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.

 003  004  005  006  007  008  009  010  011  012  013  014  015  016  017  018  019  020  021  022  023  024  025  026  027  028  029  030  031  032  033  034  035  036  037  038  039  040  041  042  043  044  045  046  047  048  049  050  051  052  053  054  055  056  057  058  059  060  061  062  063  064  065  066  067  068  069  070  071  072  073  074  075  076  077  078  079  080  081  082  083  084  085  086  087  088  089  090  091  092  093  094

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.

 095  096  097  098  099  100  101  102  103  104  105  106  107  108  109  110  111  112  113  114  115  116  117  118  119  120  121  122  123  124  125  126  127  128  129  130  131  132  133  134  135  136  137  138  139  140  141  142  143  144  145  146  147  148  149  150  151  152  153  154  155  156  157  158  159  160  161  162  163  164  165  166  167  168  169  170  171  172  173  174  175  176  177  178  179  180  181  182  183  184  185  186  187  188  189  190  191  192  193  194  195  196  197  198  199

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.

 200  201  202  203  204  205  206  207  208  209  210  211  212  213  214  215  216  217  218  219  220  221  222  223  224  225  226  227  228  229  230  231  232  233  234  235  236  237  238  239  240  241  242  243  244  245  246  247  248  249  250  251  252  253  254  255  256  257  258  259  260  261  262  263  264  265  266  267  268  269  270  271  272  273  274  275

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.

 276  277  278  279  280  281  282  283  284  285  286  287  288  289  290  291  292  293  294  295  296  297  298  299  300  301  302  303  304  305  306  307  308  309  310  311  312  313  314  315  316  317  318  319  320  321  322  323  324  325  326  327  328  329  330  331  332  333  334  335  336  337  338  339  340  341  342  343  344  345  346  347  348  349  350  351  352  353  354  355  356  357  358  359  360  361  362  363  364  365  366  367

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.

 368  369  370  371  372  373  374  375  376  377  378  379  380  381  382  383  384  385  386  387  388  389  390  391  392  393  394  395  396  397  398  399  400  401  402  403  404  405  406  407  408  409  410  411  412  413  414  415  416  417  418  419  420  421  422  423  424  425  426  427  428  429  430  431  432  433  434  435  436  437  438  439  440  441  442  443  444  445  446  447  448  449  450  451  452  453  454  455  456  457  458  459  460  461  462  463  464

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?"


CHAPTER XI.

ON THE PSYCHO-PHYSICAL VIEW OF NATURE.

 465  466  467  468  469  470  471  472  473  474  475  476  477  478  479  480  481  482  483  484  485  486  487  488  489  490  491  492  493  494  495  496  497  498  499  500  501  502  503  504  505  506  507  508  509  510  511  512  513  514  515  516  517  518  519  520  521  522  523  524  525  526  527  528  529  530  531  532  533  534  535  536  537  538  539  540  541  542  543  544  545  546  547

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.

 548  549  550  551  552  553  554  555  556  557  558  559  560  561  562  563  564  565  566  567  568  569  570  571  572  573  574  575  576  577  578  579  580  581  582  583  584  585  586  587  588  589  590  591  592  593  594  595  596  597  598  599  600  601  602  603  604  605  606  607  608  609  610  611  612  613  614  615  616  617  618  619  620  621  622  623  624  625  626

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.

 627  628  629  630  631  632  633  634  635  636  637  638  639  640  641  642  643  644  645  646  647  648  649  650  651  652  653  654  655  656  657  658  659  660  661  662  663  664  665  666  667  668  669  670  671  672  673  674  675  676  677  678  679  680  681  682  683  684  685  686  687  688  689  690  691  692  693  694  695  696  697  698  699  700  701  702  703  704  705  706  707  708  709  710  711  712  713  714  715  716  717  718  719  720  721  722  723  724  725  726  727  728  729  730  731  732  733  734  735  736  737  738  739  740

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.

 741  742  743  744  745  746  747  748  749  750  751  752

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