THE DOMAIN OF NATURAL SCIENCE




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The
Domain of Natural Science
THE GIFFORD LECTURES DELIVERED
IN THE UNIVERSITY OF ABERDEEN
IN 1921 AND 1922
BY
E. W. HOBSON, Sc.D., LL.D., F.R.S.
Sadleirian Professor of Pure Mathematics,
and Fellow of Christ's College, in the
University of Cambridge


CAMBRIDGE
AT THE UNIVERSITY PRESS
1923




PRINTED IN GREAT BRITAIN




PREFACE


THE prominence which that complex of knowledge
and ideas which is denoted by the term Natural
Science has attained in the thought and life of the
modern world is undeniable, and is indeed fully recognized by all thinking persons. But the question as to
the exact position which what is often termed the scientific view of the world should rightly occupy, in relation
to the other factors of human experience with which
Religion and Philosophy are concerned, is one which
gives rise to very divergent opinions amongst earnest
thinkers. This is a matter which causes grave perplexity
to the minds of great numbers of men and women. The
question has been, in various connections, a subject of
controversy for centuries, and it has in our time become
of inestimable importance, in view of the effects which
any generally accepted answer to it may have, not only
upon our theoretical views and mental life, but also
upon our attitude towards existence in its more practical aspects. My main object in preparing the lectures
which are here published was to provide a reasoned
contribution towards the clarification of ideas in relation
to this fundamentally important question. In order to
attain my object it was necessary to undertake an examination, as close as the circumstances permitted, of the
historical development, aims, and true characteristics,
of various departments of Natural Science, with a view
to the characterization of the proper position of Natural
Science in relation to Thought in general.
Based upon this examination of the leading features




vi                 PREFACE
of scientific theories and laws I have advocated particular
views of the essential characteristics of Natural Science,
the acceptance of which would imply that, in our more
general outlook upon the world, a position of much less
dependence upon Natural Science may rightly be taken
up than has been supposed by many men of Science to
be admissible. What is sometimes spoken of as the
"descriptive view" of the functions of Natural Science
is, in its main outlines, far from new, and has received
some measure of acceptance on the part of prominent
men of Science and other thinkers. I have endeavoured
to make this view more explicit and precise than in the
forms, often fragmentary, in which it has previously
been stated. That Natural Science is in no way concerned with questions as to the nature of reality, or with
efficient causation, and that the edifice it has reared is
independent of any special ontological assumptions,
and in particular of that set of assumptions known as
physical realism, are propositions which I have, throughout the lectures, illustrated and maintained. A position
of detachment, or neutrality, as regards the ontological
and other conceptions which divide various schools of
Philosophy would entail as a consequence that the
authority of Natural Science cannot properly be invoked, as of decisive weight, in favour of any assumptions which it does not need for its own purposes. The
freedom which would thus accrue to Religion and Philosophy from any compelling influence due to Natural
Science would be of course limited by the exigencies
involved in the admission that all questions relating to the
order of the world of physical percepts should be treated
in accordance with the canons of Natural Science alone;
a condition which has in the past by no means always




PREFACE


v..
Vil


been fulfilled. In the last two lectures I have given
some consideration to the further question, what influence Natural Science may exert upon our wider outlook
on the world, when it is supplemented by ontological
assumptions which are extraneous to it.
Apart from slight emendations and a few short insertions, the lectures are here published as they were
delivered in Aberdeen.
In the preparation of those lectures in which the
histories of special departments of Natural Science are
sketched I have been dependent upon information obtained from a large number of sources. Of these I here
mention only those of which I have made most use. In
the lecture on Time and Space, and in other lectures on
departments of Physics, I have utilized the works of
H. Poincare, especially La Science et l'Hypothese. In
the lectures on Corpuscular theories of matter, on Dynamics, and on The conservation of matter and energy, I
have made considerable use of the historical information
contained in the work of E. Myerson entitled Identite et
Realite'. In these and other lectures I have also utilized
Lange's Geschichte des Materialismus. In the lecture on
Electricity, Magnetism, and Light I have drawn much
historical information from Prof. E. T. Whittaker's
work entitled History of the theories of Aether and Electricity. In the lecture on The constitution of matter I
have utilized the work by A. E. Garrett on The periodic
law, and also Prof. Soddy's lectures in Science and Life.
In the lecture on Cosmical Theories I have drawn historical information from Miss Agnes M. Clerke's History
of Astronomy in the nineteenth century, and I have also
utilized J. H. Jeans' work on Problems of Cosmogony and
Stellar Physics.




viii                PREFACE
In the part of the lecture on Biological Science which
deals with a comparison of the living organism with a
machine, as regards relations of Energy, I have utilized
Prof. J. Johnstone's work on The Philosophy of Biology.
In a portion of this lecture I have also made use of Mery's
History of European civilization. In the preparation of
the lectures on The living organism, on Heredity, and on
the Evolution of Species, I have utilized the works of
Prof. J. Arthur Thomson, especially those entitled
Heredity and The Science of Life. In these lectures I
have also made much use of the historical information
contained in E. S. Russell's work entitled Form and
Function; and I have also drawn information from H. F.
Osborn's work From the Greeks to Darwin.
Before I wrote the two final lectures I perused Prof.
J. B. Baillie's Studies in Human Nature and was influenced in some points by his views.
To Dr F. H. A. Marshall, F.R.S., and the Rev. F. R.
Tennant, D.D., each of whom read the type-scripts of
some of the lectures, I am indebted for advice on special
points. To my friend Prof. James Ward, F.B.A., I owe
much. Without the stimulus received in the course of
conversations carried on with him during many years
I should probably never have ventured upon the task
of writing upon a subject so much wider than my main
subject of study.
E. W. H.


CHRIST'S COLLEGE, CAMBRIDGE
March 1923




CONTENTS
I. INTRODUCTION......
The choice of subject. The denotation of the terms Science and
Natural Science. Plan of procedure in the lectures. Effects of
recent scientific discovery on industrial and social life. Common and Instinctive knowledge. Animism and Magic. The
tension between Science and other elements of Thought. The
effect of the rise of the Copernican system. The effect of geological discovery. The effect of Darwin's views. Recent relaxation of the tension. Fear of the effect of extreme views of
men of Science. View expressed by Laplace. The views of
Biichner. Tyndall on the primacy of matter, and Huxley on
consciousness. The question of the representation of the
physical world as a closed system of deterministic type.
II. SCIENTIFIC LAWS AND THEORIES...    23
The ordinary dualism of matter and mind. Sense-impressions
and perception. Power of prediction of their course. Inferential knowledge of other percipients. Intersubjective communication. Formation of common knowledge. Defects of
language and other forms of symbolism. Natural Science as
a purposive continuation of common knowledge of percepts.
The classificatory stage. Element of arbitrariness in it. Formation of concepts as symbols for percepts. Rules or Laws. Laws
of Nature. Their mode of construction. Natural Science and
the element of fact. Natural laws as characterized by Mach
and Boltzmann. The construction of scientific laws. Discernment of order in percepts. Abstraction of individual differences, Classification. Development of scientific principles and
theories. Judgment of value of a scientific theory. Degrees of
abstraction in scientific theories. Different species of concepts.
Danger of overweighting theories with concepts. Irreducible
concepts. Degree of abstraction does not cause generic differences between theories. Case of Geometry. View of scientific
theories as conceptual schemes. Science as public knowledge.
Limitations of scope of a scientific theory. Sciences which deal
with the remote past. Approximativeness of actual measurements. Quantitative theories. Limited range of descriptive
power of a theory. Statements as regards the whole Universe inadmissible. Possibility of all-embracing conceptual scheme need
not be assumed. The notion of efficient causation unnecessary
to Science. Approximately isolated systems. Conditions of success of a conceptual scheme. Tests of applicability of a scheme.
Truth and falsehood of a scientific theory. Diversity of modes
of application of a theory. Characteristics of Pure Mathematics.




x                      CONTENTS
III. NATURAL SCIENCE IN RELATION TO PHILOSOPHY.........    51
Influence of philosophical views on scientific thought. Relation of Natural Science with systems of Metaphysical Philosophy. Neutral attitude of Natural Science as regards Philosophical and Psychological matters. Material objects. The
notion of substance. Locke's description of it. Primary and
secondary qualities of matter. Berkeley's idealistic position.
Natural Science incapable of deciding questions of reality.
Views of Poincare and Whitehead. Views of Lodge on the
Einstein theory. The dualism of the psychical and the physical.
Assumptions of realistic Philosophers. Natural Science independent of ontological hypotheses. Indispensability of Philosophical Thought. Services of Philosophy to Natural Science.
Poincare's view of the relation of facts to Science. The psychical
domain. The relations of the psychical with the physical.
Psycho-physical parallelism. Theories of interaction. Limitation of Natural Science in respect to a complete Philosophy of
Nature. Psychology. Difference between Psychology and
Natural Science as regards subject-matter and methods.
Partial similarity of Psychology and the Politico-social sciences
with Natural Science.
IV. CAUSATION AND DETERMINISTIC SYSTEMS                    74
The notion of causation. Efficient causation. Hume on
efficient causation. The notion of efficient causation discarded
by Natural Science, as not to be discovered in phenomena.
Cause as totality of conditions. Helmholtz's formulation.
Continuity of processes. Logical necessity exists in thought,
not in phenomena. Explanation in the strict sense is not the
function of Natural Science. The asymmetric relation of
causation. Deterministic schemes of the most general type.
Teleological conceptions. The notion of efficiency is not
necessarily illusory. Biological Science and efficiency. Purposiveness. Natural Science in relation to prediction. The
precise meaning of a deterministic scheme. Extensive magnitudes. Characterization of the state of a conceptual system.
Representation of the states of a system by a set of functions.
Does anything perceptual correspond to an ideal deterministic
scheme? Limitations on representation by deterministic
schemes. Difficulty of attaching a meaning to deterministic
psychical schemes. That the whole world of physical phenomena is capable of representation by a unified deterministic
cheme is unproved.




CONTENTS                            xi
V. NUMBER AND ITS DEVELOPMENTS..       99
Arithmetic as a department of Natural Science. Arithmetical
knowledge of the individual and the race. Pervasiveness of the
concepts of Arithmetic. Arithmetic as a language. The concepts
of unity and of number. The notion of order. Cardinal and
ordinal numbers. Scheme of arithmetical operations. Fractions. Exactness of process of counting. The importance of
notation in Arithmetic. The arithmetic of the Egyptians,
Babylonians, and Greeks. Indian invention of negative numbers. The inadequacy of rational numbers for the complete
representation of magnitudes. The modern theory of real
number. Adequacy of real numbers for the representation of
magnitudes. Necessity of the employment of irrational numbers. Extension of the conception of number to imaginary
numbers. The infinite and the infinitesimal. The conception
of a limit. The necessity of contemplation of the infinite.
Cantor's theory of the numerically infinite.
VI. TIME AND SPACE......   123
The individual intuitions of time and space. Public time, and
its measurement. Abstract or absolute time. Time in abstract
Dynamics. Newton's conception of absolute time. Individual
spatial perception. Private space and geometrical space.
Physical space as a construction. Conceptual space and
Geometry. The development of Geometry. Euclidean Geometry. The theory of parallels. The Geometry of Bolyai and
Lobachewsky. Descartes' introduction of analytical Geometry.
Projective Geometry. Non-Euclidean Geometries. The theories
of Riemann and Helmholtz. Riemann and the curvature of
space. The application of Geometry to physical space. Criteria
of applicability. Kant's view of space.
VII. CORPUSCULAR THEORIES OF MATTER.   I50
Two divergent views of the constitution of matter. The
parentage of atomic theories. The cosmology of Democritus.
The physics of Epicurus. The atomism of de Cusa and the
cosmology of Giordano Bruno. Descartes' mechanistic view
of the world. Disintegration of the Aristotelian conceptions.
Gassendi's views. The materialism of Hobbes. Boyle's
theories. The rise of the notion of action at a distance, in connection with Newton's theory of gravitation. The gravitational
theory of Le Sage. Newton's law as a genuine scientific scheme.
Dalton's conceptions of chemical action and of atoms. Avogadro's molecular theory. Realistic and other views of atomic
theories. Question as to the elasticity of atoms. The views of
Secchi. Boscowitch's theory of atoms without extension.




xii                     CONTENTS
Kelvin's theory of vortex atoms. The kinetic theory of gases.
General survey of the ideas which have inspired atomists.
Change in atomism due to conception of forces acting at a
distance.
VIII. DYNAMICS........   74
The nature of a dynamical scheme. Classical Mechanics.
Galileo's investigation of the law of a falling body. The contribution of Huygens to Dynamical Science. The Dynamics of
Newton; his Regulae Philosophandi. Newton's conceptions of
absolute space and of motion. The frame of reference in
abstract Dynamics. The relativity of motion. The choice of
frames of reference. The principle of inertia. Connection of
relativity of motion with inertia. Circular motion regarded as
a natural motion. Conservation of velocity in a straight line.
A priori proofs of the principle of inertia by Kant and Maxwell.
Notion of the body Alpha. The principle of inertia is not an
a priori principle, but in the conceptual scheme appears as a
definition. The general conception of force. Contact-action is
insufficiently explained. Action at a distance. Newton's
measure of force. The laws of motion. Central forces. The
extension of Newtonian Dynamics. Newtonian axes or inertial frames. The question of absolute rotation. Absolute
directions cannot be determined in physical space.
IX. THE CONSERVATION OF MATTER AND
ENERGY.                                            I98
The general notion of conservation. Difficulty as regards the
precise meaning of the conservation of matter. The conservation of weight. The conservation of mass. The establishment
of the principle of conservation of mass as the result of a
gradual evolution. The medieval views. The views of Descartes
as to conservation of matter. The views of Newton, Huygens,
Boyle, and Diderot. Chemistry and the conservation of matter.
Lavoisier's investigations. The modern origin of the principle of
the Conservation of Energy. The notion of work. The principle
as conceived by Descartes, Leibniz, and Huygens. Conception
of heat as a substance. The transformation of motion into heat.
Experiments of Rumford and Davy. Views of Seguin and
Sadi Carnot. Joule's experimental researches. Mayer and
Colding on the conservation of energy. Extension of the scope
of the principle by Helmholtz. Kinetic and potential energy.
The reduction of potential energy to kinetic energy. Perpetual
motion. The various forms of energy. Poincare on the difficulty of defining uniquely the energy of a system. The principles of Energetics. Limitations of scope of the principle.




CONTENTS                          xiii
X. MECHANICAL THEORIES AND THERMODYNAMICS.........   221
The aim of a mechanical theory. The ether as employed in
mechanical theories. The realistic conception of an ether.
Conceptual schemes and representation to the sensuous
imagination. Force not an independent concept in Newtonian
Dynamics. The department of Statics employs force as an
independent concept. The later developments of Dynamics.
Inexorable constraints. The deduction of d'Alembert's principle. Lagrange's analytical Dynamics. Modifications of
Lagrange's scheme. The development by Sir W. Hamilton.
The Hamiltonian principle. The scope of the principle. Irreversible systems. Helmholtz's cyclical systems. The earlier
theories of Heat as caloric. The theory of Heat as motion.
Carnot's investigations. Thermodynamics as developed by
Rankine, Clausius, and Kelvin. The conception of entropy.
The principle of increase of entropy, or dissipation of energy.
Restriction by Kelvin to non-living agencies. Willard Gibbs.
Free and bound energy. Doubts of the adequacy of theories
which only contemplate continuous transformations of energy.
Planck's theory.
XI. ELECTRICITY, MAGNETISM, AND LIGHT.   243
Triviality of the earliest known facts. Characteristics of various
theories. Discovery of polarity of magnets. The researches of
William Gilbert. Descartes' theory of Magnetism. Discovery
of conduction of electricity. Discovery of two species of
electricity. The Leyden jar. Theory of electric fluid. Induction. The law of electric force. The researches of Cavendish
and Michell. Coulomb's electrical and magnetic measurements.
Poisson's mathematical theories. The discoveries of Galvani
and Volta. Discovery of electrolysis. Theory of the voltaic
pile. Discovery of Oersted that a current is a magnet. Ampere's experiments and theory. Theories of ponderomotive
forces. Faraday's researches. Connection with optical phenomena. Electrical energy in a circuit. The researches of
Maxwell. Theories of light. Theories of luminiferous ether.
Electromagnetic theory of light.
XII. THE CONSTITUTION OF MATTER..   271
Irreducibility of the chemical elements. Atomic weights of
chemical elements. Prout's ideas. Newlands' law of Octaves.
Mendeleeff's periodic law. Crookes' theory of the genesis of
the elements. The electron theory of matter. The cathode rays.
Discovery of radio-activity. The source of energy in radioactivity. Discovery of helium. Rays emitted by radio-active
substances. Emanations. Transmutation theory of radio



xiv                     CONTENTS
activity. Models of the atom. Artificial excitation of disintegration. Radio-active minerals as geological clocks. The
existence of isotopes. Matter as built up of electrons and nuclei.
XIII. COSMICAL THEORIES.....  293
Nature of cosmical theories. Description of the solar system.
Bode's law. Discovery of asteroids. Discovery of Neptune.
Kant's speculations. Laplace's Nebular Hypothesis. Solar
radiation. The meteoric theory. Objections to the nebular
theory. Tidal action between the earth and the moon. Generation of satellites. Binary systems. The generation of double
stars. The generation of stars from nebulae. The Doppler
effect and the motion of nebulae. Primitive nebulae. The
classification of stars according to spectral type. Radio-active
substances as sources of heat. The age of the earth. General
remarks on cosmical theories.
XIV. EINSTEIN'S THEORY OF RELATIVITY..  316
General remarks on the theory. Question as to the position of
the theory in relation to Philosophy. General character of the
theory. Comparison with Newton's theory. The measurement
of space and time. Interdependence of spatial and temporal
measurements. Gravitational phenomena and light. The
metric of abstract Geometry. The spatio-temporal metric in a
gravitational field. The special theory of relativity. Motion
relatively to the ether. Experiments of Michelson and Morley.
Postulates of the special theory. Replacement of the Newtonian
Dynamics. Change in the conceptions of measurement. The
four-fold ordered manifold, or the "world" of Minkowski.
The general principle of relativity, and the theory of gravitational fields. Indefiniteness of Newton's law. The principle of
equivalence. Identity of gravitational and inertial mass.
Einstein's metrical theory as developed from Riemann's geometrical theory. The world-line of a material particle. The
motion of the Perihelion of Mercury's orbit. Deviation of rays
of light in the sun's gravitational field. Displacement of spectral lines. General remarks on the origin of the theory.
XV. BIOLOGICAL SCIENCE.....  344
The concept of a living organism. Scientific definition of living
organism, and criteria of distinction between animals and
plants. Method and limitations of Biological Science. The
character of the concepts employed. Restriction of Physiology
to physico-chemical processes and categories. Psychical and
Psychological categories in Biology. The methods of Psychology.
Science and the individual. The relation between the psychical
and physical aspects of an organism. Methodological dualism




CONTENTS                            xv
necessary in Biological Science. Vitalistic ideas. The notion
of guidance. Metabolism and mechanical processes from the
point of view of Energetics. Conservation of energy in the
organism. Comparison with a steam-engine. Transformation
of energy in the cells of the leaves of plants. The law of entropy
and the living organism. Driesch's theory of entelechy.
Loeb's theory of tropisms. Psycho-physics or Physiological
Psychology. The theory of specific energies. The investigations
of Fechner and others. Comparison with inorganic Science.
XVI. THE LIVING ORGANISM...373
Division into the two departments of Morphology and Physiology. Their characteristics. The biological work of Aristotle.
His teleological view. Galen's physiological knowledge. The
period initiated by Harvey. The systematization by von Haller.
The origination of mechanistic theories of Physiology by
Descartes. Vitalism in the eighteenth century. The Systematics of Linnaeus. The rise of Histology. The discoveries of
Malpighi. The biological studies of the eighteenth century.
The conception of a scale of beings. The views of Bichat on
animal and plant life. Cuvier's views of the harmony of
structure and function. Cuvier's division of the animal kingdom into types. The single type of Saint-Hilaire. The Philosophy of Nature. The law of parallelism. The views of Richard
Owen. Embryology in the hands of von Baer. His theory of
germ-layers. Von Baer on recapitulation. The cell-theory of
Schwann. Criticisms of the cell-theory.
XVII. HEREDITY.                                   397
Abiogenesis and its refutation. The origin of life on the earth.
The meaning of the term heredity. Aim of theories of heredity.
The concepts of theories of heredity. Germ-cells. The embryo.
Preformationist theories. Epigenesis. Theories of Pangenesis.
Criticism by Weismann of Darwin's theory of Pangenesis.
Weismann's theory of inheritance. The value of Weismann's
theory. Criticisms of it. The inheritance of acquired characters. Statistical theories of heredity. The investigations of
Galton. Galton's law of ancestral inheritance. The investigations of Mendel and his successors.
XVIII. THE EVOLUTION OF SPECIES...  425
Notions of evolution of Aristotle and other Greek thinkers.
St Augustine and freedom of scientific thought. Evolutionary
speculations after the Renaissance. The evolutionary views of
Buffon. The views of Linnaeus, Cuvier, and other naturalists.
The evolution theory of Lamarck. The period before Darwin.
The inception of Darwin's theory of descent. The effect of the




xvi                   CONTENTS
publication of Darwin's views. The arguments of Darwin.
The theory of Natural Selection. The anti-teleological tendency of the theory of Natural Selection. The evidence for the
theory. The effect of environment. Mental factors in evolution. The views of Neo-Darwinians and Neo-Lamarckians.
The theory of histonal competition. Weismann's views on
evolution. Germinal selection. Discontinuous variations. The
theory of De Vries.
XIX.    NATURAL       SCIENCE     AND     GENERAL
THOUGHT.......      452
The position of Natural Science as regards general theories of
existence and reality. Natural Science circumscribed in its
aim and restricted by its method. The conceptual knowledge
represented by Psychology and Logic. Natural Science,
Psychology, and Sociology are concerned with classes, not
individuals. The individuality of human beings. The principle
of order and the principle of individuality. Knowledge obtained by the statistical method. Knowledge obtained by direct
intuition. Cognition must be supplemented by other elements
of human experience. The main characteristics of Religion,
Philosophy, Science, and Art. The notion of values. Conservation of values connected with existential elements. The
relations between Religion and Philosophy. Theistic forms of
Religion. Theism. The relation between Science and Religion
is indirect, through Philosophy. Any acceptable view of reality
must not be incompatible with the existence of autonomous
Natural Science. Essential characteristics of Philosophical
systems. The case of solipsism. The element in the complex of
percepts that is independent of the individual mind. Effect of
assumptions as to reality.
XX. NATURAL SCIENCE AND THEISM..       478
Lord Gifford's description of the aims of his Lectureships.
The practical influence of Natural Science on theistic belief.
Theistic Philosophy. Theories of reality and the origin of
phenomena. The relations of God with the world. Pantheism.
God as transcendental. Intermediate theistic conceptions.
Idealism. The One and the Many. The ontological proof. The
distinction between revealed and other knowledge. The three
traditional proofs. The cosmological proof. The correlation of
reality with rational schemes. Miracles. The teleological
argument. The moral argument for theism. The existence of
physical evil. Limitations of the discussion. The practical relations between Philosophical Theology and Natural Science.
Conclusion.


INDEX


503




THE DOMAIN OF NATURAL SCIENCE
I
INTRODUCTION
THE Domain of Natural Science is the subject I have
chosen for the course of Gifford Lectures which
the Senatus Academicus have done me the honour of
inviting me to deliver. Any attempt to define and delimit the domain of Natural Science must concern itself
with the questions what Natural Science is, and what it
is not; what its precise functions within the general
domain of Thought may be; what the nature of its
relations is with other parts of that wider domain; in
what manner it has grown up, and what we may reasonably expect from it in the future; and to what kind of
essential limitations it is subject.
Our general mental outlook upon the world contains
a great mass of ideas and knowledge of which the origin
is so remote in the development of our race that they
may perhaps least inadequately be described as instinctive notions and knowledge. We possess also a heterogeneous mass of unsystematized ideas, in large part
traditional in the society in which we have grown up.
Besides these we possess a stock of ideas and knowledge
of a more or less systematized character, drawn from
the religious, philosophical, and scientific, thought of the
past and the present. My choice of subject has been
made in the hope of doing a little to promote clarity of
view as regards the relation which that special kind of
thought which, in accordance with established usage,
we denote by the name Natural Science, has with that
greater complex of ideas to which I have alluded, and
H GL                                     I




2


INTRODUCTION


which conditions our general mental attitude towards
the psychical and physical world.
The denotation of the term Science is not fixed by
usage with absolute precision. In the wider sense of
the term, a critical and systematic study of any clearly
marked out region of thought is described as a Science,
provided that study attains, or at least aims at, a reasonable standard of rigour, in relation to classification and
subsumption under general laws, in dealing with its
special subject matter. Thus, for example, Philology,
Ethics, Psychology, Economics, Anthropology, Textual
Criticism, and even Heraldry, are spoken of as Sciences.
On the other hand, Metaphysics is not usually spoken
of as a Science, probably because, however systematic
it may aim to be, its subject matter is too universal,
embracing in fact the whole of experience and existence.
The term Natural Science, with which I am here specially
concerned, is generally restricted to denote the group of
those special Sciences which concern themselves with
the study of what we call physical phenomena, including
the cases in which the phenomena are connected with
living organisms. In the somewhat narrow sense in
which I shall employ the term, Natural Science excludes
any direct consideration of the mental or psychical facts
in living organisms from its purview; although this restriction is not universally accepted in connection with
the group of Biological Sciences. It may be objected
against this avowedly narrow use of the term Natural
Science that it implies an undue restriction of the term
Nature, involving the relegation of the mental side of
life to a place outside Nature. This objection has undeniable weight. My employment of the term Natural
Science, in the meaning that it denotes the Science of
the physical world, is a matter of convenience only, and
is not intended to indicate the acceptance of the theory
that there exists any ultimate barrier between physical
Nature and the mental life of man.




INTRODUCTION


3


The object I have indicated, of describing and delimiting the domain of Natural Science, can only be
attained by undertaking an examination, from the inside,
of the methods of procedure, the assumptions, and the
course of development of various special parts of Natural
Science. Anything like a complete examination of the
history of the development, and an analysis of the
methods, of each one of the group of the Natural
Sciences would be a gigantic task far beyond the powers
of any single individual. To do this would require the
co-operation of an army of specialists intimately acquainted with the details of the present and past states
of the very numerous departments of Natural Science.
Happily, it does not seem altogether impossible to
attain at least some measure of success in an attempt to
define in general terms the nature of the contributions
which the Natural Sciences are fitted to make to a
general view of the world, by adopting the comparatively
modest procedure of examining the methods and principles of some typical branches of Science, and especially
of those branches which may be regarded, from a
methodological point of view, as in a more advanced
stage of development than the others. It might perhaps
be regarded as the more logical method to commence at
once this process of examination and analysis, and upon
its completion to deduce the more general conclusions
which could be extracted from the results obtained.
But it is more practicable, if less logical, first to state,
with as much precision as possible, the general conclusions which it is hoped to establish, and then later
to support and illustrate these conclusions by means of
an analysis of the history and general features of special
branches of Science. Accordingly, in the first few
lectures, a general account will be given of what may
be regarded as the foundations of the method adopted
in the various departments of Natural Science, and conclusions will be stated as to the nature of the knowledge
I-2




4


INTRODUCTIONN


of the physical world which organized Science affords.
Moreover, some indications will be given of the relation
of Science with other elements of thought, and especially
with Philosophical thought and speculation. In particular an attempt will be made to trace the character of
the limitations to which Scientific knowledge is subject.
After these earlier lectures, the examination, in some
detail, of various typical portions of Science, and of many
scientific theories, will be undertaken. If, in this, the
main part of the course, I devote what may seem a
disproportionately large amount of time to scientific
theories which have at the present time been abandoned,
modified, or subsumed under wider theories, the explanation must be taken to be that my object is not to
attempt to perform the impossible task of giving an
accurate account of the present state of Natural Science,
even in a few of its branches, but rather, by means of
an historical retrospect, to disclose the essential characteristics of all the theories and laws which go to make
up Natural Science. It is probably less difficult to discern the principles which underlie the essential methods
of Science, and thus to obtain a grasp of their true
character, and of the limitations to which they are subject, by examining and dissecting theories which have
attained a crystallized form, than by attempting a complete analysis of the tentative and rapidly changing
theories which actually direct, at the present time, the
gigantic efforts that are being made to advance our
knowledge of natural phenomena.
The state of various branches of Science, in our day,
is such as to lead to a vivid appreciation of the unwisdom
of considering any scientific theory as having attained
finality. Recent experience has shown that a wellestablished and successful theory may be liable to
fundamental change, owing to the discovery, by observation or experiment, of new facts incapable of reconciliation with the theory in the form in which it has




INTRODUCTION                   5
for long been accepted. At the end of the course an
attempt will be made to draw some general conclusions
of a kind which may be regarded as having a bearing
upon the great central problem, the unbiassed treatment
of which was indicated by Lord Gifford in his Will as
the object for which the Gifford Lectureships were to
be founded. The difficulty of the task I have undertaken
is such that, adequately to cope with it would strain the
resources of anyone who had spent his life in the consideration of the Philosophy of Science. It can be
undertaken only with much diffidence by one whose
consideration of the weighty matters involved therein
has been limited by the scant leisure available in a life in
which attention has been concentrated, in the main, upon
the absorbing technique of a single branch of Science.
During the last century and a half the life and work
of multitudes of human beings have, in our own country,
and in the civilized world in general, undergone a
revolutionary change, due to the application of Science
in all branches of Industry. Vast populations now exist
which, at a former period, could neither have been fed
nor provided with work. The use of the motive power
of steam, introduced in the eighteenth century, its
application to locomotion on land and sea, in the
nineteenth century, the invention of the oil engine with
its application to aviation, in the twentieth century; the
electric telegraph, the telephone, wireless telegraphy and
telephony, and other applications of Electromagnetism,
are all traceable to the discoveries of Pure Science. The
development of the Biological Sciences has led to applications, of the most far-reachingimportance, in Agriculture,
Hygiene, and in the healing Art. These applications of
Science give indications of future extensions, in which
the present limits maybe indefinitelytranscended. Other
results of the application of Science are embodied in the
great chemical industries, of which one of the most
ominous results is apparent in the discovery of ex



6


INTRODUCTION


plosives of vast destructive power. These results of
Natural Science, and many others too numerous for
mention, in its persistent efforts to dominate physical
nature, have furnished us with the mechanical means
of securing an indefinite improvement in the welfare of
mankind, if a wise use is made of the power with which
they endow us. They have also provided our civilization
with the material means of committing suicide, if the
increased mechanical powers which they afford are not
accompanied by a corresponding rise in the ethical
standards which actuate nations in their dealings with
one another. In conjunction with the revolutionary
industrial and economic consequences which have arisen
from modern applications of Science, vast changes have
been produced in the mental outlook of multitudes of
men, and these changes have given rise to social
problems of the most far-reaching character, of which
the solutions are not in sight. These problems are such
that, even in the view of optimists, it may need centuries
of unrest and strife before relative social stability is
restored. These matters must however here be left on
one side. I have referred to them because some of the
effects of Science upon the thought of the world are of
an indirect character, and are not simply logical consequences of the development of purely scientific
thought, but are rather due to the stimulating effect on
the imagination produced by the great increase in the
mechanical appliances at our disposal, and to the enlargement of the mental horizon consequent upon the
increased means of communication we possess having
abolished the comparative isolation in which considerable
aggregations of human beings formerly lived. However, our attention must be confined to the more direct
relations between scientific thought and mental life in
general.
Prolonged investigation of the ideas and beliefs of
existing races in a primitive state of development,




INTRODUCTION


7


together with indications, obtained from historical and
archaeological sources, of conditions in the earlier ages
of mankind, have made it possible to reconstruct in
broad outline the concurrent factors which made up the
complex of ideas about natural phenomena among
primitive peoples. The earliest conceptions of this kind
may be roughly subsumed under three heads. First,
there existed at all times a body of knowledge of the
uniformities of Nature, some of which had an origin
reaching back in the experience of the race to the simian,
or to the still earlier, ancestry of man. Much of this
knowledge involved no reflection, and it may perhaps
not unfittingly be described as instinctive knowledge.
A considerable part was obtained as the result of the
experience of the individual, or was communicated to
him by his fellows. Such knowledge, whether of the
kind I have called instinctive, of racial origin, or of the
kind dependent on individual experience, was due to
the necessity for action, of men or their ancestors in their
physical environment, for self-preservation, the obtaining of food and shelter, and generally for the maintenance of the life of the individual and the tribe. Such
knowledge, of an unsystematic character, largely unconscious, and involving little or no reflection, we may
describe as common knowledge. Its origin was empirical,
and this common knowledge, acquired empirically, was
the true parent of Science.
The second class of ideas of primitive peoples may
be brought under the head of Animism. To primitive
man, the distinctions between himself and his environment, and between the psychical and the physical, were
blurred and indistinct. His own thoughts, feelings,
appetites, and passions, he ejected outwards into the
objects by which he found himself surrounded. Thus
he peopled the world of objects with spirits and demons,
of like nature with himself, under the influence of the
same motives of action, the same passions and appetites.




8


INTRODUCTION


For him, all Nature was alive; all objects were the residences of spiritual beings, whose actions might be influenced by supplications, persuasion, sacrifices, and
threats. But the spirits which animated Nature were,
like men themselves, highly capricious, and their actions
often unintelligible. To their agency were attributed
many of the calamities, such as diseases and storms,
which fall upon men. It is to Animism and animistic
conceptions that we may trace back many of the Philosophical and Religious problems which have perplexed
mankind in later ages.
The third class of ideas which are found in primitive
man may be brought under the head of Magic. Instead
of supplications and sacrifices being employed to persuade external Nature to action of a kind favourable to
the wishes and interests of man, a similar end was
attained by a form of compulsion embodied in Magical
ritual. The essence of Magic has been described by
Carveth Read as follows1:
Magic supposes constant connexions of events due to the
agency, force, influence or virtue of charms, rites and spells;
which connexions, however, are found only to be tendencies of
some events to excite others, inasmuch as they may be frustrated
by counteracting charms, rites or spells. Magic is entirely constituted by notions of force, sometimes violent, as in the discharge of an enchanted spear; sometimes subtle, like the efficacy
of an opal; intangible, invisible, and operating at a distance
through space and time, like a witch's spells that eclipse the sun
or moon. These forces have only a one-sided relation to the
workaday world; they meet with no resistance from what we
take to be the "properties of matter," such as weight and impenetrability; but are themselves entirely exempt from natural
law; what we call the " real world " has no hold upon them; they
live in a world of their own. They are absolutely immeasurable;
and hence the causation, which is certainly implied in the notion
of their operation, is indefinite, and becomes vaguer and vaguer
as the magical system develops; and all this is the opposite of
what happens in the history of Science. In spite of having a
1 The Origin of Man and of his Superstitions, pp. 326, 327.




INTRODUCTION


9


necessary ground in the human mind, Magic and Science are
contrasted from the first, in their development grow wider and
wider apart, in their methods and ideas more and more opposed.
If either can be said to precede the other, it is Science (at least,
in its earliest and crudest form) that precedes Magic.
In the main the beginnings of scientific knowledge
arose out of that element which we have called common
knowledge, but throughout its development Science has
not completely disentangled itself from notions drawn
from the domains of Animism and Magic. Traces of
Animism and Magic in their cruder forms are still to be
found in the most civilized communities, and they are
to be observed in undiminished strength in some of the
lower races of mankind. Magic and Science have in
common with one another the recognition of certain
uniformities in the sequence of events, and in this
point both of them are sharply distinguished from
Animism; but this similarity is of small importance
compared with the deep seated differences in their outlook and methods; and the history of the development
of Science exhibits at all times the tension and conflict
between the fundamentally divergent attitudes of mind
which they represent. To a considerable extent this
tension between Science and Magic and Animism was,
in earlier times, held in check by the fact that what
elements of genuine scientific knowledge existed were
largely in the hands of the same persons who were
adepts in Magic. The Wizard, or Medicine Man, or the
Priest, had very frequently some knowledge of the
course of natural events greater than that possessed by
the ordinary members of the tribe or community. This
real knowledge assisted him in bolstering up his power,
and in increasing his reputation for possessing special
means of influencing the course of natural phenomena.
In conjunction with what we now regard as the fantastic cures for ailments that the Medicine Man prescribed, he also employed some modicum of empirical




IO0


INTRODUCTION


knowledge which can be regarded as rudimentary
Science. The magical power, possessed by the rainmaker, of influencing the weather, was often eked out
by some knowledge, obtained from observation, of the
course of the seasons. At all times, moreover, there
probably existed, here and there, individual members of
the community, whose exceptional mentality or occupation impelled them to obtain some increased empirical knowledge of the properties of material objects,
and of the course of natural events, and to leave aside,
or subordinate, the methods of Magic. The comparatively slow progress of Natural Science in early times
may with much plausibility be attributed to the fact
that Magic, combined with the prevailing hypostatization of mental and physical qualities, gave almost unrestricted rein to the imagination, and led to a luxuriant
growth which left far behind that steady progress of
knowledge which could only be made by the exercise
of the sober qualities requisite for the patient investigation and comparison of facts.
No doubt, Animism and Magic had some indirect
effect in stimulating the acquisition of knowledge, by
their encouragement of the practical arts, as adjuncts of
their ceremonies, but in the main the attitude of mind
which they encouraged was hostile to those habits of
mind which favour the growth of scientific knowledge.
But the principal driving force in the beginnings of
Science was of the practical order; the urgent necessity
for some kind of measurement, for the determination of
position in navigation by means of observation of the
stars, and for knowledge of the properties of materials
used in building and in the arts. Slowly the beginnings of
genuine scientific curiosity arose in some individuals, and
under favourable conditions in particular places. Some
account of Greek Science, and of the sharp contrarieties
within it due to the contest between genuine scientific
method and ideas of animistic and magical origin, will




INTRODUCTION


II


be given in later lectures. From the time of Leucippus
and Democritus onwards there was in Greece a real
rival to Animism; the physico-mechanical theory of the
world. In medieval times the progress due to Greek
thinkers was submerged by Scholasticism, with its belief
in occult qualities; genuine Science was only kept alive
by the Arabs, whose knowledge was partly drawn from
Indian sources. After the renaissance, involving as
it did the rediscovery of Greek thought, Science recommenced its path of progress which has continued
with unabated vigour from the sixteenth century until
our own time.
Both Animism and Magic, like the common knowledge of phenomena out of which Natural Science has
grown, have their modern descendants. If the validity
of the theories be admitted which trace back many of
the features of the more highly developed mental life of
the modern world to ancient animistic and magical ideas,
we have no warrant for treating these features with
indiscriminate disdain, on account of what appears to us
the fantastic crudity of their ancestry. It would indeed
appear that Animism, Magic, and common knowledge,
in their earliest forms and in their later development
through the ages, are all three natural growths representing persistent and normal forms of activity of the
human mind under the influence of ineradicable impulses. The cruder forms of animistic conceptions still
exist among uncivilized races to-day, and they have
never completely disappeared even in the higher races.
In the sixteenth and seventeenth centuries, in the age
in which the foundations of modern Science were being
laid by such men as Kepler, Galileo, and Newton, animistic notions took terrible shape in the employment of
the stake and the gibbet for the purpose of combating the
works of ubiquitous demons, as exhibited in witchcraft.
As the functions of the thinking part of the community became gradually more differentiated, the tension




12


INTRODUCTION


between Science and the other elements of thought
attained increased sharpness. This tension took, at
various times, the form of violent conflict; especially
when scientific ideas arose which seemed to threaten
the traditional ideas, and the higher or lower interests,
of the great religious corporations. An ancient religious
system carries with it through the centuries, in its
written or oral traditions, a great deal of material which,
either explicitly, or in its descriptive language, incorporates the views of natural phenomena which were
current at the time when the traditions first took a relatively fixed form. These views about the natural world
usually come to be regarded by most people as forming
an essential part of the system, which cannot be changed
or eradicated without entailing serious injury, or even
destruction, upon the whole religious system. This
attitude of mind is shared by the official representatives
of the system in question, who are frequently disposed
to use the influence of their corporation to combat any
attempt to replace in the popular mind the older scientific or quasi-scientific conceptions by more modern
ones due to the progress of scientific knowledge in later
times. After a longer or shorter period of strife, the
newer scientific view receives acceptance, or is at least
acquiesced in; and it is ultimately discovered that the
older elements, of a scientific complexion, which were
embodied in tradition were not really an essential part
of the system; and this even in cases in which the change
to the newer scientific view involved some modification
in the more strictly religious tradition.
I will here refer to three striking instances in which
a violent shock has been given, consequent upon the
rise of new scientific theories, to all those who had a
traditional attachment to conceptions of the physical
world which were in opposition to consequences of the
new theories. The first, and perhaps the most striking,
case to which I will refer was in the domain of As



INTRODUCTION


I3


tronomy, and involved the substitution of what is known
as the Copernican system for the older Ptolemaic
scheme. The method of representing the motions of the
bodies of the solar system which Ptolemy worked out in
detail, and parts of which were due to Hipparchus and
Apollonius, depended upon the idea that the earth is
at rest, and the sun and planets in motion. Thus the
Ptolemaic system of Astronomy was essentially geocentric. The earth being taken to be at rest, each of the
planets was supposed to describe an epicycle, by a uniform revolution in a circle, the centre of which moved
uniformly in a circle round the earth, which did not
however occupy the centre of that circle. By proper
adjustment of the radii of the circles, and of the velocities
of the planet and of the centre of the epicycle, Ptolemy
was able to give a fairly accurate representation of the
apparent motion of the planet. In this manner he gave
a systematic representation of the apparent motions of
the sun and the planets; in particular the stationary
points and the retrograde motions were represented.
He was also able to represent the principal inequalities
in the moon's motion. That the earth is a planet revolving round the sun as centre had been taught by
Pythagoras to his disciples, but the definite overthrow
of the Ptolemaic system was initiated by Copernicus,
who represented the motion of the earth and planets as
consisting of the uniform description of circular orbits
with the sun at rest within each orbit, but not at its
centre. It was only later, owing to the labours of Kepler,
that the elliptic form of the orbits, with the sun in a
focus, was established. So far, the whole change from
the Ptolemaic to the Copernican system consisted, as
we should now describe it, in the recognition of the fact
that the relative motions of the sun and planets are
much simplified by describing them as they would
appear to an observer in the sun, rather than to a
terrestrial observer; that is, by the adoption of a helio



I4


INTRODUCTION


centric description. It will be observed that, so far, the
only change consisted in setting up a simpler description
of one and the same set of facts. It was owing to the
great work of Newton that the full importance of the
change of descriptive scheme was made manifest; when
the heliocentric description of the motions of the bodies
of the solar system was seen to be consistent with a
dynamical scheme, of great formal simplicity; and one
to which the geocentric description did not lend itself.
Only after a prolonged period of bitter strife did the
substitution of the Copernican system, including the
motion of the earth round the sun, and its diurnal
rotation round its axis, for the Ptolemaic, meet with
general acceptance. The invention of the telescope contributed much to the consolidation of the new view.
The violent opposition to this change arose from the
fact that the substitution of the sun for the earth as the
body of reference in the solar system appeared to involve
a fundamental change in the importance of the earth,
its inhabitants and their affairs, relatively to the rest of
the Universe. Our world was degraded physically to
the position of a single satellite of a single star amongst
the vast numbers of bodies in the stellar Universe; and
this appeared to involve a corresponding moral and
spiritual degradation which it was thought would seriously react upon the current theological conceptions,
making it more difficult to maintain the importance
assigned by those conceptions to man in his spiritual
relations. The notion of the plurality of worlds, which
arose in this connection, was one of the chief stumbling
blocks; the teaching of Giordano Bruno, on this matter,
formed one of the chief charges on which he was condemned to death.
The second instance to which I will refer, of a shock
to traditional conceptions, was occasioned by the geological discoveries which assigned enormously greater
antiquity to the earth than was admitted by the tradi



INTRODUCTIONI


I5


tional biblical chronology, in accordance with which the
age of the earth was estimated at not more than about
six thousand years. After a shorter, and less violent,
conflict than in the case of the Copernican system, the
strength of the cumulative evidence adduced in favour
of the great antiquity of the earth led to its general
acceptance by educated persons.
Lastly I will refer to the controversies excited by the
publication, in I859, of Darwin's Origin of Species, and
in I87I, of his Descent of Man. Although the idea of
evolution of species, and even the suggestion that man
was descended from a race of lower animals, were by no
means novel, the special form which Darwin gave to
these ideas, the weight of the evidence he produced,
and the cogency of his reasoning, were such as to call
the attention of the world to these views in so striking
a manner as to produce a violent, though short, storm
of opposition from those who feared the effect upon the
traditional ideas of the spiritual nature of man, and his
dignified position in the scale of living beings. The
passionate repulsion which Darwin's theory called forth
in some quarters is illustrated by the celebrated debate
on the subject between Huxley and Bishop Wilberforce
at the Oxford meeting of the British Association in 860.
Belief in the descent of man from races of lower
animals was in direct opposition to the current views
about the Fall of man from a state of perfection, or at
least of innocence; in fact it appeared to amount to a
reversal of this idea, by substituting the conception of
a slow ascent of man from indefinitely lower conditions
of body and spirit. Here again the outcome has been
much as in the first two cases I have referred to. The
doctrine of evolution has been generally accepted by
educated people, at least as regards the physical side of
man, though not in all cases in accordance with the
detailed views of Darwin himself. By the world at large
it has been silently acquiesced in; at all events the world




i6             INTRODUCTION
has survived the shock, and the time of violent and
public opposition to the main conception of the evolution of man and other species belongs to the past.
The conflict of conceptions of which I have spoken
has lasted, with varying degrees of intensity, into our
own time. One of the maxima of this conflict occurred
about the middle of the nineteenth century, at a time
when the striking discoveries in various branches of
Science, especially in Astronomy, Chemistry, and Biology, together with the far-reaching character of such
generalizations as the law of the Conservation of Energy
and Biological evolution, had filled men of Science with
a feeling of confidence which sometimes took an aggressive form. At the present time, owing both to a
more critical examination of scientific theories, and to
the flood of newly ascertained facts which have led to
modifications of theories which were formerly supposed
to have attained final forms, and to the recognition of
distinct limitations in their scope, the attitude of men
of Science towards their theories has become much
more cautious than it was in the nineteenth century.
This change of attitude of men of Science, together
with greater openness of mind on the part of the representatives of religious thought, has led to a marked
diminution of the acrimony which has at times characterized the relations between those whose interests are
in the main scientific and those for whom religious considerations are paramount. The tension of which I have
spoken is essentially due to deep lying divergences of
mental attitude towards the world, dependent, not only
on differences due to varying types of education,
tradition, and occupation, but also to fundamental temperamental divergences. This tension is often to be
found within the complex mental make-up of one and
the same individual, impelling him either to adopt some
form of compromise between the contending conceptions and impulses in his mind, or else to set up a state




INTRODUCTION


I7


of equilibrium in which the diverging sets of ideas are
retained, so to speak, in separate compartments of his
mind, and lie side by side without coming into direct
conflict with one another. But this kind of mental dissociation is not possible for all persons; and thus in
many minds a state of mental unrest, of uncertainty, of
alternations of scepticism and belief, has been produced
which is not conducive to that contentment of mind
which is requisite for the highest efficiency of active
agents in the work of the world. Even in our own time
there are a considerable number of persons, some of
them highly cultivated and intelligent, whose feeling
towards the Natural Sciences is one of suspicion or
repulsion, sometimes conscious, but perhaps more often
unconscious or instinctive. This feeling of hostility or
repulsion is partly temperamental, due to a distaste for
the schematizing habits of mind of scientific thinkers,
which are alien to those more intuitional modes of
apprehension that are congenial to many minds. But
this feeling is also largely, and perhaps in a preponderating degree, due to a fear that the Scientific view of
the world leaves no room for the domain of freedom,
spontaneity, and values, for teleological conceptions, or
generally for the spiritual order of things.
The undeniable success of Natural Science, of the
palpable order, exhibited in the mechanical inventions
which have transformed modern life, while investing
Natural Science with a certain glamour, has sometimes
intensified this fear. That this fear is not prima facie
groundless, or due to the merely conservative habits of
timid minds, appears with clearness when the bold
attempts are taken into account which were made by
various representative men of Science, during the last
century, to erect an all-embracing World-Philosophy
on the basis of a Mechanical theory of Nature. It is
instructive to consider, in this connection, some specimens of the pronouncements of prominent men of
H GL




I8               INTRODUCTION
Science. Perhaps the most famous utterance of this
kind is that of Laplace in his essay on probability (I812),
which exhibits in the most striking way the feeling of
confidence produced by the triumphant success of
Astronomers in applying the law of gravitation to the
calculation of the motions of the bodies of the solar
system. Laplace writes:
We ought then to regard the present state of the universe as
the effect of its antecedent state and as the cause of the state
that is to follow. An intelligence, who for a given instant should
be acquainted with all the forces by which nature is animated
and with the several positions of the beings composing it, if
further his intellect were vast enough to submit those data to
analysis, would include in one and the same formula the movements of the largest bodies in the universe and those of the
lightest atom. Nothing would be uncertain for him; the future
as well as the past would be present to his eyes. The human
mind, in the perfection it has been able to give to astronomy,
affords a feeble outline of such an intelligence. Its discoveries in
mechanics and in geometry,joined to that of universal gravitation,
have brought it within reach of comprehending in the same
analytical expressions the past and future states of the system of
the world. All its efforts in the search for truth tend to approximate it without limit to the intelligence we have just imagined.
As a remarkable example of confident scientific dogmatism the following quotation from the preface to
Biichner's well-known work, Kraft und Stoff, published
in I855, is of much interest:
If these pages may venture to claim any merit or characteristic, it is that of representing a determination not to shrink
with dismal horror from the simple if unavoidable consequences
of an unprejudiced contemplation of nature from the standpoint
of empirical philosophy, but to admit the truth regardless of
what may follow. We cannot make things different from what
they are, and nothing seems to us more preposterous than the
attempts of some distinguished naturalists at introducing
orthodoxy into natural science. We do not pretend to bring
forward anything absolutely new or anything that had never
been heard of before. Similar views and views cognate to ours




INTRODUCTION


I9


have been taught in all ages, and some of them were laid down
by the oldest Greek and Indian Philosophers, but their groundwork, which is necessarily empirical, could only be supplied by
the progress of natural science in the present century. It is
therefore obvious that these views, in their present clearness and
consistency, are essentially a trophy of modern times, and closely
related with the new and gigantic achievements of empirical
science. Indeed, scholastic philosophy, ever riding the high,
though from day to day more and more emaciated horse, lays
the flattering unction to its soul that these views have long been
disposed of, and would fain consign them to the limbo of
oblivion, with which object it has labelled them "Materialism,"
"Sensationalism," "Determinism," and so on; nay, the gentlemen of that school go so far in their assumed supercilious
superiority, as to talk of having given them "the historical
quietus." But they themselves are going down day by day in
the public estimation, and losing ground in their speculative
hollowness before the rapid rise of the empirical sciences, which
are making it daily more evident that both the macrocosmic and
microcosmic worlds obey, at every stage of their genesis, existence and subsidence, the mechanical laws which lie in the very
nature of things. Starting from the recognition of the indissoluble relation that exists between force and matter as an
indestructible basis, the view of nature resting upon empirical
philosophy must result in relegating every form of supernaturalism or idealism from what may be called the hermeneutics
of natural facts, and in looking upon these facts as wholly
independent of the influence of any external power dissociated
from matter. There seems to us to be no doubt about the
ultimate victory of this realistic philosophy over its antagonists.
The strength of its proofs lies in facts, and not in unintelligible
and meaningless phrases. But in the long run there is no contending against facts, it is useless to kick against the pricks.
Again, in his chapter on Thought, Biichner writes:
That thinking is and must be a mode of motion is not merely
a postulate of logic, but a proposition which has of late been
demonstrated experimentally.
Consciousness, like thought, is a performance or action or
phenomenal activity of certain parts or tissues of the brain, and
in that capacity it is subject to all the changes which take place
in the condition, nutrition and growth of the brain.... How and
in what way the atoms, the nerve cells, or, to speak generally,
2-2




20               INTRODUCTION
matter began to produce and bring forth sensation and consciousness, is quite unimportant for the purpose of our investigation, it is sufficient to know that such is the case.... The simple
solution of the problem lies in the fact that not only physical
but also psychical energies inhere in matter, and that the latter
always becomes manifest wherever the necessary conditions are
found, or that, wherever matter is arranged in a certain manner
and moved in a certain way in the brain or the nervous system,
the phenomena of sensation and thought are produced in similar
fashion, as those of attraction and repulsion are under other
conditions.... The development of mind from matter is indeed
one of the latest, most difficult and most complicated triumphs
of physical forces, and is the product of a protracted toil, rising
from step to step, through countless centuries, till reaching the
height of humanity. Nor can we say what shall be brought
forth of similar fruit by the coming ages: we must confess that
perhaps as yet we see only the incomplete, the imperfect, and
that perchance we have no conception of what matter may yet
be able to accomplish in its further evolution in mental phenomena and faculties, by further complications and yet more
highly developed forms of motion.
The same idea of the primacy of matter over life and
mind has been expressed by Tyndall:
Divorced from matter, where is life to be found? Whatever
ourfaith may say, our knowledge shows them to be indissolubly
joined. Every meal we eat, and every cup we drink, illustrates
the mysterious control of Mind by Matter.
Huxley, in spite of the fact that he professed to hold
idealistic opinions, expressed views somewhat similar
to those of Biichner and Tyndall, as to the relation of
matter and consciousness. He writes1:
There is every reason to believe that consciousness is a
function of nervous matter, when that nervous matter has obtained a certain degree of organization, just as we know the
other actions to which the nervous system ministers, such as
reflex action and the like, to be. As I have ventured to state my
views of the matter elsewhere "our thoughts are the expression
of molecular changes in that matter of life which is the source
of our other vital phenomena.".. I really know nothing whatever,
and never hope to know anything, of the steps by which the
1 Critiques and Addresses, I873, p. 283.




INTRODUCTION


21


passage from molecular movement to states of consciousness is
effected.... All that I have to say is that, in my belief, consciousness and molecular action are capable of being expressed by one
another, just as heat and mechanical action are capable of being
expressed in terms of one another.
Whatever we may think of the crude dogmatism of
some of the pronouncements I have quoted, we must
not be blind to the enormous effect which such ideas
have produced, on the one hand, in inducing large
numbers of persons, especially among the half-educated,
to believe that these and similar views, expressed by
prominent representatives of Science, embodied completely demonstrated results of Science, and on the other
hand, in producing a repulsion to Science, often of an
undiscriminating character, among many who feared its
disintegrating influence upon cherished conceptions and
beliefs. It is for this reason that I have dwelt at some
length upon the extreme claims of complete supremacy
made by scientific thinkers in the last century. It will
be observed that these claims were made on behalf of
the special mechanical theory of matter in vogue in the
nineteenth century, so extended as to have become a
complete theory of the physical and psychical domains.
The effect of the later advances in Physics in the last
thirty years, especially in the domain of Electromagnetism, has however been such as to produce a revolutionary change in many of the older conceptions of the
properties of matter. The flood of new facts which have
been discovered in connection with radiation and radioactivity has thrown the most serious doubts upon the
range of applicability of the mechanical theories of the
eighteenth and nineteenth centuries, at least in the forms
in which they were then held. The bold extension of
those theories into a World-Philosophy has, in consequence of our extended knowledge, lost much of what
plausibility it ever possessed. Not only the electromagnetic theory of the constitution of atoms, and the




22


INTRODUCTION


quantum theory of energy, but also the latest theory,
involving fundamental changes in our conceptions of
the measurement of time and space, and of gravitation,
have shaken to their foundations the notions upon
which the older mechanistic theories were based. It
must not, however, be assumed that, however great are
the modifications which physical theories of material
phenomena may undergo in consequence of increased
knowledge of facts, the possibility has been for ever
removed that some physical theory may again arise
which may make claims of a far-reaching character
similar to those which were made on behalf of the
mechanical theory of matter in the nineteenth century.
From the point of view of general thought, the really
fundamental question in this connection is whether, or
how far, it is possible to represent the physical world as
a closed and independent system of deterministic type,
uninfluenced by the psychical world. Whether such a
closed scheme be what has been called mechanistic or
not, is, from the point of view of those who are concerned primarily with the relations between scientific
thought and the larger world of thought, a matter of
secondary importance.
As a result of modern criticism of the foundations of
Science, a view of the nature of scientific laws and
theories has arisen, principally from the ranks of men
of Science, which differs in some important respects
from the opinions formerly held on such matters. This
view, which I propose to explain in the next three
lectures, has met with acceptance by many men of
Science of eminence, and by some Philosophers. But
it would be going too far to say that the scientific world
in general has attained that degree of emancipation from
certain ideas which have held sway for ages which
would be implied in the complete acceptance of the
view of the domain of Natural Science to which I refer.




II


SCIENTIFIC LAWS AND THEORIES
A GENERAL survey of the world, from the ordinary
point of view which we call that of common sense,
makes apparent to us the existence of two great complexes or domains; that which we call the physical or
material, and that which we call the psychical or mental.
This is the ordinary dualism of matter and mind; a
dualism which we all accept for the common purposes
of everyday life, when we are not philosophizing, and
which was erected into a philosophical system by
Descartes and his followers. We regard ourselves as
sentient beings, endowed with consciousness, with the
powers of feeling, thinking, willing, and remembering,
as psychically active. We appear to receive from without, during our waking hours, a continuous stream of
sense-impressions leading to what we call the perception of external objects and events, or grouped senseimpressions. These perceptions succeed one another in
time, and change with greater or less rapidity. Although
our percepts are in a continual state of flux, the changes
in them are not entirely haphazard, but exhibit a considerable degree of regularity of sequence, of such a
kind that we possess a certain amount of power of prediction of their future course. This power of prediction
is an essential condition of the possibility of any actions
on our part, because, in its absence, we should have no
knowledge of what the effects of any of our actions
might be. We find moreover that we possess certain
powers of classifying and sorting out our percepts, of
forming by abstraction from them permanent concepts,
under which whole classes of our percepts are subsumed. Among the perceptual objects of which we are




24   SCIENTIFIC LAWS AND THEORIES
aware are the bodies of other individuals, and we
believe, on indirect evidence (leaving aside such debatable matters as the existence of telepathy), that these
individuals, like ourselves, have perceptions, and powers,
similar to our own, of classifying these perceptions, of
abstraction, of forming concepts, and of remembering.
We also believe, as a matter of inference from experience, that the actions of other individuals are influenced
by their perceptions in a manner similar to that which
we find to be the case in ourselves. Further, we believe,
also on inferential grounds, that the percepts of other
individuals have a large measure of resemblance to our
own; and that other persons discern regularities in the
sequences of their perceptions which are closely related
to those which we ourselves discern. We all alike have
the apparent power of producing, by the exercise of our
wills, changes in the perceptual domain or physical
world. Such changes in the external environment, involving the use of our voices, or the motion of parts of
our bodies, apparently produced by our wills, become
percepts for other individuals, and thus form the medium
of communication between one individual and another.
The means of communication of ideas between individuals, thus conditioned, lead to the formation of a
body of common knowledge of the perceptual world,
which is regarded as accessible to all normal individuals.
If we regard this picture as descriptive of what has been
going on during an indefinite past, and if we take into
consideration the cumulative effect of the process, combined with the memories of individuals and the traditions of the race, we have, from a non-philosophical
point of view, an account, of some plausibility, of the
way in which common or public knowledge of the
physical world may be regarded as having grown up.
It must, however, be observed that the experience of an
individual is, in its fulness of detail, unique and incommunicable; it is only incompletely made intelligible to




SCIENTIFIC LAWS AND THEORIES 25
other individuals by means of language or other forms
of symbolism. Thus language, in its very nature, involves abstraction in which the elements of actual
individual experience are replaced by symbols which
fail to represent with absolute completeness what they
are designed to describe.
It is a part of that view of the nature of Natural
Science, or the Science of physical percepts, which I
propose to develop, that Science is essentially a purposive continuation of the formation of what I have
called common knowledge, but carried out in a more
systematic manner, and to a much higher degree of
exactitude and refinement. The earliest stages in the
formation of scientific knowledge are of the kind which
may be described as classificatory. Physical objects are
arranged in classes, in accordance with observed similarities in the objects assigned to any one class; physical
events or sequences of events are classified in accordance
with observed regularities or similarities in those events
or sequences. There is, however, always a certain degree
of arbitrariness involved in the selection of the precise
similarities or regularities which form the basis of the
classification; thus at the very commencement of the
process of building up scientific knowledge, the analyzing
and generalizing powers of the mind find scope for their
activity, and are necessary factors. The result of this
process expresses itself in the formation of abstractions
or concepts, which are not identical with any of the
perceptual objects or happenings which conditioned
their formation, but which serve as a conceptual symbolization or representation of those aspects of the latter
which were regarded as alone relevant for the purpose
of classification. A rule, or law, which affords a conceptual description of a particular kind of sequence of
physical events in the perceptual world is set up in the
first instance in the manner I have indicated, on the
basis of actual observation or experiment, dealing with




26   SCIENTIFIC LAWS AND THEORIES
a selected class of sequences of physical events. It is
then extended hypothetically to the descriptive representation of sequences which may occur under conditions not in all respects identical with those in which
those observations were first made which suggested the
law. When this hypothetical extension is justified by
further experiments or observations, and especially by
its power of prediction of what will happen, or be
observed under certain conditions, the rule or law becomes what is usually called either a scientific law or
principle, or a law of nature. The term law of nature
has too frequently been taken to imply that it meant a
purely objective law, as it were inherent in the perceptual world, which natural phenomena must of necessity obey, whereas a law of nature is in reality a conceptual law set up by the activity of the mind of man,
but conditioned as regards its validity by the perceptual
world which must be taken as a datum. The law is
constructed by the mind, but not purely arbitrarily; it
has limitations other than, and besides those imposed by,
the canons of thought. In the physical complex there
is the essential element of fact, which must be taken as
a given, and unalterable, datum required for the construction of the law. Thus the complex of phenomena
or appearances which we call the physical world must
be taken to be in fact such that the law must stand the
test of applicability for the purpose of resuming certain
tracts of uniformity in it. Natural Science need not
however go beyond this recognition of the existence of
this element of fact; it is unnecessary for its purposes to
make the assumption that a single law has a precise
correspondence with a single definite set of relations
which actually subsist in Nature. Still less is it necessary
for the purposes of Natural Science to assume that the
law corresponds to a set of relations between real
entities. The formation of precise views on these
matters is the task of Philosophers; the man of Science




SCIENTIFIC LAWS AND THEORIES 27
need not go beyond the superficial view which contents
itself with the recognition of the essential element of
fact in the world of percepts.
The two opposite tendencies, either to regard natural
laws as entirely constructible by the mind, in accordance
with what were regarded as a priori necessities of thought,
or on the other hand to regard them as something
entirely external to, and only discerned by, the mind,
have both been prominent in the history of Science.
It may perhaps be the case that some of the writers who
uphold the "descriptive" view    of the character of
Natural Science have so expressed themselves that the
importance of the element of fact or "givenness" in the
realm of percepts appears to be unduly minimized. Both
the activity of the mind, and the data of perception, are
factors in the genesis of a so-called law of nature.
Natural laws have been characterized by Mach1 as
"abridged descriptions" and as "comprehensive and
condensed reports about facts." It may be doubted
whether such a characterization is quite adequate as
applied to conceptual schemes in general. The laws
always contain less than the facts, because the concepts
which the laws employ in their statements are but symbolical representations which leave out of account those
differences of individual detail which are to be found in
perceptual objects or events which fall under a particular
class. In the words2 of Boltzmann:
It has never been doubted that our ideas are merely images
of the objects (or rather symbols for them) which have a certain
relationship with the objects, and never completely correspond
to them, but are related to them as letters to sounds or notes to
musical tones. Also on account of the limitation of our intellect
they are able only to depict a small part of the objects.
There is no absolute line of demarcation between
scientific knowledge and common knowledge. The
1 Popular Scientific Lectures, Trans. McCormack, p. 193.
2 Vorlesungen ueber die Principe der Mechanik, Vol. I, pp. 1-2.




28 SCIENTIFIC LAWS AND THEORIES
knowledge which we have, that solid bodies, when unsupported, fall to the ground, would hardly be described
as scientific knowledge, but it is not in essence distinct
from other rules of the order of phenomena which we
do dignify by the name scientific knowledge, on account
of the fact that they rest upon experience which is less
readily analyzed, or separated off from the general complex of physical experience, or which requires more
refined methods of observation. But the law of falling
bodies, in accordance with the more exact observations
made by Galileo involving numerical determination,
became a genuine scientific law. By the process of
hypothetical generalization it was extended by Newton
into the general law of gravitation, the power of which
to represent to a great degree of accuracy the observed
motions of the bodies of the solar system was brilliantly
established by Newton himself and a number of eminent
researchers in the field of Gravitational Astronomy.
A law, or rule, which refers to some particular kind
of sequence of events, under certain conditions, and
which is regarded as a scientific law, or principle, is
very often constructed by means of artificial production
of the set of conditions under which the sequence of
events in question is observed to take place; the law is
then said to have been discovered by experiment. In
other cases, in which the conditions are not produced
artificially, but occur without the intervention of the
observer, the law is said to be discovered by observation.
But in all cases, the discovery, or rather the construction,
of a scientific law involves that synthetic activity of
thought which manifests itself in a constructive process
in which actual percepts are employed only as the raw
material and starting point of the mental process. In
the attempt to discover a scientific law, a selective process is requisite in regard to the percepts, some greater
or lesser part of what is perceived must be ignored, as
irrelevant to the purpose on hand; this selective pro



SCIENTIFIC LAWS AND THEORIES 29
cedure amounts to a process of abstraction, in which
some elements of our actual percepts are removed, and
not attended to. The degree of abstraction employed,
and thus the degree in which the concepts differ from
percepts, varies greatly in different departments of
Science, and also varies greatly according to the stage of
development which a particular department has reached.
A scientific law is accordingly always, in some greater
or lesser degree, abstract, in the sense that it represents
only a part of what is in any individual case actually
perceived; it describes a particular sequence of physical
events which, in an actual case, is accompanied by other
percepts or events in relation to which the law has no
application. For example, when Kepler discovered the
law that the earth and planets describe elliptical orbits
with the sun in one focus, he abstracted from all the
physical properties of the sun and planets, even from
their sizes; all the infinite details of their physical constitution being irrelevant to his particular purpose of
describing the main features of their relative motions.
A complex of physical facts, given as percepts, appears
in the first instance to be confused and irregular, on
account of its complexity; but the effect of further
scrutiny is gradually to reduce it, in some degree at
least, to order, when similarities and relatively permanent
elements are discovered within the complex; and the
ultimate result of this process is that we are enabled
mentally to reconstruct groups of facts within the complex. It must, however, be observed that in Nature
there are no precise repetitions in every detail, for conditions are never on two occasions absolutely identical.
Absolute likeness of sequence exists only in our conceptual schemes of representation, not in the world of
percepts. With the purely individual, Science cannot
deal; it operates with the typical, and a type is an
abstraction. Even in the preliminary work of classification of those groups of sense-impressions which we call




30 SCIENTIFIC LAWS AND THEORIES
physical objects, such as is undertaken in the classificatory stage of Zoology, or in systematic Botany,
abstraction is made of individual differences which
distinguish one member of a group from another. Thus
two plants of the same order, genus, species, and variety,
are never exactly alike in all respects; they are judged to
be of the same variety because they are alike in certain
particulars defined in accordance with the arbitrary
rules which are employed for the purpose of classification. A plant of a given variety, or of a given species, is
an abstract conception; the successive grades of classification represent different degrees of abstraction in the
formation of the concepts that correspond to them. It
appears then that, even in the earlier stages of scientific
thinking, what we really work with are not the percepts
themselves, but concepts which symbolize types of
percepts, and in the formation of which concepts some
of the elements of actual percepts are left out of account.
It is, however, only in the earlier stages of development of a branch of Science that the procedure consists
mainly of the classification of objects by means of the
substitution of concepts for the objects themselves, or
of the construction of the kind of rules which form the
simpler laws of sequence of events. As the Science
advances, such laws are generalized into those more
comprehensive laws which are often called scientific
principles. A law, or principle, which has reached a
high stage of generality, or a group of such laws considered as forming a single body of doctrine, is what is
known as a scientific theory, and this forms a conceptual
scheme under which a wide class of physical sequences
is subsumed. But whether such a conceptual scheme
is called a theory, a principle, or a law-and that is to
some extent a matter of usage, or historical accidentits genesis always involves a more or less extensive
constructive mental process, or synthesis. Such a conceptual scheme involves the employment of conceptual




SCIENTIFIC LAWS AND THEORIES


31


objects, the character of which, and the relations between which, are assigned as postulations of the scheme.
These postulations must always be regarded as hypothetical or tentative. The possibility of setting up such
a scheme has actual physical experience as its essential
condition, but the constructive and generalizing work
of thought is no less essential. The original function of
such a scientific theory, or conceptual scheme, is to
provide an ideal representation of some more or less
restricted range of physical phenomena as actually observed, that is of certain sequences and regularities in
percepts. But the functions of a conceptual scheme are
much wider than those of merely describing symbolically
what has actually been observed. The scheme is applied
hypothetically to predict what will be observed in
circumstances which differ in some degree, or in some
characteristics, from those in which the experiments or
observations which led up to the theory were made.
The value and the range of validity of the particular
conceptual scheme have to be estimated by its actual
success in the fulfilment of this function of prediction.
Thus a scientific theory, considered as an hypothesis,
not as a dogma, must be judged by a comparison of its
consequences with perceptual fact; and especially by
its power of forecasting the occurrence of hitherto
unknown or unobserved facts, when, by more minute
observation, or by artificial production of experimental
tests, the occasion for such comparison arises.
The different conceptual theories which are employed
in the various branches of Natural Science vary greatly
in their degrees of abstraction, and in their degrees of
precision. In the most abstract and precise theories, the
language employed is the most precise form of language
which we possess, that of arithmetic and its generalization in Mathematical Analysis. As any branch of Science
progresses to a higher stage of development, there are
to be found at least portions of it in which it becomes




32   SCIENTIFIC LAWS AND THEORIES
possible to employ theories of this character. There
exists a distinction between two species of ideal elements
of a conceptual scheme which it is important to recognize. Some concepts have direct perceptual counterparts, such concepts having been formed by a direct
process of abstraction, in which, what are, for the
purpose of formation of the scheme, irrelevant characteristics of the perceptual objects or processes have been
removed by the abstraction. Concepts of the other
species have no such direct perceptual counterparts, or
it is not assumed a priori that they have such; they are
formed by an effort of constructive imagination, for the
purposes of the representative scheme. Although concepts of this latter kind may be regarded as due to the
creative activity of thought, it must always be remembered that they would never be formed apart from a
basis of actual physical experience. In any scientific
theories in which both kinds of concepts occur, those
which have no immediate perceptual counterparts must
be regarded, at least provisionally, as purely ideal elements of the scheme, in fact as auxiliaries necessary for
the purpose of the formation of a self-contained conceptual scheme which shall serve its purpose of providing a sufficient mode of representation in thought of
the particular domain of physical events and objects. In
less advanced conceptual schemes, as for example in
those of the purely classificatory order, all the concepts
employed may be simply abstract forms, or types, of
perceptual elements. Such concepts are usually however
more precise in character than those employed in
ordinary intercourse, the definition of their characters
and relations having sharper outlines. The power which
we possess of introducing into a scientific theory, as part
of the edifice, ideal elements which do not directly
correspond to clearly defined percepts, although essential
to the more advanced work of Science, is one which has
its obvious dangers, and these dangers have by no




SCIENTIFIC LAWS AND THEORIES 33
means invariably been avoided in practice. In a workable and satisfactory scientific theory such conceptual
elements should be as few in number as possible, and
of a simple and well-defined character; otherwise the
theory has the grave defect of over-complication, or
even of vagueness, and the fundamentally important
requisites of a scientific theory which shall serve its
purposes of representation and prediction, those of
clarity and simplicity, are not satisfied. Such ideal concepts only serve their purpose provided they are subject
to precise definition. In some cases the introduction of
such elements into a theory has amounted to little more
than the employment of new words, which, on account
of the indistinctness of their denotation, have only
served the purpose of labelling our ignorance, and have
consisted of a futile attempt to disguise inability to set
up a really adequate representative scheme. If, when
each new difficulty arises, new concepts are invented for
the purpose of shelving those difficulties, the theory
becomes so overloaded that it ultimately perishes by its
own weight.
It has perhaps been in the biological sciences, which
have to deal with ranges of phenomena of the highest
degree of complexity, that this kind of danger has been
oftenest incurred; we may refer to the various vitalistic
hypotheses to illustrate the point. Some such insufficiently defined concepts have often been introduced for
the purpose of remedying, by means of an enlarged
scheme, the recognized inadequacy of some older and
more restricted descriptive scheme. In Physics, for
example, the temptation to introduce a new ether, or a
new substance with insufficiently defined properties, in
order to remove each particular inadequacy of an older
mechanical scheme, is one to which men of Science have
frequently yielded. However, in scientific theories of
the highest type, there may exist such ideal elements
without direct perceptual counterparts, the danger of
HGL                                       3




34 SCIENTIFIC LAWS AND THEORIES
the kind I have indicated being successfully avoided.
It is precisely in the proper selection and definition of
such conceptual elements of a theory that the highest
powers of the great men of Science, who are necessarily
supreme Artists, have been exhibited. Descriptive
schemes which employ concepts, the smallest in number, and the most definite in character, are the distinguishing mark of Science at its best.
In some cases it is preferable, for expository purposes,
not to reduce the concepts of a scheme to an absolute
minimum, but to retain some which are reducible to
others, as auxiliary concepts. This does not, however,
afford a dispensation from the duty of investigating
what is the smallest number of irreducible concepts. A
theory is sometimes capable of being stated in more
forms than one, according to the choice made of the
irreducible concepts, and thus some latitude arises as
to the precise form of the theory. I may refer to the
case of molar Dynamics as an illustration of this possibility. Those departments of Natural Science in which
the theories, or conceptual descriptive schemes, are
furthest removed from the region of the perceptual, and
which are thus characterized by a higher degree of
abstraction than has as yet been attained by other
branches, are Mathematics, and those parts of Physical
Science which have become in a considerable degree
amenable to mathematical treatment. It is the relative
simplicity of those aspects of the perceptual world with
which Mathematics and large parts of those Sciences
which concern themselves with non-living matter deal,
as compared with the Biological Sciences, which accounts for their relatively advanced character. Number
and Extension, with which Arithmetic, in the extended
sense, and Geometry, are alone directly concerned, can
be developed as sciences descriptive of only the most
superficial aspects of the perceptual world, since they
may leave aside almost all the properties and qualities




SCIENTIFIC LAWS AND THEORIES              35
of perceptual things as irrelevant for the purpose of
their construction, so that in these departments the
process of abstraction and idealization can take gigantic
strides at a very early period in their genesis. Other
branches of Science, in which the disentanglement from
one another of great complexes of properties, qualities,
and sequences is a much more difficult process, are on
that account such that the stages of their growth and
development are much slower, and much more difficult
than in the case of Geometry and Mathematical Analysis.
Notwithstanding the very great difference, as regards
the degrees of abstraction, in the conceptual schemes
which belong to different branches of Science, the
difference between them need not, I venture to think,
be regarded as generic. This difference seems rather to
be one of degree of advance in the scope and character
of the abstract schemes of representation which have at
the present time been devised for describing the various
regularities of sequence with which the various branches
have respectively to deal. On the assumption of the
correctness of this view, the Science of Geometry, which
on the perceptual side starts solely with the extensional
relations of bodies, forms a kind of model, to which
other branches of Science, in their further advance, will
gradually conform. This conformity with the model
may, in the cases of most branches of Science, not
become complete in any time for which we may venture
to make a forecast, but their advances will consist of
progress in the direction of such conformity. Geometry,
the Science of spatial relations, is essentially in its origin
a physical science, but it has long ago become a deductive
Science in the sense that a complete conceptual scheme
has been constructed, of such a character that all
ordinary special spatial properties are logically deducible,
in their ideal form, from the postulations of the scheme
itself. These deductions are then applicable for the determination, subject to inevitable errors of measurement,
3-2




36 SCIENTIFIC LAWS AND THEORIES
of the corresponding relations in perceptual space.
In our actual Geometry, purely experimental determinations of special spatial properties are no longer
necessary, but only serve for purposes of illustration.
It is not inconceivable that the labours of Physicists,
in a domain on which a flood of light has fallen in the
last decades, may result in the production of a schematic
representation of the composite nature of the atom, so
precise that the various possibilities as regards the
detailed structure of the atom may be worked out. If
this were accomplished it might be possible to deduce
from such a conceptual scheme the main characteristics
of the different chemical elements, and the nature and
character of their possible combinations. The Science
of Chemistry would then have become deductive in the
same sense as that in which Geometry is a deductive
Science. However far from the attainment of this goal
most branches of Science may be, the general character
of progress in them consists of the discovery of ever
larger tracts of phenomena that are capable of conceptual description by means of schemes which succeed
in predicting further phenomena which had not been
observed until such predictions were tentatively made.
This view of the nature of a scientific theory, the
most general outline of which I have traced, that it
essentially consists of a conceptual scheme, designed by
the synthetic activity of the mind, working with the data
of perception, for the purpose of representing particular
classes of sequences and regularities in our percepts,
has been powerfully advocated in recent times by
Kirchhoff and Mach on the continent, and by K. Pearson
in this country; some indications of a similar view are
to be found in the writings of Auguste Comte and other
earlier writers. I propose to follow it out in further
detail, and to illustrate its application in various special
departments of Science which I shall consider in later
lectures. It will be found that the adoption of this




SCIENTIFIC LAWS AND THEORIES 37
point of view involves changes in the older traditional
conceptions of the relations between scientific thought
and general Philosophy. These changes will be seen to
have the effect of according to Natural Science a more
independent position in relation to Philosophical theories
than it has often been supposed to occupy. But they
will also have the effect of placing limits to the functions
of Natural Science as one of the factors which contribute
to the construction of our general view of the world; and
also of assimilating in a large degree, though not wholly,
the attitude of Natural Science towards the great speculations of Philosophy with the attitude of what is usually
called common sense. The ambiguity of meaning of
various expressions employed in their writings by men
of Science often makes it difficult to be sure whether,
or how far, they are in agreement with the descriptive
view of the nature of Natural Science of which I here
give an account.
We have taken as our starting point the fact that we
all have streams of percepts which we call our physical
experience. The reduction to some kind of order and
regularity of particular kinds of such percepts, by means
of a representation of them by conceptual schemes, is
what I have maintained to be the function of scientific
theories and laws. The important question, however,
arises that, as a percept always involves a percipient,
who is the percipient whose percepts are conceptually
described in this manner?
We cannot take Science to deal merely with the
private percepts of some one particular person. The very
essence of scientific knowledge is that it shall be what
we may call public knowledge, that is, not the partially
incommunicable knowledge of some one individual. It
is true that, at a particular time, a particular scientific
theory may only be known to a single person, its discoverer; but it is essentially capable of being communicated to, and understood by, other persons whose




38 SCIENTIFIC LAWS AND THEORIES
training and previous knowledge fit them to receive it.
In the form in which a scientific theory, or law, is
usually stated it seems to describe facts or occurrences
which are independent of any and every observer, but
the presence of some percipient is always implied. This
implied percipient must be regarded as placed in circumstances which enable him to have the perceptions, and
these circumstances include the use of necessary instruments for extending the scope of his senses, such as
telescopes, microscopes, etc.; he is assumed also to be
endowed with normal powers of perception. The answer
to the question, who is the implied percipient in a
descriptive scientific theory, is then that the theory consists of the conceptual description of what would be
observed by any normally constituted observer supposed
to be placed in a proper position, and under suitable
circumstances, and provided with the necessary appliances. Such an hypothetical observer is always implied
in the statement of a theory or law, although he is
usually not explicitly referred to. I should, however,
observe that in the latest physical theory, that of general
relativity, certain special circumstances relating to the
observer are taken into account as part of the theory
itself.
No scientific theory is designed to describe at once
the whole of the perceptions which an observer would
have under given circumstances, but only particular
classes of perceptions arbitrarily separated from the rest,
in accordance with the particular kinds of percepts and
their sequences which the theory is designed to describe,
that is, according to the particular range of phenomena
with which the particular theory deals. Thus, for example, a dynamical theory of the motions of molar
bodies will be a conceptual scheme designed to describe
only the motions of actual bodies as they would appear
under assigned circumstances to an observer who perceived those motions, and the theory would have no




SCIENTIFIC LAWS AND THEORIES             39
concern with the perceptions which such observer would
have at the same time of the colours and other physical
characteristics of the moving bodies, or of any sounds
which they might emit; these would form the subject
matter of other theories. Thus the actual perceptions
of an observer are described as it were piece-meal by
different theories belonging to different branches of
Science. A vast part of our percepts has hitherto not
been found to be amenable to abstract description of a
precise character, by any conceptual scheme.
Considerable branches of Science are concerned with
attempts to trace back into the remote past complexes
of phenomena of certain special kinds, that is to give
an historical retrospect in a particular domain. Some
branches of Science also undertake to perform a similar
function as regards the future. Geology, for example,
undertakes the task of tracing the history of the earth's
crust; Cosmical Astronomy attempts to depict the history of the evolution of the solar system and of stellar
systems. Biological Science has made great efforts to
construct a general account of the evolution of living
organisms and their various species. It is obvious that
no single percipient such as we are acquainted with can
be regarded as having followed out the changes even in
a single geological period, or the evolution of a single
race of animals, let alone the evolution of the solar system.
I do not, however, think that the general scientific
schemes which purport to describe what has happened
in very long, but of course strictly finite, periods of time,
need be regarded as falling under a different category
from those which describe the short time processes observable by a single percipient. We may regard them
all as hypothetical attempts to describe what would
actually have been perceived by an observer at any time
falling within one of the long periods in contemplation,
had such an observer been present at that time. These
historical accounts have been constructed on the basis




40 SCIENTIFIC LAWS AND THEORIES
of a knowledge of present conditions and of actually
observed processes which take place during short periods.
Their construction involves the piecing together of a
large number of short time processes, and of utilizing
our conceptual knowledge in such a way as to construct
a conceptual scheme, of a strictly general character,
which serves to give a consistent account of what we
may hypothetically regard as having led up to the actual
present conditions. It is of course clear that these
histories of processes through long periods involve a
large element of hypothesis, and a degree of uncertainty,
and especially of indefiniteness as regards lengths of
time, much greater than those which attach to conceptual accounts of short time processes which can be
observed by a single percipient.
In those advanced scientific theories which are of the
quantitative kind, that is which describe conceptually
the results of actual measurements, there is always an
element of approximation in the application of the
theory, because all actual measurement is approximate
only. Absolutely exact measurement is conceptual only,
and resides in the conceptual scheme, not in the actual
measurements that can be made in the perceptual domain. Thus, even in the case of a quantitative theory,
there is an essential element of approximativeness in the
power of a scientific theory to represent actual perceptions; much more is this so in the numerous cases in
which the theory is not quantitative but qualitative only.
It is in general only in the more advanced branches of
Science that quantitative Mathematical representation
is available. But in many branches, such as Chemistry
and Physiology, there are, in ever increasing numbers,
particular ranges of phenomena which have been made
amenable to such treatment, although in some cases
such ranges of phenomena form but a very small part
of the whole complex of phenomena with which each
such particular department of Science deals.




SCIENTIFIC LAWS AND THEORIES 41
An essential characteristic of every scientific theory is
that it only serves to give a conceptual description of a
range of phenomena which is of a limited and circumscribed character, spatially and temporarily. It only
describes what an observer would perceive in some
portion of space which, however great, is limited by his
powers of perception, even when those powers are increased by the employment of appropriate instruments.
Further, it only describes what will happen in some
limited lapse of time, however great that period may be.
It also only describes those percepts which are of some
more or less limited type. Thus the percepts described
by a theory are bounded, spatially, temporarily and
generically. There always remain possible percepts beyond any particular range, to describe which the theory
is applied. As regards time, the theory is wholly inadequate to describe any absolute beginning or end of
sequences of phenomena. The period of time to which
it may be applied is capable of being extended in both
directions, at least hypothetically, either within the
scope of the particular theory, or by means of some
more comprehensive theory applicable to phenomena
which take place in a greater period of time, during
which we can assign, at least tentatively, the relevant
circumstances. Just as absolute origins and terminations
of sequences of phenomena are beyond the reach of
scientific theory, so also is unbounded perceptual space.
Even the perceptions of the Astronomer are bounded
by the limitations of the telescope and of the photographic plate; the space of his perceptions is strictly
finite, however enormous its dimensions may be, as
estimated in our ordinary systems of measurement.
Spatial perceptions have also lower limits imposed by
the limitations of the microscope.
Although conceptually we can regard portions of
actual space or of time as capable of indefinite increase,
at any actual stage of the indefinite regress in which we




42   SCIENTIFIC LAWS AND THEORIES
are thus involved, the portion attained is strictly finite.
Thus we are not entitled to regard Science as dealing
with the whole of time, or the whole of space, or with
the whole Universe. It is indeed not clear that we are
entitled to assert that either of these is a wMhole in the
sense that, to such an hypothetical infinite whole, the
same categories are applicable as in the case of a finite
whole. These considerations would appear to indicate
the existence of limits to the application of the methods
of Natural Science. The existence of such limits has,
however, frequently been disregarded, at least in appearance, in statements made by men of Science of high
distinction. It has often been said that the whole energy
of the Universe, or the whole mass of the Universe, is
constant, or that the whole entropy of the Universe
tends to increase. To such statements a meaning can
only be assigned by assuming them to be elliptical expressions to denote, for example, that in any finite region
which received no accession of mass from without, the
total mass remains constant.
We have seen that the construction of rules, relating
to sequences and regularities in Nature, the so-called
laws of Nature, and the more comprehensive schemes
which we call scientific theories, are the work of the
human mind, utilizing the raw data of perception. We
might imagine that the perceptual world could have
been so intricate and irregular in character that it would
have been impossible to set up such rules or schemes,
under which phenomena could be subsumed. Science,
as we know it, would then have been impossible, it
could never have begun to exist. But life, as we experience it, would also have been impossible, because that
common knowledge of sequences in Nature, upon which
the possibility rests of forecasting the results of our
actions, would have been absent. The possibility of the
existence of Natural Science accordingly depends upon
the fact that there is in our percepts, that is in what we




SCIENTIFIC LAWS AND THEORIES 43
call Nature, a considerable degree of regularity in the
sequences of phenomena. This fact is often described as
the Uniformity of Nature. What the scope of this uniformity may be, and whether it is subject to extraneous
interferences, are questions which cannot be answered
a priori; only experience is relevant in any attempt to
answer them. It is not necessary for the purposes of
Natural Science to assume that the complex of physical
phenomena is such that it is, even theoretically, capable
of description by a single all-embracing conceptual
scheme. It is possible that the present state of Science,
in which different conceptual schemes are employed for
resuming different tracts of the perceptual domain, is the
only possible state. Although the actual progress of
Science has involved the gradual welding together of
separate conceptual schemes into larger wholes which
embrace tracts of phenomena that were earlier treated
separately, it is not a necessary assumption for Natural
Science that this process of unification can, even theoretically, become complete. In other words it is not
necessary to assume that Nature can even theoretically
be subsumed under a single interconnected rational
scheme.
Throughout the history of Science we find that the
most potent influence on investigation has been exercised by the notion of efficient causation. The principle
of causation, or sometimes the more general principle
of sufficient reason, has been regarded as a necessary
principle to which Nature must conform, and the aim
of Science has been regarded as that of discovering the
precise modes in which the principle is realized in
natural phenomena. But in Natural Science, conceived
in the manner I have explained, such a priori principles
are replaced by the postulate, or working hypothesis,
that sequences can be found in Nature which are capable
of being described by laws or sets of laws embodied in
determinate ideal schemes, and that it is possible to an




44 SCIENTIFIC LAWS AND THEORIES
extent of which we do not know any limits to discover
what these schemes are. That these schemes can be
applied with absolute precision to concrete cases in
Nature is an illusion dependent upon the imperfections
of our senses, even when their power is extended by
using instruments, which often prevents us from perceiving all those individual peculiarities of a concrete
phenomenon which differentiate it from other phenomena
of the same class, described by the same conceptual
scheme. To construct laws we frequently isolate artificially a part of what happens from the rest; there are
always however in a given case accessory phenomena.
In a scientific laboratory elaborate precautions are frequently made in order to isolate phenomena, and to
reduce disturbing influences as much as possible, but
there remains some residual which cannot always be
completely ignored when the experiment is regarded in
relation to some simple law of an exact kind. Thus a
scientific law states what would happen under ideal
conditions, more or less imperfectly realized in actual
cases.
We have seen that all conceptual schemes are applicable only to the description of what goes on in a portion
of the physical domain limited in various ways. This
involves the isolation, for the purpose of conceptual
description, of particular domains in the perceptual
world. The success of a conceptual scheme in discharging its descriptive function in relation to a particular
perceptual complex, depends upon the fact that, for the
purpose on hand, that particular perceptual complex
may be treated as an isolated system, in which all perceptual elements that do not belong to it are ignored.
Thus the existence of approximately isolated systems in
the world of physical phenomena is a fact upon which
the possibility of the existence of Natural Science, as we
know it, depends. For example, a descriptive scheme
for the motions of the bodies in the solar system would




SCIENTIFIC LAWS AND THEORIES              45
be impossible except for the fact that the solar system
is approximately isolated, in the sense that the disturbing effects of gravitation due to the stars can be left out
of account.
A conceptual scheme which successfully serves its
purpose in the particular department of Science to
which it belongs must satisfy certain conditions. First,
it must have logical consistency or coherence, as a
scheme which is not self-contradictory; the various
conceptual elements which it employs, and the set of
definitions and postulated relations contained in it, must
form a consistent whole, in which there is a complete
logical nexus between the parts. Secondly, the scheme
must satisfy the test of providing an actually adequate
description of sequences of phenomena of the class to
which it is intended to apply; it must have applicability,
in relation to physical phenomena. It is quite possible
to set up conceptual schemes which satisfy the first of
these conditions, that of logical coherence, but which
either do not satisfy, or are not known to satisfy, the
condition of applicability. A conceptual scheme which
differs from that of Newton's law of gravitation, in
taking the law to be that of the inverse cube, or any
other power, of the distance, instead of Newton's law
of the inverse square, could be worked out, in detail,
and would satisfy the condition of logical coherence as
fully as does Newton's law; it would however not be
applicable for even the approximate representation of
the motions of the bodies of the solar system. It would
be in conflict with fact in the perceptual domain. A
particular system of abstract Geometry may be unimpeachable as a conceptual scheme, and yet it may be
inapplicable to the description of actual spatial percepts.
A third condition, that of relative simplicity, should
be satisfied by a scientific theory, if it is to be used with
effectiveness in the building up of Science. It may
happen that two or more conceptual theories satisfy the




46 SCIENTIFIC LAWS AND THEORIES
conditions of logical coherence and of applicability. In
that case that theory will be chosen amongst all those
which are applicable to the same range of phenomena
which is of the simplest character.
The most crucial test of applicability of a scientific
theory is that it should enable us actually to predict the
results of experiments or observations of sequences of
phenomena, under conditions consistent with the scope
of the theory, which are made subsequently to the constitution of that theory. If it fails to do this successfully,
either the theory must be abandoned, and a search must
be made for a more adequate one by which it can be replaced, or else the determination of disturbing elements
must be sought in the conditions under which the new
experiments were made, of such a kind that it can be
shown that the complex of conditions in these experiments was not in accord with those postulated by the
theory. In other words it must be shown that the
observed complex of percepts did not, in these experiments, possess that feature of approximate isolation
which the theory presupposes.
It has frequently happened in the history of Science
that a theory which had a very considerable measure of
success as a representative scheme failed to satisfy one
of the requisites I have described. It has then been
abandoned in favour of some other theory which appeared more nearly to satisfy the requisite conditions.
In the first place the theory may be abandoned on
account of internal defects in the conceptual scheme
itself, in the matters of logical validity and clarity of
definition of the conceptual elements. In some cases a
clear statement, in a logical form, of a conceptual theory
has only been given, as the result of critical investigation, long after the theory was first constructed and
applied for its proper descriptive purposes. Also a
theory, apart from defects in its own nature, may be
superseded by a new theory of a more comprehensive




SCIENTIFIC LAWS AND THEORIES 47
character, one which is capable of representing a wider
range of phenomena than the older one. When such
supersession takes place, the older theory does not lose
its logical consistency and whatever degree of applicability it ever possessed.
The truth, or the falsehood, of a scientific theory is
often spoken of in a manner which obscures the real
nature of such theories. It is the fact of applicability, or
the degree of applicability of one logically coherent
theory as compared with another, which is their distinguishing feature. Assuming that both of them are
logically coherent conceptual schemes, both of which
are hypothetical in their nature, that one will be preferred, for many purposes, which has the greater range of
applicability, and hence the greater power of prediction.
If the Einstein theory of gravitation, or some modification of it, finally supersedes that of Newton, as may very
likely happen, it will be because the former is capable
of representing some phenomena which the Newtonian
theory fails to represent, and also because, as part of a
wider theory which represents phenomena of a different
class from the gravitational, it is better fitted than the
Newtonian theory of gravitation to form part of a comprehensive scheme relating to a wide class of physical
phenomena. But for the purpose of describing nearly
all the relative motions of the bodies of the solar system
the Newtonian scheme is demonstrably sufficient, and it
will always be as adequate for that purpose as it has proved
itself to be in the past. Thus it is in possessing greater
range and accuracy in its power of description, and not
in what should properly be described as its truth, unless
truth be understood to mean degree of applicability,
that the Einstein theory will be regarded as superior to
that of Newton. For all we know, the Einstein theory
itself may one day be superseded by some still more
general and comprehensive scheme that is applicable to
the description of a still greater range of phenomena.




48   SCIENTIFIC LAWS AND THEORIES
In general the method of setting up hypothetical
theories to be tested by their power of representing and
predicting phenomena may be described as methodological pragmatism. That does not of course imply the
acceptance of Pragmatism as a general philosophical
system.
In some cases there may be diversity in the manner
in which a conceptual scheme can be applied to the
purpose of describing the phenomena for the representation of which it was constructed. This may especially be the case in a theory, expressed in a mathematical
form, in which some only of the conceptual elements
have direct perceptual correspondents, the remaining
elements being required to ensure the coherence and
validity of the theory itself. A principle of correspondence is then required which shall assign the nature of
the correspondence between perceptual elements and
conceptual elements of the abstract scheme, and such
principle of correspondence is not necessarily unique.
An example of this is found in the theory of the Electromagnetic field as formulated by Maxwell. There has
been general agreement that Maxwell's equations do
represent what can be observed in the field, but different
modes have been suggested of placing into correspondence what can be observed, with the vectors which
appear in the equations.
In the most abstract branch of Science, Pure Mathematics, the powers which the mind possesses, of
idealization and generalization, have been so extensively
employed that conceptual schemes have been erected
which are not known to be applicable to the description
of any physical phenomena, although such schemes
could never have come into being without an ultimate
starting point in physical perception. This tendency on
the part of Pure Mathematics to outrun the exigencies
of applications has sometimes been made a subject of
reproach to Mathematicians, especially on the part of




SCIENTIFIC LAWS AND THEORIES 49
Physicists. Mathematicians have been accused of wasting
their intellectual efforts in a region of bare formalism
which has cut itself off from all connection with the
actual world. It has been urged that Mathematicians
have an excessive tendency to occupy themselves with
ideas that are too remote from the physical order in
which Mathematics had its origin, and in which it should
still find its proper applications. Experience has however shown, in many instances, that it is exceedingly
hazardous to prophesy that a Mathematical theory,
however remote it may apparently be from the physical
order of things, will not sooner or later find an application within that order. It is quite true that, although
Pure Mathematics is in its origin a part of Natural
Science, having arisen, like every other branch of Science,
in efforts to deal with certain aspects of the perceptual
world, it has become by rapid stages a deductive and
formal Science, in a sense and to a degree which is not
the case at present in any other branch of Natural
Science. Nevertheless, Mathematical thought, as its
history very clearly indicates, is one of the normal forms
of mental activity. If it be allowed to develop itself, on
its own lines, free from all fetters imposed upon it by
the supposed necessity of keeping itself in close relationship with other departments which can make use of
it as a tool, it will not prove itself to be in default when
new demands are made upon it by other branches of
Science, as they gradually reach those stages in which
they require the help of refined Mathematical methods.
It would be easy to establish in detail that many of the
most important applications which have been made of
Mathematics were wholly unforeseen by those who
developed the Mathematical ideas and methods so employed. Had not Mathematicians, during more than a
century, investigated those systems of Geometry that
are known as non-Euclidean, and which appeared to
many Philosophers and Physicists to be an extreme


HGL


4




50 SCIENTIFIC LAWS AND THEORIES
instance of the addiction of Mathematicians to lines of
thought which could lead nowhere, the recent theory of
Einstein could not have been formulated, because the
way would not have been prepared for the conception
of the four dimensional manifold with a non-Euclidean
metric which appears in the scheme as space-time, and
the theory of tensors which Einstein has employed in
the elaboration of his scheme would not have existed.
The new Physics and Chemistry may at no distant time
require weapons, forged by Mathematicians; the theories
of functions, of differential equations, of groups; thus
providing applications for the most refined methods
which Pure Mathematics can devise. It will be seen in
later lectures that this time has in fact already arrived.




III


NATURAL SCIENCE IN RELATION
TO PHILOSOPHY
THE current views on general Philosophy have usually
exercised a considerable influence upon the course
of development of scientific thought; and this influence
has been reciprocal. The strength of this connection was
in former times increased by the fact that one and the
same man not infrequently combined the functions of
Philosopher and man of Science. The cases of Aristotle,
Descartes, Bacon, and Ieibniz may be cited as instances
of this fact. Even Kant had strong claims to be considered as a man of Science. Philosophers such as
Locke, Hume, and Berkeley were largely occupied with
the discussion of matters which have a close connection
with the epistemology of the Natural Sciences. Not
only the more systematized metaphysics of professed
Philosophers, but also the more popular and less systematized ideas on Metaphysics, current at a particular
time, had a marked influence upon the forms in which
the scientific conceptions of the time found expression,
and exercised a considerable directive power upon the
lines of scientific investigation. Popular thought is at all
times permeated by metaphysical ideas, however detached and unsystematized their form may be, and however unconscious the majority of men may be of their
presence and true character. I have already alluded to
the strong tendency which Natural Science has exhibited
during the last two centuries to extend its original scope
and to transform itself into an all-embracing WorldPhilosophy.
With a view to the delimitation of the domain of
Natural Science in relation to human experience as a
4-2




52


NATURAL SCIENCE IN


whole, it is necessary to make an attempt to clear up
ideas as regards the position of Natural Science relative
to some of the various systems of Metaphysical Philosophy
with which it has at various times found itself in contact.
Fortunately, it is, I believe, possible to do this without
considering in much detail the characteristics of the
various metaphysical systems which have in different
ages arisen, and some of which were held contemporaneously by different schools of philosophical thought.
For our purpose it will suffice briefly to refer to a very
few of the more fundamental points in which these
systems, with their innumerable points of difference in
less fundamental respects, differ from one another. In
particular, it is requisite to attempt to draw some conclusion as regards the answer that should be given to
the question what ontological assumptions are necessary
for the special purposes of Natural Science. The history
of Science shows that assumptions of this kind have in
fact played a large part, sometimes in promoting, and
sometimes in hindering, progress. But the most important question we have to consider is how far such
assumptions are indispensable for the existence and
efficiency of Natural Science, as an organized system
devised for the systematic ordering of our physical percepts, in whatever degree experience may show that this
ordering is possible. It is clear that, in accordance with
sound Methodology, all unnecessary assumptions should
be discarded as unessential to Natural Science, even
when they have in point of historical fact been employed
as part of the scaffolding used in the building up of the
various departments of the domain. The existence and
successful functioning of Natural Science cannot properly
be invoked as a decisive reason for accepting any ontological theory or assumption which cannot be shown to
be an indispensable element in scientific thought considered as a coherent body of doctrine.
In the subject-object relation which is fundamental




RELATION TO PHILOSOPHY


53


in all experience, although the phenomena with which
Natural Science concerns itself are objects for some
subject, the precise nature of the activity of the subject
as a factor in the relation is irrelevant to Natural Science
in the sense in which the term is here used. The investigation of this activity falls within the domain of
Psychology which, for reasons I shall presently explain,
is not part of Natural Science as here delimited. For
Natural Science, the role of the subject in perception
may provisionally, or methodologically, be regarded as
purely passive, that of experiencing physical perceptions. Thus Natural Science takes these perceptions as
simple data, and need not attempt to give any account
of how far the activity of the subject is necessary for
their existence, or how far they depend upon something
foreign to the percipient. On such philosophical and
psychological matters, Natural Science, as such, has
nothing to say, and may take up a neutral attitude in
relation to conflicting views concerning them. It should
however be observed that the formation of concepts
must be regarded as due to the psychical activity of
subjects, although Natural Science is not concerned
with any detailed investigation of the modes and laws of
such psychical activity. This form of psychical activity
is considered methodologically as distinct from any
activity, or acts of attention, connected with the reception of percepts; but Natural Science is committed to
no assumption that such separation of psychical function
is valid in an ultimate sense, or that it is necessarily
anything more than one of those kinds of separation
which discursive thought is compelled to make, of elements which, from a more fundamental point of view,
may come to be regarded as inseparable elements of an
essential whole.
What we call a material object is prima facie a construct, built up by the synthesis of a group of actual
sense-impressions, and of images of earlier sense-im



54


NATURAL SCIENCE IN


pressions stored up in the memory. An effect of the
memories of sense-impressions is to enable us, in perceiving an object, to dispense with some of the actual
sense-impressions which make up the construct. Thus,
when we see a stone, we know that it is hard, without
verifying the fact by touch in each instance; this is due
to the memory of earlier sense-impressions, and to a
belief in their unchangeableness. This view of the nature
of a material object implies that there is a percipient for
whom the object is a percept; and this involves the
subject-object relation. The question now presents itself,
can the object be properly regarded as existent when
there is no percipient for whom it is a percept? In
other words, can the object be separated out from the
subject-object relation, and be regarded as having an
existence independent of any percipient and of all percipients? Is there any definite meaning, and if so what,
to be attached to the assertion of such independent
existence? The kind of answer which has been given to
such questions varies greatly in different philosophical
systems. For us, the importance of the matter depends
upon whether it makes any essential difference to
Natural Science what answer is given to these questions
and to other ontological questions relating to the existence of the percipient or subject. The answer which I
suggest and advocate is that, for the special purposes of
Natural Science, it is immaterial what answer is given
to these questions; that in fact ontological hypotheses or
theories on these points are irrelevant to Natural Science;
that it is sufficient, for example, for the purposes of
Natural Science to regard a perceptual object as a construct of sense-impressions, whatever else it may for the
purposes of systematic Philosophy be regarded as being
or implying. But this answer is one which would not
have been given until recent times by any man of
Science, and probably would not be given at the present
time by the majority of men of Science. A material




RELATION TO PHILOSOPHY


55


object has most frequently been regarded as having some
kind of sub-stratum, not identical with the synthesis of
sense-impressions, but as a thing in itself, a kind of
bearer of the various properties or qualities such as extension, motion, hardness, colour, elasticity, etc., which
are regarded as giving rise to the component senseimpressions. Thus material objects are regarded as
having an entity, material substance, as their essential
foundation; the various properties or qualities being regarded as inherent in the substance. The real object, or
thing in itself, is not identified with any or all of the
properties which it possesses, and with which our senseimpressions are related; it is not itself directly perceived, although its existence is supposed to be a
necessary inference from the existence of the percept.
This point of view was clearly expressed by Locke, who
writes1:
When we talk or think of any particular sort of corporeal substances, as horse, stone, etc., though the idea we have of them
be but the complication or collection of those several simple
ideas of sensible qualities...., yet because we cannot conceive
how they should subsist alone, nor one in another, we suppose
them existing in, and supported by, some common subject;
which support we denote by the name substance.
This conception of substance, as a sub-stratum of matter,
has been held by the adherents of some realistic systems
of Philosophy, by many, or probably most, men of
Science, and it appears to be held by those who are
dominated by the set of notions usually described as
common sense, or as naive realism. The thing in itself
has been regarded as having an existence independent
of any and all percipients; thus forming an independent
Real. The doctrine of the distinction between substance
and its properties or accidents was an essential part of
Scholasticism, pressed into the service of Ecclesiasticism.
When it was pointed out by Locke and others that at
1 Essay concerning human understanding, Bk ii, Chap. xxIII, Sec. 4.




56


NATURAL SCIENCE IN


least some of the accidental properties of the thing in
itself are clearly not independent of the percipient, a
division was made of the qualities of the substance into
two sets, called respectively primary and secondary
qualities. The primary qualities, extension, and motion,
and probably inertia, were regarded as inherent in the
object itself, and thus, like the substance, independent
of any percipient; but this independence was not asserted
of the secondary qualities such as colour, sound, smell
and temperature. These last were regarded as essentially
dependent on the percipient, but also as conditioned by
the primary qualities of the substance. The substance,
or the sub-stratum of all material things, is incapable
of being directly perceived, but is regarded as something which is of necessity conceived by the mind as
existing independently of itself, as the bearer of qualities,
just as an adjective requires a substantive which it
qualifies. In Leibnizian monadism, and also in more
modern forms of spiritualistic pluralism, matter is
regarded as the manifestation of the activity of a plurality
of psychical beings; and thus substance is of a psychical
character.
The exact opposite of this notion of the necessity
of the category of substance as the sub-stratum of the
material world was advocated by Berkeley, the great
English idealist philosopher. With him, what we have
spoken of as percepts, and what he calls ideas, constitute
the whole reality of material objects. He wrote1:
It is indeed an opinion strangely prevailing amongst men,
that houses, mountains, rivers, and in a word all sensible objects,
have an existence, natural or real, distinct from their being
perceived by the understanding.
The view expressed by Berkeley that there exists no
sub-stratum, or substance, independent of perception,
is characteristic of idealistic Philosophy, and is in sharp
contrast with the opposed realistic view.
1 The principles of Human Knowledge, Collyns Simon's edition, p. 19.




RELATION TO PHILOSOPHY


57


To decide between these opposed opinions is no part
of the function of Natural Science; in fact the method
which Science employs makes it incapable of doing so.
Ex hypothesi, substance, things in themselves, if they
exist, cannot be directly known as percepts, but are
inferences from what is perceived; and thus they cannot
form part of the perceptual order of things which it is
the function of Natural Science to classify and describe.
Whether they exist, or not, is consequently quite immaterial to Natural Science, because, even on the realistic assumption, Science is actually concerned only
with the percepts to which things in themselves give
rise, and not with the real world which the realistic
Philosopher takes to exist behind phenomena. As between these two divergent philosophical views, the
position of Natural Science may be taken to be that of
a neutral. Natural Science is not compelled either to
affirm or to deny the existence of real substance, or of
monads as independent reals. It has no need to take
account of the category of real substance. The history of
Science is full of assumptions of the existence of ethers
and substances of various kinds; but these may be taken
to be concepts, not realities independent of the psychical
world. They are parts of the conceptual scaffolding
which go to build up scientific theories and laws.
In speaking of the function of Mathematical theories,
Poincare has given1 a description of their function which
might, I think, be extended to scientific theories in
general. He writes:
Mathematical theories have not as their object to reveal to us
the real nature of things; that would be an unreasonable claim.
Their sole aim is to coordinate the physical laws that experience
reveals to us, but which, without the aid of Mathematics, we
could not even enunciate.
Even to-day men of Science under the influence of
philosophic Realism, or of the unsystematized Philosophy
1 Lepons sur la theorie mathematique de la lumiere, p. I.




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of common sense, are extremely reluctant to abandon
the idea that the notion of real entities, something more
than concepts, is essential to Natural Science. Thus, for
example, Professor A. N. Whitehead writes1:
Another favourite solution, the most attenuated form which
the bifurcation theory assumes, is to maintain that the molecules
and ether of Science are purely conceptual. Thus there is but
one nature, namely apparent nature, and atoms and ether are
merely names for logical terms in conceptual formulae of calculation. But what is a formula of calculation? It is presumably a
statement that something or other is true for natural occurrences.
Take the simplest of all formulae. Two and two make four.
This-so far as it applies to nature-asserts that if you take two
natural entities, and then again two other natural entities, the
combined class forms four natural entities. Such formulae
which are true for any entities cannot result in the production
of the concepts of atoms. Then again there are formulae which
assert that there are entities in nature with such and such
properties, say, for example, with the properties of the atoms of
hydrogen. Now if there are no such entities, I fail to see how
any statements about them can apply to nature.... The current
answer is that, though atoms are merely conceptual, yet they
are an interesting and picturesque way of saying something else
which is true of nature. But surely if it is something else that
you mean, for heaven's sake say it. Do away with this elaborate
machinery of a conceptual nature which consists of assertions
about things which don't exist in order to convey truths about
things which do exist. I am maintaining the obvious position
that scientific laws, if they are true, are statements about entities
which we obtain knowledge of as being in nature; and that, if
the entities to which the statements refer are not to be found in
nature, the statements about them have no relevance to any
purely natural occurrences. Thus the molecules and electrons
of scientific theory are, so far as science has correctly formulated
its laws, each of them factors to be found in nature. The electrons
are only hypothetical in so far as we are not quite certain that
the electron theory is true. But their hypothetical character
does not arise from the essential nature of the theory in itself
after its truth has been granted.


1 The concept of Nature, pp. 44 et seq.




RELATION TO PHILOSOPHY               59
If I understand this passage aright, the ideas of the
nature and functions of a scientific theory expressed in
it are in fundamental opposition to those which I here
advocate. Whatever else they are, molecules and electrons are concepts, and even Dr Whitehead is not quite
sure that they are anything else. They are concepts in
scientific theories, and the only question about a scientific theory is not whether it is in Dr Whitehead's sense
"true," but whether it is logically coherent and how far
it is adequate for the purpose of representation; whether
it will be superseded by some other theory which employs other concepts, because that theory is applicable
to a greater range of phenomena than the theory which
employs molecules and electrons. We do not perceive
in Nature entities such as atoms and ether; we do not
perceive entities at all, if an entity be taken to be anything more than a construct of a complex of senseimpressions present and past. What Dr Whitehead
speaks of as an elaborate machinery of a conceptual
nature which consists of making assertions about things
which don't exist, is really a scheme involving things
which do exist as concepts; and conceptual atoms have
been employed because they have been found not
merely "interesting and picturesque," but because, at
all events for the time being, they have been found to
be the best available means for representing what "is
true in nature" in Dr Whitehead's phraseology; in the
sense that they resume conceptually part of what we
perceive as natural phenomena. Natural Science postulates, as a working hypothesis, only that the perceptual
complex is such that tracts of it are capable of conceptual description by scientific schemes. It does not
require any postulate as to detailed systems of relations
or of entities within that perceptual complex, or within
any supposed reality behind that complex, which shall
account for the fact that the working hypothesis has
proved successful.




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In speaking of the new conceptual scheme of spacetime which is associated with the name of Einstein, Sir
Oliver Lodge writes1:
In such a system there is no need for Reality: only phenomena
can be observed or verified: absolute fact is inaccessible. We
have no criterion for truth; all appearances are equally valid;
physical explanations are neither forthcoming nor required;
there need be no electrical or any other theory of the constitution
of matter. Matter is, indeed, a mentally constructed illusion
generated by local peculiarities of space.
And again he writes concerning the same theory:
But, notwithstanding any temptation to idolatry, a physicist
is bound in the long run to return to his right mind; he must
cease to be influenced unduly by superficial appearances, impracticable measurements, geometrical devices, and weirdly
ingenious modes of expression; and must remember that his
real aim and object is absolute truth, however difficult of attainment that may be, that his function is to discover rather than
to create, and that beneath and above and around all Appearances
there exists a universe of full-bodied, concrete, absolute, Reality.
The notion indicated by Sir Oliver Lodge that it is
the function of Natural Science to search for reality
behind physical phenomena still receives support from
the ranks of Philosophers. Thus, for example, Professor
H. Wildon Carr writes2:
The modern era of philosophy from Descartes onwards has
been dominated by the insistence of the scientific problemthat is, the problem of the ultimate nature of the reality we
study in physical science by the experimental method.
These examples may suffice to illustrate the persistence
of this view, which I believe to be erroneous, even at
the present time.
It is interesting to observe that some men of Science
who avowedly regard Metaphysics with distrust and
aversion are most insistent in claiming "reality" for
material objects, and for molecules, atoms and ether;
oblivious of the fact that they are, as it would seem
1 Nature, cvI, pp. 796 et seq.  2 Nature, cvI, p. 809.




RELATION TO PHILOSOPHY                61
unnecessarily, by using the term "reality," illustrating
the fact of the extreme persistence in thought of metaphysical conceptions of the theoretical and of the common sense order. The term "reality" has an intensely
metaphysical complexion; and its shades of meaning in
different philosophical schemes vary considerably. Many
thinkers, like Plato, have held that the domain of concepts has a claim to "reality" superior to that of the
perceptual world.
I have already spoken of the dualism which recognizes
in the world two distinct domains, the psychical and the
physical; or as is sometimes said with less accuracy,
mind and matter. The ultimate relation of these two
domains is for the Philosopher to elucidate. For the
man of Science this dualism should be regarded only
as provisional and methodological, and need not be
taken to involve any final assumption of the fundamental disparateness or separation of the two domains.
Mind and Matter have frequently been regarded as
ultimately reducible to one of the two, or to some third
basal existent more fundamental than either of them;
but Natural Science, so long as it confines itself to its
own original functions, does not stand or fall with any
hypothesis of this character. A monistic system of
Philosophy, whether it be of the kind called materialistic,
or of the kind described as spiritualistic, or whether it
be described as neutral, will no doubt make the attempt
to interpret scientific theories in terms appropriate to
the particular system. Pluralistic schemes of Philosophy,
such as the Monadology of Leibniz, or other more
modern forms of Pluralism, must also be left to find
their own modes of interpretation of Natural Science.
But scientific thought, leading to what is known as
positive knowledge, needs none of the ontological assumptions peculiar to any one of the many rival philosophical theories of the nature of reality.
The naive realist, and some philosophers, regard the




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objects of perception, such as we perceive in our normal
life when awake, as still existing very much as we perceive them, whatever that may mean, even when no
sentient being is actually perceiving them. This is also
the view of common sense; but scientific knowledge, in
accordance with the view of its nature that I am here
propounding, is independent of the acceptance or rejection of this opinion. Scientific knowledge is also
independent of the assumptions made in any other form
of realistic Philosophy. The pure phenomenalist holds
that perceptual objects exist only when, and so long as,
they are being perceived as objects by a percipient; and
thus that the only realities are mental states and the
objective constructs actually perceived. Modified statements hold good for intermediate philosophical theories,
as for example the kind of phenomenalism which
admits the existence of voovVEva behind phenomena.
The descriptive view of the functions of scientific
theories and laws, which I here adopt, has sometimes
been characterized as phenomenalism; but this view
does not necessarily involve the acceptance of any
philosophical doctrine of phenomenalism. The phenomenalism of Natural Science is methodological only.
That the distinctions which we make in order to
meet the necessities of our discursive methods of apprehension necessarily correspond to similar distinctions in
an objective reality is an assumption, of which, though
it has been persistent through the whole history of
speculative thought, Natural Science has no need, whatever its merits may be in the view of Philosophers of
various schools.
Accordingly, the view of the function and limitations
of scientific knowledge which has arisen as the result of
recent criticism, has as one of its implications the
assumption that Natural Science is independent of
purely ontological theories, as applied either to percepts
or to percipients. If, in the sense employed by any




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63


particular philosophical system, the existence of real
entities be asserted in any particular connection, Natural
Science can make no use of such assertion, since all
scientific theories can be stated in a form which is
independent of such ontological assumptions. There is
a considerable practical advantage in the rejection, as
unnecessary for the purposes of Natural Science, of all
superfluous ontological hypotheses. So far as Science
can take up the position of neutrality, or of detachment,
with respect to the ever-conflicting and ever-varying
ontological views of Philosophers, it does not share in
the vulnerability which may be held to attach to any of
the metaphysical theories that are involved in the conflict. In view of the instability of all definite metaphysical
schemes, an instability which does not show much sign
of disappearing, the greater stability with which this
position of neutrality endows Natural Science has great
advantages.
In making these remarks as to the relation of Natural
Science with Metaphysics, I have no desire to associate
myself with the views of those men of Science who
decry Metaphysics as a barren study which can lead to
nothing but fruitless sophistries. On the contrary, I
fully recognize that, in all ages, Philosophical thought
has been indispensable, in ministering to one of the
deepest and most ineradicable impulses of the human
spirit; the imperative impulse to penetrate in some
degree into the recesses of the great mystery of life and
existence, with which we are all in relation. Although
no generally accepted solution of even the fringes of
that mystery may be in sight, yet Philosophy has done
much to fix and clarify the form in which fundamental
questions can be properly stated, and has also rendered
great services in definitely removing, by its criticisms,
many irrelevant accretions which had gathered round
the main philosophical questions. The attempt to penetrate to a reality in which we may find something




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permanent and stable in, behind, or above, the impermanent and unstable elements of physical and psychical
phenomena, is one which will continue to be made as
long as the human spirit resembles what it has been in
the past, and is in the present; although it is no part of
the functions of Natural Science to take part in this
attempt. The impulse from which this attempt arises
will never for long be stifled by merely negative criticism.
But progress on the path to reality cannot be made by
the simple expedient of transmuting such concepts as
those of matter, ether, atoms, electrons, into real substances or entities.
Philosophy has in recent centuries rendered valuable
services to Natural Science in assisting it to remove
various injurious accretions derived from medievalism,
of supposed a priori knowledge, and of unnecessary and
irrelevant ontological assumptions. The attention of
scientific thinkers has thus been concentrated on phenomena themselves, and on the mental representation
of them. Science has no longer been hampered by
medieval prejudices relating to occult properties of
supposed sub-strata behind phenomena, or by conceptions of Nature as necessarily conforming to ideas
constructed by a priori thought. The advance of Natural
Science involves two main factors, first the discovery,
by observation and experiment, of facts; and secondly
the procedure of reflective thought in classifying groups
of facts, devising rules or laws to which the groups of
facts conform, and constructing general theories which
symbolize phenomena, and under which groups of laws
can be subsumed. Both these factors are essential in all
genuine Science, but the emphasis placed on the two
factors has varied at different periods and with different
thinkers. At some periods, and with some persons,
there has been a tendency unduly to subordinate one or
other of these factors. Among the Greek thinkers,
Aristotle can be distinguished by the great importance




RELATION TO PHILOSOPHY


65


he attached to the observation of facts, but he did not
thoroughly grasp the nature of the slow and painful
process by which empirical results are obtained and
used to build up genuine scientific theories. A mere
collector of facts is far from being a real man of Science,
but there is room for real division of labour in contributing to the growth of Science, and this implies that
the work of various grades of investigators can be of
value in the construction of the edifice. Among these
grades those workers who undertake the arduous duty,
under suitable direction, of collecting precise facts, in
some given experimental or observational domain, have
an honourable place. But, in the trenchant words1 of
Poincare: "Science is built up of facts, as a house is
built up of stones; but an accumulation of facts is no
more a science than a heap of stones is a house." The
really important general theories are always the work of
men endowed with very exceptional powers of imagination, insight, and generalization, which enable them to
seize instinctively upon relations of similarity between
things that have little or no superficial resemblance with
one another. These great pioneers have formed but a
small band in the history of Science.
Even after medieval conceptions of Science had waned,
the Rationalistic school, which preceded Kant and the
critical conceptions he initiated, attempted to construct
nature out of pure Thought. The critical movement
led by Hume and Kant established the impotence of
Thought alone, devoid of the essential foundation of
actual physical experience, to formulate any genuinely
scientific view of the world of physical phenomena.
In what I have said hitherto about the functions of
Natural Science, the conceptual description of the
physical world alone has been referred to, but it is clear
that no treatment of the general aspects of the subject
would be adequate without some reference to the
1 Science and Hypothesis, p. 141.


H GL


5




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psychical order ot things. That the physical and psychical
domains have, in appearance at least, very close relations
with one another is obvious to the most casual observation. Primafacie they are not independent of one another,
but closely interconnected.
Modern Physiology, on the one side, and Modern
Psychology, on the other side, have confirmed this fact
of observation in minute detail. There is evidence, of
overwhelming strength, that psychical events or processes in the mind are accompanied by physical events
or processes in the bodily organism; that at least some
kinds of physical events or processes in the body are
accompanied by psychical happenings in the mind is
also a fact not open to doubt. The question of the
nature of the relations between such physical and
psychical processes, between body and mind, or of how
such relations are to be represented, is one of the most
difficult and intractable problems of Philosophy. An
important and influential school of Physiologists has
advocated the view that physical processes in the brain
and nervous system of the bodily organism form a closed
sequence entirely independent of the psychical processes, or happenings, which accompany these physical
processes; that, despite appearances, the sequences of
events in the bodily organism are entirely unaffected
by concomitant psychical sequences. In accordance
with this view, if we possessed a sufficiently advanced
knowledge of Physiology, a complete account, involving
only the categories of Physics and Chemistry, could be
given of all the actions of a man, of all his responses to
external stimuli, without taking into account his consciousness or will, or any of the motives to which he
himself attributes his actions.
His cognition, feelings, and conation are, in this theory,
regarded as belonging to a domain which has no influence upon the world of physical phenomena, including
all the physical happenings in his own body; the former




RELATION TO PHILOSOPHY


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are regarded as epiphenomena, or Begleiterscheinungen,
which accompany, but have no influence upon the latter.
The man is a conscious automaton, a machine endowed
with consciousness, but not with the power to influence
his own actions; although he is under the delusion that
he has this power. The closeness of the correspondence
between the sequences of events in the two domains has
been formulated in the theory known as Psycho-physical
Parallelism. Whatever be the value of this theory, it
certainly makes colossal demands upon our powers of
imagination, when we attempt to represent to ourselves
how it works, and what it implies, in an individual case.
On the 7th of May, I915, multitudes of people read
telegrams conveying the news that the Lusitania had
been torpedoed. The reading of the telegram was followed, in a vast number of cases, by bodily feelings
indicating very marked disturbances in the nervous
system. That these nervous tremors were due to a conscious apprehension of the terrible meaning of the news
is the ordinary view of common sense. But a believer
in the theory of psycho-physical parallelism is bound to
assert that the meaning of the telegram, only interpretable in psychical terms, had nothing whatever to do
with these nervous disturbances. The image of the print
of the telegram on the retina of the eye, the subsequent
neural currents to the brain, certain changes in the
smallest part of the grey matter of the brain, and neural
currents from the brain to other parts of the organism,
formed a sequence which could theoretically be accounted
for as a sequence of purely physical phenomena; the
only relevant factors in the determination of this sequence were the image on the retina, and the detailed
physical constitution and condition of the individual
organism. A slight change in the words of the telegram,
such as for example the insertion of the word "not,"
with the corresponding slight change in the image on
the retina, might have led to the absence of all the




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nervous disturbances in the body of the reader of the
telegram. A similar absence of nervous disturbance
would have been observed in the case of a person who
did not understand the language in which the telegram
was written.
The change of meaning of the telegram would have
been of vital importance, but believers in the theory of
psycho-physical parallelism must assert that the slight
change in the image on the retina was the only factor
which could account for the absence, instead of the
presence, of what were, in many cases, very marked
disturbances in the nervous system. This illustration
has been given solely with the intention of making clear
some of the implications of the theory of psycho-physical
parallelism. It is not put forward as a refutation of the
theory; a much more complete discussion of the theory
would be required for such a purpose, and it is no part
of my programme to put forward even a tentative solution of the problem to which the theory is related.
Another attempt to overcome the difficulties connected with the relations between physical and psychical
phenomena in the living organism is embodied in the
theory of interaction. In accordance with this theory,
neither the physical nor the psychical phenomena form
a closed system, but each group is affected by action
from the other. So long as the two domains are regarded
as completely disparate, the difficulty of this theory
arises from inability to represent the precise nature of
such interaction, or to discover the exact points in a
physical sequence at which the effect of the psychical
domain makes itself apparent as a disturbing factor,
introduced extraneously into the physical sequence.
It has sometimes been assumed that, regarding the
physical phenomena in the living organism as forming
a dynamical system, the action of the psychical side of
the organism is of such a character that no breach of
the conservation of energy in the dynamical system takes




RELATION TO PHILOSOPHY


69


place. If it could be established that the action of the
psychical side is such that it does no mechanical work
in the dynamical system, an interesting fact would have
been discovered, but the main principle would not be
thereby affected that the dynamical system ceases to be
an independent system, the working of which can be described simply in accordance with the laws of Dynamics.
It must be remembered that the principle of conservation
of energy itself leads only to one of the equations which
describe the motions of the parts of a material system
possessing more than one degree of freedom. The effect
produced by the psychical factor on the physical system
could only be represented by an action to which there
corresponds no reaction on the physical side. Accordingly the physical part of the organism could only be
treated as a dynamical system if we had a knowledge of
the character and numerical measures of the mechanical
forces, acting on the physical system from without, by
which the action of the psychical part of the organism
on the physical system could be represented.
There is much force in the contention that the whole
problem of the nature of the relations between the
physical and psychical domains in the living organism
is essentially insoluble because the problem is a purely
artificial one, having arisen from the original assumption
made that the physical and psychical sides are disparate,
without an underlying unity. If it be held that we have,
in treating body and mind as belonging to separate
domains, set up a distinction which does not correspond
to any really fundamental difference, this may be held
to account for our inability to formulate any satisfactory
and coherent theory of the relations between the two
artificially separated domains. Having made an arbitrary
separation into two domains supposed to be disparate in
their natures, we are unable, without contradiction, to
undo our work by recombining them into one system.
In default of a purely monistic Science which should




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take up into itself all the phenomena which we now call
physical, and also those which we call psychical, and
which should be sufficiently advanced as to absorb and
unify all our present knowledge on both sides, we are
compelled to retain the methodological procedure of
considering physical and psychical phenomena separately.
Unless we are prepared to adopt, as a general principle
of unlimited scope, the theory of psycho-physical parallelism, that the physical domain is entirely independent
of the psychical, we are compelled to restrict ourselves,
on the side of Natural Science, to the procedure of
tracing out such sequences of physical events as we find
by actual experience to be capable, at least to a practically sufficient degree of approximation, of conceptual
description in which the concepts arise from the physical
domain alone. That this is possible, even in the biological
departments of Science, to some very considerable
extent, the limits of which we do not know, has been
abundantly established by modern Physiology. Nevertheless, it is impossible to regard the physical side of
the living organism as a completely independent system,
all the happenings in which are capable of being described conceptually, in accordance with the canons of
purely physical Science, and to an extent without
theoretical limits, unless we are prepared to adopt the
unproved assumption that the physical organism is
independent of the psychical side of the living being.
The amount of partial independence is just what experience shows it to be, but no number of successes in
subsuming particular chains of processes which occur
in the living organism under conceptual schemes such
as we employ in Natural Science, will warrant us in asserting the principle of the absolute independence of the
physical side of the organism to be more than a surmise.
We are thus led to what must be regarded as a
limitation upon the claims of Natural Science, in the
sense of the term which has been here adopted, to the




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power of theoretically extending itself so as to become
a complete Philosophy of Physical Nature, independent
of all psychical factors. Physiology is completely justified
in assuming this independence as a methodological
principle, and experience alone can decide how far it
will be able to extend its present far-reaching results in
accordance with that principle. But the existence of
Psychology, the Science of the normal individual mind,
with the borderland domain of Physiological Psychology,
indicates that the possibility of representing the phenomena of the bodily organism by means of conceptual
schemes of the kind which we class as belonging to
Natural Science may have limits which cannot be passed,
however far Physiological investigation may at present
be from having reached those limits.
Hitherto I have spoken only of Natural Science as the
Science of physical phenomena, but in the wider sense,
the term Science is employed in relation to the study
not only of physical, but also of psychical, phenomena;
and it is also used in the case of the study of complexes
involving both physical and psychical phenomena. The
question arises how far the methods of procedure which
Natural Science adopts are applicable where psychical
phenomena are involved. The Science of Psychology
occupies itself with the conceptual description of sequences of psychical happenings in the normal human
mind; the idiosyncrasies of particular minds being
disregarded, just as, in the conceptual schemes of
Natural Science, the irrelevant peculiarities of individual
physical objects are disregarded. There are, however,
important respects in which Psychology differs from a
department of Natural Science. The only mental phenomena, of which an observer can directly take cognisance, are those which occur in his own mind; what
occurs in the minds of other persons he can only
ascertain indirectly through physical manifestations.
Thus introspection, and inference from physical events,




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assumed to afford sufficient indications of corresponding
psychical events, are the two sources of the facts with
which the Psychologist has to deal in ascertaining laws
and in building up his conceptual schemes of representation of psychical sequences. Moreover, psychical
processes and states are not of a quantitative character
such as are accessible to the methods of measurement
employed in the physical sciences. These measurements
of quantity are dependent upon that property of extension which has no direct correlative in the psychical
domain. Psychical states and processes may lend themselves to notions of magnitude, but not to that of extensive magnitude; an intensive magnitude not being
made up of units, and therefore not being capable of
numerical representation as is an extensive magnitude.
Such measurements as are made by experimental
Psychologists are measurements of the physical concomitants of psychical phenomena, and not of those
phenomena themselves. It has moreover been pointed
out by Professor James Ward that Psychology cannot
be defined by reference to a special subject matter, as
in a department of physical science, since it deals in
some sense with the whole of experience. Not only
has Psychology to concern itself with a more complex
subject matter than a department of Natural Science,
but the position of the observer, or percipient, relative
to the phenomena is less simple in the former case than
in the latter; and it is thus difficult to assert that
psychical events and processes are phenomena, or
factors in the subject-object relation, in quite the same
sense in Psychology as in the Natural Sciences. The
Sciences of the Politico-social group, so far as they rise
above the more superficial classification of special kinds
of phenomena, and their statistical setting, are all dependent on psychological knowledge, and share in
the peculiarities I have mentioned which differentiate
Psychology from the Natural Sciences.




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The methods of Psychology and of the Sciences of
the Politico-social group must be similar in character
and spirit to those pursued in the case of the Natural
Sciences, so far as the nature of their respective subject
matter allows; and there exist in all the former Sciences
tracts in which the results and methods of Natural
Science are directly applicable, and may render great
services. Some departments of Biological Science,
especially evolutionary Biology and the theory of Heredity, have made use of psychical categories in such a
manner that they cannot, unless such psychical elements
can be eliminated, be regarded as falling wholly under
the denomination Natural Science, in the restricted
sense in which I have employed the term. For example,
Darwin, in his Origin of Species, occupies himself to a
considerable extent with mental factors, such as sexual
selection, as contributing to the causes of the natural
selection which determines the evolution of species.
Purposiveness, and all teleological conceptions, are foreign
to Natural Science; and all departments of Biology, so
far as they make use of such conceptions, must be considered as mixed Sciences, in the sense that they make
use of concepts which represent not only physical percepts but also psychical elements.
The psycho-physical parallelist may maintain that the
psychical elements in Biology can be ultimately eliminated, and the whole be reduced to a purely physical
Science; but such assertion is of a highly speculative
and contentious character. In view of what has been
said earlier in this lecture, it cannot be assumed as
established that such complete elimination of the psychical factor is possible, even in theory.




IV


CAUSATION AND DETERMINISTIC
SYSTEMS
WHEN a physical event takes place it is usually
regarded by common sense as determined by
preceding events or processes which are deemed to have
caused it to take place. Very frequently some one preceding event is singled out as the cause of the event in
question. It is assumed that the particular event, the
effect, would not have taken place in the absence of the
cause; and that cause is regarded as affording an explanation of the occurrence of the effect. A somewhat
less summary explanation of the event recognizes a
plurality of preceding events or processes, the absence
of any one of which would have entailed the nonoccurrence of the effect. The ordinary notion of causation attributes a certain contingency to what is regarded
as the cause of an event; thus an event or sequence A
is in general only considered as the cause of an event B
when the absence of A can be easily imagined; when
this is not the case, the invariable succession of B after
A does not give rise to the idea that A is the cause of
B. For example, we do not consider the night as caused
by the preceding day, although we regard both day and
night as belonging to a fixed sequence; in this case the
element of contingency is not regarded as present. In
scientific thought, the notion of causation is expanded
so as to embrace a whole complex of conditions, some
preceding in time, and others simultaneous with, the
particular event in question. That every event has a
cause, formulates the conception of the determination
of the event by a complex of preceding and present




DETERMINISTIC SYSTEMS


75


conditions. Taking into account the fact of the existence
of approximately isolated systems in the physical world,
the cause of an event is restricted to the notion that it
consists of preceding and present conditions forming a
limited complex of relevant circumstances; all other
circumstances being regarded as irrelevant in regard to
the explanation of the event. When the event under
consideration and the relevant circumstances are of such
a character that they are measurable, or in other words
capable of being correlated with numbers, the event is
frequently said to be a function of the relevant conditions. This notion of functionality may be regarded
as a more precise form of that of causation. But, in the
view of common sense, a view which has until comparatively recently been shared by men of Science, the
notion of causation contains something more than the
bare idea that the so-called effect has an invariable
relation to the cause, in point of fact. It includes the
notion that the cause, some particular event or process,
or some sub-complex within that cause, can be regarded
as an active agent which compels the effect, regarded as
passive in the transaction. When this idea of compulsion by an active agent is an essential component of the
notion of causation, the effect being regarded as forced
by the compelling action of the cause, that cause is said
to be an efficient cause. It is not difficult to recognize
the essentially anthropomorphic character of this notion
of efficient causation. When, by an exercise of his will,
a man moves a body, or one of his own limbs, he regards
himself as an active agent who, by the power of his will,
compels the body, or limb, to move; the moved body
being looked upon as purely passive in the occurrence.
The organic feeling of exertion which accompanies the
action is to him the sign of his activity. In all our interventions in the physical world this feeling of activity,
dependent upon the consciousness of motion and contraction of muscles, is present in greater or lesser degree.




76              CAUSATION AND
By ejection of this notion of activity into the physical
world in general, we tend to attribute to the cause of an
event the role of the active agent which we feel ourselves
to be when we regard an event as caused by ourselves.
We do not however usually suppose that consciousness,
or any psychical factor, is necessarily present in a cause
when that cause is not a living organism; in this respect
the original animistic conceptions which were probably
concerned in the ejection of the notion of efficient
causation into the external world have taken a modified
form. Hume,who in his Enquiry concerning human understanding has treated the subject of efficient causation
with great clearness and fullness, writes1:
No animal can put external bodies in motion without the
sentiment of a nisus or endeavour; and every animal has a
sentiment or feeling from the stroke or blow of an external
object, that is in motion. These sensations, which are merely
animal, and from which we can d priori draw no inference, we
are apt to transfer to inanimate objects, and to suppose, that
they have some such feelings whenever they transfer or receive
motion.
The indirect character of the effect of the causation
presumed to exist when a man moves a limb has also
been pointed out by Hume, who wrote2:
We learn from anatomy, that the immediate object of power
in voluntary motion, is not the member itself which is moved,
but certain muscles, and nerves, and animal spirits, and, perhaps
something still more minute and more unknown, through which
the motion is successively propagated, ere it reach the member
itself whose motion is the immediate object of volition. Can there
be a more certain proof, that the power, by which the whole operation is performed, so far from being directly and fully known
by an inward sentiment or consciousness, is, to the last degree,
mysterious and unintelligible?
The notion that causation in the physical world is
similar to the efficient causation which we feel to be
present when we act affords an explanation of the
1 Enquiry, Sect. VII, Part II. footnote.  2 Ibid. Sect. VII, Part I.




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ancient belief, one which has for ages been persistent
in scientific thought, that all physical action involves the
contact of material bodies, because our own mode of
directly intervening in the physical world involves the
placing of some part of our bodies in contact with
external matter. The principle known as the "principle
of causation" has frequently been stated in the concise
form that every natural event has a cause; and this has
usually been regarded as a necessary or axiomatic principle of thought to which Nature must conform. It
should be observed that the principle leads unavoidably
to an endless regress. For, if an event have necessarily
a cause, that cause, being considered to be a preceding
event or a plurality of such preceding events, must itself
have a cause, and consequently we have a sequence of
events in which there can be no initial, or first, cause.
A result of modern criticism of the notions which
underlie scientific thought has been that the conception
of efficient causation has been discarded, as unnecessary
and useless, for the purposes of Natural Science. Hume's
analysis of the nature of perception made it abundantly
clear that efficient causation is not to be discovered in
natural phenomena. He writes1:
When we look about us towards external objects, and consider
the operation of causes, we are never able, in a single instance,
to discover any power or necessary connexion; any quality,
which binds the effect to the cause, and renders the one an
infallible consequence of the other. We only find, that the one
does actually, in fact, follow the other.
It should be observed that the acceptance of Hume's
position in this matter does not imply an acceptance of
his views in this connection as regards the association
of ideas. It is at the present time often thought desirable
no longer to use the terms cause or causation, in connection with physical phenomena, on account of the
fact that these terms have usually in the past been taken
1 Enquiry, Sect. VII, Part I.




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CAUSATION AND


to imply the notion of efficiency. When the term cause
is retained it is taken to be synonymous with the totality
of antecedent conditions. The principle of causation,
taken in the only sense in which it can now be retained
in Natural Science, is not a logically necessary principle,
but merely the postulation, or rather working hypothesis,
that it is possible to predict the happening of particular
events when certain complexes of antecedent conditions
are known. The mode in which this prediction is made,
in any particular case, is by the employment of conceptual laws or schemes which have reference to events
of a class to which the particular predicted event belongs.
Thus Helmholtz has written1: " The principle of causal-*
ity is nothing else than the hypothesis that all the
phenomena of nature are submitted to law." I shall
however presently show that Helmholtz's statement is
capable of a too general interpretation; and that in fact
the principle of causality, as employed in Natural
Science, is of a more stringent character than would
appear from Helmholtz's formulation of it. The extent
to which the postulate of causation actually holds good,
when applied to natural phenomena, we can only find
out by experience.
In the more superficial notion of causation of which
I spoke at the beginning of the lecture, an event A,
antecedent to an event B, is under certain circumstances
regarded as the cause of B, some interval of time elapsing
between A and B. But our sense-impressions are no
longer regarded by Psychologists as atomic, in the sense
of consisting of a set of separate sense-impressions with
intervals of time, however small, between consecutive
ones, but rather as forming a continuous stream; so that
we now have to contemplate, instead of two separate
events A and B, a continuous process to which both A
and B belong. An advanced scientific theory does not
take account only of the bare fact of the invariability
1 Ueber die Erhaltung der Kraft, Wissensch. Abh. p. 68.




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79


with which B follows A, but should give a conceptual
account of the process to which A and B belong. In
some departments of Science, especially those which
have become amenable to representation by a Mechanical
theory, this has in many cases been accomplished. In
other departments the efforts to trace continuous processes, to which both the so-called cause and the socalled effect belong, have met only with partial and
fragmentary success. Sometimes, what has been accomplished, after minute scrutiny, amounts to the insertion,
between A and B, of other intermediate observed
events, between each two consecutive ones of which a
hiatus exists, of essentially the same character as the
original hiatus between A and B themselves; and thus
the nexus between A and B has not been successfully
represented as a continuous process. Moreover it must
be observed that an event at an instant of time can only
be regarded as a conceptual limit, not as an actual percept, because an actually perceived event always has
some duration, however small that duration may be;
and thus the actual events A and B can themselves be
regarded only as processes; finite parts of a single process which includes them both.
Although, as we have seen, the search for efficient
causes in natural phenomena is chimerical, the belief in
the existence and possible discovery of such efficient
causes has been of great importance in the whole history
of Science. That belief has often been advantageous in
stimulating investigators to discover a few stages in the
endless regress to which the a priori principle of causation leads; and this has often led to the establishment
of the connection between different phenomena, although
such connections have never given any lasting satisfaction to the desire to realize the notion of efficient
causation.
Just as efficient causation cannot be discovered within
physical phenomena, so also the search for any logical




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CAUSATION AND


necessity,binding together the phenomena of a sequence,
is doomed to inevitable failure. Logical necessity has
its habitat in the domain of thought alone. It binds
together the parts of a fully worked out conceptual
scheme which represents a physical sequence, but no
such logical necessity connects the successive stages of
the physical sequence itself.
In a scientific theory in which conceptual elements
are linked together by a scheme of postulations which
fixes the relations between the different elements, the
consequences which are deducible from the structure
of the scheme follow by logical necessity from the
postulations and definitions contained in the theory.
That logical necessity is of the character in accordance
with which the conclusion in a syllogism is a necessary
inference from the major and minor premisses. But,
when the scientific theory is employed for the purpose
of describing a complex of physical phenomena, there
is no justification for a transference of the logical
necessity from the conceptual theory to the perceptual
phenomena. For there exists no logical necessity that
the theory should be applicable to the description of the
phenomena; such applicability, when it exists, is a fact
of experience only, not a necessity of thought. For
example, we possess a conceptual scheme which assigns,
in accordance with the Newtonian dynamical scheme
and the law of gravitation, the relative accelerations of
the bodies of the solar system, represented conceptually
by points with definite mass-coefficients; and this scheme
is employed to deduce the orbits of the points which
represent the earth and planets about the point which
represents the sun. The forms of the orbits follow by
logical necessity from the postulations of the scheme.
Yet there is no purely logical necessity that the earth
should continue next year to describe, even approximately, the actual orbit round the sun which it has
described during the past year, in approximate accord



DETERMINISTIC SYSTEMS


8i


ance with the orbit as deduced from the conceptual
scheme. For there is no logical necessity that in the
future year the conceptual scheme will continue to serve
its descriptive purpose. The expectation which we have
that it will do so is of overwhelming strength, based
upon our past experience of the adequacy of the conceptual scheme. This expectation amounts to a subjective certainty, the existence of which in our minds is
a fact for an explanation of which reference must be
made to the Psychologist and the Epistemologist, not
to the Logician. Thus there is no purely logical necessity
that the seasons will recur as they have done in the past;
there is only an expectation of such strength that it may
be called a practical certainty, but not a theoretical or
absolute certainty. Without such practical certainty, of
this character, human action would be an impossibility;
the great survival value which this subjective feeling of
certainty as regards the continuance of a certain uniformity in Nature must have possessed during the stages
of evolution may be an important factor in the explanation of its existence. The fact that this kind of expectation, which Natural Science accepts as a working hypothesis, is found by general experience to be satisfied is
one of which Metaphysicians will take account in their
theories of the nature and structure of reality; but for
the purposes of Natural Science no such inferences of
a philosophical kind are necessary.
The very common idea thir it is the function of
Natural Science to explain physical phenomena cannot
be accepted as true unless the word "explain" is used
in a very limited sense. The notions of efficient causation, and of logical necessity, not being applicable to the
world of physical phenomena, the function of Natural
Science is to describe conceptually the sequences of
events which are to be observed in Nature; but Natural
Science cannot account for the existence of such sequences, and therefore cannot explain the phenomena
H GL                                      6




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CAUSATION AND


in the physical world, in the strictest sense in which the
term explanation can be used. Thus Natural Science
describes, so far as it can, how, or in accordance with
what rules, phenomena happen, but it is wholly incompetent to answer the question why they happen. When
a sequence of phenomena can be imitatively represented
by means of some other sequence of a more familiar
type, this latter is frequently said to afford an explanation of the former. Thus Lord Kelvin said, "when I
have made a mechanical model I understand a process."
This dictum might conceivably be understood in either
of two senses. In the first, that in which, as the context
shows, Lord Kelvin understood it himself, the model is
taken to be a concrete model in which actual material
bodies are employed to constitute its parts. Such a
model affords an explanation of the phenomena which
it pictures, in the more limited sense to which I have
alluded; it makes plain by means of a reduction of the
relatively unfamiliar to the more familiar. Such employment of a model is really only a preliminary, but
often a very important, stage in the process of gradually
arriving at a genuine scientific theory of the class of
phenomena in question. The next stage commences
with an attempt to explain the model, that is, to proceed
further, with the model as starting point; unless indeed
the working of the model has already been shown to be
representable by a conceptual scheme, in which case
the phenomena in question have, by an indirect process,
been subsumed under a scientific theory. The second
sense in which Lord Kelvin's dictum might be interpreted is that the model is of a conceptual character,
of which the parts are conceptual objects, of defined
characteristics, and related in assigned modes with one
another. In this case the subsumption of the particular
phenomena under a scientific scheme has been accomplished; to achieve this result is of the very essence of
scientific method. The predilection which has been




DETERMINISTIC SYSTEMS


83


shown during centuries, for mechanistic modes of representing all physical phenomena, that is for schemes
based upon the Mechanics of molar bodies, is to be
accounted for, in large measure at least, by the fact that
our most familiar experiences of natural phenomena are
related to the motions, and interaction by contact, of
gross bodies.
It will have been noticed that, not only in the more
popular notion of causation, but also in that more
refined and clarified form of the conception which we
call scientific causation, the cause which conditions the
effect precedes in time, or is at least not subsequent to,
that effect; and this even when the motion of efficiency
has been completely extruded. Thus, in the sequences
with which Natural Science deals, an earlier part of a
sequence is regarded as determining a part which is subsequent in time. The relation between what we call the
cause and what we call the effect is not a symmetric
relation; we do not in Natural Science regard an effect
as conditioned or determined by a cause in the future.
For the practical purposes of Science, those of prediction, it is essential that such an asymmetric determination should be the object of search. But this mode of
determination is not the most general which we can take
into account.
A deterministic scheme ot the most general type is
one for which we possess, or at least are assured of the
existence of, a principle or set of rules which is sufficient
to determine any single part of the sequence; although
a knowledge of that part of the whole sequence which
precedes in time the part to be determined may not
suffice for such determination. In such a case the whole
sequence is subject to a law, or formula, which makes it
possible, at least in theory, to calculate the particular
characteristics of any assigned part of the sequence.
Thus all parts of the sequence are mutually related with
one another, and with the sequence as a whole, in such
6-2




84


CAUSATION AND


a way that a knowledge of the principle of the whole
sequence is requisite for the purpose of calculating the
detailed nature of any part of it. In such a sequence
there is no asymmetric relationship of cause and effect.
The conceptual sequences with which Natural Science
has to do, and the search for which is essential for the
particular purposes of Natural Science, are deterministic,
but not of the most general type. When I said that
Helmholtz's definition of causation in Science is too
general I meant that it appears to refer to a deterministic scheme of the general type, and not to the particular
species with which Natural Science is concerned. The
methodological assumption of Natural Science is that
there are to be found, in natural phenomena, not
merely physical sequences which are describable by
means of deterministic schemes, but sequences capable
of description by that particular species of such schemes
as permits of the determination of the present from a
knowledge of the past, without taking into account the
future. In other words, those deterministic schemes
which Natural Science employs are such that they involve the asymmetric relation of causation.
It is possible to imagine that, to an unlimited extent,
natural phenomena could be representable by deterministic schemes of the general type, and yet that, only
to a more limited extent, were the phenomena representable by such deterministic schemes as Natural
Science, taken in the restricted sense, is capable of
employing. Determinism in this general sense does not
necessarily imply predictability of the future by means
of a knowledge of the past or the present. To make such
prediction, a certain knowledge about the whole complex, past, present and future, would be necessary. The
principle of sufficient reason, or of Ground and Consequence, may be considered as more general than the
principle of Causation. The Ground may be embodied
in the principle which underlies the whole scheme;




DETERMINISTIC SYSTEMS


85


whereas when the principle of Causation is applicable
there must be a restriction of, or specialization in, the
nature of the Ground, of such a character that asymmetric causation is realized in the scheme. A deterministic sequence of the general type does not necessarily
involve the same kind of uniformity that exists in a
scheme for which the causal relation holds. This is a
fact which admits of a simple illustration. Let us suppose that a particle had been moving, during the indefinite
long past, uniformly in a straight line; let us further
suppose that, at a certain time, it suddenly proceeds to
describe with uniform velocity a semi-circle, returning
to the original straight line at the further extremity of
the diameter of the semi-circle, and that then it proceeds to move for ever afterwards in the continuation of
the original straight line in which it was moving before
the semi-circular deviation took place. This is representable as a deterministic system, in which a single
formula can actually be given, by means of which the
position of the particle at any assigned time can be
calculated. The law, or principle, embodied in this
formula, makes provision for the single deviation from
uniformity of the motion of the particle, which occurs
during the semi-circular motion. Yet no observation of
the earlier part of the motion would have given any
means of predicting the excursion of the particle along
the semi-circle; the asymmetric law of causation would
be inapplicable in such a case.
It has sometimes been suggested that teleological
conceptions may be taken into account in deterministic
schemes of the general type. A process in which the
teleological factor is involved, whether it be completely
deterministic or not, is one in which the present cannot
be regarded as independent of the future parts of the
process. Whether this suggestion is of value in general
Philosophy is a difficult question which lies outside the
boundaries of the present discussion, and which I con



86


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sequently must refrain from discussing. It is quite
clear that, in that particular type of deterministic schemes
which Natural Science employs, there is no possibility
of any direct representation of the effects of purposiveness, regarded as introducing the element of contingency.
Although, as we have seen, the motion of efficient
causation, implying the activity of an agent that compels
or necessitates an effect, is one which forms no part of
the stock of conceptions which Natural Science can
utilize, it does not in the least follow that the conception
of such activity is illusory and must be denied all validity
in the more general domain of Thought. The consciousness of activity which we all have, whenever we act,
cannot be shown to be illusory, unless it can be conclusively shown that it can be brought under some other
category. The mere fact that the notion of efficiency is of
anthropomorphic origin is quite insufficient to establish
that this conception is illusory, or that it is necessarily
possible to dispense with it in the account which
Thought attempts to give of all the factors in experience.
All that has been maintained in the earlier part of this
lecture is that Natural Science, as distinct from a complete Philosophy of Nature, in the method it adopts of
describing conceptually that part of our experience
which we call physical, has no need of the conception of
efficient causation, and can make no use of it, for its
own special purposes. It may be a debatable point
amongst Psychologists, whether the notion of efficient
causation is necessary in Psychological Science. In any
case the reasons which have led to its extrusion from
Natural Science cannot, without much further examination, be held to be sufficient to justify its exclusion from
the categories of Psychology.
I have already referred to the fact that, in Biology,
considerable use has been made of mental factors in
relation to Evolution, and that teleological conceptions
have not always been dispensed with. Unless the theory




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of psycho-physical parallelism could be raised to something more than a surmise, or at most a postulation of
which the range of applicability is unknown, it is exceedingly difficult to conceive that Natural Science, in
the narrow sense, when applied to living organisms, can
ever succeed in giving accounts of sequences of physical
phenomena in such organisms which shall attain a degree
of completeness as great as has been reached in some
of the departments of Science which leave living
organisms out of account. Neither can it be regarded
as certain, perhaps not even as probable, that the future
progress of biological Science will ever enable it, in all
its departments, completely to dispense with the notions
of purpose and activity, although indefinitely larger
tracts of phenomena than at present may be found
capable of the descriptive treatment which Natural
Science applies. I have already pointed out that, in the
strictest sense of the term explanation, Natural Science
explains nothing; it only describes conceptually. It is
just the absence of the conceptions of efficient causation
and of logical necessity in the kind of account which
Natural Science is able to give of portions of the world
of physical phenomena that prevents us from regarding such account as explanatory in the strict sense of
the term. Philosophy has always occupied itself with
attempts to find explanations in the complete sense; and
thus to condemn a priori, as necessarily illusory, the
notions of efficient causation, activity, and purposiveness,
amounts to a dogmatic denial of all validity, and of all
possibility of success, to metaphysical thought. Some
men of Science are apparently prepared to take up this
dogmatic attitude towards Metaphysical Philosophy.
Their position however admits of no justification by
arguments of demonstrative, or even very cogent, force.
It is in opposition to what is at least prima facie the most
immediate knowledge we possess, that derived from the
experience we have when we will and act.




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Natural Science deals simply with what is observed
to happen in the physical world. By minute examination
and comparison of observations it discovers the existence of sequences of sufficient similarity with one
another, to be capable of being described by rules, which
are often called Laws of Nature. It synthesizes these
rules into conceptual schemes, called general theories.
It employs these rules and theories tentatively to predict
what will happen when a portion of a sequence which
appears to fall under one of these laws or theories is
observed; the prediction refers to that part of the
sequence which has not yet been observed. There is no
logical necessity that the prediction should be successful; but in case the rule or theory has been abundantly
verified by past observations, there is an expectation of
success, the disappointment of which, by age-long habit,
we do not usually contemplate. This expectation is often
so strong that, when we are so disappointed in the verification of our predictions, we attribute the fact to a
mistake in our belief that the sequence concerned is
actually sufficiently similar in character to those to
which the law or theory is known to be applicable. In
fact we search, usually with ultimate success, for disturbing factors in the conditions, which remove the
particular sequence in question from that particular
type which alone the law or theory is designed to
describe. As the result of such further investigations,
either the law or theory is rejected as insufficiently
established; or a wider theory is set up which takes
account of what we at first regard as disturbing factors
preventing the older theory from being applicable to the
new observed sequence, so that the wider and amended
theory suffices for application to the description of a
wider class of sequences than did the older theory.
It is no part of my programme to make a contribution
to the interminable discussions, with which Philosophers
and Theologians have for centuries occupied themselves,




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on the subject of Free Will and Determinism; whether
the world can be regarded as a deterministic system, or
whether it is subject to interventions, either continually
or sporadically, the character and amount of which are,
even in theory, incalculable. There is however one
question, relevant to these discussions, which is in the
direct line of thought that concerns itself with the true
character of Natural Science. I have already spoken of
deterministic schemes as employed in Natural Science;
but what precise meaning can be attached to the term
deterministic scheme? This question seems to require
a somewhat closer examination than I have already
given to it, especially in view of the fact that it has
frequently been assumed, without critical examination,
both by Determinists and by Voluntaryists, that the
conception of a deterministic scheme is perfectly definite.
Let us consider a finite system of ideal objects, endowed with a set of properties or qualities, and subject
to a defined set of relations with one another. We regard
this system as subject to change, as a certain variable,
usually taken to represent time, takes up a continuous
set of values. At any one time we regard the system as
being in a certain state. By a known state, we mean
that we have precise information as regards the positions,
motions, and all other relevant facts relating to the
various properties of and relations between the ideal
bodies, as they are at the time at which the state is
known. In what kind of language is it possible to express
this knowledge of the state of the system at a particular
time? The only language that we possess which would
appear to be adequate for this purpose is that of Arithmetic, in which numbers are employed. We therefore
assume that the state of the ideal system can be completely specified by a finite set of numbers. This amounts
to the assumption that all the positions, properties,
qualities, and mutual relations which constitute the
state of the system can be completely specified by means




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CAUSATION AND


of a finite set of numbers. This again involves the assumption that all these factors of the state are capable
of exact measurement; that is that each one of them can
be correlated with one number or with several numbers.
It is necessary to consider in some detail what this
correlation involves. In the perceptual domain there
are two kinds of magnitude, known as extensive and
intensive respectively. Only extensive magnitudes are
capable of direct measurement, that is, of correlation, to
a degree of approximation depending on the fineness of
our instruments, with the system of numbers. The
possibility of such correlation depends upon the fact
that an extensive magnitude can be regarded as consisting of units, all of which are identical in respect of
magnitude; this is the equivalent to the property that
two extensive magnitudes may be added together, their
sum forming another magnitude of the same kind. The
sizes, spatial positions, velocities, and accelerations of
perceptual bodies are all extensive magnitudes. On the
other hand, such qualities as temperature and colour
are not extensive magnitudes; they are not directly
additive as are extensive magnitudes, although they may
be regarded as having greater or less intensity. But
nevertheless some qualities which are not extensive
magnitudes may be correlated with such magnitudes,
and in that case they are indirectly measurable, and the
intensities can then be placed in correspondence with
numbers. The correlation of different kinds of intensive
magnitude with the number scale has actually been
effected, as in the case of temperature, or in that of
colour which has been correlated with several numbers.
Unless this correlation has been accomplished for the
case of any particular intensive magnitude, the quality
which it represents cannot be specified numerically as
regards amount. All actual measurements, direct or
indirect, are more or less inexact, being dependent on
the defects of our senses and of the instruments which




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91


we may employ to extend the scope of our senses. In
a conceptual scheme, these practical measurements are
replaced by ideally exact measurements, not subject to
errors of observation; thus the correlation of magnitudes
with numbers in a conceptual scheme becomes ideally
exact.
We now assume that, in our finite conceptual system,
all the elements which characterize the state of the
system are capable of exact correlation, each with one
number or with several numbers. On this assumption,
the complete state of the system at a given time is
specified by a certain set of numbers. In order to
represent changing states of the system, we have to
consider a set of variables whose values depend on the
time-variable; and which for a particular value of the
time-variable coincide with the numbers that represent
the state of the system at that particular time. If we consider the ideal system during a particular interval of the
time-variable, we may conceive the variables of the set,
by the specification of values of which a state of the
system is defined, to be functions of the time-variable.
If the forms of all these functions are known, we have
the means of calculating, to any assigned degree of
approximation, the state of the system at any time falling
in the time-interval for which the functions are defined;
thus the state of the system at every point of that timeinterval is determinate, and theoretically calculable. In
the case of a particular conceptual system, such as we
are here considering, the postulations and definitions
which we make as regards the relations between the
component parts of the system, and the laws governing
the changes in that system, may suffice actually to
determine these functions. When these functions are
all known to us, the conceptual scheme is not only
deterministic, but is actually determined, in the sense
that we have such a complete knowledge of the character
of the system as a whole, that we can by simple calcula



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tion determine in detail its state at any assigned time
falling in the interval for which it is a deterministic
system.
If we have not actually determined the forms of the
functions, but have convinced ourselves by sufficiently
cogent reasoning that these functions exist, as implicitly
defined by the general laws which hold for the system,
we still regard the system as deterministic, although
only theoretically determined. For values of the timevariable outside the range for which the system is known
to be deterministic, the functions employed may become
meaningless, or in any case may have no application to
the system. Without any essential difference in the
definition, it is possible to contemplate a system which
is deterministic for indefinitely great ranges of value of
the time-variable, of either sign. A conceptual system
which is deterministic, in accordance with this definition
of the meaning of the term, is deterministic in the more
general sense; there is no distinction between the relation of past or present with future states different from
the relation of future with past or present states. The
whole aggregate of states of the system for all the time
during which it is deterministic are bound together by
one common nexus. But a deterministic system of the
particular type which Natural Science can employ must
be such that the forms of the functions for the whole
time can be inferred from a knowledge of some or all
of the states of the system prior to a time falling within
the time-interval for which the functions validly represent the system. In this case, future states of the system
are determined by past states, on the assumption of the
actual, or of the theoretical, existence of the functions.
This may be stated in the form that functions determining the states of the system up to some particular
time can be continued beyond that particular time, so
as to continue to represent the states of the system for
some finite, or indefinitely great, interval of time sub



DETERMINISTIC SYSTEMS


93


sequent to the particular time considered. In this
manner we can represent to ourselves a deterministic
scheme in which the asymmetric causal relation (not
efficient causation) exists. Having determined the forms
of the functions for all the times up to a particular
time, a knowledge that the system is deterministic in
this sense, involves the postulation that it is possible to
determine, by some kind of continuation, the forms of
the functions for future values of the time-variable, and
thus by employment of these functions to determine
states of the system at future times. This involves the
postulation that the laws which regulate the system
continue to be valid in the future.
Is it possible to extend the definition of a deterministic system so as to apply to the case in which the states
of the system are only representable by means of a nonfinite set of functions? It is impossible exhaustively to
exhibit in any manner an infinite set of functions, or
even an infinite set of numbers. If such an infinite set
of functions is to be regarded as known, or determinate,
we must possess the means, when some finite part of
the set is known, of determining any one of the others
by means of some finite set of rules. Thus the states of
the system are virtually determined by a finite set of
functions together with a finite set of rules for the
determination of any other of the functions when the
former have already been determined. This amounts to
the virtual reduction of the system to a finite one, and
it is only on the assumption that this reduction is possible
that the conception of a deterministic system whose
states are specified by an infinite set of variables has any
meaning.
We are undoubtedly able to define particular conceptual schemes of a deterministic type, and in particular
such as include the causal relation, in the sense in
which that term is employed without involving the
notion of efficient causation. Does there exist in the




94


CAUSATION AND


perceptual world anything that corresponds to such an
ideal deterministic scheme? The success of physical
Science depends in a considerable degree upon the fact
that we are able to mark off in the perceptual world,
approximately isolated domains in which selected kinds
of sequences or events can be described with a very
considerable degree of approximation by means of ideal
deterministic systems to which such sequences are
correlated. It must however be observed that such correlation is confined in each case to a limited set of the
relations between the physical objects of the perceptual
domain, and never embraces the whole of their properties or relations. For one class of properties or relations we may require one ideal deterministic scheme,
and for another class we may require another such
scheme. Science cannot be said to have succeeded, even
in the case of a strictly limited, approximately isolated
group of percepts, in correlating all their observed relations and all their changes with a single ideal deterministic scheme. The notion of correlation of isolated
systems with particular deterministic ideal systems may
have its scope widened, so as to take account of the
detailed differences between one perceptual system and
another one of a similar character, by supposing that
the functional relations in the ideal scheme involve a
certain number of variable parameters, to which particular values have to be assigned when the ideal scheme
is to be applied to describe what happens in one
particular perceptual system belonging to the given
class.
In those cases in which we are able to correlate the
phenomena, of a certain class, in an approximately
isolated group of percepts, with an ideal deterministic
scheme which serves as a description of those phenomena, the correlation can never be taken to be valid
beyond some limited period of time, of greater or less
duration. That this is the case depends, at least partially




DETERMINISTIC SYSTEMS


95


upon the fact that the perceptual system is only approximately isolated; and that outside some limited period
of time the relations of the system with other systems
external to it may produce an effect upon the system
which is no longer negligible; the deviations of the
system from the conceptual scheme which represents
it may be cumulative in amount, and may produce after
a sufficiently long time an effect which makes the correlation no longer even approximately valid. It thus
appears that a group of phenomena can only be shown
to be representable by means of an ideal deterministic
scheme, to a certain degree of approximation, and only
for some limited time which may however in some cases
be very great, according to our ordinary measures of
time. The description is never applicable to represent
in one scheme all the phenomenal aspects of the perceptual objects which have to do with the phenomena
described by the scheme. It must also be remembered
that many phenomena have hitherto proved too intractable to admit of complete representation by any
kind of conceptual scheme, least of all by a scheme
which we can describe as deterministic in the precise
sense that it is expressible by relations of magnitudes.
The assertion seems to be certainly true that Natural
Science has not succeeded in showing that all that
happens in a group of objects, however small that group
may be, is deterministic, in the sense that it can be
completely represented by an ideal deterministic scheme
of the precise kind that I have depicted. Attempts,
which I shall describe in later lectures, have been
made to show that all the happenings in a group of
perceptual objects, or even in the whole perceptual
world, could be represented by certain kinds of conceptual schemes, especially of the kind called an atomic
theory, in which everything is regarded as dependent
upon the interactions of suppositive elementary particles. All such schemes have proved utterly inadequate




96


CAUSATION AND


to give a representation of more than a part of what is
observed to happen. Even the theoretical possibility of
the existence of a deterministic scheme applicable to the
whole physical world and to all that happens in it, is
open to many and grave objections; partly dependent
upon the difficulty to which I have before alluded, of
extending to the world as a whole, conceptions which
can only be regarded as significant when applied to a
finite part of the world, and partly dependent upon our
inability to estimate the cumulative effects of those
individual peculiarities of perceptual objects which have
always to be disregarded by Natural Science. Even
Laplace, when he contemplated the scheme which I
referred to in my first lecture, had to postulate the
existence of a kind of superhuman calculator; and the
complex of electrical and sub-molecular phenomena
which have been discovered since Laplace's time has
indefinitely increased the difficulty of imagining what
even the main outlines of such a scheme would have
to be.
Mioreover the possibility of regarding, even theoretically, the physical world, or a finite part of it, as capable
of complete description by means of a deterministic
scheme, cannot be contemplated without some consideration of the relations of the physical world to the
psychical world. If indeed the theory of psycho-physical
parallelism could be established completely, the psychical
world could be completely ignored, and no special
difficulty would arise from its existence as a system
which exercises no influence whatever upon physical
phenomena. If there exists complete detailed parallelism
between the two domains, it would appear to follow as
a consequence of the assumption that the physical world
is completely describable by a rigidly deterministic
scheme, that the same assertion could be made as
regards the psychical world. On the other hand, on an
interactionist theory of the relation between the two




DETERMINISTIC SYSTEMS


97


domains, the influence of the psychical world upon the
physical must be in some way included in the deterministic scheme which ex hypothesi is applicable to the
physical world. If this could not be done, the physical
world would cease to be representable with absolute
completeness by means of the deterministic scheme; the
influence of the psychical factor would then be of a
character which could only be described as an interference with the physical order of things. If the view,
to which I have referred in the third lecture, were
accepted, that the physical and the psychical worlds are
in reality only parts, or aspects, of one single fundamental system, the assumption of the deterministic
character of the physical world would appear to involve
the assumption that the whole fundamental system,
which would embrace both what we call physical and
what we call psychical, is in some sense deterministic.
At all events those processes or events in it which appear
to us to belong to the physical world would be regarded
as forming a deterministic system, and possibly other
events or processes in it might be regarded as free, or
not determined; but these latter would then have to be
considered to have no influence on the former, otherwise
we should be faced with the same difficulty as in the
case of the theory of interaction between two disparate
domains.
It is difficult, if not impossible, to attach a precise
meaning to the conception of a deterministic psychical
scheme, of a kind similar to the meaning which we
attach to a deterministic physical scheme. For psychical
qualities, processes, and events are not prima facie
measurable in a manner similar to that in which physical
qualities, processes, and events may be measurable.
What Psychologists measure are not psychical events or
processes, but their physical concomitants; and these
measurements could only be taken to represent psychical
measurements on the unproved assumption of some


HGL


7




98


DETERMINISTIC SYSTEMS


complete and exact correlation between the two domains,
extending even to the transference of numerical measurements from the physical to the psychical domain. This
transference may be, and probably is, sufficient, and of
value, for certain purposes; but the assumption that all
psychical states, processes, and events are completely
and exactly measurable, by means of a transference of
the measurements of physical concomitants, is at least
a very large assumption, and one extremely difficult to
justify.
The conclusion of the whole matter seems to be that
the conception that the whole world of physical phenomena, or that a finite part of that world, is theoretically
capable of being represented by a unified deterministic
scheme is unproved and unprovable. All that Natural
Science has established is that tracts of phenomena can
be found which are sufficiently represented for certain
purposes by means of deterministic schemes. A very
large part of Natural Science has not yet reached the
stage in which deterministic schemes, of the kind which
involve relations of number, are applicable; much of it
is in the stage in which only classification in abstract
types can be employed, and to which precise measurement is not yet applicable. But it is a working hypothesis, employed in all the more advanced departments
and stages of scientific thought, that tracts of phenomena
can be discovered, to which deterministic schemes can
be applied for the purposes of precise description and
of prediction. The justification for this postulation is to
be found in the past successes of such advanced parts
of Natural Science, and we are not acquainted with
barriers which will prevent ever larger tracts of phenomena from being correlated with deterministic descriptive schemes.




V


NUMBER AND ITS DEVELOPMENTS
ARITHMETIC, the Science of Number, taken in
Z the general sense of the term now employed by
Mathematicians, is the most advanced and also the most
purely abstract department of Natural Science. Abstract
Arithmetic is usually spoken of as a formal Science, like
Logic; and it may therefore perhaps occasion some
surprise that I should speak of it as a department of
Natural Science. However, it resembles other branches
of Natural Science in the fact that it was in connection
with physical experience that it took its origin. Its
earliest development as a Science consisted in setting
up a conceptual scheme for the representation of certain
aspects of the physical world. In this respect it does
not differ generically from other branches of Natural
Science; although the stages by which it became formal
and deductive were much more rapid than in those
departments which deal with what may perhaps be
described as more complex, and less superficial, aspects
of the world of physical phenomena. I have already
suggested that every branch of Natural Science, when
it reaches a sufficiently advanced stage of development,
tends to become purely formal and deductive. The
Science of Geometry has reached this stage of development, and the same thing may be said of Mechanics.
It is unnecessary, and would be outside the scope of
these lectures, to enter into any discussion of the
questions whether, or how far, what are known as the
truths of Arithmetic embody a priori knowledge, and
may thus be regarded as logically necessary truths, presupposed as a precondition of experience, and whether




ioo  NUMBER AND ITS DEVELOPMENTS
this accounts for the kind of apodictic certitude which
we attach to the knowledge of simple arithmetical relations. Whatever views be held as regards the much
debated subject of the philosophy of Arithmetic, the
fact remains that the knowledge of the individual has
its origin, as had that of the race, in physical experience;
although it may of course be maintained that the
function of physical experience was simply to awaken,
and make explicit, conceptions already present in the
mind in a latent form. In any case the part played by
physical experience in the development of arithmetical
concepts in an explicit form affords, I think, sufficient
justification for reckoning Arithmetic amongst the
Natural Sciences. I have spoken of Arithmetic, including the higher Mathematical Analysis which is a
development of more elementary Arithmetic, as the
most advanced branch of Natural Science; this must be
taken to mean that it has more completely than in the
case of any other department, with the possible exception of Geometry, which I shall consider in the next
lecture, reached the stage in which it consists of the
detailed development of the implications of a purely
conceptual scheme; no further recourse being required
to observation or experiment in order to test its range
of applicability to describe certain aspects of the physical
world.
Whilst Arithmetic, in the wide sense of the term, is
the most advanced and the most abstract branch of
Science, its rudimentary parts are far more popularly
understood, and used in applications, than is the case
with any other branch of Natural Science. The grocer
when he weighs out and sells his goods makes use of
conceptions which were developed only by a long process of evolution; when he enters his receipts in his
books, the notation he uses is a warrant of the great
importance of notation in a formal Science, and embodies one of the triumphs of our race, as a mode of




NUMBER AND ITS DEVELOPMENTS             Io
economizing thought. The concepts of Arithmetic in
their more elementary form, or in the higher developments to which they attain in Mathematical Analysis,
pervade all departments of Natural Science and all the
Mechanical arts. The Philosopher, in his reflections on
spatial and temporal relations, on number and quantity,
on matter and motion, is in a region of thought in which
the boundary between his own domain and that of the
Mathematical Analyst is difficult to delimit with precision. The Epistemologist has always been accustomed
to consider Mathematical knowledge as a kind of touchstone on which to test his theories of the nature of
knowledge. The dominant views in some departments
of philosophical thinking have been notably influenced
by the results of recent Mathematical research, and may
not improbably be in future further modified from the
same quarter. The universality of Arithmetic in the
Natural Sciences consists of the fact that numbers, or
variables which are interpreted numerically, enter into
every conceptual scientific scheme that has reached a
stage characterized by extreme precision of statement.
For Arithmetic, in the developed form of Mathematical
Analysis, provides the very language in which the precise
descriptions contained in such schemes are clothed.
The concepts of unity, and of number, or degree of
plurality, were in their formation occasioned by physical
perceptions. The precise mode in which these concepts
were formed is a matter for psychologists to discuss and
determine. As formal categories they lie at the base of
the Science of Arithmetic, and consequently of Mathematical Analysis, which is now regarded as essentially
no more than abstract Arithmetic carried to a higher
stage of development by the help of certain postulations
concerned with the domain of the infinite, or transfinite.
The notion of unity, the product of mental activity in
relation with an environment, is the form under which
an object is subsumed when it is the object of attention.




102 NUMBER AND ITS DEVELOPMENTS
Thus unification is the result of an act of attention which
involves a differentiation of the presentational continuum. A physical object, brought under this category
of unity, may for all other purposes be recognized as
possessing any degree of complexity. It is sufficient, in
order that the object may be subsumed under the form
of unity, that is, regarded as a single object, that it be
so far distinct, within the presentational continuum, as
to be recognized as discrete and identifiable. What
external marks are necessary, that an object may be so
recognized as discrete, is a matter for the judgment of
the mind which performs the act of unification. There
is a large degree of arbitrariness, limited only by the
powers of perception of the individual mind, in this act
of unification, or of apprehending an object under the
form of unity; no special, or uniquely definable, physical
characteristics of the object are essential for this purpose, but only some sufficient degree of differentiation
of the object from its environment.
The notion of plurality is involved when attention is
paid either successively or simultaneously to objects,
each of which is subsumed under the form of unity.
This notion, at first indefinite, takes the form of definite
plurality when a collection or aggregate of objects is
attended to. Such a group or collection is then regarded
both as a single whole to which unity is attributed, and
also as a definite plurality, consisting of a set of objects
each of which is regarded for the particular purpose as
one. The single objects which compose the collection
need not possess any parity as regards size, weight, or
any other special quality, but may be of the most diverse
characters, although in practice they usually have some
similarity of nature which forms the ground of their
being treated as a collection. In any case a certain
logical parity is ascribed to them in virtue of the fact
that each one of them is subsumed under the form of
unity. The fact has recently been pointed out and




NUMBER AND ITS DEVELOPMENTS 0o3
illustrated by Prof. James Ward1 that the earliest conception of number, as a degree of plurality, probably
arose as the result of immediate intuition of the differing
qualitative characters of very small groups of objects.
Thus a pair of objects can be intuitively recognized as
qualitatively different from a group of three objects,
without recourse to the process of counting. This immediate intuition of the number of objects in a group is
facilitated when the objects are arranged in some recognizable pattern. It is probable that not only human
beings, but also higher animals, possess the power of
discerning intuitively this qualitative characteristic of a
very small group of objects, and of recognizing that
something is changed in a group originally, say of three
objects, when one of them has disappeared. It is clear
however, that this avenue to the concept of number is
of very limited scope. All further development of the
concept was made in connection with the process of
counting, or tallying. For this process, the two notions
of order and correspondence are requisite.
In virtue of the notion of order, relative rank is
assigned to each object in a collection, so that the collection becomes an ordered aggregate. In actual counting
the order is usually assigned to the objects during the
process itself, as an order in time, and this may be done
in an arbitrary manner. The order of the elements in an
aggregate may however be assigned in a manner dependent on their sizes, weights, or other qualities, or
in a manner dependent on their relative spatial positions.
Order may however be regarded as an abstract conception, independent of any particular mode of ordering;
for an aggregate to be an ordered one, it is necessary
and sufficient that each object of the aggregate be
recognized to possess a certain rank, in virtue of which
it is definite, as regards any two of the objects which
may be selected, which of the two has the lower, and
1 Mind, Vol. xxIX, p. I37.




104 NUMBER AND ITS DEVELOPMENTS
which the higher, rank. The notion of correspondence
underlies the process of tallying, or that of counting on
the fingers. The objects of one aggregate are regarded
as standing in a logical relation with those of another
aggregate, of such a character that a definite element of
one aggregate is regarded as corresponding to a definite
element of the other aggregate. The correspondence
between two aggregates is complete, or (I, I), when, to
each object of either of the aggregates, there corresponds
one object, and one only, of the other aggregate'. The
number, or degree of plurality, of an aggregate can then
be defined as the concept of the quality of plurality
which the aggregate has in common with all aggregates
with which it can be placed in complete correspondence.
Thus a number is the concept of the quality which the
members of a family of similar aggregates have in common. The number I, although not characteristic of a
plurality, is still in formal Arithmetic regarded as one
of the integral numbers. If, in counting an aggregate,
the process is stopped before the aggregate is completely
counted, we may regard the part counted as a section of
the aggregate; such a section can then be taken to be an
aggregate having a particular number. The sections of
an aggregate, together with the aggregate itself, have an
order assigned to them, the same as that of the objects
of the aggregate itself. Thus the numbers of the sections
are themselves ordered, and we thus conceive the
numbers I, 2, 3, etc., to be arranged in a definite order,
usually called their natural order.
A number which specifies the degree of plurality of
an aggregate is called a cardinal number, but a number
which specifies the rank of a particular object in an
aggregate is called an ordinal number. Cardinal and
ordinal numbers are distinguished from one another in
that their descriptive functions are different. It has been
maintained by some writers on the foundations of
1 The aggregates are then said to be similar to one another.




NUMBER AND ITS DEVELOPMENTS o05
Arithmetic that the notion of an ordinal number is
logically prior to that of a cardinal number. This does
not, however, seem to be necessarily the case; either
concept can be employed in a systematic treatment of
the subject as fundamental, the other being then regarded as derivative.
Before Arithmetic can be considered to be a developed
Science the further step must be made of the introduction of a scheme of relations between the numbers
dependent on the operations of addition and multiplication, with the inverse operations of subtraction and
division. There has been a very prolonged discussion
amongst philosophers as to whether the judgment expressed by such a proposition as that 7 + 5 = 12 is an
analytic judgment, in the sense that the truth of the
proposition can be deduced from an analysis of the concepts of the three numbers, or whether that judgment
is synthetic, in the sense that some further extraneous
knowledge is required to warrant the judgment. There
can however be no doubt that, historically, and in the
individual, the explicit knowledge of such relations, in
simple cases, was empirical; being derived from actual
counting of the combined aggregate when two aggregates are amalgamated into one. The general conceptual
scheme of relations involving operations arose as a
generalization of knowledge obtained from such physical
experience. It seems certain that the fundamental
notions that I have specified must have been possessed
by primitive man, in an implicit form, long before the
notion of abstract number reached an explicit and
developed form. But the earliest records we possess of
ancient peoples, those of the Egyptians, the Babylonians,
and the Chinese, show that they possessed arithmetical
knowledge that had already attained a very considerable
degree of development.
The origin of fractional numbers is doubtless to be
ascribed to the necessities arising in connection with




io6 NUMBER AND ITS DEVELOPMENTS
measurement. The division of an object into equal parts,
and the representation of one or more of such equal
parts, was the empirical origin of the notion of a fraction.
But in our present theories of Arithmetic, the concept
of number, both integral and fractional, is taken to be
independent of any conceptions relating to measurement. That operation is now regarded as requiring the
application of Arithmetic, but as not connected with
the foundations of the subject, the necessary empirical
basis of which rests exclusively upon the operation of
counting. A fraction, as for example -I, may be regarded
as representing the operation of counting three objects
each of which belongs to an aggregate of five objects,
no assumption of equality of the objects in respect of
size or other quality being requisite. On this basis the
theory of the Arithmetical operations involving fractional
numbers can be made to rest. The more purely formal
theory of fractional numbers, as usually expounded,
regards each one as defined by a pair of integers forming
a single object, a couple; formal laws are then postulated
as to the relations between such couples, forming the
basis of the scheme of operations involving them.
It is very important to remark that the operation of
counting conceptual objects in which the integral numbers are employed, and the extension of that process, in
which fractional numbers are employed, are both free
from that element of approximativeness which appertains to every operation of actual measurement. The
number of an aggregate of objects is an exact description
of that aggregate in a certain respect, but number as
applied to the measurement of extensive magnitude
represents the measure only subject to the inexactitude
inherent in our sensuous perception, even when refined
instruments are employed. This unique peculiarity of
the application of the conceptual scheme of Arithmetic
to the perceptual domain depends upon the fact that
the operation of unification is independent of any




NUMBER AND ITS DEVELOPMENTS I07
special physical properties of the object to which it is
applied, and involves only the mental operation of
general differentiation of the object from the environment. Measurement on the other hand is directly concerned with one at least of such physical properties. The
application of the conceptual scheme of Arithmetic for
descriptive purposes in the physical domain, so long as
we confine that application to the original purpose for
which it exists, has a certain absolute exactitude which
does not appertain to any other conceptual scheme in
its application to the concrete, and which no longer
appertains to Arithmetic when it is applied to represent
measures of physical magnitudes.
In all schemes of a symbolic character, such as written
language, symbolic logic, or Arithmetic, the facility with
which the scheme works depends very largely upon the
choice of a suitable and simple notation. In the case of
Arithmetic, the notation which we employ, and which
is of Indian origin, represents perhaps the greatest
labour-saving invention that has ever been made. The
notation for numbers is systematic, in the sense that all
integral numbers can be denoted, on a uniform plan, by
the employment, in the decimal scale which we use, of
ten distinct symbols. This number of symbols could be
reduced to two, if we employed the dyad scale. The
principle that the place which a digit occupies, in the
set of digits used to represent a number, indicates the
mode in which it is interpreted to represent a multiple
of a power of ten, is the crucial point in the principle
upon which the notation is founded. It is a remarkable
fact that neither the Greeks nor the Romans were in
possession of a systematic notation for numbers. An
attempt to carry out even a simple addition or multiplication, in which the numbers are represented by
Roman numerals, is the simplest path to a conviction of
the vast importance of the great Indian invention which
renders arithmetical operations practicable in accord



io8  NUMBER AND ITS DEVELOPMENTS
ance with uniform rules. There is an important invention, included in our system of notation, which
appears to have been introduced later than other parts
of the system. This is the use of a symbol denoting zero,
which is employed to indicate the absence of a particular
power of ten in the representation of a number as the
sum of multiples of powers of ten, together with a digit
which is one of the first nine numbers, or is itself zero.
The most ancient account of the Arithmetic of the
Egyptians is contained in the Papyrus of Ahmes, and is
entitled Direction for attaining a knowledge of all secret
things. In this arithmetic both integral and fractional
numbers appear, but the notation is not a systematic
one. Only fractions with unity as numerator are employed, such a fraction being denoted by the integer
which represents the denominator, but with a dot placed
over it. The only exception is the fraction |, for which
a special sign was employed. Other fractions are expressed as the sums of fractions all of which have unity
for their numerators, and a table is given in the Papyrus
for expressing fractions in this manner. Ahmes also
dealt with some problems involving arithmetical and
geometrical progressions. A Babylonian table has been
discovered in which the first sixty square numbers are
given, and also some cube numbers. In this table a
sexagesimal system of notation is employed, in which
the place in order represents a power of 60. This was in
essence a systematic notation, and so far an anticipation
of the invention of a later age, but the Babylonians do
not seem to have employed a zero to represent vacant
units. Although the Greeks possessed no systematic
arithmetical notation and no sign for zero, they managed to perform arithmetical operations of some difficulty. For example, the greatest Greek Mathematician,
Archimedes, in his discussion of the quadrature of the
circle, inscribed a regular polygon of seventy-two sides
in a circle, and obtained a good approximation to the




NUMBER AND ITS DEVELOPMENTS Io9
ratio of a side to the diameter of the circle. To do this
he had to extract the square roots of several large
numbers, to a sufficient degree of accuracy for his
purpose. The nature of the method he employed in
performing these operations has been a subject of considerable discussion in our time. It has frequently been
said that the ancient Greeks were great Geometers, but
poor calculators. It is certain that their command of
Arithmetic was seriously hampered by the unsystematic
character of the notation they employed. The Greek
Mathematicians, even the arithmetician Diophantes, did
not take the important step of introducing negative
numbers, in order to remove the impossibility of carrying
out the operation of subtraction of a greater number
from a lesser. This step was, however, taken by the
Indians; the Indian Astronomer Aryabhata employed
distinct names for positive and negative numbers, which
denote respectively possession and debt. Even the
representation of positive and negative numbers by
segments of a straight line in opposite directions was
known to the Indians.
As a result of the introduction of difference of sign,
we possess what is known as the ordered aggregate of
rational numbers, a conceptual arithmetic scheme in
which the operation of subtraction is in all cases a
possible one. We have here an example of the extension
of the domain of number, in order to extend the scope
of a formal operation, and thereby to increase the utility
of the conceptual scheme in regard to its applications
to the description of magnitudes. As we shall see, this
extension is but one example of several extensions of
the domain of number, made in accordance with the
same general principle. The inadequacy of rational
numbers for the complete representation of all ideal
magnitudes was discovered by the Greeks. A rigorous
theory of the ratios of incommensurable magnitudes,
the discovery of the existence of which is attributed to




iio  NUMBER AND ITS DEVELOPMENTS
Pythagoras, was given by Euclid in the tenth book of his
treatise on Geometry. A valid proof was given by the
Pythagoreans that the ratio of the length of the diagonal
of a square to that of a side cannot be exactly represented
by any rational number. On the formal side, the restrictions, which hold within the domain of rational
numbers, upon the possibility of operations such as the
extraction of roots of numbers, pointed in the direction
of an extension of the conception of number, of such a
character that, in an extended domain of number, such
operations would no longer be impossible, when both
rational and irrational numbers are recognized as falling
within the extended domain.
The modern theory of the aggregate of real numbers
arose out of the exigency of the requirement of a complete theory of ideal magnitudes, and out of the limitations which exist in the domain of rational numbers as
regards arithmetical operations within that domain. The
term "real number" is a somewhat unfortunate one,
but it arose historically from an attempt to distinguish
between these numbers and the so-called imaginary
numbers of which I shall presently speak. The theory
of the aggregate of real numbers, as developed in a
complete form by Cantor, Dedekind, and Weierstrass,
involves a postulation related to the conception of the
infinite. The integral numbers I, 2, 3,... form an
ordered sequence which has a first term, the number i,
but no last term. The very principle of the sequence is
that any particular term is succeeded by another term,
and thus the existence of a last term would be in contradiction with this principle. The sequence is thus an
example of the unending, or indefinitely great. The
postulation is made that nevertheless the sequence can
be regarded as a single object for thought, having
definite properties. Considered as an aggregate of distinct objects, that of the integral numbers, this single
object, the aggregate, is infinite, or transfinite. In the




NUMBER AND ITS DEVELOPMENTS              I I
form given to this postulation by Cantor, the sequence
is regarded as defining a new ordinal number co, which
is not identical with any of the finite ordinal numbers
i, 2, 3,... but is of higher rank than any of them, and
is not immediately preceded by any one of them. This
new number co is called the first transfinite ordinal
number, and is made by Cantor the starting point of a
new series of transfinite ordinal numbers. The more
general form of the postulate to which I have referred
is that, if any sequence of objects P1, P2, P3,... is so
defined, by means of a finite set of rules, that a definite
object of the sequence corresponds to each integer of
the sequence i, 2, 3,..., then the unending sequence
P, P2, P3,... may be regarded as a single definite
object with definite properties. The justification for
making this postulation is two-fold. In the first place
it must not lead to contradiction, when the logical consequences of the scheme based upon it are scrutinized,
and in the second place the scheme which involves this
postulation must be of utility as a conceptual scheme
for the application of number in a general theory of
magnitude.
For the purpose of defining a real number which is
not rational, that particular kind of unending sequence
is employed which is known as a convergent sequence
of rational numbers. If we carry out, for example, the
steps of the process of extracting the square root of the
number 2, we obtain successive sections of a convergent
sequence. The rational numbers which form the elements
of the sequence cannot of course be exhibited exhaustively, but we possess a set of rules which suffice for the
calculation of any one of the numbers by means of a
definite procedure involving in each case a finite number
of applications of the rules. In this sense the sequence
is regarded, in accordance with the postulate, as a single
definite object, defined by a norm, or set of rules,
although it cannot be completely exhibited as in the




II2 NUMBER AND ITS DEVELOPMENTS
case of a finite set of objects. We observe that, when we
have carried out a large number of steps of the process,
we have determined a rational number represented by
a decimal with a large number of digits, and this differs
from any of the later rational numbers which may be
determined by a decimal in which a corresponding
large number of places are occupied by zero. This is a
particular example of a convergent sequence, the rational
numbers successively obtained representing a set of
continually closer approximations to the object which
we regard as defined by the sequence, namely the
irrational number V 2. In the general case, every convergent sequence of rational numbers is taken to define
a real number, and, as in the case of V/ 2, this real
number is not necessarily a rational number. The
establishment of the complete theory of real numbers
involves a detailed investigation which fixes the precise
theory of identity and of relative order of the real numbers so defined. It also establishes the essential fact that
the arithmetical operations which express relations between rational numbers can all be extended, with
enlarged scope, to the case of real numbers.
The result of this investigation is that we have before
us an aggregate, that of the real numbers, positive, zero,
and negative, in which any two of the real numbers have
a definite relation of rank, specifying their relative order
in the aggregate. This aggregate has two properties of
capital importance. In the first place every convergent
sequence of which the elements are real numbers itself
defines a real number, known as the limit of the sequence. This is expressed by the statement that the
domain of real numbers is closed. The domain of
rational numbers is not closed, in this sense of the term,
since a convergent sequence of rational numbers does
not necessarily define, or have as its limit, a rational
number. The second property is that every real number
can be exhibited, in an indefinite number of ways, as a




NUMBER AND ITS DEVELOPMENTS              113
sequence of real numbers. This property may be described as that of connexity, and also appertains to the
aggregate, or domain, of the rational numbers. Both
these properties, in one of which the domain of rational
numbers is lacking, are essential to the fitness of the
aggregate of real numbers for the purpose of the complete conceptual representation of linear magnitudes.
The question has frequently been asked, how can
Number, which is essentially discrete, be employed for
the complete representation of magnitude, which is
essentially continuous? In considering the answer which
should be given to this question it is necessary to
scrutinize the meaning which can be assigned to the
expression " continuous." As in all other such cases we
must consider both the perceptual and the conceptual
meanings which may be attached to the term continuum.
On the perceptual side, in which we are considering
actual measurement of a magnitude, the notion of continuity, or absence of gaps, implies that between any
two magnitudes, of the kind considered, there exist
other magnitudes, and that the process of continual
contemplation of magnitudes filling up the gaps between
any pair can be carried out to an extent limited only by
our powers of perception, even when instruments of the
utmost precision are employed for the purpose. The
notion of continuity, regarded in this way, gives rise
to what may be described as the sensible continuum of
magnitudes, and containing, as it does, an inherent
element of indefiniteness, dependent on the approximative character of our sensuous perceptions, it can
only be raised into the position of a precise conceptual
scheme by a process of abstraction and idealization.
For the representation of all actual measurements the
rational numbers are sufficient, and can be applied to
represent any actual magnitude to any required degree
of approximation, the degree of approximation which it
is worth while to aim at being dependent on an estimate


8


HGL




I4 NUMBER AND ITS DEVELOPMENTS
of the inevitable errors which the mode of observation
entails. The exigencies of our method of representing
aspects of the perceptual world by ideally exact conceptual schemes necessitate however the development
of a theory of measurement in which the characteristic
properties of a conceptual continuum are assigned by
definitions and postulations. That the employment of
rational numbers only is insufficient for the purposes of
such a conceptual scheme was known, as we have before
remarked, as early as the time of the Greek Mathematicians. The modern theory of the aggregate of real
numbers, or of the arithmetic continuum, as it is now
called, has been devised as a scheme sufficient for the
purpose of denoting the magnitudes in the conceptual
continuum, which is taken to be the idealization of the
notion of the sensible continuum of perception. The
aggregate of rational numbers, since it is not closed, is
not a conceptual continuum in the sense in which the
term is applied to the aggregate of real numbers. It has
been charged against Mathematicians that, in setting up
such a scheme as the arithmetic continuum, they have
introduced an unnecessary complication, in view of the
fact that rational numbers suffice for the representation,
to any required degree of approximation, of all sensibly
continuous actual magnitudes; that in fact an instrument has been created of an unnecessary degree of
fineness for the purposes to which it is to be applied.
The answer to the charge is that Mathematical Analysis,
which is based upon the arithmetic continuum, and
essentially consists of operations involving numbers,
would become unworkable as a conceptual scheme, or
at least much more cumbrous, if the conception of
irrational numbers were excluded from it. The results
of operations involving rational numbers constantly lead
to irrational numbers, without which the operations
would be impossible if their effects are to be regarded
as definite. But in order to appreciate the full weight of




NUMBER AND ITS DEVELOPMENTS              I15
this answer it is necessary to consider the great generalization of Arithmetic which is made when variables are
introduced which denote unspecified numbers. The
passage is then made from the primitive form of Arithmetic to Algebra, in which the formal operations of
Arithmetic are represented as relations between sets
of unspecified numbers represented by non-numerical
symbols. The result of an algebraic operation, expressed
by general formulae, such for instance as the simple case
of the solution of a quadratic equation, would not
always be interpretable when special numerical values
are assigned to the symbols, if the only admissible numbers were rational ones.
Without the employment of the conception of irrational numbers the function of Mathematical Analysis
would be degraded to that of determining only approximate results of the operations it employs, and in consequence its technique would have indefinitely greater
complication, of such a character that, at least in its
more abstruse operations, it would break down, or lead
to results which contained a margin of error difficult to
estimate.
I have described in general terms the gradual extension of the concept of number, commencing with
integral numbers, proceeding to fractional numbers,
then to negative numbers, and thus attaining to the
aggregate of rational numbers, and lastly, by the employment of an ontological postulate, the extension to
the arithmetic continuum, which is capable of describing
adequately relations of magnitude in the conceptual
domain of continuous magnitude. There is however one
further extension of the conception of number, essential
for the purposes of general Mathematical Analysis,
which has very frequently been a stumbling-block for
non-Mathematicians, who have founded upon it the
charge that it consists of a species of jugglery with
symbols that from their very nature are meaningless. I
8-2




ii6  NUMBER AND ITS DEVELOPMENTS
allude to the introduction of complex, or so-called
imaginary, numbers. The operation of determining a
square root of a negative number is not a possible one,
within the domain of real number, since the square of
every real number is a positive number. On a principle
similar to that by which the domain of numbers was
extended by the introduction of positive and negative
signs, in order that the operation of subtraction might
become always a possible one in the enlarged domain,
the domain of real number is further extended so as to
become one in which every number, whether positive
or negative, has a square root. The new domain is then
that of what is known as complex number. The existence, subject to the law of contradiction, of a new number whose square is -i is postulated; this is usually
denoted by a. It can then be shown that, in a new
domain, in which each number is the sum of a real
number and of C multiplied by a real number, where
either of the real numbers may be any number belonging
to the arithmetic continuum, it is possible to postulate a
consistent scheme of relations of operations. This scheme
is of such a character that the laws of operations are in
formal agreement with the laws which hold for operations which involve only real numbers. This extension
of the domain of real numbers to that of complex numbers has the advantage that it involves a notably enlargement of the scope of algebraical processes. For example,
every quadratic equation has solutions, which would
not be the case if the existence of real numbers only be
admitted, and thus a much greater degree of generality
is introduced into Arithmetic and Mathematical Analysis
generally by the extension. Moreover, the boldness of
the Mathematicians who ventured upon this extension
has its reward in the fact that the new set of numbers,
the complex numbers, are applicable to the specification
of the positions of points in a plane, as is illustrated by
the well-known Argand diagram. The theory of func



NUMBER AND ITS DEVELOPMENTS I 7
tions of a complex variable has become a most important
branch of Analysis, indispensable for many purposes,
among which are applications to abstract, or Mathematical, Physics. The popular prejudice against the use
of the number t, or V-I, and of the whole system of
complex numbers is based on the ground that V/-i
represents an impossible operation. When the matter
is regarded aright, there is no justification for this
prejudice. What is a possible operation, or what is an
impossible one, does not depend upon any absolute
criterion of possibility, but upon the characteristics of
the domain in which operations are carried out; the
possibility or impossibility is in fact relative to a
particular domain. So long as the domain was that of
signless number, the operation of subtraction was not
always a possible one, for example 3 - 5 represented an
impossible operation, and could only be taken to represent an " imaginary" number, in relation to the domain.
Similarly, the operation of extracting a square root of a
negative number is only impossible within the domain
of real number; it becomes a possible operation within
the enlarged domain of complex number. The so-called
imaginary numbers have just the same conceptual
reality as the so-called real numbers, provided the
domain of such numbers has been defined in accordance
with a precise scheme of definitions and postulations
subject to the law of contradiction. The validity of the
scheme having been justified, its utility is the justification for its actual construction.
In the time of the Greek Mathematicians, and also
during the progressive period of Mathematical thought
which commenced in the sixteenth century and has
lasted to the present time, the conceptions relating to
the infinite and infinitesimal, together with the related
conception of a limit, have almost continuously occupied
the attention of Mathematicians and Philosophers.
Much of the confusion of ideas on these matters which




II8  NUMBER AND ITS DEVELOPMENTS
lasted from the time of the origin of the Differential
Calculus, created by Newton and Leibniz, was due to a
failure to distinguish with sufficient clearness of outline
between the conceptual and the perceptual sides of the
measurement of magnitudes, and to the uncritical
acceptance of notions derived from sensuous intuition
as sufficient for the basis of a rigorous conceptual
scheme. On the subject of the infinite and the infinitesimal the views expressed have frequently presented
a diversity akin to that which has been exhibited in
relation to general Philosophy. There have been all
shades of believers, sceptics, pragmatists, and finitists.
Although it is a popular belief that Mathematics is of
such a character as to leave no room for differences of
opinion, it is a fact that, even at the present time, there
exist differences of opinion amongst Mathematicians
about the foundations of the Science, and more especially
about matters in which the notion of the infinite is
involved. In the earlier presentations of the Differential and Integral Calculus, frequently called the
Infinitesimal Calculus, theories concerning infinitesimal
magnitudes played a large part. The notion of an
infinitesimal was frequently complicated with ideas
about motion, derived uncritically from spatial and
temporal intuition. Resulting from the clarification of
the conceptions of the foundations of Arithmetic and
Mathematical Analysis, resting upon a purely arithmetical basis, which has taken place during the last
half century, the conception of infinitesimal numbers
has been excluded from ordinary Arithmetical Analysis,
as an unnecessary conception in the scheme. Every
number regarded, for the purposes of the Calculus, as
existent, belongs to the arithmetic continuum, and is
therefore finite, if it be not the single number zero.
The term infinitesimal is no longer used to denote a
number, or a magnitude; when it is used at all it is
employed to describe a process of change, and even for




NUMBER AND ITS DEVELOPMENTS               I9
that purpose it is better avoided, so as not to give rise
to misunderstandings.
The fundamental conception of a limit, to which I
have already referred in connection with the definition
of an irrational number, emerged first amongst the
Greeks, in a geometrical form, embodied in the method
of exhaustions. It is remarkable that their conception
of the nature of a limit attained a standard of rigour,
greater than that which obtained amongst modern
Mathematicians before our own time. This is exemplified
in the proof given by Euclid, in the eleventh book of his
Elements, that the circumference of a circle varies as
its radius. Archimedes applied the method of limits in
a rigorous manner to various problems of what we now
call integration, such as the determination of areas and
volumes. The advantages which the methods of Leibniz
and of Newton had over that of Archimedes were of a
practical kind, as they had a form which made them
readily applicable to calculations, whereas the geometrical
form in which the Greek method was clothed, together
with the absence of a convenient arithmetical scheme,
made the application of the Greek method to particular
problems decidedly cumbersome. But in the matter of
soundness of theory, the method of Archimedes was
superior to those of Leibniz and Newton. The construction of the arithmetic continuum was an absolutely
necessary requisite for a rigorous theory of limits, and
thus for the foundation, on a logical basis, of the Differential and Integral Calculus. In default of such a construction, when the notion of magnitude, as directly
given by sensuous intuition, was taken as part of the
basis of the doctrine of limits, the frequent endeavours
that were made to prove that every convergent sequence
of numbers necessarily converges to a limit were doomed
to inevitable failure.
That the contemplation of the infinite, in some form,
is indispensable to Mathematicians arises from the fact




I20  NUMBER AND ITS DEVELOPMENTS
that even comparatively simple problems, such as those
of the determination of the length of a curvilinear arc,
or of the area enclosed by a curve, can in no case be
solved by employing a finite number of the operations
of arithmetic, except when the curve consists of segments of straight lines. An approximation to the magnitude of an area can be found by division of the area
into a sufficiently large number of rectangles, leaving a
small undetermined part of the area out of account.
By increasing the number of rectangles indefinitely in
a suitable manner, the measure of the area required is
exhibited as the limit of the sequence of numbers determined by the approximations. Thus the magnitude to
be determined is only obtainable as the limit of a
sequence involving an indefinitely great set of arithmetical operations. The Integral Calculus is concerned
with methods of calculating the limits defined in such a
manner. It is clear moreover that the very conception
of an area or length, as having a definite magnitude, is
dependent upon the concept of a limit, and that it is
only defined by, and is dependent upon, the existence
of a definite limit to a sequence. The notion of the
gradient, or differential coefficient of one variable with
respect to another, is one which is essentially dependent
upon the conception of a limit, and the gradient only
has a precise meaning when a rigorous theory of limits
has been previously established. The older idea that a
gradient may be regarded as the limiting ratio of vanishing quantities, and which was justly derided by
Bishop Berkeley, affords a striking example of the hazy
conceptions of fundamental matters with which Mathematicians for a long period of time contented themselves. The abstract conceptual schemes, of an advanced
character, employed for the purpose of representing
measurable physical processes, are very frequently expressed in the form of relations between the gradients
of various magnitudes with respect to other magnitudes,




NUMBER AND ITS DEVELOPMENTS               12I
or to the time-variable. These relations, known as
differential equations, are of fundamental importance in
connection with such abstract schemes of representation;
and the mathematical theory concerned with the determination of the values of variables, under certain conditions, in terms of the time-variable, is consequently,
for the purposes of theoretical physics, of great importance. In the theory of differential equations, we
have one of the most important examples of the fact
that the abstract theories of Pure Mathematics provide
the means for utilizing conceptual scientific schemes for
the purpose of representing in their quantitative aspect
a great variety of physical phenomena.
The fact that Mathematical methods are, in a very
large class of cases, unable to deal with objects, or with
processes, except by breaking them up into parts, and
increasing indefinitely the number of those parts, is a
significant example of a limitation imposed upon us by
what appears to be a definite characteristic of our modes
of apprehension. We appear to be unable to grasp some
of the relations of a whole, without breaking it up, as it
were atomistically, and then proceeding to reconstruct
the whole by a synthetic process which is confined to a
continual approach to the whole along the path of an
endless regress, which by its very nature, is such that
the whole is never actually reached within the process,
although a scrutiny of the laws of the regress may
enable us to obtain a knowledge of the relations of the
whole.
The mathematical theory of the numerically infinite,
and especially the developments of Georg Cantor and
his school in the theory of infinite aggregates of objects,
have aroused considerable interest in the ranks of Philosophers. Some of them, such as, for example, Josiah
Royce, have suggested that the results of the theory of
transfinite numbers may throw light upon questions of
General Philosophy, such as the fundamental problem




122  NUMBER AND ITS DEVELOPMENTS
of the One and the Many. I cannot enter here into a
discussion of the extension of the theory of the transfinite into the more general region of thought, but I
would suggest that extreme caution should be exercised
in attempting to extend results of such a theory as that
of transfinite aggregates to a domain wider than its
original one. The theory has been created for a special
purpose, that of dealing with certain aspects of the
numerically infinite, and its constructions and results
are all dependent upon a set of postulations and definitions which it has been the aim of investigators to make
as precise in character as possible. The most careful
scrutiny of the meaning to be attached to the terms
employed in any extensions of the theory, such as those
to which I have alluded, is of the last importance; otherwise there is a serious danger of falling into grave errors
in setting up theories in which vague analogies, involving
the surreptitious use of such terms as the infinite, in a
sense different from that in which they are employed in
the Mathematical theory, take the place which should
be occupied by a critically explored foundation.




VI


TIME AND SPACE
THE physical experience of each individual contains
as elements what are called his intuitions of time
and of space. These elements are only separable by
abstraction out of the unity of his actual experience; and
when the separation is made, the elements represent his
own private temporal and spatial intuitions. This fact
is sometimes expressed by saying that each person has
his own time and his own space, as forms of his physical
perception. A large amount of attention has been
devoted by Psychologists to the precise character of the
individual's intuitions of time and space, and to the
mode in which his powers of judgment involved in such
intuitions are developed. Every actual presentation in
what the percipient regards as the present involves both
duration in time and spatial extension. Whatever part
of his time does not belong to this duration of an act of
attention appears to him as the past, in which he had
earlier presentations, or to the future in which he
anticipates further presentations. There is thus in his
intuition of time a qualitative distinction between the
past and the future. The present always has some actual
duration, and can only be regarded as an instant of time
without duration, by an abstraction representing the
conceptual limit of a duration indefinitely diminished.
Such a conceptual instant of time is accordingly not an
element in his actual perceptions, and thus his time
cannot, except by abstraction, be regarded as composed
of instants. There has been some discussion of the
question whether perceptual time must be regarded as
continuous or discrete. Public time, that which we




I24


TIME AND SPACE


employ in social life, and in which from the point of
view of Science physical processes are regarded as taking
place, is a concept constructed by means of intersubjective intercourse. An event perceived by different
percipients is, under certain conditions, regarded as one
and the same event capable of being perceived by any
normal percipient in suitable circumstances; it is then
regarded as occurring in time, that is in public time,
which may be correlated with the private time of any
particular percipient. I leave here out of account the
questions which have recently aroused great interest as
to the simultaneity of an event for different observers.
For all ordinary purposes, not only of every-day life,
but also of Science, the older conceptions on this matter
are sufficient. By consideration of sequences of such
events, or of processes, regarded as independent of a
particular percipient, a general correlation is set up
between this public time and the private time of a
percipient; the notions of past, present, and future being
transferred to public time. The measurement of time is
founded upon that of some standard measurable process,
sometimes that of the rotation of the earth. Thus equal
intervals of time may be defined as intervals in which
the earth turns through equal angles; and the time of a
complete rotation of the earth may be taken as a unit
of time, the sidereal day. It is a result of many forms
of empirical observation that there exist a number of
different physical processes which, when employed in
the manner I have indicated, give rise to one and the
same measure of time, at least when a suitable averaging
is resorted to. Thus for example, a nearly constant
number of oscillations of a properly constructed pendulum take place in equal intervals of time measured by
the earth as time-keeper. The ordinary units of time,
the year, the mean solar day, the minute, and the second,
are obtained by a somewhat more complicated construction dependent both on the average rotation of the




TIME AND SPACE


I25


earth round the sun, and on the rotation of the earth
relative to the sun. We possess, in spring watches,
pendulum clocks, sand-glasses, etc., the means of
measuring time which, with greater or less accuracy,
correspond with the standard time founded on the mean
solar day.
For the purposes of an abstract conceptual scheme
time, as empirically measured, must be idealized. For
this purpose, Newton suggested the conception of
absolute time, as that which flows uniformly. Such a
definition however can only be taken as generally
descriptive, and not as really defining a precise conception, because the expression "flows uniformly"
implies already an underlying conception of time. Thus
his definition is open to the fatal objection of circularity.
In abstract Dynamics, and in fact in any purely abstract
conceptual scheme, the role of time is played by an
independent variable which, on account of the function
it has in a scheme to be applied to the description of
actual physical processes, may be spoken of as the timevariable. The field of this time-variable is taken to be
the arithmetic number-continuum; the aggregate of all
real numbers. Thus a particular time, in the abstract
scheme of Dynamics, is simply a number; a particular
value of the time-variable; and an interval of time is the
difference of the numbers which represent the two ends
of the interval. In this completely abstract conception
of time the generic distinction between future and past
which exists in perceptual time has disappeared; there
exists only an ordered aggregate of a particular type, of
which the elements are conceptual instants, the relation
of elements of lower rank in the order to those of higher
rank being all that corresponds to the original distinction
of past from future. When the conceptual scheme is
applied to describe actual physical processes, as for
example, the motions of bodies, the time-variable is
correlated with the public time, as measured by the




126


TIME AND SPACE


standard physical process, in such a manner that equal
intervals of the arithmetic continuum which is the field
of the time-variable, correspond with equal measurable
amounts of the standard process. This procedure
adopted by Science, of representing abstract time by
the arithmetic continuum, has been spoken of by Bergson
as the spatialization of time, no doubt on account of the
fact that the linear spatial continuum can also be correlated with the arithmetic continuum. It should however be observed that, in doing this, a process of abstraction is applied to the space of perception of a character
similar to that which has been employed in the case of
time. Thus a more accurate account of the procedure
of Science would be to say that both time and (linear)
space are represented conceptually by one and the same
abstract scheme, that of the arithmetic continuum.
Both have in fact been represented as ordered aggregates
of the same type. It must be remembered that the
arithmetic continuum is a construction not arrived at
by idealizing the notion of linear space1, but that it is
logically independent of the notion of spatial magnitude. One of the chief applications of the arithmetic
continuum is to provide an ideally exact theory of the
measurement both of spatial and of temporal magnitudes.
As in the case of the temporal perception of an
individual, his spatial perception is only separable by
abstraction from his whole physical experience. His
spatial perceptions are dependent on a variety of sensations, partly visual, and partly tactile and motor. The
spatial perceptions which arise from these two sources
considered separately differ notably from one another
in character; the representative space of the individual
is a synthesis of the two, in which habit, that is past
spatial experience, plays a large part. The geometrical
space which we employ when we reason about the
spatial properties of ideal bodies is of a conceptual
1 The logical, not the historical, order is here referred to.




TIME AND SPACE


I27


character, and differs in important respects both from
visual space and from tactile and motor space. Geometrical space is of three dimensions, infinite, or at
least unbounded, homogeneous, and isotropic.
It is impossible for me to analyse in detail the characteristics of visual space, and of tactile and motor
space. It must suffice to say that visual space is neither
homogeneous nor isotropic, and that, in it, distance is
only indirectly appreciated by convergence of the two
eyes, with muscular sensations due to accommodation
between them. This space has three dimensions, and
the particular kind of Geometry adapted to the conceptual description of it is what is known as projective
Geometry. In tactile and motor space we make direct
experimental estimates of distances and measures. Each
muscle gives rise, when it is contracted, to a special
sensation, so that motor space may be said to have as
many dimensions as we have muscles. The conceptual
Geometry which corresponds to tactile and motor space
is what is called Metric Geometry. The private, or representative, space of an individual percipient, like his
private time, is finite, but with an indistinct boundary.
Private space, like private time, is sensibly continuous,
as any portion of it is conceived to contain lesser
portions, without any definite limit of smallness. As in
the case of time, it is not atomic, for it does not consist
of points without extension: the notion of a point of
space is, like an instant of time, a pure abstraction. In
spatial, as in temporal, presentations, there are qualitative distinctions of direction. The frame of spatial
reference of the individual is provided by his own body,
and the qualitative distinctions of up and down, right
and left, depend upon this frame of reference. The
private space of the individual has a certain absoluteness,
because the sense of effort which he has when he moves
relatively to his material environment is lacking when
he himself remains quiescent and the external objects




128


TIME AND SPACE


are moved relatively to his body; the relative motions of
his body and the objects being the same in the two cases.
The space employed in social intercourse, sometimes
called physical space, is a construction of such a character that the private spaces of individuals may be
systematically correlated with portions of it. The possibility of such correlation depends upon the fact that an
identity is attributed to a physical object of such a kind
that it is regarded by all percipients as one and the same
object. Thus physical space is the complex of the spatial
relations of physical objects regarded as a system of
objects capable of being perceived by any percipient in
suitable circumstances. It is in physical space that all
the phenomena with which Natural Science is concerned are regarded as happening. The frame of reference of the single individual having been eliminated, the
only meaning that can be ascribed to the rest or motion
of a body in physical space is conservation, or change,
of spatial relations with other bodies. For any measurement of position, or of the motion of a body, some
standard frame of reference in some standard body,
such as the earth, or the sun, or the walls and floor of
the room, must be assigned. Thus all spatial relations in
physical space are relations of extension of perceptual
objects and between different objects; physical space
cannot be regarded as empty space because, with the
disappearance of perceptual objects, the whole scheme
of relations which constitutes physical space would disappear.
The opinion has been prevalent that external objects
are localized by us in a geometrical space of a unique
character with definite properties; this geometrical space
has been frequently regarded as a kind of ready-made
framework in which we localize our physical perceptions.
When I discuss the possible forms of geometrical space,
it will, I think, appear that this is by no means the case.
To give a systematic scheme descriptive of the relations




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in physical space is the first object of the Science of
Geometry, although in some of its developments the
Science has so extended itself as to transcend this
primary object. In order to attain this object, the spatial
relations are idealized, and transformed into a precise
form, by means of a system of definitions and postulations. By this process of abstraction and idealization,
conceptual space, the space of abstract geometry, has
been created. It is in this conceptual space that all the
ideal objects of Geometry are regarded as situated, and
as subject to a scheme of relations specified by a system
of postulates. In accordance with the general method
of scientific procedure such a scheme must satisfy the
condition of logical coherence; that is of freedom from
contradiction. It must also satisfy the test of applicability to the description of the spatial relations which
are observed to hold in physical space. As a result of
the efforts of Mathematicians, prolonged through many
centuries, and supplemented by the critical examination
to which the foundations of the subject have been subjected during the last few decades, the Science of Geometry has attained to a degree of coherence which may
be held to justify the designation of it, that I made in an
earlier lecture, as a model to which other departments
of Science may tend to conform. Geometry is better
fitted than Arithmetic to be regarded as a model, in this
sense, because, although Arithmetic, as we saw in the
last lecture, has been completely conceptualized, it did
not originate from the necessity of describing conceptually any one particular class of physical properties;
whereas Geometry had its origin in the effort to describe
one particular class of physical relations, namely the
extensional properties of perceptual objects.
I propose to give some account, necessarily brief, of
the process of growth by which Geometry, in its present
highly developed state, has come into being. A remarkable illustration of the fact that a valid conceptual


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scheme, adequate for the purposes of description for
which it was devised, is not necessarily unique, is
afforded by the fact that different conceptual schemes
of Geometry have been constructed. These systems are
all logically coherent; they are inconsistent with one
another as regards their several postulations; and yet
several of them are adequate, at least when they are
suitably restricted, for the purpose of description of
percepts. The nature of our knowledge of spatial relations has been a problem for Epistemology and Psychology, which has been very widely discussed. In this
connection, the history of Geometry is of great interest,
especially since, as a result of modern investigations, a
decisive refutation has been provided, from within the
Science, of Kant's celebrated views as to our spatial
intuition. These views are still maintained in some
philosophical circles, often owing to an insufficient comprehension of the nature of the modern developments
of abstract Geometry. I reserve a discussion of the
latest theory of spatial and temporal relations, that connected with the names of Minkowski and Einstein, for
a later lecture. It is sufficient here to remark that, if
that theory be finally established, it does not affect the
validity of the previously existing theories of Geometry,
or their applicability as descriptive schemes sufficient
for ordinary purposes. The older Geometry will accordingly never be completely superseded by the more comprehensive theory to which I have referred.
The earliest rigorous treatment of the Science of
Geometry was that of the Greek Mathematicians; of
this we possess a systematic account in that great textbook of the subject, Euclid's Elements of Geometry. This
contains an account of the current Geometry in Euclid's
time; it is the most ancient text-book of Science that
we possess; and the fact that it has been used for many
centuries is a testimonial to its excellence. To a large
extent, the Greek Geometry exhibits, in dealing with




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physical extension, the true scientific method. The
simplest regularities and uniformities observable in the
shapes and spatial relations of actual bodies were singled
out and then conceptualized. Thus points, straight lines,
planes, rectilineal figures, circles, spheres, pyramids, and
other objects, transformed and idealized from percepts
into concepts, are the geometrical objects with which
Euclid deals in his abstract Geometry. Later writers,
especially Apollonius and Archimedes, treated of the
Geometry of cones and conic-sections; an example of
non-utilitarian scientific investigation which bore useful
fruit in the hands of Kepler. The type of Geometry,
developed by the Greeks, known as Euclidean Geometry, and for a long time regarded as the only possible
type of Geometry, still remains the standard Geometry
for practical purposes and for ordinary scientific purposes. It forms the conceptual basis for all our actual
spatial measurements, and notwithstanding the increased
generality of modern geometrical schemes, it will certainly continue to be employed for the more ordinary
purposes of Science, even if it be superseded for the
purposes of certain general theories relating to gravitation and Electrodynamics. In order to distinguish the
Euclidean Geometry from other more modern Geometrical schemes of a divergent character, it is frequently
spoken of as the Geometry of Euclidean space. In view
of the fact that geometrical space cannot properly be
regarded as an entity endowed with special properties,
since it in reality represents a mere possibility of spatial
determinations, it is more accurate to speak of Euclidean
Geometry as a Euclidean system of spatial relations, or
as Geometry with a Euclidean Metric. In the nominal
definitions of the geometrical objects with which his
scheme deals, Euclid gives a somewhat elaborate descriptive account of characteristics of those objects. In
his deductive treatment, many of these characteristics
are not in any way employed in the argument. In some
9-2




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of the definitions, postulates lie hidden which later
criticism has brought to light and stated explicitly. An
example of this is the fourth definition of the fifth Book,
which implicitly contains the postulate known as the
axiom of Archimedes. What we now regard as the
postulated or hypothetical scheme of relations in Euclidean Geometry appears in part, in Euclid's elements,
in his axioms, or common notions, which he regards as
self-evident; and in part in his postulates which are
taken to be facts that are unproved, but the assumption
of the truth of which is necessary for the purposes of
his theory. Other postulates, derived from intuition,
are made implicitly in the course of his deductions. In
Euclid's own form, although his Geometry has many
of the characteristics of a valid conceptual scheme, the
treatment is far from exhibiting that example of a flawless deductive scheme, for which it has frequently been
accepted. From Euclid's own time onwards there has
been much discussion and criticism relating to the true
character of the definitions, axioms, and postulates of
the scheme. This has resulted during recent decades in
a restatement of the foundations of the subject, of such
a kind that synthetic Geometry may now be regarded
as conforming to all the requisites of a conceptual
scientific scheme.
The objects with which Euclid deals, points, straight
lines, planes, etc., are obtained as the result of idealization of actual percepts, in which some constituents of
the percept are removed by abstraction. Thus a point is
the ideal limit, postulated as existing, of an object from
which its extension is completely abstracted. It retains
spatial relations with other objects, that is, position.
A line is an object in which we abstract from the thickness of perceptual lines; and a straight line is an idealized
object which arises from our observation of empirical
straightness. Euclid gives no complete conceptual representation of relations of magnitudes. Thus, equality of




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magnitude, and the relations of congruence of segments
of straight lines, of angles, and of areas, are not made
to depend upon a logically complete system of postulations; but recourse is had to intuition of magnitude in
the region of percepts. This defect is exhibited prominently in Euclid's use of the method of superimposition
employed in the theorems relating to congruent figures.
The apparent reluctance with which Euclid employs
this method, restricting it, as he does, to cases in which
its use seemed unavoidable, would appear to indicate
that he had misgivings as regards its logical validity.
As it stands, no complete defence is possible against the
charge of circularity which has been made against this
reasoning. In the modern form of synthetic Geometry,
a system of definitions and postulations relating to congruence is introduced into the foundations of the conceptual scheme itself, rendering the intuitional method
of superimposition unnecessary.
Euclid's theory of parallels has formed the starting
point, from his own day onwards, of discussions which
have ultimately led to a generalization of the whole
theory of Geometry. It is remarkable that the so-called
axiom of parallels was given by Euclid himself not as
an axiom or self-evident truth, but as a postulate, an
assumption necessary to his scheme; and thus it may
be interpreted as having a hypothetical character. Since
every actual observation of relations in physical space
is confined to some region which is necessarily finite,
whereas abstract Geometry deals with relations extended
into indefinitely great regions, it is clear that any
postulation relating to parallels must be incapable of
complete empirical verification of a direct kind. But
the postulate may have consequences, such as that the
sum of the angles of a triangle is two right angles, which
can, subject to inevitable errors of measurement, and
with certain physical assumptions, be regarded as
capable of empirical verification. From an early time




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onwards, attempts have been made to dispense with the
use of the postulate relating to parallels, as an independent assumption, by showing that it can be proved
as a deduction from the rest of Euclid's scheme.
Proclus (410-485 A.D.), in his commentary on Euclid's
Elements, gave a detailed account of attempts, made by
Ptolemy and by himself, to effect this deduction. These
attempts, and many others made in modern times, were,
as we now know, doomed to inevitable failure. The first
Geometer who appears to have contemplated the possibility of an hypothesis, relating to parallels, inconsistent
with that of Euclid, was Girolamo Saccheri (1667-1733),
a Professor at Pavia. But his invincible prejudice in
favour of the Euclidean hypothesis, as a necessary constituent of the only possible Geometry, prevented him
from recognizing the true implications of his investigations. Lambert (I728-I777) went further in the same
direction as Saccheri; he showed that the area of a
triangle is proportional to the difference between the
sum of its three angles and two right angles, in the two
cases corresponding to what we now call hyperbolic and
spherical Geometry. John Wallis (I616-I703) remarked
that Euclid's postulate may be replaced by the equivalent assumption that similar triangles of different magnitudes exist. It is in fact sufficient to assume the
existence of only two similar triangles, with different
magnitudes; as was observed by Laplace and by Carnot.
Legendre (I752-I833) carried out investigations, of
permanent value, in connection with various long-continued attempts to prove the truth of Euclid's postulate.
These investigations prepared the way for the great
change in the whole position of Euclidean Geometry
which resulted from the labours of Lobachewsky,
Bolyai, and Gauss; the last of whom was the first
Mathematician to express a definite conviction that the
postulate is incapable of proof.
The first publication of a synthetic geometrical theory




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in which a postulate was employed that differs essentially from Euclid's postulate relating to parallels was in
the form of a treatise by Lobachewsky, which appeared
in I829; this was followed in 1832 by the publication of
a similar theory discovered independently by Bolyai.
In accordance with Euclid's postulate, through a point
outside a straight line, one and only one coplanar straight
line can be constructed which does not intersect the first
straight line, but, in the theory of Lobachewsky and
Bolyai, a whole sheaf of such straight lines can be constructed. This sheaf is bounded by two straight lines
said to be parallel to the one considered. It was shown
that, when this postulate is substituted in the Euclidean
scheme, for that of Euclid, it is possible to build up a
systematic Geometry in which the properties of figures
will be in some respects very different from those in
the Euclidean Geometry. For example, the sum of the
angles of a triangle is less than two right angles, the
amount of the difference depending upon the size of the
triangle, so that, in this system, there can exist no relation of similarity between two rectilineal figures of
different dimensions. This Geometry is a species of what
is called non-Euclidean Geometry, and in view of later
discoveries, this particular non-Euclidean Geometry,
developed by Lobachewsky and Bolyai, is now called
hyperbolic Geometry. That Geometry in which no
postulate relating to parallels is employed, and which
therefore includes what is common to both Euclidean
and non-Euclidean Geometry, is frequently called absolute, or general, Geometry. A later discovery was made
by Riemann, of which I shall presently have more to
say, that a Geometry is possible, in accordance with
which all coplanar straight lines intersect one another,
so that no parallel straight lines exist. In this Geometry,
of which two distinct forms are now recognized, called
respectively spherical and elliptic, a straight line is
always a closed figure of finite length. In place of the




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well-known Pythagorean theorem relating to the sides
of a right-angled triangle, which is fundamental in
Euclidean Geometry, more general metrical relations
were developed by Lobachewsky and Bolyai for the
hyperbolic case. In spherical and elliptic Geometry the
corresponding metrical relations are identical with those
expressed by the ordinary formulae of spherical Trigonometry. The two important questions which arise as
regards the systems of non-Euclidean Geometry, sometimes called meta-Geometry, are first, that of the logical
validity of the schemes, and secondly, that of their
applicability to the description of actual spatial relations
in physical space.
Before, however, I discuss these questions, it is convenient to refer to two developments of Geometry,
different in kind, but both of the most far-reaching
importance, and both of such a character that, as a
result of them, the whole Science of Geometry can be
viewed from standpoints much more general than that
of the older traditional scheme of synthetic Geometry.
The first of these new departures in the Science was
the introduction of Analytical Geometry by Descartes,
in his method of employing coordinates to represent
the positions of points of space. An essential element in
Descartes' coordinate Geometry is that sets of three
numbers (or in plane Geometry, two numbers) are correlated uniquely with points of space. This method of
correlation, taken in conjunction with the conception
of current coordinates, in which variables are employed,
each of which is capable of taking up the values of the
elements of a number-continuum, has the effect of
reducing the statement of all geometrical relations, and
properties of figures, to purely algebraical statements.
Thus all Geometry-in Descartes' original scheme this
is restricted to Euclidean Geometry-is reduced to
arithmetical Algebra. Every theorem of synthetic Geometry is correlated with a corresponding theorem of a




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purely arithmetic nature, that is, one in which the
objects to which the theorem relates are sets of elements,
each of which consists of a triplet of numbers. We
possess a modern Mathematical theory known as the
Theory of Aggregates, in which, when an aggregate of
elements is taken as fundamental, the properties of
selected sub-aggregates, or portions of the fundamental
aggregate, are classified and developed. If the fundamental aggregate be taken to be the set of all triplets of
numbers, each of which numbers may be any number
whatever of the arithmetic continuum, this fundamental
aggregate may be taken as corresponding to all the
points of geometrical space. All geometrical constructs,
such as straight lines, planes, circles, spheres, etc.,
correspond to particular sub-aggregates, or parts, of the
fundamental aggregate; and their geometrical relations
correspond to properties of such sub-aggregates. It thus
appears that Geometry is capable of a further stage of
abstraction, in which synthetic Geometry is replaced by
a scheme of relations between sets of objects, each of
which is merely a triplet of numbers. Geometry thus
becomes, in fact, a purely arithmetic scheme. The purely
technical advantage of the consequent reduction of the
ascertainment of particular geometrical properties to
calculation by means of equations is, for most purposes,
enormous. But besides this simplification, other consequences follow from the substitution of relations of
variables and numbers for the objects and relations of
synthetic Geometry. In plane Geometry, a curve is
represented by an equation involving as variables the
coordinates of a point on the curve; this equation is a
relation which is satisfied whenever the variables take
the values of the coordinates of any particular point
whatever on the curve. When however we consider the
arithmetic number-system to be extended so as to
become the system of complex numbers, we find, in
general, that there exist pairs of complex numbers which




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satisfy the equation of a curve, and thus the equation of
a curve determines relations between complex numbers
besides those of real numbers which alone have reference to the curve as originally conceived. It is convenient so to extend the language of synthetic Geometry
as to take account of the extended interpretation of
which the equations employed in the analytical scheme
are capable. We speak therefore of a pair of complex
numbers as the coordinates of an imaginary point in the
plane. Thus, besides the original curve that is correlated
with its equation, there exists, in general, a set of
imaginary points in the plane, the coordinates of which
satisfy the equation, as do those of the real points.
Another extension of the use of geometrical language to
denote arithmetical or algebraical facts concerning the
equations of analytical Geometry is that involving the
employment of infinite numbers, pairs of which, in a
certain sense, may satisfy the equation of a curve. We
then say that the curve contains points at infinity, which
may be real or imaginary. The great advantage of this
last extension of the use of geometrical language to
denote arithmetical facts is apparent in the greater
generality of form which it enables us to give to the
statement of geometrical properties. To take a very
simple case, instead of the statement that two circles in
a plane may either not intersect one another, or may
touch at one point, or may intersect one another in two
points, we may assert that any two circles, as also is the
case for any two conics, intersect one another in four
points. The correctness of this statement implies that
two points may be coincident, and that any point may
be real or imaginary, and either in the finite part of the
plane or at infinity. The comprehensive statement is in
reality an expression of a property of the equations of
the circles or conics. Without this kind of generalization
of the use of geometrical language it is impossible to
make a general statement about the spatial relations of




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figures of certain classes without reference to a number
of special cases which may arise, some of them exceptional in character. Thus the statement that two straight
lines in a plane intersect one another holds good, for
Euclidean Geometry, without mentioning the exceptional case of parallels, because, on the algebraical
ground I have referred to, parallels intersect one another
at a point at infinity.
The full import of these extensions of the elements
with which Geometry deals appears however only in
connection with the second great development to which
I have referred, that of Projective Geometry. Any precise description of the scope of this kind of Geometry
would necessarily be of so technical a character that I
cannot attempt to give one here. It may however be
observed that a property which is distinctive of Projective Geometry is that two coplanar straight lines
always intersect one another. Euclidean Geometry is
accordingly not projective, but when the new entities
which I have spoken of as points at infinity are introduced into it, it can then be expressed in the projective
form. In the general sense of the term Geometry that
is now employed, the fundamental elements in each
Geometry consist of points, considered as a class of
primitive elements; particular sub-classes of this fundamental class defining straight lines, planes, etc. One
Geometry differs from another one in accordance with
the nature of the relations that are postulated to exist
between the fundamental elements, such as those relating
to the order of points on a straight line. A method of
introducing coordinates, independent of the ordinary
Euclidean notions of congruence, was introduced by
von Staudt; and this forms the basis of the analytical
treatment of Projective Geometry, which is essentially
independent of metrical considerations. But metrical
relations of distance and of angles have been introduced into Projective Geometry, in the form of purely




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descriptive relations, by Poncelet, and later, in a very
general form, by Cayley, in connection with the theory
of invariants. The method of Cayley has an important
bearing on the question of the validity of the nonEuclidean Geometries which we call Lobachewskyan
and Riemannian. It has in fact been shown that all the
relations in either of these systems are capable of being
represented as relations within the Euclidean scheme.
It follows that, if there arises any logical contradiction
from the postulations made in hyperbolic, or in elliptic,
Geometry, there must be exhibited a corresponding
contradiction in Euclidean Geometry. Thus, if Euclidean
Geometry be assumed to be a scheme free from contradiction, it is demonstrable that this is also true of nonEuclidean Geometry, whether hyperbolic or elliptic.
All three Geometries stand therefore on the same
footing, as conceptual schemes free from internal
contradiction.
Before I consider the question whether the nonEuclidean Geometries have the same applicability as
the Euclidean, for the purpose of describing relations
in physical space, it is necessary to refer to a mode of
considering the matter which was devised by Riemann
and Helmholtz, and which has produced an epochmaking effect upon the whole theory of the foundations of
Geometry. This new development was explained in two
memoirs, the one by Riemann, the Mathematician, bears
the title On the hypotheses which lie at the base of Geometry, the other by Helmholtz, the Physicist, is entitled
On the facts which lie at the base of Geometry. Thus,
what Riemann regards as hypotheses in a scheme of conceptual space relations correspond to what Helmholtz
regards as facts of observation in physical space. The
fact of experience which is regarded by Helmholtz as of
significance in relation to the theory of Geometry is
that freely movable rigid bodies exist in physical space;
their dimensions remaining unaltered during the motion.




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This may be stated in the more precise form that, if a
pair of particles A, B of any one such body can be
brought into coincidence with a pair A', B' of another
such body, then the coincidence of congruency remains
unaltered when the pair of bodies are moved in any
manner. Of course it is assumed that certain conditions
as regards temperature and absence of strains are
satisfied, so that approximately rigid bodies, which are
the only ones that exist, may be regarded as perfectly
rigid. The statement is equivalent to the one that a rigid
body is freely movable, and so that the measurable
distance of any pair of points of the body remains
unaltered, the measure of distance being estimated by
means of some standard body. In order that a system
of abstract Geometry may be applicable to describe
actual spatial relations, in which these facts of experience are taken into account, the measure of distance
between any two points of the geometrical space should
be such as to be an invariant for a certain set of transformations which shall represent mobility. In any
transformation of this set, points P, Q are made to
correspond to other points P', Q' respectively, and the
metric system of the Geometry should be such that the
measures of the distances PQ, P'Q' should be identical,
for every pair of points and for every transformation of
the set. Any Geometry founded upon a metric system
in which this condition is satisfied can be applied to the
representation of spatial relations in physical space, in
such wise that the numerical measure of the distance
between any pair of particles of a rigid body remains
unaltered as the body is freely moved, without strains
or changes of temperature.
Riemann's theory is based upon an extension of
Gauss' general theory of curved surfaces in Euclidean
space, that is when the ordinary Euclidean metric
system is employed. In Gauss' theory, the position of
a point on a given surface is specified by two numbers,




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the values of two variables, which may be regarded as
the coordinates of the point in a widely generalized
sense of the word. There is a large degree of arbitrariness in the choice of these variables; when to either of
these variables a constant value is assigned, the other
one remaining variable, the point represented by the
coordinates lies on a curve upon the surface. Thus
a particular coordinate-system on a surface is defined
by a mesh-system formed by two families of curves
lying upon the surface; this mesh-system being arbitrary,
but subject to certain conditions of continuity. The
element of distance of a point from a neighbouring
point of the surface can then be shown to be the square
root of a quadratic function of the differentials of the
coordinates of the point, the coefficients depending in
general upon the particular point. Gauss established
the existence, at each point, of a certain function of
these coefficients, which is invariant for all systems
of meshes, and depends only on the nature of the surface
in the neighbourhood of the point; it is called the absolute curvature of the surface at the point. He further
showed that, in order that a certain set of transformations might exist, which represent the motion of a
portion of the surface into a new position, in which
every element of length in a first position corresponds
to an equal element in a second position, the necessary
and sufficient condition is that the absolute curvature
of the surface should at all points have the same value.
Interpreted in physical language this means that a
portion of a material surface can be freely shifted along
the surface without stretching any part of the material,
bending being however permissible. This condition is,
for example, realized by the surface of a sphere, on
which all figures are movable without alteration of
dimensions; it would however not be satisfied, for
example, by the surface of an ellipsoid, for which the
property of free-mobility is accordingly lacking. In




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Riemann's abstract theory, a manifold of elements is
taken as fundamental, each of which is specified by a
set of n real numbers. Each of these n numbers may be
any number in the arithmetic continuum, or in some
specified portion of it; thus the fundamental manifold is
ordered, and has an n-fold order. The particular case
which provides a Geometry applicable to physical space
is that in which n has the value 3. The manifold of
elements is purely abstract, and free from any conception directly dependent upon spatial intuition, as is
emphasized by the fact that n may have any integral
value. The question was considered by Riemann, as
also, in the case n - 3, by Helmholtz, what system of
metric relations must be introduced into the manifold,
in order that the system which consists of the manifold
subject to this metric system may be regarded as the
space of a Geometry, capable, when n = 3, of affording
a conceptual representation of the physical space in
which measurements of physical bodies are made. Denoting an element of the manifold by the term point,
and the numbers which specify that point by the term
coordinates, both Riemann and Helmholtz consider the
question what is the most general form of an element of
distance between two points of the manifold, expressed
as an integral, of which the integrand, or element of
distance, involves the differentials of coordinates, in
order that continuous transformations may exist, in
which the distances between corresponding pairs of
points remain unaltered in any of these transformations.
The answer to this question is that, in the first place,
the square of the element of length must be defined as
a quadratic function of the differences of coordinates
of its extremities, the coefficients in this quadratic function being continuous functions of the coordinates of
the point from which the element of length is measured.
Further, there exist a certain number of functions (in
the case n = 3, this number is 6) of the coefficients in




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the quadratic function which must have one and the
same value, invariant for continuous transformations of
the coordinates. It can be proved that (in case n > 2)
if this holds good at one point it holds at all points.
Thus there must exist a certain constant which has one
and the same value at all points of the manifold. This
constant may be either positive, negative, or zero. In
the last case the metric system introduced into the
manifold is Euclidean, and the manifold is then, when
n - 3, said to be the space of Euclidean Geometry.
When the constant is negative, the space, for n = 3, is
that of the hyperbolic geometry of Lobachewsky and
Bolyai. When it is positive, the space is of the new kind
discovered by Riemann, and may be either spherical or
elliptic. It is unbounded, but in a certain sense finite,
and Riemann has drawn special attention to the fact
that a space being unbounded is quite consistent with
its being, in the sense referred to, finite. This is, for
example, the case with that two-dimensional space
which consists of the surface of a sphere.
On account of the analogy with Gauss' case of the
two-dimensional space which forms a surface in ordinary
Euclidean Geometry, Riemann termed the constant
which I have referred to, the curvature of the particular
space. This term has proved a somewhat unfortunate
one, as it has led to much misunderstanding. It has
given rise to the idea that non-Euclidean three-dimensional space is itself curved, and this notion has been
strengthened by the usual illustration of the case of a
non-Euclidean two-dimensional space regarded as embedded in a three-dimensional space, for example in
that of the spherical surface to which I have already
referred. The fact is however that the so-called curvature
is not curvature of the space, but represents only a
property of the metric system introduced, with a considerable degree of arbitrariness, into the manifold. It
is, for this reason, better to speak of this constant as the




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space-constant of the particular geometrical space. The
manifold, as such, has no curvature, and no metric
properties; these latter are introduced into it, when it
becomes a space, for the purpose of representing conceptually our system of actual measurements in physical
space which depend upon physical properties of objects
in that space. Instead of speaking of Euclidean or nonEuclidean three-dimensional space, it is more accurate
to speak of geometrical space with an imposed Euclidean, or with an imposed non-Euclidean, metric
system with an assigned space-constant, positive or
negative. Moreover it is unnecessary to regard such
space as a section of a space of higher dimensions, since
the metrical scheme introduced into it does not require
the consideration of a manifold with more dimensions
than that of the space considered. The element of length
being assigned as the square root of a quadratic function
of the differentials of coordinates, the distance between
any two points of the space is determined as the
stationary value of the integral of the element of length
taken from one of the points to the other. In the case,
taken for purposes of illustration only, of the twodimensional Geometry of a surface in ordinary Euclidean
space, the distance between two points is the length of
a geodesic which passes through the two points. The
term geodesic has been frequently used in connection
with the distance between two points in space with a
non-Euclidean metric; but again this use of the term,
although convenient, is apt to be misleading, as it suggests that the space must necessarily be regarded as a
"surface" in space of four dimensions.
From the point of view that the difference between
so-called Euclidean, and so-called non-Euclidean, geometrical space does not refer to any distinction of
property of the point-manifold itself-that manifold
being regarded as a mere field of possible metrical relations to be imposed upon it-the question whether our


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physical space is Euclidean or non-Euclidean, in itself,
would appear to have no immediate meaning. The real
question can only be taken to be whether, or under
what restrictions, if any, the actual observed relations
in physical space can be described by means of an
abstract geometrical scheme with a non-Euclidean
metric.
Euclidean Geometry, and also non-Euclidean Geometry with either a positive or a negative space-constant,
are all self-consistent conceptual schemes. The crucial
question, of very considerable theoretical interest, which
arises in connection with them is that of their applicability to the description of actual spatial relations in
physical space. Are they all so applicable, or is there
anything in physical phenomena which compels us to
assign to Euclidean Geometry a position, in relation to
such applicability, which does not attach to nonEuclidean Geometry? No doubt Euclidean Geometrv
is the simplest for the purpose, but is non-Euclidean
Geometry a possible system for application to physical
space, if we are prepared to sacrifice simplicity? It
should be observed that the question as here discussed
is considered from the pre-Einstein point of view, in
which no attempt is made to include gravitational or
electromagnetic phenomena in the geometrical scheme.
In the first place it may be observed that, in sufficiently small portions of physical space, the results of
adopting one or other of these schemes will be, in view
of the approximative character of all our measurements,
indistinguishable from one another. It would thus
appear that all our ordinary spatial measurements are
consistent with a non-Euclidean geometrical scheme,
provided the numerical measure of the space-constant,
when correlated with our ordinary scales, is sufficiently
small; no deviations being then observable from the
results of applying a Euclidean metric. For example,
in either hyperbolic or elliptic Geometry, the sum of




TIME AND SPACE


I47


the angles of a triangle of sufficiently small dimensions
is indistinguishable from two right angles.
It has however been maintained by Poincare and
many other Geometers that the applicability of nonEuclidean Geometry is theoretically possible, independently of any such restriction on the value of the spaceconstant; that it is in fact fundamentally a matter of
convention and convenience, not of absolute necessity,
which system we may employ. It is held by these
Geometers, and supported by cogent reasoning, that no
crucial experiment, consisting of the measurement of
lengths and angles, can be made which, whatever its
results may be, is inconsistent with a non-Euclidean
scheme of representation, provided a requisite readjustment of the statement of physical laws be made,
especially of the laws of Optics. If points of physical
space be suitably correlated with points of geometrical
space whose coordinates are assigned in accordance with
either of the systems in question, the measurement of
lengths by means of measuring rods will be consistent
with the assumed principle of the existence of rigid bodies
freely movable with unaltered numerical dimensions. If
angles are determined indirectly by means of formulae
connecting them with measured lengths, these formulae
will differ with the system adopted. When angles are
measured directly by methods which involve the use of
rays of light, the determinations will depend upon the
assumptions made as to the paths of such rays. For
example, as we have already seen, in either hyperbolic
or elliptic Geometry, the sum of the angles of a triangle
of sufficiently small dimensions, measured in physical
space, will be indistinguishable from two right angles.
In order to measure the angles of larger triangles, we
have to make use of rays of light; and the comparative
simplicity of the Euclidean metric system, for purposes of
application, arises from the fact that, if we employ it, we
can assert that the path of a ray of light is a straight line.
10-2




148


TIME AND SPACE


It has frequently been suggested that astronomical
observation might be employed to decide the question
which kind of Geometry is the true one as representative
of our physical space; in particular by the measurement
of the angles of a triangle with very long sides. We
might suppose, for example, that by such measurement,
a triangle was discovered for which the sum of the
angles differed from two right angles by an amount
which could not be accounted for by instrumental
errors. We should then have a choice of two interpretations of the observed fact. We might either say that
physical space is only describable by a non-Euclidean
Geometry, or we might say that it is Euclidean, but that
the paths of rays of light are not strictly straight lines,
but curved paths of such a character that the triangle
with curvilinear sides was such as to explain the observed
amount of the deviation of the sum of the angles from
two right angles. Again, if we found that the sum of
the angles was two right angles, we might either affirm
that physical space is Euclidean, or else that it is nonEuclidean, but that the path of a ray of light is not a
straight line. As an illustration we may take the fact
that, on a spherical surface, the sum of the angles of a
triangle of which the sides are geodesics exceeds two
right angles; but that there exist triangles of which the
sides are not geodesics, for which the sum of the angles
is equal to two right angles.
Whilst admitting the strength of the case in favour
of the view that physical laws are capable of being so
stated that our actual spatial measurements are capable
of being described by means of a Geometry with a nonEuclidean metric, it would certainly be more satisfying,
as a confirmation of this view, if we were in possession
of a detailed statement of the precise modification of
physical laws, of our habits in relation to spatial intentions, and of our practical modes of measurement,
which would be rendered necessary by the adoption of




TIME AND SPACE


I49


an abstract non-Euclidean Geometry as the mode of
description of actual spatial relations.
That the Euclidean metric is the simplest for all
ordinary applications to physical space, because it
admits of greater simplicity in the statement of physical
and dynamical laws, is abundantly clear. For all ordinary
purposes it will not be superseded, but the possibility
of the employment of other schemes, as theoretically
applicable, is also clear. The interest in the development
of non-Euclidean Geometries, apart from their technical
interest for Mathematicians, lies in the distinction it has
laid bare between those elements in our Geometry
which are introduced as definitions and conventions,
admitting variety of detail, and those which are fixed
by facts of observation.
It appears to have been held by Kant that our
Euclidean system of Geometry is present in the mind
a priori as a necessary presupposition of physical experience. The development of systems of Geometry, of
logical validity equal to that of the Euclidean, and
capable of being applied, although with great loss of
simplicity, to describe our actual experiences of spatial
relations, would appear to provide a definite refutation
of the Kantian view of space as an a priori form, at least
as regards so precise a form as that of the Euclidean
scheme. It has frequently been suggested that the study
of non-Euclidean Geometry, and still more general
schemes of Geometry which have occupied the attention
of Mathematicians during more than half a century, is of
purely technical interest, and can lead to nothing which
has any relation to physical phenomena. The rise of the
Einstein theory of relativity, which essentially depends
upon the ideas developed by Riemann, Helmholtz,
Minkowski, and others, affords a sufficient proof of the
hazardous character of all prophecies as to the nonapplicability of abstract theories and their generalizations
for the purpose of representing physical phenomena.




VII


CORPUSCULAR THEORIES
OF MATTER
OTH in ancient and in modern times two divergent
conceptions of the constitution of matter have been
employed in scientific theories. In the first of these,
matter is regarded as consisting of groups of discrete
entities separated from one another by empty spaces;
these are the atomic, or corpuscular, theories of matter.
In the rival view, matter is regarded as continuous, and
indefinitely divisible. So long as either of these types
of theory of the constitution of matter is regarded
realistically, as a constituent part of an ultimate philosophical view of the world, they are of course mutually
exclusive. However, in accordance with the view of the
true character of all scientific theories which I am endeavouring to explain and illustrate in these lectures,
any theory of the constitution of matter will be taken
solely as a conceptual representation of some assigned
domain of natural phenomena; thus an atomic theory
may legitimately be employed for some purposes, and a
continuous theory of matter for other purposes. Frequently a theory of mixed type, in which both atoms
and continuous substances occur as conceptual elements,
has been made the basis of an attempt to represent
classes of phenomena. The employment of different
types of theory for different purposes has been a common
procedure in modern times. For example, an atomic
theory has been long prevalent for the purpose of representing facts relating to chemical combinations; and the
kinetic theory of gases affords another example of such
a corpuscular theory. On the other hand a continuous




CORPUSCULAR THEORIES OF MATTER 15I
theory of matter is employed in the theory of Elasticity,
and in the theory of the motions of fluids; the treatment
of these subjects being thus made capable of the application to them of continuous Mathematical Analysis.
Various attempts were made, especially in the first half
of the nineteenth century, to represent these latter
phenomena by means of corpuscular theories; but all
such efforts were based upon complicated hypotheses
as to the nature of interactions between corpuscles, and
they can only be regarded as having attained a moderate
degree of success. In any such case the simpler continuous theory is to be preferred, so far as it can be
shown to have as large a range of applicability for the
purposes of representing the phenomena, as has a corpuscular theory. It is of course always possible to regard
a continuous theory as containing an idealization, by a
process of averaging, of the particles of substances that
are taken to be only sensibly continuous; and it has in
fact been maintained that a continuous theory must
necessarily be regarded in this way if it is to be considered as valid. The advantage of such idealization then
consists in the fact that the theory becomes capable of
expression in the form in which differential equations
are used. In theories of the mixed type, imponderable
substances have played a large part; we have for instance
an example of such a substance in the modern ether of
the electromagnetic theory.
The parentage of all atomic theories is to be found in
Leucippus and Democritus, or even earlier; Democritus
erected a Cosmology on the basis of the idea that the
only existing objects are atoms in empty space; that
these atoms are indestructible and eternal; and that all
change consists in the aggregation and separation of
these atoms. It may be observed that Aristotle, as an
opponent of the atomistic theory, refused to admit the
validity either of the conception of empty space, or of
that of an indivisible atom. The Cosmology of Demo



152 CORPUSCULAR THEORIES OF MATTER
critus leaves no room for contingency, or for teleology;
since all change, being due to the motion of the atoms,
is subject to an unconditional necessity. The atoms have
an endless variety of form, and are infinite in number;
in their eternal fall through infinite space, the greater
atoms strike against the smaller ones, and, since the
former have a greater velocity, the impacts give rise to
lateral movements and vortices. These form the commencement of the growth of worlds, an innumerable
series of which come into existence and perish. The
atoms act on each other only by collision or pressure,
and the variety in gross bodies is due to the variety in
the number, size, shape, and arrangement in space, of
the atoms of which they are composed. Democritus regarded such sensations as those of sweetness, bitterness,
warmth, and colour, only as deceptive opinions; nothing
but atoms and empty space were regarded by him as
real. The phenomena of life are produced by fine,
smooth, round atoms, like those of fire, which permeate
the whole body, and constitute the soul, which is thus
recognized as distinct from the body. The Physics of
Epicurus was founded upon the conceptions of Democritus, and was employed by him to remove that dualism
of mind and matter which is involved in the Philosophy
of Plato and of Aristotle. He regarded space as infinite,
and containing an infinite number of indestructible and
indivisible atoms in perpetual motion. These atoms
differ from one another only in size, shape, and weight,
and they move with equal velocities. As they move they
give rise to new worlds which perpetually tend to dissolution, and then to the production of fresh series of
worlds. The soul is a fine substance, like warm air,
distributed through the whole body. From the surfaces
of external objects there is a constant stream of fine
particles; and thus actual material copies of external
objects enter the living body, which conducts these
images to the soul, thus giving rise to sense-impressions.




CORPUSCULAR THEORIES OF MATTER 153
These materialistic views of the world, which were
taken up at a later time by Lucretius, whatever may be
thought of their general tendencies, had the effect of
eradicating from the minds of their adherents those
animistic and magical conceptions which, together with
the habit of constantly appealing to final causes, had
done much to hinder the development of scientific
methods of investigation. In the middle ages, the current
view of matter, in accordance with the Aristotelian
tradition, was that it consisted of underlying substantial
forms which possessed accidental properties; our perceptions relating only to these latter. For a long period
the attention of investigators was devoted to the occult
properties of substances, but no interestwas taken in their
quantitative aspect. Atomism, as taught by the ancients,
had been completely submerged by Aristotelianism, and
it was not until the fifteenth century that atomistic conceptions again arose in connection with the metaphysical
speculations of Nicolas de Cusa (140I-I464) and other
writers who took part in the growing criticism of the
dominant Aristotelianism. The most important influence upon later developments of Atomism was exercised
by the speculations of Giordano Bruno, although their
metaphysical character was such that their immediate
effect on Physics was negligible.
The Cosmology of Giordano Bruno (1548-1600),
which amounted to a metaphysical Monadology, was a
scheme in which Being and Thought were coincident,
so that the structure of time, space, and matter can be
discerned by the operations of Thought alone. In accordance with his Philosophy, it is necessary to conceive
time, space, and matter as composed of indivisible
minima, that is, instants, points, and atoms. That there
must exist, in each domain, a fundamental indivisible
whole, with which our conception begins, he regarded
as a necessary postulate of Thought. This notion of the
atom, Bruno regarded as having a certain relativity; the




154 CORPUSCULAR THEORIES OF MATTER
magnitude of this indivisible, or minimum, is fixed
according to circumstances. In Astronomy, for example,
the heavenly bodies are the irreducible minima, and are
thus to be regarded as atoms.
It has been the mechanistic view of the world of
physical phenomena conceived by Descartes that has
exercised the greatest influence on later conceptions of
matter. It is difficult to reconcile Descartes' view of mind
and matter as the two fundamental substances with his
idealism; his theory of the physical world is, however,
practically of a purely materialistic type. In extension
and motion he recognized the source of all physical
percepts, and thus all the occult properties with which
medieval thought endowed substance were removed, the
external world forming a purely mechanical system. He
was not an atomist in the same sense as Democritus or
Lucretius, for he regarded all space as one fundamental
substance, infinitely divisible; thus, for him, spatial extension and substance were identical. The only differentiation of substance is due to the motion of its parts;
thus a body, or portion of matter, is what can be moved
as a whole relatively to the surrounding substance.
Since space is a plenum, and since, on account of the
identity of extension and substance, there can take place
no diminution of the volume of substance in a portion
of space, it follows that a circular streaming must form
the basis of all motion. Descartes assumed that originally the substance of which the physical world consists
was broken up into particles which were in rapid motions
of rotation, and also in circular translational motion.
By means of this motion, particles which were originally
of irregular shape have become rounded, splinters
having been broken off from them in the course of the
gradual grinding. This process he regards as having
given rise to three kinds of corpuscles. The first of these
form elementary fire, and consist of splinters of various
and varying size and form, in motion with enormous




CORPUSCULAR THEORIES OF MATTER               155
velocities. The sun and the stars are composed of this
kind of elementary matter. The matter of the second
kind, elementary air, consists of imperceptibly small
spherical atoms, which move with great velocities in
vortices; they fill all interplanetary space. The matter
of the third kind, the elementary earth, consists of
larger corpuscles of various forms, and in less rapid
motion; the ordinary material bodies, and the earth and
planets, consist of atoms of this species. The conception
of ordinary matter which is in accordance with this
scheme is that it consists of atoms which are in fact, but
not in thought, indivisible. The interstices between
these atoms are filled with atoms of the other two kinds,
so that although the amount of ordinary matter in a
fixed space may vary, the total amount of matter of all
three kinds in that space is invariable. Plants and
animals, like inorganic bodies, are machines; their vital
spirits consisting of fine material in motion, as with
Democritus. Descartes described the separate stages of
a mechanism involving pressure and collision, as forming
an uninterrupted chain of effects produced by external
objects through the senses upon the brain, and back
from the brain through nerves and muscular filaments.
This view of a materially conceived world, subject to a
rigorously determined sequence of causes and effects,
would appear to be irreconcilable with Descartes'
idealistic Metaphysics.
The difficulties relating to the possibility of the
motion of atoms and bodies in a plenum filled with
impenetrable substance were so great that a return was
inevitable to the simpler conception of the ancient
atomists, that atoms are surrounded by empty space.
The writings of Pierre Gassendi (I592-I655) had an
important effect in hastening the disintegration of the
Aristotelian conceptions of matter, the authority of
which had been already to a considerable extent undermined. Gassendi regarded empty space and atoms as




156 CORPUSCULAR THEORIES OF MATTER
the only principles in nature. All atoms he regarded as
consisting of one and the same substance; they are only
distinguished by differences of magnitude, shape, and
weight. A limited number of different kinds of atoms
suffices to explain the variation of bodies by their
different groupings. In this view he was in agreement
with that of Epicurus, but not with that of Democritus,
who had regarded the number of different forms of
atoms as infinite. The weight of the atoms he regarded
as due to an inherent capability of self-determined
motion; it is in the motion of the atoms that the explanation of all physical properties of bodies is to be
sought. The main importance of Gassendi's work
depends upon the fact that he was the first definitely to
return to the ancient atomism, and thus to complete the
breach with the medieval views of matter.
The Philosopher Thomas Hobbes (1588-I679) occupies an important position in the history of Materialism.
Under the influence of Galileo, in regard to the theory
of motion, he regarded the motion of bodies in space as
the original phenomenon upon which all others depend,
if they are to be subjected to scientific treatment. He
was not an orthodox atomist, because he recognized the
existence, not only of corpuscles, but also of a continuous fluid which fills all the interspaces between
corpuscles, and in which motion is propagated. This
conception of vibratory motion, or Conatus, is an
attempt to objectify pressure or stress, and may thus be
regarded as a step in the direction of the introduction
of forces acting at a distance.
The breach with the Aristotelian conception of substantial forms, involved in the schemes of Descartes and
Gassendi, was consolidated by the great practical experimenter Robert Boyle (I626-I69I). Although his chief
interest was in the ascertainment of facts by experiment,
Boyle did not fail to recognize the necessity for a theory
which would bind together the results of his experi



CORPUSCULAR THEORIES OF MATTER I57
mental investigations in Chemistry and in relation to
the weight, pressure, and elasticity of air. In his general
Philosophy, Boyle was not a Materialist, but held a
theory, formed out of Descartes' Physics and Gassendi's
Metaphysics, which was designed not only to do justice
to his scientific views as a Chemist and Physicist, but
also to be compatible with his orthodox religious
opinions. He distinguished between two orders of
corpuscles, those of the second order, which form the
constituents of matter, being formed by the aggregation
of corpuscles of the first order. Between the corpuscles
there are pores containing various effluvia. The primary
constituents of matter, which we should now call
molecules, are not absolutely indivisible, but the primary
corpuscles of which they are composed are so firmly
fitted together that they are only with difficulty separated
from one another. Boyle set up various hypotheses
relating to the corpuscles, for the purposes of explaining
the constitution of air and other substances, and also of
chemical combinations. Of these hypotheses, varying
considerably according to the purposes for which they
were designed, he appears to have recognized the
tentative character. He attempted to explain all chemical
changes mechanically, and laid considerable stress upon
the quantitative determination of weights. He prepared
the way for the modern chemical theory of elements,
recognizing specific weight and chemical reaction as the
distinguishing marks of a particular substance. On the
theoretical side, he recognized the identity in character
of the molecules which form the mass of the substance.
The physical phenomena of heat, electricity, magnetism,
and the transformations into one another of solid, liquid,
and gaseous conditions of matter, he regarded as all
capable of mechanical explanations.
The great discovery by Newton of the law of universal gravitation led ultimately to the great change in
the nature of corpuscular and atomic theories which




158 CORPUSCULAR THEORIES OF MATTER
was produced when what is called action at a distance
became an essential element in dynamical theories of
matter. Prima facie Newton's discovery involves the
discarding of the ancient notion that all action must be
due to contact; a prejudice of which the origin is closely
connected with the conception of efficient causation.
The idea that the gravitation between two bodies is the
resultant effect of the attractions upon one another of
their individual corpuscular parts is inconsistent with
the older atomic theories in which all interaction was
regarded as due to impacts. For Newton and his contemporaries, under the influence of the notion that
impact, or contact of some kind, is the only species of
admissible explanation of physical action, the law of
gravitation as it stood was incomplete without an indication of some mechanism by which the gravitational
attraction can be deemed to be produced. A cause of
gravitation must be sought for, and discovered, before
the law could be regarded as, in the true sense, embodying a physical theory of the phenomenon. Newton
himself declared "The reason of these properties of
gravity I have not, as yet, been able to deduce"; and
again, in a letter to Bentley, he writes':
It is inconceivable that inanimate brute matter should, without
the mediation of something else which is not material, operate
upon and affect other matter, without mutual contact, as it
must do if gravitation, in the sense of Epicurus, be essential and
inherent in it. And this is the reason why I desired you would
not ascribe innate gravity to me. That gravity should be innate,
inherent and essential to matter, so that one body may act upon
another at a distance, through a vacuum, without the mediation
of anything else by and through which their action may be conveyed from one to another, is to me so great an absurdity that
I believe no man, who has in philosophical matters a competent
faculty of thinking, can ever fall into it. Gravity must be
caused by an agent acting constantly according to certain laws;
but, whether this agent be material or immaterial, I have left
to the consideration of my readers.
1 Opera, Horsley's edition, Vol. iv, p. 438.




CORPUSCULAR THEORIES OF MATTER 159
Newton's contemporaries for the most part shared
his view, fearing the reintroduction of occult causes
into Physics. Thus Huygens declared that "Newton's
principle of attraction appeared to him absurd." John
Bernoulli, who attempted to explain the motions of the
planets by means of a modified form of the Cartesian
theory of vortices, proclaimed "the two suppositions of
an attractive faculty and a perfect void " to be " revolting
to minds accustomed to receiving no principle in Physics
save those which are incontestable and evident." Euler
insisted on the necessity of supposing that gravitation is
due to some subtle material medium; and D'Alembert
regarded the real cause of gravitation as unknown, in
contradistinction to action by impact, of which we have
a clear mechanical conception. Until far into the nineteenth century this rejection of the notion of action at a
distance held sway. Thus for example1 E. Du BoisReymond writes:
Forces acting through void space are in themselves inconceivable, nay absurd, and have become familiar concepts amongst
physicists since Newton's time from a misapprehension of his
doctrine and against his express warning.
Again, Balfour Stewart and Tait2 write:
Of course, the assumption of action at a distance may be
made to account for anything; but it is impossible (as Newton
long ago pointed out in his celebrated letter to Bentley) for
anyone who has in philosophical matters a competent faculty of
thinking for a moment to admit the possibility of such action.
Since Newton's time, very numerous attempts have
been made to account for the phenomena of gravitation
by propagation through a fluid or an elastic medium, or
by means of impacts. All these theories must be pronounced to have failed in the purpose for which they
were designed; in some cases, such as in the impact
theory of Le Sage, on account of the nature of the
1 Ueber die Grenzen des Naturerkennens, p. 1.
2 The unseen Universe, 3rd ed., p. xoo.




I60 CORPUSCULAR THEORIES OF MATTER
assumptions made in them, and in other cases on account
of their inability to represent the known facts relating to
gravitation. In the first place, gravitation is propagated
instantaneously, or at least with a velocity which has
been estimated by Laplace to be at least fifty million
times that of light. Moreover all bodies appear to be
absolutely transparent to gravitational action; and it is
not subject to any kind of reflection or refraction. It
appears also to be independent of the structure, or
physical and chemical conditions, of the bodies between
which it acts; its energy is unchangeable and inexhaustible.
Of all the attempts to account for gravitation on a
corpuscular theory, that of Le Sage is the most ingenious,
and is perhaps the only one which has been so far
developed that its inherent weakness can be fully exposed. This theory is that the gravitation of bodies
towards each other is due to the impact upon them of
corpuscles or atoms moving in all directions through
space. Each of these so-called ultramundane corpuscles
is so small that collisions between pairs of them are of
rare occurrence. If a body is not in any way screened
from the bombardment of these corpuscles it would not
acquire any motion, since the effects of the bombardment on all parts of its surface would neutralize one
another. If there are two bodies in space, each acts, to
a certain extent, as a screen against the bombardment
of the other; thus, for each body, a smaller number of
corpuscles will strike it on the side which is towards the
other body than on the further side. Each body will
appear to be attracted towards the other body, owing to
the effect of the excess of the impacts it receives on the
side furthest from the other body. Leaving out of
account those corpuscles that have already struck some
mundane body, and taking account only of those that
come from infinite space, it can be shown that the force
of attraction between two bodies whose dimensions are




CORPUSCULAR THEORIES OF MATTER              I6I
small compared with the distance between them will
vary as the product of the sections of the bodies taken
normal to the distance, and inversely as the square of
that distance. In order that this may coincide with the
attraction, as given by the law of gravitation, it is necessary
that the effective areas of the bodies be proportional to
their masses. Le Sage shows that, in order that this
may be the case, whether the body be large or small, it
must be assumed that the size of the solid atoms of the
body is very small compared with the distances between
them, so that a very small proportion of the corpuscles
are stopped by even very dense and large bodies.
Maxwell has shown that the energy of the corpuscles
that is spent in maintaining the gravitation of a single
pound of matter towards the earth must be millions of
millions of foot-pounds per second. It can be shown
that, on the assumption of perfect elasticity of the corpuscles, so that they rebound from the body with the
same velocity with which they struck it, there will be no
excess of the impacts on any other body on one side
over the other side. On the other hand, if the velocity
after impact is less than that of approach, although the
attraction between the bodies will be accounted for, the
excess of the energy which the corpuscles brought with
them over that which they carry away remains to be
accounted for. If any appreciable part of this excess
appears in the form of heat in the body, it will, as is
stated by Maxwell, in a few seconds raise it, and in like
manner the whole universe, to a white heat.
I have emphasized the extreme reluctance which,
from Newton's time onwards, men of Science have
shown to regard the law of gravitation as anything but
a mere stepping-stone on the way to the construction of
a genuine scientific theory of gravitation, because, for
us, Newton's law has all the characteristics of a genuine
theory to represent a certain class of phenomena. In
the hands of Newton himself, the law of gravitation,
HGL                                      II




I62 CORPUSCULAR THEORIES OF MATTER
together with his scheme of Dynamics, proved a sufficient basis for the mathematical deduction of Kepler's
laws of planetary motion, and for a deductive treatment
of the principal inequalities in the motion of the moon.
By the labours of a large number of Mathematical
Astronomers, lasting into our own time, it has been
shown that Dynamical Astronomy, with the Newtonian
law as its base, is sufficient to afford an accurate representation of very nearly all the observed motions of the
bodies of the solar system, and that the condition of
predictability of such motions for a considerable time
is satisfied. The law itself could hardly be surpassed in
point of simplicity and of comprehensiveness of statement. It will not lose its importance as the foundation
of a valid conceptual scheme if, as seems at present not
unlikely, the new law of gravitation given by Einstein
supersedes Newton's law as a more accurate law of the
phenomenon. The detailed consequences of the two
laws are indistinguishable from one another in the
motions they assign to the bodies of the solar system,
except apparently in the single instance of the motion
of the Perihelion of Mercury's orbit. The idea that
matter cannot act where it is not has no relevant meaning
for us; since the notion of efficient action is no longer
part of the stock of notions which Natural Science
employs. The acceptance of Newton's law as a genuine
scientific law depends upon the ascertained fact that it
resumes in a formula, with sufficient degree of approximation for nearly all purposes, a whole class of observed
facts relating to the motions of actual bodies.
In the first half of the eighteenth century, owing to
the influence of Newton's theory, the conception that
atoms exert an attractive force on one another rapidly
modified the older atomic theories. The attractions
between atoms made it unnecessary to conceive of them
as having rough surfaces provided with hooks, since
impacts no longer played the all-important part which




CORPUSCULAR THEORIES OF MATTER 163
they did in the older theories. There also arose the
conception of imponderable atoms exercising repulsive
forces on one another, filling the interstices between
the gravitating atoms. These imponderable atoms were
regarded as very much smaller than the ponderable
atoms. Gradually these imponderable atoms came to be
replaced by a continuous atmosphere of light and heat
surrounding the ponderable atoms.
The views which were accepted in England, late in
the eighteenth century, when Dalton was developing
his conceptions of chemical actions, were stated by him
as follows:
These observations have tacitly led to the conclusion, which
seems universally adopted, that all bodies of sensible magnitude,
whether liquid or solid, are constituted of a vast number of
extremely small particles, or atoms of matter bound together
by a force of attraction, which is more or less powerful according
to circumstances, and which, as it endeavours to prevent their
separation, is very properly called in that view attraction of
cohesion; but as it collects them from a dispersed state (as
from steam into water), it is called attraction of aggregation,
or more simply, affinity. Whatever names it may go by, they
still signify one and the same power....Besides the force of
attraction, which, in one character or another, belongs universally to ponderable bodies, we find another force that is
likewise universal, or acts upon all matter which comes under
our cognisance, namely, a force of repulsion. This is now
generally, and I think properly, ascribed to the agency of heat.
An atmosphere of this subtile fluid constantly surrounds the
atoms of all bodies, and prevents them from being drawn into
actual contact.
Dalton was led to the fundamentally important conception of atomic weight by an interpretation, in terms
of the atomic theory, of the observed fact that chemical
combinations take place in definite and simple numerical
proportions. He regarded all the atoms of a single
substance as identical in size and weight; and thus the
weight of a body is the product of the weight of an
1 New System of Chemical Philosophy, Pt I, pp. 141-143.
III



164 CORPUSCULAR THEORIES OF MATTER
atom by the number of atoms in the body. If one atom
of the one substance always unites with one, two, etc.
atoms of the other, then the regularity in the combining
weights is made intelligible.
An important modification of Dalton's atomic theory
was made early in the nineteenth century by the molecular theory initiated by Avogadro. This theory did not
at first attain to general acceptance, on account of the
rise of the electrochemical theory, in accordance with
which the cause of affinities is to be found in the
electrical relations of the atoms. Gay-Lussac had discovered that the various gases, under equal pressures
and temperatures, combine in simple volumetric proportions. Avogadro explained this by the assumption
that the numbers of smallest particles in equal volumes
of different gases are the same, when they are under
equal pressures and temperatures. He supposed that, in
compound gases, and at least partially in simple gases,
there exist combinations of two or more atoms; thus
the smallest particle of a chemical body is not the atom
but the molecule, a group of atoms. Chemical change,
by combination or separation, he regarded as due to
a change of place of atoms which grouped themselves
into molecules of altered atomic composition.
These molecular and atomic theories of chemical
combination were inspired by a spirit of physical realism
in which sensuous images accessible to the imagination
played a preponderating part. Although atoms and
molecules are not perceptual objects, it is undoubtedly
the case that this attitude of mind has been advantageous
in facilitating the formation of theories which we may
now regard as having a purely conceptual character.
This use of the sensuous imagination attained a luxuriant
growth after the discoveries of dimorphism and of isomerism, when it was found that substances of like
chemical constitution appear in very different forms.
Elaborate schemes for the localization in the molecule,




CORPUSCULAR THEORIES OF MATTER 165
and the transposition and various groupings of atoms
arose in this connection. But there was soon a reaction
against the idea that anything more than a convenient
symbolism was implied in these formulations. Thus, for
example, Liebig1 declared in I838 that:
we know nothing as to the condition in which the elements of
two compound bodies are, so soon as they have united in a
chemical combination, and the way in which we conceive these
elements as grouped in the combination rests merely upon a
convention which has been consecrated by habit under the prevailing theory.
Again Kekule, in his manual of Organic Chemistry,
I861, speaking of the proportional numbers of combining
weights, as representing fact, says2:
If to the symbols in these formulas another meaning is assigned, if they are regarded as denoting the atoms and the atomic
weights of the elements, as is now most common, the question
arises: what are the sizes or (relative) weights of the atoms?
Since the atoms can be neither measured nor weighed, it is
obvious that we can only be led by reflection and speculation
to the hypothetical assumption of determinate atomic weights.
After the time of Dalton, who may be regarded both
as a Physicist and a Chemist, the researches of Chemists
and of Physicists led them along very divergent paths.
Physics was treated in a manner which involved Mathematical Analysis, whereas Chemistry remained for a
long time inaccessible to such methods. In our own
day the barrier between these two branches of Science
is being broken down.
Returning to the purely physical side of the atomic
theory, we observe that the plausibility which attached
to the notion that matter consists of solid, hard, impenetrable atoms which come into collision with one
another loses its cogency when we inquire closely what
happens as the result of such collisions. It seems impossible to attribute to such atoms the property which
1 See Kopp's Entwickelung der Chemie, p. 597.  2 Vol. I, p. 56.




i66 CORPUSCULAR THEORIES OF MATTER
in sensible bodies is called elasticity, because that
involves relative motions of the parts, which cannot take
place in a perfectly rigid atom. The atom must be therefore regarded as inelastic. But in that case every collision between a pair of atoms would entail a loss of
energy of motion, and the kinetic energy of an aggregation of such atoms would in consequence gradually
disappear. Efforts of various kinds have been made to
surmount this difficulty of the apparent impossibility of
attributing elasticity to the atom, in the sense in which
the term is employed in connection with gross bodies.
One method is to regard the smallest particle of matter
not as the atom, but as the molecule consisting of a
group of atoms; in fact to follow the procedure adopted
on other grounds by the Chemists. The suggestion is
then that, in a shock between two molecules, the atomic
constitution of the molecules may be such as to admit
of their behaving like elastic bodies, although the
constituent atoms are inelastic. The necessity of attributing perfect elasticity to the molecules has been
recognized by those physicists who have developed the
modern kinetic theory of gases. Both Clausius and
Maxwell have emphasized this view. Lord Kelvin1 has
asserted that:
we are forbidden by the modern physical theory of the conservation of energy to assume inelasticity, or anything short of
perfect elasticity, in the ultimate molecules, whether of ultramundane or of mundane matter.
That the introduction of the molecular scheme, for
the purpose of getting over this difficulty relating to
elasticity, is a simple shifting of the difficulty further
back to the atoms constituting the molecules, and not a
solution of it, was the opinion that led Secchi to consider
a different mode of dealing with the matter. He suggested that the apparent repulsion of the atoms and
their reciprocal collisions can be simply referred to an
1 Phil. Mag. (4), Vol. XLV, p. 329.




CORPUSCULAR THEORIES OF MATTER 167
appropriate motion; it being sufficient for this purpose
to suppose them to be in rotation. Relying on a theorem
of Poinsot relating to the reflection of a rotating body
from a resisting obstacle, Secchi attempted, without
success, to show that, taking into account both the
energy of rotational and of translatory motion, the total
energy of two atoms is unaltered by an impact. Secchi
also made unsuccessful attempts to explain the aggregation of atoms so as to form molecules, and the phenomenon of gravitation.
One of the most important attempts to evade the
difficulty of defining precisely the character of the interaction between atoms which impinge upon one another,
or come into contact, consisted of the radical step of
depriving the atoms of all extension, and supposing
them to be mere centres of attractive or repulsive force.
In the middle of the eighteenth century an atomic
theory of this kind was propounded by Boscowitch.
According to his theory matter consists of a swarm of
atoms, each of which occupies a geometrical point of
space, is capable of motion, and possesses a certain
mass, so that a certain force is required in order to give
such an atom a given acceleration. Two atoms at a
distance from one another exceeding a certain small
length attract one another with a force varying as the
inverse square of the distance. For smaller distances
the force is attractive for some distances and repulsive
for others. In order to obviate the possibility of two
atoms ever being in the same position, Boscowitch
supposes that, for all distances below a certain minimum,
the force is repulsive, and increases indefinitely as the
distance is diminished. The system of atoms which
constitutes a material body occupies a certain region of
space, by reason of the forces between the component
atoms of the system and any other atoms which may be
brought near them. No second body can come to
occupy the same region of space because, before it




I68 CORPUSCULAR THEORIES OF MATTER
could do so, the mutual actions of the atoms of the two
systems would produce a repulsion between the two
bodies, too great to be overcome by any force which we
can apply. In this scheme, all action between bodies is
action at a distance, and there is no such thing as actual
contact between two bodies, although they may be so
close to one another that the atoms of the two which
are nearest exercise a great force of repulsion. When
atoms were no longer regarded, in accordance with the
view of the earlier atomists, as acting immediately on
one another by contact, but by forces of attraction and
repulsion acting at a distance through empty space,
there seemed no longer any sufficient reason to attribute
the property of extension to the atom. Even if it was
retained, it was merely in deference to a desire to satisfy
the sensuous imagination by making the atom resemble
the bodies perceptible by our senses. Ampere, Cauchy,
and Faraday all regarded the atoms as unextended, or
as simple centres of force.
Another interesting and remarkable theory of the
nature of the atom, of quite a different character from
those to which I have referred, is Lord Kelvin's theory
of vortex atoms. He imagines all space to be filled by
an absolutely homogeneous, incompressible, frictionless
fluid. It had been shown by Helmholtz that, in such a
fluid, vortex tubes or filaments can exist in which the
fluid is in permanent rotational motion, and that such a
filament can form a closed ring, which may be called a
vortex ring. Such a ring at all times consists of the same
portions of the fluid, and is consequently of invariable
volume. It is indestructible, and could not be formed
in any portion of the fluid that is in irrotational motion.
Two such rings could never amalgamate, or come into
contact with one another. As such rings might be
knotted, and two or more of them might be linked
together by the passage of one ring through another
without contact of their cores, the possibility occurred




CORPUSCULAR THEORIES OF MATTER 169
to Lord Kelvin that a new atomic theory might be
founded on the existence of such rings; their convolutions and linkings admitting of an endless variety of
forms. If such a convoluted, or linked, ring be regarded
as an atom, such an atom would have permanence in
magnitude and strength, capability of internal vibrations,
and indestructibility. Thus, not the original fluid, although it possesses inertia, but only the vortex rings in
it are regarded as having the character of matter. Such
a vortex ring would seem to have more of the properties
requisite to the atom than any of the earlier kinds of
atoms. The difficulty of explaining the inertia of what
is only a mode of motion of a substance, and not a
substance itself, was pointed out by Maxwell. No substantial progress has been made in the direction of
showing that the phenomenon of gravitation or the
thermal and optical properties of matter are explained
by this theory of vortex atoms.
The only theory of the constitution of matter which
really comes to close quarters with the thermal and
mechanical properties of the substance, and that can be
regarded as an atomic theory, although the elementary
parts of matter with which it deals consist of molecules,
assumed to behave, on impact with one another, like
perfectly elastic spheres, is the dynamical theory dealing
with gases, known as the kinetic theory of gases. It is
impossible here to enter into the complicated details of
this theory, any adequate description of which would
involve mathematical formulae. It has had very considerable success, although it would seem not complete
success, in coordinating a large number of known facts
relating to the mechanical and thermal properties of
gases, and it may consequently be regarded as being
the most comprehensive theory, of the atomic type, that
has been proposed, for coordinating the physical properties of matter, in at least one of its forms. The theory
has even been pushed so far as to assign rough estimates




170 CORPUSCULAR THEORIES OF MATTER
for the size and mass of a molecule of a gas such as
hydrogen. As early as 1738, Daniel Bernoulli conceived
the idea that a gaseous mass consists of a large number
of perfectly elastic molecules in rapid motion of translation. On this hypothesis the pressure on the walls of
the containing vessel is due to the impact against them
of the molecules of the gas. This leads very simply to
an explanation of the law of Boyle and Mariotte, that
the pressure in the gas varies inversely as its volume,
the temperature being unchanged. The further working
out of the theory entails the use of calculation by
probabilities, or in other words the statistical method.
This method was introduced by Maxwell, and further
developed by Willard Gibbs and Boltzmann. The
molecules are divided into groups, in each of which the
molecules are in the same state of motion. It is shown
that, in every gas, every molecule has on the average, for
a given temperature, the same kinetic energy, and that
this is a fixed multiple of the absolute temperature of
the gas. The theory has been applied to explain the
diffusion of gases, and to the theory of their spectra.
If we take a general survey of the ideas which have
inspired atomists both in ancient and in modern times,
we see that the leading motive has been to reduce the
complexities of the various forms of matter, and of the
phenomena connected with them, by assuming that
all matter is made up of parts, each of which has only
the simplest of the perceptual properties of matter in
bulk, those of indestructibility and of motion through
space. The assumption that the world consists of atoms
and of empty space has in appearance the supreme merit
of simplicity, and it satisfies the instinctive craving for
unification of heterogeneous and complex elements in
our actual perceptions. The idea that all interaction requires contact has the merit of apparent accordance with
our own experience, and thus by habit it has acquired
the great advantage of picturability to the imagination.




CORPUSCULAR THEORIES OF MATTER               171
The roughnesses and hooks which atoms were often
supposed to possess served to increase the feeling of
atomists that they possessed a picture which appealed
to what was familiar in actual experience. In fact all the
assumptions as to the nature and properties of atoms
were inspired by the desire to explain the complex forms
and properties in the material world by reducing everything in the smallest parts of matter to simple processes
of the most familiar type. The extreme reluctance that
was exhibited to assign to the atoms any properties
which were not familiar features of matter in bulk is
characteristic of the thoroughgoing realistic spirit which
has dominated the minds of nearly all atomists through
the centuries. The supposed necessity that atoms, or
corpuscles, should satisfy this condition of picturability
has naturally exercised a considerable restrictive influence upon the possibility of developing atomic theories
which should really be adequate for describing the more
complex phenomena connected with matter; and this
may account for many failures. Those of us who do not
feel bound by such requirements are entitled to regard
a purely conceptual atomic theory as admissible, whatever properties are assigned to the atoms, provided the
requirements of a self-consistent scheme are satisfied,
whether these properties are directly copied, or not,
from perceptual properties of gross bodies. Subject to
this condition of applicability, the success of the scheme
in representing actual properties of matter, as the result
of a synthetic process of combining the effects of the
conceptual atoms or molecules in an aggregation, is the
only criterion which an atomic scheme need satisfy; a
reasonable degree of simplicity being presupposed. This
view of atomic theories, or one closely approaching it,
is certainly held by those Chemists who regard the
number and arrangement of atoms in molecules as
having a symbolic meaning only.
When Chemists found it necessary to regard the




172 CORPUSCULAR THEORIES OF MATTER
molecule, and not the atom, as the smallest part of a
particular substance, the question whether the atom
could be regarded as the ultimate constituent of all
matter became insistent. It seemed that it would be
necessary to assume the existence of as many different
kinds of atoms as there are chemical elements. If we
stop there, the notion that all matter is reducible to
atoms consisting of one primitive substance has to be
given up. The knowledge obtained of the relations between the atomic weights of different elements made it
difficult to stop short at the recognition of the existence
of some seventy elements. The idea then suggested
itself that the atom must be regarded as consisting of a
group of smaller atoms, of the second order. It seems
difficult to stop at any particular stage of the indefinite
regress into which we are launched, when an atom of
any order is regarded as composed of smaller atoms of
the next higher order. If, at any stage, we substitute
extensionless centres of force, the whole scheme loses
that character of picturability which was one of its chief
recommendations.
As soon as the conception of forces acting at a distance
came to be accepted, in consequence of the Newtonian
doctrine of gravitation, Atomism, in the primitive meaning of the term, underwent a profound modification.
The theory of matter became a dynamical scheme in
which the notion of central forces is introduced as a
necessary conception, on a parity with the older conceptions of atoms or corpuscles, and empty space. Of
this Dynamical scheme of relations I shall speak in a
later lecture. I must also postpone any consideration of
the modern electron theory of matter, as also of the still
more revolutionary theory associated with the name of
Einstein, in accordance with which matter and its relations are represented by a purely geometrical conceptual
scheme.
The reluctance with which the notion of force, as an




CORPUSCULAR THEORIES OF MATTER 173
independent conception, was accepted, is accounted for
by the inveterate materialistic prejudice, in accordance
with which matter is regarded as the only ultimately
real object. That this view of the unique reality of
matter is no longer tenable has been concisely expressed
by Helmholtz, who wrote1:
It is just as inaccurate to try and explain matter as something
real, and force as a mere notion to which nothing real corresponds; both are rather abstractions from the real, formed in
exactly the same way. We can perceive matter only through its
forces, never in itself.
1 Ueber die Erhaltung der Kraft, Wissensch. Abh., Vol. I, p. I4.




VIII


DYNAMICS
HE mechanical theory of natural phenomena consists
of a formulation of the idea that all such phenomena
can be viewed as essentially consisting in changes in the
motions of parts of material systems. A scheme in
which, under given conditions, all these motions can
be numerically calculated, in accordance with a set of
fundamental laws, is known as a dynamical scheme.
The first postulate of such a scheme is that matter can
be regarded as being of such a character that certain
aggregations of it, or certain parts of such aggregations,
remain unchanged through all changes of distribution
and configuration in space; thus retaining a certain
identity. The shape which this idea of conservation of
matter through all changes has taken in modern times
is formulated in the principle of the conservation of
mass. Of the gradual emergence of this principle from
less definite attempts to fix the character of the unchanging elements in the mechanical theory, I shall give
an account in the next lecture. In the present lecture
I propose to give an account, to some extent critical, of
the Classical Mechanics which is associated with the
names of Galileo and Newton, in which the fundamental concepts employed are those of Force, Mass,
Time, and Space. The Classical Mechanics, as originally
conceived, is the Mechanics of gross bodies, or Molar
Mechanics; it is non-atomic in that it does not assume
matter to have an atomic, or a molecular constitution,
although, as we shall see, it requires the assumption
that matter is indefinitely divisible into parts. The
notions of force and mass which influenced the building




DYNAMCICS


175


up of the Classical Mechanics were for the most part
realistically and concretely conceived; and the mode in
which they were employed was largely directed by the
notion of efficient causation. Consequently, a considerable amount of alteration of what have been the traditional modes of presenting the theory is requisite, in
order that it may be stated in the form of a conceptual
scheme in which it is made clear what parts of that
scheme consist of definitions, and how far the postulations made in it have been derived from direct observations of the behaviour of actual bodies in motion under
certain conditions.
The ideas as to the motion of heavy bodies which
prevailed before the time of Galileo were so confused
and conflicting that we may regard the foundation of
the Science of Dynamics as due to Galileo (I564-I642).
It should however be observed that a scientific treatment
of Statics, which is concerned with the conditions under
which bodies remain at rest, had been initiated by
Archimedes, and possessed some considerable body of
doctrine at the time of Galileo. The first problem which
Galileo set himself to solve was that of describing the
mode in which heavy bodies actually fall; and this
independently of any attempt to answer the question
why they fall. The first attempt which he made to
describe the motion of a falling body consisted of a
guess that the velocity attained by a body falling freely
from a state of rest is proportional to the height through
which it has fallen. This hypothesis Galileo abandoned,
not because of a failure to verify it experimentally, but
because he convinced himself that it led to contradiction. The idea that there is anything self-contradictory in such a motion, is however erroneous; such
motion is conceivable, although it is not that of a falling
body. Galileo then made the hypothesis that the velocity
acquired is proportional to the time of the fall, and he
correctly deduced as a consequence of this hypothesis




176


DYNAMICS


that the height fallen would be proportional to the
square of the time. In order to verify that this is the
actual law of the motion of falling bodies, he first
experimented on a smooth ball rolling down an inclined
plane. As no pendulum clock then existed he measured
the time of the motion by weighing the water which
flowed through a small orifice at the bottom of a very
large vessel full of water, during the motion of the ball.
In this manner he confirmed his surmise that the height
moved by the rolling sphere would be proportional to
the square of the time of the motion from rest.
He then made the assumption, confirmed by an
experiment in which he employed a simple filar pendulum with a heavy ball attached to it, that the velocity
acquired depends only on the vertical height through
which a body has moved from rest. He was then able
to connect the acceleration of a freely falling body with
that of a body moving along an inclined plane. In this
manner Galileo ascertained the law of the fall of a body
independently of any theory; thus obtaining a genuine
scientific law by observation. A most important new
conception introduced into Dynamics by Galileo is
that of acceleration, the gradient of velocity with respect
to time. He perceived that, when a body is in such
circumstances that it is set in motion, or has its motion
altered, it is the acceleration that characterizes the
immediate manner in which these circumstances exhibit
themselves. Before Galileo, the immediate condition of
the production of motion was recognized in pressure on
the body due to contact with another body, but it was
quite unknown that it is acceleration, and neither
velocity nor position, that is determined by the pressure.
Thus Galileo's discovery paved the way for reaching
the modern conception of force, as determining acceleration. Of great importance are Galileo's investigations
of the motion of projectiles, which led him to the conception that the projectile has two independent motions,




DYNAMICS


177


a horizontal uniform motion, and a vertical uniformly
accelerated motion. Thus he introduced in this case the
principle of compounding motions in accordance with
the parallelogram law.
The next great contributor to Dynamical Science after
Galileowas Christian Huygens (629-1695), who invented
the pendulum clock. One of his greatest discoveries is
that of the existence and magnitude of the acceleration
of a point which describes a circle uniformly directed
towards the centre of the circle; its so-called centripetal
acceleration. This is a case of acceleration in which the
magnitude of the velocity remains unchanged, the
acceleration representing the effect of the change in its
direction. Huygens was the first to ascertain the
magnitude of the acceleration due to gravity by means
of pendulum observations. In connection with his
determination of the centre of oscillation of a compound
pendulum Huygens was led to a particular case of the
principle that work is what determines velocity, in fact
to a particular case of the modern principle that the
change of kinetic energy in a system is equal to the
work done upon the system. What was afterwards
called the moment of inertia was also introduced by
Huygens in this connection.
Apart from Newton's supremely important discovery
of the universal law of gravitation, but in close connection with that discovery and his deductions from it,
he laid down in a complete form the essential principles
of Dynamics as they have been accepted by succeeding
generations, although, as I have already stated, the form
in which abstract Dynamics, as a conceptual scheme, is
now presented differs, in some important respects,
from Newton's own formulation in his laws of motion
and the accompanying scholia. Although Newton had
by no means emancipated himself from the idea that a
complete scientific theory must provide an explanation
of phenomena in accordance with the law of efficient


HGL


12




178


DYNAMI:CS


causation, as is shown by his refusal to accept the notion
of so-called action at a distance, his method of procedure
is that of the ascertainment of actual facts which he then
employs for the formulation of scientific laws. His view
of scientific method is formulated in a set of rules' for
the conduct of natural inquiry (the Regulae Philosophandi).
Rule I. No more causes of natural things are to be
admitted than such as truly exist and are sufficient to
explain the phenomena of these things.
Rule II. Therefore, to natural effects of the same kind
we must, as far as possible, assign the same causes; e.g.
to respiration in man and animals; to the descent of
stones in Europe and America; to the light of our
kitchen fire and of the sun; to the reflection of light on
the earth and on the planets.
Rule III. Those qualities of bodies that can be
neither increased nor diminished, and which are found
to belong to all bodies within the reach of our experiments, are to be regarded as the universal qualities of
all bodies.
If it universally appear, by experiments and astronomical observations, that all bodies in the vicinity of
the earth are heavy with respect to the earth, and this in
proportion to the quantity of matter which they severally
contain, that the moon is heavy with respect to the
earth in the proportion of its mass, and our seas with
respect to the moon; and all the planets with respect to
one another, and the comets also with respect to the
sun; we must, in conformity with this rule, declare, that
all bodies are heavy with respect to one another.
Rule IV. In experimental physics propositions collected by induction from phenomena are to be regarded
either as accurately true or very nearly true, notwithstanding any contrary hypotheses, till other phenomena
occur, by which they are made more accurate, or are
1 Opera, Horsley's edition, Vol. III, pp. 2-4.




DYNAMICS


I79


rendered subject to exceptions. This rule must be
adhered to, that the results of induction may not be
annulled by hypotheses.
I have already in an earlier lecture pointed out the
defect in Newton's definition of absolute time, and
urged the view that, in a purely conceptual scheme, an
independent variable which takes as its values the
numbers of the arithmetic continuum must be employed; in the application of the scheme to percepts, an
interval of this continuum must be taken to correspond
to the duration of some standard process, an interval of
public time.
Newton's views concerning space and motion he has
stated substantially in the following form1:
Absolute space, in its own nature and without regard to
anything external, always remains similar and unmovable.
Relative space is some movable dimension or measure of
absolute space, which our senses determine by its position with
respect to other bodies and which is commonly taken for immovable space.
Absolute motion is the translation of a body from one absolute
place to another absolute place: and relative motion, the translation from one relative place to another relative place. And thus
we use, in common affairs, instead of absolute places and
motions, relative ones; and that without any inconvenience.
But in physical disquisitions, we should abstract from the
senses. For it may be that there is no body really at rest, to
which the places and motions of others can be referred.
In these statements we can recognize, in a somewhat
involved form, the distinction which we now make
between conceptual space, that of abstract geometry,
and physical space, the space of perceptual bodies. The
conceptual space of ordinary Dynamics, that which
Newton calls absolute space, is a three-fold ordered
aggregate in which metrical relations of the Euclidean
type are employed. As basis of the system of measurement in this space we postulate the existence of a
Opera, Horsley's edition, Vol. II, pp. 6-i2.
12-2




i8o


DYNAMICS


definite frame, usually conceived as a set of coordinate
axes. In the conceptual scheme, a particle is regarded
as being at rest or in motion according as its coordinates
remain unaltered, or not, when the independent timevariable takes up varying values. This is then the conceptual definition of rest and motion.
The frame of reference being once for all fixed, conceptual motion may then be regarded as absolute. As
to what we are to understand by a particle or a body
in this absolute conceptual space, I shall speak later.
Abstract Dynamics consists of a scheme of rules by
which, having given certain specifications, it is possible
to calculate the positions of all the conceptual bodies of
a system of such bodies in this purely conceptual space.
The whole set of such rules is so devised that these
calculated motions may be employed for the description
and approximate determination of the actual motions
of a system of perceptual bodies in physical space. In
order that the conceptual scheme may be applicable in
this manner, it is necessary to define a mode in which
the positions of conceptual bodies in conceptual space
are to be made to correspond to positions of actual
bodies in physical space.
The only meaning which we can attach to the motion
or rest of a body in physical space is that it is in motion
or at rest relatively to some other body, taken as a
standard; in other words, motion and rest in physical
space are purely relative. In order then to employ the
conceptual scheme for the description of actual motions,
it is necessary to fix some frame in physical space which
shall be taken to correspond to the conceptual frame,
or coordinate axes, the existence of which has been
postulated. This can only be done by taking some
particular body, regarded as unchangeable in its shape
and dimensions, to correspond to a conceptual body at
rest, and defining the frame as fixed in this body. Or
some more complicated process may be employed for




DYNAIMICS


I8I


fixing upon a frame of reference, the basis of which
however always rests upon a choice of actual material
bodies. The success of the conceptual scheme of abstract
Dynamics consists in the fact that it is possible to
determine such a frame of reference in physical space
that, when the positions of the bodies whose motions
are under investigation are measured with reference to
the frame, their actual positions at different instants of
public time are found to correspond, with a sufficient
degree of approximation, to the calculated positions of
the corresponding conceptual bodies in the conceptual
scheme. What frame of reference is taken in any
particular case depends upon the particular motions to
be investigated, and upon the degree of precision that is
requisite in the determination of those motions. For
the purposes of representing motions of bodies in this
room it will often be sufficient to take a frame fixed
relatively to the walls of the room, say a vertical line
and two perpendicular lines fixed on the floor. For
more delicate observations this will give an insufficient
determination of motions; when, for example, the fact
of the rotation of the earth must be taken into account,
we take a frame determined by the directions of the
so-called fixed stars. For the determination of the
motions of the planets we take a frame of reference in
the sun, fixed relatively to the stars, to correspond to
the conceptual frame at rest in conceptual space. The
fact that such actual motions are sufficiently described
in this manner by the corresponding motions in the
conceptual scheme of Newtonian abstract Dynamics is
the only ground upon which that scheme can be accepted
as adequate for its purpose. There is no a priori reason
why that scheme may not have to be superseded, in
whole or in part, by some different scheme, in case it
fails to describe with sufficient degree of approximation
any actually observed set of motions. In particular
there is no a priori reason for assuming that Newtonian




I82


DYNAMTIICS


Dynamics is sufficient for the purpose of describing the
motions of the sub-molecular parts of bodies, as all the
direct verifications of the adequacy of the scheme have
reference to the motions of molar bodies. Any extension
of the scheme to the representation of motions in the
microcosmic region, in which the motions cannot be
directly observed, is a hypothesis, the value of which
must be judged by those deductions from it which are
capable of direct verification by actual measurements.
In the Newtonian Dynamics there is embodied a
principle, that of inertia, which is closely related to the
Newtonian conception of Force. Newton has stated this
principle in one of the definitions which precede his
laws of motion, and has further stated it in the first of
his laws of motion. His definition takes the form that:
" The resident force of matter is a power of resisting, by
which every body, so far as in it lies, perseveres in its
state of rest or of uniform motion in a straight line."
This definition may be regarded as rendered superfluous
by his later definitions of force, as they include the
notion that all accelerations are dependent upon impressed forces.
Before we examine the precise meaning that can be
attached to the principle of inertia, and its position in
the scheme of abstract Newtonian Dynamics, it is
advisable to glance briefly at the history of the principle;
bound up as it is with the conception of force. The
conception of a body persisting indefinitely in uniform
rectilineal motion was quite unknown to the ancients.
For example, Aristotle employs the impossibility of such
motion in an argument of reductio ad absurdum. He
believed that a body can only be moved by the action
of another body which is continually in contact with it.
Under ordinary conditions this continuous contact is
veiled from our eyes, because when we project a body,
we at the same time impart a certain motion to the air,
and this continues to act upon the projectile; in a




DYNAM ICS


i83


vacuum this would not take place. In accordance with
another theory, advanced by Hipparchus, a body that
is set in motion has received from another body an
impulse which continues to reside in the projected body
after the contact with the other body has ceased. This
impulse keeps it in motion, in a straight line, although
that motion is not uniform, but gradually diminishes
and finally ceases. There was no idea, at that time, that
the velocity of the projectile could maintain itself without action from without. The same conception was
expressed by Themistius, a commentator on Aristotle,
who compared the impulse received by the body to the
communication of heat to a body, which remains in the
body for some time whilst the body gradually cools
down to its original temperature. Another notion which
was commonly held by ancient physicists is that uniform
circular motion is a natural kind of movement which
persists unchanged when not interfered with.
The idea of the relativity of motion, which by later
Physicists was connected with the principle of inertia,
was distinctly conceived by Cardinal Nicolas de Cusa
(140I-I464) in the first half of the fifteenth century. De
Cusa, who tried to demonstrate that the earth can move
without our perceiving it, uses as an illustration the fact
that a boat in rapid motion may appear to be at rest to
persons in it who do not see the banks; but he failed to
connect the idea of relative motion with the principle of
inertia. Although he affirmed the possibility of indefinite
motion in a straight line, he still accepted the Aristotelian doctrine of natural circular motion. He explained
the fact that a smooth ball started in motion on a smooth
floor continues in motion, by the persistence of the
tendency of the ball to rotate in its rolling motion. It
is the perfection of rotundity that causes the perpetuity
of the motion. The persistence of the motion of the
ball is, with him, persistence of the rotation, and not
that of the translation.




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DYNAMICS


Copernicus, like Aristotle, attributed to the celestial
bodies a natural circular motion, and denied that this
gives rise to the appearance of a centrifugal force. On
the other hand, Kepler believed that the movements of
the planets were due to a material emanation from the
sun, and that each of them would stop dead in its orbit
if the sun ceased to act. Benedetti, a precursor of
Galileo, still believed that the impulse communicated
to a projectile decreases continually with the time, but
he regarded the motion of a projectile as compounded
of the motion due to the original impulse which started
it and of the natural motion due to its weight, although
he did not understand the cumulative effect of the
weight. Even Galileo in his earlier utterances seems
not to have rid himself of the idea of the gradual diminution of the impressed impulse. He never ceased to
regard the circular motion of the heavenly bodies as a
natural motion, just as Copernicus and the Greek
Philosophers had done, but he distinctly affirmed the
perpetuity of rectilineal motion in a horizontal direction,
and appealed for confirmation to the example of a ball
rolling on a plane. The principle of inertia, although
implied in his works, was not stated by him in its
general form.
The first such statement in a complete form of the
conservation of velocity in a straight line is due to
Descartes, who emphasized the fundamental importance
of the principle in the general theory of motion. It is
not unlikely that he was influenced by the writings of
Galileo which had already been published when he formulated the principle for the first time. He divided the
principle into two statements, the first referring to a body
at rest, and the second to a body in motion. Thus he says:
A body when it is at rest has the power of remaining at rest and
of resisting everything which could make it change. Similarly
when it is in motion, it has the power of continuing in motion,
with the same velocity and in the same direction.
A similar statement of the principle in two parts was




DYNAM ICS


i85


given by D'Alembert, who attempted to demonstrate it
by means of the principle of sufficient reason, without
any appeal to experience except as regards the mere
existence of motion.
Other attempts to give an a priori proof of the
principle have been made by Kant and by Maxwell.
The latter advanced what has at least the appearance of
being a proof1 of the principle by a reductio ad absurdum.
He supposes that the movement of a body left to itself
might gradually cease, in which case it would have a
negative acceleration. This would be changed into a
positive acceleration if we considered the motion relative
to some body to which an appropriate motion was
assigned. Maxwell infers that the law has no meaning
unless the possibility of defining absolute rest and
velocity be admitted, and argues that the denial of the
law contradicts the only doctrine of time and space
which we can form. The same argument might however
be applied to prove the impossibility of uniform motion.
The fact is that, in physical space, there is no meaning
in the assertion that a body moves uniformly in a
straight line, unless some material frame is specified
with respect to which the motion is measured. However, a definite meaning exists when the body is taken
to be a conceptual body moving in conceptual space, in
which all positions are assigned absolutely by numerical
specifications. The reluctance which Natural Philosophers have shown sharply to disentangle the conceptual
statement of scientific laws and theories from statements
relating to percepts has not only obscured the real
nature of Science but has introduced much confusion
into the formulation of results. It is essential for clarity
of meaning in scientific theories to distinguish quite
clearly between statements of facts of observation and
postulations of the conceptual scheme which is designed
to summarize and describe those facts.
An example of the inconvenience I have referred to
1 Matter and Motion, Larmor's ed., pp. 28, 29.




i86


DYNAMICS C


is exhibited in proposals which have been made by
C. Neumann, Streintz, and others, to do away with the
difficulties which arise in making statements about the
motion of actual bodies in physical space by assuming
the existence of some standard body. C. Neumann
assumes1 the existence, somewhere in physical space, of
a body, called the body Alpha, which is completely immovable; and all motions of other bodies are referred
to this body Alpha. Such a body is a mere figment of
the imagination, and can certainly serve no purpose in
actual measurements of motion. This notion of a body
Alpha occurred to both Newton and Euler, but was
rejected by them as unsatisfactory. Streintz gives2 the
name "fundamental body" to some body which can be
regarded as independent of the bodies that surround it,
and is ascertained by the aid of pendulum experiments
not to be in rotation. He then regards the principle of
inertia as affirming that, when any body is not subject
to external influence, it describes uniformly a straight
line when referred to "fundamental coordinates" fixed
in the standard body. It seems preferable in every way
to regard the "fundamental body," or "the body Alpha,"
merely as the postulated ideal frame with reference to
which rest and motion in conceptual space are reckoned,
instead of regarding it as an actual body in physical
space. If luminiferous ether could be regarded as an
actual substance filling all physical space, it would be
natural to regard it as affording the means of describing
the motion of actual bodies; a body at rest would then
be one which was at rest relatively to the ether. But in
accordance with the view which I have maintained in
these lectures, the ether is, if indeed it continues to
retain any place in the Science of the future, a concept
introduced for the purposes of scientific theory, and not
1 Ueber die Prinzipien der Galilei-Newton'schen Theorie, Leipzig,
I870.
2 Die physikalischen Grundlagen der Mechanik, Leipzig, 1883.




DYNAMn/ICS


i87


a percept. Moreover, as I shall explain in a later lecture,
all attempts to detect the supposed velocity of a body
relatively to a material ether have proved fruitless.
The question arises as to what is the true nature of
the principle of inertia. The view that it is an a priori
truth is untenable unless the principle of efficient
causation, or that of sufficient reason, is employed; and
the employment of such principles is not necessary for
the purposes of Natural Science. It is hardly possible to
maintain that the principle of inertia is a direct description of observed facts. We are not acquainted with any
actual bodies which satisfy the conditions of validity of the
principle. No actual body is, or can be, so isolated from
other bodies, as to be removed from conditions dependent
on those other bodies; and as we have already seen,
when an actual body is said to be at rest, that means
relatively to some other material system. Uniform
motion in a straight line relatively to, say, the earth,
would neither be uniform nor rectilinear relatively, say,
to the sun. The observation of a ball moving on a
smooth floor, which is usually appealed to as an experimental proof of the principle, does not really establish
it as an empirical law, or even as an approximate fact
of observation, partly on the ground I have just mentioned, but also because the ball, having weight, and
being in contact with the floor and the surrounding
air, does not satisfy the condition of being under no
forces. That, apart from friction, the weight and the
pressure of the ground, taken together, do not affect the
horizontal motion, can only be inferred when other
parts of the dynamical scheme are taken into account.
Thus the law of inertia is not a direct result of facts of
observation; and this probably accounts, to some extent
at least, for the length of the period during which it
was not discovered, or was not generally accepted. It
follows then that the law of inertia cannot be taken in
isolation, but must necessarily be regarded as forming a




i88


DYNAMI CS


part of a complete conceptual scheme of Dynamics. The
exact position of the principle can only be determined
when the whole of Newtonian Dynamics is taken into
account, including the doctrine of the relation of motion
with force. We shall see that the principle appears in
this conceptual scheme in the form of a definition. On
the impossibility of regarding the principle of inertia
either as an a priori truth, or as embodying a direct
result of observation, Poincare has remarked':
If it be said that the velocity of a body cannot change, if there
is no reason for it to change, may we not just as legitimately
maintain that the position of a body cannot change, or that the
curvature of its path cannot change, without the agency of an
external cause? Is, then, the principle of inertia which is not an
a priori truth, an experimental fact? Have there ever been
experiments on bodies acted on by no forces? and, if so, how
did we know that no forces were acting?
One of the main points in which Newton's theory of
Dynamics contains an advance beyond the conceptions
of Galileo or Huygens is in the generalization of the
conception of force. For the origin of the notion of
force we must go back to the feeling of exertion which
we have when our muscles are employed in changing
the position of actual bodies. The notion was extended
to cover the case of the interaction of two bodies in
contact with one another; the force then acting on one
of the bodies being regarded as the efficient cause of
the motion produced in it. The explanation of motion
as due to contact action appeared to explain the phenomenon in the sense to which I have referred in an
earlier lecture, that it reduced it to the familiar case of
contact of the human body with another object. That a
precise examination of what occurs in case of contact
between two bodies must be undertaken in order to
explain the effects of pressure was not recognized until
modern times. We now know that such an examination
1 Science and Hypothesis, 1905, p. 91.




DYNAMICS


I89


involves conceptions as to the relations between the
smallest parts of the bodies in contact, and that this
reduction to the corpuscular domain gives rise to
problems as difficult as the original one which led to
the whole investigation. The futility of the attempt to
indicate the point at which efficient causation is to be
found is exhibited by the endless regress to which we
are led when we attempt to account for the interactions
of the smallest parts of the bodies. If we refuse to
regard a body as consisting of corpuscles of any kind,
but take it to be absolutely homogeneous, we are unable
to begin to understand the nature of contact action.
Involved in the discovery by Newton of the law of
gravitation is the extension of the notion of force to
such cases as the so-called forces of attraction between
the sun and the planets. Newton himself, under the
influence of the ancient idea that all forces must be due
to contact, was, as I have already observed, never
satisfied with the bare fact of the existence of these
forces, but thought that they were inconceivable without
some mechanism, such as a homogeneous ether, which
should condition the forces by ultimate reduction to
contact action. However, the existence of forces, regarded as due to the action of one body on another body
at a distance, came to be accepted as part of the apparatus
of Dynamics, and was accepted by Newton himself with
the reservation just indicated.
Newton's mode of measuring a force depends upon
the notion of the mass of a body as a measurable quantity. The formulation by Newton of the definition of
mass as the quantity of matter in a body measured by
the product of its volume and density has the defect
that it is circular, since the conception of density is
meaningless unless that of mass has been previously
defined. Newton ascertained by means of experiments
that a body has a property, the mass, distinct from its
weight, whereby the quantity of motion of the body is




I90


DYNAMICSC


determined. He also showed that the mass of a body
may, in one and the same place, be measured by its
weight, and that the ratio of the weight to the mass is
independent of the chemical composition of bodies.
There is a lack of clarity in the deductions from these
experiments, owing to the want of a satisfactory definition of the mass, or quantity of matter which a body
contains.
Newton's first two laws of motion contain a statement
of the relation of an impressed force to the change of
motion of the body, which is really equivalent to a
definition that he had previously given; and this we
now express in the form that the force is proportional
to the acceleration it produces in its own direction. As,
according to this definition, there is no acceleration
without force to produce it, the first law is really contained in the second. Newton's third law of motion,
that the forces between two bodies which influence one
another are equal and in opposite directions, only
attains a precise meaning when the notion of mass as
having numerical measure has been made precise by
means of a definition. If we consider the action and
reaction between two bodies as equal, the ratio of their
relative masses may be taken to be the inverse ratio of
their accelerations in opposite directions. This only
amounts to the definition of the mass-ratios of the two
bodies. If now we consider a third body, we can then
define the mass-ratio of any pair of them. It was pointed
out by Mach that observation is required to establish
the fact that to each one of the bodies a number may
be assigned, such that, for each pair of the bodies, their
relative masses are given by the ratio of the two numbers
assigned to the bodies of the pair. Thus a regular and
consistent mode of assignment of mass-numbers can be
set up, such that the law of action and reaction between
any pair of bodies will hold good. That the mass-number
assigned to a body is independent of change in its form,




DYNAMICS


I91


or chemical and thermal conditions, is a fact which must
be verified by observation.
The rule that the force acting on a body is equal to
the product of its mass and its acceleration, depends
upon the possibility of measuring three magnitudes, the
force, the mass, and the acceleration. I have pointed
out that mass is not capable of measurement independently of the notion of the equality of two forces. The
impossibility of an independent measurement of force
has been trenchantly signalized1 by Poincare as follows:
When we say that force is the cause of motion, we are talking
metaphysics; and this definition, if we had to be content with
it, would be absolutely fruitless, would lead to absolutely nothing.
For a definition to be of any use it must tell us how to measure
force; and that is quite sufficient, for it is by no means necessary
to tell what force is in itself, nor whether it is the cause or the
effect of motion. We must therefore define what is meant by
the inequality of two forces. When are two forces equal? We
are told that it is when they give the same acceleration to the
same mass, or when acting in opposite directions they are in
equilibrium. This definition is a sham. A force applied to a
body cannot be uncoupled and applied to another body as an
engine is uncoupled from one train and coupled to another. It
is therefore impossible to say what acceleration such a force,
applied to such a body, would give to another body if it were
applied to it. It is impossible to tell how two forces which are
not acting in exactly opposite directions would behave if they
were acting in opposite directions. It is this definition which
we try to materialise, as it were, when we measure a force with
a dynamometer or with a balance.
The impossibility of measuring independently of one
another the three magnitudes, force, mass, and acceleration, as they are employed in Newton's scheme, and of
then verifying experimentally that force is proportional
to the product of mass and acceleration, leads to the
conclusion that force must be defined as the product of
mass into acceleration, and thus does not appear in the
scheme as an independent conception. Thus the first
1 Science and Hypothesis, I905, p. 98.




I92


DYNAM ICS


two laws of Newton, regarded as part of a conceptual
scheme, consist in reality of a definition of force as the
product of the mass-number of a body into its acceleration in a particular direction, which direction is taken to
define the direction of the force. The fact of observation
which leads to the formulation of Newton's third law
of motion is that a consistent set of mass-numbers can
be assigned to a set of bodies so that any two of the
bodies have towards each other accelerations inversely
proportional to the mass-numbers of the two bodies.
Newton gave a deduction of the principle known as
the parallelogram of forces from his second law of
motion. But in order that this deduction may be valid,
the assumption is required that the accelerations produced in a body by two other bodies are independent of
each other; so that the acceleration which B produces
in A is independent of the fact that C is also producing
an acceleration in A. That this is the case in actual
bodies is not self evident, and can only be established
by experience; in the conceptual scheme the assumption
must appear as a postulate. If it be assumed that all
the forces between bodies are central forces, depending
in magnitude only on the distances between the bodies,
Newton's deduction will be justified; but if we do not
make this assumption we have no means of resolving
the acceleration of A into components, one of which we
regard as due to B, and the other as due to C; and in
that case we have no means of assigning definite massnumbers to the three bodies, and therefore the principle
of equality of action and reaction would no longer be
valid.
It would thus appear that the validity of the Newtonian scheme is dependent on the assumption that the
forces between any two bodies are central. But this
assumption has really been implied in the statement
of the laws of motion. For the acceleration of a body
has no definite meaning unless the body be either of




DYNAMICSC


I93


negligible dimensions, or unless the acceleration be
taken to mean that of some one particular point in the
body; for in general a body can move not only translationally but also rotationally, and thus different parts
of a body may have different accelerations. In order
that Newton's laws of motion may have a precise
meaning, it must be assumed either that the bodies
referred to in them are considered as masses concentrated at points, or as bodies of non-negligible size
which are equivalent to such concentrated masses at
their centroids; the forces between any pair of such
bodies being along the straight lines joining their
centroids. This was realized with sufficient degree of
approximation in the cases of motion of bodies of the
solar system to which Newton applied his Dynamics.
The extension of Newton's Dynamics to the general
case of bodies of various shapes, acting and reacting on
one another, can only be made by a process of integration, in which the bodies are divided up into a large
number of very small parts, so that, very approximately,
any two of such parts can be regarded as two points at
which the masses of the respective parts are concentrated; and it is assumed that the forces which act
between two such parts, whether they belong or not to
one and the same body, are central, that is along the
line joining these parts. By means of the procedure of
passing to a limit, that is of integration, the dynamical
scheme can be built up in a form applicable to a system
of bodies. This scheme takes the form of a set of differential equations called the equations of motion of the
system. In the conceptual scheme, the bodies consist
of defined portions of geometrical space, which are
movable in that space, retaining in the case of a rigid
body their dimensions unaltered; the mass of each such
body appears in the equations of motion only as a coefficient, to which in any special case a numerical value
is assigned. The position for each value of the timeHGL                                      13




I94


DYNAMICS


variable, of any ideal rigid body, is denoted by a set of
numbers which represent coordinates relatively to the
frame of reference with respect to which all positions
are referred. The motion of a point is denoted by a
functional relation between its coordinates and the
time-variable. As we have seen, the principle of inertia
and the conception of force appear in the scheme as
definitions, and the masses of bodies or of parts of
bodies appear as coefficients to which numerical values
may be assigned. The hypothesis of central forces
between elementary parts of bodies, of magnitudes
dependent only on the distance, is also an essential part
of the scheme. The equations of motion of the conceptual bodies suffice to determine their motions, and
the results obtained suffice for the approximate determination of the motions of the perceptual bodies in an
approximately isolated system, provided a correlation
between measurements in physical space and in geometrical space can be set up.
To do this it is requisite to assign, for each special
problem, a frame of measurement in physical space
which may be taken to correspond with the absolute
conceptual frame by means of which all positions in the
geometrical space are measured. This frame of reference
must be fixed relatively to some actual body, and when
it consists of rectangular axes it may be spoken of as a
Newtonian set of axes. The utility of the conceptual
dynamical scheme for its purpose of describing the
motions, with sufficient accuracy, of an approximately
isolated material system, depends upon the verifiable
fact that it is possible to determine, with sufficient
degree of approximation for the purpose on hand, a
Newtonian set of axes1 suitable to the particular system.
If such a Newtonian set of axes has been determined in
any particular case, any other set of axes parallel to the
1 By some writers the terms "Galilean axes" or "inertial frames"
are employed.




DYNAMICS                    I95
former will also be a Newtonian set, provided its origin
moves with uniform velocity relatively to the origin of
the first set. This principle is subject to the test of
experience, but it is an essential principle of Newtonian
Dynamics. But a set of axes in rotation with regard to
a Newtonian set will not be itself a Newtonian set,
unless additional forces, including the so-called centrifugal forces, are introduced into the system.
In this connection, it has frequently been maintained
that, in physical space, the rotation of a body is absolute,
whilst its motion of translation is relative; and thus that
absolute directions, unlike absolute positions, can be
defined. Newton's celebrated experiment with the water
in a bucket is frequently appealed to in support of this
contention. As long as the bucket is at rest the water
in it has a flat surface, but if the bucket be made to
rotate about its vertical axis, a motion of rotation is
gradually set up in the water, when its surface is no
longer plane, but has the form of a concave surface of
revolution. It is then suggested that there is an absolute
difference between axes fixed relatively to the bucket
when it was at rest, and axes fixed relatively to it when
it is in motion. The only relevant fact would however
appear to be that it is possible to determine with sufficient
approximation, in any concrete case, the directions of a
Newtonian set of axes; and other axes in rotation relatively to these do not form a Newtonian set. It can
however be shown that the directions of the axes of a
Newtonian set have no valid claim to be regarded as
absolute.
In the first place a Newtonian set of axes, determined
for a particular isolated system, need no longer be a
Newtonian set for an enlarged system. For example, in
this room, a vertical line and two horizontal lines, fixed
relatively to the walls, would form a Newtonian set of
axes, for the purpose of describing the motions of
ordinary bodies in this room; and in particular the




i96


DYNAIMICS


rotation of Newton's bucket would mean rotation relatively to these axes. But for the purpose of describing
the motion of Foucault's pendulum, these axes would
no longer form a Newtonian set. For that purpose we
should have to take the axis of the earth and two other
axes fixed in direction relatively to the stars. Again, if
we had to take account of the slow precessional motion
of the earth's axis, these last would no longer form a
set of Newtonian axes. However far we may proceed
in this manner to fix Newtonian axes which shall be
sufficient for the motions of ever more extensive, or
complicated, systems of bodies, we are unable to assert
that the axes obtained will necessarily be a Newtonian
set for the purpose of measuring the motions of every
system of bodies whatever, which we may at any time
have to consider. Moreover the approximate determination of a Newtonian set of axes is insufficient to
determine, even approximately, absolute directions, because the smallest error in the determination will be
of a cumulative character, involving the existence of a
large, and no longer negligible, error after a sufficiently
long time. The rotation of a set of axes, however slow
that rotation may be, will, in a sufficiently long time,
produce a large deviation of the axes from their original
position. There is thus no warrant for the assertion that
it is possible to assign directions in physical space which
may, for every possible purpose, be regarded as Newtonian axes. On this ground, and also because the
Newtonian system of Dynamics, although the simplest
system for the descriptive purposes for which it was
devised, is not the only possible system that might be
employed, and may not even suffice for all purposes, it
is not correct to assert that absolute directions in
physical space can be determined by any means at our
disposal; nor are we compelled to conceive absolute
directions in physical space as existent.
To regard the earth as rotating round its axis is




DYNAMICS                   197
convenient for the purposes of Descriptive Astronomy,
and it is also convenient for the purposes of the dynamical description of the motions of bodies in the solar
system. But however convenient is the assertion that
the earth rotates, it still remains a convention, although
the opposite assertion would vastly complicate both the
astronomical and the dynamical schemes which would
then have to be employed. In connection with the new
general theory of relativity, both spatial relations and
dynamical theory have to be considered from a point of
view which necessitates a rediscussion of such matters
as the introduction of centrifugal, and other, forces
when non-Newtonian sets of axes are employed in
connection with the motions of a material system.




IX


THE CONSERVATION OF MATTER
AND ENERGY
T   HE notion that, in all the multifarious changes
which we perceive to take place in the material
world, there must be permanent elements which persist
unaltered through all these changes, has been, throughout the history of Science, one of the guiding ideas
which have ultimately given rise to such formulations
as those contained in the expressions Conservation of
Matter, Conservation of Weight, Conservation of Mass,
and Conservation of Energy. The principle of permanence expressed in such a formula as that "nothing is
created and nothing destroyed" has usually been regarded as an a priori principle closely related to the
principle of causation. The very generality of this a
priori principle has prevented it from functioning as
an efficient guide to the determination of the precise
elements in the perceptual world to which the characteristic of persistence through all transformations
appertains. Thus the actual successes of scientific investigation in this order of ideas have consisted in the ascertainment, by experiment and observation, of empirical
laws, the law of conservation of mass, and the law of
conservation of energy. Such empirical laws are ascertained to express definite facts relating to a considerable,
but limited, range of observed phenomena, the conservation which they express being of such a character
that it is expressed quantitatively, in a numerical form.
The laws are then adopted as hypothetical principles in
conceptual theories which relate to ranges of phenomena




MATTER AND ENERGY


199


wider than those to which the empirical verification has
been in the first instance applied.
The value of the laws in their general form must
depend upon their success in performing their functions
of description and prediction in relation to new classes
of phenomena to which they are tentatively applied.
The a priori principle in its general form, to which I
have referred, as a metaphysical principle expressing a
supposed necessity of thought, need not be accepted as
a part of the foundations of Natural Science, whatever
its actual influence may have been upon the minds of
scientific investigators in the past. The main difficulty
as regards the principle of the conservation of matter,
the principle that matter is neither created nor destroyed,
is that of forming a clear conception of what it exactly
is that is conserved. If we regard matter as a construct
including a complex of physical properties, extension,
colour, hardness, conductibility of heat and electricity,
etc., we have ample and obvious evidence that these
properties do not persist unchanged, but are subject to
the largest changes in what we regard as one and the
same material system. What then is to be understood
by the statement that matter can be neither created nor
destroyed; that is, by the principle of the conservation
of matter? If we assert that what persists unchanged is
a sub-stratum, substance itself, not identified with any
or all of these physical properties, but regarded as their
bearer, not only do we reduce the principle to one
dependent upon a metaphysical theory, but we remove
from it all possibility of verification. It then becomes a
bare philosophical assertion which has no direct relation
with the world of percepts, and is thus outside the
domain of Natural Science. A real scientific law of conservation must contain an indication of some measurable
quality or property of matter, which can be ascertained
to remain unaltered in magnitude during the actual
chemical and motional transformations occurring in the




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physical world. Not even what is regarded as a primary
quality of matter, that of extension, is conserved as a
measurable quantity unaltered through all transformations.
There is however one other property which we have
come to associate with all matter, that of weight; this is
estimated by the balance, the systematic employment of
which by Lavoisier brought about what has been
described as a revolution in chemical Science. If however we understand conservation of matter to mean
conservation of weight, we are at once met with the
difficulty that the weight of what we regard as one and
the same piece of matter, when estimated by the spring
balance, varies with the latitude of the place at which
it is measured. Moreover, in accordance with the theory
of gravitation, the weight would be very greatly altered
if the matter were transported to another planet. However, in the chemical transformations which take place
in any one locality, a verification of the principle of
conservation of matter consists in a verification of the
principle of conservation of weight. For different
localities the differences of weight of one and the same
object are eliminated, in accordance with the Newtonian
and Galilean Dynamics, by dividing the weight by the
acceleration due to gravity; this division yielding the
measure of the mass of the body. Thus the conservation
of matter now comes to be taken to mean the conservation of mass.
With the concept of mass I have already dealt more
fully in connection with Dynamics. The actual mass of
a body can be regarded as a quality which can be
measured as derivative from the two measurements of
weight and of acceleration. That the mass of a body is
the amount of matter in it is a tautological statement
which can only be taken to denote that the meaning
assigned to the term quantity of matter is that it is the
mass regarded as a measurable quality of the body. The




MATTER AND ENERGY


20I


principle of the conservation of matter, regarded as
mass, has however a much wider meaning than that it
is unalterable for one and the same body in whatever
position it be placed, or however it be moving. It includes the assertion that mass, as a measurable quantity,
is unchanged in amount throughout all the chemical
and thermal changes that may take place in an isolated
material system.
Thus the principle implies that matter, however it be
sub-divided actually, or conceptually, may be regarded
as having a quality, the mass or quantity of matter,
which is measurable, and remains unchanged in total
amount, during all motions, and all chemical, thermal,
or other transformations. The only means we have in
an abstract conceptual scheme of representing this
assumed quality is by the employment of numbers in
relation to conceptual bodies in geometrical space. In
its abstract form, the principle asserts an invariant
property of the sum of such numbers for the conceptual
elements of a limited system. Only to a limited extent
has this general law been verified experimentally; for
the difficulties of measurement, and of securing complete isolation of the substances which undergo chemical
transformation, are very great, being subject to errors
difficult to take completely into account. Moreover,
the conservation of mass through all motions is capable
only of indirect verification, in connection with the
verification of the adequacy of a particular dynamical
scheme.
Accordingly the principle must be regarded as an
hypothesis, which has been verified approximately in a
large number of cases and is assumed by the Chemist
and the Physicist to hold good as descriptive of relations in a large range of actual phenomena, but as
subject to possible refutation in cases in which more
refined methods of measurement are employed in connection with electromagnetic or other phenomena.




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In the new electron theory of matter, of which I shall
speak later, mass occupies a wholly different position
from that which is assigned to it in the mechanical
theory which has been here discussed. In accordance
with the electron theory, the mechanical masses of bodies
are no longer constant, but have a sensible variation
when the bodies are set in motion with velocities
comparable with the velocity of light.
A sketch of the history of the doctrine of the conservation of matter is sufficient to show that the establishment of the principle, in the modern form of that of
the conservation of mass, was the result of a gradual
evolution. The ancient atomists, not being in possession
of the principle of inertia, made no distinction between
mass and weight. In this they are in accord with the
popular view held even to-day. For example, Lucretius,
following Democritus in this order of ideas, appears to
have regarded weight as an unalterable characteristic of
all matter; in his poem the atoms are in motion on
account of their weight; and this involves an identification of their masses with their weights. On the other
hand, the school of Aristotle, radically opposed as it was
in almost all respects to the views of the atomists, did
not believe that all matter has weight. With the Aristotelians, the notions of matter and weight are kept
quite distinct from one another; weight being regarded
as an accidental quality of matter, like colour or temperature. In their view, weight is the resultant of two
opposed qualities, heaviness and lightness. Fire has no
heaviness, and earth no lightness. Water and air have
both, one being preponderant in water, and the other
in air. Plato had observed that the four elements are
constantly transformed into one another; thus air and
fire are concerned with the transformation of matter,
as when water boils or when wood burns.
In the middle ages the Aristotelian views about matter
were prevalent, although traces are to be found of the




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influence of the ancient atomists. Some of the Alchemists used reasoning founded on the consideration of
weights; but they did not attach to weight the primary
importance which it came later to possess, and they did
not believe in its unchangeableness during transformations. Indeed some of them expressly mention change
of weight as occurring in the transmutation of matter.
Thus, for example, Geber writes (? eighth century A.D.):
"By our artifice we easily form silver out of lead; in
the transformation the latter does not preserve its own
weight but changes into a new weight." This view of
the Alchemists cannot be explained away as involving
merely a reference to change of specific weight, or to
the change of weight produced by absorption of matter
from air or fire; indeed specific weight and absolute
weight were constantly confused with one another, even
as late as the seventeenth century. As long as the weight
of a body was regarded as a merely accidental quality,
like its colour, it was quite natural to suppose that it
might be changed without the addition or subtraction
of matter. Even Bacon held opinions not very different
from those of the Alchemists. In some statements he
affirmed the existence of absolutely light bodies, and
that change of weight may accompany a change of
state; but in other statements, probably following the
ideas of the atomists, he affirmed the constancy of
weight. It must always be remembered that, with the
Alchemists, and generally with those under the influence
of the Aristotelian conception of substantial forms, the
question whether the weight remained quantitatively constant or not, during a transformation, seemed a matter of
subordinate importance; the smallest change of quality
was in their eyes of much greater interest. A knowledge
of the relations of quantity being of little importance to
the adherents of the philosophical doctrine of substantial forms, it is hardly possible to regard the notion
of the permanence of mass, considered quantitatively,




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THE CONSERVATION OF


as a part of the stock of ideas of those who were under
the influence of Aristotelian conceptions. For the
notion of matter as underlying substance, differentiated
from the accidental quality of weight, did not admit of
quantitative measurement, although the substance was
regarded as in some sense persisting through all changes.
As the gradual emancipation from the Aristotelian conceptions took place, the conception of mass became
clarified. It was formulated with tolerable clearness by
Kepler, and also by Descartes, although with the latter,
as with the Aristotelians, weight remained an accidental
property, not possessed by all matter. That air has
weight was generally recognized in the time of Descartes,
but fire was still regarded as devoid of weight. With
Descartes, matter being a plenum, the quantity of
matter was indicated by its volume; thus he makes the
assertion that "when a jar is full of gold or of lead, it
does not contain more matter than when we think it
empty." But he regarded the terrestrial matter as the
only kind to be taken into account in mechanical action,
and thus in reality he made a distinction between mass
and volume, the mass appertaining to terrestrial matter
only. He did not however recognize that mass and
weight are in a fixed ratio; in fact the title of one of the
chapters of his Principles is " That their weight has not
always the same proportion to their matter," and here
matter is to be understood as terrestrial matter. For a
long period after the time of Descartes, the notion that
weight is an accidental quality of matter prevented the
general acceptance of the conservation of weight,
although opinion on the point was by no means unanimous even in Descartes' own time. Jean Rey, in his
essays, which were published before the actual appearance of Descartes' Principles, attempted to give an a
priori demonstration that weight is conserved in every
transformation. Moreover he gave an experimental
proof that air is heavy, and that in the formation of lime




MATTER AND ENERGY


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the increase of weight is due to material taken from the
air. The persistence of disbelief in the invariability of
weight is exhibited in the utterances of many writers,
even until the end of the eighteenth century. Thus
Hobbes declares that " all accidents other than greatness
or extension can be engendered or destroyed," thereby
leaving no room for the conservation of weight or of
mass. Leibniz, who had a clear conception of mechanical mass, states that "water contains in equal volume
as much matter as mercury, only to the matter belonging
to the water there is added a foreign non-heavy matter
which is between its pores" for "it is a strange fiction
to make all matter heavy."
Newton, who did not admit the existence of imponderable matter, showed experimentally that the weight
is proportional to the mass of a body; and Huygens
stated definitely that quantity of matter is measured by
its weight. The separate lines of work of Physicists and
Chemists make it difficult to ascertain the views of the
Physicists of the seventeenth and eighteenth centuries
on the nature of chemical phenomena. Almost the only
exception to this separation was the work of Robert
Boyle, both Physicist and Chemist, who appears to
admit the principle of conservation of weight, without
however explicitly formulating it. In the seventeenth
century, although it was generally admitted that air has
weight, it was not generally believed that this is the case
for fire. In the eighteenth century, however, we find
that Berkeley regarded the increase of weight of some
metals when heated, for example in the case of antimony,
as due to the fire in the sun's rays; he remarked that we
do not know the weight of a solar ray. Diderot stated
that "the fire of our furnaces considerably augments
the weight of some matter, such as calcinated lead."
The special form which the notion of imponderable
substance took in the minds of the Chemists of this
period was the theory of phlogiston, a substance in



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THE CONSERVATION OF


vented to account for thermal phenomena. Phlogiston
was endowed with negative weight, and since it intervened in all chemical reactions there was no difficulty in
conceiving that its admixture with matter prevented the
conservation of weight in a chemical transformation.
How little attention was paid by Chemists to all questions relating to quantities is illustrated by the fact that
one of the chief French Chemists of the day, Macquer,
on hearing that Lavoisier was preparing an attack upon
the theory of phlogiston, stated that he had been disquieted for a moment, but was reassured when he
learned that Lavoisier's objections were based solely on
quantitative considerations. The definite establishment
of the principle of conservation of matter, by the systematic use of the balance, is mainly due to Lavoisier,
and may be dated from his memoir on The change of
water into earth, published by the French Academy in
I773. This was followed in I774 by a work in which, by
the employment of the balance, he decided between the
rival theories of Black and Meyer as to what happens
in chemical transformations. First applying the principle without explicitly formulating it, he afterwards
gave a precise statement of it. In his elementary treatise
on Chemistry he remarks that "any matter can furnish
nothing in an experiment beyond the totality of its
weight," and further "the determination of the weight
of materials and their products before and after experiments is the basis of everything useful and exact in
Chemistry." "In every operation there is an equal
amount of matter before and after the operation."
In 1774, Lavoisier described the commencement of
his fundamental discoveries relating to combustion. He
verified that various metals, when heated in a closed
vessel, receive an increase of weight, and that the
amount of air in the vessel is diminished; he showed
that the loss of weight of the air is nearly equivalent to
the increase of weight of the metal. A slight increase




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207


in the weight of the whole vessel he properly attributes
to an exterior deposit due to the fire. In this way he
provides a refutation of the idea of the intervention of
the element fire, and shows that the increase of weight
can only come from the air. Even after the composition
of water became known, and the phenomenon we call
oxidization, the new conceptions of Lavoisier only
triumphed slowly. They do not appear to have been
completely accepted by Priestley or Cavendish. Scheele
regarded heat as a compound of phlogiston and oxygen;
both of them he thought of as heavy, but supposed that
together they give rise to an imponderable substance.
Heat united with very little phlogiston is transformed
into light, but united with a great quantity it becomes
inflammable air, that is hydrogen. Even Lavoisier shows
traces of analogous conceptions. He regarded oxygen
as resulting from a combustion of ponderable matter
and an imponderable fluid, caloric. Heat he regarded
as a material element contained in a gas, and the conception which he had of gases was related, by means of
intermediate hypotheses, with that of imponderable fluids.
After the vicissitudes which I have sketched, the
principle of the conservation of matter, regarded as
measured by dynamical mass, has come to be accepted
as an empirical law which is applicable within a large
range of phenomena, although some Chemists have
maintained that it is possible to detect deviations from
the law, which cannot be assigned to the effect of instrumental errors or to disturbing factors not easily taken
fully into account. Moreover the flood of light which
has lately been thrown upon the properties of radioactive substances, has suggested views in which dynamical mass no longer holds its former position as fundamental and irreducible. It has indeed been suggested
that, in accordance with the electrical theory of matter,
there would be nothing surprising in a change of weight
owing to chemical reactions.




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The origin of the principle of the Conservation of
Energy is much more modern than that of the principle
of the Conservation of Matter. In its general form, the
principle of the Conservation of Energy dates back only
to the middle of the nineteenth century, but in its
restricted form, as a principle of Mechanics in the
narrower sense of the term, it is in the ideas of Descartes,
Leibniz, and especially of Huygens, that we find its
origin. The notion of matter is one formed by common
sense, but the conception of energy has been created
by Science for its own special purposes. It seems therefore quite natural that the doctrine of the conservation
of energy should have arisen at a much later stage in
the history of Science than that of the conservation of
matter, at least in a crude form. The notion of work, as
measured by the product of a force into the displacement, in the direction of the force, of the body on which
it acts, is due to Galileo, who showed that in simple
mechanical machines the work of the resistance in a
displacement is equal to that of the power. He concluded that, by the aid of such machines, it is impossible
to create work, but he did not show that work cannot
be destroyed. For the case of a falling body, he gave
the formula which expresses the principle of energy.
The next step in the direction of setting up a general
principle relating to the movement of bodies was taken
by Descartes, who attempted to set up a principle of the
conservation of motion through all changes in the
physical world. In this attempt he made the mistake of
taking the sum of the products of the masses into their
velocities, instead of the squares of the velocities, as
representing the quantity which is conserved. This
error was pointed out by Leibniz in a treatise bearing
the title "A short demonstration of a Remarkable Error of
Descartes and others, concerning the Natural Law by
which they think that the Creator always preserves the
same Quantity of Motion; by which, however, the Science




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of Mechanics is totally perverted." Leibniz distinguished
between simple pressure (vis mortua) and the force of a
moving body (vis viva), but he confused the question
of the right measure of force with that of the constancy
of momentum and of the kinetic energy of a system.
Neither the Cartesian nor the Leibnizian measure of the
effectiveness of a body in motion is, as Leibniz observed,
to be identified with the Newtonian measure of force.
Leibniz, like Descartes, regarded the principle formulated by him as embracing all the phenomena of the
Universe. He justified the principle of the conservation
of vis viva by an appeal to the principle of causation, in
the form that the effect is equal to the cause.
To attempt to demonstrate this law (he writes) would obscure
it. Indeed, everyone regards it as an incontestable axiom that
every efficient cause cannot perish, either totally, or in part,
without producing an effect equal to the loss. The idea that we
have of the vis viva, as it exists in a body in motion, is something
absolute, independent, and positive; that it remains in the body
even if the rest of the Universe were annihilated. It is then
clear that, if the vis viva of a body diminishes or increases on
impact with another body, the vis viva of this other body must
change, increase, or diminish by the same quantity.
His view of the scope of the principle appears clearly
in the following passage:
I had maintained that active forces are conserved in the
world. It has been objected that two soft or inelastic bodies,
when they collide, lose part of their force. I answer that this
is not so. It is true that the "wholes " lose it in respect of their
total motion, but the parts receive it, being agitated internally
by the force of the collision. Thus the loss ensues only in
appearance. The forces are not destroyed, but dissipated amongst
the minute parts. That is not as if they were lost, but it is like
the changing of large coins into small ones.
The truth of the principle had previously been demonstrated by Huygens, who had however formulated it
without indicating the great generality of its scope. He


H GL


14




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distinguished between the conservation of vis viva and
that of momentum in his statement:
The quantity of motion possessed by two bodies may be
augmented or diminished by their encounter; but there remains
always the same quantity on the same side, if we subtract the
quantity of opposite motion. The sum of the products of every
hard body multiplied by the square of its velocity is always the
same before and after the encounter.
It must be observed that, both for Descartes and for
Leibniz, the world consists only of matter in motion,
and there exists no action at a distance. Consequently
they did not admit the existence of what we call potential
energy, so that, for them, the principle of energy consisted in the constancy of the total kinetic energy. It
should also be observed that Leibniz, in the passage I
have quoted, in speaking of the dissipation of the molar
energy amongst the smallest particles of a body, does
not seem to have considered this transformation as
equivalent to the production of heat.
In the eighteenth century, the conception of heat as
a substance gradually gained upon the Cartesian idea
of heat-motion. The amount of this substance was
supposed to be conserved in its passage from one
body to another. When it ceased to manifest itself by
means of the thermometer, heat was regarded by
Black as still present, but as latent heat, capable of
manifesting itself in certain conditions, and thus his
conception of latent heat was analogous to our conception of potential energy. Even the invention of the
steam engine produced no immediate change in this
conception of the substantiality of heat. Watt and his
successors failed to attain to the view that thermal
changes indicate any relation between heat and mechanical motion. However, towards the end of the century,
Lavoisier and Laplace tentatively related the production
of heat by friction with the conception of heat-motion,
and defined the amount of heat as the sum of the




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products of the masses of the molecules into the squares
of their velocities.
Early in the nineteenth century, direct experimental
demonstrations were obtained, by Rumford and by
Humphry Davy, of the transformation of motion into
heat. The concept of latent vis viva, called by Poncelet
work, was formulated (I803) by Lazare Carnot; this
concept is that now known as potential energy. In
I839, the engineer Seguin, in a work on the construction
of railways, remarked that:
As the theory at present adopted would lead however to this
result (perpetual motion) it appears to me more natural to suppose that a certain number of calories disappear in the act of
producing mechanical force or power, and conversely; and that
the two phenomena are bound together by conditions which
assign to them invariable relations.
It appears however that, before Seguin, the principle
of the equivalence of heat and mechanical energy had
been conceived in its generality by Sadi Carnot, who
obtained by calculation an estimate of the mechanical
equivalent of heat. In his earlier work, Carnot had
employed the material theory of heat, and his later
formulation of the modern theory was preserved only
in manuscript notes which remained unpublished until
1871. But it was the experimental researches of Joule,
published in 1843, that brought prominently before the
scientific world the theory of the equivalence of heat
and mechanical energy. The earlier estimates obtained
by Joule of the number of foot-pounds of work equivalent to the heat required to raise the temperature of a
pound of water one degree Fahrenheit were widely discordant, varying between 742 and o040; but as the
result of a later series of experiments he obtained 770
foot-pounds as the equivalent; and this is not very
different from the value now accepted. Joule did not
however doubt that the value of the equivalent exists
as a definite number, notwithstanding the considerable
14-2




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THE CONSERVATION OF


variation in his experimental determinations of it. His
certitude on the matter was derived from his conviction
of its a priori necessity.
We might reason a priori, (he writes)1 that such an absolute
destruction of living force cannot possibly take place, because
it is manifestly absurd to suppose that the powers with which
God has endowed matter can be destroyed any more than that
they can be created by man's agency; but we are not left with
this argument alone, decisive as it must be to every unprejudiced mind.
The work of J. R. Mayer on the conservation of
energy, which appeared in the year before the first
publication of Joule's experiments, was the earliest
publication on the subject in its modern form. In 1843
there also appeared a work on the same subject by a
Danish savant, A. Colding. The work of Mayer was
mainly guided by his philosophical ideas, and did not
include any experimental verifications of the principle,
such as those of Joule. He appeals to the old idea that
forces are causes, and that the cause is equal to the
effect. His determination of the equivalent of heat by a
calculation presupposes the existence of the constant
relation of equivalence, as was the case in the similar
calculation made by Sadi Carnot. Colding makes the
assumption that energy persists as a kind of indestructible, non-material substance. Thus he writes2:
Since forces are spiritual and immaterial beings, since they
are entities which are known to us only by their empire over
nature, these entities must doubtless be very superior to every
existing material thing; and as it is evident that it is by forces
alone that the wisdom that we perceive is expressed and that
we admire in nature, these powers must be in relation with the
spiritual, immaterial, and intellectual power itself which guides
the course of nature; but if this be so it is absolutely impossible
to conceive that these forces should be anything mortal or
perishable. Without doubt consequently they must be regarded
as absolutely imperishable.
1 Scientific Papers, Vol. I, p. 269.
2 Annales de Chimie et de Physique, (4), Vol. I (I864), p. 467.




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So far, the scope of the principle of the Conservation
of Energy has been confined to the mechanical domain
and to the equivalence of mechanical energy and heat,
but in the well-known treatise published by Helmholtz
in 1847, there is consistently developed the doctrine
that the conservation of energy is applicable to all
departments of Physics. This work gives ample evidence
that the author, like Joule, Mayer, and Colding, originally regarded the principle as one which follows from
the principle of causation.
In a very interesting passage in the introduction to
his treatise Helmholtz writes:
It is the object of these sciences (the physical sciences) to
seek for the laws by which the different processes in nature are
reduced to general rules, and from these rules can be re-determined. These rules, for example the laws of the refraction or
reflection of light, and that of Mariotte and Gay-Lussac for the
volume of gases, are clearly nothing but general notions by
which all the phenomena concerned are embraced. The search
for them is the business of the experimental part of our sciences.
Their theoretical part on the other hand attempts to find the
unknown causes of the processes from the visible effects; it
attempts to bring them under the law of causality. We are compelled and authorized to do this by the principle that every
change in nature must have a sufficient cause. The immediate
causes to which we attribute phenomena may be invariable or
variable; in the latter case the same principle compels us to seek
for other causes of this variability, and so on, until we have
arrived at ultimate causes which work according to an invariable
law, which consequently at every time with the same external
conditions produce the same effect. The final goal of the theoretical sciences is thus to search for the final invariable causes
of the processes in nature. Whether all processes can be reduced
to such (causes), that is, whether nature is completely comprehensible, or whether there are changes in her, which do not
obey the law of necessary causation, and which therefore fall
into the domain of spontaneity, or freedom, cannot here be
decided. It is certainly clear that science, whose aim it is to
comprehend nature, must start from the hypothesis that she is
comprehensible, and must examine and form conclusions in




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THE CONSERVATION OF


accordance witil this hypothesis until it is perhaps compelled
by irrefutable facts to acknowledge the limits of the hypothesis.
It will be observed that Helmholtz's opinions as to the
search for efficient causation in nature being the function of the theoretical parts of Natural Science are in
divergence with the view that I have maintained in these
lectures. It is however very interesting to notice that,
at a later time, Helmholtz declared that he had modified
the opinions expressed in the passage I have quoted. In
fact, in a note appended to a later edition of his work he
says:
The philosophical discussions in the introduction were more
strongly influenced by Kant's epistemological views than I at
the present time would recognize as correct. I have later made
clear to myself that the Principle of Causality is in fact nothing
else than the hypothesis that all natural phenomena are subject
to law.
It would appear from this statement that Helmholtz
emancipated himself from the idea that efficient
causation is to be found in nature by the aid of
science, and that he finally identified the term causality
with the recognition of invariability in sequences of
phenomena.
Leaving aside the supposed demonstration of the
principle of energy by means of the a priori principle of
causation, it is possible to deduce the principle from
the classical system of Mechanics, if the assumption be
made that all the phenomena of motion are governed by
central forces; in which case they form what is called a
conservative system. In accordance with this assumption the forces acting between every pair of corpuscles
of a system are along the line joining them, are equal in
magnitude and opposite in direction; and the magnitude
of such a force depends solely on the distance between
the particles. The whole energy of such a system consists then of two parts, the kinetic energy, or energy of
the motion of the system, and the potential energy, or




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215


energy of position, which represents the capacity of the
forces of the system to do mechanical work. That the
principle, in the form that the sum of the kinetic energy
and the potential energy of the system is constant for
such a system during its motion, follows as a mathematical consequence of the dynamical scheme was
demonstrated by Helmholtz in his treatise. But as
regards the application of the principle in general
physics, it should be remarked that it is exceedingly
doubtful whether, in the molecular or sub-molecular
domain, it is possible to restrict the forces to those of
the central type. For example, it does not seem possible
to regard the phenomena of permanent deformation and
of crystallization as involving such forces only.
We have already seen that the desire to form picturable images of mechanical phenomena resulted in a
reluctance to accept the notion of forces acting at a
distance as part of a mechanical scheme. The same
tendency has led to attempts to explain what is apparently
potential, or latent, energy as really reducible to the
kinetic energy of small parts of bodies, or of a medium,
and thus ultimately to abolish the distinction between
kinetic and potential energy; the latter being regarded
as less concrete or picturable than the former. But in
accordance with the view of the character of conceptual
schemes which has been adopted in these lectures, the
concept of potential energy as a measurable quantity is
really on the same footing as that of kinetic energy; and
consequently even if it still appears to be a desirable
simplification to reduce the two concepts to one, such
reduction has no longer the same urgency as with those
who feel bound to a more realistic interpretation of the
concepts of dynamical science.
The principle of energy has frequently been regarded
as a consequence of the principle of the impossibility of
perpetual motion. Very numerous attempts, many of
them very ingenious, have been made to construct




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THE CONSERVATION OF


machines which should actualize perpetual motion. The
conviction of the impossibility of perpetual motion
however became gradually very strong among men of
science. It was affirmed by Leonardo da Vinci, Galileo,
Stevinus, and by Leibniz, the last of whom employed
it to establish the principle of vis viva. In 1775, the
French Academy of Sciences directed that solutions
submitted to the Academy of the problems of the
duplication of the cube, the trisection of an angle, the
quadrature of the circle, and of the construction of
machines involving perpetual motion, should no longer
be examined. In the case of the last problem the
Academy based its decision upon a priori grounds stated
as follows:
The construction of perpetual motion is absolutely impossible:
even if friction and the resistance of the medium did not
ultimately destroy the resistance of the moving force, that force
can only produce an effect equal to its cause; ifthen it is desired
that the effect of a finite force shall continue indefinitely, it is
necessary that the force should be infinitely small in a finite
time. Making abstraction of the friction and the resistance, a
body on which motion has been once impressed will conserve
it always; but it would be by not acting on other bodies, and the
only perpetual motion possible on this hypothesis (which moreover cannot be realized in nature) would be absolutely useless
for the object at which the constructors of perpetual motion
aim.
Helmholtz showed that the principle of energy is
deducible from that of the impossibility of perpetual
motion. That impossibility he regarded however as a
fact of experience established by the numerous vain
attempts to construct a perpetuum mobile. It has however been pointed out by Poincare that it is only in the
case of reversible phenomena that the conservation of
energy follows from the impossibility of perpetual
motion. The general principle of the conservation of
the energy of an isolated system, in all its various forms,
through all the physical and chemical changes which




MATTER AND ENERGY


217


the system may undergo, can only be regarded as an
hypothetical principle to be used tentatively as a guide
in our attempts to describe conceptually the various
processes in the transformations. The fact that we have
no assurance that all the possible forms of energy which
may occur in physical phenomena are known to us
makes it impossible to conceive that the principle
should admit of anything like complete empirical
verification.
The history of science exhibits the discovery of
various forms of energy previously unrecognized. In
particular, our knowledge of the phenomena of electricity, except in merely trivial manifestations, dates
only from the investigations of Gilbert, three centuries
ago. The recent discoveries in the last few decades;
those of Hertzian waves, Roentgen rays, and of radioactive substances, have made us acquainted with forms
of energy whose existence had been previously unsuspected. Are we even now certain that we are acquainted
with all the forms of energy that may be discovered in
solar radiation, and which may be beyond the known
limits of the luminous, thermal, and actinic rays? The
discovery of the Roentgen rays is one illustration of the
fact that a form of energy may long remain undiscovered
when it is, so to speak, before our eyes; for Crookes'
tubes had been employed for a quarter of a century
before Roentgen's discovery of the rays to which they
give rise. We have seen that, in the case of a system
consisting of particles between which central forces act,
dependent only on their relative distances, the principle
of energy takes the simple form that the sum of the
kinetic energy and the potential energy is constant; the
former depending only on the velocities of the particles,
and the latter upon their positions, but not on their
velocities; so that the total energy can be resolved only
in one manner into the sum of the two components.
But if, as in the case of Weber's law of mutual action of




218


THE CONSERVATION OF


two electric molecules, the mutual action depends not
only on their distance but also on their velocities and
on their accelerations, the second part of the energy
would depend on the velocities, and it might contain
terms depending on the squares of the velocities. In
such a case we have no means of distinguishing between
terms which belong to the two parts respectively of the
total energy. Poincare has pointed out1 that we then
have no means of defining the energy of the system,
because, if the total energy of the system is constant,
so also is any function of that total energy; and such a
function might be substituted for the energy itself, and
made the basis of an amended definition of the energy
of the system. There exists in such a case no means of
fixing upon a precise definition of the energy, as such
that it may be divided into two parts, each of a specified
form. Moreover, if the principle of energy is to be of
any use, it is necessary to take account of the distinctions
between the mechanical energy of molar bodies and the
other forms of energy, such as heat, chemical, and
electrical energy. This can only be done if it is possible
to divide the whole energy of the system into parts
which are absolutely distinct in form; a part involving
only the squares of the velocities of the bodies, another
part independent of these velocities and of the thermal
and electric states of the system, and a third part independent of the velocities and the positions of the bodies,
and dependent only on their internal states. But the
case of electric energy, due to the mutual electric action
of the bodies, suffices to show the impossibility of this
division into such separate parts. For the electrostatic
energy depends, not only on the electric charges of the
bodies, but also upon their positions. If the bodies are
in motion, their electrodynamic energy depends, not
only on their states and positions, but also on their
velocities. We have therefore no obvious means of
1 Science and Hypothesis, 1905, p. 126.




MATTER AND ENERGY


2I9


selecting and separating out the different parts of the
total energy in the desired manner. The conclusion
drawn by Poincare from these considerations is that,
when an attempt is made to extend the principle of the
conservation of energy so as to embrace all the phenomena with which Physics has to deal, we are faced
with the difficulty of defining the energy of the system
in a unique manner, so that different parts of it may be
identified as referring to the different phenomena which
occur in the system. He remarks that, when this extreme
generality is aimed at, there appears to be nothing left
of the principle, except an enunciation: "there is something constant," and that, in this form, the principle
lies outside the bounds of experiment and is reduced to
a kind of tautology.
This criticism is pertinent in relation to the attempt
made by Ostwald and others to set up a science of
Energetics, based upon the Principle of Energy, and
that of Least Action (or some other similar principle),
with the view of avoiding the difficulties connected with
the hypothesis of the existence of atoms. The fundamental conception of Energetics is that every change in
an isolated system is regulated by two laws. The first is
that the sum of the kinetic and potential energies is
constant through all the transformations of the system.
The second is that, if the system passes from one configuration at one time to another configuration at another
time, the passage always takes place in such a manner
that the mean value of the difference of the two kinds of
energy in the interval of time between the two specified
times is a minimum.
The lessons to be drawn from the history of the varying conceptions that have arisen at various times, in
connection with the sustained efforts that have been
made to attain clear conceptions of what it is that is
conserved in matter and its various transformations, are
mainly those of the inadequate character of a priori




220        MATTER AND ENERGY
conceptions such as the principle of causation, and of
the partial character of the empirical verification of the
principles. It would appear that, when the utmost has
been attained as regards clearness of statement of these
principles, as conceptual laws, there remains an element
of doubt and uncertainty, and of tentativeness as regards the range of applicability, both in practice and in
theory, of these laws, in their function of describing the
actual changes and transformations in the perceptual
world.




x


MECHANICAL THEORIES AND
THERMODYNAMICS
THE aim of a Mechanical theory of a special class
of natural phenomena has been to represent the
changes, of which the phenomena consist, in terms of
mass and motion. Other concepts, those of force, work,
and energy, are also employed in most such theories;
and each one of these may be taken to be an independent
concept, or else as derivative, according to the special
form in which an abstract theory is stated, and also
according to the special class of phenomena with which
the theory deals. In the case of the changes of position
and motion of molar bodies, the Mechanical theory employed is founded upon the Classical Dynamics of
Galileo and Newton; and, as I have previously shown,
this theory may be stated in a form in which force, work,
and kinetic energy are derivative conceptions defined
in terms of the fundamental concepts of mass, space, and
time. In the more developed forms of this theory,
certain general principles have been deduced as consequences of the fundamental assumptions of the theory,
and one or other of these principles has been tentatively
assumed as the basis of more extended mechanical
theories, independently of the original assumption that
all the forces of a system of bodies are central forces. I
propose in the present lecture to give some account of
more general mechanical theories which have been set
up in this manner. Mechanical theories have been
devised to describe other physical phenomena, such as
the elastic deformations of solid bodies, the phenomena
of sound and vibrations, optical, thermal, and electro



222


MECHANICAL THEORIES


magnetic phenomena. For some purposes, not ponderable matter, but an imponderable ether of some special
type, has been assumed as the field of masses in motion.
In such cases the character of the relation between
ordinary matter and the assumed ether forces itself upon
the attention as urgently requiring elucidation, if ether
and matter are to be considered together in one mechanical scheme. But, as we shall see, it is possible to set up
an abstract mechanical theory without making a complete set of detailed assumptions as to the precise nature
of all the connections of the system. The relations
between ether and matter have been speculatively conceived in a variety of ways. For example, in Kelvin's
vortex atom theory, the basis of ordinary matter consists of vortex rings in an ether conceived as having the
properties of a perfect fluid. A view which was propounded by Riemann makes the smallest element of
matter a singular point in the ether at which a continuous annihilation of ether takes place. In Larmor's
rotational ether, an electron is a point of the ether at
which a special kind of singularity exists. In some
theories in which ether and matter are combined, the
Newtonian law of the equality of action and reaction is
not satisfied when the matter alone is taken into account,
so that the interaction of matter and ether forms an
essential part of the dynamical scheme.
In all mechanical or quasi-mechanical theories, the
masses, whether of ponderable matter, or of imponderable ethers, have usually been conceived in accordance
with the realism of common sense, not as pure concepts,
but as independently existent entities. As I have before
urged, however, the validity and the usefulness of these
theories are in no way invalidated if we refrain from all
realistic assumptions, and regard the elements employed
in the theories as concepts only. A strong underlying
motive which has usually dominated investigators who
have built up these theories has been the desire to give




AND THERMODYNAMICS


223


explanations which should make the stages of physical
processes accessible to the sensuous imagination. Any
elements in a theory which fail to satisfy this requisite
have been accepted with reluctance. I have already
referred to a striking instance of this in the unwillingness
which has been shown to regard the notion of action at
a distance as anything more than a provisional makeshift to be reduced, if possible, to the apparently more
plausible notion of action by contact, by means of some
articulated mechanism involving the propagation of
action through some medium connecting the bodies
whose interaction is to be explained. That action by contact
is itself in need of elucidation has, however, gradually
forced itself upon the minds of men of Science, though
apparently not in all cases upon the minds of scholars
for whom Greek Philosophy contains the quintessence of
all wisdom. The failure to attain to anything except an
indefinite regress, of attempts to reduce contact action
to a form which would satisfy the craving for efficient
causation, has been one of the factors which have led to
the removal of the category of efficient causation from
Natural Science. A great advantage of the modern view,
that a scientific theory is a purely conceptual scheme, is
that such a theory is emancipated from the somewhat
narrow limitations imposed by the necessity that its
form should be such that a sensuous representation of
phenomena is provided. Of this freedom, some modern
theories, especially those of Geometry, and the theory
known as the Einstein theory of relativity, have availed
themselves in a strikingly large measure. The latter
theory is not a mechanical theory, in the sense in which
I here employ the term, and I accordingly postpone
any discussion of it. In the Newtonian Dynamics, as
originally conceived, mass and force were taken as
fundamental and independent concepts; force being
regarded as a cause which produced, as its effect, change
in the amount of motion, defined as product of mass




224


MECHANICAL THEORIES


into velocity. I have, however, in an earlier lecture,
pointed out that this causal relation must be removed
from its position in the theory, in order that the theory
may be stated as a conceptual scheme which becomes
a deductive one when the requisite definitions and
postulations have been fixed in precise form. The
notion of force is then not independent of the concepts
of mass,time,and space, since force is used as a synonym
for the product of mass into acceleration, if indeed the
term continues to be employed. In fact, if the scheme
originally stated by Newton in his laws of motion, his
definitions and deductions, be taken in a revised form,
as the basis of a kinetic theory of the motions of molar
bodies, the existence of forces, and the existence of
accelerations, are not two facts of observation, but one
only. That the earth and the sun move towards one
another with accelerations in a definite ratio is the one
fact, as regards their relation to one another, which is
relevant to the dynamical theory. The supposed cause
of this, their so-called mutual attraction, is not an independent fact, but merely an assumption made in accordance with the supposed necessity of accounting for the
first fact as due to causation. Thus, in Newtonian
Dynamics, as a theory of the motion of such bodies as
those of the solar system, force is not an independent
concept; the theory operates with the three independent
concepts of time, space, and mass; motion being regarded as a transformation involving elements of space
and time.
There exists, however, the department of Statics
which is much more ancient than the Dynamics of
Galileo and Newton. In accordance with the Newtonian
Dynamics the weight of a body is the product of its
mass into the acceleration with which it falls to the
ground. But if the body be supported by a spring
balance, or in other manner, in accordance with the
principles of Statics the weight is regarded as a force




AND THERMODYNAMICS


225


still in existence, but balanced by another force due to
the support. If we refrain from formulating any theory
as to the state of the support, such as a kinetic theory of
its corpuscles, the notion of force, as a stress or pressure,
is requisite as an independent concept. Moreover, when
the small-scale phenomena which occur in the smallest
parts of bodies are taken into consideration, the notion
of force, as an independent concept, is indispensable,
in default of a complete kinetic theory of the corpuscles
or smallest parts of such bodies. Especially in the
theory of the elastic deformations of solid bodies, the
bodies are frequently treated as continuous distributions
of mass; and the conception of force, in the form of
systems of stresses between adjacent parts of a body, is
employed as a necessary element in the construction of
a theory of the strains or elastic displacements within
the body.
In order to comprehend the character of the mechanical schemes that have been tentatively applied to the
small-scale phenomena of Physics, it is necessary to
consider the later developments of Newtonian Dynamics,
and the mode in which some of these have been extended by generalization into schemes of Dynamics that
are free from some of the special restrictions involved
in the original treatment of the subject by Newton and
Galileo. In view of what I have said, the notions of
force and mass as independent conceptions, or else the
equivalent conceptions of work and mass, are in general
required in the basis of mechanical schemes. A conception which is usually regarded as requisite in the
more general formulation of Dynamics, as for example
in the system developed by Hertz, is that of inexorable
constraints in a system of particles or of gross bodies;
these constraints have the effect of diminishing the
number of possible independent modes of motion of
which a system is capable. They are usually represented,
on the perceptual side, by rigid connections, inextensible


H G L


15




226


MECHANICAL THEORIES


cords or rods, connecting different parts of a system.
A method of formulating Dynamics so that it may be
applicable to systems which include such inexorable constraints was introduced by d'Alembert. He regarded
the reversed mass-accelerations of all the elementary
parts of a system as in equilibrium with the forces acting
on the system, whether from without, or between parts
of the system. All the conditions which determine the
motions of the system are then included in one formulation, of a statical character; the condition, namely, that
no work is done by the equilibrating forces in any displacements consistent with the preservation of the
postulated set of connections of the parts of the system.
The deduction of d'Alembert's principle from Newton's
theory in its conceptual form is subject to considerable
logical difficulties. A method has been given by Boltzmann, in his lectures on Dynamics, by which these
difficulties may be overcome. He takes as the basis of
his treatment a finite set of masses concentrated at
points; between each pair of such points he considers
forces to act, of equal magnitude and in opposite
directions along the line joining them, so that each of
the points has an acceleration towards the other, inversely proportional to the assigned masses of the
points, in accordance with Newton's third law of motion.
Boltzmann then builds up the equations of motion of a
system of bodies, in which there may be inexorable
constraints, by treating each body as consisting of a
very large, but finite, number of mass-points. The
constraints are represented in the same manner by sets
of mass-points between which forces act that are functions of the distances between pairs of these mass-points,
of such a character that these functions have very great
magnitude whenever the standard distance between a
pair of the points is changed, either by excess or by
defect. External forces acting on a system he takes as
also due to external mass-points. In this manner Boltz



AND THERMODYNAMICS


227


mann deduces d'Alembert's principle, and in fact the
whole dynamical scheme for a system of bodies, in a
manner which avoids the logical difficulties of the way
in which that extension of Newtonian Dynamics has
usually been made. But the greatest advance in the
direction of setting up a unified scheme of dynamics, in
a form so general that it is applicable to a system in
which many of the details concerning the connections
of the system may be unknown, was made by Lagrange,
and published by him in 1788, in his great work, the
Mecanique Analytique.
In the history of Science it is possible to find many
cases in which the tendency of Mathematics to express
itself in the most abstract forms has proved to be of
ultimate service in connection with physical theories.
A striking example of this is to be found in Lagrange's
abstract formulation of Dynamics, as given in the
AMecanique Analytique. In order to characterize the
spirit in which this great work is conceived I cannot do
better than quote the words of Lagrange himself from
the Preface. He writes:
We have already various treatises on Mechanics, but the plan
of this one is entirely new. I intend to reduce this Science, and
the art of solving problems relating to it, to general formulae,
the simple development of which provides all the equations
necessary for the solution of each problem. I hope that the
manner in which I have tried to attain this object will leave
nothing to be desired. No diagrams will be found in this work.
The methods that I explain require neither geometrical, nor
mechanical, constructions or reasoning, but only algebraical
operations in accordance with regular and uniform procedure.
Those who love Analysis will see with pleasure that Mechanics
has become a branch of it, and will be grateful to me for having
thus extended its domain.
Lagrange's procedure was to express d'Alembert's variational equation in a form in which a certain number of
variables of the most general kind are employed. The
number of the variables is the number of independent
I5-2




228


MECHANICAL THEORIES


motions which are allowed to the system by the constraints or connections contained in it. This number is
what we now call the number of degrees of freedom of
the system, and the independent variables which specify
the configuration of the system are called the generalized
coordinates; the generalized force-component which
corresponds to each of the generalized coordinates is
defined as the coefficient of the variation of that generalized coordinate in the final expression for the virtual
work of the forces which act on the system or between
its parts. In the case in which the forces of the system
form what is known as a conservative system, that is
when they are the gradients of that single function of
the generalized coordinates which we call the potential
energy, Lagrange's equations of motion of the system
are such that only a knowledge of the forms of two
functions is required to make a determination of the
positions of the system possible when its configuration
and motion at one specified time are given. These two
functions are the kinetic energy of the system expressed
as a quadratic function of the generalized velocities,
that is of the gradients of the generalized coordinates
with respect to the time, and the potential energy,
which is a function of the generalized coordinates only.
It will be observed that a conservative system includes
as a particular case that in which all the forces are
functions of the distances only between pairs of particles
between which they act, as in the original Newtonian
scheme.
The Lagrangian equations of motion are equivalent
to the statement that the possible paths of a conservative
system are the extremals of the time-integral of the
single function which is expressed as the difference
between the kinetic energy and the potential energy of
the system. An important modification of the Lagrangian
dynamical scheme was made by Helmholtz, by E. J.
Routh, and by Thomson and Tait, in order to make a




AND THERMODYNAMICS


229


formulation suitable to the case in which the system
contains parts that are independently in rotation. By
elimination of the coordinates and velocities corresponding to these freely rotating parts of the system, their
effect is taken account of by a modification of the
Lagrangian function. In the modified form the velocities
of the rotating parts of the system give a contribution
to the potential energy of the system. This so-called
method of "ignoration of coordinates "-a technical term
which has given rise to some misunderstanding-by
indicating that part of the potential energy of a system
may be regarded as really dependent upon the kinetic
energy of motions that are concealed within the system,
has been sometimes regarded as a step in the direction
of reducing all potential energy to kinetic energy. To
those who attach paramount importance to the direct
correlation of all the conceptual elements of a scientific
scheme with sensuous intuition the ultimate reduction
of potential energy to kinetic energy has usually been
regarded as an ideal to be striven after. This aim is,
however, of subordinate importance for those who are
willing to accept as valid and satisfactory a scientific
conceptual scheme in which some of the concepts employed do not correspond directly with anything that is
accessible to sensuous intuition.
The analytical Mechanics founded by Lagrange was
extended and generalized in the two fundamentally important memoirs on the subject published by Sir
William Hamilton. In accordance with the Hamiltonian
scheme the whole of Dynamics is subsumed under what
are known as the principles of least, and of varying action.
In two alternative forms, a single function, either the
action, or the characteristic function, according to the
alternative adopted, has the property that the whole of
the possible motions of a system are disclosed from a
complete knowledge of the form of the function, by
means of the variation of an integral in which the




230


MECHANICAL THEORIES


function is the integrand. Like most such general principles, the principle of least action has a previous history:
it was originally formulated for simple cases of motion by
Maupertuis, without any adequate foundation. On
Maupertuis' discovery, Whewell1 writes:
Maupertuis conceived that he could establish a priori by
theological arguments that all mechanical changes must take
place in the world so as to occasion the least possible quantity
of action. In asserting this it was proposed to measure the
action by the product of velocity and space; and this measure
being adopted, the mathematicians, though they did not generally assent to Maupertuis' reasonings, found that his principle
expressed a remarkable and useful truth, which might be established on known mechanical grounds.
The Hamiltonian principle can only be deduced from
the principles of Dynamics, as formulated by Newton
and Galileo, by the employment of certain restrictions
on the nature of the forces and the connections in the
system. Although these restrictions are of a very general
character, they imply various restrictions upon the
nature of the motions which can be deduced from the
Hamiltonian principle. When the principles of Mechanics, as formulated by Lagrange, Hamilton, and
Jacobi, are taken as the basic principles of the Science,
it is unnecessary to assume a priori that their applications are restricted in the manner to which I have referred. The principles of energy, and of least and varying
action, may be accepted hypothetically for the purposes
of conceptual description of actual motions; the test of
the descriptive value of the principles, as in all such
cases, can only be that of experience. The fundamental
conceptions with which the scheme operates are those
of space, time, energy, and mass; the last of these appears
on the abstract side only in the form of coefficients in
the energy-function. In this scheme the concept of
force does not appear as an independent notion, but
1 History of the Inductive Sciences, Vol. II, p. 94.




AND THERMODYNAMICS                231
only as a derivative conception, that of a gradient of
potential energy. Although this has the advantage of
being free from the various difficulties connected with
the conception of force in the Dynamics of Newton and
Galileo, the identification and formal representation of
the various forms of energy that are required in connection with various physical phenomena constitute the
main difficulty in the employment of the general dynamical scheme in which the conception of energy is fundamental. The Hamiltonian principle, in either of its
equivalent forms, gives a complete account of the transformations of the energy of a system between its various
forms, by means of the employment of a single principle,
whenever we are in possession of the formal expressions
for the kinetic and potential energies of a particular
dynamical system. One great advantage of this general
dynamical scheme is that it affords the means of discovering the main features of the various motions that
occur in a system, without requiring the possession of
a complete knowledge of the details of the mechanism
of the system. In fact an indefinite variety of actual
mechanistic systems can always be imagined, for all of
which the forms of the kinetic and potential energyfunctions are identical; and the consequences of the
fundamental principle are applicable to the description
of the changes in all such systems.
Until about the middle of the last century most of
the theories which were set up for the description of
the various physical phenomena consisted of attempts
to reduce them to cases of forces acting at a distance
between material atoms. In fact the Newtonian system
of gravitating forces between the bodies of the solar
system formed the model and the inspiration of such
attempts. In the second half of the nineteenth century
the concentration of the attention of men of Science
upon the principle of the Conservation of Energy, owing
to the brilliant verification, by Joule, of that principle,




232


MECHANICAL THEORIES


in the case of transformation of mechanical work into
heat, led to the adoption of the transformations of
energy as fundamental in the newer physical theories.
In these newer theories, especially in Thermodynamics,
and in the hands of Maxwell, in Electromagnetics, the
generalized scheme of Dynamics associated with the
names of Lagrange and Hamilton found a wide field of
application. As regards the Hamiltonian principle, in
either of its forms, considered as an hypothetical scheme
for the description of physical processes, the chief
question which arises is as to its scope; that is, whether
it is capable of representing all the motions which take
place in an isolated physical system. There is one
important restriction of the principle which appears to
limit its applicability even in the case of the motion of
gross bodies. The connections of a system are expressed
by means of equations connecting the coordinates which
represent the positions of the bodies of the systems;
but in some cases these equations essentially involve
the gradients of these coordinates with respect to the
time, that is the velocities. In this latter case the
Hamiltonian principle, at least in its original form, is
not applicable, and if it be employed it may lead to
results which are not in accordance with the actual
motions which take place in such a system. Attention
has been drawn by Hertz, in the introduction to his
attempt to formulate anew the principles of Dynamics,
to a comparatively simple case in which the Hamiltonian
principle is in default. This is the case of a spherical
body rolling freely on a horizontal plane sufficiently
rough to prevent all sliding motion. If the initial and
final positions of the body are arbitrarily assigned, there
is always one mode of motion such that the Hamiltonian
integral is a minimum. But in point of fact there are
initial and final positions, such that, even with initial
velocities at our free choice, the body will not move
into its final position unless external forces are applied




AND THERMODYNAMICS                 233
to it to compel it to do so. Even if the initial and final
positions are so chosen that a natural motion from one
into the other is possible, this motion is not the one for
which the time of the motion is a minimum, as it should
be in accordance with the Hamiltonian principle. We
may attempt to explain this discrepancy by denying the
possibility of a motion of rolling, absolutely without
sliding, but there certainly exist natural motions in
which this condition is very approximately satisfied, and
we should consequently expect that the Hamiltonian
condition would give in such cases an approximation to
the actual motion, which appears not to be the case.
The general result is that, in some natural motions, the
connections of the system are of such a character that
the Hamiltonian principle is not applicable to the
description of these motions.
There is, however, another restriction on the scope
of the Hamiltonian Dynamics, at least in its original
form, which must be taken into account in attempting
to form an estimate of its range of applicability. The
kinetic energy of a system, expressed in terms of the
generalized velocities, is a homogeneous quadratic function of those velocities, which accordingly is unchanged
in value if the signs of all the velocities be reversed
without changing their magnitudes. It follows that the
motions in the system are all capable of being described
in the reverse order, without being otherwise changed;
in other words the motions described are all reversible
motions. There is, however, evidence of overwhelming
strength, which emerged originally in connection with
the theory of Heat, that some small-scale motions which
occur in nature must be regarded as irreversible, at all
events that we are unable by any means at our disposal
to realize such motions in the reverse direction. This
difficulty Helmholtz endeavoured to combat in his
investigations on cyclical systems. He showed that, by
elimination of the coordinates which represent certain




234


MECHANICAL THEORIES


concealed stationary motions in a system, the kinetic
energy is expressed in a form which involves not only
quadratic, but also linear, terms in the remaining
velocities of the system; and in that case the Lagrangian
equation would lead to motions which are irreversible,
in default of means for reversing the concealed motions
of small parts of the system. In this connection Helmholtz has pointed out the necessity of considering more
general forms, than the original one, of the expression
for what he terms the kinetic potential of a system; by
which is denoted in the original scheme the difference
between the potential and the kinetic energy. The
attempts made by Helmholtz and others to construct a
mechanics based on the conception of energy, and on
the hypotheses of the conservation of energy and the
principle of least action only, without having recourse
to atomic assumptions, which should be applicable to
thermal, electrical, and chemical phenomena, have been
only partially successful, especially as the difficulties
connected with the interpretation of the irreversibility
of many physical processes have not been overcome.
The most completely developed physical theory dependent upon the principle of the Conservation of
Energy, supplemented by another principle not immediately obtainable from Classical Mechanics, is the
theory of Heat, known as Thermodynamics. Before the
time of Rumford and Davy, heat was regarded, for the
most part, as an indestructible substance, caloric. It
was supposed that when caloric entered a body, the
effect of combination was in general an expansion of
the body; even when contraction was the result of the
combination, the analogy of certain chemical combinations, such as that of potassium and oxygen, could be
appealed to. The phenomenon of conduction of heat, as
transference of caloric, presented no difficulty. The
difference of specific heat of various substances was
explained by assuming that they required different




AND THERMODYNAMICS


235


amounts of caloric to be mixed with them in order to
produce equal changes of temperature. Black's theory
of the latent heat of water assumed that water differs
from ice at the same temperature in containing an admixture of a definite equivalent of caloric which was
represented by a molecular state of the body which does
not exhibit itself in the form of a rise of temperature.
Thus the theory of caloric provided a plausible explanation of the most prominent thermal phenomena, with,
however, the important exception of the production of
heat by friction or concussion, frequently recognized by
the adherents of the theory as not capable of satisfactory
explanation.
The theory that heat is representable as motion,
instead of as a substance, was established by the experiments on friction of which Count Rumford published
an account in the Phil. Trans. for 1798. He pointed out
that friction led to an inexhaustible supply of heat, and
that this is inconsistent with the theory of heat as a
substance, but consistent with the idea that heat consists
of motion. The theory of heat as motion was still more
lucidly developed by Davy, in a tract published in
I799, in which he gives an account of his experiment
in which two pieces of ice were rubbed together until
both were almost entirely melted. The general law of
the communication of heat was laid down by Davy in
his Chemical Philosophy, published in I812, in the
proposition that "The immediate cause of the phenomenon of heat, then, is motion, and the laws of its
communication are precisely the same as the laws of the
communication of motion."
The foundations of modern Thermodynamics were
first laid down by Sadi Carnot in his essay, Reflexions
sur la puissance motrice du feu, published in 1824. He
recognized the important fact that, in order to produce
work by heat, it is necessary to have two bodies at
different temperatures, and he pointed out the analogy




236


MECHANICAL THEORIES


of work done when there is a fall of temperature with
the case of work done by a fall of water from a higher
to a lower level. He introduced the notion of a cycle of
operations in which the initial and final states of a body
are identical as regards temperature, density, and molecular condition. Carnot's theory of the heat engine is
injuriously affected by the fact that he still held the
theory of heat as caloric, but he stated the important
result that, to obtain the maximum of work in a cycle,
that cycle must be reversible; which means that heat
must only pass from a body to another body at very
nearly the same temperature. He showed that the ratio
of the work done by a reversible engine to the heat
taken from the source is a function of the temperatures
of the source and condenser only; when this difference
is very small the ratio is that known as Carnot's function, which depends only on the temperature of the
source. In i848, this conception was made by Lord
Kelvin the basis of his absolute thermometric scale,
independent of the properties of any particular substance.
The development of Thermodynamics, based on the
rejection of the theory of caloric, was carried out by
Rankine, Clausius, and Kelvin, almost simultaneously.
Rankine based his investigations on the hypothesis that
the motion which represents the heat in a body consists
of molecular vortices or circulating streams. He supposes that the whirling matter is diffused in the form
of atmospheres round nuclei, and that radiation,
whether of light or heat, consists in the transmission of
a vibratory motion of the nuclei, by means of forces
which they exert on one another. By means of this
special hypothesis of molecular vortices he deduced
from general dynamical principles what is termed the
general equation of the mechanical action of heat. In
later works he introduced the function known as the
Thermodynamic function, and applied his principles of




AND THERMODYNAMICS


237


Thermodynamics to various practical questions relating
to the steam engine and other heat engines.
The theories of Clausius and Kelvin have the advantage of being independent of any special theory of
the character of the motion which exhibits itself as heat,
but instead of being deducible from the general principles of Dynamics they make use of a fact of observation
known as the second law of Thermodynamics, or the
principle of Clausius. The first law of Thermodynamics
is taken to be the principle of Conservation of Energy,
as applied to the equivalence of heat and mechanical
work, the amount of which was determined by Joule
and later experimenters. The principle of Clausius
asserts that, in a series of transformations in which the
final is identical with the initial state, it is impossible
for heat to pass from a colder to a warmer hody unless
some other accessory phenomenon occurs at the same
time. A more precise statement of the principle, called
by Clausius "the law of the equivalence of transformations," is to the effect that:
in all cases in which a quantity of heat is transformed into work,
and the bodies by means of which that transformation is effected
return at the end of the operation to their original condition;
another quantity of heat must at the same time pass from a hotter
to a colder body; and the proportion which this latter quantity
of heat bears to the former depends solely upon the temperatures
of the bodies between which it passes, and not upon the nature
of the intervening bodies.
A function called "entropy" was introduced by Clausius,
whose value is found by dividing the quantity of heat
expended in producing a given change in a given substance by the absolute temperature as measured by a
perfect gas thermometer. The conception of entropy is
a case of a fundamental concept, essential to the scheme
of Thermodynamics, which does not directly represent
anything accessible to sensuous perception. The second
law of Thermodynamics involves the postulation that




238


MECHANICAL THEORIES


the entropy of a thermally isolated system always tends
to increase. The change in entropy is quite distinct from
change in temperature, and from the change which
consists in loss or gain of heat. For example, in chemical
reactions, the entropy increases without any heat being
supplied to the substances. When a perfect gas expands
in a vacuum its entropy increases, and yet the temperature does not change, and the gas has neither
yielded nor received heat. A difficulty in the conception
of entropy is that it is not possible to define the equality
of entropy of two bodies chemically different, although
it is possible to compare the variations of entropy to
which the bodies are separately subject.
The principle that entropy tends continually to increase has been stated by Perrin in the form that "an
isolated system never passes twice through the same
state." It involves the postulation that the course of
physical phenomena is, so to speak, in a definite direction which is never reversed. This principle has been
generalized by Clausius and Kelvin, in a form in which
it is made to apply to the whole universe. It is said that
the entropy of the universe is continually increasing.
Thus, although the whole energy of the universe is
regarded as remaining constant through all transformations, it becomes increasingly unavailable, since the
energy is transformed gradually into heat uniformly
distributed at an everywhere identical temperature. The
final state of the universe would then be one in which
nothing would happen, because no energy would anywhere be available for the purposes of the chemical and
thermal transformations by which all change is conditioned.
This theory of the dissipation of energy is open to the
very serious criticism applicable to all statements made
about the physical universe as a whole. The extension
of a principle, asserted in the first instance to apply to
a finite isolated system, is made hypothetically to one




AND THERMODYNAMICS                239
which we are not warranted in regarding as finite. If
we consider ever larger portions of the universe, it may
be the case that the energy and the entropy of a portion
increase indefinitely as the portion is continually increased, and in that case the assertion ceases to have a
definite meaning. Besides, the range of validity of the
principle, even as applied to an isolated finite system,
has not really been ascertained. Kelvin himself expressly excluded living organisms in his statement that
" it is impossible, by means of inanimate material agency,
to derive mechanical effect from any portion of matter
by cooling it below the temperature of the coldest of the
surrounding objects." Moreover, the view is held by
those who have considered the matter from the point of
view of molecular or atomic theories that the principle
of increase of entropy is only a statistical principle, based
upon the laws of probability. The tendency of these
statistical methods, developed by Willard Gibbs and by
Boltzmann, is to regard the principle as pointing out
that a given system tends towards the configuration
presented by the maximum probability; and the entropy
of the system is expressible in terms of the numerical
value of this probability. This maximum probability
increases with the number of molecules concerned, but
does not reach absolute certainty.
Thermodynamics, in the hands of Willard Gibbs and
later investigators, has attained to very great success in
its application to the ascertainment of laws regulating
changes of chemical constitution or of physical state.
After Gibbs, Helmholtz introduced into the domain of
Chemistry the conception that energy can be divided
into two parts; the first, free energy, capable of undergoing all transformations and of producing external
action; the second, bound energy, only manifesting
itself by giving out heat. It is the variation of the free
energy, not that of the total energy, which is efficient
in determining chemical reactions. The utilization of




240


MECHANICAL THEORIES


the theory of Gibbs, expressed chiefly in what is known
as the Phase Law, was due to Van der Waals,Van t'Hoff,
and Roozeboom, in the discussion of complicated
chemical reactions.
On the whole it may be asserted that the principles
of Thermodynamics have proved a most valuable tool
for the coordination of a great number of physical and
chemical properties of matter; but the laws must still
be considered as hypotheses the precise range of
applicability of which is not known and has not yet
been definitely delimited.
Both in those theories which depend upon corpuscular hypotheses and in those in which no molecular or
atomic hypothesis is made, the conception of Energy
as a measurable quantity capable of continuous transformation has been employed as a fundamental element.
In any theory based upon the Classical Dynamics, even
when extended in accordance with the conceptions of
Hamilton and others, the transference of energy from
one form to another is regarded as essentially continuous
in amount. This is also the case when the fact of the
irreversibility, or apparent irreversibility, of certain
motions is taken into account and formulated abstractly
in a theory of entropy. Certain facts have, however,
recently emerged which throw very serious doubts upon
the adequacy of any theory in which continuous transformations of energy are admitted for the purpose of
representing certain classes of phenomena. A theory of
Quanta, in which portions of energy are transformed
by jumps, that is discontinuously, has arisen as the
result of an attempt to represent these facts. This theory,
associated with the name of Planck, arose in the first
instance from an investigation of the spectrum of blackbody radiation; and it also has bearings upon the theories
of the line-spectra of the elements and of the specific
heats of solid bodies. The result, up to the present time,
of the discussion of this matter has been to show that




AND THERMODYNAMICS


24I


there are exceedingly strong grounds for the assertion
that certain kinds of phenomena involving the transformation of energy are incapable of being described by any
conceptual scheme of the kind which we call Newtonian
Dynamics, even in a generalized form; but the subject
is still in the region of controversy. The theory of
radiation starts from the assumption of the (conceptual)
existence of an ether which must be regarded as continuous, or at least as very much more finely grained
than ordinary matter. On this assumption it can be
proved that, in accordance with the dynamical theory,
a state of thermal equilibrium between, say, a piece of
iron and the surrounding ether, can only be attained
when all, or nearly all, the energy of the motion of the
parts of the iron has been drawn from the body into the
surrounding ether. There is in fact a tendency for the
whole energy of moving systems immersed in a medium
of any type like that which must be assigned to the
ether to be transferred to the medium, and ultimately to
be found in the shortest vibrations which that medium
is capable of executing. It is now a matter for observation to determine whether this is what actually happens;
and the observed facts relating to thermal equilibrium
between a black-body and the surrounding medium
appear to give decisive evidence that what should
happen in accordance with the dynamical theory does
not actually occur. A law of thermal equilibrium
between a black-body and the surrounding medium
was obtained by Planck from thermodynamical considerations; and this law is inconsistent with the total,
or almost total, absorption of the energy by the ether.
It has been shown that Planck's law is in close agreement with the results of observation. A demonstration
has been given by Poincare, which has been widely
accepted as valid, that Planck's law of partition of
energy between the black-body and the surrounding
medium is not consistent with any scheme of continuous


HGL


I6




242


THERMODYNAMICS


transference of energy, but necessarily involves the
assumption that the energy is transferred discontinuously by jumps. This would also be the case if Planck's
law be taken to express only an approximation to the
actual law of partition. It would thus appear that the
motion of the medium must be governed by laws which
involve the quantum-theory; and this negatives the
possibility of describing such motion in accordance with
the Dynamics of Newton and Galileo, or with any extension of it in which the treatment of the transference
of energy by continuous amounts is fundamental.
The general result of these recent investigations is to
indicate that limits of the applicability of Newtonian and
Post-Newtonian Dynamics exist. The dynamical scheme
has been found sufficient, with certain reservations
indicated by the Einstein theory of relativity, for the
description of the large-scale phenomena of Physics;
but there never existed any cogent reason for assuming
that the scheme need necessarily prove adequate for the
description of all the small-scale phenomena.




XI


ELECTRICITY, MAGNETISM,
AND LIGHT
THE department of Physical Science with which the
terms Electricity and Magnetism are associated has
a history of which the interest is unsurpassed by that of
any other branch of Science. Until the sixteenth century
the knowledge both of electrical and of magnetic phenomena was exceedingly scanty, and it was not until the
nineteenth century, the birth-time of Electromagnetism,
that the two sets of phenomena were brought into relation with one another. The relative triviality of those
facts of observation in connection with which the terms
Electricity and Magnetism were employed which were
known before modern times gave no ground for any
expectation of the fundamental position which Electrical
Science now holds in relation to our theories of matter,
or for any anticipation of the fact that light itself would
one day be regarded as an electromagnetic phenomenon.
One of the greatest triumphs of the Science of the nineteenth century, directly on the theoretical side, and indirectly in relation to practical applications, consisted of
the breaking down of the barriers which previously appeared to exist between the three sets of phenomena,
those of Electricity, of Magnetism, and of Light. This
is a remarkable instance of the general law of development of Natural Science; the growing together into one
complex structure of what had previously been developed
as separate edifices; the process of gradual unification.
I propose to give an outline of the stages by which the
studies of Electricity and of Magnetism were developed,
at first separately, and later as one department, up to
I 6-2




244


ELECTRICITY, MAGNETISM,


the time when the great step was taken by Clerk Maxwell
of subsuming the phenomena of Light under an electromagnetic scheme. Of some of the later aspects of Electrical Science, in relation to theories of the constitution
of matter, I shall speak in the next lecture.
The history of these subjects contains an account of
the rise and fall of a multiplicity of theories; of these
the earlier consisted mainly of crude conceptions in
which effluvia, fluids, and corpuscles played their part.
These theories were designed to appeal to familiar images
of the sensuous imagination. Later on, the same rivalries
are to be found, as we have already observed in other
branches of Physical Science, between the notions of
action at a distance, action by contact, and propagation
through media. Such theories, imperfect, contradicting
one another, and sometimes even self-contradictory,
served in various degrees the all-important functions of
fixing the direction of future observation of facts, and
of predicting results by which the theories could be
verified or refuted. This subjection of rival theories to
the law of the survival of the fittest is an essential element
in the life of Natural Science.
In ancient times the sole knowledge of Electricity and
Magnetism was confined to the facts that amber (r/XEK-pov), when rubbed, has the power of attracting light
substances, and that a certain iron ore (Xioos /ayr)7rtL)
has the power of attracting small pieces of iron. The use
of the magnetic needle for indicating directions at sea
was known at the time of the crusades, and it appears
also to have been known in China at an early period.
One discovery of importance was made in the thirteenth
century by Petrus Peregrinus; that of the polarity of the
natural magnet, or loadstone. Experimenting with a
loadstone of globular shape, Peregrinus showed, by
placing needles in contact with it, in various positions
all over the surface, that the positions of the needles all
pointed towards two points at opposite ends of the




AND) LIGHT


245


stone; and these points he proposed to call the poles.
He then observed that the way in which magnets set
themselves, and attract each other, depends only on
the position of the poles, as if the magnetic power were
concentrated in these points. No corresponding increase of knowledge of electric phenomena was obtained
until the sixteenth century; these were known only as a
few bare facts connected with amber and one or two
other substances. The first considerable increase of knowledge, both of electric and magnetic phenomena, was due
to the researches of William Gilbert (I540-I603), of
Colchester. He made the highly important discovery
that the orientation of magnets may be accounted for
by regarding the earth itself as a great magnet, with its
poles in high northern and southern latitudes, in accordance with the general principle that the northseeking pole of every magnet attracts the south-seeking
pole of every other magnet, and repels its north-seeking
pole. In Electricity, Gilbert discovered that a whole
class of bodies, such as glass, sulphur, sealing-wax, etc.,
have the same property as amber, that of being electrified
by friction. The contrast in various respects between
magnetic and electric forces which Gilbert observed,
especially in respect of screening and polarity, led him
to set up the theory that electrical phenomena are due to
an atmosphere of effluvia round an electrified body; the
effluvia consisting of material liberated from the body
by the process of friction. As he regarded action at a
distance as inconceivable, the effluvia provided for the
necessary contact with the attracted bodies, in analogy
with his view of the phenomena of falling bodies as due
to the atmosphere acting as an effluvium by means of
which the earth draws all bodies towards itself. This
theory was generally accepted by those Natural Philosophers of the period who were interested in the subject,
but there was some difference of opinion amongst them
as to the manner in which the effluvia acted on the




246


ELECTRICITY, MAGNETISM,


neighbouring small bodies. Gilbert himself imagined
that the effluvia had an inherent tendency to return to
the body from which they emanated. Moreover the fact
that electrified bodies exercise repulsion as well as attraction did not remain unobserved. Notwithstanding
the rise of the Newtonian theory of gravitation, which
shattered that emanation theory of falling bodies which
had suggested the emanation theory of electricity, the
latter theory remained unaffected into the eighteenth
century.
The first attempt to set up a theory of Magnetism was
due to Descartes, who connected it with his general
theory of vortices. He postulated the existence of a
vortex of fluid matter round the magnet; this fluid entering the magnet at one pole and leaving it at the other.
The fluid matter was regarded as acting upon iron on
account of resistance offered by the particles of iron to
the motion of the fluid. Even in the eighteenth century,
Euler and two of the brothers Bernoulli worked out
theories of magnetism based upon the hypothesis of
vortices.
A very important discovery, that of conduction of
Electricity, was published in 1729 by Stephen Gray, a
pensioner of the Charterhouse, who showed, as he
describes in his own words:
that the Electrick Vertue of a Glass Tube may be conveyed to
other Bodies so as to give them the same Property of attracting
and repelling Light Bodies as the Tube does, when excited by
rubbing; and that this attractive Vertue might be carried to
Bodies that were many Feet distant frem the Tube.
No other mode of electrification than that by friction had
been previously known. It was found that this property
of conduction belongs only to a certain class of bodies,
especially to metals. To these bodies Desaguliers, in
1736, gave the name of non-electrics or conductors. The
view that electric effluvia are inseparably connected with
the body that has been rubbed became untenable in




AND LIGHT


247


consequence of the discovery of conduction; and thus
the notion of an electric fluid of an imponderable
character was substituted for the effluvium; and this
fluid came to be regarded as one of the chemical elements, although it was thought by some Physicists to
be closely connected with caloric, the substance of which
heat was then supposed to consist. This last idea was
refuted by an experiment of Stephen Gray, who showed
that the electrification of two similar bodies, one of which
was solid and the other hollow, produced exactly similar
effects. From this it was concluded that only the surface
of a body has to do with its electrification, whereas
caloric is diffused through the whole substance of the
body. The next important discovery, made by Charles
Francois du Fay (I698-I739), was that there are two
species of electrification. He showed that an electrified
body repels another one that has been electrified in the
same way as itself, but that two bodies whose electricities
are of different species attract one another. To the two
kinds of electricity the existence of which his experiments
proved, he gave the names "vitreous" and "resinous,"
by which they are still known. In I745, Pieter van
Musschenbroek (I692-I76I), a Professor at Leyden, as
the result of an attempt to prevent the electric charge
of a body from undergoing the decay which had been
observed to take place when the charged body is surrounded by air, was led to the discovery of the Leyden
phial, or jar, as a means of accumulating the effect of
electrification; at the same time the physiological effect
of the discharge of the phial through the human body
was discovered. Soon after this discovery, a London
apothecary, William Watson, in the course of experiments with the Leyden jar and its discharge, was led to
propound, in 1746, the theory that the phenomena are
due to the presence of an "electrical aether" which is
transferred, but not created or destroyed, during the
process of charging and discharging the Leyden jar. In




248


ELECTRICITY, MAGNETISM,


accordance with his theory the electrification of a body
is due to its receiving, at the expense of some other
body, an excess of electric fluid over the normal amount
that belongs to the body; the fluid in the other body
being correspondingly depleted. The two species of
electrification correspond, on this theory, respectively to
an excess or a defect of the electric fluid belonging to
the body that is electrified.
The same theory was independently proposed a few
months later by Benjamin Franklin (I706-I790), of
Philadelphia, as the result of various experiments. The
principle of Watson and Franklin may be stated as that
of the conservation of electric charge, indicating that,
in any isolated system, the total quantity of electric fluid
is invariable. In view of this principle, Franklin was
led to regard electrification as being either positive or
negative, according as it denoted an excess or defect of
electric fluid in the electrified body; he attributed the
positive sign to vitreous, and the negative sign to resinous,
electricity. In considering the theory of the Leyden jar,
Franklin was led to assume that the glass in the jar is
impenetrable to the electric fluid, although the attractive
effect between one electrified body and another body is
not destroyed by interposing a glass plate between them.
He was thus led to the idea that the surface of the glass
nearest the electrified body is able to influence its other
surface through the glass, and that this effect then accounts for the influence on the other body. In the case
of the jar the excess of fluid on the inner face exercises
through the glass a repulsion, which causes a defect of
fluid on the outer face. This interpretation of fact was in
accordance with a theory, the one-fluid theory of electricity, which, unlike the older conception of effluvia,
involved the notion of action at a distance. The view
that glass is impermeable to the electric fluid was extended by Aepinus (I724-1802) so as to apply to all
non-conductors. In order to account for the repulsion




AND LIGHT


249


between two resinously charged bodies, Aepinus, who
also held the one-fluid theory, set up the hypothesis
that the particles of ordinary matter repel each other,
but that, when bodies are unelectrified, this repulsive
force is balanced by the attraction which he, like Franklin,
assumed to exist between matter and electric fluid. He
suggested that gravitation might be due to a slight inequality between these attractive and repulsive forces.
He applied his theory to the explanation of the induction
of electric charges, a phenomenon which had been previously observed by others, and had been studied a little
earlier by John Canton (1718-1772) and by Wilcke.
I have spoken of the facts known at an early period
relating to electricity and magnetism as trivial. There is,
however, one phenomenon that cannot be so described,
although it fortunately occurs only occasionally. I mean
the phenomenon of lightning in thunderstorms. The
important discovery that lightning is an electric phenomenon, akin to the spark observed when a Leyden jar
is discharged, is due to Benjamin Franklin. Experimenting during thunderstorms with kites, he was able
to establish the electric character of the occurrence by
charging a Leyden jar by means of electricity conducted
from the kite.
The discovery of the law of force between two electric
charges was made by Joseph Priestley (I733-I804), the
discoverer of Oxygen. He showed experimentally that,
when a hollow metal vessel is electrified, there is no
charge on the inner surface, and no electric force in
the interior; and he inferred from this fact that the law
of force is that of the inverse square of the distance, the
same as that of gravitation. This discovery of Priestley,
taken in conjunction with Franklin's law of the conservation of electric charge, brought the phenomena of
Electrostatics for the first time to a completeness of
description sufficient to render them accessible to Mathematical calculation. The first to take advantage of the




250


ELECTRICITY, MAGNETISM,


possibility of applying calculation to electrical phenomena was the Hon. Henry Cavendish (I73I-I8I0). In
a memoir presented in I771 to the Royal Society he
adopted the one-fluid theory of Aepinus and Franklin,
which he had however discovered independently. In
this memoir he assumed the law of electric force between charges to be inversely as some less power of the
distance than the cube, and virtually introduced, under
the term intensity of electrification, the notion of the
potential, which later became fundamental in electrical
theory, although its use was hindered by the fact that
he did not definitely assume the law of force to be that
of the inverse square. Unfortunately, Cavendish's researches remained for the most part unknown until i879,
when they were published at the instance of Lord Kelvin
who had examined the manuscripts. It then appeared
that Cavendish had not only rediscovered the law of
the inverse square but had even determined the correct
value for the ratio of the electric charges carried by a
circular disc and a sphere of the same radius in metallic
connection with one another; thus he introduced the
conception of electrostatic capacity. Moreover, he
anticipated the later discovery by Faraday of specific
inductive capacity, and investigated experimentally the
conducting powers of different materials for electrostatic discharges.
Whilst the progress I have described of knowledge of
electric phenomena was made, the subject of Magnetism
had not been neglected. The law of force between magnetic poles was discovered by John Michell (I724-1793),
a Fellow of Queens' College, Cambridge, who published
his researches in 1750. It had been previously believed
that the attraction between opposite poles followed a
different law from that of the repulsion between like
poles. A theory of magnetic fluid similar to that of the
one-fluid theory of electricity was propounded in I759
by Aepinus, who regarded the two poles of a magnet as




AND LIGHT


251


places where the magnetic fluid existed in excess and
in defect of the normal amount respectively. He supposed that the particles of the fluid repel each other, but
attract particles of iron and steel; moreover he saw that
it was necessary to assume that the material particles of
the magnet repel each other. By later investigators a
two-fluid theory of magnetism was employed; these fluids
were taken to have properties of attraction and repulsion
similar to those of vitreous and resinous electricity.
Exact measurements, both electric and magnetic, were
made by Charles Augustin Coulomb (I736-I806), who
employed for this purpose the torsion balance. By this
means he verified decisively the law that the force
between two small globes charged electrically is inversely
as the square of the distance between their centres, and
is repulsive or attractive, according as the electricity is of
the same or of the opposite kinds. Instead of the onefluid theory which had been accepted as the basis of the
explanation of electrostatic phenomena by Franklin,
Aepinus, and Cavendish, Coulomb postulated the existence of two fluids, corresponding respectively to the two
kinds of electrification. He supposed that, in an uncharged body, these fluids were both present in equal
amounts, and thus neutralized each other; but when in
an electric field, a decomposition of the neutral fluid takes
place into equal amounts of the separated fluids, which
can then be separately located. The controversy which
took place between the upholders of the rival one-fluid
and two-fluid theories was in reality one between the
merits of two conceptual descriptions of the phenomena,
and no experimental evidence was forthcomingwhich was
capable of deciding between them in respect of their power
of representing the facts of observation. Coulomb also
verified the law of force of magnetic poles on one another
by means of the torsion balance. He endeavoured to
explain the fact that the two magnetic fluids, unlike the
two electric fluids, cannot be divided between different




252


ELECTRICITY, MAGNETISM,


substances, so as to obtain a magnetic pole in isolation. He propounded the view that the magnetic fluids
cannot move from one molecule of the magnetic body
to another, so that each molecule always contains equal
amounts of the two fluids, which are separated within
the molecule when the substance is magnetized, giving
rise to two poles in each molecule.
At the end of the eighteenth century the theory of
effluvia in Electricity and that of vortices in Magnetism
had been eliminated, and had in each case been replaced
by a theory which involved the postulation of the
existence of either one or two fluids, and in which the
notion of attractive and repulsive forces acting at a
distance was a fundamental element. The Sciences had
now arrived at a point which made them accessible to
Mathematical Analysis. A complete mathematical development of Electrostatics on the basis of the two-fluid
theory was published in I812 by Simeon Denis Poisson
(I78I-I840). He showed that, on the basis of the theory
of attractions and repulsions between particles of the
fluids which were supposed capable of moving freely in
conductors, there is no electric force in the interior of
the conductor; and that a charge of such a body, which
consists of an excess of one kind of fluid over the other,
distributes itself over the surface of the conductor as a
layer, of which the thickness at every point dependsupon
the shape of the surface. He showed that, in simple cases,
it is possible to determine the distribution of electricity
over the surface of the conductor; and for these purposes he transferred to electric theory various results
which had been obtained in the theory of gravitational
attraction, in which the same law of force is involved
as in the case of electricity. Of special importance was
the introduction into electrostatics of the potential function which had been previously introduced into gravitational theory by Lagrange,who showed that an attractive
force at a point can be expressed as the gradient of this




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function. In all later developments of electricity, and
in their practical applications, this notion of potential,
or of potential difference, has proved to be of fundamental importance; upon it depends the whole theory
of the distribution of electrical charges upon conductors,
and it was seen later that this is but a small part of the
function which this conception fulfils in electrical theory.
In 1824, Poisson published a corresponding complete
mathematical theory of Magnetism, of which the notion
of the potential is the basis. He showed that the effect
of a magnetized body can be represented by fictitious
surface- and volume-distributions of magnetic matter.
At the same time, he gave a theory of the temporary
magnetization induced in a body made of soft iron by
the approach of a permanent magnet. Very important
developments and extensions of the mathematical theory
of electricity and of magnetism were published in 1828 by
George Green (I793-184I), to whom the term potential
is due, which has ever since been employed to designate
the function introduced into the theory by Lagrange and
Poisson. The celebrated theorem known as Green's
theorem was established in this memoir; of this theorem
Poisson's resolution of the effect of magnetization of a
body into the sum of parts due to surface- and volumedistributions is simply a particular case.
The theories of Electrostatics and of Magnetism had
now been so far developed that they had been subsumed
under schemes of conceptual description which represented the ascertained facts, and which were sufficient
for mathematical calculation of the details of the phenomena, in cases which were sufficiently simple, and for
the ascertainment in the form of mathematical theorems
of a variety of the more general aspects of those phenomena. The stage of development of Electrical Science
in general which comes next for consideration arose from
the discovery, in the latter part of the eighteenth century,
of a set of phenomena belonging to a quite new class.




254


ELECTRICITY, MAGNETISM,


Luigi Galvani, Professor of Anatomy at Bologna, by
an accidental observation made in 1780 whilst dissecting a frog, was led to the discovery that, if the
nerves and the muscles of the frog are connected by a
metallic arc formed of more than one kind of metal, the
limbs of the frog became violently convulsed. He was
led to the conclusion that this was due to a flow of
electric fluid, and he considered the phenomenon as
essentially similar to what happens when a Leyden jar
is discharged. This view did not receive universal acceptance; some physicists thinking that this so-called
galvanism, or animal electricity, was a fluid different
from the electric fluid which was regarded as functioning
in electrostatic phenomena. In 1792, the opinion was
maintained by Alessandro Volta (I745-I827), Professor
at Pavia, that the essential element in Galvani's experiment was the connection of two different metals by a
moist body, and that the supposed animal electricity,
due to the nervous system of the frog, had nothing to do
with the phenomenon. Shortly afterwards, Fabroni, of
Florence, placed two plates of different metals in water
and observed that, when they were put in contact, one
of them became partially oxidized; from this he concluded that some chemical action is connected with the
galvanic phenomenon. In 800o, Volta showed that the
galvanic effect could be made much greater by constructing a pile in which a number of pairs of zinc and
copper discs were used, each pair being separated from
the next by a disc of moist pasteboard. This pile is the
parent of the battery employed in electric telegraphy.
A distinct shock could be felt when the highest and the
lowest discs were simultaneously touched by the fingers;
and it appeared that this shock could be repeated any
number of times. As the result of this and further experiments, Volta set up his electrical theory of the pile,
as due to the contact of each copper disc with a zinc
disc, the pasteboard discs acting merely as conductors.




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He recognised the existence of a continuous electric
current so long as the circuit is completed by joining
the highest and lowest discs. When Volta's discovery
had been communicated in I8oo to Sir Joseph Banks,
President of the Royal Society, Volta's experiment was
repeated by Nicholson and Carlisle who, having placed
a drop of water on the upper plate of the pile, in order
to make the electric contact of the highest and lowest
discs more efficient, observed that round the wire there
was at the drop of water a disengagement of gas. They
then introduced a tube of water, into which the wires
from the two terminals of the pile were immersed,
and observed that an inflammable gas, hydrogen, was
liberated at one wire, whilst the other became oxidized.
This observation of the effect of the decomposition
of the water constituted the great discovery of electrolysis, which was soon extended to the decomposition
of various metallic salts in solution. It was shown by
Woolaston that water could be decomposed by the discharge of frictional electricity, thus identifying the currents of the electricity of Volta's currents with those of
frictional electricity. There were two views as to the
mode in which the current in the pile is produced, the
so-called contact theory, that it is due to molecular action
between the two different metals in contact with one
another, and the chemical theory, that it is due to
chemical action involving oxidation of the zinc in the
pile. The chemical theory was supported by Humphry
Davy (I778-I829), Professor of Chemistry at the Royal
Institution. He showed that there is no current when
the water between the pair of plates is quite pure, and
that their power of action is in great measure proportional
to the power of the conducting fluid between the plates
to oxidize the zinc. Davy afterwards proposed a theory
of the voltaic pile in which the contact and chemical
actions were both recognized as contributing to the
effect; the contact of the metals disturbing equilibrium




256


ELECTRICITY, MAGNETISM,


whilst the chemical changes in the liquid constantly
tend to restore the conditions under which the contact
energy is exerted. He was led to make the assertion
that chemical affinity is essentially electrical in its nature.
A comprehensive chemical theory of the electric current
and of chemical combination was advanced by the
Swedish chemist Berzelius (1779-1848), dependent on
the hypothesis of the existence of electric charges
within the atoms of matter. This was an anticipation
in some respects of conceptions which have become of
fundamental importance in recent decades, although the
detailed theory of Berzelius did not survive him. He
was inclined to regard both electricity and caloric as
substances devoid of gravitation, but possessing affinity
to gravitating substances.
So far as I have at present proceeded in the account
of the gradual increase of knowledge of electrical and
magnetic phenomena no connection between the two
sets of phenomena had been exhibited. A discovery made
in I820 by the Danish physicist, Hans Christian Oersted
(1777-1851), produced in its ultimate implications a revolutionary effect upon the whole future of the Sciences
of Electricity and Magnetism, exhibiting as it did the
closest connection between the two sets of phenomena.
It had been for some time suspected that an electric
discharge has an effect upon the magnetic needle, but
Oersted was the first to demonstrate its existence by his
observation that a magnetic needle, when placed in the
neighbourhood of a continuous electric current in a
straight wire parallel to the needle, tends to set itself
at right angles to the wire. It thus appeared that the
phenomenon of an electric current could not be localized
entirely in the conducting wire, but had an effect which
spread itself through the surrounding space, and produced
an alteration in the orientation of the magnetic needle.
Oersted's discovery was described at a meeting of the
French Academy shortly after it was made, and this led




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257


to further investigation by physicists. Two of these,
Biot and Savart, shortly afterwards announced their
discovery of the law of force of the straight current upon
the magnetic pole; that the force on a pole is at right
angles to the plane through the wire and the pole, and
its intensity is inversely proportional to the distance of
the pole from the wire. It was shown by Arago that a
magnetic field produced by an electric current may be
employed in the same manner as one produced by a
magnet, to induce magnetization in iron, and thus that
an electric current is a magnet.
Almost immediately after Oersted's discovery had become known, Andre Marie Ampere (1775-I836) showed
that two parallel wires carrying currents attract each
other when the currents are in the same direction, and
repel each other when the currents are in opposite directions. Ampere set himself the work of developing a
complete theory of the pondero-motive forces which
act between circuits carrying electric currents, on the
basis of the conception of forces acting at a distance
between pairs of elements of the two currents. He showed
in particular that an electric circuit is equivalent in its
magnetic effects to what is called a magnetic shell, that
is a distribution of elementary magnets on a surface
bounded by the circuit, with the axes of the magnets
normal to that surface. He regarded magnetism as essentially an electrical phenomenon; each magnetic molecule being looked upon as having a small closed circuit
within it, in which a permanent electric current flows.
Ampere succeeded in developing upon this basis a complete mathematical theory of the mechanical interaction of
circuits carrying electric currents, that is, of electromagnets. Of this theory, Maxwell wrote1 several decades later:
The experimental investigation by which Ampere established
the laws of the mechanical action between electric currents is one
of the most brilliant achievements in Science. The whole, theory
1 Electricity and Magnetism, 3rd ed., Vol. II, p. 175.


HGL


17




258


ELECTRICITY, MAGNETISM,


and experiment, seems as if it had leaped, full-grown and full
armed, from the brain of the 'Newton of Electricity.' It is
perfect in form, and unassailable in accuracy, and it is summed
up in a formula from which all the phenomena may be deduced,
and which must always remain the cardinal formula of electrodynamics.
The theory of Ampere is, like the similar theories
developed later by Grassmann, Stefan, and Korteweg,
based upon the conception of action at a distance; no
account being taken of the medium between the electric
circuits. The essential assumption in these theories is
that the whole effect of one circuit on another can be
represented as the resultant effect of forces acting between pairs of elements of the two circuits. These
theories differ from one another in respect of the nature
of the forces which they assume to act between pairs
of elements, but they all agree in representing the actual
forces between complete circuits, which alone can be
submitted to the direct test of observation. They make
no explicit use of the principle of the conservation of
energy, and indeed that principle would imply the existence of couples between pairs of elements, which are not
taken into account in Ampere's theory. Ampere's theory
cannot accordingly be regarded as a dynamical theory,
although it, like the other theories of the same type,
affords a sufficient representation of the interaction of
complete circuits. Theories of another type, which also
take no account of the medium between the circuits,
were developed later by Gauss, W. E. Weber, Riemann,
and Clausius. These represent the action of currents by
assuming that the forces acting at a distance between
electrified bodies depend upon the velocities and accelerations of those bodies. In all these theories, except
that of Clausius, Fechner's hypothesis is adopted, that
an electric current consists of a flow of positive electricity in one direction and a flow of negative electricity
with the same velocity, and of equal amount, in the




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259


opposite direction. They differ from one another in
respect of the law of force which they assume to exist
between pairs of moving elements of electricity. The
theories of Weber and Clausius, unlike that of Gauss,
are consistent with the principle of the conservation of
energy, and consequently suffice to represent the induction of currents, as well as the mechanical forcive
between the circuits.
Another theory of electrodynamical forces was developed by F. E. Neumann, and extended by Helmholtz.
This was a dynamical theory based upon the assumption
of a law of the mutual energy between elements of currents but, like those I have hitherto mentioned, it took
no account of the dielectric medium. The power possessed by different metals to conduct electric currents
was investigated by Humphry Davy, but a complete
theory of this conduction was formulated by George
Simon Ohm (I787-I854), based upon a large amount
of experimental work. Ohm compared the flow of
electricity in a current with the flow of heat in a wire,
of which the theory had been given in a complete form
by Fourier in his Analytical Theory of Heat. He introduced the notion of electroscopic force as a conception
which plays a part analogous to that of temperature in
the conduction of heat. Tension, or difference of electroscopic force at two places in a conductor, he regarded
as effective in producing a current between those places,
just as conduction of heat is dependent on difference
of temperature. Each voltaic cell he regarded as possessing a definite tension, which is a contribution to the
driving force of a current in any circuit in which it is
placed. He did not, however, relate differences of electroscopic force with differences of potential (or as we now
say with electromotive force) as that conception appears
in Poisson's theory of electrostatics. Notwithstanding
this defect, the publication of what has since been known
as Ohm's law constituted a considerable advance in
17-2




260


ELECTRICITY, MAGNETISM,


knowledge of the conduction of electric currents, and
much of the later development of the subject up to the
middle of the nineteenth century was dependent upon it.
A complete transformation of the whole manner in
which the phenomena of Electricity and Magnetism are
conceived was the ultimate result of the researches of
Michael Faraday, whose genius as an experimental investigator, in accordance with the inductive method, has
certainly never been surpassed, and perhaps never been
equalled. Only a study of his great work, the ExperimentalResearches, can lead to ajust appreciation of the profound insight which led him to the discovery of a multitude of facts, in an orderly succession, under the guidance
of novel conceptions radically different from those which
had guided previous investigators. Upon his discoveries
rest ultimately not only the modern theory of Electromagnetic Science, but also, in its practical applications,
the Science of electrical engineering as we know it. The
first question which Faraday set himself was to discover
whether an electric current in one circuit can induce
a current in another circuit, in analogy with the known
fact of Electrostatics that a charged conductor induces
a charge in neighbouring conductors. In 1831 he published a memoir in which he gave an account of the
answer he had obtained to this question. He found that
such a current was in fact induced, but that it lasted only
for an instant, when the primary current was started or
stopped; no induced current existing whilst the primary
current flowed steadily. With a view to a formulation
of the laws of the induction of currents he took the step
of concentrating his attention on the dielectric, or nonconducting medium which surrounded the circuits, thus
initiating the breach with the older conception that the
phenomena are localized in the conductors, in accordance
with the notion of action at a distance; with Faraday
the phenomena are localized mainly in the dielectric. He
constructed for his guidance the conception of lines of




AND LIGHT


26I


force, which he conceived to fill the space in the neighbourhood of magnets or electromagnets; the direction
of each such line at any point being that of the magnetic
force which would act upon a magnetic pole at that
point. Every such line of force he regarded as a closed
curve which at some part of its length passed through
the magnet or electromagnet with which it was associated. These lines of force he conceived to form unit
tubes of force such that, for any one tube, the product
of its cross-section into the magnetic force is constant
along its whole length. In this manner he formed an
intuitional geometrical representation of the phenomena
of magnetism which assisted greatly in directing his investigations. On the basis of his experiments he obtained
a formulation of the law of induction of currents in
circuits-that the electromotive force induced in a circuit
is measured by the rate of change of the number of unit
tubes which pass through the circuit. The full import of
this important result was only understood later, when
the theory of electromagnetic induction was formulated
mathematically by Maxwell and others A few years later
this fruitful discovery was followed by that of self-induction, in which the effect of the electromotive force
in an electric circuit due to a change in the magnitude
of a current through that same circuit was established.
Faraday also completed the identification of currents, as
exhibited by an electrostatic discharge, with those due
to voltaic cells, by showing that the magnetic, calorific,
and other effects are the same in the two cases. A very
important set of Faraday's investigations were concerned
with the chemical decomposition in the cells, leading to
a statement of the quantitative laws of electrolysis. In
this connection he made the statement, of great significance at the present day, that " the atoms of matter are
in some way endowed or associated with electrical powers,
to which they owe their most striking qualities, and
amongst them their mutual chemical affinity."




262


ELECTRICITY, MAGNETISM,


The concentration of Faraday's attention on the dielectric media was rewarded by the discovery that such
a medium has a definite specific inductive capacity, the
magnitude of which is different for various dielectric
substances. He regarded electrostatic induction as consisting of a certain polarized state of the particles of the
medium, similar to that which precedes the decomposition of an electrolyte, into which they are thrown by
the inducing surfaces or particles. This state of polarization disappears when the inducing force is removed; it
can exist continuously only in insulators, because a conductor is incapable of retaining this state of its particles;
an immediate discharge taking place if it be set up in
the conductor.
Among many other investigations of Faraday were
those connected with his discovery of diamagnetism, and
his investigations of the magnetization of crystals, to
which attention had been called by the discovery made
by Pliicker, of the University of Bonn, that certain uniaxal crystals, placed between the poles of a magnet,
tend to set themselves so that the optic axis has the
equatorial position.
One surmise of Faraday is of extreme interest in view
of its later realization in the hands of his great disciple
and successor James Clerk Maxwell. In I85I, Faraday
suggested that, if the hypothesis of a luminiferous ether
be admitted, that ether may have other uses than simply
the conveyance of radiant light and heat. He had in
1845 made the important discovery that the plane of
polarization of a ray of light is rotated when it passes
through a magnetic field, when the polarized ray passes
in the direction of the lines of magnetic force. This
established a direct connection between optical and electromagnetic phenomena, and is of the highest interest in
view of the later amalgamation of the theories of light
and electromagnetism. It was found by Joule, in 1841,
that the amount of heat evolved in a given time in a wire




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263


through which a current is passing is proportional to the
resistance of the wire and to the square of the strength
of the current, that is to the product of the current into
the electromotive force on the wire. This enabled him
to perfect the theory which had been developed by
Roget and Faraday, that the chemical energy derived
from the cell has its equivalent in work done in the outer
part of the circuit. Thus the amount of energy transformed from the potential energy of chemical affinity into
an electrical form has its equivalent in the heat evolved
and dissipated in the circuit, and in any work done
otherwise in connection with the outer circuit. It was,
however, shown by Kelvin and Helmholtz that the
electrical energy furnished by a voltaic cell need not be
derived exclusively from the chemical energy of the cell,
but may also depend upon the abstraction of energy
from neighbouring bodies which is also converted into
electrical energy. Helmholtz applied also the principle
of energy to the case of systems containing electric currents, and showed that the phenomenon of magnetoelectric induction can thus be taken into account. His
theory was, however, defective in that he did not take
into account the electro-kinetic energy of the currents
themselves.
It was largely to the researches of Lord Kelvin that
the complete theories of the magnetic and electromagnetic fields are due. He introduced into Magnetism
the distinction between the vectors afterwards named
by Maxwell the magnetic force and the magnetic induction, and he extended his theory to take account of
magneto-crystallic phenomena. Ohm's theory of linear
conduction of currents was generalized by Kirchhoff
(1824-1887) to include the case of conduction in three
dimensions, the analogy with Fourier's theory of the conduction of heat being useful for this purpose. Kirchhoff
showed that, in a system of conductors, the currents so
distribute themselves as to produce the minimum amount




264


ELECTRICITY, MAGNETISM,


of heat. Hitherto there had been no complete identification between the electroscopic force of Ohm and difference of electrostatic potential; this hiatus in electric
theory was filled up in I849 by Kirchhoff.
The theoretical investigations of Electrical Science in
the last half century have in a very large degree been
dominated by the conceptions of James Clerk Maxwell,
and by the mathematical formulation of the phenomena
in the electromagnetic field to which he was led by those
conceptions. The modes of geometrical representation
of the state of the electromagnetic field which were
devised by Faraday, and which guided him in his researches, show that his mode of thinking was essentially
of a mathematical kind, and indeed he possessed constructive mathematical power of a high order. He lacked
however that command of the technique of Mathematical
Analysis, the possession of which enabled Maxwell to
follow out Faraday's conception of the localization of the
phenomena in the dielectric medium surrounding conducting substances, and to develop in mathematical
form the geometrical notion which Faraday employed
of lines of force, and, by an enrichment of Faraday's
ideas with conceptions of his own, to give a dynamical
theory of the electromagnetic field. In this work he was
appreciably influenced by the ideas and mathematical
analogies contained in the work of Lord Kelvin. In
particular he accepted Kelvin's idea that Magnetism is
essentially a phenomenon of rotation, in the form of the
conception that, in a magnetic field, there is rotation of
the medium about the lines of magnetic force, and that
electric currents are to be regarded as a phenomenon of
translation.
One of his most fundamental conceptions is that an
electrostatic field involves " electric displacement" in the
direction of the lines of force, and that, when the field is
varied, the variation of this electric displacement, whatever precise interpretation the term may receive, must




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be regarded as an electric current. This notion of
electric displacement is a development of the notion of
Faraday that in a ponderable dielectric there is an actual
displacement of electric charge on the small conducting
particles of which he assumed the dielectric to consist;
whereas with Maxwell the displacement occurs even in
free ether devoid of ponderable matter. Maxwell was
thus led to one of his most characteristic assumptions,
that every current forms a closed circuit; thus a current
employed in charging a condenser is closed, being completed by the displacement-current in the dielectric
between the coatings of the condenser. Another fundamental conception of Maxwell's scheme is that magnetic
energy is the kinetic energy of a medium occupying the
whole of space, whilst electric energy is to be regarded as
the energy of a system of strains of the same medium.
In his great memoir A Dynamical Theory of the Electromagnetic Field, published in 1864, he gave his theory in
the form of equations connecting vectors in the electromagnetic field. In his treatise, published in 1871, he
gave a fuller account of his theory of stresses in the
ether and in dielectrics, but he was not completely successful in conceiving a mechanism by which such systems
of stresses could be sustained. Space will not allow me
to give even in outline an account of Maxwell's various
investigations in this great branch of Physics, or of the
very important developments of the subject made by his
successors. I accordingly turn to the great step which
Maxwell made in the unification of Science when he
set up his electromagnetic theory of light. The theory
of Optics has a long and intensely interesting history,
but I must confine myself to a summary account of the
later stages of that history, leading up to the state of
the subject when Maxwell introduced his great unification.
The first Natural Philosopher who made any advance
upon the crude Cartesian theory of light, as consisting




266


ELECTRICITY, MAGNETISM,


of a propagation of pressure, not of motion, through a
set of globules which constitute space, was Robert Hooke
(I635-I703), one of the founders of the Royal Society.
He appears to have initiated the wave theory of light;
regarding light as consisting of a system of minute vibrations propagated in a medium, or ether. He introduced
the notion of a wave surface, as a sphere with centre at
the luminous point, and he made an attempt on these
principles to explain reflection and refraction. On the
other hand, Newton supposed light not to be constituted
by the vibrations of an ether, although he regarded such
vibrations as existing in close connection with light, but
by streams of corpuscles emitted by luminous bodies;
the various colours being due to differing corpuscles
which excite vibrations of differing types in the ether.
One side of Newton's views of the matter constituted
the celebrated emission theory which was for a long
time the rival of the wave theory. The important fact
was established by Roemer, in I675, that light requires
a finite time for its transmission.
The undulatory theory was greatly improved by
Christian Huygens (I629-I695), who gave satisfactory
accounts, on that basis, of the phenomena of reflection
and refraction, and who explained the varying velocity
of light in different substances. He concerned himself
with the study of the double refraction of light by such
crystals as Iceland spar; and explained the phenomenon
as due to the propagation of two waves in the crystal,
with different fronts, a sphere and a spheroid. The
ultimate triumph of the wave theory over the rival
emission theory was largely due to the labours of Thomas
Young (I773-I829), who showed that the former theory
gives the more satisfactory explanation of the phenomena
of reflection and refraction, of interference fringes in
shadows, of the colours of thin plates, and of the behaviour of light in crystals. The important discovery
of polarization by reflection at the surface of water, at




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a certain angle, was made by Etienne Louis Malus
(1775-1812), who showed that such a reflected ray has
the same peculiarities as one of the rays which has
suffered double refraction. The discovery of biaxal
crystals is due to David Brewster (1781-1868). The
vibrations of the ether were for the most part conceived
as longitudinal, in the direction of propagation of the
light, on the analogy of sound-waves. The important
step was taken by Augustin Fresnel (I788-1827) of conceiving the vibrations to be in a direction perpendicular
to that of the direction of propagation of the light. His
theory, the first attempt at the construction of a dynamical theory of the phenomena, was the first of a
series of theories based upon the view that the ether
behaves more like an elastic solid, and not like a compressible fluid. Fresnel supposed that no longitudinal
wave exists, and that in a polarized pencil the direction
of the vibrations is perpendicular to the plane of polarization. A celebrated instance of the power of prediction
even of an imperfect theory such as that of Fresnel is
the fact that it enabled Sir William Rowan Hamilton
to predict the occurrence of conical refraction, in which
a single ray proceeding in a crystal in a certain direction
would on emergence give rise to a whole bundle of rays
forming a conical surface; this prediction was verified
experimentally by Humphry Lloyd, of Dublin. Investigators were confronted with the difficulty that the ether
appears to behave like an elastic solid, in relation to such
rapid vibrations as those of light, but at the same time
to yield freely to such comparatively slow motions as
those of the planets; that there is no necessary inconsistency involved in this was pointed out by Stokes, who
referred to the analogy of such substances as pitch and
shoemaker's wax, which have both rigidity and plasticity.
The exigencies of the theory of the luminiferous ether
naturally led to mathematical investigation of the vibrations which can be propagated in elastic solids. Theories




268


ELECTRICITY, MAGNETISM,


of such vibrations were given by Navier, Cauchy, and
Poisson, the last of whom established the existence of
two types of wave, one transverse, and the other longitudinal, propagated with different velocities, both of
which he determined in terms of the elastic constants
of the medium. In 1828, this theory was so extended by
Cauchy as to take account of crystalline substances. The
difficulties in the way of conceiving a type of medium
in which the vibrations propagated would accord with
the known properties of light, especially the difficulties
as regards the conditions which would hold at the interface of two media, led to the development of further
theories of the matter by James MacCullagh (I809 -1847), by F. E. Neumann (I798-I895), and by George
Green. MacCullagh developed the theory of a new kind
of medium endowed with rotational elasticity; this theory
appears to be sound as a dynamical theory, and to accord
with the properties of light; Kelvin afterwards devised
a model to illustrate this kind of rotational elasticity.
From the point of view of Dynamics the theory of
George Green was superior to the pre-existing theories,
but the vibrations of Green's type do not accord very
well with optical phenomena. Cauchy, in a later theory
which he advanced in I839, introduced a type of ether
so designed that longitudinal waves are suppressed;
this type of ether was designated by Lord Kelvin labile
ether.
At this stage the difficulties in the way of conceiving
a substantial ether as the vehicle of conveyance of light,
in a manner capable of being completely represented in
accordance with a dynamical scheme, had become parallel
to the corresponding difficulties relating to electromagnetic phenomena. The great merit of Maxwell consists in his perception that the difficulties of both departments can be concentrated upon a single scheme. He
showed that, in his electromagnetic medium, electromagnetic oscillations can be propagated with a velocity




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in agreement with the known velocity of light. He
established the fact that his equations of the electromagnetic field accord with the formulation obtained by
the elastic-solid theory, and that it thus affords a general
explanation of metallic reflection. There still remained
various difficulties which Maxwell did not completely
overcome, and these have led to a great amount of
subsequent investigation; but that light is to be regarded
as an electromagnetic phenomenon, to be investigated as
a portion of the phenomena of electromagnetism, has
been generally accepted by all recent investigators. The
discovery, by Hertz, in the light of this order of ideas,
of the long waves, employed in wireless telegraphy,
identical in most respects, except their physiological
properties, with those of light, is a striking example of
the value of the unification of ideas which is due to
Maxwell.
Amidst the differences of opinion which have prevailed as to the precise manner in which electromagnetic
phenomena may be best conceived, Maxwell's equations
remain endowed with an undoubted power of representation of what can be actually observed. It would appear
that the time may have arrived at which the scaffolding
constituted by notions of a substantial ether, with
properties difficult to formulate precisely and consistently, may be removed. The theory of electromagnetism and of light would then have reached the
stage of abstraction in which the electromagnetic field
would be regarded solely as a field of vectors, distributed
and changing in accordance with definite mathematical
laws; the notion of a substantial ether having served its
purpose as a guide, and been superseded by a more
highly abstract scheme in which all such models are
discarded. If this be done the theory will have reached
the high stage of abstraction towards which all conceptual theories tend as they approach completion. The
development of such a theory necessarily involves the




270 ELECTRICITY, MAGNETISM, LIGHT
previous existence of a series of attempts to represent
the phenomena by means of sensuous images which
always contain elements of inadequacy, and often of
contradiction. The existence of a series of successive
grades of abstraction is a law of the mental evolution of
scientific theories. The question whether this gradual
increase of abstractness in scientific theories represents
a recession from, or an approach towards, "reality," in
any metaphysical sense of the term, is a question which
will receive differing answers from Philosophers of different schools. Natural Science has no direct concern
with the answer which may be given to such a question.




XII
THE CONSTITUTION OF MATTER
THE notion that all the various forms of matter are
constituted by, or evolved from, some primordial
constituent has at all times been a speculative idea which
has exercised a powerful influence upon the minds of
thoughtful men. With the alchemists of medieval times
the related idea of the possible transmutation of different
kinds of substances led to experiments undertaken with
a view to the discovery of some means of effecting such
transformation, and especially of converting baser metals
into gold. However, for a long time after the rise of
modern Chemistry, in which the atomic theory of Dalton
was fundamental, the trend of chemical investigation
was in the direction of negativing the conception that
all forms of matter may be conceived as ultimately one.
All forms of matter appeared to be constituted by combinations of some seventy or more irreducible elements
which were conceived as unchangeable and permanently
distinct from one another. This conception was expressed as late as the year 1873 by Clerk Maxwell in
his British Association address, in the words:
Natural causes, as we know, are at work which tend to modify,
if they do not at length destroy, all the arrangements and
dimensions of the earth and the whole solar system. But though
in the course of ages catastrophes have occurred and may yet
occur in the heavens, though ancient systems may be dissolved
and new systems evolved out of their ruins, the molecules out
of which these systems are built-the foundation stones of the
material universe-remain unbroken and unworn.
I propose to give some account of the steps which have
led in our time to the reversal of this view.
Although speculative doubt always persisted in rela



272 THE CONSTITUTION OF MATTER
tion to the view that all matter must be conceived as
constituted of a very considerable number of fundamentally distinct elements, until the last decade of the
nineteenth century no single definitely ascertained fact
was known which made untenable the current chemical
conception of the irreducibility and inconvertibility of
the atoms of the different chemical elements, although
some facts were known which seemed to indicate certain
family relationships within various groups of elements.
In the last 30 years a multitude of new facts have been
discovered which have made it impossible any longer to
regard all the chemical elements as fundamentally distinct from one another, and which have given new life
to the older conception of transmutation. Moreover
they have led to a theory of the constitution of the atom
which, although much remains to be done before it can
be regarded as a completely verified conceptual scheme,
has made remarkable advances in its power of representing a very considerable complex of phenomena. In
accordance with this theory, the electron, or definite unit
electric charge, is conceived as a most important part,
if not the whole, of the fundamental constituents out of
which all atoms are built up. Should this theory be fully
established, it would appear that the goal would be
reached, the instinctive desire for which had inspired all
the speculations relating to the unity which may underly
the diverse forms of matter.
In the years 1815 and 1816 the attention of chemists
was directed by Prout to some facts which tended to
show that there is a relationship between the properties
of various elements and their atomic weights. He called
attention to the fact that the three magnetic elements,
iron, nickel, and cobalt, had in accordance with his
estimate the same atomic weight, double that of nitrogen;
from this, and other instances of approximate equality
of atomic weights, he concluded that substances having
nearly the same atomic weights resemble one another in




THE CONSTITUTION OF MATTER 273
properties, and can combine more readily with one
another. From an examination of the atomic weights
he made the suggestion that all the other elements have
atomic weights which are integral multiples of the atomic
weight of hydrogen, and he suggested that they are all
compounded of hydrogen and oxygen; in his later writings
he put forward the view that hydrogen is the primordial
element. In this order of ideas Prout had various successors, amongst whom Newlands, who promulgated in
1865 his Law of Octaves, is the most prominent as a
predecessor of Mendeleeff. Newlands arranged the elements in a table in which all the elements are consecutively numbered, and appear nearly in the order of
their atomic weights; they are arranged in a number of
groups, each group containing seven members, but in
some cases two elements are coupled together so as to
occupy one place in the table. The elements in each
group are placed in a vertical column, and all the columns
are arranged in parallel, so that the first, eighth, fifteenth,
etc. elements appear in a horizontal line. In this
arrangement all the elements which are in a horizontal
line are analogous, having similar properties. In the
table the relation between elements of the same family
is the same as that which in music exists between the
notes at the extremities of one or more octaves. In
some cases analogous elements whose atomic weights
are consecutive appear with consecutive numbers in the
table, but in general the numbers attached to analogous
elements differ by 7,or a multiple of 7. Various objections
in detail were raised to this classification, one of which
was that it left no room for the inclusion of elements
which might be discovered later. The fact that caesium,
thallium, indium, and rubidium had been discovered
only a few years earlier made this objection a cogent
one. Both Newlands' arrangement of the elements, and
an earlier one embodied in the so-called telluric helix
propounded by De Chancourtois in 1863, indicated a


HGL


I8




274 THE CONSTITUTION OF MATTER
periodic variation of properties among the elements, when
arranged in the order of their atomic weights.
The Periodic Law of the elements, as it is now known,
is the system published by Mendeleeff, of Petrograd, in
I869. He showed that, when the elements are arranged
in order of their atomic weights, elements which have a
given property occur periodically; thus, for example,
chlorine, bromine, and iodine have similar fundamental
properties, although their atomic weights differ widely
from one another. Again elements which have very
similar chemical and physical properties have atomic
weights of nearly the same magnitude, unless they increase regularly. Further, when the elements are arranged in the ascending order of atomic weight, they
are in their proper order as regards valency. The elements, such as hydrogen, oxygen, nitrogen, etc., which
are most widely distributed have the smallest atomic
weight; consequently Mendeleeff called hydrogen and
those elements included in the second series of his
table "typical elements." The general principle on
which Mendeleeff insisted may be expressed in the form
that the physical and chemical properties of an element
are periodic functions of its atomic weight. The table
which Mendeleeff published in 1871 has been since
accepted as an authoritative illustration of the Periodic
Law. The gaps in his table were taken by Mendeleeff
to indicate the existence of elements which had not yet
been discovered, and he was enabled to predict the
characteristic properties of such unknown elements. The
success of these predictions was conspicuous in the cases
of Scandium, Gallium, and Germanium, which were
discovered a few years after the publication of the table;
and this success was important evidence of the value of
the Periodic Law. In the case of some elements Mendeleeff
employed the law to correct their atomic weights. Thus
the atomic weight of uranium had previously been regarded as I20, but Mendeleeff gave reasons, founded on




THE CONSTITUTION OF MATTER 275
his theory, for supposing that the number should be 240;
and this surmise was shown by Zimmermann, in I88I,
to be correct. Very important work was done by Lothar
Meyer in the establishment of the theory of the Periodic
Law; he expressed his results in the form of a curve in
which the atomic weights are plotted as abscissae, and
the atomic volumes as ordinates; in this curve there are
four well-marked maxima, and the elements arrange
themselves in six divisions. He considered in detail the
bearing of the tabulation upon the electro-positive and
the electro-negative properties of the elements, and upon
magnetic and diamagnetic properties. The connection
of the melting and boiling points of compounds with
the places of the elements of which they are formed in
the table of atomic weights was the subject of extensive
researches by Carnelley, between the years 1879 and I885.
Many applications of the Periodic Law, in connection
with chemical and physical properties, and especially
relating to the spectra of the elements, were made by a
considerable number of researchers.
In 1887, Sir William Crookes published papers dealing
with the Genesis of the Elements. He suggested the
hypothesis that all the elements, including hydrogen,
have been evolved from one original substance which
he proposed to name protyle. He suggested also that
our knowledge of any particular element is that only of
an average specimen of such element, so that, even as
regards atomic weight, there may be possible variations,
between limits, for one and the same element. The
Periodic Law and the detailed confirmation of its value
which it received from the results obtained by a great
mass of research seemed to make it highly probable that
some community of structure and of composition of all
elements must be recognized, which would account for
the law, but it was not until I896 that a direct breach
was made in the seeming fixity and inconvertibility of
the different kinds of atoms.
18-2




276 THE CONSTITUTION OF MATTER
The acceptance of the electromagnetic theory of light
naturally led to the conclusion that the radiation from
hot bodies must have its origin in the vibration of electric
systems, and thus that the atoms of such bodies must
contain electric charges capable of setting up such
vibrations. This suggested an electron theory of matter,
developed by Larmor and Lorentz, in accordance with
which matter is composed of electric charges, and its
mass is to be regarded as electromagnetic inertia. On
these lines it was suggested that the movement of the
electric charges would be changed by a magnetic field,
and that this would displace the spectrum of the substance. That this is actually the case was shown by
Zeeman in the case of the spectral lines of sodium, which
were split into two under the influence of a strong
magnetic field.
When an electric discharge is passed between electrodes through a glass vessel, so exhausted that the air
pressure is extremely small, luminous rays, known as
cathode rays, proceed from the negative electrode, or
cathode. When they strike an insulated conductor they
impart to it a negative electric charge. It was shown
by Sir J. J. Thomson that these rays can be deflected
from the rectilinear path by magnetic force, or by
electric force, in the same manner as a stream of
negatively electrified particles would be deflected. From
this it was concluded that the cathode rays consist of
a stream of negatively electrified particles in rapid
motion. By measurement of the deflections, employing
the highly probable assumption that all the particles are
equally charged, Thomson estimated that their velocity
is about one-tenth that of light, and that their mass,
which appeared to be independent of the nature of the
electrodes, is about one seventeen-hundredth part of
the mass of a hydrogen atom. The estimate has been
later the subject of revision. This amounted to a discovery of a particle of very much smaller mass than that




THE CONSTITUTION OF MATTER 277
of the lightest particle hitherto taken to exist, viz. the
atom of hydrogen; and such a particle might be taken
to be a component of all of the various substances
which could be employed as electrodes. Since electricity
in motion possesses electromagnetic energy, its effects
are comparable with those of mechanical inertia; moreover the electromagnetic energy of a moving charge is
increased as the velocity is increased. It appeared then
that the whole, or a part, of the mass of material substances might be of electromagnetic origin. It seemed
possible to regard the corpuscle as simply an electric
charge of definite amount, and to identify it with the
electron of Larmor and Lorentz. As the charges of the
corpuscles in the cathode rays are all negative, there
remained for consideration the question of the nature of
positive electrons, and the part they might play in the
phenomena of matter; further facts were required for
the elucidation of these questions.
In I896, a most important discovery, that of a radioactive substance, was made by Becquerel, in Paris. He
found that compounds of the metal uranium, which has
the highest atomic weight of all the elements, continually
emit rays capable of penetrating opaque screens, and of
affecting photographic plates. The intensity of this effect
was measured by utilizing the fact that the rays from
the uranium convert the air through which they pass
into a conductor of electricity. It was thus found that
the radio-activity of the substances depends solely upon
the amount of uranium which they contain. It was soon
afterwards shown by M. and Mme Curie that thorium,
the element of next highest atomic weight to uranium,
possesses the property of radio-activity; but other of the
known elements failed to exhibit the same property.
On experimenting with pitch-blende and other natural
minerals in which uranium occurs, M. and Mme Curie
found that the radio-activity of pitch-blende is greater
in amount than what could be attributed to the uranium




278 THE CONSTITUTION OF MATTER
that it contains, and they then succeeded in chemically
separating out of the pitch-blende compounds of a new
and intensely radio-active substance, to which the name
radium was given, only a minute amount of which is
contained in the pitch-blende. They also found other
very radio-active substances in the pitch-blende. Radium
was found to be a chemical element with a well defined
place assigned by the Periodic Law, and it resembles the
element barium in its powers of entering into chemical
compounds; it has a characteristic line spectrum, and
its radio-activity is about two million times as great as
that of uranium; its atomic weight was found to be next
less than that of thorium. The amount of energy given
off spontaneously by pure radium compounds, and
capable of being transformed into light and heat, is so
great that it was estimated to be sufficient to heat a
quantity of water equal to the weight of the radium
from the freezing-point to the boiling-point every threequarters of an hour.
The urgent question at once presented itself as to the
source of this great amount of energy which is continually being given out by the radio-active substance.
It was found that, in radium and the other radio-active
substances, the intensity of the radiation is independent
of such physical conditions as temperature and pressure,
and that the radiation cannot be inhibited by any known
means. The discovery was made by Sir E. Rutherford,
in 1899, that the radiation from thorium contains three
distinct kinds of rays, which are known respectively
by the names a-rays, f-rays, and y-rays. The investigation of the character of these rays was carried out by
Sir E. Rutherford and Professor Soddy; from their investigations, and those of others, a large number of
facts as to the radio-activity of thorium and the other
radio-active substances were disclosed. Of the detailed
investigations of this order, I can give no account, but
must confine myself to a statement of the main facts




THE CONSTITUTION OF MATTER 279
which emerged as the result of many elaborate and
difficult experiments. It was found that the a-rays, when
tested by means of magnetic and electric fields, behave
as positively electrified material particles, and that they
are stopped by a sheet of paper. These particles carry
two atomic charges of positive electricity, and they travel
with a velocity of from one-twentieth to one-fifteenth
of that of light. At first there was some doubt whether
these particles should be regarded as atoms of hydrogen
or of helium, the latter of which is always to be found
in minerals containing uranium and thorium. It was
ultimately shown by a spectroscopic examination of the
light from a few milligrams of radium that the a-rays
must be taken to consist of a stream of helium atoms.
It was later shown that the a-rays from uranium,
thorium, polonium, actinium and other radio-elements
all consist of helium atoms.
The history of the discovery of helium is a remarkable
one. It was first made in I868 by observation in the
spectrum of the solar chromosphere of a bright yellow
line which had not been observed to exist in the spectrum
of any known terrestrial substance; from this it was concluded that there exists in the sun an element hitherto
unknown, and to this the name helium was given by
Iockyer, who was one of the first to observe it. Other
lines were observed in the solar spectrum to accompany
the yellow line, and the same spectrum was observed
in the light from many stars. But it was not until I895
that helium was discovered to exist on the earth. After the
discovery, by Lord Rayleigh and Sir W. Ramsay, of the
element argon in the atmosphere, a search for argon in
mineral substances was undertaken, and in the course
of this search a gas was found to be given off by solutions
of minerals containing uranium. On examination of the
spectrum of this gas by Lockyer, it was found to be
identical with that of the helium discovered nearly thirty
years earlier in the solar spectrum.




280 THE CONSTITUTION OF MATTER
The f-rays have been shown to be negative electrons;
they are deflected both by magnetic and electric fields,
and by means of measurement of such deflections their
identity with the cathode rays in the vacuum tube has
been established. These P-rays travel with great velocity,
approaching that of light, and much more rapidly than
the cathode rays; they have great penetrative power,
being able to pass through a considerable thickness of
tinfoil or of glass, without losing their power of affecting
the photographic plate. They consist of unit charges of
negative electricity, just half as great as the positive
charges carried by the a-atoms. The effective mass of
a f-electron is only a very small fraction of the mass of
one of the a-atoms; and it is the latter which contain
most of the energy emitted in the radiation. The y-rays
are not deflected by magnetic and electric fields, but
they have much greater power of penetration than the
P-rays. They have been identified with the Roentgen rays
emitted outside the vacuum-tube when the cathode ravs
are passing through the interior of the tube.
Besides the rays given out by these radio-active
substances, radium and thorium also give out gases,
called emanations, which are themselves radio-active
substances. It was found that the radio-activity of these
emanations suffers a gradual decay, whilst the walls of
the vessels in which they are contained become radioactive. But when the walls are washed with certain
acids they lose their radio-activity, and this is transferred to the acids. The inference from this is that the
emanation gives rise to a new radio-active substance
which is deposited on the wall of the vessel, and is
soluble in the acid. It has been shown that, in a radioactive substance, chemical changes are going on which
result in the production of small quantities of new
substances. By chemical processes, minute quantities
of these new radio-active substances have been separated
out from uranium and thorium, and it is found that they




THE CONSTITUTION OF MATTER 281
have properties different from those of the original
substance. At first these new substances are highly
radio-active, whilst there is a diminution of the radioactivity of the original substance. After some time the
radio-activity of the new substances decays, whilst that
of the original substance gradually goes back to its
original amount. Rutherford and Soddy, who worked
together, by the examination of the phenomena connected with the radio-activity of thorium, were led to
the conclusion that a whole series of new chemical
substances are produced successively, the whole process
being accompanied by the liberation of energy. The
atom of uranium, in twelve successive changes, appears
to expel 7 atoms of helium and 5 electrons, one such
atom or electron at each change. The activity of a radioactive substance is found to be independent of the
presence and nature of any other substance with which
it may be in chemical combination; from this it can be
inferred that radio-activity is to be regarded as an atomic,
not a molecular, phenomenon. From the point of view
of the atomic theory the changes in the nature of a
substance are to be regarded as a series of successive
changes in the constitution of the atoms. These atoms
become dissociated into simpler parts, and the process
is accompanied by the liberation of some of the internal
energy of the atom. The amount of energy given out
in these atomic changes is enormously greater than the
amount liberated in ordinary chemical changes which
only involve molecular changes, that is alterations in the
grouping of atoms, whilst the atoms themselves are unaltered. It has been estimated that an ounce of radium,
in the course of its average life of about 2500 years,
gives out as much energy as is evolved by burning ten
tons of coal.
We have before our eyes, in the case of a radio-active
substance, an actual transmutation of elements,the possibility of which had, as I have already stated, often been




282 THE CONSTITUTION OF MATTER
speculatively asserted. This transmutation theory of
radio-activity was formulated by Rutherford and Soddy
in 1903, and soon afterwards some of its consequences
were verified in a striking manner. It was shown that,
when a small quantity of radium emanation was watched
during its slow decay, the spectrum of the element
helium could be seen; and later the presence of helium
in the a-rays was demonstrated spectroscopically. The
successive products of the disintegration of radium were
examined with minute care by Rutherford, who traced
out a long series of such disintegration products, most
of which are so small in amount that they can only be
detected by means of their radio-activities. Some of
these products give out only a-rays, one gives out both
fS- and y-rays, and one gives out all three kinds of rays;
two of them give out no rays. They show great differences
in their rates of decay; one of them decays to half its
amount in three minutes, another in 143 days, another
in 40 years, whilst radium itself takes about 2600 years
for its half decay. Results of a similar character have
been obtained as regards uranium and thorium and their
radio-active products. As the result of various experiments by R. J. Strutt and B. B. Boltwood there is strong
reason for believing that radium is a descendant of
uranium, although probably not the direct product. An
immense amount of work has been done by a large
number of Physicists in the detailed investigation of
the phenomena of radio-activity and of the properties of
the new substances discovered as the products of disintegration.
Various hypotheses have been advanced as to the
mode in which the phenomena are to be conceived;
and these may be divided into two classes. In the first
of these the energy emitted from the radio-elements is
regarded as being obtained at the expense of the internal
energy of the atom. In the second type of theory the
energy is regarded as derived from external sources, the




THE CONSTITUTION OF MATTER 283
radio-elements being regarded as mechanisms for the
transformation of such energy into the forms manifested
in radio-activity. The evidence appears to be very
strongly in favour of the theories of the first type; since
no experimental evidence appears to have been obtained
that the energy is obtained at the expense of any external
source. The theory of Rutherford and Soddy assumes
that, on an average, a definite proportion of the atoms
of each radio-active substance become unstable at a
given time, and when this instability occurs there is a
disintegration of the atom, usually of a violent character.
The disintegration is supposed to consist of the expulsion, either of an a-particle or of a f-particle, or of both
simultaneously, but in some cases the change in the
atom does not appear to involve the expulsion of either
kind of particle. When an a-particle, of atomic weight
4, is expelled, the remaining atom is lighter than before,
and a substance composed of such atoms has chemical
properties different from those of the original substance.
This new substance becoming again, as regards some of
its atoms, unstable, another a-particle may be expelled;
and this process may be repeated through a series of
stages. As the disintegration proceeds, the substance
consists of a mixture of atoms of the original type and
of the new type; that this is the case with radium is
evidenced by the fact that the original spectrum of
radium persists unchanged whilst the disintegration very
slowly proceeds. The difference between the chemical
properties of the original substance and the new substance formed by its disintegration is strikingly illustrated
by the case of radium. Radium itself is an element
closely allied to barium, and it has a definite spectrum
of bright lines similar to the spectra of the alkaline
earths; it is non-volatile at ordinary temperatures. But
the emanation is a chemically inert gas which, in its
spectrum, and in the absence of definite chemical
properties, resembles the group of inert gases to which




284 THE CONSTITUTION OF MATTER
argon belongs; it condenses at - 150~ C. This emanation
is unstable, and emits a-rays, that is atoms of helium;
the residual atoms then form a new substance, called
radium-A, which behaves like a solid, and is deposited
on the surface of bodies. The question arises, what are
the final substances which appear as the result of the
whole process of successive formation of new substances?
One of these final products is helium; this is formed
by the accumulation of the a-rays given out during the
successive transmutations. The other final product is
lead, which is always a constituent of uranium minerals,
the atomic weight of which appears to agree with the
atomic weight of the substance which would be obtained
from radium by the expulsion of 5 a-particles; one such
a-particle is known to be expelled in each of five out
of the eight successive transformations of radium into
lead. The whole line of descent of uranium through
ionium and radium, in a considerable number of successive transformations, has been traced out; the rates
at which these transformations proceed show enormous
variations ranging from a few seconds to thousands of
years.
It was natural that attempts should be made to devise
models of the atom which should be capable of representing the facts which have been discovered relating to
the instability of radio-active substances. A model of
this kind was suggested by Lord Kelvin, and investigated
in a detailed manner, in 1904, by Sir J. J. Thomson. The
atom is supposed to consist of a uniform sphere in which
is a positive electric field, and throughout which are
distributed a number of negative electrons in motion.
The total positive charge in the sphere is taken to
be balanced by the charge of the electrons, when the
atom is electrically neutral. The idea is that the properties
of the atom depend upon the number of the electrons,
upon their distributions in their orbits, and upon the
stability of such systems. The possibility presented itself




THE CONSTITUTION OF MATTER              285
of connecting in this manner the various possible types
of atoms with the Periodic Law of the elements. Sir J. J.
Thomson showed that the electrons moving under the
electric forces upon them must distribute themselves in
a number of concentric shells of differing radii. In order
to simplify the mathematical problem of determining
possible arrangements of the electrons, Thomson worked
out in detail the case in which the electrons move in
rings in one plane, the electrons being arranged at equal
angular intervals. He determined the number of rings
and the number of electrons in each ring which are such
that stability of the system is ensured. It appears that
such an atom, containing a large number of revolving
electrons, may radiate energy extremely slowly. There
must however come a time when this small but continual loss of energy from the atom results in a rearrangement of its electrons into a new system, or in
the expulsion of one or more of the electrons from the
atom. It was suggested that the disintegration of the
atoms of the radio-active elements is due to this result
of the gradual loss of energy by radiation. It was shown
that only a very small part of the mass of the atom was
due to its electrons, the rest being presumably associated
with the positive electric field. In order to account for
the scattering of the a-particles when they penetrate a
thin film of matter, this theory of the atom was modified
by Rutherford.
In accordance with the theory of the atom which was
advanced by Rutherford, the atom contains a charged nucleus, of dimensions exceedingly minute compared with
the whole volume of the atom, that is with what is called
the atomic domain; and this nucleus replaces the sphere
of positive electrification in Thomson's theory. The nucleus consists of a number of charged elements, some of
which are positive and others negative, their resultant
being a positive charge. Surrounding this nucleus there
are in the atomic domain a number of electrons; that




286 THE CONSTITUTION OF MATTER
number being equal to the number of resultant positive
unit charges in the nucleus, so that, in the neutral atom
so constituted, the total number of unit charges, positive
and negative, is zero. The resultant positive charge of
the nucleus is taken to be about half the product of the
atomic weight multiplied by the fundamental unit charge.
The existence of the small massive nucleus was suggested by the fact that the a- or helium-atoms when they
penetrate into matter are divided through a considerable
angle; the existence of an intense field of force in the
interior of the atom being deduced from this fact. The
properties of an atom are regarded as dependent mainly
on the resultant nuclear charge, and not on its mass;
this is taken to explain many facts connected with the
Periodic Law of the elements. The electrons in the outer
part of the atom are arranged at distances from the
nucleus, and are controlled by the forces due to the
resultant charge of the nucleus and their own electric
fields. The nuclear electrons form a close combination
with the positively charged units that make up the main
part of the mass of the nucleus. It is regarded as probable
that, in the region just outside the nucleus, an electron
cannot be in stable equilibrium. Rutherford makes the
statement as regards the electrons internal to, and external to, the nucleus that1:
While no doubt each of the external electrons acts as a point
charge in considering the forces between it and the nucleus, this
cannot be the case for the electrons in the nucleus itself......
Under the intense forces in the latter the electrons are much
deformed and the forces may be of a very different character
from those to be expected from an undeformed electron as in the
outer atom. It may be for this reason that the electron can play
such a different part in the two cases and yet form stable systems.
In accordance with this theory, radio-active change
originates always in the nucleus of the atom, the expelled
a- and P-particles coming from the nucleus. There is
1 "Bakerian Lecture," Proc. Roy. Soc., Vol. 97, pp. 377-8.




THE CONSTITUTION OF MATTER 287
evidence that, when the rapidly moving a-particles, or
helium atoms, pass through dry nitrogen, they give rise to
rapidly moving particles that closely resemble hydrogen.
This is shown by observations of their brilliant scintillations when they are allowed to strike a zinc-sulphide
screen. It is thus suggested that, by a breaking up of
the nitrogen atoms, hydrogen has been obtained; if this
inference is fully confirmed, a very important step has
been taken in the theory of the transmutation of the
elements. The hypothesis has naturally been made that
the atoms of all the elements are built up of hydrogen
nuclei and of electrons. In accordance with this view
the helium nucleus is composed of four hydrogen nuclei
and two negative electrons, with a resultant electric
charge of two positive units. This is a modification of
Prout's hypothesis to which I have already referred, that
all the elements are built up of hydrogen as a fundamental
constituent.
Rutherford's theory of the nuclear constitution of the
atom has been further developed by Dr Niels Bohr, of
Copenhagen, who studied the optical spectra of hydrogen
and of helium, regarded as the simplest types of atoms.
His theory takes account of, and employs, the new
quantum-theory, in accordance with which radiation
only occurs discontinuously in quanta, or definite unit
losses of energy, at moments when one stable configuration of the atom changes into another such configuration.
This theory has been further elaborated by Sommerfeld
and others, and appears to have had a considerable
amount of success in coordinating various phenomena
connected with hydrogen and helium, and especially in
accounting for the complexity of the lines of their
spectra. The mathematical theory of the hydrogen atom
has been worked out; the calculation of the spectral
lines of hydrogen, known as the Balmer series, is in full
accord with observation; and even the displacement of
the lines which occurs when the hydrogen is in an




288 THE CONSTITUTION OF MATTER
electric field has been successfully determined. It is a
remarkable fact that this theory of the hydrogen atoms
makes use of the theory of Abelian functions, a purely
mathematical theory which had been worked out without
any expectation that it would ultimately have any such
application. This is an example, which has many parallels
in Physical Science, of the fact that the most abstract
Mathematics, which of course has its ultimate roots in
the physical domain, does not really lose its connection
with its original source, however long the time may
be before the fact of such connection attains explicit
recognition.
The development of the theory of the constitution of
the atom is still in progress. Complete success in devising a model of the atom which would make possible
the calculation of the detailed varieties of configuration
of which such a model might be capable would be a
considerable step towards the goal of turning Chemistry
into a deductive science, in accordance with which all
possible elementary forms of matter might be ascertained, and the possible nature of compounds, with their
chemical and physical properties, predicted.
The known radio-active elements, in spontaneous disintegration, are few in number, most of the elements
being so stable that no such disintegration of them can
be detected. We do not at present possess any fully
established means of exciting artificially the process of
expulsion of a- or f -particles from the atom, and of thus
liberating energy of the enormous stock which appears
to be internal to the atom; but efforts are being made
to accomplish this. Sir E. Rutherford, by bombarding
the atoms of nitrogen with a-rays, has, as I have already
remarked, succeeded in producing hydrogen atoms from
a small proportion of the nitrogen atoms. If the means
of solving this problem of transmutation are in the
future discovered, the practical use of such a method
would lie more in the utilization of the internal energy




THE CONSTITUTION OF MATTER 289
of the atom as a source of energy, than in the solution
of the ancient problem of transforming one kind of
matter into another. We should be in possession of a
source of energy by which quantities of energy would
be liberated and made available, enormously in excess
of the amounts obtainable by the ordinary chemical
processes of combustion, in which only molecular energy
is utilized, and in which the vast store of energy in the
atoms remains unaffected.
The existence of the radio-active minerals is of much
importance in connection with attempts to estimate the
age of geological strata in which they are contained, as
we know the rate of change of uranium and its various
products into the final product, lead. As Soddy has
written 1:
To-day we know that the radio-active minerals are in reality
geological clocks, and they record more accurately than in any
other way the age of the stratum in which they occur. In a
uranium mineral, for example, each I per cent. of lead in terms
of the quantity of uranium signifies the lapse of 80,000,000
years. Errors of course are possible, if lead should have been
an original constituent of the mineral, but these are minimized
by taking a large number of different minerals. On the other
hand every cubic centimetre by volume of helium per gram of
uranium in a uranium mineral signifies 9,000,000 years, andas here helium, being a gas that forms no compounds, cannot
have been initially present, and as, moreover, some will have
escaped-the age of the mineral by this method is a minimum,
whereas the age by the lead content may be too high. The
carboniferous rocks tested by this new method appear to have
an age of some 350,000,000, and the oldest Archean rocks of
over,500,000,000 years.
It will be observed that the processes going on in
radio-active substances consist exclusively of the breaking
up of more complex atoms into lighter and less complex
parts, and so far as our experience goes these processes
are irreversible. Accordingly no light is thrown upon the
1 Science and Life, pp. Ioo, 11o.


HGL


19




290 THE CONSTITUTION OF MATTER
possibility of the present complex elements having been
built up out of simpler components, such as is involved
in the view that a simple primeval form of material once
existed, from which the present elements were evolved
by processes in which there was a gradual complication
of the original material. If we are to conceive that our
present forms of matter were so evolved, we must
imagine that such evolution took place under conditions
very widely different from those which we are able to
observe at present. Passage from the more complex to
less complexity is what we observe at present, and the
idea that change in the reverse order once took place
is at present a speculation resting upon no evidence
obtained from observation.
A very important discovery has been made in connection with the investigation of radio-activity, that of
the existence of isotopes, the name given to elements
which have identical chemical properties, and occupy the
same place in the Periodic Table, but have different
atomic weights. If an a-particle, or a p-particle, is expelled from an atom, the new atom has different properties from the original one. If it is the a-particle that
is expelled, the element after this expulsion passes in the
Periodic Table to the place next but one to that of the
original element, the atomic weight being diminished.
If it is the P-particle that is expelled, the change of place
is into the next place in the table, in the opposite direction. But if an a-particle and two f-particles are expelled,
in any order, the element returns to its original place in
the table; its atomic weight is diminished by four units,
but its chemical and spectroscopic properties are the
same as before the expulsions took place. The elements
in the original and final form are in fact isotopes. The
places in the Periodic Table appear to represent the
integral nett charges of electricity in the atomic nucleus.
The a-particle having a charge of two units of positive
electricity, when it is expelled from the atom, moves the




THE CONSTITUTION OF MATTER 29r
element through two places in the Periodic Table,
whereas the expulsion of a B3-particle with its unit
negative charge, moves the element one place in the
opposite direction. Isotopes such as thorium and ionium
have identical chemical and spectroscopic properties,
although their radio-active properties are different. The
lead obtained as the final product of the series which
commences with uranium has a different atomic weight,
206, from the lead which is the final product of the
thorium series, for which the atomic weight is 208.
Ordinary lead has the atomic weight 207-2, and this
suggests that it is a mixture of the two isotopes.
There appear to exist also isotopes which have the
same atomic weight as well as identical chemical properties, although the internal energy in the atoms of the
isotopes is different in the two cases. Such isobaric
isotopes are obtained when, as in the case of thorium,
there is a branching off of the successive disintegrations
into two series, each with a final product.
By means of a new method of deflecting the rays which
consist of streams of charged particles, F. W. Aston has
been able to determine the masses of a number of atoms
with a great degree of accuracy. By this method he
has been enabled to show that some of the elements,
formerly supposed to be homogeneous, consist of a
mixture of two or more isotopes with different atomic
weights. When the atomic weight of oxygen is taken to
be 32, so that the atomic weight of hydrogen is Ioo8,
it has been found that the atomic weights of many of
the elements are represented exactly by integral numbers.
In other instances, such as chlorine, in which this is
not the case, the elements have been found to consist
of a mixture of isotopes, each with an atomic weight
represented by an integer; the fractional atomic weight
of the element as a mixture being due to the fact that
the element contains portions of different atomic weights.
Thus, for example, chlorine, of which the atomic weight
I9-2




292 THE CONSTITUTION OF MATTER
is 35-46, obtained by chemical methods, is a mixture, in
the proportion of 3 to i, of two isotopes of which the
atomic weights are 35 and 37 respectively. These investigations have led to the theory, which has been to a
great extent verified, that the fractional irregularities of
the atomic weights in the Periodic Table are due to the
existence of isotopes.
The known facts appear to be consistent with the
view that all matter is built up of electrons and atomic
nuclei; all the electrons being alike, and the atomic
nuclei being constructed of fundamental units all alike,
and of electrons. The electrons and nuclei must be
regarded as postulated concepts subject to defined interrelations, and which, from the point of view of this
theory, are irreducible.
As I have already indicated in an earlier lecture, for
a long period chemical and physical investigations were
carried on independently of one another, and they had
but few points of contact. An important effect of the
great discoveries of the last quarter of a century, relating
to the composite character of the atom, has been to break
down the barrier which had long separated the sciences
of Physics and of Chemistry. The two great departments
have in our day at last joined forces; their meeting-place
has been within what was long regarded as the impregnable fortress of the atom.




XIII


COSMICAL THEORIES
I PROPOSE in the present lecture to give some account of various theories which have been designed
to describe the processes of change by which the bodies
of the solar system and the stellar universe may be conceived as having reached their present configurations
and general physical states. Such theories consist of
attempts to trace back into the remote past the history
of the solar system and of other configurations which we
observe telescopically and with the aid of the spectroscope, on the assumption that the physical laws by
means of which we describe the present motions of
bodies, and their thermal and other physical changes, are
adequate to describe what we conceive to have happened
in the immense periods of time which we must assume,
in accordance with those laws, to have elapsed since
the primeval conditions postulated by such cosmical
theories. This assumption involves a considerable element of speculation, because our knowledge of the range
of applicability of physical laws is determined and limited
by observations of the special physical processes which
we can actually observe only during strictly limited
intervals of time; and these laws can only hypothetically
be extended to embrace processes going on under conditions widely different, and during immense periods of
time.
An examination of the bodies of the solar system
reveals certain striking uniformities within that system.
In the first place, the eight great planets Mercury, Venus,
the Earth, Mars, Jupiter, Saturn, Uranus, and Neptune,
all revolve in the same direction round the sun; their




294


COSMICAL THEORIES


orbits are all nearly in the same plane, and are all approximately ellipses with small eccentricities, that is,
nearly circular. The same statement holds good of the
minor planets or asteroids, some 900 in number, which
have been discovered in the solar system. These facts
are sufficient to suggest that the uniformity in the orbits
of the bodies of the solar system may be represented as
the result of some process of formation from a common
origin. But these facts do not stand alone. The planets
rotate round their axes in the same direction, and their
satellites, with certain exceptions, rotate round the
primaries in nearly circular orbits, nearly in the same
plane, and in the same direction. Jupiter and Saturn
each has a system of satellites, forming systems like the
system of the sun and planets, but on smaller scales. It
appears that the exceptional cases arise on the outermost
parts of the solar system, and the outermost parts of the
systems of Jupiter and Saturn. Thus Neptune has only
one satellite, and this has retrograde motion; that is, its
direction of revolution is in the opposite direction to that
of the great majority of the satellites and of all the
planets. Uranus has four satellites, whose orbits are all
highly inclined to the ecliptic; Saturn has nine satellites,
of which the outermost has retrograde motion; and its
orbit has a large eccentricity. Of the nine satellites of
Jupiter, the two outermost have retrograde motion. Thus
in each case the deviation from uniformity attaches to
the planets or satellites which are at the outermost
distance from the sun or the primary.
In 1772 it was pointed out by Titius, a Professor at
Wittenberg, that the distances of the six planets at that
time known, from the sun, were very approximately
represented by numbers which increase according to a
very simple law; but that there was one interruption,
the planet that should in accordance with the law have
its orbit between that of Mars and that of Jupiter not
being known to exist. The matter was taken up by Bode




COSMICAL THEORIES


295


of Berlin, who suggested that an unknown planet remained to be discovered which would occupy the vacant
place in the table of distances indicated by the law of
Titius, which has since been known as Bode's law. The
discovery of the planet Uranus was made about this
time, and it appeared that its distance from the sun was
in fair conformity with the law. This discovery lent
further weight to the prediction that a planet would be
found to occupy the vacant place, and this prediction was
ultimately verified in an unexpected manner. A number
of small planets or asteroids were discovered in the
space between the orbits of Mars and Jupiter, the first
of these being discovered on the first day of the nineteenth
century, by Piazzi, a Professor at Palermo. The idea then
presented itself that these asteroids were fragments of a
single planet which had been broken in pieces by an
explosion. It was indeed shown that three of these
asteroids really conformed closely to Bode's law. Since
that time the number of asteroids observed in the region
between the orbits of Mars and Jupiter has steadily
increased, until some 9oo of them have been found, but
the notion that they are all fragments of one original
planet is for various reasons regarded as untenable.
Nevertheless the discovery is remarkable as the first
instance of a successful prediction of the existence of
bodies which had at the time not been observed. The
story of the still more remarkable verification of the
prediction made by J. C. Adams and Leverrier, that a
planet would be found outside the orbit of Uranus, has
often been told. This discovery of the planet Neptune
was predicted mainly as the result of observation that
the motion of the planet Uranus exhibited disturbances
which could only be accounted for on the hypothesis
that they were due to an unobserved planet at greater
distance from the sun. Whereas the prediction which
led to the discovery of the first asteroids was inspired
by the purely empirical rule known as Bode's law, in




296


COSMICAL THEORIES


the prediction which led to the discovery of Neptune,
that law was only employed to afford ground for a
probable value of the distance of the planet from the
sun; calculations based upon the law of gravitation,
taken in conjunction with the data afforded by the irregularities in the motion of Uranus, forming the chief
basis of the prediction.
After this survey of some of the main features of the
solar system, and of the considerable degree of the
regularities in its structure, I now turn to the consideration of the theories that have been proposed as to the
mode in which it has been evolved, and in particular
to the celebrated Nebular hypothesis, first suggested by
the Philosopher Immanuel Kant, and developed more
precisely by Laplace.
Kant's speculation as to the origin of the heavenly
bodies was published in I755 in his Allgemeine Naturgeschichte und Theorie des Himmels, and was applied by
him not only to the solar system but to the stellar
universe. He supposed the solar system to have been
developed from an initial condition in which a cold
nebula at rest consisted of a vast mass of evenly diffused
particles. This nebula he supposed to commence to concentrate under the mutual gravitation of its parts, and
to become hot in doing so, owing to the consequent compression. He made the quite arbitrary and unwarranted
assumption that this process would give rise to a rotation of the whole mass. He assumed further that the
matter would concentrate into rings which would be
in rotation, and he illustrated this by the case of the
rotating rings of Saturn which he regarded as giving
evidence of the correctness of his views. He supposed
that in course of time these rings would become unstable,and would form planets by a process of agglomeration; thus leading to a system of planets revolving round
the densest and hottest core, the sun. The satellites he
supposed to be formed by a repetition of similar processes




COSMICAL THEORIES


297


on a smaller scale, as the planets gradually contracted
under their own gravitation.
The Nebular hypothesis, as we now understand it,
was first put forward by Laplace,in I796,in his Exposition
du Systeme du Monde. Unlike Kant, of whose speculations Laplace appears not to have had any knowledge,
Laplace limited his theory to the case of the solar
system. He assumed as the original form from which
the solar system was evolved an immense nebula of a
flat or lens-shape, consisting of extremely attenuated, but
gaseous, material. Unlike Kant he assumed the nebula
to be originally in a state of slow rotation round an axis,
thus avoiding the dynamical difficulty in Kant's theory
as to how the rotation came into being. The mass falls
in upon itself, owing to the gravitation of its parts, and
as this occurs, the central portion becomes hotter as the
whole mass shrinks, whilst the outer parts are cooled by
radiation at the surface. Since the angular momentum
of the mass remains constant in amount, the shrinkage
of dimensions of the nebula produces a gradual increase
in the angular velocity of rotation. It was assumed by
Laplace that this increase of angular velocity led to the
separation of an outer ring of matter from the main mass.
As the shrinkage of the main mass proceeded, this process of shedding a ring of matter would be repeated, and
as the result, there would be a central core surrounded
by a series of concentric rings, all in rotation about a
central axis. The planets were then supposed to be
formed in the manner indicated by Kant, the rings
becoming unstable, and ultimately forming planets.
In accordance with this theory, the rings of Saturn
represent a stage in the general process of the formation
of planets and of satellites. The theory does not explain
why, in accordance with dynamical theory, the successive rings should become unstable, or why the rings
of Saturn should form an exception to this instability.
It was pointed out by Sir G. H. Darwin that, in case




298


COSMICAL THEORIES


a ring became unstable, the resulting planet would not
be formed on the perimeter of the ring, but at its centre
of gravity. The fact that a continuous rotating ring of
matter would be unstable was established by Maxwell
in 1857, who found that the stability of the rings of
Saturn could only be accounted for by the supposition
that they consist of an aggregation of separate small
fragments, or a heap of stones.
The highly speculative character of Laplace's theory
makes it clear that, before it can be accepted as a
probable account of the manner in which we may conceive the solar system to have arisen, the various stages
which it includes must be scrutinized in the light of the
principles of Dynamics. The difficulties of a mathematical kind which are involved in an investigation of
the forms taken up by rotating masses of tenuous gas,
of the stability of such forms, and in the ascertainment
of what happens when the forms become unstable, are
so great that nearly all the mathematical investigators
who have concerned themselves with these questions
have found it necessary to treat the nebulae as consisting of homogeneous incompressible fluid. Although
the results of these investigations throw some considerable light upon the dynamical possibility of the
processes involved in the theory in its original form,
they cannot be regarded as having an absolutely decisive
weight in our judgment of the theory. The details of
that theory have received considerable modification at
the hands of later investigators, but the notion that the
solar system originated from a nebulous mass of gas,
and the conception that increased rotation of that mass
owing to shrinkage has been a main factor in the change
from the primitive nebula to the present system, remain
as at least highly probable hypotheses; and they have
not been superseded by any other theory.
By later Astronomers the nebular hypothesis has been
considered more generally than by Laplace, in relation




COSMICAL THEORIES


299


to the idea that stars in general may be conceived to
have been formed out of nebulae. The enormous increase of our knowledge of the nature of heavenly bodies,
due to vastly increased powers of observation, in which
photography plays an essential part, to the employment
of the spectroscope, not only for the determination of the
chemical nature of the composition of the sun, stars, and
nebulae, but also of their motions, and the discovery that
double and multiple stars are extremely numerous in the
stellar system, have brought Laplace's theory into relation
with a number of fundamental problems of cosmology.
After the discovery, in 1842, of the mechanical equivalence of heat and motion, the attention of Physicists
was directed to the question of the means by which
the solar radiation of heat and light is maintained. It
was pointed out by Mayer that this radiation would
have long since ceased if the sun were a body simply
cooling, or in a state of combustion, and without any
source of energy from which heat could be obtained.
As a source from which the solar radiation might be
sustained he advanced the meteoric hypothesis. In
accordance with this hypothesis a swarm of meteoric
bodies is constantly falling into the sun, and by the
stoppage of their motion an immense amount of heat
is evolved. Mayer calculated that, from this source,
from 4600 to 9200 times as much heat would result, as
would result from the burning of equal masses of coal.
The difficulty of this theory is that the amount of matter
which must fall into the sun, in order to maintain its
radiation, must be enormous, amounting to a mass equal
to that of the moon every half year. So great an accession
to the mass of the sun would materially affect the gravitation of the planets towards it, producing a diminution
in the periods of their revolution round the sun. In the
case of the earth, Mayer reckoned that this would involve an annual diminution of the length of the year by
a not very small fraction of a second; and he postulated




300


COSMICAL THEORIES


an emission of material from the sun to neutralize this
continual reinforcement of material from the meteorites.
Independently of Mayer, this theory was advanced by
Waterton at the Meeting of the British Association in
1853, and was fully discussed by Lord Kelvin in the following year. Kelvin pointed out that, although the influx
of meteorites "is the only one of all conceivable causes
of solar heat which we know to exist from independent
evidence," that source of heat is entirely insufficient.
An intraplanetary supply of meteorites would be much
too scanty to achieve any perceptible result, and it was
estimated by Kelvin that a sufficient supply from extraplanetary space would have resulted in a shortening of
the year by six weeks since the beginning of the Christian
era. This theory of the maintenance of solar heat led to
the hypothesis that the earth and planets had been formed
by an agglomeration of meteorites, which must have
been of amount much greater than they at present receive.
But the evidence of geological deposits negatives this
theory, since no considerable parts of these deposits can
possibly have been of meteoric origin.
A theory which has been much more widely accepted than the meteoric hypothesis was put forward
by Helmholtz in I854. In accordance with this theory
the maintenance of the solar heat and light is due to
the shrinkage of the material of which the sun is composed. As the envelope of the sun is continually cooled
by radiation, a condensation towards the centre occurs
which develops heat. In order to provide for the present
rate of emission, a diminution of the solar diameter by
380 feet a year would suffice, and in five million years
the sun would have shrunk to less than half its present
bulk. Helmholtz estimated that radiation may have gone
on at its present intensity for about twenty-two million
years, an estimate which geologists find insufficient. This
theory is in accordance with Laplace's view that the
material of which the sun is composed was once diffused




COSMICAL THEORIES


30I


through an immense portion of space, and has gradually
been condensed, with the production of heat, to the
present dimensions.
To that part of the nebular theory which relates to
the formation of planets many objections have been
raised. Among these is the failure to account for the
retrograde motion of the satellite of Neptune and of the
satellites of Uranus. Again, one of the two satellites of
Mars revolves round the planet in a shorter time than
that of the rotation of the planet round its axis, a fact
which it is difficult to reconcile with Laplace's views as
to the generation of satellites. An objection to the theory
of Laplace is that since, under gravitation towards the
sun, the velocity diminishes as the distance from the
centre increases, the rings could only coalesce into globes
with a backward rotation round their axes. Laplace himself was aware of this difficulty, which he endeavoured
to meet by the assumption that the rings had sufficient
cohesion to allow them to rotate in one piece, as if they
were solid. Thus the outer part of a ring would move
faster than the inner part, and would give rise to planets
spinning forwards, as is actually the case. It is difficult
to conceive that the materials of nebulae can have had
the amount of cohesion which is involved in this explanation. To meet these difficulties a theory was propounded
by Faye, in accordance with which all the planets except
Uranus and Neptune were formed in a practically homogeneous nebula, before its condensation had formed the
sun. This nebula revolved like a solid body, with velocity
increasing with the distance from the axis. The planets
were then formed by agglomeration, and consequently
their rotation was direct. The sun was subsequently
formed by contraction of the nebula, and Uranus and
Neptune were formed at a time when this contraction
was far advanced; in accordance with this theory the
order of formation of the planets is the reverse of what
it is in accordance with Laplace's theory.




302


COSMICAL THEORIES


Attention was called to a factor of great importance
in connection with the origin of satellites by the investigations of Sir G. H. Darwin on the effects of tidal action
between the Earth and the Moon. The moon, by producing tides in the ocean, which act as a kind of brake
upon the rotation of the earth round its axis, produces
a very slow lengthening of the day, and this effect must
have been much greater than it is at present when the
earth was liquid or plastic. A precisely similar effect was
produced by the tidal action of the earth on the moon.
The result of this tidal action in the case of the moon
is the present state of the rotation of the moon round
its axis in the same period as its rotation round the earth,
so that it always shows the same half of its surface to
the earth. Darwin proved that a consequence of the
tidal action of the moon on the earth, in accordance with
the laws of Dynamics, is that the moon must slowly
recede from the earth while its time of rotation round
the earth is very slowly increased. He showed that the
ultimate result will be that the day and the month will
become equal to one another, each being then about
I400 hours, a state of tidal equilibrium being then
reached, with the moon at much greater distance from
the earth than at present. If this process be traced back
from the present time, at some epoch not less than
54 million years ago, the length of the day was between
2 and 21 hours; the moon was then almost in contact
with the earth, and rotated round the earth in the same
period as that of the earth round its axis. The period of
rotation of the moon round the earth when they were
nearly in contact was, in accordance with Kepler's
law, between 2 and 21 hours. It can be shown mathematically that the most rapid period of rotation of a
fluid mass, of spheroidal shape, of density equal to the
earth's average density, which is consistent with stability,
is two hours and twenty minutes. It was consequently
suggested that the moon was generated by the separation




COSMICAL THEORIES


303


of a mass from the earth, owing to instability due to too
rapid rotation of the earth, a breakage into two masses
being the consequence. That the general character of the
changes in the rotation of the earth and in the length
of the month is in accordance with the laws of Dynamics
is not open to doubt, but serious criticism has been
directed to the theory that the initial stage of this process
is connected with the genesis of the moon. It was
pointed out that, when the moon was very near the
earth, a disruptive strain would act on the moon, sufficient to prevent it holding together as one continuous
mass; to meet this objection Darwin suggested that at
that time the moon may have been an aggregation of
separate masses. But it would appear that the effect of
tidal action would have caused a dispersion of these
separate masses, and to this objection there does not
appear to be any adequate answer, so that this part of
Darwin's theory cannot be regarded as established.
The questions were discussed by Darwin whether
satellites of the other planets in the solar system could
have originated in the same manner as the moon, and
whether the present relations between the primaries and
their satellites could be accounted for as effects of tidal
friction. The answers to both questions were in the
negative. The circumstances in these cases are different
from those of the earth and moon, in the important
respect that the ratio of the mass of the moon to that
of the earth is much greater than the ratio of the mass
of any other satellite to that of its primary. Consequently
tidal friction has been a much more important factor in
determining the relations of the earth and moon than
in any other case in the solar system. Darwin showed
that satellites, such as those of Jupiter, Saturn, or Mars,
in all probability never revolved round the primaries in
much smaller orbits than at present; thus in these cases
no such great tides ever existed as were raised on the
earth when the moon's orbit was much smaller than at




304


COSMICAL THEORIES


present. Darwin also considered the effect which the
friction of the tides raised on the planets by the sun may
be taken to have had in altering the periods of their
orbits and their distances from the sun. He showed that
no large effects upon the planetary orbits can have arisen
from the solar tides, and thus that the planets cannot
have arisen by separation of portions of the plastic mass
of the sun, in a manner similar to that by which he
supposed the moon to have been detached from the
earth. This removes one possible alternative to Laplace's
theory of the formation of the planets. It was, however,
pointed out by Roche, in 1872, that the tides produced
on the planets by the sun may have been a decisive
factor in relation to the formation of satellites. According
to Darwin's view, a satellite will be formed when the
rotation of a fluid or plastic mass comes to exceed a
certain magnitude, when the figure becomes unstable.
There would then be no production of a satellite in case
the solar tidal friction were sufficient to prevent the
attainment of this critical angular velocity of rotation;
this may be held to account for the fact that Mercury
and Venus have no satellites, their nearness to the sun
involving a large effect of solar tides in reducing what
would otherwise have been their greatly increased periods
of rotation as their masses contracted. In the case of the
earth and the outer planets, the friction of the solar
tides may be supposed not to have been sufficient to
prevent the instability arising owing to too rapid rotation, and thus to have prevented the formation of
satellites by division into two parts.
It was formerly supposed that our solar system was
typical of what was regarded as the ordinary state of
things in the universe; that in fact every star was a sun
with its attendant satellites, forming for each star a
system with a character resembling our solar system.
The discovery of the existence in the heavens of a large
number of binary stars has modified this view, since it




COSMICAL THEORIES


305


has exhibited the existence of systems of a character
differing widely from the solar system. When William
Herschel commenced his great exploration of the heavenly
bodies, his attention was early directed to double stars,
that is pairs of stars apparently very close to one another.
At first he shared the general opinion that the connection between such a pair was purely optical; that in
fact they might be simply two stars at widely different
distances, but of which the directions happened to be
nearly the same. But he was later led to the conclusion
that, in a very considerable number of cases, double
stars really consist of binary combinations in which the
two stars rotate round one another. In several cases the
periods of these rotations were determined. It has since
been shown that such binary combinations exist in large
numbers. Of the nineteen stars which are at the present
time nearest to us it is definitely known that eight, no
less than 42 per cent. of the whole, are binary stars.
A scrutiny of various parts of the heavens makes it
probable that not less than one-third of all the stars that
can be observed are binaries. As long ago as 1764 it was
observed by John Goodricke, a deaf mute, that the
brilliancy of the star Algol was subject to periodic variation, and he suggested that this phenomenon was due
to periodic eclipses by an invisible companion star. The
correctness of this theory was verified in 1889, when it
was proved that the star was moving in an orbit of such
a character as would result from gravitation with a dark
companion star, by which it would be partially eclipsed.
By means of the spectroscope it has been proved that
many stars which are single in appearance really consist
of two stars in orbital motion round one another. It was
from the first surmised that the motions of the stars in
a binary combination are such as would be in accordance
with the law of gravitation, and that this is actually the
case has been established in a number of instances. The
first of such cases of gravitating binaries was ascertained


HGL


20




306


COSMICAL THEORIES


in 1827, by Savary of Paris, who proved that, for a binary
star in the Great Bear, the orbits of the components
were ellipses with a period of 58~- years, in full accord
with the gravitational scheme. In a number of binaries
it has been found possible to determine the ratio of the
masses of the two stars which form the combination, and
in all such cases the masses have been found to be not
very unequal. It has been found, for example, that in the
visual binaries in which the ratio of the masses is pretty
accurately determined, one of the stars is never less in
mass than one-third the mass of the other star, and a
similar result has been obtained in the case of nineteen
spectroscopic binaries. On the other hand, in the solar
system, the mass of the greatest planet, Jupiter, is less
than a thousandth part of the mass of the sun. It follows
that the sun and Jupiter cannot be regarded as forming
a binary system similar to the many binary systems which
have been observed. Thus a binary star with its attendant
planets, if such exist, forms a system which cannot but
be radically different in character from the solar system.
Moreover the existence of triple, and of multiple, systems
has been observed. It has been shown that triple systems
consist normally of a pair close together, with a third star
revolving at a distance from the centre of gravity of the
pair, about ten times the distance of the stars of the pair
from one another.
A theory has been worked out by Sir G. H. Darwin
which gives at least an indication of the mode in which
a double star may be conceived of as generated from a
single original mass. The mathematical difficulties of a
theory of this kind are so great that they have to be
simplified by means of assumptions which are of such a
character that the conditions in the mathematical investigation are widely different from what we must assume them to be in actual cases, and yet not so widely
different that they destroy all the value of the results of
the investigation as an indication of what may be con



COSMICAL THEORIES


307


ceived to have happened in such actual cases. An actual
star, consisting of liquid or viscous matter in rotation,
radiates its heat, and shrinks gradually as it cools. Its
density will be at any one time very various, as we proceed
from the outer surface to the interior parts, and its
average density will increase as the whole mass contracts.
Darwin considered what would happen in the case of a
mass of liquid which at any one time is homogeneous
and incompressible. The shrinkage is assumed to be so
slow as to be consistent with this assumption, although
that shrinkage of course involves an increase of density
of the fluid. It has long been known, from mathematical
investigation, that such a mass of liquid in rotation is
in relative equilibrium if it have the form of a spheroid
with its axis of symmetry as the axis of rotation, this
axis being the smaller axis of the elliptic sections of the
spheroid through that axis. It is further known that this
form is a stable one, so long as the rate of rotation of
the mass is sufficiently slow. As the mass slowly shrinks,
its rate of rotation increases, in accordance with the
dynamical principle of the constancy of its angular
momentum, and the stability of the form gradually
diminishes. As it shrinks, and the angular velocity of
rotation increases, its shape changes; in fact it becomes continually more flattened at the poles. When it
attains a certain shape, in which the equatorial and
polar axes are in the ratio of Iooo to 583, the stability
entirely disappears; it has reached what has been called
by Poincare a figure of bifurcation, a point at which the
series of spheroids through which it has passed during
the gradual increase of its rate of rotation passes over
into a new series of figures of a different kind. This new
series of figures consists of a set of ellipsoids with all
their axes unequal. As the increase of the rate of rotation
proceeds, the axes of the equatorial ellipse on the plane
perpendicular to the axis of rotation become continually
more unequal until, when a certain angular velocity has
20-2




308


COSMICAL THEORIES


been attained, and the longest axis has become about
three times the shortest, a new figure of bifurcation is
reached, when the stability of the motion has again disappeared. It had been shown mathematically byPoincare
that the new forms which the liquid will take up, after
this form of bifurcation is passed, will consist of a series
of pear-shaped figures, in which the figure is blunted at
one end and prolonged into a sort of snout at the other
end. The question whether these pear-shaped figures are
stable is a mathematical problem of great difficulty, and
the answer given to the question has been different by
different investigators. Darwin came himself to the
conclusion that they are stable, but Liapounoff came to
the opposite conclusion, and further investigations by
Jeans appear to confirm the answer given by Liapounoff.
The suggestion made by Darwin was that, as the rotation
still further increases, the inequality between the two
parts of the pear becomes accentuated, a furrow being
formed; and that in the end a new figure of bifurcation
is reached, after which the mass splits into two parts
which separate, the subsequent history being that of
the two separate masses which represent the two stars
in a binary combination. Great doubt is thrown upon
the applicability of this theory to the case of the genesis
of double stars by the uncertainty as to the stability of
the pear-shaped figures, but even if the fact that they
are unstable be finally admitted, they do not wholly
lose their importance in connection with theories of
cosmogony.
Various mathematical investigations have been carried
out in another direction, with a view to throwing light
upon the double star problem. Instead of attempting
to trace out a series of changes in a single continuous
mass, with a view to the ascertainment whether, and
under what conditions, it may divide into two separate
masses, as in Darwin's theory, the problem has been
approached from the other end. In fact the problem of




COSMICAL THEORIES                 309
determining the tidal action upon one another of two
separate masses rotating round one another without
change of relative position has been taken as the starting
point. This problem was studied in considerable detail
by E. Roche, and also by Sir G. H. Darwin. By the
former of these investigators the problem was simplified
by considering one of the bodies to be a rigid sphere,
and the series of forms of the other body, which was
taken to consist of a fluid mass, was traced out. It was
found that no relative equilibrium of the two masses is
possible if the distance of the two bodies from one
another is less than a certain minimum depending cipon
the ratio of their masses, so that they cannot rotate in
close contact with one another. The more difficult
problem connected with the double star question was
discussed by Darwin, in which both masses are fluid,
so that each is distorted under the tidal forces generated
by the other. As the result of intricate investigations of
the stability of the figures of the two rotating masses,
it appears to be impossible to trace back the states of
these masses to a time when they were in contact, or
nearly so, with one another. In fact it appears that in
those cases there are no figures which are stable, except
ellipsoids and spheroids, whereas, if the masses are to
be regarded as coalescing, they must pass through forms
which are far from being ellipsoidal. When instability
is reached it would seem that the whole character of the
motion of the system must be changed. These problems
are so difficult that no complete solution of them has
as yet been obtained. Jeans, who has continued the investigations of Poincare and Darwin in various directions, has expressed the opinion that it is highly probable
that tidal action may produce systems such as are seen
in the solar system and in the systems of Jupiter and
Saturn; that increasing rotation may produce systems
such as are seen in ordinary binary stars, and that the
close approach of two stars revolving about one another




3Io


COSMICAL THEORIES


may produce systems such as Saturn's rings, and possibly
also the asteroids.
I turn now to some consideration of that part of the
theory of Laplace and Kant which has to do with the
generation of stars from nebulae. As discerned by the
telescope, reinforced by the photographic plate, there
exist nebulae of several distinct forms; some I5,000
nebulae have been hitherto investigated. The most frequent of these are the spiral nebulae which consist of
a nucleus with two arms emerging from opposite points;
these two arms have each a spiral form, approximately
that of the equiangular spiral. The so-called planetary
nebulae are comparatively very few in number; they
have an ellipsoidal shape. There are also ring-shaped
nebulae, and also elongated or spindle-shaped nebulae.
Besides these are irregular nebulae such as the great
nebula in Orion. All the knowledge we have of nebulae,
apart from their apparent shapes, as disclosed by the
telescope, has been obtained by analysis of their spectra.
In this manner, information is obtained in the first place
of their chemical constituents. In I862, Sir W. Huggins
and Miller in London, and Father Secchi in Rome, commenced the work of studying the stars and nebulae by
the new means which the spectroscope provided. In
1864, a bright planetary nebula in Draco was found by
Huggins to consist of a mass of glowing vapour which
showed the characteristic bright line of nitrogen, and a
fainter line showed itself to be the F line of hydrogen.
By I868, Huggins had examined the spectra of about
70 nebulae, about one-third of which turned out to be
of a gaseous character, and showed the nitrogen line.
It was found that all the planetary and annular nebulae,
as well as the irregular nebulae in the region of the milky
way, have the character of a glowing vapour. Besides
the knowledge of the constitution of stars and nebulae
which has been obtained by the use of the spectroscope
and has built up the department of stellar chemistry,




COSMICAL THEORIES


31I


spectroscopic analysis has also been applied to determine
the motions of stars and nebulae relatively to the earth,
in the line of sight. In 1842, Christian Doppler, of
Prague, enunciated the principle that the colour of a
luminous body must be changed when the body approaches or recedes from the observer with sufficient
velocity to make the change sensible. In an amended
form, this principle is that the velocity of approach or
recess of a star or nebula will exhibit itself by means of
a displacement of the lines of the spectrum of the body
from the position of the lines of the spectrum of the
same substance in the laboratory, and that this velocity
may be estimated by measuring the amount of this displacement in one direction or the other. This very delicate
operation of measurement of displacement was first successfully carried out by Huggins, in I868, in the case
of the star Sirius, leading to the result that the star is
receding from the solar system at the rate of twenty-nine
miles a second. A more accurate observation made in
1872 diminished this rate to about twenty miles per
second, and estimates were made by Huggins for other
stars, some of which are receding from, and others approaching the solar system. Since that time the measurement of such velocities has formed part of the regular
work of Astronomers. The earlier observations of nebulae
did not disclose any traces of their motion in the line of
sight, but later observations have shown that the spiral
nebulae have very great velocities. For example, the Andromeda nebula has been estimated to have a velocity of
approach of 300 kms. per second, or perhaps somewhat
more. Other spiral nebulae have been estimated to have a
still greater velocity of recession, exceeding i oo kms.
per second. In fact the average velocity of spiral nebulae
is some twenty times greater than the average velocity
of a star belonging to the system of which the sun is
a member. But besides these velocities of nebulae as
wholes, the spectroscopic method has been applied to




312


COSMICAL THEORIES


detect the rotation of nebulae. A nebula in Virgo was
discovered, in I914, by Slipher, to be in rotation; and
the velocity of rotation has been since estimated to be
330 kms. a second, at a distance of 2' from the centre,
the velocity increasing proportionally to the distance
from the centre. In the cases of other nebulae similar
observations have been made, but in one case at least
the rotation does not appear to be one in which the
mass rotates like a rigid body, for the angular velocity
diminishes with the distance from the centre. The irregular nebulae are found to be almost at rest relatively
to the stars of our system, and the planetary nebulae
have, with certain exceptions, velocities much smaller
than the spiral nebulae. The evidence appears to show
that the spiral nebulae are quite outside our system of
stars, and they are tentatively regarded as island universes, each comparable in scale with the system of stars
of which the sun is a member.
The opinion that the origin of stars may be traced to
nebulae has been very widely held, but there has been
some considerable difference of opinion as to whether
a primitive nebula should be regarded as a mass of gas
or as a cloud of dust. Lord Kelvin made the suggestion
that a collection of meteoric stones, vaporized by collisions, would give rise to a gaseous nebula from which
a star might be generated by contraction, in accordance
with the hypothesis of Laplace. The study of stars by
the spectroscopic method led to a classification of them
into types which were supposed to be indicative of
different stages of development, after the nebular stage
had been passed. Five such successive stages have been
distinguished, the one first after the formation from
nebulae representing the hottest and least dense stars,
and the last, the red stars, being the coolest, and nearest
to extinction. But recent discoveries of H. N. Russell
have thrown great doubt, not so much upon the theory
of the nebular origin of stars as upon the notion that




COSMICAL THEORIES


3I3


these five spectral types represent the successive ages
of stars. There is strong reason to think that a star of
the first, or hottest, type is not at the beginning of its
history as a star, but half way through it. In accordance
with this later view a nebula is, or becomes, nonluminous, and remains so until the mass becomes incandescent as a giant red star. It then passes in order
through the successive stages in which it becomes hotter,
until it becomes of the hottest type, and then proceeds
in the reverse order, until it finally becomes again a red
star, previously to its extinction. It is thought that only
the most massive stars ever go through all the different
stages, many of them turning back before the type of
the hottest star is reached. This is in accordance with
the observed fact that all the stars of the hottest type
are of exceptionally great mass. Russell's theory is
chiefly based upon the evidence afforded by the observed
absolute magnitudes of stars. The stars in the three
redder and cooler spectral classes were found to fall
into two detached groups. In one of these groups the
absolute magnitude was found to be nearly independent
of spectral type. In the other group, consisting of stars
of smaller magnitude, that magnitude varies with the
spectral type, being smallest in the reddest type. The
stars of these two groups have been called "giant" and
"dwarf" stars respectively. A mathematical investigation, by Jeans, of the changes to be expected in a mass
of gas when there is a continual emission of radiation
from its surface yields results which are in accordance
with Russell's theory.
A new factor has been brought into the discussion
of all questions connected with the production and radiation of heat by the recent discovery of the presence of
radio-active substances which provide sources of heat
that had previously not been recognized. It has been
maintained recently that radio-active substances in the
interior of the earth may have provided a source of heat




3I4


COSMICAL THEORIES


of sufficient amount to form an important factor in
estimates of the age of the earth based upon tie length
of time it may have required to cool down to its
present condition. The estimates of such age formed by
Physicists, such as Lord Kelvin, which have been regarded by Geologists as inadequate, may, it is held,
have to be completely revised when the emission of heat
from radio-active substances is taken into account. As
regards the effect of radio-active substances in the sun,
it has been pointed out by Lindemann that the radioactive energy in the sun must be regarded as insignificant
in amount compared with the gravitational energy. It
is however possible that some more effective means of
production of energy, of a sub-atomic character, perhaps
involving the actual destruction of matter, may have to
be recognized as contributory to the solar radiation. Of
the efforts made in connection with the great Science
of Geology to give a conceptual account of the mode in
which the earth may be regarded as having reached its
present condition, I am unable to give any account. The
few fundamentally important cosmical problems that I
have discussed may perhaps afford a sufficient illustration of the highly speculative character of those parts
of Natural Science which concern themselves with the
ideal reconstruction of the physical conditions of the
remote past. It is clear that the scientific theories relating to such matters are tentative and hypothetical, in
a sense and in a degree which marks them off from
theories which refer to short time processes, that are
capable of verification or refutation of a more direct
kind than is possible in the case of theories which have
essential reference to immense periods of time. The investigation of cosmical theories involves a completely
justifiable attempt, by an extension of our conceptions
of the actual processes which we can observe, to represent such processes as but portions of processes which
we conceive to have proceeded during periods belonging




COSMICAL THEORIES               315
to a vastly greater time-span than that of any actual
observer. To make the attempt, in a certain sense to
understand the present by means of the past, we are
impelled by ineradicable impulses of the human mind.
The mental satisfaction obtained by imbedding processes
which we can actually observe in long time processes beyond our direct reach is one which we shall never forego.
Needless to say, when we attempt to push back physical
processes ideally, as far as may be, we involve ourselves
in an indefinite regress. Of absolute origins, Science
knows nothing, and we can form no conception. The
so-called primordial state, such as is postulated in the
nebular hypothesis, presents a problem which we do
not attempt to solve. With some such postulation, behind
which we do not go, every attempt at an historical
scientific construction must commence.




XIV


EINSTEIN'S THEORY OF RELATIVITY
THE theory of space, time, and gravitation, propounded by Albert Einstein, is one of the small
class of scientific theories which have at once succeeded
in captivating the attention, not only of the scientific
world, but also, in a marked degree, of a very large
number of persons whose primary interests are not
scientific. The interest of Physicists and Mathematicians
is sufficiently explained by the ambitious character of a
theory of which the aim is to combine in a single scheme
temporal and spatial measurements together with gravitational phenomena, and by the fact that the theory includes a new law of gravitation, and a new Mechanics
involving a breach with certain assumptions formerly
supposed to be axiomatic, if indeed they were ever
explicitly recognized. But in the case of Philosophers
and of others to whom Natural Science is only of mediate
or of secondary interest, another factor enters into the
explanation of the amount of attention they have given
to the theory. This is the prevalence of the idea that
Einstein's theory of relativity has implications which
reach beyond the purely scientific domain; and that it
may serve to throw light upon, and perhaps to lead to
changes in, our general philosophical views of the nature
of reality. The term "relativity" is a very general one,
capable of being employed in various directions in
philosophic thought. Meanings may be assigned to it,
of more general scope, and perhaps of a less definitely
circumscribed character, than the rigidly defined meaning
which is assigned to the term by Einstein and other
Physicists who have concerned themselves with the




EINSTEIN'S THEORY OF RELATIVITY 317
development and exposition of the theory. There has
thus been exhibited in some quarters a disposition to
make this theory a starting point for the development
of relativistic views outside and beyond the scope of the
scientific theory itself.
It is a question of general epistemological interest
whether, apart from the undoubtedly great importance
of the theory in its purely scientific aspect, there is anything in the nature of Einstein's theory which should
properly give it, in the eyes of Philosophers and of the
educated public, a unique position in relation to general
thought, of a kind which other previously existing
physical theories do not occupy. Is Einstein's theory not
only technically, but also generically, different from earlier
physical theories? Does it rest upon a philosophically
different basis? The answer to be given to this question
cannot be considered in complete independence of the
diverging views which are held as to the true character
and functions of scientific theories in general. To those
who regard Science as a means of penetrating to the
inner nature of reality, the theory, so far as it is regarded
as established, will appear to have given new knowledge
of the inner nature of the real world, at least as regards
spatial, temporal, and material relations. But some of
those who hold this view of the functions of Science
have been disconcerted by the highly abstract mathematical form of the theory, in accordance with which
matter, at least in connection with its gravitational
phenomena, exhibits itself no longer as substance, but
only in the guise of specializations in a spatio-temporal
metric. This attitude of mind is well illustrated by the
utterance of Sir Oliver Lodge, a propos of this theory,
which I quoted (p. 60) in my third lecture. The apparent
impossibility of denying the descriptive efficiency of the
abstract scheme, and the admiration excited by the
constructive genius manifested in its creation, struggle
with a reluctance to being drawn away from what, in




318 EINSTEIN'S THEORY OF RELATIVITY
accordance with pre-conceived ideas, and especially the
old hankering after an underlying sub-stratum of matter,
is regarded as concrete reality. On the other hand, to
those whose view of the character and functions of
scientific theories is in general agreement with that which
I have advocated, Einstein's theory of relativity, however highly its scientific importance may be estimated,
will appear to be merely a new conceptual theory, much
more comprehensive no doubt in its scope than older
theories which it aims to replace, but epistemologically
on the same footing. Its extremely abstract character
will occasion no great surprise, since that must be a
feature of every scientific theory which lays such a claim
to precision and comprehensiveness of descriptive power
as does the theory in question. This theory proposes
certain changes in scientific hypotheses formerly accepted as a basis of physical theories, of so striking a
character that it may without exaggeration be described
as revolutionary in its tendency. But if this revolution
became stabilized, and its prospects of becoming so
appear to be bright, although not at present fully
assured, it will, so far as I can see, be a revolution purely
internal to Natural Science, and will in no sense radically
affect the external relations of Science with general
Thought. In fact, my answer to the question I have
raised as to whether the Einstein theory is generically, or
epistemologically, different from other scientific theories
is that no such difference exists: that it has in fact the
same independence of all special ontological assumptions
and theories as has Natural Science in general.
One of the advantages which may be expected ultimately to accrue from the widespread discussion of the
theory by Physicists, Mathematicians, and Philosophers,
is that it will lead to a clarification of ideas as to the
nature and scope of scientific theories in general. The
theory of Einstein is peculiarly well suited for this purpose because, when it is clearly presented, and its founda



EINSTEIN'S THEORY OF RELATIVITY 319
tions are scrutinized, the fact that its basal conception
consists of an ideal scheme,abstract in the highest degree,
incapable of representation by the sensuous imagination,
and in which nearly all elements derived from perceptive
intuition have been removed by abstraction, forces itself
upon the attention in such wise that its essential character
is less easy to disguise than in the case of many current
physical theories. There can be no pretence that a fourfold ordered manifold in which the ordering is neither
spatial nor temporal, with a suitably chosen metric imposed upon it, but not intrinsic to it, is anything else
than a most highly abstract conception. The metrical
relations imposed upon this manifold can be analysed
mathematically, but the manifold has neither in whole
nor in part perceptual actuality, or presentability to the
sensuous imagination. It can be conceived but not
imagined.
I propose to explain in general terms the main characteristics of Einstein's theory of space, time, and gravitation, so far as is possible in the short time at my disposal
for the purpose. Extensions of the theory have been
suggested, -with a view to combining in one whole not
only a theory of space, time, and gravitation, but also
a theory of electromagnetic phenomena in general; but
with such extensions I do not propose here to deal.
I must say, however, at once that the theory is only
capable of complete expression in mathematical terminology, and is of such a character that only a trained
Mathematician who is prepared to spend a considerable
amount of time and energy upon the detailed study of
its foundations can obtain a complete grasp of it.
Fortunately, however, it is possible, without making such
a complete study of the theory, to obtain a general knowledge of the points in which it differs from the older
physical theories. These older theories have proved
adequate for the purpose of measurement of spatial and
temporal magnitudes and of gravitational effects, not




320  EINSTEIN'S THEORY OF RELATIVITY
onlyin ordinarylife, but for nearly all scientific purposes;
and they will always retain that adequacy. It is only for
certain special scientific purposes that Einstein's theory
will, in case it survives the tests which are being applied
to it, be applied in future; it will never be applied in
any of the measurements made in ordinary life.
When it is said that Einstein's theory of gravitation
has overthrown and superseded Newton's theory, that
is in any case only true in a very limited sense. In all
ordinary astronomical cases Einstein's theory gives the
same results as Newton's; it is only in very special cases
that the difference between the results they lead to is
sufficiently large to be discernible by the means of
measurement at our disposal. That Newton's theory has
its limits of applicability will not surprise those of us
who contemplate the discovery of such limits in the case
of all conceptual theories. The claim is made on behalf
of Einstein's theory that the limits of its applicability are
wider than those of Newton's theory. Before Einstein's
theory can be regarded as an established and indispensable part of our stock of scientific conceptual apparatus, it must not only be shown to be logically selfconsistent and to be capable of being applied for the
purpose of providing an adequate symbolical description
of the range of percepts to deal with which it has been
constructed, but it must further be shown that no conceptual scheme of a simpler character is adequate for
the representation of the same range of physical percepts.
The first point in which Einstein's theory has made
a new departure has reference to the measurement of
space and time only. In order to explain the character
of this new departure I must refer to the observations
I made in my lecture on "Time and Space," having
reference to the pre-Einstein theory of their measurements. The private spaces of individuals were, I pointed
out, correlated with a single public, or physical space,
in which all physical objects with which we have to deal,




EINSTEIN'S THEORY OF RELATIVITY 321
either in ordinary life, or in Science, are regarded as
located, and in which all actual spatial measurements are
made. Similarly, the private times of individuals were
correlated with a single public time, in which all measurements of time, by clocks, by the rotation of the earth,
and by other sufficiently approximatively equivalent
processes, are made. That these constructs, of a single
physical space, and a single public time, each independent of the other, are sufficient to represent, by
a process of correlation, the private spatial and temporal
experiences of all individuals, whatever might be their
relative positions and motions, has been the universal
assumption made before the rise of the theory of
relativity; and it has been regarded as axiomatic. For
all the purposes of ordinary life, and also for the purposes
of Science, it had always been found sufficient, until
attention was directed to certain facts of observation to
which I shall presently refer. The first great breach of
the Einstein theory with the previously universally accepted tradition consists in a denial of the sufficiency of
these constructs, a single physical space, and a single
public time, independent of each other, as affording the
basis of a system of spatial and temporal measurements
which will completely describe the spatio-temporal experiences of all observers. The measurements in the
single physical space were all made in approximate accordance with an abstract geometrical scheme, of which
the basis was a manifold of elements with a three-fold
order, into which manifold a Euclidean metric was
introduced. This three-fold ordered manifold with the
imposed Euclidean metric is the space of abstract
Euclidean Geometry, each element, or point, of which
manifold is representable by a triplet of numbers. Its
properties were developed deductively as a scheme of
Euclidean Geometry, and all its metric properties correspond to facts of measurement which can be verified in physical space, with a degree of approximation
HGL                                       31




322 EINSTEIN'S THEORY OF RELATIVITY
dependent on the fineness of our senses when reinforced
by instruments of precision. In the same manner, public
time was correlated with an abstract scheme which consists of a singly ordered manifold of elements represented
by the numbers of the arithmetic continuum. Thus the
theory of the measurements of physical space and public
time rested upon the postulation of two abstract ordered
manifolds, one with a three-fold order, and the other
singly ordered, entirely independent of one another. In
accordance with the theory developed by Minkowski
and Einstein, instead of these two abstract manifolds a
single manifold with a four-fold order of its elements is
postulated, to serve as the theoretical basis for all actual
spatial and temporal measurements. In order that it
may serve both for spatial and for temporal measurements it must be, whenever it is applied, in some manner
split up into two manifolds, the one with a three-fold
order, and the other with a single order. An essential
element in the theory is that this cannot be done in a
unique manner; indeed, if the division were unique, the
system would be reduced to the former scheme, in which
there are two independent manifolds, one to be correlated
with physical space, and the other with public time. An
essential point of the theory is that, although each element of the fundamental manifold is represented by
four numbers, or coordinates, it cannot be said straight
off that three of these are for the purpose of spatial, and
the other for the purpose of temporal, representation.
The mode in which the sets of four numbers are to be
applied to represent both spatial and temporal measurements will depend upon the observer, that is upon the
physical frame of reference which he employs. Thus
space and time cannot be immediately separated out
from one another in any absolute way, in the fundamental four-fold manifold. An element of this manifold,
specified by four numbers, may be regarded as an abstract
event, that is an extensionless object at an instant of




EINSTEIN'S THEORY OF RELATIVITY 323
abstract time, but the actual event to which it may be
made to correspond will be represented by three spatial
measurements representing the position of a point, and
a clock-time, both of which will be dependent upon the
physical frame of reference employed by the observer.
Thus the position and time of a single elementary actual
event will be measured in wholly different ways temporally and spatially, when referred to different physical
frames of reference. In order that the fundamental fourfold manifold may be effective for its purpose, a suitable
metric must be introduced into it; and the whole value
of the theory depends upon making such a choice of
this metric that the manifold may serve its purpose of
providing an abstract representation of spatio-temporal
measurements which will succeed in resuming the actual
facts in the physical domain, as measured with reference
to any actual physical frame which a particular observer
may employ. Einstein's success, so far as it is established,
consists in his having discovered, by the employment of
a refined mathematical Analysis involving the use of the
Calculus of Tensors, how this choice could be made.
On the observational side, the leading feature of the new
scheme is that there is no longer a single physical space
and a single public time, common to all observers, but
that two physical frames of reference in motion relatively
to one another will have different physical spaces and
different time-measurements. As I have already remarked, for ordinary purposes not only of everyday life,
but of Science, the differences in question are negligible;
it is only for certain purposes, to which I shall presently
refer, that the Einstein scheme gives results which are
sensibly different from those obtained on the basis of
the older theory, and thus becomes effective.
It is sometimes said that Einstein's theory involves an
obliteration of all distinction between time and space,
and between past and future, since their measurements
cannot be disentangled from one another in any absolute
21-2




324 EINSTEIN'S THEORY OF RELATIVITY
way. It must, however, be remembered that the original
qualitative distinctions in our spatial and temporal intuitions are untouched by Einstein's theory or by any
other theory of the measurement of space and time. These
qualitative distinctions, as also the qualitative intuitional
distinction between future and past, are removed by
abstraction, even in accordance with the older traditional
scheme, when we pass from intuitional space and time
to abstract space and abstract time; moreover all our
measurements of public time, by means of clocks or by
any other method in which some regular physical process
is employed, are spatial measurements; and thus the
qualitative distinction between time and space has already
been removed by abstraction. This has been spoken of
by Bergson as the spatialization of time. The obliteration
of the distinction, in Einstein's scheme, consists in an
abstraction of all qualities of intuitional space and time
except the element of order, which appertains to both,
a three-fold order in the one case, and a linear order in
the other. The fundamental four-fold manifold involves
the notion of order, that is of an abstract order, neither
specifically spatial nor specifically temporal, but obtained
by abstraction from both, and generalized into a single
four-fold order. Both extension and duration have been
removed by abstraction. That this abstract manifold is
spoken of by many expositors of the theory as the
"world" may perhaps have a certain convenience, but
it is, I think, somewhat unfortunate, especially when it
is stated or suggested that this abstract construct is the
"real world," because such terminology has at least the
appearance of involving a prejudgment of metaphysical
theories.
The next point in which Einstein's theory has introduced new conceptions is connected not only with the
measurement of space and time, but also brings those
measurements into relation with physical phenomena,
especially with the gravitational phenomena of matter,




EINSTEIN'S THEORY OF RELATIVITY           325
and with the electromagnetic phenomenon of light. The
ancient idea that space and time are objects with definite
metric properties of their own, independent of matter,
but forming a kind of framework into which material
bodies, at rest or in motion, could be fitted, and that
these metric properties of space and time are independent
of all physical laws, had become moribund after the
investigations of the foundations of Geometry due to
Riemann and Helmholtz. It was shown by them that
the abstract Geometry that is applicable to describe our
measurements in physical space has a metric which is
not fixed a priori, but is dependent on the fact of experience that there exist bodies which are approximately
rigid and are freely movable in physical space; the
metric is then so determined that the numerical measures
of the distances of pairs of points of such bodies remain
unaltered during their motions. So far the choice was
left open either of employing a Euclidean metric, or a
non-Euclidean metric, with either positive or negative
space-constant, as the basis of measurements in physical
space. But, as I described in my lecture on "Time and
Space," in connection with the suggested experiment of
measuring the angles of a triangle with very large sides
in order to determine whether a Euclidean or a nonEuclidean metric would accord with facts of observation
in physical space, the interpretation of any result obtained
in such an experiment would depend upon the mode in
which we formulate the laws of Optics. This shows that
the system of Geometry adapted to form the basis of
measurements in physical space is dependent upon the
form in which we state physical laws; and thus that
Geometry and Physics are in our experience inseparable
from one another. The choice of the particular system
of metrical Geometry best adapted to describe our spatial
measurements will be such as to be consistent with the
simplest formulation of physical laws, especially those
of Optics. This fact of the interdependence of spatial




326 EINSTEIN'S THEORY OF RELATIVITY
measurements and physical laws is not a convention
but an unalterable datum of observation. In Einstein's
theory, this interdependence is developed to a further
degree, so that spatial and temporal measurements are
brought into relation with the phenomenon of gravitation. In accordance with the theory, any portion of
matter exhibits itself in the presence of a gravitational
field surrounding the matter, in which spatio-temporal
measurements must be made in accordance with a
metric varying from one part to another of the field, and
differing from the metric which would be applicable at
places very remote from gravitating matter. This gravitational phenomenon, considered as involving a special
metric distribution in the field of the body, will exhibit
itself in the orbital motion of a particle, or in the path of
a ray of light, in the field. Thus, for example, the orbital
motion of a planet round the sun is represented, not as
due to a supposed attractive force towards the sun, but
simply as exhibiting the spatio-temporal metric throughout the gravitational field of the sun. Thus, in Einstein's
theory, what we have been accustomed to regard as the
effect of gravitational forces is included in a scheme of
spatio-temporal measurements. In a sense, gravitation
is included in Geometry, only that Geometry is fourdimensional, non-Euclidean, and it has a metric of a
kind more complicated than in the older schemes of nonEuclidean Geometry; moreover matter, at least in its
gravitational aspect, only exhibits itself in and through
the spatio-temporal metric. The fact that measurements
of time and space are not independent of the physical
phenomena of gravitation, and of light, leads to the idea
that all the laws of physics and the laws which govern
spatio-temporal measurements must be regarded as belonging to one interconnected whole, and should be
conceptually represented by a single unitary scheme.
Einstein's theory is an attempt to attain to such a
unitary scheme, at least so far as the gravitational phe



EINSTEIN'S THEORY OF RELATIVITY 327
nomena are concerned. Extensions of Einstein's conception have been already suggested by Weyl and others
with a view to embracing electromagnetic phenomena
in general, as well as gravitation, in a single conceptual
scheme.
In order to describe in more detail the nature of the
theory of relativity it is necessary to sketch the history
of its origin. The theory has been developed in two
stages, the first of these culminating in what is known
as the special theory of relativity, and the second in the
general theory of relativity. The special theory of relativity is accepted by some Physicists and Mathematicians
who are sceptical as regards the general theory, at least
in the particular form in which it has been developed
by Einstein. The special theory takes no account of the
phenomenon of gravitation; it is applicable strictly only in
localities very remote from large gravitating masses; but
it applies also very approximately to optical phenomena
in weak fields of gravitation such as that of the earth.
It had its origin in experimental investigations undertaken in connection with the electromagnetic theory of
light, in accordance with which light was regarded as an
electromagnetic disturbance propagated through the
ether. After various attempts to elucidate the structure
of the ether and its relations with material bodies, it
came to be regarded for the most part as a substance
which is undisturbed by the motion of material bodies
through it, and as freely interpenetrating such bodies.
On this theory the ether would form a natural frame of
reference with respect to which all motions of material
bodies might at least ideally be measured; and it became
a matter of importance, as a test of the theory, to detect
by experimental observation the existence of the velocities
of material bodies relative to the ether. It is known that
light is propagated with a velocity approximately of
300,000 kms., or I86,ooo miles, a second. If a body is
moving in the direction of, or in the opposite direction




328 EINSTEIN'S THEORY OF RELATIVITY
to, a ray of light, with a velocity through the ether which
is an appreciable fraction of the velocity of light, the
ray of light would appear to have a velocity relative to
the body, less in the first case, and greater in the second,
than the velocity of propagation of light through the
ether; the defect or excess being the velocity of the
body through the ether. The earth, in its orbital motion
round the sun, has a velocity relatively to a frame fixed
by the sun and stars of about 30 kms., or I8 miles, a
second. Although the velocity of the earth with respect
to the ether was unknown, if it were assumed that it is
of the same order of magnitude as the orbital velocity
of the earth, it seemed certain that the velocity relative
to the ether might be detected, and its magnitude determined, by comparing the velocity, relatively to the earth,
of a ray of light in the direction of the earth's orbital
motion with that of a ray in the perpendicular direction,
provided sufficiently precise measurements were made
in an apparatus devised for the purpose. The celebrated
experiment of Michelson and Morley, carried out in
1887, was devised in order to establish the existence of
the expected effect due to the velocity of the earth
relative to the ether; the effect of such velocity was
expected to be apparent if the velocity of the earth in
the ether was only a quarter of its orbital velocity.
A beam of light from a single source was divided by
partial reflection at a mirror into two portions, one in
the original direction of the beam, and the other at right
angles to the first. These two portions were reflected
back by mirrors, and struck the first mirror again, when
portions of them would be re-united. The whole apparatus could be rotated into any position, and could be
fixed so that the original beam was in any required
direction. The two portions of the beam which are reunited, having required different times to pass from the
first mirror to the reflecting mirrors and back again,
exhibit interference fringes. A rotation of the instru



EINSTEIN'S THEORY OF RELATIVITY 329
ment was expected to show a displacement of these
fringes, owing to a change in the retardation of one
portion relatively to the other. But when various positions
of the apparatus were explored, no trace of such displacement of the fringes was observed, whereas the
expected displacements would have been easily capable
of measurement had they existed. The same negative
result was reached by a still more refined experiment
conducted by Morley and Miller in I905. This and other
experiments have led to the conclusion that it is impossible to detect by any observation the motion of a
material body relatively to the ether. The first attempt
to account for this inability to detect any effect of the
motion of matter through the ether was made by
Fitzgerald, and independently by Lorentz. It was suggested that, when a material body is in motion through
the ether, its dimensions in the direction of the motion
are shortened by an amount depending upon the square
of the ratio of its velocity to the velocity of light; and
thus that the whole apparatus in the Michelson-Morley
experiment is shortened in this way in the direction of
its velocity through the ether, thus annulling the displacements of the interference fringes. This so-called
Fitzgerald contraction of a body moving through the
ether could only be conceived as due to some interaction of the ether with the constituents of the moving
body; the hypothesis of its existence was much discussed, and led to various difficulties.
The interpretation of the observed facts that was given
by Einstein was of a radically different character; it was
in this connection that he propounded the theory which
is now called the special theory of relativity. The first
postulate of this theory is that all physical phenomena,
as observed from a material body as frame of reference,
appear to be completely independent of any uniform
translational motion which that body may have relatively
to another physical frame of reference. The second




330 EINSTEIN'S THEORY OF RELATIVITY
postulate of the theory is that the velocity of light is
independent of the motion of the source of light. If
these postulates be accepted, the negative result of the
Michelson-Morley experiment is immediately accounted
for. In accordance with these postulates the velocity
of light is the same when referred to any two frames
of reference in uniform translational motion with respect
to one another, and is independent of the origin of the
light. It should be observed that the traditional Newtonian Dynamics is in full accord with the foregoing
postulate of relativity, because the Newtonian equations
of motion of a dynamical system referred to a given
material frame of reference are unaltered when a new
frame of reference is employed which is in uniform
translational motion with respect to the given frame.
But that is not the case with Maxwell's equations for the
representation of electromagnetic phenomena relatively
to a frame of reference fixed in the ether. It was shown
by Larmor and Lorentz that these equations are unaltered in form when the coordinates and the time are
transformed by means of a certain linear transformation, of a less simple character than the transformation
of the coordinates in a dynamical system, from one
frame of reference to another in uniform translational
motion with respect to the first. In this so-called Lorentztransformation, the new coordinates are expressed in
terms of the old, in a form which involves the time as
well as the translational velocity of the new frame of
reference, and the new time-measurement involves not
only the original time-measurement but also the original
spatial coordinates. It thus appears that the Newtonian
dynamical scheme and the Maxwellian electrodynamical
scheme which represents optical phenomena, when taken
together, do not satisfy the postulates of the restricted
principle of Relativity; and consequently one of these
must be changed if that principle is to be accepted. The
bold step taken by Einstein consists of a rejection of the




EINSTEIN'S THEORY OF RELATIVITY 331
Newtonian system of Dynamics, and the substitution of
a new dynamical scheme in which the Lorentz-transformation is applicable, not only in the electromagnetic
equations, but also in Dynamics. The consequences and
implications of this step are of a far-reaching character.
A complete revision of the old ideas about spatial and
temporal measurements is involved in the change. When
we pass from the measures of space and time which an
observer with his physical frame of reference employs to
the corresponding measurements of time and space employed for the same physical event by another observer
with a frame of reference in uniform translational motion
with respect to that of the first observer, the scales both
of spatial and temporal measurements of one and the
same event are different for the two observers. The
measure of time for the one observer depends not only
upon the measure of time of the other observer, but also
upon his spatial measurements, as well as upon the
relative velocities of the two observers; thus there exists
no single system of measurement of time which is
common to all observers. Neither can the two observers
employ one and the same system of spatial measurement.
If two events occur at different places, the interval of
time between them will be measured differently by two
observers in motion relatively to one another; for one
observer the two events may be simultaneous whilst for
the other observer the same events may occur at different
times. The distance between the places at which the two
events occur will be in general different for the two
observers. It might appear that, as the intervals both
of time and of space which distinguish two events depend
upon the observer, there is no invariant relation between
the two events; that is no relation which is common to
all the observers. When, however, the complete scheme
in an abstract form is set up, it appears that this is not
the case. The important step was taken by Minkowski
of establishing that the Lorentz-transformation is capable




332 EINSTEIN'S THEORY OF RELATIVITY
of simple representation and interpretation if an abstract
four-fold ordered continuous manifold, which he called
the "world," with a certain metric system imposed upon
it, is taken as the conceptual basis upon which all actual
spatio-temporal measurements are made to rest. In view
of the use which is to be made of the elements of this
manifold, any one of which is to be regarded as correlated with an actual observable event which for any
observer is extremely small both in extension and in
time-measure, these elements of the manifold may be
spoken of as abstract elementary events; they are frequently also spoken of as points in a four-dimensional
geometrical space, but such language ought not to be
taken to imply that our intuitional notions of space and
of time are applicable to this fundamental manifold. The
notion of continuous order, taken in the abstract, is the
sole remaining element of our spatial and temporal intuitions which is a constituent of the conception of this
manifold. The order of the elements is assigned by correlating each element with four real numbers, often spoken
of, analogically, as the coordinates of the element. The
metric is imposed upon the manifold by means of a
definition of what may be called the " separation of two
elements." It is taken to be the square root of the sum
of the squares of the differences of the corresponding
coordinates of the two elements; but in this sum three
of the squared differences are taken to have the negative
sign, and the metric is not therefore strictly an extension
of the ordinary Euclidean metric. If the velocity of
light is not taken to be unity, the positive squared difference in the expression for the "separation" must be
multiplied by the square of the constant numerical
measure of the velocity of light. If now, by means of
the linear Lorentz-transformation, a new system of coordinates is introduced into the manifold, so that each
element is represented by a new quadruplet of numbers
determined from the original quadruplet by means of




EINSTEIN'S THEORY OF RELATIVITY             333
the Lorentz-transformation, it will be found that the
"separation" between two elements has exactly the same
value when expressed in terms of the new coordinates
as it had originally. It is this fact upon which the utility
of the abstract scheme depends. The "separation"
between two particular events is an invariant for all
observers. A change of coordinates in the manifold, of
the kind described, corresponds to, and represents, a
change from the spatio-temporal measurements of one
observer, with his own scale of measurements, to the
measurements of another observer in uniform motion relatively to the former one. Thus this fundamental manifold forms the ultimate conceptual basis for Einstein's
restricted theory of relativity which satisfies the two
postulates I have already specified. Of the four coordinates of an element of the manifold, for a particular
observer in the physical domain, that one of the coordinates which appears with the positive sign in the
expression for the "separation" of two events may be
taken to represent by correlation his measure of time,
and the other three his spatial measurements relatively
to his rectangular frame of reference. But we cannot
say that, of the four coordinates of an element of the
fundamental manifold, one is to be always correlated
with temporal measurements and thus represents abstractly a time, and the other three represent abstractly
spatial coordinates. This is the case for one particular
observer, but another observer in motion relatively to
the former will have to employ an entirely different set
of four coordinates in the manifold for correlation with
his spatio-temporal measures; this set being related with
the former set by means of the Lorentz-transformation.
Thus in general each one of the four coordinates in the
manifold is correlated with a mixture of spatial and
temporal measures in the physical domain.
A set of events in the physical domain, represented
by a material particle which never impinges on any




334 EINSTEIN'S THEORY OF RELATIVITY
other particle, will be correlated with a continuous
set of elements in the fundamental manifold, forming
what may by analogy be called a straight line, and this
is spoken of by Minkowski as the world-line of the
particle. This purely statical abstract object in the manifold represents conceptually the whole history of the
material particle, past and future; its interpretation
spatially and temporally will vary according to the circumstances of the observer, involving the way in which
he judges from his own relative standpoint the spatial
and temporal circumstances of the particle of matter in
question. The principle of relativity, as embodied in the
two postulates, is incompatible with the conception of
ether as a substantial medium for the transmission of
light. There have at all times been great difficulties in
formulating the properties of an ether which should
satisfy the conflicting demands which fact and theory
seemed to require. Notwithstanding these difficulties
there has been much reluctance on the part of Physicists
to give up a conception which was designed to afford
a pictorial representation of electromagnetic phenomena,
and which appealed strongly to those who regard the
ether as a concrete, though not directly perceptible,
reality. The idea has however gained ground that it is
in the equations of the electrodynamic theory that the
really effective formulation of that theory is to be found,
and thus the ground has been prepared for that final
removal of the formerly useful notion of the ether which
is involved in the acceptance of Einstein's theory, even
in its restricted form.
The restricted principle of relativity, which was completely stated by Einstein in 1905, changes radically the
notions of the measurement of time and space which
were employed in the Newtonian Dynamics and in
ordinary affairs, but the principle suffers from the defect
that the relativity is applicable only to material frames
of reference in uniform motion of translation with




EINSTEIN'S THEORY OF RELATIVITY 335
respect to one another, and that it takes no account of
the phenomena of gravitation. The decade after I905 was
spent by Einstein in an endeavour to remedy these
defects; and in I9I5 he found himself able to propound
a complete principle, in which the relativity is applicable
to all frames of reference in motion of any kind with
respect to one another. The general principle of relativity includes the conception that all actual spatial and
temporal measurements are dependent upon, and vary
in a definite manner with, a material frame of reference
employed by an observer. The scheme includes a
mode of taking into account, and measuring by means
of spatio-temporal measurements, gravitational phenomena as exhibited in gravitational fields. This mode of
treating gravitation is fundamentally different from that
of Newton; there is in it not even a suggestion of anything that could be regarded as a causal law of gravitation, in accordance with the older traditional meaning
attached to the term causation. The special spatiotemporal peculiarities of a field of gravitation are taken
to give the only theory of gravitation that Science
requires, or can attain. Newton's law of gravitation, in
accordance with which the gravitation between two
material particles is represented by a stress proportional
in magnitude to the product of their masses and inversely as the square of their distance from one another,
being independent at any instant of the motions of the
particles and of all other matter, had become quite indefinite in meaning. In the first place, the mass of a
given particle has, in modern Physics, lost the characteristic property of having a constant value independent
of the motion of the particle relative to an observer. In
accordance with the electron theory of matter, which
rests upon the observed facts of radiation and radioactivity, a material particle is constituted in part at least
of electrons, of which the effective mass increases when
their velocity is increased. Thus the mass of a particle




336 EINSTEIN'S THEORY OF RELATIVITY
may be sensibly changed if it is set in motion as a whole,
with a velocity comparable with that of light, or when
the motions of the electrons within it are considerably
altered by the receipt of energy from without. In fact,
in accordance with the special principle of relativity,
the mass of a system is the total energy in it divided by
the square of the velocity of light; this energy being
measured relatively to the observer. It is then no longer
clear how the magnitudes of the masses in Newton's
law are to be fixed. Again, the distance between the two
particles has no longer an absolute measure independent
of their motions relative to an observer. Further, the
fact that gravitational force, in Newton's theory, depends at any instant only on the positions of the particles
at that instant is not consistent with the conception of
the propagation of gravitation with finite velocity through
a medium; and thus Newtonian gravitation was never
linked up with other physical phenomena in any unitary
scheme.
As I explained in my lecture on Dynamics, in the
Newtonian system all actual phenomena of motion are
described upon the basis of a conceptual scheme in
which there is an absolute frame of reference; but in
physical space, it is possible with a degree of approximation sufficient for the purposes of any special case, to
determine so-called Newtonian frames of reference which
can be correlated with the absolute conceptual frame.
Any frame of reference which is in rotational motion, or
in accelerated translational motion, with respect to a Newtonian frame can only be employed if certain " fictitious
forces," among which is the so-called centrifugal force in
the case of rotation, are introduced into the equations of
motion. An essential element in Einstein's scheme consists in his principle of equivalence, which involves a
denial of any distinction between such fictitious forces
and gravitational forces. All of them alike are regarded as
due to the gravitational field, and the mode in which this




EINSTEIN'S THEORY OF RELATIVITY 337
gravitational field exhibits itself is in the spatio-temporal
measurements suitable to the particular frame of reference
employed. In any sufficiently small region it is impossible
by any experiment to distinguish between a fictitious
and a gravitational field of force. The Einstein scheme
does not depend, as does the Newtonian, upon the selection of any specially suitable frames of reference; all
such frames are on a parity, and in this the completeness
of the relativity consists.
This independence of any particular frame of reference
is essentially connected with the fact that all actual processes of measuring space or time consist in the establishment of the coincidence of two points belonging to two
material bodies; and such a coincidence is a fact which
is unaltered by any change in the frame of reference
employed in making the measurements.
In Newtonian Dynamics the equality of the inertial
mass and the gravitational mass of a material body remained as a bare fact, derived from observation, which
appeared from the theoretical point of view to be merely
an accident. In Einstein's conceptual scheme the two
are indistinguishable, or rather identical, in accordance
with the fundamental postulates of the theory. An essential consequence of Einstein's scheme is that any
phenomena which an observer perceives in his neighbourhood to be due to a gravitational field would be
perceived unaltered if the gravitational field were not
present, provided that the observer makes his frame of
reference move with the acceleration characteristic of the
gravitational field at the place at which he makes the
observation. This amounts to the assertion that it is
possible to eliminate a gravitational field, at a particular
place, by a proper choice of the frame of reference. The
fundamental four-fold ordered manifold is employed as
the conceptual basis of the general theory of relativity,
as in the case of the special theory, except that the metric
imposed upon it is of a more complicated character,
HGL                                       22




338 EINSTEIN'S THEORY OF RELATIVITY
designed to take into account, by correlation, the existence
of gravitational fields, and to exhibit their presence in
spatio-temporal measurements. The determination of
the precise mode of correlating the conceptual scheme
with actual measurements would appear to be one of
the most difficult points in the whole theory.
The precise mode in which the theory of this metric was
developed by Einstein is of a so highly technically mathematical character, that I can only give the briefest and
most superficial indication of its nature. In this matter
the geometrical investigations of Gauss and Riemann
were the original source of inspiration. Riemann's theory
of a continuous ordered manifold, designed by him for
the purpose of investigating the foundations of Geometry,
led in the hands of Ricci, Christoffel, and others, to
a mathematical development known as the theory of
tensors, and this theory was utilized by Einstein, and
developed for his purpose of determining the nature of
a metric system to be imposed upon the fundamental
manifold, of such a character that it could be applied to
the spatio-temporal metrical characterization of observed
gravitational fields. Two elements of the fundamental
manifold are regarded as neighbouring elements when
each of the four coordinates of one of them differs by a
very small number (more exactly a differential) from the
corresponding coordinate of the other. The "separation" of two such neighbouring elements is defined as
the square root of a quadratic function of the differentials
of the four coordinates. The ten coefficients in this
quadratic function are in general functions of the coordinates of the element, and are taken to be the
potentials characterizing a gravitational field. In order
that this definition of the differential "separation" may
be the basis of a definite metric in the manifold, these
coefficients must satisfy a certain number of conditions
which involve also the gradients of these coefficients.
The determination of these conditions is made in ac



EINSTEIN'S THEORY OF RELATIVITY 339
cordance with the developed Riemannian theory, which
had its origin in Gauss' theory of the curvature of surfaces in three-dimensional space. Einstein succeeded in
overcoming the great difficulty of showing how all these
conditions could be satisfied so as at the same time to
make the metric available for the representation of actual
spatio-temporal measurements in gravitational fields,
such as those which are given us by experience.
The history of a material particle which does not
impinge upon any other particle is completely represented by a set of elements in the fundamental manifold
which may, by analogy, be spoken of as a geodesic in
that manifold, and is obtained as an extremal of the
integral of the differential separations. This geodesic
is the world-line of the particular particle, and it is
analogous to a curved line in ordinary geometry whenever the particle is in a gravitational field; when there
is no such field the world-line is the analogue of a
straight line, as in the special theory. The world-line
forms the conceptual basis of a spatio-temporal description of the motion of the particle, as estimated by an
observer who chooses arbitrarily his material frame of
reference. All such frames may be employed indifferently,
the observed path of the particle being relative to the
particular frame employed; but the world-line is absolute for a particular material particle.
One of the most important applications which Einstein
has made of his theory is to the motion of a planet in
the gravitational field of the sun. His result contains a
correction to the Newtonian law of force upon the planet
which is in any actual case very small, but the effect of
which in one case, that of the planet Mercury, is sufficient to render it capable of being observed. This has
formed one of the tests of the applicability of Einstein's
theory, and the theory appears to have proved itself able
to stand the test. In accordance with Newton's theory
of gravitation, a planet would move over and over again
22-2




340 EINSTEIN'S THEORY OF RELATIVITY
in a fixed ellipse with the sun in one of the foci, in accordance with Kepler's law, if it were not subject to the
disturbing effects due to the other planets. Thus the
line joining the sun to the planet, when in perihelion,
that is when nearest to the sun, would preserve for all
time a direction fixed relatively to the stars. However,
one of the effects upon the orbit produced by the gravitation of the other planets is to give this line a very small
change of direction which steadily accumulates, so as to
make the change in a century sufficient to be observed.
The orbit of the planet Mercury is more elongated than
in the case of other planets, so that it can be observed
at what times it is in perihelion more accurately than
in the case of a planet whose orbit is more nearly circular. The motion of the line due to the disturbance
of the other planets was calculated by astronomers to
amount to 532" in a century; but the actually observed
amount was found to be 574", so that the excess of 42"
remained to be accounted for. It was calculated by
Einstein that the effect of his amendment to Newton's
law would be that there would be an advance of the line
of perihelion amounting to 43" in a century, which
differs only by a very minute amount from that of the
discrepancy to be accounted for. It thus appears that
Einstein's law leads in this crucial case to a result which
is in close agreement with observation, and in which
Newton's law is in default. It is not possible to apply
a similar test in the case of the other planets, either
because the corresponding advance is too small, or because the orbits are too nearly circular to make sufficiently
accurate observations possible.
Another crucial test of the theory was provided by
observation of the deviation of direction in a ray of
light which takes place when the ray passes very near
to the sun. It has for some time been known that radiation has inertia, as manifested in radiation-pressure; in
accordance with Einstein's theory it consequently has




EINSTEIN'S THEORY OF RELATIVITY 341
gravitational mass, or weight. As the result of a
calculation made by Einstein, the effect of the intense
gravitational field near the surface of the sun would be
that a ray of light passing very near that surface would
appear to an observer on the earth to be deflected through
an angle of i".74. If Newton's theory of gravitation
were applicable the deflection would be just half this
amount, o"'87. It was decided to put to the test of
observation the question which of these two values represents the deflection that actually takes place. At the
time of a total eclipse of the sun, a deflection of the light
from a star very near the sun would exhibit itself in an
apparent displacement of the star from its true position
in a direction away from the sun. The very delicate
operation of measuring the displacements of position of
stars near the sun at the time of the total eclipse on
May 29, I919, was undertaken by astronomers in two
expeditions, one in Brazil, and the other in the Gulf of
Guinea. Although the observations of the latter expedition were very seriously hindered by cloudy weather,
in both cases the results of observation of several stars,
after the elaborate process of correction for various errors
of observation had been carried out, were found to be
very fairly in accordance with Einstein's prediction, and
definitely to rule out the correctness of the deflection as
calculated by Newton's theory. Thus the result of the
observations was distinctly in favour of Einstein's theory
as against the Newtonian. It must, however, be remembered that such observations are of an extremely
delicate character, and involve various sources of error.
Accordingly it is necessary to await the results of further
observations of the same kind before complete confidence
may be placed upon the results of this test.
The third practical test of the theory which has been
made has not as yet been brought to a decisive conclusion. It was expected that, in accordance with the
theory, when the spectral lines of a chemical substance




342 EINSTEIN'S THEORY OF RELATIVITY
in the sun are observed, they should show a displacement from the position of the spectral lines of the same
substance on the earth. The results obtained by different
observers in the course of their attempts to verify or
disprove this predicted effect are discordant and contradictory. It appears therefore that, up to the present
time, no decisive evidence has been obtained by means
of which it can be definitely decided whether the theory
satisfies this test or not.
Whatever be the ultimate fate of Einstein's theory, and
to whatever modifications it may in the future be subjected, it is, as it stands, of the highest interest, not only
on account of its comprehensive character, and on account of the novelty of its conceptions, but also as a
chapter in the history of Science. It could not have
arisen apart from its two great roots, the one the physical
theory of Electromagnetism, including the theory of light,
and the other the highly abstract theory of Geometry in
its most generalized form. The latter includes a whole
series of investigations into the foundations of Geometry
which reach far back into the past, culminating in the
work of Riemann and Helmholtz, who were the first to
perceive that Geometry, as applicable to actual measurements, is not really independent of Physics, and which
have been continued in Mathematical detail by others.
All this line of mathematical work was carried out
almost entirely by thinkers who had no hope that their
labours would ever form an essential groundwork of a
great physical theory. This piece of history illustrates in
a most striking manner the fact that there is never any
certainty that the most abstract mathematical theory
which, although it of course has its ultimate roots in the
perceptual domain, appears to have no direct relations
with that domain, may not turn out to be of the last
importance in relation to some unforeseen theory of
physical phenomena. That the further development of
Einstein's theory, so that it may embrace all the phe



EINSTEIN'S THEORY OF RELATIVITY 343
nomena with which Physics has to deal, may be subject
to limitations which, in its present form at least, it cannot
avoid without undergoing at least some modification, is
indicated by the rise and development of the theory of
Quanta. It would appear that sub-atomic Dynamics is
essentially dissimilar in character to that of systems
which consist of large aggregations of atoms. This would
appear to indicate that Einstein's scheme which, like the
Classical Dynamics, is a continuous theory, may prove
to be inapplicable as the conceptual basis for the representation of sub-atomic phenomena. Suggestions have
even been made recently that a discrete manifold may
be requisite as the conceptual basis of spatio-temporal
measurements for this purpose, instead of the continuous
manifold employed at present by the Einstein theory.
It is interesting to observe that Riemann himself made,
at the conclusion of his celebrated dissertation on the
foundations of Geometry, a remark which showed that
he had a prophetic insight into the possibility of his conceptions being one day linked up with the physics of
matter. The passage to which I refer states that':
The question as regards the validity of Geometry in the
region of the infinitesimally small is connected with the question
relating to the inner ground of the metric of space. In connection with this question which can surely be reckoned as
belonging to the doctrine of space, the above remark can be
applied, that, in a discrete manifold, the principle of the metric
is contained in the conception of the manifold itself, but that,
in a continuous manifold, it must come from outside. Thus the
reality upon which space is based must either form a discrete
manifold, or else the basis of the metric must be sought outside,
in binding forces that act upon it.
1 Gesamm. Werke, 2nd ed., Vol. I, p. 285.




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BIOLOGICAL SCIENCE
AMONG the manifold objects which we perceive
1in the physical world, those which we call living
organisms, whether animals or plants, are universally
recognized as forming a class, the members of which are
distinguished in a variety of ways from other objects in
the whole physical complex. The term Biology I here
take in the most general sense, as denoting the whole
group of sciences which concern themselves with the
study of the physical characteristics of living organisms,
their forms and parts, the various processes in which
their parts are involved, and with the physical relations
which they have with one another and with their environment, their genetic relations, and their geographical
distribution. As all Science is a higher development of
the kind of knowledge which we associate with the expression common sense, it may be worth while to inquire
what are the conceptions of a living organism possessed
by those persons who are not scientific biologists, but
who possess sufficient powers of reflection to be able to
formulate some kind of answer to the question what is
to be understood under the term living organism, as
distinct from objects that are not regarded as living. The
concept of a living organism, as formed by such persons,
cannot be expected to be capable of definition in clear
cut terms, in which any single criterion of a simple
character is employed as a decisive test which could be
applied to any and every special instance, to determine
whether a particular object is, or is not, a living organism.
The descriptive account of the meaning of the term
animal which might be given by a non-Biologist would




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refer to a group of characteristics, all of them present
in ordinary cases, but some of which might be absent
or indiscernible in a special instance. In fact such a
definition would not be applicable to cases on the borderline. Perhaps the most general descriptive statement
that might be made is that an animal is an object which
appears to have an active power of self-maintenance as
against its surroundings; that this involves a behaviour
on its part which has at least the appearance of what
can only be described in psychical terms as purposive
activity, with a view to the preservation of its individuality.
It would be added that, although it has a relative permanence of form, an animal takes nutriment, grows, and
reproduces itself. Reference would be made to the fact
that an animal is distinguished by the complex, and apparently purposive, character of its reaction to external
stimuli; and that these reactions often depend upon the
past history of the individual, to a degree much greater
than in the case of non-living objects. In psychical
terminology, in fact, an animal can learn from experience.
A person who has not studied Biology would find it
much more difficult than in the case of an animal to
describe in general terms what he understands by a plant,
because a much closer examination is required before
the main characteristics of a plant can be brought to
light, especially as regards its relations with its environment. At least some simulation of what appear to be
characteristic properties of living organisms may be found
in the inorganic domain. For example, the growth of
crystals, and the phenomena of elastic fatigue, and of
hysteresis, present analogies with the growth of organisms
and with their dependence upon past history.
The questions now present themselves, what is the
scientific definition of a living organism? What are the
scientific criteria of the distinction between an animal
and a plant? Is it possible to give such definitions without the employment of psychological categories? It will,




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I think, be admitted that it is unreasonable to expect
that these questions can receive answers that are more
than tentative, until Biological Science has reached a very
advanced stage. During the earlier stages, only amplifications, qualifications, and modifications, of common-sense
criteria and distinctions can be expected. In fact, final
answers to such questions must be regarded as goals to
be attained, if at all, only when Biological Science has
reached a very high stage of development. We must be
prepared to contemplate as at least a possibility, the
answer to the question, what is the ultimate distinction
between living matter and non-living matter? to be that,
as long as we remain within the categories of Natural
Science, in the restricted meaning of the term which I
have adopted in these lectures, that is, excluding all
psychical and psychological categories, and employing
only those of physico-chemical schematism, there is no
such ultimate distinction. That, in fact, the difference
between what is called living matter and what is called
non-living material can be represented only as a difference
of degree in complexity of structure and of the physicochemical processes associated with the two types of
material. In this connection it is well to remember that
the range of organisms with which modern Biological
Science deals has been enormously extended by the use
of the microscope; and that thus, the border-line cases
of organisms of comparatively simple structure and functions, inaccessible to ordinary observation, form a most
important part of the study, in relation to such questions
as those I have indicated.
However different the biological sciences, in respect
of procedure and history, when regarded superficially,
may appear to be from the inorganic sciences, the method
applied in dealing with the phenomena in which living
organisms are involved is fundamentally the same as that
which is applicable to the inorganic domain. Based upon
the observation, sifting, and classification, of the facts




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of perception, concepts and conceptual schemes are constructed which suffice to represent sequences of phenomena of certain classes in the domain under consideration. All that I have said in the earlier lectures as to the
total inability of scientific method to attain to explanations, in the full sense of the term, of any happenings
in the perceptual domain, is as valid in Biology as in
Physics and Chemistry. All modern biological theories,
in their descriptions, have reference to processes of the
physico-chemical order, and are consequently for this
reason, as well as others, subject to the same limitations
as Physics and Chemistry. Thus Biology, as a great department of Natural Science, is, and from the nature of
its methods must always remain, unable to discover an
answer to the metaphysical question what life is, in what
its essence consists. What it can do is to give conceptual
descriptions of what living organisms do, of the phenomena of which the parts of the organism are the seat,
and of the interactions of the organism with its environment, which includes other living organisms of similar
and dissimilar kinds. It can endeavour to ascertain the
physical conditions subject to which what we call life
manifests itself. As in the case of the inorganic Sciences,
the limitations imposed by the character of its methods
have by no means always been fully recognized by those
who have built up the edifice of Biology; and this has
frequently led to the employment of language, in the
statement of theories and laws, which, taken as it stands,
implies that efficient causation has been discovered, immanent in the processes and sequences that are described.
The assumptions of physical realism have been accepted
by many, probably most, men of Science; but of these
assumptions Biological Science is really as independent
as are Physics and Chemistry.
A fundamental question in relation to the character
of Biological Science is that as to the nature of the concepts which it must employ in formulating its conceptual




348


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schemes and laws. In the first place, it may be asked
how far Biological Science can confine itself to the concepts employed in Physics and Chemistry; such concepts
as are sometimes spoken of as mechanical, although for
historical reasons it is probably better to avoid the expression. The fruitful work done by modern Physiology
has been carried out subject to the assumption that the
categories of Physics and Chemistry are sufficient to
form the basis of that work. The striking success of
investigations of this order gives ample warrant for belief
in the utility of the assumption as a working hypothesis,
and is such as to afford a justification for the hope that
a continuation of investigations on the same lines may
lead to an indefinite extension of physiological knowledge. It would however appear that these physiological
investigations are confined to special processes in the
organism, and require integration before they can be
applied to give an account of the coordinated happenings
in which the organism as a whole is involved. Moreover,
it should be remarked that the concepts of Physics and
Chemistry cannot be regarded as once for all fixed; at
the present time they are very noticeably in a state of
flux. Further, it has to be taken into account that, in
accordance with the discoveries of the modern Science
of Biochemistry, the chemical processes in the living
organism exhibit marked peculiarities which differentiate them from those which take place in non-living
matter.
It may next be asked whether Biological Science requires, in addition to the concepts of Physics and
Chemistry, further concepts of a kind not specifically
psychical. Attempts have been made to supplement the
physico-chemical concepts by others, for the purpose
of attaining a more complete and satisfactory account of
what goes on in the living organism, than can be provided
with the help of the former concepts alone. The difficulties encountered in such attempts have been great.




BIOLOGICAL SCIENCE                349
It has proved difficult, or as some would say, impossible,
to define and delimit such concepts with a precision of
the kind they must have if they are to perform a useful
function in scientific theories. That the want of such
concepts is widely felt is clear, but however desirable
their construction may be, without adequate definition
and circumscription, and without precise postulations
as regards their relations with the physico-chemical concepts, they will represent little more than words employed
to cloak ignorance. Besides the difficulties of precise
definition and of correlation with existing concepts, and
assuming that these difficulties can be overcome, there
remains the question as to the utility of such concepts
in formulating and extending physiological knowledge.
It has, for example, been maintained by Dr J. S. Haldane
that the concept of the organism as a whole, as distinct
from all the parts and their physico-chemical relations,
but having relations with those parts, is a necessary one,
if we are to have any understanding of the living body.
This concept of an entity which is concerned with the
unification of the whole complex of phenomena in the
living body is an exceedingly plausible one, and from
various points of view commends itself in a high degree
to the mind. But has it been formulated, and have its
relations with the parts been formulated, in such wise
as to enable Physiologists really to increase their knowledge of the living organism? Is such a concept, from
the point of view of Physiology, more than an aspiration,
an idea of great cogency? Does it assist in solving the
problem of the integration of all the processes in the
body? Has not all the precise knowledge which has
actually been obtained by Physiologists as to these processes dispensed with this conception, because it has
lacked the requisite distinctness of outline? I gather
that, up to the present time, however much its need
may be felt, such a purely physiological concept has
not been constructed so as to satisfy the criteria I have




350         BIOLOGICAL SCIENCE
indicated. There can of course be no a priori objection
to the introduction of such a concept in Physiological
Science, provided it is properly defined and proves itself
useful in furthering the aims of that Science. Physiologists appear for the most part to think that the true
line of progress of the knowledge of the organism which
they seek to obtain must, at the present time, be in
accordance with the methodological assumption that
the living being is to be regarded, for their purposes,
as a physico-chemical mechanism, to be investigated by
methods in which quantitative chemical processes and
measurable physical processes are alone dealt with. It
may be true, and in the judgment of most of us it is
true, that the physico-chemical categories will always
prove to be inadequate to do more than describe conceptually greater or smaller, but certainly limited, tracts
of phenomena in the living organism. Nevertheless,
the Physiologist who, in the light of present conditions,
accepts as his present policy that of restricting his investigations to the tracing out of physico-chemical, and
consequently measurable, processes has full justification for his attitude; and he is the best judge of the
prospective value of that policy. It is only when his
policy is developed into a dogma that he gives an opening
for legitimate criticism. It must, I think, be admitted
that the problem of relating all the phenomena in the
organism with a concept which represents the aspect of
the whole organism, not as a mere sum of parts, but as
a unified individual, has not been solved. The problem
of the relation of the one and the many has proved as
intractable in Physiology as in the wider domain with
which Philosophy concerns itself. In recent times, the
discovery that a living organism, animal or plant, contains, except in the case of the simple uni-cellular beings,
an association of cells, looser or closer according to the
particular kind of organism, each of which cells has its
independent life, but each of which also provides its




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351


specialized contribution to the life and maintenance of
the organism as an individual whole, has emphasized the
urgent character of that special form of the problem of
the relation between the one and the many, to which
the single organism gives rise.
Suggestions have also been made as to the introduction
of concepts which should be capable of employment in
those parts of Biology in which the individual organism
is transcended, and which should play a useful part in
the formulation of racial relations. Such concepts, perhaps in even greater degree than in the case of the
single organism, suffer from a vagueness of definition
which unfits them for use in precise scientific theory,
however great may be the need for them felt by the
speculative mind.
The question whether psychical, or at least psychological, concepts are to be admitted in Biological Science
is one which, besides being a methodological question
of great importance, is related to general issues of farreaching import as regards our general views of the
world. The existence of a psychical side of at least the
higher organisms is now universally recognized, and in
the case of lower organisms some rudimentary psychical
elements such as bare sensation, indistinct awareness of
changes in the environment, and even some rudimentary
form of memory of past experience, are most frequently
assumed to be present. In so far as such concepts, of
a psychical or psychological character, are employed
as an essential element in the conceptual descriptive
schemes of Biology, that Science does not wholly belong to Natural Science, in accordance with the meaning
to which I have restricted the term in these lectures,
but which restriction I have admitted to be open to
very pertinent criticism, and have adopted only for
convenience. Biology may, so far as such conceptions
form part of it, be described as a mixed Science, partly
physical and partly psychological. A survey of the history




352         BIOLOGICAL SCIENCE
of Biological Science shows that many, or most, biological theories of a general kind have employed in their
statement psychological concepts; but whether, and how
far, these can be eliminated, is a crucial question which
has led to much diversity of opinion in recent times. In
the special department of Physiology, as I have already
observed, the recent tendency has been to employ a
methodological restriction to physico-chemical concepts,
and thus to resume the processes within the living organism in conceptual schemes which involve these concepts only. This however does not, or should not,
involve the dogmatic assumption that there are no limits
to what can be attained by this method; and I shall
presently give reasons which tend to prove that such
limits must actually exist.
I have already, in one of my earlier lectures, observed
hat Psychology, of which the object is to describe conceptually the mental processes of the typical individual,
has two methods open to it. The first is that of introspection, in which the observer contemplates the processes in his own mind, and accepts similar descriptions
from others as regards their own minds. The second
method is that of inference from the behaviour of other
persons, that is from the physical phenomena exhibited
by them, and assumed to be in correlation with the
psychical side of such persons. In the case of the psychology of animals the second method is the only one at
our disposal. The first method is the one which affords
the only basis for the interpretations of physical concomitants, in man or in other animals, which we make
in drawing conclusions as to the psychical processes
associated with them. We have direct knowledge of the
psychical processes in our own minds, and we have also
knowledge of the perceptual or physical processes which
we connect with the former. By what has been termed
ejection we transfer the psychical side, or at least some
elements of it, to other living organisms which appear




BIOLOGICAL SCIENCE                353
to originate physical sequences of a kind similar to those
which we regard ourselves as originating. Thus our
knowledge of the psychical side of living organisms in
general, when that side is presumed to exist, is essentially
indirect and inferential. The question whether psychological concepts should be included, or not, in the conceptual laws and schemes of Biology, would appear to
depend largely, if not wholly, upon whether or not such
concepts are regarded as necessary for the purpose of
introducing an element of contingency into the sequences
to which the laws and schemes have reference; and thus
of, at least to some extent, impairing the value of such
laws and schemes as instruments for predicting what
will happen in concrete cases. If it is believed that no
such element of contingency is involved in the use of
psychological concepts, it would appear that the psychical
side of the living organism is relegated to the position
of a mere epi-phenomenon which does not really affect
the physical events; this is the position of the thoroughgoing psycho-physical parallelist. For those who hold
this view, the psychical concepts can then play no essential role in the conceptual theories and laws in
question, and might without real loss be eliminated
from them.
I have at the beginning of these lectures emphasized
the fact that it is not in the province of Science to deal
with the purely individual; its role is to resume conceptually what a class of individuals have in common,
to extract the universal. When a conceptual scheme is
applied in any individual case, there is always some part,
greater or less in amount and importance, of what
happens or is observed, which lies outside the scope of
the scientific generalization employed. It may even be
held that in general a scientific theory is essentially of
a statistical character, the individual peculiarities of particular percepts or trains of percepts, to the description
of which it is applied, being left out of account, but being


HGL


23




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BIOLOGICAL SCIENCE


always in some degree present. In the case of those
percepts and trains of percepts with which Biological
Science is concerned, this residuum, of an individual
character, is often, and perhaps usually, of much greater
importance than in the cases in which no living organisms
are involved. It is certainly true that, the more highly
organized a living being is, the greater is the importance
of the individual peculiarities which, escaping all schematic description, remain outside the purview of scientific
schematism.
The contemplation of the problems presented by the
living organism and its relations, especially in the case
of man and the higher animals, brings us face to face
with the question of how the relation between the
psychical and the physical domains is to be conceived;
of the relation between body and mind. It is a question
which cannot be simply ignored in connection with any
general view of the nature and scope of Biological Science,
although it may be very properly ignored by the investigators in many special departments of that Science.
That the psychical side of a human being, and his body,
which represents the construct of what we directly perceive, exercise an apparent influence upon one another
is a matter of common knowledge. A change in the
moral character of a man is sometimes the apparent
effect of a blow on his head, which may be ascertained
to be accompanied by a lesion in his brain. Conversely,
a psychical disturbance, such as that produced by bad
news, is apparently the cause of marked physical disturbance in the body, temporary or permanent, sometimes even of death. One thing seems certain; that, in
any comprehensive view of the matter, we can leave out
of account neither the conceptual knowledge of the set
of percepts which we call the physical side of a man or
animal, as resumed in its representation by physicochemical descriptive schemes, nor the direct apprehension which we have in our own cases of the psychical




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side of our being, and which we are forced to admit by
inference as being present, at least to some degree of
development, in other animals; although its form may
be of lesser complexity, shading down to bare awareness
or sensation, as we descend the scale of animal existence.
For Natural Science, as applied to man and other living
organisms, the question takes the form, what limitations,
if any, does the presumed presence of the psychical
factor introduce into the scope of that scientific method
which attempts to describe, in terms of physico-chemical
mechanism, the inner processes in living organisms, and
their relations with the environment? One answer to this
question, to which I have already referred in an earlier
lecture, that given by the thorough-going psycho-physical
parallelist, is that there is no such limitation, because
the psychical side is a mere Begleiterscheinung, an epiphenomenon, of which the physical organism is really
quite independent. Now it cannot possibly be maintained that the correctness of this view has been proved.
The actual successes which have been attained in
the representation of particular tracts of physiological
phenomena as physico-chemical processes, great as they
undoubtedly have been, are still at an immeasurable
distance from the attainment of such proof, in relation
to the organism as a whole. The assertion of this view,
in its absolute form, is then merely a dogma resting on
nothing but an illegitimate extension to a whole, of what
may have been shown to be true of some parts. It is
in direct contradiction with the immediate deliverance
of our consciousness as to the real efficiency of the will,
the determinations of which we regard as completely
dominated by nothing outside our own spiritual nature.
Primafacie, the validity of this primary intuitive apprehension is quite as much entitled to recognition as the
more indirect conceptual knowledge provided by Natural
Science. Nothing short of the most cogent evidence, far
greater than any that has hitherto been adduced, would
23-2




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be necessary before we could properly admit that this
intuitive apprehension is wholly illusory.
The attitude of the pure Idealist, or of the Panpsychist,
is a quite different one. Their philosophical position is
that of psychical monism, just as the system of the
psycho-physical parallelist amounts in practice, though
not necessarily in terms, to materialistic monism. The
view of psychical monism is that there is in reality but
one system concerned in the living organism, and that
that one system is fundamentally psychical; that the
separation into mind and body is artificial, and if made,
cannot render possible the treatment of either part in
complete independence of the other. Very much the
same statement applies to the views of the neutral monist
who does not attribute either to the physical or to the
psychical elements a fundamental role, but regards them
both as aspects or modes of some reality more fundamental than either. However much there may be to be
said, from a philosophical point of view, in favour of a
monistic system of a kind which refrains from completely subordinating one side of the nature of the living
organism to the other, such a system has not yet been
shown to be capable of being so developed in detail as
to constitute an articulated monistic Science, into which
all the manifold particulars of organic life, both on the
physical and the psychical sides, can be fitted. It follows
that, unless we are prepared to remain in a nebulous
region of generalities, we must necessarily, for scientific
purposes, as in ordinary life, remain content with a
methodological dualism which provisionally recognizes
in separation the two domains of the psychical and the
physical. The investigation of these two domains are
respectively the functions of Psychology and of Biology.
That there are relations between the two domains is
manifest, and cannot be ignored on either side, whenever a general view of the scope of either Science is to
be formulated. The problem of formulating these rela



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tions is one of such great difficulty, on account of the presumed, or provisional, disparateness of the two domains,
that no even tolerably complete solution is in sight. For
those phenomena connected with living organisms in
which mechanical or rather physico-chemical categories
seemed to be insufficient, or in which it seemed to be
impossible to ignore the relations between the physical
and psychical sides of the organism, some method of provisionally representing those relations in a very general
way has frequently been attempted in various vitalistic
theories. The attempts made in the older vitalism to
localize and delineate the nature of the interaction
between body and mind were crude, and had the
character of makeshifts introduced at various points ad
hoc. They frequently involved the use of such vague and
hybrid expressions as "vital force": and their inability
to rid themselves of the nebulosity of view which the
use of such expressions indicates has had the result that
they have fallen into general discredit amongst Biologists.
Renewed attempts have been made in recent times to
deal with the problem of the relations between the
physical and psychical sides of the living organism. On
the one hand, efforts have been made to conceive, or
rather to represent, the nature of the influence which
the psychical side exerts upon the physical side of the
living being; and on the other hand, the investigations
which fall under the head of the department known as
Psycho-physics or as Physiological Psychology have
resulted in the construction of a mass of detailed empirical knowledge of the relations between physical sensedata and the sensations or the perceptions to which they
are regarded as giving rise. I propose to give some
indications of both these lines of thought.
A widely spread idea is that the action of the psychical
upon the physical can be represented as of the nature
of guidance, which must manifest itself as physical
guidance. This guidance is regarded as, on the psychical




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side, teleological in character. It is not always assumed
that the consciousness of the individual is necessarily
involved in it; it is often thought of as immanent in all
the parts of the organism, and manifesting itself in
guidance of the motions of the molecular or submolecular constituents of the living body. Now it is
clear that, if the psychical side of a living being is to
be regarded as having any intelligible relation with the
physical organism, and through that organism with the
physical environment, the organism cannot be regarded
as simply and solely a machine, or physico-chemical
complex, in which all the processes can be completely
described as a dynamical system in which all the changes
are completely determined by the inner relations of its
parts and its relations with the external physical environment. The effect of the psychical side can only be represented as an actual interference with what would be the
course of the physical system if it were independent of
the psychical side. It is a matter for consideration what
form this interference can take, how it should be conceived as manifesting itself in the physical organism.
This question can only be considered in the light of
facts obtained by accurate observation of the actual
working of the living body as a whole, in relation to its
environment, and of the presumably connected processes in the various parts of the body. These facts of
observation may be expected to throw light upon the
points of similarity and of difference between a living
body and a machine, or system composed of non-living
materials.
A living organism is a system of complex structure
composed essentially of a mixture of substances of very
complex chemical constitution which fall under the
term protoplasm, and also a large amount of water and
some small admixture of other substances. The protein
substances are relatively stable, and yet, in the living
organism, they are continually breaking down into




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chemical substances of simpler atomic constitution, a
process known as katabolism; and they are constantly
being built up, this latter process being known as
anabolism. To the totality of physico-chemical changes
in the living substance of the organism the term metabolism is applied. In an anabolic process energy is
required for the purpose of transforming substances of
comparatively small chemical energy into more complex
substances of much higher chemical potential. On the
other hand, in katabolic processes energy is liberated
by chemical changes of the reverse kind. In the higher
animals, the proteins, carbohydrates, and fats, contained
in the food, are broken up into simpler constituents, in
the process of digestion, and are then synthesized into
more complex constituents by an anabolic process, and
become capable of being transported by the tissue fluids
to all parts of the body. In the animal body an important
part is played by the purely physical processes of osmosis
and diffusion of liquids in the circulation of food
materials, secretory, and excretory substances, from
blood to lymph, and from lymph to cell-substance, or
to glandular cavities. All such processes, anabolic, katabolic, and mechanical, may be regarded from the point
of view of Energetics, of which the concept of energy
and its transformations is the basis; and the question
arises whether, or how far, the principles of Energetics
apply to the living body and to its various parts? As
regards the law of Conservation of Energy, known also
as the first law of Thermodynamics, it appears to have
been established, as the result of various investigations
conducted with great care, that the law holds good for
the living body of an animal. The energy of the food and
oxygen absorbed by the animal is shown to have its
equivalent in the mechanical work done by the animal,
together with the energy lost from its body by conduction and radiation. In general, the metabolism of the
animal as a whole, and the chemico-physical changes in




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detail, conform to the law of Conservation of Energy.
The forms which energy takes in the living body are
mechanical, chemical, thermal, and electrical; in fact
the same forms as are found in the inorganic domain.
The existence of no form of energy peculiar to living
matter has to be postulated as a condition that the
law of Conservation of Energy should hold good for
the organism and its parts. The idea that some such
special form of energy, often denoted by the term Biotic
Energy, is required to account for the phenomena in the
living organism can thus, in all probability, be rejected.
The whole weight of evidence goes to show that the
influence of the psychical side upon the physical side
of the organism cannot be conceived of as taking the
form of a supply of energy, and it must consequently be
such as to be consistent with the law of Conservation
of Energy; the various forms of the energy in the living
organism being the same as in the non-living domain.
But it by no means follows from this that the physical
side of the living being must be regarded as independent
of the psychical side. For the changes in a system are
not determined by the law of Conservation of Energy
alone; the nature and amounts of the transformations of
one form of energy into equivalent amounts in other
forms have to be known before the changes in the
system are completely determinate. It is just in these
transformations from one form of energy to another
that striking differences are found between the cases of
the living organism and of the non-living domain.
With a view to the consideration of the nature of these
differences, let us compare, in respect of the transformations of energy, the processes which go on in the
working of a steam engine and in a warm-blooded
animal. In a steam engine, the source of the energy
from which the work done by the engine is derived is
chemical potential energy of the coal. When the coal is
burned in the boiler-furnace, this energy is transformed




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36I


into heat which vaporizes water in the boiler, producing
steam at high temperature. By the expansion of the
steam in the high-pressure cylinder, mechanical work is
done in driving forward the piston. The steam is then
cooled in the condenser. During the whole process a
certain amount of heat is converted into an equivalent
amount of mechanical work. The high chemical potential
energy of the coal and oxygen is transformed in large
part into the energy of heat, the remainder being the low
chemical potential energy of the residual products. Of
the heat produced, a large part is dissipated, or rendered
unavailable for use, by radiation from the boiler, steampipes, and other parts of the machine, and by friction, or
remains as low-temperature heat in the water of the
condenser. The nett result is that only a comparatively
small part of the energy derived from the chemical transformation of the coal and oxygen has been made available
by the engine for the performance of mechanical work.
In the warm-blooded animal, in which the temperature
is maintained at a constant level, usually higher than
that of the medium in which it lives, the source of energy
is the chemical energy of the food and oxygen which is
taken into the body. These substances undergo chemical
transformations in the alimentary canal and in the tissues
of the body, in the processes of digestion and assimilation. A certain proportion of the food taken into the body
has become a part of the muscles, nerves, etc. of the
body. At the same time, portions of the substances of
high chemical potential, the proteins, carbohydrates,
and fats, are transformed into water, urea, and carbondioxide, which are substances of relatively low chemical
potential. The energy taken into the animal reappears
in large part as mechanical work, that used in bodily
movements, in the motion of the heart, lungs, blood,
etc.; and a part appears as heat, sufficient to compensate
the loss of heat by radiation and conduction from the
body; also a portion is employed in the formation of




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digestive juices and in the propagation of nerve currents.
A chief point in which this process is distinguished from
that which goes on in the steam engine consists in the
much smaller part played by the production and dissipation of heat. It appears that the chemical transformations proceed without the production of anything approaching the amount of heat developed when chemical
transformations occur in connection with non-living
matter. The changes from chemical to mechanical energy
appear to take place more directly than in inorganic
processes; a comparatively small amount of heat is produced, and the dissipation of energy is accordingly
greatly reduced. This feature of the transformation of
chemical energy, with only an insignificant production
of heat, is still more marked in the case of the coldblooded animals, whose temperature is nearly the same
as that of the medium, and rapidly adjusts itself to
changes in the temperature of the medium. In their case
the rate of the metabolic changes is dependent on the
varying temperature of the environment, whereas the
corresponding changes in the warm-blooded animal are,
within limits, almost independent of the temperature of
the environment. It thus appears that the living animal,
considered as a chemico-physical mechanism, in which
energy is received from outside and is converted into
mechanical work done by the organism, is very much
more efficient than is the steam engine or any other
machine that can be constructed for doing mechanical
work; in the sense that the proportion of the energy
supplied that is transformed into such work is much
greater in the former case than in the latter.
The transformation of energy which goes on in the
cells of the green leaves of plants is of a still more
peculiar character. Whilst the animal receives substances
of high chemical potential in its food, the green plant
receives carbon-dioxide and water, substances of relatively low chemical potential, and transforms them in




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the tissues of the leaves into starch, a substance of much
higher chemical potential. The energy required to produce this transformation is derived from the solar radiation; in some manner, the details of which we are unable
at present completely to describe, the chlorophyll in the
cells of the green plant absorbs these radiations and
utilizes their energy in the anabolic process of building
up starch. This starch is converted into soluble sugar
which is circulated through the vessels of the plant. The
plant at the same time absorbs nitrates from the soil,
and the soluble sugar together with the nitrogenous
salts is employed in building up and accumulating
proteins and other organic substances in the plant.
During these processes there is no dissipation of heat,
and only a little mechanical work is done by the plant
in connection with the circulation of protoplasm, in the
movements of the tendrils, and in other ways. The
general effect of the processes which go on in the living
plant is to build up, retain, and accumulate, a store of
energy in a form in which it is available for doing
mechanical and other work when it is liberated. This
effect is produced by transforming energy from the sun
which would otherwise be dissipated and become unavailable. In the general economy of living organisms
the energy of the organic materials in vegetable foods,
built up in the manner I have described, is essential
for the existence of animal life. Moreover the stores of
energy in wood, and the energy derived in past ages from
the solar radiation and stored up in coal, form the chief
sources from which we draw the thermal energy which
we require for the practical purposes of human life.
The transformations of energy which occur in machines,
and generally in the inorganic domain, are irreversible,
always in one direction tending to the diminution of
available energy. This fact of observation has been formulated in the second law of Thermodynamics, the most
precise form of which is the law of Clausius that, in any




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isolated system, the entropy tends to a maximum. The
question arises whether a living organism, considered
as a physico-chemical mechanism, is such that its transformations of energy are in accordance with this law?
It is probably difficult to show that there is in the
organism a direct infringement of the law; that the
processes are reversible; but it is clear that, as a matter
of degree, the amount of dissipation of energy in the
processes connected with living organisms is very much
less than in any corresponding processes connected with
a non-living mechanism alone. The second law of
Thermodynamics is usually regarded as a statistical
result which holds for the vast swarms of particles, in
motion in all directions, of which material systems are
conceived to consist. From this point of view, either the
law would not hold good, or the consequences to which
it leads would be modified in degree, in a system in
which the number of molecules involved in a particular
transformation of energy was so small as to make the
purely statistical method inapplicable. A similar default
in, or modification in the effect of, the law, might occur
if we conceived that, by some selective process applied
to the individual particles, without any alteration in the
energy of their motions, but affecting the directions of
these motions, the statistical result obtained by considering all directions as alike in reference to the motions,
were invalidated. In either of these ways it might be
possible to account for the striking difference which I
have described between the transformations of energy
in the living organism and in the mechanisms to which
the second law of Thermodynamics applies in its most
precise form. These considerations give rise to the suggestion that the peculiarities of the processes which
occur in the living organism may be regarded as due to
some control of the transformations of energy in the
organism which is not present in the case of the processes
of a corresponding kind which occur in the inorganic




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domain. As we have already seen, unless the supposed
influence of the psychical side of the living organism is
merely an illusion, there must be somewhere or somehow what can only be represented as an interference
manifesting itself within the physical processes themselves; and it would appear that some regulation or
control of the transformations of energy is the way in
which this interference can be best conceived; for the
facts of observation appear to negative the idea that any
supply of energy which would nullify the law of conservation is involved. The processes in the living organism
may thus be represented as controlled by an agency, of
which the effect is to favour the anabolic building up
of stores of chemical energy of high potential, and, by
release of such energy, to permit of sudden transformations of it into mechanical work.
A view of the character of the processes in the living
organism in this order of ideas has been developed, both
on the biological and on the philosophical side, by
Prof. Driesch, in his theory of entelechy. He regards
the mode of operation of entelechy to consist in suspension of such transformations of energy as would be
possible on the basis of pre-existing differences of intensity, for example of chemical potential, and further, in
the relaxation of such suspension. He regards entelechy
as non-material and non-spatial, but acting, so to speak,
into space. Although it regulates the transformations
in the organism, its effects are strictly limited by the
possibilities afforded by the structure of the organism;
its action must be subject to given, preformed, conditions
of a physical kind, and it does not itself involve the supply
to, or the extraction from, the organism, of energy in any
form. A full account has been given, in his Gifford
lectures, by Driesch of the "proofs" and arguments
upon which his vitalistic theory rests. I must rest content with having indicated, in the briefest manner, a
possible mode in which the purposive activity of a living




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organism may be held to manifest itself in the physical
processes in the organism, and consequently indirectly
in some degree in physical sequences in the physical
world in general.
It is no doubt possible to exaggerate the part which
the mental side of an organism should be regarded as
playing in what happens in the body. In this connection,
the experiments of Loeb, and his deductions from them,
leading to his theory of tropisms, are of much interest;
he has endeavoured with apparent success to account
for such phenomena as the turning of organisms of
certain species towards the light, or in other cases away
from the light, on purely physico-chemical principles,
without taking account of the existence of sensations, or
of will, as mental factors in the phenomena. He attempts
to show that, in such cases, neither the notion of will
nor that of blind instinct prompting the organisms to
their actions is necessary. The following utterance shows
that he is sanguine as to the extension of his substitution
of physico-chemical description for psychical factors, to
a degree which but few will regard as warranted by any
evidence which we possess at present. He writes1:
Our wishes and hopes, disappointments and sufferings, have
their source in instincts which are comparable to the light
instinct of the heliotropic animals. The need of the struggle for
food, the sexual instinct with its poetry and its chain of consequences, the maternal instincts with the felicity and the
suffering caused by them, the instinct of workmanship and
some other instincts, are the roots from which our inner life
develops. For some of these instincts the chemical basis is at
least sufficiently indicated to arouse the hope that their analysis,
from the mechanistic point of view, is only a question of time.
However far Loeb's mode of viewing the responses to
stimuli, in the case of lower organisms, may be justified,
in the case of man and higher animals it very soon reaches
limits which cannot be passed. The lack of uniformity
The mechanistic conception of life, p. 30.




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in the responses of men to external stimuli is no doubt
in part traceable to differences in their organic constitutions, dependent on differences of past history, but, at
least in the case of the more complicated complexes of
stimuli, when all possible allowance has been made for
individuality of physical constitution, there remains a
residuum which renders futile the hope, or the fear, that
the notion of an individual psychical character, exhibiting itself in the nature of the response to such
stimuli, can ever be dispensed with. So far as this
individual psychical element is operative, it escapes, not
only physical analysis, but also schematic psychological
analysis. In it consists the freedom of will of the individual, not necessarily to be regarded as ultimately purely
arbitrary, but as not completely determined by, or
capable of being linked up in a determinate scheme with,
anything foreign to itself. The recognition of this fact
does not make all phenomena in which that element is
concerned wholly inaccessible to scientific description;
for a determination of the will is preceded and succeeded
by physical sequences of such a character that they, or
at least portions of them, are representable by psychophysical conceptions; and on the psychical side a somewhat similar statement holds as regards psychological
sequences.
The physiologists of the eighteenth century, especially
von Haller, were the first to demonstrate the importance
of the properties of irritability and sensibility in the
nervous system, and to emphasize the function of the
central organ of the nervous system, the brain, in
synthesizing the elements which represent sensation on
the physical side. But the emergence of the Science of
Psycho-physics or Physiological Psychology as a distinct
department may be regarded as due to Cabanis (I757 -i808), who laid its foundations in his Rapports du
Physique et du Moral de l'Homme. His leading idea,
related with the Philosophy of Locke, was that the




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function of the brain in relation to thought is parallel
to the physical functions of other organs of the body;
as he expresses himself:
In order to arrive at a correct idea of those operations from
which thought arises, we must consider the brain as a particular
organ, destined specially to produce it in the same way as the
stomach and the intestines are there to produce digestion, the
liver to filter the bile, the parotid, maxillary, and sublingual
glands to prepare the salivary juice.
This amounts apparently to a practical subordination of
thought to its physical concomitants.
At the end of the eighteenth, and the beginning of the
nineteenth, centuries, many vague and fanciful theories
arose in the shape of speculations connected with animal
electricity, animal magnetism, mesmerism, and Phrenology, which retarded the progress of the Science, and
for a time produced a deflection from those really
fruitful lines of investigation which were approached by
the school of Johannes Miller, and were later put on
a firm basis by Helmholtz and Emil Du Bois-Reymond.
The distinction between sensory and motor nerves, that
the anterior nerves of the spine as efferent, or motor,
nerves carry the nervous stimulus to the different organs,
whilst the posterior nerves, as sensory or afferent nerves,
carry the peripheral stimuli of the senses to the nervous
centres, was established by the labours of Charles Bell,
Magendie, and Johannes Miiller. The last of these
investigators, in his doctrine of "specific energies,"
introduced another important distinction into the theory
of the sensory nervous apparatus. This theory was
afterwards adopted, and its importance emphasized, by
Helmholtz, who brought it into connection with the
theory of colour first advanced by Thomas Young. In
accordance with the doctrine of specific energies, by the
stimulus of any single nerve-fibre, only such sensations
can be produced as belong to the order of one definite
sense, and every stimulus which affects this nerve pro



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duces only sensations belonging to this definite order.
For example, any effective stimulus of the optic nerveapparatus produces always the sensation of light, whereas
the same stimulus would, if effective, produce in the
auditory nerve-apparatus the sensation of sound. Thus
the quality of our sensations depends, not on the
stimulus, but on the nervous apparatus. A prodigious
amount of physiological and anatomical work has been
done with a view to elucidate the functions of the
external or terminal organ relating to a particular sense,
of the connecting nerves, and of the central organ,
situated in the brain, which we must regard as related,
on the physical side, with perception of the particular
kind. Of first rate importance, in this order of investigation, is the work of Helmholtz in Physiological Optics
and Physiological Acoustics. He showed, for example,
that the ear, when subjected to anatomical and acoustical
analysis, exhibits itself as a delicate resonator which
absorbs the different elementary periodic movements
into which musical sounds can be analysed harmonically,
and that different nerve-fibres carry them separately to
the central organ in the brain with which we must
regard perception of sound as connected. He was
enabled to relate the quality of musical notes known
as "timbre," which is different for the same note
produced by different instruments, with the physical
production of the notes. Helmholtz accepted Young's
hypothesis that there exist in the eye three distinct kinds
of nerve-fibres, related to three distinct modes of colour
sensation, corresponding to the three simple colours,
red, green, and violet, of which all colours are compounded. As regards sound, he regarded differences of
pitch and character in notes as dependent upon the
differences of sensitive nerve-fibres; each nerve-fibre
exhibiting only difference of intensity of the stimulus.
That this should be so is in accordance with the doctrine
of specific energies.


H GL


21




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The investigations of the brothers Weber, and especially of E. H. Weber of Leipzig, were commenced
even before those of Johannes Miiller. They had as their
object the establishment of the relation between the subjective side of sensation and its physiological side. This
experimental work was continued by Fechner, to whom
the term Psycho-physics is due. His treatment of the
subject, as represented in his Elemente der Psycho-physik,
published in I860, was an investigation of the relations
of mind and body involving the measurement of psychical
quantities and the establishment of their correlation with
physical quantities. The philosopher Herbart had already
attempted to subject psychical phenomena to exact
methods of calculation, and Lotze had suggested the
existence of a definite and constant connection between
sensation and stimulus. The fundamental difficulty of
such quantitative correlations consists in the fact that
sensation is not an extensive magnitude consisting of
equal units, although a particular sensation may have
intensive magnitude; and this fact has naturally led to
much criticism of all attempts at correlation in a mathematical form between the psychical and physical sides
of sensation. Of the investigations of Wundt and many
other investigators in this important field it is impossible
for me to speak. A flood of light has been thrown by
these investigators upon the physical processes within
the organisms which are related with sensations and
perceptions. But great as is the scientific importance of
the detailed investigations of Physiological Psychology,
and invaluable in their application to practical problems
of life as the results obtained, and to be obtained, may
prove to be, there remains a gap between the physical
and psychical sides of the organism, unbridged, and
perhaps from its very nature impassable for our intelligence.
In many respects the progress of the Science of living
organisms has been much slower and more difficult than




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in the cases of those branches of Science which deal
with inorganic processes. This can be accounted for in
a large measure by the fact that the extreme complexity
of structure and of function of the living organism makes
the process of isolating those elements which should be
examined as separate objects or processes a much more
intricate and difficult one than in the class of phenomena
with which onlyPhysics and Chemistry are concerned. In
the latter Science a particular phenomenon which it is
desired to examine can be more readily submitted to
laboratory experiments in which the requisite conditions
of isolation from other, and disturbing, elements are
artificially produced, than in biological Science, where
the phenomena to be examined occur in connection with
an animal or plant simply as it is given as a whole, with
all its complex inter-relations, and in which the complete
isolation of a particular process is either impossible or
can only be made with some approximation by the employment of a very high degree of technical skill. In
Biology, the problems of mere classification are more
complex, and occupy a much larger place in the history
of the development of the Science than is the case in
the inorganic Sciences. That stage of Biological Science
which is resumed under the term Natural History is one
of vastly greater extent than what corresponds to it in
such sciences as Physics and Chemistry; moreover it
must be regarded as still far from complete.
The method of abstraction and generalization by which
scientific laws and theories, descriptive of processes, are
set up is of slower and more difficult application in
Biology than in the inorganic Sciences. Great as have
been the difficulties of reaching conceptual laws which
resume extensive tracts of phenomena in the inorganic
Sciences, and however incomplete the results of this
process may still be, in the case of the Biological Sciences
the difficulties of such achievement have been still
greater, and the stages at present reached must be in all,
24-2




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or most, departments regarded as less advanced than
in the sciences which are not concerned with life. The
degrees of abstraction and concreteness are very various
in different departments of Science, but the ultimate
aims of all branches of Biological Science are parallel
with those of inorganic Sciences. Much of Biology remains purely descriptive, with a low degree of abstractness, and a large amount of detail remaining to be filled
up.
Many of the great generalizations of Biological Science
have been due in large measure to the breadth of view
attained by those who examined living organisms, their
habits, distribution, and environment, on a large scale,
as travellers; and who thus avoided the narrowing effect
of too exclusive occupation with work in the laboratory.
Such work in the laboratory is largely of an anatomical
and morphological character in which the dead organism
is examined; and even in the physiological work the
living organism is torn asunder from the environment
in which its race was developed.
In the next three lectures I propose to sketch the
origin and development of some of the great general
theories and special departments of Biological Science.




XVI


THE LIVING ORGANISM
FROM the earliest times in which animals and plants
formed objects of study, a certain bifurcation in
Biological Science is discernible which corresponds to
two different points of view in relation to such objects.
This cleavage of the Science into two divergent lines of
advance has persisted throughout its history, and has
manifested itself in the division of Biology into several
special departments, each of which is connected with
one or other of these two modes of regarding the organism. Many investigators have concerned themselves
exclusively or mainly with one or other of these two
divisions of the subject; and many others who have not
exclusively cultivated either side of the subject have
shown a tendency to emphasize the greater relative importance of one or other of the two aspects, especially
in relation to the problem of classification of animals
and of plants. In the first of these modes of regarding
the organism the attention is directed to the study of
" form"; this is the point of view of the Anatomist; and
the term Morphology, invented by Goethe, is now employed to designate the study of form in the most general
sense; the term being not even always confined to the
study of the forms of living organisms. Throughout the
history of the Biological Sciences, Morphology has advanced from the study of the forms of animals and
plants as wholes, to that of the structure, connections,
and spatial positions, of the various organs of which
they are composed, and their comparison in different
animals or plants; then to the structure of the tissues;
and, in the nineteenth century, to the study of the




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forms and structure of the cells of which the body of
the organism is built up; and finally to the study of the
chemistry of protoplasm, consisting of a mixture of
complex substances, which is the fundamental material
that distinguishes the living organism from non-living
matter.
Whilst Morphology has to do with the statical aspect
of the organism, it is the province of the other great
division of the Biology of the single organism, Physiology,
to deal with the kinetic aspect, as expressed by "function." Taking as its starting point the activities of an
animal as a whole, Physiology has advanced, through the
study of the functional activities of the various organs,
to that of the activities of the tissues; later to the study
of the active life of the individual cells, and finally to
the metabolism of the protoplasm. It seems as if the
two great departments of Morphology and Physiology
had found their meeting point in the study of the protoplasm, of which the structure and metabolism appear to
represent the physical aspect of what we call life. To the
department of Morphology there belongs that study of
the anatomy and histology of extinct species which is
denoted by the term Palaeontology. The study of Embryology, that of the early stages in the growth of the
organism, its organs and tissues, has belonged chiefly
to Morphology, but in recent times the Physiology of
the processes at work during the development has also
been studied. Both Embryology and Palaeontology are
in close relation with those racial and evolutionary
aspects of Biology of which I shall speak more directly
in the two following lectures. The distinction between
the Morphological and the Physiological points of view
has exhibited itself in the history of the Science in relation to the problem of the classification of animals and
plants. It is in fact clear that the element of arbitrariness
which appertains to such classification leaves ample room
for differences in the emphasis laid upon form and upon




THE LIVING ORGANISM                375
function, in relation to the similarities and divergences
upon which the classification is based.
Although there existed some knowledge both of the
anatomical and of the functional aspects of animals, and
especially of man, before the time of Aristotle (384 -322 B.C.), as recorded in the writings of Hippocrates and
his school, Aristotle may be regarded as the founder of
Comparative Anatomy. That great thinker and observer
was much less of an Aristotelian, and reached much more
nearly a true conception of scientific method, than the
medieval thinkers who based their views upon his
Philosophy. He had a knowledge of over 500 different
animals, and had an extensive acquaintance with the
structure of many of them. He studied not only the
more ordinary beasts, birds, and fishes, but also cuttlefish,
snails, oysters, crabs, crawfish, lobsters, sea-anemones,
sponges, fish-lice, and even intestinal worms. Extensive
as was his anatomical knowledge, his interest in it was
however secondary to his interest in the functions of the
various organs and parts of animals. He appears to have
been the first to study in any detail the development of
the chick; and he made a commencement of the study
of comparative Embryology. The results of his study
of form are contained in his great work the Historia
Animalium, but that work also contains the results of
this observations in comparative physiology, and in the
distribution and behaviour of many species of animals.
His later book De Partibus Animalium deals with what
he regarded as the causes of the form and structure of
animals, and thus amounts to a discussion of the functions
of their parts, of the relations of form with function, and
of the adaptedness of structure. In the comparison of an
animal of one species with that of another, a clear distinction has been made in later times between homologies
andanalogies. By homology is understood the correspondence between the organs and parts of animals of different
species, in relation to a common spatial order; whereas




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analogy denotes a correspondence between organs and
parts which have the same function in the two animals.
Aristotle made use of both these notions, although he
did not state quite clearly the distinction between them.
He made the first attempt at a scientific classification of
animals into a number of groups based upon similarities
of structure. He distinguished between back-boned and
back-boneless animals, but held the erroneous view that
only the former have blood. He recognized various
modes of classifying animals, not only in accordance
with their structure, but also in manners dependent on
function, such as their manner of life, their mode of
reproduction, their food, etc. Aristotle recognized clearly
that animals belonging to the same one among the great
groups into which he divided them have a unity in the
plan of their construction. In his Historia Animalium,
taking man as the standard, after describing his external
and internal parts in detail, he compared viviparous
quadrupeds with man, and traced out the unity of
plan in the structure of man and all such quadrupeds.
Although this constitutes a definite contribution to
Morphology, what he really sought for were not homologies, but parts with the same functions; his interest
being mainly in functioning organs, and not in merely
spatial relationship of parts. Aristotle foreshadowed
various ideas and distinctions which came to be of great
importance in Science in later times. Among these is the
principle of division of labour amongst different organs,
a point which was emphasized in modern times by M/ilneEdwards. Thus he writes in De Partibus Animalium:
"Whenever, therefore, Nature is able to provide two
separate instruments for two separate uses she does so,
instead of acting like a coppersmith who, for cheapness,
makes a spit and lamp-holder in one." He recognized
the distinction between tissues and organs, the homogeneous and heterogeneous parts of the body. Although
he was unable to describe the structure of tissues, as




THE LIVING ORGANISM                    377
does the modern Histologist, he described their distribution in the body. Aristotle also foreshadowed the
notion of a scale of beings, to the development of which
in modern times I shall presently refer. It is of interest to
observe that, in Aristotle's view, the gradation of organic
forms is the consequence, not the cause, of gradation in
their activities. Thus, in the Historia Animaliium, he writes:
Plants have no work to do beside nutrition, growth, and reproduction; they possess only the nutritive soul. Animals possess
in addition sensation, and the sensitive or perceptive soul.
Again he writes in De Partibus Animalium:
Plants, again, inasmuch as they are without locomotion, present
no great variety in their heterogeneous parts. For where the
functions are but few, few also are the organs required to effect
them.... Animals, however, that not only live but feel, present
a great multiformity of parts, and this diversity is greater in
some animals than in others, being most varied in those to
whose share has fallen not mere life, but life in high degree.
Now such an animal is man.
It thus appears that, in Aristotle's teleological view, the
pre-determined character of the activities of organisms
is to be regarded as the cause of their structural characteristics. Aristotle's Physiology was to some extent
based upon observation, but contained a large element
of erroneous assumption. All the functions are connected with animal heat, associated with the blood, and
centralized in the beating heart, which is the seat of
the soul. The brain is bloodless, and produces mucus,
and the sense-organs are in the head, so that they may
not be overheated by the blood.
After the time of Aristotle, and before the commencement of the modern epoch, the only progress of
importance in knowledge of Physiology was that made
by the celebrated Physician Galen (132-200 A.D.), who
recognized that the art of medicine should rest upon
the basis of a physiological knowledge which must be
dependent upon a groundwork of Anatomy. By his




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experiments on monkeys and swine, he showed that the
arteries contain blood, not air; and he attained to an
understanding of the meaning of the brain and nervous
system. He was the first to show that the nerves connected with sensation are different from those which
have to do with motion, and that they form separate
parts of the nervous system. He elaborated a pathological theory which dominated the theory and practice
of medicine until the sixteenth century.
The inauguration of the modern period, involving a
breach with the dominant physiological tradition of
Aristotle and Galen, is due to Harvey (1578-I657). As
in other branches of Science, the fuller recognition of
the importance of observation and experiment, as the
basis of all theory, differentiates this period from the
medieval. Harvey's introduction of precise methods of
observing and experimenting into physiological Science had
an importance, in directing the attention of others to the
true line of progress, as great as his own great discovery of
the circulation of the blood. The results obtained, owing
to the new impetus, were collected and systematized
by Albrecht von Haller (I708-I777) in his Elementa
Physiologiae Corporis Humani. Among von Haller's own
researches were those on respiratory movements, the
contractility of muscles, and the irritability of nerves.
The origin of all modern mechanistic and physicochemical theories of the living organism may be traced
back to Descartes (I596-I650). The astronomical discoveries of Copernicus, Tycho, and Kepler, the mechanics
of Galileo, and Harvey's discovery of the circulation of
the blood, suggested to him the idea that mechanical
laws could be applied to the purpose of explaining the
phenomena of life in the bodies of man and other
animals. His Physiology was mechanistic, in the strict
sense, and included no chemical conceptions; everything was made to depend upon heat, hydraulics, tubes,
and valves. He believed it possible to account, on these




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lines, for all the phenomena of organic life in animals
and in man. His physiology was based upon that of
Galen, supplemented by Harvey's discovery of the circuital motion of the blood, but he did not accept the
idea that the heart acts as a propulsive apparatus. The
food in the intestine was absorbed by the blood, and
carried to the liver, where it became charged with the
"natural spirits," and then passed to the heart, which
charged it with "vital spirits," in virtue of the innate
heat of the heart and the action of the lungs. The blood
became rarefied, owing to the flame of the heart, fed by
the natural spirits; and this expansion of the fluid produced the circulation, when directed by the valves of
the heart and great vessels. The finer and rarefied parts
of the blood pass off in two directions, one to the organs
of generation, and the more important to the cavities of
the brain, where they not only serve to nourish that
organ, but also give rise to a fine ethereal flame or wind,
through the action of the brain upon them, and thus
form the "animal spirits." From the brain these spirits
are conveyed by means of the nerves, which are regarded
by Descartes as pipes, to various parts of the body,
where they act upon the muscles. The impressions of
the sense-organs are also conveyed by these tubular
nerves to the brain.
In Descartes' view, all animals except man are pure
automata, simple mechanisms, devoid of any element
of feeling or consciousness. "The animals," he says,
"act naturally by springs, like a watch." In the case of
man, there is added to the bodily mechanism the
rational soul, spiritualistic and immortal, which is located
in the pineal gland, situated in the middle of the brain.
By this assumption, Descartes attempted to reconcile
his mechanistic conceptions with his idealistic philosophy, but it is safe to assert that he failed to give any
intelligible account of the relations between the rational
soul and the purely mechanistic body, a failure which




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is characteristic of all the later theories of which his
may be regarded as the parent. In the latter part of the
seventeenth century,Physiology continued for some time
to be purely mechanistic, but in the modified sense
that chemical discoveries were utilized in the description
of the life of the organism.
The eighteenth century was a period in which vitalistic
hypotheses dominated physiological conceptions, especially owing to the influence of the chemical and vitalistic
ideas of Stahl and his followers. During this time the
progress of Physiology was much retarded by the lack
of progress in Chemistry, which was probably the result
of Stahl's phlogistic hypothesis. Not until the great
chemical discoveries of Lavoisier and his successors were
adequate chemical conceptions made available in Physiological Science. Physico-chemical conceptions, without
vitalistic hypotheses, became again dominant in Physiology, and lasted throughout the nineteenth century, culminating in the writings of Huxley and Max Verworn.
But little progress was made in the classification of
animals during the eighteen centuries subsequent to the
time of Aristotle. During the fifteenth and sixteenth
centuries of our era, considerable additions were made
to the list of known animals, but hardly any improvement was made in their classification. The first to make
any such advance was John Ray (I628-1705), the predecessor of Linnaeus. He was the first to define the
use of the term "species," as denoting a group of similar
individuals exhibiting constant characteristics from one
generation to another; and he was the first to emphasize
the anatomical characteristics as a basis of classification.
The greater systematizer Linnaeus (I707-I778), who
has been described as "a classifying, coordinating, and
subordinating machine," in his great work, the Systema
Naturae, introduced a system of classification of plants
and animals which formed the starting point of modern
Systematics. Linnaeus employed a binomial system of




THE LIVING ORGANISM


38I


nomenclature, and graded his classification into classes,
orders, genera, species, and varieties. He recognized six
classes of animals,-Mammals, birds, amphibians (including reptiles), fishes, insects, and vermes; this classification was afterwards made more precise by Lamarck,
who established sixteen classes instead of six. Linnaeus
believed that each species was descended from a pair
originally created, each expressing an idea in the divine
mind. Moreover, in his doctrine of continuity, taken
in a loose sense of the term, he maintained that the
species could be arranged in series with no hiatus between two consecutive series. Thus, unlike the present
view, he recognized no genetic relations between closely
allied species, and left no room for the occurrence of the
discontinuous variations which many persons at the
present time believe to be an important factor in evolution.
The advances made in human anatomy from the time
of Aristotle until the end of the sixteenth century were
accompanied by but little advance in the knowledge of
comparative Anatomy. A great impetus was given to
anatomical studies by the invention of the microscope, at
the beginning of the seventeenth century. One of the first
effects of the use of the microscope was the discovery
of the complex structure of tissues. Up till that time
they had been regarded as little more than inorganic
substances, possessing, however, some organic properties
such as contractility. Thus the study known as Histology came into being, and in particular, important discoveries relating to muscle fibres were made. Among
other applications of the microscope was the study of
the comparative Anatomy of lower animals, and that of
the metamorphoses of insects, with which the name of
Swammerdam is associated. One of the first to make
extensive use of the new instrument was Malpighi, who
studied by its aid the development of the chick. He also
ascertained the minute structure of the lungs, and
demonstrated the connection of the arteries with the




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veins; further he described the histology of the spleen,
the kidney, the liver, and the cortex of the brain; he
showed that the liver is really a conglomerate gland, and
discovered the Malpighian bodies in the kidney.
The biological studies of the eighteenth century were
largely in the direction of general natural history and,
under the influence of Linnaeus, of the problem of
classification. The minute study of insects was continued by Reaumur and Bonnet, who attached, however,
more importance to their habits and physiology than to
their anatomy. The general conception, to which I have
already alluded, of a scale of beings was first put into
a detailed and systematic form by Bonnet (I720-1793),
and was in fact extended by him to all the objects in the
Universe. He constructed a long table, headed "Idee
d'une Echelle des etres naturels," which begins with
Man, the Orang-outang, the Ape, Quadrupeds, and ends
with earth, water, fire, and more subtle matter. The
scale is not based upon any definite principle, either
morphological or functional, but is supposed to represent a gradation involving all possible orders of perfection. This conception of a gradation of beings was
also held by Buffon (I707-I788), but in his hands it
takes more consistent form as a functional gradation.
He pointed out the fact that the groups of Invertebrates
are very different in structural plan from those of the
Vertebrates, and had a clear conception of the unity of
plan which is characteristic of the Vertebrates. Moreover he, for the first time, expressed the idea that community of origin might be at the base of the unity of
plan, although he was far from being a consistent
Evolutionist. He pointed out the difficulty of supposing
that one species may arise from another by a process of
degeneration, and oscillated between the ideas that
species are definitely discontinuous with one another,
and that they can be united in larger groups.
Xavier Bichat (I77I-I802), who was mainly a human




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anatomist, worked out in detail the Aristotelian distinction between the animal and the vegetative parts and
functions of animals, to which Buffon had also drawn
attention. The animal life, which does not appertain to
plants, he described as the order of functions which
connect the animal with its environment; these organs
are the afferent and efferent nerves, the brain, the senseorgans, and the voluntary muscles; the brain being the
central organ. The organic or vegetative life has for its
central organ the heart, and includes the processes of
digestion, circulation, respiration, exhalation, absorption,
secretion, nutrition, and calorification. He regarded the
plant and animal as standing for two different modes of
life. The only relation which the plant has with the environment is that involved in nutrition; the animal has
in addition to this organic life, a life of active relation
with surrounding things. He observes1 that:
One might almost say that the plant is the framework, the
foundation of the animal, and that to form the animal it sufficed
to cover this foundation with a system of organs fitted to establish
relations with the world outside. It follows that the functions
of the animal form two quite distinct classes. One class consists
in a continual succession of assimilation and excretion; through
these functions the animal incessantly transforms into its own
substance the molecules of surrounding bodies, later to reject
these molecules when they have become heterogeneous to it.
Through this first class of functions the animal exists only within
itself; through the other class it exists outside; it is an inhabitant
of the world, and not, like the plant, of the place which saw
its birth. The animal feels and perceives its surroundings, reflects
its sensations, moves of its own will under their influence, and,
as a rule, can communicate by its voice its desires and its fears,
its pleasures and its pains. I call organic life the sum of the
functions of the former class, for all organized creatures, plants
or animals, possess them to a more or less marked degree, and
organized structure is the sole condition necessary to their
exercise. The combined functions of the second class form the
"animal" life, so named because it is the exclusive attribute of
the animal kingdom.
1 Recherches Physiologiques sur la Vie et la Mort, 3rd ed., p. 2.




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Bichat contrasts the symmetry of the nerves and
muscles of the animal life with the asymmetrical arrangement of the visceral muscles and the sympathetic nerves
which belong to the organic life. He points out that
habit is all-important in the animal life, but denies that
habit has any influence upon the organic life. He states
that the organs of the organic life attain their full perfection independently of use; whereas the organs of the
animal life require education in order to reach perfection. These views as to the independence of the organic
life of habit and use would probably no longer be accepted without considerable modification.
A very similar distinction between animal and vital
functions was emphasized by the great Comparative Anatomist Cuvier (I769-I832), who studied both structure
and function, and even regarded the latter as the more
important in that it determined the former. Following
Aristotle he asserted that a plant is an animal that sleeps.
HIe was aware, owing to the recent progress of Chemistry,
that the material of the body is principally composed of
combinations of Carbon, Nitrogen, Hydrogen, and Phosphorus, forming albumen, fibrine, etc. Although the
discovery of the cellular nature of tissues was not made
until after his death, he observed that the organism can
be resolved into small flakes and filaments which form
a " cellulosity." Cuvier was the first completely to recognize as a definite principle the harmony between structure
and function. " It is," he writes, " on this mutual dependence of the functions and the assistance which they lend
one another that are founded the laws that determine
the relations of their organs; these laws are as inevitable
as the laws of metaphysics and mathematics, for it is
evident that a proper harmony between organs that act
one upon another is a necessary condition of the existence
to which they belong." We have here an attempt to form
a concept of the coordinated organism as distinct from
the sum of the parts considered either in relation to




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structure or function; but by the conditions of existence
he meant adaptations of functions and organs within the
organism, and he hardly considered the external conditions or the environment. In accordance with his wellknown principle of correlation, from one part of an
animal, having given a model of the group to which it
belongs, the whole may be constructed, for as he says:
'All the organs of an animal form a single system, the
parts of which hang together, and act and react upon one
another; and no modifications can appear in one part
without bringing about corresponding modifications in
all the rest." From the shape of one organ the shape of
the other organs can be inferred, having given sufficiently
extensive knowledge of functions, and of the relation of
structure to function in each kind of organ. The functional dependence of the parts he interpreted in terms
of what is later known as the general metabolism of the
organism, that is the constant chemical changes in all
the parts, and the accompanying interchanges with the
outside.
One of the most important results of Cuvier's work
is his division of the animal kingdom into four principal
types of form, the Vertebrates, Molluscs, Articulates,
and Radiates. The first three have bilateral symmetry,
and the last radial symmetry. In formulating these four
divisions, each of which is built upon one plan, Cuvier
was influenced by the idea that the characters of the two
sets of organs, the vegetative and the animal, and the
correlations within each, must form the basis of the
classification.
In contradistinction with Cuvier it was held by his
adversary Geoffroy Saint-Hilaire (I772-I844) that the
structure of all animals may be referred to a single type.
He went so far as to assert that " There is, philosophically
speaking, only a single animal." By that single animal
he meant an abstract generalized type to which, by the
principle of homology, all actual animal structures could


H G L


25




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be made to conform. With him, homology was not so
much homology of organs as of parts and connections.
His principle of connections was his guide in tracing an
organ through all its functional transformations, for as
he says: "an organ can be deteriorated, atrophied,
annihilated, but not transposed." As a pure Morphologist for whom "form" was everything, his difference
from Cuvier's attitude, in which the stress is laid on
"function," was fundamental. His attempts to prove,
by means of what appear to be very far-fetched homologies, that animals, as far apart as Vertebrates and
Cephalopods, conform to the same fundamental type,
were criticized and demolished by Cuvier. Cuvier
showed that, although Saint-Hilaire had discovered
many hidden homologies, especially by his important
discoveries concerning foetal structure, the unity of plan
and composition, as conceived by Saint-Hilaire, does not
exist in actuality. Cuvier further maintained that the
whole principle of homology, so far as it is valid, is subordinate to the principle of the functional coordination
and adaptation of the parts.
Biologists in Germany, and also to a large extent in
France, were, during the early part of the nineteenth
century, under the influence of a number of ideas which
formed part of what is known as the Philosophy of
Nature. The principal conceptions of this school were
the existence of a unique plan of structure, the idea of
the scale of beings, and the notion of parallelism between
the stages of individual development and the stages of
the scale of beings, which latter has for us an obvious
connection with theories of evolution. A further theory
of this school was that of the repetition or multiplication
of parts within the individual, of which the vertebral
theory of the skull is the most striking example. The law
of parallelism, first laid down by Kielmeyer, and afterwards, in a more developed form, by Oken (I779-I851),
asserted that the embryo of every animal passes during




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its development through all stages of the animal kingdom,
or at least through the stages of one or more classes
lower down in the scale. It was held that the animal
kingdom is a dismemberment of the highest animal,
man, and thus that animals are only the persistent foetal
stages or conditions of man, who contains within himself
all the animal kingdom. The embryo of higher animals
was compared with the adults of lower animals. It was
stated, for example, by Tiedemann that "Every animal
before reaching its full development, passes through the
stage of organization of one or more classes lower in the
scale, or, every animal begins its metamorphosis with
the simplest organization." A detailed account of facts
which support this theory was given by Meckel, but his
treatment contains very imaginative comparisons between organs of animals of widely differing groups, and
involves a mixture of morphological homologies with
physiological analogies. Meckel admitted that man does
not pass in his development through the whole animal
series, but asserted that, at least as regards single organs
or organ-systems, the embryo of man passes through
many animal stages. Although he was not a thoroughgoing evolutionist, he held that the higher animal, in his
gradual evolution, passes through the permanent organic
stages which lie below it. As he says: " The development
of the individual organism obeys the same laws as the
development of the whole animal series." An adherent
of this school, K. G. Carus, asserted as a general law
of Nature that the higher formations include the lower;
that the animal includes the vegetable; that it is by a
rational necessity that the development of a perfect
animal repeats the series of antecedent formations. The
theory of the repetition or multiplication of parts within
the organism was pushed to the absurd extreme of
attempting to demonstrate that the whole organization
is repeated in certain of its parts; for example Oken
asserted that in the head the whole trunk is repeated,
25-2




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the upper jaw corresponding to the arms, the lower to
the legs, and that in each jaw the same bony divisions
exist as in the limbs. Nevertheless the recognition of
serial homologies was a real contribution to morphology,
although many absurd or arbitrary homologies were
maintained. It was maintained by Carus that the whole
skeleton is only a repeated vertebra. The influence of the
Philosophy of Nature, as an a priori theory, is indicated
in the statement of Carus that, in respect of the formation of the skeleton throughout the animal kingdom, he
wishes to know "how such and such a formation is
realized in virtue of the eternal laws of reason." He
held that all forms of skeletons can be deduced from the
hollow sphere, so that every skeleton can be represented
schematically by a number of hollow spheres suitably
modified in shape and suitably arranged; and he endeavoured to work out this idea as applied both to vertebrates and invertebrates. Although he was strongly under
the influence of the a priori ideas of the German
anatomists of the period, he was in many respects less
a disciple of Saint-Hilaire than of his great opponent
Cuvier; he held that the connections of bones and
muscles change in accordance with functional requirements, and he did not accept the "law of connections"
in its rigorous form.
The leading ideas of Cuvier in relation to functions,
of Saint-Hilaire in relation to the principle of connections, and Oken's notion of the serial repetition of
parts, were combined, modified, and reduced to clearer
forms, by Richard Owen. The main idea developed in
his work On the Archetype and Homologies of the Vertebrate Skeleton is that the vertebrate skeleton is composed
of a series of segments, each of which he calls a vertebra.
His archetype is a scheme of what is usually constant
in the vertebrate skeleton, both the number and arrangement of the bones in any actual vertebrate being subject
to variation. He defined the object of his work to be




THE LIVING ORGANISM               389
to deduce the relative value and constancy of the different
vertebral elements, and to trace the kind and extent of
their variations within the limits of a plain and obvious
maintenance of a typical character. He accepted in a
modified form Oken's vertebral theory of the skull,
which was afterwards demolished by Huxley. In the
determination of homologies he followed Saint-Hilaire's
principle of connections, and rejected the method of
their determination by the mode of development. In his
view, comparative anatomy explains embryology, and
not the reverse. Relations of homology he analysed into
three kinds, special, general, and serial homology. Special
homology consists of the correspondence of a special
part or organ, determined by its relative position and
connections, with a part or organ in a different animal;
thus involving reference to a common type. Owen's
general view of the nature of living things was that
organic forms result from the competitive working of
two principles, the first of which brings about a vegetative repetition of structure, while the other, involving
a teleological factor, shapes the living organism to its
functions. The first of these principles, "a general
polarizing force" illustrating the archetype of the vertebrate skeleton, is the same principle which produces
repetition of the forms of crystals in the inorganic
domain. The second principle, the" adaptive " or" special
organizing force," produces the diversity of organic
beings.
Although observations of the development of the
embryo had been made from the time of Aristotle,
Embryology attained its position as a Science in the
hands of von Baer (1792-1876). The first volume of his
great treatise on the subject, published in I828, contains
a full and adequate account of the development of the
chick, as obtained by minute and accurate observations,
and a discussion of the laws of development in general.
In the Scholia at the end of his description of the




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development of the chick he refutes the notion of preformation; and defends the idea that the essential nature
of the animal determines its differentiation, and that
no stage of development is solely determined by the
antecedent stage. He holds the vitalistic conception that
a guidance, involving the idea of the completed whole,
is active at each stage of development. He shows that
the process of the development of the embryo is one of
differentiation, by which the germ becomes increasingly
individualized, and the developing animal increasingly
independent. In describing the stages of development,
he lays down his theory of germ-layers. In his account
of the process he states that first of all the germ separates
out into heterogeneous layers, which with advancing
development acquire ever greater individuality, and on
their first appearance show rudiments of the structures
which will characterize them later. In the germ of the
bird, at the beginning of incubation, there can be distinguished an upper smooth continuous surface and a
lower, more granular, surface. The blastoderm then
separates into two layers, of which the lower develops
into the plastic body-parts of the embryo, the upper into
the animal parts; the lower shows clearly a further
division into two subsidiary layers, the mucous-layer
and the vessel-layer; the original upper layer also shows
a division into two, which form respectively the skin and
the muscle-layer, the latter of which contains, in an undifferentiated state, the skeletal and muscular systems,
the connective tissues, and the nerves belonging to them.
Thus the process of determination results in the formation
of four layers. The uppermost layer will form the outer
covering of the embryo; from it there differentiates out
at an early stage the rudiment of the central nervous
system, forming a more or less independent layer. Below
this outermost layer is the one from which the muscular
and skeletal system is developed, then the vessel-layer
which gives rise to the main blood vessels. The inner



THE LIVING ORGANISM               39I
most layer will form the mucous membrane of the
alimentary canal and its dependencies; it is originally,
like the other layers, a flat plate. From all these layers
tubes are developed by the bending round of their
edges; the process of primary differentiation is then complete. He then describes the later histological and other
morphological differentiations which occur concurrently.
Through morphological differentiation the various parts
of the fundamental organs become specialized through
unequal growth, first into the primitive organs, and then
into the functional organs of the body. At the same time
there is a differentiation in the substance of the layers,
whereby cartilage, muscle, and nerve separate out, and
a part of the mass becomes fluid. Through histological
differentiation, the texture of the layers and incipient
organs becomes individualized. Originally the germ
consists of an almost homogeneous mass, with clear or
dark globules in suspension. This homogeneity gradually
gives place to heterogeneity, the structureless mass
becomes fibrous to form muscles, hardens to form
cartilage and bone, liquifies to form blood, and differentiates in a large number of other ways.
Von Baer showed that the extreme views of those who
were dominated by the law of parallelism, as stated by
Meckel and Serres, that the higher animals repeat in
their development the adult stages of the lower, went
far beyond the facts. He pointed out, for example, that
the developing chick is at a very early stage demonstrably
a vertebrate, and does not recapitulate the organization
of a polyp, a worm, or a mollusc. By means of various
examples he showed that the recapitulation is never that
of the whole organization of a lower animal, but only
that of particular parts or simple organs. Like Cuvier,
von Baer recognized four main types of animals, the
vertebrate, the molluscan, the longitudinal, and the
radiate. He held that the manner of development is
determined by the type of organization; the type being




392


THE LIVING ORGANISM


observable in the very earliest stages of development.
In his view the development of the individual is always
from the general to the special; the general characters
of the group to which an embryo belongs appear earlier
in development than the special characters of the particular animal. There is, during the development, a progressive gradation of generality, the less general structural
features always appearing after the more general. So far
from endorsing Meckel's law of Parallelism, he held that
the embryo, instead of passing through the states of
other definite forms, separates itself from them; and
that the embryo of a higher animal never resembles the
adult of another animal form, but only its embryo. Thus,
not even within the group, is there a real scale which
the higher forms must mount. The apparent resemblance
between the embryos of the higher, more differentiated,
members of the group with the lower, less differentiated,
adult forms arises from the fact that these latter diverge
but little from the generalized type. Embryology is of
great assistance to comparative anatomy, of which the
aim is to discuss the general type, or the common plan
of structure of the animals of a group. For, as the
embryo develops from the general to the special, the
state in which each organ first appears should represent
the typical state of that organ within the group. Thus
the true homologies of parts will be best determined by
studying their earliest developments. Homologies should
be restricted to a single type; organs with similar functional relations belonging to members of different types
should not be regarded as homologous, on account of
their different modes of development. The classification
should not depend on the adult structure but on the
characters shown in early development, because these
latter show the characters of the type to which an animal
belongs in their more generalized form.
Von Baer, like Cuvier, rejected the notion of the scale
of beings, not only by recognizing the existence of four




THE LIVING ORGANISM                393
totally different types, but by showing that, even within
one type, any serial arrangement only held good for
separate organs or sets of organs, and not for the whole
complex of organs; so that there is no actual scale of
beings even within one and the same type. It has been
said that man is only the highest animal in respect of
his nervous system.
The publication by Theodor Schwann, in I839, of the
discovery that the tissues in the animal body are composed of cells, constituted a most notable and farreaching advance upon the discovery of the complex
structure of the tissues. In accordance with the great
conceptual scheme, the cell theory, the cell is both a
morphological, or statical, unit, and also a physiological,
or dynamical, unit, in the formation of the complete
body. It is a statical unit, inasmuch as all plants and
animals are built up of cells and modifications of cells.
It is a dynamical unit, inasmuch as the life of the
organism is dependent upon the activities of the cells
of which it is composed. In the words of Schwann:
"The whole organism subsists only by means of the
reciprocal action of the single elementary parts." It was
in the domain of Botany that the existence of cells was
first observed, but Schwann was the first to announce
the identity of the cellular structure of all living beings,
animal or vegetable. In the seventeenth century, cells
in plants had been discovered by Robert Hooke,
Malpighi, and Leeuenhoek, but they were then not
regarded as living independent structural units. In the
nineteenth century, when great improvements in the
microscope had been made, it was established by the
work of various observers that the tissues of plants are
composed of elements which can be reduced, for the
most part, to spherical closed cells. By the coalescence
and elongation of the cells the vessels and fibres of the
plants are formed. At first, attention was concentrated
on the cell-walls, but in 1838 Schleiden pointed out the




394           THE LIVING ORGANISM
importance of a discovery, that had been made a few
years earlier, of the existence in every cell of a small body
which he called the cytoblast, and which is now usually
called the nucleus; this nucleus is embedded in a gummy
substance called the cytoplasm. Schleiden showed that
plants are built up of cells and their modifications, and
that a single cell, or ovum, is the original form of the
plant embryo; further he gave a theory of the mode in
which, by cell division, new cells are formed. The conception that the cell is a partially independent living
unity was expressed by him in the statement that
"Each cell carries on a double life; one a quite independent and self-contained life, the other a dependent
life, in so far as the cell has become an integral part of
the plant." Here we have, in a distinct form, the view,
in relation to the plant organism, that it is to be regarded
as a multiplicity in unity, a notion which was extended
by Schwann to the animal organism. The occurrence of
small globules in animal tissues had been observed
earlier, for example by von Baer in the embryo of the
chick; and that cells were to be found in animal tissues
was well known to Johannes Miller and other investigators, but a complete theory of cells in animal
tissues was the work of Schwann'. In the first part of
1 Professor J. Arthur Thomson has drawn my attention to an article
entitled " The Dawn of the Cell Theory," by Professor J. H. Gerould,
published in The Scientific Monthly, vol. xiv (1922), in which it is
pointed out that the cell theory is older than has generally been supposed.
The beginning of the theory is traced back to Lamarck, who wrote in
his Philosophie Zoologique published in I809: "No body can possess
life if its containing parts are not a cellular tissue, or formed by a
cellular tissue.... Thus every living body is essentially a mass of
cellular tissues in which more or less complex fluids move more or
less rapidly; so that, if this body is very simple, that is, without special
organs, it appears homogeneous, and presents nothing but cellular
tissue containing fluids which move within it slowly; but, if its organization is complex, all its organs without exception, as well as their most
minute parts, are enveloped in cellular tissue, and even are essentially
formed by it." In the same year, 1809, Mirbel, in the second edition
of his Exposition de la Theorie de l'Organisation Vegetale, reaches the
conclusion that "The plant is wholly formed of a continuous cellular
membranous tissue....Plants are made up of cells, all parts of which




THE LIVING ORGANISM                     395
his book Microscopic Investigations Concerning the Agreement in Structure and Growth of Animals and Plants he
dealt with cartilage cells and with the cells of the notochord, showing that the nucleus, the nucleolus, and the
cell-walls, play the same part, and behave in the same
manner, as in the plant cells, in accordance with
Schleiden's theory. It was already known that all
animals are developed from an ovum; the fundamentally
important discovery of the existence of the ovum in
mammals had been made by von Baer, in 1827. The
position of the ovum in relation to the cell theory was
a subject which Schwann carefully investigated; he came
to the conclusion that the ovum is itself a cell, but not
one of the simplest kind. Every individual organism
begins its life as a single cell; and in the simplest or
uni-cellular organism, such as the Amoeba, it remains
uni-cellular; in other cases the gradual formation of the
body is due to the division and successive re-division
of the fertilized ovum into a coherent mass of cells.
Although there are considerable differences between the
cells in any one organism, and in the cells of different
organisms, there appears to be a considerable amount of
agreement in the structure of all cells of animals and
plants. The cell theory has been the basis of the modern
theories of heredity of which I shall speak in the next
lecture. The study of the cleavage of the fertilized ovum,
and the gradual formation of cells by segmentation,
leading to the formation of tissues, was commenced by
Schwann, and carried on by various investigators, of
form one and the same membranous tissue." Both Mirbel and Lamarck
were associated with the Musee de l'Histoire Naturelle. With Mirbel
and Lamarck the emphasis was laid upon the membranous cellular
tissue as fundamental, rather than upon the cell as a vital unit. The
idea of the individuality of the cell is due to Dutrochet, in I824. Up
to this point the existence of the nucleus had not been discovered;
the general occurrence of nuclei was recognized by Robert Brown,
in 1833, and his work inspired that of Schleiden. It would thus appear
that the cell theory had been developed to a considerable extent
many years before the time of Schwann.




396


THE LIVING ORGANISM


whom von K6lliker was the most eminent, on account
of the vast amount of detailed embryological and
anatomical work he accomplished. Although the cell
theory has been for the most part accepted as a representation of the facts, a certain amount of criticism has
been directed against its all-sufficiency, based upon some
exceptional cases which have been observed, in which
cell-walls appear to be absent, the nuclei dividing
without cellular division, the result being a mass of
protoplasm containing many nuclei, but without cell
boundaries. The fact that, both in animals and plants,
intercellular filaments, composed of protoplasm, have
been discovered connecting the cells, may also not be
without significance. The complete generality of the
process of cell-formation in organic life has been challenged; and that process has been held by some investigators, such as Sachs on the botanical side, and
Sedgwick on the zoological side, to be a secondary process, the growth of the protoplasmic mass itself being
the primary factor in the development of the plant or
animal. It has in fact been held that the plant forms
cells, but the cells do not form plants. However this
may be, the cell theory, as a working scheme, has been
of inestimably great assistance in the development of
biological science.




XVII


HEREDITY
THAT some living organisms arise by what is called
spontaneous generation from non-living matter was
the universal belief of men of Science until about three
centuries ago. That invertebrate animals at least are
generated spontaneously was held by Aristotle, who
remarked the occurrence of small animals in putrifying
substances. Until the seventeenth century this belief
was firmly held both by zoologists and botanists; even
Bacon believed in the spontaneous origin of plants, such
as thistles, from dead earth. The first doubter of the
truth of this conception of abiogenesis appears to have
been Francisco Ridi (I626-I697), the Florentine contemporary of Harvey, who retained the traditional belief.
Ridi showed by experiment that no grubs or insects
appeared in the flesh of a dead animal if it was protected
with sufficient care from intrusion from outside. With
the problem of the origin of parasites he had less success.
By some it was supposed that parasites were generated
spontaneously from the juices of their hosts, by others
that they had been originally created with their hosts;
Adam being supposed to have harboured all the human
parasites from the beginning. Not until the nineteenth
century was the germinal origin of internal parasites
established by careful experimenters.
During the nineteenth century various experimenters
demonstrated the possibility of preventing the appearance of animalcules in infusions, by a process of
sterilization which prevented the entry into the infusions
of living germs from the external air. But even as late
as I858, Ponchet asserted, before the Paris Academy of




398


HEREDITY


Sciences, that he had proved that microscopic organisms
arise apart from pre-existing germs. However, the investigations of Louis Pasteur, who showed that all
putrefaction and many kinds of fermentation are due to
microscopic living organisms, and the work of Tyndall,
who showed that absolute sterilization of infusions could
be attained by intermittent applications of heat, led to
the general acceptance of the law of biogenesis. As at
present accepted, the law of biogenesis is of a negative
character, asserting that there is no known evidence
that living organisms at present arise from non-living
matter. As regards the further question whether we
should conceive that, under conditions widely differing
from those which obtain at present, there has been continuity between living and non-living matter, the answer
must be entirely a matter of speculation, because, from
the nature of the case, the only mode of obtaining a
decisive answer, that of observation and experiment, is
not available. It is somewhat remarkable that Huxley,
in stating before the British Association his conviction
that the law of biogenesis had been fully established,
expressed as his opinion that he would have witnessed
the origin of protoplasm from non-living matter if he
could have been present at the beginning of organic
evolution.
There have been various speculations made as to the
origin of life on the earth; among these is the suggestion
of Lord Kelvin that living organisms reached the earth
in meteorites. The persistence of the conception of the
continuity of living with inorganic matter is exhibited
in the views of prominent men of Science such as
Haeckel, Ray Lankester, the Botanist von Nageli, and
the Physiologist Pfluiger, all of whom accept the idea
that this continuity has at some former time been
effective in relation to the origin of living matter.
The term "heredity" implies the fact that living organisms can produce their like, the resemblance, though




HEREDITY


399


never absolutely perfect, extending to the most minute
details of construction and function. The modern theories
of heredity have been constructed upon the basis of an
enormous amount of work in the collection and sifting
of facts. Their aim, in accordance with the view developed in these lectures, is to represent in one or more
conceptual schemes, abstract in some greater or less
degree, the main facts of the general likeness of parent
and offspring, of the occurrence of variations in the
offspring, that is of characters not exhibited in the same
degree by the parent; of specific similarities in the offspring with characters in one or both parents; of the
fact that characteristics may occur in the offspring which
are not to be found in either parent, but which were
exhibited by a grandparent or by a remoter progenitor;
and lastly, of the fact that characters acquired by a
parent in the course of his or her life, as the apparent
result of interaction with the environment, seem, in
some cases, to reappear in the offspring. The last of
these facts has given rise to much controversy; and its
proper interpretation is one of the most crucial questions
which have arisen in connection with theories of heredity
and in the more general theories of evolution.
I have pointed out in my earlier lectures the fact that
a conceptual theory frequently employs, not only concepts to which there are directly corresponding percepts,
but also concepts which have not, or at any given time
are not known to have, any percepts to which they
directly correspond. The modern theories of heredity
make use of concepts which belong to both these classes.
Not only concepts of the first class, but also those of
the second class, have almost invariably been regarded
by those who have originated or developed the theories
as denoting really existing entities, in the sense attributed
to the term by physical realists. But, as in the case of
the theories of Physics and Chemistry, the value of these
theories may be estimated quite independently of the




400


HEREDITY


acceptance of the views of physical realists; they may
be judged exclusively in relation to their coherence as
conceptual schemes, and to their effectiveness in representing the perceptual facts and sequences to which they
are applied.
Most of these theories have as their starting point the
group of perceptual facts resumed in the cell theory of
the construction of the living organism, applicable to
both animals and plants. There are, in the case of most
multi-cellular organisms, amongst the cells of which they
are constituted, special kinds of cells known as germcells, distinct in various respects from the somatic, or
body-cells, which constitute the greater part of the
organisms. Prolonged investigation has shown that the
embryo of a new individual organism appears to begin,
in the case of the majority of animals and plants, in the
union of two minute cells, the sperm-cell and the ovum,
or egg-cell, which combine to form a single cell, the
fertilized ovum. The embryo is then observed to arise
by a process of successive formation of new cells obtained
by division of the fertilized ovum into two cells, followed
by a series of successive divisions of cells into two parts
each of which is a complete cell. Although this sexual
mode of reproduction holds for the great majority of
living organisms, it is not the only process by which
reproduction occurs. In some organisms, particularly
amongst those which are uni-cellular, there is no fertilization of the ovum, which divides without previous fertilization into two cells; so that the reproduction is
asexual; and in some cases there are no special germcells. It will be sufficient for our purpose to leave aside
such special modes of reproduction which occur in
various lower organisms, and to confine our attention
to the sexual mode of reproduction which holds good
for the higher animals and for many plants.
Since the seventeenth century, the ovum and the
sperm have been regarded, or at least one of them has




HE REDIITY


40I


been regarded, as in some sense the physical basis of the
new organism. A type of theory called preformationist
was held from the seventeenth century until the beginning of the nineteenth. In accordance with preformationism the ovum, or the sperm, contained a complete
preformed miniature organism which only required to
be unfolded or evolved, and increased in size, in order
to become the new animal. Moreover, either the ovum
or the sperm, according to the views of two contending
parties, not only contained the miniature copy of the
animal of the next generation, but this contained a still
more minute copy which would ultimately develop into
an individual of the second generation; and this contained a still more minute copy, and so on for all future
generations. This theory of emboitement, sometimes
known as the pill-box theory, contained as its essence
the principle that nothing new arises, that nothing is
generated, but that everything pre-existed in an invisible
form; as it was expressed by Albrecht von Haller, "Es
gibt kein Werden"; there is no such thing as becoming,
there is only an unfolding or evolution. The opposite
theory of Epigenesis, that the evolution of the animal
consists of a gradual increase of complexity from what
at first appears to be comparatively simple, and that
thus something essentially new arises, was shown by
Wolff to be a fact of perception in the case of the
embryonic development of the chick, and ultimately got
the upper hand over preformationism. However arbitrary
the preformationists' theories may appear to be, we
must, I think, recognize, when it is remembered that
the successive miniature organisms were never supposed
to be discernible to the senses, that they have some of
the essential elements of a genuine scientific theory,
considered as a purely conceptual representation of
what can be observed to happen. The crudity of preformationism as a conceptual scheme exhibits itself in
the very restricted range of the perceptual facts to the
HGL                                      26




402


HEREDITY


representation of which it lends itself. It fails, for
example, to distinguish the specific characters which
may be inherited from either parent, and gives no
suggestion as to how the origin of congenital variations
is to be represented. That it did not account for the
preformation is not a decisive objection to the theory,
for every scientific theory necessarily suffers at some
point or other from a similar defect. Moreover, some
of the modern theories, especially that of Weismann,
are closely akin to preformationism.
The next type of theory designed to represent facts
of inheritance is the pangenetic theory. The general
notion of pangenesis was adumbrated by Democritus
and Hippocrates, and in later ages by Maupertuis and
Buffon. The general idea of pangenesis is that the
germinal cells contain samples contributed by all parts
of the body, and that these samples give rise in the
embryo to parts similar to those from which the samples
came. The modern forms of the pangenetic theory are
exhibited in the theory of Physiological units, propounded by Herbert Spencer in 1863, and in the wellknown theory of Pangenesis, suggested as a provisional
hypothesis by Charles Darwin in i868. The essential
principle of Darwin's theory is that every cell of the
body, not too highly differentiated, throws off at each
stage of its development characteristic gemmules, or
small particles, which can later multiply by fissure, and
give rise to cells like those from which they originated.
These gemmules are continually given off and conveyed
in the blood; they become specially concentrated in the
germ-cells of both sexes, or in buds. In the development of the embryo these gemmules unite with others
like themselves, and being aggregated in the germ-cells,
they invest these latter with the power of developing
into a new complete organism; but some may remain
latent during the development of embryos through
several generations before they become active. Each




HEREDITY


403


gemmule reproduces the cell from which it was derived,
and they become active in the same order as that in
which the corresponding cells follow each other in the
ontogeny of the part to which they belong.
If Darwin's theory be regarded as a purely conceptual
theory, in which the gemmules are not regarded as
representing perceptual objects, it possesses the essentials
of a genuine scientific theory. The suggestion that the
gemmules might be replaced by a theory of transmission of force, as in the theory of Herbert Spencer,
would seem to lead to no improvement on the theory
in its original form, because it would not increase the
capabilities of the theory as a scheme for the representation of observed facts. For force-centres have no advantage over conceptual gemmules in depictive power,
and the importation of the term " force," in an undefined
sense, is open to serious objection. As a scientific theory
Darwin's seems to be a marked advance, in point of
effectiveness for its purpose, over the preformationist
theory; because it lends itself to the representation, not
only of the simpler facts of heredity with which the
latter theory was alone concerned, but also to that of
the transmission of characters which may remain latent
for several generations, and further to the transmission
of characters special to the male offspring through the
female parent, without becoming manifest in her. Also,
as has been pointed out by Sir Ray Lankester, the theory
can be made to account for the appearance, at a particular
period of life, of characters inherited and remaining
latent in the young organism.
It is interesting to quote some remarks made by
Weismann, in reference to Darwin's theory of Pangenesis. Weismann1 wrote in his Keimplasma:
I certainly consider even now that Darwin's theory must be
looked upon, and that he probably considered it, rather as an
inquiry into the problem of heredity than as a solution of the
1 See The germ-plasm, p. 5.




404


HEREDITY


problem. His assumptions do not, properly speaking, explain
the phenomena. They are to a certain extent a mere paraphrase
of the facts, an explanation of a purely formal nature, based on
speculative assumptions, which were made not because they
seemed possible, or even likely, but because they provided a
formal explanation of all the phenomena on one principle. If
we suppose that each cell arises from a special gemmule, and
that these gemmules are present wherever they are wanted, it is
easy to see how that structure, the origin of which we wish to
explain, may appear in any given position. Further, when a
large number of cells is to arise in regular succession from one
egg-cell, the desired sequence of cells must of course result if
we assume that the gemmules present become active in the
required order. But this supposition does not really explain
the phenomena. Even at the present day our explanations are
imperfect enough, and are far from going to the bottom of the
matter, but they differ from Darwin's provisional hypothesis in
that they attempt to find out the actual facts concerned in the
process, and to arrive at a real, and not a merely formal, solution
of the problem.
This criticism is, I think, based upon a radical misapprehension of the nature, scope, and possibilities, of
a scientific theory. Weismann's own theory, of which I
shall presently give an account, founded, as it no doubt
is, on a much more extensive range of observational
knowledge than that of Darwin, is open to criticism
on similar lines, more especially in its latest and most
developed form. Neither Weismann's theory, nor any
other theory, does or can "go to the bottom of the
matter," or "really explain the phenomena," or find a
"real" solution of the problem. Modified theories of
pangenesis have been propounded by Francis Galton,
and by W. K. Brooks; also a general theory of intracellular pangenesis has been developed later by De
Vries; but it is not necessary for our purpose to consider
these in detail. As in the case of Darwin's theory, the
scope of the first two of these is limited, because they
do not come to close quarters with the known detailed
facts relating to the inner constitution of germ-cells.




HEREDITY


405


The later theories, especially that of Weismann, make
full use of such knowledge, and the scope of their power
of representing facts of heredity is correspondingly increased. A most important part of Weismann's views on
the mechanism of heredity is contained in his theory
of the continuity of germinal protoplasm. This is not
the only theory of the kind that has been suggested; the
theories of Jager, Galton, and Nussbaum are of a somewhat similar character, although in some of them the
continuity asserted is rather that of germ-cells than of
the protoplasmic contents of such cells. As Weismann's
system has been worked out in great detail, and was
gradually developed by him into a form of increased
complication, designed to increase its scope and to meet
objections raised to earlier forms of it; and as it has
become not merely a theory of heredity but also of racial
evolution, I propose here to consider it in some detail,
as an example of a biological theory developed with consummate skill and with most ambitious aims, especially
as it has become very prominent in relation to various
matters connected with the most crucial questions of
Biology, in particular the question relating to the inheritance of acquired characters.
Weismann's theory, like other theories I have mentioned, is based upon the idea that the continuity of
characters in heredity is to be depicted by means of a
continuity of material between parent and offspring,
that continuity holding through a complete series of
generations. In order to understand the nature of
Weismann's theory it is necessary to state a number
of observed facts, accumulated by many workers in the
subject, relating to the fertilized ovum, its organization,
and what happens in it on maturation, and in the events
leading up to the development of the embryo. Every
cell in the body of an animal of given species contains
a general cell-substance, or cytoplasm, consisting in part
of protoplasm, that is of living matter, and partly of




406


HEREDITY


non-living material of various kinds. Within the cytoplasm is a small body called the nucleus of the cell, and
within this nucleus there is contained a substance known
as chromatin, because it can readily be stained and thus
distinguished from the rest of the material contained in
the nucleus. In many animal cells there are small
bodies known as centrosomes, which appear to play a
very important part in the operation which results in
the division of the cell into two cells; in higher plants
there appear to be no centrosomes. The chromatin, contained in the nucleus, in certain conditions of the cell,
consists of a number of separate masses, called chromosomes or idants; these are rod-like, looped, or granular
bodies, and in any one species the number of these
chromosomes is a definite even number, the same in all
the cells. In man this number is now said to be 48; but
it was for some time thought to be 24; in some other
organisms the number is smaller, and in some larger.
It was shown, in 1883, by Van Beneden, and has been
since confirmed, for the cases of many animals and
plants, that the male sperm-cell and the female ovum,
when prepared for conjugation, each contain only half
the number of chromosomes which each somatic cell of
the species contains. This process of maturation, which
results in the halving of the number of chromosomes in
a germ-cell, either male or female, is carried out in a
complicated manner; and, in the case of higher animals,
it usually commences two cell-generations earlier than
the formation of the gametes, that is of the matured
sperm-cell and egg-cell which conjugate. When a spermcell reaches an ovum, it bores its way into it, and
leaves behind its cytoplasmic substance; the nuclei of
the sperm-cell and of the ovum then move towards
each other and unite as a single nucleus which contains the chromosomes contributed by the two cells.
The number of the chromosomes is thus brought up
to the full number characteristic of the cells of the




HEREDITY


407


species. The walls of the ovum then become harder, and
this prevents the entrance of any other sperm-cell. There
is no centrosome in the mature ovum, but the spermcell brings its centrosome with its nucleus. This centrosome then divides into two, which appear to play an
important part in the subsequent division of the now
fertilized ovum into two cells. The ovum is very much
larger than the sperm-cell, and thus provides the bulk
of the material necessary for the initial stage of the
development of the embryo. Both parents contribute
alike the chromosomes, and in Weismann's theory it is
these chromosomes upon which rest the foundations of
the inherited organization of the offspring; the male
parent contributing the centrosome, which appears to
play the part of directing the process of division of
the fertilized ovum. In accordance with the view of
Weismann and his school, in any given species, the
nuclei of the two gametes contain the hereditary substance; and more particularly this is contained in the
chromosomes; the hereditary substance contains the
primary constituents of the whole organism. After the
single nucleus of the fertilized ovum has been formed,
which contains the paternal and maternal chromosomes
in equal numbers, the process of sub-division of the
cell begins. Each chromosome divides longitudinally
into two halves, and these halves ultimately arrive at
the poles of the nucleus, one half at each pole, where
one of the paternal centrosomes is situated. From each
centrosome there radiates a system of rays which seem
to be associated with the direction by the centrosomes
of the movements of the half-chromosomes. Near each
pole there comes to be a group of semi-chromosomes,
half of paternal, and half of maternal origin. Each of
these groups forms a new nucleus; the cytoplasm of the
cell divides into two halves, across the equatorial plane,
and thus two new cells are composed, each of which
has a nucleus, of which the chromatic material is half




408


HEREDITY


of paternal, and half of maternal, origin. In certain cases
this process of division of the cell into two, each containing a nucleus, half of paternal, and half of maternal,
origin, has been observed through several successive
divisions. The development of the embryo then proceeds
by continual cell division, starting with the pair of cells
into which the fertilized ovum divides. The swarm of
new cells which is thus developed consists partly of
somatic cells, of various constructions, which form the
various parts and organs of the developing embryo, and
partly of undifferentiated cells which ultimately become
the germ-cells of the new individual. Investigation has
shown that, in certain cases, these young germ-cells contain equal numbers of paternal and maternal chromosomes; in later stages it is thought probable that there
is a separation of the paternal and maternal chromosomes, such that each cell contains some of each, but
not necessarily in equal numbers. This is regarded as
giving rise to the possibility of various combinations of
the paternal and maternal hereditary characters, as represented in the final germ-cells of the new individual.
We are now in a position to consider, in some detail,
Weismann's theory of inheritance; a theory which, it
should be observed, takes for granted the essential
phenomena of life, nutrition, assimilation, and growth.
In the first place, the assumption, based on evidence of
observation to which I have already alluded, is made,
that there is a definite hereditary material located in the
chromosomes of the nuclei of the sperm-cell and the
egg-cell. Weismann next assumes, also on evidence based
upon observation, that the germ-cell contains not only
the primary constituents of a single individual of the
species, but also those of several, often even of many,
individuals. The whole material of which the chromosomes, or idants, are composed, is called the germ-plasm;
as Weismann writes: " I call the idioplasm of the germcells, germ-plasm, or the primary constituent substance




HEREDITY


409


of the whole organism, and the complex of primary
constituents necessary to the production of a complete
individual I call ids." Thus a chromosome, or idant, is
made up of several ids, each of which is the bearer of
a complete inheritance. The fundamental conception
of the theory of the continuity of the germ-plasm, as
stated by Weismann in 1885, is that, in each development of the new embryo, a portion of the germ-plasm
contained in the fertilized ovum is not used up in the
formation of the offspring, but is reserved, and remains
unchanged in the body of the offspring, for the formation of the germinal cells of the following generation.
Thus what is continuous is usually not germinal cells
but the germ-plasm "of definite chemical and special
molecular constitution." This idea of a physical nexus
between the successive generations of a race, in the
handing on from one generation to another of the germplasm, has sometimes been imaginatively described as
the immortality of the germ-plasm. As an embryo is
developed by continual bifurcation of cells, starting
from the fertilized ovum, gradual differentiation of these
cells sets in, and various body-cells, with specialized
structure and function, appear; these form the various
parts and organs of the developing embryo. But some
cells remain undifferentiated, and gradually lead up to
the germ-cells of the new organism. Ultimately, on the
maturity of the new organism, these germ-cells become
liberated, and each of them contains chromosomes both
of paternal, and of maternal, origin. It is then assumed,
as a generalization of a number of observed facts, that
each such chromosome contains germ-plasm derived
from the ancestors of both parents; and thus there comes
to be in the chromosomes an accumulation of material
derived from earlier ancestors, both on the paternal and
maternal sides. It should be observed that, up to a certain
point, namely the existence of ids, the concepts employed
are of the kind which correspond to perceptual objects.




4IO


HEREDITY


But when, as Weismann assumes, each id is regarded
as containing in itself, in some sense, all the generic,
specific, and individual characters of a new organism,
a concept is created to which no perceptual element
corresponds; in fact the theory passes beyond the stage
in which it is exclusively descriptive of perceptual facts
that can be directly observed. But Weismann does not
stop at this point; since the dissimilarity of parts of
the organism must be represented in the ids, he regards
each id as consisting of many invisible constituents,
which he calls determinants. Each determinant is concerned with the formation of some special organ in the
embryo. It is not necessary to assume that there are in
the germ-plasm as many determinants as there are cells
to be determined in the individual at every stage of its
development. There must however be as many of these
as there are regions in the fully formed organism capable
of independent and transmissible variation, including
all the stages of development. Thus a single determinant
can represent a group of cells which can vary en bloc.
In order to account for the mode in which the determinants give rise to cells and tissues, the special characteristics of which are represented by the different
determinants, Weismann regards them as composed of
groups of biophors, which are to be regarded as the
minutest vital units. These biophors exhibit the primary
vital characteristics, assimilation, metabolism, growth,
and multiplication by fissure; each such biophor is supposed to have a molecular constitution. Whilst the id
represents the complete individual, the determinants
represent its different parts and groups of cells; the
biophors represent characters. A germ-cell contains as
many biophors as the individual which this cell is to
produce possesses of elementary individual characters;
and each biophor may vary independently, so as to
produce a corresponding modification in the character
it represents. The biophors are supposed to be liberated




HEREDITY


4II


in the cytoplasm of the cells of the embryo. As the
fertilized ovum divides and re-divides, the blastomeres,
that is the resulting cells, become heterogeneous, if not
at the very earliest stages of re-division, at all events
at an early stage; that is they become suited to form
certain parts only of the embryo. This fact is represented
in Weismann's theory by the conception that the determinants, bearing various characters, become gradually
distributed amongst different cells during the process
of successive segmentation of cells. This process of distribution must be regarded as of an orderly character;
and thus the various determinants must be looked upon
as being definitely localized, in accordance with a definite
structure of the germ-plasm. In order that this distribution may take place so that each determinant may reach
the proper locality in the new individual, there must be
qualitative differences, as regards the distribution of
determinants, between the first two cells which appear
on the first division of the ovum. The blastomere which
contains the determinants corresponding to one class of
organs will then at the next division split into two cells
which again differ from one another in respect of the
determinants they contain, and so on; the ontogenesis
depending at every stage upon cleavage of a cell into
dissimilar cells, and producing in the end dissimilar
structures. As the organs and tissues become differentiated, the germ-plasm becomes less complex, owing
to the release of continually more of its determinants,
and ultimately transforms itself into the idioplasm to be
found in each cell, and which only contains determinants
of one cell and of parts of it. The biophors then break
off from the determinants, and these scatter themselves
through the cytoplasm, and thus impart to the cell its
specific character. Some cells may contain determinants
which do not break up into biophors, but which remain
in reserve until a later time when the cell may become
more differentiated. The biophors require, for the actual




412


HEREDITY


production of the characters they represent, the cooperation of the cytoplasm which constitutes the body
of the cell in which they operate.
Weismann speaks of the determinants as being kept
in relation with one another by "vital affinities"; and
this can only be taken to mean that it must be accepted
as a postulation of his theory that they do keep in relation with one another, and that he does not attempt to
set up a definite theory as to the modes of such relations.
When "vital affinities" and the like make their appearance in a biological theory, it is a sure sign that a
point has been reached beyond which the descriptive
power of the theory cannot pass. The whole secret of
the phenomena may be regarded as having been concentrated in these determinants and biophors with their
vital affinities, and in principle, the distance from an
" explanation," in any ultimate sense, is precisely where
it was before the theory was developed. The spatial
scale of the phenomena has been reduced, and that is
all. This does not however imply that such a theory is
useless; quite the contrary. The determinants and
biophors do not represent perceptual objects, although
definite properties are assigned to them. They are
regarded by Weismann as having a definite chemical
constitution, and this may differ in different kinds of
biophors; or there may be a difference of the arrangement of the atoms in the matter even with one and the
same chemical constitution. The biophors may play a
similar part in a conceptual scheme to that which atoms
or electrons play in physico-chemical schemes, and may
be equally indispensable. Thus, I take it, Weismann's
scheme, when regarded as a conceptual scheme, is not
open to objection, any more than was Darwin's theory
of Pangenesis, merely on the ground that it employs
concepts such as biophors. It should however be remarked that Weismann himself was far from accepting
the view that biophors are purely conceptual entities.




HEREDITY


4I3


His physical realism appears clearly in his statement:
"The biophors are not, I believe, by any means mere
hypothetical units; they must exist, for the phenomena
of life must be connected with a material unit of some
kind."
The actual value of Weismann's theory, as a descriptive scheme worked out in great detail, and with consummate skill, is one which only experts can estimate;
and the opinions among them seem to be sharply divided.
Besides the objection that the primary constituents, determinants and biophors, are purely ideal, the further
objection has been raised that these concepts are unnecessary, and introduce an undue complication. The
weight of this latter objection can perhaps best be
estimated by contrasting Weismann's theory with other
theories of the germ-plasm, such as that advanced by
Delage. This does not employ such units as determinants
or biophors, but relies upon the variety of chemical
substances contained in the germ-cells. In Simon's
mnemonic hypothesis, the basis of heredity is the "unconscious memory" of the organism, transmitted in the
germ-plasm.
Very serious objections to the correctness, or at least
to the generality, of the assumptions of Weismann's
theory, have been made upon the basis of experimental
observations. Some experiments tend to show that the
importance of the chromatic matter of the nucleus as
the bearer of heredity is exaggerated in Weismann's
theory; that, in some cases at least, the cytoplasm plays
a larger part than that merely of nutrition of the embryo.
Experiments carried out by Delage on the eggs of seaurchins have shown that it is possible to fertilize a
fragment of the ovum which contains no nucleus, and
that some measure of development may occur in this
fertilized cytoplasm. Doubt has also been thrown upon
the essential character of the effect of the sperm-cell in
fertilizing the ovum. It was shown by Loeb that the




414


HIEREDITY


eggs of the sea-urchin may be fertilized by adding to
the water in which they live certain chemical substances.
This artificial parthenogenesis suggests that, in conjugation, the essential part played by the sperm-cell is
that of introducing some chemical substance which
stimulates the ovum to activity. That part of Weismann's
theory which has to do with the segregation of the
determinants was supported by experiments carried out
by Roux. He found that the first division of the fertilized
ovum of a frog marked out the right and left halves of
the body, the one blastomere developing into the right
half of the embryo, and the other into the left half.
Roux succeeded in destroying one of the blastomeres,
and found that the other one developed into a halfembryo. This seemed to be decisive proof that, in the
segmentation, there was a differentiation of the two
blastomeres as bearers of heredity. But a later experiment of Driesch showed that if, after one of the blastomeres has been destroyed, the egg be turned upside
down, the uninjured blastomere develops into a whole
embryo, only smaller than if the whole egg had been
allowed to develop. He showed also that, if the uninjured egg, when in the two-cell stage, be turned upside
down, two whole embryos are developed. This and
similar results obtained appear to show that, up to the
two-cell stage, there can be no such segregation of the
determinants as is indicated by Weismann's theory.
Upon these experiments Driesch has based one of his
proofs of the existence of entelechy.
In close connection with Weismann's theory of
heredity, and with other theories of a more or less
similar type, is the great controversial question as to the
inheritance of acquired characters. Not only on account
of its importance in Biological Science, especially in
connection with theories of Evolution, but also on
account of the consequences which any answer to it
entail in relation to social questions reaching far beyond




HEREDITY


4I5


the narrower sphere of Biology, this question has given
rise to acute and lasting controversy. By the term
"acquired character," or "somatic modification," is to
be understood any structural change in the body of a
multi-cellular organism, of a kind which involves some
change from the normal structure of the species to which
the individual belongs, and which is acquired and remains
permanent during the lifetime of the individual, and can
be shown to be traceable to a change of environment,
such as climate, or to functional use or disuse, such as
is involved in specialized habits, or in mutilations. From
the point of view of any theory of germ-plasm, the
question whether a somatic modification of this kind is
heritable or not is equivalent to the question whether
such modification is accompanied by a specific change
in the germ-cells, such that the offspring will inherit, in
some degree or other, the modification which the parent
acquired. It is quite clear that this question can only
be answered empirically, after an abundant amount of
observation and experiment specially directed to elucidate
the matter. No answer can be accepted as decisive which
depends upon a theoretical deduction from a special
theory of heredity such as that of Weismann; on the
contrary, a decisive answer to the question, obtained by
means of fully sifted observations, would be a most
crucial test of the value of such a theory as descriptive
of the facts of heredity. In the empirical investigation
of the matter, it must in the first instance be shown that
what purports to be an inherited modification is really
an acquired character, in accordance with a precise
definition of the meaning of the term. The mere fact
that a bodily peculiarity reappears in several generations
is not sufficient proof that it has been inherited as such
an acquired somatic modification; it may be a congenital
germinal variation which has not been originally produced in the manner described. It must further be
shown that the apparent recurrence of an acquired




416


HEREDITY


modification in a later generation is not connected with
a recurrence of the environmental change, or of the
specialized habits of life, which were the origin of the
parental modification. Lastly, it must be shown that
the somatic modification of the parent is not accompanied by a modification of the germ-cells. If such
modification of the germ-cells of the parent can be
shown to occur, as may happen for instance when the
whole body, including the reproductive cells, is poisoned,
there will be some effect produced upon the offspring,
not necessarily owing to the somatic modification of the
parent, but owing to the accompanying modification of
the germ-cells. This last restriction naturally introduces
an element of difficulty into the investigation, because
it cannot be easily possible to ascertain the circumstances
in which a somatic modification is accompanied by a
change in the germ-cells. The negative answer given by
Weismann and others to the question whether acquired
somatic modifications are heritable depends upon the
assumption that, at least in the case of the more ordinary
somatic modifications, there is no corresponding specific
modification in the germ-cells.
In accordance with the theory that the basis of inheritance is the germ-plasm which is separate, and
remains segregated, from the somatic cells, everything
turns upon the relations between the body-cells and the
germ-cells. Weismann's view is that an acquired modification in general affects only the somatic cells, and has no
influence on the germ-cells, at least in the direction of
producing in them such specific modification that the
acquired somatic modification becomes heritable. It
should be observed that Weismann does not assert that
the germ-cells remain absolutely unaffected by the
modification in the body-cells, but only that they are
not so specifically affected that the offspring will thereby
exhibit the same modification that was acquired by the
parent, or even a tendency to it. Some of the later




HEREDITY


417


developments of the theory of Weismann I shall have
occasion to refer to in connection with theories of
Evolution.
The theories of heredity to which I have referred all
depend upon the procedure of employing facts obtained
by observation and experiment to suggest a descriptive
scheme, or as some would say a mechanism, which will
enable us to trace out in detail the processes which lead
up to the facts that are observed, and to use such descriptive scheme as an instrument for predicting occurrences, not yet observed, which would follow in appropriate circumstances, on the assumption that the
descriptive scheme is sufficient for this purpose. The
verification, or lack of verification, of the potency of the
scheme in such predictions is then to be regarded as
a test of the value and scope of the particular theory
in question.
There exists however another method of procedure
which may be described as the statistical method, that
has been in recent years applied to matters relating to
the study of heredity. The essence of this method consists in the ascertainment, by direct observation, of the
frequency of occurrence of a certain character or group
of characters in a large number of individuals of a
particular species, compared with the occurrence of the
same character or group of characters or related characters,
or of certain conditions, in the parents, or in the remoter
ancestors, of those individuals of the species in which
they are found. The purely statistical facts obtained are
then analysed by the mathematical methods of statistics
with a view to the determination of correlations, the
existence of at least some of which may have been
previously unsuspected, and in particular of obtaining
numerical estimates of the average frequency with which
parental or ancestral characters reappear in the offspring.
The application of this method does not require the
employment of any theory as to the genetic modes in


HGL


27




418


H~EREDITY'


which the correlations between different generations of
a race are set up; but the results of statistical theory
may be used to assist in the setting up of such theories,
or in the work of discriminating between alternative
theories that have been suggested by workers who
proceed by the non-statistical method.
The first person who seems to have fully grasped the
possibility of applying the statistical method to problems
of heredity, and to the general problem of variation in
connection with evolutionary conceptions, was Francis
Galton, who applied the method to a variety of questions
in this order of ideas. The method has been followed
up by a band of researchers, of whom the most prominent
has been Professor K. Pearson, and their work has been
incorporated in the department known as Biometrics;
this department of research is represented by an important periodical appearing under the name Biometrika.
The particular characters amenable to this method are
often of a measurable character, and are thus capable
of being correlated with numbers. In a given race, the
average measure of a particular character of the kind
considered has, for a very large number of individuals,
at any one time an average value known as the mean
value of the character for the race. It may happen that
the mean value of the character may change from
one generation to another, but in point of fact many
characters preserve their mean value unchanged for
many generations. When the mean value changes, it
cannot be concluded that the character is necessarily
heritable; the change may occur without any individual
heredity in the particular character. Galton himself
conducted a most careful statistical inquiry, partly by
using the records of about a hundred and fifty families,
relating to stature, colour of the eyes, some kinds of
disease, and artistic faculty. It will be observed that
the last two characters are not measurable in the same
sense as the others, but their occurrence in different




HEREDITY


419


members of the families can be counted; moreover
Galton applied his method, as exhibited in the last case,
to mental as well as physical characters. He also made
observations on characters in sweet-peas and moths, as
well as various measurements in his anthropometric
laboratory. One of the simpler problems which can be
treated by this method is that of ascertaining whether
a deviation of a particular character, such as stature,
from its mean value in the race is heritable or not. On
the basis of a large series of observations, it was shown
by Galton, and more precisely by Pearson, that the
average stature of a son is, reckoned in inches, 3 I I inches
plus nine-twentieths of the number of inches in the
stature of his father. This warrants the conclusion that
deviation of stature from the mean of the race is heritable.
Thus if a father differs in stature from the average of
the race, the same is on the average true of his son. It
also shows that, on the average, the deviation of stature
of the son is in the same direction as that of the father
but, on the average, smaller in amount. This last fact is
a particular case of a generalization propounded by
Galton, after careful statistical inquiry, and known as
the law of filial regression. This law is most simply
illustrated by the case of stature, although it has been
verified in its general features in the case of other
characters. If, for example, fathers whose height is
72 inches be taken, the mean height of their sons is
70-8 inches; these are still, on the average, taller than
the average of the general population, but differ less
from it than the fathers, and thus they regress towards
the mean of the stock. A similar result holds in the case
of fathers who have some height less than that of the
average of the stock; their sons are, on the average, taller
than the fathers, but less tall than the average of the
stock; they have progressed towards the mean. Galton
also showed by his statistical studies that the average
of human stature is very constant from generation to
27-2




420


HEREDITY


generation, although there is statistical evidence that
there is no correlation between the statures of people
who marry. These facts Galton attempts to formulate
by means of a principle of organic stability of the race,
in accordance with which there is a stable type, or
average of the particular character, which is preserved
unchanged through successive generations. He does not
however connect this principle of organic stability with
natural selection, in virtue of which the stability of the
average value of the character and the fact of regression
towards it might be accounted for on the principle that
the average value was more suited to the environment
than a different average value would be. The explanation
of this specific stability given by Galton is that each
child inherits in part only from the parents, and in part
from the more remote ancestors, and since what he calls
the mid-parentage is, on the average, nearer than the
exceptional parents to the mean for the race, the children
of selected parents are, on the average, more mediocre
than their parents. The fact that the mid-parentage is
nearer to the average of the stock than the exceptional
parents is supposed to be due to the fact that, when the
ancestors are counted back for many generations, they
consist of so many and such varied elements that they
become, as regards their average characters, indistinguishable from the general population. The theory has
been criticized by W. K. Brooks on the ground that,
while the child is descended from a long line of ancestors,
it inherits from none but the parents, and that it can only
be said in a figurative sense to inherit from more remote
ancestors. To estimate the force of this criticism, it
should be observed that Galton establishes at most
merely the bare fact that there is a correlation between
the characters of the child and those of its more remote
ancestors, and that when this has been done the power
of his method, at least as regards this particular application, is exhausted; the method establishes no inter



HEREDITY


42r


mediate nexus between the remote ancestors and the
child. Galton's opinion that the correlation is due to
the organic stability of the race is in no sense an explanation of the correlation, but consists merely of the introduction of an expression to denote the ascertained fact
of constancy of type through many generations. Only
such a theory as that of Weismann's theory of the continuity of the germ-plasm is in a position to exhibit by
a pictorial representation how the nexus between the
child and its more remote ancestors is to be regarded
as being through the parents; in that the germ-plasm
of the parents contains elements derived from the
ancestors. It would thus appear that Galton's ascertainment of the fact of correlation between the characters
of the child and those of its more remote ancestors
affords some confirmation of the descriptive potency of
the theory of the continuity of germ-plasm. The criticism
of Galton's theory seems then to be valid only as pointing
out a necessary limitation in the scope of the method,
so long as that method is considered, as it should be,
only as a mode of establishing facts of correlation, and
not as providing a description of any mechanism of the
correlation. Criticism has also been directed against
the view that the remote ancestors are so numerous
that, in the bulk, they may be taken to be equivalent,
for the purposes of discussion, to the general population.
A man has for example theoretically 4096 twelfth grandparents, but it is argued that this number must, in
practically all cases, be markedly diminished, since the
descendants of the largest part of a more or less isolated
population, as it exists and intermarries at any one time,
die out after a few generations; it is thus argued that
a tolerably isolated population must actually be descended from a relatively small number of individuals
who existed and intermarried at some former period,
and that, this being the case, the force of Galton's
reasoning is much diminished. But this consideration




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would seem merely to point out that caution is required
in pushing Galton's argument too far, rather than to
provide a complete refutation. In any case the ascertained facts of correlation stand on a much firmer basis
than any attempt to explain them theoretically.
Galton went further than merely to ascertain the fact
that there is a correlation between the characters, such
as the stature, of an individual and the corresponding
characters in his ancestry. The law of ancestral inheritance, which he based upon his observations and
statistics, includes a numerical estimate of the amount
of correlation. The law is that:
The two parents between them contribute on the average one
half of each inherited faculty, each of them contributing one
quarter of it. The four grandparents contribute one quarter, or
each of them one sixteenth; and so on, the sum of the series
-2 + i -+ i- — l +... being equal to i, as it should be. The
pre-potencies or sub-potencies of particular ancestors, in any
given pedigree, are eliminated by a law that deals only with
average contributions, and the varying pre-potencies of sex in
respect to different qualities are also presumably eliminated.
It must always be remembered that such a law is purely
statistical, dealing only with averages in a large population; its correctness cannot be refuted by production of
single or a few individual cases in which it can be shown
not to represent the facts; the only refutation possible
must itself be based upon statistics dealing with a very
large number of individual cases.
A very different class of investigations relating to the
laws of inheritance in hybrid varieties have been of great
prominence in the biological work of the last two
decades. These investigations are dependent upon the
original discoveries of G. J. Mendel, Abbot of Briinn,
who, after many years of experimental investigation,
chiefly of varieties of peas, published them in i866. The
law known as Mendel's law was rediscovered by several
botanists in 19oo, and the attention of Biologists was




HEREDITY


423


drawn to Mendel's results by Mr Bateson, who extended
them by a series of his own experiments. The theory
of the Mendelian school is based upon statistical results
relating to the descendants of hybrid varieties obtained
by crossing two varieties of a species which differ from
one another in respect of some particular character.
Mendel found that, in investigating the effect of crossing
two varieties, for example of peas, differences in certain
specific characters could be separately investigated, and
this gave rise to the conception that the difference
between the two varieties is compounded of differences
in respect of separate unit characters. When a variety
with one particular character, such as tallness, is crossed
with a variety which differs in respect of the corresponding character, such as dwarfness, in the offspring,
one of the two characters, in particular tallness, is found
to occur in all the offspring. That character, tallness, is
said to be dominant, and the character which does not
appear in the offspring is said to be recessive. The
hybrids, all of which possess the dominant character,
were then crossed with one another, and it was found
that amongst the offspring both characters occurred, but
the number with the dominant character (tallness) was
three times the number of those with the recessive
character (dwarfness). When the recessives are crossed
with one another, they give rise only to recessives, and
they breed true for any number of generations. But
when the dominants fertilized themselves, they produced
one-third pure dominants, and two-thirds cross-bred
dominants. The former of these breed true, and the
latter give rise to a mixture of dominants and recessives
in the proportion of three to one.
The law of dominance, that is of the fact that one of
the two characters is dominant, in that all the hybrids
possess it, is the first part of Mendel's scheme. The
second part is the law of segregation, which expresses
in numerical form the facts, in relation to the suc



424


HEREDITY


cessive generations, as to the segregation of the two
characters in sets of individuals which thereafter breed
true. Mendel suggested as an explanation of his experimental results a theory of the existence of two kinds
of germ-cells which exist in all the hybrids in equal
numbers-the theory of gametic segregation; and he
made an hypothesis on this basis as to how the segregation of the characters is effected. This theory of gametic
segregation and combination has been applied by Mendelians to represent and to predict, in accordance with
the law of averages, the results of crossing varieties,
and it has been extended to cases in which the characters
considered are not distinguishable into dominant and recessive characters.
In certain cases the laws of Mendel have been verified,
but in the long series of experiments which have been
made in this order of investigations various complications have been introduced in the interpretation of the
results. Important practical applications of the theory
have been made, especially in the production of wheat
which possesses the quality of immunity to the disease
known as rust. Hopes have been entertained that investigations on Mendelian lines might throw light upon
the general theory of evolution of species, but it appears
to be doubtful whether there is at present much prospect
that such hope will be fulfilled.




XVIII


THE EVOLUTION OF SPECIES
T   HE general notion of evolution, that in some sense
the present state of things has arisen from earlier
states by some kind of development, is as old as recorded
thought; and traces of the application of the idea to the
case of the evolution of living organisms, by a process
of growth and modification of primordial germs, are to
be found in the utterances of the earliest Greek thinkers.
Even the idea of the survival of the fittest was indicated,
in a crude form, by Empedocles (495-435 B.C.), who
depicted the gradual origin, in accordance with abiogenesis, first of plants, and then of animals, through
the chance play of combining and separating forces
acting on the elements fire, water, earth, and air. After
elimination of the earliest forms, fitter ones were, he
thought, produced, but still fortuitously. Both the
atomists, typified by Democritus, and their rivals, such
as Anaxagoras, who entertained teleological conceptions,
had some general ideas as to the occurrence of fitness
or adaptation. With Aristotle, the notion of evolution
attained to greater precision than with his predecessors.
He followed Plato in regarding the creation of the cosmos
as a process of descent from the more to the less perfect,
but his dualistic interpretation of this process permitted
him to hold a teleological theory of organic evolution.
Although he regarded types as the realization of an
original formative principle, and in this sense fixed, he
appears to have entertained the possibility of the spontaneous generation of the lowest organisms. He regarded
organs as fashioned by nature in the order of their
necessity, those essential to life coming first. Aristotle




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THE EVOLUTION OF SPECIES


clearly stated the conception of the survival of the fittest,
but only to reject it. He definitely refused to admit that
adaptation is due to the elimination of the unfit, but
regarded the process of successive adaptation as due to
an immanent principle striving to attain a certain end;
and that end he believed to be the production of man.
During the centuries between the close of the Greek
period and the Renaissance, under the influence of the
Church, the theory of special creation of species, resting
upon a literal interpretation of the Mosaic cosmogony,
became dominant. But there were not wanting, among the
Fathers of the Church, and later among the Scholastics,
those who, like St Augustine, by accepting a less literal
interpretation, combined it with the general conception
of evolution. The following utterance1 of St Augustine
in favour of freedom of scientific thought is of much
interest, as showing how far he was in advance of the
later attitude of the Church for many centuries in this
matter.
It very often happens (he writes) that there is some question
as to the earth or the sky, or the other elements of this world... respecting which one who is not a Christian has knowledge
derived from most certain reasoning or observation, and it is
very disgraceful and mischievous and of all things to be carefully
avoided, that a Christian speaking of such matters as being
according to the Christian Scriptures, should be heard by an
unbeliever talking such nonsense that the unbeliever perceiving
him to be as wide from the mark as east from west, can hardly
restrain himself from laughing.
Unfortunately this more liberal interpretation of the
Mosaic cosmogony died out amongst theologians, and
the doctrine of special creation became completely
dominant until the nineteenth century. When it was
discovered that many species had become extinct, and
been replaced by new ones, the doctrine was extended
to embrace a whole series of special creations of species.
1 The passage is quoted by Osborn in his work From the Greeks to
Darwin, p. 19.




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427


In the period immediately after the Renaissance the
attention of scientific workers was devoted rather to
detailed work in Anatomy and Physiology than to the
working out of so general a conception as that of organic
evolution. It was not until the seventeenth century that,
largely in the hands of Philosophers, the notion of
organic evolution, as a speculative idea, again came into
some prominence in the modern world. One of the first
to suggest the transmutation of species by accumulated
variations, and to advocate the experimental investigation of the subject, was Francis Bacon (I56I-I626).
Descartes (I596-1650), although he was much hampered
in the expression of his opinions by what he regarded
as the necessity of paying lip-service to the orthodox
doctrine of special creation, clearly showed, in his Traite
de l'homme, and in his essay Sur les passions, that he
believed himself to have found an explanation of the
universe, and in particular of the phenomena of life, on
purely physical principles, and also of the mode in which
the universe had been evolved. Leibniz, in his doctrines
of monadism, and of continuity, developed the view that
each monad is the focus of an endless process of evolution and involution; and that all natural orders of beings
form a single chain along which progress is made continuously. Spinoza also expressed a belief in evolution
in accordance with invariable laws. Immanuel Kant
should also be mentioned as having, at least in his earlier
period, advanced the conceptions of Selection, Adaptation, Environment, and Inheritance; but he appears later
to have abandoned his evolutionary views, as unconfirmed by observation.
The first Naturalist who clearly expressed the idea
that the unity of plan in the structure of animals may
be due to community of origin was Buffon (I707-I788).
His statements on the subject are vacillating, perhaps
owing to difficulties which he felt in breaking with the
orthodox conceptions, and he had no complete or con



428     THE EVOLUTION OF SPECIES
sistent theory of evolution; but some of his utterances
are quite explicit as to the possibility of the descent of
all the species of animals from one stock. For example,
in his account of the Ass, he argues that: "Once admit
that within the bounds of a single family one species may
originate from the type species by degeneration, then
one might reasonably suppose that from a single being
Nature could in time produce all the other organized
beings." Elsewhere he speaks of the reduction of the
two hundred species of quadrupeds which he described
to a small number of families "from which it is not
impossible that all the rest are derived." Within each
of the families the species branches off from a parent
species. Buffon anticipated various ideas, such as pangenesis, the struggle for existence, artificial and natural
selection, and geographical isolation; all of which became
later of fundamental importance in the theory of evolution. Being originally in accord with Linnaeus, his
views gradually changed, but there were, in his statement
of them, continual indications of indecision as regards the
opposed conceptions of special creation and evolution.
The great systematist Linnaeus (I707-I778) reckoned
at first "as many species as issued in pairs from the
hands of the Creator," but later he modified his views
so far as to admit that although "all the species of one
genus constituted at first one species they were subsequently multiplied by hybrid generation, that is by
intercrossing with other species."
It is remarkable that Cuvier (1769-I832), although
he recognized very clearly a succession of forms in time,
was a most resolute opponent of the evolutionary view
of descent, mainly on account of lack of sufficient
evidence for it. The course of the changes in his opinions
was diametrically opposite to that of Buffon and Linnaeus.
Starting with the most advanced views of Buffon as to
mutability of species, he gradually arrived at a point
such as that from which Linnaeus started, insisting on




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429


the fixity not only of species but also of varieties.
Although evolutionary ideas had some influence over
the minds of such men as Charles Bonnet (I720-I793),
Robinet (I735-I820), Oken (I779-I851), and Geoffroy
Saint-Hilaire, none of them developed a definite and
comprehensive theory of the matter. Some of them,
such as Bonnet and Oken, appear to have regarded
evolution as a purely ideal conception of the development of types of structure, and not to be regarded as an
historical account of the genesis of species. Others,
such as Robinet, Treviranus, Tiedemann, and Meckel,
held the view that it described an actual historical
process. Under the influence of such ideas as that of
the scale of beings, evolution was most frequently
regarded as taking place through a linear series, as it
were along a straight ladder, and not as figured by a
genealogical tree, as it came to be by Lamarck and
Darwin, and by all later evolutionists. The necessary
preliminaries of a theory of evolution based upon
scientific induction, and not merely upon philosophical
speculation, consisted in the development of the studies
of Embryology, Palaeontology, Comparative Anatomy,
and Distribution. This preliminary work was carried
out with much vigour in the eighteenth century, but
not until the nineteenth was it sufficiently completed to
provide a sufficiently solid basis of facts and ideas.
Jean Baptiste Pierre Antoine de Monet, known as the
Chevalier de Lamarck (1744-1829), first a soldier, then
a medical man, commenced his investigations as a
Botanist, and afterwards passed on to Zoology. He did
important work in systematic Botany and Zoology; and
these studies led him to the conception of the mutability
of species, and to the theory of the origin of species by
descent. In this domain he may be regarded as the most
important figure previous to Darwin; and certain of his
opinions which differ from those of Darwin have not
lost their importance in relation to the more recent dis



430     THE EVOLUTION OF SPECIES
cussions of the problems of evolution. In one of his
earliest works, written in i766, presented to the Paris
Academy of Sciences in I780, and published in I794,
he affirmed his belief in the immutability of species, and
his disbelief in abiogenesis; he asserted that all organic
individuals descend from other individuals entirely
similar, which taken together constitute the whole
species. His change of opinion did not become manifest
until I802, when he sketched out his evolution theory
in his Recherches sur l'organisation des corps vivants.
About the same time he proposed the term "Biology"
for the sciences of life; it is remarkable that in the same
year an identical proposal was made by Treviranus, also
one of the pioneers of the theory of evolution, largely
from the point of view of the Philosophy of Nature. The
highest point of development of Lamarck's views was
exhibited in his Philosophie Zoologique, published in
I809; in which his whole scheme of Evolution is
gradually built up in logical form. His scheme includes
the main ideas that species vary under changing external
influences; that there is a fundamental unity in the
animal kingdom; and that there exists a progressive and
perfecting development. Lamarck's theory of evolution
is closely allied in some points with the theory given
somewhat earlier by Erasmus Darwin, the grandfather
of Charles Darwin. The fundamental assumption made
by Lamarck is that changes acquired by means of the
functional reactions of an animal with its environment,
and changes produced in a plant by means of changes
in its environment, are transmissible by heredity to the
offspring. He believed that changes in the environment
produce no direct changes in the animal organism, but
that the indirect changes are due to induced changes of
habit, or of functional reaction, of the animal. The
changed habits, involving the new or changed use of
parts, result in structural changes which are transmissible to offspring, and thus form the main factor




THE EVOLUTION OF SPECIES                 43r
in evolution. In the case of plants, changes of soil and
climate have a direct influence upon the structure; and
the changes so produced are transmissible.
The essence of Lamarck's theory is contained in four
laws; the law of growth; the law of functional reaction;
the law of use and disuse; and the law of use-inheritance.
In accordance with the law ofgrowth, the size of any living
body, and the dimensions of its parts, are increased by
the body's own activities, up to some limit imposed by
the nature of the living body itself. The law offunctional
reaction asserts that the production of a new organ results
from a new need which continues to be felt, and from
the new movements originated and sustained by this
need. The law of use and disuse asserts that the development and power of action of organs are in proportion
to the use made of those organs. The law of useinheritance asserts that all that has been acquired, imprinted, or changed, in the organization of an individual
during the course of its life is transmitted to the new
individuals that descend from the individual so modified.
The difficulty of understanding the exact nature of
Lamarck's conceptions is much increased by the fact
that, although he propounded the most thorough-going
materialistic ideas, his view of life and evolution is
expressed in psychological terms; and thus contradictions arise within his theory, owing to the irreconcilability of his materialism with his vitalism. His
apparently materialistic conceptions are expressed in
the statements:
No kind or particle of matter can have in itself the power of
moving, living, feeling, thinking, nor of having ideas; and if,
outside of man, we observe bodies endowed with all or one of
these faculties, we ought to consider these faculties as physical
phenomena which Nature has been able to produce, not by
employing some particular kind of matter which itself possesses
one or other of these faculties, but by the order and state of
things which she has constituted in each organization and in
each particular system of organs...Every animal faculty, of




432


THE EVOLUTION OF SPECIES


whatever nature it may be, is an organic phenomenon, and
results from a system of organs or an organ-apparatus which
gives rise to it and upon which it is necessarily dependent.... The
more highly a faculty is developed the more complex is the system
of organs which produces it and the higher the general organization; the more difficult also does it become to grasp its mechanism.
But the faculty is none the less a phenomenon of organization,
and for that reason purely physical.
Lamarck regarded evolution as a process of gradual complication of organisms, by which new organs and therefore new faculties arise. On the other hand he divided
animals into three groups; insensitive animals; sensitive
animals; and intelligent animals. The first group have
no principle of reaction to external excitations, but
passively prolong them into actions, in accordance with
purely mechanical principles; they possess merely irritability. The second group have, in addition to irritability,
a power which Lamarck calls the " sentiment interieur."
This is a power of reaction to external stimuli which
involves the feeling of a need, and results in instant
action; it is usually called instinct in animals, and does
not imply either consciousness or will, but acts by transformation of external into internal excitations. The third
group, consisting of vertebrates, have the faculties of
intelligence and will, in addition to the irritability of the
other two groups and the "sentiment interieur" possessed by the second group. In Lamarck's opinion,
intelligence and will have little or nothing to do with
evolution. The "sentiment interieur" produces not only
instinctive actions, but also the formation of new organs
as due to needs experienced by this inner feeling. It
is not easy to apprehend the precise meaning which
Lamarck attached to his first law, that of growth. The
extension is not produced simply by nutrition, an immanent power is required to produce it; and this power
he regards as dependent on a subtle fluid, somewhat in
accordance with the ancient conception of a soul consisting of a very subtle form of matter. In his second




THE EVOLUTION OF SPECIES


433


law, that of functional reaction, the psychological conception of a felt need is the essential factor; and this it
would appear cannot be reconciled with his materialistic
conceptions, unless indeed it is capable of some other
interpretation; for, although the need is not a conscious
one, it is felt by the "sentiment interieur." Moreover,
at this point, a definite hiatus appears between the case
of animals of the lowest group, and those of the two
higher groups. For animals of the insensitive kind
possess no "sentiment interieur" which can experience
needs; in their case, their behaviour, and consequently
their evolution, depends upon the purely mechanical
action of fluids set in motion by the direct physical
action of the environment; whereas as Lamarck states
"this is not the case with the more highly organized
animals which possess feeling." As soon as a need is
felt, the inner feeling directs the fluids and forces to the
part of the body which can, by its action, satisfy the
need. If the requisite organ exists it is stimulated to
action; if no such organ exists, and the need is of a
sustained character, the required organ is gradually
produced and developed in accordance with the third
law, that of use and disuse.
It is in the third and fourth laws, which admit of
verification or refutation by observation, that the main
present interest of Lamarck's theory lies. The third law
asserts the priority of function to form, and the fourth
law asserts the heritability of acquired characters, a
proposition which has become in our day one of the
most keenly debated questions. Great as was the work
of Lamarck as a Naturalist, his reputation as a man of
Science has suffered, owing to his unrestrained tendency
towards speculation of a philosophical kind.
In the decades immediately preceding the new epoch
which commenced with the publication of Darwin's
Origin of Species, the idea of the derivation of species
from one another, or from primordial forms, was familiar


HGL


28




434     THE EVOLUTION OF SPECIES
to Biologists, but no general theory of the evolution of
species was developed. By such men as Cuvier and von
Baer, the evidence in favour of these current notions of
evolution was regarded as insufficient; although, in
I834, von Baer expressed his belief in a limited amount
of evolution. The facts of variability and those of palaeontology led him to believe that many species have been
evolved from parent stocks; but the absence of sufficient
evidence led him to reject any comprehensive doctrine
of descent from a primordial stock. Both in England
and in France the fixity of species was the creed of the
great majority of Biologists, including Richard Owen,
who never accepted Darwinian views; they spent their
energies in the detailed study of special departments of
Biological Science, without taking much interest in any
wide generalizations. Those of them who had read the
works of Lamarck were for the most part unconvinced.
The French Naturalist Geoffroy Saint-Hilaire, a colleague
of Lamarck, who was by nature more a Philosopher than
a Naturalist, belonged more to the Buffon-Lamarck
school of thought than to that of Cuvier, the great opponent of the evolutionary idea, although Saint-Hilaire
held views very different from those of Lamarck as to
the principal factors in evolution, and was less radical in
his views. Saint-Hilaire denied the inherited influences
of habit, which Lamarck regarded as all-important, and
considered that the transformations of organisms were
due to the direct influence of the environment, the role
of the organisms being comparatively passive. In the
celebrated debate between him and Cuvier which was
held in the Academy of Sciences in I830, the superior
knowledge of Cuvier gave the victory to the antievolutionist side; the basis of fact necessary to build up
a solid theory of evolution by induction being at that
time insufficient. It is interesting to remark that Goethe,
who was himself a philosophical evolutionist, expressed
much greater interest when he received the news of this




THE EVOLUTION OF SPECIES                 435
discussion than he did as regards the news of the Paris
revolution which arrived at the same time.
Robert Chambers published his well known work
Vestzges of the Natural History of Creation in 1844. The
evolutionary views which it contained led to its being
badly received by reviewers, who charged the author with
irreligious tendencies. Like Saint-Hilaire and Buffon,
Chambers regarded modifications of animal structure as
due to the direct action of the environment, in opposition
to the opinion of Lamarck; but he combined with this
the Aristotelian conception of a perfecting principle.
The state of opinion in the period just before the
Darwinian has been described by Weismann in his
Descendenz-Theorie; he writes1:
It is impossible to estimate the effect of Darwin's book on
the Origin of Species unless we fully realize how completely the
biologists of that time had turned away from general problems.
I can only say that we, who were then the younger men, studying
in the fifties, had no idea that a theory of evolution had ever
been put forward, for no one spoke of it to us, and it was never
mentioned in a lecture. It seemed as if all the teachers in our
Universities had drunk of the waters of Lethe, and had utterly
forgotten that such a theory had ever been discussed, or as if
they were ashamed of these philosophical flights on the part of
Natural Science, and wished to guard their students from
similar deviations.
It should however be observed that there were some
exceptions to this attitude of indifference to general conceptions. In I830 there appeared Lyell's Principles of
Geology, in which the author rejected the notion of Cuvier
that sudden catastrophes have been an important element
in producing terrestrial changes. Various palaeontological discoveries, from the time of Cuvier, paved the
way for the conception of evolution. From a philosophical point of view, Herbert Spencer, in 1852, advocated the transmutation theory. But neither the
cell-theory nor the great advance of knowledge of
1 See The Evolution theory, Vol. I, pp. 27-28.




436     THE EVOLUTION OF SPECIES
Embryology succeeded in overcoming the hostility of
the Naturalists of the period to such ideas, and indeed
to the discussion of any general conceptions.
The inception of the idea of theory of Descent in the
mind of Darwin was due to the varied observations he
made during the five years of the voyage of the Beagle
to South America, commenced in I831. He took with
him Lyell's Principles of Geology, in which Lamarck's
doctrines are fully discussed. He himself stated in later
years that three classes of facts had brought the matter
strongly before his mind; the manner in which closely
allied species replace species in going southward; the
close affinity of the species inhabiting the islands near
South America to those proper to the continent; and the
relation of certain living species to the extinct species.
From 1835 onwards Darwin devoted his time to meditation on the theory and mode of transmutation, and to the
collection and sifting of an enormous mass of facts,
especially relating to domestic animals and cultivated
plants, until then mostly ignored by scientific men. The
idea of the survival of the fittest was first suggested to
him by a study of Malthus' Principles of Population.
Malthus had pointed out that a competition amongst
individuals arises, in relation to the food supply, from
the fact that man tends to increase in geometrical ratio,
whilst the increase of the food supply is only in arithmetical ratio; thus bringing about the disappearance of
individuals less suited than others to sustain the contest.
Darwin saw that an extension of this idea might explain
the adaptation of all living organisms to the environment. Until 1858, Darwin published no account of his
new theory; he was determined to hold it back until
the verification from fact should afford evidence of
irresistible weight. In that year, Alfred Russel Wallace,
who afterwards became his friend, had independently
reached a similar theory, and the result of a communication by Wallace of his own manuscript embodying his




THE EVOLUTION OF SPECIES              437
views to Darwin was that two short papers were published
in the Journal of the Linnean Society, on June 30, 1858;
the first, by Darwin, consisted of an abstract of manuscripts written in I839 and I844: "On the variation of
Organic Beings in a state of Nature; on the Natural
Means of Selection; on the Comparison of Domestic
Races and True Species "; together with a letter written
in I857 to Asa Gray. The second paper, by Wallace,
consisted of an essay written in February I858, " On the
Tendency of Varieties to depart independently from the
Original Type." There is much similarity, together with
striking differences, in the views therein expressed by
the two investigators.
The publication of the Origin of Species in I859,
followed in I87i by the Descent of Man, had an effect
upon Biological Science, and ultimately upon other departments of Science, such as Sociology and Anthropology, which can only be described as revolutionary. This
was not due mainly to the novelty of Darwin's conceptions, for, as we have seen, not only had the idea of
organic evolution been from the earliest times amongst
the speculative conceptions of Philosophers, but it had
also been developed in detail, especially by Lamarck and
Erasmus Darwin. Even the special conception of Natural
Selection as a factor in Evolution, although it arose
quite independently in Darwin's mind, had been suggested by several earlier writers, as Darwin himself
acknowledged. The vast influence of Darwin's work was
due in the first place to the fact that he, for the first time,
firmly established the fact of the transmutation of species,
as an induction resting upon a vast accumulation of
facts obtained by observation and experiments of very
various kinds. In the second place he, for the first time,
worked out the special theory of Natural Selection as a
prime factor in Evolution, and adduced a great and
varied array of facts illustrating the modes of its operation. The first of these elements in Darwin's work was




438


THE EVOLUTION OF SPECIES


ultimately decisive in its effects. After a period of bitter
strife, the fact of organic evolution has been accepted
by all Biologists, and by the educated part of the general
public. After Darwin's work, it was no longer a speculative hypothesis, but a well attested deduction from
observation. As regards the position of Natural Selection
as the chief factor in Evolution, it is not possible to
speak so positively. On this matter the opinions of
Biologists have been, and still are, much divided; a fact
to which I shall refer later in some detail.
The arguments employed by Darwin in support of his
first main thesis, the relation of species to one another
by descent, are drawn from various departments of
Science; and he shows that they all point to the same
conclusion. In Darwin's opinion the most conclusive
evidence is drawn from the facts disclosed by Embryology. The resemblance between the embryos of various
animals is much closer than the resemblance between
the adults; the fact that the embryos of vertebrates, such
as birds and snakes, are almost indistinguishable from
one another at the earliest stage of their development;
the similarity in embryos of homologous parts which
become later on differentiated; all these point back to
an ancestor common to a whole group of different
animals. A strong confirmation is afforded by the survival
of vestigial organs. From the morphological point of
view, Darwin deduces that unity of type represents
actual relationship between the species which possess
that unity. From Palaeontology he produced cogent
evidence dependent on the close similarity of the fossil
remains of two successive geological formations. He
shows that more highly organized forms of life have
been developed successively and gradually from parents
closely resembling them. The facts of geographical distribution of species and fauna form an important part
of his argument. The fact that the faunae of regions
in which the geographical and climatic conditions are




THE EVOLUTION OF SPECIES


439


not very dissimilar diverge widely from one another is
taken to indicate the local development of those faunae.
The various barriers to migration are to be regarded as
an important factor; valuable indications being given
by an examination of the fauna of islands. Thus, when
an island is sufficiently far from a continent, it can be
observed that certain genera of animals which exist on
the continent are entirely lacking on the island. It will
be observed that the theory of descent, or of such a
relationship involving the transmutation of species, is
a descriptive theory of the most general kind; a wide
generalization starting from the perceptual fact of the
relation between individual animals and plants and their
offspring, rising to the conception that the existence of
the most diverse species of animals and plants can be
unified by universalizing this one kind of relationship.
It takes the fact of heredity as the fundamental fact, but
assumes, on the basis of induction from a great mass of
observed facts, that the relation of similarity between
parent and offspring is not so close but that it is consistent with extreme dissimilarity between individual
organisms, which are to be conceived as connected with
one another by a complex of relationships (not a single
chain) of which the unit is the single relation which
holds between parent and offspring. Subject to this
restriction, the theory is independent of any special conception, such as that of germ-plasm, as to the details of
the relation between parent and offspring; that is as to
any mechanism by which this relation of heredity can
be described. It is also independent of any theory of
the origin, nature, or magnitude, of the variations, or dissimilarities, between parent and offspring, which, by their
accumulation in the whole complex of which the relation
between individuals of different species consists, amount
to the actual dissimilarity between such individuals.
The second main thesis of Darwin is that the chief
factor in Evolution is natural selection. Setting out from




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THE EVOLUTION OF SPECIES


the existence of small heritable spontaneous variations
in the individual organism, that is of small deviations
from the average of the stock to which the individual
belongs, the structural variations of a particular character
being in the directions both of excess and defect, in
accordance with the theory, those of the variations which
give some advantage to the individual over his fellows
in the struggle for existence lead to the preservation
of the individuals possessing and propagating them, in
preference to those who do not possess them. This
selective process going on through many generations, in
which the originally small variations are increased by
accumulation, at every stage giving an advantage to those
individuals in which the variations are in the profitable
direction, leads ultimately to the survival of a stock which
differs from the original stock in respect of the particular characters; thus a new stock more fitted to the
environment than was the original stock comes into
being. In Darwin's own words: " Natural Selection acts
only by the preservation and accumulation of small inherited modifications, each profitable to the preserved
being." It should be observed that the doctrine does not
include any account of the mode in which the variations
arise or are inherited; their existence as variations, of
small but varying magnitude, in every direction, and
both in excess and defect relatively to the mean, and
heritable, is simply accepted as a datum derived from
observation. These variations are frequently described
as "fortuitous," or as "spontaneous," which can only
be taken to express the fact that they are not assumed
to be variations in any particular direction, or to be the
outcome of any assigned process.
From a general point of view, this theory is remarkable
on account of its anti-teleological tendency; for it is
designed to account for what has prima facie the appearance of a process of progressive purposive adaptation to the environment; it substitutes an account of the




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441


mode in which such adaptation can be conceived to arise
which does not involve, at any stage, any element which
requires the notion of guidance towards an end, due
either to conscious or unconscious agency. When such
an expression as "struggle for existence" is employed
in connection with this theory, the expression must be
taken in a very broad sense. The struggle may last for
many generations, and does not necessarily imply violent
individual contests in which the less favoured individuals
succumb. A slow racial elimination of the less fit, owing
to their lesser ability than the more fit to cope with their
fellows, with their racial enemies, to obtain sustenance,
and to propagate their kind, must be included in the
denotation of the expression. Natural selection provides,
not only for the progressive adaptability to the environmental conditions, but for the transmutations of the
adaptations when they have reached a maximum.
It is impossible, and indeed for my purpose unnecessary, to give any account of the wealth of the evidence
which Darwin gave in support of his theory, or of the
many illustrations and applications of it which he gave.
Many objections to it were seen and dealt with by
Darwin himself; others have been since advanced in
the course of the ceaseless discussion of the factors in
evolution which has taken place since the theory was
advanced, and during which the attention of researchers
has been largely devoted to the origin and mode of
inheritance of variations, which remained entirely open
in the original theory. In the course of his work,
Darwin paid a great deal of attention to the artificial
selection of varieties amongst domesticated plants and
animals, and especially in the breeding of pigeons. In
this case the purpose of the gardener or breeder plays
an effective part in selecting for propagation those
individual plants or animals which "spontaneously,"
according to a common expression, exhibit those variations which the gardener or breeder selects as according




442


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with his purpose. This is quite consistent with the antiteleological character of Darwin's theory, for the purposiveness is not immanent in the plants and animals,
but is in the mind of the gardener or breeder. The
selection in the case of undomesticated animals and
plants differs from this only in the fact that no gardener
or breeder interferes in its free play.
Darwin believed that natural selection is sufficient to
account for the evolution of the most complicated organs,
but he at all times admitted the existence of other contributory factors of evolution, especially of what may be
described as the Lamarckian factor, the dependence on
inherited effects of use and disuse. Thus he writes':
It is generally acknowledged that all organic beings have been
formed on two great laws-Unity of Type, and the Conditions
of Existence. By unity of type is meant that fundamental agreement in structure which we see in organic beings of the same
class, and which is quite independent of their habits of life. On
any theory, unity of type is explained by unity of descent. The
expression of conditions of existence, so often insisted on by the
illustrious Cuvier, is fully embraced by the principle of natural
selection. For natural selection acts by either now adapting the
varying parts of each being to its organic and inorganic conditions
of life; or by having adapted them during past periods of time;
the adaptations being aided in many cases by the increased use
or disuse of parts, being affected by the direct action of the
external conditions of life, and subjected in all cases to the several
laws of growth and variation. Hence, in fact, the law of the
Conditions of Existence is the higher law; as it includes, through
the inheritance of former variations and adaptations, that of
Unity of Type.
It has been however pointed out by E. S. Russell that
Cuvier meant by "Conditions of Existence" not environmental conditions, as Darwin assumed, but the
coordination of parts to form the whole of an organism.
It has been held that Darwin gives little weight in
his theory to, and does not account for, the correlation
1 Origin of Species, 6th ed., Pop. Imp., pp. 260-I.




THE EVOLUTION OF SPECIES             443
of variations in different parts of the organism. In the
course of time Darwin somewhat modified his belief in
the relative weight of the factor of natural selection as
compared with the factor due to the direct action of the
environment. Thus, in i862, in a letter to Lyell, he
writes: " I hardly know why I am a little sorry, but my
present work is leading me to believe rather more in the
direct action of physical conditions"; again in 1876, he
wrote to Moritz Wagner: "When I wrote the Origin,
and for some years afterwards, I could find little good
evidence of the direct action of the environment; now
there is a large body of evidence." In view of the great
influence which Darwin's theories have exercised upon
modern views of mental evolution, and upon Sociology,
it is important to observe that Darwin recognizes psychological factors as contributory to organic evolution. In
the Descent of Man, he advocated the view "that there
is no fundamental difference between man and the higher
mammals in their mental faculties." In this matter he
differed from A. R. Wallace, who held that the higher
mental faculties of man had been derived from a special
stream of spiritual influence at some period in the
evolution of the race.
Darwin had no theory of mental evolution, but took
into account the fact that conscious experience, with
concomitant physiological processes, occurs in connection with some kinds of animal behaviour. The close
correlation of psychological and physiological processes
he accepts without entering upon the difficult and highly
controversial philosophical discussions which arise in
this connection. If the theory of psycho-physical parallelism be accepted, a psychological description can be
regarded as a merely convenient form in which the
precisely corresponding organic processes in the brain
and nervous system can be denoted. There is no evidence
that Darwin accepted this theory in its complete sense,
but he appears to have accepted, at least implicitly, the




444     THE EVOLUTION OF SPECIES
methodological hypothesis that mental evolution is correlated with organic evolution through heredity. Only by
the assumption of this regulative idea can psychological
factors be recognized as within the domain of Natural
Science; without it no account of the behaviour of the
higher animals would be in any degree adequate which
was not based in large part upon a developed mental
science as an independent department. The nature of the
responses of an organism to stimuli, arising from the
environment, depends upon the detailed structure of the
organism; Darwin held that, in the main, natural selection was the mode in which the organism had been
adapted through heredity to respond fittingly to such
stimuli. Especially what is known as instinctive behaviour he regarded as the result of racial preparation,
transmitted through organic heredity, principally in
accordance with natural selection, but in many cases
arising partly or wholly from the inheritance of modifications due to use and disuse. But Darwin recognized that
instincts can be modified in the course of the individual
life. Intelligent modification of behaviour Darwin regarded as due to the combination of incipient variation
and acquired modification; under natural selection this
combination has survival value. A factor of evolution
which Darwin considered to have been of importance in
certain cases is sexual selection; and this involves the
prima facie recognition of a psychological element, since
it includes the conception of the choice or preference
of the females for males with certain characteristics, as
the origin of secondary sexual characters other than
weapons of offence and defence; these latter being of
importance in this connection only in the struggle
between males for the possession of the females. Darwin
states as regards the female that "it is not probable that
she consciously deliberates; but she is most excited or
attracted by the most beautiful, or melodious, or
gallant, males." This part of Darwin's theory has been




THE EVOLUTION OF SPECIES             445
the subject of much criticism on the part of later
Naturalists.
The great cleavage of opinion as regards the relative
importance in evolution of the two principles, that of
natural selection, and the Lamarckian factor of the inheritance of acquired characters, has been represented
in the time subsequent to Darwin by the two schools
of Neo-Darwinians, for whom natural selection is the allimportant factor in evolution, and the Neo-Lamarckians,
who regard natural selection as of little importance, and
lay the chief stress upon the inheritance of acquired
characters. This divergence of opinion began in Darwin's
own time; A. R. Wallace laying more exclusive stress
than did Darwin himself upon Natural Selection, and
Herbert Spencer being an advocate of the Lamarckian
view. A large part of the work of Biologists since the
time of Darwin has been devoted to elucidation of the
mode in which variations may be conceived to arise in the
organism, and of the conditions under which modifications of the individual organism become hereditary. The
question has also arisen, whether the small modifications,
the existence of which Darwin presupposed as the basis
of his theory, are as important in relation to evolution
as larger variations which arise as discontinuous variations, or mutations. The notion of the struggle for
existence has been extended to embrace the conception
of competition of tissues, cells, and smaller units, within
the organism.
The notion that, in addition to the personal selection
which depends upon the struggle between an individual
and the other members of his race, and also against foes
of different race from himself, and against the environment, there is also a struggle of parts and cells within
the organism was introduced by Roux in I881. This
histonal struggle, or competition between the tissues
for sustenance, may give rise to local modifications which
produce acquired adaptations to the environment during




446


THE EVOLUTION OF SPECIES


the individual lifetime. The question whether such
modifications are hereditary, and thus whether they have
evolutionary value or not, falls under the general question
of the heritability of acquired characters.
The chief inspiration of the Neo-Darwinian school
has been derived from the teaching of Weismann, who
rejected almost completely the Lamarckian idea of the
inheritance of acquired character as a factor in evolution,
on the grounds that there has been discovered no
sufficient proof that such inheritance in fact takes place,
and that, in accordance with his theory of the germplasm as the inherited substance, such inheritance is
on theoretical grounds impossible. I have in the last
lecture given a necessarily summary account of the
detailed scheme which was gradually developed by
Weismann for the purpose of representing, or as he
would have said, of explaining, the facts of inheritance.
The chief contribution which Weismann has made,
based on his theory of germ-plasm, to evolutionary
theory, is the introduction of the idea of germinal
selection. This consists in an extension of the notion of
natural selection to the sphere of the germ-plasm, with
its representative units of ids, determinants, and biophors.
This extension is employed to supplement and complete
Darwin's theory, by providing a theory of the mode in
which the hereditary variations may be conceived to
arise. According to Weismann there is not only the
struggle between organs, tissues, and cells, going on in
the organism, as pointed out by Roux, but also a
struggle between the determinants. When the determinants multiply by cleavage, the new determinants
differ in size and in power of assimilation of nutrition.
Those determinants which have greater power of assimilation become stronger than the others, and increase
their superiority over the others in this respect. This
struggle is resumed in every successive generation, as
each generation receives its germ-plasm with its deter



THE EVOLUTION OF SPECIES


447


minants from the preceding one. The result is that
gradually the parts of the organism which are represented
by the more vigorous determinants which are successful
in the struggle become more strongly marked. This is
held to explain the accumulation of small modifications
in certain specific directions. The explanation of the fact
that useful variations are always to be found, and are
increased, is that the corresponding determinants are
better nourished and their offspring are stronger than
those of other determinants. Whether the original appearance of useful modifications, apart from their increase
when already present, can be thus accounted for has been
doubted by critics of the theory. Weismann also regards
his theory as explaining the correlation of different parts
of the organism in the product of adaptations; and also
the degeneracy of useless organs he regards as due to
the fact that the determinants corresponding to them
obtain less nutriment than the others, and have consequently weaker offspring; this process of gradual
weakening of the particular kind of determinants leads
gradually in the course of many generations to the
obliteration of the organ corresponding to them, or to
its only remaining as a vestige. The fact of the reappearance of ancestral characters is explained by Weismann
as due to a struggle between the various ids, determinants,
and biophors, in the fertilized egg-cell. Since these are
derived from the germ-plasm of both parents, and that
germ-plasm in each case contains ancestral germ-plasm,
there is occasion for a contest in which certain selected
representative particles become effective in performing
the function of forming the embryo, whilst others remain
only in germ-plasm which is transmitted unchanged to
the offspring. Weismann's theory of germinal selection
is intended to explain why variations in a fixed direction
take place, as well as why complex organs with many
correlated parts appear, whilst these facts remain unexplained by Darwin's theory of selection in its original




448


THE EVOLUTION OF SPECIES


form. Although Weismann was strongly opposed to the
Lamarckian view, he has admitted that in some cases
nutritive and other environmental conditions may produce heritable modifications by direct action upon the
germ-plasm within the body, but he regards direct
action upon the somatic plasm as quite insufficient to
produce such heritable variations.
Weismann's theory is a very complicated one which
he gradually evolved in various stages; it has elements
in common with other theories, but these elements he
adapted for his particular purposes. Of all the theories
which work with representative bodies in the germplasm it is the most complete, and it aims, whether successfully or not is a matter of controversy, at resuming
a larger complex of facts than any other such theory.
Like every theory dealing with evolution, or with vital
phenomena in general, it ends up at a point in which
the real difficulties of explanation of the facts of the
existence and activities of living organisms are pushed
back and concentrated on certain living elementary beings
such as biophors, whose existence and activities as living
beings are simply postulated without further analysis.
This characteristic of such theories is precisely parallel
to the analogous case of physical and chemical theories
in which the phenomena of inorganic matter are made
to depend upon properties assigned to postulated conceptions such as electrons, corpuscles, or atoms, in accordance with certain postulations. In both cases, when
a theory has reached at least a provisionally definite
form, the theory does not analyse further the ultimate
postulations upon which it rests. Thus, in the rigorous
sense of the term explanation, such a scientific theory
is no nearer an explanation of the phenomena than at
its starting point; and yet when the necessary character
of this restriction, as inherent in Natural Science, has
been grasped, it will be seen that this does not detract
from the utility of the theory as a descriptive scheme.




THE EVOLUTION OF SPECIES


449


During the time which has elapsed since Darwin's investigations led to the complete acceptance by the
Scientific world of the fact of evolution of species, the
scrutiny of the factors of evolution has led to an enormous
amount of discussion and of detailed work of observation
with a view to its elucidation. A large number of theories
have been propounded, of which it is impossible for me
to give any account. I must confine myself to an indication of the main features of a theory which recognizes
the importance of a class of facts which were regarded
by Darwin as of little or no importance in relation to
evolution. Besides the originally small variations accumulating by slow changes, and consequently known as
continuous variations, the role of which in Darwin's
theory was a fundamental one, there exists another mode
of variation which consists of a sudden or discontinuous
variation. The characters of a species, or of a variety,
sometimes undergo a sudden modification not due to
the accumulation of continuous variations. These discontinuous variations, some cases of which were mentioned by Darwin himself, have since been the subject
of attention by many naturalists who considered them
to be of importance in relation to evolution. A developed
theory of evolution regarded as dependent upon these
discontinuous variations, or mutations, has been propounded by the Dutch Botanist De Vries, and rests
upon the basis of a large number of experiments on
transplanted wild plants and on various cultivated plants
in a botanical garden at Amsterdam. For the slowly
accumulated variations in the Darwinian theory, which
may require an enormous amount of time to produce
such changes as the evolution of species contemplates,
De Vries proposes to substitute periodical, but sudden
and quite noticeable steps. He observes that "this
assumption only requires a limited number of mutative
periods which might well occur within the time allowed
by physicists and geologists for the existence of animal


HGL


29




450      THE EVOLUTION OF SPECIES
and vegetable life on the earth." This theory of periodical
mutations is regarded as consonant not only with the
fact that species or varieties change, but as reconciling
that fact with the constancy of species for long periods
of time, subject to individual fluctuations:
Mutability (he writes) is not a permanent feature but a
periodic phenomenon, producing at times new qualities and at
other times leaving the plants unchanged during long successions
of generations. All lines of the genealogic tree show alternating
mutating and constant species. Some lines may be mutating
at the present moment; others may momentarily be constant....In a complete and systematic enumeration of the real units
of nature, the elementary species and varieties are thus observed
to be discontinuous and separated by definite gaps. There is no
reason to suppose that the world is reaching the end of its
development, and so we are to infer that the production of new
species and varieties is still going on. In reality, new forms
are observed to originate from time to time, both wild and in
cultivation, and such facts do not leave any doubt as to their
origin from other allied types, and according to natural and
general laws.
De Vries, in formulating his theory of the laws of mutation, maintains that new elementary species appear suddenly without intermediate steps, springing laterally
from the main stem; and several such new species may
arise from the parental form at once, in accordance with
his experiments. The new elementary species attain their
full constancy at once, and transmit their characters to
their progeny independently of any external conditions.
Some of the new strains are evidently elementary species,
while others are to be considered as retrograde varieties.
The term species is used by De Vries in a sense not
identical with the Linnaean species; his elementary
species is more nearly what is denoted by variety, in
the ordinary classification. De Vries attributes an important part in evolution to natural selection, but he
regards it as operative between species, and not between
individuals of the same species. The external environ



THE EVOLUTION OF SPECIES              451
ment he regards as influencing this interspecific selection,
and as also probably determining the appearance of a
period of mutation, but otherwise of no influence. The
origin of the mutations he regards as germinal, congenital, and depending on changes within the sexual cells.
Many cogent criticisms have been made of the theory
of De Vries. The importance of the mutations he
describes in evolution in general has been the subject
of much controversy, but his experiments show clearly
that the mutations play some part as a factor in evolution; how great that part is, or how general, remains
for determination in the future.


29-2




XIX


NATURAL SCIENCE AND
GENERAL THOUGHT
I HAVE now completed my survey of the methods
and implications of Natural Science, as exhibited in
its various departments. Necessarily incomplete as this
survey has been, I venture to hope that it may serve as
a basis for the formation of an estimate of any bearings
which Natural Science, as a special domain of Thought,
may have upon our more general views of the world,
and especially upon the spiritual aspects of experience.
In the earlier lectures of the course I stated and discussed
in some detail the main conclusions, relating to the true
characteristics and the limitations of scope of Natural
Science, which I conceive to be borne out and illustrated
in the course of the survey of special departments of
Science which is made in the later lectures. It seems
clear that the position of Natural Science in relation to
the more general conceptions of existence which fall
within the province of Philosophy and Religion will be
largely affected by the nature of our conceptions of the
basic characteristics of Natural Science, and of its scope.
The nature of any ontological assumptions which may
be regarded as necessary for the existence of Natural
Science as a systematic scheme of thought, or which
may be held to be implicit in that scheme, will be of
the highest importance in this regard. The view which
I have maintained and illustrated, that the life of Natural
Science consists essentially, when it is rightly understood,
of an organized attempt, or rather a series of efforts continued through the centuries, to provide conceptual
representations of our physical percepts, will, if it be




GENERAL THOUGHT


453


accepted, have a marked effect upon the external relations of this department of thought. This view tends to
limit and circumscribe the influence which Natural
Science will have upon the wider views of the world
with which Philosophy and Religion concern themselves.
If it be admitted that Natural Science, when reduced
to its essential elements, is independent of any opinions
which may be held as regards a reality behind phenomena, and if the notions of final causes and of efficiency
be regarded as extraneous to it, it would seem to follow
that the existence and the special results of Natural
Science cannot be employed in any very direct manner
for the purpose of throwing light upon the nature of an
assumed reality, or of exercising any decisive influence
in the contest between rival views as to the nature of
reality.
By adopting phenomenalism as a methodological assumption, independently of any dogmatic assumption
that it represents an ultimate philosophical view of
existence on its physical side, Natural Science cuts itself
off from the possibility of providing criteria which shall
be logically effective in relation to metaphysical theories
of the nature of the "real," or of "existence." This
question of the character of any assumptions about
reality which Natural Science may be thought to need
is of practical importance because the fact that Natural
Science has achieved successes of a kind and degree
which cannot be ignored will lead to a presumption, of
irresistible weight for many minds, in favour of any
existential proposition which Natural Science can show
to be really necessary as a foundation for the structure
which it has reared. Such existential propositions, if
accepted on the authority of Natural Science, and
apparently evidenced by its successes, would have a
far-reaching and vital effect upon all conceptions,
philosophical, or religious, of the nature of reality. As
an illustration, we may take the influence, outside




454


NATURALI SCIENCE


Science itself, of physical realism, frequently supposed
to be a necessary assumption for Natural Science.
In one of my earlier lectures I have emphasized the
advantages which accrue to Natural Science by taking
up a position of complete independence in relation to
Philosophical theories as to the origin of the perceptual
complex with which Natural Science has to do. These
advantages entail the restriction that Natural Science
cannot exercise so important an influence in relation to
our general attitude towards the world as has sometimes
been assigned to it, especially by those who have attempted so to extend its scope that it was developed
into a complete and dominant World Philosophy. If
the view of the character and scope of Natural Science
which I have maintained be correct, Natural Science is
circumscribed in its aim and restricted by its method.
No doubt the majority of men of Science in all ages
have believed that their aim was to penetrate to what
they regarded as the reality behind phenomena, and that
successful scientific investigations might be expected to
give them some detailed information as to the inner
relations within that reality. Certainly many, and probably most, men of Science in recent times have been
dominated by the conception known as physical realism,
in accordance with which the material world not only
comes under the category of the real, but is articulated
in a manner corresponding closely to the distinctions
and constructions which the mind makes, by abstraction and idealization, in the process of analysing and
symbolizing its physical percepts. However, the reflections of many of those who have studied the general
characteristics of scientific procedure and theories, some
of them active investigators in some special department
of Science, have led them to the conviction that the
adoption of physical realism is an otiose opinion, which,
whatever its merits as a philosophical theory may be, is
essentially unnecessary as a presupposition of scientific




AND GENERAL THOUGHT


455


method. I think there is evidence that this emancipatory
movement is gaining ground, and is likely in the future
to progress much further in the minds of those who find
leisure to devote some thought to the underlying assumptions and implications relating to the methodology of
Natural Science.
We possess other kinds of knowledge besides that with
which Natural Science is prima facie alone concerned.
There exists a body of conceptual knowledge of the
mode of working of the human mind, represented by
the results of Psychology and Logic. One of the functions of Psychology is to investigate the process of
perception, and to trace out the relations between
perception and conception, the bare recognition of which
elements, in separation due to abstraction, is sufficient
for the purposes of Natural Science, which does not
utilize the light thrown by Psychology upon their relations within the fundamental unity of the mind. Psychological investigations appear to fall into two distinct
departments. On the one hand, the subject is a purely
psychical science in which the various functions and
processes in the mind are analysed, and the relations
between them investigated. In the other branch, on the
border line between Mental Science and Physiology, the
functions of the mind are studied by means of their
physical manifestations, subject to assumptions as to
the correlation of psychical and physical events and
processes. On this latter side, Psychology must be regarded as a mixed Science, in the sense in which for the
sake of distinction I have already employed the term.
Pure logic, the study of the formal side of thought, may
be regarded as closely connected with the more abstract
side of Psychology. By analogy, Logic may be described
as the Grammar of Thought.
Such a Science as Sociology, in its various departments, including Anthropology, Politics, and Economics,
may be regarded as a mixed Science, depending, as these




456


NATURAL SCIENCE


departments do, upon investigations and classifications
which involve both physical and psychical categories.
Psychology and the Sciences of the sociological group
have one point in common with Natural Science. They
are all concerned not with the individual as such, but
with classes of individuals. To all of them, the individual
is of concern only so far as it is a member of some more
or less extensive class; and only those features of the
individual which it has in common with all the members
of such a class are relevant to these branches of knowledge. This transcendence of the purely individual is
characteristic of all knowledge which, in the widest sense
of the term, can be designated as scientific knowledge.
Of an individual object, taking the term object in a wide
sense, we may have knowledge which, when sufficiently
extensive, always differentiates that object from all others,
and may be considered collectively as knowledge of the
history of that individual object. The history of a particular object, including its present characteristics, is
unique, so far as elements in that history are not referred
to a system of classification in which the individual is for
a specific purpose merged in a class. That history is
never identical in all respects with the history of any
other object. So far as the individual object is unique,
our knowledge of it fails to be subsumed under scientific
knowledge, in however general a sense we understand
that term; it remains as historical, not as scientific,
knowledge. That element in the genesis of an object
or complex, in its relations during its existence with its
environment, and in its past and present states, which
eludes all scientific classification and subsumption under
scientific laws and theories, may vary enormously in
importance and amount in different cases. No two stones
are ever found, upon close examination, to be absolutely
indistinguishable from one another in respect of shape,
size, and structure. The stories of the past buffetings
of the two stones, if it were possible or worth while to




AND GENERAL THOUGHT                 457
obtain knowledge of them, would be full of details unlike in the two cases; this divergence being held to
account for their present dissimilarities. The scientific
interest in the stones is usually confined to an interest
in the heap to which they belong, the individuals being
regarded only as samples of that heap; and their irresoluble individual differences are for scientific purposes
neglected, unless some specially distinctive features are
observed in them which may give rise to some additional
scientific inquiry.
When we turn to the case of living organisms, and
especially to the case of a particular man, as an historical
being, the same considerations hold good, with greatly
increased force and import. At each juncture in the life
of a man, physical and psychical Science can be applied
to give some partial accounts of the physiological processes in his body and nervous system, of his reactions
to external stimuli, and of the psychical processes in his
mind which are related to the determinations of his will;
but these accounts are never complete in all details, and
they cannot be welded together into a single coherent
whole. Not only his past history, but also the individual
peculiarities which distinguish him from every other
being of his kind, have some greater or less significance
at every juncture of his life. Neither his history nor
his character are identical with those of any other being.
I have emphasized this fact of the ubiquity of the element
of individuality in the perceptual world, because it is
one of which the bias of the man of Science, who for
his special purpose does not, and cannot, concern himself with the purely individual, except so far as it is
assumed to be an instance of the general, tends to
minimize the importance in relation to a general view
of the world. For the struggles of the man of Science
are in the direction of a constant attempt to diminish
the importance of the purely individual element, by
showing that it is but an instance of the general, and it




458


NATURAL SCIENCE


is in doing this that his successes consist. There remains
however in the world the irresoluble element of individuality, of the complete removal of which there is no
prospect. The principle of order in the world requires
to be supplemented, if not limited, by the principle of
individuality. The fact that the history and characteristics of a human being are unique does not however
make it impossible for other persons to have a knowledge
of that individual, sufficient for the purpose of predicting,
with some greater or less degree of assurance and precision, what the actions of that individual will be in
given circumstances. This implies the assumption that
the determinations of his will are not so wholly irregular
and incalculable but that his conduct in a given situation
is predictable in some considerable degree. In other
words, it is possible to have a knowledge of his character
based upon knowledge of his past history, and subject
to the assumption that his character has a functional
relation with the elements of his past behaviour. The
difficult questions how far what is called self-determination goes, and whether indeed it is an ultimately valid
conception, including the question of the validity of
what is known as psychological determinism, lie quite
outside the scope of these lectures, and accordingly
cannot be here discussed. The kind of knowledge that
we may have of a particular man is essentially knowledge
of an individual, not of a class, although a very considerable part of it has no doubt reference to our knowledge
of classes to which the individual belongs, and so far
is equivalent in kind to scientific knowledge.
Science has however made much use in quite recent
times of one method of eliminating the effects of what
I have called the principle of individuality. That method
is the statistical, by means of which valuable scientific
knowledge may be obtained as to the behaviour of
groups of individuals; it consists of a process of elimination of the effects of the purely individual characteristics




AND GENERAL THOUGHT


459


of the individuals of which the groups are composed.
By this method, upon the basis of measured facts of
observation relating to a large group of individuals,
tabulated in numerical form, it is possible to correlate
the behaviour of the members of the group with certain
kinds of motives, not necessarily conscious, relating to
that behaviour, in such wise that, by statistical analysis,
a knowledge can be obtained of the average effect of the
motive upon the members of the group. I have used
the term "motive" in a very general sense, to denote
any circumstance or set of circumstances which can be
correlated with a particular kind of behaviour. Sometimes the statistical method has been applied to detect
the existence of such correlation, without previous
assumption that a particular set of circumstances was
a relevant motive. This method is of special value in
the case of a motive of which the intensity is variable;
it may then be shown by statistical analysis that the
average behaviour, of the particular kind, of the members
of a group has a functional relationship with the intensity
of the motive. By this method a considerable amount
of knowledge may be obtained of how a large group of
individuals will behave in certain circumstances, and
especially as to the variability of behaviour when those
circumstances change in a known manner. If, for
example, it has been shown that there is, in a given
community, and for a considerable period, a correlation between the number of marriages in a year and
the average price of corn in that year, a definite piece
of information has been obtained, depending for its
establishment upon a process of elimination of most of
the many individual characteristics of members of the
community which, in any single instance, will be factors
in deciding for or against marriage. The effect of all
other motives has been eliminated in the statement of
the fact of correlation between the number of marriages
and the one particular kind of circumstance, the price




460


NATURAL SCIENCE


of corn. A large amount of knowledge of partial correlations, obtained by this method, may afford scientific information about the average behaviour of the members
of a community; and this knowledge may have value as
a means of forecasting the future. But its value, as
enabling us to predict the behaviour of a particular
individual, in assigned circumstances, is evanescent, because no particular person is the average individual;
neither has he in all respects a close resemblance to the
average. For predictions of any value as to the behaviour
of a particular person recourse must be had to the purely
individual knowledge of those who have sufficient acquaintance with his past history, and his present character
and circumstances, to enable them to form an estimate;
and such individual knowledge is not in the main
scientific knowledge.
Both scientific knowledge and the individual knowledge of which I have spoken have in common the fact
that they are of the discursive kind, obtained by an
analysis and subsequent synthesis of particular elements.
The two are distinguished from one another by the large
divergence in the degrees of systematization which they
involve; the former is obtained by systematic schematization; the latter by a process of unsystematic synthesis,
although it may contain elements in which systematic
schematization is not entirely absent. Both these kinds
of knowledge are abstract, in very different degrees. But
besides these kinds of knowledge, there exists a kind
of apprehension which is more immediate and direct,
although it is often inextricably combined with knowledge of the other kinds. This is knowledge as given by
direct intuition, in which the object in the subjectobject relation is apparently apprehended all at once,
as a whole, and not by a conscious synthesis of its parts
and their relations. This intuitive knowledge, of which
the highest example is to be found in the apprehension,
in some moments, of the artist, or of the mystic, is




AND GENERAL THOUGHT


46I


transitory and fleeting; for a process of analysis into
parts and relations, and resolution into discursive knowledge, is incipient in it. In its purity it is incommunicable; as soon as an attempt is made to describe
it in language, the stage of abstraction has already been
reached, and the description fails to represent with
absolute completeness the unique individuality of the
whole, as given by the original intuitive apprehension.
In the individual who has been the subject in the intuition, the impression of the whole, as grasped at once
intuitively, may remain more or less vividly in the
memory, after the inevitable process of dissection has
commenced, but only that process gives him the power
to communicate what he has experienced to others, and
in the act of communication the object, as a unique
indivisible whole, becomes merged in systems of classification in which those aspects of it which constitute its
uniqueness are largely lost. An exceptional power of
obtaining an intuitional grasp of a complex as a whole
is an essential element in the mental outfit of a man of
Science of the highest order. Such intuitional and
imaginative apprehension precedes and conditions any
striking success in the process of discovery of the inner
relations within the complex.
That element of human experience and life which
may be summed up in the word cognition includes
common knowledge, which becomes in its developed
form scientific knowledge, and also what I have spoken
of as individual knowledge and immediate or intuitional
knowledge. But when we have taken all these, not completely separable, kinds of knowledge into account, we
have still to remember the fact that cognition, however
generally the term is understood, is but one among other
elements which make up the whole of human experience
and life. It would not be necessary for my purpose to
give a rigorous analysis of the elements of human
experience such as might satisfy the psychologist or




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philosopher, even if I were competent to attempt such
a task. It is sufficient to refer to the elements of feeling
and desire which include the fundamental springs of
will and activity. These take explicit form in the apprehension of values, spiritual, moral, intellectual, aesthetic,
and material. In actual experience, these elements, with
the cognitive element, enter as components separable
from one another only by abstraction; the fundamental
unity of the stream of experience is such that it is not
merely a sum of such elements. The qualitative difference
between various kinds of experience may be to some
extent represented by differences as regards prominence
or intensity in these abstractly separable elements.
The main characteristics of the great departments of
thought and activity, Religion, Philosophy, Science, and
Art, rest upon distinctions in the emphasis which they
place, in their aims and procedure, upon the various
elements of mental experience to which I have alluded.
It may be said that the interests of Philosophy and of
Pure Science are cognitive, or as is often said, intellectual;
that the object of Philosophy is to obtain systematic
knowledge and understanding of experience as a whole,
and that the object of Science is to represent certain
kinds of experience conceptually. It may also be said
that the interests of Religion and of Art are in the main
connected with values, spiritual and moral values in the
one case, and aesthetic values in the other. But such
statements, however correct they may be, if they are
understood as applying to ultimate aims and results,
cannot be applied without qualification to the processes
and activities by means of which the aims are realized
and the results obtained. That the aims of Philosophy
and of Science are to attain to truth, independently of
the specific character of the valuations of that truth
when obtained, is doubtless correct; their direct concern is with cognition, and not with valuation. But for
a Philosopher or a man of Science, truth is itself a value




AND GENERAL THOUGHT


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of the highest kind, even if the truth contain unpleasant
features; a recognition of its immediate ideal value, or
in some cases of its mediate value as a means for the
attainment of practical ends, is an essential spring of
action in the mind of the genuine Philosopher or man
of Science. The sustained emotion which we call the
love of truth as a value is essential to the pursuit of
Philosophical and of Scientific knowledge. At every
stage in the age-long struggle to reach philosophical
or scientific truth the combatants have been animated
and sustained by a consciousness of the value of their
goal. It is however true that the feeling for values, or
rather for specific values, is one which has to be kept in
severe restraint by the investigator, in subordination to
the cognitive side of his mind, for otherwise it may
distort his vision in a manner which may be very
detrimental to the attainment of his aims.
That, for the domains of religion and morality, the
apprehension of values is the fundamental factor can
hardly be denied. In relation to religion this has been
formulated by H6ffding in the thesis that the fundamental axiom of religion is the conservation of values.
But it must nevertheless be recognized that, to found
religion exclusively on the basis of feelings involving
apprehension of specific values and their conservation,
and without any elements of cognition, that is of knowledge, actual or speculative, is an impossibility. Conservation of values implies the existence of forms and
modes in which they are conserved; and this fact
necessarily brings values and their conservation into
connection with conceptions of existence and reality.
All valuations arise originally in close connection with
ordinary experience of the physical and psychical world;
and even in their most clarified form, as they appear in
the religious and moral consciousness, they are dependent
for their imagery, their concrete expression, and their
concepts, upon elements derived from that experience.




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Thus the domains of value and of existence are ultimately inseparable. All historical religions have been
in close connection with the cognitive side of human
experience, and in some periods they have exercised
an almost complete domination over that side. In the
middle ages Religion represented the unity in which all
conceptions not only of the spiritual world, but also of
the physical world, were embraced; in which Art and
in fact all forms of spiritual activity were contained, and
by which all their activities were regulated. All values
and all conceptions of existence and reality were bound
together in a unified system which, only at the time of
the Renaissance, began to dissolve, when Science, Art,
and Philosophy set up claims to independence and
development on their own lines, unfettered by restrictions imposed upon them by the supposed necessities
of the unified system in which they had previously been
combined. The advent of this separation of different
forms of spiritual life and activity led to the state of
tension, manifested in the contest, lasting into our own
time, between the conceptions of Natural Science and
those older conceptions of the physical world which
were for a long time considered to be essentially connected with Religion. The complete cessation of this
contest is dependent upon a full recognition of the
validity of the claims both of Religion and of Science
to independent life and activity, each in its own sphere;
and this requires as its condition a more precise formulation of the boundaries of their respective spheres than
had formerly been made.
A somewhat similar remark may be made as regards
the relations between Religion and Philosophy, but, as I
venture to think, the difficulties in this case are essentially
greater than in the case of the relations between Religion
and Natural Science. Philosophy and Religion are both
of them, in different senses, concerned with the whole
of human experience. The interests of Philosophy are




AND GENERAL THOUGHT


465


mainly intellectual, being concerned with the cognitive
side of experience; those of Religion are more concerned
with the practical side of experience, with the spiritual
needs of man, with his hopes and fears, his moral values,
and his emotional attitude towards existence. Philosophy,
in its eclectic view of human experience, does not, and cannot, leave out of account the nature and specific character
of values, and especially of that species of values which
we call moral; and here it has a specially close plane of
contact with Religion. The fact, to which I have already
alluded, that Religion, which is primarily concerned with
values, cannot wholly divorce itself from conceptions of
existents in which those values are realized, brings it also
in this regard into close contact with Philosophical
Thought. In point of fact, Religion has at all times
found itself impelled, at all events in the minds of more
cultivated persons, to ally itself with some form of
Philosophy, in which it might find the existential elements which might warrant an ascription of objectivity
and of permanence to the carriers of its primary elements
of value. The difficulty of combining religious conceptions with a coherent Philosophical scheme has been
increased by the historical fact, to which Professor
Sorley has recently drawn attention in an emphatic
manner, that, in the development of Philosophical
systems, the cognitive side of experience has formed
almost exclusively the material out of which the systems
have been built up; these systems have then been under
the necessity of adapting themselves ex post facto to
include that element of experience which we call the
apprehension of values. Although it has been recognized,
at least in some quarters, that Religion does not supply
a World-Philosophy, and that it should not be expected
to fulfil that function, and further that it is not in
essence bound up with any philosophical theory of the
nature of existence, it remains nevertheless true that,
in so far as Religion seeks to relate its concepts of spiritual


HGL


30




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value with existential forms in which those values are
realized, it is impossible to treat Religion as completely
independent of philosophical implications. Nearly all
actual forms of Religion have been theistic, whether
monotheistic or polytheistic, and even Buddhism which,
in its esoteric form, is regarded as an exception, is not
solely a Religion, but is also a most pronounced form
of World-Philosophy. All forms of theism involve ideas,
differing widely in their nature, mode of formulation,
and degree of precision, which are concerned with the
relations between God and the world. Thus the assertion
of the existence of God does not simply and solely express
a religious belief, or an attitude of faith, but takes also
the form of a philosophical proposition, or hypothesis,
of an ontological character. This brings with it a series of
philosophical questions, some of which at least are of
so urgent a character, not merely from a purely cognitive
point of view, but in relation to the needs of the religious
consciousness, that some answers to them, more or less
precise, are an imperative necessity. This is so, not
merely or mainly on account of the inherent philosophical
importance of these questions, but because the whole
character of Religion, and its efficiency in providing for
the primary spiritual needs which Religion is to satisfy,
are very fundamentally affected by the nature of the
answers given to those questions.
Theism has been the fundamental basis of the most
multifarious forms of Religion; and these have exhibited,
in different ages and in different countries, and even
among different groups of persons in one and the same
age and country, the most diverse ideas as to the conception of God, and of the place which theism should
fill in a general view of existence and life. In fact the
particular kind of theism in a religious system is of most
fundamental importance in relation to the actual functioning of the creed, as an expression of the spiritual
character of its adherents, and as influencing all their




AND GENERAL THOUGHT


467


activities. There exist, and have always existed, many
persons who, whilst dominated in their innermost being
by theistic belief, do not feel any need for the kind of
support which may be afforded to their belief by philosophical conceptions. Many others, whilst employing
such conceptions as an adjunct to their belief, do not
feel the need of evidential support of their faith from
the side of reasoned Philosophy. To such persons, their
theistic belief appears to them in the form of direct
intuitive knowledge, obtained in the course of actual
experience, and having the character of a certainty which
can dispense with any proofs of the sort which reasoned
Philosophy can be expected to furnish. Such persons
may employ some kind of philosophical or rational
scheme, more or less consciously, as a kind of framework into which their faith, resting as it does on
intuitive knowledge, may be fitted. This framework
is usually supplied by the traditional conceptions of
the society in which they live, or of the particular
group in which they have been educated, or by which
they have been most effectively influenced. The most
highly developed form of this attitude of mind towards
theistic belief is to be found amongst the mystics of all
periods, many of whom may be regarded as specialists
in relation to religious experience of the directly intuitional kind. The accounts of this kind of experience
given in such a work as William James' Varieties of
Religious Experience are of extreme interest. It is there
shown that mystical phenomena are to be found amongst
the adherents of the most diverse creeds, and yet present
in all such cases an essential similarity of general
character. The accounts given of experiences of this
order, by those who have been their subjects, take forms
which are coloured in a high degree by their preconceived
views; and this appears to indicate the presence of a large
element of subjectivity in the interpretation of the actual
experience. It is accordingly difficult to assess highly
30-2




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the value of such experience as evidence of the truth of
any special cognitive views which may be held by persons
who have such experience, and who interpret it in terms
of their personal and traditional beliefs. It seems clear
that all attempts that may be made to base theistic, or
more general, religious belief upon intuitional knowledge obtained directly in actual experience, either of
the mystical order or of a more ordinary kind, will
ultimately prove insufficient as a basis for such belief
amongst a very large number, and probably the majority,
of men. The history of religion shows that this is the
case. Belief based upon direct intuition will, for those
who have it, remain unaffected by discursive thought
in relation to Philosophy and Science; and it is not for
the sake of such persons that it is necessary to treat of
the relations which theistic belief may have with philosophical or scientific views. For the more thoughtful
members of the community, these relations have in
our time an inestimable importance; and the influence
of the views formed of their specific character has
an ultimate effect upon the attitude towards theistic
belief of multitudes of men who do not consciously
concern themselves with such relations on the more
theoretical side. Accordingly, the nature and extent of
any influence which Natural Science exerts, or ought to
exert, upon theistic belief, both in its general and its
more specific characters, presents a problem, the importance of which can hardly be overestimated, in view
of the effect which solutions of it may have, directly
upon the cognitive side, and indirectly upon other sides,
of the religious consciousness.
Speaking broadly, the main concern of Religion is
with values, and with existence as embodying values,
whilst Natural Science, in its results, has no concern
with values. On the other hand, Philosophy is concerned
with cognition related both to values and to existence.
It would then appear that the relation of Natural Science




AND GENERAL THOUGHT


469


with Theism, so far as such relation exists, is in a sense
indirect, as it is through the mediation of Philosophy
that it becomes effective. The influence which Natural
Science may have upon Theism may be taken to depend,
first upon any assumptions of a metaphysical kind which
may be held to be necessary for Natural Science to make
for its own purposes; and secondly upon the amount
and nature of the support which the existence and
success of Natural Science may afford to particular
theories of a philosophical or metaphysical character.
As regards the influence of the first kind, the acceptance
of that view of the essential character of Natural Science
which I have advocated, and which view I have attempted
to establish as correct, leads I think to the inference that,
subject to one limiting condition, of which I shall speak
presently, Natural Science, taken by itself, does not
directly affect the theistic position, either positively by
providing support, or negatively by giving rise to objections. In fact, if no philosophical assumptions are made
which lie outside the necessities of Natural Science, the
position of Natural Science in relation to theism, as
in relation to ontological theses generally, is one of
neutrality or independence. It is important to emphasize
the fact that this position of independence only appertains to Natural Science when all conceptions not
strictly necessary for its existence as a schematization of
physical percepts are excluded. When Natural Science
is taken in combination with metaphysical views which
are, in accordance with the opinion here adopted,
extraneous to it, its position in relation to theistic, or
other ultimate, ideas relating to the nature of reality
becomes very different, and its influence upon such
conceptions may become of great importance.
It may perhaps seem to be the case that the position
of independence here assigned to Natural Science is too
absolute; that such a position of apparent isolation in
the more general domain of thought and existence is




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untenable. It may be objected that the fact that it has
been found possible to develop such a system as Natural
Science, which undoubtedly possesses a certain kind
and amount of efficiency in relation to life and activity,
must lead to some inferences which show their effect
in introducing limitations upon the characteristics of
a general philosophy of the world, and in particular
upon any theistic, or other, mode in which ultimate
reality is conceived. Such an objection would be valid,
were it not for one important limitation which the
existence of Natural Science, as we know and possess it,
places upon any theistic, or other, view of the world.
Stated shortly, that limitation consists in the fact that
any acceptable view of the world, whether theistic or
other, must be such as not to be incompatible with the
existence of Natural Science. Any such general Philosophy must provide within itself a place which Natural
Science may occupy as an autonomous system. The
principle of order in physical phenomena, with the
limiting principle of individuality in these phenomena,
provide principles with which any Philosophy and any
theistic view must not be incompatible, if direct contest
between Natural Science and Philosophical or Theistic
thought is not to arise. Subject to completely adequate
satisfaction of this condition, Natural Science offers no
obstacle to the free development of Theistic, or other,
Philosophies on their own lines, in the sense that no
other purely logical consequences follow from the acceptance of Natural Science, which are effective in the
wider domains of Thought. I have already observed
that any further influence which Natural Science may
have upon general views of the world depends upon the
nature of any ontological hypotheses or postulations
which may be made in attempts to explore the nature
of reality, but which go beyond any assumptions which
Natural Science itself needs.
Before discussing the nature of the bearings which




AND GENERAL THOUGHT


47I


Natural Science may have upon theories of reality, and
in particular upon theistic theories, when it is supplemented by ontological postulations which go beyond its
own special requirements, I propose to make a few
remarks upon what I take to be an essential characteristic
of all Philosophical systems. Metaphysical Philosophy
is not constructed upon a basis deduced by the canons
of pure Logic from any set of axioms, assumptions, or
presuppositions, which are accepted by all human
minds as possessing self-evidence, or even definiteness
of meaning. To the constructor of, and to the adherents
of, a particular kind of Philosophy, the assumptions and
presuppositions of the system, when indeed they are
all explicitly recognized, usually appear to possess
apodictic certainty, to be woven in the very web of the
mind. To the adherents of a rival system, and to critics
in general, these same assumptions or presuppositions
may appear to lack this quality of self-evidence, to be
only probable in some degree, or in no degree, and often
to be either meaningless, or to have only a hazy and illdefined meaning. In this region of thought, what seems
obvious and certain to some minds, appears to be neither
obvious nor true, at least without much qualification,
to other minds, at the same epoch, or in a different age.
If Philosophers were not only logic-machines working
in accordance with a single logical canon, but also in
possession of a single universally accepted set of premisses or postulations to be employed in the logical
processes, the state of philosophical thought would be
very different from what it actually is, and always has
been. We might then at least hope to attain to a Philosophy which would receive general assent from all those
persons who made a sufficient amount of effort, and
were possessed of sufficient mental grasp, to enable them
to understand it; but this is very far from being the case.
The presuppositions which commend themselves to
different minds diverge in the widest manner from one




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another. These presuppositions which serve as premisses,
necessary before deductive logic can function, depend
upon the widely varying characteristics of particular
minds and groups of minds. The causes which make
a particular premiss appear to a particular mind, or to
a particular class of minds, to have either irresistible
cogency, or at least a high degree of probability, are not
in their nature purely intellectual. They depend upon
the education, traditions, and idiosyncrasies of such
individuals or classes; upon irreducible peculiarities of
particular minds. The reaction of the individual, as an
inseparable whole, to his experience, is involved in their
selection. In this whole, understanding, feelings, desires,
and tradition, are only in abstraction separable from
one another. It is the undifferentiated individuality that
is really effective in determining the choice of the axioms
and premisses which commend themselves to the individual as the fundamental elements in his Philosophy.
The reasoning faculty we distinguish as a separate
faculty by abstraction only; its function is coordinative,
and it is operative upon data which it does not, and
cannot, alone originate.
When an explanation is offered of the fact that there
has always existed a large degree of divergence between
different philosophical views and creeds, emphasis is
often laid upon misunderstandings as to the terminology
in which philosophical ideas are expressed. No doubt
this factor is a real one in this connection; as language
was originally developed chiefly to serve much more
purely practical purposes than to provide the means
of expressing subtle philosophical distinctions. The
process of adaptation of language for the latter purpose
is certainly far from complete, and leaves ample scope
for shades of variation in the precise meaning that can
be attached to the terms employed; and this naturally
leads to controversies in which these differences in the
interpretation of language are far from negligible. Never



AND GENERAL THOUGHT


473


theless, the reason I have before given to account for
the extreme divergences in philosophical opinions is,
I think, more fundamental than what is due to the
imperfect functioning of language. The particular axioms
and postulations, performing the function of premisses,
of a particular system of Philosophy are the true characteristics of that system; they form the essential element
which distinguishes it from rival views. It is in differences of these characteristic premisses that the everlasting divergence of philosophical views which flourish
in the same age, or have been prevalent in different ages,
is to be found.
An important consequence of the a-logical character
of such a premiss, that is of the fact that it is not a
purely logical deduction from other premisses of which
the validity is admitted by all normal minds, is that an
absolute denial of its truth cannot be refuted by any
process of reasoning which rests upon a basis recognized
by all men, or even by all Philosophers. Every such
presupposition rests in reality upon a judgment of
probability, ranging from faith of various degrees up to
moral certainty. Some presuppositions of this class are
of such a character that they do actually receive the
assent of all, or nearly all, men. An example of this is
the belief in the existence of other selves besides our
own selves. If however a person chooses to take up the
position of a solipsist, it would appear that his view
cannot be refuted by any process which would show
that his opinion leads to logical contradiction. His
view appears to us however absurd, and almost insane,
because we all have an irresistible belief, that seems to
us consonant with our whole experience, that his view
is wrong. Although I have described scientific knowledge as essentially public knowledge, it is not necessarily
absolutely impossible for a solipsist, if there be such a
person, to set up an account of the world, regarded
exclusively as a series of psychical happenings in his




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own consciousness. As regards other ontological hypotheses, there is no such practical unanimity of opinion
as in the case of the hypothesis of the existence of other
persons with a psychical being resembling our own; and
this lack of unanimity exhibits itself in the presence
of all the manifold realistic and idealistic species of
Philosophy with which the history of Philosophy is conversant.
I have emphasized the fact that the influence of
Natural Science upon our general views of the world
depends upon the nature of ontological postulations
which go beyond any assumptions which Natural Science
needs for its own purposes. The complex of physical
percepts, taken as an appearance, into the ground of
which Natural Science does not need to inquire, cannot
be regarded simply as a product of the individual mind,
although the activities of the individual are clearly a
factor in determining physical presentations. As the fact
of the existence of Science as public knowledge testifies,
the physical complex contains a large element which
appears to be independent of any particular percipient,
and in this the objectivity of the physical world for
collective mankind consists. For individual minds, this
given element can only be taken as a datum which is
accepted in perception. The relations between sensations and perception form a domain which the psychologist seeks to describe conceptually, but he, like the
worker in Natural Science, accepts as a datum the fact
that we have a stream of sensations which we do not
appear to originate. The fact that there is, in the complex
of percepts, an element independent of any individual
mind, of a character consistent with its description and
symbolization by rational schemes, to an extent of which
we do not know the limits, is the fundamental fact which
emerges from the results and history of Natural Science.
Scientific laws and theories are the product or creation
of mental activity, but are dependent, for the raw material




AND GENERAL THOUGHT


475


from which they are constructed, upon given characteristics of the perceptual complex. These characteristics
are not free creations of the mind, but data without
which Science could not even begin to exist. Thus
Natural Science exists only in virtue of the fact that the
physical domain, the perceptual complex, is of such a
character as to render possible, to the extent which we
find by experience is actually the case, its conceptual
representation by laws and schemes.
When a metaphysical theory or hypothesis is set up
as to the fundamental character of this element of the
physical complex that is apparently not dependent on
the individual mind; that is when some ontological
assumption is made having reference to it, there is then
provided a bridge by means of which Natural Science
is connected with general Philosophy. The precise
nature of such an ontological assumption varies widely
in different species of metaphysical Philosophy. All
forms of realistic Philosophy agree in making the assumption that the element in the physical domain which
appears to be independent of the individual mind has
a real and independent existence, which does not wholly
depend upon its being a co-factor in the subject-object
relation. The assumption is that there exists a real complex which is not, as it were, exhausted in the subjectobject relation, but can be separated out of that relation
not merely in abstraction, so that it has an existence
independent of that relation. Different forms of realism
vary widely in the conceptions they adopt as to the
nature of that reality. In some forms of realism this
element is taken to be psychical in its nature, and thus
in some degree akin to the human mind; and some
systems of thought assume it to consist of a multitude
of psychical individuals, or monads. The realism of
common sense conceives this element as materialistic,
and articulated in the same way as it appears to be in
our perceptions, although a distinction between primary




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and secondary qualities of material objects receives some
recognition; the independent real element being more
closely related with the former. The ordinary realism
of Science, physical realism, regards the real complex
as articulated in accordance with the distinctions introduced by scientific theories, and of a non-psychical
character; atoms, electrons, etc., are regarded as real.
Neither scientific realism, nor that of common sense, is
necessarily materialistic in the sense of assuming the
psychical to be reducible to, or entirely subordinate
to, the non-psychical, although, as we have seen, this
assumption has been made by various influential exponents of Natural Science. Some forms of realism
have been agnostic, in the sense that, although they
have recognized the existence of this real element in the
world, they have regarded any precise characterization
of it as outside the domain of possible knowledge. All
forms of what may be called critical realism, to distinguish it both from physical realism, and naive realism,
assume the existence of a reality which in some sense
manifests itself in our sense-perceptions. But critical
realism refrains from identifying the real with any of
the conceptual constructions of Natural Science. In
particular, Monadism refuses to identify Monads with
atoms or electrons or biophors. Idealism, even when it
does not stop short at purely subjective idealism, does
not take the step of separating the object out of the
subject-object relation and regarding it as existing independent of that relation. Whilst holding fast to that
relation, without which it regards objectivity as meaningless, it usually regards the postulation of a Universal
Mind, for which the world is a realm of objectivity, as
essential for a Philosophy which shall suffice to give any
adequate account of human experience.
In the next lecture I propose to discuss the bearings
which Natural Science may have upon the Philosophy
of Theism, when some ontological hypothesis is made




AND GENERAL THOUGHT                477
which will suffice to establish the nexus, of which I have
spoken, with an outlook on existence wider than that
which Natural Science need adopt for its own special
purpose. Such a discussion can naturally deal with but
a small part of the whole subject of Theism, but it is
concerned with an aspect of the problem of Religion
which is of great importance in relation to contemporary
thought on the subject.




XX


NATURAL SCIENCE AND THEISM
IN describing the subject of which the Gifford
Lecturers are to treat, Lord Gifford spoke of it as:
the true knowledge of God, that is of the Being, Nature, and
Attributes of the Infinite, of the All, of the First and the Only
Cause, that is, the One and Only Substance and Being, and the
true and felt knowledge (not merely nominal knowledge) of the
relations of man and the universe to Him, and of the true
foundations of all ethics and morals....
Lord Gifford proceeded to make it clear that the most
absolute freedom of opinion is to be accorded to the
Lecturers; he said that:
they may be of any religion or way of thinking, or as is sometimes
said, they may be of no religion, or they may be so-called
sceptics or agnostics or freethinkers, provided only that the
" patrons" will use diligence to secure that they be able reverent
men, true thinkers, sincere lovers of and earnest enquirers after
truth....
He says further:
I wish the Lecturers to treat their subject as a strictly natural
science, the greatest of all possible sciences, indeed, in one sense,
the only science, that of Infinite Being, without reference to or
reliance upon any supposed exceptional or so-called miraculous
revelation..... The lecturers shall be under no restraint whatever in their treatment of their theme.
It is clearly in accord with the broad philosophic
spirit in which these words are conceived, and which I
have quoted on account of their intrinsic interest, that
a particular Lecturer should be free to treat the subject
of Theism from a special point of view which may lead
up only to one special aspect of Theism. The particular




NATURAL SCIENCE AND THEISM 479
point of view which I have chosen is the one which I
conceive to be that of Natural Science. I have already
given some indication of the existence of other points
of view, which may be of equal, and some of them
probably of greater, importance than that of Natural
Science, from which the great central problem of reality
may be regarded. The path has been prepared for the
brief and fragmentary treatment of the subject which
I here give by the discussions in the last lecture.
Besides the influence which may accrue to Natural
Science in relation to theoretical Theism when existential
assumptions are made which are not themselves any
part of the necessary basis of Natural Science, there is
another kind of influence, more of a practical kind,
which Natural Science exerts upon general views of the
nature of reality. I have referred to the manifold
character of the influences which cause a predisposition
to accept or reject ontological hypotheses or assumptions
which form the bases of particular views of reality, and
have emphasized their a-logical character. The habits
of mind induced by the study of Natural Science, and
even by acquaintance with the general character of the
practical and other results of that study, often play a
large part in producing a selective predisposition to
accept fundamental postulations of the kind to which
I refer. Natural Science has in the past undoubtedly
had the effect of producing, in circles wider than those
of scientific investigators, a bias in favour of certain forms
of realistic philosophy, and even probably of some forms
of theism as against other forms. I have maintained
throughout that this is a bias, or predisposing influence,
and not a logical consequence of the acceptance of
Natural Science and its special results as possessing a
certain kind of validity. In fact, familiarity with Natural
Science may be a cause of belief in the truth of certain
premisses employed in philosophical schemes, without
providing in any proper sense a reason for such belief.




480 NATURAL SCIENCE AND THEISM
Theistic Philosophy is dependent upon the two concepts of existence and value, since Theism, in any
ordinary sense of the term, is not only concerned with
the existence of God, but with His relation to our conceptions of value and its conservation. The estimation
of value is, as we have seen, not a matter with which
Natural Science has any direct concern; any bearings
which Natural Science may be thought to have upon
the estimation of values can accordingly be chiefly in
and through its bearings upon questions of existence.
Any general theory of reality must be expected to give
some account of the origin of the phenomena with which
Natural Science has to deal; of the relation of those
phenomena with the real, of whatever nature that real
may be held to be. Since the real must be such as in
some manner to give rise to the phenomena of which
Natural Science has to give a conceptual description, it
would appear that the kind of knowledge to which
Natural Science attains may lead to inferences as to
some of the characteristics of the real.
No Theistic theory would be of the slightest value
which failed to give some account of the relations of
God with finite spirits and with the world of phenomena.
Neither on the theoretical nor on the practical side could
such a theory serve any useful function; it would set up
a conception without any real content. The very various
types of theism which exist, and have existed, are mainly
differentiated from one another by the character of
this relation and the modes in which it is conceived. At
one extreme we have Pantheism, in accordance with
which God is the only existent, the All, who is identified
with the world, or at least with all that can be said to
be real in the world. The world contains nothing that
is existentially distinct from God, and this statement
must include all finite spirits. The great philosophical
problem of the One and the Many, pantheism attempts
to solve by means of a suppression of the Many. Our




NATURAL SCIENCE AND THEISM              481
actual experience of plurality being reduced to an
illusory appearance, the fundamental difficulty of pantheistic or absolutist systems of philosophy consists in
their inability to provide a satisfactory account of the
origin of this illusion; to account for the apparent
differentiation of the One into an apparent multiplicity
of phenomenal forms. Pantheistic theories have taken
various forms; there exists spiritualistic Pantheism, as
for example in the system of Spinoza, and materialistic
Pantheism, practically indistinguishable from Atheism;
there exists also the absolutism of some forms of Idealistic
Philosophy, in which God is the Absolute.
At the other extreme of theistic theory is the view in
accordance with which God is purely transcendental; a
Being essentially external to, although in relation with,
the world of which He may have been the Creator; such
Creation being conceived as the calling into existence by
His will of a world, external to Himself, existentially
separate from Himself, and subject to laws which He
has prescribed, but which render the world, at least to
a considerable extent, autonomous. In its most pronounced form this view was that of the Deism implicit
in the theology of the eighteenth century; it was held
not only by some of the writers known as deists, but
in at least equal measure by many of their more orthodox
opponents. The controversies of that time turned to a
large extent upon the degree of autonomy which the
Creator had granted to the world, and on questions as
to whether or not He had on specific occasions interfered
with that autonomy. The analogy of the relation of a
watchmaker to the watch he makes, or of an artificer to
a machine he constructs, was often uncritically appealed
to, as illustrating the relation of the Creator to the world.
The fact that the watchmaker, or the artificer, has to
deal with materials which he does not make, and with
given material properties over which he has no complete
control, did not appear to be of sufficient significance


HGL


3I




482   NATURAL SCIENCE AND THEISM
to destroy the value of such analogies for exponents of
this order of ideas.
Although there still remain in the popular mind
distinct traces of this conception of a purely transcendental Deity, conceptions intermediate between the two
extremes of which I have spoken are probably dominant
in the theistic thought of the present day. God is conceived of as immanent in the world, and more especially
in finite spirits, which live and move and have their
being in Him. He is also usually conceived to be transcendent, but varying emphasis is placed by different
thinkers upon the two elements of immanence and
transcendence. All views of this species, however wide
their differences may be, agree in the one respect in
which they all differ from Pantheism, in refusing to
assert that God is the only reality; although they may
regard reality other than that of God as derivative,
having its ultimate origin in Him, and dependent upon
Him for its continued existence, but possessing, at least
in some degree, a relative independence. The particular
mode in which this kind of Theism is conceived depends
to some considerable extent upon whether the general
Philosophy adopted is conceived in accordance with a
realistic or an idealistic attitude of mind. The theist who
combines with his theism some form of realism, naive,
physical, or critical, usually admits real existence to
appertain to a domain which is not of a purely psychical
character; for him there are real, non-mental, things
which may manifest themselves as objects in the subjectobject relations of psychical beings, but they exist independently of being mere factors in those relations.
For him, there really exists besides God, a world not
wholly of a spiritual or mental character; reality consists
of God and the world. This is not only the view of the
great majority of theists who are not philosophers, but
it is also the view of some philosophical theists. On the
other hand, for the Idealist, material objects and events




NATURAL SCIENCE AND THEISM                483
have no existence independent of their forming a factor
in the subject-object relations of a mind. All reality is
taken to be essentially bound up with this form of relation, and consists of minds for which alone objects exist.
The existence of God, the Universal Mind, is frequently
held to be a necessary inference from this view of the
nature of existence. The fact that the objectivity of the
world of things is, for the single finite mind, temporary
and intermittent, and that we are compelled to look
back to a time when minds such as we know did not
yet exist, is held to lead to the conclusion that a Universal
Mind exists, for whom all objects are eternally present.
The conception of the merely potential existence of
objects which no one perceives, and at times when there
is no sentient being for whom they are objects, as would
have been the case when the earth was unfitted for the
existence of living beings such as we know, is regarded
as inadmissible. It is held that, for the complete objectivity of the world, a Universal Mind is necessary. As
Dr Rashdall has written:
We cannot understand the world of which we form a part
except upon this assumption of a Universal Mind for which,
and in which, all that is exists. Such is the line of thought
which presents itself to some of us as the one absolutely convincing and logically irrefrageable argument for establishing the
existence of God.
In accordance with this view, the totality of existence
consists of God, the Universal Mind, and derivatively
of finite spirits; the world exists solely as eternal object
or idea for the Universal Mind, and as, in some partial
or fragmentary form, it appears in the subject-object
relations of finite minds.
The difficulty of Theism of these types, whether
combined with a realistic or an idealistic philosophy,
is in some sense the converse of that which presents itself in connection with Pantheism. Commencing with
the Many, these types of Theism attempt to provide
31-2




484 NATURAL SCIENCE AND THEISM
adequately for the existence of the One, whilst maintaining some degree of real independence for the Many,
whereas Pantheism, commencing with the One, finds
its crucial difficulty in making any real provision for the
appearance of multiplicity. Some of the adherents of
Theistic systems which commence with, and maintain,
the real existence of the Many do not hesitate to admit
that to do this involves the recognition of some limitation in the Being of God. This limitation is often represented as being a willed self-limitation. The position of
the world in relation to the Universal Mind is often
held to be a necessary one, since a subject without an
object is as unthinkable as an object without a subject;
it is conceived that it is only in connection with the
subject-object relation that either subject or object has
a meaning. It was in fact said by T. H. Green that
"the world is as necessary to God as God is to the
world."
This mode of argument for the existence of God is
of a purely metaphysical kind, as indeed are all ontological arguments, whether they rest upon an idealistic
or a realistic basis; and this is the case in particular
for the traditional "Ontological Proof" of the existence
of God. It is clear that Natural Science has no bearings
upon arguments of this kind, the validity of which must
be estimated upon metaphysical grounds alone. It is
only when a theistic theory, based upon such an argument, passes to a later stage, beyond that of the assertion
of existence, that the special characteristics of the world
of phenomena become relevant in relation to the Nature
of God; that the possibility can be contemplated that
Natural Science may have a part in the development
of Theism.
It may be remarked that, for thinkers of the present
day who hold the view that the Divine is immanent in
the human spirit, it is not so easy as it appeared to be
to the older exponents of Natural Religion to draw a




NATURAL SCIENCE AND THEISM            485
hard and fast line between revealed knowledge and such
knowledge as may be obtained by the continued use of
the rational faculty of man. For all knowledge obtained
by the activities of human spirits in which the Divine
Spirit is immanent may be regarded as in some sense
a divine revelation; all the activities of finite spirits being
conditioned by the immanent presence of the Divine
Spirit. The quite sharp distinction between revealed
knowledge and other knowledge would thus seem to
be characteristic of that conception of a purely
transcendental Deity which was commoner amongst
the thinkers of the eighteenth century than it is in our
time.
The three traditional proofs of the existence of God,
the ontological, the cosmological, and the teleological,
are, since the famous destructive criticisms of them by
Immanuel Kant, no longer regarded as proofs in the
sense that they consist of deductions, in accordance with
the canons of logic, from premisses which no man can,
or does, refuse to admit. They remain, however, in
broadened and extended forms, as lines of argument
which are still employed, rather inductively than deductively; but the validity of such lines of argument is
dependent upon foundations which cannot escape the
fullest scrutiny. The moral argument, stated by Kant,
has, in modified forms, come to represent the line of
thought which, in its insistence upon the fundamental
importance of that aspect of existence which we associate
with the terms "value" and "valuation," is regarded
by most Theists of the present day as outweighing in
importance and cogency all other aspects of the subject.
Theism is now very frequently regarded as finding its
main support in the existence of the domain of moral
standards; whereas in much of the thought of the
eighteenth century, and even later, this relation of de
pendence was taken in the reverse order.
The cosmological proof, adopted by Thomas Aquinas




486 NATURAL SCIENCE AND THEISM
from Aristotle, is founded upon the conception that the
world is due to a sequence of causes which may be continued in the backward direction, each cause of the
sequence being taken to be the effect of the preceding
cause. In accordance with this conception it is argued
that we arrive at a first cause, which is taken to be God,
for otherwise the causal sequence would constitute an
indefinite regress, in which case all explanation of the
present existence of the world would be in default. It
is accordingly assumed that, in order that the world
may be intelligible to us, it must have commenced with
a first cause. To this argument there is in point of logic
a fatal objection. It is assumed as the law of the sequence
that every cause is the effect of a preceding cause, and
thus the assumption that, through the sequence, we
arrive at a member which is a cause but not an effect is
a breach of the law of the sequence. Moreover, the
assumption that an indefinite regress is inconceivable,
or inconsistent with the existence of an intelligible
world, is groundless; in fact it is possible to define such
sequences, in accordance with an intelligible law. On
these grounds, the proof in its original form has been
generally rejected as invalid; but, as we shall see, it is
related with a line of argument which is still of importance, and to which the results of Natural Science
have made a contribution of great weight. There are
two points to notice about the proof, apart from the
fatal contradiction which it involves. In the first place,
it assumes, as perfectly definite, the conception of causation, without examination of the question whether the
causes are to be considered as efficient causes, or whether
they are to be considered simply as denoting totalities
of conditions which experience shows to precede certain
effects. In either case, an examination of the origin of
the nexus between cause and effect would appear to be
necessary before a proof could be accepted which professes to lead to an explanation of the existence of the




NATURAL SCIENCE AND THEISM              487
world. Again it takes, in accordance with an uncritical
notion of causation, a temporal succession of causes, and
it professes to arrive at an explanation of the whole
temporal series by means of a first cause, but not of a
law, or final cause, of the whole sequence, which shall be
operative or effective at all times. In fact it strongly
suggests the conception of a purely transcendental
Deity, who as a first cause starts the whole sequence of
causes, but then leaves it to itself, not being immanent
as the ground of the world. Moreover the proof
assumes the validity of the concept of the world as a
whole, which in our experience can never be a completed
concept.
If we consider the perceptual world from the point
of view of Natural Science, we see that one fact about
it has been established which may be made the basis of
a cosmological argument free from the defects of the
proof in its original form, although the fact to which I
allude is subject to limitations as to its scope, which can
only be removed by means of a postulation or assumption which may commend itself to the mind as appearing
probable. The whole history of Natural Science tends
to extend the scope of the ascertained fact that the
perceptual domain is such that whole tracts of it, and
processes in it, are capable of description by rational
schemes. Whatever then be taken to be the nature of
the reality which the perceptual world manifests, or of
the ground of phenomena, that reality or ground must
be of such a character that it has some correlation with
the rational processes of our minds. This fact may have
to be adapted, in the precise form of its interpretation,
to whatever philosophical formulation may be adopted
of the relation of phenomena with reality or with
their ground. Whatever such formulation be employed,
whether of a realistic or idealistic type, it will express
the fact that the phenomena of perception have as their
ground a reality, mental or non-mental, which is an




488 NATURAL SCIENCE AND THEISM
ordered system apprehended by us as rational. This line
of thought does not proceed by means of a principle
of causation to the recognition of the existence of a
first cause, from which the world as we know it has
originated, but fixes attention upon the principle of
rational order, as in some sense immanent in the world,
not as an external cause, but as the ground of the whole
life and movement of a reality of which the phenomenal
domain is the manifestation. This line of thought has
led to the postulation that reality is fundamentally
rational and unitary. It is rational, as exhibited in its
correlation with human reason, and unitary, as consisting of a completely interconnected system. The coincidence of this conception of a completely unified
rational reality with the religious concept of God will
require for its realization further elements, outside the
purview of Natural Science, in which the conception
of values will be the chief factor. It may be held that
Natural Science provides a most important part of the
justification for the ascription of complete rationality to
the real ground of the phenomenal world. But the
limitation must be fully recognized, that this postulation
of complete rationality of the real goes far beyond anything that has been, or can be, unimpeachably established by Natural Science. Apart altogether from the
passage from the phenomenal domain to the real ground,
a passage outside the scope of Natural Science itself, the
evidence that phenomena can be described by, and correlated with, rational mental processes is incomplete;
and thus the complete rationality of the Universe is an
hypothesis, not a known fact. It is the methodological
axiom of Science that the correlation of phenomena with
rational schemes can be carried out to an unlimited
extent, but the actual amount of verification which the
axiom has received is at all times strictly limited. It
cannot be demonstrated that no limits exist, to which
the procedure of Natural Science, in accordance with




NATURAL SCIENCE AND THEISM 489
this axiom, may be subject. Accordingly, what Natural
Science provides is indicative evidence, and not demonstrative proof, of the unlimited accessibility of
natural phenomena to rational schematization. But undoubtedly the actual achievements of Natural Science
have been sufficient to cause in many minds a belief in
the unlimited rationality of the ground of the real world.
With the difficult questions which arise in connection
with this line of thought, especially as regards the
relation of the wills of sentient beings with the real,
conceived as a unitary rational being or principle, I
cannot here deal; nor can I discuss the related question which arises as to the significance of what I have
called in the last lecture the principle of individuality.
I must be content with having briefly indicated a line
of thought which has largely supplanted the older mode
of thinking embodied in the traditional cosmological
proof.
The foregoing discussion of the significance of order
or uniformity in Nature, as forming a part of the possible
basis of a theistic Philosophy, would be incomplete without some reference to the fact that it is historically far
from true that the chief evidence of the existence of
God, derived from a contemplation of the phenomena
of perception, has always been found in the order and
uniformity discernible therein. On the contrary, the
religious mind has very frequently fixed its attention,
not so much upon that order, as upon the occurrence
of supposed breaches of that order, as exhibited in
miracles. It is not too much to say that the evidences
of divine Power have often been sought rather in miracles,
regarded as breaches of natural order, than in the
existence of that order itself. As Goethe has said: " Das
Wunder ist des Glaubens liebstes Kind." This view of
the evidential value of miracles appears to be in close
connection with the conception of God as transcendent,
as influencing the world not from within but from with



490 NATURAL SCIENCE AND THEISM
out. Just as the arbitrary and sporadic acts of an absolute
sovereign appeal to many minds as more striking, and
more dramatic, evidences of power than are exhibited
in the more even course of the orderly working of a
constitutional government, the incalculable acts of a
transcendental Deity, conceived as a despotic personality,
appear to provide more cogent evidence of divine power
than does the orderly and ubiquitous working of an
immanent Deity. The decay of the belief in miracles
which has taken place progressively in modern times is
undoubtedly due in large part to the progress of Natural
Science, with the emphasis which it places upon order
or uniformity in phenomena. It must however be distinctly recognized that there exists, and can exist, no
a priori proof of the impossibility of what are called
miracles. If that impossibility has been sometimes
asserted by exponents of Natural Science, the assertion
is merely a piece of a priori dogmatism, quite incapable
of substantiation on scientific grounds. We have no
a priori knowledge of what can, and what cannot, occur
in Nature. We have only presumptions, psychologically
explicable as expectations due to habits of thought
founded on our past experience. The decay of belief of
which I have spoken depends in large part upon a
change of attitude towards an unusual or unexpected
occurrence, and also consists in large part of a more
critical attitude towards the evidence that such alleged
events have actually occurred. To the modern man of
science, an event which does not appear to happen in
accordance with known laws is an occurrence which
suggests to him the inadequacy of those laws, or the
presence of some disregarded factor. He is incited to
attempt, by means of the extension of known laws, to
subsume the occurrence under a more complete set of
laws, or to determine the character of the particular
disturbing factor which had in the first instance been
left out of account. What would perhaps formerly have




NATURAL SCIENCE AND THEISM 49I
been regarded as a miracle is, for the modern man, not
a case of breach of order, but an occasion for extending
his knowledge of it. We now know that the evidence of
witnesses of miracles which took place in an uncritical
age, in which the belief in their actuality was dominant,
requires the most rigid scrutiny before it can be accepted
as of any value. Such scrutiny is most difficult to apply
in the case of events which happened in a remote age
and in a distant country. We know how difficult it is
to apply crucial tests to the evidence of alleged events
of an abnormal kind in our own time. Miracles are
always most plentiful at a time, or in a community, in
which the belief in their occurrence is most prevalent.
Moreover the possibility must be taken into account
that superior knowledge of physical laws, or exceptionally
great powers, normal in kind, on the part of a supposed
worker of miracles, may account for the exceptional
occurrences. This may particularly be the case in miracles
of healing; in that matter the modern study of Psychotherapeutics may be relevant. There is a further question
to be considered in this connection. Supposing it were
assumed that a particular occurrence were a breach of
natural order, what would the miracle prove, in relation
to its assumed agent? Even in the ages of faith, when
miracles were commoner than anyone supposes them to
be now, this question was one of importance. Some
scrutiny of the source of the miraculous intervention
was required, as it was by no means always assumed
that a miracle was of divine origin. A person who appeared to have powers of influencing natural phenomena
which we were wholly unable to bring into line with our
scientific knowledge, or with any readily conceivable
extension of it, would not now by any means necessarily
be accepted as of infallible authority on other matters,
such as Philosophy and Theology.
The view of miracles, in their relation to Religion and
Natural Science, which is widely held by modern thinkers




492 NATURAL SCIENCE AND THEISM
is aptly and trenchantly expressed by the Danish
Philosopher H6ffding1 in the following passage in his
Philosophy of Religion:
Even if, in spite of all these circumstances, we were to believe
in any particular case that we had here before us a real miracle,
i.e., a deviation from the law-abiding order of nature, the concept
of God which could be based on this fact would necessarily bear
the stamp of imperfection; for a miracle is a makeshift, a way
out, something which has to make up for a want in the order
of nature. The ordering of nature has not been so effected
that by it all the divine ends can be attained. God encounters
an obstacle within his own order of nature. It is as if there were
two gods-one who is active during the ordinary course of
things, and another who, in particular cases, corrects the work
of the former. Hence the concept of miracle is dangerous from
the religious as well as from the scientific standpoint. It is a
bastard which neither parent can afford to own. The Church
is wise in not acceding to the re-awakened desire for miracles.
It is true of increasingly large circles that miracles, which in
former times were a proof and support of religion, are now
rather a stumbling-block which its apologists have to defend,
and which in their hearts they must often wish themselves
well rid of. The less we think of the relation between God and
the world as a purely external one, analogous to the relation
between a clock-maker and his clock, the less there is room
for, or possibility of, miracles. The happenings of the world
differ widely in value, and excite our admiration in very differing
degrees; the highest does not take place every day. But there
is nothing to prevent all events being subject to the same great
law. It is large enough to embrace an infinite number of things
and of problems. May we not assume that that which is of
highest value may be reconcilable with the principle of natural
causation? The concept of miracle really arises from the negative
answer to this question. From whence the right to negate is
derived is not easy to discover. The fact that something is of
the highest value does not preclude a purely natural origin.
The concept of miracle rests on an identification of estimation with explanation, an identification to which is largely due
the confusion which at present characterises the religious
problem.
1 English translation, pp. 29-30.




NATURAL SCIENCE AND THEISM              493
Closely connected with the cosmological argument in
favour of a theistic view of the Universe is the teleological argument, which rests upon the fact that there
is much in the world of phenomena which has at least
the appearance of purposiveness or design. The older
forms of this argument have consisted in tracing out in
Nature special cases of adaptation, taken as indicating
the effects of design on the part of an intelligent Being.
By such exponents of the teleological argument as
Archdeacon Paley, much emphasis was placed upon the
existence, in a great variety of special cases, of intricate
arrangements adapted to the attainment of useful ends;
it was held that the existence of such contrivances could
only be explained as due to intelligent design. The kind
of purposiveness taken to be exhibited in such cases was,
in accordance with the prevailing view, assumed to be
analogous to that of the artificer who constructs a
machine which shall fulfil some purpose which he has
in view. The argument depended upon the idea that
when we find such a machine actually working, so as to
attain an end, we are entitled to infer that it has been
constructed by an intelligent being who had that end in
view. It must be observed that the purposiveness so
conceived is not immanent in the machine, but in its
designer; the relation of the designer to the machine is
conceived as an external relation. The designer works
with given materials possessed of given properties; his
intelligence is exhibited in the ingenuity with which he
works under strictly limited conditions which are for
him unalterable; he has to do the best he can with the
materials at his disposal. The shortcomings of this line
of argument are clear; it might with some plausibility
be employed by a polytheist to infer the existence of a
number of gods of considerable, but limited, power and
intelligence, operating in a world which they did not
create. The inference to a God of unlimited power and
intelligence would present greater difficulties. The very




494 NATURAL SCIENCE AND THEISM
necessity for contrivance or ingenuity implies limitation
of power.
The progress of Natural Science during the time that
has elapsed since this form of the teleological argument
was commonly employed, and especially the great progress in Biological Science, has resulted in a radical
change of the mode in which teleology in relation to
natural phenomena is conceived, and in the whole aspect
of the place which a teleological view of the world
occupies in general Thought. By modern teleologists
who regard the matter from the points of view of Natural
Science and of Psychology, purposiveness is regarded
no longer as indicating a purely external relation, but
as in some sense immanent in living organisms, and
possibly in a wider sense in the phenomenal world
generally. In this connection two questions are of
fundamental importance; the question of the existence
and scope of natural selection, and the general question
of the nature of vital processes and of the mode in which
the relation of the physical with the psychical side of the
living organism is to be conceived. As I have already
pointed out in my lecture on Evolution, the theory of
Natural Selection is anti-teleological in its tendency. It
is an attempt to give an account of what appears prima
facie to be progressive adaptation in relation to ends,
by means of a scheme in which no teleological factor
is admitted. As we have seen, there is at the present
time much difference of opinion as to the sufficiency
and scope of this theory of Natural Selection, so that
it must be regarded as very far from having been established that it gives such an adequate account of the
evolution of species, and their adaptations to environment, that a teleological factor can be dispensed with.
When the theory is extended to mental evolution, it
seems, to say the least, extremely doubtful whether it
can, with any plausibility, be appealed to as giving a
credible account of the evolution of the higher mental




NATURAL SCIENCE AND THEISM 495
and spiritual faculties of man; in default of any sufficient
evidence that these faculties have survival value in the
biological sense of the term. The view has been expressed,
and is held by many persons, that the development of
the higher faculties of man, so far from being an evolution, subject to the law of struggle for existence and the
survival of the fittest in the biological sense, involves
really a reversal of that law. The difficult question as
to the relations between the psychical and physical sides
of living organisms I have already spoken of in an earlier
lecture. I have contended that there is no sufficient
ground for the assertion that what we believe to be our
power of purposively influencing natural phenomena is
an illusion. If this view be accepted, it is difficult not
to recognize a teleological factor as present in all higher
organisms at least. But on this matter there is at present
much diversity of opinion amongst those who are best
able to form a judgment. Natural Science must therefore be taken to speak with an uncertain voice at the
present time, as regards the question whether it is compelled or not to recognize a teleological factor, as a
supplement to the physico-chemical categories, in its
efforts to give an account of vital processes. I think it
may fairly be said that there exists no sufficient ground
for negativing the hypothesis that something of the
nature of entelechy is a necessary factor in all vital
processes, even if the positive arguments in its favour
that have been given by Driesch and others are regarded
as insufficient. But, even on this side, admitting the
necessity of conceiving entelechy, or purposive directivity, as exhibited in all vital processes, the teleological
argument based upon such admission does not take us
very far. It would be a long step to pass on to a unitary
purposiveness which should be immanent in, or associated with, all natural phenomena; and indeed, if
the argument could be carried so far, it might become
pantheistic in its result. The independence of finite




496 NATURAL SCIENCE AND THEISM
spirits which allows their purposiveness to exhibit its
effect in the phenomenal domain would appear to be
inconsistent with, or at least very difficult to reconcile
with, the conception of a completely unified purposiveness which pervaded all natural phenomena. If purposiveness is not to be completely unified, it is difficult,
without very arbitrary assumptions, to trace limits to
the degree of its diffusion.
I have already referred to the fact that the argument
in favour of theism which is in our time generally
regarded as the most important and most convincing is
the Moral Argument which is based upon the human
conception of moral values. In its earliest form, this
argument was developed by Kant, who made the belief
in the existence of God depend upon a postulation of
the practical reason, as distinct from the pure reason
which he regarded as impotent to establish the truth of
theism. In our day this argument has been developed
by various exponents, amongst whom I may allow
myself to refer to Professor Sorley, who in his Gifford
Lectures on Moral Values and the Idea of God, delivered
here in Aberdeen, has presented the argument with
great force and admirable skill. Any general discussion
of this argument would be quite outside the scope of
these lectures, even if time allowed me to attempt it.
There is however one aspect of the argument to which
I must refer, because it is one in which facts of the
phenomenal domain, brought to light by Natural Science,
have significance. In dealing with the argument from
design, the older writers on Natural Religion laid great
stress upon the beneficent results for man of many of
the contrivances exhibited in Nature. The evidences of
design which are discernible in natural phenomena were
conceived to afford a proof not only of the intelligence
and power of the Designer, but also of His goodness, as
exhibited in the benevolence with which Nature has
been adapted to serve the needs, and further the well



NATURAL SCIENCE AND THEISM 497
being, of the human race. The reverse side of this
picture of Nature was referred to by Hume, and has
been strongly emphasized by later thinkers. J. S. Mill,
in his essay on "Nature," framed a terrible indictment
against the callousness, cruelty, and injustice exhibited
in the ordinary course of natural phenomena.
In sober truth (he writes) nearly all the things which men are
hanged or imprisoned for doing to one another, are nature's
every day performances.... Nature impales men, breaks them
as if on the wheel, casts them to be devoured by wild beasts,
burns them to death, crushes them with stones like the first
Christian martyr starves them by the quick or slow venom of
her exhalations, and has hundreds of other hideous deaths in
reserve, such as the ingenious cruelty of a Nabis or a Domitian
never surpassed.
Mill drew the conclusion that the God who is responsible
for Nature, as he describes it, cannot be a Being both
of unlimited power and unlimited goodness. In a later
essay, Mill maintained that all the evidences from design,
and from the characteristics of Nature, point to a Deity
whom he describes as:
A Being of great but limited power, how or by what limited
we cannot even conjecture; of great, and perhaps unlimited
intelligence but perhaps, also, more narrowly limited than his
power; who desires, and pays some regard to, the happiness of
his creatures, but who seems to have other motives of action
which he cares more for, and who can hardly be supposed to
have created the universe for that purpose alone. Such is the
Deity whom Natural Religion points to; and any idea of God
more captivating than this comes only from human wishes, or
from the teaching of either real or imaginary Revelation.
The general picture of organic evolution painted by
modern Biology is one in which the struggle for existence
and nutriment, involving pain and death, is an important
and possibly a dominant feature, although its repulsive
aspect may perhaps sometimes have been drawn in too
lurid colours. The ruthless sacrifice of multitudes of
individuals appears to be a feature of the ordinary
HGL




498 NATURAL SCIENCE AND THEISM
course of evolution; it has been said that Nature cares
nothing for the individual, but much for the race. The
facts brought to light by Bacteriology and Parasitology
have disclosed the existence of many organisms, and of
what have the appearance of being most ingenious contrivances, the apparent purpose of which is to inflict
torture and death upon other, and usually higher,
organisms. The result of modern knowledge of this
kind has been to heighten the impression produced by
the widespread and intricate character of what from our
point of view we describe as physical evil. A consistent
theist must regard as, in some sense, God's creatures,
those living organisms whose presence and activities
condition the existence of tetanus, cholera, typhus, and
many other diseases. The contemplation of this aspect
of phenomena gives rise to a very real problem, not only
for Theologians, but for great numbers of thoughtful
persons.
Of the problem presented by the existence of moral
evil I cannot here speak, but for modern theistic Philosophy, in which the human spirit, with its needs, its
rationality, and above all, its moral values, constitutes
the basic point of departure, the existence of physical
evil gives rise to difficulties which have never been overcome, although many earnest attempts have been made
to soften their asperity. Some of these attempts proceed
in the direction of indicating the possibility of a resolution of the difficulties, if they could be regarded from
a point of view higher than that which the limitations
of the human mind, and of its knowledge of the circumstances of the Universe, enable it to occupy. Some
expositors lay great stress upon the necessity of an imperfect world as providing a field of activity such as is
needed for the progressive perfectibility of free spirits.
I must confess that many such attempts to cope with
the difficulties of the kind I have indicated impress me
as having the unsatisfying character of special pleading.




NATURAL SCIENCE AND THEISM 499
However this may be, it is, I think, admitted by candid
exponents of Philosophical Theism that there do remain
very real difficulties in reconciling Theism, of a type
such as will completely satisfy the religious consciousness,
with some aspects of the actual world of our experience.
This does not by any means necessarily entail the consequence that the views of the philosophical theist must
be rejected, but it does, I think, necessitate a recognition
of the fact that a complete synthesis of the conceptions
of existence, and of value, in a unitary theistic view of
the Universe, has proved so far to be beyond the reach
of the human mind. Many persons think it highly
probable, or even morally certain, that such a unified
view will prove unattainable so long as the present
limitations of the human mind remain.
My main aim has been, by means of a delineation of
the domain of Natural Science, to vindicate the perfect
freedom of Religious and Philosophical thought from
any fear of destructive interference from the side of
Natural Science, subject to the sole condition that no
encroachment is made upon the autonomy of Natural
Science in its own proper domain. It has been no part
of my aim in these lectures to indicate the use which
Religious and Philosophical Thought may make of this
freedom, or to state any results to which I might conceive it to lead. To have attempted to do this would
have been to open out a field of discussion of an extent
far beyond anything that could possibly have been joined
on to the special subject of my course. Successful
vindication of this freedom would serve to allay the fears,
often perhaps only half conscious, of numbers of thinking
men who, intimidated by the striking triumphs of Natural
Science in extending our knowledge of the order of
phenomena, and in turning that knowledge to practical
account, imagined that Natural Science, as a rigidly
deterministic world-philosophy, was perhaps destined
to exercise a complete domination over the spiritual
32-2




500 NATURAL SCIENCE AND THEISM
domain, and to leave no possibility of any real content
being assignable to the conception of human freedom.
I need hardly emphasize the fact that the removal of
the destructive criticism which grounds itself upon
Natural Science does not suffice to refute criticism which
rests upon philosophical or psychological grounds.
There was a time when Theology claimed to occupy
the whole territory of Natural Science, and held it in
complete thraldom. The history of the prolonged struggle
of Science for autonomy on its own territory has been one
in which Theology has lost every battle. Unfortunately,
Science has not always remained content with the
vindication of its freedom, but has attempted to extend
its dominion into territory which is not its own. There
are happily at the present time hopeful signs pointing
to a cessation, or at least a mitigation, of the conflict. On
both sides, the prevailing temper is markedly different
from what it has been within living memory. There is
greater readiness than formerly to admit that the conditions of life as we experience it are such that different
methods are requisite for dealing with differing aspects
of our life and experience; and that this involves the
necessity of granting freedom to those who pursue the
different lines of thought and investigation appropriate
to these different parts or aspects of our whole experience.
The discursive modes of thinking to which the limitations of the human mind bind us compel us to treat in
separation, even on the purely intellectual side, the
different aspects of our experience with which Philosophical Theology and Natural Science are concerned.
How much more is this seen to be the case when we
consider how very different in kind are the needs which
are to be satisfied by these two departments of thought.
Religion, which is very far from being the same thing
as Theology, always sooner or later feels the need of
some support, on the intellectual side, from reasoned
theological conceptions. Theology, I refer to here on




NATURAL SCIENCE AND THEISM 501
its philosophical side; the philosophical Theologian is
simply a Philosopher who pays special attention to those
aspects of Philosophy which have a specifically religious
bearing. Theology, as distinct from general Philosophy,
has as its function the provision of a cognitive basis for
Religion, of which the essence is not mainly cognitive,
but is mainly concerned with the moral and emotional
sides of human nature. If we were in possession of, and
able to grasp, a unified view of the Universe, in which
all the elements of existence and valuation were completely synthesized, the division of labour of which I
have spoken would be unnecessary; we should not require
to mark out frontiers between Science and Philosophy
or Theology; but of such a synthesis there is not the
remotest prospect in view. The secret of the Universe
has revealed itself neither to the Theologian nor to the
Philosopher. The man of Science, as such, is not even
concerned with that secret. The untrammelled freedom
which must be allowed to workers in all departments of
the great cultural work of humanity, to Philosophers and
Theologians, to Historians, to the cultivators not only
of Natural Science, but of Science of all kinds, should
not however involve the erection of rigid impassable
barriers which shall mark off domains which hold no
communication with one another. On the contrary,
workers in one department will often receive the most
valuable enlightenment, and most important suggestions,
-from quarters outside their own special line.
The summary discussions in the present lecture, and
in the one immediately preceding it, deal professedly
only with partial aspects of the great central questions
which have in all ages been the subject of unending
scrutiny on the part of thinkers, great and small; related
as these questions are, and always have been, with the
deepest thoughts, hopes, and feelings, of multitudes of
human beings. The tentative character of much of
what I have said in this connection may appear to




502 NATURAL SCIENCE AND THEISM
many persons so devoid of sharply defined results as
to be eminently unsatisfactory. I can only plead that
anything like a dogmatic statement of personal opinion
on most weighty matters, only a few partial aspects
of which I have been able to discuss, would have been
irrelevant and inappropriate, since it would have been
foreign to what I placed before myself as the aim of
these lectures.




INDEX


Abiogenesis, 397 et seq.
Abstraction, 29, 30
Acceleration, 176; in circular motion,
177
Acquired characters, 314 et seq.,
414 et seq., 446
Action, at a distance, 158; contact,
159, I88
Activity, 75, 86, 87
Adams, J. C., 295
Aepinus, 248, 250, 251
Aggregates,  112;  Infinite,  121;
Theory of, 137
Agriculture, 5
Ahmes, his papyrus, 108
Algebra, 115
Ampere, I68, 257
Analogy, 376
Anatomy, 375, 377, 381; Comparative, 384
Anaxagoras, 425
Ancestral inheritance, 422
Animal, Definition of, 345
Animism, 7 et seq.
Anthropology, 2, 437, 455
Antiquity of the earth, 15, 289, 314
Apollonius, 13
Approximate isolation, 46, 95
Aquinas, Thomas, 485
Arago, 257
Archimedes, 108, 119, 131, 175;
axiom of, 132
Argon, 279
Aristotelianism, 153
Aristotle, 51, 64, 151, 152, 202,
375 et seq., 384, 397, 425, 486
Arithmetic, 99 et seq.
Aryabhata, 1og
Asteroids, 295
Aston, F. W., 291
Atomic theory, 15 et seq., 163 et seq.,
172, 271, 284 et seq., 288; Boscowitch's, 167 et seq.
Atomic weights, 272, 274, 291
Atoms, 166, 170 et seq.; interaction
between, I67; vortex, 168, 222
Avogadro, 164
Bacon, F., 51, 203, 427
Balmer, 287
Banks, Sir J., 255


Bateson, 423
Becquerel, 277
Bell, Charles, 368
Benedetti, 184
Bergson, 126, 324
Berkeley, Bishop, 51, 56, 120, 205
Bernoulli, Daniel, 170, 246
Berzelius, 256
Bichat, Xavier, 382, 384
Binary stars, 304 et seq.
Biogenesis, 398
Biology, 344 et seq., 430; Concepts of,
349
Biophor, 410 et seq., 446 et seq.
Biot, 257
Black, 206, 21o, 235
Bode, 294
Body Alpha, 186
Bohr, Niels, 287
Boltwood, B. B., 282
Boltzmann, 27, 170, 226, 239
Bolyai, 134 et seq., 144
Bonnet, 382, 429
Boscowitch, 167
Boyle, Robert, 156, 157, 205
Brewster, David, 267
Brooks, W. K., 404, 420
Brown, Robert, 395
Bruno, Giordano, 14, 153
Biichner, 18-20
Buffon, 383, 402, 427, 428, 435
Cabanis, 367
Canton, John, 249
Cantor, G., II0, III, 121
Cardinal numbers, 104
Carlisle, 255
Carnelley, 275
Carnot, L., 211
Carnot, S., 134, 211 et seq., 235,
236
Carr, H. Wildon, 60
Carus, K. G., 387, 388
Cathode rays, 276
Cauchy, 168,268
Causality, Principle of, 78 et seq.
Causation, efficient, 43, 74 et seq.,
86, 87, 93; and logical necessity,
80; asymmetric relation of, 85
Cavendish, 207, 250, 251
Cayley, 140




504


INDEX


Cell, Galvanic, 263
Cell-theory, 393 et seq., 400, 435;
criticism of, 396
Centrosomes, 406
Chambers, Robert, 435
Chemical elements, irreducible, 271;
inconvertible, 272; Periodic law of,
274, 285; Genesis of, 275
Chemistry as a deductive science, 36,
40
Christoffel, 338
Chromosomes, 406 et seq.
Classification of animals and plants,
380 et seq.
Clausius, 116, 236 et seq., 258
Cognition, 461, 468
Colding, A., 212
Combustion, 206
Comte, Auguste, 36
Concepts, 23, 30, 33 et seq., 59; of
two species, 32
Conceptual schemes, 31 et seq., 44 et
seq., 88; inadequacy of, 95, 96
Conservation, of energy, I98, 208 et
seq., 23; of mass, 198, 200 et seq.;
of weight, 198 et seq.; of matter,
198 et seq.; in living organisms,
359 et seq.; of values, 463
Continuity, 113
Continuous magnitude, 15
Continuum, Arithmetic, 114, II9
Copernican system, 13, 14, 184
Copernicus, 378
Corpuscles of different orders, I57
Correlation of parts of animals and
plants, 385
Correspondence, Notion of, 103
Cosmical Astronomy, 39; Theories
of, 293 et seq.
Cosmogony, Mosaic, 426
Coulomb, 251
Counting, 104, Io6
Crookes, Sir W., 275
Crystals, biaxal, 267
Curie, 277
Currents, Conduction of, 263
Cusa, Nicolas de, 153, 183
Cuvier, 384 et seq., 388, 391, 392,
428, 434, 435, 442
Cyclical systems, 233
Cytoplasm, 394, 405
d'Alembert, 159, I85, 226
Dalton, I63, I65
Davy, Humphry, 211, 234, 235, 255,
259


Darwin, Charles, I5, 73, 402, 429,
430, 433, 436 et seq., 449
Darwin, Erasmus, 430
Darwin, Sir G. H., 297, 302 et seq.,
306 et seq.
De Chancourtois, 273
Dedekind, Io
Deism, 481 et seq.
Delage, 413
Democritus, I, I5 1 et seq., 202, 402,
425
Desaguliers, 246
Descartes, 51, 184, 204, 2o8 et seq.,
246, 378, 379, 427; on Geometry,
136; view of the world, 154 et seq.
Descriptive character of natural laws,
27
Design, 496 et seq.
Determinants, 410 et seq., 446 et seq.
Deterministic schemes, 83 et seq.,
89 et seq.; and teleology, 85, 98
De Vries, 404, 449, 450, 451
Diamagnetism, 262
Diderot, 205
Diophantes, I09
Dissipation of energy, 238
Dominance, 423
Doppler, 311
Driesch, 365, 414, 495
Du Bois-Reymond, E., I59, 368
Duration, 123
Dutrochet, 395
Dynamics, 175; Abstract, I77; of
systems, 193; Newtonian, 225,
330 et seq., 337; Hertz's formulation of, 225, 232; Lagrange's, 227;
Hamilton's formulation of, 229,
232, 233
Economics, 2
Efficient causation, 43, 75 et seq.
Einstein, Albert, 47, 50, 6o, I30,
I62, 242, 316, 318, 320, 327, 329,
330, 334, 335, 338 et seq.
Electric, law of force, 249, 250; distribution, 252; potential, 253; currents, chemical theory, 255; contact theory, 255; action of, 257,
258; induction of, 260; displacement, 264, 265
Electricity, 243 et seq.; one-fluid
theory, 248, 250; animal, 254; two
species, 247
Electrification, by friction, 245; by
conduction, 246; by induction,
249




INDEX


505


Electrodynamical forces, 259
Electrodynamics, 330 et seq.
Electrolysis, 255
Electromagnetic field, 263, 264, 269
Electromotive force, 259
Electron, 272, 280, 284, 286; theory
of matter, 276, 277
Electrostatics, 252
Emanations, 280
Embryo, 387, 400; Nutrition of, 413
Embryology, 389 et seq.
Empedocles, 425
Energetics, 219
Energy, potential, 2o0 et seq.; of a
system, 217 et seq.; conservation
of, 231; Dissipation of, 238, 362 et
seq.; free and bound, 239; Transference of, 240; Partition of, 241;
Biotic, 360
Entelechy, 365, 495
Entropy, 237 et seq., 364
Environment, 430 et seq., 443
Epicurus, I52, 156
Epigenesis, 401
Epistemology, o10
Ether, 222, 262, 267, 327 et seq.;
Stresses in, 265; Waves in, 268
Euclid, Iio, 119; Elements of
Geometry, 130 et seq.; theory of
parallels, 133, 135
Euler, i86, 246
Event, 322 et seq.
Evolution, 5, 425 et seq., 497; Mental factors in, 86; Mental, 443
Explanation, 81, 87; by models, 82
Extension, 34
Fabroni, 254
Fall of man, 15
Falling bodies, 175
Faraday, I68, 250, 260 et seq.
Fay, du, 247
Faye, 301
Fechner, 258
Fermentation, 398
Fertilization of ovum, 414
Fitzgerald, 329
Force, Conception of, I76, 182,
I88, 223, 224; of attraction, 189;
Measure of, 191, 209; Central, 192
et seq.
Form and Function, 375 et seq.,
386
Foucault, 196
Fourier, 259
Fractional numbers, 105, io6


Frames, of reference, I80, i8I, 194,
336 et seq.; Inertial, 194
Franklin, Benjamin, 249, 250
Free Will, 89
Fresnel, 267
Functional reaction, law of, 43 I et seq.
Functionality, 75
Galen, 377 et seq.
Galilean axes, I94
Galileo, 11,28,156, 175,176, I84,208,
216, 221, 224, 225, 230, 231, 378
Galton, F., 404, 405, 418 et seq.
Galvani, 254
Galvanism, 250
Gametes, 406, 424
Gases, Kinetic theory of, 169, 170
Gassendi, Pierre, 155, I56
Gauss, 134, 142, 258, 338
Gay-Lussac, 164
Geber, 203
Geology, 39, 314
Geometry, 34 et seq., 45, 129 et seq.,
I40 et seq., 321 et seq.; Projective,
I27, 139; Euclidean, I31 et seq.;
non-Euclidean, 135 et seq.; Analytical, I36 et seq.; Applicability of
non-Euclidean, 148 et seq.
Germinal selection, 446 et seq.
Germ-layers, 390
Germ-plasm, 415 et seq., 446; continuity of, 409
Gerould, J. H., 394
Gibbs, Willard, 170, 239
Gifford, Lord, 478
Gilbert, 217, 245
Goethe, 373, 434, 489
Goodricke, John, 305
Grassmann, 258
Gravitation, Newton's law of, 157 et
seq., I6I, 231; Propagation of, I59;
le Sage's theory of, I6o et seq.;
Einstein's theory of, 320, 335
Gravitational fields, 326 et seq., 335 et
seq.; of the sun, 339
Gray, Asa, 437
Gray, Stephen, 246, 247
Green, George, 253, 268
Green, T. H., 484
Guidance, 357
Haeckel, 398
Haldane, J. S., 349
Hamilton, Sir W. R., 229, 230, 232,
267
Harvey, 378, 397




506


INDEX


Heat, as a substance, 210; Mechanical
equivalence of, 211 et seq.
Heliotropic animals, 366
Helium, 279, 284
Helmholtz, 149, 173, 213 etseq., 228,
233, 234, 239, 259, 263, 300, 325,
342, 368, 369; on causality, 78, 84;
on Geometry, 140, 143; on vortex
rings, i68
Herbart, 370
Heredity, 397 et seq., 439; Weismann's theory of, 405 et seq.; Statistical study of, 417 et seq.
Herschel, Sir W., 305
Hertz, H., 225, 232, 269
Hertzian waves, 217
Hipparchus, 183
Hippocrates, 375, 402
Histology, 377, 381
Hobbes, Thomas, 156, 205
H6ffding, 463, 492
Homology, 375, 385, 389
Hooke, Robert, 266, 393
Huggins, Sir William, 310, 311
Hume, David, 51, 65; on efficient
causation, 77
Huxley, 15, 380, 389, 398
Huygens, I59, 177, 205, 208, 209
Hygiene, 5
Hysteresis, 345
Idants, 409
Ids, 409 et seq., 446 et seq.
Idealism, 356, 476
Ignoration of coordinates, 229
Individual, 354, 367
Individuality, Principle of, 458
Inertia, Principle of, I82, 184, 185, I87
Inertial frames, I94
Infinitesimal, Concept of, I 17
Infinitesimal Calculus, 1 I8
Infinity, Concept of, 17, I9, 120
Integral Calculus, I20
Irreversible motions, 233
Isotopes, 290, 291
Jacobi, 230
Jager, 405
James, William, 467
Jeans, J. H., 309, 313
Joule, 211 et seq., 231, 262
Kant, Immanuel, 51, 65, 185, 296,
297, 310, 427, 485, 496; on spatial
intuition, 130, 149
KekulI, 165


Kelvin, Lord, 166, 236 et seq., 250,
263, 264, 268, 284, 300, 312, 314,
398; on explanation, 2; on vortex
atoms, I68, 222
Kepler, II; his laws, 29, 131, I84,
204, 378
Kielmeyer, 386
Kinetic energy, 209, 2I0
Kirchhoff, 36, 263, 264
Knowledge, Common, 24; Scientific,
28, 455 et seq.; Individual, 460 et
seq.
Korteweg, 258
Lagrange, 227,229,230,232,252,253
Lamarck, 381, 394, 395, 429 et seq.
Lambert, 134
Language, 25, 472
Laplace, 96, 134, i6o, 210, 296 et
seq., 301, 310, 312; his view of the
world, 18
Larmor, Sir J., 222, 276, 277, 330
Lavoisier, 200, 206, 207, 210, 380
Laws, 28, 30, 44; as descriptive of
percepts, 25; of Nature, 26, 42, 88;
Natural, 27
Least action, 230, 234
Leeuenhoek, 393
Legendre, 134
Leibniz, 5I, 216, 427; on Monadism,
56, 6i, 118, 119, 205, 208 et seq.
Le Sage, 159 et seq.
Leucippus, I, 151
Leverrier, 295
Leyden jar, 247
Liapounoff, 308
Liebig, 165
Life, Origin of, 398
Light, Polarized, 262; and mass or
weight, 341
Lightning, 249
Limit, Concept of, 117, I19
Lindemann, F. A., 314
Linnaeus, 380 et seq., 428
Living organism, 373 et seq.
Lobachewsky, 134 et seq., 144
Locke, John, 51, 367; on substance,
55
Lockyer, Sir Norman, 279
Lodge, Sir Oliver, on reality, 60, 317
Loeb, 366, 413
Logic, 455
Lorentz, H. A., 276, 277, 330
Lotze, 370
Lucretius, I53, 154, 202
Lyell, 435, 436, 443




INDEX


507


MacCullagh, 268
Mach, 27, 36, 190
Macquer, 206
Magendie, 368
Magic, 8 et seq.
Magnetic, law of force, 250; potential, 253; force, 263; induction, 263
Magnetic needle, 244; Polarity of,
244, 252
Magnetism, 243 et seq.; terrestrial,
245; Vortical theory of, 246; Onefluid theory of, 250; Two-fluid
theory of, 251
Malpighi, 381, 393
Malthus, 436
Malus, 267
Manifold, fourfold ordered, 319,
332 et seq., 338
Mass, measurement of, 189, 335;
Conservation of, 200 et seq.
Material objects as constructs, 53 et
seq.
Materialism, 156
Mathematical Analysis, Ioo, 115
Mathematical theories, 57
Matter, Constitution of, 150; Continuous theory of, 151; as plenum,
154
Maturation, 406
Maupertuis, 230, 402
Maxwell, James Clerk, 48, i6i, i66,
169, 170, 232, 244, 257, 261 et
seq., 268, 269, 271, 298; on the
principle of inertia, I85
Mayer, J. R., 212 et seq., 299, 300
Measurement, Io7; Spatial, 145 et
seq., 320 et seq.; Temporal, 124 et
seq., 320 et seq.
Mechanical theory of Nature, I7; of
living organisms, 378 et seq.
Mechanics, Classical, 174
Mechanistic theory, 154 et seq., 221
Meckel, 387, 391, 429
Medicine, 377
Mendel, G. J., 422 et seq.
Mendeleeff, 273, 274
Mercury's perihelion, 340
Metabolism, 359
Metaphysical Philosophy, 471 et seq.
Meteors, 299, 300
Methodological dualism, 356
Meyer, 206
Meyer, Lothar, 275
Michell, John, 250
Michelson, 328
Mill, J. S., 497


Miller, 310, 329
Milne-Edwards, 376
Minkowski, 130, 149, 322, 331, 334
Miracles, 489 et seq.
Mirbel, 394, 395
Mnemonic hypothesis, 413
Momentum, 209
Monadism, 476
Monadology, 153
Monism, 356
Moon, Genesis of, 302 et seq.
Morley, 328, 329
Morphology, 373 et seq.
Motion, of projectiles, 176; circular,
177, 184; relativity of, 183, 196;
and force, 190; perpetual, 215,
216
Muller, Johannes, 368, 370, 394
Mutations, 449
Mysticism, 467
Natural Science, and ontological assumptions, 52, 54, 62, 63; and
facts, 65; Limitations of, 70 et seq.
Natural selection, 428, 437, 439 et
seq,, 445, 494
Nature, Uniformity of, 43; Laws of,
88; Philosophy of, 86, 386
Navier, 268
Nebulae, 310 et seq.
Nebular hypothesis, 296 et seq.
Neptune, Discovery of, 295
Nerves, sensory and motor, 368
Neumann, C., i86
Neumann, F. E., 259, 268
Newlands, 273
Newton, Sir Isaac, ii, i82, i86,
I89, 192, I93, 195, 205, 221, 224
et seq., 230, 231, 266, 320, 335;
on gravitation, 28, 45, xi8, 119,
157, 158, 161, 177 et seq.; on
absolute time, 125; on absolute
space, 179
Newtonian axes, 194 et seq.
Nicholson, 255
Nucleus, 394 et seq., 406 et seq.
Number, 99 et seq., ioi; Notation
for, 107; Negative, o09; Rational,
o09 et seq.; Real, I o et seq.; Discreteness and continuity of, 113;
Complex, Imaginary, 16, 117;
Transfinite, 121
Nussbaum, 405
Oersted, 256
Ohm, 259, 264




508


INDEX


Oken, 386 et seq., 429
Ontology, 52, 452, 469 et seq., 474 et
seq., 479
Optics, 265, 325; Wave theory of,
266
Order, Notion of, 103
Ordinal number, 104
Organic stability, 420
Organism, living, 345 et seq.; Concept of, 349
Ostwald, 219
Ovum, 400, 406 et seq.
Owen, Richard, 388, 389, 434
Palaeontology, 438
Paley, Archdeacon, 493
Pangenesis, 400 et seq.
Pantheism, 480 et seq.
Parallelism, Law of, 386 et seq., 391,
392
Parallelogram of forces, 192
Parallels, Theory of, I33, 134
Parasites, 397
Partition of energy, 241
Pasteur, Louis, 398
Pearson, K., 36, 4I8, 419
Percepts, 29
Percipients, 37, 38
Peregrinus, 244
Perrin, 238
Pfluger, 398
Phenomenalism, 62, 453
Philosophy and Natural Science, I5
et seq., 63, 64, 452 et seq.; Idealistic, 56; Realistic, 57 et seq.; Pluralistic, 6
Philosophy of Nature, 86, 386
Physiological Psychology, 71, 357,
367
Physiology, 40, 66, 71, 348, 455
Piazzi, 295
Planck, 240, 241
Plant, Definition of a, 345
Plato, 152, 202, 425; on reality, 6I
Plucker, 262
Pluralism, 61
Plurality, Notion of, 102; Discernment of, 103
Poincar6,H., 28, 219, 241, 307 et seq.;
on Mathematical theories, 57; on
facts, 65; on Geometry, 147; on
inertia, I88; on force, I9I
Poisson, 252, 253, 268
Poncelet, I40, 211
Ponchet, 397
Ponderomotive forces, 257


Postulations, of Science, 59; Hypothetical, 31
Pragmatism, 48
Preformationism, 401
Priestley, 207, 249
Primordial states, 315
Proclus, 134
Protoplasm, 358, 398, 405
Prout, 272, 273, 287
Psychical order, 66; Relations with
physical, 68 et seq., 96, 97; Relations with Biology, 73; Measurements, 370
Psychological concepts, 351 et seq.
Psychology, 53, 66, 71 et seq.,
444, 455; Methods of, 352
Psycho-physical parallelism, 67 et seq.,
96, 355, 443
Psycho-physics, 357, 367
Ptolemy, 14, I34
Pure Mathematics, 48 et seq.
Pythagoras, IIo
Quantum, 240, 343
Radiation, 241; Solar, 299
Radio-activity, 277 et seq., 288
Radium, 278, 283
Ramsey, Sir W., 279
Rankine, 236
Rashdall, 483
Ray, John, 380
Ray Lankester, Sir, 398, 403
Rayleigh, Lord, 279
Read, Carveth, on Magic, 8
Realism, 475 et seq.
Reality, 60, 6i
Reaumur, 382
Refraction, 266; Conical, 267
Regulae Philosophandi, Newton's,
178
Relativity, Einstein's theory of, 316
et seq.; Special theory of, 327 et
seq.; Generalprinciple of, 355 etseq.
Religion, 462 et seq.
Rey, Jean, 204
Ricci, 338
Ridi, 397
Riemann, Bernhard, 135, I40, 143,
144, 149, 222, 258, 325, 338, 342,
343
Robinet, 429
Roche, 304, 309
Roemer, 266
Roentgen rays, 217
Roget, 263




INDEX


509


Roozeboom, 240
Rotation, Absolute, 195
Routh, E. J., 228
Roux, 414, 445, 446
Royce, Josiah, on transfinite numbers,
121
Rumford, 2II, 234, 235
Russell, E. S., 442
Russell, H. N., 312, 313
Rutherford, Sir E., 278, 28x et seq.,
285 et seq.
Saccheri, 134
Sachs, 396
St Augustine, 426
Saint-Hilaire, Geoffroy, 385, 386,
388, 429, 434, 435
Saturn's rings, 297
Savart, 257
Savary, 306
Scale of beings, 382, 392
Scheele, 207
Schleiden, 393 et seq.
Scholasticism, I I
Schwann, Theodor, 393 et seq.
Science, Denotation of term, 2;
Natural, 2 et seq.; Tension produced by, I2 et seq.; as World
Philosophy, 21, 22; of physical percepts, 25; as descriptive schemes, 36;
as public knowledge, 37; and the
typical, 29
Scientific theories, Quantitative, 40;
Limitations of, 41, 42, 70 et seq.;
Truth and falsehood of, 47
Secchi, i66, 310
Sedgwick, A., 396
Sdguin, 211
Sense-impressions, 23
Sentiment interieur, 432 et seq.
Sequences, Convergent, III, I 12
Serres, 391
Sexual selection, 444
Slipher, 312
Sociology, 455
Soddy, 278, 28i et seq., 289
Solar system, 293 et seq.
Solipsism, 473 et seq.
Sommerfeld, 287
Sorley, W. R., 465, 496
Space, Intuitional, 123; Geometrical,
126 et seq.; Private, 127, 320 et seq.;
Physical, 128, 320 et seq.; Curvature of, 144; constant, I44; Kant's
view of, 149; Absolute and relative, 179, I8o


Specific energies, Doctrine of, 368
Specific inductive capacity, 250, 262
Spencer, Herbert, 402, 435, 445
Sperm-cell, 406
Spinoza, 427
Stahl, 380
Statics, 224
Statistical view of scientific theories,
353; analysis, 459 et seq.
Stefan, 258
Stevinus, 216
Stewart, Balfour, 159
Stokes, Sir G. G., 267
Streintz, I86
Strutt, R. J. (Lord Rayleigh), 282
Substance, 55; Primary and secondary qualities of, 56, 57
Sufficient reason, Principle of, 43
Tait, P. G., 159, 228
Teleology, 85, 377, 425, 440, 493 et
seq.
Tensors, Calculus of, 323 et seq.
Theism, 466, 476, 478 et seq.; Proofs
of, 485 et seq.
Themistius, 183
Theory of Evolution, 15, 425 et seq.
Thermal equilibrium, 241
Thermodynamics, 232, 234 et seq.;
Laws of, 237, 363 et seq.
Thomson, J. Arthur, 394
Thomson, Sir J. J., 276, 284, 285
Tidal action, 302, 304
Tiedmann, 387, 429
Time, Intuitional, 123; Private, 321
et seq.; Public, 124, 32i et seq.;
Measurement of, 124; Abstract,
I25, I26; Spatialization of, 126,324
Titius, 294
Transmutation of substances, 271,
28I, 282; of species, 427, 435, 437
et seq.
Treviranus, 429
Tropisms, 366
Tycho, 378
Tyndall, 398; on primacy of matter,
20
Unity, Concept of, o10
Values, 462 et seq., 485; Conservation
of, 463
Van Beneden, 406
Van der Waals, 240
Van Musschenbroek, 247
Van t'Hoff, 240




510


INDEX


Variation, 437; Inheritance Of, 441,
449; Discontinuous, 449 et seq.
Vegetative life, 383
Vertebrate skeleton, 388 et seq.
Verworn, Max, 380
Vinci, Leonardo da, 21,6
Vital affinities, 412
Vitalism, 357, 380, 431
Volta, 254, 255
Von Baer, 389, 391, 392, 394, 395,
434
Von Haller, 367, 378, 401
Von Kdlliker, 396
Von Nilgeli, 398
Wagner, Moritz, 443
Wallace, Alfred Russel, 436, 437,
443, 445
Wallis, John, 134
Ward, James, 72, 103
Watson, William, 247
Watt, 21I0
Weber, E. H., 370


W~eber, W. E., 217, 258
Weierstrass, 110o
Weismann, 402 et seq., 407 et seq.,
412, 4i6, 435, 446 et seq.
WeYl, 327
Whewell, W., 230
Whitehead, A. N., 58, 59
Wilberforce, Bishop, I 5
Wilcke, 249
Wireless telegraphy, 269
Witchcraft, i i
Wolff, 40I
Woolaston, 255
World, 324, 332
World-line, 334
World Philosophy, 17, 21, 51, 454,
465, 499
Wundt, 370
Young, Thomas, 266, 368, 369
Zeeman, P., 276
Zimmermann, 275


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