A HISTORY
OF
EUROPEAN THOUGHT
IN THE
NINETEENTH CENTURY
BY
JOHN THEODORE MERZ
In Four Volumes
1907-1914
VOLUME II
William Blackwood and Sons
Edinburgh and London.
This electronic edition prepared by Dr.
David C. Bossard
from original documents in his personal library.
August, 2006.
Copyright © 2006 by David C. Bossard.
CONTENTS.
PREFACE iii
iv
CHAPTER
VI.
ON THE KINETIC OR MECHANICAL VIEW
OF NATURE.
The idea of motion in ancient philosophy, 3; Descartes' development of
the kinetic view, 6; Huygens and Newton, 7; Revival of the kinetic
view in the nineteenth century, 7; Young and Fresnel, 8; Undulatory and
emission theories, 11 ; Both theories kinetic, 11; Undulatory theory
prepared by acoustics, 12; Newton's authority on the side of the
emission theory, 14; But also suggests the other theory, 15; Biot,
Brewster, and Laplace against the undulatory theory, 16; Euler, the
successor of Huygens, 16; Young, 16; His "general law of the
interference of light," 18; Theory of the luminiferous ether, 18;
Brougham's attack on Young, 19; Augustin Fresnel, 21; Difficulties
presented by the polarisation of light, 22; Fresnel's Memoir on
Diffraction, 25; Young and Fresnel introduce the conception of
transverse vibrations, 28; Mechanical difference between light and
sound, 30; The properties of the ether, 31; Other kinetic theories, 34;
Kinetic theory of gases, 34; Vortex motion, 35; Faraday's researches,
35; Problems as to the nature of the ether, 36; The theory of
elasticity, 40; The problem of the ether may be treated mathematically,
44; or experimentally, 44; Necessity of combining the two methods, 44;
Spectrum analysis, 45; The clue furnished by the phenomena on which it
depends, 47; Sir G. Stokes, 47; Gustav Kirchhoff, 48; Explanation of
fluorescence, 52; View of the ether as an "elastic solid," 54; Lord
Kelvin's researches, 56; Tyndall's 'Heat,' 57; Lord Kelvin's vortex
theory of matter, 57; Helmholtz's investigations, 58; Earlier
researches on vortex motion, 61; Influence of Helmholtz's
investigations in England, 62; Difficulties of the vortex ring theory,
64 ; Modern view of electrical phenomena: Faraday, 66; "Lines of
force," 68; Development of the conception by Lord Kelvin, 71;
Clerk-Maxwell, 76; His series of works on the theory of electricity, 78
; His conception of "tubes of force," 80; "Electrotonic state" of
matter, 81; Correspondence between velocities of light and electricity,
84; "Elastic disturbances" of the same medium, 85; Consequences on the
lines of the theory of Energy, 87; Destructive effect of the new
theories on the astronomical view, 89 ; Lord Kelvin on the vibrations
of the ether, 91; Indefiniteness of the electro-magnetic theory, 93.
[003] It was a favourite idea with the
philosophers of antiquity that everything is in motion, that rest is to
be found nowhere in nature, and that the entire process of life and
sensation in particular is brought about by the communication and
transference of minute movements of a purely mechanical kind. Out of
the deep conviction that everything around us and in us is in
perpectual flux two distinct problems resulted:... the problem of
explaining the apparent rest and permanency of many observable
phenomena and properties of natural objects, and the higher ethical
problem of fixing upon that which is lastingly real and important in
the continuous change of sensation and opinion.
[006] The kinetic view of nature, however useful and suggestive it may
have shown itself to be in recent times, did not yield any fruits of
real knowledge either in the hands of the ancients or even in those of
the first great philosophers of modern times, in those of Descartes.
Just like attraction and atomism, the kinetic theory had to be worked
out by the instruments of measurement and calculation, by the exact
method, before it led to any actual results.
CHAPTER
VII.
ON THE PHYSICAL VIEW OF NATURE.
Recapitulation, 95; Insufficiency of the astronomical, atomic, and
kinetic views, 96; The conception of energy, 96; The term first used by
Young, 98; Watt introduces the term "power," 99; Poncelet introduces
the term "mechanical work," 101; Black, Rumford, and Davy, 102;
Correlation of forces, 105 ; Liebig, 105; Johannes Müller, 106; F.
Mohr, 107; Mayer, 108; Joule, 110; Helmholtz, 112; "Work" and "energy"
introduced by Clausius and Thomson, 115; Sadi Carnot, 117; Carnot
introduces the idea of "availability," 119; Thomson introduces the idea
of "dissipation," 119; Fourier, 120; His influence on Carnot, 122;
Clapeyron's graphical method, 123; Perpetual motion impossible, 124;
Application by William and James Thomson, 126; The two laws of
thermodynamics, 128; Summary statement of Thomson (Lord Kelvin), 132;
Rankine, Zeuner, and Hirn, 133; Revolutions brought about by idea of
energy, 137 ; Helmholtz on "tension," 138 ; "Potential" and "actual"
energy, 139; The Scotch school, 141; Thomson and Tait, 144 ;
Clerk-Maxwell, 145; Faraday, 146; Helmholtz on electro-dynamics, 149;
Ostwald's physical chemistry, 153 ; The factor of "cost" in industry,
155; Berthelot and Ostwald, 157; Arrhenius, 159; Graham and Andrews,
161 ; Dissociation, 163; Hittorf and Kohlrausch, 164; Victor Meyer on
change of chemical views, 165 ; Ostwald's journal, 166; Willard Gibbs,
167; Entropy, 169; Horstmann, 170; Helmholtz's "free energy," 173;
Kelvin's "available energy," 174; Ostwald's 'Allgemeine Chemie,' 176;
"Kinetics" and "energetics," 180; Criticism of mechanical view, 183;
The outcome, 187; Recent triumphs of atomic view, 188; Modern
electrical researches, 189; The term "electron," 193 ; Difficulties of
Clerk-Maxwell's theory, 194 ; What are electric charges? 195 ; Dr
Larmor's position, 195; Objections raised by atomists, 198;
Artificial character of modern dynamical explanations, 199; The
philosophic problem raised, 199.
[Three great generalizations: of
attraction, of atoms, and of undulations.
Astronomical view -- action at a distance
Atomic view -- particle nature of matter
Kinetic or Mechanical view -- constant motion]
[095] I have already remarked that none of the three great
generalisations ... have been creations of the philosophers of the
nineteenth century. Their first enunciation belongs to antiquity,
though they have only within the last three hundred years been
expressed in sufficiently precise terms to permit of practical
measurements and mathematical deductions. The first step towards a
scientifically comprehensive employment of the familiar but vague terms
of attractions, of atoms and of undulations came, as we have seen, in
each case from some solitary thiinker of this country [England -ed]:
from Newton, from Dalton, from Thomas Young. The systematic elaboration
belongs to the combined scientific exertions of all the civilised
nations of the world. In books on astronomy, physics, and chemistry, up
to the middle of the century, we can hardly find any theoretical
expositions which are not based upon one or more of these three ideas.
[095] None of these three principles, however, appeared
sufficient to cover the whole field. The law of gravitation
embraced cosmical and some molar [= "human sized", visible]
phenomena, but led to vagueness when applied to molecular actions. The
atomic theory led to a complete systematisation of chemical compounds,
but afforded no clue to the mysteries of chemical affinity. And the
kinetic or mechanical theories of light, of electricity, and megnetism,
led rather to a new dualism, a division of science into sciences of
matter and of the ether. The unification of scientific thought which
was gained by any of these three views, the astronomical, the atomic,
and the mechanical, was thus only partial. A more general term had to
be found ... One of the principal performances of the second half of
the nineteenth century has been to find this more general term... the
conception of energy.
[107 - quoting the 1837 memoir of F. Mohr "On the Nature of Heat"]
"Besides the known fifty-four chemical elements there exists in nature
only one agent more, and this is called 'Kraft'; it can under suitable
conditions appear as motion, cohesion, electricity, light, heat, and
magnetism."
CHAPTER
VIII.
ON THE MORPHOLOGICAL VIEW OF
NATURE.
The abstract sciences, 200; Convenience and usefulness of the process
of abstraction, 201 ; Interest opposed to the spirit of abstraction,
202; The descriptive sciences, 203; The breaking down of old landmarks,
204; The spirit of exploration, 206; The medical interest, 207;
Physical science applied to medicine, 208 ; Schwann, 209; Darwin, 209;
Herbert Spencer, 210; Whewell's divisions abandoned, 210; Divisions of
natural history, 211; Morphology and genetics, 213; Other aspects, 215;
Life and mind, 216; Vitalistic and psycho-physical aspects, 218;
Morphology defined, 219; Artificial and natural systems, 220; Linnaeus
and Buffon, 221; Morphology of crystals, 222; Morphology on a large
scale, 224; Humboldt, 225; Morphology on a minute scale, 227; Its
improvement, 230; Morphology and classification, 231; Study of separate
organs, 283; Outdoor studies, 234; Jussieu, 235 ; Problem of
organisation, 236; Cuvier, 237; "Types," 238; De Candolle, 239;
Regularity and symmetry, 241; Goethe's metamorphosis, 243; The ideal
type, 245; Palaeontology, 247; Cuvier's catastrophism, 250; Study of
analogies, 250; Geoffroy Saint-Hilaire, 253; Cuvier and Geoffroy, 255;
Richard Owen, 257; Study of homology, 258; The cellular theory, 260;
Hugo von Mohl, 262; Schleiden and Schwann, 263; Transition to the study
of development, 264; Affinity, 267; Insufficiency of the morphological
view, 270; Herbert Spencer's "physiological units," 272; Change of
scientific interests, 273 ; The morphological period, 274.
[215] Were the real world only one out
of many possible worlds which the mathematical mind can imagine ...;
were the actual forms of nature only some of the infinitely possible
states of equilibrium ...; were the actual course of things -- as
mathematicians since Laplace have fancifully put it -- only one
particular solution of the general differential equations of the world
motion, -- then the two great domains of morphology and genesis would
exhaust the subject, and satisfy all the interests by which natural
history has been created. Unfortunately for the pure mathematician, but
fortunately for the rest of mankind, notably the poet and the artist,
it is not so. An enormous gulf separates the creations of nature from
the most perfect machine; and the fact that, with all the most delicate
methods at her command, her most perfect machines, like the human eye,
do not come up to the demands of the optician, shows us that other
agencies and interests are at work than we have as yet been able to
grasp. So long as astronomy was content to observe the orbits and
motions of the heavenly bodies from a distance, it indeed appearred
possible to define that science as merely "une question d'analyse"; but
in astronomy even, spectroscopy has brought distant objects near to us
and opened out endless vistas into a purely descriptive branch of the
science, a natural history of the heavens. Still more so is this the
case when we fix our gaze on the world immediately surrounding us -- on
the things and events in which we ourselves take an active part. Here
two phenomena attract our attention -- the problem of life, and the
problem of consciousness or mind.
[274] The period from 1800 to 1860 can be termed the morphological
period of natural science. It succeeded the period of the simpler
natural history, which had been mainly occupied with classification and
description of specimens. During the morphological period the knowledge
of the existing things and forms of nature was not only largely
extended by excursions into distant lands and periods of history, but
forms were also studied in situ,
and the living things visited in their habitats. A deeper knowledge of
the connection and interdependence of natural things and events was
thus gained.
CHAPTER
IX.
ON THE GENETIC VIEW OF NATURE.
Statics and dynamics of living forms, 276; "Evolution," 278; "Genesis,"
279; Leibniz's 'Protogaea,' 280; Kant's nebular theory, 282; Laplace,
284; "Cyclical" view, 236; Supplanted by genetic view, 290; Geology,
290; Hutton, 292; Lyell, 293; Embryology, 296; Epigenesis and
evolution, 298; C. F. Wolff, 298; Pander and K. E. von Baer, 299; Von
Baer's comprehensive views, 302; Von Baer's views in modern terms, 306;
Phylotaxy and phylogenesis, 308; Lamarck, 309; The term "Biology," 312;
"Environment," 314; The "Natur-philosophie," 315; Lamarck and Von Baer,
316; The 'Vestiges,' 318; Popular influence, 320; Genetic view in
Germany and France, 321; Apologetic literature in England, 324; Mausel
and Darwin, 326; Triumph of the genetic view, 328; Humboldt's 'Kosmos'
and the 'Origin of Species,' 329; "Variation," 331; Malthus, 332;
"Struggle for existence," 333; Outdoor
studies, 334; "Natural selection" and "sexual selection," 336; Meaning
of natural classification, 336; Fertilisation of plants and "Mimicry,"
338; The judicial method, 339; Darwin and Newton compared, 341;
Unsolved problems, 343; Genetic view on a large scale, 345;
Philosophical theories, 346; Herbert Spencer, 346; Haeckel, 347;
Combines Darwin and Lamarck, 350; Philosophical problems, 352; Problem
of life, 352; Genetic view strengthened by physics and chemistry, 355;
The heat of the sun, 357; Spectrum Analysis, 359; Genesis of the
cosmos-Faye and Lockyer, 360; Palaeontology and geophysics, 363;
Dissipation of energy, 364; Mystery of the actual processes of Nature,
366.
[293 - Regarding Lyell's work in
'Principles of Geology'] When he entered upon his geological
researches, which were conducted during his very extensive travels all
over Europe, a new element had already been introduced into science, of
which neither Hutton nor Werner had beeen able to avail themselves
extensively. This was the identification of geological strata according
to the fossil remains which were contained in them, -- a realization of
the plan of work already dimly foreshadowed in Leibnitz's 'Protogaea,'
but nevertheless accepted even by Humboldt as only a doubtful
indication. This valuable branch of geological science had been started
by William Smith in his 'Tabular View of the British Strata' in 1790,
and further elaborated in his geological map of England (1815), which
was the fruit of his own unaided labours....
[294 - quoting Lyell] "....the classification of the secondary
formations belongs to England, where the labours of Smith were steadily
directed to these objects; the foundation of the third brandh, that
relating to the tertiary formation, was laid in France by the splendid
work of Cuvier and Brongniart." To these words of Lyell we can now add
that the theoretical explanations were first suggested, and the correct
line of reasoning on this accumulated evidence initiated, by Sir
Charles Lyell himself.
The key to the doctrines of Lyell was the study of existing causes --
the attempt to show how the slow agencies which we now see at work in
nature around us are sufficient to explain the successive changes which
the recognisable strata of the earth's crust with their fossil remains
indicate as having occurred in former ages. ... This was to break with
the idea of great and general convulsions ... and it also meant
upsetting the vague notions which set a limit to the time which should
be allowed for the operations of natural causes.
[356 regarding the source of the sun's heat -- a dilemma not resolved
until the discovery of nuclear fusion processes, long after this book's
publication - ed] Where does the heat of the sun come from, and how is
it maintained? These were some of the questions which began to be
asked. ...Shortly before the pioneers of the mechanical theory of heat
published their first essays, experimets had already been maid by Sir
John Herschel and independently by Pouillet in France, with the object
of measuring the annual expenditure of heat by the sun. They had found
it to be an enormous quantity. ... Mayer was the first who seems to
have put the question definitely: How is this enormous expenditure of
heat defrayed, which would, if not in some way compensated, have
resulted, even in historical times, in a great lowering of the
temperature of the sun, and hence also of that on the surface of our
globe, such as is contradicted by all historical evidence?
[363 continuing the previous remarks] The stories of nature, as written
from the point of view of the embryologist, the systematic zoologist
and botanist, and the palaeontologist, seemed more and more to confirm
and support each other. The same cannot be said if we write the history
of our earth from the point of view of the geological record on the one
side and from that of the purely physical data afforded by
thermodynamics on the other. Lord Kelvin has shown [footnote citing his
1868 address to the Geological Society of Glasgow] that the untold ages
to which geologists, since the time of Lyell, have been accustomed to
reckon, are not supported by our present knowledge of the periods
during which the so-called secular cooling of the earth has been going
forward.
CHAPTER
X.
ON THE VITALISTIC VIEW OF NATURE.
The cosmical and the terrestrial views, 369; Vagueness of biological
theories, 370; Impossibility of prediction, 372; Oscillation of
biological thought, 374; The unknown factor, 375; The purely scientific
aspect, 377; Influence of medicine, 379; Practice urges the question:
What is life? 381; Bichat, 381; His Vitalism, 383; His definition of
life, 383; Vitalism and Darwinism, 386; The extreme vitalism, 388;
Attack from the side of chemistry, 389 ; Change in organic chemistry,
393; Influence of Liebig, 394; "Stoffwechsel" and "Kreislauf des
Lebens," 395; "Autonomy of the Cell," 395; "Division of Physiological
Labour," 396; Johannes Müller, 397; Influence of doctrine of
energy,
399; Mechanism, 399; Lotze and Du Bois-Reymond, 401; Liebig's vitalism,
405; Darwin, 406; Lotze and Claude Bernard, 409; Darwinism and final
causes, 411; "Natural result" against "purpose," 413; Organisation and
individuation, 415; Biology and economics, 415; The cellular theory,
417; Schwann, 419; Circulation of matter and energy, 420; "Metabolism,"
422; Structural analysis of morphological elements, 423; Synthesis of
organic substances, 425; The "physical" method, 42S; Properties of the
living substance, 429; Environment, 430; The "internal medium," 432;
Natural selection within the organism, 435; Mobility of living matter,
438; Anabolism and Catabolism, 442; Reproduction, 443; The protoplasmic
theory, 444; Spencer's law of limit of growth, 445; Fusion of two
elements, 446; New problems, 448; Weisinanu on heredity, 450;
Biogenesis, 451 ; The ubiquity of life, 452; The continuity of living
forms, 453; "Pangenesis," 454: Germ-substance and body-substance, 457 ;
Germ-plasma and bodyplasma, 458; Differentiation of germplasma, 459;
Weismann v. Laznarck, 460; Two aspects of the problem of life, 462;
Transition to psycho-physics, 464.
[374] The history of biological thought
... presents us with the spectacle of a repeated oscillation between
two extreme views: on the one side the conviction that the problem of
life is insoluble, and on the other, the assertion that it is soluble,
though it is admittedly as yet unsolved. ... We are quite satisfied
that purely mechanical and physical and possibly chemical processes
make up the whole of the weather problem, and that the difficulty is
simply one of complexity and intricacy. A similar attitude has in the
course of our century frequently been taken up with regard to the
problem of life [footnote: see, for instance Huxley's comparison of an
organism with a clock], but it has always been abandoned again. ...
Biological knowledge has
become purely chemical, physical, and mechanical, but not so biological
thought. The question "What is
life?" still haunts us. Let us see what position the foremost
representatives of modern biological research have taken up to this
question. We find that they can be divided into two classes.
First, there are those who have studied the
phenomena of living matter solely by the means which the advancing
sciences of dynamics, physics, and chemistry have placed at their
command. To them biology is an applied science. The question "What is
life?" is, according to their view of method, only to be solved by
degrees, by bring the forms and processes manifested in the living
world more and more under the sway of observation, measurement, and
possibly calculation. ...The stronghold in which life is intrenched is
to be conquered by ... the attacking forces of dynamics, physics, and
chemistry. It will have to yield some day, though that day may be far
off.
[Second], there is a larger class of students --
those who study biology as a basis of the art of healing, the medical
profession. To them the question of life and death, of the normal or
abnormal co-operation of many processes in the preservation of health
or the phenoma of disease, is of prime interest: the knowledge of the
mechanical, physical and chemical ... is only the means to an end.
[By the first class] we are being continually told
that these questions are premature or metaphysical, and that the answer
which we may give to them is of no scientific importance and of no
scientific value. The question, "What is electricity? What is the the
ether?" cannot yet be answered; nevertheless the sciences which deal
with the properties of the ether or of electrical bodies are advancing
daily. So also -- we are told -- does the science of biology progress,
even though we leave the question "What is life?" unanswered. This
would be a tenable position if the living organism were like an
electrical or an optical apparatus... but [in dealing with] the living
organism we are again and again tempted to ask, "What is life? On
what does the normal and healthy co-operation of all parts in the
living organism depend? In what does it consist?" Fragmentary knowledge
may be well enough so far as it goes, but every medical practitioner
must painfully feel it to be altogether insufficient.... Thus the
question will again and again be asked, "What is life?"
[435 footnote, quoting du Bois-Reymond's Address "Darwin versus
Galiani"] Here is the knot, here the great difficulty that tortures the
intellect which would understand the world. Whoever does not place all
activity wholesale under the sway of Epicurean chance, whoever gives
only his little finger to teleology, will
inevitably arrive at Paley's discarded 'Natural Theology,' and much the
more necessarily, the more clearly he thinks and the more independent
his judgment. ...The physiologist may define his science as the
doctrine of the changes which take place in organisms from internal
causes. ...No sooner has he, so to speak, turned his back on himself
than he discovers himself talking again of functions, performances,
actions, and purposes of the organs. The possibility, ever so distant,
of banishing from nature its seeming purpose, and putting a blind
necessity everywhere in the place of final causes, appears therefore as
one of he greatest advances in the world of thought's from which a new
era will be dated in the treatment of these problems. To have somewhat
eased the torture of the intellect which ponders over the world-problem
will, as long as philosophical naturalists exist, be Charles Darwin's
greatest title to glory".
CHAPTER
XI.
ON THE PSYCHO-PHYSICAL VIEW OF
NATURE.
Abstract and concrete sciences, 465; Their different methods, 466;
Inner experience, 468; Psycho-physics, 469; Cabanis's simile, 470;
Prepared
by Locke and HaIler, 471; Berkeley's 'Theory of Vision,' 472; Bernoulli
and Euler, 474; Animal electricity, 475; Phrenology, 476; Dr Young's
colour theory, 480; Charles Bell, 481 ; Miller's "specific energies,"
482; Helmholtz, 485; "Timbre" defined, 488; Analogy between sound and
colour, 489; Helmholtz and Kant, 491; The brothers Weber, 492;
Fechner's Psycho-physics, 493; Influence of Herbart, 494; His attack on
the "faculty-psychology," 495; Unity of mental life, 496; Mathematical
psychology, 498; Lotze's physiology of the soul, 500; Two sides of
Lotze's doctrine, 502; The psycho-physics of vision, 504; Wheatstone's
stereoscope, 505; Localisation of sensations, 507; Lotze's "local
signs," 508; Fechner, 508; Wundt, 511; Physiological psychology, 512;
Wundt, Fechner, and Lotze compared, 515; The unity of consciousness,
516; Doctrine of parallelism, 518; Münsterberg, 521; Phenomenon of
centralisation, 524; Externalisation and growth of mind, 525; Wundt's
treatment of central problem, 525; Introspective method, 527; The
"objective mind," 529; Its study prepared by Herder, 531 ; His 'History
of Mankind,' 534; Separation of natural and mental sciences, 534; The
problem of language, 536; The exact treatment, 538; Phonetics, 539; The
dividing line between man and brute, 541; Summary, 543; The three facts
impressed by psycho-physics, 545; Transition to statistics, 546.
CHAPTER
XII.
ON THE STATISTICAL VIEW OF NATURE.
Life and Mind as limiting conceptions, 548; Results of abstract
science, 550; Uncertainty in the concrete, 552; Scientific spirit in
business, 553; The science of large numbers, 555; Belief in general
order, 556; Bacon's "Method of Instances," 557; General idea underlying
enumeration, 561; Doctrine of averages, 561 ; Statistics in France,
Germany, and England, 562; John Graunt and Halley, 564; Probability,
Co-operation, Equitable Distribution, 566; The Science of Chances, 568;
Condorcet, 570; Laplace, 572; Four applications, 574; Theory of Error,
574; Method of Least Squares: Gauss, 576; Laplace, 578; Quetelet, 579;
The "mean man," 580; Social statistics and freewill, 583; Buckle, 584;
Criticism of pretension of statistics, 586; Historical criticism,
588; Application in physics, 589; Clausius and Clerk-Maxwell, 590;
Mathematical representation of experimental laws, 592; Irreversibility
of natural processes, 593; Lord Kelvin, 594; "Availability" a theorem
in probability, 597; "Selection" as conceived by Clerk-Maxwell, 598;
Statistical knowledge of nature, 600; As opposed to historical and
mechanical knowledge, 603; Sameness and variation, 607; Darwin, 608;
Galton, 609; "Pangenesis," 610; Lends itself to statistical treatment,
611; Problem of Heredity, 613 ; Mr Bateson's historical treatment, 615;
"Particulate" descent, 615; Application of theory of error, 618;
Difference in application to living and lifeless unite, 620; Professor
Pearson: The mathematical problem, 621; Statistical knowledge
one-sided, 624; Critical methods, 626; The instrument of exact
research, 626.
CHAPTER
XIII.
ON THE DEVELOPMENT OF
MATHEMATICAL THOUGHT DURING
THE NINETEENTH CENTURY.
History of thought, 627; Difference between thought and knowledge, 628;
Popular prejudices regarding mathematics, 628; Use of mathematics,
630; Twofold interest in mathematics, 632; Origin of mathematics, 634;
Gauss, 636; Cauchy, 636; Process of generalisation, 638; Inverse
operations, 639; Modern terms indicative of modern thought, 643;
Complex quantities, 644 ; The continuous, 644; The infinite, 644;
Doctrine of series: Gauss, 645; Cauchy's Analysis, 647; Revision of
fundamentals, 649; Extension of conception of number, 650; The
geometrical and the logical problems, 651; Quaternions, 654;
Foundations of geometry, 656; Descriptive geometry, 658; Poncelet,
659; Character of modern geometry, 662; Method of projection, 663; Law
of continuity, 664 ; Ideal elements, 664; Principle of duality, 665;
Reciprocity, 666; Steiner, 667; Mutual iufluence of metrical and
projective geometry, 668; Plucker, Chasles, Cayley, 671; Historical and
logical foundations, 671; Generalised co-ordinates, 673; Ideal
elements, 674; Invariants, 676; Theory of forms, 678; Theory of
numbers, 680; Symmetry, 681; Determinants, 682; Calculus of operations,
684; Principle of substitution, 686; General solution of equations,
687; Theory of groups, 689; Continuous and discontinuous groups, 691;
Theory of functions, 693; Physical analogies, 696; The potential, 698;
Riemann, 700; Weierstrass, 702; Riemann and Weierstrass compared, 707;
Examination of foundations, 709; Non-Euclidean geometry, 712; Curvature
of space, 715; Generalised conceptions, 717; Klein's exposition,
718; Sophus Lie, 719; Theory of numbers, 721; Gauss's theory of
congruences, 723; Generalised conception of number, 726; Process of
inversion, 727; Kummer's ideal numbers, 728; Modern algebra, 730;
Algebraical and transcendental numbers, 780; Counting and measuring,
732; Georg Cantor's theory of the transfinite, 735; Correspondence,
736; Arithmetising tendency in mathematics, 738.
RETROSPECT
AND PROSPECT.
Order and unity, 742; Philosophical problems, 743; Individuality, 746;
Practical interests attaching to Order and Unity, 748; The geographical
centre of philosophic thought, 750.
INDEX 753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768 769
770
771
772
773 774
775
776
777
778
779
780
781
782 783
784
785
786
787
788
789
790
791
792
793
794 795
796
797
798
799
800
801
802
803
804
805
806
807