Production Note
Cornell University Library pro-
duced this volume to replace the
irreparably deteriorated original.
It was scanned using Xerox soft-
ware and equipment at 600 dots
per inch resolution and com-
pressed prior to storage using
CCITT Group 4 compression. The
digital data were used to create
Cornell's replacement volume on
paper that meets the ANSI Stand-
ard Z39.48-1984. The production
of this volume was supported in
part by the Commission on Pres-
ervation and Access and the Xerox
Corporation. Digital file copy-
right by Cornell University
Library 1991.®owll	Jitatg
~P ruSCi'ZrfJL S cu^n cc .b^rn.......
SjUXMVh^piAjjA bo C -U, 'fcSsUtAij injmJiS'
is.o.iA ig...........^n,BOHN’S PHILOSOPHICAL LIBRARY
THE POSITIVE PHILOSOPHY OF
AUGUSTE COMTE
VOL. I.GEORGE BELL & SONS
LONDON : YORK STREET, COVENT GARDEN
AND NEW YORK, 66, FIFTH AVENUE
CAMBRIDGE :	DETGHTON, 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
1896
'r:CHISWICK PRESS .—CHARLES WHITTINGHAM AND CO.
TOOKS COURT, CHANCERY LANE, LONDON.INTRODUCTION.
“If it cannot be said of Comte that he has created a science, it
may be 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 generally admitted to have been the most emi-
nent and important of that interesting group of thinkers whom
the overthrow of old institutions in France turned towards social
speculations. ’?—John Morley.
HE foregoing quotations from the two English
authorities who have most severely criticized the
“ Positive Polity ” of Auguste Comte, bear witness to
the profound impulse 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 speculations, 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,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 was born at Montpellier, in the south of
France, 19tli Jan., 1798, the eldest son of Louis Comte,
treasurer of taxes for the department of Herault, and of
Rosalie Boyer, whose family produced some eminent
physicians. 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 courage 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 Xapoleon 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 he 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 Poly technique, the first on the list of
candidates for the south and centre of France.
N 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, whetherINTRODUCTION.
Yll
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 support-
ing himself by 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 Saint-Simon, with
whom he remained in relations for four or five years. The
vague, optimistic, and humanitarian dreams of this singular
reformer did undoubtedly exercise a certain fascination
over the youthful mind of Comte, and gave his genius and
character an inflexible 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 and moral re-organization could
only proceed from the authority of government. Saint-
Simon claimed as his own the work of his young 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 work was a pamphlet in 191
pages, published in May, 1822, with an introduction byINTRODUCTION.
viii
Saint-Simon. It was entitled a “Prospectus of the
scientific works required for the reorganization of Society,
by Auguste Comte, former pupil of the Ecole Poly technique ”
He republished his pamphlet with 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 “ Bevue Occidentale ” (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 rthat which 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 been treated by
Comte and by his adherents as the the first sketch of the
“ Positive Philosophy.”
Between 1816 and 1826 (aetat. 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, Bonnin, Poinsot, Carnot, Guizot, J. B.
Say, Dunoyer, Professor Buchholtz of Berlin, de Yillele,
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 (aetat. 28), lie
opened in his own rooms a course of public lectures 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 general, Cerclet,INTRODUCTION.
IX
Montgery, and other young students. The series was in
fact that which was subsequently published. At the
fourth lecture the course was abruptly broken oft*. Intense
mental strain, together with domestic misery, brought on
an attack of insanityNL 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. Esquirol 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 Philosophy, and he had
the satisfaction of seeing the same eminent men amongst
his audience, with the exception of Humboldt, who was 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 course at the Athenee. He also
gave other gratuitous public lectures, including the series
on Popular Astronomy which he repeated during eighteen
years, from 1830 to 1848^( In 1832, Comte was appointed
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 < candi-
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 separately, with
a dedication to Baron Fourier and M. de Blainville. A
brief note described it as the result of the author’s laboursX
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 1885, 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 be 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 repudiation of Saint-Simon, and his no less
vehement condemnation of M. Arago and the official
directors of the Ecole Polytechnique.
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 distraction. No wish for pre-
mature publication was suffered to lead his mind off 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 work. In the history of
men who have devoted their lives to great thoughts, I
know nothing nobler than that of these twelve years.” 1
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.
1 “Auguste Comte, et la Philosophic Positive,” 1863, p. 188.INTRODUCTION.
XI
His method of composition 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 pages and was written in twenty-eight days,
although 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 pathos, 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 gradually became a confirmed man-
nerism.
Tedious and even repulsive 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
Yoltaire, Hume and Berkeley. But his long-drawn andXll
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 general
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 often adopted his generous sug-
gestion.
“ The Philosophic 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. lxvii., 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 44 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 historyINTRODUCTION.
xin
of philosophical evolution. In 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 finally, after some interruptions, in the
beginning of November, 1853, and it was received with a
chorus of approval by the French philosopher and by his
English readers. Comte’s own opinion is set forth in the
letters of his printed by M. Littre in his biographical work,
to which we shall presently return. George Grote, 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. Avezac-Lavigne, a Bordeaux disciple of Comte, appeared
in May, 1871. The correspondence between him and Miss
Martineau is set out in the “ Autobiography ” (vol. iii.,
p. 310).
The outspoken language of the “ personal preface ” to
the sixth volume of the “ Philosophie ” 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 theXIV
INTRODUCTION.
sixth volume, his 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
“Bevue Occidental ” (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 great 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 part, 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
ground 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. Baikes Currie,
and Sir W. Molesworth, provided, in 1844, the salary ofINTRODUCTION.
XV
<£200 of which he was deprived; but to the surprise, and
even the indignation, of Comte, thev declined to make this
permanent. Mr. G-rote and other friends made some
further contributions; and ultimately, by the help of M.
Littre, Dr. Charles Robin, Dr. Segond, and others, a
regular subsidy was established in 1849. It began with
8,000 francs (£120), ultimately rose to 8,000 francs (£320),
and it has been continued until the present 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 by
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 £400, but 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 most
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 Yaux, 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 veuvageXVI
INTRODUCTION.
kernel. 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 he died of
cancer on September 5th, 1857. He was buried in Pere 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 directions of his last will.
This is not the place to enter on the complex 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 biography 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 “ Philosophy,” 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 theINTRODUCTION.
XVll
point of view of the unity of Comte’s career these ten pages are
crucial, for they contain the entire scheme of Comte’s future
philosophical labours as he designed them in 1842. paid 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 impression 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 difficult task.” “ The important undertaking
that you so happily conceived and have so worthily accom-
plished will give my ‘ Positive Philosoj)hy ’ a competent
audience greater than I could have hoped to find in my
i.	6INTRODUCTION.
xviii
own lifetime.” 44 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 Martineau 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 44 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 popular exposition, and admiring the
energy and skill with 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
pages to something 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 been presented in outline and not in detail.
Nor can it be denied that there are points, and even pointsINTRODUCTION.
XIX
of great importance, in 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 attempt to modify or even to
indicate them. It has been no part 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.
1895.
For biographies and criticisms of Comte the following
works may be consulted:
G. H. Lewes. “History of Philosophy.” 5th edition, 1880,
vol. ii.
J. Stuart Mill. “System of Logic,” vol. ii., and “Auguste Comte
and Positivism,” 1865.
John Morley. Comte, “ Encyclopaedia Brit.,” vol. vi. ; “ Critical
Miscellanies,” vol. iii.
Littre. “Auguste Comte et la Philosophie Positive,” 1863.
Herbert Spencer. “ Essays,” vol. iii. T. Huxley. “Essays.’
“Revue Occidentale,” passim, 1878-1895, and especially three
articles by Lonchampt, vol. xxii. p. 271, vol. xxiii. pp. 1, 135
(1889).
Robinet. “Vie d’Auguste Comte,” 1860.
Dr. Bridges. “ Unity of Comte’s Life and Doctrine,” 1866.
Professor E. S. Beesly. “ Comte as a Moral Type,” 1885.
“The Positivist Review,” 1893-1895 (monthly), W. Reeves, 185,
Fleet Street.
Dr. Congreve. “Collected Works.”
“Lettres d’Auguste Comte,” a Valat (1870), a John Stuart Mill
(1877). “Testament ’ (1884).PREFACE
BY
HARRIET MARTINEAU.1
IT may appear strange that, in these days, when the
French language is almost 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, but
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 foundation of all that is systematic in our
knowledge. This being the case, it may appear to be a
needless labour to render into our own tongue what is
clearly existing in so many of the minds which are guiding
and forming popular views. But it was not without reason
that I undertook so serious a labour, while so much 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
1 To the first edition published by John Chapman in 1853, in two
volumes 8vo, of which the present edition is a reprint.XXII
PREFACE.
of receiving due honour, sooner or later. Before the end
of the century, society at large will have become aware
that this work is one of the chief honours of the century,
and that its author’s name will rank with those of the
worthies who have illustrated former ages: but 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 awakened
and extended in proportion to the spread among us of a
popular knowledge of what M. Comte has done: and this
hope was one reason, though, as I have said, not the chief,
for my undertaking to reproduce his work in England in a
form as popular as its nature admits.
A stronger reason was that M. Comte’s work, in its
original form, does no justice to its importance, even in
France ; and much less in England. It is in the form of
lectures, the delivery of which was spread over a long
course of years; and this extension of time necessitated
an amount of recapitulation very injurious to its interest
and philosophical aspect. M. Comte’s style is singular.
It is at the same time rich and diffuse. Every sentence is
full fraught with meaning ; yet it is overloaded with words.
His 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 years, becomes weakness
when those instructions are presented as a whole; and it
appeared to me worth while to condense his work, if I
undertook nothing more, in order to divest it of the dis-
advantages arising from redundancy alone. My belief isPREFACE.
xxm
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 researches 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 one can inquire into the mode of life
of young men of the middle and operative classes without
being struck with the desire that is shown, and the 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 borne, if a year
or two of humble toil could help, more or less, to supply
the need.
In close connection with this was another of my reasons.
The supreme dread of every one who cares for the good of
nation or race is that men should be adrift for want of an
anchorage for their convictions. I believe that no one
questions that a very large proportion of our people are
now so adrift. With pain and fear, we see that a multi-XXIV
PREFACE.
tude, who might and should be 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 effort that
has been made to obviate this kind of danger; and my
deep persuasion is that it will be found to retrieve a vast
amount of wandering, of unsound speculation, of listless
or reckless doubt, and of moral uncertainty and depression.
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 filiation
of the sciences within the boundaries 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 question the general soundness of the exposi-
tion, or of the relations of its parts, are of another school,
and will simply neglect the book, and occupy themselves
as if it had never existed. It is not for such that I have
been working, but for students who are not schoolmen;
who need conviction, and must best know when their need
is satisfied. When this exposition of Positive Philosophy
unfolds itself in order before their eyes, they will, I am
persuaded, find there at least a resting-place for their
thought,—a rallying-point of their scattered speculations,
—and possibly an 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 be 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 hand 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
build 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. The
last words of his last letter were an entreaty that I wouldXXVI
PREFACE.
let him know if more money would, in any way, improve
the quality of my version, or aid the promulgation of the
book. It was a matter of deep concern to me that he died
before I could obtain his opinion as to the manner in
which I was doing my work. All that remained was to
carry out his wishes as far as possible ; and to do this, no
pains have been spared by myself, or by Mr. Chapman,
who gave him the information that called forth his bounty.
As to the method I have pursued with my work,—there
will be different opinions about it, of course. Some will
wish that there had been no omissions, while others would
have complained of length and heaviness, if I had offered
a complete translation. Some will ask why it is not a
close version as far as it goes; and others, I have reason
to believe, would have preferred a brief account, out of my
own mind, of what Comte’s philosophy is, accompanied by
illustrations of my own devising. A wider expectation
seems to be that I should record my own dissent, and that
of some critics of much more weight, from certain of
M. Comte’s views. I thought long and anxiously of this ;
and I was not insensible to the temptation of entering my
protest, here and there, against a statement, a conclusion,
or a method of treatment. I should have been better
satisfied still to have adduced some critical opinions of
much higher value than any of mine can be. But my
deliberate conclusion was that this was not the place nor
the occasion for any such controversy. What I engaged to
do was to present M. Comte’s first great work in a useful
form for English study: and it appears to me that it would
be presumptuous to thrust in my own criticisms, and out
of place to insert those of others. Those others can speak
for themselves, and the readers of the book can criticize it
for themselves. Ho 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.
XXV11
This by no means implies that 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 mv 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
Glasgow, was kind enough to undertake this service.
After two careful readings, he suggested nothing material
in the way of alteration, in the case of the first two
sections, except the omission of Comte’s speculation on the
possible mathematical verification of Laplace’s Cosmogony.
But more had 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 be unfair to present what he wrote
under that head in 1835 as what he would have to say
now. The choice lay therefore between almost re-writing
this portion of M. Comte’s work, or so largely abridging it
that only a skeleton presentment of general principles
should remain. But as the 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 Mchol suggested these, asXXV111
PREFACE.
notes. 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 in
Mental Philosophy, or assents to other conclusions held of
main importance by the disciples of the Positive Philo-
sophy. The contrary, 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 been offered.
With the reservation I have made, I am bound to take the
entire responsibility,—the Work being absolutely and
wholly my own.
It will be observed that M. Comte’s later works are not
referred to in any part of this book. It appears to me
that they, like our English criticisms on the present Work,
had better be treated of separately. Here his analytical
genius has full scope; and what there is of synthesis is, in
regard to social science, merely what is necessary to render
his analysis possible and available. For various reasons, I
think it best to stop here, feeling assured that if this Work
fulfils its function, all else with which M. Comte has
thought fit, to follow it up will be obtained as it is
demanded.
During the whole course of my long task, it has appeared
to me that Comte’s work is the strongest embodied rebuke
ever given to that form of theological intolerance which
censures Positive 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 will not
slacken for its appearing among us in an English version.
It cannot be otherwise. The theological world cannot but
hate a 1 >ook which treats of theological belief as a transient
state of the human mind. And again, the preachers and
teachers, of all sects and schools, who keep to the ancient
practice, once inevitable, of contemplating and judging of
the universe from the point of view of their own minds,
instead of having learned to take their stand out of them-
selves, investigating from the universe inwards, and not
from within outwards, must necessarily think ill of a work
which exposes the futility of their method, and the worth-
lessness of the results to which it leads. As M. Comte
treats of theology and metaphysics as destined to passPREFACE.
XXIX
away, theologians and metaphysicians must necessarily
abhor, dread, and despise his work. They merely express
their own natural feelings on behalf of the objects of their
reverence and the purpose of their lives, when they charge
Positive Philosophy with irreverence, lack of aspiration,
hardness, deficiency of grace and beauty, and so on. They
are no judges of the case. Those who are—those who have
passed through theology and metaphysics, and, finding
what they are now worth, have risen above them—will
pronounce a very different judgment on the contents of
this book, though no appeal for such a judgment is made
in it, and this kind of discussion is nowhere expressly pro-
vided for. To those who have learned the difficult task of
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 stimulating to his intellectual taste.
We find ourselves suddenly living and moving in the midst of
the universe,—as a part of it, and not as its aim and object.
We find ourselves living, not under capricious and 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 of no instruction so favourable to
aspiration as that which shows us how great are our
faculties, how small our knowledge, how sublime the
heights which we may hope to attain, and how boundless
an infinity may be assumed to spread out beyond. We
find here 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 who insist on belief without
evidence and on a philosophy derived from their own in-
tellectual action, without material and corroboration from
without, and not with those who are too scrupulous andXXX
PREFACE.
too humble to transcend evidence, and to add, out of their
own imaginations, to that which is, and may be, referred
to other judgments. If it be desired to extinguish pre-
sumption, to draw away from low aims, to fill life with
worthy occupations and elevating pleasures, and to raise
human hope and human effort to the highest attainable
point, it seems to me that the best resource is the pursuit
of Positive 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 are
those which are more eminently fostered. The habit of
truth-seeking and truth-speaking, and of true dealing with
self and with all things, is evidently a primary requisite;
and this habit once perfected, the natural conscience, thus
disciplined, will train up all other moral attributes to some
equality with it. To all who know what the study of
philosophy really is,—which means the study of Positive
Philosophy,—its effect on human aspiration and human
discipline is so plain that any doubt can be explained only
on the supposition that accusers do not know what it is
that they are calling in question. My hope is that this
book may achieve, besides the purposes entertained by its
author, the one more that he did not intend, of conveying
a sufficient rebuke to those who, in theological selfishness
or metaphysical pride, speak evil of a philosophy which is
too lofty and too simple, too humble and too generous, for
the habit of their minds. The case is clear. The law of
progress is conspicuously at work throughout human 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 every sect; and therefore must that philo-
sophy be favourable to those virtues whose repression
would be incompatible with progress.CONTENTS.
INTRODUCTION.
CHAPTER I.
ACCOUNT OF THE AIM OF THIS WORK.—VIEW OF THE NATURE
AND IMPORTANCE OF THE POSITIVE PHILOSOPHY.
PAGE
Preliminary survey...................................1
Law of human development.............................1
First stage ...	 2
Second stage.........................................2
Third stage..........................................2
L ltimate point of each ....	...	2
Evidences of the Law.................................3
Actual...............................................3
Theoretical..........................................3
Character of the Positive Philosophy ....	5
. History of the Positive Philosophy.................6
New department of Positive Philosophy................7
Social Physics ..........	8
Secondary aim of this work...........................8
To review the philosophy of the sciences	....	8
Glance at speciality.................................0
Proposed new class of students......................10
Advantages of the Positive Philosophy...............11
1.	Illustrates Intellectual function................11
2.	Must regenerate Education........................13
3. Advances Sciences by combining them	.	.	.	.14
4.	Must reorganize Society..........................15
No hope of reduction to a single Law .	.	.	.	.17
/ "
CHAPTER IL^
VIEW OF THE HIERARCHY OF THE POSITIVE SCIENCES.
Failure of proposed classifications
True principle of classification .
Boundaries of our field .
19
20
20xxxii	CONTENTS.
PAGE
Theoretical Inquiry............................22
Abstract science...............................23
Concrete science...............................23
I Hfficulty of classification..................23
Historical and Dogmatic Methods ......	24
True principle of classification...............26
('liaracters...................................26
1.	Generality..................................26
2.	Independence................................26
I norganic and Organic phenomena.............27
1.	Inorganic .	..............................27
1.	Astronomy ..........	28
2.	Physics ‘.................................28
3.	Chemistry.................................28
II. Organic..................................28
1.	Physiology..................................29
2.	Sociology...................................29
Five Natural Sciences : their filiation......29
Filiation of their parts.....................30
Corroborations..........	30
1.	This classification follows the order of disclosure of sciences 30
2.	Solves heterogeneousness..................30
3.	Marks relative perfection of sciences.....31
4.	Effect on Education ........	32
Effect on Method...............................32
Orderly study of sciences. .......	33
Mathematics....................................34
A department...................................34
A basis ...........	34
An instrument................................34
A double science.............................34
Abstract Mathematics, an instrument..........35
Concrete Mathematics, a science..............35
Mathematics pre-eminent in the scale.........35
BOOK I.
MATHEMATICS.
CHAPTER I.
mathematics, abstract and concrete.
Description of Mathematics......................37
Object of Mathematics...........................37
General Method..................................38
Examples........................................38
True Definition of Mathematics..................40
Its two parts...................................41CONTENTS.	xxxiii
PAGE
Their different objects................................42
Their different natures................................43
Concrete Mathematics...................................43
Abstract Mathematics	........	44
Extent of its domain...................................45
Its Universality.......................................45
Its limitations........................................46
CHAPTER II.
GENERAL VIEW OF MATHEMATICAL ANALYSIS.
Analysis...................................................50
True idea of an equation...................................50
Abstract functions .	.	.	.	.	.	.	.	.51
Concrete functions.........................................51
Two parts of the Calculus .......	52
Algebra....................................................53
Arithmetic ................................................53
Its extent.................................................53
Its nature.................................................54
Algebra....................................................55
Creation of new functions..................................55
Finding equations between auxiliary quantities ...	56
Division of the Calculus of functions......................57
Section 1. Ordinary Analysis, or Calculus of Direct Func-
tions .	  58
Its object.................................................58
Classification of Equations................................58
Algebraic equations........................................59
Algebraic resolution of equations .	.	.	.	.59
Our existing knowledge ........	60
Numerical resolution of equations .	.	.	.	.	.61
The Theory of equations....................................62
Method of indeterminate coefficients.......................62
Section 2. Transcendental Analysis, or Calculus of Indirect
Functions.............................................63
Three principal views......................................63
History....................................................63
Method of Leibnitz.........................................64
Generality of the formulas.................................66
Justification of the Method................................67
Newton’s Method............................................68
Method of Limits...........................................68
Fluxions and fluents. .....................................69
Lagrange’s Method..........................................70
Identity of the three methods..............................71
Their comparative value....................................72
The Differential and Integral Calculus ...	75
Its two parts..............................................75
i.	cxxxiv	CONTENTS.
PAGE
Tlieir mutual relations.......................................75
Cases of union of the two.....................................76
Cases of the Differential calculus alone .....	77
Cases of the Integral calculus alone. .....	77
The Differential Calculus. .......	77
Two portions ........	78
Subdivisions..................................................78
Reduction to the elements.....................................79
Transformation of derived functions for new variables .	.	79
Analytical applications.......................................80
The Integral Calculus ........	81
Its divisions.................................................81
Subdivisions..................................................81
One variable or several ........	81
Orders of differentiation.....................................82
Quadratures ..........	83
Algebraic functions...........................................83
Transcendental functions......................................83
Singular Solutions .........	84
Definite integrals............................................84
Prospects of the Integral Calculus............................85
Calculus of Variations ........	87
Problems giving rise to this calculus .....	87
Other applications............................................89
Relation to the ordinary calculus.............................90
CHAPTER III.
GENERAL VIEW OF GEOMETRY.
Its Nature.........................................93
Definition.........................................93
Idea of Space......................................94
Kinds of extension.................................94
Geometrical measurement............................95
Measurement of surfaces and volumes................95
Of curved lines..........	95
Its illimitable field..............................96
Properties of lines and surfaces ......	97
Two general Methods................................98
Special or ancient, and general or modern geometry .	.	98
Geometry of the ancients...........................99
Geometry of the right line .......	100
Graphical solutions................................100
Descriptive Geometry...............................101
Algebraic solutions................................102
Trigonometry ..........	102
Modern, or Analytical Geometry.....................104
Analytical representation of figures..............104
Position...........................................105CONTENTS.	XXXV
PAGE
Position of a point........................................106
Plane curves...............................................106
Expression of lines by equations...........................106
Expression of equations by lines...........................107
Change in the line changes the equation.	.	.	.	.107
Every definition of a line is an equation .	.	.	.	.108
Choice of co-ordinates.....................................109
Determination of a point in space .	.	.	.	.	.111
Determination of Surfaces by Equations, and of Equations hv
Surfaces..............................................112
Curves of double curvature.................................112
Imperfections of Analytical Geometry.......................113
Imperfections of Analysis..................................113
CHAPTER IV.
RATIONAL MECHANICS.
Its nature.........................................115
Its characters.....................................116
Its object.........................................116
Matter not inert in Physics........................117
Supposed inert in Mechanics........................118
Field of Rational Mechanics........................118
Three Laws of Motion...............................119
Law of inertia .	  119
Law of equality of action and re-action .	.	.	.	.119
Law of co-existence of motions.....................120
Two Primary divisions .	.	.	.	.	.	.	.123
Statics and Dynamics...............................123
Secondary divisions................................123
Solids and Fluids..................................123
Section 1. Statics.................................125
Converse methods of treatment......................125
First method.......................................125
Statics by itself..................................125
Second method......................................125
Statics through Dynamics...........................125
Moments............................................126
Want of unity in the method........................127
Virtual Velocities.................................127
Theory of Couples .	.	.	.	.	.	.	.130
Share of equations in producing equilibrium ....	130
Connection of the concrete with the abstract question .	.	132
Equilibrium of fluids..............................134
Hydrostatics.......................................134
Liquids............................................135
Gases..............................................135
Section 2. Dynamics .	  137
Object ........... 137XXXY1
CONTENTS.
PAGE
Theory 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 II.
ASTRONOMY.
CHAPTER I.
GENERAL VIEW.
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 STUDY OF ASTRONOMY.
Section 1. Instruments...................157
Observation..............................157
Shadows..................................157
Artificial methods .........	158
The pendulum.............................159
Measurement of angles....................159
Requisite corrections....................161CONTENTS.	xxxvii
PAGE
Section 2.	Refraction..................................161
Section 3.	Parallax....................................163
Sections	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...............................180
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...........................................185
Kepler............................................185
His three laws....................................186
First law.........................................186
Second law ..........	186
Third law.........................................187
Three problems....................................188
Prediction of Eclipses............................188
Transit of Venus .........	189
Foundation of Celestial Mechanics.................189
Section 2. Dynamical Phenomena....................190
Gravitation .	  190
Character of laws of Motion.......................190
Their history .	  191
Newton’s demonstration............................193
Old difficulty explained..........................194
Term A ttraction inadmissible.....................194
Extent of the demonstration.......................195CONTENTS.
Term Gravitation unobjectionable .	.	.	.	.	.196
Gravitation is that of Molecules......................197
Secondary gravitation.................................198
Domain of the law.....................................198
CHAPTER IV.
CELESTIAL STATICS.
Consummation by Newton
Statical considerations
First method of inquiry into masses
Second method....................
Third method.....................
Section 1. Weight of the earth
Section 2. Form of the planets
Difficulty of the inquiry .
Geometrical estimate
Estimate from perturbations .
Indirect estimate of the earth’s form
Hydrostatic theory of planetary forms
Section 3. The Tides
Question of the tides
Theory of the tides ....
Influence of the Sun ....
Of the moon......................
Composite influence ....
Requisites for exactitude.
200
200
200
201
201
202
204
204
204
205
205
206
206
206
207
208
208
208
209
CHAPTER V.
CELESTIAL DYNAMICS.
Perturbations................................211
Instantaneous................................211
Gradual perturbations........................213
Perturbations of translation .......	213
Problem of three bodies ........	213
Centre of the Solar System...................214
Problem of the Planets.......................215
Of the Satellites............................215
Of the Comets ..........	216
Perturbations of rotation....................218
The Planets..................................218
The Satellites...............................219
Device of an Invariable Plane................219
Stability of our system......................220
Resistance of a Medium.......................221
Independence of the solar system.............222
Achievements of celestial dynamics...........223CONTENTS.
XXXIX
CHAPTER VI.
SIDEREAL ASTRONOMY AND COSMOGONY.
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.....................231
Its domain.............................................232
Compared with Chemistry................................232
Its generality.........................................232
Dealing with masses or molecules.....................  233
Changes of arrangement or composition of molecules .	.	233
Description of Physics...............................  234
Instruments............................................235
Methods of inquiry...................................  235
Observation............................................235
Experiment.............................................235
Application of Mathematical analysis .....	236
Encyclopedic rank of Physics...........................237
Relation to Astronomy..................................237
To Mathematics.........................................238
To the other sciences..................................238
To human progress......................................239
Human power of modifying phenomena.....................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..................................................248xl
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 gases................................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 thermological influence ....	260
Two parts..........................................260
Mutual influence...................................260
Radiation of Heat.................................261
Propagation by contact.............................262
Conductibility.....................................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 equilibrium .	.	.	.	.	.	•	269
Section 3. Thermology connected with analysis .	.	. 270
Section 4. Terrestrial temperatures................271
Interior heat ..........	272
Temperature of the planetary intervals.............272
Conditions of the problem..........................273CONTENTS.
xli
CHAPTER IV.
ACOUSTICS.
PAGE
Its nature............................................274
Relation to the study of Inorganic bodies .	.	.	.274
Relation to Physiology................................275
To Mathematics........................................275
Divisions....................................... .	.278
Section 1. Propagation of sound........................  279
Effect of atmospheric agitation .	.	.	.	.	.279
Section 2. Intensity of sounds........................280
Section 3. Th eory of tones...........................  .281
Composition of sounds ........	283
Recapitulation.........................................  283
CHAPTER V.
OPTICS.
Hypotheses on the nature of Light...........................285
Excessive tendency to systematize...........................288
Divisions of Optics .........	289
Irrelevant matters .........	289
Theory of Vision............................................289
Specific Colour of bodies .	.	.	.	.	.	.	.291
Section 1.	Study of direct light..........................292
Optics proper.............................................  292
Imperfections...............................................292
Photometry..................................................293
Section 2.	Catoptrics	........	294
Great law of reflection.....................................294
Law of absorption not found .......	294
Section 3.	Dioptrics......................................295
Great law of refraction.....................................295
Newton’s discoveries on elementary colours ....	297
Section 4.	Diffraction....................................298
CHAPTER VI.
ELECTROLOGY.
History.................................................300
Condition...............................................300
Arbitrary hypotheses....................................300
Relation to Mathematics.................................301
Unsound application.....................................302
Sound application.......................................302
Limits..................................................302
Divisions...............................................302xlii	CONTENTS.
PAGE
Section 1. Electric production.........303
Causes of electrization ........	303
Chemical action......................  303
Thermological action...................303
Friction...............................304
Pressure	304
Contact ........	•	•	304
Other causes .	.........	304
Instruments .........	304
Section 2. Electric statics............306
Great law of Distribution .......	306
Electric equilibrium.........	306
Section 3. Electric dynamics...........307
Ampere’s experiments...................307
Conclusion of Physics..................310
BOOK IV.
CHEMISTRY.
CHAPTER I.
Its nature........................................  312
Great imperfection .	.	.	.	.	.	.	•	.313
Capacities..........................................313
Object of Chemistry.................................314
Specific character of its action....................315
Condition of action.................................315
Definition..........................................316
Elements ...........	317
Combination.........................................317
Rational definition .........	318
Means of investigation..............................318
Observation.........................................318
Experiment..........................................318
Comparison ..........	319
Chemical analysis and synthesis.....................320
Rank of the science.................................321
Relation to Mathematics.............................322
To Astronomy........................................322
To Physiology.......................................323
To Sociology........................................324
Degree of possible perfection .......	324
Intrusion of hypotheses ........	325
Actual imperfection.................................326
Comparative imperfection .......	326
Relation to human progress..........................327
Art of Nomenclature.................................327CONTENTS.
xliii
PAGE
State of chemical doctrine...................................329
Divisions of the science.....................................330
No organic chemistry.........................................331
Principles of composition and decomposition .	.	.	.331
CHAPTER II.
INORGANIC CHEMISTRY.
Mode of beginning the study...............................333
Plurality of elements.....................................334
Classification of elements. ......	336
Classification of Berzelius...............................337
Premature effort..........................................338
Requisite preparation as to method........................338
As to doctrine............................................339
First condition...........................................339
Second condition..........................................340
Method of analysis........................................340
Chemical dualism..........................................340
Law of double saline decompositions.......................344
Chemical theory of air and water..........................345
Air.......................................................346
Water.....................................................347
CHAPTER III.
DOCTRINE OF DEFINITE PROPORTIONS.
Scope of the doctrine ........	349
As to Doctrine...........................................  350
As to Method............................................350
Its history.............................................350
Richter’s law...........................................350
Berthollet’s extension..................................  .351
Dalton’s further extension..............................352
Atomic theory...........................................352
Extension by Berzelius..................................353
Extension by Gay-Lussac.................................354
Wollaston’s verification.................................  354
Scope of application of Numerical Chemistry ....	355
Objection of dissolution................................357
Of metallic alloys......................................358
Of organic substances....................................  358
Application of the principle of dualism.................359xliv
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..........................
Becquerel’s..................................
Study of Combustion..........................
Lavoisier’s theory...........................
First division...............................
Berthollet’s limitation......................
Second division..............................
Difficulties of the theory...................
Its position and powers......................
PAGE
363
363
363
364
364
365
365
366
366
367
367
368
370
373
CHAPTER V.
ORGANIC CHEMISTRY.
Confusion of two kinds of fact................375
Relation of Chemistry to Anatomy..............377
To Physiology ..........	378
Partition of Organic Chemistry................381
Application of dualism to organic compounds....	383
Summary of the Chapter........................384
Summary of the Book...........................384THE POSITIVE PHILOSOPHY OF
AUGUSTE COMTE.
INTRODUCTION.
CHAPTER I.
ACCOUNT OF THE AIM OF THIS WORK.-YIEW OF THE NATURE
AND IMPORTANCE OF THE POSITIVE PHILOSOPHY.
AG-ENERAL 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 especially important
that the field of research should be marked out with all
possible accuracy. For this purpose, I will glance at the
considerations which have originated this work, and which
will be fully elaborated in the course of it.
4 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 f
human intelligence, in all directions, and p^ress.Uman
through all times, the discovery arises of a
great fundamental law, to which it is necessarily subject,
and which has a solid foundation of proof, both in the
facts of our organization and in our historical experience.
The law is this :—that each of our leading conceptions,—2
POSITIVE PHILOSOPHY.
each branch 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 opposed: 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/
First sta^e	In ^ke°l°gical state, the human mind,
3 * seeking the essential nature of beings, the
first and final causes (the origin and purpose) of all effects,
—in short, Absolute knowledge,—supposes all phenomena to
be produced by the immediate action of supernatural beings.
«	In the metaphysical state, which is only a
econ ace. mo^j£ca^jon 0f the 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
tnere reference of each to its proper entity.
Third sta^e 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 no\y
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.	j
TT1 .	The Theological system arrived at the
of each.G P°mt 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 sameGROUNDS OF THE LAW OF PROGRESS.
3
way, in the last stage of the Metaphysical system, men
substitute one great entity (Nature) as the cause of all v
phenomena, instead of the multitude of entities at first
supposed. In the same way, again/the ultimate perfection/
of the Positive system would be (if such perfection could*
be hoped for) to represent all phenomena as particular aspects
of a single general fact;—such as Gravitation, for instance. J
The importance of the working of this general law wilF
be established hereafter. At present, it must suffice to
point out some of the grounds of it.
There is na science which, having attained
the positive stage, does not bear marks of th™awCeS °
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	Actual
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 point of departure of the individual and of
the race being the same, the phases of the mind of a man
correspond to the epochs of the mind of the race. Now,
each of us is aware, if he looks back upon his own history,
that he was a theologian in his childhood, a 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 Theoretical
from the necessity that always exists for
some theory to which to refer our facts, combined with the
clear impossibility that, at the outset of human knowledge,
men could have formed theories out of the observation of
facts. All good intellects have repeated, since Bacon’s
time, that there can be no real 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 it4
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 could 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 confirmation or rejection of a theory. But it
could not be so in the earliest days; and it is to the
chimeras of astrology and alchemy that we owe the long
series of observations and experiments on which our positive
science is based. Kepler felt this on behalf of astronomy,
and Berthollet on behalf of chemistry. Thus was a 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 about6
POSITIVE PHILOSOPHY.
origin 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, because it connects under
one head the whole immense variety of astronomical facts;
exhibiting the constant tendency of atoms towards each
other in direct proportion to their masses, and in inverse
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 phenomena are identical; which is
the point from which the question set out. Again, M.
Fourier, in his tine 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, which is above all made up of facts
that are general, simple, and independent of other sciences,
arrived first; then terrestrial Physics; then Chemistry;
and, at length, Physiology.
It is difficult to assign any precise date to this revolution
in science. It may be said, like everything else, to have
been always going on; and especially since the labours of
Aristotle and the school of Alexandria; and then from the
introduction of natural science into the West of Europe by
the Arabs. But, if we must fix upon some marked period,
to serve as a rallying point, it must be that,—about two
centuries ago,—when the human mind was astir under the
precepts of Bacon, the conceptions of Descartes, and the
discoveries of Galileo. Then it was that the spirit of the
Positive philosophy rose up in opposition to that of the
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 physio-
logical,—there was an omission which will have been
noticed. Nothing was said of Social jffieno- New depart-
mena. Though involved with the physio- ment of Posi-
logical, Social phenomena demand a distinct tive philoso-
elassification, both on account of their im- h^y-
portance and of their 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 treatmentPOSITIVE 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 present work to establish.
S ‘ 1 Ph ’	would be absurd to pretend to offer this
^	* new science at once in 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 superiority, and leave to them only an
historical existence.
,	. We have stated the special aim of this
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 The purpose of this work is not to give an
philosophy of account of the Natural Sciences. Besides
the Sciences, that it would be endless, and that it wouldDESULTORY DIVISION OF RESEARCH.
9
require a scientific preparation sucli as no one man possesses,
it would be apart 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 philosophy without
a basis of social science, without which it could not be all-
comprehensive ; and, on the other hand, we could not pursue
Social science without having been prepared by the study
of phenomena less complicated than those of society, and
furnished with a knowledge of laws and 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 present; 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 supposed that our course Snecialitv
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 particular study. It is inevitable that each should be
possessed with exclusive notions, and be therefore in-
capable of the general superiority of ancient students,ib
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.	the spirit 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 prepared 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 results obtained by each.
We see some approach already to this arrangement. Once
established, there would be nothing to apprehend from any
extent of division of employments. When we once have a
class of learned men, at the disposal of all others, whose
business it shall be to connect each new discovery with the
general system, we may dismiss all fear of the great whole
being lost sight of in the pursuit of the details of know-
ledge. The organization of scientific research will then beFIRST 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 Positive
must now glance at the chief advantages Philosophy,
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 Illustrates the
affords the only rational means of exhibiting Intellectual
the logical 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 would be
needed to see the image. In the same manner, the mind
may observe all phenomena but its own. It may be said
that a man’s intellect may observe his passions, the
seat of the reason being somewhat apart from that of
the emotions in the brain; but there can be nothing
like scientific observation of the passions, except from
without, as the stir of the emotions disturbs the 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 pause 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 pursuit, no one proposition
is established to the satisfaction of its followers. They are
divided, to this day, into a multitude of schools, still dis-
puting 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 observingSECOND BENEFIT.
13
the progress of the human mind in the light of science;
that is, by ceasing, for the moment, to be psychologists.
The view just given in relation to logical Science becomes
yet more striking when we consider the logical Art.
The Positive Method can be judged of only in action.
It cannot be looked at by itself, apart from the work
on which it is employed. At all events, such a 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, without knowing half so much about it as the man
who has once put it in practice upon a single particular of
actual research, even without any philosophical intention.
Thus it is that psychologists, by dint of 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 applications. 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 manifestation 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 regene-
Philosophy, an effect not less important and rate Educa-
far more urgently wanted, will be to regene- t^on-
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 present 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 conceptions 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	III. 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 several special points of view;
and for want of this, very important problems wait for
their solution much longer than they otherwise need do.
To go back into the past for an example: Descartes* grandTHIRD 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 ; and it issued from the union of two sciences which
had always before been separately regarded and pursued.
The case of pending questions is yet more impressive; as,
for instance, in Chemistry, the doctrine of Definite Propor-
tions. Without entering upon the discussion of the funda-
mental principle of this theory, we may say with assurance
that, in order to determine it—in order to determine
whether it is a law of nature that atoms should 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 simple or a compound
body. It was concluded by almost all chemists that azote
is a simple body; the illustrious Berzelius hesitated, on
purely chemical considerations; but he was also influenced
by the physiological observation that animals which receive
no azote in their food have as much of it in their tissues as
carnivorous animals. 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
IY. The Positive Philosophy offers the
only solid basis for that Social Reorganiza- nize society1"
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 he 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;
and, this ascertained, every man, whatever may ha ve 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 general revolution of the human mind
is nearly accomplished. We have only to complete the
Positive Philosophy by bringing Social phenomena within
its comprehension, and 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 takePRECAUTIONARY OBSERVATION.
17
place spontaneously, and will re-establish 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 abundantly discredited by the process. It is
time now to be doing something more effective, without
wasting our forces in needless controversy. It is time to
complete the vast intellectual operation begun by Bacon,
Descartes, and Galileo, by constructing the system of
general ideas which must henceforth prevail among the
human race. This is the way to put an end to the 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 duction 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
well to secure this Work at once from any imputation of
the kind, though its development will show how undeserved
such an imputation would be. Our intellectual resources
are too narrow, and the 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 with
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 proveable (which it cannot be while we are
i.	c18
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 attempt, in all future
time, to reduce them rigorously to one.
Having thus endeavoured to determine the spirit and
influence of the Positive Philosophy, and to mark the goal
of our labours, we 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 phenomena, and consequently the corre-
sponding positive sciences.19
CHAPTER II.
VIEW OF THE HIERARCHY OF THE POSITIVE SCIENCES.
IN proceeding to offer a Classification of the Sciences, we
must leave on one side all others that have as yet been
attempted. Such scales are those of Bacon and D’Alembert
are constructed upon an arbitrary division of Failure of pro-
file faculties of the mind ; whereas, our posed classic-
principal faculties are often engaged at the cati°ns.
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 failure 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 homogeneousness in the different parts of the
intellectual system,—some having successively become posi-
tive, while others remain theological or metaphysical.
Among such incoherent materials, classification is of course
impossible. Every attempt at a distribution has failed
from this cause, without the distributor being able to
see why;—without his discovering that a radical 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 has20
POSITIVE PHILOSOPHY.
arrived for laying down a sound and durable system of
cientific order.
We may derive encouragement from the 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 affinities 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 blie mutual dependence of the sciences,—
of classifica- a dependence resulting from that of the corre-
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.
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 he declared to
be an extract from the whole of Science, and which has
been so differently and so strangely interpreted by his
metaphysical commentators.
There can be no doubt that Man’s study of nature must
furnish 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
tion.
sponding phenomena,—must determine the
Boundaries of
our field.
The field of human labour is either specula-
tion or action: and thus, we are accustomed
to divide our knowledge into the theoreticalTHE 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 any phenomenon seems to arise in violation
of the familiar laws of nature. This need of disposing
facts in a comprehensible order (which is the 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 pursued22
POSITIVE PHILOSOPHY.
during long ages with a purely scientific intention. For
instance, the ancient Greek geometers delighted themselves
with beautiful speculations 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 use Condorcet’s
illustration, “ the sailor who is preserved from shipwreck
by the exact observation of the longitude, owes his life to
a theory conceived two thousand years before by men of
genius who had in view simply geometrical speculations.’*
Our business, it is clear, is with theoretical researches,
letting alone their practical application altogether. Though
we may 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 conceptions, 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 department of Positive
philosophy may arise : but it will be in a distant future.
If we remember that several sciences are implicated in
every important art,—that, for instance, a true theory of
Agriculture requires a combination of physiological, 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 development of these
constituent sciences.
One more preliminary remark occurs, before we finish theTHE 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 our business is with them
alone, as the second are derived, and however important,
not rising into the rank of our subjects of contemplation.
We shall treat of physiology, but not of botany and
zoology, which are derived from it. We shall treat of
chemistry, but not of 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 preliminary knowledge of
several sciences, and an advance of those sciences not yet
achieved; so that, if there were no other reason, we must
leave these secondary classes alone. At a future time
Concrete Physics will have made progress, according to
the development of Abstract Physics, and will afford a
mass of less 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	0f
easy a matter as it may appear. However Pia«eifiPifi0Ti
natural it may be, it Will always involve
t
-ti24
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
in 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. the Dogmatic. These are wholly distinct from
each other, and any other method can be nothing but some
combination of these two. By the first method know-
ledge is presented in the same order in which it was actu-
ally obtained by the human mind, together with the way
in 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
point of view, should begin to reconstitute the science as a
whole. A new science must be pursued historically, the
only thing to be done being to study in chronological 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
more 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 different parts of geometry.
The writings of Archimedes and Apollonius were, in fact,
about all. On the contrary, a modern geometer commonly
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 position in science. If every mind
had to pass through all the stages that every predecessor
in the study had gone through, it is clear that, however
easy it is to learn rather than invent, it would be impossibleHISTORICAL AND DOGMATIC METHODS OF STUDY. 25
to effect tlie 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 be 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 proper 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 under
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 what26
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 were, 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 popular
language, changes. Thus we have to choose the one right
order (and there can be but one right) out of 720 possible
ones. Very few of these have ever been proposed ; yet we
might venture to say that there is probably not one in favour
of which some plausible reason might not be assigned; for
we see the wildest divergences among the schemes which
have been proposed,—the sciences which are placed by some
at the head of the scale being sent by others to the further
extremity. Our problem is, then, to find the one rational
order, among a host of possible systems.
True principle Now we must remember that we have to
of classifica- look for the principle of classification in the
ti°n*	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 phenomena may be included within a very
few natural categories, so arranged as that the study of
each category may be grounded on the principal laws of the
preceding, and serve as the basis of the next ensuing.
This order is determined by the degree of simplicity,
Generality. or, 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.FUNDAMENTAL 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 determines the neces-
sary connection of the sciences by the successive depen-
dence of their phenomena. It is worthy of remark in this
place 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 proceed to our
classification.
We are first struck by the clear division of Inorganic and
all natural phenomena into two classes—of Organic phe-
inqrganic and of organic bodies. The orga- nomena.
nizei-are evidently, in fact, more complex and less general
tlikn the inorganic, and depend upon them, instead of
being depended on by them. Therefore it is that physio-
logical study should begin with inorganic phenomena;
since the organic include all the qualities belonging to
them, with a special order added, viz. the vital phenomena,
which belong to organization. We have not to investigate
the nature of either; for the positive philosophy does not
inquire into natures. Whether their nature be supposed
different or the same, it is evidently necessary to separate
the two studies of inorganic matter and of living bodies.
Our classification will stand through any future decision as
to the way in which living bodies are to be regarded; for,
on any supposition, the general laws of inorganic physics
must be established before we can proceed with success to
the examination of a dependent class of phenomena.
Each of these great halves of natural j inorganic
philosophy has subdivisions. Inorganic phy-
sics' must, in accordance with our rule of generality and the28
POSITIVE PHILOSOPHY.
order of dependence of phenomena, be divided into two
1 Astronom	sections—of celestial and terrestrial pheno-
mena. Thus we have Astronomy, geometri-
cal and mechanical, and Terrestrial Physics. The neces-
sity of this division is exactly the same as in the former
case.
Astronomical phenomena are the most general, simple,
and abstract of all; and therefore the study of natural
philosophy must clearly begin with them. They are them-
selves independent, while the laws to which they are subject
influence all others whatsoever. The general effects of
gravitation preponderate, in all terrestrial phenomena, over
all effects which may be peculiar to them, and modify the
original ones. It follows that the analysis of the simplest
terrestrial phenomenon, not only chemical, but even purely
mechanical, presents a greater complication than the most
compound astronomical phenomenon. The most difficult
astronomical question involves less intricacy than the
simple movement of even a solid body, when the determin-
ing circumstances are to be computed. Thus we see that
we must separate these two studies and proceed to the
second only through the first, from which it is derived.
In the same manner, we find a natural division of Terres-
trial Physics into two, according as we regard bodies in
their mechanical or their chemical character.
2.	Physics. Hence we have Physics, properly so called,
3.	Chemistry. an<^ Chemistry. Again, the second class
must be studied through the first. Chemi-
cal phenomena are more complicated than mechanical, and
depend upon them, without influencing them in return.
Every one knows that all chemical action is first submitted
to the influence of weight, heat, electricity, etc., and pre-
sents moreover something which modifies all these. Thus,
while it follows Physics, it presents itself as a distinct
science.
Such are the divisions of the sciences relating to in-
tt oprAYir organic matter. An analogous division arises
in the other half of Natural Philosophy—the
science of organized bodies.
Here we find ourselves presented with two orders of
phenomena; those which relate to the individual, andORGANIC PHYSICS.
29
those which relate to the species, especially when it is
gregarious. With regard to Man, especially, this distinc-
tion is fundamental. The last order of phenomena is
evidently dependent on the first, and is more complex.
Hence we have two great sections in organic physics—
Physiology, properly so called, and Social
Physics, which is dependent on it. In all Physiology.
Social phenomena we perceive the working 2. Sociology,
of the physiological laws of the individual;
and moreover something which modifies their effects, and
which belongs to the influence of individuals over each
other—singularly complicated in the case of the human
race by the influence of generations on their successors.
Thus it is clear that our social science must issue from that
which relates to the life of the individual. On the other
hand, there is no occasion to suppose, as some eminent
physiologists have done, that Social Physics is only an
appendage to physiology. The phenomena of the two
are not identical, though they are homogeneous ; and it is
of high importance to hold the two sciences separate.
As social conditions modify the operation of physiological
laws, Social Physics must have a set of observations of
its own.
It would be easy to make the divisions of the Organic
half of Science correspond with those of the Inorganic, by
dividing physiology into vegetable and animal, according
to popular custom. But this distinction, however im-
portant in Concrete Physics (in that secondary and special
class of studies before declared to be inappropriate to this
work), hardly extends into those Abstract Physics with
which we have to do. Vegetables and animals come alike
under our notice, when our object is to learn the general
laws of life—that is, to study physiology. To say nothing
of the fact that the distinction grows ever fainter and
more dubious with new discoveries, it bears no relation
to our plan of research ; and we shall therefore consider
that there is only one division in the science of organized
bodies.
Thus we have before us Five fundamental Natural
Sciences in successive dependence,—Astro- gciences
nomy, Physics, Chemistry, Physiology, and30
POSITIVE PHILOSOPHY.
finally Social Physics. The first considers the most general,
simple, abstract, and remote phenomena known to ns, and
those which affect all others without being affected by them.
The last considers the most particular, compound, concrete
phenomena, and those which are the most interesting to
Man. Between these two, the degrees of speciality, of
complexity, and individuality are in regular proportion to
the place of the respective sciences in the scale exhibited.
T1W filiation This—cashing out everything arbitrary—we
must regard as the true filiation of the
sciences ; and in it we find the plan of this work.
Filiation of As we Procee(^’we shall find that the same
their parts. principle which gives this order to the whole
body of science arranges the parts of each
science ; and its soundness will therefore be freshly attested
as often as it presents itself afresh. There is no refusing
a principle which distributes the interior of each science
after the same method with the aggregate sciences. But
this is not the place in which to do more than iiylicate what
we shall contemplate more closely hereafter.//We must
now rapidly review some of the leading properties of the
hierarchy of science that has been disclosed.
Corroborations. This gradation is in essential conformity
with the order which has spontaneously
taken place among the branches of natural
philosophy, when pursued separately, and
without any purpose of establishing such order.
Such an accordance is a strong presumption
that the arrangement is natural. Again, it coincides with
the actual development of natural philosophy. If no lead-
ing science can be effectually pursued otherwise than
through those which precede it in the scale, it is evident
that no vast development of any science could take place
prior to the great astronomical discoveries to which we owe
the impulse given to the whole. The progression may
since have been simultaneous; but it has taken place in
the order we have recognized.
This consideration is so important that it
is difficult to understand without it the
history of the human mind. The general
law which governs this history, as we have already seen,
1. This classi-
iication follows
the order of
disclosure of
sciences.
2. Solves hete-
ro^eneousness.CLASSIFICATION CORROBORATED.
31
cannot be verified, unless we combine it with the scientific
gradation just laid down: for it is according to this grada-
tion that the different human theories have attained in
succession the theological state, the metaphysical, and
finally the positive. If we do not bear in mind the law
which governs progression, we shall encounter insurmount-
able difficulties: for it is clear that the theological or meta-
physical state of some fundamental theories must have
temporarily coincided with the positive state of others
which precede them in our established gradation, and
actually have at times coincided with them; and this
must involve the law itself in an obscurity which can be
cleared up only by the classification we have proposed.
Again, this classification marks, with pre- 3. Marks rela-
cision, the relative perfection of the different tive perfection
sciences, which consists in the degree of pre- 111 sciences,
cision of knowledge, and in the relation of its different
branches. It is easy to see that the more general, simple,
and abstract any phenomena are, the less they depend on
others, and the more precise they are in themselves, and
the more clear in their relations with each other. Thus,
organic phenomena are less exact and systematic than
inorganic; and of these again terrestrial are less exact and
systematic than those of astronomy. This fact is completely
accounted for by the gradation we have laid down; and
we shall see as we proceed, that the possibility of applying
mathematical analysis to the study of phenomena is exactly
in proportion to the rank which they hold in the scale of
the whole.
There is one liability to be guarded against, Defects are in
which we may mention here. We must beware us» not in
of confounding the degree of precision which science*
we are able to attain in regard to any science, with the cer-
tainty of the science itself. The certainty of science, and
our precision in the knowledge of it, are two very different
things, which have been too often confounded ; and are so
still, though less than formerly. A very absurd proposition
may be very precise ; as if we should say, for instance, that
the sum of the angles of a triangle is equal to three right
angles; and a very certain proposition may be wanting in
precision in our statement of it; as, for instance, when we82
POSITIVE PHILOSOPHY.
assert that every man will die. If the different sciences
offer to us a varying degree of precision, it is from no want
of certainty in themselves, but of our mastery of their
phenomena.
4 Effect on The 111 os^ interesting property of our
Education formula of gradation is its effect on educa-
tion, both general and scientific. This is its
direct and unquestionable result. It will be more and
more evident as we proceed, that no science can be effectually
pursued without the preparation of a competent knowledge
of the anterior sciences on which it depends. Physical
philosophers cannot understand Physics without at least a
general knowledge of Astronomy; nor Chemists, without
Physics and Astronomy ; nor Physiologists, without
Chemistry, Physics, and Astronomy; nor, above all, the
students of Social philosophy, without a general knowledge
of all the anterior sciences. As such conditions are, as yet,
rarely fulfilled, and as no organization exists for their
fulfilment, there is amongst us, in fact, no rational scientific
education. To this may be attributed, in great part, the
imperfection of even the most important sciences at this
day. If the fact is so in regard to scientific education, it
is no less striking in regard to general education, i Our
intellectual system cannot be renovated till the natural
sciences are studied in their proper order^ Even the highest
understandings are apt to associate their ideas according
to the order in which they were received: and it is only
an intellect here and there, in any age, which in its utmost
vigour can, like Bacon, Descartes, and Leibnitz, make a
clearance in their field of knowledge, so as to reconstruct
from the foundation their system of ideas.
Effect on	Such is the operation of our great law
Method	upon scientific education through its effect
on Doctrine. We cannot appreciate it duly
without seeing how it affects Method.
As the phenomena which are homogeneous have been
classed under one science, while those which belong to
other sciences are heterogeneous, it follows that the Positive
Method must be constantly modified in an uniform manner
in the range of the same fundamental science, and will
undergo modifications, different and more and more com-EFFECT ON METHOD.
33
pound, in passing from one science to another. Thus,
under the scale laid down, we shall meet with it in all its
varieties; which could not happen if we were to adopt a
scale which should not fulfil the conditions we have ad-
mitted. This is an all-important consideration ; for if, as
we have already seen, we cannot understand the positive
method in the abstract, but only by its application, it is
clear that we can have no adequate conception of it but by
studying it in its varieties of application. No one science,
however well chosen, could exhibit it. Though the Method
is always the same, its procedure is varied. For instance,
it should be Observation with regard to one kind of
phenomena, and Experiment with regard to another; and
different kinds of experiment, according to the case. In
the same way, a general precept, derived from one funda-
mental science, however applicable to another, must have
its spirit preserved by a reference to its origin; as in the
case of the theory of Classifications. The best idea of the
Positive Method would, of course, be obtained by the study
of the most primitive and exalted of the sciences, if we
were confined to one ; but this isolated view would give no
idea of its capacity of application to others in a modified
form. Each science has its own proper advantages; and
without some knowledge of them all, no conception can be
formed of the power of the Method.
One more consideration must be briefly
adverted to. It is necessary, not only to have Qf sciences™ ^
a general knowledge of all the sciences, but
to study them in their order. What can come of a study
of complicated phenomena, if the student have not learned,
by Nthe contemplation of the simpler, what a Law is, what
it is to Observe; what a Positive conception is; and even
what a chain of reasoning is P Yet this is the way our
young physiologists proceed every day,—plunging into the
study of living bodies, without any other preparation than
a knowledge of a dead language or two, or at most a super-
ficial acquaintance with Physics and Chemistry, acquired
without any philosophical method, or reference to any true
point of departure in Natural philosophy. In the same
way, with regard to Social phenomena, which are yet more
complicated, what can be effected but by the rectification
i.	d84
POSITIVE PHILOSOPHY.
of the intellectual instrument, through an adequate study
of the range of anterior phenomena ? There are many who
admit this: but they do not see how to set about.the
work, nor understand the Method itself, for want of the
preparatory study ; and thus, the admission remains barren,
and social theories abide in the theological or metaphysical
state, in spite of the efforts of those who believe themselves
positive reformers.
These, then, are the four points of view under which we
have recognized the importance of a Rational and Positive
Classification.
Mathematics.
It cannot but have been observed that in
our enumeration of the sciences there is a
prodigious omission. We have said nothing of Mathe-
matical science. The omission was intentional; and the
reason is no other than the vast importance of mathe-
matics. This science will be the first of which we shall
treat. Meantime, in order not to omit from our sketch a
department so prominent, we may indicate here the general
results of the study we are about to enter upon.
A de Dartment -^n Presen^ of our knowledge we
P	’ must regard Mathematics less as a constituent
part of natural philosophy than as having been, since the
, ,	.	time of Descartes and Newton, the true basis
J\ [)Q ^ls;
of the whole of natural philosophy ; though
it is, exactly speaking, both the one and the other. To us.
it is of less value for the knowledge of which it consists,
substantial and valuable as that knowledge is, than as
Vn instrument	the mos^ powerful instrument that the
’ human mind can employ in the investigation
of the laws of natural phenomena.
In due precision, Mathematics must be
divided into two great sciences, quite distinct
from each other—Abstract Mathematics, or
the Calculus (taking the word in its most extended sense),
and Concrete Mathematics, which is composed of General
Geometry and of Rational Mechanics. The Concrete part
is necessarily founded on the Abstract, and it becomes
in its turn the basis of all natural philosophy; all the
phenomena of the universe being regarded, as far as
possible, as geometrical or mechanical.
A doubleMATHEMATICS A DOUBLE SCIENCE.
35
The Abstract portion is the only one which Abstract ma-
is purely instrumental, it being simply an thematics an
immense extension of natural logic to a eer- instrument,
tain order of deductions. G-eometry and Concrete ma-
mechanics must, on the contrary, be re- thematics a
garded as true natural sciences, founded, science,
like all others, on observation, though, by the extreme
simplicity of their phenomena, they can be systematized
to much greater perfection. It is this capacity which
has caused the experimental character of their first
principles to be too much lost sight of. But these two
physical sciences have this peculiarity, that they are now,
and will be more and more, employed rather as method
than as doctrine.
It needs scarcely be pointed out that in placing Mathe-
matics at the head of Positive Philosophy, we are only ex-
tending the application of the principle which has governed
our whole Classification. We are simply carrying back our
principle to its first manifestation. Geometrical and Mechani-
cal phenomena are the most general, the most simple, the
most abstract of all,—the most irreducible to others, the
most independent of them ; serving, in fact, as a basis to all
others. It follows that the study of them is an indispen-
sable preliminary to that of all others. Therefore must
Mathematics hold the first place in the Mathematics
hierarchy of the sciences, and be the point pre-eminent
of departure of all Education, whether general *n tlie scale-
or special. In an empirical way, this has hitherto been the
custom,—a custom which arose from the great antiquity of
mathematical science. We now see why it must be
renewed on a rational foundation.
We have now considered, in the form of a philosophical
problem, the rational plan of the study of the Positive
Philosophy. The order that results is this; an order
which of all possible arrangements is the only one that
accords with the natural manifestation of all phenomena.
Mathematics, Astronomy, Physics, Chemistry, Physio-
logy, Social Physics.36
BOOK I.
MATHEMATICS.
CHAPTER I.
MATHEMATICS, ABSTRACT AND CONCRETE.
E are now to enter upon the study of the first of the
Six great Sciences ; and we begin by establishing
the importance of the Positive Philosophy in perfecting the
character of each science in itself.
Though Mathematics is the most ancient and the most
jierfect science of all, the general idea of it is far from
being clearly determined. The definition of the science,
and its chief divisions, have remained up to this time vague
and uncertain. The plural form of the name (grammati-
cally used as singular) indicates the want of unity in its
philosophical character, as commonly conceived. In fact,
it is only since the beginning of the last century that it
could be conceived of as a whole; and since that time
geometers have been too much engaged on its different
branches, and in applying it to the most important laws of
the universe, to have much attention left for the general
system of the science. How however the pursuit of its
specialities is no longer so engrossing as to exclude us from
the study of Mathematics in its unity. It has now reached
a degree of consistency which admits of the effort to reduce
its parts into a system, in preparation for further advance.
The latest achievements of mathematicians have prepared
the way for this by evidencing a character of unity in its
principal parts which was not before known to exist. Such
is eminently the spirit of the great author of the Theory of
Functions and of Analytical Mechanics.NATURE AND OBJECT OF MATHEMATICS.
The common description of Mathematics, j}egcr- o£
as the science of Magnitudes, or, somewhat mathematics
more positively, the science which relates to
the Measurement of Magnitudes, is too vague and unmean-
ing to have been used but for want of a better. Yet the
idea contained in it is just at bottom, and is even suffi-
ciently extensive, if properly understood; but it needs
precision and depth. It is important in such matters not
to depart unnecessarily from notions generally admitted ;
and we will therefore see how, from this point of view, we
can rise to such a definition of Mathematics as will be
adequate to the importance, extent, and difficulty of the
science.
Our first idea of measuring a magnitude	^
is simply that of comparing the magnitude mathematics.
in question with another supposed to be
known, which is taken for the unit of comparison among
all others of the same kind. Thus, when we define mathe-
matics as being the measurement of magnitudes, we give a
very imperfect idea of it, and one which seems to bear no
relation, in this respect, to any science whatever. We seem
to speak only of a series of mechanical procedures, like a
superposition of lines, for obtaining the comparison of
magnitudes, instead of a vast chain of reasonings, inex-
haustible by the intellect. Nevertheless, this definition
has no other fault than not being deep enough. It does
not mistake the real aim of mathematics, but it presents as
direct an object which is usually indirect; and thus it mis-
leads us as to the nature of the science. To rectify this,
we must attend to a general fact, which is easily estab-
lished ; that the direct measurement of a magnitude is
often an impossible operation ; so that if we had no other
means of doing what we want, we must often forego the
knowledge we desire. We can rarely even measure a right
line by another right line ; and this is the simplest measure-
ment of all. The very first condition of this is that we
should be able to traverse the line from one end to the
other; and this cannot be done with the greater number of
the distances which interest us the most. We cannot do
it with the heavenly bodies, nor with the earth and any
heavenly body, nor even with many distances on the earth;38
POSITIVE PHILOSOPHY.
and again, the length must be neither too great nor too
small, and it must be conveniently situated; and a line
which could be easily measured if it were horizontal be-
comes impracticable if vertical. There are so few lines
capable of being directly measured with precision, that
we are compelled to resort to artificial lines, created to
admit of a direct determination, and to be the point of
reference for all others. If there is difficulty about the
measurement of lines, the embarrassment is much greater
when we have to deal with surfaces, volumes, velocities,
times, forces, etc., and in general with all other magni-
tudes susceptible of estimate, and, by their nature, difficult
of direct measurement. It is the general fact of this diffi-
culty, inherent in almost every case, which necessitates the
formation of mathematical science; for, finding direct
measurement so often impossible, we are compelled to
devise means of doing it indirectly. Hence arose Mathe-
matics.
General
method.
The general method employed, and the
only conceivable one, is to connect the mag-
nitudes in question with some that can be
directly determined, and thus to ascertain the former,
through their relations with the latter. Such is the precise
object of Mathematics, regarded as a whole. To form any-
thing like a worthy idea of it, we must remember that the
indirect determination of magnitudes may have many
degrees of indirectness. It often happens that the magni-
tudes to which undetermined magnitudes are to be referred
cannot themselves be measured directly, and must them-
selves be made the subject of a prior process, and so on
through a whole series; and thus, the mind is often
obliged to establish a long course of intermediaries between
the one and the other point of the inquiry—points which
may appear at the outset to have no connection whatever.
Examples	^ this appears too abstract, it may become
1	* plain by a few examples. In observing a
falling body, we are aware that two quantities are involved:
the height from which the body falls, and the time
occupied in its descent. These two quantities are connected,
as they vary together, and together remain fixed. In the
language of mathematicians, they are functions of eachMETHOD OF APPLICATION.
39
other. The measurement of one being impracticable, it is
supplied by that of the other. By observing the time
occupied by a stone in falling down a precipice, we can
ascertain the height of the precipice as accurately as if we
could measure it with a horizontal line. In another case,
we may be able to know the height whence a body has
fallen, and unable to observe the time with precision, and
then we must have recourse to the inverse question,—to
determine the time by the distance; as, for instance, if we
were to inquire how long it would take for a body to fall
from the moon. In these cases, the question is very
simple, supposing we do not complicate it with considera-
tions of intensity of gravity, resistance of a fluid medium,
etc. But, to enlarge the question, we must contemplate
the phenomenon in its greatest generality by supposing the
fall to be oblique, and taking into account all the principal
circumstances. Then, instead of two variable quantities,
simply connected, the phenomenon will present a consider-
able number,—the space traversed, whether in a vertical or
horizontal direction; the time employed in traversing it:
the velocity of the body at each point of its course; and
even the intensity and direction of the impulse which sent
it forth ; and finally, in some cases, the resistance of the
medium, and the intensity of gravity. All these quantities
are so connected that each in its turn may be determined
indirectly by means of the others, and thus we shall have
as many mathematical inquiries as there are magnitudes
co-existing in the phenomenon considered. Such a very
simple change as this in the physical conditions of a
problem may place a mathematical question, originally quite
elementary, in the rank of those difficult questions whose
complete and rigorous solution transcends the power of
the human understanding.
Again,—we may take a geometrical example. We want
to determine a distance not directly measurable. We
shall conceive of it as making a part of some figure, or
system of lines of some sort, of which the other parts are
directly measurable; let us say a triangle (for this is the
simplest, and to it all others are reducible). The distance
in question is supposed to form a portion of a triangle, in
which we are able to determine directly, either another40
POSITIVE PHILOSOPHY.
side and two angles, or two sides and one angle. The
knowledge required is obtained by the mathematical
labour of deducing the unknown distance from the ob-
served elements, by means of the relation between them.
The process may, and commonly does, become highly com-
plicated by the elements supposed to be known being them-
selves determinable only in an indirect manner, by the aid
of fresh auxiliary systems, the number of which may be
very considerable. The distance, once ascertained, will
often enable us to obtain new quantities, which will offer
occasion for new mathematical questions. Thus, when we
once know the distance of any object, the observation,
simple and always possible, of its apparent diameter, may
disclose to us, with certainty, however indirectly, its real
dimensions ; and at length, by a series of analogous in-
quiries, its surface, its volume, even its weight, and a mul-
titude of other qualities which might have seemed out of
the reach of our knowledge for ever. It is by such labours
that Man has learned to know, not only the distances of the
planets from the earth and from each other, but their actual
magnitude,—their true form, even to the inequalities on
their surface, and (what seems much more out of his reach)
their respective masses, their mean densities, and the lead-
ing circumstances of the fall of heavy bodies on their
respective surfaces, etc. Through the power of mathema-
tical theories, all this and very much more has been
obtained by means of a very small number of straight
lines, properly chosen, and a larger number of angles. We
might even say, to describe the general bearing of the
science in a sentence, that, but for the fear of multiplying
mathematical operations unnecessarily, and for the conse-
quent necessity of reserving them for the determination of
quantities which could not be measured directly, the know-
ledge of all magnitudes susceptible of precise estimate
which can be offered by the various orders of phenomena,
would be finally reducible to the immediate measure-
ment of a single straight line, and of a suitable number of
angles.
True defini- We can now define Mathematical science
tion of mathe- with precision. It has for its object the
maties.	indirect measurement of magnitudes, and itACTUAL SCOPE OF MATHEMATICS,
41
proposes to determine magnitudes by each other, according to
the precise relations which exist between them. Preceding
definitions have given to Mathematics the character of an
Art; this raises it at once to the rank of a true Science.
According to this definition, the spirit of Mathematics con-
sists in regarding as mutually connected all the quantities
which can be presented by any phenomenon whatsoever, in
order to deduce all from each other. Now, there is evi-
dently no phenomenon which may not be regarded as
affording such considerations. Hence results the naturally
indefinite extent, and the rigorous logical universality of
Mathematical science. As for its actual practical extent,
we shall see what that is hereafter.
These explanations justify the name of Mathematics,
applied to the science we are considering. Bv itself it
signifies Science. The Greeks had no other, and we may
call it the science ; for its definition is neither more nor less
(if we omit the specific notion of magnitudes) than the
definition of all science whatsoever. All science consists in
the co-ordination of facts; and no science could exist among
isolated observations. It might even be said that Mathel-
matics might enable us to dispense with all direct observal-
tion, by empowering us to deduce from the smallest possibly
number of immediate data the largest possible amount of
results. Is not this the real use, both in speculation and in
action, of the laws which we discover among natural pheno-
mena ? If so, Mathematics merely urges to the ultimate
degree, in its own way, researches which every real science
pursues, in various inferior degrees, in its own sphere.
Thus it is only through Mathematics that we can thoroughly
understand what true science is. Here alone can we find
in the highest degree simplicity and severity of scientific
law, and such abstraction as the human mind can attain.
Any scientific education setting forth from any other point,
is faulty in its basis.
Thus far, we have viewed the science as a whole. We
must now consider its primary division. The secondary
divisions will be laid down afterwards.
Every mathematical solution spontaneously jTSTW0 PARXS
separates into two parts. The inquiry being,
as we have seen, the determination of unknown magnitudes,42
POSITIVE PHILOSOPHY.
through their relation to the known, the student must, in
.	the first place, ascertain what these relations
objects1 er6n are>	case un(^er notice. This first
is the Concrete part of the inquiry. When it
is accomplished, what remains is a pure question of num-
bers, consisting simply in the determination of unknown
numbers, when we know by what relation they are con-
nected with known numbers. This second operation is the
Abstract part of the inquiry. The primary division of
Mathematics is therefore into two great sciences:—Abstract
Mathematics, and Concrete Mathematics. This divi-
sion exists in all complete mathematical questions what-
ever, whether more or less simple.
Recurring to the simplest case of a falling body, we must
begin by learning the relation between the height from
which it falls and the time occupied in falling. As
Geometers say, we must find the equation which exists be-
tween them. Till this is done, there is no basis for a com-
putation. This ascertainment may be extremely difficult,
and it is incomparably the superior part of the problem.
The true scientific spirit is so modern, that as far as we
know, no one before Galileo had remarked the acceleration
of velocity in a falling body, the natural supposition having
been that the height was in uniform proportion to the
time. This first inquiry issued in the discovery of the law
of Galileo. The Concrete part being accomplished, the
Abstract remains. We have ascertained that the spaces
traversed in each second increase as the series of odd num-
bers, and we now have only the task of the computation of
the height from the time, or of the time from the height;
and this consists in finding that, by the established law, the
first of these two quantities is a known multiple of the
second power of the other ; whence we may finally deter-
mine the value of the one when that of the other is given.
In this instance the concrete question is the more difficult
of the two. If the same phenomenon were taken in its
greatest generality, the reverse would be the case. Take
the two together, and they may be regarded as exactly
equivalent in difficulty. The mathematical law may be
easy to ascertain, and difficult to work ; or it may be diffi-
cult to ascertain, and easy to work. In importance, in ex-CHARACTER OF THE TWO DIVISIONS.
43
tent, and in difficulty, these two great sections of Mathe-
matical Science will be seen hereafter to be equivalent.
We have seen the difference in their Their different
objects. They are no less different in their natures,
nature.
The Concrete must depend on the character of the
objects examined, and must vary when new phenomena
present themselves: whereas, the Abstract is wholly inde-
pendent of the nature of the objects, and is concerned only
with their numerical relations. Thus, a great variety
of phenomena may be brought under one geometrical
solution. Cases which appear as unlike each other as
possible may stand for one another under the Abstract pro-
cess, which thus serves for all, while the Concrete process
must be new in each case. Thus the Concrete process is
Special, and the Abstract is General. The character of the
Concrete is experimental, physical, phenomenal: while the
Abstract is purely logical, rational. The Concrete part of
every mathematical question is necessarily founded on con-
sideration of the external world; while the Abstract part
consists of a series of logical deductions. The equations
being once found, in any case, it is for the understanding,
without external aid, to educe the results which these
equations contain.
We see how natural and complete this main division is.
We will briefly prescribe the limits of each section.
As it is the business of Concrete Mathe- £oncrete
matics to discover the equations of pheno- Mathematics
mena, we might suppose that it must com-
prehend as many distinct sciences as there are distinct
categories of phenomena ; but we are very far indeed from
having discovered mathematical laws in all orders of
phenomena. In fact, there are as yet only two great
categories of phenomena whose equations are constantly
known:—Geometrical and Mechanical phenomena. Thus,
the Concrete part of Mathematics consists of Geometry
and Rational Mechanics.
There is a point of view from which all phenomena
might be included under these two divisions. All natural
effects, considered statically or dynamically, might be re-
ferred to laws of extension or laws of motion. But this44
POSITIVE PHILOSOPHY.
point of view is too high for ns at present; and it is only
in the regions of Astronomy, and, partially, of terrestrial
Physics, that this vast transformation has taken place.
We will then proceed on the supposition that G-eometry
and Mechanics are the constituents of Concrete Mathe-
matics.
A1 A ^	The nature of Abstract Mathematics is
Mathematics. precisely determined. It is composed ot
what is called the Calculus, taking this word
in its widest extension, which reaches from the simplest
numerical operations to the highest combinations of trans-
cendental analysis. Its proper object is to resolve all
questions of numbers. Its starting-point is that which is
the limit of Concrete Mathematics,—the knowledge of the
precise relations—that is, the equations—between different
magnitudes which are considered simultaneously. The
object of the Calculus, however indirect or complicated the
relations may be, is to discover unknown quantities by the
known. This science, though more advanced than any
other, is, in reality, only at its beginning yet; but it is
necessary, in order to define the nature of any science,
to suppose it perfect. And the true character of the
Calculus is what we have said.
From an historical point of view, Mathematical Analysis
appears to have arisen out of the contemplation of geome-
trical and mechanical facts ; but it is not the less indepen-
dent of these sciences, logically speaking. Analytical ideas
are, above all others, universal, abstract, and simple; and
geometrical and mechanical conceptions are necessarily
founded on them. Mathematical Analysis is therefore
the true rational basis of the whole system of our positive
knowledge. We can now also explain why it not only gives
precision to our actual knowledge, but establishes a far
more perfect co-ordination in the study of phenomena
which allow of such an application. If a single analytical
question, brought to an abstract solution, involves the
implicit solution of a multitude of physical questions, the
mind is enabled to perceive relations between phenomena
apparently isolated, and to extract from them the quality
which they have in common. To the wonder of the student,
unsuspected relations arise between problems which, insteadEXTENT OF THE DOMAIN OF MATHEMATICS. 45
of being, as they appeared before, wholly unconnected, turn
out to be identical. There appears to be no connection
between the determination of the direction of a curve at
each of its points and that of the velocity of a body at
each moment of its variable motion ; yet, in the eyes of the
geometer, these questions are but one.
When we have seized the true general character of Mathe-
matical Analysis, we easily see how perfect it is, in com-
parison with all other branches of our positive science.
The perfection consists in the simplicity of the ideas con-
templated ; and not, as Condillac and others have supposed,
to the conciseness and generality of the signs used as instru-
ments of reasoning. The signs are of admirable use to
work out the ideas, when once obtained; but, in fact, all
the great analytical conceptions were formed without any
essential aid from the signs. Subjects which are by their
nature inferior in simplicity and generality cannot be raised
to logical perfection by any artifice of scientific language.
We have now seen what is the object and	.
what is the character of Mathematical Science, dcmurin ° 1 b
It remains for us to consider the extent of
its domain.
We must first admit that, in a logical view, T, . v,
this science is necessarily ana rigorously
universal. There is no inquiry which is not finally reducible
to a question of Numbers ; for there is none which may
not be conceived of as consisting in the determination of
quantities by each other, according to certain relations.
The fact is, we are always endeavouring to arrive at
numbers, at fixed quantities, whatever may be our subject,
however uncertain our methods, and however rough our
results. Nothing can appear less like a mathematical
inquiry than the study of living bodies in a state of disease;
yet, in studying the cure of disease, we are endeavouring
to ascertain the quantities of the different agents which
are to modify the organism, in order to bring it to its
natural state, admitting, as geometers do, for some of these
quantities, in certain cases, values which are equal to zero,
negative, or even contradictory. It is not meant that such
a method can be actually followed in the case of complicated
phenomena; but the logical extension of the science, which46
POSITIVE PHILOSOPHY.
is what we are now considering, comprehends such instances
as this.
Kant has divided human ideas into the two categories of
quantity and quality, which, if true, would destroy the
universality of Mathematics ; but Descartes’ fundamental
conception of the relation of the concrete to the abstract in
Mathematics abolishes this division, and proves that all
ideas of quality are reducible to ideas of quantity. He had
m view geometrical phenomena only; but his successors
have included in this generalization, first, mechanical
phenomena, and, more recently, those of heat. There are
now no geometers who do not consider it of universal ap-
plication, and admit that every phenomenon may be as
Jogically capable of being represented by an equation as a
curve or a motion, if only we were always capable (which
we are very far from being) of first discovering, and then
resolving it.
Its limitations.
The limitations of Mathematical science are
not, then, in its nature. The limitations are
in our intelligence : and by these we find the domain of
the science remarkably restricted, in proportion as pheno-
mena, in becoming special, become complex.
Though, as we have seen, every question may be con-
ceived of as reducible to numbers, the reduction cannot be
made by us except in the case of the simplest and most
general phenomena. The difficulty of finding the equation
in the case of special, and therefore complex phenomena,
soon becomes insurmountable, so that, at the utmost, it is
only the pheuomena of the first three classes,—that is,
only those of Inorganic Physics,—that we can even hope
to subject to the process. The properties of inorganic
bodies are nearly invariable ; and therefore, with regard to
them, the first condition of mathematical inquiry can be
fulfilled : the different quantities which they present may
be resolved into fixed numbers; but the variableness of
the properties of organic bodies is beyond our management.
An inorganic body, possessing solidity, form, consistency,
specific gravity, elasticity, etc., presents qualities which are
within our estimate, and can be treated mathematically;
but the case is altered when Chemical action is added to
these. Complications and variations then enter into theTHE LIMITATIONS OF MATHEMATICS.
47
question which at present baffle mathematical analysis.
Hereafter, it may be discovered what fixed numbers exist
in chemical combinations: but we are as yet very far from
having any practical knowledge of them. Still further are
we from being able to form such computations amidst the
continual agitation of atoms which constitutes what we
call life, and therefore from being able to carry mathe-
matical analysis into the study of Physiology. By the
rapidity of their changes, and their incessant numerical
variations, vital phenomena are, practically, placed in
opposition to mathematical processes. If we should desire
to compute, in a single case, the most simple facts of a
living body,—such as its mean density, its temperature,
the velocity of its circulation, the proportion of elements
which at any moment compose its solids or its fluids, the
quantity of oxygen which it consumes in a given time, the
amount of its absorptions or its exhalations,—and, yet
more, the energy of its muscular force, the intensity of its
impressions, etc., we must make as many observations as
there are species or races, and varieties in each ; we must
measure the changes which take place in passing from one
individual to another, and in the same individual, according
to age, health, interior condition, surrounding circum-
stances perpetually varying, such as the constitution of the
atmosphere, etc. It is clear that no mathematical precision
can be attained amidst a complexity like this. Social
phenomena, being more complicated still, are even more
out of the question, as subjects for mathematical analysis.
It is not that a mathematical basis does not exist in these
cases, as truly as in phenomena which exhibit, in all clear-
ness, the law of gravitation: but that our faculties are too
limited for the working of problems so intricate. We are
baffled by various phenomena of inorganic bodies, when
they are very complex. For instance, no one doubts that
meteorological phenomena are subject to mathematical
laws, however little we yet know about them; but their
multiplicity renders their observed results as variable and
irregular as if each cause were free of all such conditions.
We find a second limitation in the number of conditions
to be studied, even if we were sure of the mathematical
law which governs each agent. Our feeble faculties could48
POSITIVE PHILOSOPHY.
not grasp and wield such an aggregate of conditions, how-
ever certain might be our knowledge of each. In the
simplest cases in which we desire to approximate the
abstract to the concrete conditions, with any completeness,
—as in the phenomenon of the flow of a fluid from a given
orifice, by virtue of its gravity alone,—the difficulty is such
that we are, as yet, without any mathematical solution of
this very problem. The same is the case with the yet
more simple instance of the movement of a solid projectile
through a resisting medium.
To the popular mind it may appear strange, considering
these facts, that we know so much as we do about the
planets. But in reality, that class of phenomena is the
most simple of all within our cognizance. The most com-
plex problem which they present is the influence of a third
body acting in the same way on two which are tending to-
wards each other in virtue of gravitation ; and this is a
more simple question than any terrestrial problem what-
ever. We have, however, attained only approximate solu-
tions in this case. And the high perfection to which solar
astronomy has been brought by the use of mathematical
science is owing to our having profited by those facilities
that we may call accidental, which the favourable constitu-
tion of our planetary system presents. The planets which
compose it are few; their masses are very unequal, and
much less than that of the sun; they are far distant from
each other; their forms are nearly spherical; their orbits
are nearly circular, and only slightly inclined in relation to
each other ; and so on. Their perturbations are, in conse-
quence, inconsiderable, for the most part; and all we have
to do is usually to take into the account, together with the
influence of the sun on each planet, the influence of one
other planet, capable, by its size and its nearness, of occa-
sioning perceptible derangements. If any of the condi-
tions mentioned above had been different, though the law
of gravitation had existed as it is, we might not at this day
have discovered it. And if we were now to try to investi-
gate Chemical phenomena by the same law, we should find
a solution as impossible as it would be in astronomy, if the
conditions of the heavenly bodies were such as we could
not reduce to an analysis.MATHEMATICS: FINAL CONSIDERATIONS.
49
In showing that Mathematical analysis can be applied
only to Inorganic Physics, we are not restricting its domain.
Its rigorous universality, in a logical view, has been estab-
lished. To pretend that it is practically applicable to the
same extent would be merely to lead away the human mind
from the true direction of scientific study, in pursuit of an
impossible perfection. The most difficult sciences must
remain, for an indefinite time, in that preliminary state
which prepares for the others the time when they too may
become capable of mathematical treatment. Our business
is to study phenomena, in the characters and relations in
which they present themselves to us, abstaining from intro-
ducing considerations of quantities, and mathematical laws,
which it is beyond our power to apply.
We owe to Mathematics both the origin of Positive
Philosophy and its Method. When this method was in-
troduced into the other sciences, it was natural that it
should be urged too far. But each science modified the
method by the operation of its own peculiar phenomena.
Thus only could that true definitive character be brought
out, which must prevent its being ever confounded with
that of any other fundamental science.
The aim, character, and general relations of Mathema-
tical Science have now been exhibited as fully as they could
be in such a sketch as this. We must next pass in review
the three great sciences of which it is composed,—the
Calculus, Geometry, and Rational Mechanics.
i.50
CHAPTER II.
GENERAL VIEW OF MATHEMATICAL ANALYSIS.
Analysis 'T'HE historical development of the Ab-
A stract portion of Mathematical science
has, since the time of Descartes, been for the most part
determined by that of the Concrete. Yet the Calculus in
all its principal branches must be understood before pass-
ing on to Geometry and Mechanics. The Concrete portions
of the science depend on the Abstract, which are wholly
independent of them. We will now therefore proceed to a
rapid review of the leading conceptions of the Analysis.
Tr idea of Eirst, however, we must take some notice
an'equation	^e general idea of an equation, and see
how far it is from being the true one on
which geometers proceed in practice; for without settling
this point we cannot determine, with any precision, the
real aim and extent of abstract mathematics.
The business of concrete mathematics is to discover the
equations which express the mathematical laws of the
phenomenon under consideration; and these equations are
the starting-point of the calculus, which must obtain from
them certain quantities by means of others. It is only by
forming a true idea of an equation that we can lay down
the real line of separation between the concrete and the
abstract part of mathematics.
It is giving much too extended a sense to the notion of
an equation to suppose that it means every kind of relation
of equality between any two functions of the magnitudes
under consideration; for, if every equation is a relation of
equality, it is far from being the case that, reciprocally,
every relation of equality must be an equation of the kind
to which analysis is, by the nature of the case, applicable.
It is evident that this confusion must render it almost im-
possible to explain the difficulty we find in establishing theFUNCTIONS OF A TWO-FOLD NATURE.
51
relation of the concrete to the abstract which meets ns in
every great mathematical question, taken by itself. If the
word equation meant what we are apt to suppose, it is not
easy to see what difficulty there could be, in general, in
establishing the equations of any problem whatever. This
ordinary notion of an equation is widely unlike what
geometers understand in the actual working of the science.
According to my view, functions must themselves be
divided into Abstract and Concrete; the first of which
alone can enter into true equations. Every equation is a
relation of equality between two abstract functions of the
magnitudes in question, including with the primary mag-
nitudes all the auxiliary magnitudes which may be con-
nected with the problem, and the introduction of which may
facilitate the discovery of the equations sought.
This distinction may be established by both the a priori
and a posteriori methods; by characterizing each kind of
function, and by enumerating all the abstract functions yet
known,—at least with regard to their elements.
A priori; Abstract functions express a mode .
of dependence between magnitudes which may	unc
be conceived between numbers alone, without
the need of pointing out any phenomena in whichitmay be
found realized : while Concrete functions are ~	, ,
,11.	•	-n i Concrete ranc-
tfiose whose expression requires a specified t-ons
actual case of physics, geometry, mechanics,
etc.
Most functions were concrete in their origin,—even those
which are at present the most purely abstract; and the
ancients discovered only through geometrical definitions
elementary algebraic properties of functions, to which a
numerical value was not attached till long afterwards,
rendering abstract to us what was concrete to the old
geometers. There is another example which well exhibits
the distinction just made—that of circular functions, both
direct and inverse, which are still sometimes concrete, some-
times abstract, according to the point of view from which
they are regarded.
A posteriori; the distinguishing character, abstract or
concrete, of a function having been established, the ques-
tion of any determinate function being abstract, and there-52
POSITIVE PHILOSOPHY.
fore able to enter into true analytical equations, becomes a
simple question of fact, as we are acquainted with the
elements which compose all the abstract functions at
present known. We say we know them all, though analy-
tical functions are infinite in number, because we are here
speaking, it must be remembered,—of the elements—of the
simple, not of the compound. We have ten elementary
formulas; and, few as they are, they may give rise to an
infinite number of analytical combinations. There is no
reason for supposing that there can never be more. We
have more than Descartes had, and even Newton and
Leibnitz; and our successors will doubtless introduce
additions, though there is so much difficulty attending
their augmentation, that we cannot hope that it will pro-
ceed very far.
It is the insufficiency of this very small number of analy-
tical elements which constitutes our difficulty in passing
from the concrete to the abstract. In order to establish
the equations of phenomena, we must conceive of their
mathematical laws by the aid of functions composed of
these few elements. Up to this point the question has
been essentially concrete, not coming within the domain of
the calculus. The difficulty of the passage from the con-
crete to the abstract in general consists in our having only
these few analytical elements with which to represent all
the precise relations which the whole range of natural
phenomena afford to us. Amidst their infinite variety,
our conceptions must be far below the real difficulty; and
especially because these elements of our analysis have been
supplied to us by the mathematical consideration of the
simplest phenomena of a geometrical origin, which can
afford us a priori no rational guarantee of their fitness
to represent the mathematical laws of all other classes of
phenomena. We shall hereafter see how this difficulty of
the relation of the concrete to the abstract has been
diminished, without its being necessary to multiply the
number of analytical elements.
Thus far we have considered the Calculus as a whole.
Two arts of	mus^ now cons^er its divisions. These
the Calculus divisions we must call the Algebraic Calculus,
or Algebray and the Arithmetical Calculus, orALGEBRAIC AND ARITHMETICAL CALCULUS.
53
Arithmetic, taking care to give them the most extended
logical sense, and not the restricted one in which the terms
are usually received.
It is clear that every question of Mathematical Analysis
presents two successive parts, perfectly distinct in their
nature. The first stage is the transformation of the pro-
posed equations, so as to exhibit the mode of formation of
unknown quantities by the known. This	,
constitutes the algebraic question. Then en-	®
sues the task of finding the values of the formulas thus ob-
tained. The values of the numbers sought are already
represented by certain explicit functions of given numbers :
these values must be determined; and this .	,.
is the arithmetical question. Thus the alge- n me 1C*
braic and the arithmetical calculus differ in their object.
They differ also in their view of quantities,—Algebra con-
sidering quantities in regard to their relations, and Arith-
metic in regard to their values. In practice, it is not
always possible, owing to the imperfection of the science of
the calculus, to separate the processes entirely in obtaining
a solution; but the radical difference of the two operations
should never be lost sight of. Algebra, then, is the
Calculus of Functions, and Arithmetic the Calculus of
Values.
We have seen that the division of the Calculus is into
two branches. It remains for us to compare the two,
in order to learn their respective extent, importance, and
difficulty.
The Calculus of Values, Arithmetic, appears . . ,	,.
at first to have as wide a field as Algebra, n me 1C*
since as many questions might seem to arise from it as we
can conceive different algebraic formulas to be valued.
But a very simple reflection will show that it is not so.
Functions being divided into simple and compound, it is
evident that when we become able to deter-	Extent
mine the value of simple functions, there
will be no difficulty with the compound. In the algebraic
relation, a compound function plays a very different part
from that of the elementary functions which constitute it;
and this is the source of our chief analytical difficulties.
But it is quite otherwise with the Arithmetical Calculus.54
POSITIVE PHILOSOPHY.
Thus, the number of distinct arithmetical operations is
indicated by that of the abstract elementary functions,
which we have seen to be very few. The determination of
the values of these ten functions necessarily affords that of
all the infinite number comprehended in the whole of
mathematical analysis: and there can be no new arith-
metical operations otherwise than by the creation of new
analytical elements, which must, in any case, for ever
be extremely small. The domain of arithmetic then is, by
its nature, narrowly restricted, while that of algebra is
rigorously indefinite. Still, the domain of arithmetic is
more extensive than is commonly represented; for there
are many questions treated as incidental in the midst of a
body of analytical researches, which, consisting of determi-
nations of values, are truly arithmetical. Of this kind are
the construction of a table of logarithms, and the calcula-
tion of trigonometrical tables, and some distinct and higher
procedures ; in short, every operation which has for its
object the determination of the values of functions. And
we must also include that part of the science of the Calculus
which we call the Theory of Numbers, the object of which
is to discover the properties inherent in different numbers,
in virtue of their values, independent of any particular
system of numeration. It constitutes a sort of transcen-
dental arithmetic. Though the domain of arithmetic is
thus larger than is commonly supposed, this Calculus of
values will yet never be more than a point, as it were, in
comparison with the calculus of functions, of which mathe-
matical science essentially consists. This is evident, when
we look into the real nature of arithmetical questions.
Determinations of values are, in fact, nothing else than
Its nature real transformations of the functions to be
valued. These transformations have a special
end; but they are essentially of the same nature as all
taught by analysis. In this view, the Calculus of values
may be regarded as a supplement, and a particular appli-
cation of the Calculus of functions, so that arithmetic dis-
appears, as it were, as a distinct section in the body of
abstract mathematics. To make this evident, we must ob-
serve that when we desire to determine the value of an un-
known number whose mode of formation is given, we defineALGEBRA.
55
and express that value in merely announcing the arith-
metical question, already defined and expressed under a
certain form ; and that, in determining its value, we merely
express it under another determinate form, to which we are
in the habit of referring the idea of each particular number
by making it re-enter into the regular system of numera-
tion. This is made clear by what happens when the mode
of numeration is such that the question is its own answer;
as, for instance, when we want to add together seven and
thirty, and call the result seven-and-thirty. In adding
other numbers, the terms are not so ready, and we trans-
form the question; as when we add together twenty-three
and fourteen : but not the less is the operation merely one
of transformation of a question already defined and ex-
pressed. In this view, the calculus of values might be re-
garded as a particular application of the calculus of func-
tions, arithmetic thereby disappearing, as a distinct section,
from the domain of abstract mathematics.—And here we
have done with the Calculus of values, and pass to the
Calculus of functions, of which abstract mathematics is
essentially composed.
We have seen that the difficulty of esta-	.. ,
j	A ifffthrA
blishing the relation of the concrete to the
abstract is owing to the insufficiency of the very small
number of analytical elements that we are in possession
of. The obstacle has been surmounted in a great number
of important cases: and we will now see how the esta-
blishment of the equations of phenomena has been
achieved.
The first means of remedying the difficulty	.
of the small number of analytical elements ne^functions.
seems to be to create new ones. But a little
consideration will show that this resource is illusory. A
new analytical element would not serve unless we could
immediately determine its value: but how can we deter-
mine the value of a function which is simple; that is,
which is not formed by a combination of those already
known ? This appears almost impossible: but the intro-
duction of another elementary abstract function into
analysis supposes the simultaneous creation of a new
arithmetical operation; which is certainly extremely diffi-56
POSITIVE PHILOSOPHY.
cult. If we try to proceed according to the method which
procured us the elements we possess, we are left in entire
uncertainty; for the artifices thus employed are evidently
exhausted. We have thus no idea how to proceed to create
new elementary abstract functions. Yet, we must not
therefore conclude that we have reached the limit appointed
by the powers of our understanding. Special improve-
ments in mathematical analysis have yielded us some
partial substitutes, which have increased our resources:
but it is clear that the augmentation of these elements
cannot proceed but with extreme slowness. It is not in this
direction, then, that the human mind has found its means
of facilitating the establishment of equations.
Finding equa- This first method being discarded, there
tions between remains only one other. As it is impossible
auxiliary to find the equations directly, we must seek
< plan titles.	f or corresponding ones between other auxiliary
quantities, connected with the first according to a certain
determinate law, and from the relation between which we
may ascend to that of the primitive magnitudes. This is
the fertile conception which we term the transcendental
analysis, and use as our finest instrument for the mathema-
tical exploration of natural phenomena.
This conception has a much larger scope than even pro-
found geometers have hitherto supposed ; for the auxiliary
quantities resorted to might be derived, according to any
law whatever, from the immediate elements of the ques-
tion. It is well to notice this ; because our future im-
proved analytical resources may perhaps be found in a new
mode of derivation. But, at present, the only auxiliary
quantities habitually substituted for the primitive quanti-
ties in transcendental analysis are what are called—
1st, infinitely small elements, the differentials of different
orders of those quantities, if we conceive of this analysis in
the manner of Leibnitz : or
2nd, the fluxions, the limits of the ratios of the simulta-
neous increments of the primitive quantities, compared
with one another ; or, more briefly, the prime and ultimate
ratios of these increments, if we adopt the conception of
Newton: or
3rd, the derivatives, properly so called, of these quanti-THE TWO BRANCHES OF ALGEBRA.
57
ties; that is, the coefficients of the different terms of their
respective increments, according to the conception of La-
grange.
These conceptions, and all others that have been pro-
posed, are by their nature identical. The various grounds
of preference of each of them will be exhibited hereafter.
We now see that the Calculus of func- Division of
tions, or Algebra, must consist of two distinct the Calculus
branches. The one has for its object the °f 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 independent 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-
culus of Direct 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 Indirect 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.
Its object.
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.
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
of Equations must evidently be determined by the nature
of the analytical elements of which their
members are composed. Accordingly, analysts first divide*
equations with one or more variables into two principal
classes, according as they contain functions of only the first
three of the ten couples, or as they include also either 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
Algebraic
equations.
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
equations only. In the first place, we must
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 different inferior
degrees, as must also the formula which determines the
unknown quantity: and therefore, however slight we
may, a 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 striven
to advance the resolution of equations. The general equa-
tion of the fifth degree has itself, thus far, resisted all
attempts. The formula of the fourth degree is so difficult
as to be almost inapplicable; and analysts, while by no
means despairing of the 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 be 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 treated only a very small
part of algebra, properly so called; that is, of the calculus
of direct functions, comprehending the resolution of all the
equations that can be formed by the analytical functions
known to us at this day. Again, we must 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 known, 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-
knowled^e1” plete resolution of the equations of the first
four degrees; of any binomial equations ; of
certain special equations of the superior degrees ; and of a
very small number of exponential, logarithmic, and circular
equations. These elements are very limited ; but 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 algebra; 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 numerical resolution is strictly sufficient, it is
not the less a very 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 must be carefully distinguished
in our minds, on account of their different objects, and62
POSITIVE PHILOSOPHY.
therefore of the different ways in which quantities are con-
sidered by them. Moreover, there is, in regard to their
methods, an entirely different procedure in their rational
distribution. In the first part, we have nothing to do
with the values of the unknown quantities, and the division
must take place according to the nature of the equations
which we are able to resolve; whereas in the second, we
have nothing to do with the degrees of the equations, as
the methods are applicable to equations of any degree
whatever; but the concern is with the numerical character
of the values of the unknown quantities.
These two parts, which constitute the im-
equations^ ° mediate object of the Calculus of direct-
functions, are subordinated to a third, purely
speculative, from which both derive their most effectual
resources, and which has been very exactly designated by
the general name of Theory of Equations, though it relates,
as yet, only to algebraic equations. The numerical 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 equations, and then those that relate to
their transformation; the business of these last being to
modify the roots of an equation without knowing them,
according to any given law, provided this law is uniform in
relation to all these roots.
One more theory remains to be noticed, to complete our
rapid exhibition of the different essential parts of the cal-
Method of cuius of direct functions. This theory, which
indeterminate relates to the transformation of functions
Coefficients. into series by the aid of what is called the
Method of indeterminate Coefficients, is one of the most
fertile and important in algebra. This eminently 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; soCALCULUS OF INDIRECT FUNCTIONS.
63
that, by the useful combination of the two theories, the
employment of the method of indeterminate coefficients
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. 5(2?) in a former section to	.
the views of the transcendental analysis pre- views ^pnncipa
sented by Leibnitz, Newton, and Lagrange.
We shall see that each conception has advantages of its
own, that all are finally equivalent, and that no method
has yet been found which unites their respective charac-
teristics. Whenever the combination takes place, it will
probably be by some method founded on the conception of
Lagrange. The other two will then offer only an historical
interest; and meanwhile, the science must be regarded as
in a merely provisional state, which requires the use of all
the three conceptions at the same time; for it is only by
the use of them all that an adequate idea of the aualysis
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	,
(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 in64
POSITIVE PHILOSOPHY.
substituting for the curve the auxiliary consideration of a
polygon, inscribed or circumscribed, by means of which
the curve itself was reached, the limits of the primitive
ratios being suitably taken. There is no doubt of the
filiation of ideas in this case; but there was in it no 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,
which was impossible to them.
Lagrange justly ascribes to the great geometer Fermat
the first idea in this new direction. Fermat 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
undergone certain suitable transformations. After some
modifications of the ideas of Fermat 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 more 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.
Method of The me^0(i Leibnitz consists in intro-
Leibnitz. 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 the quantities whose relations we are seeking.
There are relations between these differentials which are
simpler and more discoverable than those of the primitive
quantities; and by these we may afterwards (through a
special calculus employed to eliminate these auxiliary in-
finitesimals) recur to the equations sought, which it would
usually have been impossible to obtain directly. This
indirect analysis may have various degrees of indirectness ;
for, when there is too much difficulty in forming the equa-
tion between the differentials 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 being repeated any number of times, pro-
vided the whole number of auxiliaries be finally 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 being either the
successive powers of the same primitive infinitely small
quantity, or as being quantities which may be regarded as
having finite ratios with these powers ; so that, for instance,
the second or third or other differentials of the same variable
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 simple to treat, only observing the con-
i.	f66
POSITIVE PHILOSOPHY.
dition that the new elements shall differ from the preceding
only by quantities infinitely small in relation to them. It
is thus that it becomes possible in geometry to treat curved
lines as composed of an infinity of rectilinear elements, and
curved surfaces as formed of plane elements; and, in
mechanics, varied motions as an infinite series of uniform
motions, succeeding each other at infinitely small intervals
of time. Such a mere hint as this of the varied application
of this method may give some idea of the vast scope of the
conception of transcendental analysis, as formed by Leibnitz.
It is, beyond all question, the loftiest idea ever yet attained
by the human mind.
It is clear that this conception was necessary to complete
the basis of mathematical science, by enabling us to 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-
thTforinulas pressing in 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 positive approximations
between classes of wholly different phenomena, through theJUSTIFICATION 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, imper-	.fi	.
feet in its logical basis. At first, geometers	Metli©? °
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 difficult
problems. It became necessary, however, to recur to the
basis of the new analysis, to establish the rigorous exact-
ness of the processes employed, notwithstanding their
apparent breaches of the ordinary laws of reasoning,
Leibnitz himself failed to justify his conception, giving,
when urged, an answer which represented it as a mere
approximative calculus, the successive operations of which
might, it is evident, admit an augmenting amount of error.
Some of his successors were satisfied with showing that its
results accorded with those obtained by ordinary algebra,
or the geometry of the ancients, reproducing by these last
some solutions which could be at first obtained only by the
new method. Some, again, demonstrated the conformity
of the new conception with others; that of Newton 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 Carnot who furnished this at last, by
showing that the method was founded on the principle 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 quantities 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. Carnot’s theory is
doubtless more subtle than solid; but it has no other
radical logical vice than that of the infinitesimal method
itself, of which it is, as it seems to me, the 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 exhibited to allow of my giving
only the principal idea of the other two methods.
,	Newton ottered his conception under several
Newtons	,n .	mu ± u- u •
Method difterent torms m succession. That which is
now most commonly adopted, at least on the
continent, was called by himself, sometimes the Method of
\prime and ultimate Ratios, sometimes the Method of Limitst
by which last term it is now usually known.
Under this Method, the auxiliaries intro-
duced are the limits of the ratios of the
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
Method of
limits.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 offered by the
analysis of Leibnitz, which are merely considered from
another point of view. Thus, curves will be regarded as
the limits of a series of rectilinear polygons, and variable
motions as the limits of an aggregate of 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 flints1*8 ^
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 transcendental analysis. I
mean the Calculus of fluxions and of fluents, founded on the
general notion of velocities.
To facilitate the conception of the fundamental idea, let
us 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 curve, the abscissa, the
ordinate, the arc, the area, etc., will be regarded as simul-
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 fluent. Henceforth,
the transcendental analysis will, according 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. What 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, the70
POSITIVE PHILOSOPHY.
method of prime and ultimate ratios, which alone is redu-
cible to a calculus. It therefore necessarily admits of the
same general advantages in the various principal applica-
tions of the transcendental analysis, without its being
requisite for us to offer special proofs of this.
T	, Lagrange’s conception consists, in its ad-
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 coefiicient 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
between 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 simplify the equations, and even to obtain them
more easily. Successive derivation is a general artifice of
the same nature, only of greater extent, and commanding,
in consequence, much more important resources for this
common object.
But, though we may easily conceive, d 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 have 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.COMPARISON 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 three just exhibited. We must next com-
pare and estimate these methods ; and in the first place
observe their perfect and necessary conformity.
Considering the three methods in regard
to their destination, independently of pre- three methods6
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 with those under investigation; the
auxiliaries being substituted for the express object of
facilitating the analytical expression of the mathematical
laws of phenomena, though they must be finally eliminated
by the help of a special calculus. It was this which 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 conceiviug 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
will 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 preliminary operation before the
calculus can be of any use, and one which could not by its
nature be subjected to abstract and invariable rules :—it is
that which has for its object the establishment of equations,
which are the indispensable point of 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 finding the equations between auxiliary magnitudes,
whence the Calculus leads to equations directly relating to
the proposed magnitudes, which had formerly to be estab-
lished directly. Whether these indirect equations are72
POSITIVE PHILOSOPHY.
differential equations, according to Leibnitz, or equations
of limits, according to Newton, or derived equations, accord-
ing to Lagrange, the general procedure is evidently always
the same. The coincidence is not only in the result but in
the 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 are necessarily identical in these different methods,
which must therefore, under any application whatever, lead
to rigorously uniform results.
.	If we endeavour to estimate their compara-
tiv^vahie^1^ ^ve va^ue» we shall find in each of the three
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 magnitudes. 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 tin* 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 false idea: and it is in
fact impossible to conceive of them clearly, though we may
sometimes fancy that we do. This false idea bears, to my
mind, the characteristic impress of the metaphysical age
of its birth and tendencies of its originator. By the in-
genious principle of the compensation of errors, we may, as
we have already seen, explain the necessary exactness ofCOMPARISON OF THE THREE METHODS.
73
the processes which compose the method; but it is a
radical inconvenience to be obliged to indicate, in Mathe-
matics, two classes of reasonings so unlike, as that the one
order are perfectly rigorous, 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 irreproachable. This is eluding the difficulty, and
not resolving it; and besides, the advantages of this
method, its ease and rapidity, are almost entirely lost
under such a transformation. Finally, the infinitesimal
method exhibits the very serious defect of breaking the
unity of abstract mathematics by creating a transcendental
analysis founded upon 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 limits, though clear and exact, is not
the less a foreign idea, on which analytical theories ought
not to be dependent.
This perfect unity of analysis, and a purely abstract
character in the fundamental ideas, are found in the con-
ception of Lagrange, and there alone. It is therefore the
most philosophical of all. Discarding every heterogeneous
consideration, Lagrange reduced the transcendental analysis74
POSITIVE PHILOSOPHY.
to its proper character,—that of presenting 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 philosophical superiority
marks it for adoption as the final theory of transcendental
analysis ; but it presents too many difficulties in its appli-
cation, in comparison with the others, to admit of its
exclusive preference at present. Lagrange himself had
great difficulty in rediscovering, by his own method, the
principal results already obtained by the infinitesimal
method, on general questions in 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 employ them all (and es-
pecially the first and last) almost indifferently, in the
solution of all important questions, whether of the calculus
of indirect functions in itself, or of its applications. In all
the other departments of mathematical science, the con-
sideration 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 facility in the
use of this powerful instrument.THE TWO CALCULI OF INDIRECT FUNCTIONS.
7 5
THE DIFFERENTIAL AND INTEGRAL CALCULUS.
The Calculus of Indirect functions is Itg two 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 corresponding primitive magnitudes ; by
the other we seek, conversely, these direct equations by
means of the indirect equations first established. This is
the double object of the transcendental analysis.
Different names have been given to the two systems,
according to the point of view from which the entire
analysis has been regarded. The infinitesimal method,
properly so called, being most in use, almost all geometers
employ the terms Differential Calculus and Integral 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, approaching as nearly as
may be to that of Lagrange.
The differential calculus is obviously the . .
rational basis of the integral. We have seen relations1 ^
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 those 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 infinitesimals76
POSITIVE PHILOSOPHY.
or the derivatives, introduced as auxiliaries, being the final
object of the calculus of indirect functions, it is natural to
think that the calculus which teaches us to deduce the
equations between 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 departure of any calculus whatever. The very
use of the differential calculus is enabling us to differentiate
the various equations; and it cannot therefore be the
process for establishing them. This common error arises
from confounding the infinitesimal calculus with the 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 prepare an indispensable basis for the integral
calculus. A few words will show that a primary direct
and necessary office is always assigned to the differential
r i . calculus. In forming differential equations,
oflthe°tw()11()n we 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. How
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 this is merely
a question of differentiation. This is the way in which the
differential calculus not only prepares a basis for theEXAMPLES OF THE TWO CALCULI.
77
integral, but makes it available in a multitude of cases
which could not otherwise be treated. There Cases of the
are some questions, few, but highly impor- Differential
tant, which admit of the employment 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 primitive 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 infinitesimals, without the question giving rise to
any integration. There are also questions, few, but highly
important, which are the converse of the last, requiring the
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,
differentially, 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. The differential calculus is, in such cases, not
essential to the solution of the problem, which will depend
entirely on the integral calculus. Thus, all questions to
which the analysis is applicable are contained in three
classes. The first class comprehends the problems which
may be resolved by the differential calculus alone. The
second, those which may be resolved by the integral cal-
culus alone. These 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 of78
POSITIVE PHILOSOPHY.
either calculus, which are quite a different
tiaf Calcufus 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 be studied.
The first division of the differential calculus is grounded
on the condition whether the functions to be differentiated
T r	are explicit or implicit; the one giving rise
wopoi 1011s. ^ ^he differentiation of formulas, and the
other to the differentiation of equations. This classifica-
tion is rendered necessary by the imperfection of ordinary
analysis; for if we knew how to resolve all equations
algebraically, it would be possible to render every implicit
function exjfiicit; and, by differentiating it only in that
state, the second part of the differential calculus would
be immediately included in the first, without giving rise
to any new difficulty. But the algebraic resolution of
equations is, as we know, still scarcely past its infancy,
and unknown for the greater number of cases; and we
have to differentiate a function without knowing it, though
it is determinate. Thus we have two classes of questions,
the differentiation of implicit functions being a distinct
case from that of explicit functions, and much more
complicated. We have to begin by the differentiation of
formulas, and we may then refer to this first case the
differentiation of equations, by certain analytical 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 preponderant importance.
c i r • .	Each of these parts of the differential
calculus is again divided: and this sub-
division exhibits two very distinct theories, according as
we have to differentiate functions of a single variable, orSUBDIVISIONS 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
implicit. One more distinction remains, to complete this
brief sketch of the parts of the differential calculus. The
case in which it is required to differentiate at once different
implicit functions combined in certain primitive equations
must be distinguished from that in which all these functions
are separate. The same imperfection of ordinary analysis
which prevents our concerting every implicit function into
an equivalent explicit one, renders us unable to separate
the functions which enter simultaneously into any 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 proper
elimination among them.
We have now seen the different parts of
this calculus in their natural connection and tll®
rational distribution. The whole calculus is
finally found to rest upon the differentiation of explicit
functions with a single variable,—the only one which is
ever executed directly. Now, it is easy to understand that
this first theory, this necessary basis of the whole system,
simply consists of the differentiation of the elementary
functions, ten in number, which compose 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 fundamental differentials, and to that of
the two general principles, by one of which the differentia-
tion of implicit 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 Transforma-
new variables is a theory which must be just tionof derived
mentioned, to avoid the omission of an indis- functions for
pensable complement of the system of diffe- ne>v variables.80
POSITIVE PHILOSOPHY.
rentiation. It is as finished and perfect as the 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 deferential equations, that system of indepen-
dent variables which may appear to be most advantageous,
though it may afterwards be relinquished, as an inter-
mediate step, by which, through this theory, we may pass
to the final system, which sometimes could not have been
considered directly.
1	Though we cannot here consider the con-
apphcations Crete applications of this calculus, we must
glance at those which are analytical, because
they are 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 most beautiful and the most important application of
the differential calculus to general analysis, and which
comprises, besides the fundamental series discovered by
Taylor, the remarkable series discovered by Maclaurin,
John Bernouilli, Lagrange and others. Secondly, the
general theory of maxima and minima values for any
functions whatever of one or more variables: one of the
most interesting problems that analysis can present, how-
ever elementary it has become. The third 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 appearance, for certain hypotheses made on
the values of the corresponding variables. In every view,
the first question is the most eminent; it is also the most
susceptible of future extension, especially by conceiving,
in a larger manner than hitherto, of the employment of
the differential calculus for the transformation of functions,
about which Lagrange left some valuable suggestions
which have been neither generalized nor followed up.
It is with regret that I confine myself to the generalities
which are the proper subjects of this work ; so extensive
and so interesting are the developments which might other-
wise be offered. Insufficient, and summary as are the views
of the Differential Calculus just offered, we must be noTHE 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 Cahuhis”1^
principle of distinguishing the integration of
explicit differential formulas from the integration of implicit
differentials, or of differential equations. The (jjvjsjons
separation of these two cases is even more
radical in the case of integration than in the other. In
the differential calculus this distinction rests, as we have
seen, only on the extreme imperfection of ordinary analysis.
But, on the other hand, it is clear that even if all equations
could be algebraically resolved, differential equations would
nevertheless constitute a case of 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, a v . .
and for analogous reasons, each of these two
branches of the integral calculus is divided again, accord-
ing as we consider functions with a single n	.
variable or functions with several mdepen- or severay
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.	G82
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.
Orders of	^ new distinction, highly important here,
differentiation though not 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 butALGEBRAIC AND TRANSCENDENTAL FUNCTIONS. 83
Algebraic
functions.
by reducing them to this elementary case, which is the only
one capable of being treated directly. This , ,
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 complicated, 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 special 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
of algebraic functions and transcendental func-
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- m	,	, ,
gration of transcendental functions is quite functi0ns
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 Solutions (some-
tions1 ^ S° U times called 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 theory remains to be noticed, to
te^rals6111 complete our review of that collection of
analytical researches which constitutes the
integral calculus. It takes its place outside of the system,
because, instead of being destined for true 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 proposed differential formula, haveDEFINITE 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, under our inability to
obtain the true integrals, it is of the utmost importance to
have been able to obtain this solution, incomplete and in-
sufficient as it is. This has now been attained for all cases,
the determination of the value of definite integrals having
been reduced to entirely general methods, which leave
nothing to be desired, in many cases, but less complexity
in the calculations; an object to which analysts are now
directing all their special transformations. This kind of
transcendental arithmetic being considered perfect, the
difficulty 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 of86
POSITIVE PHILOSOPHY.
a wider generality. The transcendental analysis is yet too
near its origin, it has too recently been regarded in a truly
rational manner, for us to have any idea what it may 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 urging further our present calculus of in-
direct functions ; and I am persuaded that when geometers
have exhausted the most important 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 knowledge 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
abstract 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
corresponds to a multitude of concrete researches, Man
having no other resource 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 his 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
blems; a name which is truly applicable to t^lis Calculus,
only a very small number of them. They consist in the
investigation of the maxima and minima of certain inde-
terminate 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 separately 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 maximum and a positive for the
minimum. Such are the fundamental conditions belonging88
POSITIVE PHILOSOPHY.
to tlie 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, according to the condition of the maximum or
minimum of a certain definite integral, merely indicated,
which depended on that function. We cannot here follow
the history of these problems, the oldest of which is that
of the solid of least resistance, treated by Newton in the
second book of the 4 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 solutionsSOME OF ITS APPLICATIONS.
89
previously obtained being of any essential aid. The part
common to all questions of this class had not been dis-
covered; and no abstract and general treatment was
therefore provided. In his endeavours to bring all isoperi-
metrical problems to depend on a common analysis,
Lagrange was led to the conception of a new kind of diffe-
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 trans-
formation, and though the Method of Varia- tions.1^1
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 In the section on the Integral Calculus, I
ordinary Cal- noticed D’Alembert’s creation of the Calculus
cuius.	0j partial differences, in which Lagrange
recognized a new calculus. This new elementary idea in
transcendental analysis,—the notion 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 use. A recognition of such a derivation as this for the
method of variations may exhibit its philosophical characterCONSIDERATIONS 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 ns 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
directly. The Method of Variations forms, then, the most
sublime part of that vast system of mathematical analysis,
which, setting out from the simplest elements of algebra,
organizes, by an uninterrupted succession of ideas, general
methods more and more potent for the investigation of
natural philosophy. This is incomparably the noblest and
most unquestionable testimony to the scope of the human
intellect. If, at the same time, we bear in mind that the
employment of this method exacts the highest known
degree of intellectual exertion, in order never to lose sight
of the precise object of the investigation in following
reasonings which offer to the mind such uncertain resting-
places, and in which signs are of scarcely any assistance,
we shall understand how it may be that so little use has
been made of such a conception by any 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 only92
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
be to show how,—supposing the general science of the
Calculus to be in a perfect state,—it has been possible to
reduce, by invariable procedures, to pure questions of
analysis, all the problems of Geometry and Mechanics ;
and thus to invest these two great bases of natural
philosophy with that precision and unity which can only
thus be attained, and which constitute high perfection.93
CHAPTER III.
GENERAL VIEW OF GEOMETRY.
Its nature.
WE 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 were regarded as immovable,
geometry would still exist; whereas, for the phenomena of
Mechanics, motion is required. Thus Geometry is the
more general of the two. It is also the more simple, for
its phenomena are independent of those of Mechanics,
while mechanical phenomena are always complicated with
those of geometry. The same is true in the comparison of
abstract thermology with geometry. For these reasons,
geometry holds the first place under the head of Concrete
Mathematics.
Instead of adopting the inadequate ordi-
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
Definition.94
POSITIVE PHILOSOPHY.
aim,—sometimes direct, though oftener indirect,—of geo-
metrical labours.
The rational study of geometry could
Idea of Space.
never have begun if we must have regarded
Kinds of ex-
tension.
at once and together all the physical properties of bodies,
together with their magnitude and form. By the character
of our minds we are able to think of the dimensions and
figure of a body in an abstract way. After observation
has shown us, for instance, the impression left by a body
on a fluid in which it has been placed, we are able to
retain an image of the impression, which becomes a ground
of geometrical reasoning. We thus obtain,—apart from
all metaphysical fancies,—an idea of Space. This abstrac-
tion, now so familiar to us that we cannot conceive the
state we should be in without it, is perhaps the earliest
philosophical creation of the human mind.
There is another abstraction which must
be made before we can enter on geometrical
science. We must conceive of three kinds of
extension, and learn to conceive of them separately. We
cannot conceive of any space, filled by any object, which
has not at once volume, surface, and line. Yet geometrical
questions 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- r . .
ment 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
difficult, 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.	lines1™6
When we consider curved lines, it is evident
that their measurement must be indirect, since we cannot
conceive of curved lines being laid upon each other with
any precision or certainty. The procedure is first to reduce
the measurement of curved to that of right lines; and
consequently to reduce to simple questions of right lines
all questions relating to the magnitude of any curves 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 the96
POSITIVE PHILOSOPHY.
circumference; and again, as the length of an ellipse
depends on that of its two axes.
Thus, the science of Geometry has for its object the
final 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 mn a 6 lately unbounded. There may be as many
questions as there are conceivable figures;
and the variety of conceivable figures is infinite. As to
curved Lines, if we regard them as generated by the motion
of a point governed by a certain law, we 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 we may say that if lines have
one infinity of possible change, surfaces have two. As
for Volumes, they are distinguished from one another only
by the surfaces which bound them ; so that they partake
of the variety of surfaces, and need no special considera-
tion under this head. If we add the one further remark,
that surfaces themselves furnish a new means of con-
ceiving of new curves, as every curve may be regarded asSCOPE 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- *aces-
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
curve 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 Greek geometers,
were needed as preparation for this discovery, that Kepler
might select from them the characteristic which was the
true key to the planetary orbit. In the same way, the
spherical figure of the earth could not have been discovered
if the primitive character of the sphere had been the only
one known ;—viz. the equidistance of all its points from an
interior point. Certain properties of surfaces were the
means used for connecting the abstract reasoning with the
concrete fact. And others, again, were required to prove
that the earth is not absolutely spherical, and how much
otherwise. The pursuit of these 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 properties 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
finished. 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 points
of view from which each figure may be regarded,
i.	h98
POSITIVE PHILOSOPHY.
Two "eneral There are two general Methods of treating
Methods. * geometrical questions. These are commonly
called Synthetical Geometry and Analytical
Geometry. I shall prefer the historical titles of Geometry
of the Ancients and Geometry of the Moderns. But it is,
in my view, better still to call them Special Geometry and
General Geometry, by which their nature is most accurately
The Calculus was not, as some suppose,
unknown to the ancients, as we perceive by
their applications of the theory of propor-
tions. The difference between them and us
is not so much in the instrument of deduction
as in the nature of the questions considered. The ancients
studied geometry with reference to the bodies under notice,
or specially: the modems 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
by the general method used since the great revolution
under 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
conveyed.
Special or
ancient, and
general or
modern Geo-
metry.SPECIAL AND GENERAL GEOMETRY*
99
modern. The experience of the 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 procedure. 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 Qeometr Gf
ancients, in its character of an indispensable the°£nicients
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
are few and simple, and the structure of reasoning erected
upon them vast and complex. The only elementary
materials, obtainable by direct study alone, are those which
relate to the right line for the geometry of lines; to the
quadrature of rectilinear plane areas ; and to the cubature of
bodies terminated by plane faces. The beginningof 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 careful we must
be 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 to100
POSITIVE PHILOSOPHY.
carry logical demonstration into a region where direct
observation will serve ns better.
There are two ways of studying the right
thTri^lnline ^ne—^he 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
is,	first, to 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
Graphical
solutions.
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 employed it still for a great number of in-
vestigations for which we now consider the use of the
Calculus indispensable.
While the graphic or constructive method answers well
when all the parts of the proposed figure lie in the same
plane, it must receive additions before it can be applied to
figures whose parts lie in different planes. Hence arises a
new series of considerations, and different systems of Pro-
jections. Where we now employ spherical trigonometry,
especially for problems relating to the celestial sphere, the
ancients had to consider how they could replace construc-
tions in relief by plane constructions. This was the object
of their analemmas, and of the other plane figures which
long supplied the place of the Calculus. They were 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 appli-DESCRIPTIVE GEOMETRY.
101
cation, I may observe that Descriptive Geo-	. .
metry, formed into a distinct system by Geometrv6
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 towards a philo-
sophical 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 endeavoured to sketch out a
philosophical system of mechanical arts, and at least suc-
ceeded in pointing out the direction in which the object
must be pursued. Of Descriptive Geometry, it may
further be said that it usefully exercises the students’
faculty of Imagination,—of conceiving of complicated 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 moderns. Yet, as destined
to an industrial application, Descriptive Geometry has102
POSITIVE PHILOSOPHY.
here been treated of only in the way of digression. Leaving
inventors of trigonometry,—spherical as well as recti-
linear,—though it necessarily remained imperfect 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 polygonometrv to simple trigonometry.
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 propor-
tional to them: or they are introduced indirectly, by the
chords of these arcs, which are hence called their trigono-
metrical lines. The second of these methods was the first
adopted, because the early state of knowledge admitted of
its working, while it did not admit 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 preferred, as the more simple, the
equations existing only between right lines, instead of
between right lines and arcs of circles.
To meet the probable objection that it is rather a com-
plication than a simplification to introduce these lines,
which have at last to be eliminated, we must explain a
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
the subject of graphic solution, we have to notice the other
branch,—the algebraic.
Some may wonder that this branch is not
treated as belonging to General Geometry.
But, not only were the ancients, in fact, the
Trigonometry.
The difficulty lies in forming three distinct
equations between the angles and the sides
little.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 position 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 previous
resolution of the corresponding algebraic question. But
for this, the ancients could not have obtained trigono-
metry. When Archimedes was at work 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 any line relating to an arc which is a
very simple function of the first. Thus, we may say that
the number of trigonometrical lines actually employed by
modern geometers is unlimited through the augmentations
we may obtain by analysis. Special names have, however,
been given to those indirect lines only which refer to the
complement of the primitive arc,—others being in much
less 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 doue is evidently to
representation find and fix a method for expressing analyti-
<>f figures. cally the subjects which afford the phe-
nomena. If we can regard lines and surfaces analytically,MODERN OR ANALYTICAL GEOMETRY.
105
we can so regard, henceforth, the accidents of these
subjects.
Here occurs the difficulty of reducing all geometrical
ideas to those of number: of substituting considerations of
quantity for all considerations of quality.—In dealing with
this difficulty, we must observe that all geometrical ideas
come under three heads:—the magnitude, the	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 points of which 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
in space, the co-ordinates cannot be less than three. They
may be multiplied without limit: but whether few or many,
the ideas of position will have been reduced to those of
magnitude, so that we shall represent the displacement of
a point as produced by pure numerical variations in the
values of its co-ordinates.—The simplest case of all, that of
plane geometry, is when we determine the position of a point
on a plane by considering its distances from two fixed right
lines, 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
point1011 ° a ^e Positi°n of 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 that
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 learn what it was that he effected.
Plane Curves Descartes treated only geometry of two
dimensions in his analytical method : and we
will at first consider only this kind, beginning with Plane
Expression Curves. Lines must be expressed by equa-
of lines by tions ; and again, equations must be expressed
Equations. by lines, 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 expressed by an appropriate equation. If the point
describes no certain line, its co-ordinates must be two
variables iu dependent 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 ofANALYTICAL 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, hne*-
in a determinate system of co-ordinates. In the absence
of any other known property, such a relation would be a
very characteristic definition ; and its scientific effect would
be to fix the attention immediately upon the general course
of the solutions of the equation,which will thus be noted in
the most striking and simple manner. There is an evident
and vast advantage in this picturing of equations, which
reacts strongly upon the perfecting of analysis itself. The
geometrical locus stands before our minds as the 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.
Eecurring to the representation of lines by Change in the
equations, which is our chief object, we see line changes
that this representation is, by its nature, so ^ie equation,
faithful, that the line could not undergo any modification,
even the slightest, without causing a corresponding change
in the equation. Some special difficulties arise out of this
perfect exactness; for since, in our system of analytical
geometry, mere displacements of lines affect equations as
much as real variations of magnitude or form, we might be
in danger of confounding the one with the other, if geo-
meters had not discovered an ingenious method expressly
intended to distinguish them always. It must be observed
that general inconveniences of this nature appear to be
strictly inevitable in analytical geometry; since, ideas of108
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 simple
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. out that not only must every defined line give
rise to a certain equation between the two co-ordinates
of any one of its points, but every 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-
perty 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 what 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 what a system of co-ordinates is, and if we
admit all possible systems, it is clear that such a relation
can be nothing else than the equation of the proposed line,
in a system of co-ordinates of a corresponding nature to
that of the mode of generation considered; as in the case
of the circle, the common definition of which may be re-ANALYTICAL GEOMETRY.
109
garded 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 equation of a curve by
means of some of its characteristic properties, the difficulty
must proceed from our taking up a designated system of
co-ordinates, instead of admitting indifferently all possible
systems. These systems are not all equally suitable ; and,
in regard to curves, geometers think that they should almost
always be referred, as far as possible, to rectilinear co-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-ordinates,
to the case—is the remaining point which we
have to notice.
First, we must distinguish very carefully the two views,
the converse of each other, which belong to analytical
geometry, viz. the relation of algebra to geometry, founded
on the representation of lines by equations, and, 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 of110
POSITIVE PHILOSOPHY.
method which we are considering is far from being the
same under the 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 arriving 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 possible, rather than with any
other inclination. In representing lines by equations, we
must take any inclination of the axes which may best suit
the particular question ; but, in the converse case, it is easy
to see that rectangular axes permit us to represent equations
in a more simple, and even in a more faithful manner.
For if we extend the geometrical locus of the equation into
the several unequal regions marked out by oblique axes, we
shall have differences of figure which do not correspond to
any analytical diversity; and the accuracy of the represen-
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 nextRECTILINEAR AND POLAR CO-ORDINATES. Ill
to it, both because its opposite character enables it to solve
in the simplest way the equations which are too compli-
cated for management under the first; and because polar
co-ordinates have often the advantage of admitting of
a more direct and natural concrete signification. This is
the case in Mechanics, in the geometrical questions arising
out of the theory of circular movement, and in almost all
questions of celestial geometry.
Such was the field of the labours of Descartes, his
conception of analytical geometry being designed only
for the study of Plane Curves. It was Clairaut who,
about a century later, extended it to the study of Surfaces
and Curves of double curvature. The conception having
been explained, a very brief notice will suffice for the
rest.
With regard to Surfaces, the determination Determination
of a point in space requires that the values of of a point in
three co-ordinates should be assigned. The sPace-
system generally adopted, which corresponds with the
rectilinear system of plane geometry, is that of distances
from the point to three fixed planes, usually perpendicular
to each other, whereby the point is presented as the inter-
section of three planes 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
dimensions. Instead of the intersection of two lines, it
must be that of three surfaces which determines the point;
and each of the three surfaces has, in the same way, all its
conditions constant, except one, which gives rise to the
corresponding co-ordinates, whose peculiar geometrical
effect is thus to compel the point to be situated upon that
surface. Again, if the three co-ordinates of a point are
mutually independent, that point can take successively all
possible positions in space ; but, if its position on any 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 indetermination in the position of the
point.
Determination In expression of Surfaces by Equations,
of Surfaces by and again in the expression of Equations by
Equations, and Surfaces, the same conception is pursued as
analytical definition of the proposed surface, since it must
be verified for all the points of this surface, and for them
only. If the surface undergoes any change, the equation
must, as in the case of changing lines, be modified accord-
ingly. All geometrical phenomena relating to surfaces may
be translated by certain equivalent analytical conditions,
proper to equations of three variables : and it is in the
establishment and interpretation of this harmony that the
science of analytical geometry of three dimensions essentially
consists. In the second and converse case, every equation
of three variables may, in general, be represented geometri-
cally by a determinate surface, defined by the characteristic
property that the co-ordinates of all its points always pre-
serve the mutual relation exhibited in this equation.
Thus we see in this application the complement of the
original idea of Descartes ; and it is enough to say this, as
every one can extend to surfaces the other considerations
which have been indicated with regard to lines. I will
only add that the superiority of the rectilinear system of
co-ordinates becomes more evident in analytical geometry
of three dimensions than in that of two, on account of the
geometrical complication which would follow the choice of
any other.
curvature. 0f analytical geometry of three dimensions,
—the same principle is employed. According to it, it is
clear that when a point is required to be situated upon
some certain curve, a single co-ordinate is enough to deter-
mine its position completely, by the intersection of this
curve with the surface resulting from this co-ordiuate.
The two other co-ordinates of the point must thus be
regarded as functions necessarily determinate, and distinct
In the first case, the equation will be the
Curves of
double
In determining Curves of double curvature,
■which is the last elementary point of viewIMPERFECTIONS OF ANALYTICAL GEOMETRY. 113
from the first. Consequently, every line, considered in
space, is represented analytically no longer by a single
equation, but by a system of two equations between the
three co-ordinates of any one of its points. It is evident,
indeed, from another point of view, that the equations
which, considered separately, express a certain surface,
must in combination present the line sought as the inter-
section of two determinate surfaces. As for the difficulty
occasioned by the infinity of the number of couples of
equations, through the infinity of couples of surfaces which
can enter the same system of co-ordinates, and by which
the line sought may be hidden under endless algebraical
disguises, it must be got rid of by giving up the facilities
resulting from such a variety of geometrical constructions.
It is sufficient in fact, to obtain from the analytical system
established for a certain line, the system corresponding to
a single couple of surfaces uniformly generated, and which
will not vary except when the line itself shall change.
Such is a natural use of this kind of geometrical combina-
tion, which thus affords us a certain means of recognizing
the identity of lines in spite of the extensive diversity
of their equations.
Analytical Geometry still presents some Imperfections
imperfections on the side both of geometry of Analytical
and of analysis.	Geometry.
In regard to Geometry, the equations can as yet represent
only entire geometrical loci, and not determinate portions of
those loci. Yet it is necessary, occasionally, to be able to
express analytically a part of a line or surface, or even a dis-
continuous line or surface, composed of a series of sections
belonging to distinct geometrical figures. Some progress
has been made in supplying means for this purpose, to
which our analytical geometry is inapplicable; but the
method introduced by M. Fourier, in his labours on discon-
tinuous functions, is too complicated to be at present intro-
duced into our established system.
In regard to analysis, we are so far from T . ^
°	J	ii.*i Imperfections
having a complete command or analytical Analysis
geometry, that we cannot furnish anything
like an adequate geometrical representation of analytical
processes. This is not an imperfection in science, but in-
i.	i114
POSITIVE PHILOSOPHY.
herent in the very nature of the subject. As Analysis is
much more general than geometry, it is of course impos-
sible to find among geometrical phenomena a concrete
representation of all the laws expressed by analysis: but
there is another evil which is due to our own imperfect
conceptions; that, in our representations of equations of
two or of three variables by lines or surfaces, we regard
only the real solutions of equations, without noticing any
imaginary ones. Yet these last should, in their general
course, be as capable of representation as the first. Hence
the graphic representation of the equation is always im-
perfect ; and it fails altogether when the equation admits
of only imaginary solutions. This brings after it, in ana-
lytical geometry of two or three dimensions, many incon-
veniences of less consequence, arising from the want of
correspondence between various analytical modifications
and any geometrical phenomena.
We have now seen what Analytical Geometry is. By
this science we determine what is the analytical expression
of such or such a geometrical phenomenon belonging to
lines or surfaces: and, reciprocally, we ascertain the geo-
metrical interpretation of such or such an analytical con-
sideration. It would be interesting now to consider the
most important general questions which would exemplify
the manner in which geometers have actually established
this beautiful harmony: but such a review is not neces-
sary to the purpose of this Work, and would occupy too
much space. We have seen what is the character of gene-
rality and simplicity inherent in the science of Geometry.
We must now proceed to ascertain what is the true philo-
sophical character of the immense and more complex science
of Rational Mechanics.115
CHAPTEK IV.
RATIONAL MECHANICS.
MECHANICAL phenomena are by their	nature
nature more particular, more compli-
cated, and more concrete than geometrical phenomena.
Therefore they come after geometry in our survey; and
therefore must they be pronounced to be more difficult to
study, and, as yet, more imperfect. Geometrical questions
are always completely independent of Mechanics, while
mechanical questions are closely involved with geometrical
considerations,—the form of bodies necessarily influencing
the phenomena of motion and equilibrium. The simplest
change in the form of a body may enhance immeasurably
the difficulties of the mechanical problem relating to it, as
we see in the question of the mutual gravitation of two
bodies, as a result of that of all their molecules; a question
which can be completely resolved only by supposing the
bodies to be spherical; and thus, the chief difficulty arises
out of the geometrical part of the circumstances.
Our tendency to look for the essences of things, instead
of studying concrete facts, enters disastrously into the
study of Mechanics. We found something of it in geo-
metry ; but it appears in an aggravated form in Mechanics,
from the greater complexity of the science. We encounter
a perpetual confusion between the abstract and the con-
crete points of view; between the logical and the physical;
between the artificial conceptions necessary to help us to
general laws of equilibrium and motion, and the natural
facts furnished by observation, which must form the basis
of the science. Great as is the gain of applying Mathe-
matical analysis to Mechanics, it has set us back in some
respects. The tendency to a priori suppositions, drawn by
us from analysis where Newton wisely had recourse to
observation, has made our expositions of the science less116
POSITIVE PHILOSOPHY.
clear than those of Newton’s days. Inestimable as mathe-
matical analysis is for carrying the science on and upwards,
there must first be a basis of facts to employ it upon; and
Laplace and others were therefore wrong in attempting to
prove the elementary law of the composition of forces by
analytical demonstration. Even if the science of Mechanics
could be constructed on an analytical basis, it is not easy
to see how such a science could ever be applied to the
actual study of nature. In fact, that which constitutes the
reality of Mechanics is that the science is founded on some
general facts, furnished by observation, of which we can
give no explanation whatever. Our business now is to
point out exactly the philosophical character of the science,
distinguishing the abstract from the concrete point of view,
and separating the experimental department from the
logical.
T, i ,	We have nothing to do here with the causes
or modes ot production ot motion, but only
with the motion itself. Thus, as we are not treating of
Physics, but of Mechanics, forces are only motions pro-
duced or tending to be produced; and two forces which
move a body with the same velocity in the same direction
are regarded as identical, whether they proceed from
muscular contractions in an animal, or from a gravitation
towards a centre, or from collision with another body, or
from the elasticity of a fluid. This is now practically
understood; but we hear too much still of the old meta-
physical language about forces, and the like ; and it would
be wise to suit our terms to our positive philosophy.
T , . ,	The business of Rational Mechanics is to
S ° jeC ’ determine how a given body will be affected
by any different forces whatever, acting together, when we
know what motion would be produced by any one of them
acting alone: or, taking it the other way, what are the
simple motions whose combination would occasion a known
compound motion. This statement shows precisely what
are the data and what the unknown parts of every me-
chanical question. The science has nothing to do with the
action of a single force; for this is, by the terms of the
statement, supposed to be known. It is concerned solely
with the combination of forces whether there results fromSCOPE OF RATIONAL MECHANICS.
117
that combination a motion to be studied, or a state of
equilibrium, whose conditions have to be described.
The two general questions, the one direct, the other
inverse, which constitute the science, are equivalent in
importance, as regards their application. Simple motions
are a matter of observation, and their combined operation
can be understood only through a theory: and again, the
compound result being a matter of observation, the simple
constituent motions can be ascertained only by reasoning.
When we see a heavy body falling obliquely, we know
what would be its two simple movements if acted upon
separately bv the forces to which it is subject,—the direc-
tion and uniform velocity which would be caused by the
impulsion alone ; and again, the acceleration of the vertical
motion by its weight alone. The problem is to discover
thence the different circumstances of the compound move-
ment produced by the combination of the two,—to deter-
mine the path of the body, its velocity at each moment,
the time occupied in falling; and we might add to the two
given forces the resistance of the medium, if its law was
known. The best example of the inverse problem is found
in celestial mechanics, where we have to determine the
forces which carry the planets round the sun, and the
satellites round the planets. We know immediately only
the compound movement:	Kepler’s laws give us the
characteristics of the movement; and then we have to go
back to the elementary forces by which the heavenly bodies
are supposed to be impelled, to correspond with the
observed result: and these forces once understood, the
converse of the question can be managed by geometers,
who could never have mastered it in any other way.
Such being the destination of Mechanics, we must now
notice its fundamental principles, after clearing the ground
by a preparatory observation.
In ancient times, men conceived of matter ,,
as being passive or inert,—all activity being ert ppygicg
produced by some external agency,—either
of supernatural beings or some metaphysical entities. Now
that science enables us to view things more truly, we are
aware that there is some movement or activity, more or
less, in all bodies whatever. The difference is merely of118
POSITIVE PHILOSOPHY.
degree between what men call brute matter and animated
beings. Moreover, science shows us that there are not
different kinds of matter, but that the elements are the
same in the most primitive and the most highly organized.
If we knew of any substance which had nothing but
weight, we could not deny activity even to that; for in
falling it is as active as the globe itself,—attracting the
earth’s particles precisely as much as its own particles are
attracted by the earth. Looking through the whole range
of substances, up to those of the highest organization, we
find everywhere a spontaneous activity, very various, and
at most, in some cases, peculiar; though physiologists are
more and more disposed to regard the most peculiar as a
modification of antecedent kinds. However this may be,
it would be purely absurd now to regard any portion of
matter whatever as* inert, as a matter of fact, or under the
head of Physics. But in Mechanics it must
ii^JvlechanicT^ so regar(led, because we cannot establish
any general proposition upon the abstract
laws of equilibrium or motion without putting out of the
question all interference with them by other and inherent
forces. What we have to beware of is mixing up this
logical supposition with the old notion of actual inertia.
Field of Ha- As for how this is to be done,—we must
tional Me- remember what has been just said,—that in
chanics.	Mechanics, we have nothing to do with the
origin or different nature of forces; and they are all one
while their mechanical operation is uniform. It is impos-
sible to conceive of any substance as devoid of weight, for
instance ; yet geometers have logically to treat of bodies
as without an inherent power of attraction. They treat of
this power as an external force; that is, it is to them
simply a force; and it does not matter to them whether it
is inherent or external,'—whether it is attraction or impul-
sion,—while it is the fall of the body that they have to
study. And so on, through the whole range of properties
of bodies. When we have so abstracted natural properties,
in our logical view, as to have before us an unmixed case
of the action of certain forces, and have ascertained
their laws,—then we can pass from abstract to concrete
Mechanics, and restore to bodies their natural active pro-THE THREE LAWS OF MOTION.
119
perties, and interpret their action bv what we have learned
of the laws of motion and equilibrium. This restoration is
so difficult to effect,—the transition from the abstract to the
concrete in Mechanics is so difficult,—that, while its theo-
retical domain is unbounded, its practical application is
singularly limited. In fact, the application of rational
mechanics is limited (accurately speaking) not only to
celestial phenomena, but to those of our own solar system.
One would suppose that the single property of weight was
manageable enough ; and that of a given form intelligible
enough: but there are such complications of physical cir-
cumstances,—as the resistance of media, friction, etc.,—
even if bodies are conceived of as in a fluid state, that their
mechanical phenomena cannot be estimated with any
accuracy. And when we proceed to electrical and chemical,
and especially to physiological phenomena, we are yet more
baffled. General gravitation affords us the only simple and
determinate law; and even there we are perplexed, when
we come to regard certain secondary actions. It may be
doubted whether questions of terrestrial mechanics will
ever admit—restricted as our means are—of a study at
once purely rational and precisely accordant with the
general laws of abstract mechanics,—though the knowledge
of these laws, primarily indispensible, may often lead us to
frequent and valuable indications and suggestions.
Bodies being supposed inert, the general
facts, or laws of motion to which they are motion s °
subject, are three ; all results of observation.
The first is that law discovered by Kepler, T f
which is inaptly called the law of inertia. ^ ° mer ia’
According to it, all motion is rectilinear and uniform ; that
is, any body impelled by a single force will move in a right
line, and with an invariable velocity. Instead of resorting
to the old ways of pronouncing or imagining why it must
be so, the Positive Philosophy instructs us to recognise the
simple fact that it is so ; that, through the whole range of
nature, bodies move in a right line, and with a uniform
velocity, when impelled by a single force.
The second law we owe to Newton. It is Law of equa-
that of the constant equality of action and lity of action
reaction; that is, whenever one body is moved an(^ reaction.120
POSITIVE PHILOSOPHY.
by another, the reaction is such that the second loses pre-
cisely as much motion, in proportion to its masses, as the
first gains. Whether the movement proceeds from impul-
sion or attraction, is, of course, of no consequence. Newton
treated this general fact as a matter of observation, and
most geometers have done the same; so that there has been
less fruitless search into the why with regard to this second
law than to the first.
Law of co-ex- The third fundamental law of motion
istence of involves the principle of the independence or
motions.	coexistence of motions, which leads immediately
to what is commonly called the composition of forces. Galileo
is, strictly speaking, the true discoverer of this law, though
he did not regard it precisely under the form in which it is
presented here:—that any motion common to all the
bodies of any system whatever does not affect the particular
motions of these bodies with regard to each other; which
motions proceed as if the system were motionless. Speak-
ing strictly, we must conceive that all the points of the
system describe at the same time parallel and equal straight
lines, and consider that this general motion, whatever may
be its velocity and direction will not affect relative motions.
No a priori considerations can enter here. There is no
seeing why the fact should be so, and therefore no antici-
pating that it would be so. On the contrary, when Galileo
stated this law, he was assailed by a host of objections that
his fact was logically impossible. Philosophers were ready
with plenty of a priori reasons that it could not be true :
and the fact was not unanimously admitted till men had
quitted the logical for the physical point of view. We now
find, however, that no proposition in the whole range of
natural philosophy is founded on observations so simple,
so various, so multiplied, so easy of verification. In
fact, the whole economy of the universe would be over-
thrown, from end to end, but for this law. A ship impelled
smoothly, without rolling and pitching, has everything
going on within it just the same as if it were at rest; and,
in the same way, but on the grandest scale, the great globe
itself rushes through space, without its motion at all affect-
ing the movements going on on its surface. As we all
know, it was ignorance of this third law of motion whichTHIRD LAW OF MOTION.
121
was the main obstacle to the establishment of the Coper-
nican theory. The Copernicans struggled to get rid of the
insurmountable objections to which their doctrine was
liable by vain metaphysical subtleties, till Galileo cleared
up the difficulty. Since his time, the movement of the
globe has been considered an all-sufficient confirmation of
the law. Laplace points out to us that if the motion of
the globe affected the movements on it, the effect could not
be uniform, but must vary with the diversities of their
direction, and of the angle that each direction would make
with that of the earth: whereas, we know how invariable
is, for instance, the oscillating movement of the pendulum,
whatever may be its direction in comparison with that of
the travelling globe.
It may be as well to point out that rotary motion does
not enter into this case at all, but only translation, because
the latter is the only motion which can be, in degree and
direction, absolutely common to all the parts of a system. In
a rotating system, for instance, all the parts are not at an
equal distance from the centre of rotation. When the
interior of a ship is affected, it is by the rolling and pitch-
ing, which are rotary movements. We may carry a watch
any distance without affecting its interior movements ; but
it will not bear whirling.
And, again, the forward motion of the globe could be
discovered by no other means than astronomical observation;
whereas, the changes which occur on the surface of the
earth, produced by the inequality of the centrifugal force
at its different points, are sufficient evidence of its rotation,
independently of all astronomical considerations whatever.
The law or rule of the composition of forces, which is
involved in the general fact just stated, is, in fact, identical
with it. It is only another way of expressing the same
law. If a single impulsion describes a parallelogram of
forces, as the scientific term is, the effect of a second will
be to describe the diagonal of the parallelogram. This is
nothing more than an application of the law of the inde-
pendence of forces; since the motion of any body along a
straight line is in no way disturbed by a general motion
which carries away, parallel with itself, the whole of this
right line along any other right line whatever. This con-122
POSITIVE PHILOSOPHY.
sideration leads immediately to the geometrical construction
expressed by the rule of the parallelogram of forces. And
thus it appears that this fundamental theorem of Rational
Mechanics is a true natural law; or, at least, a direct appli-
cation of one of the greatest natural laws. And this is the
best account to give of it, instead of looking to logic for a
fallacious a priori deduction of it. Any analytical demon-
stration, too, must suppose certain portions of the case to be
evident; and to talk of a thing being evident is to refer
back to nature, and to depend on observation of nature.
It is worthy of remark that those who wish to make a
separate law of the composition of forces, in order to avoid
introducing the third law into the prolegomena of Mechanics,
and to dispense with it in the exposition of Statics, are
brought back to it when entering upon the study of
Dynamics. Upon this alone can be based the important
law of the proportion of forces to velocities. The rela-
tions of forces may be determined either by a statical or
dynamical procedure. No purpose is answered by the
transposition of the general fact of the independence of
forces to the dynamical department of the science: it is
equally necessary for the statical; and a world of meta-
physical confusion is saved by laying it down as the broad
basis that in fact it is.
These three laws are the experimental basis of the
science of Mechanics. From them the mind may proceed
to the logical construction of the science, without further
reference to the external world. At least, so it appears to
me ; though I am far from assigning any a priori reasons
why more laws may not be hereafter discovered, if these
three should prove to be incomplete. There cannot, in the
nature of things, be many more ; and I would rather incur
the inconvenience of the introduction of one or two, than
run any risk of surrendering the positive character of the
science and overstraining its logical considerations. We
cannot however conceive of any case which is not met by
these three laws of Kepler, of Newton, and of Galileo; and
their expression is so precise, that they can be immediately
treated in the form of analytical equations easily obtained.
As for the most extensive, important, and difficult part of
the science, the mechanics of varied motion or continuousDIVISIONS OF RATIONAL MECHANICS.
123
forces, we can perceive the possibility of reducing it to
elementary Mechanics by the application of the infinitesimal
method. For each infinitely small point of time, we
must substitute a uniform motion in the place of a varied
one, whence will immediately result the differential equa-
tions relative to these varied motions. We may hereafter
see what results have been obtained in regard to the abstract
laws of equilibrium and motion. Meantime, we see that
the whole science is founded on the combination of the
three physical laws just established; and here lies the
distinct boundary between the physical and the logical
parts of the science.
As for its divisions, the first and most im- ^wo primary
portant is into Statics and Dynamics; that divisions,
is, into questions relating to equilibrium and Statics and
questions relating to motion. Statics are the Dynamics.
easiest to treat, because we abstract from them the element
of time, which must enter into Dynamical questions, and
complicate them. The whole of Statics corresponds to the
very small portion of Dynamics which relates to the theory
of uniform motions. This division corresponds well with
the facts of human education in this science. The fine
researches of Archimedes show us that the ancients, though
far from having obtained any complete system of rational
Statics, had acquired much essential knowledge of equili-
brium—both of solids and fluids—while as yet wholly
without the most rudimentary knowledge of Dynamics.
Galileo, in fact, created that department of the science.
The next division is that of Solids from Secondary
Fluids. This division is generally placed divisions,
first, but it is unquestionable that the laws	Solids and
of statics and dynamics must enter into the	Fluids.
study of solids and fluids, that of fluids requiring the
addition of one more consideration,—variability of form.
This however is a consideration which introduces the neces-
sity of treating separately the molecules of which fluids are
composed, and fluids as systems composed of an infinity of
distinct forces. A new order of researches is introduced
into Statics, relative to the form of the system in a state of
equilibrium ; but in Dynamics the questions are still more
difficult to deal with. The importance and difficulty of the124
POSITIVE PHILOSOPHY.
researches under this division cannot be exaggerated.
Tlieir complication places even the easiest cases beyond our
reach, except by the aid of extremely precarious hypotheses.
We must admit the vast necessary difficulty of hydrostatics,
and yet more of hydrodynamics, in comparison with statics
and dynamics, properly so called, which are in fact far more
advanced.
Much of the difficulty arises from the mathematical
statement of the question differing from the natural facts.
Mathematical fluids have no adhesion between their par-
ticles ; whereas natural fluids have, more or less ; and
many natural phenomena are due to this adherence, small
though it be in comparison with that of solids. Thus,
the result of an observation of the quantity of a given
fluid which will run out of a given orifice will differ widely
from the result of the mathematical calculation of what it
should be. Though the case of solids is easier, yet there
perplexity may be introduced by the disrupting action of
forces, of which abstraction must be made in the mathe-
matical question. The theory of the rupture of solids,
initiated by Galileo, Huyghens, and Leibnitz, is still in a
very imperfect and precarious state, great as are the pains
which have been taken with it, and much needed as it is.
Not so much needed however as the mechanics of fluids,
because it does not affect questions of celestial mechanics ;
and in this highest department alone can we, as I said
before, see the complete application of rational mechanics.
There is a gap left between these two studies, which
should be pointed out, though it is of secondary importance.
We want a Mechanics of semi-fluids, or semi-solids,—as of
sand, in relation to solids, and gelatinous conditions of
fluids. Some considerations have been offered with regard
to these “ imperfect fluids,” as they are called; but their
true theory has never been established in any direct and
general manner.
Such is the general view of the philosophical character
of Rational Mechanics. We must now take a philosophical
view of the composition of the science, in order to see how
this great second department of Concrete Mathematics has
attained the theoretical perfection in which it appears in
the works of Lagrange, who has rendered all its possibleMETHODS OF STATICS.
125
abstract questions capable of an analytical solution, like
those of geometry. We must first take a view of Statics,
and then proceed to Dynamics.
SECTION I.
STATICS.
There are two ways of treating Eational Converse
Mechanics, according as Statics are regarded methods of
directly, or as a particular case of Dynamics. treatment.
By the first method we have to discover a principle of
equilibrium so general as to be applicable to the conditions
of equilibrium of all systems of possible forces. By the
second method, we reverse the process,—ascertaining what
motion would result from the simultaneous action of any
proposed differing forces, and then determining what rela-
tions of these forces would render motion null.
The first method was the only one possible First method,
in the early days of science; for, as I have Statics by
said before, G-alileo was the creator of the itself,
science of Dynamics. Archimedes, the founder of Statics,
established the condition of equilibrium of two weights
suspended at the ends of a straight lever; that is, he
showed that the weights must be in an inverse ratio to
their distances from the fulcrum of the lever. He endea-
voured to refer to this principle the relations of equilibrium
proper to other systems of forces; but the principle of the
lever is not in itself general enough for such application.
The various devices by which it was attempted to extend
the process, and to supply the remaining deficiences, were
relinquished when the establishment of Dynamics permitted
the use of the second method,—of seeking geconci
the conditions of equilibrium through the method,
laws of the composition of forces. It is by Statics
this last method that Yarignon discovered through Dy-
the theory of the equilibrium of a system namics-
of forces applied upon a single point; and that D’Alembert
•afterwards established, for the first time, the equations of
equilibrium of any system of forces applied to the different126
POSITIVE PHILOSOPHY.
points of a solid body of an invariable form. At this day,
this is the method universally employed. At the first
glance, it does not appear the most rational,—Dynamics
being more complicated than Statics, and precedence being
natural to the simpler. It would, in fact, be more philoso-
phical to refer dynamics to statics, as has since been done;
but we may observe that it is only the most elementary
part of dynamics, the theory of uniform motions, that we
are concerned with in treating statics as a particular case
of dynamics. The complicated considerations of varied
motions do not enter into the process at all.
The easiest method of applying the theory of uniform
motions to statical questions is through the view that,
when forces are in equilibrio, each of them, taken singly,
may be regarded as destroying the effect of all the others
together. Thus, the thing to be done is to show that any
one of the forces of the system is equal, and directly
opposed, to the resulting force of all the rest. The only
difficulty here is in determining the resultant force; that
is, in mutually compounding the given forces. Here comes
in the aid of the third great law of motion, and having
compounded the two first forces, we can deduce the com-
position of any number of forces.
After having established the elementary laws of the
composition of force, geometers, before applying them to
the investigation of the conditions of equilibrium, usually
subject them to an important transformation, which, with-
out being indispensable, is of eminent utility, in an
analytical view, from the extreme simplification which it
introduces into the algebraical expression of the conditions
Moments	The transformation consists
in what is called the theory of Moments, the
essential property of which is to reduce, analytically, all
the laws of the composition of forces to simple additions
and subtractions. Without going into an examination of
this theory, it is necessary simply to say that it considers
statics as a particular case of elementary dynamics, and
that its value is in the simplicity which it gives to the
analytical part of the process of investigation into the con-
ditions of equilibrium. Simple, however, as may be the
operation, and great as may be the practical advantageLAGRANGE’S DEVELOPMENT OF RATIONAL MECHANICS. 127
gained through the treatment of statics as a particular
case of elementary Dynamics, it would be satisfactory to
return, if we could, to the method of the
ancients,—to leave Dynamics on one side, -n ^ethocf
and proceed directly to the investigation of
the laws of equilibrium regarded by itself, by means of a
direct general principle of equilibrium. Geometers strove
after this as soon as the general equations of equilibrium
were discovered by the dynamic method. But a higher
motive than even the desire to place statics in a more
philosophical position impelled them to establish a direct
Statical method: and this it was which caused Lagrange
to carry up the whole science of Rational Mechanics to the
philosophical perfection which it now enjoys.
D’Alembert made a discovery (to be treated of here-
after), by the help of which all investigation of the motion
of any body or system might be converted at once into a
question of equilibrium. This amounts, in fact, to a vast
generalization of the second fundamental law of motion;
and it has served for a century past as a permanent basis
for the solution of all great dynamical questions; and it
must be so applied more and more, from its high merits of
simplification in the most difficult investigations. Still, it
is clear that this method compels a return into statics; and
Statics as independent of Dynamics, which are altogether
derived from Statics. A science must be imperfectly laid
down, as long as it is necessary thus to pass backwards and
forwards between its two departments. In order to estab-
lish the necessary unity, and to provide scope for D’Alem-
bert’s principle, a complete reconstitution of Rational
Mechanics was indispensable. Lagrange effected this in
his admirable treatise on “ Analytical Mechanics,” the lead-
ing conception of which must be the basis of all future
labours of geometers upon the laws of equilibrium and
motion, as we have seen that the great idea of Descartes is
with regard to geometrical speculations.
The principle of Virtual Velocities,—the y|rtuaj vej0
one which Lagrange selected from among the cities.
properties of equilibrium,—had been dis-
covered by Galileo in the case of two forces, as a general
property manifested by the equilibrium of all machines.128
POSITIVE PHILOSOPHY.
John Bernouilli extended it to any number of forces, com-
posing any system. Yarignon afterwards expressly pointed
out the universal use that might be made of it in Statics.
The combination of it with D’Alembert’s principle led
Lagrange to conceive of the whole of Rational Mechanics
as deduced from a single fundamental theorem, and to give
it that rigorous unity which is the highest philosophical
perfection of a science.
The clearest idea of the system of virtual velocities may
be obtained by considering the simple case of two forces,
which was that presented by Galileo. We suppose two
forces balancing each other by the aid of any instrument
wdiatever. If we suppose that the system should assume
an infinitely small motion, the forces are, with regard to
each other, in an inverse ratio to the spaces traversed by
their points of application in the path of their directions.
These spaces are called virtual velocities, in distinction
from the real velocities which would take place if the
equilibrium did not exist. In this primitive state, the
principle, easily verified with regard to all known machines,
offers great practical utility; for it permits us to obtain
with ease the mathematical condition of equilibrium of any
machine whatever, whether its constitution is known or not.
If we give the name of virtual momentum (or simply of
momentum in its primitive sense) to the product of each
force by its virtual velocity,—a product which in fact then
measures the effort of the force to move the machine,—we
may greatly simplify the statement of the principle in
merely saying that, in this case, the momentum of the two
forces must be equal and of opposite signs, that there may
be equilibrium, and that the positive or negative sign of
each momentum is determined according to that of the
virtual velocity, which will be considered positive or nega-
tive according as, by the supposed motion, the projection
of the point of application would be found to fall upon the
direction of the force or upon its prolongation. This abridged
expression of the principle of virtual velocities is especially
useful for the statement of this principle in a general
manner, with regard to any system of forces whatever. It
is simply this: that the algebraic sum of the virtual
moments of all forces, estimated according to the preced-VIRTUAL VELOCITIES.
129
ing rule, must be null to cause equilibrium : and this con-
dition must exist distinctly with regard to all the elemen-
tary motions which the system might assume in virtue of
the forces by which it is animated. In the equation, con-
taining this principle, furnished by Lagrange, the whole
of Rational Mechanics may be considered to be implicitly
comprehended.
While the theorem of virtual velocities was conceived of
only as a general property of equilibrium, it could be veri-
fied by observing its constant conformity with the ordinary
laws of equilibrium, otherwise obtained, of which it was a
summary, useful by its simplicity and uniformity. But, if
it was to be a fundamental principle, a basis of the whole
science, it must be underived, or at least capable of being
presented in its preliminary propositions as a matter of
observation. This was done by Lagrange, by his ingenious
demonstration through a system of pulleys. He exhibited
the theorem of virtual velocities very easily by imagining a
single weight which, by means of pulleys suitably con-
structed, replaces simultaneously all the forces of the
system. Many other demonstrations have been furnished;
but, while more complicated, they are not logically superior.
From the philosophical point of view it is clear that this
general theorem, being a necessary consequence of the
fundamental laws of motion, can be deduced in various
ways, and becomes practically the point of departure of the
whole of Rational Mechanics. A perfect unity having been
established by this principle, we need not look for any
others ; and we may rest assured that Lagrange has carried
the co-ordination of the science as far as it can go. The
only possible object would be to simplify the analytical
researches to which the science is now reduced ; and nothing
can be conceived more admirable for this purpose than
Lagrange’s adaptation of the principle of virtual velocities
to the uniform application of mathematical analysis.
Striking as is the philosophical eminence of this prin-
ciple, there are difficulties enough in its use to prevent its
being considered elementary, so far as to preclude the con-
sideration of any other in a course of dogmatic teaching.
It is for this reason that I have referred to the dynamic
method, properly so called, which is the only one in general
I.	K130
POSITIVE PHILOSOPHY.
use at present. All other considerations must however be
only provisional. Lagrange’s method is at present too new;
but it is impossible that it should for ever remain in the
hands of a small number of geometers, who alone shall be
able to make use of its admirable properties. It must
become as popular in the mathematical world as the great
geometrical conception of Descartes : and this general pro-
gress would be almost accomplished if the fundamental
ideas of transcendental analysis were as widely spread as
they ought to be.
Theor of	The ^rea^es^ acquisition, since the regene-
Couples°	ration of the science by Lagrange, is the
conception of M. Poinsot,—the theory of
Couples, which appears to me to be far from being sufficiently
valued by the greater number of geometers. These Couples,
or systems of parallel forces, equal and contrary, had been
merely remarked before the time of M. Poinsot, as a sort
of paradox in Statics. He seized upon this idea, and made
it the subject of an extended and original theory relating to
the transformation, composition, and use of these singular
groups, which he has shown to be endowed with properties
remarkable for their generality and simplicity. He used
the dynamic method in his study of the conditions of
equilibrium: but he presented it, by the aid of his theory
of couples, in a new and simplified aspect. But his con-
ception will do more for dynamics than for statics ; and it
has hardly yet entered upon its chief office. Its value will
be appreciated when it is found to render the notion of the
movements of rotation as natural, as familiar, and almost
as simple as that of forward movement or translation.
Share of equa- One more consideration should, I think, be
tions in pro- adverted to before we quit the subject of
during equili- statics as a whole. When we study the nature
hr mm.	0f the equations which express the conditions
of equilibrium of any system of forces, it seems to me not
enough to establish that the sum of these equations is in-
dispensable for equilibrium. I think the further statement
is necessary,—in what degree each contributes to the result.
It is clear that each equation must destroy some one of the
possible motions that the body would make in virtue of
existing forces ; so that the whole of the equations mustEQUATIONS OF EQUILIBRIUM.
131
produce equilibrium by leaving an impossibility for the
body to move in any way whatever. Now the natural
state of things is for movement to consist of rotation and
translation. Either of these may exist without the other •
but the cases are so extremely rare of their being found
apart, that the verification of either is regarded by geo-
meters as the strongest presumption of the existence of the
other. Thus, when the rotation of the sun upon its axis
was established, every geometer concluded that it had also
a progressive motion, carrying all its planets with it, before
astronomers had produced any evidence that such was
actually the case. In the same way we conclude that
certain planets, travelling in their orbits, rotate round
their axes, though the fact has not yet been verified.
Some equations must therefore tend to destroy all progres-
sive motion, and others all motion of rotation. How many
equations of each kind must there be P
It is clear that, to keep a body motionless, it must be
hindered from moving according to three axes in different
planes—commonly supposed to be perpendicular to each
other. If a body cannot move from north to south, nor the
reverse; nor from east to west, nor the reverse; nor up,
nor down, it is clear that it cannot move at all. Movement
in any intermediate direction might be conceived of as
partial progression in one of these, and is therefore impos-
sible. On the other hand, we cannot reckon fewer than
three independent elementary motions ; for the body might
move in the direction of one of the axes, without having
any translation in the direction of either of the others.
Thus we see that, in a general way, three equations are
necessary, and three are sufficient to establish the absence
of translation ; each being specially adapted to destroy one
of the three progressive motions of which the body is
capable. The same view presents itself with regard to the
other motion,—of rotation. The mechanical conception
is more complicated; but it is true, as in the simpler case,
that motion is possible in only three directions,—in three
co-ordinated planes, or round three axes. Three equations
are necessary and sufficient here also; and thus we have
six which are indispensable and sufficient to stop all motion
whatever.132
POSITIVE PHILOSOPHY.
When, instead of supposing any system of forces what-
ever as the subject of the question, we particularize any,
we get rid of more or fewer possible movements. Having
excluded these, we may exclude also their corresponding
equations, retaining only those which relate to the possible
motions that remain. Thus, instead of having to deal with
six equations necessary to equilibrium, there may be only
three, or two, or even one, which it will be easy enough to
obtain in each case. These remarks may be extended to
any restrictions upon motion, whether resulting from the
special constitution of the system of forces, or from any
other kind of control, affecting the body under notice. If,
for instance, the body were fastened to a point, so that it
could freely rotate but not advance, three equations would
suffice: and again, if it is fastened to two fixed points, two
equations are enough; and even one, if these two fixed
points are so placed as to prevent the body from moving on
the axis between them. Finally, its being attached to three
fixed points, not in a right line, will prevent its moving at
all, and establish equilibrium without any condition, what-
ever may be the forces of the system. The spirit of this
analysis is entirely independent of any method by which
the equations of equilibrium will have been obtained: but
the different general methods are far from being equally
suitable to ihe application of this rule. The one which is
best adapted to it is, undoubtedly, the Statical one, pro-
perly so called, founded, as has been shown, on the prin-
ciple of virtual velocities. In fact, one of the characteristic
properties of this principle is the perfect precision with
which it analyses the phenomena of equilibrium, by dis-
tinctly considering each of the elementary motions per-
mitted by the forces of the system, and furnishing im-
mediately an equation of equilibrium specially relating to
this motion.
Connection of When we come to the inquiry how
the concrete geometers apply the principles of abstract
with the ah- Mechanics to the properties of real bodies, we
stract question. must state that the only complete application
yet accomplished is in the question of terrestrial gravity.
Now, this is a subject which cannot, logically, be treated
under the head of Mechanics, as it belongs to Physics. ItCONNECTION OF CONCRETE AN1) ABSTRACT QUESTION. 133
is sufficient to explain that the statical study of terrestrial
gravity becomes convertible into that of centres of gravity ;
and that all confusion between the two departments of
research would be avoided if we accustomed ourselves to
class the theory of centres of gravity among the questions
of pure geometry. In seeking the centre of gravity as
(according to the logical denomination of the ancient
geometers) the centre of mean distances, we remove all
traces of the mechanical origin of the question, and convert
it into this problem of general geometry:—Given, any
system of points disposed in a determinate way with regard
to each other, to find a point whose distance to any plane
shall be a mean between the distances of all the given
points to the same plane.—The abstraction of all considera-
tion of gravity is an assistance in every way. The simple
geometrical idea is precisely what we want in most of the
principal theories of Rational Mechanics, and especially
when we contemplate the great dynamic properties of the
centre of mean distances; in which study the idea of
gravity becomes a mere encumbrance and perplexity. It
is true that, by proceeding thus, we exclude the question
from the domain of Mechanics, to place it in that of
Geometry. I should have so classed it but for an un-
willingness to break in upon established customs. How-
ever it may be as to the matter of arrangement, it is highly
important for us not to misapprehend the true nature of
the question.—The integral calculus offers the means of
surmounting those difficulties in determining the centre of
gravity which are imposed bv the conditions of the ques-
tion. But, the integrations in this case being more com-
plicated than those to which they are analogous,—those of
quadratures and cubatures,—their precise solution is, owing
to the extreme imperfection of the integral Calculus, much
more rarely obtained. It is a matter of high importance,
however, to be able to introduce the consideration of the
centre of gravity into general theories of analytical
mechanics.
Such is, then, the relation of terrestrial gravitation to
the science of abstract Statics. As for universal gravita-
tion, no complete study has yet been made of it, except in
regard to spherical bodies. What we know of the law of134
POSITIVE PHILOSOPHY.
gravitation would easily enable us to compute the mutual
attraction of all known bodies, if the conditions of each
body were understood by us; but this is not the case. For
instance, we know nothing of the law of density in the
interior of the heavenly bodies. It is still true that the
primitive theorems of Newton on the attraction of spherical
bodies are the most useful part of our knowledge in this
direction.
Gravity is the only natural force that we are practically
concerned with in Rational Statics : and we see, by this,
how backward this science is in regard to universal gravi-
tation. As for the exterior general circumstances, such as
friction, resistance of media, and the like, which are alto-
gether excluded in the establishment of the rational laws
of Mechanics, we can only say that we are absolutely igno-
rant of the way to introduce them into the fundamental
relations afforded by analytical Mechanics, because we have
nothing to rely on, in working them, but precarious and
inaccurate hypotheses, unfit for scientific use.
As for the theory of equilibrium in regard
<>f*fluids1Um	bodies,—the application which it
remains for us to notice,—those bodies must
be regarded as either liquid or gaseous.
-PP t ,	...	Hydrostatics may be treated in two ways.
} ros a ics.	may seek the laws of the equilibrium of
fluids, according to statical considerations proper to that
class of bodies: or we may look for them among the laws
which relate to solids, allowing for the new characteristics
resulting from fluidity.
The first method, being the easiest, was in early times
the only one employed. Till a rather recent time, all
geometers employed themselves in proposing statical prin-
ciples peculiar to fluids; and especially with regard to the
grand question of the figure of the earth, on the supposi-
tion that it was once fluid. Huyghens first endeavoured
to resolve it, taking for his principle of equilibrium the
necessary perpendicularity of weight at the free surface of
the fluid. Newton’s principle was the necessary equality
of weight between the two fluid columns going from the
centre,—the one to the pole, the other to some point of the
equator. Bouguer showed that both methods were bad,EQUILIBRIUM OF FLUIDS.
135
because, though each was incontestable, the two failed, in
many cases, to give the same form to the fluid mass in
equilibrium. But he, in his turn, was wrong in believing
that the union of the two principles, when they agreed in
indicating the same form, was sufficient for equilibrium.
It was Clairaut who, in his treatise on the form of the
earth, first discovered the true laws of the case, setting out
from the evident consideration of the isolated equilibrium
of any infinitely small canal; and, tried by this criterion,
he showed that the combination required by Bouguer might
take place without equilibrium happening. Several great
geometers, proceeding on Clairaut’s foundation, have carried
on the theory of the equilibrium of fluids a great way.
Maclaurin was one of those to whom we owe much ; but it
was Euler who brought up the subject to its present point,
by founding the theory on the principle of equal pressure
in all directions. Observation of the statical constitution
of fluids indicates this as a general law; and it furnishes
the requisite equations with extreme facility.
It was inevitable that the mathematical theory of the
equilibrium of fluids should, in the first place, be founded,
as we have seen that it was, on statical principles peculiar
to this kind of bodies : for, in early days, the	T .
characteristic differences between solids and	^
fluids must have appeared too great for any geometer to
think of applying to the one the general principles appro-
priated to the other. But, when the funda-	Gases
mental laws of hydrostatics were at length
obtained, and men’s minds were at leisure to estimate the
real diversity between the theories of fluids and of solids,
they could not but endeavour to attach them to the same
general principles, and perceive the necessary applicability
of the fundamental rules of Statics to the equilibrium of
fluids, making allowance for the attendant variability of
form. But, before hydrostatics could be comprehended
under Statics, it was necessary that the abstract theory of
equilibrium should be made so general as to apply directly
to fluids as well as solids. This was accomplished when
Lagrange supplied, as the basis of the whole of Bational
Mechanics, the single principle of Virtual Velocities. One
of its most valuable properties is its being as directly186
POSITIVE PHILOSOPHY.
applicable to fluids as to solids. From that time, Hydro-
statics, ceasing to be a natural branch of science, has taken
its place as a secondary division of Statics. This arrange-
ment has not yet been familiarly admitted; but it must
soon become so.
To see how the principle of Virtual Velocities may lead
to the fundamental equations of the equilibrium of fluids,
we have to consider that all that such an application
requires, is to introduce amoDg the forces of the system
under notice one new force,—the pressure exerted upon
each molecule, which will introduce one term more into the
general equation. Proceeding thus, the three general
equations of the equilibrium of fluids, employed when
hydrostatics was treated as a separate branch, will be
immediately reached. If the fluid be a liquid, we must
have regard to the condition of incompressibility,—of
change of form without change of volume. If the fluid be
gaseous, we must substitute for the incompressibility that
condition which subjects the volume of the fluid to vary
according to a determinate function of the pressure; for
instance, in the inverse ratio of the pressure, according to
the physical law on which Mariotte has founded the whole
Mechanics of the gases. We know but too little yet of
these gaseous conditions ; for Mariotte’s law can at present
be regarded only as an approximation,—sufficiently exact
for average circumstances, but not to be rigorously applied
in any case whatever.
Some confirmation of the philosophical character of this
method of treating hydrostatics arises from its enabling us
to pass, almost insensibly, from the order of bodies of
invariable form to that of the most variable of all, through
intermediate classes,—as flexible and elastic bodies,—
whereby we obtain, in an analytical view, a natural filiation
of subjects.
We have seen how the department of Statics has been
raised to that high degree of speculative perfection which
transforms its questions into simple problems of Mathema-
tical Analysis. We must now take a similar review of the
other department of general Mechanics,—that more ex-
tended and more complicated study which relates to the
laws of Motion.THEORY OF RECTILINEAR MOTION.
137
SECTION II.
DYNAMICS.
Object.
The object of Dynamics is the study of the
varied motions produced by continuous forces.
The Dynamics of varied motions or continuous forces
includes two departments,—the motion of a point, and that
of a body. From the positive point of view, this means
that, in certain cases, all the parts of the body in question
have the same motion, so that the determination of one
particle serves for the whole ; while in the more general
case, each particle of the body, or each body of the system,
assuming a distinct motion, it is necessary to examine
these different effects, and the action upon them of the re-
lations belonging to the system under notice. The second
theory being more complicated than the first, the first is the
one to begin with, even if both are deduced from the same
principles.
With regard to the motion of a point, the question is to
determine the circumstances of the compound curvilinear
motion, resulting from the simultaneous action of different
continuous forces, it being known what would be the recti-
linear motion of the body if influenced by any one of these
forces. Like every other, this problem admits of a converse
solution.
But here intervenes a preliminary theory, T] .
which must be noticed before either of the ^linear motion,
two departments can be entered upon. This
theory is popularly called the theory of rectilinear motion,
produced by a single continuous force acting indefinitely in
the same direction. It may be asked why we want this,
after having said that the effect of each separate force
is supposed to be known, and the effect of their union the
thing to be sought. The answer to this is, that the varied
motion produced by each continuous force may be defined
in several ways, which depend on each other, and which
could never be given simultaneously, though each may be
separately the most suitable ; whence results the necessity
of being able to pass from any one of them to all the rest.
The preliminary theory of varied motion relates to these138
POSITIVE PHILOSOPHY.
transformations, and is therefore inaptly termed the study of
the aetion of a single force. These different equivalent defini-
tions of the same varied motions result from the simultaneous
consideration of the three distinct but co-related functions
which are presented by it,—space, velocity, and force, con-
ceived as dependent on time elapsed. Taking the most
extended view, we may say that the definition of a varied
motion may be given by any equation containing at once
these four variables, of which only one is independent,—
time, space, velocity, and force. The problem will consist
in deducing from this equation the distinct determination
of the three characteristic laws relating to space, velocity,
and force, as a function of time, and, consequently, in
mutual correlation. This general problem is always re-
ducible to a purely analytical research, by the help of the
two dynamical formulas which express, as a function of
time, velocity and force, when the law of space is supposed
to be known. The infinitesimal method leads to these
formulas with the utmost ease, the motion being considered
uniform during an infinitely small interval of time, and as
uniformly accelerated during two consecutive intervals.
Thence the velocity, supposed to be constant at the instant,
according to the first consideration, will be naturally
expressed by the differential of the space, divided by that
of the time; and, in the same way, the continuous force,
according to the second consideration, will evidently be
measured by the relation between the infinitely small incre-
ment of the velocity, and the time employed in producing
this increment.
Lagrange’s conception of transcendental analysis ex-
cluding him from this use of the infinitesimal method for
the establisment of the two foregoing dynamic formulas,
he was led to present this theory under another point
of view, more important than seems to be generally sup-
posed. In his Theory of Analytical Functions, he has
shown that this dynamic consideration really consists in
conceiving any varied motion as compounded, each moment,
of a certain uniform motion and another motion uniformly
varied,—likening it to the vertical motion of a heavy body
under a first impulsion. Lagrange has not given its due
advantage to this conception, by developing it as he mightTHE TWO CASES OF RATIONAL DYNAMICS. 139
have done. In fact, it supplies a complete theory of the
assimilation of motions, exactly like the theory of the con-
tacts of curves and surfaces, in the department of geometry.
Like that theory, it removes the limits within which we sup-
posed ourselves to be confined, by disclosing to us, in an
abstract way, a much more perfect measure of all varied
motion than we obtain by the ordinary theory, though
reasons of convenience compel us to abide by the method
originally adopted.
The first case or department of rational ,,	,
■j •	,1	,	£ x , • n . ,	Motion of a
dynamics,—that ot the motion or a point, or point
of a body which has all its points or portions
affected by the same force,—relates to the study of the
curvilinear motion produced by the simultaneous action of
any different continuous forces. This case divides itself
again into two,—according as the mobile point is free, or
as it is compelled to move in a single curve, or on a given
surface. The fundamental theory of curvilinear motion
may be established in either case, in a different way; each
being susceptible of direct treatment, and of being con-
nected with the other. In the first case, in order to deduce
the second, we have only to regard the active or passive re-
sistance of the prescribed curve or surface as a new force to
be added to the others proposed. In the other way, we
have only to consider the moving point as compelled to
describe the curve which it must traverse; and this is
enough to afford the fundamental equations, though this
curve may then be primitively unknown.
The other, more real and more difficult
case, is that or the motion of a system of system
bodies in any way connected, whose proper
motions are altered by the conditions of their connection.
There is a new elementary conception about the measure-
ment of forces which some geometers declare to be logically
deducible from antecedent considerations, and to which
they would assign the place and title of a fourth law of
motion. For the sake of convenience we may make it into
a fourth law of motion; but such is not its philosophical
character. The idea is, that forces which impress the
same velocity on different masses are to each other
exactly as those masses; or, in other words, that the140
POSITIVE PHILOSOPHY.
forces are proportional to the masses, as we have seen them,
under the third law of motion, to be proportional to the
velocities. All phenomena, such as the communication of
motion by collision, or in any other way, have tended
to confirm the supposition of this new kind of proportion.
It evidently results from this, that when we have to com-
pare forces which impress different velocities on unequal
masses, each must be measured according to the product of
the mass upon which it acts by the corresponding velocity.
This product is called by geometers quantity of motion;
and it determines the percussion of a body, and also
the pressure that a body may exercise against any fixed
obstacle to its motion.
Proceeding to the second dynamical case, we see that
the characteristic difficulty of this order of questions con-
sists in the way of estimating the connection of the
different bodies of the system, in virtue of -which their
mutual reactions will necessarily affect the motions which
each would take if alone; and we can have no a priori
knowledge of what the alterations will be. In the case of
the pendulum, for instance, the particles nearest the point
of suspension, and those furthest from it, must react on
each other by their connection,—the one moving faster
and the other slower than if they had been free; and no
established dynamic principle exists revealing the law
which determines these reactions. Geometers naturally
began by laying down a principle for each particular case ;
and many were the principles thus offered, which turned
out to be only remarkable theorems furnished simul-
taneously by fundamental dynamic equations. Lagrange
has given us, in his “ Analytical Mechanics,” the general
history of this series of labours: and very interesting it is,
as a study of the progressive march of the human intellect.
This method of proceeding continued till
principle61 S the	D’Alembert, who put an end to
all these isolated researches by seeing how to
compute the reactions of the bodies of a system in virtue
of their connection, and establishing the fundamental
equations of the motion of any system. By the aid of the
great principle which bears his name, he made questions
of motion merge in simple questions of equilibrium. TheD’ALEMBERT’S PRINCIPLE.
141
principle is simply this. In the case supposed, the natural
motion clearly divides itself into two,—the one which subsists
and the one which has been destroyed. By D’Alembert’s
view, all these last, or, in other words, all the motions that
have been lost or gained by the different bodies of the system
by their reaction, necessarily balance each other, under the
conditions of the connection which characterizes the pro-
posed system. James Bernouilli saw this with regard to
the particular case of the pendulum; and he was led by it
to form an equation adapted to determine the centre of
oscillation of the most simple system of weight. But he
extended the resource no further ; and what he did detracts
nothing from the credit of D’Alembert’s conception, the
excellence of which consists in its entire generality.
In D’Alembert’s hands the principle seemed to have a
purely logical character. But its germ may be recognized
in the second law of motion, established by Newton, under
the name of the equality of reaction and action. They are,
in fact, the same, with regard to two bodies only acting
upon each other in the line which connects them. The one
is the greatest possible generalization of the other; and
this way of regarding it brings out its true nature, by
giving it the physical character which D’Alembert did not
impress upon it. Henceforth therefore we recognize in it
the second law of motion, extended to any number of
bodies connected in any manner.
We see how every dynamical question is thus convertible
into one of Statics, by forming, in each case, equations of
equilibrium between the destroyed motions. But then
comes the difficulty of making out what the destroyed
motions are. In endeavouring to get rid of the embarrass-
ing consideration of the quantities of motion lost or gained,
Euler, above others, has supplied us with the method
most suitable for use,—that of attributing to each body a
quantity of motion equal and contrary to that which it
exhibits, it being evident that if such equal ana contrary
motion could be imposed upon it, equilibrium would be
the result. This method contemplates only the primitive
and the actual motions which are the true elements of the
dynamic problem,—the given and the unknown ; and it is
under this method that D’Alembert’s principle is habitually142
POSITIVE PHILOSOPHY.
conceived of. Questions of motion being thus reduced to
questions of equilibrium, the next step is to combine
D’Alembert’s principle with that of virtual velocities.
This is the combination proposed by Lagrange, and de-
veloped in his “Analytical Mechanics,” which has carried
up the science of abstract Mechanics to the highest degree
of logical perfection,—that is, to a rigorous unity. All
questions that it can comprehend are brought under a>
single principle, through which the solution of any problem
whatever offers only analytical difficulties.
D’Alembert immediately applied his principle to the
case of fluids—liquid and gaseous, which evidently admit
of its use as well as solids, their peculiar conditions being
considered. The result was our obtaining general equa-
tions of the motion of fluids, wholly unknown before. The
principal of virtual velocities rendered this perfectly easy,
and again left nothing to be desired, in regard to concrete
considerations, and presented none but analytical diffi-
culties. We must admit however that our actual know-
ledge obtained under this theory is extremely imperfect,,
owing to insurmountable difficulties in the integrations
required. If it was so in questions of pure Statics, much
more must it be so in the more complex dynamical ques-
tions. The problem of the flow of a gravitating liquid
through a given orifice, simple as it appears, has never yet
been resolved. To simplify as far as they could, geometers
have had recourse to Daniel Bernouilli’s hypothesis of the
parallelism of sections, which admits of our considering
motion in regard to horizontal laminae instead of particle
by particle. But this method of considering each horizontal
lamina of a liquid as moving altogether, and taking the
place of the following, is evidently contrary to the fact in
almost all cases. The lateral motions are wholly abstracted,
and their sensible existence imposes on us the necessity of
studying the motion of each particle. We must then con-
sider the science of hydrodynamics as being still in its
infancy, even with regard to liquids, and much more with
regard to gases. Yet, as the fundamental equations of the
motions of fluids are irreversibly established, it is clear that
what remains to be accomplished is in the direction of
mathematical analysis alone.RESULTS OF RATIONAL MECHANICS.
Results.
Such is the Method of Rational Mechanics.
As for the great theoretical results of the
science,—the principal general properties of equilibrium
and motion thus far discovered,—they were at first taken
for real principles, each being destined to furnish the
solution of a certain order of new problems in Mechanics.
As the systematic character of the science has come out
however, these supposed principles have
shown themselves to be mere theorems,—	theorems
necessary results of the fundamental theories
of abstract Statics and Dynamics.
Of these theorems, two belong to Statics. T f
The most remarkable is that discovered by ^awoIrePose*
Torricelli with regard to the equilibrium of heavy bodies.
It consists in this; that when any system of heavy bodies
is in a situation of equilibrium, its centre of gravity is
necessarily placed at the lowest or highest possible point,
in comparison with all the positions it might take under
any other situation of the system.—Maupertuis afterwards,
by his working out of his Law of Repose, gave a large
generalization to this theorem of Torricelli’s, which at once
became a mere particular case under that law ; Torricelli’s
applying merely to cases of terrestrial gravitation; while
that of Maupertuis extends throughout the whole sphere
of the great natural attractive forces.
The other general property relating to Stability and
equilibrium may be regarded as a necessary instability of
complement of the former. It consists in equilibrium,
the fundamental distinction between the cases of stability
and instability of equilibrium. There being no such thing
in nature as abstract repose, the term is applied here to
that state of stable equilibrium which exists where the
centre of gravity is placed as low as possible ; while un-
stable equilibrium is that which is popularly called equi-
librium ; and it exists when the centre of gravity is placed
as high as possible. Maupertuis’s theorem consisted in
this,—that the situation of equilibrium of any system is
always that in which the sum of vires vivce (active forces)
is a maximum or a minimum ; and the one under notice,
developed by Lagrange, consists in this,—that in any
system equilibrium is stable or unstable according as144
POSITIVE PHILOSOPHY.
the sum of vires vivce is a maximum or a minimum.
Observation teaches the facts in the most simple cases;
but it requires a large theory to exhibit to geometers that
the distinction is equally applicable to the most compound
systems.
Proceeding to the theorems relative to dynamics, the
most direct way of establishing them is that used by
Lagrange,—exhibiting them as immediate consequences
of the general equation of dynamics, deduced
theorems*1 from the combination of D’Alembert’s prin-
ciple with the principle of virtual velocities.
The first theorem is that of the conservation of the motion
of the centre of gravity, discovered by Newton. Newton
showed that the mutual action of the bodies of any system,
Conservation whether of attraction, impulsion, or any
of the motion other,—regard being had to the constant
of the centre equality between action and reaction,—can-
of gravity. not in any way affect the state of the centre
of gravity; so that if there were no accelerating forces
besides, and if the exterior forces of the system were
reduced to instantaneous forces, the centre of gravity
would remain immovable, or would move uniformly in a
right line. D’Alembert generalized this property, and ex-
hibited it in such a form that every case in which the
motion of the centre of gravity has to be considered may
be treated as that of a single molecule. It is seldom that
we form an idea af the entire theoretical generality of such
great results as those of rational Mechanics. We think of
them as relating to inorganic bodies, or as otherwise cir-
cumscribed ; but we cannot too carefully remember that
they apply to all phenomena whatever; and in virtue of
this universality alone are the basis of all real science.
The second general theorem of dynamics is
areas^ & °	principle of areas, the first perception of
which is attributable to Kepler. In its sim-
plest form it is this ; that if the accelerating force of any
molecule tends constantly towards a fixed point, the vector
radius of the moving body describes equal areas in equal
times round the fixed point; so that the area described at
the end of any time increases in proportion to the time:
and the reciprocal fact is clear,—that the evidence of theDYNAMICAL THEOREMS.
145
areas and the times proves the action upon the body of a
force directed towards the fixed point. This discovery of
Kepler’s is the more remarkable for having been made
before dynamics had been really created by Galileo. Its
importance in astronomy we shall see hereafter. But
though, in its simplest form, it is one of the bases of celes-
tial Mechanics, it is, in fact, only the simplest particular
case of the great general theorem of areas, exhibited in the
middle of the last century by D’Arcy, Daniel Bernouilli,
and Euler. Kepler’s discovery related only to the motion
of a point, while the later one refers to the motion of any
system of bodies, acting on each other in any manner what-
ever ; which constitutes a case, not only more complex, but
different, on account of the mutual actions involved. It
yields proof, however, that though the area described by
the vector radius of each molecule may be altered by recip-
rocal actions, the sum of the areas described will remain
invariable in a given time, and will increase therefore in
proportion to the time. As the theorem of the centre of
gravity determines all that relates to motions of transla-
tion, this determines all that relates to motions of rotation:
and the two together are sufficient for the complete study
of the motion of any system of bodies, in either direction.
And here comes in the facility afforded by M. Poinsot’s
conception—referred to under the head of Statics. By
substituting for the areas or momentum of the geometers,
the couples engendered by the proposed forces, a philoso-
phical completeness is given to the theory, and a concrete
value, and proper dynamic direction, to what was before a
simple geometrical expression of a part of the fundamental
equations of motion.
Laplace elicited from the theory of areas
that dynamic property which he called the piane*™8^ 6
invariable plane, the consideration of which
is highly important in celestial mechanics. It is in the
study of astronomy that the importance fully appears of
the determination of a plane, whose direction is unaffected
by the mutual action of different bodies in our own solar
system; for we thus obtain a point of reference, a neces-
sarily fixed term of comparison, by which to estimate the
variations of the heavenly bodies. We are far from having
I.	L146
POSITIVE PHILOSOPHY.
Moment of
inertia.
Principal axes.
yet attained precision in the determination of the situation
of this plane ; hut this does not impair the character of the
theorem in its relation to rational Mechanics. Again, we
are indebted to Poinsot, who, by simplifying, has once
more extended the process to which his method is applied:
and he has repaired an important omission made by Laplace,
in taking into the account the smaller areas described by
satellites, and their rotation, and that! of the sun itself;
whereas Laplace has attended only to the larger areas de-
scribed by the planets in their course round the sun.
Finally, there are Euler’s theorems of the
moment of inertia, and the principal axes,
which are among the most important general
results of rational Mechanics. By means of
these we are able to arrive at a complete
analysis of the motion of rotation.—By means of all the
theorems just touched upon, we are put in the direct way
to determine the entire motion of any body, or system of
bodies whatever.—Besides them, geometers have discovered
some which are less general, but, though by no means in-
dispensable, yet very important from the simplification
they introduce into special researches. Students will re-
cognize their functions, when their mere names are pre-
sented ; which is all that our space allows :—I refer to the
theorem of the conservation of active forces,—singularly
important in its applications to industrial Mechanics: the
theorem, improperly called the principle of the least action,
as old as Ptolemy, who observed that reflected light takes
the shortest way from one point to another,—an observa-
tion which was the basis of Maupertuis’ discovery of this
property: and lastly, a theorem not usually classed with
the foregoing, yet worthy of no less esteem,—the theorem
of the co-existence of small oscillations, of Daniel Bernouilli.
This discovery is as important in its physical as its logical
bearings; and it explains a multitude of facts which,
clearly known, could not be referred to their principles. It
consists in showing that the infinitely small oscillations
caused by the return of any system of forces to a state of
stable equilibrium coexist without interference, and can be
treated separately.
This reference to the principal general theorems hithertoCONCLUSION.
147
discovered in Rational Mechanics concludes our review of
the second branch of Concrete Mathematics.
As for our review of the whole science, I Conclusion
wish I could better have communicated my
own profound sense of the nature of this immense and
admirable science, which, the necessary basis of the whole
of Positive Philosophy, constitutes the most unquestionable
proof of the compass of the human intellect. But I hope
that those who have not the misfortune to be wholly igno-
rant of this fundamental science may, according to the
process of thought which I have indicated, attain some
clear idea of its philosophical character.
To preserve complete the philosophical arrangement of
Mathematics in its present state, I ought to consider here
a third branch of Concrete Mathematics ;—the application
of analysis to therm ological phenomena, according to the
discoveries of Fourier. But, to avoid too great a breach of
customary arrangement, I have reserved the subject, and
shall place Thermology among the departments of Physics.
Mathematical philosophy being now completely character-
ized, we shall proceed to examine its application to the study
of Natural Phenomena, in their various orders, ranked ac-
cording to their degree of simplicity. By this character
alone can they cast light back again upon the science which
explains them ; and under this character alone can they be
suitably estimated. According to the natural order laid
down at the beginning, we now proceed to that class of
phenomena with which Mathematics is most concerned,—
the phenomena of Astronomy.148
BOOK II.
ASTRONOMY.
CHAPTER I.
GENERAL VIEW.
Its nature T T is easy to describe clearly the character
T of astronomical science, from its being
thoroughly separated, in our time, from all theological
and metaphysical influence. Looking at the simple facts of
the case, it is evident that though three of our senses take
cognizance of distant objects, only one of the three per-
ceives the stars. The blind could know nothing of them ;
and we who see, after all our preparation, know nothing of
stars hidden by distance, except by induction. Of all
objects, the planets are those which appear to us under the
I least varied aspect. We see how we may determine their
forms, their distances, their bulk, and their motions, but we
can never know anything of their chemical or mineralogical
structure; and, much less, that of organized beings living
; on their surface. We may obtain positive knowledge of
their geometrical and mechanical phenomena; but all
j physical, chemical, physiological, and social researches, for
I which our powers tit us on our own earth, a,re out of the
question in regard to the planets. Whatever knowledge is
obtainable by means of the sense of Sight, we may hope to
attain with regard to the stars, whether we at present see
the method or not; and whatever knowledge requires the
aid of other senses, we must at once exclude from our
expectations, in spite of any appearances to the contrary.
As to questions about which we are uncertain whether they
finally depend on Sight or not,—we must patiently wait,DEFINITION AND LIMITATIONS OF ASTRONOMY. 149
for an ascertainment of their character, before we can
settle whether they are applicable to the stars or not. The
only case in which this rule will be pronounced too severe
is that of questions of temperatures. The mathematical
thermology created by Fourier may tempt us to hope that,
as he has estimated the temperature of the space in which
we move, we may in time ascertain the mean temperature
of the heavenly bodies : but I regard this order of facts as
for ever excluded from our recognition. We can never
learn their internal constitution, nor, in regard to some of
them, how heat is absorbed by their atmosphere. Newton’s
attempt to estimate the temperature of the comet of 1680
at its perihelion could accomplish nothing more, even with
the science of our day, than show what would be the tem-
perature of our globe in the circumstances of that comet.
We may therefore define Astronomy as the Definition
science by which we discover the laws of the
geometrical and mechanical phenomena presented by the
heavenly bodies.
It is desirable to add a limitation which is
important, though not of primary necessity. Restriction.
The part of the science which we command
from what we may call the Solar point of view is distinct,
and evidently capable of being made complete and satis-
factory ; while that which is regarded from the Universal
point of view is in its infancy to us now, and must ever be
illimitable to our successors of the remotest generations.
Men will never compass in their conceptions the whole of
the stars. The difference is very striking now to us who
find a perfect knowledge of the solar system at our com-
mand, while we have not obtained the first and most simple
element in sidereal astronomy—the determination of the
stellar intervals. Whatever may be the ultimate progress
of our knowledge in certain portions of the larger field, it
will leave us always at an immeasurable distance from
understanding the universe.
Throughout the whole range of science, there exists a
constant and necessary harmony between our needs and
our knowledge. We shall find this to be true everywhere.
The fact is, we need to know only what, in some way or
other, acts upon us; and the influence which acts upon us150
POSITIVE PHILOSOPHY.
becomes, in turn, our means of knowledge. This is evi-
dently and remarkably true in regard to Astronomy. It
is of the highest importance to us to know the laws of the
solar system: and we have attained great precision with
regard to them; but, if the knowledge of the starry
universe is forbidden to us, it is clear that it is of no real
consequence to us, except as a gratification of our curiosity.
The interior mechanism of each solar system is essentially
independent of the mutual action of distant suns ; as it
may well be, considering the distance of these suns from
each other, in comparison with the distance of planets from
their suns. Our tables of astronomical events, constructed
in advance, proceed on the supposition of there being no
other system than our own ; and they agree with our direct
observations, precisely and necessarily. This is our proper
field; and we must remember that it is so. We must keep
carefully apart the idea of the solar system and that of the
universe, and be always assured that our only true interest
is in the former. Within this boundary alone is astro-
nomy the supreme and positive science that we have deter-
mined it to be; and, in fact, the innumerable stars that
are scattered through space serve us scientifically only
as providing positions which may be called fixed, with
which we may compare the interior movements of our
system.
We shall find, as we proceed through the
pi oration eX'	w^°le gradation of the science, that the more
complex the science, the more various are the
means of exploration; whereas, it does not at all follow, as
we shall see, that the completeness of the knowledge ob-
tained is in any proportion to the abundance of our means.
Our knowledge of astronomy is more perfect than that of
any of the sciences which follow it; yet in none are our
means of exploration so few.
The means of exploration are three:—direct observation;
observation by experiment; and observation by comparison.
In the first case, we look at the phenomenon before our
eyes ; in the second, we see how it is modified by artificial
circumstances to which we have subjected it; and in the
third, we contemplate a series of analagous cases, in which
the phenomenon is more and more simplified. It is onlyMEANS OF ASTRONOMICAL INVESTIGATION. 151
in the case of organized bodies, whose phenomena are
extremely difficult of access, that all the three methods can
be employed; and it is evident that in astronomy we can
use only the first. Experiment is, of course, impossible;
and comparison could take place only if we were familiar
with abundance of solar systems, which is equally out of
the question. Even simple observation is reduced to the
use of one sense,—that of sight alone. And again, even
this sense is very little used. Reasoning bears a greater
proportion to observation here, than in any science that
follows it; and hence its high intellectual dignity. To
measure angles and compute times are the only methods
by which we can discover the laws of the heavenly bodies ;
and they are enough. The few incoherent sensations con-
concerned would be, of themselves, very insignificant; they
could not teach us the figure of the earth, nor the path of
a planet. They are combined and rendered serviceable by
long-drawn and complex reasonings; so that we might
truly say that the phenomena, however real,	rank
are constructed by our understanding. The
simplicity of the phenomena to be studied, and the diffi-
culty of getting at them, constitute, by their combination,
the eminently mathematical character of the science of
astronomy. On the one hand, the perpetual necessity of
deducing from a small number of direct measures, whether
angular or horary, quantities which are not themselves
immediately observable, renders the use of abstract mathe-
matics indispensable; while, on the other hand, astro-
nomical questions being always, in themselves, problems of
geometry, or else of mechanics, must fall into the depart-
ment of concrete mathematics. Again, the regularity of
astronomical forms admits of geometrical treatment; and
the simplicity of astronomical movements admits of me-
chanical treatment with a very high degree of precision.
There is perhaps no analytical process, no geometrical or
mechanical doctrine, which is not employed in astronomical
researches, and many of them have as yet had no other aim.
Considering the simple nature of astronomical investi-
gations, and the easy application to them of mathematical
means, it is evident why astronomy is, by common consent,
placed at the head of the natural sciences. It deserves152
POSITIVE PHILOSOPHY.
this place, first, by the perfection of its scientific character;
and, next, by the preponderant importance of the laws
which it discloses.
Passing over, for the present, its utility in the measure-
ment of time, the exact description of the globe, and the
perfecting of navigation, which are not circumstances that
could determine its rank, we may just observe that it
affords an instance of the necessity of the loftiest scientific
speculations to the satisfaction of the most ordinary wants.
Hipparchus began to apply astronomical theory to the find-
ing the longitude at sea. A prodigious amount of geome-
metrical science has gone to improve our tables of longitude
up to their present point; and if we cannot now get within
half-a-dozen miles of a true estimate in the seas under the
line, it is for want of more science still.
Those who say that science consists in an accumulation
of observed facts may here see how imperfect is their
account of the matter. The Chaldeans and Egyptians
When it he collected facts from observation of the
cameYscience. heavens;; but there was no astronomical
science till the early G-reek philosophers re-
ferred the diurnal movement to geometrical laws. The
aim of astronomical researches was to establish what would
be the state of the sky at some future time; and no accu-
mulation of facts could effect this, till the facts were made
the basis of reasonings. Till the rising of the sun, or of
some star, could be accurately predicted, as to time and
place, there was no astronomical science. Its whole progress
since has been by introducing more and more certainty and
precision into its predictions, and in using smaller and
smaller data from direct observation for a more and more
distant prevision. No part of natural philosophy manifests
more strikingly the truth of the axiom that all science has
• prevision for its end: an axiom which separates science
from erudition, which relates the events of the past, with-
out any regard to the future.
„ ,	, However impossible may be the aim to
a single law	reduce the phenomena of the respective
sciences to a single law, supreme in each,
this should be the aim of philosophers, as it is only the
imperfection of our knowledge which prevents its accom-PRE-EMINENCE AS A SCIENCE.
153
plishment. The perfection of a science is in exact propor-
tion to its approach to this consummation; and, according
to this test, astronomy distances all other sciences. Sup-
posing it to relate to our solar system alone, the point is
attained; for the single general law of gravitation compre-
hends the whole of its phenomena. It is to this that we
must recur when we wish to show what wq mean by the
explanation of a phenomenon, without any inquiry into its
first or final cause; and it is here that we learn the true
character and conditions of scientific hypothesis,—no other
science having applied this powerful instrument so ex-
tensively or so usefully. After having exhibited these
great general properties of astronomical philosophy, I shall
apply them to perfect the philosophical character of the
other principal sciences.
Regarding astronomical science, apart from its method,
and with a view to the natural laws which it discloses, its
pre-eminence is no less incontestable. I have always ad-
mired, as a stroke of philosophic genius, Newton’s title of
his treatise on Celestial Mechanics,—‘ The Mathematical
Principles of Natural Philosophy; ’ for it would be im-
possible to indicate with a more energetic conciseness that
the general laws of astronomical phenomena are the basis
of all our real knowledge.
We may see at a glance that astronomy is , .
independent of all the natural sciences, de- other Sciences.
- pending on Mathematics alone : and though,
philosophically speaking, we put Mathematics at the head
of the whole series, we practically regard it less as a natural
science of itself (from the paucity of phenomena which it
presents to observation) than as the repository of principles
by which the natural sciences are interpreted and investi-
gated. Philosophically speaking, astronomy depends on
Mathematics alone, owing nothing to Physics, Chemistry,
or Physiology, which were either undiscovered, or lost in
theological and metaphysical confusion, while astronomy
was a true science in the hands of the ancient geometers.
But the phenomena of the other sciences are dependent,
naturally as well as systematically, on astronomical facts,
and can be perfectly studied only through astronomy. We
cannot thoroughly understand any terrestrial phenomenon154
POSITIVE PHILOSOPHY.
without considering what our globe is, and what part it
bears in the solar system, as its situation and motions
affect the conditions of everything upon it; and what would
become of our physical, chemical, and physiological ideas,
without consideration of the law of gravitation ? In the
remotest case of all, that of Social phenomena, it is certain
that changes in the distance of the earth from the sun, and
consequently in the duration of the year, in the obliquity
of the ecliptic, etc., which in astronomy would merely
modify some coefficients, would largely affect or completely
destroy our social development. It is no exaggeration to
say that Social physics would be an impossible science, if
geometers had not shown us that the perturbations of our
solar system can never be more than gradual and restricted
oscillations round a mean condition which is invariable.
If astronomical conditions were liable to indefinite varia-
tions, the human existence which depends upon them could
never be reduced to laws.
Not less important is the influence of astronomical
science on our own intelligence. It has done much more
than relieve us from superstitious terrors and absurd
notions about comets and eclipses,—notions which, as
Laplace observed, would spring up again immediately if
our astronomy were forgotten. This science has done much
more for our understandings than that. It has done more
than any other pursuit—simply because it is the most
scientific of all—to expose and destroy the doctrine of final
causes, which is generally regarded by the moderns as the
basis of every religious system, though it is in fact a con-
sequence and not a cause. The knowledge of the motion
of the earth has overthrown the very foundation of the
doctrine, which supposed the universe to be subordinated
to our globe, and therefore to Man. Since Newton’s time,
the development of celestial Mechanics has deprived theo-
logical philosophy of its principal intellectual office, by
proving that the order maintained throughout our system
and the whole universe is by the simple gravitation of its
parts. If we took an a priori view, we should say that, as
we exist, our system must be such as to admit of our
existence; and one necessary condition of this is such a
degree of stability in our system as we actually find. ThisTWO GREAT DIVISIONS OF THE SCIENCE.
155
stability we scientifically perceive to be a simple conse-
quence of mechanical laws working among the incidents of
our system,—the extremely small planetary bodies in their
relation to the larger sun; the small eccentricity of their
orbits, and moderate inclination of their planes ; which in-
cidents, again, are necessary consequences of the mode of
formation of the entire system. The stability by virtue of
which we hold our existence is not found in the case of
comets, whose perturbations are not only great, but liable
to indefinite increase; and their being inhabited is incon-
ceivable. Thus, the doctrine of final causes would be
reduced to the truism that there are no inhabited bodies
in our system but those which are habitable. This brings
us back to the principle of the conditions of existence,
which is the true positive transformation of the doctrine
of final causes, and of far superior scope and profit in every
way.
We have next to consider the divisions of	. .
the science. These arise immediately out of	the sconce*
the fact, now familiar to us, that astronomical
phenomena are either geometrical or mechanical. They are
Celestial Geometry, which is still called Astronomy, from
its having possessed a scientific character before the other ;
and Celestial Mechanics, of which Newton was the immortal
founder. Though our business is with our own system,
the same division extends to Sidereal astronomy, supposing
that kind of exploration to be within our Celestial
power. As before, we see geometry to be Geometry,
more simple in its phenomena than me-
chanics, and that mechanics is dependent on geometry,
without reciprocity. In fact, men were successfully in-
quiring into the forms and sizes of the heavenly bodies, and
studying their geometrical laws, before anything was known
of the forces which changed their positions. Whereas, the
province of Celestial Mechanics is to analyse Celestial
the motions of the stars, in order to refer Mechanics
them, by the rules of Rational Mechanics, to
the elementary motions regulated by a universal and in-
variable mathematical law;—thence, again, departing to
perfect the knowledge of real motions by scientifically
determining them a priori, taking from observation the156
POSITIVE PHILOSOPHY.
necessary data—the fewest possible—for the calculations
of general mechanics. This is the link by which astronomy
and physics are connected, and connected so closely that
some great phenomena render the transition almost in-
sensible ; as in the theory of the Tides. But it is evident
that the whole reality of celestial mechanics consists in its
having issued from the exact knowledge of true movements,
furnished by celestial geometry. It was for want of this
point of departure that all attempts before the time of
Newton, even Descartes’, however valuable in other ways,
failed to establish systems of celestial mechanics. This
division of the science into two parts has therefore nothing
arbitrary in it, nor even scholastic : it is derived from the
nature of the science, and is at once historical and dog-
matic. As for the subdivisions, we need not trouble our-
selves with them now.
In regard to the point of view from which the science
should be regarded, Lacaille thought it would simplify
matters extremely to place his observer on the surface of the
sun. And so it would, if the thing could be done in accor-
dance with positive knowledge ; but undoubtedly the solar
station should be the ultimate and not the original one,
under a rational system of astronomical study. And when,
as in the case of this work, the object is the analysis of the
scientific method, and the observation of the logical filia-
tion of the leading scientific ideas, it matters less to obtain
a clearer exposition of general results than to adhere to the
positive method.
I suppose my readers to be well acquainted with the two
fundamental facts of the diurnal and annual rotation of our
globe, as data without which nothing could be clearly
understood of the essential methods and general results
of astronomical science. I am not giving a treatise on
astronomy, nor even a summary; but a series of philo-
sophical considerations upon the different parts of the
science, in which any extended special exposition would be
misplaced.
We must first see what methods of observation astro-
nomers need, and are possessed of.157
CHAPTER II.
METHODS OF STUDY OF ASTRONOMY.
Observation.
SECTION I.
INSTRUMENTS.
ALL astronomical observation is, as we
have seen, comprehended in the measure-
ment of times and of angles. The two considerations
concerned in attaining the great perfection we have
reached are the perfecting the instruments, and the appli-
cation by theory of certain corrections, ^without which their
precision would be misleading.
The observation of shadows was the first ol ,
resource of astronomers, when the rectilinear
propagation of light was established. Solar shadows, and
also lunar, were very valuable in the beginning; and much
was obtained from the simple device of a style, so fixed as
to cast a shadow corresponding with the diurnal rotation to
be observed : but the alterations rendered necessary by the
annual motion, and impossible to make on that apparatus,
rendered the instrument unfit for precise observations.
Again, by comparing the length of the shadow cast by
a vertical style with the height of the style, the correspond-
ing angular distance of the sun from the zenith was com-
puted : and a valuable method this was : but the penumbra
rendered the accurate measurement of the shadow impos-
sible. The difficulty, aggravated by its unequal amount at
different distances from the zenith, was partly removed by
the use of very large gnomons; but not completely.
These imperfections determined astronomers to get rid as
soon as possible of the process of gnomonic measurement.
Shadows will always be at hand to measure by when better
means are wanting: and one application of this instru-158
POSITIVE PHILOSOPHY.
meat remains in our observatories,—as the basis of the
meridian line, regarded as dividing into two equal parts
the angle formed by the horizontal shadows of the same
length which correspond to the two equivalent parts of
the same day. In this case, the penumbra is harmless,
as it affects the two parts equally; and as for the obli-
quity of the sun’s motion, that may be mainly got rid of
by choosing the period of either solstice,—especially the
summer one. It is easy, too, to rectify the observation by
the stars.
Proceeding to more exact methods, and, first, with regard
to measurement of time, it is clear that the most perfect of
all chronometers is the sky. It seems as if it would be
enough, after knowing precisely the latitude of one’s obser-
vatory, to measure the distance of any star from the zenith,
and learn its horary angle, and, as an immediate conse-
quence, the time that has elapsed, by resolving the spherical
triangle formed by the pole, the zenith, and the star. If a
sufficiently wide observation of this kind had been made,
and numerical tables formed for certain selected stars,
great results might have been obtained from this natural
method ; but it is insufficient; and it has the defect of
making the measure of time depend on that of angles,
which is the least perfect of the two, in our day. This
method is therefore used only in the absence of a better, as
in nautical astronomy; and its commonest service is in
regulating other chronometers, by a comparison with that of
the heavens themselves. Artificial methods of measuring
time are therefore indispensable in astronomy.
Artificial	Every phenomenon which exhibits con-
methods	tinuous change might serve, in a rough way,
to mark time : various chemical processes, or
even the beating of our own pulses, might afford a measure,
more or less inaccurate: astronomical phenomena are ex-
cluded, because they are what we want to measure: and
we therefore have recourse to physical means, and find
weight the best. The ancients tried it in the form of the
flow of liquids; and to water clocks succeeded the hour-
glass ; but the uncertainty of these led to solids being
preferred; and in the form of weight having a vertical
descent. By no care, however, could the disturbancesMEASUREMENT OF TIME AND OF ANGLES. 159
caused by natural forces be remedied, till Galileo, by bis
creation of rational dynamics, suggested the pendulum.
Whether it is or is not correct to assign to T1 , 1
Galileo the idea of using the pendulum as a ie pen u um‘
measurer of time, it is certain that his discoveries sug-
gested it, and that Huvghens enabled us to use it. He had
recourse to the highest principles of science to render this
service, and discovered the principle of vires vivse, which,
besides being scientifically indispensable, afforded to art
new means of modifying oscillations without changing the
dimensions of the apparatus. Considered as a collection
of discoveries for a single aim, Huyghens’ treatise De
Horologio oscillatorio, is perhaps the most remarkable
example of special researches that the history of the human
mind has yet exhibited. From that time, the perfecting
of astronomical clocks became merely a matter of art. In
regard to fixed clocks, two things have to be attended to ;—
the diminution of friction, by improved methods of sus-
pension, and the correction by a compensating apparatus
of irregularities caused by variations of temperature. As
for portable chronometers, worked by a spiral spring, they
are a marvellous invention; but they belong to the pro-
vince of art, and not science.
In regard to the measurement of angles,
. -i .-i,	♦	,	i i • "u u Measurement
it is clear that an instrument which would Qf ano.jes
admit of an allowance for minutes and
seconds, must be of a size incompatible with minute pre-
cision. It must always be that large apparatus must be
so affected in its weight and temperature as to be impaired
in its accuracy. The large telescopes of modern times are
intended to show us stars otherwise invisible: and no one
thinks of using them for purposes of precise measurement.
It is generally agreed now that instruments for measuring
angles should not be more than ten feet in diameter when
we are dealing with an entire circle; and they are usually
not more than six or seven. The wonder then is how we
are to estimate angles to a second, as we do every day,
with circles whose size would scarcely indicate minutes.
It is done by the concurrent use of three methods,—the
eye-piece, the use of the vernier (so called after its
inventor), and the repetition of angles.160
POSITIVE PHILOSOPHY.
It was long before it occurred to astronomers to use
their lenses for any other purpose than the discovery of
new objects : but at last it occurred to them to replace the
ancient transoms and modern sights by an eye-piece which
should secure the advantages without the inaccuracies of a
large instrument. Morin first made this use of a lens.
Auzout followed with his invention of the reticle; and, a
century after, Dollond gave us a power of absolute pre-
cision by his invention of the achromatic object-glass.
Vernier proposed in 1631 to divide intervals into parts
much more minute than could be marked. He enabled us
to ascertain angles, within half a minute, or circles divided
only into sixth-parts of a degree. The precision obtainable
by his simple apparatus is indefinite, being limited only
by our difficulty in detecting the coincidence of the line of
the vernier with that of the limb. The union of the third
method with these two gives us the perfection we have
attained. It is strange that we should have been so long
in perceiving that, the imperfection of angular instruments
having nothing to do with the dimensions of the angle to
be measured, we should gain much by increasing, in fixed
proportions, the magnitude of the angles, which is equiva-
lent to diminishing the imperfection of the instrument.
The repetition of angles served every purpose immediately,
with regard to terrestrial objects, on account of the steadi-
ness of the point of view; but there was the difficulty,
with regard to the heavenly bodies, of their perpetual
change of place. Borda applied himself to measure the
distance from the zenith of the stars when they crossed
the meridian; and the star then remains sensibly at the
same distance from the zenith long enough to allow the
operation of the multiplication of the angle. By these
means, angular instruments are matched with horary in
regard to precision. They require from the observer a
diligent patience in applying all the minute precautions
and rectifications which experience has proved to be indis-
pensable to the fullest use of these instruments.
Then, we have Roemer’s meridional eye-glass, which fixes
the instant of the passage of a star over the meridian. The
plane of the meridian is made in this case purely geome-
trical, by being described by the optic axis of a simple eye-DIFFICULTIES FROM REFRACTION.
161
glass, properly disposed ; which is enough when all we wan.
to know is the precise moment of the star’s passage. Then
there are the micrometrical instruments, by which we mea-
sure the diameters of stars, and, generally all small an-
gular intervals. These are the material instruments of
observation,—horary and angular. We must now advert
to the intellectual means,—that is, to the corrections which
astronomers must apply to the results exhibited by their
instruments. There would be little use in perfecting our
instruments, if refraction and parallax introduced as much
error into, our observations as we had got rid of by the im-
provement of our apparatus.
The corrections required are of two kinds.	.
The first relate to the errors caused by the corrections
position of the observer,—the ordinary re-
fraction and parallax. No deep astronomical knowledge is
required for the correction of these. The second class,
arising from the same cause, since they proceed from the
observer being on a moving planet, are founded on primary
astronomical theories: they are the annual parallax, the
precession of the equinoxes, aberration, and nutation. Our
business now is with the first and most important class.
SECTION II.
REFRACTION.
The light which comes to us from any star ^ ,	..
x-u	i x j -j-l xi	Refraction,
must be more or less turned aside by the
action of the terrestrial atmosphere. We must estimate
the amount of this deviation before our observations can
answer any theoretical purpose. The star is, by this re-
fraction, made to appear too near the zenith, while left in
the same vertical plane. Only at the zenith is the error
absent, while it increases as the star descends to the horizon.
This error, primarily affecting distances from the zenith,
must affect, indirectly, all other astronomical measure-
ments, except azimuths: but it would be easy to calculate
them, if we once knew the law of diminution and increase
of refraction at different distances from the zenith. Philo-
i.
M162
POSITIVE PHILOSOPHY.
sophers have tried the logical way and the empirical, and
have ended by combining the two.
If our atmosphere were homogeneous, the refraction of
light would be uniform and calculable. But our atmo-
sphere is composed of strata; and the consequent refrac-
tions are excessively unequal, and increasing as the light
penetrates a denser stratum, so that its passage constitutes
a curve of the last degree of complication. Even this would
be calculable, with more or less pains, if we knew the law
of variation of these atmospheric densities : but we do not
and cannot know that law. We have no exact knowledge
of the laws of temperature, and cannot estimate atmo-
spheric changes, either as to number or degree: and all
mathematical processes founded on laws of pressure, etc.,
may be good as exercises, but are of no value in estimating
refraction. As to the empirical method, if the refraction
remained always constant at the same height, we might
construct tables ; and, by extending our observations, and
instituting various comparisons, we might hope to obtain
such a mass of materials as would afford us some certain
results. This is what astronomers have, in fact, patiently
and laboriously done, by the help of the improved instru-
ments we have spoken of. They have used whatever geo-
metrical help they could make applicable: but the results
are discouraging enough. There is nothing like uniformity
in the results: for the changes in the atmosphere are beyond
our calculation and measurement. We study the barometer,
the thermometer, and the hygrometer, at the right moment;
we can learn from them only the changes taking place on
the spot in which we are ; and our tables of refraction vary
as our observatories, and even in one observatory at diffe-
rent times. Delambre found differences of four or five
minutes between one day and another, after taking all
imaginable pains. All that we can do is to confine our
observations to the nearest possible approach to the zenith,
and to place no reliance on what we attempt near the
horizon. By doing this, we shall find our astronomical ob-
servations less affected by the unmanageable difficulties of
refraction than might be anticipated.DIFFICULTIES FROM PARALLAX.
163
SECTION III.
PARALLAX.
The difficulty of the parallaxes can be dealt Parallax
with much more easily and satisfactorily than
that of the refractions. Observations of the heavenly
bodies made in different places could not be exactly com-
pared without a reference, in idea, to those which would be
made from an imaginary observatory, situated in the middle
of the earth, which is besides the true centre of apparent
diurnal motions. This correction, which is called the
parallax, is analogous to that which is constantly made in
measurements of the earth’s surface, under the more
logical name of reduction to the centre of the station.
The effect of the parallax, like that of refraction, is
upon the distance of stars from the zenith alone, leaving
the star in the same vertical plane, and placing it too far
from the zenith, instead of too near, as in the case of re-
fraction. In this instance too, as in the other, though not
according to the same law, the deviation increases as the
star descends to the horizon. In like manner, too, there
must be secondary modifications for all the other astro-
nomical quantities, except with regard to the azimuths.
The rectification is easy in comparison with the other case,
from the absence of the hopeless difficulties caused by our
ill-understood atmosphere. The similar course of the two
difficulties, producing counteracting effects, has, we may
observe, relaxed the attention of astronomers to the facts
of refraction and parallax, by partly concealing their in-
fluence on actual observations.
The parallax does not, like refraction, affect all the stars
alike, but, on the contrary, affects all unequally, and each
according to its position. It is insensible with regard to
all which lie outside the limits of our system, on account of
their immense distance ; and it varies extremely within our
system, from the horizontal parallax of Uranus, which can
never reach a half-second, to that of the moon, which may
at times exceed a degree. Here lies the radical distinction,
in astronomical calculations, between the theory of paral-164
POSITIVE PHILOSOPHY.
laxes and that of refractions. The determination of ques-
tions of parallaxes does not wholly depend, like that of re-
fraction, on methods of observation in astronomy, but is
truly a portion of science. Depending as it does, ultimately,
on the estimate of the distances of the stars from the earth,
it pertains to celestial geometry, through the necessity of
knowing the law of motion of each star. Thus, it consti-
tutes a part of the science itself; though, in the absence of
direct knowledge of the distances of stars, an empirical
method of determining the coefficients, analogous to that
employed in the case of refraction, may be adopted. The
method which will suffice is to choose a place and time
which will show the proposed star passing the meridian
very near the zenith: then to measure, for several consecu-
tive days, its polar distance, so as to know pretty nearly the
amount of this distance at any moment of the process: and
this being laid down, then to calculate, for this instant,
according to the horary angle and its two sides, the true
distance from the star to the zenith, when it is considerably
remote from it, without being too near the horizon (say
from 75° to 80°) : and then, the comparison of this dis-
tance with that which is actually observed at the moment,
will evidently disclose the corresponding parallax, and
therefore the horizontal parallax, provided the due correc-
tion for the refraction has been made. This is the method
by which it is most easily established that the parallax of
all the stars is absolutely insensible.
It is a serious inconvenience in this method, that all the
uncertainty of the case of refraction is introduced into that
of parallax. In regard to a body whose parallax is very
great, as the moon, the uncertainty is of small consequence;
but in regard to the sun, or other distant body, an error of
one-third, or even one-half, in the value of its horizontal
parallax, might be occasioned. The method is absolutely
inapplicable to the remotest of our planets ; and not only
to Uranus, but to Saturn and Jupiter. The rational
method must be resorted to, in the case of these. The
empirical method has been mentioned here from the philo-
sophical interest which attaches to the fact that, up to a
certain point, the true distances of stars from the earth, at
least in proportion to its radius, may be ascertained by ob-CATALOGUE OF STARS.
165
servations made in one place; a thing which appears, at
first sight, geometrically impossible.
SECTION IV.
CATALOGUE OF STARS.
I am disposed to give a place here, con-
trary to custom, to the Catalogue of stars, gt^rg °”ue 0
which I think should be reckoned among
our necessary means of observation in astronomy. This
catalogue is a mathematical table of directions by which
we find the different stars. Such a determination is a basis
of direct knowledge in regard to Sidereal astronomy : while,
in regard to our own system, it is simply a valuable means
of observation, which supplies us with terms of comparison
indispensable for the study of the interior movements of
the system. Such has been the essential use of catalogues
of stars, from Hipparchus, who began them, to this day.—
In order to fulfil their purposes, these catalogues should
contain the greatest possible number of stars, spread over
every region of the sky. Astronomers have done their
duty well; for it is a settled habit with them to determine,
as far as they can, the co-ordinates of every new star which
they observe; and thus our catalogues are very voluminous,
and for ever augmenting. Our business here is not with
the system of classification and nomenclature adopted in
these catalogues. The nomenclature, bearing as it does
the marks of the primitive theological state of astronomy,
might be easily replaced by one of a methodical character,
—the objects to be classified being of the simplest nature,
and the distinctions being, in fact, only those of position.
But it is this very simplicity which prevents the need from
being felt as it would among more complex elements,—
useful as a rational system would no doubt be in finding
and assigning the places of stars. The change will be made
in time, no doubt, and the need is not urgent. Stars are
not known by their names, for astronomical purposes, but
by their descriptions; and the classification and nomencla-
ture in the catalogue, resulting from the fundamental divi-166
POSITIVE PHILOSOPHY.
sion of the circle, are as perfect as possible; and all else is
of little importance. I would only ask that we should
cease to speak of the magnitudes of stars, as marking their
rank, and substitute the word brightness, in order to avoid
all risk of supposing stars to be large or small in proportion
to their brightness or dimness. The word brightness would
be a simple declaration of the fact, without judging the
causes, which we are far from understanding.
By viewing these methods as I have brought them to-
gether, we may trace the progress of the science from its
earliest days. With regard to angular measurement, for
instance, the ancients observed with exactness a degree at
the utmost; Tycho Brahe carried up the precision to a
minute, and the moderns to a second;—a perfection so
recent, that observations which lie more than a century
behind our time are considered, from their want of pre-
cision, inadmissible in the formation of astronomical
theories.
My object has been, chiefly, to show the harmony which
exists among these different methods of observation ; a
harmony which, while it tends to perfect them all, up to a
certain point, still restricts them all, by making each a
limit to the rest. ~No improvement in horary or angular
instruments, for instance, could carry us far, while our
knowledge of refraction remains as imperfect as it is. But
there is no reason to suppose that we have approached the
limits imposed by the conditions of the subject.167
CHAPTER III.
GEOMETRICAL PHENOMENA OF THE HEAVENLY BODIES.
SECTION I.
STATICAL PHENOMENA.
THE phenomena of our solar system divide T
themselves into two classes,—the Stat ical phenomena °
and the Dynamical. The first class compre-
hends the circumstances of the star itself, independent of
its motions; as its distance, magnitude, form, atmosphere,
etc.: the other comprehends the facts of its displacements,
and the mathematical considerations belonging to its diffe-
rent positions. According to the usual analogy, the first
is independent of the second ; while the second could have
no existence without the first. The Statical phenomena
would exist if the system was immovable: while the
dynamical are wholly determined by the statical conditions.
The first thing necessary to be known
about any heavenly body is its distance from dances ^ 1S"
the earth : and the difficulty of obtaining
this ground for further observations is extremely great,—
the smallness of the base of our triangle, and the immensity
of the distance of the planet, rendering all accuracy hopeless
in very many cases. Towards the middle of the last century,
when it was desired to determine the horizontal parallax of
the moon,—the most manageable of the heavenly bodies,—
Lacaille went to the Cape of Good Hope, and Lalande to
Berlin, to observe its distance at the same moment from
the zenith,—that moment being appointed,—as the middle
of an anticipated eclipse. The stations were so chosen as
to afford a pretty accurate knowledge of the extent of the
line of the base,—which was about as long a one as our
globe could afford. The observations of the two distances168
POSITIVE PHILOSOPHY.
of the moon from the zenith must thus afford the necessary
data for the resolution of the triangle which must give the
distance sought: and thus we have obtained a very exact
knowledge of the moon’s distance, which, at its mean, is
about sixty terrestrial diameters, and about which we are
sure that we cannot be mistaken to the extent of more than
twelve miles. The same method might serve to give us,
though with much less precision, the distance of Venus
and even Mars, if the observation was made when they
were nearest to the earth; but it becomes too uncertain
with regard to the sun. It would leave an uncertainty of at
least an eighth, or about twelve millions of miles. Of course,
it is of no avail with regard to yet more distant bodies.
The method used by astronomers under this difficulty is
to measure, first, distances for which our small terrestrial
bases will serve ; and on these, according to their related
phenomena, to erect other calculations ; thus making of the
first a basis for the support of new estimates. Aristarchus
of Samos conceived of an ingenious method of discovering
the distance of the sun through that of the moon; but the
uncertainty about seizing the exact moment of the quad-
rature of the moon introduced fatal inaccuracy into the
calculation. Halley’s method, by means of the passage of
Mercury and Venus over the sun, is more circuitous, and
suitable only to an advanced state of geometrical science ;
but it is far more accurate, and the only one now admissible,
for determining the parallax of those planets and of the
sun, and therefore the distance of the sun from the earth,
through the differences in the transit observable at two
very distant stations. By this method, we can estimate,
within a hundredth part, the distance of the sun from the
earth. This distance being ascertained, we have it for a
basis for other calculations. We have only to observe the
angular distance from the sun to the proposed body, at two
periods separated by six months,— that is, from opposite
points of the earth’s orbit. This gives us an immense
triangle, the base of which is twice the length of the dis-
tance of the earth from the sun: and thus it is that our
knowledge of the earth’s motion has helped us to a base
twenty-four thousand times longer than the longest that
can be conceived on our own globe. It is true, the planetPLANETARY DISTANCES.
169
observed will have changed its place in the interval; but
the remoter planets,—which alone are in question here,—
move very slowly;—Saturn’s circuit, for instance, occupying
thirty years ; and our times of observation being practically
reducible to a shorter time than six months,—even to two
or one, with regard to those planets of our system which
move more rapidly; while the slower ones may be con-
sidered almost stationary, during such short periods of
time ; and again, allowance can be made for this small
change of place, according to the geometrical theory of its
proper motion. It is in this way that astronomers have
attained to their knowledge of the positions of the remotest
bodies of our system. The numbers by which we express
their relations to the distance of the earth from the sun,
are now certain to the third decimal at least.
The vast increase of the basis of observation afforded us
by our knowledge of the earth’s movement is clearly the
greatest that we can attain. If we have cleared the bounds
of our globe, we certainly cannot go beyond its orbit.
Great as this distance appears to us, it vanishes when we
want to ascertain the distances of stars outside our system.
All measurement is here so out of the question that the
most we can do is to fix a limit within which they cer-
tainly are not,—saying, for instance, that the nearest star is
at least two hundred thousand times more remote than the
sun, or ten thousand times further off than the remotest
planet of our system ; which is quite sufficient to establish
the independence of our system.
When we have ascertained the distance of the planets
from the earth, it is easy to understand how we may find
their distances from each other, since, in the triangle in
which each is contained, two sides are already given, and
the angle to the earth can always be measured. It is only
with regard to the sun and the moon that the distances to
the earth are of importance. It is enough to know the
distances of the planets from the sun, and of the satellites
from their planets, which involve little variation. These
are our means for ascertaining astronomical distances. As
we might anticipate, our assurance is in proportion to the
nearness; and great remoteness baffles us entirely. We
see here again, as everywhere, that the most simple and170
POSITIVE PHILOSOPHY.
elementary determination depends on the most delicate
and complex scientific theories. This first case exhibits so
much of the spirit of astronomical procedure, that we may
go more rapidly through the other statical heads of celestial
geometry.
11	. The distances of the stars from our globe
being once ascertained, we can learn what-
ever we desire about their form and size by observation, if
it be but precise enough. Their very distance is favour-
able to this ; for, while their motion or ours displays in
turn all their possible aspects, our distance enables us to
see at once the whole of each aspect. With regard to the
most distant and the smallest, however,—to the stars
outside our system, and the satellites of Uranus, and the
small planets between Mars and Jupiter,—they can appear
to us only as points of vivid light, and their sphericity is
concluded upon only through a bold induction. But, in
observing the larger planets of our system, we have only to
measure their apparent diameter in all directions, after
allowing for refraction and parallax. It is much easier to
us to learn the form and size of sun and moon than of our
own globe, since we have had the aid of glasses. The only
case of difficulty is that of Saturn’s rings; as it once was
with the moon, whose changing aspects greatly puzzled the
ancients. The most simple geometry now solves the last
difficulty, and Huyghens has helped us over the first.
With these exceptions, direct observation assures us that
the planets are all round, with more or less flattening at
the poles and bulging at the equator, in proportion to the
rapidity of their rotation.
As for the size of the heavenly bodies, it is easily calcu-
lated from the measurement of the apparent diameter
combined with the determination of the distance; and the
only reason why men were so long and so widely mistaken
about the dimensions of the planets was that their real
distances were unknown. No rule as yet appears which
connects these results with the order of the distance of the
planets from the sun. All we know is that the sun is
larger than all the other bodies of the system put together,
and in general that the satellites are much smaller than
their planets, as the laws of celestial mechanics require.PLANETARY ATMOSPHERES.
171
With regard to the bodies outside of our system, as we
have no knowledge of their distances, we are, of course,
ignorant of their dimensions.
It is by the occultation of stars, as starry
eclipses are called, that we make observations atmospheres
on the atmospheres of the planets, by seeing
what deviation their atmospheres cause in the light of the
remote stars which they eclipse. As the sun’s light is pro-
longed to us by the refraction of our atmosphere, the
atmosphere of a planet defers (only in a much greater
degree) the occultation of the star, and also shortens it;
and the comparison of the apparent duration of the eclipse
with that which it would otherwise be, gives us data for
the calculation of the atmosphere which causes the devia-
tion. It is thus that we learn that the moon has no
appreciable atmosphere. The horizontal refraction which,
on our globe, would reach thirty-four minutes, does not in
the moon amount to a single second. And the inference
that an atmosphere is wanting there is confirmed by
M. Arago, who in a different path of inquiry, about the
polarization of light reflected from liquid surfaces, has
established the fact that there are not, on the surface of
the moon, any great liquid masses, fitted to form an
atmosphere. The next best-known case is that of Venus,
which exhibits a horizontal refraction of thirty minutes,
twenty-four seconds. As for the extent of the atmo-
spheres, it may be roughly conjectured from the cessation
of the refracting power; but such conjectures must be
very loose, as the refracting power may become imper-
ceptible to us, far within the limits of an atmosphere
becoming attenuated towards its verge. The strangest
phenomenon is that of the telescopic planets, with the ex-
ception of Vesta; the atmosphere of Pallas, for instance,
being more than twelve times the diameter of the planet.
The usual condition, however, appears to be that shared by
our globe,—of an atmosphere which is very shallow in pro-
portion to the dimensions of the planet: and this is nearly
all we know.
The remaining statical topic is that of the _	. , .
form and size of the earth, which has been left	01 m
to the last, on account of its special nature.172
POSITIVE PHILOSOPHY.
Means of dis-
covery.
No glance of the eve will aid us here, nor
any direct observation whatever. A long
accumulation of indirect observations, serv-
ing as a basis for complex mathematical reasonings, are
our only means. The geometrical aspect of the question
must be taken first, though it depends on the highest
mechanical theories, and arises from a mechanical begin-
ning. In the infancy of mathematical astronomy, the
variations exhibited in different places by the diurnal
movement furnished the first geometrical proof of the
earth being round. It was enough to establish its evi-
dently and exclusively spherical character, that the change
exhibited by the height of the pole on each horizon was
always in exact proportion to the length traversed accord-
ing to any meridian whatever: and this remains the source
of all our geometrical knowledge of the form and dimen-
sions of our planet. Astronomers reached their knowledge
of its precise form through that of its size ; for it was long
before its deviation from the perfect spherical form was
understood. In this, as in every case, the form of any body
is appreciable only by measuring its dimensions in various
directions ; and here the only difficulty is in the measuring.
The first principles of the discovery were given by Eratos-
thenes, in the early days of the school of Alexandria; but
his method was never effectually employed till the middle
of the seventeenth century, when Picard undertook to
measure the degree between Paris and Amiens. This was
the great starting point of the measuring operations, which
must have revealed, as they became more perfect, the truth
that the earth is not a perfect sphere; but Newton, by his
theory of gravitation, and with his one fact of the shorten-
ing of the seconds pendulum at Cayenne, settled the
matter, by deciding that our globe must necessarily be
flattened at the poles, and bulge at the equator, in the
relation of 229 to 230. The astronomers could not at once
pronounce against the evidence of direct measurement,
while the geometers saw the fact to be certain; and the
controversy between these two orders of philosophers, for
half a century, led to those scientific operations which have
brought us all to one mind. The question was settled by
the great expedition sent forth, above a century ago, byPLANETARY MOTIONS.
173
the French Academy, to measure, at the equator and the
pole, the two extreme degrees of latitude which must
exhibit the widest variation from each other: and the
comparison of these with each other, and with Picard's
degree, terminated the controversy, and established, not
only the truth of Newton's discovery, but the very near
accuracy of his calculation. All the experiments made
since, in various countries, have united in confirming the
fact of the continual lengthening of degrees in approaching
the pole. It does not follow that the figure of the earth
has been ascertained with absolute precision. There are
slight discrepancies which may either be from imperfection
in our estimates, or from the earth not being precisely an
ellipsoid of revolution ; but whatever may be the result of
future labours, we know that we are near enough to the
truth for all practical purposes, unless in questions of
extreme delicacy. We have no absolute knowledge here,
any more than in any other department; and we must be
content to make our approximations more complex as new
phenomena arise to demand it. Such is the true character
of the advances that have been made in this science from
the beginning. Superficial observers may call its theories
arbitrary, from the incessant changes of view that have
arisen: but the knowledge gained has always been positive ;
every scientific opinion has corresponded with the facts
which gave rise to it; and such opinions remain therefore
useful and sound at this day, within their own range. The
science has thus always exhibited a character of stability,
through all incidents of progression, from the earliest days
of the Alexandrian school till now.
Such are the statical aspects of the planets of our system.
We have now to look at the geometrical theory of their
Planetary
motions.
motions.
Like all other bodies, the planets have a
motion composed of translation and rotation.
The connection of these two motions is so
natural, that when we know of the one we look for the
other. Yet they present very different degrees of difficulty,
and require separate consideration.
The progression of the stars was observed long before
their rotation,—the unassisted eye being enough for the174
POSITIVE PHILOSOPHY.
Rotation.
first; yet the geometrical study of their rotations is
easier, because the motions of the observer have no effect
upon them ; whereas they largely affect questions of trans-
lation. And again, the question of orbits is the chief
difficulty of the study of translations ; and it does not
enter into that of rotations. The latter nearly approaches
to the character of statical questions ; and therefore it ought
to be taken first in the exposition of celestial geometry.
Galileo introduced the study of rotations
by discovering that of the sun, which was sure
to follow closely on the invention of the telescope. The
method used is obvious enough, and the same in all cases ;—
to observe any marks that may exist on the surface of the
body, their displacement and return. The more such
points of observation are multiplied, the more accurate and
complete will be the calculations of time, magnitude,
uniformity of movement, etc., deducible from them. There
is no more delicate task than this, except with regard to
the sun and moon ; and none that more absolutely requires
a special training of the eye. It is a proof of this, that a
careful and honest observer, Bianchini, supposed the rota-
tion of Venus to be twenty-four times slower than it is.
Some bodies, as Uranus, are too remote, and others, as the
satellites and new planets, too small, to have their rotation
established at all, though it is concluded from analogy and
induction. We, as yet, know of no law determining the
time of these rotatios : they are not connected with dis-
tances, nor with magnitudes ; and they seem only to have
some general, but not invariable, connection with the
degree of flattening at the poles.1 But if the duration is,
though regular in each case, altogether irregular as regards
the different bodies, the case is much otherwise with the
direction; for it is always, throughout our system, from
west to east, and on planes slightly inclined to that of the
solar equator: and this constitutes an important general
datum in the study of our globe.
The study of translations, much more com-
Translation.
plex, is also much more important, if we con-
1 The rotations of some of the satellites are known. They all
follow the law of the moon’s rotation, namely, the time corresponds
with the orbital periods.—J. P. N.PLANETARY MOTION OF TRANSLATION.
175
sider the great end of astronomical pursuit—the exact
prevision of the state of the heavens at some future time.
Besides that the movement of the earth constitutes an
important elemeut in such a study, it must make a differ-
ence with regard to other stars, whether the observer is
fixed or moving, as his own movement must affect his
observations of other motions. We might indeed decide
with certainty, without this introductory knowledge, that
the sun and not the earth is the true centre of the motions
of all the planets, as Tycho Brahe did when he denied our
own motion; for it is enough, with this view, to establish
that the distances from the planets to the sun scarcely
vary at all, while their distance from us varies excessively ;
and again, that the solar distance between each inferior
planet and the sun is less, and between each superior planet
and the sun is greater, than our distance from the sun.
But we cannot go further than this,—we cannot determine
the form of the planetary orbits, or the mode in which they
are traversed, without making a careful and exact allowance
for the displacement of the observer. Deferring for the
present the subject of the earth’s motion, we will briefly
notice some important data connected with the planetary
motions, which may be obtained without reference to our
own movement, and which are so simple as to rank among
statical researches. I mean particularly the knowledge of
the planes of orbits, and of the duration of the sidereal
revolutions, which has nothing to do with the form of the
orbits or the variable velocity of the planets. A plane being
determined by three points, it is enough to observe three
positions of a star to draw a geometrical conclusion about
the situation of the plane of its orbit. Astronomers do
not now use, in these operations, the declinations and right
ascensions, which are the only co-ordinates directly observed,
but, for the sake of convenience, two other spherical co-
ordinates, improperly called astronomical latitude and lon-
gitude, which are analogous, with regard to the ecliptic, to
the others with regard to the equator. After having de-
termined the latitude and longitude of the planet in the
three positions, its nodes are found; that is, the points at
which its orbit meets the plane of the ecliptic, and the
inclination of the orbit to this plane. It is evident that176
POSITIVE PHILOSOPHY.
confirmation may be obtained by observing other positions
of the body, if they are chosen sufficiently remote from
each other; and thus we may obtain a far greater precision
than in the case of rotations. It is thus that we have
learned that the planes of all the planetary orbits pass
through the sun ; and the same with regard to the satellites
of any planet; and that these planes are in general slightly
inclined to the ecliptic, and more slightly still to the plane
of the solar equator, except the newly-discovered planets,
in whose case we find the inclination much more con-
siderable.
The duration of the sidereal revolutions
luticma reV°" may> course, be directly observed, in the
first instance, by looking for the return of the
star to the same spot in relation to the centre of its motion.
If we suppose its motion to be uniform, which we may for
a first approximation, we can estimate its course by observ-
ing the time required between any of the three positions,
without waiting for the total revolution, which is sometimes
very slow. The geometrical law of this motion permits us
to determine, from this kind of observation, the exact time
of the planetary revolution. The values of these periodic
times are not irregularly divided among the bodies of our
system, like the other data that we have noticed. The
shorter the course, the more rapid the motion; and the
duration increases more rapidly than the corresponding
distance; so that the mean velocity diminishes in propor-
tion as the distance increases. We owe to Kepler the
discovery of the harmony between these two essential
elements, and it is one of the most indispensable bases of
celestial mechanics.
Such is the spirit of the methods by which celestial
geometry is made to yield us the elementary data which
characterize the bodies of the solar system. We have still
to consider those of our own planet, before we proceed to
the geometrical laws of the planetary motions.
Motion of the Earth.
We are accustomed to think of the motions of transla-
tion and rotation as inseparable ; but, in the transitionMOTION OF THE EARTH.
177
from supposing tlie earth to be motionless, to the present
state of our knowledge, a theory existed that it whirled
round its axis, but was stationary in space.	Evidences of
We now perceive that, in addition to the	the Earth’s
general evidence of the double motion of the Motion,
planetary bodies, we have special evidence about our own
globe,—that the annual motion could not exist without the
diurnal, though we might logically suppose beforehand
that it could.
As the rotation of the earth cannot be absolutely uniform
in all parts of its surface, some indications of its course
must exist among terrestrial phenomena. We must there-
fore distinguish between the celestial and terrestrial proofs
of our diurnal motion, while the annual motion admits
only of the former.
Immediate appearances go for nothing in
this case; for it is clear that, to our eyes (as Ceptions C°n"
we do not feel the rotation), it must be
exactly the same thing whether we move round among the
heavenly bodies, or whether they, fixed in a system, move
round us in a contrary direction. There was nothing
absurd in the latter supposition, in the old days when men
had no doubt of the stars being very near, and not much
larger than they appear to the eye, while they exaggerated
the size of the earth. They could not avoid supposing that
such a mass must be immovable, while the small stars,
with their little intervals, were seen moving every day.
Even when the Greek astronomers had sketched out the
true geometrical theory of the movements of the planets,
they treated only of the directions, and had no idea of
measuring distances ; and it required the whole strength of
positive evidence of dimensions and distances to uproot
men’s strong and natural persuasion of the stability of their
globe. From the moment of our obtaining an idea of the
proportions of the universe, the old conception became too
revolting to reason to be sustained. When it was under-
stood that the earth is a mere point in the midst of pro-
digious intervals, and that its dimensions are extremely
small in comparison with that of the sun, and even of other
bodies of our own system, it was absurd to suppose that
such a universe could travel round us every day. What
I.	N178
POSITIVE PHILOSOPHY.
How they
gave way.
velocities would be required to enable the outlying stars to
complete such a daily circuit,—making allowance for their
being twenty-four thousand times nearer the earth, if the
earth describes no orbit,—and how small the movement of
the earth, while those prodigious masses were travelling at
such speed! On mechanical grounds, the centrifugal force
would be seen to be unmanageable. In every way, the
supposition was perceived to be monstrous. Again, the
passage of stars before each other, and in a contrary direc-
tion to that of the general movement of the sky, showed
that they were at different distances from each other, and
not bound into an unvarying fabric. Hence arose the
notion of Aristotle and Ptolemy, of a system of solid and
transparent firmaments. But the existence of comets alone
was enough to confute this, appearing as they
do in all regions of the sky in turn. As
Fontenelle said, this theory put the universe
in danger of being fractured. It was, curiously enough,
Tycho Brahe, the most illustrious opponent of the Coper-
nican system, who provided for the overthrow of his own
arguments by first j>resentiDg the true geometrical theory
,	of comets. Long before modern precision
tion 1S 10 a was	men had been prepared by such
considerations as the above to conclude upon
the rotation of the earth. Long before Copernicus, a rough
conception of the truth existed. Even Tycho Brahe felt
the astronomical superiority of the true theory; but it
seemed to be contradicted bv what is before our eyes,—the
fall of heavy bodies, etc. Copernicus himself could not
remove the objections which arose out of men’s ignorance
of the laws of Mechanics. These objections held their
ground for a century, till Galileo established the great law
which we have recognized as one of the three on which
Bational Mechanics is based,—that the relative motions of
different bodies are independent of the common motion of
the whole. Till this was established, the supposition of the
rotation of the earth was inadmissible. It is a curious fact,
•casting much light upon the action of the human mind, that
the opponents of Galileo taunted him with the so-called fact
that a ball let down from the top of the mast of a ship in
motion would not fall at the foot of the mast, but some wayPROOFS OF THE EARTH’S ROTATION.
179
behind,—neither they nor anybody else having tried the
experiment, which would have shown them that their sup-
posed fact was a mistake. The followers of Copernicus did
worse,—they admitted the so-called fact, but tried to
reason away its bearings with fantastic subtleties. The
matter was not settled even by the demonstrations of
Galileo, nor till Gassendi compelled observation by a public
experiment in the port of Marseilles.
That order of experiments has been carried on, and
would be of high value if we could obtain perpendicular
stations of sufficient height for the purpose. It is clear
that a lofty tower must describe a larger circle in the same
time at the top than at the base; and that any body
dropped from it must share the higher rate of velocity,
having a slight horizontal velocity in the direction of the
earth’s rotation,—falling therefore a little to the east of
the base of the tower. Omitting the consideration of the
resistance of the air, this amount is calculable in the func-
tion of the height of the tower and of its latitude ; but ex-
periment would also be valuable; and it is to be hoped
that it will be tried at the equator, where the deviation
must be greater than anywhere else.
The most certain terrestrial proof of the Influence of
earth’s rotation is found by tracing the in- centrifugal force
fluence of the centrifugal force upon the uPon gravity,
direction and intensity of weight. This has been done by
that observation of Richer, on the shortening of the seconds-
pendulum at Cayenne, which has been mentioned as having
emboldened Newton to declare the true figure of the earth.
The deviation from the spherical form is too small to ac-
count for more than one-third of the effect observed; and
the other two-thirds are precisely what would be required,
at the equator, where the centrifugal force is greatest, on
the supposition of the earth’s rotation. Wherever the de-
licate observation can be made with sufficient precision on
other points of the globe’s surface, the result answers to
the theory. Thus, we should have sufficient assurance, in
the absence of the abundant astronomical proofs that we
possess of the rotation of the earth. Probably no one fact
has ever, in the history of our race, produced such conse-
quences as that observation of Richer’s,—two-thirds of the180
POSITIVE PHILOSOPHY.
estimated effect having completely established the rotation
of our globe, and the other third having led Newton to the
ascertainment of its form.
The movement of translation is ascer-
lation S ianS tainable only by astronomical proofs, for the
difference in velocity of the various parts of
the globe, in virtue of this motion, is too slight to be sen-
sible to us, or to produce any effect on terrestrial pheno-
mena. When the circuits of other planets were known,
men’s minds were prepared for that of the earth,—the
question then being whether the earth was in analogy
with Venus, Mars, Jupiter, etc., or whether, while they
continued their courses round the sun, the sun made a
yearly circuit round the earth. Reason must declare, in
such a case, that any uncertainty must arise simply from
the position of the observer, who, placed on any other
planet, would have doubted whether he was not the centre
of the heavenly motions. Any observation of mere appear-
ances must evidently go for nothing in this case; as ap-
pearances must be exactly the same,—the parallelism of
its axis of rotation being unaltered,—whether the earth or
the sun is in the ecliptic, and the other in the centre. The
proofs must be derived from better testimony than mere
appearances; and they so abound that we have only to
choose among those which are presented by the whole
range of the heavens.
_	.	, The phenomenon called the precession of
prpepcci on at	J-
the equinoxes the equinoxes was observed by Hipparchus,
who was struck by the difference of two de-
grees which he observed between the longitudes of stars in
his time and those which had been recorded a century and
a half before. To account for such a phenomenon, suc-
cessive astronomers imagined other heavens; a process
that they repeated with regard to nutations, which was a
phenomenon too minute for their observation. To account
for it, on the supposition of the earth being stable, a third
general movement of the whole heavens must be supposed.
Newton indicated, and Bradley afterwards proved, that
very slight alterations in the parallelism of the earth’s
axis—such alterations as must result from the influence of
the sun, and yet more of the moon, upon the equatorialPROOFS OF THE EARTH’S TRANSLATION.
181
bulge,—precisely account for the perturbations which
create such confusion under the ancient view of the earth’s
stability. The most unquestionable proof of all, however,
is in that class of phenomena called the retro- Retrogradations
gradations and stations of the planets, which and stations of
are perfectly explained by the annual circuit ^ie planets,
of our globe, and are otherwise quite incomprehensible. If
two boats are gliding down a river, at different rates of
speed, the one must appear to the other advancing,
stationary, or retrograde, according as its own speed is
smaller, equal, or greater. With regard to the heavenly
bodies, their velocities and other circumstances are known
to us, so that we can calculate what their courses ought to
be to our eyes, on the supposition of our own annual move-
ment. The appearances answering to our scientific expec-
tation, the proof is practically complete. If the earth
moves, the retrogradation of the larger planets ought to
happen, as it does, when they are in opposition, and that
of Venus and Mercury when they are in inferior conjunc-
tion. The regular occurrence of this coincidence was not
even attempted to be explained by the ancients.
We have called these proofs practically complete ; and
they were held to be so by Copernican philosophers before
the time of Kepler and G-alileo : but our age is not satisfied
without a more strict mathematical evidence, amounting to
demonstration.
The one demonstration on which modern Aberration
science rests is that derived from the various ^ Ho-ht
phenomena of the aberration of light, which
are quite incompatible with the stability of the globe.
Roemer’s observations of the satellites of Jupiter suggested
to him the use of light as a measure of distance. Know-
ing what changes must be taking place at various distances
from us in the heavens, and knowing the velocity of light,
the variations in time at which the changes become visible
to us will be a measure of our change of place and distance.
For instance, the first satellite of Jupiter is eclipsed every
forty-two hours and a half. The eclipse will take place in
a shorter or a longer time than this to our eyes, in propor-
tion as we are removed to the one side or the other of our
mean distance from Jupiter, on account of the smaller or182
POSITIVE PHILOSOPHY.
greater space that the light will have to travel through.
By extending our observation, not only to the other satel-
lites of Jupiter, but to those of Saturn and Uranus, we have
obtained further verifications of the relation of our orbit to
theirs, and also, proof of the uniformity in the passage of
light,—at least within our own system. If the earth were
immovable, we might have an error of time, with regard
to distant stars, but not of place : but, by compounding the
velocity of the earth in its orbit with that of light, which is
about ten thousand times greater, we can calculate how far
any star ought to appear to deviate from its position. This
deviation is found not to exceed, at its maximum, twenty
seconds in any direction ; and therefore forty seconds is the
greatest deviation which can appear in the position of
any star in the course of the year. It was the striking
periodicity of these deviations which led Bradley to seek
for the true theory in the combination of the motion of the
earth with that of light, and to work it out with the mathe-
matical exactness permitted by modern science : and there
is nothing in the case to prevent the direct application of
the mathematical process to the visible phenomena. The
result is an unquestionable demonstration of the annual
movement of the earth, with which all the phenomena of the
case precisely agree, and without which they could not exist.
It is evident that this knowledge of aberrations compels
us to add another correction to those of refraction and
parallax, and the same is the case with regard to the pre-
cession of the equinoxes and nutation. Thus, as science
advances, the preparation of a phenomenon, observed with
the best instruments, for scientific use, becomes a delicate
and laborious operation.
Influence of These are the considerations which have
scientific fact led men to the knowledge of the double
upon Opinion. motion of the planet we inhabit. No other
intellectual revolution has ever so thoroughly asserted the
natural rectitude of the human mind, or so well shown the
action of positive demonstration upon definitive opinions ;
for no other has had such obstacles to surmount. A very
small number of philosophers, working apart, without any
other social superiority than that which attends positive
genius and real science, have overthrown, within two cen-KNOWLEDGE OF THE EARTH’S MOTION.
183
turies, a doctrine as old as our intelligence, directly estab-
lished upon the plainest and commonest appearances, inti-
mately connected with the whole system of existing opinions,
general interests, and dominant authorities, and supported
moreover by human pride, powerful in the recesses of each
individual mind. The whole system of theological belief
rested on the notion that the entire universe was ordained
for Man, a notion which appears truly absurd the moment
it is seen that our globe is only a subaltern star,—not any
centre whatever, but circulating in its place and season,
among others, round the sun, whose inhabitants might,
with more reason, claim the monopoly of a system which is
itself scarcely perceptible in the universe. The notion of
final causes and providential laws undergoes dissolution at
the same time; for, the once clear and reasonable idea of
the subordination of all things to the advantage of Man
being exploded, no assignable purpose remains for such
providential action. As the admission of the motion of
the earth overthrows the whole theory founded on the
human destination of the universe, it is no wonder that re-
ligious minds revolted from the great disclosure, and that
the sacerdotal power maintained a bitter rage against its
illustrious discoverer.
The Positive philosophy never destroys a doctrine with-
out instantly substituting a conviction, adequate to the
needs of our human nature. If the vanity of Man was
grievously humbled when science disabused him of his
notion of his supreme importance in the universe, to this
vanity at once succeeded a lofty sentiment of his true in-
tellectual dignity, when he saw what means were in his
power, under such difficulties as his position imposed upon
him, for the discovery of such a truth as he had attained.
Laplace has pointed this out, showing how to the fantastic
and enervating notion of a universe arranged for Man has
succeeded the sound and vivifying conception of Man dis-
covering, by a positive exercise of his intelligence, the
general laws of the world, so as to be able to modify them,
for his own good, within certain limits. Which is the
nobler lot ? Which is most in harmony with our highest
instincts P Which is the most stimulating to our facul-
ties ? And which is the most animating to our feelings ?184
POSITIVE PHILOSOPHY.
One more remark suggested by these discoveries is that
a clear distinction is for ever established between our
system and the universe at large. The old notion of the
universe as a single system was founded on the error of the
stability of the earth as its centre. The discovery of the
earth’s revolution at once transported all the external stars
to distances infinitely more considerable than the greatest
planetary intervals, and has left no place for the idea of
system at all, beyond the limits of our sun’s influence. We
do not know, more or less, and men will probably never
know, whether the innumerable suns that we see compose
a general system, or any number, large or small, of partial
systems entirely independent of each other. The idea of
the universe therefore is excluded from positive philosophy ;
and that philosophy is, strictly speaking, bounded by the
limits of the solar system, in regard to definite results ;
and this circumscription is, as elsewhere, to be regarded as
real progress. This restriction is further justified by the
knowledge we have obtained of all really universal pheno-
mena being essentially independent of the interior pheno-
mena of our system, since the astronomical tables of the
state of our system, prepared without reference to any other
sun than our own, invariably coincide with the minutest
direct observations. The theory of the earth’s revolution
has not as yet exerted its due influence on our views, and
especially in regard to this last consideration. This is
doubtless owing to the imperfections of our education,
which keep back these high philosophical truths till even
the best minds have been possessed with an opposite doc-
trine : so that the positive knowledge which they after-
wards attain commonly does little more than modify and
restrain the bad tendencies of their education, instead of
ruling and guiding their highest faculties.
Kepler's Laws.
The first idea that occurs to us when we are once satisfied
of the revolution of the earth is that our point of view
ought henceforth to be the centre of the sun.
This transformation of our observation is
called the annual parallax, and follows the
Annual
parallax.KEPLER’S LAWS.
185
Circles.
same rules as tlie diurnal parallax, allowance being made
for the much greater distance. Whether our observations
of the sidereal heavens are geocentric or heliocentric,—
from the middle of the earth or of the sun,—is of no
appreciable consequence ; but within our system the annual
parallax is of sensible importance. When, from the central
point of view, the orbits of the planets are determined, we
can proceed to that great aim and end of the science,—the
prevision of future conditions of the heavens at appointed
times.
The earliest supposition was that the
motions of the planets were uniform and
circular. The ancients had a superstition, as their writings
abundantly show, that the circle was the most perfect of
all forms, and therefore the most suitable for the motion
of such divine existences as the stars. Their choice of the
form was wise : they had to suppose some form, while that
of the circle answered best to what they saw ; and we our-
selves now take it provisionally in forming the theory of a
new star. But the superstitious attachment of the ancients
to this form was a serious impediment to the advance of
astronomy. For every deviation and new appearance a
new circle was supposed, till all the simplicity of the
original hypothesis was lost in a complication of epicycles.
By the end of the sixteenth century the number of circles
supposed necessary for the seven stars then known
amounted to seventy-four, while Tycho Brahe was dis-
covering more and more planetary movements for which
these circles could not account. Thus it is that men cleave
to old ideas and methods till they are utterly worn out,
and proved beyond recal to be ineffectual, under all addi-
tions that can be made to them.
Then came Kepler, the first man for twenty
centuries who had the courage to go back to
the beginning, as if nothing had been done in the way of
theory. He took for his materials the complete system of
exact observations which were the result of the life of his
illustrious precursor, Tycho Brahe. Notwithstanding the
natural hardihood of his genius, his works reveal to us how
strenuously he had to maintain his enthusiasm, in order to
support the toils of so bold and difficult an enterprise—
Kepler.186
POSITIVE PHILOSOPHY.
rational as it was. He chose the planet Mars for study;
and it was a happy choice ; because the marked eccentricity
of that planet was most apt to suggest the true law of
irregularity. Mercury is more eccentric still; but it does
not admit of continuous observation. He discovered three
great laws, which, extended from the case of Mars to that
of all the other planets in our system, constitute the founda-
His three laws ^on	Mechanics. The first law
regulates the velocity ; the second determines
the figure of the orbit: the third establishes harmony
among all the planetary motions.
First law	^ had l°ng been remarked that the angular
velocity, (that is, the larger or smaller angle
described, in a given time, by its vector radius,) of each
planet increases constantly in proportion as the body ap-
proaches the centre of its motion ; but the relation between
the distance and the velocity remained wholly unknown.
Kepler discovered it by comparing the maximum and
minimum of these quantities, by which their relation
became more sensible. He found that the angular velocities
of Mars at its nearest and furthest distance from the sun
were in inverse proportion to the squares of the corre-
sponding distances. Another way of expressing this law
is used by himself ; that the area described in a given
time by the vector radius of the planet is of a constant
magnitude, though its form is variable : or, again, in other
words, that the areas described increase in proportion to
the times. Thus he destroyed the old notion of the
uniformity of the planetary motions, and showed that the
uniformity was not in the arcs described, but in the areas.
a , i	The second law was less difficult to dis-
feecond law.	,	.
cover, when once Kepler had surrendered his
attachment to the circle. The next figure that presented
itself must naturally be the ellipse, which is the simplest
form of closed curve, after the circle. The Greek geometers
had advanced the abstract theory of this curve some way.
Kepler could not long hesitate where to place the sun in it:
it must be either in the centre or in one of the two foci.
No mathematical labour was needed to show him that it
could not be in the centre: and thus, in constructing
elliptic orbits, Kepler was necessarily led to place the sunKEPLER’S LAWS.
187
in the focns for all the planets at once. His hypothesis
once formed, it was easy to verify it by comparison with
observations, the first principles of the required calculations
being laid down beforehand. The second law of Kepler ,
then is that the planetary orbits are elliptical, having the
sun for their common focus.
These two laws determined the course of Third law
each planet; but the movements of all round
their common focus seemed to be purely arbitrary, till
Kepler discovered his third law. Being distinguished by
the most remarkable genius for analogy ever seen in man,
Kepler sought, and successfully, to establish some kind of
harmony among all these various movements. He spent
much time in pursuing the old metaphysical ideas of
certain mystic harmonies which must exist in the universe :
but, beyond the general conception of harmony, he obtained
no assistance from these vague notions. The ground on
which he proceeded was, in fact, the observation of astro-
nomers that the planetary revolutions are always slow in
proportion to the extent of their orbits. If he had confined
himself to this ground, this discovery would certainly not
have occupied seventeen years of assiduous toil. At last
his labour issued in the discovery that the squares of the
times of the planetary revolutions are proportional to the
cubes of their mean distances from the sun: a law which
all subsequent observations have verified. One important
result of this law is that we may determine the periodic
times and mean distances of all the planets by any one.
By it, for instance, we have determined the duration of
the year of Uranus, when once we knew its distance from
the sun: and, conversely, if we discovered a new planet
very near the sun, we need only observe its short revolution,
to be able to calculate its distance, which, in that position,
we could not effect by other means. Astronomers are
every day using this double facility, afforded them by
Kepler’s third law.
These are the three laws which will for ever constitute
the basis of celestial geometry, in regard to planetary
motions. They answer for all the bodies in our system,
regulating the satellites, by placing the origin of areas and
the focus of the ellipse in the centre of the respective planets.188
POSITIVE PHILOSOPHY.
Three pro-
blems.
Since Kepler’s time, the number of bodies in our system
has more than trebled ; and all have in turn verified these
laws. By them, motions of translation require for their
determination nothing more than a simple geometrical
problem, which demands from direct observation only a
certain number of data,—six for each planet. And thus
is a perfectly logical character given to astronomy.
The application of these laws, restricted to
our own system, is naturally divided into
three problems ; the problem of the planets ;
that of the satellites ; and that of the comets. These are
the three general cases of our system; and, by the applica-
tion to them of Kepler’s laws, we may assign to every body
within the system, its precise position, in all time past and
all time to come: and thence again, we can exhibit all the
secondary phenomena, past and future, which must result
from such relative positions. The next striking fact of
. this kind to the general mind is the predic-
Eclipses	tion °* echpses, absolutely conclusive as it
is, with regard to the accuracy of our geo-
metrical knowledge. This kind of prediction, quite apart
from the vague prophesying of ancient times, when eclipses
occurred, as they do now, necessarily from the planetary
orbits being all closed curves, and which men’s experience
told them must return,—began in the immortal school of
Alexandria; and its degree of precision, to the hour, then
to the minute, then to the second, faithfully represents the
great historical phases of the gradual perfecting of celestial
geometry. It is this which will, apart from all other con-
siderations, for ever make the observation of eclipses a
spectacle as interesting for philosophers as for the public,
and on grounds which the spread of the positive spirit will
render, we may hope, more and more analogous, though
unequally energetic.
We are learning to make more use of this class of phe-
nomena, and to make out new uses from them, as time
goes on. Independently of their practical utility in regard
to the great problem of the longitudes, they have been
found, within a century, very important in determining
with more exactitude the distance of the sun from our
earth. Whether it be an eclipse by the moon, or theECLIPSES: THEIR USES.
transit of Venus or Mercury, the difference
in duration of the phenomenon, observed in
different parts of the earth, will furnish the
relative parallax of that body and the sun, and conse-
quently the distance of the sun itself. Some bodies are
more fit than others for this experiment, certain conditions
being necessary, which are not common to all. Of the three
known bodies which can pass between us and the sun, two
—the Moon and Mercury—are excluded by these condi-
tions ; and there remains only Venus. Halley taught us
how to conduct and use the observation. The parallax, in
such a position, offers suitable proportions, being nearly
three times that of the sun; and the angular velocity is
small enough to allow the phenomenon, (lasting from six
to eight hours) to present differences of at least twenty
minutes between well chosen observatories. I have specified
this case, on account of its extreme importance to the
whole system of astronomical science; but it would be
quitting our object and plan to notice any other secondary
cases.
I must remark upon one very striking truth which be-
comes apparent during the pursuit of astronomical science ;
—its distinct and ever-increasing opposition as it attains a
higher perfection to the theological and metaphysical spirit.
Theological philosophy supposes every thing to be governed
by will; and that phenomena are therefore eminently vari-
able and irregular,—at least virtually. The Positive phi-
losophy, on the contrary, conceives of them as subjected to
invariable laws, which permit us to predict with absolute
precision. The radical incompatibility of these two views
is nowhere more marked than in regard to the phenomena
of the heavens ; since, in that direction, our prevision is
proved to be perfect. The punctual arrival of comets and
eclipses, with all their train of minute incidents, exactly
foretold, long before, by the aid of ascertained laws, must
lead the common mind to feel that such events must be
free from the control of any will, which could not be will
if it was thus subordinated to our astronomical decisions.
The three laws of Kepler form the founda- Foundations of
tion of the higher conception to which we are Celestial
next to pass on; the mechanical theory of Mechanics.
189
Transit of
Venus.190
POSITIVE PHILOSOPHY.
astronomical phenomena. By this ulterior study, we
obtain new determinations; but a more important office of
the Mechanical theory is to perfect celestial geometry it-
self, by giving more precision to its theories, and establish-
ing a sublime connection among all the parts of our solar
system, without exception. The laws of Kepler, inesti-
mable as they are, have come to be regarded as a sort of
approximation,—supposing, as they do, various elements to
be constant, while they are subject to more or less altera-
tion. The exact knowledge of the laws of these variations
constitutes the principal astronomical result of celestial
mechanics, independently of its own high philosophical
importance.
SECTION II.
DYNAMICAL PHENOMENA.
Gravitation.
The laws of Motion, more difficult to dis-
laws of Motion cover than those of extension, and later in
’ being discovered, are quite as certain, uni-
versal and positive in character; and of course it is the
same with their application. Every curvilinear displace-
ment of any kind of body,—of a star as well as a cannon
ball,—may be studied under the two points of view which
are equally mathematical: geometrically, in determining
by direct observation the form of the trajectory and the
law by which its velocity varies, as Kepler did with the
heavenly bodies; and mechanically, by seeking the law of
motion which prevents the body from pursuing its natural
straight course, and which, combined with its actual
velocity, makes it describe its trajectory, which may hence-
forth be know a priori. These inquiries are evidently
equally positive, and in like manner founded upon phe-
nomena. If we find still in use some terms which seem
to relate to the nature and cause of motion, they are only
vestiges of a mode of thinking long gone by; and they do
not affect the positive character of the research.
The two motions which constitute the course of the
cannon ball are perfectly known to us beforehand; but weHISTORY OF THE LAWS OF MOTION.
191
have not the geometrical knowledge of its trajectory. With
regard to the star, our knowledge of its trajectory compen-
sates exactly for the difficulty of our preliminary ignorance
about its elementary motions. If the law of the fall of
weights had not been directly established, we should have
learned it, indirectly, but no less surely, from the observa-
tion of the curvilinear motions produced by weight.
Celestial Mechanics was then founded on
a firm basis, when through Kepler’s laws, Their history,
and by the rules of rational Dynamics, dis-
covery was made of the law of direction and intensity of
the force which must act upon the planet to divert it from
the tangent which it would naturally describe. This
fundamental law once discovered, all astronomical researches
enter into the domain of Mechanics, in which the motions
of bodies are calculated from the forces which impel them.
This was the course philosophically and perseveringly pur-
sued by Newton.
It does not detract from Newton’s merits that Kepler
had some foresight of the results of his great laws. He
carried their dynamic interpretation as far as the science
of his day permitted ; and, seeking for what could not yet
be found, he wandered off among fantasies. The true pre-
cursors of Newton, as founders of dynamics, were Huyghens
and G-alileo,—especially the last: yet history tells of no
such succession of philosophical efforts as in the case of
Kepler, who, after constituting celestial geometry, strove to
pursue that science of celestial mechanics which was, by
its nature, reserved for a future generation. As the means
were wanting, he failed; but the example is not the less
remarkable.
The first of Kepler’s laws proves that the accelerating
force of each planet is constantly directed towards the sun.
The accelerating force, however great it may be supposed,
does net at all affect the magnitude of the area which
would be described in a given time by the vector radius of
the planet, in virtue of its velocity, if its direction passes
exactly through the sun, while it would inevitably change
it on any other supposition. Thus, the permanence of this
area,—the first general datum of observation,—discloses
the law of direction. The great difficulty of the problem,192
POSITIVE PHILOSOPHY.
gloriously solved by Newton, lies in tbe discovery, by
means of Kepler’s other two theorems, of the law of the
intensity of this action, which we speak of as exercised by
the sun on the planets.
When Newton began to work on this conception, he
took Kepler’s third law as his basis, supposing the orbits,
as he might do for such a purpose, to be circular and
uniform. The solar action, equal, and opposed to the
centrifugal force of the planet, thus became necessarily
constant at the different points of the orbit, and could not
vary but in passing from one planet to another. This
variation between one planet and another was provided for
by the theorems of Huyghens relating to the centrifugal
force in the circle. This force being in proportion to the
relation between the radius of the orbit and the square of
the periodic time, must vary from one star to another in-
versely to the square of its distance from the sun, in virtue
of the permanence which Kepler showed to exist of the
relation between the cube of this distance and this same
square of the periodic time, for all the planets. It was
this mathematical consideration which put Newton in the
way of his great discovery, and not any metaphysical
reasonings, such as prevailed before it, and which probably
never entered his mind, one way or another.
There remained the difficulty of explaining how this law
of the variation of the solar action agreed with tho geome-
trical nature of the orbits, as exhibited by Kepler. The
elliptical orbit presented two remarkable points,—the aphe-
lion and the perihelion, in which the centrifugal force was
directly opposed to the action of the sun, and consequently
equal to it; and the change in this action there must be at
the same time more marked. The curve of the orbit was
evidently identical at these two points ; the action then had
simply to be measured, according to Huyghens’ theorems,
by the square of the corresponding velocity. Thence, it
was easily deduced, from Kepler’s first law, that the de-
crease of the solar action, from the perihelion to the
aphelion, must be inversely to the square of the distance.
Here was a full confirmation of the law which related to
the different planets by an exact comparison between the
two principal positions of each of them. Still, however,newton’s demonstration.
193
the elliptical motion had not been considered. Any other
curve would, thus far, have served as well as the ellipse,
provided its two extremities had shown an equal curvature.
The remaining portion of the demonstration,—the measure-
ment of the solar action throughout the extent of the orbit,
—is to be obtained only by transcendental analysis. The
process is necessary for carrying on the comparison of the
solar action and the centrifugal force; and the theory of
the curvature of the ellipse is required. Huyghens made a
near approach to the principle of this great process ; but
it could not be completed without the aid of the differential
analysis, of which Newton was the inventor, as well as
Leibnitz. By the aid of this analysis, the force of the
solar action in all parts of the orbit is easily T , ,
estimated, m various ways; and it is round demonstration,
to vary inversely to the square of the distance,
and that it is independent of the direction. Furthermore,
the same method shows, in accordance with Kepler’s third
law, that the action varies in proportion to distance alone;
so that the sun acts upon all the planets alike, whatever
may be their dimensions, their distance only being the cir-
cumstance to be considered. Thus Newton completed his
demonstration of the fundamental law that the solar action
is, in every case, proportionate, at the same distance, to
the mass of the planet; in the same way that, by the
identity of the fall of all terrestrial bodies in a vacuum, or
by the precise coincidence of their oscillations, proof had
already been obtained of the proportion between their
weight and their masses. We thus see how the three laws,
of Kepler have concurred in establishing, according to the
rules of rational mechanics, this fundamental law of nature.
The first shows the tendency of all the planets towards the
sun; the second shows that this tendency, the same in
every direction, changes with the distance from the sun,
inversely to its square ; and the third teaches that this
action is always simply proportionate, the distance being
equal, to the mass of each planet. In accordance with the
laws of Kepler, which relate to the whole interior of our
system, the same theory applies to the connection between
the satellites and their planets.
Newton thought it necessary to complete his demonstra-
i.	o194
POSITIVE PHILOSOPHY.
tion by presenting it in an inverse manner; that is, by
determining a priori the planetary motions which must
result from such a dynamic law. The process brought him
back, as it must do, to Kepler’s laws. Besides furnishing
some means of simplifying the study of these motions, this
labour proved that, whereas, by Kepler’s laws, the orbit
might have had more figures than one, the ellipse was the
only one possible under the Newtonian law.
nia rffi if T It was once a great perplexity to some
explained1 ^ people, which others could not satisfactorily
explain, that when the planet is travelling
towards its aphelion we cannot say that it tends towards
the sun. But the difficulty arose out of the use of inap-
propriate language. The question is, not whether the
planet is nearer to the sun than it lately was, but whether
it is nearer than it would have been without the force that
sends it forward. It is always tending towards the sun to
the utmost that is allowed by the other force to which it is
subjected. The orbit is always concave towards the sun;
and it would evidently have been insurmountable if the
trajectory could have been convex. In the same way, when
a bomb ascends, its weight is not suspended or reversed:
it always tends towards the earth, and is, in fact, falling
towards it more rapidly every moment, even if ascending,
because it is every moment further below the point at
which it would have been but for the action of the earth
upon it ; and its trajectory is always concave to the
ground.
Term Attrac- I have thus far carefully avoided giving
tion inadmis- any name to the tendency of the planets
sible-	towards the sun, and of the satellites towards
the planets. To call it attraction would be misleading;
and we, in truth, can know nothing of its nature. All that
we know is that these bodies are connected, and that their
effect upon each other is mathematically calculable. It is
by quite another property of Newton’s great discovery
that this effect is explained, in the true sense of the word,—
that is, comprehended from its conformity with the ordinary
phenomena which gravity continually produces on the sur-
face of our globe. Let us now see what this property of
the discovery is.EXTENT OF THE DEMONSTRATION.
195
We owe a great deal to the moon. If the earth had no
satellite, we might calculate the celestial motions by the
rules of dynamics, but we could not connect them with
those which are under our immediate observation. It is
the moon which affords this connection by enabling us to
establish the identity of its tendency towards the earth
with weight, properly so called ; and from this knowledge,
we have risen to the view that the mutual action of the
heavenly bodies is nothing else than weight properly
generalized; or, putting it the other way, that weight is
only a particular case of the general action. The case of
the moon is susceptible of the most precise testing. The
data are known; and by dynamical analysis, the intensity
of the action of the earth upon the moon is exactly ascer-
tainable. We have only to suppose the moon close to the
earth, with the due increase of this intensity, inversely to
the square of the distance, and compare it with the inten-
sity of weight on the earth, as manifest to us bv the fall of
bodies, or by the pendulum. A coincidence between the
two amounts to proof; and we have, in fact, mathematical
demonstration of it. It was in pursuing this method of
proof that Newton evinced that philosophical severity which
we find so interesting in the anecdote of his long delay,
because he could not establish the coincidence, while con-
fident that he had discovered the fact. He failed for want
of an accurate measurement of a degree on the earth’s sur-
face ; and he put aside this important part of his great
conception till Picard’s measurement of the earth enabled
him to establish his demonstration.
The identity of weight and the moon’s ten-	. ,
deucy towards the earth places the whole of demonstration,
celestial mechanics in a new light. It shows
us the motions of the stars as exactly like that of projectiles
which we have under our immediate observation. If we
could start our projectiles with a sufficient and continuous
force, we should, except for the resistance of the air, find
them the models of the planetary system: or, in other
words, astronomy has become to us an artillery problem,
simplified by the absence of a resisting medium, but com-
plicated by the variety and plurality of weights.—If our
observation of weight on our globe has helped us to a196
POSITIVE PHILOSOPHY.
knowledge of planetary relations, our celestial observations
have in turn taught us the law of the variation of weight,
imperceptible in terrestrial phenomena. Men had always
conceived weight to be an inalterable property of bodies,
finding that no metamorphosis,—not even from life to
death,—made any change in the weight of a body, while it
remained entire. This was the one particular in which
men might suppose they had found the Absolute. In a
moment, the Newtonian demonstration overthrew this
fast-rooted notion, and showed that weight was a relative
quality,—not under the circumstances in which it had
hitherto been observed, but under the new one,—the
position of the observed body in the system,—its distance
from the centre of the earth. The human mind could
hardly have sought out this fact directly: but, once revealed
in the course of astronomical study, the verification easily fol-
lowed ; and experiments on our globe, in the vertical direc-
tion, and yet more in the horizontal, have established the
reality of the law, by experiments too delicate, from the
necessity of the case, to be appreciable, if we had not
known beforehand what differences must be found to exist.
. It is to express briefly the identity between
tation nnob' weight and the accelerating force of the
jectionable. planets that the happy term Gravitation has
been devised. This term has every merit.
It expresses a simple fact, without any reference to the
nature or cause of this universal action. It affords the
only explanation which positive science admits ; that is, the
connection between certain less known facts and other
better known facts. Since the creation of this term, there
has been no excuse for the continued use of the word
attraction. It is desirable to avoid pedantry in language;
but it is of high importance to preserve pure the positive
character of so fundamental a conception as this, by using
a term which expresses exactly what we know, and dis-
missing one which assumes what is purely fanciful, and
wholly incorrect. Attraction is a drawing towards. Now,
when we draw anything towards us, the distance is of no
importance : the same force draws the same body with
equal ease three feet or thirty feet, which is directly con-
tradictory to the facts of gravitation. Our business is withPRIMARY AND SECONDARY GRAVITATION. 197
the fact of the action, and not at all with its nature. It
was the use of this metaphysical term, it now appears,
which occasioned the opposition that the Newtonian theory
encountered so long, and especially in France. Descartes
had, by laborious efforts, banished the notions of occult
qualities, which he perceived to be so fatal to science; and
in this theory of attraction, his followers saw a falling back
into the old metaphysical delusions. We perceive this in
the writings of John Bernouilli and Fontenelle: and it
appears that the clear and positive scientific intellect of
France did good service in stripping off from the sublime
discovery of Newton the metaphysical appearance which
obscured its reality for a time.
One more consideration remains to be ad- Gravitation is
verted to. We have regarded the heavenly that of mole-
bodies thus far as points, without reference cules-
to their forms and dimensions. But as it is proved that
the intensity of the action of the sun on the planets, and
of the planets on their satellites, is proportioned to the
mass of the body acted upon, it is clear that the force
operates directly only on molecules, which are all indepen-
dently affected by it; and equally, their distance being the
same. The gravitation of molecules is therefore the only
real one ; and that of masses is simply its mathematical
result. In the mathematical study of motions however it
is necessary to have a conception of a single force, instead
of such an infinity of elementary actions: and hence arises
that preliminary part of celestial mechanics which consists
in compounding in one result all the mutual gravitation
of the molecules of two stars. Newton founded this portion,
with all the rest; and the two theorems which he esta-
blished for the purpose still remain the commonest expres-
sion of this important theory. He proved that if the stars
were truly spherical, and their strata were homogeneous,
the gravitation of their particles would be so balanced that
the bodies might be treated as points, in the study of their
motions of translation. But the irregularity of their forms,
however slight, must be considered in the theory of their
rotations, to which these theorems cease to be applicable.
For any other form than the sphere, the problem becomes
very complicated; and the analytical difficulties can be198
POSITIVE PHILOSOPHY.
surmounted only by approximation, notwitbstauding all
the perfections introduced into the theory in recent times.
And unless we could also learn what is the law of density
in the interior of the stars,—a kind of knowledge which
seems to be for ever beyond our reach,—we cannot attain
a perfect solution.
~	.	The fundamental law of Rational Me-
"ravitation. chanics, which declares the necessary equality
of action and reaction, shows that gravitation
must be mutual,—that the sun must tend towards the
planets, and the planets towards their satellites. The
extreme inequality of the masses renders the ascertainment
of the inverse gravitation extremely difficult; yet its reality
is established by various secondary phenomena. The
gravitation of the planets towards each other is a necesary
part of the whole conception ; but it was not mathe-
matically demonstrated till Newton s successors deduced
from it an exact explanation of the perturbations observed
in the principal motions of the planets. Their labours
have established secondary gravitation as positively as the
primary.
Thus has every kind of proof concurred to establish that
great fundamental law which is the noblest result of our
aggregate studies of nature. All the molecules of our
system gravitate towards each other, in proportion to their
masses, and inversely to the squares of their distances.
I dare not, as many do, confidently extend
lawliam° 6 ^e aPplication of this law to the entire
universe. There can be no objection to enter-
taining it analogically till we obtain some knowledge of
the mechanism of the sidereal heavens ; but we must
remember that we have not yet that knowledge, and that
we cannot promise ourselves that we ever shall. Without
the phenomena of our own system, the theory of its motions
would be only an intellectual exercise and sport: there can
be no positive science apart from phenomena, and of the
phenomena of the universe beyond our own system we are
not in scientific possession.1 It must be understood that
1 M. Comte omits here all notice of such positive applications as
we are able to make in Sidereal astronomy. He takes no notice
of the fact that the motion of the multiple stars in elliptical orbits,OPERATION OF NEWTON’S DISCOVERY.
199
I advocate simply a suspension of judgment where there
is no ground for either affirmation or denial. I merely
desire to keep in view that all our positive knowledge is
relative ; and, in my dread of our resting in notions of
anything absolute, I would venture to say that I can con-
ceive of such a thing as even our theory of gravitation
being hereafter superseded. I do not think it probable ;
and the fact will ever remain that it answers completely
to our present needs. It sustains us, up to the last point
of precision that we can attain. If a future generation
should reach a greater, and feel, in consequence, a need to
construct a new law of gravitation, it will be as true as it
now is that the Newtonian theory is, in the midst of inevit-
able variations, stable enough to give steadiness and confi-
dence to our understandings. It will appear hereafter
how inestimable this theory is in the interpretation of the
phenomena of the interior of our system. We already see
how much we owe to it, apart from all specific knowledge
which it has given us, in the advancement of‘ our philoso-
phical progress, and of the general education of human
reason. Descartes could not rise to a mechanical concep-
tion of general phenomena without occupying himself with
a baseless hypothesis about their mode of production.
This was, doubtless, a necessary process of transition from
the old notions of the absolute to the positive view; but
too long a continuance in this stage would have seriously
impeded human progress. The Newtonian discovery set
us forward in the true positive direction. It retains
Descartes’ fundamental idea of a Mechanism, but casts
aside all inquiry into its origin and mode of production.
It shows practically how, without attempting to penetrate
into the essence of phenomena, we may connect and assimi-
late them, so as to attain, with precision and certainty, the
true end of our studies,—that exact prevision of events
which a priori conceptions are necessarily unable to supply.
and in accordance with Kepler’s law of the velocities, demonstrates
the existence of a law of force, according to the inverse square of
the distance.—J. P. N.200
CHAPTER IY.
CELESTIAL STATICS.
KEPLER’S laws connected celestial phe-
nomena to a certain degree, before
Newton’s theory was propounded: but they
left this imperfection,—that phenomena which ranked
under two of these laws had no necessary connection with
each other. Newton brought under one head all the three
classes of general facts, uniting them in one more general
still; and since that time we have been able to perceive
exactly the relation between any two of the phenomena
which are all connected with the common theory. As far
as we can see, there is nothing more to gain in this
direction.
Consumma-
tion by
Newton.
We have seen what this great conception is in itself.
We have now to observe its application to the mathe-
matical explanation of celestial phenomena, and the per-
fecting of their study. Eor this purpose,
we will recur to our former division of sub-
Statical con-
siderations.
jects, and contemplate the phenomena of
planets as immovable first, and of planets in motion
afterwards; the statical phenomena first, and the dyna-
mical afterwards.
To know the mutual gravitation of the heavenly bodies,
we must know their masses. Such knowledge once ap-
peared inaccessible from its very nature; but the New-
tonian theory has put it within our power, and furnished
us with a wholly new set of ideas about these bodies.
There are three ways in which the inquiry has been pro-
First method secuted, all differing from each other, both
of inquiry in generality and in simplicity. The first
in o masses. method, the most general, the only one in
fact which is applicable to all cases, is the most difficult.
It consists in analyzing the special share of each body inSTATICAL INQUIRIES.
201
the perturbations observed in the principal motions of
another,—both of translation and rotation. Here two
elements are concerned,—the distance, and the mass of
the star in question. The first is well known, the other
is not; and only an approximate determination is possible.
It is difficult to apportion the shares in the action; and
geometers place little dependence on the computation of
masses obtained by this method, in comparison with that
obtained by either of the others.
Nearest in generality to this first method t ,,	,
is that which Newton employed with regard ^conclmethod,
to planets that had a satellite; that of comparing the
motion of the satellite round the planet with that of the
planet round the sun. The law which determines the
action by the distance being compared, in its results, in
the two cases, gives the relation of the masses of the sun
and the planet. The mass of Jupiter, determined by
Newton in this way, has undergone little change of state-
ment by methods since employed; and what difference
there is is almost wholly owing to the data of the process
being now better known.
The third method is the most direct and T1 . , f] i
simple of all; but it is the most restricted, 111 111 e 10< *
as it is necessarily confined to the planet inhabited by the
observer. It consists in estimating the relative masses by
the comparison of the weights which they produce. If we
knew the mass of any planet, we should know what would
be the weight of things on its surface, or at a given dis-
tance ; and reciprocally, the weight being known, we are
able to estimate the mass. With the pendulum, we have
measured terrestrial weight with absolute precision ; and,
diminishing it, inversely to the square of the distance, we
shall know its value at the distance of the sun. We have
then only to compare it with the amount, before well
known, which expresses the sun’s action upon the earth, to
find immediately the relation of the mass of the earth to
that of the sun. With regard to every other planet, on
the contrary, it must be the estimate of its mass which
would yield that of its corresponding gravity. All these
methods being practicable in the case of the earth, its
mass, in comparison with that of the sun, must be con-202
POSITIVE PHILOSOPHY.
sidered the best known of all within our system. The
mass of the moon, and that of Jupiter, are now estimated
almost as perfectly ; and those of Saturn and Uranus come
next. We are less sure about the other three which have
been calculated,—Mercury, Venus, and Mars; though the
uncertainty about them cannot be very great. Of the
telescopic planets and the comets we know scarcely any-
thing, owing to their extreme smallness, which precludes
their exerting any sensible influence on perturbations.
Comets pass, during their prodigious course, near very
small stars, such as the satellites of Jupiter and Saturn,
without producing any perceptible derangement. As for
the satellites, we have no knowledge except of the moon,
and approximately, of those of Jupiter. No comparison
of results has as yet exhibited any harmony whatever
between them. The only essential circumstance which
they present is the vast superiority of the size of the sun
to the whole contents of the system. Those entire con-
tents, if thrown together, would scarcely amount to a
thousandth part of the mass of the sun. Looking abroad
from the sun, we see alternating, without any visible order,
here decreasing, there increasing masses. We might have
supposed, a priori, as Kepler did, that the masses were
regularly connected with the volume* (which are them-
selves irregular however), so that the mean densities
should be continually less in mathematical proportion to
their distances from the sun. But, independently of this
numerical law, which is never exactly observed, the simple
fact of the decrease of density presents some exceptions, in
regard to Uranus, among others. No rational ground can
be assigned for this.
SECTION I.
WEIGHT OF THE EARTH.
. . f . These are the means by which the masses
earth * °	16 of the bodies of our system are ascertained.
The remaining process is to bring them into
relation with our estimates of weight, by ascertaining theWEIGHT OF THE EARTH.
203
total weight of the earth. Bouguer was the first who
distinctly perceived the possibility of such an estimate,
during his scientific expedition to Peru, when he found
that the neighbourhood of vast mountains slightly affected
the direction of weight. We see how, in accordance with
the law of gravitation, a considerable mass, regarded as
condensed in its centre of gravity, may affect the plumb-
line, however slightly, if it be brought close enough, sub-
jecting it to a secondary gravitation, which affords data
for a comparison between the action of the earth and that
of the mountain. By this, some estimate may be formed
of the proportion of the mountain to the globe. In the
time of Bouguer science was not advanced enough to admit
of more than the conception of how the thing could be
done. Half a century later, Maskelyne observed the
mountain Schehallion in Scotland, and found that it occa-
sioned an alteration in the natural direction of weight of
from five to six seconds; and Hutton deduced from this
that the weight of the earth is equal to four and a half
times that of a similar volume of distilled water at its
maximum of density. Anything like exactness, however,
is out of the question while there must be so much uncer-
tainty about the weight of the mountain, which can be
calculated only from its volume.
When Coulomb had invented his Torsion Balance, in-
tended to measure the smallest forces, Cavendish saw how
the earth might be weighed by comparing it, by means of
this balance, with artificial masses which might be com-
puted. By his immortal experiments, he discovered the
mean density of our globe to be five and a half times
equal to that of water; whence we can, if we think proper,
deduce the weight of the earth in cwts. and tons.—We
thus obtain, among other advantages, some insight into
the constitution of our globe, which by its positivity, puts
to flight many fanciful notions. The density of the parts
near the surface is so far below the average,—water occu-
pying much space, for instance,—that the density nearer
the centre must be much above the average. This is in
accordance with the indications of Celestial Mechanics;
and it furnishes us with one condition of the interior of
the globe. There can be no void there. What there is we204
POSITIVE PHILOSOPHY.
know not, further than that it must be something con-
sistent with the condition of superior density.
SECTION II.
FORM OF THE PLANETS.
.	The next great statical inquiry relates to
planets 6	^orm the heavenly bodies, as deduced
from, the theory of their equilibrium.
G-eometers suppose the planetary bodies to have been
originally fluid, because their equilibrium can thus consist
with only one form; whereas, if they had been always
solid, as our earth is now, their equilibrium might have
been compatible with any form whatever. Several pheno-
mena indicate this supposition, and it agrees remarkably
with the whole of our direct observations.
If the planets had no motion of rotation, their being
perfectly spherical would accord with the equilibrium of
their molecules: but the centrifugal force engendered by
the rotation must necessarily modify the primitive form,
-ry^ lf f by altering, more or less, the direction, or
the inquiry.	intensity of weight, properly so called.
Huyghens established this with regard to
the direction, and Newton with regard to the intensity.
We thus become easily assured of the general fact of the
nearly spherical form of all the planets, and of their being
slightly flattened at the poles: but, when we go further,
and attempt to estimate their forms mathematically, and
learn the precise degree of the flattening at the poles, the
question becomes one of transcendental analysis, and is
involved in difficulty which can never be entirely sur-
mounted. The inquiry involves a sort of vicious circle,
which does not admit of a logical issue. In order to form
an equation of the surface, we ought, by the law of equili-
brium of fluids, to know the weight of the
estimate1^ molecules concerned; whereas, by the law
of gravitation, this can be ascertained only
through the knowledge of the form of the planet, and even
of the mode of variation of its interior density. All thatPLANETARY FORMS.
205
can be done is to discover whether the proposed form
fulfils such and such conditions. Maclaurin discovered a
theorem, highly valued bv geometers, which has become
the basis of all our inquiries on this subject, and which
shows that the ellipsoid of revolution precisely fulfils the
conditions of equilibrium. But this supposes the structure
of the body to be homogeneous; which it is not, in any
case. The labours of geometers have however brought
within very narrow limits the possible variations of the
polar flattening. The result with regard to the earth is
that the mathematical rule perfectly agrees with direct
observation.
In the case of the planets we have another
resource. Their flattening affects certain perturbations!
phenomena of perturbation, by the study of
which we obtain materials for an estimate. Altogether,
the calculations and measurements agree more closely than
we could have ventured to hope. The only case which
seems to present a real exception is that of Mars, which,
by its magnitude, its mass, and the time of its rotation,
should be little more flattened than the earth ; whereas, if
the observations of Herschell are exact, it is almost as
much so as Jupiter.—We must observe, moreover, that
though, as Maclaurin has shown, equilibrium is compatible
with the ellipsoid form, this form is not to be supposed
the only one:—witness, in our own system, the rings of
Saturn, which are a remarkable example to the contrary:
and Laplace has demonstrated how these rings could, even
in a fluid state, be in equilibrium.
The most useful consequence of the mathe- Indirect esti-
matical theory of the planetary forms is that mate of the
it has established an important relation be- earth’s form,
tween the value of the different degrees on the earth’s
surface and the intensity of the corresponding gravity,
measured by the length of the seconds pendulum in different
latitudes. We can thus, with great ease, multiply our
indirect observations about the form of our globe ; whereas
the geometrical estimate of degrees is a long and laborious
operation, which cannot be often repeated with due care.
But, generally speaking, the more indirect a measurement
is, cceteris paribus, the more uncertain it is: and there206
POSITIVE PHILOSOPHY.
remains the uncertainty arising from our ignorance of the
law of interior density in our earth; so that our chief
reliance should still be on mathematical measurement,
eonducted with due care.
Hydrostatic An interesting question belonging to the
theory of pla- hydrostatic theory of the planetary forms is
netary forms. 0f conditions of stability of equilibrium
of the fluids which are collected on a part or the whole of
the surface of the planets. Laplace shows this stability to
depend, under all circumstances, on the density of the
fluid being less than the mean density of the planet; a
view established with regard to the earth by Cavendish’s
fine experiment.
SECTION III.
THE TIDES.
There remains the question of the tides,—the last im-
portant inquiry under the head of celestial statics. Under
the astronomical point of view, this is evidently a statical
question,—the earth being, in that view, regarded as
motionless: and it is not less a statical question in a
mathematical view, because what we are looking at is the
.	figure of the ocean during periods of equili-
the^tides ° brium, without thinking of the motions
which produced that equilibrium. More-
over, this inquiry naturally belongs to the study of the
planetary forms.
A particular interest attaches to this question, from its
being the link between celestial and terrestrial physics,—
the celestial explanation of a great terrestrial phenomenon.
—Descartes did much for us in establishing this. He
failed to explain the phenomenon; but he cast aside the
metaphysical conceptions which had prevailed before, and
showed that there was a connection between the change of
the tides and the motions of the moon; and this certainly
helped to put Newton in the way of the true theory. As
soon as it was known that the cause of the tides was to be
looked for in the sky, the theory of gravitation was certain
to afford its true explanation. Newton therefore gave outTHE TIDES.
207
the simple principle that the unequal gravitation of the
different parts of the ocean towards any one of the bodies
of our system, and particularly towards the sun and moon,
was the cause of the tides: and Daniel Bernouilli after-
wards perfected the theory. The same theory answers for
the atmosphere: but we had better study it in the case of
the seas alone ; on account of the uncertainty of our know-
ledge of the vast gaseous covering of our globe, whose
diffused mass almost defies precise observation.
Suppose the earth joined to any heavenly
body by a line passing through the earth’s tides ^ °	6
centre. It is clear that the point of the
earth’s surface which is nearest the other body will gravi-
tate towards it more, and the remoter point less, than the
centre, inversely to the squares of their respective distances.
The first point tends away from the centre : and the centre
tends away from the second point; and in each case the
fluid surface must rise; and in nearly the same degree in
both cases. The effect must diminish in proportion to the
distance from these points in any direction: and at a dis-
tance of ninety degrees it ceases. But there the level of
the waters must be louvered because of the exhaustion in
that place caused by the overflow elsewhere. And here
enters a new consideration, difficult to manage:—the
changes in the terrestrial gravity of the waters, occasioned
by their changes of level.—Thus the action of any heavenly
body causes the ocean to assume the form of a spheroid,
elongated in the direction of that body. Newton calcu-
lated the chief part of the phenomenon of the tides on the
supposition of an ellipsoid of homogeneous structure, as
he had done in estimating the effect of the centrifugal
force on the earth’s figure, substituting for the centrifugal
force the difference between the gravitation of the centre
of the globe and that of its surface next the proposed
body. After that, Maclaurin’s theorem served Daniel
Bernouilli for a basis of an exact theory of the tides.
Thus far, we have regarded the tides only as if they
were a fixed accumulation of waters under the proposed
star. This is the mathematical basis of the whole ques-
tion ; but the most striking part has yet to be considered,—
the periodical rise and fall. It is the diurnal motion of208
POSITIVE PHILOSOPHY.
our globe which causes this rise and fall, by carrying the
waters successively into all the positions in which the other
body can raise or depress them. Hence arise the four
nearly equal periodical alternations, when the two greatest
elevations take place during the two passages of the
heavenly body over the meridian of the place, and the
lower levels at its rising and setting; the total period
being precisely fixed by combining the terrestrial rotation
with the proper daily movement of the heavenly body.
The last indispensable element of the question is the
valuation of the powers of the different heavenly bodies.
This calculation is easily made from the difference between
the gravitation of the centre of our globe and that of the
extreme points of its surface next the observed body.
G-uided by the law of gravitation, we can determine which,
among all the bodies of our system, are those which can
participate in the phenomenon, and what is
the sun06 ° ^ie share taken by each. We thus find that
the sun by its immense mass, and the moon
by its proximity, are the >nly ones which produce any
,,	appreciable tides: that the action of the
ie moon. inoon js fr0IQ two and a half to three times
more powerful than that of the sun ; and that, conse-
quently, when they act in opposite directions, that of the
moon prevails; which explains the primary observation of
Descartes about the coincidence of the tidal period with
the lunar day.
Thus far, we have considered only the
effect of a single heavenly body upon the
tides; that is, the case of a simple and
abstract tide. The complication is very great, when the
action of two such bodies has to be considered. But the
resources of science are sufficient to meet this case,—even
deriving from it new means of estimating the mass of the
sun and moon ;—and also of calculating the modifications
arising out of the various distances of the earth from
either body; and again, of tracing the changes of direction
caused by the diurnal movement of the proposed body,—
whether in accordance with the earth’s axis of rotation, or
parallel with the equator, which makes the difference
between the tides of our equinoctial and solstitial lunar
Composite
influence.PROOF OF THE THEORY.
209
months. As for the difference of the phenomenon in
various climates, the consideration of latitude is the only
one which affords much result. At the poles, there can of
course be no other tides than such as are caused by the
flux and reflux of waters elsewhere, as the earth has no
rotation there. The equator must exhibit the tides at
their extremes, not only on account of the diminished
gravitation there, but yet more on account of the more
complete diversity of the successive positions occupied by
the waters during the daily rotation. Elsewhere the great-
ness of the tide must vary in proportion to the force of the
rotation.
The mathematical theory of the tides
accords with direct observation to a degree exactitude °T
which is really wonderful, considering how
many hypotheses geometers must have recourse to, to
make the questions calculable at all, and how many inac-
cessible data would be required to make an estimate
thoroughly logical. It would not even be enough to know
the extent and form of the bed of the ocean. Something
beyond that in difficulty is required,—the true law of
density, in the interior of the earth, as with regard to the
figure of the planets. We ought to know too whether the
interior strata are solid or fluid, in order to know whether
they participate in tidal phenomena, and whether they
therefore modify those at the surface or not. These con-
siderations show the soundness of the advice given by one
who was full of the true mathematical spirit, consisting
above all in the relation of the concrete to the abstract,
Daniel Bernouilli, who recommended geometers “ not to
urge too far the results of formulas, for fear of drawing
conclusions contrary to truth.”
The comparison between mathematical theory and direct
observation has never been carried out to any advantage,—
all the measurements having been taken in the ports, or
near the shore. The tides in such places are very indirect;
and they cannot properly represent the regular tides from
which they issue, their force being chiefly determined by
the form of the soil,—at the bottom as well as on the
surface,—and even perhaps affected by its structure. These
are incidents which cannot enter into mathematical esti-
i.
p210
POSITIVE PHILOSOPHY.
mates; and to them we must doubtless refer the vast
differences in the height of the tides at the same time, and
in nearly the same place,—as, for instance, the tides of
Bristol and Liverpool, of Granville and Dieppe. The only
way of making an effectual direct observation would be to
note the phenomena of the tides in a very small island, at
the equator, and thirty degrees at least from any conti-
nent, for such a course of years as would allow of repeated
record of variations as repeatedly foreseen. In this way,
and in no other, might the mathematical theory of the
tides be verified and perfected.
Whatever may be the uncertainty with regard to some
of the data of this great theory, it has that conclusive
sanction,—the fulfilment of its previsions ;—a fulfilment
so exact as to guide our conduct; and this, as we know, is
the true end of all science. The principal local circum-
stances, except the winds, being calculable, it has been
found practicable to assign for each port the mean height
of the tides and their times ; and thus have mathematical
determinations been proved to be sufficiently conformable
to reality, and a class of phenomena which, a century ago,
were regarded as inexplicable, have been referred to invari-
able laws, and shown to be as little arbitrary as anything
else.
Such are the philosophical characteristics of the three
great questions which compose the statical department of
Celestial Mechanics. We must next look into the dyna-
mical department, as represented by the phenomena of our
system.211
CHAPTER Y.
CELESTIAL DYNAMICS.
THE principal motion of the planets is, as p ,	.
we have seen, determined by the gravita- ei m a 10ns’
tion of each of them towards the focus of its orbit. The
regularity of this movement must be impaired by the
mutual gravitation of the bodies of the system. The most
striking of these derangements were observed by the
School of Alexandria, in the first days of Mathematical
Astronomy ; others have been observed, in proportion as
our knowledge became more precise; and now, all are ex-
plained with such completeness by the theory of gravitation,
that the smallest perturbations are known before they are
observed. This is the last possible test and triumph of the
Newtonian system.
There are, as Lagrange pointed out, two principal kinds
of perturbations, which differ as much in their mathe-
matical theory as in the circumstances which constitute
them; instantaneous changes, from shocks or explosions,
and gradual changes or perturbations, properly so called,
caused by secondary gravitation, requiring time. The first
kind may never have taken place in our T ,	,
, i ,	^	Instantaneous,
system; but it is necessary to consider it,
not only because it is of possible occurrence, but because
it is a necessary preliminary to the study of the other kind,
—the gradual perturbations being treated theoretically as
a series of little shocks.
The first case is easy of treatment. No collision or explo-
sion would affect Kepler’s laws: and, if the form of the
orbit was altered, the accelerating forces would remain the
same ; and thus, the new variation once understood, our
calculations might proceed as before. Supposing a collision
between two planets, or the breakage of one planet into
several fragments by an internal explosion; there might
be any variations whatever in the astronomical elements of212
POSITIVE PHILOSOPHY.
their elliptical movement; but there are two relations
which are absolutely unalterable, and which might, in my
opinion, generally enable us to establish the reality of such
an event at any period whatever: these are the essential
properties of the continuous motion of the centre of gravity,
and the invariableness of the sum of the areas,—both
resting on that great law of the equality of action and re-
action to which all changes must conform. From these
must result two important equations between the masses,
the velocities, and the positions of the two bodies, or the
two fragments of the same body, considered before and
after the event. No indication at present leads us to sup-
pose that the case of collision has ever occurred in our
system; and it is evident that such an encounter, though
not mathematically impossible, would be very difficult.
But it is far otherwise with regard to explosions.
The little planets discovered between Mars and Jupiter
have mean distances and periodic times so nearly identical,
that Mr. Olbers has conjectured that they once formed a
single planet, which had exploded into fragments. Lagrange
added a supposition, from the irregularity of their form,
that the event must have happened after the consolidation
of the primitive planet. When their masses become known,
I think this conjecture may be subjected to mathematical
proof,—in this way. By calculating the positions and suc-
cessive velocities of the centre of gravity of the system of
these four planets, we might, if they had such an origin,
retrace the principal motion of the primitive planet. If
we should then find this centre of gravity describing an
ellipse round the sun as a focus, and its vector radius
tracing areas proportioned to the times, this event would
be as completely established as any fact that we have not
witnessed. We have not yet the materials for such a test;
but it is interesting to see how celestial mechanics may
establish, in a positive manner, events like these which
appear to have left no evidence behind them. It is obvious
that the instantaneous character of such a change must
preclude our fixing any date for it, since the pheno-
mena would be precisely the same, whether the explosion
were recent or long ago. It is otherwise with regard to
perturbations, properly so called.PLANETARY PERTURBATIONS.
213
Lagrange believed that these explosions had been frequent
in our system, and that this was the true explanation of
comets, judging from the greatness of their eccentricity
and inclination, and the smallness of their masses. We
have only to conceive that a planet may have burst into
two very unequal fragments, the larger of which would
proceed pretty nearly as before, while the smaller must
describe a very long ellipse, much inclined to the ecliptic.
Lagrange showed that the amount of impulsion necessary
for this change is not great; and that it is less in proportion
as the primitive planet is remote from the sun. This ojrinion
is far from having been demonstrated; but it appears to
me more satisfactory than any other that has been proposed
on the subject of comets.
The important and difficult subject of per- ~	,
, t , • 1 .	i i • x £ i	Gradual per-
turbations is the principal object or celes- turbations
tial mechanics, for the perfecting of astrono-
mical tables. They are of two classes ; the one relating to
motions of translation, the other of rotation. The latter
are, as before, the most difficult: but the motions of rota-
tion are less altered than the other class, within our own
system ; and they are less important to be known.
In the study of motions of translation, the ^
planets must be treated as it they were con- 0f translation
densed in their centres of gravity.
The direct method, the only rational one, of calculating
the differential equations of the motion of any one planet,
under the influences of all the rest, is impracticable, from
the unmanageable complication of the problem. It would
make an inextricable analytical enigma. Geometers have
therefore been obliged to analyze directly the motion of
each planet round that which is its focus, taking for
modification only one at a time. This is pro|qem 0f
what constitutes in general the celebrated three boilie^
problem of three bodies, though this denomi-
nation was at first employed only for the theory of the
moon. It is easy to see what circumvolutions are involved
in this method, since the modifying body, being in its
turn modified by others, compels a return to the study of
the primitive body, to understand its perturbations. The
determination of the motions of the whole of our system214
POSITIVE PHILOSOPHY.
must, by its very nature, be a single problem. It is the
imperfection of our analysis which obliges us to divide it
into detached problems, and to overload our formulas with
multiplied modifications. The elementary problem of two
bodies,—one of these even being regarded as fixed,—is the
only one that we are capable of bringing to a solution; the
problem of the elliptical motion, represented by Kepler’s
laws; and here the calculations are extremely laborious.
It is to this type that geometers have to refer the motions
of the planets, by extremely complicated approximations,
accumulating the perturbations separately produced by
every body that can be supposed to exert any influence;
and these perturbations prescribe the series required for
the integration of the equations belonging to the case of
the three bodies.
Then follows the task of choosing the perturbations
which have to enter into the estimate. The law of gravita-
tion enables us to compare the secondary influences involved
in each case,—the masses of all within our own system
being supposed to be known. It is a favourable circum-
stance to mathematical research that our system is consti-
tuted of bodies of very small mass in comparison with the
sun (making the perturbations extremely small) ; moreover,
very few, very far from each other, and very unequal in
mass ; the result of all which is that, in almost every case,
the principal motion is modified by only one body. If the
contrary had been the case, the perturbations must have
been very great, and extremely varied, since a great number
of bodies must have powerfully acted in each disturbance.
Celestial Mechanics must then, we should think, have pre-
sented an inextricable complication, being incapable of
reduction to the problem of three bodies.
The study of modified motions divides itself into three
parts, answering, as in a former case, to the planets, satel-
lites and comets. Rigorously speaking, we ought to make
a fourth case of the sun, which cannot here be regarded as
motionless, because the planets react upon it. In fact, we
cannot allow ourselves to consider any point within the system
as motionless, except the centre of gravity of
Solar System the sJstem itself, which is the true focus of
planetary motion, and round which the sunTHE THREE PROBLEMS.
215
itself must oscillate, in directions which varv according to
the positions of the planets. This point is always between
the centre and the surface of the sun. But we cannot
approach nearer to the fact than this: we shall probably
never be able to indicate this centre precisely; and it is
enough for practical purposes, and necessary to them, to
consider the sun as fixed, except as to its rotary motion.
The same conclusion must be come to with regard to the
planets and their satellites,—even in the case of the earth
and moon, where the variations of the primary body are
greatest. The centre of gravity falling within the mass of
the primary body, its variations from that centre may be
neglected as having no appreciable influence on the motion
of translation ; and thus, celestial mechanics presents, in
this branch, no other problems than those treated, under
another point of view, by celestial geometry.
The simplest problem is here, as before, probjem 0f
that of the planets, and for the same reasons, tjie pianets
—the smallness of their eccentricities, and of
the inclinations of their orbits. There is also a considerable
uniformity of perturbations, since each planet remaining
in the same regions of the sky, continues in the same
mechanical relations, though their intensity varies within
certain limits. The least privileged of these bodies in
these matters is unhappily our own planet, on account of
the heavy satellite which escorts it so closely, and to which
its chief perturbations are due ; though this does not save
it from being sensibly troubled by others, at the period of
opposition, and especially by such a mass as that of Jupiter.
No other planet with satellites, not even Jupiter, is in so
unfavourable a case; for Jupiter’s motion could not be
very much deranged by the action of his satellites, however
near in position, since the mass of the largest is less than
a ten-thousandth part of his, while the mass of our moon is
a sixty-eighth part of that of the earth. Jupiter’s circula-
tion is sensibly affected by Saturn alone. The simplest
case of all seems to be that of Uranus, from its being the
last planet, and very remote from the next; and its six
satellites do not appear to trouble its motion.
The problem of the satellites is necessarily Problem of
more complicated than that of the planets, the Satellites.216
POSITIVE PHILOSOPHY.
on account of the instability of the focus of the princi-
pal motion, as in celestial geometry. Besides their own
perturbations, the satellites have reflected upon them
all those to which their planet is liable. The founders
of Celestial Mechanics were long perplexed, for instance,
by the perpetual acceleration of the mean motion of the
moon ; it vras considered inexplicable, till Laplace discovered
its cause in the slight variation to which the eccentricity
of the earth’s orbit is subject. In regard to the direct per-
turbations of the satellites, there is an essential distinction
between the case of one, and that of several satellites. In
the first,—the single case of our moon,—the disturbing
body is the sun, on account of its unequal action on the
planet and the satellite. If the difficulties arising out of
this position are greater than in the case of any other satel-
lite, it is partly because the case more immediately concerns
us, and because our opportunities of observation disclose
more fully the imperfection of our means. For, in the
mathematical point of view, there must be more complexity
in the case of several satellites; all that is true in regard
to one being true in regard to each one, with the addition
of the mutual action of the members of the group. Their
perturbations are reduced by the preponderating size of
their planet; but from there being so many of them, of
such nearly equal sizes and direction, and all so close
together, the difficulty of calculating their motions is so
great that the only theory as yet established is that of
the satellites of Jupiter. For the motions of three of them,
Laplace found means completely to account. Those of
Saturn and Uranus are known only geometrically, we having
not even an approximate estimate of their masses. It is to
be remembered, however, that we do not need so perfect a
knowledge of them as of the moon; and that a much less
exact theory will suffice for them than for the moon, whose
slightest irregularity is very evident to us.
f	The comets intervene to increase our diffi-
the Comets culties about the satellites. From the ex-
treme prolongation of their orbits, and their
inclination in all directions, comets are in a state of ever
variable mechanical relations, from the number of bodies
that they approach in their course ; whilst the planets, andPROBLEM OF THE COMETS.
217
eve a the satellites, have always the same relations, the
variation being only in the intensity. The perturbation
which, in every other case, bears a very small proportion
to the gravitation, may, in the case of comets, exceed it:
so that it is conceivable that a comet might be diverted
from its orbit, and become a satellite, when it passes near
so considerable a body as Jupiter, Saturn, or even Uranus.
Besides the eccentricities of comets, there are other circum-
stances, such as their small weight, and their possible loss
of weight by parting with some of their atmosphere to the
bodies they approach, which tend to perplex the study of
their perturbations. These are the incidents which make
it so difficult to foresee exactly the return of these little
bodies. When we have studied them so long and so
laboriously as to have, to the best of our belief, mastered
their case, we find that their periods are entirely changed
through one omitted circumstance. A memorable example
of this was the comet of 1770, calculated by Lexell. This
comet had then a revolution of less than six years : but it
has never appeared since, having been entirely deranged
by passing too near Jupiter. The imperfection of our
knowledge about these small bodies is from the same
cause that renders them of very little consequence to us.
From their vast distances, their action upon any one body
of the system is little more than momentary ; and their
lightness prevents even the satellites from being affected
by their passage. The passage of the comet of 1770
among the satellites of Jupiter proved this, in a striking
manner. Their tables, constructed beforehand, without
any idea of such an incident, perfectly agreed with direct
observations ; a proof that the intrusion of the comet did
not sensibly affect their motions. There is, therefore, no
more occasion for the puerile fears of our day than for the
religious terrors of former times, in regard to the passage
of comets. Their collision with the earth is all but im-
possible ; and they could not otherwise be felt at all.
Their mere approach, however near, could have no other
effect than to raise somewhat the corresponding tide. If
a comet could pass two or three times nearer to us than
the moon (which no known comet could do) its very small
mass could produce no other effect than an imperceptible218
POSITIVE PHILOSOPHY.
rise of the tides. We have therefore no immediate and
practical reason to regret the imperfection of our cometary
theories.
Passing from the perturbations proper to
oferotatk)n0nS m°fi°ns of translation, we must notice those
belonging to rotation.
The ellipsoid bodies of our system must, whether they
began or not, have ended, sooner or later, with turning
round one of their axes,—and that one the most stable,—
that of their smallest diameter: for, as we have seen, it is
their rotation that has produced their deviation from a
perfectly spherical form, and determined the direction
favourable to stability. The regularity of this rotation is
evidently so indispensable to the existence of living bodies
The lanets on sur^ace a planet, that we might
1 ‘a priori assert this stability wherever life is
possible, from the time when it became possible. But,
stable as each planet is in itself, its mutual gravitation
with others must introduce certain secondary modifica-
tions, the bearing of which must be upon the direction
of its axis in space. It is only with regard to the earth
that these modifications concern us; for however great
they might be in any other body, they could in no way
affect us.
If the planets were perfect spheres, the total gravitation
of their particles must pass through their centres of gravity;
and thus, it is only through their slight failure in sphericity
that they can act at all upon one another’s rotation; that
failure being caused by the rotation itself. We see here
how the same necessity which secures the stability of the
rotations, with regard to their duration and their poles,
determines, from another point of view, the inevitable
alteration of the parallelism of their axes.—In our own
planet the precession of the equinoxes, modified by the
nutation, results from the action of the other bodies of our
system,—especially of the sun and moon,—upon our equa-
torial protuberance. The power of each body is, as in the
case of the tides, in the direct ratio of its mass, and
inversely to the cube of its distance; so that the sun and
moon are the only bodies whose influence need be con-
sidered. Further, the extent of the deviation depends onPERTURBATIONS OF ROTATION.
219
the mass and magnitude of the earth, on the time of its
rotation, on its degree of flattening, and on the obliquity
of the ecliptic. The intensity of the influence must vary,
as in the case of the tides, with the variable distance of the
sun from the earth, and yet more of the moon ; but the
want of uniformity is too slight to be perceptible to direct
observation.—These are the general causes which doWmine
the small changes which the rotation of our globe under-
goes, in regard to the direction of its axis in space.—The
case of the other planets bears a general likeness to that
of the earth, varied according to the different inclinations
of their axes to their orbits, their position, their mass,
their size, the duration of their rotation, and the degree of
their flattening at the poles. On all these grounds, the
perturbations of Mars are the most remarkable.
The rotation of the satellites presents one
consideration of the highest interest,—that	ie sa e 1 es-
remarkable equality between the duration of this rotation
and that of their circuit round their planet, by which they
present always the same hemisphere, except from those
verv small oscillations called librations, whose law is well
understood. The fact is absolutely certain only with
regard to the moon ; but our mechanical principles -justify
our erecting it into a general law of all the satellites.
Lagrange has shown that it results from the preponderance
that, by the action of the planet, the nearer hemisphere
must acquire at the outset, whence arises a natural ten-
dency in the satellite to return perpetually to the same
position. If it is thus with the moon, there is every reason
to suppose the same fact with regard to satellites belonging
to heavier planets, to which they are proportionally nearer.
Such are the various kinds of perturbations produced in
the movements of the bodies of our system, by their
mutual action. This study may be simplified and rendered
much more exact, bv the device of referring all these move-
ments to a plane whose position must neces- Device of an
sarily be independent of all their variations. invariable
—Among several planes which have been plane,
proposed, differing in their degrees of variableness,
M. Poinsot has discovered one which is the only truly
invariable one, but which is extremely difficult to deter-220
POSITIVE PHILOSOPHY.
mine, since it requires not only an estimate of the planetary
masses, but data dependent on the mathematical law of
the interior density of the heavenly bodies,—a law which
is still very hypothetical. The theory is complete; but its
precise application is at present impossible. Whatever
may be the practical difficulties, we cannot but feel a deep
interest in seeing how Celestial Mechanics has accom-
plished the fixing of an invariable plane in the midst of
all the interior perturbations of our system, as Newton
had first recognized an inalterable velocity,—that of the
centre of general gravity. These are the only two elements
in our system which are rigorously independent of all the
events that can occur in its interior;—of even the vastest
commotions that our imagination can suggest. Such
variations as they can be conceived to have could relate
only to the most general phenomena of the universe, pro-
duced by the mutual action of different suns, of which
they would afford us the clearest manifestation, if such
knowledge were within our reach.
We end this study of perturbations with
our system a recognition of the stability of our own
system, in regard to all its most important
constituent bodies. Setting aside the comets, all the
variations whatever of any perceptible value are periodical;
and their period is usually very long, while their extent is
very small; so that the whole of our planetary sy stem can
only oscillate with extreme slowness round a mean state,
from which it deviates very little. Through all starry
changes the translations of our planets present the almost
rigorous invariableness of the great axes of their elliptical
orbits, and of the duration of their sidereal revolutions:
and their rotation shows a regularity even more perfect, in
its duration, in its poles, and even, though in a somewhat
smaller degree, in the inclination of its axis to the corre-
sponding orbit. We know, for instance, that from the
time of Hipparchus, the length of the day has not varied
the hundredth part of a second. Amidst all this general
regularity, we perceive a special and most marked stability
with regard to the elements which are concerned in the
continued existence of living beings.—Such are the sublime
theorems of natural philosophy for which humanity isSTABILITY OF THE SOLAR SYSTEM.
221
indebted to the sum of the great works executed in the
last century by the successors of Newton.
The general cause of these important results lies in the
small eccentricity of all the principal orbits, and the small
divergence of their planes. If the planets had had come-
tary orbits and planes, there would have been no regularity
—no periodicity,—and, we may add, no life upon their
surface. No planets can be habitable but such as have
their oscillations restricted within very narrow limits.
The Mathematical theory of celestial me- .	f
chanics has taken no notice, thus far, of the a^ediunf °
resistance of any general medium, in which
these motions are proceeding. The conformity of our mathe-
matical tables with observed facts shows that the resistance
is imperceptible in degree; yet, as it is manifestly impos-
sible that it should be null, the geometers have endeavoured
to prepare beforehand a general analysis of it. Considered
apart from its intensity, this action is of a totally different
nature from that of perturbations, though gradual like
them : for it cannot be periodical, and must always be
exercised in the same direction, so as continually to diminish
all velocities, and the more the greater they are. It cannot
alter the positions of the orbits, but can by possibility
affect only their dimensions, and periodic times, and the
duration of rotations : that is, it affects the elements which
are spared by the perturbations. Thus, the rotations! must
become slower, the orbits must grow smaller and rounder,
and their periodic times shorter; because, as velocity dimi-
nishes, the solar action must become more powerful, and
these effects are not only continuous, but always increasing
in rapidity. So, in a future too remote to be assigned, all
the bodies of our system must be united to the solar mass,
from which it is probable that they proceeded: and thus
the stability of the system is simply in relation to the per-
turbations properly so called. These are among the incon-
testable indications of Celestial Mechanics.
As yet, we practically fail to recognize the effect of a
resisting medium. We neither trace its operations, nor
should know how to calculate it if we could trace it.
Whenever we do, it will be by the study of comets ; for
their small mass, and the great surface which they present222
POSITIVE PHILOSOPHY.
to the action of the medium when their atmospheres are
widely diffused, must render its resistance much more
appreciable than in the case of planets,—their velocity
being besides naturally at its maximum at the moment of
this expansion. Some contemporary astronomers believe
that they have established the effect of this resistance in
regard to one or two comets. Hitherto the study of these
bodies seems to be only negatively useful, to prevent the
return of the absurd terrors which they formerly occasioned.
We now see that there is no body in our system, however
insignificant, whose theory may not offer to us a direct and
positive interest, since we may owe to comets the knowledge
of one of the most important general lawrs of the system to
which we belong, and that which, in a remote future, must
chiefly rule its destinies.1
Independence In our geometrical review we saw, by the
of the solar agreement of astronomical tables with direct
system.	observation, that our system is independent
of all that lies outside. This incontestable truth is con-
firmed by the mechanical view. If our system gravitated
towards any of the suns outside, the action of other suns
would nearly neutralize the tendency. Again, it would be
only by an unequal action of those suns upon our piantets
that any change could be occasioned. Again, the vast
distances would, according to our law of gravitation, make
the action of remote suns imperceptible. The nearest body,
if a million times heavier than our system, would produce
an effect incalculably smaller than the action which occa-
sions our tides. We may therefore pronounce the indepen-
dence of our system to be perfectly certain. I notice this
because we seem to find here the only exception to the
great encyclopedical law which is the basis of this work,—
that the most general phenomena rule the most particular,
without being in any degree reciprocally influenced. Thus
our astronomical phenomena regulate those of our own
globe,—whether physical, chemical, physiological, or social.
iTet here we find that the phenomena of the universe have
no influence over those of the solar system. There is no
difficulty about this to persons who, like myself, admit
1 M. Comte estimates too lightly the indications of a medium
given by Encke's comet.—J. P. N.ACHIEVEMENTS OF CELESTIAL DYNAMICS. 223
that our researches are limited by the boundaries of our
own system, and that positive knowledge cannot go beyond
it. The study of the universe forms no part of natural
philosophy: a truth which will become more apparent,
and be seen to be more important the further our studies
extend.
At the close of this brief review of celestial dynamics,
we see that, great as are the achievements since Newton’s
time, we are reminded in many directions of the imperfec-
tion which results from the insufficiency of our mathe-
matical analysis. In the execution of astronomical tables
it has to borrow from celestial geometry other aid than the
estimate of indispensable data, derived from direct observa-
tion ; and this in regard not only to bodies whose mechani-
cal theory is but just initiated, but with regard to some
with which we are best acquainted.
We see however that besides the sublime Achievements
direct knowledge afforded to us, celestial of Celestial
dynamics has powerfully contributed to per- Dynamics,
feet the whole body of astronomical theories in regard to
their definitive aim,—the exact prevision of the state of the
heavens at any period whatever, past or future. Kepler’s
laws might suffice to determine the state of our system fur
a short time, proper data being chosen; but if we wish to
extend the inquiry, back or forwards, to any considerable
period, we find the most perfect theory of perturbations
absolutely necessary. It is to celestial dynamics that we
owe our power of ranging up and down the centuries, to
fix the precise moments of various celestial phenomena,
such as eclipses, with certainty, and with a minuteness
only inferior to that which is possible in the case of present
events.
Though we have, according to my view, completed our
consideration of astronomical science, it would be felt to
be a great omission if we passed over altogether what is
now called Sidereal Astronomy. We will therefore see
how much there is that we can conceive to be positive in
regard to cosmogony.224
CHAPTER VI.
SIDERIAL ASTRONOMY AND COSMOGONY.
,, ... .	I HE only branch of Sidereal Astronomy
A which appears to admit ot exact study
is that of the relative motions of the Multiple Stars, first
discovered by Herschell. By multiple stars astronomers
understand stars very near each other, whose angular dis-
tance never exceeds a half minute, and which, for this
reason, appear to be one, not only to the naked eye, but
to ordinary telescopes, only the most powerful lenses being
able to separate them. The relative movements of these
stars tend to deceive us as to their precise multiple cha-
racter, as, for instance, by mutual occultations, which do
not permit us to separate them. Among some thousands
of multiple stars registered in the catalogues, before the
southern heavens had been really explored, almost all were
only double, and we have found none which are more than
triple,—a circumstance which may be owing solely to the
imperfection of our telescopes, as we knew of none but
single stars before Herschell’s time. However interesting
the study of them is, they constitute only a particular case
in the universe, as the intervals of the stars which compose
them are probably much smaller than those which divide
the suns of the universe, so that the study of their relative
motions does not lead us up to any of the great general
phenomena of the heavens, and the speciality would be
more conspicuous if astronomers did as I think they ought,
—form their catalogues of those double stars only whose
motions they have fully established. With regard to
others, we cannot be sure whether their duality is a real
relation or an accident. Knowing nothing whatever of
their interval, or of the distance of either of them from us,
we cannot be sure whether they form a system any more
than any other two stars combined by chance in theMULTIPLE STARS.
225
heavens. Because a few incontestable examples are before
us of a binary system, in which the smaller circulates
round the larger, it is anything but philosophical to con-
clude the same to be the case with the whole multitude of
double stars, some of which may appear so merely through
an accident of position, apparent only to our own system.
Analogy is not applicable here ; as what looks like analogy
is merely the imperfection of our investigations. No astro-
nomer would venture to assert that if our telescopes were
what they may one day become, we might not find between
stars now apparently independent a multitude of clustered
intermediate stars which should render the case of duality
almost general. The apparent nearness would not then
be a sufficient ground for presuming their mutual revolu-
tions, because it is in virtue of their very small number
that analogy now suggests that presumption. The only
positive study in sidereal astronomy is that of the known
relative motions of certain double stars, at present not
more than seven or eight in number. We could never
hope to assign with accuracy their orbits, or their periodic
times, or any solid basis for dynamical conclusions.1 The
importance of such inquiries is much diminished by the
consideration that our system, which, in such a case, means
our sun, belongs to no groups of the kind,—either investi-
gated or merely pointed out. This circumstance seems to
me not at all accidental; for, if our system made a part of
a double star, which it is not difficult to imagine, it would
probably be impossible for us ever to be aware of the other
part of such a duality, because in the direction of the sun
it would be so near that its light would be lost to us in
that of our sun. Such a case might, however, have a
scientific interest for us, not only as elucidating the dis-
placements of our system, but as allowing such great pre-
cision, as might arise from the position of the inquirer on
one of the stars of the couple.
The first of the few orbits of double stars known to us
1 M. Comte quite underrates the importance of the phenomena
of the multiple stars. The orbits of a very considerable number
are now distinctly ascertained, and the laws of motion in their
orbits. The existence of a motion of revolution is fixed, with re-
gard to the far greater number.—J. P. N.
I.	Q226
POSITIVE PHILOSOPHY.
was investigated by Savary. They all present a very con-
siderable eccentricity, the smallest of which is double, and
the greatest four times greater than that of the most eccen-
tric in our system. Of their periodic times, the shortest
slightly exceeds forty years, and the longest six hundred.
We cannot perceive that the eccentricity and the duration
bear any fixed relation to each other, and neither seems to
depend at all on the angular distance of the respective pairs
of stars. This is the sum of what we know about the
double stars; and unless we could learn something of their
linear distance from our system and from each other, our
conceptions can neither be accurate, nor of great importance.
M. Savary has proposed a method, founded on the known
velocity of light, by which these distances will, if ever, be
estimated:1 but the uncertainty of some of the elements
which must enter into the question is so great that the
most that can be hoped for is the fixing of certain limits
within which the real distance may be supposed to lie:
and this is all that M. Savary himself proposed. At
present, we know only the nearer limit, beyond which, not
only the double stars, but the whole starry host, are known
to lie.
Our Cosmo-
gony.
Proceeding now to ascertain what we may
rationally conceive of our own cosmogony, I
need hardly say that we must put aside
altogether any notion of creation, as unintelligible,—all
that we are able to conceive of being successive transfor-
mations in the sky; and of these, only such as have pro-
duced its present state. Here, again, we find our own
system to be the only subject of knowledge. We are in
possession of some facts in regard to it which may bear
testimony to its immediate origin; but we can form no
reasonable conjectures about the formation of the suns
themselves. The phenomena necessary for such a purpose
are not only not explored but not explorable. Whatever
may be the interest of Herschell’s curious observations on
the progressive condensation of the nebulae, they do not
warrant his conclusion of their transformation into stars ;2
1	They are calculated ; hut by strictly geometrical methods.
2	This portion of Herscliell’s speculation must he abandoned.COSMOGONY OF LAPLACE.
227
for from such a conclusion must flow consequences about
form and motion which must be in harmony with estab-
lished phenomena; and of these we have absolutely
none.
The beginning of positive cosmogony was Origin of
when geometers, pursuing the mathematical Positive
theory of the figures of the planets, showed Cosmogony,
that they were originally in a state of fluidity. We cannot
go further bach than this: and we must set out with an
existing sun, turning on its axis with an indeterminate
velocity, admitting, for the formation of the planetary
system, no agencies which we do not now see at work, in
the phenomena which we habitually witness, though they
may have wrought formerly on a larger scale. These
restrictions are indispensable to the scientific character of
the inquiry; and, after all, our cosmogonic theories, how-
ever guarded, must remain essentially conjectural, if ever
so plausible. No mathematical principles can enter here
as into celestial mechanics, leading us up to a definite
theory and excluding every other. No abstract theory of
formations is possible ; and the utmost we can do is to
collect such information as can be had, construct hypo-
theses from it, and compare them carefully and con-
tinuously with the whole of the phenomena that we ex-
plore. Such hypotheses, whatever degree of consistency
they may attain, can never, like the law of gravitation,
take rank among general facts: for we can never be sure
that some other hypothesis may not turn up which would
equally well answer the present purpose, and some others
besides.
The cosmogony of Laplace seems to me to
present the most plausible theory of any yet Laplace0**^ °
proposed. It has the eminent merit of re-
quiring, for the formation of our system, only the simple
agents, weight and heat, which meet us everywhere, and
which are the only two principles of action which are
absolutely general. The point in which I differ from
Laplace is with regard to comets, which he regards as
strangers in our system whereas Lagrange’s view of
What he fancied to he instances of nebulous matter turn out to
be galaxies, or vast groups of stars.—J. P. N.228
POSITIVE PHILOSOPHY.
them, before cited, appears to be preferable, as being con-
sistent with the independence of our solar group.
The hypothesis of Laplace tends to explain the general
circumstances of our system, viz., the common direction of
all the planets from west to east; that of their rotations; and
that of all the satellites: also, the small eccentricity of all
the orbits ; and finally, the small inclination of their planes,
especially in comparison with that of the solar equator.
It is supposed by this theory that the solar atmosphere
was originally extended to the limits of our system, in
virtue of its extreme heat; that it was successively con-
tracted by cooling; and that the planets were formed by
this condensation. The theory rests on two mathematical
considerations. The first involves the necessary relation
between the successive expansions or contractions of any
body whatever and the duration of its rotation ; by which
the rotation should be quickened as the dimensions lessen
and becomes slower as they increase, so that the angular
and linear variations sustained by the sum of the areas
become exactly compensated. The other consideration
relates to the connection between the angular velocity of
the sun’s rotation and the possible extension of its atmo-
sphere, the mathematical limit of which is at the distance
at which the centrifugal force, due to this rotation, be-
comes equal to the corresponding gravity: so that if any
portion of the atmosphere should be outside of this limit,
it would cease to belong to the sun, though it must con-
tinue to revolve with the velocity it had at the moment of
separation. From that moment, it ceases to be involved
in any further consequences from the cooling of the solar
atmosphere. It is evident, from this, how the solar atmo-
sphere must have diminished, as to its mathematical limit,
without intermission, in regard to the parts situated at the
solar equator, as the cooling was for ever accelerating the
rotation. Portions of the atmosphere, thus parted with,
must form gaseous zones, situated just beyond the respec-
tive limits and this constituted the first condition of our
planets. By the same process the satellites were formed
out of the atmospheres of their respective planets. Once
detached from the sun, our planets must become first
liquid and then solid, in the course of their own cooling,CONCLUSION.
229
without being further affected by solar changes: but the
irregularity of the cooling, and the unequal density of
parts of the same body must change, in almost every case,
the primitive annular form, which remains in the rings of
Saturn alone. In most cases, the whole gaseous zone has
gathered, in the way of absorption, round the preponderat-
ing portion of the zone as a nucleus: thence the body
assumed its spheroidal form, with a revolving motion in
the same direction as its movement of translation, on
account of the excess of the velocity of the upper molecules
in comparison with that of the lower.
This theory answers to all the appearances of our system,
and explains the difficulty of the primitive impulsion of the
planets. It shows, also, that the formation of the system
has been successive, the remotest planets being the most
ancient, and the satellites the most modern.1
If from points of view like these the stability of our
system can scarcely be regarded as absolute, what it may
lead us to suspect is that, by the continuous resistance of
the general medium, our system must at length be re-
united to the solar mass from which it came forth, till a
new dilatation of this mass shall occur in the immensity
of a future time, and organize in the same way a new
system, to follow an analogous career. All these pro-
digious alternations of destruction and renewal must take
place without affecting the most general phenomena, occa-
sioned by the mutual action of the suns; so that these
revolutions of our system, too vast to be more than barely
conceived of by our minds, can be only secondary, even
local events, in relation to really universal transformations.
It is not less remarkable that the natural history of our
system should be, in its turn, as certainly independent of
the most prodigious changes that the rest of the universe
can .undergo: so that whole systems are, perhaps fre-
quently, developed or condensed in other regions of space,
1 The author subjoins a proposed mathematical verification of
Laplace’s cosmogony, which is not given in the text, as it does not
seem to rest on adequate foundations. If an arithmetical verifica-
tion be ever obtained, it will probably be in connection with the
periods of the rotations of the different planetsperiods already
in so far connected with the nebular hypothesis by the investiga-
tions of an American inquirer—Mr. Kirkwood.—J. P. N.230
POSITIVE PHILOSOPHY.
without our attention being in any way drawn towards
these immense events.
T> ., , .. The end I had in view in this exposition
of astronomical philosophy will be attained
if I have clearly exhibited, in regard both to method and
to doctrine, the true general character of this admirable
science, which is the immediate foundation of the whole of
Natural Philosophy. We have seen the human mind, by
means of geometrical and mechanical researches, and with
the help of constantly improving mathematical aids,
attaining to a precision of logical excellence superior to
any that other branches of knowledge admit of. We see
the various phenomena of our system numerically esti-
mated, as the different aspects of the same general fact,
rigorously defined, and continually reproduced before our
eyes in the commonest terrestrial phenomena; so that the
great end of all our positive studies, the exact prevision of
events, has been attained as completely as could be desired,
in regard alike to the certainty and extent of the prevision.
We have seen how this science must operate in liberating
the human intellect for ever from all theological and
metaphysical thraldom by showing that the most general
phenomena are subjected to invariable relations, and that
the order of the heavens is necessary and spontaneous.
This last consideration belongs more particularly to a sub-
sequent part of this work ; but it has been our business to
point out as we went along how the development of astro-
nomical science has shown us that the universe is not
destined for the passive satisfaction of Man; but that
Man, superior in intelligence to whatever else he sees, can
modify for his good, within certain determinate limits, the
system of phenomena of wdiich he forms a part,—being
enabled to do this by a wise exercise of his activity, dis-
engaged from all oppressive terror, and directed by an
accurate knowledge of natural laws. Lastly, we have seen
that the field of positive philosophy lies wholly within the
limits of our solar system, the study of the universe being
inaccessible in any positive sense.1
1 As before remarked, M. Comte speaks much too absolutely
here, in oversight of what modern astronomical researches have
really accomplished.—J. P. N.231
BOOK III.
PHYSICS.
CHAPTER I.
GENERAL VIEW.
ASTRONOMY was a positive science, in Imperfect
its geometrical aspect, from tire earliest condition of
days of the School of Alexandria; but tlie science.
Physics, which we are now to consider, had no positive
character at all till GTalileo made his great discoveries on
the fall of heavy bodies. We shall find the state of
Physics far less satisfactory than that of Astronomy, not
only on account of the greater complexity of its pheno-
mena, but under its speculative aspect, from its theories
being less pure and systematized, and, under its practical
aspect, from its previsions being less extended and exact.
The precepts of Bacon and the conceptions of Descartes
have advanced it considerably in the last two centuries, in
its character of a positive science; but the empire of the
primitive metaphysical habits is not to be at once over-
thrown; and Physics could not be immediately imbued
with the positive spirit, which Astronomy itself, our only
completely positive science, did not assume in its mechanical
aspect till the middle of that period. The further we go
among the sciences, the more we shall find of the old un-
scientific spirit, and not only in their details, but impairing
their fundamental conceptions. If we now compare the
philosophy of Physics with the perfect model offered to us
by astronomical philosophy, I hope we shall perceive the
possibility of giving to it, and afterwards to the other
sciences in their turn, the same positivity as the first,232
POSITIVE PHILOSOPHY.
though, their phenomena are far from admitting of an
equal perfection of simplicity and generality.
First, we must see what is the domain of Physics,
properly so called.
r, ,	.	Taken together with Chemistry (for the
s i omain. present), the object of the two is the know-
ledge of the general laws of the Inorganic world. This
study has marked characters, to be analysed hereafter,
distinguishing it from the science of Life, which follows it
in our encyclopedic scale, as well as from that of astronomy
which precedes it. The distinction between
Chemistry. Physics and Chemistry is much less easy to
establish; and it is one more difficult to
pronounce upon from day to day, as new discoveries
bring to light closer relations between them. Though the
division between these sciences is less obvious than between
any other two in the scale, it is not the less real and indis-
pensable, as we shall see by three considerations which,
perhaps, might be insufficient apart, but which, when
united, leave no uncertainty.
Its "eneralitr First, the generality which characterises
” ° c ’ physical researches contrasts with the spe-
ciality inherent in the chemical. Every physical con-
sideration is applicable to all bodies whatever, while
chemistry studies the action appropriate to a particular
substance. If we look at their classes of phenomena, we
find that gravity manifests itself in the same way in all
bodies; and the same with phenomena of heat, of sound,
of light, and even electrical effects. The difference is only
in degree. But, in the compositions and decompositions
of Chemistry, w'e have to deal with specific properties,
which vary not only in elementary substances, but in their
most analogous combinations. The only exception which
can be alleged, in the whole domain of Physics, is that of
magnetic phenomena; but modern researches tend to prove
that they are a mere modification of electrical phenomena,
which are unquestionably general. The general properties
of Physics were, in the metaphysical days of the science,
regarded as consisting of two classes; those which were
necessarily, and those which were contingently universal.
But the false distinction arose from the notion of that age,DISTINCTION BETWEEN PHYSICS AND CHEMISTRY. 233
that the business of science was to inquire into the nature
of bodies,—the study of their properties being a mere
secondary affair. Now that we know our business to be
with the properties alone, we see the error, and need only
ask whether we can conceive of any body absolutely devoid
of weight, or of temperature.
In the second place, Physics relates to Dealing with
masses, and Chemistry to molecules; inso- masses or
much that chemistry was formerly called molecules.
Molecular Physics. But, real as this distinction is, we
must not carry it too far, but remember that purely
physical action is often as molecular as chemical action;
as in the case of gravity. Physical phenomena observed
in masses are usually only the sensible results of those
which are going on among their particles: and, at most,
we can except from this only phenomena of sound, and
perhaps of electricity. As for the necessity of a certain
mass, to manifest physical action, that is equally indispens-
able in chemistry. The best way of expressing the general
fact which lies at the bottom of this distinction is, perhaps,
that in chemistry, one at least of the bodies concerned
must be in a state of extreme division ; while this is so far
from being a necessary condition of physical action that it
is rather an impediment to it. This is a proof of a real dis-
tinction between the two sciences, though it may not be a
very marked one.
In the third place, the constitution of changes of
bodies,—the arrangement of their molecules, arrangement or
—may be changed, in exhibiting physical composition
phenomena; but the composition of theirof molecules-
molecules remains unchangeable: whereas, in Chemistry,
not only is there always a change of state in one of the
bodies concerned, but the mutual action of the bodies alters
their nature; and it is this alteration which constitutes
the phenomenon. Many physical agents can, no doubt,
work changes of composition and decomposition, if their
operation be very energetic and prolonged; and it is this
which forms such connection as there is between the two
sciences: but, at that point of activity, physical agencies
pass the boundary, and become chemical.
Positive philosophy requires that we should draw off234
POSITIVE PHILOSOPHY.
altogether from the study of agents, to which it may be
imagined that phenomena are to be referred. Any number
of persons may discover a supposed agent,—as, for instance,
the universal ether of modern philosophers, by which a
variety of phenomena may be supposed to be explained;
and we may not be able to disprove such an agency. But
we have no more to do with modes of operation than with
the nature of the bodies acted upon. We are concerned
with phenomena alone; and what we have to ascertain is
their laws. In departing from this rule, we leave behind
us all the certainty and consistency of real science.
Keeping within our true limits, then, we see that if chemical
phenomena should be reduced by analysis into the form of
purely physical actions,—an achievement very possible to
the present generation of scientific men,—our fundamental
distinction between the two sciences will not be shaken.
It will still be true that in a chemical fact something more
is involved than in a simply physical one: namely, the
characteristic alteration undergone by the molecular com-
position of the bodies, and therefore by the whole of their
properties. Such a distinction is secure amidst any
scientific revolution that can ever happen.
^	. .	From these three considerations, taken
o/physics!*	together, we derive our description of Physics.
This science consists in studying the laws
which regulate the general properties of bodies, commonly
regarded in the mass, and always placed in circumstances
which admit of their molecules remaining unaltered, and
generally in their state of aggregation. With a view to
the great end of all science, we must add that the aim of
physical theories is to foresee, as exactly as possible, all
the phenomena that will be exhibited by a body placed in
any set of given circumstances, excluding, of course, such
as could alter its nature. This is not the less true because
we can rarely attain the prescribed aim. The imperfection
is in our knowledge alone. In estimating the true cha-
racter of any science, the only way is, first, to suppose the
science perfect, and then to study the fundamental difficul-
ties presented by this ideal perfection.
Our description shows us how much more complexity we
shall find in physical than in astronomical inquiries. InMETHODS OF PHYSICAL INQUIRY.
235
astronomy we study bodies, known to us only T ,
by sight, under two aspects only, their forms
and motions. All considerations but these are excluded.
But in Physics, on the contrary, the bodies we have to
study are recognized by all our senses, and are regarded
under an aggregate of general conditions, and therefore
amidst a complication of relations. It is clear, not only that
this science is inferior to astronomy, but that it would be
impracticable if the group of fundamental obstacles was
not compensated for, up to a certain point, by the extension
of our means of exploration. We meet here the law,
before laid down, that in proportion as phenomena become
complicated they thereby become explorable under a pro-
portionate variety of relations.
Methods of
inquiry.
Observation.
Of the three procedures which constitute
our art of observing, the last, Comparison,
is scarcely more applicable here than with
regard to astronomical phenomena. Its proper application
is, in fact, to the phenomena of organized bodies, as we
shall see hereafter. But the other two methods are en-
tirely suitable to Physics. Observation was,
in astronomy, restricted to the use of a single
sense; but in Physics, all our senses find occupation.
Yet would Observation effect little without the aid of
Experiment, the regulated use of which is the great
resource of physicists in all questions that involve any
complexity. This procedure consists in ob- F ,	,
serving beyond the range of natural circum- ‘1
stances;—in placing bodies in artificial conditions, expressly
instituted to enable us to examine the action of the pheno-
mena we wish to study under a particular point of view.
We can see at once how eminently this art is adapted to
physical researches; and how it must there find its
triumphs: since there are hardly any bounds to our power
of modifying bodies, for the purpose of studying their
phenomena. In chemistry, experiment is commonly sup-
posed to be more complete than in any other department:
but I think it is of a higher order in -physics, for the reason
that in chemistry the circumstances are always artificially
arranged, while in physics we have the choice of natural236
POSITIVE PHILOSOPHY.
or artificial circumstances; and the philosophical character
of experimentation consists in choosing the freest possible
case that will show us what we want. We have a wider
range, and a choice of simpler cases, in physics than in
chemistry; and in physics, therefore, is experiment
supreme.
Application of The next great virtue of physics is its
mathematical allowing the application of mathematical
analysis.	analysis, which enters into this science, and
at present goes no further;—not yet, with real efficacy,
into chemistry. It is less perfect in physics than in astro-
nomy ; but there is still enough of simplicity and fixedness
in physical phenomena to allow of its extended use. Its
employment may be direct or indirect:—direct when we
can seize the fundamental numerical law of phenomena, so
as to make it the basis of a series of analytical deductions ;
as when Fourier founded his theory of the distribution of
heat on the principle of thermological action between two
bodies being proportionate to the difference of their tem-
peratures : and indirect, when the phenomena have been
referred to some geometrical or mechanical laws; when,
however, it is not properly to physics that the analysis is
applied, but to geometry or mechanics. Such are the
cases of reflection or refraction in the geometrical relation;
and in the mechanical, the investigation of weight, or of a
part of acoustics. In either case extreme care is requisite
in the first application, and the further development should
be vigilantly regulated by the spirit of physical research.
The domain of physics is no proper field for mathematical
pastimes. The best security would be in giving a geo-
metrical training to physicists, who need not then have
recourse to mathematicians, whose tendency it is to despise
experimental science. By this method will that union
between the abstract and the concrete be effected which
will perfect the uses of mathematical, while extending the
positive value of physical science. Meantime, the uses of
analysis in physics are clear enough. Without it we should
have no precision, and no co-ordination ; and what account
could we give of our study of heat, weight, light, etc. ?
We should have merely series of unconnected facts, in
which we could foresee nothing but by constant recourseRELATIONS TO OTHER SCIENCES.
237
to experiment; whereas, they now have a character of
rationality which fits them for purposes of prevision.
From the complexity of physical phenomena, however, the
difficulty of the mathematical application is great. In
some, a part of the essential conditions of the problem
must be thrown out, to admit of its transformation into a
mathematical question; and hence the necessity for reserve
in the employment of analysis. The art of combining
analysis and experiment, without subordinating the one to
the other, is still almost unknown. It constitutes the last
advance of the true method of physical study; and it will
be developed when physicists, and not geometers, conduct
the analytical process, and not till then.
Having seen what is the object of Physics, Encyclopedi-
and what the means of investigation, we cal Rank of
have next to fix its position in the scientific Physics,
hierarchy.
The phenomena of Physics are more complicated than
those of Astronomy; and Astronomy is the scientific basis
and model of Physics, which cannot be effectually studied
otherwise than through the study of the more simple and
general science. In this, we individually follow the course
of our race. It was by Astronomy that the T> . ,.
positive spirit was introduced into natural Astronomy
philosophy, after it had been sufficiently
developed by purely mathematical investigations. Our
individual education is in analogy with this: for we have
learned from astronomy what is the real meaning of the
explanation of a phenomenon, without any impracticable
inquiry about its cause, first or final, or its mode of produc-
tion. Physics should, more than the other natural sciences,
follow closely upon astronomy, because, after astronomy,
its phenomena are less complex than any.
Besides these reasons belonging to Method, there is the
grand consideration that the theories of astronomy afford
the only data for the study of terrestrial physics. Our
position in the solar system, and the motions, form, size,
and equilibrium of the mass of our world among the other
planets, must be known before we can understand the
phenomena going on at its surface. What could we make
of weight, for instance, or of the tides, without the data238
POSITIVE PHILOSOPHY.
afforded by astronomical science ? These phenomena indeed
make the transition from astronomy to physics almost
insensible. In this way Physics is indi-
Mathematics rec^7 connected with Mathematics. There
is also a direct connection, as some physical
phenomena have a geometrical and mechanical character,—
as much as those of astronomy, though under a great com-
plication of the circumstances. The abstract laws of space
and motion must prevail as much in the one science as in
the other. If the relation is thus unquestionable in the
doctrine, it is not less so in the spirit and method which we
must bring to the study of physics. It must be ever re-
membered that the true positive spirit first came forth
from the pure sources of mathematical science; and it is
only the mind that has imbibed it there, and which has
been face to face with the lucid truths of geometry and
mechanics, that can bring into full action its natural posi-
tivity, and apply it in bringing the most complex studies
into the reality of demonstration. No other discipline
can fitly prepare the intellectual organ. We might further
say that, as geometrical ideas are more clear and funda-
mental than mechanical ideas, the former are more neces-
sary, in an educational sense, to physicists than the latter,
though the use of mechanical ideas is the more immediate
and extended in physical science. Thus we see how we
must conclude that the education of physicists must be
more complicated than that of astronomers. Both have
need of the same mathematical basis, and physicists must
also have studied astronomy, at least in a general way.
And this, again, assigns the position of their science.
Its rank is equally clear, if we look in the opposite direc-
tion,—at the sciences which come after it.
,	It can hardly be by accident that, in all
the rest1 °	languages of thinking peoples, the word which
originally indicated the study of the whole
of nature should have become the name of the particular
science we are now considering. Astronomy is, in fact, an
emanation from mathematics. Every other natural science
was once comprehended under the term Physics ; and that
to vphich it is now restricted must be supreme over the
rest. Its relation to the rest is just this: that it investi-RELATION TO HUMAN PROGRESS.
239
gates the general properties common to all bodies ; that is,
the fundamental constitution of matter; while the other
sciences exhibit the modifications of those properties peculiar
to each: and the study of those properties in the general
must, of course, precede that of their particular cases. In
regard to Physiology, for instance, it is clear that organized
bodies are subject to the general laws of matter, those laws
being modified in their manifestations by the characteristic
circumstances of the state of Life. The same is the case
with Chemistry. Without admitting the questionable
hypothesis under which some eminent men of our time
refer all chemical phenomena to purely physical action, it
is yet evident that the concurrence of physical influences
is indispensable to every chemical act. What could we
make of any phenomenon of composition or decomposition,
if we left out all data of weight, heat, electricity, etc. ? And
how could we estimate the chemical power of these various
agents without first knowing the laws of the general influ-
ence proper to each? Chemistry is closely dependent on
Physics ; while Physics is wholly independent of Chemistry.
As for the direct operation of this science Relation to
on the human intellect, it is less marked than human pro-
that of the two natural sciences which occupy gress.
the extremities of the scale,—astronomy and physiology,—
which immediately contemplate the two great objects of
human interest,—the Universe and Man: but one striking
fact with regard to Physics is that it has been the great
battle-ground between the old theological and metaphysical
spirit and the positive philosophy. In Astronomy the
positive philosophy took possession, and triumphed almost
without opposition, except about the earth’s motion ; while
in the domain of Physics the conflict has gone on for cen-
turies ; a circumstance attributable to the imperfection of
physical, in comparison with astronomical science.
With this science begins the exhibition of Human power
human power in modifying phenomena. In of modifying
Astronomy, human intervention was out of phenomena,
the question: in Physics it begins; and we shall see how
it becomes more powerful as we descend the scale. This
power counterbalances that of exact prevision which we
have in astronomy, through its extreme simplicity. The240
POSITIVE PHILOSOPHY.
one power or the other,—the power of foreseeing or of
modifying,—is necessary to our outgrowth of theological
philosophy. Our prevision disproves the notion that phe-
nomena proceed from a supernatural will, which is the
same thing as calling them variable: and our ability to
modify them shows that the powers under which they
proceed are subordinated to our own. The first is the
higher order of proof ; but both are complete in their way,
and certain to command, sooner or later, universal assent.
The proof which Franklin afforded of human control over
the lightning destroyed the religious terror of thunder as
effectually as the superstition about comets was destroyed
by the prevision of their return; though the experiments
bv which Franklin established the identity of the lightning
with the common electric discharge could be decisive only
with physicists, while the generality of men could under-
stand how the return of comets was foreseen. As the
opposition between the theological and the positive philo-
sophies becomes less simply evident, our power of interven-
tion becomes more varied and extended; the amount of
proof yielded in the two cases being equal in the eyes of
men in general, though not strictly equivalent.
. .	.	In regard to the speculative rank of Phy-
perfect011 im s^cs> ^ *s c^ear ^ d°es n°t admit of pre-
vision to any extent at all comparable to that
of astronomy, because it consists of numerous branches,
scarcely at all connected with each other, and concurring
only in a feeble and doubtful way in its chief phenomena.
We can therefore see only a little way forward; often
scarcely beyond the experiment in hand: but we shall see
its speculative superiority to the sciences which come after
it, when, in studying chemistry and physiology, we find
another kind of incoherence existing among their pheno-
mena, making prevision more imperfect still. The great
distinction of Physics is that which has been referred to
before,—that it instructs us in the art of Experiment.
Philosophers must ascend to this source of experimentation,
whatever their special objects may be, to learn what are the
spirit and conditions of true experimentation, and what the
necessary precautions. Each of the sciences in the scale
presents, besides the characters of the positive methodTHE SCIENTIFIC USE OF HYPOTHESIS.
241
which are common to them all, some indication appropriate
to itself, which ought to be studied at its Characteris-
source, to be duly appreciated. Mathema- tics of each
tical science exhibits the elementary con- science,
ditions of positivity: astronomy determines the true study
of Nature: physics teaches us the theory of experimen-
tation : chemistry offers us the art of nomenclature : and
physiology discloses the true theory of classification.
I have deferred till now what I have to offer on the
important subject of the rational construction and scientific
use of hypotheses, regarded as a powerful and indispensable
auxiliary to our study of nature. It is in the p, ..	. f
region of astronomy that I must take my hypothesis °
stand in discussing this subject, though it
was not necessary to advert to it while we were surveying
that region. Hypothesis is abundantly employed in astro-
nomy ; but there it may be said to prescribe the conditions
of its own use,—so simple are the phenomena in question
there. From thence do I think it necessary to derive,
therefore, our conceptions of the character and rules of this
valuable resource, in order to its employment in the other
departments of natural philosophy.
There are only two general methods by which we can get
at the law of any phenomenon,—the immediate analysis of
the course of the phenomenon, or its relation to some more
extended law already established; in other words, by induc-
tion or deduction. Neither of these methods would help
us, even in regard to the simplest phenomena, if we did
not begin by anticipating the results, by making a pro-
visional supposition, altogether conjectural in the first
instance, with regard to some of the very notions which are
the object of the inquiry. Hence the necessary introduc-
tion of hypotheses into natural philosophy. The method of
approximation employed by geometers first suggested the
idea; and without it all discovery of natural laws would
be impossible in cases of any degree of complexity ; and in
all, very slow, But the employment of this instrument
must always be subjected to one condition, the neglect of
which would impede the development of real knowledge.
This condition is to imagine such hypotheses Necessary con -
only as admit, by their nature, of a positive dition.
I.	R242
POSITIVE PHILOSOPHY.
and inevitable verification at some future time,—the pre-
cision of this verification being proportioned to what we
can learn of the corresponding phenomena. In other
words, philosophical hypotheses must always have the
character of simple anticipations of what we might know
at once, by experiment and reasoning, if the circumstances
of the problem had been more favourable than they
are. Provided this rule be scrupulously observed, hypo-
theses may evidently be employed without danger, as often
as they are needed, or rationally desired. It is only sub-
stituting an indirect for a direct investigation, when the
latter is impossible or too difficult. But if the two are not
employed on the same general subject, and if we try to
reach by hypothesis what is inaccessible to observation and
reasoning, the fundamental condition is violated, and
hypothesis, wandering out of the field of science, merely
leads us astray. Our study of nature is restricted to the
analysis of phenomena in order to discover their laws; that
is, their constant relations of succession or similitude; and
can have nothing to do with their nature, or their cause,
first or final, or the mode of their production. Every
hypothesis which strays beyond the domain of the positive
can merely occasion interminable discussions, by pretending
to pronounce on questions which our understandings are
incompetent to decide.—Every man of science admits this
rule, in its simple statement; but it cannot be practically
understood,—so often as it is violated, and to such a degree
as to alter the whole character of Physics. The use of
conjecture is to fill up provisionally the intervals left here
and there by reality; but practically we find the two mate-
rials entirely separated, and the real subordinated to the
conjectural. It is necessary, therefore, to ascertain and
explain the actual state of the question with regard to
Physics.
The hypotheses employed by physical in-
hypothesisS° quirers in our day are of two classes: the
first, a very small class, relate simply to the
laws of phenomena: the other, and larger class, aim at de-
termining the general agents to which different kinds of
natural effects may be referred. Now, according to the
rule just laid down, the first kind alone are admissible:THE TWO CLASSES OF HYPOTHESES.
243
the second have an anti-scientific character, are chimerical,
and can do nothing but hinder the progress of science.
In Astronomy, the first class only is in use,
because the science has a wholly positive indispensable
character. A fact is obscure; or a law is
unknown: we proceed to form a hypothesis, in agreement,
as far as possible, with the whole of the data we are in
possession of; and the science, thus left free to develop
itself, always ends by disclosing new observable conse-
quences, tending to confirm or invalidate, indisputably, the
primitive supposition. We have before noticed frequent
and happy examples of this method of discovering the
primary truths of astronomy. But, since the establish-
ment of the law of gravitation, geometers and astronomers
have put away all their fancies of chimerical fluids causing
planetary motions; or have, at least, indulged in them
merely as a matter of personal taste, and not of scientific
investigation. It would be well if, in a study so much
more difficult as Physics, philosophers would imitate the
astronomers. It would be well if they would
confine their hypotheses to the yet unknown chimericalSS
circumstances of phenomena, or their yet
hidden laws, and would entirely let alone their mode of
production, which is altogether beyond the limit of our
faculties. What scientific use can there be in fantastic
notions about fluids and imaginary ethers, which are to
account for phenomena of heat, light, electricity and
magnetism ? Such a mixture of facts and dreams can
only vitiate the essential ideas of physics, cause endless
controversy, and involve the science itself in the disgust
which the wise must feel at such proceedings. These
fluids are supposed to be invisible, intangible, even im-
ponderable, and to be inseparable from the substances
which they actuate. Their very definition shows them to
have no place in real science; for the question of their
existence is not a subject for judgment: it can no more be
denied then affirmed: our reason has no grasp of them at
all. Those who, in our day, believe in caloric, in a lumi-
nous ether, or electric fluids, have no right to depise the
elementary spirits of Paracelsus, or to refuse to admit
angels and genii. We find them spurning Lamarck’s notion244
POSITIVE PHILOSOPHY.
of a resonant fluid; but the misfortune of this hypothesis
was that it came too late,—long after the establishment of
acoustics. If it had been put forth as early in the days of
science as the hypotheses about heat, light, and electricity,
this resonant fluid would no doubt have prospered as well
as the rest. Without going into the history of more of
these baseless inquiries, it is enough to point out that they
are irreconcileable with each other; and when superficial
minds witness the ease with which they destroy each other,
they naturally conclude the whole science to be arbitrary,
consisting more in futile discussion than in anything else.
Each sect or philosopher can show how untenable is the
hypothesis of another, but cannot establish his own; and
it would generally be easy to devise a third which might
agree with both. It is true, physicists are now eager to
declare that they do not attribute any intrinsic reality to
these hypotheses ; and that they countenance them merely
as indispensable means for facilitating the conception and
combination of phenomena. But we see here the working
of an incomplete positivity, which feels the inanity of such
systems, and yet dares not surrender them. But besides
that it is scarcely possible to employ a fictitious instrument
as a reality without at times falling into the delusion of
its reality, what rational ground is there for proceeding in
such a way, when we have before us the procedure and
achievements of astronomical science, for a pattern and a
promise ? These hypotheses explain nothing. For instance,
the expansion of bodies by heat is not explained,—that is,
cleared up,—by the notion of an imaginary fluid interposed
between the molecules, which tends constantly to enlarge
their intervals; for we still have to learn how this sup-
posed fluid came by its spontaneous elasticity, which is, if
anything, more unintelligible than the primitive fact.
And so on, through the whole range. These hypotheses
clear away no difficulties, but only make new ones, while
they divert our attention from the true object of our
inquiries. As for the plea that habit has so taken hold of
the minds of inquirers, that they would be adrift if de-
prived of all their moorings at once, and that their language
must be superseded by a wholly new one, I think this kind
of difficulty is very much exaggerated. We have seen,HISTORY OF THE SECOND CLASS.
245
within half a century, how often men have contrived to
pass from some physical systems to their opposites without
being much hindered by obstacles of language. There
would be scarcely more difficulty in casting aside futile
hypotheses; and we might, as we see by existing examples,
gradually substitute the real and permanent meaning of
scientific terms for the fanciful and variable interpre-
tation.
These fluids are nothing more than the old „ f
entities materialized. Whichever way we second-class16
look at it, what is heat apart from the warm
body, light apart from the luminous body,—electricity
apart from the electric body P Are they not pure entities,
like thought apart from the thinking body, and digestion
apart from the digesting body ? Here we have, instead of
abstract beings, imaginary fluids deprived, by their very
definition, of all material qualities, so that we cannot even
suppose in them the limit of the most rarefied gas. If the
descent of these from the old entities be not recognized,
what filiation of ideas can ever be admitted ? The essential
character of metaphysical conceptions is to attribute to
properties an existence separate from the substance which
manifests them. WTiat does it matter whether we call
these abstractions souls or fluids ? The origin is always the
same ; and it is connected with that inquisition into the
essence of things which always characterizes the infancy
of the human mind, occasioning, first, the conception of
gods, which grew into that of souls, which became in time
imaginary fluids. In all positive science, our understand-
ings, unable to pass abruptly from the metaphysical to the
positive stage, have travelled through this transition state
of development. Metaphysics itself is the transition stage
from theology to positive science; but a secondary tran-
sition is also necessary, as we see by the fact; a transition
from metaphysical to positive conceptions. The mathe-
maticians and astronomers have attained the positive basis.
The physicists, the chemists, the physiologists, and the
social philosophers, are now in the last period of tran-
sition ; the physical inquirers, ready to pass up to the level
of the astronomers and geometers, and all the others held
back for a while by the complexity of their respective sub-246
POSITIVE PHILOSOPHY.
jects; as we shall see hereafter. This bastard positivism
was the way out of the old metaphysical condition, in
which men would, but for it, have been imprisoned to this
day. Nascent science first humoured the constitutional
need, and then led us on by offering to our minds, in the
place of the old scholastic entities, new entities, more
tangible, which must by their nature introduce into our
studies the contemplation of phenomena and their laws,
restricting us to these more and more. This seems to have
been the important temporary use of this system of hypo-
theses ; to enable us to pass from the metaphysical to the
positive stage.
In Astronom Astronomy has not been exempted from
this transition state, any more than the other
sciences ; but it was over so long ago that it is forgotten,—
so few are those who are interested in the history of
philosophy ! If we look back to the action of the human
mind in the seventeenth century, we shall see how geo-
meters and astronomers were preoccupied with hypotheses
of the kind we are considering. There is no better example
of them than that famous conception, the Vortices of
Descartes; for it presents clearly the three stages of exis-
tence common to them all; the creation of the hypothesis,
its temporary use, and its rejection when its purpose is
answered. These vortices, so ridiculed by men who believe
in caloric, ether, and elecipc fluids, helped us to a sound
philosophy by introducing the idea of mechanism, where
even Kepler had imagined only the incomprehensible action
of souls and genii. When the discussion had attained the
firm ground of Celestial Mechanics, founded upon the
Newtonian theory, the influence of the Cartesian hypothesis
ceased to be progressive, and became retrograde. To the
last, the Cartesian philosophers insisted, in arguments as
plausible as those of our existing physicists, that it was
impossible to philosophize without such a hypothesis.
They were answered in the only effectual way, by philo-
sophizing in another mode: and the vortices were heard
of no more when geometers and astronomers apprehended
the true object of scientific studies. The Cartesian hypo-
thesis contributed to the education of the human mind by
leading it to see that we have nothing to do with theAPPLICATION IN PHYSICS.
247
primitive agents, or mode of production of phenomena,
but only with their laws. If, in the other sciences, we
have, as their professors assert, reached the stage of posi-
tivity, hypotheses like that of the vortices may be dismissed,
as no longer needed to bring us out of the metaphysical
state. As soon as they are needless, they become per-
nicious.
The transition has been obvious elsewhere jn pilvsjcs
than in astronomy. It has taken place in	^
the most advanced departments of Physics; and espe-
cially with regard to Weight. There was scarcely a philo-
sopher, even long after Galileo’s time, who had not some
system to offer about the causes of the fall of bodies. At
that time, such hypotheses appeared the only method of
studying weight; but who hears of them now ? Acoustics
was emancipated about the same time. The labours of
Fourier will evidently release thermology; and then there
will remain only the study of light and of electricity; 1
and no reason can be assigned for their exclusion from the
general rule. The question will be regarded as settled
henceforward by all who believe that the historical develop-
ment of the human mind is subject to natural laws, deter-
minate and uniform ; and such will admit, as the principle
of the true theory of hypotheses, that every scientific hypo-
thesis, to be a matter of judgment, must relate exclusively
to the laws of phenomena, and never to their mode of
production.
Here we find, as in every analogous case of difficulty,
the use of the comparative historical method which I have
just employed. We shall enlarge on this hereafter: mean-
time, I must offer the observation that the philosophy of
the sciences cannot be properly studied apart from their
history; and, conversely, that the history apart from the
philosophy, would be idle and unintelligible.
In reviewing the different departments of Rule of ar-
Physics, I shall follow the rule which deter- rangement in
mines the order of the sciences themselves: Physics,
that is, I shall take them in the order of the generality of
their phenomena, their simplicity, the relative perfection
1 In Electricity, the hypothesis of fluids is rapidly yielding
before the rational idea of Polarity.—J. P. N.248
POSITIVE PHILOSOPHY.
of our knowledge of them, and their mutual dependence.
Under this rule, we shall find that the departments that
offer themselves first border upon Astronomy, and those
that come last upon Chemistry.
First will come Weight, in solids and fluids, regarded
statically and dynamically. About this assignment, there
are no two opinions; weight being absolutely universal.
Its phenomena are simple, independent of others, and so
exactly understood that science is here almost as positive
as in astronomy, to which it is very nearly allied. Electric
phenomena being the most opposite of all to those of
weight, in all these particulars, will come last; and they
are closely allied to chemistry. Between them will come
thermology, acoustics, and optics. Fourier put away, in
his study of heat, all fantastic notions about imaginary
fluids, and brought his subject up to such a point of
positivity as to place it next to the study of gravity.
Acoustics might, perhaps, contest its place with thermology,
but for the generality of the phenomena of the latter. In
regard to positivity, there is little to choose between them:
but there are gaps in our knowledge of acoustics which
Qr(jer	also indicate the lower place for it. Our
order then is,—Barology, or the science of
weight: thermology, or the science of heat: acoustics,
optics, and electricity. But we must beware of attaching
too much importance to this arrangement, which is really
little better than arbitrary, though as good as our present
knowledge admits. We shall now proceed to a philo-
sophical review of them, exempt from details; having, in
this chapter, analysed the proper object of Physics; the
modes of investigation appropriate to it; its position in
regard to the other sciences ; its influence upon the educa-
tion of human reason; its degree of scientific perfection:
its incomplete positivity at present; the means of remedy-
ing this by a sound institution of hypotheses; and finally
the rational distribution of its different departments.249
CHAPTER II.
BAROLOGY.
NOTWITHSTANDING the advanced state of our means
for the study of Barology, we have no complete theory
of weight, but only fragmentary portions of a theory, dis-
persed through treatises on rational mechanics or physics.
It will be of great advantage to bring them together.
The division of the subject is into two Divisions
principal sections, subdivided into three;
the Statical and Dynamical consideration of weight, in its
application to solids, liquids, and gases. Both philoso-
phically and historically, this division is indicated.
SECTION I.
STATICS.
Taking the statics of gravity first, we must statics of
point out that we owe our elementary notions Gravity,
of positive barology to Archimedes. He first
clearly established that the statical effort produced in a
body by gravity,—that is, its weight,—is entirely inde-
pendent of the form of the surface, and	H' to v
depends only on the volume, as long as the	*
nature and constitution of the body remain unchanged.
This may appear to us very simple; but it is not the less
the true germ of a leading proposition in natural philo-
sophy, which was perfected only at the end of the last
century; namely, that not only is the weight of a body
independent of its form, and even of its dimensions, but
of the mode of aggregation of its particles, and of all
variations which can occur in their composition, even by
the different vital operations; in a word, that this quality
is absolutely unalterable, except by the circumstance of its250
POSITIVE PHILOSOPHY.
distance from the centre of the earth. Archimedes could
take none but geometrical circumstances into the account;
but, in this elementary relation, his work was complete.
He not only discovered that, in homogeneous masses, the
weight is always in proportion to the volume; but he dis-
closed the best means' of measuring, in solid bodies, by his
famous hydrostatic principle, the specific co-efficient which
enables us to estimate, according to this law, the weight
and volume of the body, by means of each other. We owe
to him too the idea of the centre of gravity, together with
the first development of the corresponding geometrical
theory. Under this view, all problems respecting the equi-
librium of solids are included in the domain of rational
mechanics : so that, except the important relation of weight
to masses, which could be fully known only to the moderns,
Archimedes ought to be regarded as the true founder of
statical barology, in relation to solids. There is, however,
another leading idea which was not clear in the time of
Archimedes, though it became so, soon afterwards; that
of the law of the direction of gravity, which men spon-
taneously considered to be constant, and which the school
of Alexandria ascertained to vary from place to place,
always being perpendicular to the surface of the terrestrial
globe; a discovery which is evidently due to astronomy,
by which alone the means are offered of manifesting and
measuring, by comparison, the divergence of verticals.
Cases of	The anc^en^s ^ia(^ n0 accurafe ideas about
liqnids°	the equilibrium of liquids; for Archimedes
contemplated only the equilibrium of solids
sustained by liquids. His principle did not result, as with
us, from an analysis of the pressure of the liquid, against
the vessel containing it, thus disclosing the total force
exercised by the liquid in sustaining the weight. The
theory of gravitating liquids is to be ascribed to the
moderns.
The mathematical characters of fluids is that their mole-
cules are absolutely independent: and the geometrical
character of liquids is that they are absolutely incompres-
sible. Neither of these is strictly true. The mutual ad-
herence of fluid molecules now forms an interesting section
of physics; and, as for the compressibleness of liquids, itCASE OF LIQUIDS.
251
was indicated by several phenomena,—especially the trans-
mission of sound by water,—and it is now proved by un-
questionable experiments. The contraction as yet produ-
cible is very small; and we do not know what law the
phenomenon follows, in its relation to variation of pressure.
But this uncertainty dues not affect the theory of the equili-
brium of natural liquids, owing to the extreme smallness of
the condensation. In the same way, imperfect fluidity is
no hindrance, provided the mass has a certain extension.
We may therefore put aside these exceptions, and proceed
to consider the equilibrium of gravitating liquids, in the
two cases in which they are studied : in a mass so limited
that the verticals are parallel, which is the ordinary case;
or in a great mass, as that of the sea, in which we have to
allow for the variable direction of gravity.
In the first case, there is clearly no difficulty Yirst ca^e
about the surface : and the whole question is
of the pressure against the enclosing vessel. Stevinus,
following the principle of Archimedes, showed that the
pressure upon a horizontal boundary, or floor, is always
equal to the weight of the liquid column of the same base
which should issue at the surface of equilibrium: and he
afterwards resolved into this the case of an inclined boun-
dary, by decomposing it into horizontal elements, as we
now do. From this it appeared that the pressure is always
equal to the weight of a vertical column which should have
the proposed boundary for its base ; and for its height that
of the surface of equilibrium above the centre of gravity of
this boundary. According to that, the infinitesimal analysis
enables us easily to calculate the pressure against any de-
finite portion of any curved surface. The most interesting
physical result is the estimate of the total pressure sup-
ported by the whole of the vessel, which is necessarily equi-
valent to the weight of the liquid it contains. The equili-
brium of floating bodies is only a simple application of this
rule of measurement of pressure. The immersed part of
the solid is a boundary ; and it is clear that the pressure of
the liquid to sustain the body is equivalent to a vertical
force equal to the weight of the displaced fluid, and applied
to the centre of gravity of the immersed portion. This
rule is precisely the principle of Archimedes. The main252
POSITIVE PHILOSOPHY.
problem was geometrically treated by him. The only really
difficult research in this matter relates to the conditions of
stability of this equilibrium, and the analysis of the oscil-
lations of the body floating round its stable position; and
this is one of the most complex applications of the dynamics
of solids. If the question was of the vertical oscillations
of the centre of gravity, the study would be easy, because
we can estimate at once the way in which the pressure in-
creases as the body is further immersed, and diminishes as
it rises, tending always to a return to the primitive state.
But it is otherwise with oscillations from rotation, whether
of rolling or pitching, the theory of which is a matter of
much interest in naval art. Here, the mathematical diffi-
culties of the problem can be met only by abstracting the
resistance and agitation of the liquid; and the labours of
geometers become merely mathematical exercises, of no
practical use.
Second case The	the equilibrium of the vast
liquid masses which compose the greater part
of the earth’s surface is clearly connected with the general
theory of the form of the planets: but difficulties, uncon-
nected with the figure of the planets, intervene, and cannot
be entirely surmounted. Rational hydrostatics shows us that
equilibrium is possible when there is the same density at all
points equally distant from the earth’s centre ; a condition
which is impossible under our variety of temperatures in
different positions. There is no rational result from any
practicable study of currents, of varying temperatures,
of the compressibleness of liquids, all of which, though
following unknown laws, are necessary to the solution of
the problem. We have no better resource, at present, than
in empirical studies : and these, which belong more to the
natural history of the globe than to physics, are very im-
perfect.
The theory of tides will hereafter, when sciences and
their arrangements are more perfected, take its place in the
department of barology. The periodical disturbances of the
equilibrium of the ocean are a proper subject for study in
connection with terrestrial gravity; and it can make no
difference that the cause of those perturbations is found in
the planetary system.CASE OF GASES.
253
In studying the third question of equili- r ,
brium, that of gases, we meet with a difficulty ase 0 gases*
which does not pertain to that of solids and liquids;—we
have to discover the gravity of the general medium in
which we live. In the case of liquids, we have only to
weigh an empty vessel, and then the same vessel filled;
whereas, in the case of the atmosphere, a vacuum cannot be
created but by artificial means, which must themselves be
founded on a knowledge of the weight of the medium to be
weighed. The fact can be ascertained only by indirect
means ; and those are derived from the theories of pressure
which we have been noticing. Stevinus was not thinking
of the atmosphere in elaborating his theory; but, as it
answered for heterogeneous liquids, it must answer for the
atmosphere. From that date, the means for H. ,
ascertaining the equilibrium of the atmo-	5
sphere, in a positive manner, were provided. Galileo pro-
jected the work, in his last years ; and it was well executed
by his illustrious disciple, Torricelli. He proved the exis-
tence and measurement of atmospheric pressure by showing
that this pressure sustained different liquids at heights in-
versely proportioned to their densities. Next, Pascal estab-
lished the necessary diminution of this pressure at increas-
ing heights in the atmosphere; and Guericke’s invention of
the air-pump, an inevitable result of Torricelli’s discovery,
gave us direct demonstration in the power of making a
vacuum, and consequently of estimating the specific gravity
of the air which surrounds us, which had hitherto been
only vaguely computed. The creation and improvement of
instruments of observation is an invariable consequence of
scientific discovery ; and, in this case, the fruits are the
barometer and the air-pump.
One condition remained, before we could apply the laws
of hydrostatics to atmospheric equilibrium. We had to
learn the relation between the density of an elastic fluid
and the pressure which it supports. In liquids (supposed
incompressible) the two phenomena are mutually indepen-
dent ; whereas, in gases they are inevitably connected: and
herein lies the essential difference between the mechanical
theories of the two fluids. The discovery of this elementary
relation was made about the same time by Mariotte in254
POSITIVE PHILOSOPHY.
France and Boyle in England. These illustrious philoso-
phers proved by their experiments that the different
volumes successively occupied by the same gaseous mass
are in an inverse ratio to the different pressures it receives.
This law has since been verified by increasing the pressure
to nearly that of thirty atmospheres; and it has been
adopted as the basis of the whole Mechanics of gas and
vapours. But we must beware of accepting it as the
mathematical expression of the reality; for it is evidently
the same thing as regarding elastic fluids as always equally
compressible, however compressed they may already be;
or, conversely, as always equally dilatable, however dilated
they may already be :—suppositions which cannot be inde-
finitely extended. But thus it is, more or less, with all the
laws we have ascertained in our study of nature. They
approximate, within narrow limits, we must suppose, to
the mathematical reality: but they are not that reality
itself, even in the grand instance of gravitation. These
laws are sufficient for our use and guidance; and that is
all the result that positive science pretends to.
Under the law of Mariotte and Boyle, the
tli^problem theory of atmospheric equilibrium falls into
the department of Rational Mechanics. We
see at once that the air can no more be in a state of real
equilibrium than the ocean ; and so much the further from
it as heat expands air more than water. Yet we must
conceive of the partial equilibrium of a very narrow atmo-
spheric column, to form a just general idea of the mode of
diminution in regard to the density and pressure of the
different strata. Putting aside considerations of heat, and
the small effects of gravitation in such a case, we see that
density and pressure must diminish in a geometrical pro-
gression for altitudes increasing in arithmetical progres-
sion: but this abstract variation is retarded by the
diminishing heat of the loftier atmospheric strata, which
makes each stratum more dense than it would be from its
position. Here, therefore, we are stopped by the inter-
vention of a new element which we do not understand,—
the law of the vertical variation of atmospheric tempera-
tures—our ignorance of which can be supplied only by
inexact and uncertain expedients. G-reat caution is neces-DYNAMICS OF BAROLOGY.
255
sary in using Bouguer’s method of measuring altitudes by
the barometer; a method very ingenious, but depending on
such complex and uncertain conditions, and requiring some-
times so much delay, that it is even preferable, when cir-
cumstances permit, to enter upon a geometrical measure-
ment, which has so greatly the advantage in certainty.
Yet, considered by itself, the method of measurement by
the barometer is valuable for its contributions to our know-
ledge of the surface reliefs of our globe.
SECTION II.
DYNAMICS.
We have now to consider the laws of notion of gravitat-
ing bodies; and of Solids, in the first place.
The last elementary notion about gravity,—that of the
necessary proportion between weight and mass,—which
was still wanting to statical barology, was established by
the admirable observation that all bodies in a vacuum fall
through the same space in the same time. Proceeding
from this, we must examine the discovery of the laws of
motion produced by gravity. We shall find	,
herein, not only the historical origin of
physics, but the most perfect method of philosophizing of
which the science admits. Aristotle observed the natural
acceleration of bodies in their fall; but he could not
discover the law of the case, for want of the elementary
principles of rational dynamics. His hypothesis, that the
velocity increases in proportion to the space traversed, was
plausible, till Galileo found the true theory of varied
motions. When Galileo had discovered the law that the
velocity and the space traversed were necessarily propor-
tioned, the one to the time, and the other to its square, he
showed how it could be verified in two ways; by the
immediate observation of the fall, or by retarding the
descent at will by the aid of a sufficiently inclined plane,—
allowance being made for the friction. An ingenious in-
strument, which affords a convenient verification, was
afterwards offered by Attwood: it retards the descent,256
POSITIVE PHILOSOPHY.
while leaving it vertical, by compelling a small mass to
move a very large one in that direction.
By this one law of G-alileo, all the problems relating to
the motions of falling bodies resolve themselves into ques-
tions of rational dynamics. They, indeed, compelled its
formation, in the seventeenth century; as, in the eighteenth,
questions of celestial mechanics thoroughly developed it.
In all that relates to the motion of translation of a body
in space, this study is due to G-alileo, who established the
theory of the curvilinear motion of projectiles—allowance
being made for the resistance of the air. All attempts,
however, to ascertain the effect of this resistance have
hitherto been in vain ; and therefore the study of the real
motion of projectiles is still extremely imperfect.
As for the motions that gravity occasions in bodies that
are not free in space,—the only important case is that of a
body confined to a given curve. It constitutes the problem
of the pendulum, which we have already considered, as the
immortal achievement of Huvghens. Its practical interest,
as the basis of the most perfect chronometry, presented
itself first: but it has besides furnished two general con-
sequences, very essential to the progress of barology.
First, it enabled Newton to verify the proportion of weight
to mass with much more exactness than would have been
possible by the experiment of the fall of bodies in a
vacuum; and, in the second place, the pendulum has
enabled us, as we saw before, to observe the differences of
the universal gravity at different distances from the earth’s
centre. It is by the use of this method that we are con-
tinually adding to our knowledge of the measure of gravity
at various points on the globe, and, therefore, of the figure
of the earth.
In these different sections of dynamical barology, solid
bodies are regarded as points—considerations of dimensions
being discarded: but a new order of difficulties arises
when we have to consider the particles of which the body
is really formed. With regard to cases of restricted
motion, as the pendulum, the thing to be done is to find
out under what laws the different points of the body
modify the unequal times of their respective oscillations,
so that the whole may oscillate as one point, real or ideal.CASE OF FLUIDS.
257
This law, discovered by Huyghens, and afterwards obtained
by James Bernouilli in a more scientific manner, easily
transforms the compound into the simple pendulum, when
the moment of inertia of the body is known. The study
of the pendulum is involved in all the questions of the
dynamics of solids. To give it the last degree of precision,
it is necessary to consider the resistance of the air, though
that resistance is small in comparison with the case of
projectiles. This is done, with ease and certainty, bv
comparing theoretical oscillations with real ones exposed
to the resistance of the air; when, of course, the difference
between the two gives the amount of that resistance.
We have seen enough of the difficulties of	Fluids
hydrodynamics to understand that the part
of dynamical barology which relates to fluids must still be
very imperfect. In the case of the gases, and especially of
the atmosphere, next to nothing has been attempted, from
the sense of the impracticable #nature of the inquiry. The
only analysis which has been proceeded with, „	£1.	.,
in regard to liquids, is that of their flow by ‘ ff •
very small orifices in the bottom or sides of a vessel: that
is, the purely linear motion, mathematically presented by
Daniel Bernouilli, in his celebrated hypothesis of the
parallelism of laminae. Its principal result has been to
demonstrate the rule, empirically proposed by Torricelli,
as to the estimate of the velocity of the liquid at the orifice,
as equal to that of a weight falling from the entire height
of the liquid in the vessel. Now, this rule has been recon-
ciled with observation, even on the supposition of an in-
variable level, only by an ingenious fiction, suggested by
the singular phenomenon of the contraction of the liquid
filament. The case of a variable level is scarcely entered
upon yet; or any which involves the form and size of the
orifice. As for more complex cases,—their theory is yet
entirely in its infancy.1
1 The subjects here spoken of by M. Comte have recently received
remarkable elucidation through experiments entered upon chiefly
by the instrumentality of the “ British Association for the advance-
ment of Science.” There cannot be much hope, in the present
state of our knowledge, of ascending otherwise than empirically
towards any general law or rules which will either comprehend
phenomena, or be of use in practice.—J. P. N.
i.	s258
POSITIVE PHILOSOPHY.
From this cursory review of Barology we may carry
away some idea of its spirit, and of the progress of which
it admits. Imperfect as our survey has been, we may per-
ceive that this first province of Physics is not only the
purest, but the richest. We may observe in it a character
of rationality, and a degree of co-ordination which we shall
not meet with in other parts of the science. It is because
we look for a consistency and precision almost like those
that we find in astronomy that we consider barology so
imperfect as we do. It has long attained its position of
positivity: there is no one of its subdivisions which is
.	not at least sketched out: all the general
ofXBarofo«y. 6 means investigation, Observation, Experi-
ment, and Comparison, have been succes-
sively applied to it; and thus its future progress depends
only on a more complete harmony between these three
methods, and on a more uniform and close combination
between the mathematical spirit and the physical.259
CHAPTER III.
THEEMOLOGY.
TVTEXT to the phenomena of gravity, those of heat are,
- ^ unquestionably, the most universal in the province of
Physics. Throughout the economy of terrestrial nature,
dead or living, the function of heat is as important as that
of gravity, of which it is the chief antagonist. T,
n	^	j.ts nature
The consideration of gravity presides over
the geometrical and mechanical study of bodies: while
that of heat prevails in its turn, when we investigate deeper
modifications, relating to either the state of aggregation or
the composition of molecules: and finally, vitality is sub-
ordinated to it. The intelligent application of heat consti-
tutes the chief action of man upon nature. Thus, after
barologv, thermologv is, of all the parts of physics, the
one which most deserves our study.
The earliest scientific observations in thermologv are
almost as old as the discoveries of Stevinus and Galileo on
gravity, the invention of the thermometer having taken
place at the beginning of the seventeenth century: but,
owing to the complication of its phenomena, it has always
been far distanced by barologv. At the end Itg jlistorv
of the seventeenth century, the indications
of the thermometer could not be compared, for want of
two fixed points, the necessity for which was, at that time,
nointed out by Hewton. The greatest difference between
the two studies, however, was in their spirit. While philo-
sophers were already inquiring, with regard to weight, what
were its phenomena and their laws, those who were studying
heat were looking for the nature of fire, and reducing the
facts of the case to something merely episodical. It was
only at the end of the last century, that the great discovery
of Black imparted anything of a scientific character to
thermology, while barology was almost as much advanced260
POSITIVE PHILOSOPHY.
as it is now. Our philosophers still entertain some of the
old chimeras; but now very loosely, and, as they say, to
facilitate the study of the phenomena. The labours of
Fourier, however, must soon establish a thoroughly scientific
method; and this result cannot but be aided by the fact
that the two great modern hypotheses about the nature of
heat are in direct collision. It is certain that, of all the
provinces of physics, thermology is the nearest to a complete
emancipation from the anti-scientific spirit.
Of all the branches of Physics which admit
Mathematics Mathematical Analysis, this is the one which
exhibits the most sjiecial application of it.
Barology enters into the province of rational mechanics;
and so, in a less degree, does acoustics. The analytical
theory of heat now offers a scientific character as satisfac-
tory as that of gravity and sound; and it may be treated
as a dependency of Abstract Mechanics, without any resort,
to chimerical hypotheses. Our present business is, however,
with the purely physical study of Heat.
SECTION I.
MUTUAL THERMOLOGICAL INFLUENCE.
Two jarts	Physical thermology consists of two parts,
*	’ distinct, but nearly connected. The first
relates to the mutual influence of two bodies in altering
their respective temperatures. The second consists in the
study of those alterations : that is, of the modifications or
entire change which the physical constitution of bodies
may undergo in consequence of their variations of tempera-
ture, stopping at the point at which chemistry intervenes,
—that is, where the molecular composition becomes affected.
The first of these consist in the theory of warming and
cooling.
First* Mutual	^ermo^°^cal e^ecf is produced be-
influence	tween bodies of precisely the same tempera-
ture. The action begins when the tempera-
tures become unequal. The warmer body then raises the
temperature of the cooler; and the cooler depresses that ofPROPAGATION OF HEAT.
261
the warmer, till, sooner or later, they reach a common
temperature. Though the final state may usually be at an
unequal distance from the two extremes, action is not, pro-
perly speaking, the less truly equivalent to reaction in a
contrary direction. This case again divides itself into two.
The bodies may act at a distance, greater or smaller; or
they may be in contact. The first case constitutes what is
called the radiation of heat.
The direct communication of heat between two isolated
bodies was long denied by philosophers, who regarded the
air, or some other medium, as indispensable to the effect.
But there is now no doubt about it; as thermological
action takes place in a vacuum: and the small density
and conducting power of the air could not account for the
effects observed in the majority of ordinary cases. This
action, like that of gravity, extends, no doubt, to all dis-
tances, in conformity to the fundamental approximation
between these two great phenomena, pointed out by Fourier:
for we can conceive of the planets of our system as exerting
an appreciable mutual influence in this respect: and it
seems as if the temperature of the whole solar system were
attributable to the thermometrical equilibrium to which
all the parts of the universe are for ever tending.
The first law relating to such an action con-	f
sists in its rectilinear propagation. Though Heat1 ^ °
the term radiation has been connected with
untenable hypotheses, we may retain it, provided we care-
fully restrict it to the meaning that it is in a right line
that two points act thermologically on each other. It thus
implies that in placing bodies to prevent this mutual
action between two others, the absorbing body must be
placed in a right line.—This radiating heat can be reflected
like light, and in conformity with the same rule: and it
undergoes the same refractions as light, with some modifica-
tions.—Another question about this action relates to the
influence of the direction of radiation considered in regard
to the surface of either the warming or the warmed body.
The experiments of Leslie, confirmed by the mathematical
results of the case, have established that, in either case,
the intensity of the action is greater as the rays approach
the perpendicular, and that it varies in proportion to the262
POSITIVE PHILOSOPHY.
sine of the angle that they form with each surface.—The
last consideration, and the most important, is the difference
of temperatures between the two bodies. When this differ-
ence is not very great, the intensity of the action is in
jjrecise proportion to it: but this relation appears to cease
when the temperatures become very unequal.
When the radiation is not direct, but transmitted through
an intermediate body, the conditions just noticed become
complicated with new circumstances relating to the action
of the interposed body. We owe to Saussure a fine series
of experiments—hardly varied enough however,—upon the
effect of a set of transparent coverings in changing remark-
ably the natural mode of accumulation or dispersion of
heat whether radiant or obscure. More recently, M. Melloni
has pointed out an essential distinction between the trans-
mission of heat and that of light, in proving that the most
translucent bodies are not always those through which
heat passes most easily ; as was previously supposed.
It is well for physicists to separate the
l)v°Contactn radiation of heat from its propagation by
contact, for analytical purposes; but it is
evident that the separation cannot be found in nature. They
are always found in connection, however unequally. Besides
that the atmosphere can hardly be absent, and is always
establishing an equilibrium of heat between bodies that are
apart, it is clear that it is only the state of the surface that
can be determined by radiation. The interior parts, which
have as much to do with the final condition as the surface,
can grow warmer or cooler only by contiguous and gradual
propagation. Thus, no real case can be analysed by the
study of its radiation alone. And again, the action by
contact of two bodies can take place only in those small
portions which are in contact, while the bodies are acting
upon each other by the radiation of all the rest of their
surfaces. Thus, though the two modes of action are really
distinct, the analysis of either is rendered extremely difficult
by their perpetual combination.
Of the three conditions noticed above, relating to the
action exerted at a distance, the only one which applies
equally to the propagation of heat by contact is the differ-
ence of temperatures. The temperature of the parts inCO^DUCTIBILIT Y.
263
question can have so little inequality, that the law which
makes the action increase in proportion to the difference
may be regarded almost as an exact expression of the fact.
As for the law relating to direction, it probably subsists
here too, though we cannot be perfectly assured of it. But
that law which relates to the distance must be totally
changed ; for, on the one hand, the action of the almost con-
tiguous particles cannot be nearly so great as the variations
which we are able to estimate would lead us to suppose;
and, on the other hand, when we compare the various small
intervals between them, the diminution is certainly much
more rapid than in the case of distant bodies.
Whatever may be the mode of the warming and cooling
of respective bodies, the final state which is established
under the laws just noticed is numerically determined by
three essential coefficients, proper to every natural body, as
its specific gravity is in barology.
Under the old term conductibility, two	...^
,.	£ -i -j	Conductibilitv.
properties were confounded, which houner
separated, giving them the names of penetrability and
permeability; the first signifying that by which heat is
admitted at the surface, or dispersed from it, and the
other that by which the changes at the surface are propa-
gated through the interior. Permeability p	^
depends altogether on the nature of the body ermea 11 > •
and its state of aggregation. The differences of bodies in
this respect have always been open to observation; for
instance, the rapid propagation of heat in metallic bodies
in comparison with coal, which may be burning at one
point and scarcely warm a few inches off, while the heat
rapidly pervades the whole body of metal. It varies with
the physical constitution of bodies, so diminishing in fluids
that even Eumford went so far as to deny permeability in
them altogether, ascribing the propagation of heat in them
to interior agitation. This was a mistake; but permea-
bility is very weak in liquids, and weaker p ,
still in gases. As to penetrability, while	y’
partly depending on the state of aggregation of bodies, it
depends much more on the state of their surfaces,—on
colour, polish, and the regularity in which radiation in
various directions can take place, and divers other modifi-264
POSITIVE PHILOSOPHY.
cations : and it changes in the same surface as it is exposed
to the action of different media.
Strictly speaking, the different degrees of these two
kinds of conductibility cannot affect the final thermological
state of the two bodies, but only the time at which it is
reached. Yet, as real questions often become mere questions
of time, it is clear that if the inequalities are very marked,
they must affect the intensity of the phenomena under
study; for instance, where the permeability is so feeble
that the requisite interior heat cannot be obtained in time,
but by applying such heat to the surface as will break or
burn it; in which case, the phenomenon cannot take place ;
or, not within any practicable time. In general, the more
perfect both kinds of conductibility, the better the bodies
will obey the laws of thermological action, at a distance or
in contact. It would therefore be very important to
measure the value of these two coefficients for all bodies
under study: but unhappily such estimates are at present
extremely imperfect. The business was vague enough, of
course, when the two kinds of conductibility were con-
founded ; but Fourier has taught us how to estimate per-
meability directly, and, of course, penetrability indirectly,
bv subtracting the permeability from the total of conducti-
bility. But the application of his methods is as yet hardly
initiated.
Specific heat.
One consideration, that of specific heat, re-
mains to be noticed, as concurring to regu-
late the results of thermological action. Whether under
conditions of equal weight, or of equal volume, the different
substances consume distinct quantities of heat to raise
their temperature equally. This property, of which little
was known till the latter half of the last century, depends,
like permeability, only in a less degree, on the physical
constitution of bodies, while it is independent of the state
of their surfaces. It must considerably affect the equalized
temperature common to two bodies, which cannot be equally
different from the primitive temperature of each, if they
differ from each other in the point of their specific heat.
Physicists have achieved a good deal in the estimate of
specific heat. The best method is that of experiment with
the calorimeter, invented by Lavoisier and Laplace, for itsCONSTITUENT CHANGES BY HEAT.
265
measurement. The quantity of heat consumed by any
body at a determinate elevation of temperature, is estimated
by the quantity of ice melted by the heat it gives out in its
passage from the highest to the lowest temperature. The
apparatus is so contrived as to isolate the experiment from
all thermological action of the vessel and of the medium;
and thus the results obtained are as precise as can be
desired.
These are the three coefficients which serve to fix the
final temperatures which result from the thermological
equilibrium of bodies. Till we know more of the laws of
their variations, it is natural to suppose them essentially
uniform and constant: but it would not be rational to
conceive of conductibility as identical in all directions, in
all bodies, however their structure may vary in different
directions; and specific heat probablv undergoes changes
at extreme temperatures, and especially in the neighbour-
hood of those which determine a new state of aggregation,
as some experiments already seem to indicate. However,
these modifications are still so uncertain and obscure, that
physicists cannot be blamed for not keeping them, at this
day, perpetually in view.
SECTION II.
CONSTITUENT CHANGES BY HEAT.
The second part of thermology is that Second part,
which relates to the alterations caused by Constituent
heat in the physical constitution of bodies, changes by
These alterations are of two kinds; changes ^eat.
of volume, and the production of a new state of aggrega-
tion ; and this is the part of thermology of which we are
least ignorant.
These phenomena are independent of those of warming
and cooling, though they are found together. When we
heat any substance, the elevation of temperature is deter-
mined solely by the portion of heat consumed, the rest of
which (often the greater part), insensible to the thermo-
meter, is absorbed to modify the physical constitution.266
POSITIVE PHILOSOPHY.
Latent heat.
This is what we mean when we say that a
portion of heat has become latent; a term
which we may retain, though it was originally used in con-
nection with a theory about the nature of heat. This is
the fundamental law discovered by Black, by observation
of the indisputable cases in which a physical modification
takes place without any change of temperature in the
modified body. When the two effects co-exist, it is much
more difficult to analyse and apportion them.
Considering first the laws of change of
volume ° volume, it is a general truth that every
homogeneous body dilates with heat and
contracts with cold; and the fact holds good with hetero-
geneous bodies, such as organized tissues especially, in
regard to their constituent parts. There are very few
exceptions to this rule, and those few extend over a very
small portion of the thermometrical scale. The principal
anomaly however being the case of water, it has great
importance in natural history, though not much in abstract
physics, except from the use that philosophers have made
of it to procure an invariable unity of density, always at
command. These anomalies, too rare and restricted to
invalidate any general law, are sufficient to discredit all
a priori explanations of expansions and contractions,
according to which every increase of temperature should
cause an expansion, and every diminution a contraction,
contrary to the facts.
Solids dilate less than liquids, under the same elevation
of temperature, and liquids than gases; and not only
when the same substance passes through the three states,
but also when different substances are employed. The
expansion of solids proceeds, as far as we know, with
perfect uniformity. We know more of the case of liquids,
which is rendered extremely important from its connection
with the true theory of the thermometer, without which
all thermological inquiries would be left in a very dubious
state. Experiments, devised by Dulong and Petit, have
shown that for above three hundred centigrade degrees
the expansion of the mercury follows an exactly uniform
course,—equal increase of volume being produced by heat
able to melt equal weights of ice at zero. This is the onlyCHANGE IN STATE OF AGGREGATION.
267
case fully established; but we have reason to believe that
the rule extends to that of all liquids. The most marked
case of such regularity is that of gases. Not only does
the expansion take place by equal gradations, as usually in
liquids and solids, but it affects all gases alike. Gases
differ from each other, like liquids and solids, in their
density, their specific heat, and their permeability; yet
they all dilate uniformly and equally, their volume in-
creasing three-eighths, from the temperature of melting-
ice to that of boiling water. Vapours are like gases in this
particular, as in so many others. These are the simple
general laws of the expansion of elastic fluids, discovered
at once by Gay-Lussac at Paris, and Dalton at Manchester.
Next, we have to notice the changes pro- Change in
duced by heat in the state of aggregation of state of
bodies.	aggregation.
Solidity and fluidity used to be regarded as absolute
qualities of bodies; whereas, we now know them to be
relative, and are even certain that all solid bodies might
be rendered fluid if we could apply heat enough, avoiding-
chemical alteration. In the converse way, we used to sup-
pose that gases must preserve their elasticity, through all
degrees of cooling and of compression ; whereas Bussy and
Faraday have shown us that most of them easily become
liquid, when they are seized in their nascent state; and
there is every reason to believe that by a due combination
of cold and pressure, they may be always liquefied, even in
their developed state. Under this view therefore, different
substances are distinguished only by the different parts of
the indefinite thermometrical scale to which their succes-
sive states, solid, liquid, and gaseous, correspond. But
this simple inequality is an all-important characteristic,
which is not yet thoroughly connected with any other
fundamental quality of each substance. Density is the
relation which is the least obscure and variable,—gases
being in general less dense than liquids, and liquids than
solids. But there are striking exceptions in the second
case, and might be in the first, if we knew more of gases
in regard to compression, and in varied circumstances of
other kinds. As for the three states of the same substance,
there is always, except in some cases of scarce anomaly,268
POSITIVE PHILOSOPHY.
rarefaction in the fusion of solids and in the evaporation
of liquids. All these changes have been brought by the
illustrious Black under one fundamental law, which is,
both from its importance and its universality, one of the
Law of en- finest discoveries in natural philosophy. It
gagement and is this: that in the passage from the solid
disengage- to the liquid state, and from that to the
ment of heat. gaseous, every substance always absorbs a
more or less distinguishable quantity of heat, without
raising its temperature ; while the inverse process occasions
a disengagement of heat, precisely correspondent to the
absorption. These disengagements and absorptions of heat
are evidently, after chemical phenomena, the principal
sources of heat and cold. In an experiment of Leslie’s, an
evaporation, rendered extremely rapid by artificial means,
has produced the lowest temperatures known. Eminent
natural philosophers have even believed that the heat
which is so abundantly disengaged in most great chemical
combinations could proceed only from the different changes
of state which commonly result from them. But this
opinion, though true in regard to a great number of cases,
has too many exceptions to deal with to become a general
principle.
v	We have now done with physical thermo-
apours.	logy. But the laws of the formation and
tension of vapours now form an appendix to it; and also
of course hygrometry. The theory is, in fact, the necessary
complement of the doctrine of changes of state; and this
is its proper place.
Before Saussure’s time, evaporation was regarded as a
chemical fact, occasioned by the dissolving action of air
upon liquids. He showed that the action of the air was
adverse to evaporation, except in the case of the renovation
of the atmosphere. The test was found in the formation
of vapour in a restricted space. Saussure found that, in
such a case, with a given time, temperature, and space, the
quantity and elasticity of the vapour were always the
same, whether the space was a vacuum or filled with gas.
The mass and tension of the vapour increased steadily with
the temperature; whereas it appears that no degree of cold
suffices to stop the process entirely, since ice itself producesVAPOURS.
269
a vapour appreciable by very delicate means of observation.
We do not know by what law the increase of temperature
accelerates the evaporation, at least while the liquid remains
below boiling point; but the variations in elasticity of the
vapour produced have been successfully studied.
One term common to all liquids is the boiling point.
At that point, the rising tension of the vapour formed has
become equal to the atmospheric pressure; a fact which
can be ascertained by direct experiment.
Proceeding from this point, Dr. Dalton dis- o^eladlftioi^
covered the important law that the vapours
of different liquids have tensions always equal between
themselves to temperatures equi-distant from the corre-
sponding boiling points, whatever may be otherwise the
direction of the difference. Thus, the boiling of water
taking place at one hundred degrees, and that of alcohol
at eighty degrees, the two vapours, having at that point
the same tension, equal to the atmospheric pressure, will
then have equal elasticities, superior or inferior to the pre-
ceding, when their two temperatures are made to vary in
the same number of degrees. The many new liquids dis-
covered by chemists since this law was found have all
tended to confirm it. It is very desirable that some
harmony should be discovered between the boiling tem-
peratures of different liquids and their other properties;
but this remains to be done, and these temperatures appear
to us entirely incoherent, though there is every reason to
believe that they are not so.
It is evident that this law of Dalton’s simplifies pro-
digiously the inquiry into the variation of the tension of
vapours, according to their temperatures ; since the analysis
of these variations in one instance will serve for all the
rest. The experiments undertaken for this purpose by
Dr. Dalton and his successors have not fully established
the rule of proportion between the tension and the tem-
perature : but an empirical law proposed by Du long has
thus far answered to the observed phenomena. All a priori
determinations of the law have utterly failed.
The study of hygrometrical equilibrium TT	, . .
v	v o	j.	H v^oTOinptriO/i I
between moist bodies seems a natural ad- equilibrium,
j unct to the theory of evaporation. Saussure270
POSITIVE PHILOSOPHY.
and Deluc have given ns a valuable instrument for this
inquiry; but we know scarcely anything of the laws which
regulate the equilibrium of moisture. Prevision, which is
the exact measure of science of every kind, is almost non-
existent in the case of hygrometry. The small part that it
plays in the inorganic departments of nature is, no doubt,
the reason of the little attention that physicists have de-
voted to it: but we shall hereafter find how important is its
share in vital phenomena. According to M. de Blainville,
hygrometrical action constitutes the first degree and ele-
mentary mode of the nutrition of living bodies, as capil-
larity is the germ of the most simple organic motions.
In this view, the neglect is much to be regretted. It is
one instance among a multitude of the mischiefs arising
from the restricted training of natural philosophers. In
this case, two important studies, which can be accomplished
only by physical inquirers, are neglected, merely because
their chief destination concerns another department of
science.
After this survey, we can form some idea of the charac-
teristics of this fine section of Physics. We see the
rational connection of the different questions comprised
in it; the degree of perfection which each of them has
attained ; and the gaps which remain to be filled up. A
vast advance was made when, by the genius of Fourier,
the most simple and fundamental phenomena of heat were
attached to an admirable mathematical theory.
SECTION III.
CONNECTION WITH ANALYSIS.
Thermology This theory relates to the first class of
connected	cases,—those in which an equalization of
with Analysis. heat takes place between bodies at a distance
or in contact; and not at all to those in which the
physical constitution is altered by heat. It is only by an
indirect investigation that we can learn how heat, once
introduced into a body from the surface, extends through
its mass, assigning to each point, at any fixed moment, aTHERMOLOGY CONNECTED WITH ANALYSIS. 271
determinate temperature ; or the converse—how this in-
terior heat is dissipated, by a gradual dispersion through
the surface. As direct observation could not help us here,
we must remain in ignorance, if Fourier had not brought
mathematical analysis to the aid of observation, so as to
discover the laws by which these processes take place.
The perfection with which this has been done opens so
wide a field of exploration and application, unites so strictly
the abstract and the concrete, and is so pure an example
of the positive aim and method, that future generations
will probably assign to this achievement of Fourier’s the
next place, as a mathematical creation, to the theory of
gravitation. Many contemporaries have hastened into the
new field thus opened; but most of them have used it
only for analytical exercises which add nothing to our
permanent knowledge; and perhaps the labours of M.
Duhamel are hitherto the only ones which afford really
any extension of Fourier’s theory, by perfecting the analy-
tical representation of thermological phenomena.
According to the plan of this work, we ought not to
quit the limits of natural philosophy to notice any con-
crete considerations of natural history,—the secondary
sciences being only derivatives from the primary. It is
departing from our rule, therefore, to bring forward the
important theory of terrestrial temperatures: yet this
most important and difficult application of mathematical
thermology forms so interesting a part of Fourier’s doc-
trine, that I cannot refrain from offering some notice
of it.
SECTION IV.
TERRESTRIAL TEMPERATURES.
The temperature of each point of our „	, . ,
globe is owing (putting aside local or acci- temperatures,
dental influences) to the action of three
general and permanent causes variously combined: first,
the solar heat, affecting different parts unequally, and
subjected to periodical variations: next, the interior heat
proper to the earth since its formation as a planet: and272
POSITIVE PHILOSOPHY.
thirdly, the general thermometrical state of the space
occupied by the solar system. The second is the only one
of the three which acts upon all the points of the globe.
The influence of the two others is confined to the surface.
The order in which they have become known to us is that
in which I have placed them. Before Fourier’s time, the
whole subject had been so vaguely and carelessly regarded,
that all the phenomena were ascribed to solar heat alone.
It is true, the notion of a central heat was very ancient;
but this hypothesis, believed in and rejected without suffi-
cient reason, had no scientific consistency,—the question
having never been raised of the effect of this original heat
on the thermological variations at the surface. The theory
of Fourier afforded him mathematical evidence that at the
surface the temperatures would be widely different from
what they are, both in degree and mutual proportion, if
T .	,	. the globe were not pervaded by a heat of its
n enor lea . own^ independent of the action of the sun ;
a heat which tends to dispersion from the surface, by
radiation towards the planets, though the atmosphere must
considerably retard this dispersion. This original heat
contributes very little, in a direct way, to the temperatures
at the surface; but without it, the solar influence would
be almost wholly lost, in the total mass of the globe ; and
it therefore prevents the periodical variations of tempera-
ture from following other laws than those which result
from the solar influence. Immediately below the surface,
the central heat becomes preponderant, and soonest in the
parts nearest the equator; and it becomes the sole regu-
lator of temperatures, and in a rigorously uniform manner,
Temperature in proportion to the depth.—As to the third
of the plane- cause, Fourier was the first to conceive of
tary intervals, ft He was wont to give, in a simple and
striking form, the results of his inquiries in the saying
that if the earth left a thermometer behind it in any part
of its orbit, the instrument (supposing it protected from
solar influence) could not fall indefinitely: the column
would stop at some point or other, which would indicate
the temperature of the space in which we revolve. This is
one way of saying that the state of the temperatures on
the surface of the globe would be inexplicable, even con-CONCLUDING REMARKS.
273
sidering the interior heat, if the surrounding space had
not a determinate temperature differing but little from
that which we should find at the poles, if we could pre-
cisely estimate it. It is remarkable that, of the two new
thermological causes discovered by Fourier, one may be
directly observed at the equator and the other at the
poles; whilst, for all the intermediate points, our observa-
tion must be guided and interpreted by mathematical
analysis.
New as this difficult inquiry is, our progress in it
depends only on the perfecting of the observations which
Fourier’s theory has marked out for us. When the data
of the problem thus become better known, this theory will
enable us to lay hold of some certain evidences of the
ancient thermological state of our globe, as well as of its
future modifications. We have already learned one fact
of high importance ; that the periodical state of the earth’s
surface has become essentially fixed, and cannot undergo
any but imperceptible changes by the continuous cooling
of the interior mass through future ages. This rapid
sketch will suffice to show what a sudden scientific con-
sistency has been given, by the labours of one man of
genius, to this fundamental portion of natural history,
which, before Fourier’s time, was made up of vague and
arbitrary opinions, mingled with incomplete and inco-
herent observations, out of which no exact general view
could possibly arise.
i.
T274
CHAPTER IV.
ACOUSTICS.
Its nature.
THIS science had to pass, like all the rest,
through the theological and metaphy-
sical stages; but it assumed its positive character about
the same time with Barology, and as completely, though
our knowledge of it is, as yet, very scanty, in comparison
with what we have learned of gravity. The exact in-
formation which was obtained in the middle of the seven-
teenth century about the elementary mechanical properties
of the atmosphere, opened up a clear conception of the
production and transmission of sonorous vibrations. The
analysis of the phenomena of sound shows us that the
doctrine of vibrations offers the exact expression of an in-
contestable reality. Besides its philosophical interest, and
the direct importance of the phenomena of Acoustics, this
department of Physics appeals to special attention in two
principal relations, arising from its use in perfecting our
fundamental ideas regarding inorganic bodies, and Man
himself.
Relation to	By studying sonorous vibrations, we obtain
the study of some insight into the interior mechanical con-
inorganic stitution of natural bodies, manifested by the
bodies.	modifications undergone by the vibratory
motions of their molecules. Acoustics affords the best, if
not the only means for this inquiry; and the small present
amount of our acquisitions seems to me no reason why we
should not obtain abundant results when the study of
acoustics is more advanced. It has already revealed to us
some delicate properties of natural bodies which could not
have been perceived in any other way. For instance, the
capacity to contract habits,—a faculty which seemed to be-
long exclusively to living beings (I mean the power of con-
tracting fixed dispositions, according to a prolonged seriesSCIENTIFIC RELATIONS OF ACOUSTICS.
275
of uniform impressions),—is clearly shown to exist, in a
greater or smaller degree, in inorganic apparatus. By
vibratory motions, also, two mechanical structures, placed
apart, act remarkably upon each other, as in the case of two
clocks placed upon the same pedestal.
On the other hand, acoustics forms a basis p . .
to physiology for the analysis of the two Ph^ioio^y
elementary functions which are most impor-
tant to the establishment of social relations,—hearing and
the utterance of sound. Putting aside, in this place, all
the nervous phenomena of the case, it is clear that the in-
quiry rests on a knowledge of the general laws of acoustics,
which regulate the mode of vibration of all auditory appa-
ratus. It is remarkably so with regard to the production
of the voice,—a phenomenon of the same character with
that of the action of any other sonorous instrument, except
for its extreme complication, through the organic variations
which affect the vocal system. Yet, it is not to physicists
that the study of these two great phenomena belongs. The
anatomists and physiologists ought not to surrender it to
them, but to derive from physics all the ideas necessary for
conducting the research themselves : for physicists are not
prepared with the anatomical data of the problem, nor yet
to supply a sound physiological interpretation of the results
obtained. Science has indeed suffered from the prejudices
which have grown out of the introduction into physics of
superficial theories of hearing and phonation, from physical
inquirers having intruded upon the province of the phy-
siologists.
After Barology, there is no science which _ .
admits of the application ot mathematical Mathematics
doctrines and methods so well as Acoustics.
In the most general view, the phenomena of sound
evidently belong to the theory of very minute oscillations
of any system of molecules round a situation of stable
equilibrium ; for, in order to the sound being produced,
there must be an abrupt perturbation in the molecular
equilibrium ; and this transient derangement must be fol-
lowed by a quick return to the primitive state. Once pro-
duced, in the body directly shaken, the vibrations may be
transmitted at considerable intervals, by means of an elastic276
POSITIVE PHILOSOPHY.
medium, by exciting a gradual succession of expansions and
contractions which are in evident analogy with the waves
formed on the surface of a liquid, and have given occasion
to the term sonorous undulations. In the air, in particular,
so elastic as it is, the vibration must propagate itself, not
only in the direction of the primitive concussion, but in all
directions, in the same degree. The transmitted vibra-
tions, we must observe, are always necessarily isochronal
with the primitive vibrations, though their amplitude may
be widely different.
It is clear from the outset that the science of acoustics
becomes, almost from its origin, subject to the laws of
rational Mechanics. Since the time of Newton, who was
the first to attempt to determine the rate of propagation of
sound in the air, acoustics has always been more or less
mixed up with the labour of geometers to develope abstract
Mechanics. It was from simple considerations of acoustics
that Daniel Bernouilli derived the general principle relat-
ing to the necessary and separate co-existence, or indepen-
dency, of small and various oscillations occasioned at the
same time in any system, by distinct concussions. The
phenomena of sound afford the best realization of that law,
without which it would be impossible to explain the com-
monest phenomenon of acoustics,—the simultaneous exis-
tence of numerous and distinct sounds, such as we are every
moment hearing.
Though the connection of acoustics with rational Me-
chanics is almost as direct and complete as that of Barology,
this mathematical character is far less manageable in the
one case than the other. The most important questions in
barology are immediately connected with the clearest and
most primitive mechanical theories; whereas the mathe-
matical study of sonorous vibrations depends on that diffi-
cult and delicate dynamical theory,—the theory of the per-
turbations of equilibrium, and the differential equations
which it furnishes relative to the highest and most imper-
fect part of the integral calculus. Vibratory motion of one
dimension is the only one, even in regard to solids, whose
mathematical theory is complete. Of such motion of three
dimensions we are, as yet, wholly ignorant.
To form any idea of the difficulties of the case, we mustSCIENTIFIC RELATIONS OF ACOUSTICS.
277
remember that vibratory motions must occasion certain
physical modifications of another nature in the molecular
constitution of bodies ; and that these changes, though
affecting the vibratory result, are too minute and transient
to be appreciable. The only attempt that has been made
to analyse such a complication is in the case of the thermo-
logical effects which result from the vibratory motion. La-
place used this case to explain the difference between the
velocity of sound in the air as determined by experiment
and that prescribed by the dynamic formula, which indi-
cated a variation of about one-sixth. This difference is
accounted for by the heat disengaged by the compression
of the atmospheric strata, which must make their elas-
ticity vary in a greater proportion than their density,
thereby accelerating the propagation of the vibratory
motion. It is true, a great gap is left here; since, as it is
impossible to measure this disengagement of heat, we must
assign to it conjecturally the value which compensates for
the difference or the two velocities. But we learn from
this procedure of Laplace the necessity of combining
thermological considerations with the dynamical theory of
vibratory motions. The modification is less marked in the
case of liquids ; and still less in that of solids ; but we are
too far behind with our comparative experiments to be able
to judge whether the modification is or is not too incon-
siderable for notice.
Notwithstanding the eminent difficulties of the mathe-
matical theory of sonorous vibrations, we owe to it such
progress as has yet been made in acoustics. The forma-
tion of the differential equations proper to the phenomena
is, independent of their integration, a very important acqui-
sition, on account of the approximations which mathe-
matical analysis allows between questions, otherwise hetero-
geneous, which lead to similar equations. This fundamental
property, whose value we have so often to recognize, applies
remarkably in the present case; and especially since the
creation of mathematical thermology, whose principal equa-
tions are strongly analogous to those of vibratory motion.
—This means of investigation is all the more valuable on
account of the difficulties in the way of direct inquiry into
the phenomena of sound. We may decide upon the neces-278
POSITIVE PHILOSOPHY.
sity of the atmospheric medium for the transmission of
sonorous vibrations ; and we may conceive of the possibility
of determining by experiment the duration of the propaga-
tion, in the air, and then through other media; but the
general laws of the vibrations of sonorous bodies escape
immediate observation. We should know almost nothing
of the whole case if the mathematical theory did not come
in to connect the different phenomena of sound, enabling
us to substitute for direct observation an equivalent ex-
amination of more favourable cases subjected to the same
law. For instance, when the analysis of the problem of
vibrating chords has shown us that, other things being
equal, the number of oscillations is in inverse proportion to
the length of the chord, we see that the most rapid vibra-
tions of a very short chord may be counted, since the law
enables us to direct our attention to very slow vibrations.
The same substitution is at our command in many cases in
which it is less direct. Still, it is to be regretted that the
process of experimentation has not been further improved.
Acoustics consists of three parts. We
Divisions. might perhaps say four, including the timbre
(ring or tone) arising from the particular
mode of vibration of each resonant body. This quality is
so real that we constantly speak of it, both in daily life and
in natural history: but it would be wandering out of the
department of general physics to inquire what constitutes
the ring or tone peculiar to different bodies, such as stones,
wood, metals, organized tissues, etc., whose properties lie
within the scope of concrete physics. But, if we regard
this quality as capable of modification, by changes of cir-
cumstances, then we bring it into the domain of acoustics,
and recognize its proper position, though we know nothing
else about it. That part of the science presents a mere
void.
The three parts referred to are, first, the mode of propa-
gation of sounds: next, their degree of intensity; and
thirdly, their musical tone. Of these departments, the
second is that of which our knowledge is most imperfect.DIVISIONS: THE PROPAGATION OF SOUND. 279
SECTION I.
PROPAGATION OF SOUND.
As to the first, the propagation of sound,
the simplest, most interesting, and best known	o/sound ^
question is the measurement of the duration,
especially when the atmosphere is the medium. Newton
enunciated it very simply, apart from all modifying
causes :—that the velocity of sound is that acquired by a
gravitating body falling from a height equal to half the
weight of the atmosphere,—supposing the atmosphere
homogeneous. In an analogous way, we may calculate the
velocity of sound in the different gases, according to their
respective densities and elasticities. By this law the speed
of sound in the air must be regarded as independent of
atmospheric vicissitudes, since, by Mariotte’s rules, the
density and elasticity of the air always vary in proportion;
and their mutual relation alone influences the velocity in
question. Of Laplace’s rectification of Newton’s formula,
we took notice just now.—One important result of this law
is the necessary identity of the velocity of different sounds,
notwithstanding their varying degrees of intensity or of
acuteness. If any inequality existed, we should be able to
establish it, from the irregularity which must take place in
musical intervals at a certain distance.
All mathematical calculations about the Effect of at-
velocity of sound suppose the atmosphere to mospheric
be motionless, except in regard to the vibra- agitation,
tions under notice, and it is one of the interesting points of
the case to ascertain what effect is produced by agitations
of the air. The result of experiments for this purpose is
that, within the limits of the common winds, there is no
perceptible effect on the velocity of sound when the direc-
tion of the atmospheric current is perpendicular to that in
which the sound is propagated; and that when the two
directions coincide, the velocity is slightly accelerated if the
directions agree, and retarded if they are opposed: but the
amount and, of course, the law of this slight perturba-
tion are unknown.—It is only in regard to the air that the
velocity of sound has been effectually studied.280
POSITIVE PHILOSOPHY.
SECTION II.
INTENSITY OF SOUNDS.
We cannot pretend to be any wiser about
sounds1 ^ °	intensity of sounds,—which is the second
part of acoustics. Not only have the pheno-
mena never been analysed or estimated, but the labours of
the student have added nothing essential to the results of
popular experience about the influences which regulate the
intensity of sound; such as the extent of vibrating surfaces,
the distance of the resonant body, and so on. These sub-
jects have therefore no right to figure in our programmes
of physical science; and to expatiate upon them is to miscon-
ceive the character of science, which can never be anything
else than a special carrying out of universal reason and
experience, and which therefore has for its starting-point
the aggregate of the ideas spontaneously acquired by the
generality of men in regard to the subjects in question.
If we did but attend to this truth, we should simplify our
scientific expositions not a little, by stripping them of a
multitude of superfluous details which only obscure the
additions that science is able to make to the fundamental
mass of human knowledge.
With regard to the intensity of sound, the only scientific
inquiry,—a very easy one,—which has been accomplished,
relates to the effect of the density of the atmospheric
medium on the force of sounds. Here acoustics confirms
and explains the common observation on the attenuation of
sound in proportion to the rarity of the air, without inform-
ing us whether the weakening of the sound is in exact
proportion to the rarefaction of the medium, as it is natural
to suppose. In my opinion, we know nothing yet of a
matter usually understood to be settled,—the mode of de-
crease of sound, in proportion to the distance of the
sounding body; as to which science has added nothing to
ordinary experience. It is commonly supposed that the
decrease is in an inverse ratio to the square of the distance.
This would be a very important law if we could establish
it: but it is at present only a conjecture; and I prefer
admitting our ignorance to attempting to conceal a scien-THEORY OF TONES.
281
tific void, by arbitrarily extending to this case the mathe-
matical formula which belongs to gravitation. A natural
prejudice may dispose us to find it again here; but we
have no proof of its presence.
It would be strange if we had any notion of the law of
the case, when we have not yet any fixed ideas as to the
way in which intensity of sound may be estimated; nor
even as to the exact meaning of the term. We have no
instrument which can fulfil, with regard to the theory of
sound, the same office as the pendulum and the barometer
with regard to gravity, or the thermometer and electro-
meter with regard to heat and electricity. We do not even
discern any clear principle by which to conceive of a sono-
meter. While the science is in this state, it is much too
soon to hazard any numerical law of the variations in
intensity of sound.
SECTION III.
THEORY OF TONES.
The third department of acoustics,—the
theory of tones,—is by far the most interest- Tories^ °
ing and satisfactory to us in its existing state.
The laws which determine the musical nature of different
sounds, that is, their precise degree of acuteness or gravity,
marked by the number of vibrations executed in a given
time, are accurately known only in the elementary case of
a series of linear, even rectilinear, vibrations produced
either in a metallic rod, fixed at one end and free at the
other, or in a column of air filling a very narrow cylindrical
pipe. It is by a combination of experiment and of mathe-
matical theory that this case is understood. It is the most
important for the analysis of the commonest inorganic
instruments, but not for the study of the mechanism of
hearing and utterance. With regard to stretched chords,
the established mathematical theory is that the number of
vibrations in a given time is in the direct ratio of the square
root of the tension of the chord, and in the inverse ratio of
the product of its length by its thickness. In straight and282
POSITIVE PHILOSOPHY.
homogeneous metallic rods this number is in proportion to
the relation of their thickness to the square of their length.
This essential difference between the laws of these two
kinds of vibrations is owing to the flexibility of the one
sounding body and the rigidity of the other. Observation
pointed it out first, and especially with regard to the effect
of thickness. These laws relate to ordinary vibrations,
which take place transversely; but there are vibrations in
a longitudinal direction much more acute, which are not
affected by thickness, and in which the difference between
strings and rods disappears, the vibrations varying recipro-
cally to the length; a result which might be anticipated
from the inextensibility of the string being equivalent to
the rigidity of the rod. A third order of vibrations arises
from the twisting of metallic rods, when the direction be-
comes more or less oblique. It ought to be observed
however that recent experiments have shown that these
three kinds are not radically distinct, as they can be mutu-
ally transformed by varying the direction in which the
sounds are propagated. As for the sounds yielded by a
slender column of air, the number of vibrations is in inverse
proportion to the length of each column, if the mechanical
state of the air is undisturbed: otherwise, it varies as the
square root of the relation between the elasticity of the air
and its density. Hence it is that changes of temperature
which alter this relation in the same direction have here an
action absolutely inverse to that which they produce on
strings or rods : and thus it is explained by acoustics why
it is impossible, as musicians have always found it, to
maintain through a changing temperature the harmony at
first established between stringed and wind instruments.
Thus far the resonant line has been supposed to vibrate
through its whole length. But if, as usually happens, the
slightest obstacle to the vibrations occurs at any point, the
sound undergoes a radical modification, the law of which
could not have been mathematically discovered, but has
been clearly apprehended by the great acoustic experimen-
talist, Sauveur. He has established that the sound pro-
duced coincides with that which would be yielded by a similar
but shorter chord, equal in length to that of the greatest
common measure between the two parts of the whole string.COMPOSITION OF SOUNDS.
283
The same discovery explains another fundamental law,
which we owe to the same philosopher,—that of the series
of harmonic sounds which always accompanies the principal
sound of every resonant string, their acuteness increasing
with the natural series of whole numbers ; the truth of
which is easily tested by a delicate ear or by experiment.
The phenomenon is, if not explained, exactly represented
by referring it to the preceding case; though we cannot
conceive how the spontaneous division of the string takes
place, nor how so many vibratory motions, so nearly simul-
taneous, agree as they do.
These are the laws of simple sounds. Of
the important theory of the composition of 0f°sounds10n
sounds we have yet very imperfect notions.
It is supposed to be indicated by the experiment of the
musician Tartini, with regard to resulting sounds. He
showed that the precisely simultaneous production of any
two sounds, sufficiently marked and intense, occasions a
single sound, graver than the other two, according to an
invariable and simple rule. Interesting as this fact is, it
relates to physiology, and not to acoustics. It is a pheno-
menon of the nerves ; a sort of normal hallucination of the
sense of hearing, analogous to optical illusions.
The vibrations of resonant surfaces have exhibited some
curious phenomena to observation, though the mathematical
theory of the case is still in its infancy: and M. Savart’s
observations on the vibratory motions of stretched mem-
branes must cast much light on the auditory mechanism,
in regard to the effects of degrees of tension, the hygrome-
trical state, etc.
The study of the most general and most complicated
case, that of a mass which vibrates in three dimensions, is
scarcely begun, except with some hollow and regular solids.
Yet this analysis is above all important, as without it it is
clearly impossible to complete the explanation of any real
instrument; even of those in which the principal sound is
produced by simple lines, the vibrations of which must
always be more or less modified by the masses which are
connected with them. We may say that the state of
acoustics is such that we cannot explain the fundamental
properties of any musical instrument whatever. Daniel284
POSITIVE PHILOSOPHY.
Bernouilli worked at the theory of wind instruments; a
subject which may appear very simple, but which really
requires the highest perfection of the science, even putting
aside those extraordinary effects, far transcending scientific
analysis, which the art of a musician may obtain from any
instrument whatever, and restricting ourselves to influences
which may be clearly defined and durably characterized.
Imperfect as is our review of Acoustics, I hope we now
understand something of its general character, the impor-
tance of its laws, as far as we know them, the connection of
its parts, the development that they have obtained, and the
intervals which are left void, to be filled up by future know-
ledge.285
CHAPTER Y.
OPTICS.
THE emancipation of natural philosophy from theological
and metaphysical influence has thus far gone on by
means of a succession of partial efforts, each isolated in in-
tention, though all converging to a final end,
amidst the entire unconsciousness of those JJyPotJiesis °n
who were bringing that result to pass. Such Lhdit.
an incoherence is a valuable evidence of the
force of that instinct which universally characterizes
modern intelligence ; but it is an evil, in as far as it has
retarded and embarrassed and even introduced hesitation
into the course of our liberation. Ho one having hitherto
conceived of the positive philosophy as a whole, and the
conditions of positivity not having been analysed, much
less prescribed, with the modifications appropriate to diffe-
rent orders of researches, it has followed that the founders
of natural philosophy have remained under theological and
metaphysical influences in all departments but the one in
which they were working, even while their own labours
were preparing the overthrow of those influences. It is
certain that no thinker has approached Descartes in the
clearness and completeness with which he apprehended the
true character of modern philosophy ; no one exercised so
intentionally an action so direct, extensive, and effectual on
this transformation, though the action might be transitory;
and no one was so independent of the spirit of his contem-
poraries ; yet Descartes, who overthrew the whole ancient
philosophy about inorganic phenomena, and the physical
phenomena of the organic, was led away by the tendency of
his age in a contrary direction, when he strove to put new
life into the old theological and metaphysical conceptions of
the moral nature of man. If it was so with Descartes, who
is one of the chief types of the progress of the general de-286
POSITIVE PHILOSOPHY.
velopment of humanity, we cannot be surprised that men
of a more special genius, who have been occupied rather
with the development of science than of the human mind,
should have followed a metaphysical direction in some
matters, while in others not very remote they have mani-
fested the true positive spirit.
These observations are particularly applicable to the philo-
sophical history of Optics,—the department of Physics in
which an imperfect positivism maintains the strongest con-
sistence,—chiefly through the mathematical labours which
are connected with it. The founders of this science are
those who have done most towards laying the foundations
of the Positive Philosophy,—Descartes, Huyghens, and
Newton ; yet each one of them was led away by the old
spirit of the absolute to create a chimerical hypothesis on
the nature of light. That Newton should have done this
is the most remarkable, considering how his doctrine of
gravitation had raised the conception of modern philosophy
above the point at which Descartes had left it, by estab-
lishing the radical inanity of all research into the nature
and mode of production of phenomena, and by showing
that the great end of scientific effort is the reduction of a
system of particular facts to one singular and general fact.
Newton himself, whose favourite saying was, “ O ! Physics,
beware of Metaphysics ! ” allowed himself to be seduced by
old habits of philosophizing to personify light as a sub-
stance distinct from and independent of the luminous
body: a conception as metaphysical as it would have been
to imagine gravity to have an existence separate from that
of the gravitating body.
After what has been said about the philosophical theory
of hypotheses, there can be no occasion to expose the ficti-
tious character of the respective doctrines of philosophers
on the nature of light. Each one has exposed the unten-
ableness of those of others; and each explorer has confined
himself to the evidence which favoured his own conception.
Euler brought fatal objections against the doctrine of emis-
sion ; yet, at the present day, our instructors conceal the
fact that the advocates of the emission doctrine have offered
equally fatal objections to that of undulations. To take the
most simple instance—Has the fact of propagation in allHYPOTHESES ABOUT LIGHT.
287
directions, characteristic of the vibratory motion, ever been
reconciled with the common phenomenon of night; that is,
of darkness produced by the interposition of an opaque
body ? Does not the fundamental objection of the New-
tonians about this matter hold its ground against the sys-
tem of Descartes and Huyghens, untouched at this hour as
it was above a century ago, after all the subterfuges that
have been in use ever since P The case is made clearer by
the fact that there are phenomena which the two theories
will suit equally well. If the laws of reflection and refrac-
tion issue with equal ease from the hypotheses of emission
and undulation, it is pretty clear that our business is with
the laws, and not with the hypotheses. The mathematical
labours expended on the opposite theories will not have
been thrown away ; they will show, in a very short time,
that the ana]ytical apparatus is no certain instrument of
truth, as it has served the purpose of both hypotheses
equally well; as it would, quite as easily, of many others,
if the progress of positivity was not excluding, more and
more, this vicious method of philosophizing. It is true, the
most enlightened advocates of both systems are ready to
give up the reality of emission and of undulation, and hold
to them only as a matter of logical convenience,—as a rally -
ing-point of ideas. But if we can pass from the one hypo-
thesis to the other without affecting the science at all, it is
clear that such an artifice is needless. We must admit, as
we before said, that the combination of scientific ideas
would be extremely difficult to minds trained under the
prevalent habits of thought, if they were suddenly deprived
of such a mode of connection as they here contend for; but
it is not the less true that the next generation of scientific
thinkers would combine their ideas more easily, and much
more perfectly, if they were trained to regard directly the
relations of phenomena, without being troubled by artifices
like these, which only obscure scientific realities.
The history of Optics, regarded as a whole, seems to
show that these hypotheses have not sensibly aided the
progress of the theory of light, since all our important
acquisitions have been entirely independent of them. This
is true not only of the laws of reflection and refraction,
which were discovered before these hypotheses were created,288
POSITIVE PHILOSOPHY.
but with regard to all the other leading truths of Optics.
The hypothesis of emission no more suggested to Newton
the notion of the unequal refrangibility of the different
colours, than that of undulation disclosed to Huyghens the
law of double refraction proper to certain substances.
Great discoverers like these observe a connection of facts,
and then create a hypothesis to account for the connection;
and then those who come after them conclude that the
chimerical conceptions must be inseparable from the im-
mortal discoveries. There is a use, as I have before
asserted, in these imaginary conceptions, which, in regard
to their one function, are indispensable. They serve,
transiently, to develope the scientific spirit by carrying us
over from the metaphysical to the positive system. They
can do this and nothing more, and they accomplished their
task some time ago. Their action can henceforth be only
injurious, and especially in the case of Optics, as any one
may see who will inquire into the state of this science,—
particularly since the almost universal adoption of the un-
dulatory in the place.of the emissive system.
Excessive ten- One more error must be noticed before we
clency to sys- leave the subject of the unscientific pursuit
tematize. 0f Optics. Some enlightened students ima-
gine that the science acquires a satisfactory rationality by
being attached to the fundamental laws of universal
mechanics. The emission doctrine, if it means anything,
must suppose luminous phenomena to be in analogy with
those of ordinary motion ; and if the doctrine of undulation
means anything, it means that the phenomena of light and
sound are alike in their vibratory agitation; and thus the
one party likens optics to barology and the other to
acoustics. But not only is nothing gained by the supposi-
tion, but if either was the case, there would be no room for
imagination or for argument. The connection would be at
once apparent to all eyes on the simple view of the pheno-
mena. Such a reference of phenomena to those general
laws has never been a matter of question or of conjecture.
The only difficulty has been to know those laws well enough
to admit of the application. No one doubted the mechanical
nature of the principal effects of gravity and sound long
before the progress of rational dynamics admitted of theirTENDENCY TO SYSTEMATIZE.
289
exact analysis. The application powerfully tended, as we
have seen, to the perfecting of barology and acoustics ; but
this was precisely because there was nothing forced or
hypothetical about it. It is otherwise with Optics. Not-
withstanding all arbitrary suppositions, the phenomena of
light will always constitute a category sui generis, necessarily
irreducible to any other: a light will be for ever hetero-
geneous to a motion or a sound.
Again, physiological considerations discredit this con-
fusion of ideas, by the characteristics which distinguish the
sense of sight from those of hearing, and of touch or pres-
sure. If we could abolish such distinctions as these by
gratuitous hypotheses, there is no saying where we should
stop in our wanderings. A chemical philosopher might
make a type of the senses of taste and smell, and proceed
to explain colours and tones by likening them to flavours
and scents. It does not require a wilder imagination to do
this, than to issue as a supposition, now become classical,
that sounds and colours are radically alike. It is much
better to leave such a pursuit of scientific unity, and
to admit that the categories of hetereogeneous phenomena
are more numerous than a vicious systematizing tendency
would suppose. Natural philosophy would no doubt be
more perfect if it were otherwise ; but co-ordination is of
no use unless it rests on real and fundamental assimilation.
—Physicists must then abstain from fancifully connecting
the phenomena of light and those of motion. Ail that
Optics can admit of mathematical treatment is with rela-
tion, not to mechanics, but to geometry, which is eminently
applicable to it, from the evidently geometrical character of
the principal laws of light. The only case in which we can
conceive of a direct application of analysis is in certain
optical researches in which observation would immediately
furnish some numerical relations: and in no case must the
positive study of light give place to a dynamical analysis.
These are the two directions in which geometers may aid
the progress of Optical science, which they have only too
effectually impeded by prolonging the influence of anti-
scientific hypotheses through inappropriate and ill-conceived
analyses.
The genius of Fourier released us from the necessity of
i.	u290
POSITIVE PHILOSOPHY.
applying the doctrine of hypotheses, as previously laid
down, to the case of thermology: and neither barology nor
acoustics required it. As to electrology, there are abun-
dance of chimerical conceptions preponderant in that depart-
ment : but their absurdities are so obvious, that almost all
their advocates acknowledge them. It is in Optics that
the plausibility and consistence of such chimeras give
them the most importance ; and I have therefore chosen
that department as the ground on which they should be
judged.
We will now pass from these useless hypotheses to the
real knowledge that we are in possession of about the
. .	. theory of light. The whole of Optics is
Op\ics°nS ° naturally divided into four departments, as
light, whether homogeneous or coloured, is
direct, reflected, refracted, or diffracted. These elementary
effects usually co-exist in ordinary phenomena; but they
are distinct, and must therefore be separately considered.
These four parts comprehend all optical phenomena which
are rigorously universal; but we must add, as an indispens-
able complement, two other sections, relating to double
refraction and polarization. These orders of phenomena
are proper to certain bodies ; but, besides that they are a
remarkable modification of fundamental phenomena, they
appear in more and more bodies, as the study proceeds, and
their conditions refer more to general circumstances of
structure than to incidents of substance. For these reasons
they ought to be exactly analysed. As for the rest, it is not
our business to classify the application of these six depart-
ments either to natural history, as in the
beautiful Newtonian theory of the rainbow,
or to the arts, as in the analysis of optical
instruments. These applications serve as the best measure
of the degree of perfection of the science; but they do not
enter into the field of optical philosophy, with which alone
we are concerned.
For the same reasons which have led us to
condemn theories of hearing and utterance,
in connection with Physics, we must now
refuse to include among optical phenomena the theory of
vision, which certainly belongs to physiology, When phy-
Irrelevant
matters.
Theory of
vision.THEORIES OF VISION AND COLOUR.
291
sicists undertake the study of it, they bring only one of the
special qualifications necessary, being otherwise on a level
with the multitude; and, however important their one
qualification may be, it cannot fulfil all the conditions. It
is in consequence of so many conditions being unfulfilled,
that the explanations hitherto offered have been so incom-
plete, and therefore illusory. There is scarcely a single
law of vision which can be regarded as established on
a sound basis, even where the simplest and commonest
phenomena are in question. The elementary faculty of
seeing distinctly at unequal distances remains without any
satisfactory explanation, though physicists have attempted
to refer it to almost every part of the ocular apparatus in
succession. This humbling ignorance is no doubt owing
to scientific men, both physiologists and physicists, having
left the theory of sensations in the hands of the metaphy-
sicians, who have got nothing out of it but some deceptive
ideology : but before this time we should have approached
to something like positive solutions, but for the bad
organization of scientific labour among us. If, from the
time of these questions beginning to assume a positive
character, anatomists and physiologists had occupied them-
selves with a theory of vision grounded on the materials
furnished bv Optical science, instead of looking to physi-
cists for solutions which they could not furnish, our con-
dition in regard to this important subject would be some-
what less deplorable than it is.
Another study which must be excluded
from Optics, and from all natural philosophy,	p0(pes
is the theory of the colour of bodies. I need
not explain that I am not referring to the admirable
Newtonian experiments on the decomposition of light,
which have supplied a fundamental idea, common to all
the departments of Optics. I refer to the attempts made
to ascertain, now through the theory of emission, and now
through that of undulation, the inexplicable primitive
phenomenon of the elementary colour proper to every
substance. The so-called explanations, about the supposed
faculty of reflecting or transmitting such and such a kind
of rays, or of exciting such and such an order of ethereal
vibrations, in virtue of certain supposed arrangements of292
POSITIVE PHILOSOPHY.
the molecules, are more difficult to conceive than the fact
itself, and are, in truth, as absurd as the explanations that
Moliere puts into the mouth of his metaphysical doctors. It
is lamentable that we should have such comments to make
in these days. Nobody now tries to explain the specific
gravity proper to any substance or structure: and why
should wre attempt it with regard to specific colour, which
is quite as primitive an attribute P—In physiology, the
consideration of colours is of high importance, in connection
with the theory of vision ; and in natural history, it may
prove a useful means of classification: but, in optics, the
object of the true theory of colours is merely to perfect the
analysis of light, so as to estimate the influence of structure
or other circumstance upon transmitted or reflected colour,
without entering into the causes of specific colouring. The
field of inquiry is vast enough, without any such illusory
research as this.
SECTION I.
STUDY OF DIRECT LIGHT.
n .	,	, The first department is that of Optics,
p ics piopei. pr0perjy so called or the study of direct light.
This and catoptrics are the only part of the science culti-
vated by the ancients ; but this branch is as old as the
knowledge of the law of the rectilinear propagation of light
in every homogeneous medium. This primary law makes
purely geometrical questions of problems relating to the
theory of shadows; questions difficult to manage in many
cases, but not in the most important,—those of very distant
luminous bodies, or bodies of extremely small dimensions.
The theory depends, both for the shadow and the penumbra,
on the determination of an extensible surface, circumscribed
at once by the luminous and the illuminated body.—What-
ever its real antiquity may be, this first part of Optics is
still very imperfect, regarded from the second point of
T ,	view; that is, with regard to the law^s of
mpei ec ions. in-tensity 0f light, or what is called pho-
tometry. Important as it is to have a clear knowledge, our
notions are as yet either vague or precarious as to how thePHOTOMETRY.
293
intensity of light is modified by such circumstances as its
direction, whether emergent or incident; its distance ; its
absorption by the medium ; and, finally, its colour.
We are met by a grand difficulty at the ,
outset. We have no photometrical instru- 0 °me ry‘
ments that can be depended on for enabling us to verify
our conjectures on the different modes of gradation of light.
All our photometers rest on a sort of vicious circle, being
devised in accordance with the laws which they are destined
to verify, and generally according to the most doubtful of
all, in virtue of its metaphysical origin,—that which relates
to distance. We have called light an emanation; have
calculated its intensity by the square of its distance; and
then, without confirming this conjecture by any experiment
whatever, we have proceeded to found the whole of photo-
metry upon it. And when this conjecture was replaced by
that of undulations, we accepted the same photometry,
neglecting the consideration that it must require revision
from its very basis. It is clear what our present photo-
metry must be, after such treatment as this. The law
relating to direction, in the ratio of the sine of the angle of
emergence or of incidence, is no better demonstrated than
that of distance, though it comes from a less suspicious
source. It has nothing about it at present like Fourier’s
labours on radiating heat; and yet it seems as if it would
admit of an analogous mathematical elaboration. The only
part of photometry which has, as yet, any scientific con-
sistency is the mathematical theory of gradual absorption
of light by any medium. Bouguer and Lambert have given
us some interesting knowledge about this: but even here
we are on unstable ground, for want of precise and unques-
tionable experiments. Again, the photometrical influence
of colour has been the subject of some exact observations ;
but we are not yet in possession of general and precise con-
clusions, unless it be the fixing of the maximum of bright-
ness in the middle of the solar spectrum. Thus, to sum up,
in this first, oldest, and simplest department of optics, phi-
losophers have scarcely outstripped popular observation,—
leaving out what belongs to geometry, and the measurement
of the velocity of the propagation of light, which is fur-
nished by astronomy.294
POSITIVE PHILOSOPHY.
SECTION II.
CATOPTRICS.
It is otherwise with regard to catoptrics, and yet more,
dioptrics, if we discard questions about the first causes of
reflection and refraction. Scientific studies have largely
extended and perfected universal ideas about those two
orders of general phenomena; and the varied effects be-
longing to them are now referred with great precision to a
very small namber of uniform laws, of remarkable sim-
plicity.
, The fundamental law of catoptrics, well
rettectionV ° known by the ancients, and abundantly con-
firmed by experiment, is that whatever may
be the form and nature of the reflecting body, and the
colour and intensity of the light, the angle of reflection is
always equal to the angle of incidence, and in the same
normal plane. Under this law, the analysis of the effects
produced by all kinds of mirrors is reduced to simple
geometrical problems, which might, it is true, involve some
long and difficult calculations, according to the forms of
some bodies, if it were not usually sufficient to examine the
simple forms of the plane, the sphere, and, at most, the
circular cylinder. If we pretended to absolute precision
in the analysis of images, we might encounter considerable
geometrical difficulties: but this is not necessary. This
analysis depends, in general, mathematically speaking, on
the theory of caustic curves, created by Tschirnhausen. But
even in the application of this theory, some conjectures are
hazarded; and the want of direct and exact experiments, and
the uncertainty which attends almost all the parts of the
theory of vision, prevent our depending too securely on the
reality of the remote results of any general principle that
we can yet employ.
Law of absor Every luminous reflection upon any body
tion no^ffiund! w^a^ever is accompanied by an absorption of
more or less, but always of a great part of
the incident light; and this gives rise to a second interest-
ing question in catoptrics. But our knowledge about itDIOPTRICS.
295
amounts to very little, from our backwardness in photo-
metry ; so that we have not yet laid hold of any law. We
do not know whether the loss is the same in all cases of
incidence: nor whether it is connected with the degree of
brightness: nor what is the influence of colour upon it:
nor whether its variations in different reflecting bodies are
in harmony with other specific, and especially optical
characters. These questions are not only untouched : they
have never been proposed. All that we know is simply
that the absorption of light appears to be always greater
(but to what degree we are ignorant) by reflection than by
transmission. From this has resulted, in recent times, the
use of lenticular beacons, introduced by Fresnel.
A more advanced kind of inquiry belongs to the study of
transparent substances; but here, again, the laws are ill
understood. In these bodies, reflection accompanies refrac-
tion, and we have the opportunity of inquiring by what
laws, general or special, the division between transmitted
and reflected light takes place. We only know that the
last is more abundant in proportion as the incidence is
more oblique; and that reflection begins to become total
from a certain inclination proper to each substance, and
measured exactly with regard to several bodies. The incli-
nation appears to be less in proportion as the substance is
more refracting : but the supposed law of the case is con-
nected with chance conjectures upon the nature of light,
and requires to be substantiated by direct experiment.
SECTION III.
DIOPTRICS.
Of all the departments of Optics, dioptrics is at present
the richest in certain and exact knowledge, reduced to a few
simple laws, embracing a large variety of phenomena. The
fundamental law of refraction was wholly ~	.
unknown to the ancients, and was discovered refraction.
at the same time, under two distinct and
equivalent forms, by Snellius and Descartes. It consists
of the constant proportion of the sines of the angles that296
POSITIVE PHILOSOPHY.
the refracted ray and the incident ray, always contained in
the same normal plane, form with the perpendicular to the
refracting surface, in whatever direction the refraction may
be. The fixed relation of these two sines, when the light
passes from a vacuum into any medium whatever, consti-
tutes the most important optical coefficient of every natural
body, and holds a real rank in the aggregate of its physical
characteristics. The philosophers have laboured at its
determination with much care and success, by ingenious
and exact processes: they have prepared very extensive
tables, which may rival, as to precision, our tables of
specific gravity—the uncertainty not exceeding a hundredth
part of the numerical value of the refracting power. If
the light passes from one medium to another, the relation
of the refraction depends on the nature of both: but in
every case, the inverse passage gives it always a precisely
reciprocal value; as experiment has constantly shown.
Again, while a body undergoes no chemical change, and
becomes only more or less dense, the relation of refraction
which belongs to it varies in proportion to the specific
gravity; as may be easily shown, especially with regard to
liquids, and yet more to gases, in which we can so exten-
sively modify density by temperature and pressure. This
is why philosophers have adopted, in preference to the
proper relation of refraction, its quotient by the density,
which they have named refracting power; in order to obtain
more fixed and specific characters in the dioptric com-
parison of different substances. There is substantial ground
for this distinction, though its origin was suspicious. But
it must be observed that the refracting power varies when
the substance does not undergo any chemical change, but
passes, as we have seen in the case of water, through diffe-
rent states of aggregation. These variations in the refract-
ing power have given occasion to conflicts between the
advocates of the two hypothetical systems,—each of which
requires an invariability in the refracting power which we
do not know to exist: and the difficulty of separating what
is really established from what they require is one of the
mischievous consequences of anti-scientific hypotheses, and
one which may well render the actual character of the
science itself doubtful to impartial minds.newton’s discoveries on elementary colours. 297
Newton’s discoveries of the unequal re- Newton’s dis-
frangibility of the different elementary coveries on
colours form an indispensable complement elementary
of the law of refraction. From the fact of colours*
the decomposition of light in a prism, it clearly follows
that the relation of the sine of incidence, though constant
for each colour, varies in the different portions of the solar
spectrum. The total increase which it undergoes from the
red rays to the violet measures the dispersion proper to each
substance, and must complete the determination of its re-
fracting power in the common tables, where only the mean
refraction can be inserted. This estimate constitutes, from
its minuteness, one of the most delicate operations of optics,
and does not admit of so much exactness as that of the re-
fracting action properly so called, especially in bodies which
bend the light but little, as the gases ; but it is ascertained
for a considerable number of substances, solid or liquid.
In comparing the changes of the dispersive power as we
pass from one body to another, we discover that the varia-
tions are not, as Newton supposed, in proportion to the re-
fracting power : and indeed we find, in more than one case,
that the light is least dispersed by substances which refract
it most. The discovery of this discrepancy between two
qualities which appear to be analogous was made by
JDollond, about the middle of the last century. It is an
idea of high importance in Optics, as it indicates the possi-
bility of achromatism by the compensation of the opposite
action pertaining to two different substances which, with-
out that, could not cease to disperse the light but by ceas-
ing to bend it.
The laws of refraction show us that there can be none
but purely geometrical difficulties in the analysis of the
effects of homogeneous media upon the light which traverses
them. The great complication which might arise from the
form of the refracting body is diminished in ordinary cases
by our satisfying ourselves with plane, spherical, or cylin-
drical surfaces: but we should yet find the inquiry embar-
rassing, and especially in regard to the dispersion, if we
did not confine it to an approximate estimate of the few
commonest circumstances.298
POSITIVE PHILOSOPHY.
SECTION IV.
DIFFRACTION.
The modification called diffraction has now become one
of the essential parts of Optics. It was entered upon by
Grimaldi and Newton, advanced by the researches of Dr.
Young, and completed by those of Fresnel. It consists of
the deviation, always accompanied by a more or less marked
dispersion, that light undergoes, in passing close by the
edges of any body or opening. Its simplest way of mani-
festing itself is by the unequal and variously-coloured
fringes, some exterior and some interior, which surround
the shadows produced in a darkened room. The famous
general principle of interferences, discovered by Dr. Young,
is the most important idea connected with this theory. It
was not appreciated, remarkable as it is, till Fresnel made
use of it to explain several interesting phenomena, diffi-
cult to analyse; and, among others, the celebrated pheno-
menon of the coloured rings, which were by no means
fully accounted for by Newton’s admirable efforts. The
law of interferences is this : that when two luminous
cones emanate from the same point, and follow, for any
reason, two distinct courses, but little inclined towards
each other, the intensities proper to the two lights neu-
tralize and augment each other alternately, increasing by
equal and minute degrees, the value of which is deter-
mined, the difference in length between the entire paths
traversed by the two cones. It is a pity that this im-
portant principle should have suffered, like the rest, from
being implicated with chimerical conceptions on the nature
of light.
We have done all that the nature of this Work admits,
in regard to Optics ; and we must pass over the subjects of
the double refraction proper to various crystals, the general
law of which was discovered by Huyghens. We must also
omit the phenomena of polarization, disclosed by Malus.
In what I have brought forward, I hope that, while I have
pointed out the gaps in this science, of which we are tooDIFFRACTION.
299
little conscious at present, I have also placed in a clear
light the great and numerous results obtained during the
last two centuries, notwithstanding the disastrous prepon-
derance of vain hypotheses about the nature of light over
the spirit of rational experimentation.300
CHAPTER VI.
ELECTROLOGY.
History.
THIS last branch of Physics, relating as it
does to the most complex and least
manifest phenomena, could not be developed till after the
rest. The electrical machine indeed is as old as the air-
pump ; but it was not till a century later that the study
assumed a scientific character, through the distinction of
the two electricities, Muschenbroek’s experiments with the
Leyden jar, and then through Franklin’s great meteoro-
logical discovery, which was the first manifestation of the
influence of electricity in the general system of nature. TJp
to that time, the isolated observations of philosophers had
only suggested the character of generality inherent in this
part of Physics, as in all others, by continually adding to
the number of substances susceptible of electrical pheno-
mena : and it was not till the end of the last century that
this department of Physics presented anything like the
rational character which belongs to the others. It is owing
to the labours of Coulomb that it takes its place, and still
an inferior place, with the rest.
Ho other science offers so great a variety
of curious and important phenomena; but
facts do not constitute science, though they are its founda-
tion and material. Science consists in the systematizing
of facts under established general laws: and, regarded in
this way, Electrology is the least advanced of all the
branches of Physics, imperfect as they all are. In the ab-
sence of ascertained laws, arbitrary hypo-
hypotheses thesis has run riot. The simple confidence
with which students have explained all
phenomena by endowing imaginary fluids with new pro-
perties for every fresh occurrence, reminds us of the old
metaphysical explanations,—the ancient entities being
Condition.HYPOTHESES ABOUT ELECTRICITY.
301
merely replaced by supposed fluids. But the delusion is
less mischievous here than in Optics, where the arbitrary
conjectures are closely connected with real laws, and share
their imposing character. In electrology the hypotheses,
standing alone, exhibit their barrenness; and everybody
can see that they have borne no share in the great dis-
coveries of the last half-century, though the discoveries,
once made, have been afterwards attached to the hypo-
theses. Most people regard them now as a sort of
mnemonic apparatus, useful for connecting facts in the
memory, though originally designed for a very different
purpose. They are a bad apparatus for even this object,
which would be much better answered by a system of
scientific formulas especially adapted to that use. And,
though less mischievous than in Optics, hypotheses of this
order do harm in electrology, as everywhere else, by con-
cealing from most minds the real needs of the science. It
should be remembered, moreover, that anti-scientific action
like this extends its influence over the succeeding and more
complex sciences, which, on account of their greater diffi-
culty, require the severest method, the type of which will
naturally be looked for in the antecedent sciences. It is a
serious injury to transmit to them a radically vicious model.
While physicists are using these hypotheses as having
avowedly no intrinsic reality, their very use leads students
of the successive sciences, and especially physiologists, to
consider them the very sublimity of physics, and to proceed
to take them for the bases of their own labours. We see
how the notion of magnetic and electric fluids tends to
confirm that of a nervous fluid, and to encourage wild
dreams about the nature of what is called animal mag-
netism, in which even eminent physicists have shared.
Such consequences show how a study which is naturally
favourable to the positive development of human intelli-
gence may, by vicious methods of philosophizing, become
fatal to our understandings.
From the complex nature of the pheno-
mena, there can be but little application of Mathematics
mathematics in electrology. It has as yet
borne only a small share in the progress of the science :
but it is as well to point out the two ways,—the one302
POSITIVE PHILOSOPHY.
illusory, the other real,—in which the application of mathe-
matics has been attempted.
Those who have occupied themselves with
application imaginary fluids as the causes of electrical
and magnetic phenomena, have transferred
the general laws of rational mechanics to the mutual action
of their molecules; thus making the body under notice a
mere substratum, necessary for the manifestation of the
phenomenon, but unconcerned in its production; with
which office the fluid is charged. It is clear that mathe-
matical labours so baseless can serve no other purpose than
that of analytical exercise, without adding a
cation aPP 1 particle to our knowledge. In the other case,
—of a sound application,—the mathematical
process has been based on some general and elementary
laws, established by experiment, according to which the
study of phenomena proper to the bodies themselves has
been pursued,—all chimerical hypotheses being discarded.
This is the character of the able researches of M. Ampere
and his successors, on the mathematical investigation of
electro-magnetic phenomena, in which the laws of abstract
dynamics have been efficaciously applied to certain cases of
mutual action between electric conductors or magnets.
In examining the principal parts of electrologv, we must
exclude all that belongs to the chemical or physiological in-
fluence of electricity, and all connection of electricity with
concrete physics ; and especially with meteorology.
T.. . .	Thus limited to the physical and abstract,
electrology at present comprehends three
orders of researches. The first relates to the production,
manifestation, and measurement of electrical phenomena:
the second, to the comparison of the electric state proper
to the different parts of the same mass, or to different con-
tiguous bodies: the third, to the laws of the motions which
result from electrization: we may add, as a fourth head,
the application of the results under the other three to the
special study of magnetic phenomena, which can never
henceforth be separated from them.CAUSES OF ELECTRIZATION.
303
SECTION I.
ELECTRIC PRODUCTION.
The sum of our observations leads us to regard the elec-
tric condition of bodies as being, more or less evidently, an
invariable consequence of almost all the modifications they
can undergo: but the chief causes of elec-
trization offer themselves, in the order of trization & ^
their power and scientific importance, thus :
chemical compositions and decompositions: variations of
temperature : friction: pressure : and, finally, simple con-
tact. This distribution differs widely from that first indi-
cated by inquiry,—friction being long supposed the only,
and then the most powerful means of producing the electric
condition. The comparison of means is very far from being
exhausted ; but we may be assured that the order specified
above will never be radically changed.
There is no doubt that chemical actions
are the most general sources of electricity, as tlon™10*1 a°*
well as the most abundant; as they are with
regard to Heat. In the most powerful electrical apparatus,
and especially in the Voltaic pile, the chemical action, which
at first passed unnoticed, is now recognized, thanks to the
labours of Wollaston and others, as the principal source of
electrization, which becomes indeed almost insensible when
care is taken to exclude chemical action.—After this, the
next most powerful cause is thermological
action, though, till recently, it was recognized ac]fon10 °glCa
only in the single case of heated tourmalin.
We now know that marked differences of temperature be-
tween consecutive bars of different kinds, whether homo-
geneous or otherwise in the particular case, suffice to induce
a marked electrical condition, the more intense as the
elements are more numerous,—the thermometrical condi-
tions remaining the same.—These two causes are so power-
ful, and so difficult to exclude, that the estimate of the
others becomes a very delicate matter. It is difficult to
determine how much influence to ascribe to any cause after
these two, while yet they are almost unavoidably present.304
POSITIVE PHILOSOPHY.
Friction.
Pressure.
Thus, even about friction, which used to be
regarded as so powerful a cause, it is now
doubtful whether the friction itself has any influence, and
whether the electrization is not due to the thermometrical,
and even the chemical effects which always accompany
friction, but which used to be altogether overlooked in this
instance.
The case is nearly the same with Pressure,
the electric influence of which however is, if
less marked, more unquestionable, from our being able to
isolate it more. But the remark is above all applicable to
the production of the electric state by the simple contact of
r ,	,	heterogeneous bodies. It was by this con-
on ac * tact that Yolta brought out the power of
his wonderful instrument, while it is well known now that
chemical action bears a chief part in it, and that contact
contributes to it in only a secondary manner, if even it be
not altogether doubtful.
Besides these leading causes of electriza-
tion, there are many less important,—as
changes in the mode of aggregation, the fusion of solids,
and the evaporation of liquids. Even simple motion
suffices, under special conditions, to induce an electric
state, as M. Arago has shown in the experiment of the in-
fluence of the rotation of a metallic disc upon a magnetized
needle, near but not contiguous. Our philosophers how-
ever must beware of passing into the other extreme from
that with which they justly reproach their predecessors.
It is, no doubt, prejudicial to electro logy to neglect all
sources of electrization but the most conspicuous: but it
may be not less so to carry analysis too far, and see causes
of electrization in all sorts of minute phenomena.1
r ,	.	A special instrument, or class of instru-
ments, naturally corresponds to each ot the
general modes of electrization, in order to realize the most
favourable conditions for the production and support of the
electric state. However important these may be, it is clear
that we cannot here enter upon the consideration of them.
1 In this paragraph, M. Comte alludes to the now most fertile,
but when he wrote, the comparatively unknown subject of the
development of Electricity by Induction.—J. P. N.ELECTRICAL INSTRUMENTS.
305
But we must not pass over the instruments invented
for the manifestation and measurement of the electric con-
dition,—the electroscope and the electrometer. The most
eminent philosophers have always attached the highest im-
portance to the perfecting of these instruments, in the
invention of which real genius has often been exhibited.
Their perfection is of more consequence than that of
electric producers; because very weak electric powers often
answer best in delicate experiments, from their simplicity;
while the utmost ingenuity is required in instituting
means of manifesting and measuring the minutest electric
effects.—Though the electric condition cannot be measured
without being first manifested, and the manifestation leads
to some sort of estimate, there is a real distinction between
electroscopes and electrometers. Among simple electro-
scopes, the most remarkable for use in very delicate re-
searches, is that kind called condensers, which render
feeble electrical effects sensible through their gradual
accumulation : and all these instruments are so arranged
as to show, by the method of experimentation itself,
the positive or negative character of the electricity under
notice.—Coulomb’s electrical balance is certainly the most
perfect of electrometers. It was by its means that he dis-
covered, and that we every day demonstrate, the funda-
mental law of the variation of electric action, repulsive
or attractive, inversely to the square of the distance; a law
which could not be unquestionably obtained by any other
means. As we have advanced in the science of electro-
magnetism, a new class of electrometers has been intro-
duced, for purposes of measurement, for which Coulomb’s
balance would not answer. These are the class of multi-
pliers. Valuable and delicate as they are, they have not
yet been applied, with so much certainty as the balance,
to exact measurements, from the difficulty of propor-
tioning the graduation to the intensity of the observed
phenomenon.
i.
x306
POSITIVE PHILOSOPHY.
SECTION II.
ELECTRICAL STATICS.
The second part of electrology includes what is impro-
perly called electrical statics; a term imputable to illusory
hypotheses about the nature of electricity: yet it is not
a wholly absurd title, as it relates, in fact, to the distribu-
tion of electricity in a mass, or in a system of bodies, the
electric state of which is regarded as invariable. We may
therefore continue to use this abridged term, if we carefully
keep clear of all mechanical notions of the equilibrium of
any supposed electric fluid, and attach to it a sense analo-
gous to that of Fourier, when he spoke of an equilibrium of
heat, and of economists when they speak of an equilibrium
of population.
.	, Considering first the case of an isolated
distribution b°dy, Coulomb has established a funda-
mental law which is (metaphorically ex-
pressed) the constant tendency of electricity to the surface,
or, in rational language, that after an inappreciable instant
of time electrization is always limited to the surface, how-
ever it may have been in the first place produced. As for
the distribution of the electric state among the different
parts of the surface, it depends on the form of bodies,
being uniform for the sphere alone, unequal for all other
forms, but always subject to regular laws. The analysis of
these may be supposed to present insurmountable difficul-
ties ; nevertheless, Coulomb has established a general fact
of great importance, by comparing the electric states proper
to the extremities of an ellipsoid gradually elongated: he
has perceived that their electrization increases rapidly as
the figure is elongated, diminishing in the rest of the body ;
whence he deduced an explanation of that remarkable power
of points, disclosed by Franklin.1
Electric	The ^aws °f electric equilibrium between
equilibrium. several contiguous bodies afford a yet more
difficult and extensive inquiry. Coulomb
1 Much lias since been added to this class of investigations.—
J. P. N.ELECTRICAL STATICS AND DYNAMICS.
307
studied them only in the limited and insufficient single
case of spherical masses. However, we learn from his
labours that the nature of substances exercises no influence
over the electric distribution established among them, the
mode depending merely on their form and their magnitude ;
only, the electric state assumed by each surface is more
or less persistent, and manifests itself with more or less
rapidity, according to the degree of conductibility in the
body. Coulomb analysed completely the mutual action of
two equal spheres; discovering that the electric condition
is always null at the point of contact, scarcely sensible at
20 degrees from that point, fast increasing from 60 to 90
degrees, and then more slowly increasing up to 180 degrees,
which is its maximum. If the globes are unequal, the
smallest is the most strongly affected: and it makes no
difference whether they are electrized together, or the one
before the other. The question becomes more complex
when more than two bodies are concerned. Coulomb
examined only a series of globes ranged in a straight line;
but if they had been so placed as that each should touch
three or four others, the mode of electric distribution
would inevitably have undergone great changes. The sub-
ject must be regarded as merely initiated by this great
philosopher : and no one has added anything to it since his
time. It offers to electricians a subject of almost inex-
haustible research.1
SECTION III.
ELECTRICAL DYNAMICS.
The third part of electrology is very pro-	,	,
perly called Electrical Dynamics, because it experiments,
relates to the motions which result from
electrization. Recent as is its origin, it is superior to the
others in its scientific condition, through the labours of
M. Ampere; always supposing conjectures about the
nature of electric phenomena to be discarded. M. Ampere
1 These specific facts are now comprehended within general
laws.—J. P. N.308
POSITIVE PHILOSOPHY.
has referred the analysis of the effects observed in this
branch of electrology to one great and general phenomenon,
the laws of which he has fully ascertained; the direct
and mutual action of two threads, charged with electricity
by voltaic piles, habitually reduced to their greatest sim-
plification ; that is, almost always composed of a single
element.
M. Ampere so arranged his experiment as to guard the
conducting threads from the perturbing influence of the
earth’s electricity ; and this done, he could easily seize the
elementary laws of the phenomenon under his notice. He
found that when the twro conductors are sufficiently mobile,
they tend to place themselves in directions parallel to each
other; and that they then attract or repel each other,
according to the conformity or contrariety of the two
electric currents. In looking for the laws of the case, it is
necessary, for the sake of generality and simplicity, to
keep in view only infinitely small portions of the different
conductors. These laws, mathematically considered, relate
either to the influence of the direction, or to that of the
distance.
As to the direction, there are the two cases to be
considered of the conducting elements being in the same
plane, or in different planes. In the first case, the
intensity of the action depends only on the angle formed
by each of the two elements with the line which joins their
middle points: it is null at the same time with this angle,
and increases with it, attaining its maximum when it
becomes right. All phenomena, direct or indirect, appear
to be exactly represented if this intensity is made to vary in
proportion to the sine of the inclination, according to the
formula adopted by all the successors of M. Ampere. In
the other case,—of the conductors not being in the same
plane,—the action depends moreover on the mutual inclina-
tion of the planes indicated by each of them, and by the
common line of their middle points; and the result of this
second relation is wholly different. The perpendicularity
of the two planes determines the absence of all action:
there is attraction while the angle is acute, and it increases
as the angle diminishes, its maximum taking place at the
moment of coincidence; when the angle is obtuse, theampere’s experiments.
309
action becomes repellent, and increases as each plane
approaches towards the prolongation of the other, a
situation which produces the maximum of repulsion. The
supposition which arises in this case is that the action is in
proportion to the cosine of the angle of the two planes ;
but we have not yet attained such certainty as in the former
case.
As for the influence of distance, M. Ampere supposed
that, in analogy with Coulomb’s law of common electric
attraction and repulsion, the action of two conducting
elements is always reciprocal to the square of the distances
of their middle points. But analogy is not sufficient to
conclude upon; and direct observation is out of the ques-
tion when the parts taken are infinitely small, and the
result sought must be affected by the form and magnitude
of the conductors. However, it may be mathematically
demonstrated that, in the hypothesis adopted by M.
Ampere, the action of a rectilinear conductor, of an
indefinite length, upon a magnetized needle, must vary
exactly in the inverse ratio of their shortest distance.
This consequence has been precisely verified by experi-
ment ; and it places beyond a doubt the reality of the pro-
posed law.
Under this law, electric action would seem to be, mathe-
matically, in analogy with that of gravitation. But this
case affords a lesson against incaution in transferring to the
study of these singular movements the ordinary procedure
of abstract dynamics. Gravitation is independent of
mutual direction, which is the determining influence in
electrical dynamics : and thus the parallel fails. We see,
further, how many more difficulties are in the way of the
analysis of electric forces than in that of molecular gravita-
tion. If this last is, from its complexity, unmanageable
except in the simplest cases, it is no wonder that electrical
dynamics has not been mathematically studied further
than in one dimension, and never at all in surface. Even
this much would be hardly effected but for a last funda-
mental idea, established by M. Ampere; that in an infi-
nitely small extent, and as long as the distance is not
sensibly changed, the electric action is identical for two
conducting elements issuing at the same extremities, what-310
, POSITIVE PHILOSOPHY.
ever may be otherwise their difference of form. Such a pro-
perty must introduce valuable analytical simplifications,
tending to establish a remarkable analogy between electric,
and ordinary dynamic decompositions.
These are the grounds on which the study of the various
action of electrized threads proceeds. Among the many
dispositions of these conductors, the most interesting case is
that of the spiral form; and especially when the turns
are very close together. M. Ampere has shown the high
importance of this form, in order to imitate, as exactly
as possible, the jDhenomena characteristic of magnetized
bodies.1
# # # % # #
We have now reviewed the philosophy of
Physics1011 ° Physics, noticing in turn the aspects pre-
*	sented by the study of the properties com-
mon to all substances and all structures. These are not so
much branches of a single study as distinct sciences. Part
of our business has been to carry on a philosophical
operation, hardly necessary in astronomy, but becoming
more and more so as we descend to the more complex
sciences;—that of disengaging real science from the in-
fluence of the old metaphysical philosophy, under which it
still suffers deplorably, and which manifests itself in
Physics through illusory and arbitrary conceptions about
the primitive agents of phenomena. I have been able only
to indicate the mischief, and where it resides; and I must
leave the work of purification to rational philosophers,
whose attention will, we must hope, be more and more
drawn to this vital question. It is with the same view that
I have endeavoured to assign the true application of
mathematical theories to the principal branches of physics,
pointing out by the way the danger of the excessive sys-
tematization which is too often sought by carrying the use
1 M. Comte concludes the section on Electricity by a slight
reference to the discoveries of Oersted, Arago and others, regarding
its virtual identity with all we term the magnetic forces. But as
the whole of this most interesting and important part of Physics
has taken a new form since the date of his work, it has not, for
reasons assigned in the Preface, been thought necessary to repro-
duce his remarks in this place.—J. P. N.CONCLUSION OF PHYSICS.
311
of this powerful instrument further than the complex
nature of the corresponding phenomena would fairly allow.
While giving my chief attention throughout to the method,
I have pointed out, in brief, the principal natural laws
relating to each department of science, discovered by
human effort during the two centuries which have elapsed
since the birth of Physics, properly so called : and I have
shown what gaps are disclosed in the course of such a
survey.
Our next study will be of the last science which belongs
to the class of general knowledge, or that of inorganic
nature. Chemistry relates to the molecular and specific
reactions which different substances exert upon each other.
It is a more complex, and consequently more imperfect
science than those which we have reviewed : but its general
character may be perfected, through the means afforded by
its subordination to the anterior sciences.312
BOOK IV.
CHEMISTRY.
CHAPTER I.
,	, , W TE have now to review the last of the
s na me. V V sciences which relate to the inorganic
world. Chemistry has for its object the modifications that
all substances may undergo in their composition in virtue
of their molecular reactions. Without this new order of
phenomena, the most important operations of terrestrial
nature would be incomprehensible to us; and there is no
other class of phenomena so intimate and so complex.
Inert bodies can never appear so nearly like vital ones as
when they produce in each other those rapid and profound
perturbations which characterize chemical effects. We shall
see hereafter that the spirit of all theological and metaphy-
sical philosophy consists in conceiving of all phenomena as
analogous to the only one which is known by immediate
consciousness,—Life: and we can easily understand that
the primitive method of philosophizing must have exerted
a more powerful and obstinate dominion over chemical
phenomena than any other, in the inorganic world.—We
must consider, too, that direct and spontaneous observa-
tion must have been applied in the first place only to very
complicated phenomena, such as vegetable combustions,
fermentations, etc., the analysis of which now requires all
the resources of our science : and that the most important
chemical phenomena are produced only in artificial circum-
stances, which were long in being devised, and very difficult
at first to institute. Easy as it is now for even the most
ordinary inquirers to use known substances for the dis-
closure of new relations, we can hardly imagine the difficultyIMPERFECTION AND CAPACITIES OF CHEMISTRY. 313
there must have been, in the infancy of chemistry, in creat-
ing suitable subjects for observation : and we cannot suppose
that the ancient investigators of nature could have had
energy and perseverance to discover the principal pheno-
mena of the science if they had not been constantly stimu-
lated by the unbounded hopes arising from their chimerical
notions of the constitution of matter. The complex and
doubtful nature of the phenomena, in the	.
first place, and next the difficulty of getting fectfon1111^1
at them, are quite enough to account for the
tardy and incomplete positivity of chemical conceptions, in
comparison with all others in the inorganic region of
nature. If, as we have seen, Physics is defective in several
respects, much more must that science be so which, being
at once more difficult and more recent, seeks the laws of
composition and decomposition. Whichever way we look
at it, whether speculatively, as to the value of its explana-
tions, or actively, as to the previsions which they admit of,
this science is evidently the least advanced of all the
branches of inorganic philosophy. Indeed, it is hardly
possible to call chemistry a science at all while it scarcely
ever leads to that precise prevision which is the criterion
of perfection in speculative knowledge. We can rarely tell
what will be the result of the smallest and fewest modifi-
cations introduced among the best explored chemical opera-
tions ; and while that is the case, however important and
numerous may be the facts collected, we are in possession
of only erudition, and not science. To suppose otherwise
is to mistake a quarry for an edifice.
It is not to be hoped that chemistry can ^	...
ever attain a state of rationality so satisfac-	1
tory as that of the sciences which relate to phenomena of a
more simple character; and especially that of the eternal
type of natural philosophy,—Astronomy. But so much of
its inferiority seems to be due to a vicious philosophy, and
to the defective education of philosophers, that I cannot
but hope that a judicious philosophical analysis may con-
tribute to a speedy perfecting of so important a science.
This is the conviction that I desire to awaken by the rapid
sketch which I propose to offer of chemical philosophy, re-
garded in all its essential aspects. Little as can be done314
POSITIVE PHILOSOPHY.
within the bounds of this section, it is possible that
some one eminent inquirer may be impressed by the
necessity of submitting to a new and more rational elabo-
ration the fundamental conceptions which constitute the
science.
Object of
Chemistry.
First,—what is the general object of
Chemistry ? Vast and complex as is its sub-
ject, the definition of Chemistry is easier
than that of Physics. We are already prepared for it, in-
deed, by having contrasted that of Physics with it. It is
easy to characterize the phenomena of chemistry, in a direct
and marked manner; for all indicate an alteration, greater
or smaller, in the constitution of bodies: that is, a compo-
sition or decomposition, and generally both, taking into
the account the whole of the substances which participate
in the action. Thus, at all epochs of scientific development,
since chemistry first became an object of speculative study,
chemical researches have steadily manifested a remarkable
originality, which has prevented their being confounded
with other parts of natural philosophy ; even while Physics
itself was was mixed up, as its title shows, with physiology;
which was the case up to a very recent time.—It is by this
general character of its phenomena that Chemistry is dis-
tinguished from Physics which precedes it, and Physiology
which follows it. The three sciences may be considered as
having for their object the molecular activity of matter, in
all the different modes of which it is susceptible. Each
corresponds to one of three successive degrees of activity,
which are essentially and naturally distinguished from
each other. The chemical action obviously presents some-
thing more than the physical action, and something less
than the vital. The physical activity modifies the arrange-
ment of particles in bodies; and these modifications are
usually slight and transient, and never alter the substance.
The chemical activity, on the contrary, besides these altera-
tions in the structure and the state of aggregation, occa-
sions a profound and durable change in the very composi-
tion of the particles: the bodies which occurred in the
phenomenon are no longer recognizable,—so much has the
aggregate of their properties been disturbed.—Again, phy-
siological phenomena show us the molecular activity in aCHARACTER AND CONDITION OF CHEMICAL ACTION. 315
much higher degree of energy ; for, as soon as the chemical
combination is effected, the bodies become, once more, com-
pletely inert; whilst the vital state is characterized, over
and above all physical and chemical effects, by a double
continuous motion of composition and decomposition,
adapted to maintain, within certain limits of variation and
of time, the organization of the body by incessantly renew-
ing its substance. This is the gradation, which no sound
philosophy can ever confound, of the three modes of mole-
cular activity.
Two more characteristics of this science must be pointed
out: one relating to its nature, and the other to its general
conditions.
Chemistry would not be classed among the Specific cha*
inorganic sciences unless its phenomena were	racter of its
general; that is, unless every substance werex action,
susceptible of chemical action, more or less. And it is be-
cause chemistry is thus radically different from physiology
that it ranks as the last of the inorganic sciences,—physio-
logical phenomena being, by their nature, peculiar to cer-
tain substances, organized in certain modes. Nevertheless,
it is incontestable that chemical phenomena present, in
every case, something specific, or, to use Bergmann’s ener-
getic expression, elective. Not only does each material
element produce chemical effects which are altogether
peculiar to it, but it is the same with their innumerable
combinations of different orders, among the most analogous
of which certain fundamental differences are observable,
even so as to be adopted as their characteristics. While
therefore physical differences among different bodies are
those of degree only, chemical properties are specific. Phy-
sical properties afford the common foundation of material
existence; and it is by chemical properties that individu-
ality is manifested.
The other characteristic relates to the ~	,
£	,	-|	,. m, .	, Condition of
mode ot chemical action. The immediate action
contact of antagonistic particles is absolutely
necessary to chemical action; and therefore one at least
of the substances concerned must be fluid or gaseous.
When this condition does not already exist, it must be arti-
ficially procured by liquefying the substance. It is the316
POSITIVE PHILOSOPHY.
earliest axiom in the science, that combination cannot take
place, except under this condition ; and there is not an in-
stance upon record of chemical action between two solids,
unless at a temperature which obscures the true state of
aggregation of substances; and the action is never so
powerful as when both substances are liquid. These facts
establish the eminently molecular character of chemical
effects, and especially in comparison with physical effects.
The distinction from physiological effects is, though less
marked, as real, the latter requiring, as we shall see here-
after, the junction of solids with fluids.
n ...	The definition of Chemistry, then, is that
e m ion.	re}a^es |pe ]aws 0f the phenomena of
composition and decomposition, which result from the
molecular and specific mutual action of different substances,
natural or artificial.
It will be long, we must fear, before a more precise defi-
nition than this can be given. Meantime, however incom-
plete, the most rational that can as yet be offered is of
importance as far as it goes. In this view, and connecting,
as usual, the consideration of science with that of prevision,
the aim proposed should be this:—the characteristic pro-
perties of substances, simple or compound, being given,
and those properties being placed in a chemical relation in
well-defined circumstances, to determine in what their
action will consist, and what will be the chief properties of
the new products. This problem is, at all events, determi-
nate ; and nothing contained in it could be omitted without
its ceasing to be so ; and the formula therefore contains
nothing superfluous. On the other hand, if we could
obtain such solutions as are indicated, the application of
chemistry to the three great objects, vital phenomena, the
natural history of the globe, and industrial operations,
would be rationally organized, instead of being, as now,
the almost accidental result of the spontaneous development
of science. Each question would at once be referred to
our formula, the data of which would be supplied by the
circumstances peculiar to the application. Far distant as
we are from being able thus to conduct our inquiries, this
is the end to be kept in view: and chemists all agree that
the most advanced portions of their science are those fewELEMENTS AND THEIR COMBINATIONS.
317
and simple questions in which this aim has been more or
less completely attained.
By a continued application of this method, all the data
must finally be reducible to the knowledge of the essential
properties of simple substances, which would lead to that
of the different immediate principles: and consequently,
to the most complex and remote combination. As for the
study of the elements, that must, of course, Elements
be a matter of direct, experimental elabora-
tion, divided into as many parts as there are undecomposed
substances. Whether or not it may be possible to discover,
by rational methods, relations between the chemical pro-
perties of each element and its aggregate physical proper-
ties, we must lay down as indispensable a direct exploration
of the chemical characters of each element. This general
basis once obtained from experiment, all other chemical
problems must be susceptible of a rational solution, under
a small number of invariable laws.
The classes of combinations naturally divide ^	,.	,.
themselves into two, according, first, to the
simplicity, or the greater or less degree in which the im-
mediate principles are compounded : and, secondly, the
number of elements combined. Chemical action is observed
to become more difficult the more substances are com-
pounded : the greater part of compound atoms belong to
the first two orders; and beyond the third their composition
seems almost impossible: and, in the same way, in regard
to the number of elements, combinations lose their stability
in proportion as the elements are multiplied: there are
usually only two ; and scarcely any body involves more than
four. Thus, the number of chemical classes must always
be very small in regard to the distinction under notice : and
each of them must have a corresponding law of combination,
according to which the result might be certainly anticipated
through a knowledge of the data. This would be the scien-
tific perfection of chemistry. Our prodigious remoteness
from such a state is ascribable to the feebleness of our
faculties, and, in an accessory way, to their vicious direction.
We must remember that the great aim has begun to be
fulfilled in one secondary department of chemical research,
—the study of proportions, as we shall see hereafter.318
POSITIVE PHILOSOPHY".
What has been done in that one category makes ns ask
why an analogous perfection should not be attained in other
departments. We may sum up this account of the requi-
Rational	sites, with the fully rational definition of
definition. Chemistry, that it has for its object,—the
properties of all simple bodies beipg given, to find those of
all the compound bodies which may be formed from them.
Every science falls short of its definition: but a real defi-
nition is the first evidence that a science has attained some
consistency: it then measures its own advancement from
one epoch to another; and it always keeps inquirers in a
right direction, and supports them in a philosophic
progress.
Observation.
f .	Looking now at our means of investigation,
destination1 we s^ia^ find that in chemistry the law holds
good that the complication of phenomena
coincides with the extension of our means of inquiry.
Here Observation begins to find its full
development. IJp to this time it has been
more or less partial. In astronomy, it is confined to the
sense of sight: in physics we use hearing and touch also;
and chemistry employs, besides these, taste and smell.
How much is thus gained we may know by imagining what
would become of chemistry, if we were without taste
and smell, which are often the only means by which
we can recognize effects produced. The important thing
to observe under this head is that there is nothing acci-
dental, nor even empirical, in such a correspondence; for,
as we shall hereafter see, the sound physiological theory of
sensations shows that the apparatus of taste and smell,
unlike that of the other senses, operates in a chemical
manner, and thus shows these two senses to be specially
adapted for the perception of phenomena of composition
and decomposition.
As for Experiment, it is enough to say that
the greater number of chemical phenomena,
and especially the most instructive, are of artificial pro-
duction. Still, we must remember that the essential
character of experimentation consists in the institution, or
the choice of the circumstances of the phenomenon, in order
to a more evident and decisive investigation. This pro-
Experiment.MEANS OF INVESTIGATION.
319
cess is more difficult in chemistry than in physics, because
it is more difficult to institute two parallel cases, undis-
turbed by the intrusion of irrelevant influences; and yet
this is the fundamental condition of experimentation. On
this account, I dissent from the ordinary supposition that
the experimental method is more appropriate to chemical
than to physical researches. Though this is mv view, and
though the greater advancement of physics gives it the
advantage over chemistry, in the use of experiment, I can
have no doubt of the powerful influence of experimentation
in chemistry, independently of its having supplied new
subjects of observation. From the early days of the
science, the immortal series of Priestley’s experiments, and
yet more, those of Lavoisier, have offered admirable models,
almost comparable to the most perfect researches in physics,
and quite enough to prove that there is nothing in the
nature of chemical phenomena to prevent the extended
and luminous employment of the experimental method.
The third means, Comparison, which we ~
have before seen to be inapplicable in Astro-
nomy, and of especial use in Phvsology, begins to have a
real use in Chemistry. The essential condition of this
valuable method is that there shall be an extended series
of cases, analogous but distinct, in which a phenomenon
shall be modified more and more, whether bv successive
simplifications or gradations. It is evident that this can
take place fully only with regard to vital phenomena;
accordingly, it is only by physiological analysis that a clear
idea of its value can be obtained. But chemical phenomena
approach those of physiology nearly enough, not only to
demand this method, but to indicate that without it the
science can never find the road to perfection. The existence
of natural families in chemistry is now admitted by the
best inquirers: but the classification remains to be made.
The need of the classification must lead to the use of the
comparative method, both being based on the common
consideration of the uniformity of certain preponderant
phenomena in a long series of different bodies. There is
even such a connection between the two orders of ideas
that the construction of a natural chemical classification is
impossible without a large application of the comparative320
POSITIVE PHILOSOPHY.
art, as the physiologists understand it; and conversely,
comparative chemistry cannot be regularly cultivated with-
out the guidance of some sketch of a natural classification.
Chemistry is at present only a nascent science; general
methods are as yet scarcely recognized in connection with
it; and only a very few researches afford an example of
the comparative method ; but I am persuaded not only of
the fundamental suitability of that method in chemistry,
but of its application, before very long, to the perfecting of
the science. Such an anticipation, somewhat preceding the
spontaneous development of any science, may be a contri-
bution to its actual progress.
Chemical ana- All the means employed are subject,—
lysis and syn- especially, but not solely in chemistry,—to a
thesis.	verification by the precise collation of the
two procedures of analysis and synthesis;—or, (as these
terms have been corrupted by metaphysical uses) compo-
sition and decomposition.—Every substance which has
been decomposed must evidently be capable of recompo-
sition, whether the process be otherwise practicable or not.
If the inverse operation reproduces precisely the primitive
substance, the chemical demonstration is complete. Unfor-
tunately, the vast extension of chemical resources in this
century has had a much stronger bearing on analytical
powers than synthetical means ; so that there is at present
little proportion and harmony between the two methods.—
Such harmony is indispensable to the establishment of
certainty in some cases, as we see when we duly distinguish
two wfidely differing kinds of chemical analysis: the pre-
liminary, consisting of the simple separation of the imme-
diate principles ; and the final, leading to the determination
of the elements, properly so called. Though both are
essential to chemical research, the first is of the most im-
portant and extensive use. The elementary analysis might
be spared a synthetical verification,—because the compo-
sition of the reacting substances may be compared with the
results obtained, thus indicating the composition of the
proposed substance, the different elements of which will in
this way have been in some sort separated. The impossi-
bility of recombining the elements, to reproduce the primi-
tive body, ought not to throw any doubt on the solution,ANALYSIS AND SYNTHESIS.
321
unless there is some special reason for suspecting the
simplicity of any one of the elements. Synthesis can, in
this case, only add a valuable confirmation to what was
before not doubtful. But the case is very different when
we have to determine only the immediate principles. As
the elements concerned can produce combinations of diffe-
rent orders, we can never be sure that one or more of the
supposed immediate principles obtained does not result
from the reactions caused by the analysis itself. It is only
synthesis which, by reconstructing the proposed substance
with the materials concerned, can decide the question con-
clusively, though in some cases of feeble agency in the
reactives, and strong analogical induction, there is no room
left for reasonable doubt. In immediate analysis of great
complexity, when the agreement of various analytical means
strongly corroborates the conclusions obtained, we cannot
rely on real chemical demonstration without the synthetical
confirmation. This maxim of chemical philosophy is abun-
dantly exemplified in the analysis of mineral waters, and
yet more of organic substances.—It is noticeable that
synthesis is easiest where it is most necessary, and would
be most difficult in the case of elementary analysis, where
it can, as we have seen, be dispensed with. This is owing
to the combinations becoming less tenacious as the order of
composition of the constituent particles is higher; and if
the decomposition is easy, so is the re-composition. The
cases of immediate analysis require only feeble antagonisms,
offering no great obstacles to the synthetical operations
indispensable for their demonstration.
We have next to consider the encvclo-
pedical position of Chemistry, to justify the sci^nce 16
rank assigned to it in our scale.
It is from no vain and arbitrary consideration that
Chemistry is placed between Physics and Physiology in
our scale. By the important series of electro-chemical
phenomena Chemistry becomes, as it were, a prolongation
of Physics: and at its other extremity, it lays the founda-
tions of physiology by its research into organic combinations.
These relations are so real that it has sometimes happened
that chemists, untrained in the philosophy of science, have
been uncertain whether a particular subject lay within
I.	Y322
POSITIVE PHILOSOPHY.
their department, or ought to be referred either to physics
or to physiology.
The phenomena of Chemistry are more complex than
those of Physics, and are certainly dependent on them.
Their degree of generality is inferior,—chemical effects
requiring a much more extended concurrence of varied con-
ditions. Physical properties belong not only to all sub-
stances, but, with simple modifications, to all the states of
aggregation, and even of combination, of each of them:
whereas, it is only in a more or less determined and
restricted condition that each body manifests its chemical
properties. In a word, nature often shows us physical
effects apart from the chemical, while there can be no
chemical effects apart from certain physical phenomena.
Thus Chemistry cannot be rationally studied without a
previous knowledge of physics. Besides, the most powerful
chemical agents are derived from physics, which presents,
in its different orders of phenomena, the first distinctive
characters of different substances. It is impossible in our
day to conceive of scientific chemistry without giving it
the whole of physics for its basis: and thus is its first rela-
tion in the scale established. And, as physics is dependent
on astronomy and mathematics, so must its own dependent
be. But it must be owned that, with regard to doctrine,
the connection of Chemistry with the first two sciences is
neither extensive nor very important.
Every attempt to refer chemical questions to mathema-
tical doctrines must be considered, now and always, pro-
foundly irrational, as being contrary to the nature of the
phenomena. In the case of physics, the mischief would be,
as we have seen, merely from the misuse of
an instrument which, properly directed, may
be of admirable efficacy : but if the employ-
ment of mathematical analysis should ever become so pre-
ponderant in chemistry (an aberration which is happily
almost impossible) it would occasion vast and rapid retro-
gradation, by substituting vague conceptions for positive
ideas, and an easy algebraic verbiage for a laborious inves-
tigation of facts. The direct subordination of chemistry
T .	to astronomy is also slight, but more marked,
o s ^h0111^ jt is almost insensible in regard to abstractRELATIONS OF THE SCIENCE.
323
chemistry, which alone is cultivated in our day. But,
when the time shall come for the development of concrete
chemistry,—that is, the methodical application of chemical
knowledge to the natural history of the globe,—astrono-
mical considerations will no doubt enter in where now there
seems no point of contact between the two sciences. Geo-
logy, immature as it is, hints to us such a future necessitv,
some vague instinct of which was probably in the minds
of philosophers in the theological age, when they were
fancifully and yet obstinately bent on uniting astrologv and
alchemy. It is, in fact, impossible to conceive of the
great intestinal operations of the globe as radically inde-
pendent of its planetary conditions.—Inconsiderable as are
the relations of chemistrv with mathematics and astronomv,
in regard to doctrine, it is far otherwise with regard to
method. It is easy to see how the perfection of chemistry
might be secured and hastened bv the training of the
minds of chemists in the mathematical spirit and astro-
nomical philosophy. Besides that mathematical study is
the necessary foundation of all positive science, it has a
special use in chemistry in disciplining the mind to a wise
severity in the conduct of analysis : and daily observation
shows the evil effects of its absence. Yet, it can never be
said that chemists have so much need of a mathematical
education as physicists, because they do not need it as an
instrument in daily use, but as an intellectual preparation
for the rational study of nature. As to astronomy, we have
seen that it constitutes the most perfect type of the study
of nature; and this at once establishes its relation of
superiority to chemistrv. The more complex the pheno-
mena, the more important is the influence of such a model;
and it is only by having always before their eyes such an
exemplification of the true spirit of natural philosophy, that
chemists can rightly estimate the inanity of the metaphy-
sical explanations which vitiate their doctrine, and can
acquire an adequate sense of the true character, conditions,
and destiny of chemical science. Under this point of view,
astronomy is more useful to chemists than even physics, in
proportion to the superiority of its method.
So much for the sciences which precede T p, . ,
chemistry. As for those that follow, physi- ° ysi° '324
POSITIVE PHILOSOPHY.
siology depends upon chemistry both as a point of departure
and as a principal means of investigation. If we separate
the phenomena of life, properly so called, from those of
animality, it is clear that the first, in the double intestinal
movement which characterizes them, are essentially
chemical. The processes which result from organization
have peculiar characteristics ; but apart from such modifi-
cations, they are necessarily subjected to the general laws
of chemical effects. Even in studying living bodies under
a simply statical point of view, chemistry is of indispens-
able use in enabling us to distinguish with precision the
different anatomical elements of any organism.—We shall
see hereafter that the new science of Social Physics is
To Socioloo*r subordinated to chemical science. In the
o ocio og}.	place, it depends on it by its immediate
and manifest connection with physiology: but, besides
that, as social phenomena are the most complex and par-
ticular of all, their laws must be subject to those of all the
preceding orders, each of which manifests, in social science,
its own peculiar influence. In regard to Chemistry especi-
ally, it is evident that among the conditions of man’s social
existence several chemical harmonies between man and
external circumstances are involved. Even if individual
existence could be sustained, society could not, if these
harmonies were destroyed, or even only somewhat dis-
turbed,—as by changes in the atmospheric medium, or in
tlie waters or the soil.
T v -	The position of Chemistry among the sci-
Decree oi dos	^	u
sibfeperfection. ences beingtllus determined, the next inquiry
is about the degree of scientific perfection
that its nature admits, in comparison with others. As for
tlie method, if physics suffers from the intrusion of hypo-
theses, we may say that chemistry has been their absolute
prey, through its more difficult and tardy development.
The doctrine of affinities appears to me more ontological
than that of fluids and imaginary ethers. If the electric
fluid and the luminous ether are, as I called them before,
materialized entities, affinities are at bottom pure entities,
as vague and indeterminate as those of the scholastic
philosophy of the Middle Age. The pretended solutions
that they yield are of the usual character of metaphysicalHYPOTHESES.
325
explanations,—a mere reproduction, in ab-
stract terms, of the statement of the pheno- hypotheses
menon. The advance of chemical knowledge
which must at last discredit for ever such vain philosophy
has as yet only modified it, so far as to disclose its radical
futility. While affinities were regarded as absolute and
invariable, there was at least something imposing in them ;
but since facts have compelled the belief of their being
variable according to a multitude of circumstances, their
use has only tended to prove, more and more, their utter
inanity. Thus, for instance, it is known that at a certain
temperature, iron decomposes water, or protoxide of hydro-
gen : and yet, it has been since discovered that, under the
influence of a higher temperature, hydrogen in its turn
decomposes oxide of iron. What signifies, in this case, any
order of affinity that we may ascribe to iron and hydrogen
with oxygen? If we make the order vary with the tem-
perature, we have a merely verbal, and therefore pretended
explanation. Chemistry affords us now many such cases,
apparently contradictory, independently of the long series
of decisive considerations that have made us reject absolute
affinities,—the only ones, after all, that have any scientific
consistency whatever. The old habit is, however, so strong
that even Berthollet, in the very work in which he over-
throws the old doctrine of invariable or elective affinities,
proposes vague affinities under many modifications. The
strange doctrine of predisposing affinity is to be found in
the work, among others, of the most rational of recent
chemists, the illustrious Berzelius. When, for instance,
water is decomposed by iron through the action of sulphuric
acid, so as to disengage the hydrogen, this remarkable
phenomenon is commonly attributed to the affinity of the
sulphuric acid for the oxide of iron which tends to become
formed. How, can anything be imagined more metaphy-
sical, or more radically incomprehensible, than the sympa-
thetic action of one substance upon another which does
not yet exist, and the formation of the last by virtue of
this mysterious affection ? The strange fluids of physicists
are rational and satisfactory in comparison with such
notions. These considerations justify the desire that
chemists should have a sufficient training in mathematical,326
POSITIVE PHILOSOPHY.
astronomical, and then in physical philosophy, which have
already put an end to such chimerical researches within
their own domain, and would discard them speedily from
the more complex parts of natural philosophy. It is only
by having witnessed the purification in the anterior sciences
that chemists could realize it in their own: and there could
not be complete positivity in chemistry if metaphysics
lingered in astronomy or physics. This, again, justities the
place assigned to chemistry among the sciences. The indi-
vidual must follow the general course of his race in his
passage to the positive state. He must find that true
science consists, everywhere, in exact relations, established
among observed facts, allowing the deduction of the most
extensive series of secondary phenomena from the smallest
possible number of original phenomena, putting aside all
vain inquiry into causes ana essences. And this is the
spirit which has to be made preponderant in chemistry,
—dissolving for ever the metaphysical doctrine of afri-
nities.
The inferiority of chemistry to physics, in
fection im^ei regard to method and doctrine, explains its
relative imperfection with regard to actual
science. We have only to compare with the formula which
told us what chemistry ought to be, what it actually is, to
see that it is at an immense distance—much further than
physics—from its true scientific aim. Chemical facts
are at this day essentially incoherent, or, at best, feebly
co-ordinated by a small number of partial and insufficient
relations, instead of those certain, extended, and uniform
laws of which physics is so justly proud. As for prevision,
if it is imperfect in physics in comparison with astronomy,
it can hardly be said to exist in chemistry at all: the issue
of each chemical event being usually known only by speci-
ally consulting the immediate experiment, when, as it were,
the event is already accomplished.
Imperfect as chemistry is, in regard to
imperfection method and doctrine, it is yet superior to
physiology, and still more, to social science,
not only because, from the comparative simplicity of its
phenomena, the facts and investigations are clearer and
more decisive, but because it has a few, though very few,EDUCATIONAL FUNCTION.
327
real theories, capable of affording complete previsions; a
thing as jet impracticable, except in a general manner, with
living bodies. We shall have occasion to notice the theory
of proportions, the equivalent of which is not, in any sense,
to be looked for in physiology. We must remember, while
estimating the comparative imperfection of the sciences,
that the importance to us of their perfection is in proportion
to their simplicity; our available means being always found
to correspond with our reasonable wants. I hope, too, that
this severe estimate of the actual state of each science will
stimulate rather than discourage the student; for it is more
gratifying to our human activity to conceive of the sciences
as susceptible of vast, varied, and indefinite progress, than
to suppose them perfect, and therefore stationary, except
in their secondary developments.
This leads us to consider the function of Chemistry in
the education of the human mind.
It may be said to train us in the great art
of experimentation : not as being our exclu- nian progress^
sive teacher, for, as we have seen, physics is
superior to it in this: and it is more the art of observing
than of experimenting that Chemistry is chiefly distin-
guished for. But there is an important part of the positive
method which chemistry seems destined to carry to the
highest perfection. I do not mean the theory of classifi-
cations, of which chemists know too little at f
present; but the art of rational nomencla- cloture10™611"
tures, which is quite unconnected with classi-
fications. Since the reform in chemical language, attempts
have been incessantly made, to this hour, to form a syste-
matic nomenclature in anatomy, in pathology, and especi-
ally in zoology: but these endeavours have not had, and
never can have, any success to compare with that of the
reformers of chemical language; for the nature of the
phenomena does not admit of it. It is not by accident
that the chemical nomenclature is alone in its perfection.
The more complex phenomena are, and the more varied
and less restricted the comparisons of objects, the more
difficult it becomes to subject them to a system of deno-
minations, at once rational and abridged, so as to facilitate
the habitual combination of ideas. If the organs and328
POSITIVE PHILOSOPHY.
tissues of the living body differed only from one point of
view; if maladies were sufficiently defined by their seat;
if, in zoology, genera, or at least families could be estab-
lished by a homogeneous consideration, the corresponding
sciences might at once admit of systematic nomenclatures
as rational and as efficacious as that of Chemistry. But
the diversity of aspects, rarely reducible to one head,
renders such an arrangement extremely difficult and not
very advantageous.
The case of chemistry is the only one in which, by its
nature, the phenomena are simple, uniform and determinate
enough to allow of a rational nomenclature at once clear,
rapid and complete, so as to contribute to the general pro-
gress of the science. The idea of composition, the great
end of the science, is always preponderant. Thus, the
systematic name of each body, expressing its composition,
indicates first a correct general view, and then, the sum of
its chemical history; and, by the nature of the science, the
more it advances towards perfection, the more must this
double property of the nomenclature be developed. In
another view, dualism being the commonest constitution
in chemistry, and the most essential, and that to which all
other modes of composition are more and more referred by
science, we see that the conditions of the problem are as
favourable as possible to a rapid and expressive nomen-
clature. Thus, there has always been some system of
nomenclature, more or less rough, though none to be com-
pared to that so happily founded by Gruyton-Morveau.
Though the art can manifest its excellence only in propor-
tion to the advance of chemistry, it is in such harmony
with the nature of the science that, in its present imperfect
state, it upholds it, by provisionally supplying, as it were,
the almost absolute deficiency of true rationality. Thus
chemistry may be regarded as specially adapted to develope
one of the few fundamental means, the aggregate of which
constitutes the general power of the human mind. The
formation of a similar aid in the more complex sciences
offers a real and strong interest: and I have only desired
to show that we must resort to chemistry for the true
principles and general spirit of the art of nomenclature,
according to the rules so often set forth in this work,STATE OF DOCTRINE*
329
that each great logical artifice should be directly studied
in the department of natural philosophy where it is
found in the greatest perfection, that it may be after-
wards applied in aid of the sciences to which it less specially
belongs.
The high philosophical properties of Che- State of
mistry are more striking in regard to doctrine chemical
than to method. However imperfect our doctrine,
chemical science is, its development has operated largely
in the emancipation of the human mind. Its opposition
to all theological philosophy is marked by the two general
facts in which it has a share with all the rest of positive
philosophy,—first, the prevision of phenomena, and next,
our voluntary modification of them. We have already
seen that the more the complexity of phenomena baffles
our prevision, the greater becomes our power of modifying
them, through the variety of resources afforded by the
complexity itself; so that the anti-theological influence of
science is infallible, in the one way or the other. In
chemistry, our modifying power is so strong that the
greater part of chemical phenomena owe their existence to
human intervention, by which alone circumstances could
be suitably arranged for their production: and if the phe-
nomena of physiology and social science admit of modifi-
cation in a yet greater degree, chemistry will always, in
this particular, hold the first rank, since the highest order
of modifications is that which we here find,—those which
are most important for the amelioration of the condition of
Man. In the system of the action of man upon nature,
chemistry must ever be regarded as the chief source of
power, though all the fundamental sciences participate in
it more or less.
In this way, chemistry effectually discredits the notion of
the rule of a providential will among its phenomena. But
there is another way in which it acts no less strongly ; by
abolishing the idea of destruction and creation in nature.
Before anything was known of gaseous materials and pro-
ducts, many striking appearances must inevitably have in-
spired the idea of the real annihilation or production of
matter in the general system of nature. These ideas could
not yield to the true conception of decomposition and com-330
POSITIVE PHILOSOPHY.
position till we had decomposed air and water, and then
analysed vegetable and animal substances, and then finished
with the analysis of alkalies and earths, thus exhibiting the
fundamental principle of the indefinite perpetuity of matter.
In vital phenomena, the chemical examination of not only
the substances of living bodies, but their functions,—im-
perfect as it yet is,—must cast a strong light upon the
economy of vital nature by showing that no organic matter
radically heterogeneous to inorganic matter can exist, and
that vital transformations are subject, like all others, to
the universal laws of chemical phenomena. Chemical
analysis seems to have fulfilled its function in this direc-
tion : henceforth it must be by the more difficult, but
more luminous method of synthesis that this great philor
sophical revolution must be completed: and attempts
enough have been successfully made to prove the possi-
bility of it.
...	The divisions of the science have not been
the science clearly and permanently settled, partly be-
cause of its very recent origin, and partly on
account of its nature. In the first place, students have
been more occupied in multiplying observations than in
classifying them; and in the next, the homogeneous
character of chemical phenomena causes essential differ-
ences to be less profound, and therefore less marked, than
in any other of the fundamental sciences. In astronomy,
there can be no question of a division into geometrical and
mechanical phenomena. Physics is less a unique science
than a group of almost isolated sciences ; and they indicate
their own arrangement. We shall see hereafter that nearly
the same thing happens, though from a different cause, in
physiology. But in chemistry, the conditions are less
favourable, the distinctions being scarcely more marked
than those which exist in a single department of physics,—
as thermology, and yet more, electrology. The imperfec-
tion and small importance of its present divisions are
easily explained: and there are strong symptoms of an
approaching discussion of this great subject; for the
majority of eminent chemists are more or less dissatisfied
with the provisional division which they have been hitherto
obliged to accept as guidance in their labours.PRINCIPLES OF COMPOSITION.
331
No organic
chemistry.
The general division of organic and inor-
ganic chemistry cannot be sustained, on ac-
count of its evident irrationality. What is
at present called organic chemistry has an essentially bas-
tard character, half chemical, half physiological, and not,
in fact, either the one or the other, as we shall have occa-
sion to see. The division cannot even be sustained under
another form, as equivalent to the general distinction be-
tween cases of dualism and of other composition. For if
inorganic combinations are usually binary, there are some
which are composed of three elements and even of four ;
while, conversely, we very often meet with a true dualism
in bodies which are called organic. For a genuine division
we must look to general ideas relating to Principles of
composition and decomposition ; and in this composition
form, attending to the rule of following the an4 4ecom'
gradual complication of phenomena: first, Posltlon-
the growing plurality of constituent principles (mediate or
immediate), according as the combinations are binary, ter-
nary, etc. ; and secondly, the higher or lower degree of com-
position of the immediate principles, each of which may (as
in the case of a continual dualism) be decomposable into
two others, for a greater or smaller number of consecutive
times. Though each of these two points of view is of high
importance, the preponderance of the one or the other must
be agreed upon before the rational division of chemistry can
be organized. Though this is not the place to discuss this
new question of high chemical philosophy, it may be well
to state that I regard it as solved; and that the considera-
tion of the degree of composition is, in my eyes, evidently
superior to that of the number of elements, inasmuch as it
affects more profoundly the aim and spirit of chemical
science, as they have been characterized in this chapter.
As for the rest, whatever the decision may be, we may re-
mark that the two classifications differ from each other
much less than we might at first be tempted to suppose ;
for they necessarily concur, whether in the preliminary
or in the final case, and diverge only in the intermediate
parts.
We have now reviewed the nature and spirit of chemical
science ; the means of investigation proper to it; its true332
POSITIVE PHILOSOPHY.
encyclopedical position; tlie kind and degree of perfection
of which it is susceptible; its philosophical properties in
regard to method and to doctrine; and, finally, the mode
of division which would be suitable to it. We must com-
plete the survey of the science by a special and direct
notice of the few essential doctrines which have been dis-
closed by the spontaneous development of chemical philo-
sophy. It must be remembered that the object of this
work is not to present a treatise on each science, or to
enlarge upon it in proportion to its proper importance, or
the multiplicity of its facts: but to ascertain its relative
importance, as one head of positive philosophy. Ho one
will expect that chemical philosophy, in its present state,
can be examined here as fully or satisfactorily as, for in-
stance, astronomical philosophy, the perfection of which
admits of a methodical analysis, clear and complete,
though summary, such as befits that immutable type of
natural philosophy.333
CHAPTER II.
INORGANIC CHEMISTRY.
WHATEVER may be the principles of ,	.,	.
division and classification preferred the study"
in the general system of chemical studies, it °
is agreed by almost all chemists that the preliminary and
fundamental study should be the successive and continuous
history of all the simple bodies. The plan of M. Chevreul
is an exception to this, his method being to proceed at once
from the study of each element to all the combinations,
binary, ternary, etc., that it can form with those already
examined; confining himself, however, to compounds of
the first order. This plan has the advantage that simple
bodies are more completely known from the beginning
than by the usual method, which scatters through the
different parts of the science the most important chemical
properties of each of them. But, on the other hand, the
history of any element remains incomplete; a factitious
inequality is established among chemical researches into
different elementary substances; and the didactic incon-
venience which M. Chevreul proposed to escape seems to
me to be unavoidable, under any method. On no plan can
any chemical history be completed by a first study. The
provisional information obtained by a first study must be
followed by a revision which allows us to take into con-
sideration the whole series of phenomena relative to each
substance. The question is merely a didactic one, only of
secondary importance in this work, though of great prac-
tical interest. On any scheme, it remains certain that the
preliminary study of elementary substances is, by the
nature of the science, the necessary foundation of chemical
knowledge.
On account of the considerable and always increasing
number of substances regarded as simple, some modernPOSITIVE PHILOSOPHY.
philosophers, possessed with the notion of
the simplicity and economy of nature, have
concluded a priori that most substances
must be the various compounds of a much smaller number
of others. But, while endeavouring to conceive of nature
under the simplest aspect possible, we must do so under
the teaching of her own phenomena, not substituting for
that instruction any thoughtless desires of our own. We
have no right to presume beforehand that the number of
simple substances must be either very small or very large.
Chemical research alone should settle this; and all that
we are entitled to say is that our minds are disposed to
prefer the smaller number, even, if it were possible, so far
as there being but two, But not the less are we bound to
suppose all substances which have never in any way been
decomposed to be simple, though we should not pronounce
them to be for ever undecomposable. All chemists now
admit this rule as the first axiom of sound chemical
philosophy.
Aristotle first saw this rule, though he did not conceive
of its rational grounds. His doctrine of the four elements,
popularly cried down in our time, should be judged of as
the first attempt of the true philosophical spirit to con-
ceive of the composition of natural bodies, amidst the then
existing deficiency of all suitable means of research. To
appreciate it we must compare it with anterior notions.
Now, up to that time, all the schools, however they might
differ about other things, agreed that there was only one
elementary substance; and their dispute was about the
choice of the principle. Aristotle, with his rational
character of mind, put an end to all those barren contro-
versies by establishing the plurality of elements. This
immense progress must be considered the true origin of
chemical science, which would be radically impossible on
the supposition of a single element, excluding all idea of
composition and decomposition. Whatever appearances
may be, there is no doubt that it must be much more
difficult for the human mind to pass from the absolute
idea of unity of principle to the relative idea of plurality,
than to rise gradually, by means of research, from the four
elements of Aristotle to the fifty-six simple bodies of our
334
Plurality of
elements.PLURALITY OF ELEMENTS.
335
chemistry of this day. Our Naturists, who are all for sim-
plicity and economy without caring much for reality, have
no right to appeal to the authority of Aristotle, who had
so much reverence for reality as to infringe the notion of
simplicity which he found prevailing. They should go
back further than Aristotle,—to Empedocles or Heraclitus,
and attain the utmost simplicity at once, by admitting only
a single principle.
Other philosophers, among whom was Cuvier, have ob-
jected to the simplicity of most of the elements now ad-
mitted by chemists, that some of them seem to be extremely
abundant in nature, while others are scantily and partially
distributed: whereas, it seems natural to presume that the
different elements must be almost equally diffused through-
out the globe, and that therefore chemical analysis will
sooner or later prove the rare ones to be compound sub-
stances, requiring peculiar and rare influences for their
formation. It would be enough to say that the presump-
tion, though plausible, is nothing more than a presump-
tion : but it may be added that we know nothing of our
planet beyond the upper strata ; and we can form no pre-
judgment of the composition of the whole. It would be
too much to say that there should be an equality of
elements on the surface, even the probability being the
other way; for the heaviest elements are the rarest at the
surface, and the commonest are those which go to the com-
position of living bodies ; and the probability is strong that
the preponderance is reversed in the interior of the globe,
to make up the mean density, which is not to be found
among the solids, liquids, and gases which are required for
the existence of life. Thus the objection seems to be
converted by chemical analysis into a sort of confir-
mation.
Since the time,—recent, it is true,—of the decomposition
of the elements of Aristotle, there has not been a single in-
stance of a substance having passed from the class of simple
to that of compound bodies, while the inverse case has been
frequent. Yet, no chemist disputes the possibility of a re-
duction of the elements by a more thorough analysis ; for
chemical simplicity, as it is to us, is a purely negative
quality, not admitting of those irreversible demonstrations336
POSITIVE PHILOSOPHY,
proper to positive compositions and recompositions. The
great general example of substances called organic, the
chemical theory of which is so complex, notwithstanding
the small number of their elements, might lead us to sup-
pose that such a reduction would not be, after all, so very
great an advantage: but in this case, the difficulty seems
to me to be referable to the deficiency of duality. Not-
withstanding this example, we cannot but think that
chemistry would become more rational and more sys-
tematic, if the elements were fewer, from the closer and
more general relation which must then subsist among the
different classes of phenomena. But the apparent perfec-
tion could be only barren and illusory if we were to assume
it by conjecture anticipating the real progress of chemical
analysis.
r,1 .fi	This profusion of elements has naturally
of elements^11	endeavours to classify them. The
high importance of the question has become
manifest through the deep persuasion that the rational
classification of simple bodies must determine that of com-
pound substances, and therefore that of the whole chemical
system. The first principle to be laid down is that the
hierarchy of elementary substances is not to be determined
only by their proper essential characters, but by the less
direct consideration of the principal phenomena of the
compounds which they form. Without this requisition,
the classification would have little use or interest; for
it would be of small consequence in what conventional
order we studied fifty-six bodies all independent of
each other: whereas, with its proper condition, this ques-
tion is as important as any that chemical philosophy can
present.
The old division of the elements into the eomburent and
combustible, (those which bum in the active and in the
neuter sense,)—and the subdivision of these into metallics
and non-metallics, are evidently too artificial to be main-
tained, except provisionally. For many years, endeavours
have been made to supersede it; but no irreversible classifi-
cation has been yet obtained. M. Ampere seems to have
been the first who pointed out the necessity ; and he pro-
posed a system in 1816; but it was not one which inducedBerzelius’s classification.
337
the chemists to abandon, their ancient distribution, the
binary structure of which made it easy of application,
whatever might otherwise be its defects. A few years
after, Berzelius offered, in a simple and almost incidental
form and manner, a far superior system of classification.
He first understood the necessity of rising finally to a
unique series, constituting, by a uniform and preponderant
character, a true hierarchy; whereas, M. Ampere saw only
the importance of natural groups, which might be arbi-
trarily co-ordinated. Both conditions are imposed by the
general theory of classifications ; but that which Berzelius
had chiefly in view is unquestionably superior to the other ;
and especially in the present case, when the small number
of objects to be classified renders the formation of groups
a matter of secondary importance, provided the series be
naturally ordained.
M. Berzelius’s conception is grounded on
the consideration of electro-chemical pheno- 0f Berzehus11
mena. Its simple and lucid principle is that
the elements are to be so disposed as that each shall be
electro-negative to those which precede it, and electro-
positive to those which follow it. The series thus derived,
appears, thus far, to be in conformity with the whole of the
known properties of both the elements themselves and
their principal compounds. It is too soon however to
speak decisively of this: and, on the other hand, the
chemical preponderance of electric characters is by no
means so logically established as to compel us to seek the
bases of a natural classification in that order of phenomena.
It must, it seems to me, be clearly proved, at the outset,
that the point of departure is a real one,—that is, that a
constant order of electrization exists among the different
elements, which is maintained under all conditions of
exterior circumstances, of aggregation and decomposition:
but, not only has this never been adequately undertaken,
but there is some reason to apprehend that its result would
be opposite to the proposed principle. Whatever may be
the issue of future labours, Berzelius has secured the
eternal honour of having first exhibited the true nature of
the problem, and the aggregate of its principal conditions,
and perhaps the order of ideas in which its solution is to be
i.	z338
POSITIVE PHILOSOPHY.
sought. Whenever this solution is obtained, chemistry
will have made a great stride towards a truly rational
state: for, under a hierarchy of the elements, the sys-
tematic nomenclature of compound substances will almost
suffice to give a first indication of the general issue proper
to each chemical event; or, at least, to restrict the uncer-
tainty within narrow limits. Yet, through this very con-
nection of such a research with the whole of chemical
studies, I do not think it can be efficaciously pursued
while we separate it, as has hitherto been done, from the
question about the establishment of a complete system of
chemical classification for all bodies, simple and compound.
Now, this great question seems tome at present premature.
The preliminary conditions, both of method
and of doctrine, are, as we have seen, far
from being completed. As such a general
system of classification must constitute both the summing
up and the fundamental view of the whole of chemical
philosophy, I shall further expand my idea about it in this
place.
Premature
effort.
Requisite pre-	As for the method, it requires perfecting
paration as to in two ways, for which chemists must resort
method.	to physiology. They must understand the
fundamental theory of natural classifications, which can be
obtained nowhere else: and they must, for the same
reason, study in the same school the general spirit of the
comparative method, of which chemists have very little
idea, and without which they can never proceed properly in
search of a rational classification. These two improve-
ments must be derived from biological philosophy ; the one
to lay down the problem of chemical classification, and the
other to undertake its solution. It will be by perceiving
these harmonies and mutual applications among the
sciences commonly treated as isolated and independent,
that philosophers in all departments will at length become
aware of the reality and utility of the fundamental concep-
tion of this work ; the cultivation of the different branches
of natural philosophy under the impulsion and direction of
a general system of positive philosophy, as a common basis
and uniform connection of all scientific labours. We have
little idea what we lose by the narrow and irrational spiritSYSTEM OF CLASSIFICATION.
339
in which the different sciences are cultivated, and especially
with regard to method. When the great scientific relations
of the future shall be regularly organized, men will scarcely
be able to imagine, otherwise than historically, that the
study of nature could ever have been conceived and directed
in any other way.
As to the doctrine, we have seen that the to doctrine
desired classification cannot take place till we
have settled the preponderance of the one or the other con-
sideration,—the order of composition of the immediate
principles, or their degree of plurality. Now, such a
problem has not yet been rationally proposed. If we
suppose it resolved, adopting the rule which I think almost
incontestable, as I explained before, of treating the first
point of view as necessarily superior to the second, we
must still attend to two special conditions, before we can
proceed to the rational construction of the system of
chemical substances.
By the first of these conditions, we must
,.	-j i • , • . • p i L irst condition,
dismiss the irrational distinction of sub-
stances into organic and inorganic. We shall see hereafter
that organic chemistry must soon dissolve, parting with
some of its questions to chemistry proper, and others to
physiology. When any combination is susceptible of a
chemical examination, it must be subjected to a fixed order
of homogeneous considerations, whatever may have been
its origin and mode of concrete existence, with which
chemistry has nothing to do, unless as a source of informa-
tion. As long as any classification must be adapted to the
strange conception of a sort of double chemistry, established
upon a false division of substances, it must be precarious
and artificial in its details, because it is vitiated in
principle. The evil is felt, as is shown more and more by
the tendency to refer organic combinations to the general
laws of inorganic combinations: but it would not be
enough, as might be supposed, that a distinguished
chemist should take the initiative, in a large and direct
manner, to accomplish this important reform. Such a
work demands a special and difficult operation, requiring a
delicate combination of the chemical and physiological
point of view, in order to make a true division of what340
POSITIVE PHILOSOPHY.
should remain with chemistry, and what should return to
physiology.
The second condition is closely connected
condition with the first. It requires that all com-
binations should, if possible, be submitted
to the law of dualism, erected into a constant and necessary
principle of chemical philosophy. Great as would be such
an improvement in the way of simplification of chemical
conceptions, it must however be admitted that it is not
so indispensable to classification as the preceding. Without
the first condition, rational classification would be impos-
sible : whereas, it might take place, with imperfection and
difficulty, without the second. As for the prospects of the
case, the tendency to improvement is as real and marked
in the one case as the other; as any one may observe for
himself.
Method of	more importance to set the
analysis0 consideration of the order of composition of
immediate principles above that of their
degree of plurality, as before proposed, because the first is,
Chemical by in&ture, clear and incontestable, while
dualism	the other is always more or less obscure and
dubious. The one is, in fact, the simple
appreciation of an analytical or synthetical fact: the
second has always a certain hypothetical character, since
we then pronounce upon the mode of agglomeration of
elementary particles; which is a thing radically inacces-
sible to us. Thus, for example, a chemist may establish
with certainty that such or such a salt is a compound
of the second order, and that certain acids and alkalies are,
on the contrary, of the first order; for analysis and syn-
thesis can demonstrate that each of the last bodies is com-
posed of two elementary substances, and that, on the con-
trary, the immediate principles of the salt are decomposable
into two elements. But, in another view, when the
analysis of any substance has established the existence in
it of three or four elements, as in the case of vegetable or
animal matters, we cannot, without resort to hypothesis,
pronounce that this combination is really ternary or
quaternary, instead of being simply binary: for we can
never assert that we could not, by a preliminary analysisCHEMICAL DUALISM.
341
less violent than this final one, resolve the proposed
substance into two immediate principles of the first order,
each of which should be further susceptible of a new binary
decomposition.
If an unskilled chemist should at this day apply unduly
strong means to the analysis of saltpetre, the results might
authorize him, following our present erroneous procedure,
to conceive of this substance as a ternary combination of
oxygen, azote, and potassium: and yet we know that such
a conclusion would be false, as the substance may be easily
reconstructed by a direct combination between nitric acid
and potash, which might have been separated by a less
disturbing analysis, without occasioning their decomposi-
tion. How do we know that it may not be so with every
combination habitually classed as ternary or quaternary ?
Immediate analysis being as yet so imperfect in comparison
with elementary analysis, especially with regard to these
substances, would it be rational to proclaim, for the time
to come, its necessary and eternal impotence with regard
to them? Such judgments seem to be founded on a
confusion between these two kinds of analysis, so really
different in themselves, and so characterized in their opera-
tions by delicacy in the one case and energy in the other.—
One important consideration, relating to the synthetical
point of view, is evidence of this confusion between the
two analyses: and that is, the extreme difficulty, if not
impossibility, of verifying by synthesis the analytical
results proper to these substances. We have seen that
immediate synthesis is usually very easy, while elementary
synthesis is scarcely practicable. Thus, reciprocally, it
seems to me rational to suppose that when the recomposi-
tion cannot be effected, the analysis has not been imme-
diate,—there being no other objection to such a conclusion.
For example, we exhibit the impossibility of reproducing
by synthesis vegetable and animal substances: and this
has been even set up as a sort of empirical principle. But
is not this impossibility owing to our persisting in an ele-
mentary synthesis when we ought to proceed by an imme-
diate synthesis, the materials of which might in many
cases be discovered beforehand? This remark is true
with regard to a multitude of combinations the dualism of342
POSITIVE PHILOSOPHY.
which is, however, very certain, with the sole difference
that the immediate principles are better known. If we
tried to recompose saltpetre by directly combining oxygen,
azote, and potassium, we should succeed no better than in
reproducing organic substances by throwing together their
three or four elements: the obstacles which we admit in
the last case apply equally to the first. The most striking
achievement is that of M. Woehler, in producing the animal
substance urea. He could not have done this if he had
tried, according to the common prejudice, to combine
directly oxygen, hydrogen, carbon, and azote, which concur
in the elementary constitution of this substance, instead of
uniting only its two immediate principles, till then un-
known in this quality. Is there any reason to suppose
that it is otherwise in any other case ?—It appears then
that chemists will be safe in attributing an entire gene-
rality to the fundamental principle of the dualism of all
combinations, under the one easy condition of regarding as
still very imperfect the analysis of substances exceeding
the binary composition; and especially the substances
called organic, the true immediate principles of which
would thus remain to be discovered. These principles can
be conceived of only by imagining a considerable number
of new binary combinations, of the first and second orders,
between oxygen, hydrogen, carbon, and azote: and the
realization of this may seem, in the present state of our
knowledge, almost impossible. But we have no right to
conclude it to be so, while our analytical procedures are
what they are; and there is no scientific objection to our
supposing that there may be many more direct and
binary combinations among the elements of ternary or
quaternary substances than chemistry has yet estab-
lished.
It must be observed, however, that universal and inde-
finite dualism cannot be maintained unless chemists will
scientifically determine the sense of the word substance;
that is, restrict it to mean real combination: for it would
be easy to cite, and especially in physiological chemistry,
very marked cases of the defect of dualism. But we can-
not regard as a true chemical substance an accidental
assemblage of heterogeneous substances, whose agglomera-MEANING OF SUBSTANCE.
343
tion is evidently mechanical, such as sap, blood, a biliary
calculus, etc., unless we confound the notion of dissolu-
tion, and even of mixture, with that of combination. If
we extend in this way the use of the term substance, so
valuable in chemistry, we might as well treat, as so many
chemical substances, the waters of different seas, different
mineral waters, soils, etc. : and even more, artificial mix-
tures of a variety of salts dissolved together in water or
alcohol. We shall see hereafter that all difficulties in this
subject may be disposed of by our learning that they pro-
ceed from our not having clearly and rigorously separated
the chemical from the physiological point of view. We
may be assured that the most elementary notions of
chemical philosophy cannot be rationally established, in
their due clearness, generality, and stability, without being
founded on a full comparison with biology; a comparison
which can be organized only under a complete system of
positive philosophy.
Meantime, there is a marked tendency in the present
movement of chemical ideas, towards a complete dualism.
The increasing assimilation attempted between organic and
inorganic substances is an indirect advance in that road:
but much more striking, in this view, are experiments like
those of M. Woehler, which refer the most refractory com-
pounds to dualism, either by analysis or synthesis. A
binary formula is adopted, too, to represent the proportion
of elements proper to the most complex substances: and,
though this is not a true dualism, it helps to prepare
minds for the establishment of a real and general one.
The sum of what has been said on this important subject
of chemical dualism is this :—the real mode of agglomera-
tion of elementary particles is, and ever must be, unknown
to us, and therefore no proper object of our study :—our
positive researches being thus circumscribed, we may
rationally conceive of the immediate composition of any
substance as binary; but so as to represent all the phe-
nomena that chemistry can offer to us, in any future state
of perfection. Thus, I do not propose universal dualism
as a law of nature; for this we could never establish : but
I declare it to be a fundamental artifice of time chemical
philosophy, destined to simplify our elementary concep-344
. POSITIVE PHILOSOPHY.
tions, by using our optional intellectual liberty in accord-
ance with the true end and aim of positive chemistry.
These are the conditions necessary to the institution of
a system of natural classification, answering in chemistry
to the universal hierarchy of living bodies in biology, if the
complication of phenomena would admit of our obtaining
such a system. Up to this time, perhaps no one has
formed an adequate idea of the nature and spirit of such
an operation: but, in my view, chemical classification, thus
conceived of, is the science itself, condensed into the most
substantial summary. All I claim to have done is to have
introduced into chemical science the special kind of philo-
sophical spirit which is naturally developed by biological
science, as it has been conceived of by all its great masters,
from Aristotle downwards.
It is because I have high expectations of what Chemistry
will become, that I attach so much importance to the pre-
ceding discussion. The science is now weak and desultory,
notwithstanding its rich collection of facts: but, extended
and complex as it is, there is no fundamental science,
except astronomy, whose phenomena are so homogeneous,
and therefore so fit for a true systematization, in the
positive spirit. Now, this future constitution of chemical
science must, it seems to me, consist in a complete system
of natural classification, which cannot be obtained till all
combinations, whatever their origin, are subjected to a fixed
order of homogeneous considerations, and, on the other hand,
constantly referred to a fundamental dualism.
We cannot form any certain expectation of the future
condition of Chemistry from its present state : but, before
proceeding to examine the two doctrines which at this day
approach nearest to positive rationality,—that of definite
proportions and the electro-chemical theory,—I will indi-
cate two points of doctrine which seem, by their nature, to
indicate with precision the true dogmatic formation towards
which the science, as a whole, must tend.
Law of double First, there is the great law of double
saline decom- saline decompositions, discovered by Ber-
positions. thollet, and completed by M Dulong’s inves-
tigations on the reciprocal action of soluble and insoluble
salts. The case of double solubility, considered by Ber-CHEMICAL THEORY OF AIR AND WATER. 345
thollet, is this: two soluble salts, of any kind, mutually
decompose each other whenever their reaction may produce
an insoluble salt, or one less soluble than either of the two.
This theorem stands first among general propositions in
chemistry, and is the only one which can as yet give an
exact idea of what, in chemistry, constitutes a true law.
It has all the characters of a law : it relates to the proper
subject of chemical science; it establishes a relation be-
tween two classes of phenomena before independent; and,
above all, it admits of prevision of phenomena according
to their positive relations. In establishing this law,
Berthollet escaped some metaphysical snares, rejecting
hypotheses of affinities ; but he fell into one when he
attempted to explain the law which he had just discovered.
No law can be explained otherwise than by showing that it
enters into another, more general than itself: but this law
of Berthollet’s is alone of its kind ; and it therefore admits
of no explanation. It may hereafter be attached to a
fundamental theory of the reciprocal action of all com-
pounds of the second order ; and such a relation will truly
explain it: but at present it is simply a general fact which,
inexplicable itself, serves to explain each of the particular
facts which it comprehends.
The influence of air and water in the pro- Chemical
duction of chemical phenomena is another of theory of air
the most perfect doctrines of chemistry as it an(i water*
stands. The importance of the action of air and water in
the terrestrial economy has induced some G-erman philo-
sophers irrationally to set up the system of these two fluids
into a sort of third reign, between the inorganic and the
organic: but abstract chemistry has nothing to do with
natural history, and regards the study of air and water
from a different point of view, while aware of its funda-
mental importance.
All chemical phenomena take place in the presence of
air ; and they almost invariably require the intervention of
water: it is clear therefore that before we study any
chemical reaction, we must be able to analyse the partici-
pation of these two fluids. Thus the chemical theory of
air and water is a sort of necessary introduction to the
system of chemistry, properly so called, as belonging more346
POSITIVE PHILOSOPHY.
to method than to doctrine, and as immediately following
the study of simple bodies. It is an historical fact that
the double analysis of air and water marked the first great
advance in modern chemistry.
^•r	The influence of the air, not less important
than that of water in chemical phenomena,
was less difficult to characterize: for the air is simply a
mixture, and its chemical action is merely that of the gases
which compose it, each of which acts as it were isolated,
allowing for the diminution of intensity from its diffusion,
and for the very few cases in which the accomplishment of
the proposed phenomenon determines the combination of
the gases in an accessory way. Chemistry has only to
analyse it, leaving all other study of it to the department
of natural history. This analysis was effected in the early
days of modem chemistry, except that there is still some
uncertainty about the proportion of carbonic acid gas, and
perhaps of some other more considerable principles, as, for
instance, hydrogen, the existence of which begins to be
generally suspected. Though no appreciable change in
the composition of the atmosphere has taken place within
half a century, it is impossible to conceive that some altera-
tion must not happen, in some direction, in course of time,
among the many perturbing influences which act upon the
mixture. Their antagonism, and that of vegetable and animal
action, partly neutralizes them: but the equilibrium cannot
be precise and continuous. Geological considerations and
botanical fossils lead us to suppose that at some remote
periods the composition of the air must have been sensibly
different: and chemists themselves have actually established
some slight periodical variations, dependent on the propor-
tion of carbonic acid at different seasons. Our analytical
resources are, however, very imperfect with regard to the
accessory principles of the atmosphere ; for chemists can
ascertain nothing of the distinctions which are proved to
exist in the best-marked localities, by their influences on
living beings. The study of these variations, all-impor-
tant in its way,—even as possibly indicating the limits of
human life in a remote future,—belongs to natural history;
and that is probably the reason why chemists trouble them-
selves so little about it: and if there is neglect, it shouldWATER.
347
be charged upon the naturalists. It is true that a prepara-
tion is required for their order of study, like all others,—a
provision of knowledge, rising from physiology to as-
tronomy itself: but the research is not especially a matter
of chemical duty.
The study of water requires much more ex-	Water
tended and complex researches than that of
the air; and it is indispensable to the general system of
chemical science: for water being a real combination, and
perhaps the most perfect known to us, ma,y exercise
chemical effects proper to itself, independently of those
attributable to its elements, and apart from its importance
as a solvent,—to say nothing of it as a simple mixture.
Thus there are three aspects under which water must be
considered by chemists, all distinct and all essential; and
the appreciation of them has been slow and difficult, if
even we may say that this fundamental examination is yet
complete.
The analysis of water, represented by a quantity of hydro-
gen double in volume that of oxygen, and unquestionably
confirmed by synthesis, is the finest of the early discoveries
of modern chemistry, not only from the light it casts upon
the whole of chemical phenomena and the general economy
of nature, but also from its conquest of prodigious difficul-
ties. In regard to the first view, chemical science leaves
nothing to desire. Yet, a notion has arisen, in recent
times, of the existence of a new and more highly oxy-
genated combination between the two elements of water,
which may raise some interesting questions, not about the
irreversible composition of water, but about the kind of
chemical influence which is taken for granted in its decom-
position and recomposition in a multitude of phenomena ;
and especially, about the true mode of union of oxygen
and hydrogen in all substances, and above all in liquids,
which cannot be obtained without water. Some doubts
have lately been proposed about this, which seem to me to
deserve mature examination.
The dissolving action of water has been the subject of a
long series of laborious researches, much less difficult, and
not far from complete. Yet more attention ought to be
paid than is paid to the first experiment of Yauquelin, in348
POSITIVE PHILOSOPHY.
which it is shown that water, saturated with one salt,
remains capable of receiving another, and even acquires
by that the singular property of dissolving a new quantity
of the first. This experiment, which has been in a manner
despised, seems to me of the first order in its way, and a
fit basis of a series of interesting researches about the
apparently capricious laws of solubility, the study of which
is yet essentially empirical.
Chemists were long in conceiving that water, besides being
a solvent, might act in a really chemical manner, otherwise
than by its elements. It seemed as if a combination so
eminently neuter must be inoffensive, and inoperative, ex-
cept by its decomposition. It was Proust who thought
that this neutrality itself afforded a presumption of certain
chemical affections, independently of its composition. This
was the rational consideration which led him to create the
important study of the hydrates, regarded as a sort of new
salts, in which water plays the part, with regard to the
alkalies, of a kind of hydric acid.
The examination of these combinations, and of all others
that water can form with any substances without being
decomposed, constitutes the third and last part of the fun-
damental study of water, regarded as an indispensable
preliminary to the general system of chemical studies.349
CHAPTEE III.
DOCTRINE OF DEFINITE PROPORTIONS.
HEEE are two general doctrines in chemistry, as it
now exists, which present a systematic appearance,
and invest the science with such rationality as it has at-
tained. The first of these is the important doctrine of
Definite Proportions.
Even if this doctrine were complete, it
could exert only a secondary influence on the doctrine *16
solution of the great problem of the science,
—the study of the laws of the phenomena of composition
and decomposition. The essential question is, what separa-
tions and new combinations must take place under deter-
minate circumstances; and the theory of definite propor-
tions affords no assistance to this kind of prevision. It
proceeds, indeed, on the supposition that the question is
already solved; and that it is to be taken as the point of
departure for the estimate of each of the new products,—
of their quantity and the proportion of their elements.
Thus, the theory of definite proportions presents the
singular scientific character of rendering rational, in its
numerical details, a solution which usually remains empi-
rical in its most important aspect.
It was natural that the founders of modern chemistry
should have attended to the laws of composition and de-
composition, in preference to a study which they regarded
as subordinate; and it was natural also that, as the ad-
vance of science disclosed to them the vast difficulties of
the main problem, they should attend more and more to
the secondary study, which promised an easier and more
speedy success. But the most important office of this sub-
ordinate theory,—that of supplying the defect of imme-
diate experiment,—can be but very imperfectly fulfilled,
while it is regarded apart from the principal theory; and350
POSITIVE PHILOSOPHY.
thus, the doctrine of definite proportions will never acquire
its full scientific value till it is connected with an unques-
tionable basis of chemical laws, of which it will be the in-
dispensable numerical complement.
Meanwhile, however, it affords a real, though secondary
assistance to chemists, in rendering their analyses more
easy and more precise. Moreover, it restricts the number
of cases of combination logically possible, by exhibiting
the very small number of distinct proportions; and by
thus diminishing the uncertainty in cases of chemical
action, it is, in fact, a natural preliminary to the establish-
ment of those chemical laws to which it will be, under
another view, a necessary supplement.
A	, ,	, .	In regard to doctrine, this theory offers a
perfect type of the precise kind of rationality
which must hereafter belong to Chemistry as a whole. In
A	,	,, , regard to method, the inquirers who have
s o me o devoted themselves to establish the theory
have advanced chemical science while appearing to diverge
from it; simplifying the vast problem which their suc-
cessors will solve, and preparing for the disclosure of the
great laws of composition and decomposition, which would
be undiscoverable amidst the infinity of products, if sub-
stances could combine, within certain limits, in all ima-
ginable proportions. Such are the claims of this theory,
as to doctrine and to method.
Tf	...	It assumed its existence and present form
s ns ory.	during the first quarter of this century: and
it arose from a phenomenon discovered by Richter, and a
speculative discussion established by Berthollet.—During
the latter half of the last century, several chemists had
observed that, in the mutual decomposition of two neutral
salts, the two new salts thus formed are always equally
neuter. Bergmann, among others, had steadily and spe-
cially dwelt upon this. Yet the fact was neglected or
underrated till Richter, at the end of the century, gene-
ralized the observation, saw what it imported, and derived
from it the fundamental law which bears his name. The
tj. h, ,1 law is this: that the ponderable quantities
ic er s aw. ^ the different alkalies requisite to neutralize
a given weight of any acid are always proportionate toHISTORY OF THE THEORY.
351
those required for the neutralization of the same weight
of every other acid. This is, in fact, evidently the imme-
diate consequence of the maintenance of neutrality after
the double decomposition. Such a transformation would
appear almost spontaneous if it related to a simpler and
more developed science than Chemistry; but amidst its
complications and the imperfection of our intellectual
habits, the closest deductions are difficult if they have any
character of generality, and therefore of abstraction; and
this achievement of Eichter’s is, in consequence, eminently
meritorious, on other grounds than its high utility.—His
law, with the complements it has since received, is the
original basis of the general doctrine of definite propor-
tions. It exhibited, in the case of a considerable number
of compounds, the great end of this doctrine; viz. the
assignment to every substance of a certain chemical co-
efficient, invariable and specific, indicating the proportions
in which it can combine with each of those that have been
similarly characterized. When it had been determined, by
a double series of trials, what was the numerical composi-
tion of all the salts that may be formed bv any one acid
with the different alkalies, and any one alkali with the
different acids, Eichter’s law enabled us to deduce imme-
diately the proportions relating to all the compounds that
can result from the binary combination of these two orders
of substances. Eichter himself brought his discovery up
to this result, and prepared (but on a basis of experiment
too narrow and imperfect) the first table of what were
afterwards called chemical equivalents.
These neutral salts constituted a particular
case, which could hardly have led on to a extension S
general theory of definite proportions. The
idea of perfect neutralization must probably, at all times,
have suggested to chemists that of a single proportion, on
either side of which the neutrality must be destroyed; and
thus the neutral salts were a natural first stage of the
general theory ; but they could not in themselves involve
such a theory. It was Berthollet who extended the con-
sideration of proportions to the whole of chemical pheno-
mena. Some years after Eichter’s discovery he established
as a fundamental principle, in his “ Chemical Statics,” the352
POSITIVE PHILOSOPHY.
necessary existence of definite proportions for certain com-
pounds of all orders; and he assigned the essential con-
ditions of this characteristic property, which he attributed
to all causes which can release the product of chemical
reaction, as it forms, from the ulterior influence of the
primitive agents. He thus added to Richter’s restricted
case the idea of a great number of cases subjected to the
same principle, and able to lead on to its entire generaliza-
tion. It is assigning much too little honour to Berthollet
to recognize only the influence of his controversy with
Proust, eminent as was the service rendered by Proust in
that conflict, in establishing directly the general principle
of determinate and invariable proportions.
Such was the double origin, experimental and specula-
tive, of numerical chemistry. The next development had
_	.	. also a double character, arising from the
Extension 1 harmony between the conception of Dr.
Dalton and the experimental researches of
Berzelius, G-ay-Lussac, and Wollaston. The. inquiry was
in a nascent state when Dalton’s philosophic mind dis-
cerned its possible generality. He proposed the great
»,	. ,, r Atomic theory, under which the doctrine of
onnc ieor>. ^e£np.e pr0p0rti0ns was developed to the
whole extent that it has reached, and which serves as the
basis of its daily application. The general principle of the
theory is this: all elementary bodies are conceived of as
formed of individual atoms, the different species of which
unite, generally by twos, in a small number of groups,
constituting compound atoms of the first order, always
mechanically indivisible, but thenceforth chemically di-
visible, and, in their turn, constituting all the other orders
of composition by a series of analogous combinations. The
principle is in such harmony with scientific conceptions in
all departments, that it appeared like a happy generaliza-
tion of the most familiar ideas of scientific men in every
province of natural philosophy; and its universal and
immediate admission took place as a matter of course.
It was observed by Berzelius that the deduction of the
existence of definite proportions from this principle would
be illusory if the combinations were not restricted to a
very small number of atoms: for otherwise,—if the numberTHE ATOMIC THEORY.
353
was, though, limited, very great,—the binary assemblages
would be so multiplied that we might as well have combi-
nations in any proportions whatever; and then the atomic
theory might almost equally well represent the opposite
doctrines of definite and indefinite proportions. Dalton
was well aware of this; and the restrictions that he enun-
ciated were presently declared too narrow by his successors,
who found that they would not comprehend all existing
combinations. His assertion was, that, in every combina-
tion, one of the immediate principles always enters for a
single atom, and the other generally for a single atom
also, and always for a very small number, rarely exceeding
six. Taken with the expansion proposed by his successors,
the atomic conception evidently represents the entire doc-
trine of definite proportions. But it is the theory of suc-
cessive multiples, derived from the primary doctrine, which
especially distinguishes Dr. Dalton’s influence upon nu-
merical chemistry. From the ground of his doctrine he
easily saw that if two substances can combine in various
distinct proportions, the ponderable quantities of the one
which correspond, in the different compounds, to the same
weight in the other, must naturally follow the series of
whole numbers, since these compounds will have resulted
from the union of one atom of the second substance with
one, two, three, etc., of the first: and this constitutes a
principal element, then perceived for the first time, of the
theory of chemical proportions.
Berzelius followed, with his vast experi-	.
mental study of the whole of the important Berzelius1 ^
points concerned in numerical chemistry, the
different parts of which he has done more than any other
chemist to develope and systematize. He first perfected
Richter’s law, so as to connect it closely with the atomic
theory; by which it became susceptible of the extension
given to it by Berzelius himself, to all compounds of the
second order. But the most important new knowledge
has arisen from his numerical study of compounds of the
first order. By comparing the composition of the metallic
sulphurets and that of the corresponding oxides, he dis-
covered a law, analogous to Richter’s in regard to the
This law,—that the quantity of sulphur of the first354
POSITIVE PHILOSOPHY.
is always proportionate to the quantity of oxygen com-
bined with a like weight of the base in the second,—is
now regarded, by induction, as applicable to all the com-
pounds of the first order to which the same degree of
chemical neutrality is assignable. And again, the luminous
series of the analyses of Berzelius have precisely verified
in another direction the law of successive multiples dis-
covered by Dalton in pursuance of his atomic theory.
Gay-Lussac followed, with the valuable
Gay-Lussac ^ numerical analyses he effected by having
recourse to gaseous combinations, considered,
not as to weight, but to volume. He thus not only verified,
in a special manner, the general principle of definite pro-
portions, but presented it under a new aspect, which, by a
wise induction, comprehends all possible cases,—showing
that all bodies in a gaseous state combine in invariable
and simple numerical relations of volume. An accessory
advantage of this achievement was that the specific gravity
of the gases might be obtained with a precision often com-
parable to that of experimental estimate. It is necessary
however to warn inquirers not to be led away, in their
application of the theory of volumes to substances which
have never been vaporized, from the point of view which
in Gay-Lussac’s application is equivalent to Dalton’s, as
adopted by Berzelius.
,	The labours of Wollaston bore a great
verification* Part Li establishing the doctrine of definite
proportions. I do not refer chiefly to his
transformation of the atomic theory into that of chemical
equivalents, though it has a more positive character, and
tends to restrain the student from wandering after inac-
cessible objects, to which the first might tempt him, if not
judiciously directed. The substitution would be of high
value, no doubt, if it were not less a change of conception
than an artifice of language. Nor have I in view the
ingenious expedients by which Wollaston popularized
numerical chemistry by rendering its use more clear and
convenient. A greater service, in our present view, was
his furnishing us with the indispensable complement of
Richter’s discovery, by establishing the theory in regard
to the acid salts, since extended by analogy to the alkalineTHE APPLICATIONS OF NUMERICAL CHEMISTRY. 355
salts. The case of the acid salts was perhaps the most
unfavourable possible for the ascertainment of the prin-
ciple of invariable proportions. Wollaston effected the
proof in the most satisfactory manner; and this special
confirmation of the principle is considered, from its nature,
the most decisive of all.
Such has been the logical and historical progress of the
researches which have constituted numerical chemistry as
it is now. We can represent by an invariable number,
appropriated to each of the different elementary bodies,
their fundamental relations of chemical equivalence, whence,
by very simple formulas, immediately expressing the laws
just indicated, we easily pass to the numerical composition
proper to each combination. No further evidence of the
truth of the doctrine is needed, than the fact of so many
illustrious inquirers having attained the same view by
ways which each one opened for himself, and all agreeing
as to its positive application to all cases of importance,
differing only as to the mode of expression of the results,
in as far as the atomic theory left it indeterminate, and
therefore optional. But we must glance at the difficulties
thrown in the way of its application by a consideration of
the aggregate of chemical phenomena, in order to form a
clear idea of the final improvement of which this doctrine
vet stands in need.
Among the points which are beyond dis- Scope of appli-
pute, it is, first, evident, and no chemist has cation of Nu-
ever doubted it, that substances differ as merical Che-
much in the proportion as in the nature of mlstry*
their constituent principles. It is an axiom of chemical
philosophy that any change whatever in the numerical
composition causes a change in the whole of the specific
properties, in a more marked degree as the alteration is
greater. Varied and gradual above all others as are the
proportions produced by the chemical phenomena proper
to living bodies, they afford a striking confirmation to this
universal maxim. Therefore, in the lowest stages of
chemical analysis, chemists have always endeavoured to
assign, as a characteristic property, the proportion of the
elements of each substance, as far as was possible: and
when this was omitted, it was on the understanding that356
POSITIVE PHILOSOPHY.
the proposed combination admitted of only a certain pro-
portion ; as in the case of the neutral salts.
Again, it has long been acknowledged that there always
exists, between any two substances, a certain minimum
and maximum of reciprocal saturation, beyond or short of
which all combination becomes impossible. At the utmost,
certain variations, themselves restricted, have been sup-
posed procurable. Berthollet established, more directly
than any one else, the general and necessary existence of
these limits of combination,—one of the principal cha-
racters which distinguish it from simple mixture. It is
clear that the two extreme degrees of all combination must
be subject to special and invariable proportions: and, as
all agree in this, all argument about the opposite doctrines
of indefinite and definite proportions is reduced to the
question whether the passage from the minimum to the
maximum of saturation can be effected gradually and
almost imperceptibly, or whether it takes place always
abruptly, through a small number of well-marked degrees.
Thirdly, the possibility and actual existence of interme-
diary definite proportions are admitted by all chemists,
who can have no other dispute than about the greater or
smaller generality of such a property. We have seen that
the idea of neutrality must, sooner or later, bring after it
that of a determinate and unchangeable proportion; and
the gradual development of chemical knowledge has ex-
tended this character to more and more varied cases.
Berthollet disclosed several other causes of definite pro-
portions, which were entirely misconceived before his time,
and which may meet in almost all combinations, modifying
certain circumstances of the phenomenon. The precise
question now is, therefore, whether, besides these deter-
minate compounds, subject to fixed proportions, within the
two limits of possible combination, there does or does not
exist, in general, a continuous series of other intermediate
compounds of a less marked character; in a word, whether
definite proportion constitutes the rule, as is now generally
supposed, or, as Berthollet endeavoured to establish, the
exception. This is now the only dispute. It is no deroga-
tion from the interest of the doctrine of definite proportions
to say, as some preceding considerations compel us to do,VALUE OF THE THEORY.
357
that the decision of this disputed point is not of the
importance commonly supposed. The doctrine has tended
to simplify the general problem of chemistry; but it must
not be supposed that the solution would have been impos-
sible without this aid:—it would have been simply more
difficult and less precise. The eminent chemists who con-
curred in establishing the doctrine were naturally engrossed
by that labour; but their successors, who find numerical
chemistry constituted to their hand, must beware of losing
sight in it of the true scientific aim of chemistry. They
must not linger in this vestibule of the science, to the
neglect of the direct construction of Chemistry itself,—an
enterprise scarcely begun, and to which it is high time
that attention should be once more fully directed.
If we inquire, as we must do, how far the doctrine of
definite proportions is irrevocably established, we shall
bear in mind that the founders of numerical chemistry
have accomplished that chief part which depends on an
investigation of all known compounds, leaving only the
question whether the doctrine is compatible with certain
chemical phenomena, neglected during its formation, and
remaining to be since referred to it.
The first general objection relates to the oyection of
important phenomenon of dissolution, evi- dissolution0
dently possible in an infinity of different pro-
portions. It must be acknowledged that the distinctions
between the state of dissolution and that of combination,
by which the difficulty has been met, afford little satisfac-
tion. In my opinion the only effectual reply must consist
in the extension of the principle of definite proportions to
the phenomena of dissolution; and, difficult as it may be
to do it, it does not seem to me impossible. The way is
by the use of an hypothesis already proposed for other
cases in which it might appear less admissible. All the
successive degrees of concentration of the liquid must be
regarded as simple mixtures of the small number of definite
dissolutions which shall have been established, either
between themselves or with the dissolvent, in the manner
of habitual mixtures of water with alcohol, with sulphuric
acid, etc. In any case, the positive verification of this
hypothesis must be extremely delicate. Furthermore, to358
POSITIVE PHILOSOPHY.
Of metallic
alloys.
render the study of dissolutions fully rational, in this
point of view, it is necessary to combine with it that of
other analogous chemical phenomena, relating to the
absorption of gases by liquids or by porous solids. All
these different modes of molecular union are often energetic
enough to resist influences able to destroy certain combina-
tions, properly so called: why should they not be, like
them, subject to the rule of definite proportions, if that
rule is truly a fundamental law of nature ?
The next case, that of various metallic
alloys, is very extensive, though more par-
ticular. The difficulty lies in the question
whether these are cases of combination or of mixture.
The state of combination has been taken for granted in
the case of alloys; whereas the general application of the
principle of numerical chemistry requires that they should
be mixtures ; while, again, it is difficult to conceive of such
a mixture of solids as could resist perturbing influences
which would appear to be necessarily destructive; as great
changes of temperature, the influence of crystallization, etc.
The question can be decided only by a series of special
experiments, devised to find the general limits of the
permanence of unquestionable mixtures; and the results
might be extended to other questions of numerical
chemistry, as of certain oxides, on which explanations have
been hazarded too freely. When a true chemical theory
of mixtures is established on a proper basis of experiment,
and we leave off referring to an hypothesis of mixture all
cases in which combination seems susceptible of an inde-
terminate proportion, in order to bring them under the law
of definite proportions, we shall get rid of a formidable
objection to the principle of numerical chemistry.
The remaining case constitutes the greatest
substances obstacle of all in the way of the generaliza-
tion of the law of definite proportions: and
if it cannot be surmounted, the law sinks to the rank of
an empirical rule, fit for nothing more than facilitating a
certain order of chemical analyses. I refer to the class,
anomalous in this view, of substances called organic. And
this is the effect, in fact, of the declaration of the chemists
of our time, that organic substances do not come underTHE PRINCIPLE OF DUALISM.
359
the principle of definite proportions. It amounts to saying
that the law rules all the elements, except oxygen, hydrogen,
carbon, and azote. The division between inorganic and
organic chemistry is merely scholastic; for all chemistry
is, by its nature, homogeneous,—that is, inorganic. And
thus, if we admit of the enormous exception of the nume-
rical composition of so-called organic substances, the
doctrine of definite proportions is overthrown as a rational
theory. As it evidently cannot be founded on any a priori
considerations, it is only by a strict generality that it can
become a rational theory.
If we could not hold at once the grand Application of
principle of the dualism which pervades the principle
chemistry, and constitutes its homogeneous dualism,
character, and the doctrine of definite proportions, I should
not hesitate to sacrifice the latter: for it is more important
for chemical progress to grasp the great principle of syste-
matic dualism, than to advance our investigations by the
use of the numerical rule. But there is not, in fact, any
incompatibility between these two means of progress : and
such a brief sketch of my conception on this subject as my
limits allow may show how the doctrine of definite propor-
tions can be duly generalized only by discarding organic
chemistry as a separate body of doctrine, and extending
the principle of dualism to all organic compounds.
If we are to include all organic compounds under one
uniform system of chemistry, properly so called, we must
refer to physiology, vegetable and animal, the study of the
numerous secondary substances which owe their transient
and variable existence to the development of vital
phenomena, and which have no scientific interest except
under the head of biology. We shall see, under that head,
hereafter, what the precise classification is; and all that
we have to do with it now is to show that it proceeds from
the fundamental distinction between the state of death
and that of life. The second, and most extended class of
organic substances is chiefly composed of mixtures which,
as such, admit of all imaginable proportions, within the
limits of vital conditions. As for the substances which
exhibit real combinations, we must conceive of them as
subject to the law of definite proportions; but the com-360
POSITIVE PHILOSOPHY,
plexity, and yet more the instability of such compounds
will probably for ever forbid their being successfully
studied under the numerical point of view, which is indeed
of very inferior interest in biology.—Even after this clear-
ing of the field, we could not accomplish the desired
generalization if we had not taken a new stand with regard
to the ternary and quaternary substances contemplated by
ordinary chemistry. The rigorous dualism which I have
before, and in a higher view, shown to be necessary, seems
to supply, naturally and finally, the needs of the doctrine
of definite proportions.
As long as chemists persist in regarding organic com-
binations as ternary and quaternary,—that is, in con-
founding their elementary with an immediate analysis;—
while oxygen, hydrogen, carbon and azote are regarded as
immediately united, the compounds from them which
must be recognized as distinct, after the severest sifting,
will be enough to constitute an invincible objection to the
principle of a numerical chemistry. But if they become
binary compounds of the second, or at most the third
order, whose principles are formed by the direct and binary
combination of those three or four elements, we find our-
selves able to represent all the actual numerical varieties
established by an elementary analysis, conceiving, for each
degree of combination, a very small number of distinct and
entirely definite proportions.—In the ternary case,—appro-
priate to compounds of a vegetable origin,—their three
elements may be united in three kinds of binary combina-
tions. Combining these again, still employing at once
oxygen, hydrogen, and carbon, we have three principal
classes of compounds of the second order. But then again,
each term of the new compounds really corresponds to two
distinct substances : and thus, while admitting only one
proportion for the binary composition of these bodies, we
have already provided for the numerical composition of
twelve substances at present called ternary. But, further,
we are compelled to suppose at least three different pro-
portions for each binary combination : one producing per-
fect neutralization, and the other two the extreme limits
of the reciprocal saturation : and chemical analogies indi-
cate a much larger number of compounds. Putting thoseAPPLICATION OF THE PRINCIPLE.
361
aside, we have thirty-six compounds, without going beyond
the second order, by the known combination of three
elements on the principle of dualism. We are also entitled
now to conceive of a third possible combination between
oxygen and carbon, or between carbon and hydrogen, etc.,
which already furnish two, after being long supposed to
admit of only one. Hence, and in view of all these con-
siderations, we may be assured that by dualism we might
completely and naturally subject to the law of definite
proportions eighty-one compounds of the second order,
formed from oxygen, hydrogen, and carbon; and this
would unquestionably more than suffice to represent the
elementary analysis of all distinct substances in the range
of vegetable chemistry.
Passing on to the quaternary case,—characterizing what
is called Animal Chemistry,—it seems as if the principal
class of compounds of the second order must be more
numerous than in the ternary case : but the indispensable
condition of employing all the four elements at once
restricts the classes to three. But when we examine the
terms of the secondary compounds, we find that while two
represent only one compound each, a third represents five.
Thus, the three pairs of compounds yield fourteen of one
proportion, and forty-two of the three proportions indi-
cated in the last case. But applying, at each degree, the
rational rule of a triple binary combination, without stop-
ping at the inevitable gaps of our existing chemistry, we
find ourselves in possession of ninety-nine compounds of
the second order, now regarded as quaternary. This is
probably a larger number than a rational analysis of
animal substances will be found to require. Moreover, as
animal substances have undergone a greater degree of vital
elaboration than vegetable matters, it would be philo-
sophical to admit, with respect to them, the possibility of
a higher order of composition, such as physiological com-
binations must eminently tend to realize. On such an
hypothesis, without going beyond the third order, we
might obtain ten thousand perfectly distinct compounds
from these four elements, all formed by an invariable
dualism, and strictly subject to the law of definite propor-
tions. It is true, nature would not permit the realization362
POSITIVE PHILOSOPHY.
of more than a small part of these speculative combina-
tions; but I have pursued the consequences of my con-
ception to this extreme ideal limit, to show how abundant
are the rational resources supplied by this new theory for
the generalization of the laws of numerical chemistry. If
this view is not followed up, or some equivalent one pro-
posed, it is evident that we must give up the doctrine of
definite proportions as a law of natural philosophy, and
return to Berthollet’s theory, merely enlarging the cases of
fixed proportions which he admitted. In the present state
of the question there is no other choice. But my theory
having been, not instituted for this destination, but natu-
rally arrived at by another way, and with higher views,
and proceeding from established principles, to meet the
needs of chemical philosophy, seems to me to be presump-
tively entitled to a future, and perhaps speedy realization.
This account of the present aspects of the doctrine of
definite proportions will enable any one to judge of its real
progress from its institution to this day; of the conditions
which must be fulfilled before its principle can be con-
verted into a great law of nature; and of the rational
course which alone can lead to such a final constitution of
numerical chemistry.363
CHAPTER IV,
THE ELECTRO-CHEMICAL THEORY.
ROM the beginning of modern chemistry, Relation of
the chemical influence of electricity Electricity to
manifested itself unequivocally in many im- Chemistry,
portant phenomena, and above all, in the grand experi-
ment of the recomposition of water by the direct combina-
tion of oxygen with hydrogen, effected by the aid of the
electric spark. But the special attention of chemists was
not strongly drawn towards this agency till Volta’s im-
mortal discovery disclosed its principal energy, in render-
ing the electric action at once more complete, more pro-
found, and more continuous. Since that time, various
series of general phenomena have taught us that electricity
is a chemical agent more universal and irresistible than
heat itself, both for decomposition and combination. The
danger now is of exaggerating the relation it bears to the
general system of chemical science. Though chemistry
is united to physics by this agency more than by any
other, it must yet be remembered that the two sciences
are distinct, and that there should be no confounding of
chemical with electrical properties. In order to ascertain
with precision what are the relations of chemistry with
electrology, we must briefly review the gradation of ideas
which have led up to the present electro-chemical theory,
as systematized by Berzelius.
The first important chemical effect of the History of
voltaic influence was the decomposition of the case,
water, established by Nicholson in 1801. It Nicholson’s
was a necessary result of examination into discovery,
the action of the pile, without any chemical intention. It
confirmed a truth before well known: but it had a high
chemical value, as revealing the chemical energy of the
instrument; and it thus constituted the starting-point of364
POSITIVE PHILOSOPHY.
electro-chemical research.	We may even refer to this
origin the first attempts at founding a general theory of
electro-chemical phenomena ; for the conception offered hy
G-rothuss, to explain Nicholson’s observation by the electric
polarity of molecules, contains the germ of all the essential
ideas which have expanded, according to the requisitions
of the phenomena, into the present electro-chemical theory.
The analytical power of the voltaic pile having been
once discovered, it was natural for the chemists to apply
the new agent to the decomposition of substances which
had hitherto resisted all known means. The first series of
,	attempts produced, after a few years, the
covery	brilliant discovery by the illustrious Davy,
—of the analysis of the alkalies, properly so
called, and the earths. Lavoisier’s theory, which showed
that every salifiable base must be a result of the combina-
tion of oxygen with some metal, had foreshown this
analysis; but no chemical means had sufficed to effect it.
It was believed in, in spite of Berthollet’s discovery of the
true composition of ammonia; and the brilliant result
obtained by Davy was an easy consequence of a discovery
completely prepared for. M. Gray-Lussac soon followed,
with a more difficult but less striking achievement,—the
confirmation, by a purely chemical process, of the electrical
analysis of potash.
Nicholson’s observation having originated electro-
chemistry, and Davy’s given it a great impulse, the next
step was to investigate the chemical influence of elec-
tricity, in a scientific view. This was effectually, though
indirectly, determined by Davy’s great feat; for by it
chemistry was proved to have achieved the most im-
portant, and hitherto inaccessible analyses: and in fact,
the science has not since made any essential acquisition.
Electro-chemical action was presently and permanently
subjected to direct and regular study; and it was irrevoc-
ably constituted a fundamental part of chemical science
when Berzelius accomplished his series of
investigations on the voltaic decomposition
of all the salts, and then of the principal
oxides and acids. It was in consequence of these researches
that the habitual consideration of electric properties has
Berzelius’s
extension.SYNTHESIS.
365
Synthetical
process.
assumed a growing importance in the chemical study of
all substances, which are now scientifically divided into
the classes of electro-negatives and electro-positives. It
was thus the privilege of Berzelius to be the first to con-
ceive of the electro-chemical theory under an entirely
systematic form.
One condition remained to be fulfilled, to
give its due scientific character to this new
branch of chemistry. The voltaic action had
thus far been regarded only analytically :—it must be re-
garded synthetically also. This was done by the labours
of Becquerel, who fully established the synthetical in-
fluence of electricity, fitly applied ; and who, moreover,
employed it in effecting new and valuable combinations,
hitherto impracticable. From this procedure arose the
necessity of modifying the method of experimentation.
The apparatus which was powerful enough to decompose
was much too powerful to combine, because it would pro-
bably decompose the immediate principles which were in-
tended to be combined; and thence arose the method of
employing the protracted action of very feeble electric
powers,—every advantage being given to the disposition
of the substances to be acted upon. M. Becquerel did
this very successfully by operating with a single voltaic
element, and seizing each substance in the B
state called nascent, which is agreed upon	^
as most favourable to combination. This change of pro-
cedure is the distinctive honour of M. Becquerel. He not
only determined direct combinations, not before obtainable,
but exhibited in others, which were obtainable, the re-
markable property of clearly manifesting their geometrical
structure, through the slowness and regularity of their
gradual formation;—a character especially marked in the
case of certain metallic sulphurets, some oxides, and
several salts. It does not lie within our province to point
out the results of this method in regard to the natural
history of the globe, in explaining a great number of
mineral origins, when the time for such concrete questions
shall have arrived. It is more in our way to observe the
importance of these labours in bringing up chemical syn-
thesis to something like harmony with the progress of3 66
POSITJVE PHILOSOPHY.
analysis,—favoured as the latter pursuit had been by the
ease with which we destroy, in comparison with the diffi-
culty with which we recreate. Once more, these researches
of M. Becquerel have completed the general constitution of
electro-chemistry, which, being henceforth at once syn-
thetical and analytical, can only expand in one of these two
directions, however great the improvements remaining to
be attained.
Such has been the filiation of electro-chemical discoveries,
since the beginning of our century. A little attention to
the great phenomenon which was the original subject of
the electro-chemical theory will show how this study has
gradually led to a new fundamental conception for the
whole of Chemistry.
It has been said, through all periods of
combustion chemical research, that the study of Com-
bustion must be the central point of the
science. So it was thought in the ancient theological
period of the science; and also in the more recent meta-
physical stage, when combustibility was called phlogiston,
and regarded as an intangible materialized entity: and the
advent of the positive period of chemistry was marked by
the establishment of Lavoisier’s new theory of combustion.
In our day it is the recognized necessity of modifying this
theory that has especially led to the electric conception of
chemical phenomena.
The pneumatic theory of combustion of
Lavoisier had two entirely different objects
in view; objects which are too commonly
confounded, but which we must be careful to keep apart.
First, the analysis of the general phenomenon of combus-
tion : and, secondly, the explanation of the effects of heat
and light, which is, in the eyes of the vulgar, the more im-
portant of the two. Both were treated in the most ad-
mirable manner possible in the existing state of knowledge;
and in the characteristics of postivity and rationality La-
voisier’s theory has not since been surpassed. All combus-
tion, abrupt or gradual, was regarded as consisting in the
combination of the combustible body with oxygen, whence,
when the body was simple, must result an oxide, generally
susceptible of becoming the basis of a salt, and, if the oxygen
Lavoisier’s
theory.THEORY OF COMBUSTION.
3 67
was preponderant, a true acid, the principle of a certain
kind of salts. As for the disengagement of heat and light,
it was attributed, in a general way, to the condensation of
the oxygen, and in an accessory way, to that of the combus-
tible body, in this combination.
The first part of this anti-phlogistic theory First division
has a much more philosophical character than
the second. It was eminently rational to analyse the pheno-
menon of combustion, so as to seize whatever was common
to all cases. The conclusions drawn might be too general;
and were afterwards proved to be so ; but whatever would
stand the test of time must form a body of indestructible
truth, constituting an essential part of chemical science
through all future revolutions. The case is different with
the explanation of heat and light. The question is not,
like the first, of a chemical, but a physical nature; and,
whatever may be its final solution, it cannot affect chemical
conceptions. It would have been wiser to abstain from
offering any general explanation of the effects of heat and
light through such a supposition as that of a condensation,
which does not necessarily take place, and which is, in fact,
found to be often absent. Lavoisier hoped to attach the
thermological effect to the great law discovered by Black,
of the disengagement of heat proper to the passage of any
body from one state to another more dense; but such a
connection could not be established on the ground of
phenomena not invariably present, or indisputably mani-
fested. However, it would be too much to expect a perfect
scientific reserve in discoverers who bring out scientific
truths from a region of metaphysical fantasies. It is from
their followers that we have a right to demand it; and we
are compelled to charge upon the chemists who have been
eager to substitute the electro-chemical theory for the anti-
phlogistic theory, properly so called, a want of care in con-
structing explanations analogous to those which are dis-
missed as insufficient. To justify this charge, we must
review the proofs of the imperfection of Lavoisier’s theory,
—still regarding it under the two divisions just exhibited.
Berthollet saw presently that Lavoisier’s Berthollet’s
method of analysis of combustion must be limitation."
modified. One of the chief consequences of368
POSITIVE PHILOSOPHY.
this analysis was that every acid and every salifiable base
must be a result of combustion ; that is, of the combina-
tion of any element with oxygen ; whereas, Berthollet dis-
covered that one of the most marked of the alkalies,
ammonia, is formed of hydrogen and azote alone, without
any oxygen; and soon after, he proved that sulphuretted
hydrogen gas, in which also there is no oxygen, neverthe-
less presents all the essential properties of a real acid.
These facts have since been confirmed in every possible
way, and especially by the electric method ; and the excep-
tions, both as to alkalies and acids, have become so multi-
plied, that the investigation and comparison of them have
given that high character of generality to the study of
alkalies and acids which belongs to it in our time.
Moreover, the primitive theory of combustion has been
gradually modified by the discovery that a rapid disen-
gagement of heat and light is not always an indication of a
combination with oxygen. Chlorine, sulphur, and several
other bodies, even non-elementary, have been found to
occasion true combustion. And again, the phenomenon of
fire is no longer attributed exclusively to any special com-
bination, but, in general, to all chemical action at once very
intense and vivid.
It does not follow that because Lavoisier’s discoveries
have parted with some of their character of generality,
they have lost any of their direct value: and such altera-
tion of views as there is relates chiefly to artificial pheno-
mena, while the natural facts remain securely established.
Thus, though there are acids and alkalies without oxygen,
it is unquestionable that the greater number of them, and
especially the most powerful, are oxygenated: and again,
if oxygen be not indispensable to combustion, it remains
the chief agent, and especially in natural combustions. In
natural history, the theory is applicable, almost without
reserve, though it is insufficient for the severe conditions of
abstract science. If the universal sovereignty of oxygen
has been overthrown, it will yet be for ever the chief
element of the whole chemical system,
v, , i. . . As for the second aspect of the discovery,
the explanation of fire,—it was destroyed by
the first direct examination of it. No new facts were re-THEORY OF COMBUSTION.
369
quired for its overthrow, but merely a more scientific
appreciation of common phenomena. It will not even
serve naturalists for their concrete purposes in any degree,
having never really explained the most ordinary effects.
The required condensation is found to be only occasionally
present, and often absent in the most important cases ; so
that if it were not for the connection of this aspect of the
theory with a sounder one, it would be inconceivable how
it could have held its ground up to a recent period,—busy
as the chemists were with other theoretical speculations.
After finding that in cases where condensation was sup-
posed, expansion exists instead ; and that where we find
vivid combustion, there should, by the theory, be a great
cooling, we are brought to the reflection that if the fire on
our hearths was not a matter of daily fact to us, its exis-
tence must become doubtful, or be disbelieved, through
those very explanations by which the phenomenon has
been proposed to be established. To my mind, this is a
clear indication that the chemical production of fire does
not admit, in a general wav, of any rational explanation.
Otherwise it appears incomprehensible that men of such
genius and such science, at a time so near our own, should
have been so deluded. The electric fire, now proposed for
an explanation, must have been sufficiently known to La-
voisier, Cavendish, Berthollet, and others, to have served
as a basis to their theory, if its preponderance over the
merits of their hypothesis had been so great as is now
commonly supposed. This consideration however, striking
as it may be, is no dispensation from the duty of examin-
ing the electro-chemical conception, for which we have been
prepared by this short account of its antecedents.
According to this theory, the fire produced in the greater
number of strong chemical reactions must be attributed to
a real electric discharge which takes place at the moment
of combination (by the mutual neutralization, more or less
complete, of the opposite electric conditions) of the two
substances under consideration,—one of which must be
electro-positive and the other electro-negative. There is
every reason to fear, however, that when this theory has
been effectually examined, it will be found as defective
in rationality as its predecessor. If electric effects are
I.	B B370
POSITIVE PHILOSOPHY.
concerned in all chemical phenomena, as seems to be now
agreed npon by most chemists and physicists, they must
oftener be supposed than found: and the electric symptoms
are most impossible to detect in precisely those chemical
phenomena which have been most relied on for overthrow-
ing the old theory. And in the cases in which electriza-
tion is evident, its chemical influence is so equivocal that
some regard it as the cause, and others as the effect, of the
combination. The explanation is not yet positively estab-
lished for any phenomenon whatever that has been duly
analysed : while its vague nature leaves room for fear that
it will not be so radically or so speedily destroyed as its
predecessor. It could be plainly shown whether the
requisite condensation did or did not exist: but there is
always a resource, in the more recent case, in the faint or
fugitive character of the electric condition, which defies
our means of positive exploration,—qualities which are
far from being a ground of recommendation of a theory
which is to account for very striking and intense effects. I
do not desire to say that the disengagement of light and heat
can never have an electric origin, any more than that it can
never proceed from the condensation proposed : but I think
an impartial observation would decide that in most cases
of combustion there is neither condensation nor electriza-
tion. My own view is that these vain attempts to explain
the chemical production of fire proceed from the lingering
metaphysical tendency to penetrate into the nature and
mode of being of phenomena: and that chemical action is
one of the various primitive sources of heat and light,
which cannot, from their nature, usually admit of any
positive explanation; that is, of being referred, in this
relation, to any other fundamental influence.
If our chemical science were more advanced than it is,
we should not have to point out that the consideration of
fire, which, however important, is only a physical accessory
of chemical phenomena, cannot afford a rational ground
for a radical change in our conceptions of chemical action.
. When our predecessors regarded heat as the
the theory^ °	physical agent in composition and de-
composition, they did not pervert such a
consideration to the point of assimilating chemical toDIFFICULTIES OF THE THEORY.
371
thermological effects. We are less cautious at the present
day: we confound the auxiliary, or the general physical
agent of the phenomenon, with the phenomenon itself, and
pervert chemistry by confounding it with electrology, by
irrationally assimilating chemical to electrical properties,
as is seen especially in the theory of M. Berzelius. How
can there be any scientific comparison between the tendency
of two bodies to a mechanical adhesion after a certain mode
of electrization, and the disposition to unite all their mole-
cules, external and internal, by a true chemical action:
M. Berzelius has frankly declared that cohesion, properly
so called, admits of no electric explanation. Nothing is
gained towards explaining the molecular connection, in-
dissoluble by any mechanical force, in contrast with the
magnetic union so easily overcome, by talking of voltaic
elements with their positive or negative pole, and their
connection by the electric antagonism of the opposite poles.
Such inventions give no idea whatever of molecular co-
hesion. Nor is affinity, or the tendency to combination,
any better explained by the electro-chemical theory. Elec-
trical phenomena, in physics, are eminently general, offering
only differences of intensity in different bodies; whereas,
chemical phenomena are essentially special or elective : and
therefore every attempt to make chemistry, as a whole,
enter into any branch of physics, is thoroughly anti-
scientific. This would be enough: but we see besides that
the smallest changes in the mode of electrization reverse
the electric antagonism, and destroy the proposed electrical
order of elementary bodies: we find ourselves unable to
deduce the new electric properties which the theory bids
us look for in the compounds of different orders; we do
not know by what laws they derive their positive or nega-
tive character from the electric condition of each of the
two elements; nor are we able to approach the great end
of chemical science,—the prevision of the qualities of com-
pounds by those of their constituent elements. Moreover,
in any case, the great body of chemical phenomena opposes
insurmountable obstacles; as when oxygen, the most nega-
tive element, when entering largely into certain oxides,
finds them positive towards certain acids, into which it
enters much more sparingly,—the radicals of the first372
POSITIVE PHILOSOPHY.
being often as negative as those of the last. According to
M. Berzelius’s own frank declaration, organic compounds
cast insuperable difficulties in the way of his arrangement
of electric relations; and he alleges the transience of the
combinations in that class of cases as an explanation of
the anomaly: but chemical science would be impossible if
compounds were not throughout considered stable till
causes of decomposition arise: and if organic compounds
are guarded from these, they remain chemically stable,
like inorganic substances. The fundamental obstacle of
the whole case,—the identity of the elements, in opposition
to the electric variety,—cannot by any means whatever be
got over.
Leaving all these difficulties on one side, we learn nothing
about chemical phenomena by likening them to electric
action, for we establish thus no harmony between the pre-
tended causes and the real effects. Every attempt seems
to prove simply the auxiliary influence of electricity on
chemical effects,—acting as heat does, only with a different
intensity. In fact, there are scarcely any electro-chemical
combinations which cannot be effected by ordinary chemical
processes, without any electric indications; and the few
exceptional cases still existing may, by analogy, be expected
to be brought under the rule. If, in the face of all this,
we were to persist in investing the electrical influence with
the specific and molecular attributes of chemistry, we
should merely be restoring the old entity of affinity, decked
out with some hypothetical material attributes, which would
be far from rendering it more positive: and such a pro-
cedure would be as hurtful to physics as to chemistry, by
infusing new vagueness into our notions of electricity,
which are at present far from being sufficiently distinct.
And then might follow, as likely as not, the founding on
electrology, not only the whole of chemistry, but the
theories of heat, of weight, and probably, as a conse-
quence, that of celestial mechanics. And then, if we
added to this heterogeneous assemblage a confounding of
the supposed nervous fluid with the pretended electric
fluid, we should have attained to the show of an universal
system, devoid of all scientific use, which would fall to
pieces as soon as tested by real study, parting off intoRESULTS OF THE THEORY.
373
categories of independent doctrine, and encumbering natural
l^hilosophy with insoluble questions, which must be dis-
carded, to enable us to begin afresh.
Thus, to sum up, the great chemical influence of elec-
tricity, like that of weight, and yet more of heat, is un-
questionable ; and I have endeavoured to exhibit the high
importance of electro-chemistry to the improvement of
chemical science, of which it is one of the essential
elements. But I must, once for all, reject the conception
through which it has been attempted to transform all
chemical into electrical phenomena. In a philosophical
point of view, Lavoisier’s theory appears to me, notwith-
standing its serious imperfections, very superior as a
scientific composition to that to which it has given place.
It related directly to the aim of chemical science,—the
establishment of general laws of composition and decom-
position ; whereas the newer theory draws attention away
from it, in a vain inquiry into the intimate nature of
chemical phenomena. Thus the anti-phlogistic conception
has suggested numerous and important chemical dis-
coveries, while it is very doubtful whether as much will
ever be said for the electric theory. In an indirect way
however it may operate favourably for	^ osition
chemical science, by its binary antagonism	and powers,
suggesting the extension of dualism among
compounds which are as yet supposed to be more than
binary. Berzelius appears to have felt this connection;
and he would probably have erected dualism into a funda-
mental principle, but for his subjection to the old division
of chemistry into organic and inorganic. Others, who are
free from this entanglement, may be prepared by the
electro-chemical theory for general dualism ; though it is
not, in principle, desirable to recur to faulty means to
attain good results in an indirect way. In another view,
this theory may be useful,—in fixing the attention of
chemists on the influence of time in the production of
chemical effects ;—an influence remarkably manifested by
many phenomena, but not, as yet, directly analysed. Not
only does time naturally increase the mass of the products
of chemical re-action; it also causes formations which374
POSITIVE PHILOSOPHY.
would not otherwise exist. The chemical theory of time
is at present a blank in science ; and the phenomena of
electro-chemistry seem likely to enlighten us on this head,
—so all-important in its connection with chemical geology,
while constituting an indispensable element in the concep-
tions of abstract chemistry.
To complete our survey of the philosophy of Chemistry,
we must turn to some considerations already suggested, on
the subject of what is commonly called Organic Chemistry.375
CHAPTER Y.
ORGANIC CHEMISTRY.
ORGANIC Chemistry comprehends, it cannot be denied,
two kinds of researches, chemical and physiological,
which are perfectly distinct. For instance, the study of
organic acids, and especially the vegetable Confusion of
acids, and alcohol, ethers, etc., has a character two kinds of
as purely chemical as that of any inorganic ^act-
substances whatever; and, on the other hand, there is no
doubt of the biological character of inquiries into the com-
position of sap or blood, and of the analysis of the products
of respiration, and many other matters included in organic
chemistry. The confusion of the two orders is prejudicial
to both sciences, and especially to physiology. The division
of Chemistry conceals or violates essential analogies, and
hinders the extension of dualism into the organic region,
where it is seldom found, though, as I have shown, it is
optional, in fact, throughout the whole range of chemistry;
and thus the arbitrary arrangement is the chief obstacle in
the way of the entire generalization of the doctrine of de-
finite proportions. And whenever a true chemical theory
shall fitly replace the anti-phlogistic conception, it must
necessarily comprehend all organic, as well as inorganic
compounds. The most philosophical chemists are tending
more and more to a recognition of the identity of the two
departments; and there can be no doubt that the establish-
ment of that identity will be an immediate consequence of
a rational classification of chemical knowledge.
The confusion is more mischievous, but less felt by
chemists, under the other view,—the comprehension of
biological phenomena among those of organic chemistry.
The confusion arose from the need of chemical researches
in very many physiological questions; and these chemical
researches, being usually extensive and difficult, were out376
POSITIVE PHILOSOPHY.
of the range of the physiologists, and were taken posses-
sion of by the chemists, who annexed them to their own
domain. Both classes were to blame for the vicious
arrangement, and both must amend their scientific habits
before a division can be effected in entire conformity to
natural analogies. The physiologists have the most difficult
task before them, having to qualify themselves for inquiries
from which the chemists have only to abstain.
It is scarcely possible to characterize or to circumscribe
the physiological part of organic chemistry, formed as it is
by successive encroachments. It comprehends the chemical
analysis of all the anatomical elements, solid or fluid, and
that of the products of the organism ; and if its usurpations
remained unchecked, it would soon include the phenomena
of what Bichat called the organic life; that is, the functions
of nutrition and secretion, the only ones common to all living
bodies, and in which the chemical point of view might well
appear the natural one. In such a state of things, physio-
logy would be reduced to the study of the functions of
animal life, and to that of the development of the living
being. It is easy to see what would become of biological
science, if it were reduced to this fragmentary stale.
Chemists cannot but be unfit for the rational examination
of the important questions of anatomy and physiology,
vegetable and animal, because the research requires that
comprehension of view which their studies, as chemists,
preclude them from obtaining. In the anatomical relation,
they are perpetually overlooking the fundamental division,
established by M. de Blainville, between the true elements
of the organism, and its simple products; and they take
for one another, almost indifferently, the tissues, the paren-
chyma, and the organs. The spirit of biological investiga-
tion being unknown to them, they can neither choose their
subject well, nor direct their analysis wisely. If these are
grave inconveniences in anatomical questions, they are
much more serious in physiological problems, properly so
called, the essential conditions of which are not understood
by chemists. The rational direction of physiological analysis
can take place only by the subordination of the chemical to
the physiological view ; and therefore by the employment
of chemistry by the physiologists themselves, as a simpleRELATION OF CHEMISTRY TO ANATOMY.
377
means of investigation. It is an analogous case to that
before exhibited, of the application of mathematical analysis
to physical questions. If it is important that physicists
should employ the instrument of analysis, instead of de-
livering over physical subjects to the mathematicians, to
be a mere theme for analytical exercises, much more im-
portant is it that the greater diversity of view in chemistry
and physiology should not be lost sight of. The irrational
and incoherent studies comprised in the organic chemistry
of our day give us no idea of the true nature of the aids
that biology may derive from chemistry. A few instances
will show the high importance of an improved organization
of scientific labour.
In the anatomical order, almost all the re- Relation of
searches of chemists have to undergo an Chemistry to
entire revision by the physiologists, before Anatomy,
they can be applied to the studies of the elements or the
products of the organism. The fine series of researches of
M. Chevreul on fatty bodies are perhaps the only important
chemical study immediately applicable to biology, animal
or even vegetable. In the chemical analysis of blood or
sap, or almost any other element, a single case, taken at
hazard, is usually presented as a satisfactory type, without
any comparative investigation either of each species of
organism in its normal state, or of the degree of develop-
ment of the living being,—its sex, its temperament, its
mode of alimentation, the system of its exterior conditions
of existence, etc., and other modifications which physio-
logists alone can duly estimate. Such analyses correspond
to nothing in anatomy but the single case observed; and
even that is seldom sufficiently characterized. Hence inevit-
able divergences among chemists, who choose different
types, and discussions of no scientific use, as the discor-
dance is attributed to the different analytical methods em-
ployed, instead of to the variations which physiology would
have led them to anticipate. The case is the same with
regard to products first secreted and then excreted, as bile,
saliva, etc., which offer a still greater complication. The
chemists make no inquiry about the parts from which these
products are taken, or the modifications which may have
been occasioned by their remaining some time after their378
POSITIVE PHILOSOPHY.
production, etc. ; and therefore the analyses of these pro-
ducts, however often renewed, are still incoherent and
thoroughly defective. We owe to M. Paspail a full expo-
sure of the practice of the chemists of multiplying organic
principles almost without limit, from differences of character
which imply no distinction of nature, but are merely marks
of various degrees of elaboration of the same principle in
different developments of vegetation ; and even from con-
founding the observed substances with their anatomical
envelope. It is to be regretted that M. Paspail did not
complete his great service to science by founding rationally
the physiological portion of organic chemistry, instead of
vainly attempting to systematize organic chemistry, under
the bias of our crude chemical education,
rp pi ,	If we turn from the anatomical order of
o lysio . qUes^.*ons jqie physiological, we shall find
yet stronger evidences of the inaptitude of chemists for
biological inquiries. All endeavours have yet failed to
establish any point of general doctrine in biology ; and we
find ourselves merely with simple materials, which must be
newly elaborated by physiologists, under the view of
vitality, before they can be put to use. To give an example
or two:—the experiments of Priestley, Sennebier, Saussure,
and others, on the mutual chemical action of vegetables
and atmospheric air, were of the highest value, as insti-
tuting positive knowledge of the vegetable economy ; but
the inquiry is by no means so simple as its founders natu-
rally supposed, after having analysed one separate portion
of the phenomenon of vegetation. The absorption of car-
bonic acid, and the exhalation of oxygen, though very
important in relation to the action of leaves, are only one
aspect of the double vital motion, and cannot be understood
but in the physiological view of both. This action cannot
explain the elementary composition of vegetable substances,
or determine the kind of alteration sustained by the air
through vegetation, because it is, in other ways, partially
compensated by the precisely inverse action produced by
the germination of seeds, the ripening of fruits, etc., and
even, as regards the leaves, by the mere passages from
light to darkness. It is much to have indicated the true
nature of the requisite research, and to have supplied someRELATION TO PHYSIOLOGY.
379
materials for it. The rest is the business of the the physio-
logists. If we turn to animal physiology for examples, the
case is yet more striking.
After all the inquiry that has been made into the chemi-
cal phenomena of respiration, no fixed point is yet estab-
lished. It was long supposed that the absorption by the
lungs of atmospheric oxygen, and its transformation into
carbonic acid, would explain the great phenomenon of the
conversion of venous into arterial blood. But the problem
is much more complicated than was supposed by the
chemists who established this essential part of the phe-
nomenon, and whose labours present the most contra-
dictory conclusions in regard to the facts under their
notice. We do not know, for instance, whether the quantity
of carbonic acid formed really corresponds to the quantity
of oxygen absorbed ; and even the simple general difference
between the inhaled and exhaled air, which is the first
point to be ascertained, is far from being positively estab-
lished. The atmospheric azote appears to some to be in-
creased after respiration, while others say it is diminished,
and others again that it remains the same. The disagree-
ments about the changes in the composition of the blood
are yet more marked. Perhaps the inaptitude of chemists
and physicists for physiological researches is more striking
still in the case of animal heat. In the early days of
modern chemistry, this phenomenon seemed to be suffi-
ciently accounted for by the disengagement of heat corre-
sponding to the decarbonizing of the blood in the lungs,
which the chemists regarded as the focus of a real combus-
tion ; but this explanation was soon found to be inadequate,
even in a normal condition; and much more in various
pathological cases. Uncertain as we still are as to the
pulmonary influence in the process, we know that all
the vital functions must concur in it, in a greater or
smaller degree. There is even some reason to suppose, in
direct opposition to the opinion of the chemists, that
respiration, far from aiding to produce animal heat, con-
stantly tends to diminish it. ~No doubt, the chemical effects
occasioned by vital action must always be taken into the
account in the study of animal heat; but it is only the
physiologists who can deal with them in the light of the380
POSITIVE PHILOSOPHY.
whole subject. When we have learned to combine the
chemical and the physiological view, we may proceed to
the formation of positive doctrine, without having to deal
with preliminary obstacles ; as, in regard to such questions
as the general agreement between the chemical composition
of living bodies, and that of the whole of their aliments;
a case which constitutes one of the principal aspects of the
vital state.
It is evident, at the outset, that every living body, what-
ever its origin, must be, in the long run, composed of the
different chemical elements concerned in the substances,
solid, liquid, and gaseous, by which it is habitually
nourished; since, on the one hand, the vital motion sub-
jects its parts to a continual renovation, and, on the other,
we cannot without absurdity suppose it capable of spon-
taneously producing any real element. This consideration
is so far trom involving any difficulty, that it might lead
us to divine the general nature of the principal elements
of living bodies ; for animals feed in the first place on
vegetables, or on other animals which have eaten vege-
tables ; and in the second place, on air and water, which
are the basis of the nutrition of plants: and thus, the
organic world evidently admits of those chemical elements
only which are furnished by the decomposition of air and
water. When these two fluids have been duly analysed,
physiologists can, in a manner, forsee that animal and
vegetable substances must be composed of oxygen, hydro-
gen, azote, and carbon, as chemistry taught them. It is
true, such a prevision must be very imperfect, as it indicates
nothing of the difference between animal and vegetable
substances, nor why the latter usually contain so much
carbon and so little azote. But this first glimpse, though
it suggests some of the difficulty of the problem, yet indi-
cates the possibility of establishing such a general harmony.
But, when we proceed with the comparison, we encounter
important objections, which are at present insoluble. The
chief difficulty is that azote appears to be as abundant in
the tissues of herbivorous as of carnivorous animals, though
the solid aliments of the former contain scarcely any azote.
The opinions of chemists, as of Berzelius and Raspail, as
to the nature of azote, do not solve the difficulty, as theyDIVISIONS OF ORGANIC CHEMISTRY.
381
cast no light upon its origin. This is one of a multitude
of cases in which we cannot at all explain the chemical
composition of anatomical elements by that of the exterior
substances from which they are unquestionably derived.
Another striking case is that of the constant presence of
carbonate, and, yet more, phosphate of lime in the bonv
tissue, though the nature of the aggregate of aliments
appears to afford no room for the formation of those salts.
This system of investigations, considered in its whole
range, constitutes one of the most important general ques-
tions that can arise from the chemical study of life; and.
if it is at present hardly initiated, the backwardness is
owing, not only to its eminent difficulty, but to the bio-
logists having abandoned to the chemists a task which,
under a wise organization of labour, would have belonged
to themselves alone.
In order to effect a rational division of Partition of
organic chemistry, and to assign its portions Organic
to chemistry on the one hand and physiology Chemistry,
on the other, we must take our stand on the separation
between the state of life and that of death ; or, what comes
to nearly the same thing, between the stability and in-
stability of the proposed combinations, submitted to the
influence of common agents. Among the compounds in-
discriminately called organic, some owe their existence only
to vital motion; they exhibit perpetual variations, and
usually constitute mere mixtures : and these cannot belong
to chemistry, but must enter into the domain of biology,—
statical or dynamical, according as we study their fixed
condition, or the vital succession of their regular changes.
Such, for instance, are blood, lymph, fat, etc. The others,
which form the more immediate principles of these, are
substances essentially dead, admitting of a remarkable
permanence, and presenting all the characters of true com-
binations, independent of life: their natural pla.ce is
evidently, in the general system of chemical science, among
the substances of inorganic origin, from which they differ
in no important respect. Of these, the organic acids,
alcohol, albumen, etc., are examples. These are the sub-
stances which truly belong to organic chemistry; and no
reason exists for their separation from analogous inorganic382
POSITIVE PHILOSOPHY.
substances, even if no injury was done by sucb an arbitrary
division ; and there is more reason for giving the title of
organic to them than to the theory of oxygen, hydrogen,
carbon, and azote, or to the study of many other substances,
acid, alkaline, saline, etc., without which chemical anatomy
and physiology would be unintelligible. As for chemical
phenomena truly common to all compounds of this class,
in consequence of the necessary identity of their chief
elements, it is certainly important to assign to them their
precise relations. The most important of these phenomena
relate, at present, to the interesting and very imperfect
theory of the different kinds of fermentation. But the
consideration of these properties does not constitute a
different order from that which results from the same
ground in the case of many other compounds, purely in-
organic. The property of fermentation, however important,
has not a higher scientific value than that of burning, and
has no more right to an exclusive classification. It is
admitted that too much was attributed to combustion
formerly, in regard to inorganic substances; and we may
be attributing too much now to fermentation, or any other
common property, among so-called organic bodies. We
cannot yet assign their proper place to these compounds in
the rational system of chemical science; but we are able
to affirm that, in that system, they must be more or less
separated from each other, and interposed among sub-
stances called inorganic. Nothing more than this is needed
to settle the question of the maintenance or the suppression
of organic chemistry as a distinct body of doctrine. In
applying the principle which I have proposed, to ascer-
tain to which science any question belongs, it is enough to
inquire whether chemical knowledge will serve the pur-
poses of the research, or whether any biological considera-
tions enter into it. The proposing such an alternative is,
in fact, making the classification.
It is not our business to treat of any special application
of this principle; but it is desirable to point out that in
this partition of organic chemistry, its two portions are
very unequally divided—the study of vegetable substances
contributing most to chemistry, and that of animal sub-
stances to biology. At the first glance, we might supposeAPPLICATION OF DUALISM TO ORGANIC COMPOUNDS. 383
the difference to be the other way; for the importance of
chemical considerations is really much greater with regard
to living vegetables than animals, whose chemical functions
are, except among the lowest orders of the zoological
hierarchy, subordinated to a superior order of new vital
actions. Yet, in virtue of the higher degree of vital elabo-
ration that matter undergoes in the animal than in the
vegetable organism, the chemical part of animal physio-
logy presents a much greater extent and complexity than
the vegetable, in which, for instance, the whole series of
the phenomena of digestion is absent, and in which assimi-
lation and secretion are much simplified. But, on account
of the superior elaboration, and of the greater number of
elements, animal substances are much less stable than
vegetable: they rarely remain separate from the organism;
and, at the same time, the new immediate principles proper
to them are so few that their very existence has been ques-
tioned. Vegetation is evidently the chief source of true
organic compounds, which are derived thence by the
animal organism, and modified by it, either through their
mutual combinations or new external influences. Thus,
the true domain of chemical science must necessarily be
more extended by the study of vegetable than by that of
animal substances.
Enough has been said about the necessity of subjecting
organic compounds to the law of dualism ; but there is a
particular aspect, under which the importance of this
conception in improving chemical theories is worth a brief
notice.
In considering substances as ternary or Application of
quaternary, their multiplicity is accounted dualism to
for only by the difference in the proportions organic
of their constituent elements,—their com- comPoun s*
ponent principles being identical. Very great differences
are sometimes explained by inequalities of proportion so
small as to shock the spirit of chemical analysis: and in
other cases, the proportions being the same, the differences
remain unaccounted for;—as, for instance, in the cases of
sugar and gum, in which we find the same elements, com-
bined in the same proportions. If we extend dualism to
organic compounds, this class of anomalies disappears ;384
POSITIVE PHILOSOPHY.
for the distinction between immediate and elementary
analysis enables us to resolve by dualism, in the most
natural manner, the general paradox of the real diversity
of two substances composed of the same elements, in the
same proportions. In fact, these substances, identical in
their elementary, would differ in the immediate analysis,
as we may understand from what was offered in my
chapter on the law of definite proportions. In another
connection, chemists have remarked the possibility of
exactly representing the numerical composition of alcohol
or ether, etc., according to several binary formulas, radi-
cally distinct from each other, and yet finally equivalent
with regard to the elementary analysis. Now, if such
fictitious combinations should ever be realized, they would
produce highly distinct substances, which might differ
much in the aggregate of their chemical properties, and
yet coincide by their elementary composition. It is only
necessary to transfer the same spirit into the study of
organic combinations, by the establishment of a universal
dualism., to dissipate all these anomalies: and the re-
source may thus be happily prepared, before the cases of
isomerism (as Berzelius calls this fact) have become very
numerous.
We have now seen how heterogeneous is
tl^chapter	body °f doctrine included under the
name of organic chemistry ; how it should
be divided; what is the duty of physiologists with regard
to their share of it; and how the extension of dualism will
establish a natural agreement between the composition of
substances and their collective characters.
With regard to Chemistry at large, I have
theBoolf ° pointed out the true spirit of the science,
under a philosophical view of its present
aspects, and of the indispensable conditions of its advance-
ment. We do not want new materials so much as the
rational disposition of the details which already abound:
and I have offered two prominent ideas, in my survey of
chemical philosophy; the fusion of all genuinely chemical
studies into one body of homogeneous doctrine; and the
reduction of all combinations to the indispensable concep-
tion of a dualism always optional. These two conditions,SUMMARY.
385
distinct but connected, have been presented as necessary
to the definitive constitution of chemical science. The
application of such a conception to the only part of
chemical research which yet exhibits anything of a posi-
tive rationality has removed all doubt about its general
fitness, by showing its spontaneous aptitude to resolve the
anomalies of numerical chemistry.
With this division closes our survey of the whole of
natural philosophy that relates to universal or inorganic
phenomena. In the order of phenomena to which we next
proceed, there is at once much more complexity, and much
less established order. The study of them is scarcely yet
organized; and yet, out of the speciality of the pheno-
mena arises the most indispensable part of natural philo-
sophy,—that of which Man is first the chief object, and
then Society.
i.
CHISWICK PRESS CHARLES WHITTINGHAM AND CO.
TOOKS COURT, CHANCERY LANE, LONDON.
C CALPHABETICAL LIST
OF
BOHN’S LIBRARIES.
November. 1895.* I may say in regard to all manner of books, Bohn’s Publication Series is the
usefullest thing I know.’—Thomas Carlyle.
‘ The respectable and sometimes excellent translations of Bohn’s Library have
done for literature what railroads have done for internal intercourse.’—Emerson.
‘ An important body of cheap literature, for which every living worker in this
country who draws strength from the past has reason to be grateful.’
Professor Henry Morley.
BOHN'S LIBRARIES.
STANDARD LIBRARY .
HISTORICAL LIBRARY
PHILOSOPHICAL LIBRARY
ECCLESIASTICAL LIBRARY
ANTIQUARIAN LIBRARY .
ILLUSTRATED LIBRARY .
SPORTS AND GAMES.
CLASSICAL LIBRARY .
COLLEGIATE SERIES.
SCIENTIFIC LIBRARY.
ECONOMICS AND FINANCE
REFERENCE LIBRARY
NOVELISTS’ LIBRARY
ARTISTS’ LIBRARY .
CHEAP SERIES .
SELECT LIBRARY OF STANDARD
343 Volumes.
23 Volumes.
17 Volumes.
15	Volumes.
36 Volumes.
75 Volumes.
16	Volumes.
108 Volumes.
10 Volumes.
46 Volumes.
5 Volumes.
30 Volumes.
13 Volumes.
10 Volumes.
55 Volumes.
WORKS 31 Volumes.
' Messrs. Bell are determined to do more than maintain the reputation of
11 Bohn’s Libraries.’”—Guardian.
' The imprint of Bohn’s Standard Library is a guaranty of good editing. ’
Critic (N.Y.)
*	This new and attractive form in which the volumes of Bohn’s Standard
Library are being issued is not meant to hide either indifference in the selection of
books included in this well-known series, or carelessness in the editing.’
St. James's Gazette.
*	Messrs. Bell & Sons are making constant additions of an eminently acceptable
character to “Bohn's Libraries.”'—Athenaum.ALPHABETICAL LIST OF BOOKS
CONTAINED IN
BOHN'S LIBRARIES.
748 Vols., Small Post 8vo. cloth. Price £160.
Complete Detailed Catalogue will be sent on application.
Addison's Works. 6 vols. 3*. 6d.
each.
Aeschylus. Verse Trans, by Anna
Swanwick. 5J.
---- Prose Trans, by T. A. Buckley.
3s. 6d.
Agassiz & Gould’s Comparative Phy-
siology $s.
Alfleri’s Tragedies. Trans, by Bowring.
2 vols. 35. 6d. each.
Alford’S Queen’s English, is. & is. 6d.
Allen’S Battles of the British Navy.
2 vols. 5J. each.
Ammlanus Marcellinus. Trans, by
C. D. Yonge. 7s. 6d.
Andersen’s Danish Tales. Trans, by
Caroline Peachey. 5*.
Antoninus (Marcus Aurelius). Trans.
by George Long. 3s. 6d.
Apollonius Rhodius. The Argonautica.
Trans, by E. P. Coleridge. $s.
Apuleius, The Works of. $s.
Ariosto’s Orlando Furioso. Trans, by
W. S. Rose. 2 vols. 5s. each.
Aristophanes. Trans, by W. J. Hickie.
2 vols. 5-r. each.
Aristotle’s Works. 5 vols, 5s. each;
2 vols, 3s. 6d. each.
Arrian. Trans, by E. J. Chinnock. 5s.
Ascham’s Scholemaster. (J. E. B.
Mayor.) is.
Bacon’s Essays and Historical Works,
3s. 6d.; Essays, is. and is. 6d.;
Novum Organum, and Advancement
of Learning, 5s.
Ballads and Songs of the Peasantry.
By Robert Bell. 3s. 6d.
Bass’s Lexicon to the Greek Test. as.
'■ Bax’s Manual of the History of Philo-
j sophy. 5 s.
Beaumont & Fletcher. Leigh Hunt s
Selections. 3L 6d.
Bechstein’s Cage and Chamber Birds.
5J-
Beckmann’s History of Inventions.
2 vols. 3s. 6d. each.
Bede’s Ecclesiastical History and the
A. S. Chronicle. 5J.
Bell (Sir C.) On the Hand. 5*.
---- Anatomy of Expression. 5*.
Bentley’s Phalaris. 5s.
Bjomson’s Arne and the Fisher Lassie.
Trans, by W. H. Low. 3s. 6d.
Blair’s Chronological Tables. ioj.
Index of Dates. 2 vols. 5*. each.
Bleek’s Introduction to the Old Testa-
ment. 2 vols. 5s. each.
Boethius’ Consolation of Philosophy,
&c. 5-r.
Bohn’s Dictionary of Poetical Quota-
tions. 6s.
Bond’s Handy-book for Verifying
Dates, &c. 5s.
Bonomi’s Nineveh. 5j.
Boswell’s Life of Johnson. (Napier),
6 vols. 3s. 6d. each.
---- (Croker.) 5 vols. 2or.
Brand’s Popular Antiquities. 3 vols.
51. each.
Bremer’s Works. Trans, by Mary
Howitt. 4 vols. 3J. 6d. each.
Bridgewater Treatises. 9 vols. Various
prices.
Brink (B. Ten). Early English Litera-
ture. 2 vols. 3s. 6d. each.
----Five Lectures on Shakespeare 3s.6d>4
ALPHABETICAL LIST OF
Browne’s (Sir Thomas) Works. 3 vols.
y. bd. each.
Buchanan’s Dictionary of Scientific
'I enns. 6j.
Buckland’s Geology and Mineralogy.
2 vols. 15J.
Burke’s Works and Speeches. 8 vols.
3*. 6d. each. The Sublime and
Beautiful, is. & is. 6d. Reflections on
the French Revolution, ij.
---- Life, by Sir James Prior. 3*. 6d.
Burney’s Evelina. 3J. 6d. Cecilia
2 vols. y. 6d. each.
Burns’ Life by Lockhart. Revised by
W. Scott Douglas, y. 6d.
Burn’s Ancient Rome. 7s. 6d.
Butler’s Analogy of Religion, and
Sermons, y. 6d.
Butler’s Hudibras. y.; or 2 vols.,
y. each.
Caesar. Trans, by W. A. M’Devitte. 5*.
Camoens’ Lusiad. Mickle’s Transla-
tion, revised. 3J. 6d.
Caraias (The) of Maddaloni. By
Allred de Reumont. 3J. 6d.
Carpenter’s Mechanical Philosophy sj.
Vegetable Physiology. 6s, Animal
Physiology. 6s.
Carrel’s Counter Revolution under
Charles II. and James II. y. 6d.
Cattermole’s Evenings at Haddon
lialL y.
Catullus and Tibullus. Trans, by
W. K. Kelly. 5s.
Cellini's Memoirs. (Roscoe.) y. 6d.
Cervantes’ Exemplary Novels. Trans.
by W. K. Kelly. 3.,. 6d.
— Don Quixote. Motteux’s Trans.
revised. 2 vols. 3*. 6d. each.
—— Galatea. Trans, by G. W. J.
Gyll. y. 6d.
Chalmers On Man. y.
Chaim ing’s The Perfect Life. is. and
IJ. 6d.
Chaucer’s Works. Bell’s Edition, re-
vised by Skeat. 4 vols. y. 6d. ea.
Chess Congress of 1862 By J.
Luwenthal. y.
Chevreul on Colour, y. and 7s. 6d.
Chillingworth’s The Religion of Pro-
testants. y. 6d.
China: Pictorial, Descriptive, and
Historical, y,
Chronicles of the Crusades. 55.
Cicero's Works. 7 vols. 55. each.
1	vol., 3s. 6d.
----- Friendship and Old Age. is, and
is. 6d.
Clark's Heraldry. (Planch6.) y. and
iSJ-
Classic Tales. 3J. 6d.
Coleridge’s Prose Works. [Ashe.)
6 vols.	y. 6d. each.
Comte’s Philosophy of the Sciences.
(G. H. Lewe6.) 5s.
-----Positive Philosophy. 3 vols. 5*.
each.
Condi’s History of the Arabs in Spain.
3 vols. y. 6ci. each.
Cooper’s Biographical Dictionary.
2	vols. 5J. each.
Cowper’s Works. (Southey.) 8 vols.
3i. 6d. each.
Coxe’s House of Austria. 4 vols. y. 6d.
each. Memoirs of Marlborough.
3	vols. 31. 6d. each. Atlas to
Marlborough’s Campaigns, ioj. 6d.
Craik’s Pursuit of Knowledge, y.
Craven’s Young Sportsman’s Manual.
5J*
Crulkshank’s Punch and Judy, y.
Three Course and a Dessert, y.
Cunningham’s I ,i ves of British Painters.
3 vols. y. 6d. each.
Dante. Trans, by Rev. H. F. Cary.
3-r. 6d. Inferno. Separate, is. and'
is. 6d. Purgatorio. is. and is. 6d.
Paradiso. is. and is. 6d.
-----Trans, by I. C. Wright (Flax-
man’s Illustrations.) 5*.
----- Inferno. Italian Text and Trans.
by Dr. Carlyle. 5*.
----- Purgatorio. Italian Text and
Trans, by W. S. Dugdale. y.
De Commines’ Memoirs. Tranis. by
A. R. Scoble. 2 vols. y. 6d. each.
Defoe’s Novels and Miscel. Works.
6 vols. y. 6d. each. Robinson1
Crusoe (Vol. VII). 3s. 6d. or 5*.
The Plague in London, is. and
IS. 6d.
Delolme on the Constitution of Eng-
land. y. 6 d.
Demmins’ Arms and Armour. Trans*
by C. C. Blacx. 7;. 6d.BOfflTS LIBRARIES.
5
Demosthenes’ Orations. Trans, by
C. Rann Kennedy. 4 vols. 5J., and
1 vol. 3-f. 6d.
-----Orations On the Crown, ij. and
is. 6d.
De Stael’s Corinne. Trans, by Emily
Baldwin and Paulina Driver. 31. 6d.
Devey’s Logic. 51.
Dictionary of Greek and Latin Quota-
tions. 55.
—	of Poetical Quotations (Bohn). 6s.
——of Scientific Terms. (Buchanan.) 6s.
—— of Biography. (Cooper.) 2 vols.
5-f. each.
—	of Noted Names of Fiction.
(Wheeler.) 5 s.
—	of Obsolete and Provincial Eng-
lish (Wright.) 2 vols. 5J. each.
Didron’s Christian Iconography. 2 vols.
5-f. each.
Diogenes Laertius. Trans, by C. D.
Yonge. 5s.
Dobree’s Adversaria. (Wagner). 2 vols.
5-y. each.
Dodd’s Epigrammatists. 6j.
Donaldson’s Theatre of the Greeks. 5s.
Draper’s History of the Intellectual
Development of Europe. 2 vols. 5J.
each.
Dunlop’s History of Fiction. 2 vols.
5s. each.
Dyer's History of Pompeii. 7s. 6d.
----- The City of Rome. 55.
Dyer’s British Popular Customs. 5s.
Early Travels in Palestine. (Wright.) 5J.
Eaton's Waterloo Days. ij. and ij. 6d.
Eller’s Egyptian Princess. Trans, by
E. S. Buchheim. 3J. 6d.
Edgeworth’s Stories for Children.
3J. 6d.
Ellis’ Specimens of Early English Me-
trical Romances. (Halliwell.) 5J.
Elze’S Life of Shakespeare. Trans, by
L. Dora Schmitz. 5J.
Emerson’s WTorks. 3 vols. 3J. 6d. each,
or 5 vols. ij. each.
Ennemoser’s History of Magic. 2 vols.
5J. each.
Epictetus. Trans, by George Long. 5J.
Euripides. Trans, by E. P. Coleridge.
2 vols. 5J. each.
Eusebius’ Eccl. History. Trans, by
C. F. Cruse, 5J.
Evelyn’s Diary and Correspondence.
(Bray.) 4 vols. 5J. each.
Fairholt's Costume in England.
(Dillon.) 2 vols. 51. each.
Fielding’s Joseph Andrews. 3J. 6d.
Tom Jones. 2 vols. 3J. 6d. each.
Amelia. 5J.
Flaxman’s Lectures on Sculpture. 6j.
Florence of Worcester’s Chronicle.
Trans, by T. Forester. 55.
Foster’s Works. 10 vols. 3J. 6d. each.
Franklin’s Autobiography, is.
Gesta Romanorum. Trans, by Swan
& Hooper. 5J.
Gibbon’s Decline and Fall. 7 vols.
3J. 6d. each.
Gilbart’s Banking. 2 vols. 55. each.
Gil Bias. Trans, by Smollett. 6j.
Giraldus Cambrensis. 5J.
Goethe’s Works and Correspondence,
including Autobiography and Annals,
Faust, Elective affinities, Werther,
Wilhelm Meister, Poems and Ballads,
Dramas, Reinecke Fox, Tour in Italy
and Miscellaneous Travels, Early and
Miscellaneous Letters, Correspon-
dence with Eckermann and Soret,
Zelter and Schiller, &c. &c. By
various translators. 16 vols, 3J. 6d.
each.
---- Faust. Text with Hayward’s
Translation. (Buchheim.) 5J.
---- Faust. Part I. Trans, by Anna
Swanwick. ij. and ij. 6d.
---- Boyhood. (Part I. of the Auto-
biography.) Trans, by J. Oxenford.
ij. and ij. 6d.
---- Reinecke Fox. Trans, by A.
Rogers, ij. and ij. 6d.
Goldsmith’s Works. (Gibbs.) 5 vols.
3J. 6d. each.
---- Plays, ij. and ij. 6d. Vicar of
Wakefield, ij. and ij. 6d.
Grammont’s Memoirs and Boscobel
Tracts. 5J.
Gray's Letters. (D. C. Tovey.)
[In the press,
Greek Anthology. Trans, by E. Burges.
5j.
Greek Romances. (Theagenes and
Chariclea, Daphnis and Chloe, Cli-
topho and Leucippe.) Trans, by Rev,
R. Smith. 5J.6
ALPHABETICAL LIST OF
Greek Testament. 5?.
Greene, Marlowe, and Ben Jonson’s
Poems. (Robert Bell.) 3s. 6d.
Gregory’s Evidences of the Christian
Religion. 3s. 6d.
Grimm’s Gammer Grethel. Trans, by
E. Taylor. 3s. 6d.
---- German Tales. Trans, by Mrs.
Hunt. 2 vols. 3s. 6d. each.
Grossi’s Marco Visconti. 3^. 6d.
Guizot’s Origin of Representative
Government in Europe. Trans, by
A. R. Scoble. 3s. 6d.
---- The English Revolution of 1640.
Trans, by W. Hazlitt. 35. 6d.
---- History of Civilisation. Trans, by
W. Hazlitt. 3 vols. 3J. 6d. each.
Hall (Robert). Miscellaneous Works.
3s. 6d.
Handbooks of Athletic Sports. 8 vols.
3s. 6d. each.
Handbook of Card and Table Games.
2 vols. 3-r. 6d. each.
---- of Proverbs. By H. G. Bohn. $s.
---- of Foreign Proverbs. 5J.
Hardwick’s History of the Thirty-nine
Articles. 5^.
Harvey’s Circulation of the Blood.
(Bowie.) is. and is. 6d.
Hauff’s Tales. Trans, by S. Mendel.
3J. 6d.
---- The Caravan and Sheik of Alex-
andria. is. and is. 6d.
Hawthorne’s Novels and Tales. 4 vols.
3s. 6d. each.
Hazlitt’s Lectures and Essays. 7 vols.
3s. 6d. each.
Heaton’s History of Painting. (Cosmo
Monkhouse.) 51.
Hegel's Philosophy of History. Trans,
by J. Sibree. 5*.
Heine’s Poems. Trans, by E. A. Bow-
ring. 3J. 6d.
----Travel Pictures. Trans, by Francis
Storr. 3s. 6d.
Helps (Sir Arthur). Life of Thomas
Brassey. is. and is. 6d.
Henderson’s Historical Documents of
the Middle Ages. 5*.
Henfrey’s English Coins. (Keary.) 6s.
Henry (Matthew) On the Psalms. $s.
Henry of Huntingdon’s History. Trans,
by T. Forester. 5s.
Herodotus. Trans, by H. F. Cary.
3s- 6d.
---- Wheeler’s Analysis and Summary
of. 5J-. Turner’s Notes on. $s.
Hesiod, Callimachus and Theognis.
Trans, by Rev. J. Banks. 5^.
Hoffmann’s Tales. The Serapion
Brethren. Trans, by Lieut.-Colonel
Ewing. 2 vols. 3s. 6d.
Hogg’s Experimental and Natural
Philosophy. 5^.
Holbein’s Dance of Death and Bible
Cuts. 5j.
Homer. Trans, by T. A. Buckley. 2
vols. 5-f. each.
---- Pope’s Translation. With Flax-
man’s Illustrations. 2 vols. 55. each.
---- Cowper’s Translation. 2 vols.
3s. 6d. each.
Hooper’s Waterloo. 3s. 6d.
Horace. Smart’s Translation, revised,
by Buckley. 3s. 6d.
Hugo’s Dramatic Works. Trans, by
Mrs. Crosland and F. L. Slous. 3s. 6d.
---- Hemani. Trans, by Mrs. Cros-
land. is.
---- Poems. Trans, by various writers.
Collected by J. H. L. Williams. 3s. 6d.
Humboldt’s Cosmos. Trans, by Ott6,
Paul, and Dallas. 4 vols. 3s. 6d. each,
and 1 vol. 5s.
---- Personal Narrative of his Travels.
Trans, by T. Ross. 3 vols. 5s. each.
■--- Views of Nature. Trans, by Ott6
and Bohn. 5s.
Humphreys’ Coin Collector’s Mauual.
2 vols. 5s. each.
Hungary, History of. 3s. 6d.
Hunt’s Poetry of Science. $s.
Hutchinson’s Memoirs. 3*. 6d.
India before the Sepoy Mutiny. 5s.
Ingulph’s Chronicles. 5j.
Irving (Washington). Complete
Works. 15 vols. 3s. 6d. each ; or
in 18 vols. is. each, and 2 vols. is. 6d.
each.
---- Life and Letters. By Pierre E.
Irving. 2 vols. 3J. 6d. each.
Isocrates. Trans, by J. H. Freese.
Vol. I. ss.
James’ Life of Richard Coeur de Lion.
2 vols. 3J. 6d. each.
---- Life and Times of Louis XIV,
2 vols. 3J\ 6d. each.BOHN'S LIBRARIES.
7
Jameson (Mrs.) Shakespeare’s Hero- |
ines. 3-f. 6d.	I
Jesse(E.) Anecdotes of Dogs. 55. j
Jesse (J. H.) Memoirs of the Court of j
England under the Stuarts. 3 vols.
5s. each.
---- Memoirs of the Pretenders. $s.
Johnson’s Lives of the Poets. (Napier).
3	vols. 3s. 6d. each.
Josephus. Whiston’s Translation, re-
vised by Rev. A. R. Shilleto. 5 vols.
3-r. 6d. each.
Joyce’s Scientific Dialogues. 5J.
Jukes-Browne’s Handbook of Physical
Geology. 7s. 6d. Handbook of His-
torical Geology. 6j. The Building
of the British Isles. 75. 6d.
Julian the Emperor. Trans by Rev.
C. W. King. 5*.
Junius’s Letters. Woodfali s Edition,
revised. 2 vols. 35. 6d. each.
Justin, Cornelius Nepos, and Eutropius.
Trans, by Rev. J. S. Watson. 55.
Juvenal, Persius, Sulpicia, and Lu- ;
cilius. Trans, by L. Evans. 5J.
Kant’S Critique of Pure Reason. Trans,
byj. M. D. Meiklejohn. 55.
---- Prolegomena, &c. Trans, by E.
Belfort Bax. 3s.
Keightley’s Fairy Mythology. 5s. i
Classical Mythology. Revised by Dr. j
L. Schmitz. 5s.
Kidd On Man. 31. 6d.
Kirby On Animals. 2 vols. each.
Knight’s Knowledge is Power. 55.
La Fontaine’s Fables. Trans, by E. ;
Wright. 3s. 6d.
Lamartine’s History of the Girondists.
Trans, by H. T. Ryde. 3 vols. 3s. 6d. J
each.
— Restoration of the Monarchy in j
France. Trans, by Capt. Rafter, j
4	vols. 3s. 6d. each.	:
---- French Revolution of 1848. 35.6d.
Lamb’s Essays of Elia and Eliana.
3s. 6d.f or in 3 vols. is, each.
----Memorials and Letters. Talfourd’s
Edition, revised by W. C. Hazlitt.
2 vols. 3s. 6d. each.
----Specimens of the English Dramatic
Poets of the Time of Elizabeth. 3s. 6d.
Lanzi’s History of Painting in Italy,
Trans, by T. Roscoe. 3 vols. 3s. 6d.
each.
Lappenberg’s England under the
Anglo-Saxon Kings. Trans, by B.
Thorpe. 2 vols. 3J. 6d. each.
Lectures on Painting. By Barry, Opie
and Fuseli. 5^
I Leonardo da Vinci’s Treatise on Paint-
I ing. Trans, by J. F. Rigaud. 5s.
Lepsius’ Letters from Egypt, &c. Trans.
by L. and J. B. Horner. 5s.
Lessing’s Dramatic Works. Trans, by
Ernest Bell. 2 vols. 3s. 6d. each.
Nathan the Wise and Minna von
Barnhelm. is, and is, 6d. Laokoon,
Dramatic Notes, &c. Trans, by E. C.
Beasley and Helen Zimmern. 3s. 6d.
Laokoon separate, is, or is. 6d.
Lilly’s Introduction to Astrology.
(Zadkiel.) $s.
Livy. Trans, by Dr. Spillan and others.
4 vols. 5J. each.
Locke’s Philosophical Works. (J. A.
St. John). 2 vols. 3s. 6d, each.
---- Life. By Lord King. 3J. 6d.
Lodge’s Portraits. 8 vols. 55. each.
Longfellow’s Poetical and Prose Works.
2 vols. 5s. each.
Loudon’s Natural History. 5s.
Lowndes’ Bibliographer’s Manual. 6
vols. 5s. each.
Lucan’s Pharsalia. Trans, by H. T.
Riley. $s.
Lucian’s Dialogues. Trans, by H.
Williams. 5J.
Lucretius. Trans, by Rev. J. S.
Watson. 5 s.
Luther’s Table Talk. Trans, by W.
Hazlitt. 3J. 6d.
---- Autobiography. (Michelet).
Trans, by W. Hazlitt. 3s. 6d.
Machiavelli’s History of Florence, &c.
Trans. 3s. 6d.
Mallet’s Northern Antiquities. 5^.
Mantell’s Geological Excursions
through the Isle of Wight, &c. 5s.
Petrifactions and their Teachings.
6s. Wonders of Geology. 2 vols.
7s. 6d. each.
Manzoni’s The Betrothed. 5J.
Marco Polo’s Travels. Marsde i’s Edi-
tion, revised by T. Wright. 5J.8
ALPHABETICAL LIST OP
Martial's Epigrams. Trans. 7s. 6d.
Mar tine au’s History of England,
1800-15. y. 6d,
—	History of the Peace, 1816-46.
4	vols. y. 6d. each.
Matthew Paris. Trans, by Dr. Giles.
3 vols. 5-r. each.
Matthew of Westminster. Trans, by
C. D. Yonge. 2 vols. y. each.
Maxwell's Victories of Wellington, y.
Menzel's History of Germany. Trans.
by Mrs. Horrocks. 3 vols. y. 6d. ea.
Michael Angelo and Raffaelle. By
Duppa and Q. de Quincy. 5J.
Michelet’s French Revolution. Trans.
by C. Cocks. 3J. 6d.
Mignet's French Revolution, y. 6d.
Mill (John Stuart). Selected Essays.
[/«the press.
Miller's Philosophy of History. 4 vols.
y. 6d. each.
Milton’s Poetical Works. (J. Mont-
gomery.) 2 vols. y. 6d. each.
----- Prose Works. (J. A. St. John.)
5	vols. y. 6d. each.
Mitford’s Our Village. 2 vols. y. 6d.
each.
Moli^re’s Dramatic Works. Trans, by
C. H. Wall. 3 vols. 3s. 6d. each.
—	The Miser, Tartuffe, The Shop-
keeper turned Gentleman, is. & is. 6d.
Montagu’s (Lady M. W.) Letters
and Works. (Wharncliffe and Moy
Thomas.) 2 vols. 5^. each.
Montaigne’s Essays. Cotton's Trans,
revised by W. C. Hazlitt. 3 vols.
3s. 6d. each.
Montesquieu’s Spirit ol Laws. Nu-
gent’s Trans, revised by J. V.
Prichard. 2 vols. 3J. 6d. each.
Morphy’s Games of Chess. (Lowen-
thal.)	55.
Mot'ey’s Dutch Republic. 3 vols. 3s.6d.
each.
Mudie’s British Birds. (Martin.) 2 vols.
y. each.
Naval and Military Heroes of Great
Britain, 6j.
Neander’s History of the Christian Re-
ligion and Church. 10 vols. Life of
Christ. 1 vol. Planting and Train-
ing of the Church by the Apostles.
2 vols. History of Christian Dogma.
2	vols. Memorials of Christian Life
in the Early and Middle Ages.
16 vols.	3s. 6d. each.
Nicolinfs History of thejesuits. y.
North’s Lives of the Norths. (J essopp.)
3	vols. y. 6d. each.
Nugent’s Memorials of Hampden. 5J.
Ockley’s History of the Saracens, y. 6d.
Ordericus Vitalis. Trans, by T.
Forester. 4 vols. y. each.
Ovid. Trans, by H. T. Riley. 3 vols.
5j. each.
Pascal’s Thoughts. Trans, by C.
Kegan Paul. 3s. 6d.
Pauli’S Life of Alfred the Great, &c.
5*
---- Life of Cromwell, is. and is. 6d.
Pausanlas’ Description of Greece.
Trans, by Rev. A. R. Shilleto. 2 vols.
5.J. each.
Pearson on the Creed. (Walford.) 55.
Pepys’ Diary. (Braybrooke.) 4 vols.
55. each.
Percy's Reliques of Ancient English
Poetry. (Prichard.) 2 vols. y. 6d. ea.
Petrarch’s Sonnets. 5*.
Pettigrew’s Chronicles of the Tombs.
5J*
Philo-Judaeus. Trans, by C. D. Yonge.
4	vols. 5*. each.
Pickering’s Races of Man. 51.
Pindar. Trans, by D. W. Turner. 5J.
Planche’s History of British Costume.
5*
Plato. Trans, by H. Cary, G. Burges,
and H. Davis. 6 vols. y. each.
—— Apology, Crito, Phaedo, Prota-
goras. is. and is. 6d.
---- Day’s Analysis and Index to the
Dialogues. 5-r.
Plautus. Trans, by H. T. Riley.
2 vols. 5*. each.
---- Trinummus, Menaechmi, Aulu-
laria, Captivi. u. and ij. 6d.
Pliny's Natural History, Trans, by
Dr. Bostock and H. T. Riley. 6 vol&
5s. each.
Pliny the Younger, Letters of. Mel-
moth’s trans. revised by Rev. F. C. T,
Bosanquet. y.
Plotinus : Select Works of. 5J.BOHN'S LIBRARIES.
9
Plutarch’s Lives. Trans, by Stewart
and Long. 4 vols. 3s. 6d. each.
---- Moralia. Trans, by Rev. C. W.
King and Rev. A. R. ShiUeto. 2 vols.
5*. each.
Poetry of America. (W. J. Linton.)
3s. 6d.
Political Cyclopaedia. 4 vols. 3s. 6*f.ea.
Polyglot of Foreign Proverbs. 3s.
Pope’s Poetical Works. (Carruthers.)
2 vols. 5J. each.
---- Homer. (J. S. Watson.) 2 vols.
5s, each.
---- Life and Letters. (Carruthers.) 5*.
Pottery and Porcelain. (H. G. Bohn.)
5;. and 10s. 6d.
Propertius. Trans, by Rev. P. J. F.
Gantillon. 3s. 6d.
Prout (Father.) Reliques. 3s.
Quintilian’s Institutes of Oratory.
Trans, by Rev. J. S. Watson. 2 vols.
5j. each.
Racine’s Tragedies. Trans, by R. B.
Boswell. 2 vols. 3s. 6d. each.
Ranke's History of the Popes. Trans.
by E. Foster. 3 vols. 3s. 6d. each.
— Latin and Teutonic Nations.
Trans, by P. A. Ashworth. 3J. 6d.
---- History of Servia. Trans, by
Mrs. Kerr. 3s. 6d.
Rennie’s Insect Architecture. (J. G.
Wood.) 5J.
Reynold’s Discourses and Essays.
(Beechy.) 2 vols. 3s. 6d. each.
Ricardo’S Political Economy. (Gon-
ner.) 5s.
Richter’s Levana. 3*. 6d.
---- Flower Fruit and Thorn Pieces.
Trans, by Lieut.-C ol. Ewing. 3s. 6d.
Roger de Hovenden’s Annals. Trans.
by Dr. Giles. 2 vols. 3s. each.
Roger of Wendover. Trans, by Dr.
Giles. 2 vols. 5J. each.
Roget’S Animal and Vegetable Phy-
siology. 2 vols. 6s. each.
Rome in the Nineteenth Century. (C. A.
Eaton.) 2 vols. 5* each.
Roscoe’s Leo X. 2 vols. 3s. 6d. each.
---- Lorenzo de Medici. 3J. 6d.
Russia, History of. By W. K. Kelly.
2 vols. 3s. 6d. each.
Sallust, Floras,and Velleius Paterculus.
Trans, by Rev. J. S, Watson. 3s.
Schiller’s Works. Including History of
the Thirty Years’ War, Revolt of the
Netherlands, Wallenstein, William
Tell, Don Carlos, Mary Stuart, Maid
ofOrleans, Bride of Messina, Robbers,
Fiesco, Love and Intrigue, Demetrius,
Ghost-Seer, Sport of Divinity, Poems,
Aesthetical and Philosophical Essays,
&c. By various translators. 7 vols.
3s. 6d. each.
---- Mary Stuart and The Maid of
Orleans. Trans, by J. Mellish and
Anna Swanwick. is. and is. 6d.
Schlegel (F.). Lectures and Miscel-
laneous Works. 5 vols. 3s. 6d. each.
---- (A. W.). Lectures on Dramatic
Art and Literature. 3J. 6d.
Schopenhauer’s Essays. Selected and
Trans, by E. Belfort Bax. $s.
---- On the Fourfold Root of the
Principle of Sufficient Reason and
on the Will in Nature. Trans, by
Mdme. Hillebrand. 3s.
Schouw’s Earth, Plants, and Man.
Trans, by A. Henfrey. 5*.
Schumann’s Early Letters. Trans, by
May Herbert. 3s. 6d.
---- Reissmann’s Life of. Trans, by
A. L. Alger. 3s. 6d.
Seneca on Benefits. Trans, by Aubrey
Stewart. 3s. 6d.
---- Minor Essays and On Clemency.
Trans, by Aubrey Stewart. 5?.
Sharpe’s History of Egypt. 2 vols*
5s. each.
Sheridan’s Dramatic Works. 35. 6d.
---- Plays, is. and is. 6d.
Sismondi’s Literature of the South of
Europe. Trans, by T. Roscoe. 2
vols. 3s. 6d. each.
Six Old English Chronicles. 5J.
Smith (Archdeacon). Synonyms and
Antonyms, 5*.
Smith (Adam). Wealth of Nations*
(Belfort Bax.) 2 vols. 3s. 6d. each.
---- Theory of Moral Sentiments.
3s. 6d.
Smith (Pye). Geology and Scripture. 5*
Smollett’s Novels. 4 vols. 3s. 6d.
each.
Smyth’s Lectures on Modern History.
2 vols. 3.*. 6d. each.ALPHABETICAL LIST OP BOHN'S LIBRARIES.
to
Socrates’ Ecclesiastical History. 5s.
Sophocles. Trans, by E. P. Coleridge,
B.A. 55.
SOuthey’s Life of Nelson. 55.
---- Life of Wesley. 55.
---- Life, as told in his Letters. By J.
Dennis. 35. 6d.
Sozomen’s Ecclesiastical History. 55.
Spinoza’s Chief Works. Trans, by
R. H. M. Elwes. 2 vols. 55. each.
Stanley’s Dutch and Flemish Painters,
5s-
Starling’s Noble Deeds of Women. 55.
Staunton’s Chess Players’ Handbook.
55. Chess Praxis. 55. Chess Players’
Companion. 55. Chess Tournament
of 1851.	55.
Stockhardt’s Experimental Chemistry.
(Heaton.) 5 5.
Strabo’s Geography. Trans, by
Falconer and Hamilton. 3 vols.
55. each.
Strickland’s Queens of England. 6
vols. 55. each. Mary Queen of
Scots. 2 vols. 55. each. Tudor
and Stuart Princesses. 55.
Stuart & Revett’s Antiquities of
Athens. 55.
Suetonius’ Lives of the Caesars and of
the Grammarians. Thomson’s trans.
revised by T. Forester. 55.
Bully’s Memoirs. Mrs. Lennox’s
trans. revised. 4 vols. 35. 6d. each.
Tacitus. The Oxford trans. revised.
2 vols. 55. each.
Tales of the Genii. Trans, by Sir.
Charles Morell. 5s.
Tasso’s Jerusalem Delivered. Trans.
by J. H. Wiffen. 5*
Taylor’s Holy Living and Holy Dying.
3s. 6d.
Terence and Phaedrus. Trans, by H. T.
Riley. 55.
Theocritus, Bion, Moschus, and
Tyrtaeus. Trans, by Rev. J. Banks.
5s'
Theodoret and Evagrius. 5/.
Thierry’s Norman Conquest. Trans,
by W. Hazlitt. 2 vols. 35. 6d.
each.
Thucydides. Trans by Rev. H. Dale.
2 vols. 35. 6d. each.
---- Wheeler’s Analysis and Summary
of. 5*.
Trevelyan’s Ladies in Parliament, is.
and is. 6d.
Ulrici’s Shakespeare’s Dramatic Art.
Trans, by L. Dora Schmitz. 2 vols.
3s. 6d. each.
Uncle Tom’s Cabin. 3s. 6d.
Ure’s Cotton Manufacture of Great
Britain. 2 vols. 5s. each.
----Philosophy of Manufacture. 7s. 6d.
Vasari’s Lives of the Painters. Trans,
by Mrs. Foster. 6 vols. 3s. 6d. each.
Virgil Trans, by A. Hamilton Bryce,
LL.D. 3s. 6d.
Voltaire’s Tales. Trans, by R. B.
Boswell. 3s. 6d.
Walton’s Angler. 5s.
---- Lives. (A. H. Bullen.) 55.
Waterloo Days. By C. A. Eaton.
15. and 15. 6d.
Wellington, Life of. By ‘An Old
Soldier.’ 55.
Werner’s Templars in Cyprus. Trans.
| by E. A. M. Lewis. 35. 6d.
Westropp’s Handbook of Archaeology.
5s-
Wheatley. On the Book of Common
Prayer. 35. 6d.
Wheeler’s Dictionary of Noted Names
of Fiction. 55.
White’s Natural History of Selbome.
5s-
Wieseler’s Synopsis of the Gospels.
5*.
William of Malmesbury’s Chronicle.
5J*
Wright’s Dictionary of Obsolete and
Provincial English. 2 vols. 55. each.
Xenophon. Trans, by Rev. J. S. Wat-
son and Rev. H. Dale. 3 vols. 5s. ea.
Young’s Travels in France, 1787-89.
(M. Betham-Edwards.) 35. 6d.
---- Tour in Ireland, 1776-9.	(A. W.
Hutton.) 2 vols. 35. 6d. each.
Yule-Tide Stories (B. Thorpe.) 55.New Editions, fcap. 8vo. 2s. 6d. each net.
THE ALDINE EDITION
OF THE
BRITISH POETS.
‘This excellent edition of the English classics, with their complete texts and
scholarly introductions, are something very different from the cheap volumes of
extracts which are just now so much too common.’—St. James’s Gazette,
* An excellent series. Small, handy, and complete.’—Saturday Review,
Akenside. Edited by Rev. A. Dyce.
Beattie. Edited by Rev. A. Dyce.
•Blake. Edited by W. M. Rossetti.
♦Burns. Edited by G. A. Aitken.
3 vols.
Butler. Edited by R. B. Johnson.
2 vols.
Campbell. Edited by His Son-
in-law, the Rev. A. W. Hill. With
Memoir by W. Allingham.
Chatterton. Edited by the Rev.
W. W. Skeat, M.A. 2 vols.
Chaucer. Edited by Dr. R. Morris,
with Memoir by Sir H. Nicolas. 6 vols.
Churchill. Edited by Jas. Hannay.
2	vols.
•Coleridge. Edited by T. Ashe,
B.A. 2 vols.
Collins. Edited by W. Moy
Thomas.
Dryden. Edited by the Rev. R.
Hooper, M.A. 5 vols.
Goldsmith. Edited by Austin
Dobson.
•Gray. Edited by J. Bradshaw,
LL.D.
Herbert. Edited by the Rev. A. B.
Grosart.
•Herrick. Edited by George
Saintsbury. 2 vols.
•Keats. Edited by the late Lord
Houghton.
Milton. Edited by Dr. Bradshaw.
3	vols.
Parnell. Edited by G. A. Aitken.
Pope. Edited by G. R. Dennis.
With Memoir by John Dennis. 3 vols.
Prior. Edited by R. B. Johnson.
2 vols.
Raleigh and Wotton. With Se-
lections from the Writinsrs of other
COURTLY POETS from 1540 to lb50.
Edited by Yen. Archdeacon Hannah,
D.C.L.
Rogers. Edited by Edward Bell,
M.A.
Scott. Edited by John Dennis.
5 vols.
Shakespeare’s Poems. Edited by
Rev. A. Dyce.
Shelley. Edited by H. Buxton
Forman. 5 vols.
Spenser. Edited by J. Payne Col-
lier. 5 vols.
Surrey. Edited by J. Gregory
Foster.
Swift. Edited by the Rev. R.
Hooper, M.A. 3 vols.
Thomson. Edited by the Rev. D.
C. Toovey. 2 vols.
Vaughan. Sacred Poems and
Pious Ejaculations. Edited by the
Rev. H. Lyte.
Wordsworth. Edited by Prof.
Dowden. 7 vols.
Wyatt. Edited by J. Gregory
Foster.
To be followed by
Cowper. Edited by John Bruce,
F.S.A. 3 vols.
Kirke White. Edited by J. Potter
Briscoe.
Young. 2 vols.
These volumes may also be had bound in Irish linen, with design in gold cm Side
and back by Gleeson White, and gilt top, 3s. 6d. each net.THE ON’LY AUTHORISED AN'D COMPLETE 'WEBSTER.1
WEBSTER’S
INTERNATIONAL DICTIONARY.
Medium \to. 2118 pages, 3500 illustrations.
Prices: Cloth, £1 ns. 6d.; half-calf, £2 2s.; half-russia, £2 5s.;
full-calf, £2 8s.; full-russia, £2 12s.;
half-morocco, with Patent Marginal Index, £2 8s.
Also in 2 vols. cloth, £1 14s.; half-calf, £2 12s.; half-russia, £2 18s.
full-calf, £3 3s.
In addition to the Dictionary of Words, with their pronunciation, ety-
mology, alternative spellings, and various meanings, illustrated by quotatioifS
and numerous woodcuts, there are several valuable appendices, comprising a
Pronouncing Gazetteer of the World; Vocabularies of Scripture, Greek, Latin,
and English Proper Names; a Dictionary of the noted Names of Fiction; a
Brief History of the English Language ; a Dictionary of Foreign Quotations,
Words, Phrases, Proverbs, &c. ; a Biographical Dictionary with 10,000
Names, &c.
SOME PRESS OPINIONS ON THE NEW EDITION.
‘ We believe that, all things considered, this will be found to be the best
existing English dictionary in one volume. We do not know of any work
similar in size and price which can approach it in completeness of vocabulary,
variety of information, and general usefulness.’—Guardian.
‘A magnificent edition of Webster’s immortal Dictionary.’—Daily Telegraph.
Prospectuses, with Specimen Pages, on application.
WEBSTER’S
BRIEF INTERNATIONAL DICTIONARY.
With 800 Illustrations. Demy 82/0., 3s.
A Pronouncing Dictionary of the English Language,
Abridged from Webster’s International Dictionary.
With a Treatise on Pronunciation, List of Prefixes and Suffixes, Rules
for Spelling, a Pronouncing Vocabulary of Proper Names in History,
Geography and Mythology, and Tables of English and Indian Money,
Weights, and Measures.
London : GEORGE BELL & SONS, York Street, Covent Garden.