the degrees of the equations, as the methods are applicable to equations of any degree whatever ; but the concern is with the numerical character of the values of the unknown quantities. These two parts, which constitute the im- e, nations!"^' mediate object of the Calculus of direct functions, are subordinated to a third, purely speculative, from Avhich both derive their most eiiectual 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 resolu- tion of equations has, on account of its generality, special need of this rational foundation. Two orders of questions divide this important depart- ment of algebra between them ; first, those which relate to the composition of eqiiations, and then those that relate to tlieir transformation ; the business of these last being to modify the roots of an ec{uation without 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 complete our rapid exhibition of the different essential parts of the cal- IVlethod of cuius of direct functions. This theory, which indeterminate relates to the transformation of functions Coefticients. j^^q series by the aid of what is called the IMethod of indeterminate Coefficients, is one of the most fertile and important in algebra. This eminently analyti- cal method is one of the most remarkable discoveries of Descartes. The invention and development of the in- finitesimal 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 resources; so CALCULUS OF INDIRECT FUNXTIONS. 63 that, by the useful combiuatiou of the two theories, the employmeut of the method of indetei-miuate coefticieuts has become much 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 Functions. We must next pass on to the more im- portant and extensive branch of our science, the Calculus of Indirect Functions. SECTION II. TRANSCENDENTAL ANALYSIS, OR CALCULUS OF INDIRECT FUNCTIONS. We referred (p. 53) in a former section to „. .• • i the views of the transcendental analysis pre- view^^^'"^^^^'*^ sented by Leibnitz, ISewton, and Lagrange. We shall see that each concejjtion has advantages of its own, that all are finally equivalent, and that uo method has yet been found which unites their respective charac- teristics. Whenever the combination takes place, it will 2)robably be by some method founded on the conception of Lagrange. The other two will then offer only an historical interest ; and meanwhile, the science must be regarded as in a merely provisional state, which requires the use of all the three conceptions at the same time ; for it is only by the use of them all that an adequate idea of the analysis and its applications can be formed. The vast extent and difficulty of this part of mathematics, and its recent forma- tion, should prevent our being at all surprised at the existing want of system. The conception which will doubt- less give a fixed and uniform character to the science has come into the hands of only one new generation of geo- meters since its creation ; and the intellectual habits re- quisite to perfect it have not been sufficiently formed. The first germ of the infinitesimal method tt-^^ (which can be conceived of independently of the Calculus) may be recognized in the old Greek Method of Exhaustions, employed to pass from the properties of straight lines to those of curves. The method consisted in 64 POSITIVE PHILOSOPHY. substituting for the curve the auxihary consideration of a jiolygou, inscribed or circumscribed, by means of which the curve itself was reached, the limits of the primitive ratios being suitably taken. There is no doubt of the filiation of ideas in this case ; but there was in it no equiva- lent for our modern methods ; for the ancients had no logical and general means for the determination of these limits, which was the chief difficulty of the question. The task remaining for modern geometers was to generalize the conception of the ancients, and, considering it in an ab- stract manner, to reduce it to a system of calculation, Avhich was impossible to them. Lagrange justly ascribes to the great geometer Fermat the first idea in this new direction. Permat may be re- garded as having initiated the direct formation of tran- scendental analysis by his method for the determination of maxima and minima, and for the finding of tangents, in which process he introduced auxiliaries which he afterwards suppressed as null when the equations obtained had imdergone certain suitable transformations. After some modifications of the ideas of Format in the intermediate time, Leibnitz stripped the process of some complications, and formed the analysis into a general and distinct cal- culus, having his own notation : and Leibnitz is thus the creator of transcendental analysis, as we employ it now. This pre-eminent discovery was so ripe, as all great con- ceptions are at the hour of their advent, that Newton had at the same time, or rather earlier, discovered a method exactly equivalent, regarding the analysis from a different point of view, much moi'e logical in itself, but less adapted than that of Leibnitz to give all practicable extent and facility to the fundamental method. Lagrange afterwards, discarding the heterogeneous considerations which had guided Leibnitz and Newton, reduced the analysis to a purely algebraic system, which only wants more aptitude for application. We will notice the three methods in their order. The method of Leibnitz consists in intro- L^BNiTZ °^ ducing into the calculus, in order to facilitate the establishment of equations, the infinitely small elements or differentials which are supposed to con- METHOD OF LEIBNITZ. 65 stitute tlie quantities whose relations we are seeking. There are relations between these differentials which are simpler and more discoverable than those of the [ivimitive quantities ; and hy these we maj afterwards (through a special calculus employed to eliminate these auxdiary in- finitesimals) recur to the equations sought, which it would usually have been im])Ossible to obtain directly. This indirect analysis may have various degrees of indirectness ; for, when there is too miich difiiculty in forming the equa- tion between the diiferentials of the magnitudes under notice, a second application of the method is required, the differentials being now treated as new primitive quantities. and a relation being sought between their infinitely small elements, or second differentials, and so on ; the same transformation beiug repeated any numlier of times, pro- vided the whole number of auxiliaries be fiually eliminated. It may be asked by novices in these studies, how these auxiliary 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 affect 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, obtaining a precise idea of this by considering them as l»eing either the successive powers of the same primitive infinitely small quantity, or as being quantities which may be regarded as having finite ratios \v'itli these powers ; so that, for instance, the second or third or other differentials of the same variable are classed as infinitely small quantities of the second, third or other order, because it is easy to exhibit in them finite multiples of the second, third, or other powers of a certain first differential. These preliminary ideas being laid down, the spirit of the infinitesimal analysis consists in constantly neglecting the infinitely small quantities in comparison with finite quantities ; and generally, the infinitely small quantities of any order whatever in comparison with all those of an inferior order. We see at once how such a power must facilitate the formation of equations between the differentials of quantities, since we can substitute for these differentials such other elements as we may choose, and as will be more simjjle to treat, only observing the con- I. F 66 POSITIVE PHILOSOPHY. dition that the new elements shall differ from the precediniij 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 plane elements ; and, in mechanics, varied motions as an infinite series of uniform motion'', succeeding each other at infinitely small intervals of time. Such a mere hint as this of the varied application of this method may give some idea of the vast scope of the conception of transcendental analysis, as formed by Leibnitz. It is, beyond all question, the loftiest idea ever yet attained by the human mind. It is clear that this conception was necessary to complete the basis of mathematical science, by enabling us to estab- lish, in a broad and practical manner, the relation of the concrete to the abstract. In this respect, we must regard it as the necessary complement of the great fundamental idea of Descartes on the general analytical representation of natural phenomena ; an idea which could not be duly estimated or put to use till after the formation of the infinitesimal analysis. This analysis has another property, besides that of facili- tating the study of the mathematical laws of all phenomena, and perhaps not less important than that. The differential ^ ,. . formulas exhibit an extreme generality, ex- Generalibv of • • • i ,• i i j. • x the formulas pressing m a single equation each determinate phenomenon, however varied may be the subjects to which it belongs. Thus, one such equation gives the tangents of all curves, another their rectifications, a third their quadratures ; and, in the same way, one invariable formula expresses the mathematical law of all variable motion ; and one single equation represents the distribution of heat in any body, and for any case. This remarkable generality is the basis of the loftiest views of the geometers. Thus this analysis has not only furnished a general method for forming equations indirectly which could not have been directly discovered, but it has intro- duced a new order of more natural laws for our use in the mathematical study of natural phenomena, enabling us to rise at times to a perception of ])ositi\ e approximations between classes of wholly different phenomena, through the JUSTIFICATION OF THE LEIBNITZIAN METHOD. 67 analogies presented by the differential expressions of their mathematical laws. In virtue of this second property of the analysis, the entire system of an immense science, like geometry or mechanics, has submitted to a condensation into a small number of analytical formulas, from which the solution of all particular problems can be deduced, by in- variable rules. This beautiful method is, however, iniper- . , . feet in its logical basis. At first, geometers ^.j^^ Method were naturally more intent upon extending the discovery and multiplying its applications than upon establishing the logical foundation of its processes. It was enough for some time to be able to produce, in answer to objections, unhoped-for solutions of the most difiicult problems. It became necessary, however, to recur to the basis of the new analysis, to establish the rigorous exact- ness of the jirocesses employed, notwithstanding their apparent breaches of the ordinary laws of reasoning, Leibnitz himself failed to justify his conception, giving, when urged, an answer which represented it as a mere approximative calculus, the successive opei'ations of which might, it is evident, admit an augmenting amount of error. Some of his successors were satisfied with showing that its results accorded with those obtained by ordinary algebra, or the geometry of the ancients, reproducing by these last some solutions which could be at first obtained only by the new method. Some, again, demonstrated the conformity of the new conception with others ; that of Newton espe- cially, which was unquestionably exact. This afforded a practical justification : but, in a case of such unequalled importance, a logical justification is also required, — a direct proof of the necessary rationality of the infinitesimal method. It was Caruot who furnished this at last, by showing that the method was founded on the pi'inciple of the necessary compensation of errors. We cannot say that all the logical scaffolding of 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, Carnot's principle of the necessary compensation of errors is of more importance, in legitimating the analysis of Leibnitz, than is even yet commonly supposed. His reason- 68 POSITIVE PHILOSOPHY. ing is founded on the conception of infinitesimal quantities indefinitely decreasing, while those from which they are derived are fixed. The infinitely small errors introduced with the auxiliaries cannot have occasioned other than infi- nitely small errors in all the equations ; and when the relations of finite c^uantities are reached, these relations must be rigorously exact, since the only errors then pos- sible must be finite ones, which cannot have entered : and thus the final equations become perfect. Caruot's theory is doubtless more subtle than solid ; but it has no other radical logical vice than that of the infiaitesimal method itself, of which it is, as it seems to me, the natural develop- ment and general explanation; so that it must be adopted as long as that method is directly employed. The philosophical character of the transcendental analysis has now been sufficiently exhil>ited to allow of my giving only the principal idea of tlie other two methods. ^ ,^ Newton offered his conception under several Method different forms in succession. That which is now most commonly adopted, at least on the continent, was called by himself, sometimes the Mdhod of prime and tdtimate Ratios, sometimes the Method of Limits, by which last term it is now usually known. Under this Method, the auxiliaries intro- simultaneous increments of the primitive quantities ; or, in other words, the final ratios of these in- crements ; limits or final ratios which we can easily show to have a determinate and finite value. A special calculus, which is the equivalent of the infinitesimal calculus, is afterwards employed, to rise from the equations between these limits to the corresponding equations between the primitive quantities themselves. The power of easy expression of the mathematical laws of phenomena given by this analysis arises from the calculus applying, not to the increments themselves of the proposed quantities, but to the limits of the ratios of those incre- ments ; and from our being therefore able always to sub- stitute for each increment any other magnitude more easy to treat, provided their final ratio is the ratio of equality ; or, in other words, that the limit of their ratio is unity. It Newton's method of limits. 69 is clear, in fact, that the calculus of limits can he in no way affected by this substitution. Starting from this principle, we find nearly the equivalent of the facilities offei'ed by the analysis of Leibnitz, which are merely considered from another point of view. Thus, curves will be regarded as the limits of a series of rectilinear polygons, and variable motions as the limits of an aggregate of uniform motions of continually nearer approximation, etc., etc. Such is, in substance, Newton's conception; or rather, that which Maclaurin and d'Alembert have offered as the most rational 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 fj^p^Ig"^ ^"' the special form of the calculus of indirect functions commonly adopted by English geometers ; and also, on account of its ingenious clearness in some cases, and of its having furnished the notation best adapted to this manner of regarding the ti'anscendental analysis. I mean the Calculus oi fluxions and oifltients, founded on the general notion of velocities. To facilitate the conception of the fundamental idea, let us conceive of every curve as generated by a point affected by a motion varying according to any law whatever. The different quantities presented by the (turve, the abscissa, the ordinate, the arc, tlie area, etc., will be regarded as simiil- taneously produced by successive degrees during this motion. The velocity with which each one will have been described will be called the fluxion of that quantity, which inversely would have been called its fluted Henceforth, the transcendental analysis will, accoi'ding to this concep- tion, consist in forming directly the equations between the fluxions of the proposed quantities, to deduce from them afterwards, by a special Calculus, the equations between the fluents themselves. Wliat has just been stated respect- ing curves may evidently be transferred to any magnitudes whatever, regarded, by the help of a suitable image, as some being produced by the motion of others. This method is evidently the same with that of limits complicated with the foreign idea of motion. It is, in fact, only a way of representing, by a comparison derived from mechanics, the 70 POSITIVE PHILOSOPHY. luetliod of prime aud ultimate ratios, which alone is redu- cible to a calculus. It therefore necessarily admits of the same genei'al advantages in the various principal applica- tions of the transcendental analysis, without its being requisite for us to offer special proofs of this. Lagrange's conception consists, in its ad- LAGRANGES • 11 • 1 -i, • -1 • ii j- Method mirable simplicity, in considering the trans- cendental analysis to be a great algebraic artifice, by which, to facilitate the establishment of equa- tions, we must introduce, in the place of or with the primi- tive functions, their derived functions ; that is, according to the definition of Lagrange, the coefficient of the first term of the increment of each function, arranged according to the ascending powers of the increment of its variable. The Calculus of indirect functions, properly so called, is destined here, as well as in the conceptions of Leibnitz and Newton, to eliminate these derivatives, employed as auxiliaries, to deduce from their relations the corresponding equations 1>etween the primitive magnitudes. The transcendental analysis is then only a simple, but very considerable exten- sion of ordinary analysis. It has long been a common practice with geometers to introduce, in analytical investi- gations, in the place of the magnitudes in question, their different powers, or their logarithms, or their sines, etc., in order to simjjlify the equations, and even to obtain them more easily. Successive derivation is a general artifice of the same nature, only of greater exteutT and commanding, in consequence, much more important resources for this common 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 explain why it must be so under this method of derivation, rather than any other transformation. This is the weak side of Lagrange's great idea. We liave not yet become able to lay hold of its pre- cise advantages, in an abstract manner, and without recur- rence to the other conceptions of the transcendental analysis. These advantages can be established only in the separate consideration of each principal question ; and this verifica- tion becomes laborious, in the treatment of a complex problem. COxMPAEISON OF THE THREE METHODS. 71 Other theories have been proposed, such as Euler's Cal- culus of vanishing quantities : but they are merely modifi- cations of the thi'ee just exhibited. We must next com- pare and estimate these methods ; and in the first j)lace observe their perfect and necessary conformity. Considering the three methods in regard . •, , r , j to their destination, independently of pre- three m'etliod^ liminary ideas, it is clear that they all con- sist in the same general logical artifice ; that is, the in- troduction of a certain system of auxiliary magnitudes uniformly correlative Avith those under investigation; the auxiliaries being substituted for the express object of facilitating the analytical expression of the mathematical laws of phenomena, though they must be finally eliminated by the help of a special calculus. It was this which deter- mined me to define the transcendental analysis as the Calculus of indirect functions, in order to mark its true philosophical character, while excluding all discussion about the best manner of conceiving and applying it. Whatever may be the method employed, the general effect of this analysis is to bring every mathematical question more speedily into the domain of the calculus, and thus to lessen considerably the grand difficulty of the passage from the concrete to the abstract. We cannot hope that the Calculus Avill ever lay hold of all questions of natural philosophy — geometrical, mechanical, thermological, etc. — from their birth. That would be a contradiction. In every problem there must be a certain ])reliminai"y 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 departure for all ana- lytical investigations. But this preliminary elaboration has been remarkably simplified by the creation of the transcen- dental analysis, which has thus hastened the moment at which general and abstract processes may be uniformly and exactly applied to the solution, by reducing the opera- tion to fiudiug the equations between auxiliary magnitudes, whence the Calculus leads to equations directly relating to the proj^osed magnitudes, which had formerly to be estab- lished directly. Whether these indirect equations are 72 POSITIVE PHILOSOPHY. differential equations, according to Leibnitz, or equations of ZimiVs, according to Newton, ov 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 process ; for the auxiliaries introduced are really iden- tical, being only regarded from different points of view. The conceptions of Leibnitz and of Newton consist in making known in any case two general necessary proper- ties of the derived function of Lagrange. The transcen- dental analysis, then, examined abstractly and in its principle, is always the same, whatever conception is adopted; and the processes of the Calculus of indirect func- tions ai^e necessarily identical in these different methods, which must therefore, under any aj^plication whatever, lead to rigorously uniform results. „. . If we endeavour to estiinate their compara- i\l^ r.r.1,',1^ '' tive value, we shall find in each of the three live Veil lit;, , , _ conceptions advantages and inconveniences which are peculiar to it, and which prevent geometers from adhering to any one of them, as exclusive and final. The method of Leibnitz has eminently the advantage in the rapidity and ease with which it effects the formation of equations between auxiliary magnitiides. We owe to its use the high perfection attained by all the general theories of geometry and mechanics. Whatever may be the specu- lative opinions of geometers as to the infinitesimal method, they all employ it in the treatment of any new question. Lagrange himself, after having reconstructed tht^ analysis on a new basis, rendered a candid and decisive homage to the conception of Leibnitz, by employing it exclusively in the whole system of his "Analytical Mechanics." Such a fact needs no comment. Yet are we obliged to admit, with Lagrange, that the conception of Leibnitz is radically vicious in its logical relations. He himself declared the notion of infinitely small quantities to be a faJse 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 in- genious princij^le of the compensation of errors, we may, as we have already seen, explain the necessary exactness of COMPARISON OF THE THREE METHODS. 78 the processes which compose the method ; but it is a radical inconvenience to be obhged to indicate, in Mathe- matics, two clashes of reasonings so unlike, as tliat the one order are perfectly rigoi'ons, while by the others we de- signedly commit errors which have to be afterwards com- pensated. 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 irrepr<.)achable. 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 sucli a transformation. Finally, the infinitesimal method exhibits the very serious defect of breaking the unity of abstract mathematics by ci'eating a transcendental analysis founded upon principles widely different from those which serve as a basis to ordinary analysis. This division of analysis into two systems, almost wholly inde- pendent, tends to prevent the formation of general analy- tical conceptions. To estimate the consequences duly, we must recur in thought to the state of the science before Lagrange had established a general and complete harmony 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, regarded as exact from their origin ; and the general rules of reasoning are as constantly observed as in ordinary analysis. But it is weak in resources, and embarrassing in operation, compared with the infini- tesimal method. In its applications, the relative inferiority of this theory is very strongly marked. It also separates the ordinary and transcendental analysis, though not so conspicuously as the theory of Leibnitz. As Lagrange re- marked, the idea of liiidts, though clear and exact, is not the less a foreign idea, on which analytical theories ought not to be dependent. This jierfect unity of analysis, and a ])\irely abstract character in the fundamental ideas, are found in the con- ception of Lagrange, and there alone. It is therefore the most philosophical of all. Discarding every heterogeneous consideration, Lagrange reduced the transcendental analysis 74 POSITIVE PHILOSOPHY. to its proper character, — that of pi'esenting a very extensive class of analytical transformations, which facilitate in a re- markable degree the expression of the conditions of the various problems. This exhibits the conception as a simple extension of ordinary analysis. It is a superior algebra. All the different parts of abstract mathematics, till then so incoherent, might be from that moment conceived of as forming a single system. This jjhilosophical superiority marks it for adoption as the final theory of transcendental analysis ; but it presents too many difficulties in its appli- cation, in comj^arison with the others, to admit of its exclusive preference at present. Lagrange himself had great didiculty 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- portant. Though Lagrange, stimulated by difficulty, ob- tained results in some cases which other men would have despaired of, it is not the less true that his conception has thus far remained, as a whole, essentially unsuited to applications. The result of such a comparison of these three methods is the conviction that, in order to understand the tran- scendental analysis thoroughly, we should not only study it in its principles according to all these conceptions, but should accustom ourselves to emj^loy them all (and es- peci.dly the first and last) almost indiffei'ently, in the solution of all important questions, whether of the calculus of indirect functions in itself, or of its applications. In all the other dej^artments of mathematical science, the con- sideratii)n of different methods for a single class of ques- tions may be useful, apart from the historical interest which it presents ; but it is not indispensable. Here, on the contrary, it is strictly indispensable. Without it there can be no philosophical judgment of this admirable creation of the human mind; nor any success and facilit}" in the use of this powerful instrument. THE TWO CALCULI OF INDIRECT FUNCTIONS. THE DIFFERENTIAL AND INTEGRAL CALCULUS. The Calculus of Indirect functions is j, .^ parts necessarily divided into two parts ; or rather, it is composed of two distinct calculi, having the relation of converse action. By the one we seek the relations between the auxiliary magnitudes, by means of the rela- tions between the corresj^onding primitive magnitudes ; by the other we seek, conversely, these direct equations by means of the indirect equations first established. This is the doable object of the transcendental analysis. Different names have been given to the two systems, according to the point of view from which the entire analysis has been regarded. The intiuitesimal method, properly so called, being most in use, almost all geometers employ the terms Differential Calculus and Integral Cal- culus established by Leibnitz. Newton, in accordance with his method, called the first the Calculus of Fluxions, and the second the Calculus of Fluents, terms which were till lately commonly adopted in England. According to the theory of Lagrange, the one would be called the Calculus of Derived Functions, and the other the Calculus of Primitive Functions. I shall make use of the terms of Leibnitz, as the fittest for the formation of secondary ex- pressions, though we must, as has been shown, employ all the conceptions concurrently, apj)roaching as nearly as may be to that of Lagrange. The dilferential calculus is obviously the rr,i • . x i rational basis of the integral. We have seen velations. that ten simple functions constitute the elements of our analysis. We cannot know how to in- tegrate directly any other differential expressions than those produced by the differentiation of tliose ten func- tions. The art of integration consists therefore in bringing all the other cases, as far as possible, to depend wholly on this small number of simple functions. It may not be apparent to all minds what can be the proper utility of the differential calculus, independently of this necessary connection with the integral calculus, which seems as if it must be in itself the only directly indis- pensable one ; in fact, the elimination of the infinitesimals 76 POSITIVE PHILOSOPHY. or the derivatives, introduced as auxiliaries, beinc^ the final object of the calculus of indirect functions, it is natural to think that the calculus which teaches us to deduce the equations betAveen the primitive magnitudes from those between the auxiliary magnitudes must meet all the gene- ral needs of the transcendental analysis, without our seeing at first what special and constant part the solution of the inverse question can have in such an analysis. A common answer is assigning to the differential calculus the office of forming the differential equations ; but this is clearly an error ; for the primitive formation of differential equations is not the business of any calculus, for it is, on the contrary, the point of departin*e of any calculus whatever. The very use of the differential calculus is enabling us to differentiate the var'ous equations; and it cannot therefore be the process for establishing them. This common error arises from confounding the infinitesimal calculus with the in- finitesimal method, which last facilitates the formation of equations, in every application of the transcendental analysis. The calculus is the indispensable complement of the method ; but it is perfectly distinct from it. But again, we should much misconceive the peculiar impor- tance of this first branch of the calculus of indirect func- tions if we saw in it only a preliminary process, designed merely to pi-epare an indisi-tensable ba^is for the integral calculus. A few words will show that a jirimary direct and necessary office is always assigned to the differential . . calculus. In formings differential equations, ( ases or union , .-,' i ^ • i. j • of the two '"'*^ rarely restrict ourselves to introducing differentially only those magnitudes whose relations are sought. It would often be impossible to establish, equations without introducing other magnitudes whose relations are, or are supposed to be, known. Now in such cases it is necessary that the differentials of these intermediaries should be eliminated before the equations are fit for integration. This elimination belongs to the differential calculus ; for it must be done by determining, by means of the equations between the intermediary func- tions, the relations of their differentials ; and Ihis is merely a question of differentiation. This is the way in which the differential calculus not only prepares a basis for the EXAMPLES OF THE TWO CALCULI. 11 integral, Liit makes it available in a multitude of cases which could not otherwise be treated. There Cases of the are some questions, few, but highly imjjor- Differential tant, wliicli admit of the emjjloymeut of the calculus alone, differential calculus alone. They are those in which the magnitudes sought enter directly, and not by their diffe- rentials, into the jirimitive differential equations, which then contain differentially only the various known func- tions employed, as we saw just now, as intermediaries. This calculus is here entirely sufficient for the elimination of the intinitesimals, without the question giving rise to any integration. There are also questions, few, but highly important, which are the converse of the last, requiring the employment of the integral calculus alone. Cases of the In these, the differential equations are found Integral cal- to be immediately ready for integration, cuius alone, because they contain, at their first formation, only the in- finitesimals which relate to the functions sought, or to the really independent variables, without the introduction, differential y, of any intermediaries being required. If intermediary functions are introduced, they wi 1, by the hypothesis, enter directly, and not by their differentials ; and then, ordinary algebra will serve for their elimination, and to bring the question to depend on the integral calcu- lus only, 'ihe differential calculus is, in such cases, not essential to the solution of the problem, which will depend entirely on the integral calcidus. Thus, all questions to which the analysis is applicable are contained io three classes. The first class comprehends the problems which may be resolved by the differential calculus alone. The second, tho-e which may be resolved by the integral cal- culus alone. 'J hese are only exceptional; the third con- stituting the normal case ; that in which the differential and integral calculus 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 the applications of 78 POSITIVE PHILOSOPHY. either calculus, which are quite a different tial Calculus study from that of the abstract principles of differentiation and integration. The con- sequence of the common practice of confounding these principles with their application, especially in geometry, is that it becomes difficult to conceive of either analysis or geometry. It is in the department of Concrete Mathe- matics that the applications should he studied. The first division of the differential calculus is grounded on the condition whether the functions to be differentiated rp ^ ... , are explicit or implicit; the one giving rise to the differentiation of formulas, and the other to the differentiation of equations. This classifica- tion is rendered necessary by the imperfection of oi'dinary analysis ; for if we knew how to resolve all equations algebraically, it would be possible to render every implicit function explicit ; and, by differentiating it only in that state, the second part of the differential calculus would be immediately included in the first, without giving rise to any new difficulty. But the algebraic resolution of equations is, as we know, still scarcely past its infancy, and unknown for the greater numV>er of cases ; and we have to differentiate a function without knowing it, though it is determinate. Thus we have two classes of questions, the differentiation of implicit functions being a distinct ease from that of explicit functions, and much more complicated. We have to begin by the differentiation of formiilas, and we may then refer to this first case the differentiation of equations, by certain analytical considera- tions which we are not concerned with here. There is another view in which the two general cases of differentia- tion are distinct. The relation obtained between the differentials is always more indirect, in comparison with that of the finite quantities, in the differentiation of im- plicit, than in that of explicit functions. We shall meet with this consideration in the case of the integral calculus, where it acquires a preponderaut im]iortance. Q TV • • Each of these parts of the diff'erential hubuivisions. ,, . .^T.,, T,i- 1 calculus IS again divided : and this sub- division exhibits two very distinct theories, according as we have to differentiate functions of a single variable, or SUBDIVISIONS OF THE DIFFERENTIAL CALCULUS. 79 functions of several independent variables, — the second branch being of far greater complexity than the first, in the case of explicit functions, and. much more in that of imi^iicit. One more distinction remains, to complete this brief sketch of the parts of the differential calculus. The case in which it is required to differentiate at once different implicit functions combined in certain primitive equations must be distinguished from that in which all these functions are separate. The same imperfection of ordinary analysis which prevents our converting every implicit function into an equivalent explicit oue, renders us unable to separate the functions which enter simultaneously into auv system of equations ; and the functions are evidently still more implicit in the case of combined, than of separate functions : and. in differentiating, we are not only unable to resolve the primitive equations, but even to effect the j^roper elimination among them. We have now seen the different parts of this calculus in their natural connection and tl\e elenients rational distribution. The whole calculus is finally found to rest upon the differentiation of explicit functions with a single varialile, — the only one which is ever executed dii'ectly. Now, it is easy to understand that this first theory, this necessary basis of the whole system, simply consists of the differentiation of the elementary functions, ten in number, which C(nnpose all our analytical combinations ; for the differentiation of compound functions is evidently deduced, immediately and necessarily, from that of their constituent simple functions. We find, then, the whole system of differentiation reduced, to the know- ledge of the ten fundnmental differentials, and to that of the two general princi]>les, by one of which the differentia- tion of impHcit functions is deduced from that of explicit, and by the other, the differentiation of functions of several variables is reduced to that of functions of a single variable. Such is the simplicity and perfection of the system of the differential calculus. The transformation of derived Functions for Transfornia- new variables is a theory which must be just tionof dt'iived mentioned, to avoid the omission of an indis- functions tor pensable complement of the system of diffe- "'^^ variables. 80 POSITIVE PHILOSOPHY. rentiatiou. It is as finished and perfect as tlie other parts of this calculus; and its great importance is in its increasing our resources by permitting us to choose, to facilitate the formation of differential equations, that system of indepen- dent variables which may appear to be most advantageous, though it may afterwards be relinquished, as an inter- mediate stiq>, by which, through this theory, we may pass to the final system, which sometimes could not have been considered directly. .... Though we cannot hei'e consider the con- apnhcatioas crete applications of this calculus, we must glance at those which are analytii-al, because they a.re of the same nature with the theory, and should be looked at in connection with it. These questions are reducible to three essential ones. First, the development into series of functions of one or more variables ; or, more generally, the transformation of functions, which constitutes the mo>t beautiful and the most important application of the differential calculus to general analysis, and which comprises, besides the fundamental ser es discovered by Taylor, the remarkable series discovered by Maclaurin, John Beruouilli, Lagrange and others. Feeondly, the general theory of maxima and minima values for any functions whatever of one or more variables: one of the most iutex-esting problems that analysis can present, how- ever elementary it has become. The th rd is the least im- portant of the three : — it is the determination of the true value of functions which present themselves under an in- determinate ap])earance, for certain hypotheses made on the values of the corresponding variables. In every view, the first question is the most eminent ; it is also the most susce[)tible of future extension, especially by ccmceiving, in a larger manner than hitherto, of the employment of tlie differential 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 aie the proper subjects of this work ; so extensive and so interesting are the developments wliicli might other- wise be offered. Insufficient and summary as are the views of the Differential Calculus just offered, we must be no THE INTEGRAL CALCULUS. 81 less rapid in our survey of the Integral Calculus, properly so called ; that is, the abstract subject of integration. The Integral Calculus. The division of the Integral Calculus, like that of the Differential, 'proceeds on the Sjcu"us°''^' principle of distinguishing the integration of explicit differential formulas from the integration of itnplicit differentials,or of differential en nations. The t^ t • . , • e ,x J. • Its divisions, separation ot these two cases is even more radical in the case of integration than in the other. In the differential calculus this distinction rests, as we have seen, only on the extreme imperfection of ordinary analysis. But, on the other hand, it is clear that even if all equations could be algebraically resolved, differential equations would nevertheless constitute a case of integration altogether dis- tinct from that presented by explicit differential formulas. Their integration is necessarily more complicated than that of explicit differentials, by the elaboration of which the integral calculus was originated, and on which the others have been made to depend, as far as possible. All the various analytical processes hitherto proposed for the inte- gration of differential equations, whether by the separation of variables, or the method of multipliers, or other means, have been designed to reduce these integrations to those of differential formulas, the only object which can be directly undertaken. Unhappily, imperfect as is this neces- sary basis of the whole integral calculus, the art of reducing to it the integration of differential equations is even much less advanced. As in the case of the differential calculus, „ , ,. . . T £ T 1 i! J.1 i. subdivisions, and tor analogous I'easons, each ot these two branches of the integral calculus is divided again, accord- ing as we consider functions with a single ^ . , . variable or functions with several indepen- ^^ several ^' dent variables. This distinction is, like the preceding, even more important for integration than for differentiation This is especially remarkable with respect to differential equations. In fact, those which relate to several independent variables may evidently present this I. G 82 POSITIVE PHILOSOPHY. characteristic and higher difficulty — 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 commonly called the Integral Calculus of partial differences, created by D'Alembert, in which, as Lagrange truly perceived, geometers should 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 ordinary differential equations. A new distinction, highly important here, differentiation tl'oi^igli ^lot in the differential calculus, where it is a mistake to insist upon it, is drawn from the higher or lower order of the differentials. We may regard this distinction as a subdivision in the integra- tion of explicit or implicit differentials. With regard to explicit differentials, whether of one variable or of several, the necessity of distinguishing their different orders is occasioned merely by the extreme imperfection of the in- tegral calculus ; and, with reference to implicit differentials, the distinction of orders 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 which is our object. And in the second case, there is the additional difficulty that the higher order of the differential equations necessarily gives rise to ques- tions of a new kind. The higher the order of differential equations, the more implicit are the cases which they pre- sent ; and they can be made to depend on each other only by special methods, the investigation of which, in conse- quence, forms a new class of questions, with regard to the simplest cases of which we as yet know next to nothing. The necessary basis of all other integrations is, as we see from the foregoing considerations, that of explicit diffe- rential formulas of the first order and of a single variable ; and we cannot succeed in effecting other integrations but ALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. 83 by reducing them to this elementary case, Avhich is the only one capable of being treated directly. This ,. l- t • simple fundamental integration, often con- veniently called quadratures, corresponds in the differential calculus to the elementary case of the differentiation of ex- plicit functions of a single variable. But the integral ques- tion is, by its nature, quite otherwise comjilicated, and much more extensive than the differential 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 si:)ecial difficulties with respect to the integral calculus. Hence the indefinite extent and varied complication of the question of quadra- tures, of which we know scarcely anything completel} after all the efforts of analysts. The question is divided into the two cases ai • i . • of algebraic functions and transcenclental func- functions tions. The algebraic class is the more ad- vanced of the two. In relation to irrational functions, it is true, we know scarcely anything, the integrals of them having been obtained only in very restricted cases, and particularly by rendering them rational. The integration of rational functions is thus far the only theory of this calculus which has admitted of complete treatment ; and thus it forms, in a logical point of view, its most satisfac- tory part, though it is perhaps the least important. Even here, the imperfection of ordinary analysis usually comes in to stop the working of the theory, by which the integra- tion finally depends on the algebraic solution of equations; and thus it is only in what concerns integration viewed in an abstract manner that even this limited case is resolved. And this gives us an idea of the extreme imperfection of the integral calculus. The case of the inte- ^ , . , ,• Pi -\ i. ^ £ J.' • -x Iranscendental gration or transcendental functions is quite functions in its infancy as yet, as regards either ex- ponential, logarithmic, or circular functions. Very few cases of these kinds have been treated ; and though the simplest have been chosen, the necessary calculations are extremely laborious. 84 POSITIVE PHILOSOPHY. The theory of Singular Sohdions (some- lonT '"'''' ^^'^^^ ^^^^^^ Particular Solutions), fully de- veloped by Lagrange in his Calculus of Functions, but not yet duly appreciated by geometers, must be noticed here, on account of its logical perfection and the extent of its applications. This theory forms im- plicitly a portion of the general theory of the integration of differential equations ; but I have left it till now, be- cause it is, as it were, outside of the integral calculus, and I wished to preserve the sequence of its parts. Clairaut first observed the existence of these solutions, and he saw in them a paradox of the integral calculus, since they have the property of satisfying the differential equations without being comprehended in the corresponding general integrals. Lagrange explained this paradox by showing how such solutions are always derived from the general integral by the variation of the arbitrary constants. This theory has a character of perfect generality ; for Lagrange has given invariable and very simple processes for finding the singular solution of any differential equation which admits of it ; and, what is very remarkable, these processes require no integration, consisting only of differentiations, and being therefore always applicable. Thus has differentiation become, by a happy artifice, a means of compensating, in certain circumstances, for the imperfection of the integral calculus. ^ , ., . One more theorv remains to be noticed, to Ueiinite m- i , •• j; j.i j. n a- £ tecrals complete our review ot that collection or analytical researches which constitutes the integral calciilus. It takes its place outside 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 de- finite integrals. These definite integrals are the values of the required functions for certain determinate values of the corresponding variables. The use of these in transcendental analysis corresponds to the numerical resolution of equa- tions in ordinary analysis. Analysts being usually unable to obtain the real integral (called in opposition the general or indefinite integral), that is, the function which, differen- tiated, has produced the proj^osed differential formula, have DEFINITE INTEGRALS. 85 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 with- out having first resolved the corresponding algebraic one, which is generally the most important ; and such an analysis is, by its nature, as imperfect as that of the numerical resolution of equations. Inconveniences, logical and practical, result from such a confusion of arithmetical and algebraic considerations. But, uuder our inability to obtain the true integrals, it is of the utmost importance to have been able to obtain this solution, incomplete and in- sufiicient as it is. This has now been attained for all cases, the determination of the value of definite integrals having been reduced to entirely general methods, which leave nothing to be desired, in many cases, but less complexity in the calculations ; an object to which analysts are now directing all their special ti'ansformations. This kind of transcendental arithmetic being considered perfect, the difiiculty in its applications is reduced to making the pro- posed inquiry finally depend only on a simple determina- tion of definite integrals ; a thing 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 differen- Prospects of tial calculus constitutes by its nature a limited the Integral and perfect system, the integral calculus, or Calculus, the simple subject of integration, offers inexhaustible scope for the activity of the human mind, independently of the indefinite applications of which transcendental analysis is evidently capable. The reasons which convince us of the impossibility of ever achieving the general resolution of algebraic equations of any degree whatever, are yet more decisive against our attainment of a single method of inte- gration 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 86 POSITIVE PHILOSOPHY. a widei' generality. The transcendental analysis is yet too near its origin, it has too recently been regarded in a truly rational manner, for ns to have any idea what it may here- after 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 deriva- tion 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 re- sources than in ixrging further our present calculus of in- direct functions ; and I am persuaded that when geometers have exhausted the most imjjortant applications of our present transcendental analysis, they will turn their atten- tion 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 enter- taining a false idea of the general resources of the trans- cendental analysis to attach too much importance to this consideration. As in ordinary analysis, we find here that a very small amount of fundamental Icuowledge respecting the resolution of equations is of inestimable use. However little advanced geometers are as yet in the science of inte-' grations, they have nevertheless derived from their few absti'act notions the solution of a multitude of questions of the highest importance in geometry, mechanics, thermology, etc. The philosophical explanation of this double general fact is found in the preponderating importance and scope of abstract science, the smallest portion of which naturally corresjDonds to a multitude of concrete researches, Man having no other resoui'ce for the successive extension of his intellectual means than in the contemplation of ideas more ■ and more abstract, and nevertheless positive. Lagrange's method of variations. 87 Calculus of Variations. By Ms Calculus or Method of Variations, Lagrange im- proved the capacity of the transcendental analysis for the establishment of equations in the most difficult problems, by considering a class of equations still more indirect than differential equations properly so 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 theory ; and it is therefore necessary to indicate briefly the special nature of the problems which have given rise to this hyper-transcendental analysis. These problems are those which were long Problems known by the name of Isoperimetrical Pro- giving rise to Hems ; a name which is truly applicable to *"*** Calculus, only a very small number of them. They consist in the investigation of the maxima and minima of certain inde- tex'minate 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 particular 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 precede 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 that 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 distinguishing the maximum from the minimum, which consists, in the case of a function of a single variable, for example, in the derived function of the second order taking a negative value for the maximvim and a positive for the minimum. Such are the fundamental conditions belonging 88 POSITIVE PHILOSOPHY. to the majority of cases ; and where modifications take place, they are equally subject to invariable, though more complicated abstract rules. The construction of this general theory having destroyed the chief interests of geometers in this kind of questions, they rose almost immediately to the consideration of a new order of problems, at once more important and more diffi- cult, — those of isoperimeters. It was then no longer the values of the variables proper to the maximum or the mini- mum of a given function that had to be determined. It was the form of the function itself that had to be dis- covered, accoi'ding to the condition of the maximum or minimum of a certain definite integral, merely indicated, which depended on that fvmction. We cannot here follow the history of these problems, the oldest of which is that of the solid of least resistance, ti'eated by Newton in the second book of the ' Principia,' in which he determines what must be the meridian curve of a solid of revolution, in order that the resistance experienced by that body in the direction of its axis may be the least possible. Mechanics first furnished this new class of problems ; but it was from geometry that the subjects of the principal investi- gations were afterwards derived. They were varied and complicated almost infinitely by the labours of the best geometers, when Lagrange reduced their solution to an abstract and entirely general method, the discovery of which has checked the eagerness of geometers about such 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 maximum or a minimum, with regard to all those which it would take if the required function had any other form whatever. In treating these problems, the predecessors of Lagrange proposed, in substance, to reduce them to the ordinary theory of maxima and minima. But they pro- ceeded by applying special simple artifices to each case, not reducible to certain rules ; so that every new question reproduced analogous difficulties, without the solutions SOME OF ITS APPLICATIONS. 89 previously obtained being of any essential aid. The part common to all questions of tbis class had not been dis- covered ; and no abstract and general treatment was therefore provided. In his endeavours to bring all isoperi- metrical problems to depend on a common analysis, Lagrange was led to the conception of a new kind of diif e- rentiation ; and to these new Differentials he gave the name of Variations. They consist of the infinitely small increments which the integrals receive, not in virtue of analogous increments on the part of the corresponding variables, as in the common transcendental analysis, but by supposing that the form of the function placed under the sign of integration undergoes an infinitely small change. This abstract conception once formed, Lagrange was able to reduce with ease, and in the most general manner, all the problems of isoperimeters to the simple common theory of maxima and minima. Important as is this great and happy ti'ans- ,. formation, and though the Method of Varia- ^[qi^^ ' tions had at first no other object than the rational and general resolution of isoperimetrical problems, we should form a very inadequate estimate of this beautiful analysis if we supposed it restricted to this application. In fact, the abstract conception of two distinct natures of differentiation is evidently applicable, not only to the cases for which it was created, but for all which present, for any reason whatever, two different ways of making the same magnitudes vary. Lagrange himself made an immense and all-important applicatiom of his Calculus 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 different 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 consecutive instants, or as we pass from one point of the body to another in the same instant. One of these comparisons produces the common differentials ; the other occasions variations which are, there as elsewhere, only differentials taken from a new point of view. It is in such a general acceptation as this that we must conceive of the Calculus of Variations, to appreciate fitly the impor- 90 POSITIVE PHILOSOPHY. tance of this admirable logical instrument ; the most power- ful as yet constructed by the human mind. This Method being only an immense extension of the general trancendental analysis, there is no need of proof that it admits of being considered under the different pri- mary points of view allowed by the calculus of indirect functions, as a whole. Lagrange invented the calculus of variations in accordance with the infinitesimal conception, properly so called, and even some time before he undertook the general reconstruction of the transcendental analysis. When he had effected that important reform, he easily showed how applicable it was to the calculus of variations, which he exhibited with all suitable development, according to his theory of derived functions. But 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 considered, the more important it is to husband the powers of our minds in its application, by adopting the most direct and rapid analytical conception, which is, as we know, that of Leibnitz. Lagrange himself therefore constantly pre- ferred 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. Relation to the III the section on the Litegral Calculus, I ordinary Cal- noticed D'Alembert's creation of the Calcvlvs culns. Qj' partial differences, in which Lagrange recognized a new calculus. This new elementary idea in transcendental analysis, — the iiotion of two kinds of incre- ments, distinct and independent of each other, which a function of two variables may receive in virtue of the change of each variable separately, — seems to me to estab- lish 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 than transfer to the independent variables themselves the view already adopted for the functions of those variables ; a process which has remarkably extended its lase. A recognition of sucli a derivation as this for the method of variations may exhibit its i^hilosophical character CONSIDERATIONS ON THE METHOD OF VARIATIONS. 91 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 instrument for the mathematical study of natural phenomena, inasmuch as it introduced the consideration of auxiliary magnitudes, so chosen as that their relations were necessarily more simple and easy to obtain than those of the direct magnitudes. But we had not any general and abstract rules for the formation of these differential equa- tions ; nor were such supposed to be possible. Now, the Analysis of Variations brings the actual establishment of the differential equations within the reach of the Calculus ; for such is the general effect, in a great number of impor- tant and difficult questions, of the varied equations, which, still more indirect than the simple differential equations, as regards the special objects of the inquiry, are more easy to form: and, by invariable and complete analytical methods, employed to eliminate the new order of auxiliary infinitesi- mals introduced, we may deduce those ordinary differential equations which we might not have been able to establish dii'ectly. 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 uninterrujited succession of ideas, genei'al methods more and more potent for the investigation of natural j)hilosophv. 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 employment of this method exacts the highest known degree of intellectual exertion, in order never to lose sight of the precise object of the investigation in following reasonings which offer to the mind such uncertain I'esting- places, and in which signs are of scarcely any assistance, we shall understand how it may be that so little use has been made of such a conception by any philosophers but Lagrange. We have now reviewed Mathematical analysis, in its bases and in its divisions, very briefly, but from a philoso- phical point of view, neglecting those conceptions only 92 POSITIVE PHILOSOPHY. which are not organized with the great whole, or which, if nrged to their limit, would be found to merge in some which have been examined. I must next offer a similar outline of Concrete Mathematics. My particular task will 1)6 to show how, — supposing the general science of the Calculus to be in a perfect state, — it has been possible to reduce, by invariable procedures, to pure questions of analysis, all the problems of Geometry and Mechanics ; and thus to invest these two great bases of natural philosophy with that precision and unity which can only thus be attained, and which constitute high perfection. 93 W CHAPTER III. GENERAL VIEW OF GEOMETRY. E have seen that Geometry is a true ^ natural science ; — only more simple, ' ^^* and therefore more perfect than any other. We must not suppose that, because it admits the application of mathe- matical analysis, it is therefore a purely logical science, independent of observation. Every body studied by geo- meters presents some primitive phenomena which, not being discoverable by reasoning, must be due to observation alone. The scientific eminence of Geometry arises from the extreme generality and simplicity of its phenomena. If all the parts of the universe wei'e regarded as immovable, geometry would still exist ; whereas, for the j^henomena of Mechanics, motion is required. Thus Geometry is the more general of the two. It is also the more simple, for its phenomena are independent of those of Mechanics, while mechanical phenomena are always complicated with those of geometry. The same is true in the comparison of abstract thermology with geometry. For these reasons, geometry holds the first place under the head of Conci'ete Mathematics. Instead of adopting the inadequate ordi- d r >• nary account of Geometry, that it is the science of extension, I am disposed to give, as a general description of it, that it is the science of the measurement of extension. Even this does not include all the operations of geometry, for there are many investigations which do not appear to have for their object the measurement of extension. But regarding the science in its leading ques- tions as a whole, we may accurately say that the measure- ment of lines, of surfaces, and of volumes, is the invariable 94 POSITIVE PHILOSOPHY. aim, — sometimes direct, though ofteiier indirect, — of geo- metrical labours. T , r o The rational study of geometry could never have begun if we must have regarded at once and together all the physical properties of bodies, together with their magnitude and form. By the character of our minds Ave are- able to think of the dimensions and figure of a body in an abstract way. After observation has shown us, for instance, the impi'ession left by a body on a fluid in which it has been placed, we are able to retain an image of the impression, which becomes a ground of geometrical reasoning. We thus obtain, — apart from all metaphysical fancies, — an idea of Space. This abstrac- tion, now so familiar to us that we cannot conceive the state we should be in withoiat it, is perhaps the earliest ])hilosophical creation of the human mind. ,.. , . There is another abstraction which must Kuids of ex- 1 1 1 £ 1. J. • 1 tension "^ made beiore we can enter on geometrical science. We must conceive of three kinds of extension, and leani to conceive of them separately. We cannot conceive of any space, filled by any object, which has not at once volume, surface, and Ime. Yet geometrical ([uestions often relate to only two of these ; frequently only to one. Even when all three are to be finally con- sidered, it is often necessary, in order to avoid complica- tion, to take only one at a time. This is the second abstrac- tion which it is indispensable for us to practise, — to think of surface and line apart from volume ; and again, of line apart from surface. We effect this by thinking of volume as becoming thinner and thinner, till surface appears as the thinnest possible layer or film : and again, we think of this 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 point 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 portion of it is left. This point indicates, not extension, of course, but position, or the place of extension. Surfaces have clearly the property of circumscribing volumes ; lines, again, circumscribe sur- faces ; and lines, once more, are limited by points. METHODS OF MEASUREMENT. 95 The Mathematical meaning of measure- vient is simply the finding of the value of measurement^ the ratios between any homogeneous magni- tudes : but geometrically, the measurement is always in- direct. The comparison of two lines is direct ; that of two surfaces or two volumes can never be direct. One line may be conceived to be laid upon another : but one volume cannot be conceived of as laid upon another, nor one sur- face upon another, with any convenience or exactness. The question is, then, how to measure surfaces and volumes. Whatever be the form of a body, there Measurement must always be lines, the length of which of surfaces and will define the magnitude of the surface or volumes, volume. It is the business of geometry to use these lines, directly measurable as they are, for the ascertainment of the ratio of the surface to the unity of surface, or of the volume to the unity of volume, as either may be sought. In brief, the object is to reduce all comparisons of surfaces or of volumes to simple comparisons of lines. Extending the process, we find the possibility of reducing to questions of lines all questions relating to surfaces and volumes, regarded in relation to their magnitude. It is true that when the rational method becomes too complicated and difiicult, direct comparisons of surfaces and volumes are employed : but the procedure is not geometrical. In the same way, the consideration of weight is sometimes brought in, to determine volume, or even surface ; but this device is derived from mechanics, and has nothing to do with rational geometry. In speaking of the direct measurement of . lines, it is clear that right lines are meant. lines" ^^ When we consider curved lines, it is evident that their ineasurement must be indirect, since we cannot conceive of curved lines being laid upon each other with any precision or certainty. The pi'ocedure is first to reduce the measurement of curved to that of right lines ; and consequently to reduce to simj^le questions of right lines all questions relating to the magnitude of any curves what- ever. 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 96 POSITIVE PHILOSOPHY. circumfei'ence ; and again, as the length of an ellipse depends on that of its two axes. Thus, the science of G-eometry has for its object the final reduction of the comparisons of all kinds of extent to comparisons of right lines, which alone are capable of direct comparison, and are, moreover, eminently easy to manage. I must just notice that there is a primary distinct branch of Geometry, exclusively devoted to the right line, on account of occasionable insurmountable difficulties in making the direct comparison ; its object is to determine certain right lines from others by means of the relations proper to the figures resulting from their assemblage. The importance of this is clear, as no question 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 surfaces ; and, finally, that of volumes. . The field of geometrical science is abso- field * l"*^^y unbounded. There may be as many questions as there are conceivable figures ; and the variety of conceivable figvires is infinite. 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 generating new ones, and those again others. Surfaces, again, are conceived of as motions of lines ; and they not only partake of the variety of lines, but have another of their own, arising from the possible change of nature in the line. There can 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 surfaces may vary : and Ave may say that if lines have one infinity of possible change, surfaces have two. As for Volumes, they are distingi;ished from one another only by the surfaces which bound them ; so that they partake of the variety of surfaces, and need no special considera- tion under this head. If we add the one further remark, that sui'faces themselves furnish a new means of con- ceiving of new curves, as every curve may be regarded as SCOPE OF GEOMETRICAL SCIENCE. 97 produced by the intersection of two surfaces, we shall per- ceive that, starting from a narrow ground of observation, we can obtain an absolutely infinite variety of forms, and therefore an illimitable field for geometrical science. The connection between abstract and con- Properties of Crete geometry is established by the study lines and sur- of the properties of lines and surfaces. With- faces, out multiplying in this way our means of recognition, we should not know, except by accident, how to find in nature the figure we desire to verify. Astronomy was recreated by Kepler's discovery that the ellipse was the cuiwe which the planets describe about the sun, and the satellites about their planet. This discovery could never have been made if geometers had known no more of the ellipse than as the oblique section of a circular cone by a plane. All the properties of the conic sections brought out by the speculative labours of the G-reek geometers, were needed as preparation for this discovery, that Kepler might select from them the characteristic which was the true key to the planetary orbit. In the same way, the sjiherical figure of the earth could not have been discovered if the primitive character of the sphere had been the only one known ; — viz. the equidistance of all its points from an interior point. Certain properties of surfaces Avere the means used for connecting the abstract reasoning with the concrete fact. And others, again, were required to prove that tlie earth is not absolutely spherical, and how much otherwise. The pursuit of these labours does not interfere with the definition of Geometry given above, as they tend indirectly to the measurement of extension. The great body of geometrical researches relates to the properties of lines and surfaces ; and the study of the proj^erties of the same figure 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 far as ever from being fi.nished. And here opens another infinity. We 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 j)oints of view from which each figure may be regarded. I. H 98 POSITIVE PHILOSOPHY. There are two general Methods of treating Methods geometi'ieal questions. These are commonly called Synthetical Geometry and Analytical Geometry. I shall jirefer the historical titles of Geometry of the Ancients and Geometry of the Moderns. But it is, in my view, better still to call them Sj^ecial Geometry and General Geometry, by w^hich their nature is most accurately conveyed. Special or "^^^ Calculus was not, as some suppose, ancient, and unknown to the ancients, as we perceive by general or their applications of the theory of propor- modern Geo- tions. The difference between them and us me ly. ^^ ^^^ ^^ much in the instrument of deduction as in the nature of the questions considered. The ancients studied geometry with reference to the hodies under notice, or specially : the moderns study it with reference to the phenomena to be considered, or generally. The ancients extracted all they could out of one line or surface, before passing to another ; and each inquiry gave little or no assistance in the next. The moderns, since Descartes, employ themselves on questions which relate to any figure whatever. They 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 superiority 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 economized liy the general method \ised since the great revolution Tinder Descartes. The benefit to Concrete Geometry is no less than to the Abstract ; for the recognition of geome- trical figures in nature was merely embarrassed by the study of lines in detail; and the application of the con- templated figure to the existing body could be only acci- dental, and within a limited or doubtful range : whereas, by the general method, no existing figure can escape application to its true theory, as soon as its geometrical features are ascertained. Still, the ancient method was natural ; and it was necessary that it should precede the SPECIAL AND GENERAL GEOMETRY. 99 modern. The experience of tlae ancients, and the materials they accumulated by their special method, were indispen- sable to suggest the conception of Descartes, and to furnish a basis for the general procediu-e. It is evident that the Calculus cannot originate any science. Equations must exist as a starting-point for analytical operations. No other beginning can be made than the direct study of the object, pursued up to the point of the discovery of precise relations. We must briefly survey the geometry of the ^, ^ r . . (TponiPii'V or ancients, in its chai-acter of an indispensable ^j^^ ancients introduction to that of the 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 founded upon observation, though the materials furnished by observation eive few and simple, and the structure of reasoniug erected upon them vast and complex. The only elementary materials, obtainable by direct study alone, are those wliich relate to the right line for the geometry of lines ; to the quadrature of rectilinear plane areas ; and to the cuhature of bodies terminated by plane faces. The beginning of geometry must be from the observation of lines, of flat surfaces angu- larly bounded, and of bodies which have more or less bulk, also angularly bounded. These are all ; for all other figures, even the circle, and the figures belonging to it, now come under the head of analytical geometry. The three elements just mentioned allow a sufficiency of equa- tions for the calculus to proceed upon. More are not needed ; and we cannot do with less. Some have endea- voured to extend analysis so as to dispense with a portion of these facts ; but to do so is merely to return to meta- physical practices, in presenting actual facts as logical abstractions. The more we perceive Geometry to be, in our day, essentially analytical, the more cai-eful we must lie not to lose sight of the basis of observation on which 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 100 POSITIVE PHILOSOPHY. carry logical demonstration into a region where direct observation "vvill serve us better. There are two ways of studying the right thTri'^'h/Hne ^^"® — ^^^® graphic and the algebraic. The thing to be done is to ascertain, by means of one another, the different elements of any right line what- ever, so as to understand, indirectly, a right line, under any „ circumstances whatever. The way to do this solutions ^^' fi^^t' t^ study the figure, by constructing it, or otherwise directly investigating it ; and then, to reason from that observation. The ancients, in the early days of the science, made great use of the graphic method, even in the form of Construction ; as when Aristarchus of Samos estimated the distance of the sun and moon from the earth on a triangle constructed as nearly as possible in resemblance to the right-angled triangle formed by the three bodies at the instant when the moon is in quadrature, and when therefore an observa- tion of the angle at the earth would define the triangle. Archimedes himself, though he was the first to introduce calculated determinations into geometry, frequently used the same means. The introduction of trigonometry lessened the practice ; but did not abolish it. The Greeks and Arabians emjiloyed it still for a great number of in- vestigations for which we now consider the use of the Calculus indispensable. While the grajjhic or constructive method answers well when all the parts of the proposed figure lie in the same plane, it must receive additions before it can be applied to figures whose parts lie in different planes. Hence arises a new series of considerations, and different systems of Pro- jections. Where we now employ sj^herical trigonometry, especially for problems relating to the celestial sphere, the ancients had to consider how they could replace constrnc- tions in relief by plane constructions. This was the object of their analemvnas, and of the other plane figures which long su])plied the place of the Calculus. They were ac- quainted with the elements of what we call Descriptive Geometry, though they did not conceive of it in a distinct and general manner. Digressing here for a moment into the region of ai)pli- DESCRIPTIVE GEOMETRY. 101 cation, I may observe that Descriptive G-eo- . metry, formed into a distinct system by Geometry^ 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 in- dustrial arts ; — its few abstract problems, capable of invariable solution, relating essentially to the contacts and intersections of surfaces ; so that all the geometrical questions which may arise in any of the various arts of construction, — as stone-cutting, carpentry, perspective, dialling, fortification, etc., — can always be treated as simple individual cases of a single theory, the solution being certainly obtainable through the particular circum- stances 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 towai-ds a philo- sop>hical 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 most perfect, and the most ancient of the sciences. It must extend, in time, though less readily, to all other industrial operations. Monge, who understood the philosophy of the arts better than any one else, himself indeed endeavoui'ed to sketch out a philosophical system of mechanical arts, and at least suc- ceeded iu 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 geo- metrical combinations 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. Consist- ing, 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 modei'ns. Yet, as destined to an industrial aj^plication, Descriptive Geometry has 102 POSITIVE PHILOSOPHY. }iere been treated of only in the way of digression. Leaving the subject of graphic solution, we have to notice the other branch ,^ — the algebraic. . . , . Some may wonder that this branch is not Solutions treated as belonging to G-eneral Geometry. But, not only were the ancients, in fact, the inventors of trigonometry, — spherical as well as recti- linear, — though it necessarily remained imj)erfect in their hands ; but algebraic solutions are also no part of ana- lytical geometry, but only a complement of elementary geometry. Since all right-lined figures can be decomposed into triangles, all that we want is to be able to determine the different elements of a triangle by means of one another. This reduces polygonometiy to simple trigonometry. Ti • , The difficulty lies in form iup- three distinct '^ equations between the angles and the sides of a triangle. These equations being obtained, all trigono- metrical problems are reduced to mere questions of analysis. — There are two methods of introducing the angles into the calculation. They are either introduced directly, by themselves or by the circular arcs which are j)ropor- tional to them : or they are inti'oduced indirectly, by the chords of these arcs, which ai-e hence called their trigono- metrical lines. The second of these methods was the first adopted, because the early state of knowledge admitted of its woi'king, 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 trigono- metrical lines. — The method which employs the trigono- metrical lines is still j^ref erred, 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 com- plication than a simplification to introduce these lines, which have at last to be eliminated, we must explain a little. Their introduction divides trigonometry into two parts. In one, we pass from the angles to their trigonometrical lines, or the converse : in the other we have to determine the sides of the triangles by the trigonometrical lines of TRIGONOMETRY. 103 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 logarithms, by which all imaginable arithmetical operations are decomposed into two parts — • the first and most difficult of which is done in advance. We must remember, too, in considering the jjosition of the ancients, the remarkable fact that the determination of angles by their trigonometrical lines, and the converse, admits of an arithmetical solution, without the jirevious resolution of the corresponding algebraic question. But for this, the ancients could not have obtained trigono- metry. When Archimedes was at Avork upon the rectifi- cation 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 trigo- nometry, he had only to complete 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 to one, as the ancients did. The Arabians, and after them the moderns, attained to only four or five direct trigonometrical lines altogether ; whereas it is clear that the number is not limited. — Instead, how- ever, 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 am' line relating to an arc which is a very simple function of the first. Thus, we may say that the number of trigonometrical lines actually employed by modern geometers is unlimited through the augmentations we may obtain by analysis. Special names have, however, been given to those indirect lines only which refer to the complement of the primitive arc, — others being in much less frequent use. Out of this device arises a third section of trigono- 104 POSITIVE PHILOSOPHY. metrical knowledge. Having introduced a new set of lines, — of auxiliary magnitudes — we have to determine their relation to the first. And this study, though pre- paratory, is indefinite in its scope, while the two other departments are strictly limited. The three must, of course, be studied in just the reverse order to that in which it has been necessary to exhibit them. First, the student must know the relations between the indirect and direct trigonometrical lines : and the re- solution of triangles, properly so called, is the last process. Spherical trigonometry requires no special notice here, (all-important as it is by its uses,) — since it is, in our day, simply an application of rectilinear trigonometry, through the substitution of the corresponding trihedral angle for the spherical triangle. This view of the philosophy of trigonometry has been given chiefly to show how the most simple questions of elementary 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, beginning with the great original idea of Descartes, on which it is wholly founded. Modern, or Analytical Geometry. General or Analytical Geometry is founded upon the transformation of geometrical considerations into equiva- lent analytical considerations. Descartes established the constant possibility of doing this in a uniform manner : and his beautiful conception is interesting, not only from its carrying on geometrical science to a logical perfection, but from its showing us how to organize the relations of the abstract to the concrete in Mathematics by the ana- lytical representation of natural phenomena. Analytical The first thing to be done is evidently to representation find and fix a method for expressing analyti- of ligures. cally the subjects which aft'ord the phe- nomena. If we can regard lines and surfaces analytically. MODERN OR ANALYTICAL GEOMETRY. 105 we can so regard, heiieefortli, the accidents of tliese suhjects. Here occurs the difficulty of reducing all geometrical ideas to those of number : of substituting considerations of quantity for all considerations of quality. — In dealing with this difficulty, we must observe that all geometrical ideas come under three heads : — the magnitude, the Position figure, and the position of the extensions in question. The relation of the first, magnitude, to numbers is immediate and evident: and the other two are easily brought into one ; for the figure of a body is nothing else than the natural position of the jjoints of whirh it is com- posed : and its position 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 Geometry. 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 are known, by assigning the magnitude of the different geometrical elements which connect it with known objects. Those elements are what Descartes, and all other geo- meters after him, have called the co-ordinates of the point considered. If we know in what plane the point is situated, the co-ordinates are two. If the point may be anywhere m 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 pure numerical variations in the values of its co-ordinates. — The simplest case of all, that of plane geometry, is when we determine the ^^osition of a point on a plane by considering its distances from two fixed right lines, supposed to be known, and generally concluded to be perpendicular to each other. These are called axes. Next, there may be the less simple process of determining the position by the distances from fixed points ; and so on to greater and greater complications. But, from some system or other of co-ordinates being always employed. 106 POSITIVE PHILOSOPHY. the question of position is always reduced to that of magnitude. ,^ .,. . It is clear that our only way of marking •jj^ the position or a point is by the inter- section of two lines. When the point is determined by the intersection of two right lines, each parallel to a fixed axis, that is the system of rectilinear co- ordinates, — the most common of all. The polar system of co-ordinates exhibits the point by the travelling- of a right line round a fixed centre of a circle of variable radius. Again, two circles may intersect, 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 our- selves in this necessary method of conceiving of position : and their geometrical loci were founded upon it. It was in endeavouring to form the process into a general system tliat Descartes created Analytical Geometry. — Seeing, as we now do, how ideas of position, — and, through them, all elemen- tary geometrical ideas, — can be reduced to ideas of number, we learu what it was that he effected. „i ^ Descartes treated only geometry of two Plane Curves. -,. . ■ ■, ■ i .• i ii / t dimensions m his analytical method : and we will at first consider only this kind, beginning with Plane Expression Curves. Lines must be expressed by equa- of Hnes by tions ; and again, equations must be expressed Equations. \)y Hues, when the relation of geometrical con- ceptions to numbers is established. — It comes to the same thing whether we define a line by any one of its properties, or supply the corresponding equation between the two variable co-ordinates of the point which describes 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 exj^ressed 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 ANALYTICAL GEOMETRY. 107 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 require 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 Expression geometrical necessity of representing by a of equations certain line every equation of two variables, ^^y lines, in a determinate system of co-ordinates. In the absence of any other known property, such a relation would be a very chai'acteristic definition ; and its scientific effect would be to fix the attention immediately uj^on the general course of the solutions of the equation, which will thus be noted in the most striking and simple manner. There is an evident and vast advantage in this picturing of ecj^uations, which reacts strongly upon the perfecting of analysis itself. The geometrical loc^ls stands before our minds as the represen- tation of all the details that have gone to its preparation, and thus renders comparatively easy our conception of new general analytical views. This method has become entirely elementary in our day ; and it is employed when we want to get a clear idea of the general character of the law which runs through a series of particular observations of any kind whatever. Recurring to the representation of lines by Chanf,^e in the equations, which is our chief object, we see line changes that this representation is, by its nature, so ^1^*^ equation, faithful, that the line could not undergo any modification, even the slightest, without causing a corresponding change in the equation. Some special difiiculties arise out of this perfect exactness ; for since, in our system of analytical geometry, mei'e displacements of lines affect equations as much as real variations of magnitude or form, we inight be in danger of confounding the one with the other, if geo- meters had not discovered an ingenious method exj^ressly intended to distinguish them always. It must be observed that general inconveniences of this nature appear to be strictly inevitable in analytical geometry ; since, ideas of 108 POSITIVE PHILOSOPHY. position being the only geometrical ideas immediately re- ducible 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 should not at first confound phenomena of form with simjDle phenomena of position ; which are the only ones that equations express directly. Every defini- To complete our description of the basis of tion of a line analytical geometry, it is necessary to point IS an equation, q^^ iha,t not only must every defined line give rise to a certain equation between the two co-ordinates of any one of its points, but ever'y definition of a line is itself an equation of that line in a suitable system of co- ordinates. Considering, first, what a definition is, we say it must distinguish the defined object from all others, by assigning to it a property which belongs to it alone. But this pro- ]')erty 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 generation, and in that case the definition is explanatory. For instance, if we say that the circle is the line which in the same form contains the largest area, we offer a characteristic definition ; whereas if we choose its property of having all its points equally distant from a fixed point, we have an explanatory defini- tion. It is clear moreover that the characteristic definition always leaves room for an explanatory one, which further study must disclose. It is to explanatory definitions only that Avhat has been said of the definition of a line being an equation of that line can apply. We 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 decom- posed at each moment. Now, if we form the most general conception of Avhat a system of co-ordinates is, and if we admit all possible systems, it is clear that such a relation can be nothing else than the equation of the proposed line, in a system of co-ordinates of a corresponding nature to that of the mode of generation considen^d ; as in the case of the circle, the common definition of which may be re- ANALYTICAL GEOMETRY. 109 gardecl as being the polar equation of that curve, taking the centre of the circle for the pole. This view not only exhibits the necessary representation of every line by an equation, but it indicates the general difficulty which occurs in the establishment of these equa- tions, and therefore shows us how to proceed in inquiries of this kind which, by their nature, do not admit of in- variable 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 ec|uation of a curve by means of some of its characteristic properties, the difficulty must proceed from our taking up a designated system of co-ordinates, instead of admitting indifferently all possible systems. These systems are not all equally siiitable ; and, in regard to curves, geometers think that they should almost always be referred, as far as possible, to rectilinear co-ordi- nates. 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 may be most suitable co-ordinate'^ to the case — is the remaining point which we have to notice. First, we must distiuguish vei'y 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, recipro- cally, 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 alternately, and almost insensibly, we must be able to separate them here, for the answer to the question of 110 POSITIVE PHILOSOPHY. metliod wliicli we are considering is far from being the same under tlie two relations : so that without this distinc- tion we could not form any clear idea of it. In the case of the representation of lines 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 aiTiving at it. There can be no constant preference here of one system of co-ordinates. The recti- linear system itself, though often advantageous, cannot be always so, and may be, in turn, less so than any other. But it is far otherwise in the converse case of the repre- sentation of equations by lines. Here the rectilinear system is always to be preferred, as the most simple and trustworthy. If we seek to determine a point by the inter- section of two lines, it must be best that those lines should be the simplest possible ; and this confines our choice to the rectilinear system. In constructing geometrical loci, that system of co-ordinates must be the best in which it is easiest to conceive the change of place of a point resulting from the change in the value of its co-ordinates ; and this is the case with the rectilinear system. Again, there is great advan- tage in the common usage of taking the two axes perpen- dicular to each other, when 2:)ossible, rather than with any other inclination. In rej^resenting lines by equations, we must take any inclination of the axes which may best suit the particular question ; but, in the converse case, it is easy to see that rectangular axes permit iis to represent equations in a more sinijile, and even in a more faithful manner. For if we extend the geometrical lociis of the equation into the several imequal regions marked out by oblique axes, we shall have differences of figure which do not correspond to any analytical diversity ; and the accuracy of the x'epresen- tation will be lost. On the whole then, taking together the two points of view of analytical geometry, the ordinary system of recti- linear 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 rectilinear system. The polar system is preferred next RECTILINEAR AND POLAR CO-ORDINATES. Ill to it, both because its opposite character enables it to solve iu the simplest way the equations which are too compli- cated for management under the first ; and because j^olar co-ordinates have often the advantage of admitting of a more direct and natural concrete signification. This is the case in Mechanics, iu 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 conception of analytical geometry being designed only for the stady of Plane Curves. It was Clairaut who, about a century later, extended it to the study of Surfaces and Curves of double curvature. The conception having been explained, a very brief notice vsdll suffice for the rest. With regard to Surfaces, the determination Determination of a jjoint in space requires that the values of of a point in three co-ordinates should be assigned. The •'^pace. system generally adoj^ted, which corresj^onds with the rectilinear system of plane geometry, is that of distances from the point to three fixed planes, usually perpendicular to each other, Avhereby the point is presented as the inter- section of three ])lanes whose direction is invariable. Be- yond this, there is the same infinite variety among possible systems of co-ordinates, that there is in geometry of two tlimensions. Instead of the intersection of two lines, it must be that of three surfaces which determines the point ; and each of the three surfaces has, in the same way, all its conditions constant, except one, which gives rise to the corresponding co-ordinates, whose peculiar geometrical effect is thus to compel the point to be situated upon that sm-face. 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 sur- face is defined, two co-ordinates suffice for determining its situation at any moment ; as the proposed surface will take the place of the condition imposed by the third co-ordi- nate. This last co-ordinate then becomes a determinate function of the two others, they remaining independent of each other. Thus, there will be a certain equation between the three variable co-ordinates which will be permanent, 112 POSITIVE PHILOSOPHY. and which will be the only one, in order to correspond to the precise degree of in determination in the position of the point. Determin