LIBRARY

UNIVERSITY OF CALIFORNIA,

Class

THE AIM AND ACHIEVEMENTS

OF

SCIENTIFIC METHOD:

AN EPISTEMOLOGICAL ESSAY.

BY

T. PERCY NUNN, M.A., D.Sc.

VICE-PRINCIPAL OF THE LONDON DAY TRAINING COLLEGE.

MACMILLAN AND CO., LIMITED,
1907.

All Rights Reserved,

GENERAL

LONDON !

HARRISON AND SONS, ST. MARTIN'S LANE, W.C.,
PRINTERS IN ORDINARY TO HIS MAJESTY.

PREFACE.

THE following essay is an expansion of a paper read before
the Aristotelian Society in February, 1906. I have to acknow-
ledge the kindness of the Committee, which has permitted me
not only to embody here most of my original matter, but also
to make use of the type in which it was standing. The work
in its present form (with the exception of one or two trifling
emendations) was printed in September, 1906, and presented
to the University of London as a thesis for the degree of Doctor
of Science.

The results described in the essay were reached in the
course of a study of the problems of Science teaching in schools,
and are believed to have very definite pedagogical applications.
These applications are not considered in the present volume,
but, in view of the paucity of writings in English which treat
of the pedagogy of Science upon a philosophical basis, I have
thought it legitimate to direct the attention of the interested
reader to the places where I have discussed some of my topics
from the professional point of view. For the same reason I
venture to refer to the chapter on Science Teaching in Professor
J. W. Adamson's book on The Practice of Instruction.

The customary acknowledgements that have been made
whenever I have appealed consciously to the work of other
writers should be supplemented in two particulars. Even my
numerous footnote references to the Principles of Mathematics
hardly suggest the full amount of the inspiration which I have
drawn from the work of the Hon. Bertrand Eussell. I am

18(5759

iv PREFACE.

conscious of the influence of his mode of thought throughout
the book. The fact that I have mentioned Professor Karl
Pearson only in connexion with expressions of dissent from his
views makes me wish the more to own that the day on which
I looked into The Grammar of Science was the day when I first
surveyed as from a " peak in Darien " the attractive waters
which I have sought a little further to explore.

May, 1907.

NTENTS.

CHAPTEE I.

§ 1. Introduction. The term " scientific " has a general and a
special application. The determination of the latter is
aimed at in Chapters I and II 1

§ 2. Science is a conative process directed towards the Objective.
The Objective includes those objects of thought which
must be considered " prior " to our thinking. Sameness
for all and relevance to purpose are secondary marks of
them 4

§ 3. The Objective contains physical exis tents, psychical
existents and subsistents. Comparison with Meinong's
Objekte and Objective ... ... ... ... ... ... 7

§ 4. The " plain man's " view of the physical world. How far

are its elements Objective ? 10

§ 5. The secondary qualities are equally Objective with the
primary qualities ; but the recognition of this fact
renders necessary a rectification of the common view oO
the Thing ... .,... ... 11

§ 6. Nevertheless the Thing is Objective, being a whole com-
posed of Objective qualities in an Objective relational
nexus ... —^T 15

§ 7. There is no question of the Objectivity of psychical

existents ... ... ... ... ... ... ... 20'

§ 8. We turn therefore to the consideration of the fact that the
elements of the Objective occur in series — sometimes
discrete, sometimes compact, sometimes continuous ... 21

§ 9. The importance of the number series is due to the ease
with which its terms can be correlated with those of
other series ; but when the correlation is to be with a
compact series the original discrete number series must
be supplemented ... ... ... ... ... ... 24

§ 10. These supplemental numbers (vulgar or decimal fractions)
were invented in connection with the problem of measure-
ment— a problem which we must analyse briefly ... ... 26

§11. The spatial and temporal series present special character-
istics : the former, for example, is tri-dimensional ... 29

A 2

VI CONTENTS.

PAGE

§ 12. Both are continuous series, The compact series of rational
numbers is inadequate for correlation with their terms.
A new mode of correlation must be adopted which
involves the irrational numbers ... ... ... ... 30

§ 13. A brief inquiry into the question of the Objectivity of

Space and Time 35

CHAPTER II.

§ 14. The Objective, in actual thinking, enters into a secondary
synthesis whose object is to render the primary or
Objective data intelligible. This characteristic appears
in other conative processes besides Science 46

§ 15. For example, in the magic of savage races(; the differentia
being the nature of the secondary construction by which
the primary facts are apperceived ... ..." ... ... 48

§ 16. The same holds good of much of the so-called " science " of

the Greeks ' , •• 51

§ 17. And of the earlier modern philosophers ... ... ... 53

§ 18. The writings of Keppler shew an interesting transition
from non-scientific to scientific conations dealing with the
same primary facts. The characteristic of the secondary
constructions of Science is that they aim at making the
Objective intelligible for its own sake ... ... ... 54

§19. Many individuals before the "age of science" must have
turned to the Objective in the scientific spirit, notably
Leonardo da Vinci 59

CHAPTER III.

§ 20. Objective facts may be made intelligible by the aid of con-
cepts drawn from any context of experience, provided
that the foregoing test of the scientific character of the
secondary construction is satisfied ... ... ... ... 62

§ 21. Thus different scientific investigators may invoke, one the
concept of purpose, the other the concept of physical law
to interpret the behaviour of living organisms ... ... 64

§ 22. Bunge and Jennings defend their respective methods upon

general grounds ... ... ... ... ... ... 66

§ 23. But while there is, ultimately, a question of fact involved,
at the present stage the divergency of method really
illustrates the freedom of choice of the investigator for
which we contend 67

§ 24. The discussion leads us to regard " causal " and " final "

explanations from a new point of view 68

CONTENTS. vii

PAGE

§ 25. Hypotheses are to be defended against Ostwald, who
confuses their real with their psychological value.
Hypotheses are indispensable means of reaching "a com-
plete synoptic inventory " of the Objective 71

§ 26. This function is well illustrated by the history of the atom.
Bacon recognised that the use of this concept was an
attempt to render some facts intelligible by means of
others ... ... ... ... ... ... ... ... 74

§ 27. But Gassendi actually reintroduced the atom into modern

philosophy 75

§ 28. Kobert Boyle applied it in a genuinely scientific treatment

of chemical facts 76

§ 29. Newton indicated the manner in which the concept might
be employed throughout the whole range ol physical
science... ... ... ... ... ... ... ... 78

§ 30. It was from Newton that John Dalton drew his inspira-
tion. He applied the notion of atoms of varying weight
first to the elucidation of a physical problem, and only
subsequently to the facts of chemical combination ... 78

^31. The atom as used by Boyle and Dalton led to the precise
definition of two of the most important problems of
chemistry. With this its functions had, so far, been
completed, and its further use was,- strictly, unnecessary 81

§ 32. Objective facts are not fully determined until they have
been correlated with terms of the number series. The
attempt to do this in some cases (e.g., the hotness of
bodies) leads to a new concept of the "objective condi-
tion " of the body .-„. ... 83

§ 33. This condition is not an Objective fact (in our sense), but
has " pragmatic " value only. The consideration of the
problems involved here leads to the completion of the
theory of the Thing begun in Chapter T, § 6 85

§ 34. The early development of the science of Heat illustrates
the mode in which a hypothesis leads to the discovery of
a mathematical formula which may be regarded as an
expression of the fact that the elements concerned "take
note " of one another's condition 90

$ 35. Failure to find such a formula leads to the discovery that
other elements are involved in the scheme of relations
under consideration 91

§ 36. The elements of the Objective are brought by this process
into some form of unity. This form may, in certain
cases, simulate the Objective character of the Thing ... 92

§ 37. Moreover, there appears to be no more than a practical
difficulty in the way of an inclusion of psychical elements
in these forms of unity :. 93

Vlll CONTENTS.

CHAPTER IV.

PAGE

§ 38. Much of modern science consists of attempts to " explain "
the whole of the facts of the physical world by reducing
them to particular cases of the behaviour of matter in
motion '. 94

§ 39. Thus the facts which Black succeeded in rendering intelligible
up to a certain point by means of the special concept of
heat 95

§ 40. Eventually proved intelligible only when taken in relation

with facts of a mechanical order 96

§ 41. The development of the main ideas of Mechanics must,
therefore, be discussed from the critical standpoint. The
earliest stage may be taken to be a process of " objectifi-
cation"of the force which alters the shape- or size of a
body, and the recognition that it can be measured in terms
of weight 96

§ 42. When the case of moving bodies is reached another concept
of force is necessary. We begin by elaborating the notions
of mass and momentum as descriptive apparatus which can
be successfully applied to the analysis of the phenomena
of collision 97

§ 43. Using these notions, all the phenomena of impact may be
summarised in the formula known as the Principle of the
Conservation of Momentum. It remains to show that this
formula holds good in all cases when a force is thought of
as acting between two moving bodies. Thus we reach the
" dynamical measure of force " 99

§ 44. Newton first showed that the resultant force on a particle
(measured as rate of change of momentum) could be
regarded as the vector- sum of forces which represent the
u notice " which it is taking of the behaviour of all other
particles in the universe 102

$ 45. Since the velocity of a particle is not properly a state of
the particle, it follows, that this "notice" refers to any
two (and, therefore, to all) previous configurations of the
particles of the universe 103

§ 46. That this is not an accidental result depending upon the
peculiar character of the Newtonian descriptive apparatus
is seen from the fact that it reappears in a system of
mechanics so fundamentally different as that of Hertz ... 104

§ 47. Moreover, it can be shown to characterise other provinces
of fact besides the mechanical. It appears to be the con-
dition that changes of one order may be regarded as
"notice " of changes of another order 105

CONTENTS. IX

PAGE

§ 48. So far we have followed a more or less hypothetical develop-
ment of mechanics, because we do not know precisely the
route by which Newton reached his concept of force.
With the concept of vis viva it is otherwise 107

§ 49. Huygens openly derives it from the "instinctive perception"
that the centre of gravity of a system left to itself cannot
rise. This was developed into the Principle of Vis Viva
by the Bernoullis 108

§ 50. Upon this principle Helmholtz based his wider Principle of
the Conservation of Energy which first brought the facts
of Heat into a line with those of Mechanics 109

§ 51. But this doctrine, though it connects temperature changes
with mechanical facts, does not effect a reduction of the
former to the latter 110

§ 52. Nor is this reduction effected by Lord Kelvin's absolute

therm ody namic scale ... ... ... ... ... ... Ill

§ 53. For the latter makes no attempt to deduce from dynamical
data the experiences to which the notion of temperature
refers 112

§ 54. Even the theory of Helmholtz (which is the most complete
of such attempts) only achieves very partial success in this
direction 114

§ 55. Meanwhile dynamical formulae may be extended to include
primary facts in different physical sciences, without any
attempt at a mechanical reduction of these facts 117

§ 56. It is doubtful, however, whether temperature changes can

properly be dealt with in this way 120

§ 57. Thus the attempted reduction has not, up to the present,
been effected. The modern science of Energetics contents
, itself with attempting to bring physical facts into a form
of unity without reducing them to one type. The
Principle of the Thermodynamic Potential plays here the
part which 'the Principle of Virtual Velocities plays in
dynamics 120

§ 58. A summary of the results of Chapters III and IV ... ... 123

CHAPTER V.

§ 59. The question of the actual achievements of Science now
arises. How far is the pre-critical view of the real
validity of hypotheses to be accepted ? 126

>5 60. There are three distinct forms of hypotheses. The first class
includes attempts to supplement the data so as to produce
a complete spatio-temporal chain 128

§ 61. An important sub-class contains hypotheses of an " ejective "

type ... 129

X CONTENTS.

PAGE

$ 62. In the second class the proposed additions to the data

consist of relations 130

§ 63. The third class contains the typical hypotheses of Science,
such as those which have been considered in detail in this
essay ... ' 130

8 64. Hypotheses of the first and second classes undoubtedly
claim validity, since they claim recognition as verifiable
Objective facts. The hypotheses of the third class are not
verifiable and ultimately disappear 131

$ 65. Opposed to the " realistic " view of Science are the various
" descriptive " views which are related to Humanism.
Foincare admits the Objectivity of relations alone ... 133

§ 66. Le Roy appears to deny even this amount of Objectivity.
The whole body of scientific facts and laws has pragmatic
value only 134

§ 67. Mach's well-known principle of " economy " claims to
explain the genesis of much that we have claimed as
primary Objective fact. He regards Science as a process
which is continuous with the process by which the "plain
man's :' world was generated. In this he is in agreement
with the Humanists 135

§ 68. On the contrary, we must maintain that there is a complete
break between judgments which deal with primary facts
alone and " scientific " judgments 139

$ 69. A summary of the whole work. The Aim of Science is to
render the Objective intelligible : its Achievements are
the further Objective facts and determinations which its
hypotheses bring to light ... 142

*V OF THE

UNIVERSITY

OF

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC

METHOD.

CHAPTEE I.

§1- i

THE words " Science " and " Philosophy " seem not unhappy
examples of a distinct class of terms — terms that are
recognised everywhere as indispensable and yet can find
few who will accept the responsibility of giving them pre-
cise and exhaustive definitions. The phrase " Science and
Philosophy" is felt probably by most educated persons to
express a certain antithesis; but the vague and unanalysed
character of this antithesis is shown (for example) by the
" oddly alternative way " in which the two words are used in
such names as " Mental Philosophy " and " Mental Science."*

The causes of this confusion are mainly historical. The
modern spirit of inquiry in its first youth addressed itself
naively and indifferently to all the problems which the world
presented to it, and only learnt gradually to distinguish
essential differences of character between the various questions
which it raised and to elaborate distinct methods of dealing
with them. In England, in particular, where speculation has
rarely wandered far from the needs of practice, the antithesis
alluded to above was, until some years into the nineteenth
century, recognised only confusedly and imperfectly in the
distinction between " moral " and " natural philosophy." The

* Sidgwick, Philosophy : its Scope and Relations, London, 1902, p. 2.

B

2 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

comparative independence and isolation in which the great
English scientists of the eighteenth century worked tended to
keep them to the universal point of view which seems always to
have been an important part of the content of the notion of
*' philosophy," and hindered the division of labour which ends
in the development of the well differentiated lines of inquiry
and interpretation to which we apply nowadays the plural,
" sciences." Thus we may say (in the opinion of Mr. Merz)*
that the word " Science " did not reach its present meaning
among us until about the time ef the foundation of the British
Association (1831) and that its displacement of the older
" philosophy " was due partly to the fact that this term was
beginning to bear in England the special significance which it
had gained through the work of the great German meta-
physicians ; and partly to the dominant influence of France in
all the " sciences." In France the co-operation of investigators
under the patronage and with the assistance of the State had led
to just those results for which England during the eighteenth
century lacked the conditions, and the " Academie des
Sciences " had, practically since 1666, used the \^ord in its
present familiar sense. We may conclude, perhaps, that only
since about 1830 has European thought been fully, conscious
\of the existence of the antithesis to which the opening
paragraph drew attention.

On^account, doubtless, of this imperfect apprehension of
the distinction between Science and Philosophy the adjectives
" scientific " and " philosophical " have followed in the develop-
ment of their application courses different from those of the
corresponding nouns. Thus, while there can be no doubt that
the word " Science " refers most commonly to a certain body of
knowledge, " scientific " refers more frequently to the method or
procedure by which this knowledge has been established. By

* History of European Thought in the Nineteenth Century, 1904, i,.
p. 89.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 3

i an easy extension of its meaning the adjective is applied to all
processes by which knowledge is reached having the security
. that is thought to characterise " Science." Hence " scientific i1
comes to be wider in its application than " Science " — to have)
in fact much the same range and implication as the German
"Wiflsenschaft" which, as Merz points out,*' has acquired
a moral as well as an intellectual significance, and implies an
ideal embracing at once the highest aims of the " exact " (i.e.
" scientific " in the narrower sense precisely correlative to
" Science "), the historical, and the philosophical lines of thought.
^7 To distinguish it from this use of " scientific " as implying an
ideal of exact knowledge of things as they really are, free from
conjecture and subjective distortion, we may perhaps assert
that the typical use of " philosophical " is to imply a certain
breadth of knowledge and of view, an ability to deal with new
problems by the help of considerations drawn from a large-
mass of well-ordered experience and instructed thought.
Thus, while, according to modern usage, - the1 substantives
" Science " and " Philosophy " undoubtedly refer to different
provinces on the map of intellectual effort, however difficult
it may be to delimitate their frontiers, the corresponding
adjectives are, in general, not so restricted in their application,
but are commonly used to characterise aspects of intellectual
effort wherever it is exercised. Nevertheless they have special
senses in which they correspond precisely to their nouns, and
it should be understood that it is in this narrower sense
precisely correlative to the noun " Science " that the term
" scientific " is used in these pages— a sense which it will be
necessary to determine with some care. The first and second
of the following chapters will be devoted to this task of
determining the aims of Science, while in the remaining
chapters a critical estimate of the achievements of Science will
be attempted.

* Op. tit., i, p. 223. See also index s.v. " Wissenschaft."

B 2

THE AIM 'AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

Recent psychology, recent logic, and recent speculation are
at one in laying stress upon the solidarity between man's
" theoretical " and " practical " activities. Without the impli-
cation of acceptance or rejection of the metaphysical con-
tentions of " Pragmatism " we may usefully fall in with the
/ prevailing fashion in Thought so far as to replace the current
/ static conception of Science as a body of truths by a dynamic
(^conception of it as a definite pursuit. Such a conception of it
is adopted in this essay. Science is here conceived as a definite
secular conative process which may be distinguished in and
traced through the conscious life of civilisation. Only when a
scientific " result " is thus considered in connection with the
whole conscious process of which it is the " end " can we
hope (as Mach taught us lon^ago) to submit it to profitable
criticism^/ Since some such criticism is aimed at in this
paper, it follows that either an attempt must be made to
characterise that process or some current characterisation must
be adopted as satisfactory. As I do not know one which
I can accept as altogether suitable for my purpose, the
former alternative must be embraced.

The statement that the conative process with which Science
is identified reaches its end only in the enunciation of
judgments of a certain class will probably be received without
demur. Nor, if I say that these judgments refer to the
Objective in experience will it be complained that I am
ungenerously narrowing their field. The whole " furniture of
earth and choir of heaven," " the starry heavens without an
the moral law within " are but items in the inventory of the
Objective. At the same time, although the Objective is here
conceived as containing much more than " physical nature,"
it has its limits, and does not include everything that (in
Mr. Bradley's phrase) can be set over against the self, and so
become an "object" of attention; not everything that (because it

TflE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 5

can be thought of) is maintained by Mr. Moore and Mr. liussell
to have Being. A round square and Colonel' Newcome are
examples of objects of thought which are to be excluded
from the Objective. We can, it would seem, characterise them
by saying that these objects lack a certain priority to and
independence of our thinking which is the necessary mark and
guarantee of undoubted items of the Objective. It may be
difficult to refute the argument that Colonel Newcome must have
had being before 1854 or Thackeray could not have thought
of him, but it will hardly be maintained that Thackeray
discovered him "in just the same sense in which Columbus
discovered the West Indies." At most it could be claimed (pre-
sumably) that " the elements so mixed in him " subsisted
before Thackeray, by selecting them, brought them into a special
relation in which they were not related prior to the act, and so
" created " the Colonel.

The same distinction might have been indicated by saying
that the Objective contains everything that must be " reckoned
with," everything that must be considered as a datum for
human action. From this point of view it is clear that
although Colonel Newcome forms no part of the Objective,
Thackeray's conception of him does, being a thing that has
influenced human action on a comparatively large scale, and
being precisely the conception it was and no other in virtue
of its particular content, which must be distinguished from
its object. Similarly, the Objectivity of my thought of a
round square is demonstrated if it moves me to mirth or
becomes so irresistibly attractive as to make me a " case " for
the alienist.

But I am not disposed — at least without a struggle —
to accept the position that this relevance to action is
the essence of the Objective, and not merely a property
of it coordinate with others. The present essay is in a
sense a contribution to the discussion of this burning
question, so that I will make here only one suggestion — a

-6 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

suggestion that has probably occurred to many. If to be
Objective means to have a relevance to purpose and action,
how is it that we recognise material things and thoughts
as having that relevance before the course of events has
revealed it? Why do I ascribe Objectivity to the hundred
thalers on the table, and deny it to the content of my thought
of a hundred thalers before I have attempted to spend or even
to touch them ? The reply that it is because the former are
like in all respects to thalers which have been things " to
reckon with" — or upon, while the latter are not, is plausible
in this special case, but does not seem sufficient to meet the
general case of the recognition of Objective existents before
experience of their relevance to action. Surely, we may retort,
the perception of this similarity which is the signal of
subsequent relevance to action is the direct and simple
perception of the presence of Objectivity as such — a property
which as a matter of fact is accompanied by the property of
relevance to action.

In a somewhat similar way we may meet the contention
that the Objective is that which is " the same for all." Upon
this view the " finite centres " in which all experience occurs,
find it at once possible, and necessary for the development of
intercourse with other centres, to " pool " a large part of their
experience, and this common matter becomes the objective
world, exterior to all and the same for all. Some writers
(e.g., M. Poincare') attribute a very great importance in this
connection to language, which they seem to regard as actually
the means by which a " same for all " comes to- exist, and not
merely the means of our becoming aware that it does exist.
Here we may repeat our objection that the Objective is known
directly as such prior to the discovery that it is the same
for all. We may add in this case the further objection that
in the inventory of the Objective we include net only unique
experiences in the world of physical existents — such as
astronomical observations incapable of repetition — but also the

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 7

whole world of psychical existents, whose very nature it is
to appear in the " panorama " surrounding a single centre
only. Not a single feature of such facts can be excluded
from the operations of Science, yet how can they be regarded
as "the same for all"? Only, I submit, by a kind of extra-
polation from that part of the Objective — " physical reality " —
which, as a matter of fact, is the same for all. We must say,
that is, of such experiences, not that being the same for all
they become Objective, but that being Objective (simpliciter)
they are regarded as the same for all, and therefore, part of
the proper subject matter of Science. Being experiences
whose content announces itself as independent of the self
of the moment over against which they are set, they may be
thought of as occurring with an identical character in any
centre. They become, that is, objects whose features, like
those of " material objects " are capable of exact determination
without reference to their presentation at all — although, of
course, their position as a class of existents is fixed by their
peculiar relation to the " finite centre " in which they occur.

§ 3.

The actual contents of the Objective must be reviewed very
briefly. " Everyone except a philosopher," says Mr. Eussell,*
" can see the difference between a post and my idea of a post."
I ignore this uncomplimentary reservation and assume that
we are all prepared to admit not only that they are existents
of different orders, but also that both have (like all existents)
the character of Objectivity — the post, because it would be the
particular thing which it is, even if I did not happen to see it ;
the idea because it would be an idea with just that particular
content, even if I did not happen to perceive that I had " had "
it. Difficulty only arises in the absence of the plain guarantee
of "priority" which the perception of existence itself gives.
In this case, to quote Mr. Eussell again. " there exists

* Russell, Principles of Mathematics, i, p. 451.

8 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

everywhere the greatest confusion " — confusion that can only
be removed by the frank recognition of another type of
Objectivity which we may call Objective subsistence. If we
think of the number 100, or of TT, or of the tangent to an
ellipse it must be recognised that the object of our thought has
a priority to our thinking, that entitles it to be called Objective
in the same sense as existents must be called Objective. Such
objects of thought present themselves as features of experience
which must be "reckoned with," and are not subject to our
caprice. They* may not be obvious to untrained inspection
any more than the finer details of a microscopic section are,
but when once envisaged by the competent mental eye they are
observed to have their peculiar features as a matter of fact,
quite apart from the observation. In Mr. Kussell's forcible
phrase such " subsistents " must be " discovered in just the
same sense in which Columbus discovered the West Indies " : —
they are Objective subsistents.

Attention should, perhaps, be called to the fact that the
foregoing use of the term Objective differs from the technical
use of the same term recently introduced into Philosophy by
Meinong and his school. (Meinong's works, Ueber Annahmen,
1902, and Untersuchungen zur Gegenstandstheorie und Psycho-
logic, 1904, have been the subject of reviews and discussions by
Mr. Kussell, in Mind, N.S., Nos. 50, 51, 52 and 56). In the
complete " Theory 'of Objects " an object (Gegenstand, " an
object of discourse ") is either an Objekt or an Objektiv. Thus
(to borrow one of Mr. Kussell's examples) if I pronounce the
judgment " There was no disturbance," although I deny the
existence of a certain Objekt — a disturbance at a particular
time and place — I yet assert something positive, namely, the
fact that there was no disturbance. This fact is the Objektiv of
the judgment. So, if I assert that " A is the father of B," my
judgment concerns the JJbjekte, A, B, and . the relationship
between them, while the Objektiv of the judgment is, once
more, the fact of relationship asserted.

ilVERSlTY

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 9

It is clear from this description that an Objcktiv can never
exist: it can at best have subsistence. On the other hand,
Objekte may either exist, or have subsistence only, according to
circumstances. The question whether all non-existent objects
have subsistence is one of great difficulty as well as of great
importance, but does not concern our present inquiry. It only
arises with the consideration of the standing of such objects as
the Objektiv of a false judgment, or an impossible Objekt such
as the much discussed round square. Meinong denies the
subsistence of such objects outside the psychical process whose
content they form : (Mr. Russell, on the other hand, asserts the
subsistence of all possible objects, but defends himself against
the necessity of admitting the subsistence of impossible
objects by a theory of denoting which robs them of the
character of objectsj By this later development of his doctrine
Mr. Russell appears to be able to bring himself into line with
common-sense, freeing himself from the necessity of admitting
the Being of imaginary entities like Colonel Newcome, or
impossibilities like the round square.* Thus he leaves no
Objekte except those which I have already described as pre-
senting that priority to our thinking which the term Objective
(as I have used it) is intended to connote. But Mr. Russell
finds himself obliged to maintain the subsistence of all
Objektive, whether of true judgments or of falsef — even, ap-
parently, in the cases where the " subject " of the proposition,
in accordance with the theory of denoting, is not an Objekt.
Thus, although the round square is not an Objekt, " das
Nichtsein des runden Viereckes " is still an Objektiv^ and as-
such has subsistence. But subsistence must be granted also to
" das Sein des runden Viereckes " ; the only distinction which
it is possible to draw between this Objektiv and the former
being the (so to speak) external distinction of truth from

^ * Mind, No. 56, p. 491.

t Mind, No. 52, pp. 521 et seq. ; No. 56, p. 532.
| Ameseder, in Oegenstandstheorie, p. 55.

10 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

falsity. It hardly needs remark that Science deals with
Objective as well as with Objckte — not only, for example, with
zinc, sulphuric acid, and hydrogen, but also with the fact that
zinc displaces hydrogen from sulphuric aci<y But it will be
recognised just as readily that it concerns itself only with
Objektive which are believed to be true. Finally, then, the
Objective, as here conceived, includes (1) all Otyekte (according
to Mr. Eussell), or all that have existence or subsistence
(according to Meinong) ; and (2) all true Oljcktive.

§4.

In the last section only the broadest outline was given of
the classification of its objects which is implicitly made by
common thought. But the Objective contains — and Science
accepts at the outset of its task — much more than these very
general distinctions. The most important of the details —
which may be said to constitute the " plain man's " view of the,
world — are implied in the common structure of languages.
This structure was elaborated doubtless in order to meet the
exigencies of our transactions with the world of physical
existence ; primarily, therefore, it affords information about the
view which mankind has universally taken of that order of
rreality. Briefly,* that view is that the physical world is a
collection of Things, possessed of Qualities, occupying cor-
related positions in the great " continuous receptacles " of Space
and Time, and capable of Activities, in the course of which the
things may " change " in respect of their qualities, their mode
of occupation of space or their spatial relations to other
things, and may "cause" similar .changes in other things.
The critical examination of the notions indicated — notions
which everywhere permeate ordinary expressions of experience
— is part of the special business of Philosophy,f and cannot be

* Sigwart, Logic, I, ch. i. Cf. Sweet, History of Language, 1900,
pp. 47 et seq.

t See, e.g., Hodgson, Proc. Arist. Soc., N.S., IV, p. 2.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 11

pursued here farther than is necessary to make it clear to what
precise extent they are, in my view, to be accepted as having
Objective reference.

§5.

It will be most profitable to pay attention first to the
•qualities of things, simply because subsequent criticism has
tampered here so constantly and for so long a time with the
unsophisticated deliverances of the " plain man's " conscious-
ness that it is not easy to ascertain precisely what those
deliverances are. This criticism has not merely erected a
distinction between the " primary " and " secondary " qualities
of things but has gone on to regard it as evident to all who
take the trouble to avoid " a natural fallacy of ordinary
thinking " that the former alone really belong to things, while
ithe latter are, strictly, qualities of the sensations which things
produce in us. In opposition to this I wish to maintain that
the view actually held by the enlightened plain man is sub-
stantially the one presented and vindicated by Professor
Stout.* Eedness, sweetness, hotness, &c., belong to the body
that " has " them in the same sense as extension, hardness and
the other primary qualities belong to it. A hot piece of iron
\ remains as hot even if I move away from it, just as it remains
of the same length in spite of the diminution in the space
which' its image covers on my retina. It becomes hotter or
•colder only if it yields a different sensation under the same
conditions of perception. In fact the plain man makes here —
implicitly of course — the same distinction between changes of
sensible appearance which have "representative value"! and
those which have not, as he does in the case of changes which
he interprets as changes either in the " position " or in the
41 form " of the object according as he is able or unable to

* Stout, "Primary and Secondary Qualities," Proc. Arist. Soc., N.S.,
IV, pp. 141 et seq.

t Stout, op. cit., p. 145.

12 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

reinstate the original sensible appearance by voluntary move-
ment.*

It is true that a certain amount of difficulty is presented
by commonly occurring cases in which the plain man's lack of
interest in analysis leads him into apparent inconsistencies.
Thus when it is said of a plate in a certain position that
although it " looks " elliptical it is " really " circular, or of two
coloured materials that though they " appear " not to match
(by gaslight) they " really " do so ; or, again, that the table-
cloth which (in the dusk) " looks " black is " really " red ; the
well-known explanation that the word " really " refers here to-
assumed normal or standard conditions of perception is too
often taken as carrying the further implication that none of
the qualifications are Objective. In the case of the change
of shape this destructive suggestion is successfully resisted by
" everyone except a philosopher." Since the various shapes
are so completely under our control and can be made to melt
continuously into one another, we have no difficulty in
admitting that they all are simply different representations of
the same thing which actually is round, or elliptical, with an
endless succession of eccentricities, according to the way in
which it is looked at. Now the " plain man " will often make
similar statements in the other cases : he will say naively that
the materials which "are" of the same colour at noon "are'*
differently coloured by gaslight ; that the tablecloth which
" is " red in the daytime " is " black in the late twilight, and
this view of the matter is, I believe, the one which the " plain
man " would stand by if he were challenged ; while he would
be willing to admit the conventional character of the former
statements. Further, it is only (I submit) in the light of
special scientific attempts to " explain " these different mani-
festations of the same thing that it becomes regarded as

* PoincarS, Science et Hypothese, p. 76. La Valeur de la Science,
pp. 82 et seq.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 13

" a natural fallacy of ordinary thinking " to credit such
•changes in colour to the things themselves.

But it is possible to confront the view that secondary
qualities exist otherwise than as the contents of our per-
ceptions with greater difficulties than these. It is a familiar
fact that if a cyclist, A, rings his bicycle bell, B, whom he is
approaching, and C, from whom he is separating, hear different
notes — notes which are respectively higher and lower in pitch
than the note which A hears constantly, and all three persons
would hear if he were stationary. Can it be said that the
bell is actually " emitting " these three notes simultaneously
(not to mention the indefinite number of other notes which
might be heard at the same moment by different passers by)
but that only one of them is heard by each of the persons
concerned ? It must be admitted that the " plain man " who\
had found no offence in the thought that the same substance
might be of different colours at different times under different
circumstances would probably be alarmed at finding himself
\ thus committed to the opinion that the same bell was giving
•out to different observers different notes at the same time.

The moment at which this discovery was made would be the
moment at which a philosopher, cynically contemptuous of the
•" plain man's " confidence in his view of the world, would choose
to point out to him that the thing which is hot or heavy to one
man may be cold or light to another and would ask him whether
^ he supposed that the hotness (or heaviness) could exist at the
.same time in the same place as the coldness (or lightness).

To meet these difficulties the " plain man " would have to
become, himself, something not very different from a philosopher,
.and reply that these apparently inconsistent objects of sensation
^ do all exist and, for that matt erK in the same place, just as the
different shapes of the plate all exist in the same place, although
from each single point only one of them can be perceived.
Here an indubitable philosopher might take up the "plain
man's " case. Things which " have " the primary and secondary

14 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

qualities are not isolated reals, each one wearing the qualities it
v^ owns without regard to any other. Eather do they all form
part of one spatial real, " the physical order," which includes
our own bodies. Then it may be said in general that the
qualities perceived to exist in any one part of the real are
correlated with those perceivable in other parts — including that
part (or " body ") with which the perceiving " centre " has a
special connection. Whether the same part of the physical real
will be perceived as occupied by the same quality at the same
time by two " centres of experience " or by the same centre at
different times cannot be decided in any given case by a priori
reasoning. Thus the colours seen in the same surface by
daylight and gaslight are different, while the sounds heard from
the same bell would not perceptibly change with the change
of light. If a bell is sounded near different observers the sound
heard will depend upon the velocities with regard to the air of
the bell and the percipient's body, and it seems probable that it
will depend only upon these velocities, so that if these are
identical in the case of two observers, identical sounds will be
heard. In the case of the hotness or coldness felt in a
particular part of the real (for example, a basin of water), the
quality perceived to exist depends upon other conditions which
involve the percipient's body in what may be called an internal
way. It is clear that this account may be extended to include
such extreme cases as " colour-blindness," or complete absence
of certain forms of sensibility. The all-important point is that
\ve cannot deny to the qualities perceived the Objectivity, the
" priority " to our perception that they claim to have in each
case; scientific "explanations" of the phenomena consisting
largely, I repeat, in bringing to light .other Objective facts
which exist as well as the primary facts from which the
explanations start, and in no sense exist instead of these.*

* I need hardly point out how much I am indebted to Mr. G. E. Moore
in this discussion. Mr. Moore dealt with the subject in a paper read before
the Aristotelian Society in December, 1905 — some months after, under his

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 15

Such an analysis as this undoubtedly demands a modifica-
tion of the ordinary notion of a " thing " in accordance with
which all its qualities occupy the same place. Thus, to revert
to an example taken from Professor Stout's paper, if I move
v. away from the fire I do not suppose that the hotness of the fire
changes. At the same time my judgment that " it is not so hot
here as there " does (I hold) refer to an Objective hotness which
has "priority" to my perception, even if (as is conceivable) I
am the only person who could perceive just that hotness at that
place and time ; it is a hotness which (in Mr. Moore's words)
" exists just as my perception of it exists." We may interpret
the " plain man's " implicit recognition that these differences of
hotness " have no representative value," do not (that is) imply
any changes in the actual " hotness of the fire," not by supposing
that the different perceptions are really different perceptions of
the same hotness, but that they are different hotnesses which
exist at different places around " the fire " — that is around the
place where the other (chiefly primary) qualities of the fire are
to be found — and are correlated with the particular hotness which
characterises " the fire itself." Upon such a view as this
" thing itself " is only the place where some of its qualities exist,
while other qualities of the thing may be found throughout the^
region round about this place.*

§6-

It would be great presumption to hope to do more in a
few pages than indicate quite broadly the manner in which
the " plain man's " view of the Objectivity of the qualities
of things, secondary as well as primary, might be defended

inspiration and that of Mr. Russell, I had reached the views given above.
It is, I suppose, in accordance with precedent that a follower should
prove " plus royaliste que le roi," — a formula which seems to cover many
of my divergencies from Mr. Moore's doctrine.

* This notion is, of course, not new. See Lotze, Metaphysics, ii, p. 34 ;
Ostwald, Vorlesungen uber Natur-philosophie, p. 193.

16 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

against philosophical attacks. The question of the thing
which the " plain man " regards as " possessing " these qualities
must be treated still more briefly and (unfortunately) in a
form that may, on account of its brevity, seem dogmatic.
The difficulties that have prevented philosophers from accepting
the Thing as Objective are well known and need not be
described.* It has not been found possible to conceive

\ anything of the .nature of a " core " which should at once be
without qualities, yet be able to " have " or " support " the
qualities which we know, and to persist through their changes.
, The attempt of materialism or of mechanistic science to account
for the whole body of experience in terms of " matter " and
" force " is so easily shown by metaphysical criticism to be a
failure that.it would not be mentioned here if it were not for
its "special relevance to the aim of my essay. From the point of
view which I have adopted the capital fault of materialism lies,
riot in its belief that the primary qualities — or some of them —
are Objective, but in its denial that the secondary qualities are
so. It seeks to replace the presented facts of experience by
other alleged facts of which the former are explained to be
only appearances.

The hopelessness of this kind of solution of the problem of
Thinghood (which has been known to Philosophy from the time
of Berkeley) has led to a search for a better one in other
directions. Typical of the kind of solution that has commended
itself of late is Lotze's pronouncement that " it is not in virtue
of a substance contained in them that things are ; they are,

i when they are qualified to produce the appearance of there
being a substance in them."f The "core of substance" has
disappeared and we are left with an "empty shell" or (more
soberly) a complex of sensible qualities.

* See Bradley, Appearance and Reality, Ch. VIII ; Lotze, Metaphysics,
i, Bk. I, esp. Ch. VII.
t Metaphysics, i, § 37.

THE AIM AND ACHIEVEMENTS OP SCIENTIFIC METHOD. 17

When, after the destructive process has been carried to this
point, the question how the qualities " hang together " so as to
" present the appearance of there being a substance in them " is
again taken up, the Objectivity of the thing — its priority to out
perception — has, in the view of most philosophers, seemed
untenable. Either the connexion between the qualities is
regarded as sharing the inconsistency and, therefore* the
unreality of other forms of " appearance " ; or it is merely
psychological. Upon the latter view " the concept of perma-
nently existing things" is a great "common-sense achieve-
ment " ;* it is " a thought-symbol for a compound sensation of
relative fixedness " — " the whole operation [being] a mere affair
of economy."')'

The view which is held without suspicion upon the plane
of common sense can be saved upon the plane of metaphysical
reflexion only by the recognition of /the Objectivity that has7
already been claimed for at least some relations. If that'
claim be admitted it becomes possible to regard a whole which'
is a complex of elements in relation to one another as having
Objectivity apart from the fact of the Objectivity of its parts.
\ The " melody " which is heard when a succession of notes is
played upon a musical instrument is such a whole ; it consists
of the notes in definite Objective relations and has to be
" heard " as a presentation distinct from, though based upon,
the presentations of the single notes.J In fact, it is possible

* James, "Humanism and Truth," Mind, N.S., No. 52, p. 461.
t Mach, Science of Mechanics, 2nd Eng. ed., p. 483.
| Ameseder (in Gegenstandstheorie, p. 498) calls this process " Pro-
duhion" See KusselPs review, Mind, N.S., No. 56, p. 537 ; cf. Browning's
lines : —

And I know not if, save in this, such gift be allowed to man,

That out of three sounds he frame, not a fourth sound, but a star.
Consider it well : each tone of our scale in itself is nought ;
It is everywhere in the world — loud, soft, and all is said :
Grive it to me to use ! I mix it with two in my thought :

And, therel Ye have heard and seen : consider and bow the head !

—Abt Vogler, VII.
C

18 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

for a musician to have an " imageless apprehension " even of
a long piej^e in the process of composition, before the separate
notes or chords have been presented at all.* There is little
need to dwell upon the psychological aspects of this process,
which have been treated in detail by Professor Stout.f It
is necessary only to point ,out that by the admission of the
possible Objectivity of a whole as a distinct entity subsisting
as well as its parts and their relations, the Objectivity which
common sense finds in fyg Thing, apart from individual
qualities, can be conceded by the ^ philosopher. The Thing
" has " its qualities just as a melody has its notes. '

This attempt to justify common sense, Jike the former one,
is confronted by difficulties which cannot be overcome without
taking the " plain man " through philosophic by-ways where
he may feel that he has lost touch with the "distinctions that
are plain and few " among which he moved at the beginning
of his journey. Not only may a vast number of objects of
sensational processes have to be thought of as belonging to the
Thing although they lie outside the space within which its
"primary qualities" manifest themselves^ (the space which
we call " the thing itself ") ; not only must the network of
real relations which individualises the Thing present points
of attachment for all the various Objekte which characterise
different " states " of the Thing ;§ there is the greater theoretical
difficulty of deciding, since no part of the presented world
^appears to be independent of any other part, what precise
relational network so isolates a group of presentations as to
constitute them into a Thing.|| It must suffice here to

* The description given by Mozart (quoted by James, Pr. of Psych., \,
p. 255) is well known : " I can see the whole of it at a single glance in my
mind."
' t Stout, Analytic Psychology, i, pp. 95-6.

} P. 15, supra.^r

§ Lotze, Metaphysics, i, p. 77.

j| Of. Bradley, Appearance and Reality, p. 71.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 19

recognise these difficulties and to remark that though the
second and third may make it impossible to apply the concept
of Thing quite unambiguously throughout the field of physical
existence, yet in the cases in which the " plain man " uses
the notion in its primary sense, his confidence that he is
applying it to Objective elements of experience is (upon the
general view here adopted) abundantly justified.

A few words may be added upon the question of the
perception of a Thing as distinguished from the apprehension
of the sensational data upon the basis of which the Thing is
" constructed " at the moment of perception. As described by
psychologists* the perception of a Thing involves an " acquire-
ment of meaning " due to the " complication " of the presented
sensational ttata with other objects of the same kind, not at
the moment presented. In terms of the explanation given
above, the object of the whole psychical state is the complex
whole of these past and present sensational data in their
Objective relations. The recognition of the Thing is, then,
.quite analogous to the recognition of a melody from its
opening bars, or of a poem from its opening words — with the
difference that the whole in the latter cases has Objectivity
only in a secondary sense as the actual content of an artist's
invention. « \

But since the objects of the past sensations and a great
part of the relational nexus, are not actually presented at 'the
moment, " illusions " or misinterpre^itions of the presented
data are, possible. 'This fact does not destroy the Objectivity
of the Thing, any more than the incorrect identification of a
melody from the! opening notes implies want of Objectivity in
the melody. The "object of the whole percept is Objective,
but it is not the object (it turns out) that ought to have, been
presented, but one which, as far as the part immediately
presented is concerned, happens to be not easily distinguish-

* E.g., Stout, Manual of Psychology, 2nd ed., p. 93.

C 2

20 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

able from the latter. This leaves outstanding the difficulty
that two really different objects are not always distinguished ;
a difficulty which, though it has quite rightly been indicated*
as a serious one for Realist philosophies of this type, is one
which in a treatment that aims at little more than indicating
clearly the philosophical basis adopted in this essay, I may
perhaps ask leave to hope is not insuperable — while we
pass on.

' - ' §7. ' -

When we turn from the world of physical existences we
find the^ plain man^\ applying with -less confidence -the distinc-'
tions which are clear and easy throughout at least the greater
part of the material realm. Interest in psychical existences
arose long after language, "under the influence of interest' in the
material environment, had developed its permanent essential
forms. When attention became directed' towards objects of a
fundamentally different character the existing machinery of
expression was utilised, with as little modification as possible,
to deal ;with them. The result is that it is difficult to
determine in the case of " things " which are not undoubted
substantive material unities, how far the use of the term is
simply conventional, implying no more than a possible
"subject of discourse." Thus when the poet says of sleep

that—

" it is a blessed tKing
Beloved from pole to pole,"

it seems evident that the word is used conventionally beyond
the original limits of its application, just as in algebra the
notion of indices, which, in the original definition, must be
integers, is modified to include fractional and negative numbers.
How far the same thing would be regarded by the plain man
as true in the case of psychical existences, thoughts, feelings,
and the like, it would be hard to determine. Since, however,

* See Mind, N.S., No. 58, p. 231.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 21

no question of the Objectivity of these elements of experience
can arise, it is unnecessary to discuss the implications of the
plain man's terminology in this region.

On the other hand we cannot pass from this part of our
subject without noting the fact — of fundamental importance
for Science — that the constituents of all three of the "orders
of the Objective which we have recognised may be^considered
as forming series in respect of many of their various character-
istics.* Thus the pitches of a number of notes, their loudness,
their purity, their "reediness," their squeakiness, the time
during which they are prolonged, the distress they cause to
various listeners are all examples of series. The peculiarity
of a series is that (with the exception of two, the " first "
and the " last ") each term has to at least two others the
relation which is understood when we say that it is between
them. If a term x is between the terms m and n and is also
between p and q it will happen that two of these four
(e.g., m and p) will both have the same asymmetrical relation
to x, while x will have that same relation to n and q. In that
case we -may say that p and m precede x while n arid q follow
it. For example, if the asymmetrical relation in question be
" more squeaky," all notes which are more squeaky than
a given note will precede it ; all notes than which the
given note is more squeaky will follow. By considering each
in turn we can determine the order of the series, the " first "
term being that which precedes every other but does not
follow any, the " last " term being that which does not precede
any but follows all. It may happen that one or both of these
special terms or ends cannot be found. In the former case
the series must be infinite and in the latter it must be either

* The terminology and the main ideas of this section are borrowed
from Mr. Russell's admirable and profound treatment ; op.cit., Parts III,
IV, V, VI.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

infinite or closed — like the series of palings round a field.
In addition, it may happen that two or more terms are identical
in respect of the terms that they lie between — which implies,
that none of thes» particular terms is between another of them
and any one of the other terms. In this case the terms in
question occupy the same place in the series — it being carefully
noted that no- spatial ideas are intended by this or by similar
expressions. In the case we have considered two notes which
are "equally squeaky " occupy the same place in the series.

If we consider two terms a\y a2 of any of the above series
there will in general be a number of terms between them :
these terms, together with one of the two a\, a2, constitute
what may be called the 'stretch from &i to a2. To be distin-
guished from .this notion is that of the distance of a2 from «i,
an idea which must be kept free from spatial associations,
although, of course, it takes its name from its analogy with
distance in space. Thus in the case of two notes in a scale
the distance would be in the " interval " between them while the
stretch would consist of all the notes that could be intercalated.
Distances, then, are asymmetrical quantitative relations between
the terms of a series such that one and only one belongs to
any given pair. In all the series suggested above there are
distances as well as stretches. There are, similarly, distances
between shades of colour, degrees of hotness, &c., but it is
curious that, if after the consideration of such cases one
returns to the series of positions in space or time, it is doubtful
whether distance can still be detected in them !*

Again, it is possible to divide each' of the suggested series
into two parts so that one of the original terms becomes the
" last " term of the first part, and another (the " next " term)
becomes the "first" term of the second part; for between these
two terms no further terms of the series occur. Series which
can be treated in this way are discrete, and consist of a succession

* Kussell, op, cit.y p. 255.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 23

of consecutive terms. But a great many series which the
Objective presents to us have a structure which will not submit
to this simple operation. s Thus in the case of the series
composed of all sounds considered in respect of pitch it is
impossible even to conceive the series as divided into two parts
in such a way that one note is the highest of the first part, and
another the lowest of the other part. It is true that w^e may
determine that a given note — say " middle C " — shall be the
highest (or last) note of the former part ; but, whatever note is
now taken as the lowest (or first) of the latter part, it is always
possible to think of yet another note as intercalated between
this no.te and the middle C. In other words no note can claim
the title of the next note to C.

It is clear that this statement^could never be directly
verified, but is rather of the nature of a postulate to which
common sense seems to have been led by considerations which
amount to the following.* It is possible to have a note B
which seems to 'be in unison both with A and with C when
tried separately, while, notwithstanding this, A and C are
perceptibly discordant. In order to avoid a contradiction
analogous to the supposition that things, which are equal to the
same, are not equal, to one another, it is concluded that the
note B is "really '^intermediate in pitch between A and C but
cannot be directly perceived to be so. There is no reason to
suppose that in the- same way other notes might not be
intercalated without limit" between A and B. Thus we reach
the notion that Poincare has called iliQ, physical continuum.

It seems possible that this name is not the best that could
be chosen.' If the above analysis is correct the special charac-
teristic of these physical series as they are conceived is "that
it is always possible to find another term of the series between
any two. Thus starting with two terms TI and T2 the inter-
mediate terms will be obtained by the constantly repeated
\1 :

* Poincare, La Science et VBypotkese, p. 34.

24 THE AIM AND ACHIEVEMENTS OFV SCIENTIFIC METHOD.

addition of a new one. This process cannot lead to the
introduction of an infinite number of terms — in the accurately ^
defined sense in which philosophical mathematicians. use the
word.* As a matter of fact such a series is called by
Mr. Eussell compact] and must be distinguished from the
continuous series which implies an infinite number of terms
between any two.

§9.

The most important single series for our purposes is the
series of numbers. Among all the classes of things in th-e
Objective we can detect classes of similar classes — that is
classes between whose terms a " one-one correlation " can be
established. This is true, for example, of the days of the week,
the Hills of Rome, the Champions of Christendom, the Wise
Men of Greece and the Wonders of the World. These and
the other classes similar to them owe their similarity to the
possession of a common property — or, as Mr. Eussell puts it,
to their relation to a common term — their number. This is
the logical definition of the cardinal numbers and probably
indicates also the psychological process by which they we're
discovered. It is obvious that they form a discrete, series
infinite in number, and generated by the constantly repeated
addition of one to the first of them ; while no place in this
series is occupied by more than one terra. .It is clear that
such- a series may be correlated with any other discrete series.
This correlation — the system of counting or marking with
consecutive numbers — must at a very early stage of civilisation
have replaced simpler methods of correlation used for the same
purpose. A time came when the series of integers proved to
be insufficient for the practical purposes to which numbers
were applied, and had to be extended by the addition of the
terms with which we are familiar as " fractions." There can

* Kussell, op. cit., pp. 356-7.
t By Cantor " iiberall dicht."

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 25

loe no doubt that this extension was actually effected in con-
nexion with ^he problem of measurement, which we have yet
to consider ; but it will be instructive to conceive a method
which might have been followed which involves no reference to

this problem.
r /

Let us take a concrete example. We will suppose that we
have undertaken the task of compiling a " subject index " to
a library — a catalogue in which a reference is given to the
books dealing with each subject. Separate sections of ' the
catalogue will be devoted to the main divisions of the subjects
represented, and it will be necessary to interpolate between
these main headings, as the library developes, sub-headings
referring to the portions of the catalogue which, are devoted to
the sub-divisions of the subjects. Thus Section 518 may
contain the books on Analytical Geometry, Section 519 those
on Aids to Calculation. Later it becomes necessary to dis-
tinguish sub-sections devoted to Plane Loci, Trilinears, Conic
Sections, Loci in Space, Conicoids, &c. To mark the position
of these smaller divisions we need to interpolate between the
numbers 518 and 519 other symbols that will continue in this
new field the correlative properties of the integers. We arrive
at the idea of employing for this purpose the symbols 518*1,
518'2, 518*3, &c., — symbols which are obviously not in
possession, directly, of! all the properties of the integers, but
can just as obviously be used to continue their unambiguous
correlative function. Thus the symbol 518*3 fixes .the position
of the sub-section dealing with Conic Sections quite unambig-
uously between the sub-sections dealing with Trilinears (518*2)
and Loci in Space (518*4) respectively. When the books con-
cerned with Conic Sections are themselves numerous enough
for classification, we can arrange the correlation of the
secondary sub-divisions without disturbing the existing dis-
tribution of symbols. It is necessary only to repeat the same
device at a further stage, and to correlate the new sub-sections
devoted (say) to the Ellipse, the Hyperbola, and the Parabola

OF THE

26 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

with the symbols 518-31, 518-32, 518'33. 1$ is clear that we
have here a method capable of indefinite expansion ; that we
have hit, in fact, upon a device that enables us to convert the
numbers, extended in this conventional way, into a compact
series, in which it is possible to insert a new term between any
two given terms without limit.*

Evidently such a series could be correlated with the terms
of any other compact series — for example, the series of notes
arranged in order of pitch. The cardinal numbers could be
assigned in order to the notes of the diatonic scale based upon
the lowest of the series, while the new members could be
assigned in order as intermediate notes were identified — the
numbers being attached, for example, to the strings or pipes
which yielded the notes. If the numbers were assigned in
the manner explained above, it would be impossible that
a note identified as being between two already recognised
notes should fail to receive a number indicating unambiguously
its position.

§ 10.

,XBut the invention of these subsidiary numbers actually
/ took place, of course, in connection with quite a different
problem — the problem of measurement Among the contents
of the Objective which we have considered as capable of
arrangement in order or series, some -are recognisable as
quantities, — that is as possessing magnitude. Such quantities^
can be judged to be equal to other quantities when they
possess the same magnitude, greater or less when they possess
greater or less magnitudes. In the case of some series — e.g.,
the series of tone-pitches or colours — the terms are not
quantities ; in the case of others — e.g., the loudness and the

* This illustration is taken from a page of the Subject Catalogue of
the Science Library in the Victoria and Albert Museum. The formulae in
Professor Peano's Formulaire de Mathematique* are arranged on the same
plan.

X '

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 27

disagreeableness of the notes — they are quantities. In all cases
the distances between terms of the series (if they possess distance)
have magnitude, while the whole composed of a stretch is also
a quantity. The correlating of magnitudes — whether of terms
or of distances — with the number series will present no new
feature except in the remarkable cases in which it is possible
to say that one magnitude is double of another.

It is not difficult to see that such a statement applies in
the first instance only to numbers, and contains a reference to
the process of logical addition. The books on the top two
shelves in front of me as I write form similar classes : I could
" tell off " a book from the top shelf against one from the
second until the whole of the books were simultaneously
exhausted. Both classes of books possess, then, the same
number — which happens to be 18. By logical addition I can
regard both sets of books as constituting one class, " the books.
on the top two shelves." The number of this class, which is
the logical sum of the original similar classes, is 36. This
number is called the "double" of 18. This simple case
illustrates the property of numbers which, besides their ready
accessibility to all, and the ease with which they lend them-
selves to correlation, is the most important which they possess
for the purposes alike of practical life and of science. The
essence of this property is that when correlation has been
effected between any" terms of series and appropriate members
of the number series, it is possible to predict the effect of
carrying out a given operation on these numbered objects by
mere consideration of the properties of the terms of the
number series — the result of the operation in question being
always indicated in an interpretable way by its correlation
with the single number which is the result of the operations
that have been performed on the numerical data. Thus we
can predict without actual trial the number of pieces of paper,
each 1 foot square, which it is possible to place on a rectangular
area measuring 4 feet long and 3 feet wide.

\

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

The simplest case of measurement is that of stretches.
Even when the terms are not quantities, if the number of
terms in one stretch is double that in another it is natural to
consider the first stretch as double of the second. This is our
meaning, for example, when we say that one man's flock of
sheep is double, or twice as great as, another's.

In the case of terms which are quantities, measurement
would seem possible only when each term can properly be
regarded as the sum of parts. This is not true, for example, of
pleasures. If it gives me equal pleasure to entertain A and to
entertain B, to entertain both together gives either two
pleasures — tke former two — or else a single new pleasure — the
N .pleasure of entertaining A and B together — which cannot be
considered as the sum of the former pair. Nevertheless when,
as in this case, the physical source of a quantity is duplicated
there is a great and ^natural tendency to consider the quantity
to be doubled also. Thus most people would consider the
loudness of two equal notes to be twice the loudness of one.
It is hardly necessary to quote Leibniz's doctrine of petites
perceptions to show how strong the tendency is, and we shall
find important examples of its influence in science.*

In the case of distances we meet with a similar difficulty ;
for the distance from a term A to a term C cannot strictly be
regarded as the sum of the distances from A to an intermediate
term B and from B to 0. Here, however, it seems even more
natural than in the last case to consider the distance AC twice
the distance AB or BC if these are equal. Thus the interval
from the note C on the piano to G is equal to the interval from
G to D in the next octave. This being the case, we may con-
sistently regard the interval from C to D as twice the interval
from C to G.

There remain the cases of Space and Time, in which with-
out doubt we have wholes composed of parts. Here, though

* Infra, p. 97.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 29

an element of convention is still present (for the magnitude of
the whole can hardly be regarded as the sum of the magnitudes
of the parts), we judge one magnitude to be double another if
the quantity which has the former magnitude can be divided
into two quantities whose magnitude is the same as that of the
latter. N

In the cases considered, then, we can, by the aid of certain
more or less easy conventions, solve the problem of measure-
ment. That is, we can so correlate the members of the
number series with the members of the series ill question that
whenever the numbers are twice, thrice, four times one another,
and so on, the corresponding quantities or stretches or
distances will be in the senses indicated, twice, thrice, four
times one another, and so on. In every such case we may
make use of the correlated numbers for the purposes of
practice or theory in the manner already indicated.* It is
clear, moreover, that by conventions which contain the familiar
rules for the manipulation of decimal " fractions " we may
apply to this purpose the whole of the extended " compact "
series of numbers that was reached in the last section.

It is a commonplace that the series of Time and Space play
an all-important part in Science ; it will be desirable, therefore,
to consider briefly certain special characteristics and special
difficulties which they present. "We can dismiss in a few
words the property possessed by the spatial series of having
more than one " dimension." From the critical standpoint it is
chiefly important to note that this is not a unique property.
The notes played by a performer on the pianoforte may be

* It should be noted that measurement of "magnitudes of divisi-
bility" such as Space and Time and (conventionally) Weight require a
unit — a part which is repeatedly applied in order to measure the whole.
Such measurement involves the assumption that during the successive
applications of the unit its magnitude remains unchanged.

30 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

considered to present us with an example of a two-dimensional
series with which the three-dimensional spatial series may
be profitably compared. Each of such notes would be
characterized at once by a definite loudness and a definite pitch,
and could be described in an unambiguous way, so as to dis-
tinguish it from other notes of the same pitch but of different
loudness, or of the same loudness but of different pitch, only
"by some device that would express at the same time its
position in both the simple one-dimensional series of pitch and
intensity. This could be done either by assigning to the note
two distinct numbers, or " co-ordinates," one to fix its position
in each scale, or else by extending the notion of number still
further to meet such cases. Thus such a symbol as 256P + 8L
might conventionally mean that the note had the pitch
corresponding to a " frequency " of 256 vibrations, while on
a certain definite scale its loudness was 8. It is clear that
both of these devices may be used to mark the position of
points in space with regard to three mutually rectangular
planes. The former is the device of rectangular Cartesian
co-ordinates, the latter gives the series of " complex numbers."

§12.

At once of more importance and of more difficulty are the
questions connected with the continuity of Space and Time.
It is clear that both these series are compact, that is that we find
ourselves bound to think that between any two points of space
another point exists, and between any two moments of timi
another moment. No special problem arises here, for we have
already seen how the original number series may be extended
so as to provide for the correlation of any such point or
moment with an unique number. Difficulty first appears when
we discover that the number of points between any two points,
A and B, must be thought of as infinite in a sense in which we
cannot apply the term to the number of points which we could
conceive as correlated with numbers of the extended number

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 31

series. This sense, which it has been reserved for the modern
philosophical mathematicians to discover, must be briefly
explained.

A P Q K S T

-B

P

The straight line AB may be supposed to be the road
which an Oriental despot has driven across his empire from
one great city to another, while the shorter line ab may be
taken as a map or plan of this road showing at the points
p, q, r, s, t, the relative positions of the towns and villages,
P, Q, E, S, T, to be met with on the way. It will be
admitted that though the imperfection of human powers
would make it practically impossible to represent every feature
of the road AB on the plan ab, yet the difficulty is only a
practical one, and that there is no detail of the road that could
not conceivably be represented on the plan. The obvious con-
sequence of this admission is that every point on the line AB
•can be correlated with a point on the line ab. According to
the idiom which we have employed before, the two series of
points are similar. But the line ab could be placed so that its
points would become actually a part of the points of the
line AB. Thus we should be faced by the disturbing conX
•elusion that a part of these points may be similar to the whole/
The number of points, then, on the line AB is such that
a part of these points is capable of one-one correlation with
the whole. But however far we carried the assignment
of terms of our compact number series to the points on
the line AB, it is perfectly certain that some of these
numbers could not be correlated one by one with the whole.
We are driven, then, either to the conclusion reached by
such philosophers as Mr. Bradley that space is an irrational
idea, " riddled with contradictions/' or else to the provisional
adoption of the concept of a new type of numbers whose

32 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD,

distinctive property is to be that they are the numbers to be
assigned to the terms of a series when the part is similar to the-
whole. Fqr such numbers we may reserve the term infinite.
Fortunately we are not compelled at this stage in the progress-
of science to accept the former of the alternatives just offered.
The consequences of the admission of a type of " infinite "
numbers having the property explained have been carefully
traced out by competent thinkers, and the contradictions which
would, on Mr. Bradley's principles, condemn the concept have-
not appeared ; while, on the other hand, large provinces of fact
have fallen into order under the application of the, at first
sight, startling concept which we have reached.

But before the ideas which we have just been considering
had shaken off their scarcely reputable association with
Eleatic paradox, and had entered into the sober service of the
modern mathematician, it had been found that the compact
number series was inadequate to the demands that geometrical
theory put upon it. Thus, if the two sides of. a right-angled
triangle are equal in length, then the number which geometrical
theory compels us to correlate with the length of the hypo-
thenuse is not one of the compact series. If the number one
is correlated with the magnitude of the side, the number
which should be correlated with the magnitude of the hypo-
thenuse is one whose " square " is two. It is easy to satisfy
oneself that no such number is included in the compact series.
Two alternatives arise. Either we must conclude that
numbers are inadequate to the concept of space which our
experience forces upon us, or else we must arrive at some
means of supplementing the number . series still further by
means of clearly definable " Objective subsistences " which will
be available for correlation with the newly discovered magni-
tudes.

As a matter of fact, it seems doubtful whether we can define
with logical rigidity numbers with the required characteristics
that can be regarded as interpolated between the terms of the

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 33

existing compact series of " rational " numbers.* It appears to
be necessary to have recourse to a totally new series of concepts
which, though derived from the rational numbers, are not
identical with them. These are the " real " numbers of the
mathematician. Their nature may be indicated^ follows : —

123456 7 8 9 10 11 12 13

In the line above are printed the symbols of the integers
from 1 to 13, irregularly spaced so as to emphasise the fact
that we are required to attend only to their order, not to any
notion of distance of a spatial kind. Hitherto such a number
as nine has been regarded as the common property of an
indefinite number of " similar classes," each of which would, in
ordinary parlance, be said to contain liine terms. But, given
the definite ordinal series of integers, thus defined, it becomes
possible to regard nine as defining a class of integers, namely,
the class containing the integers 1 to 8 which precede it. In
this way each of the integers can be correlated with a class of
integers, so that " counting," say, of the persons round a dinner
table, may be conceived as correlating them, one by one, with
the classes of integers with which the symbols 1, 2, 3, &c., are
also correlated.

It is clear that nothing in this will become invalid if we
include in the classes of numbers, which are henceforward to be
our instruments of correlation, the 'whole of the rational numbers
which have received rigorous definition. Thus we can suppose
our dining guests to be correlated, one by one, with the
successive classes of rationals less than one, less than two, less
than three, &c. ; and there will be no danger lest the correlation
should be incomplete or ambiguous. It is true that our
ingenuity might run the risk of comparison with that shown
by the White Knight if we deliberately employed so far-fetched
a method in so simple a case as the counting of guests round

* Kussell, op. cit.j p. 270.

.'\ 34 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

I a dinner table ; but we should reap the reward of it when we

^ came to deal with such questions as the correlation of the

\ members of some convenient series with such spatial magni-

, tudes as the sides of the triangle recently under consideration.

For it will refdily be .seen that if we correlate the magnitude

of the sides with the -class of rationals less than one, we may

quite unambiguously correlate the magnitude of the hypo-

"~fchenuse with the class of ratibnals whose squares are less than

two — since no doubt can ever arise as to whether a given

rational number falls in this class, nor confusion of limits with

another class be feared. In this way the " irrational" numbers

which are necessary to enable us to correlate the terms of the

number series with all the points of space that anyone has

- ever found himself led to postulate can be supplied.

It need hardly be added that by an appropriate symbolism —
Jwhich is strictly a non-logical matter — these new members can
xbe made to perform the services in which we have seen the
^ great practical importance of numbers to lie.

• The difficulties that beset the measurement of Space were
imported into Time when attempts were made to consider the
question of Motion philosophically. Hence arose the famous
paradoxes of Zeno. But there is no reason why Time should not
be conceived as having the same kind of " continuity " as Space, so
that for the correlation of the moments of Time, as for the con-e-
lation of points of Space, we must have recourse to the " real "
as distinguished from the "rational" numbers. Motion then
loses its "contradictory" character; for we can regard a
" material particle " as simply a means by which correlation is
set up between (1) a single point of Space and a number of
successive moments of Time, if the particle is " at rest " ; and
i, • (2) a series of points and moments, one by one, if the particle is
"in motion."*

For details see Euasell, Ch. LIV.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 35

§ 13.

On the fascinating subject of the Objectivity of Space and
Time, our remarks must be very brief^ and, as before, limited in
the main to indicating the grounds upon which it would seem
that the existence of absolute Space and Time must be admitted,
without any serious attempt to elaborate "the arguments in
favour of this view, or to annihilate those which make
against it.

The latter may be considered under three headings. The
first of these would contain the philosophical arguments which
aim at discrediting continuous Space and Time by showing that
they are self-contradictory concepts. Mr. Bradley's attack is
too well known tjp need description here, and is typical of this
type of objection. (The essence of it is that Space and Time
demand terms and relations between these terms, while any
attempt actually to exhibit these terms involves the investigator
in a hopeless "infinite regress." But in the light of the
mathematical theories that we have been sketching, it becomes
clear that the attempt in question fails because it is undertaken
with inadequate means, not because it is essentially hopeless.
In the successive dissection of portions of Space in the hope of
finding at last the indissoluble point, one is doomed to failure
by the fact that in this way one can never arrive at an infinite
number of operations — taking the term in the sense that has
been explained. Nevertheless, we are able in the manner we
have indicated to conceive an infinite number of atomic points,
in which we can recognise the final terms which we seek, and to
do so without finding ourselves involved in the contradictions
which beset us when we embarked upon the quest armed only
with an improper concept of an " infinite number." « Thus the-
classical a priori arguments against points are successfully
resisted.

The other two classes of argument aim at showing that the
assumption of the existence of absolute Space is unprovable and

D 2 N

36 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

unnecessary rather than that it is demonstrably false. First
will come the psychologists who point to the manner in which
our concept of Space as " a single continuous receptacle " arises
from our perceptual experiences as indicative of its purely
psychological value as a great "economical invention/' An
interesting argument of this type is that developed by Professor
Poincare, who holds that our three-dimensional space is really
nothing more than a conventional reduction — now hereditary
in the race — of the indefinitely numerous "dimensions" of
an extensive character which our muscular sensations must
originally give us.*

Finally come the arguments based on the "descriptive"
view of Science. Thus to M. Poincare, faced with the old
question whether the earth revolves or whether the firmament
undergoes a diurnal revolution about it, the two propositions
" ' la terre tourne/ et ' il est plus commode de supposer que la
terre tourne/ ont un seul et meme sens ; il n'y a rien de plus
dans Tune que dans I'autre/'f

When faced with the fact that results of scientific research
— such as Newton's law of mutual accelerations, the law of
"centrifugal force," and certain results of modern electrical
theory — seem to be meaningless in the absence of absolute
space, these epistemologists reply that such theoretical results
are simply means by which we can analyse and describe the
actual behaviour of the universe as it is given to us : and that
we have no warrant for applying these descriptive instruments
to hypothetical cases in which alone they could be thought of
as proving the existence of an absolute space. Thus (they
argue, in effect) although you may apply with success to the
behaviour of the water in a rotating bucket the descriptive rules
which you reached by analysis of the behaviour of the moon, yet
your success is dependent upon the fact that you worked in the

* Science et Hypothese, p. 73.
t Op. cit., p. 141.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 37

same universe on each occasion. It does not warrant a pre-
diction of the behaviour of the water in the bucket if the whole
of the rest of the universe were removed — even supposing that
it is legitimate to assume that such a thing could happen.*

It is plain that such arguments cannot disprove the existence
of absolute space. They cannot, that is to say, prove that it is
meaningless to suppose the whole universe to be moved, each
point of it, 1000 miles from its present place, along lines
parallel to the line joining, at a given moment, given points on
the earth and the sun. It seems, then, that we may fairly
apply an argument which Mr. Bradley has on more than one
occasion claimed the right to use, and has expressed in the
epigrammatic form, "What is possible and what a general
principle compels us to say must be, that certainly &."f
The arguments of Newton, and the similar modern ones
to which I have alluded, are reasons why absolute space
M must be/' which, as Mr. Russell remarks, still await refutation.
Meanwhile fresh difficulties may be accumulated for the
philosophers who take up, under the distinguished leadership
of Mach, the extreme empiricist position. We have already
seen that these writers maintain that Newton's concept of
action between mass-points is merely a useful mode of analysing
for descriptive purposes certain events that are in the habit of
taking place in our corner of the actual presented universe ;
and that we have no right to suppose that it would prescribe
the behaviour of another material universe, nor even of the
present one if its configuration were to be modified to a serious
extent. This doctrine of the economical function of scientific
concepts is not in itself necessarily incompatible with absolute
space, but the temptation to extend it so as to give an account
of the origin of the two " single continuous receptacles " which
contain the whole manifold of human experience is one which

* Cf. Stallo, Concepts of Modern Physics, p. 200.
t Op. cit., p. 196.

38 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

this school makes no attempt to resist. The argument is
presented in different forms. Thus Professor Karl Pearson
teaches that perceptual space is simply "one of the ways in
which we perceive things apart. There is nothing in the sense-
ijnpressions themselves which involves the notion . of space."
Space has, then, the kind of existence that belongs to the order
of an arbitrary arrangement^ letters, such as the alphabet —
>a kind of existence that must be carefully distinguished from
the 'real existence, in their possession of which the letters may
be compared with the groups of sense-impressions which we
ternt pjbjects.*

It is- manifest that Professor Pearson's " mode of per-
ception " is little, if any, more than Kant's " form, of intuition "
rebaptised. As is well known, Professor James has pointed
out the entirely gratuitous assumption involved in this account
of the manufacture of perceptual space in a mythological
" Kantian machine-shop " of the mind,f and has devoted one of
his^pnest chapters to the defence of a genuine psychological
theory of the perception of space. According to this theory
ther,e is an element of crude " extensity " or volurninousness
present in many, if not in all, sensations — an undifferentiated
root from which the spatial order of perception of adult life is
developed. The actual problem, then, is to understand how
we arrange these at first chaotically given spaces into the one
regular aird orderly "world of space" which we know.J Theie
is no reason why these spatial data, merely because they are
given and many, should piece themselves together into -one
continuous whole ; and, indeed, nothing is easier than to con-
vince oneself, by observation of one's total state of conscious-
ness at any moment, that the spatial elements of the various
contributaries to that total state — " the sound of the brook, the
odour of the cedars, the comfort with which my breakfast has

* Grammar of Science, 1st ed., 1892, p. 185.
t Pr. of Psych., II, p. 273.
J Op. cit., p. 146.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 39

filled me, and my interest in this paragraph — all lie distinct
in my consciousness, but in no sense outside or alongside of
each other."* They enter into this special relation with one
another only when they come to be " discriminated as parts
from out . of a Ikrger enveloping space:"f ^ This discrimination
is rendere4 possible in many cases by a d^ference^ in the
" immanent sensibility " of the different parts of the sense
organs, so that each " carries its special sign/'f while it is
greatly aided by movements, either of the object or of the
sense-organ. Given that we have in this way analysed a
number of original vague vastnesses into a mass of correlated
spatial details, the problem now arises of summing these sense-
spaces into what we commonly think of as " real space." This
process occurs through the mediation of constant external
things, with which different sense-organs, or different parts of
the same sense-organ, come into relation from time to, time. {
Through the long-continued commerce of the sense-organs with
these constant external things we come either to regard different
spatial feelings as equal to one another, or to select certain
ones as feelings of the real size and shape of things, and to'
degrade others to the status of being merely signs of these. §
When, as in the case of the mouth, we have what we may think;
of as an external thing with which a special set of sense-
organs — those of the tongue — has dealings, tiie thing in questioii
comes to form " almost a little world in itself,"|| whose spatial
values have never been reduced to agreement with those
yielded by the other sense-organs. This fact is readily
exemplified by the simple experiment of exploring the teeth
with the tip of the tongue, and then repeating the investigation
with a finger-tip.

* P. 146.
t P. 147.

I P. 167.

§ P. 181 etseq. P. 269.

II P. 181.

40 THE AIM AXD ACHIEVEMENTS OF SCIENTIFIC METHOD.

However vigorously we maintain the principle that logical
values are independent of psychological genesis, it is difficult
to hold that this theory of spatial perception has no relevance
to the question of the status of conceptual space. Even if we
regard the process described by Professor James (or the similar
one described by Professor Ward) as merely the elaboration of
a crude " nativistic " spatial perception,* we seem to be obliged
to admit that adult perception gives us not space but spaces,
which in some cases obstinately refuse to merge their
individuality in our one " continuous receptacle." What, then,
is the precise relation of conceptual space to these diverse
perceptual spatial entities ? Granting with Professor James
that Kant's account of -the genesis is " pure mythology," are we
compelled by the better founded theory which we owe to
modern psychologists not only to recognise, contrary to Kant's
opinion, that the space of geometry is an empirical concept
drawn from experience ; Jatit to abandon also his argument that
space is not a general concept formed by abstraction from
particular presentations because it is " essentially one " and
claims to refer to that which contains all perceptual spaces as
part of itself ?f May not the notion of a " continuous
receptacle " after all be only an abstraction, a result of con-
ceptual analysis performed upon the relatively isolated and self-
contained spatial wholes which psychological inspection
reveals; while the unique character which we ascribe to
conceptual or geometrical space represents a pragmatical
simplification of the actual data which is justified only by
its fruits ?

These questions involve the consideration of the wider
question of the nature of concepts and their relation to percepts.
Here one of the essential points appears to be that the concept
and the percept are alternative ways in which the Subject deals

* Stout, Manual, 2nd ed., p. 377.

t Critique of Pure Reason, pp. 23, 25.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 41

with the same Objective elements. Thus my concept of the
Hazing fire in the next room enables me to maintain practical
.and theoretical relations with it at times when none of its
qualities are contents of a percept of mine. There is, then, in
^•every concept a reference to Objective elements, though it is
only in perception that these elements present themselves with
a guarantee of their Objective character. To employ Mr.
Bradfey's well-known idiom, in perception reality is qualified
by an ideal content, while in conception the ideal content is a
•" floating adjective " of reality — floating, however, as a captive
balloon floats, with a subtle line still holding, it fast to the
ground from which it arose. In acts of imagination — when I
picture a giant whose favourite food is bread and butter
sprinkled with light brown horses,* or when with circumstantial
detail I present the life of , Colonel Newcome; in play — when
the make-believe messenger arrives fiery red with haste astride
of the nursery chair ;f in authentic narrative of the past ; in
•expressions of command ; in plans or resolutions for future
behaviour ; in all these we have a synthesis of conceptual
•elements which, though it follows perceptual models, appears to
owe its particular form to the initiative of the thinker. And
by this ideal content reality is in each case qualified just so
far as the primary facts of the moment will permit the
qualification.

If this doctrine is true, then there is no such definite gulf
between concepts and percepts as is frequently supposed.
There is always a reference of the ideal content to the
Objective world. In perception the Objectivity of the concept
is self-announced and self -guaranteed : in mere conception
there is no such guarantee of the Objectivity of the ideal
•content, but simply absence of rejection by the presented
Objective of the proffered addition ; while in negation there

* An achievement (I believe) of the late Professor Clifford,
t Stevenson, Child's Play (in Virgimlus Puerisque).

42 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

is a rejection of conceptual elements which have their proper
home in other Objective contexts.

In particular this doctrine seems to throw doubt upon the
completeness of the distinction between perceptual and con-
[ .ceptual space. Conceptual space alone, we are told,* contains
points, curves, and surfaces, and is infinitely divisible. These
notions are reached as the result of carrying to an ideal limit
either suggestions of perception or operations possible only to
a certain extent with perceptual material. Upon our view the
relation between spatial percepts and concepts should be stated
very differently. Our concepts of the relations which we
afterwards know as continuous curves and surfaces neither are
suggested as ideal limits of perceptual data which are self
complete, nor are introduced ab extra to render the data
intelligible (as we shall afterwards maintain of the conceptual
hypotheses of science) ; they are actual constitutive elements
of the percept, and can be thought of apart from the -per-
ceptual whole only in the same waj» as I may think of the
greenness of the apple that I declined yesterday, or the
loudness of the cry which disturbed me an hour ago. Again,
such a concept as the straight line is not, upon the view here
held, gained as the limit of the process represented by the
whittling of a stick. On the contrary,' it is the concept of
the straight line that gives to the perceptual data the unity
which makes them stages of one process. Finally, we may ask
whether the process (which Professor James describes) of
analysis of a vague given spatial whole into its relational
details does not imply in the same way the presence of con-
ceptual elements isolated afterwards as the straight ^Mne and
point. The fact that jbhe point cannot be perceptually isolated
is no reason why it should not have a kind of priority to the
perception of areas which makes it in a certain real sense the
means by which areas are perceived. The concept is employed,
so to speak, in its intension, not in its extension.

* Pearson, op. cit., p. 203.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 43

The rejection of some such view as this seems to entail the
^adoption of one which makes an almost complete divorce
between the conceptual and perceptual worlds. It must, in
effect, be held that man was in the beginning thrown naked
and helpless into an unintelligible and intractable world which
as yet he had no means of reducing to his will :

e/3\67rov

K\VOVT€$ ovfc rjfcovov, a\\ oveipdrcov
a\i<yKiot, fjiop<f)alcrL rov /Jbaicpbv ftiov
ecpvpov ei/cfj Travra.

So it was until at length Prometheus (disguised, perhaps, as
Natural Selection) brought relief in the shape of the gift of
a set of concepts linked each to all in the bonds of apodeictic
certainty, and having the curious property that by the mere
consideration of the relations of these concepts man could
master, at least as far as his practical n^eds required, much of
"the bewildering flux of the perceptual world.

In a well known article,* Mr. Bussell has, in 'effect,
shewn that the denial of absolute space is equivalent to the
acceptance of views of which the above is at most an'
exaggerated statement. In the relational theory conceptual
space consists solely of the concepts of the relational nexus in
which 1 the "things" of perception are imbedded. The
"points" which are thought of as the terms of these relations
are only symbols of the things themselves. For this reason
(possibly) Mr. Eussell calls them "material points." He
proceeds to point out that " in order to give an ac.cou.nt, which
will consist with the facts, of the intersection of figures,~the
order of lines and planes, and the nature of areas and volume,"
we must be prepared to assert either that " material points "
are as a matter of empirical fact distributed just as spatial

* "Is Position in Space and Time Absolute or Relative?" Mind,
N.S., No. 39.

44 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

points are distributed in the absolute theory, or else to supple-
ment the existing distribution of these by a supply of
" possible material points" which are indistinguishable from
the real spatial points which it is sought to exclude. If we
are to avoid both horns of the dilemma we must fall back upon
that severance of the worlds of thought and perception which
has already engaged our attention. The truths of geometry
will then hold only of conceptual space, and their approximate
validity in the perceptual world is really *> matter of lucky
coincidence, however it may have come about.*

As in other cases to be considered in the sequel the
attitude which a thinker will take up on such a question as
this will depend very largely upon his intellectual tempera-
ment. If he shares with Professor James and his fellow
^Humanists a constitutional distrust of " all noble, clean-cut,
fixed, eternal, rational, temple-like systems " of thought he will
probably hold by the relational theory of space, and glory in
the name of empiricist which is hurled at him as a reproach.
If, on the other -hand, his faith in the organic unity of all
truth is sufficiently robust to push logical arguments daunt-
lessly to their ultimate conclusions even when they seem to
the Humanist " oddly personal and artificial," or even " bureau-
cratic and professional iti an absurd degree,"! he will be an
absolutist. Such a thinker will prefer to leave the difficulties
presented by the psychological analysis of space perception to
be reconciled with his main convictions by some theory of
error.

In the article to which reference has just been made,
Mr. Eussell disposes of the question of the status of moments

* Our argument is not affected by the consideration that, after all,
our space may be really of a non-Euclidean pattern, so that it is only
approximately represented by the usual homaloidal model. This would
mean that we fail to discriminate correctly the actual spatial relations
which are presented to us. It could not negative the arguments in favour
of the Objectivity of points.

+ James, Humanism and Truth, p. 467.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 45

of Time by an argument the force of which depends still more
directly upon the conditions that we have discussed in the last
paragraph. There can be no doubt (so runs the argument)
that events occur simultaneously ; that is, that they occupy
occasionally the same place in the temporal series. But it is
impossible to conceive that this may be so unless their place in
the series is fixed by their correlation with a single term of
some independent series. This independent series with which
the events of the World are correlated (just as the matter of
the world is correlated with points of space) is the series of
absolute moments of time.

46

CHAPTER II.

§ 14.

In attempting to exhibit the main outline of the Objective
as it appears to the "plain man" before the advent of
scientific interpretations, one runs the risk of an accusation
of merely adding to the inhabitants of the shadowy land,
where the " economic man " and the " natural man " who
enters into " social contracts " already dwell. At the least
one may be met by the objection that many or all of the plain
man's " views " are, after all, interpretations — interpretations
which themselves at one time represented the high water mark
of "scientific" investigation. The objection undoubtedly has
force and we must return to it later,* but the accusation may
, be evaded by the admission that the plain man as such is a
fiction in so far as he is an Abstraction from within the wider
self of each of us. Much as the total outlook of mankind
upon the world varies from China to Peru, there seems to
be a solid core of agreement everywhere which alone truly
answers to the description which we have given of the
Objective. The scientific traveller on a high plateau of the
Andes and his native guides view in different ways the im-
possibility of getting their potatoes to cook.f To the latter
the impossibility is due to the simple fact that " the cursed pot,"
doubftess owing to the devil in it, " did not wish to cook
potatoes " ; to the former it is an interesting example of the
dependence of the boiling point upon the pressure. But
although the whole " situation " may be very different in the
two cases, there is yet a common basis of inevitable fact upon

* Vide infra, p. 138.

t Darwin, The Voyage of the "Beagle."

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 47

which the scientist and the native (if he is intelligent enough)
. can see that their " animistic " or " scientific " interpretations
are simply embroideries. If (remembering at this point that
there exists a science of psychology) we say that the " things "
before our travellers — the fire, the pot, the lukewarm yet
boiling water, the unsoftened potatoes — are all of . them
" constructs," we must admit at the same time that they are
inevitable or primary syntheses which mankind everywhere
would make from the same sensational data, while the whole
situation as it exists for the two men is a secondary synthesis
which, when one's attention is called to the matter, is seen
not to be inevitable. Wherever the " objects " of attention
dealt with in the former chapter must be held to have a
synthetic character, only these primary syntheses were
intended. Adopting this distinction we may say that the
scientific process is one out of several possible alternative
processes by means of which primary facts may be submitted
to further construction ; and it will be recognised as true that
the object of this secondary synthesis is to make the primary
facts intelligible. But this characteristic, though of funda-
mental importance, does not suffice to distinguish the scientific
from all the alternative processes contemplated. To assert
that a thing is intelligible or that it has meaning is to imply
/ that it forms aft element in a system of terms in relation.
Thus a word — for example the word " button " — standing alone
has meaning chiefly in so far as it is recognised as belonging to
the English vocabulary within which it may be either a verb
or a noun. When I say, " Pray you, undo this button," the
fact that the word is now brought into relation with other
words in a definite system gives it a fuller but still incomplete
meaning : I may mean a coat button or a door button. The
doubt can be resolved only by the context, that is, by the
position of the sentence in a still wider synthesis. In this
way the request, " Pray you, undo this button," may have all
manner of meanings from the trivial one which a common

,48 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

domestic context would give it, to the profound and pathetic
significance it has on the dying lips of King Lear.*

The point in question could be illustrated indefinitely,
but it seems necessary to note only that in every case the
" system ?- in which an element finds its meaning must ulti-
mately be an apperceptive system. This term, although it"
appears to have lost its forjner vogue in psychology, *s
yet, perhaps, the best available to suggest the integral, the
. vital connection of such systems with the past experience and
present interests of an individual consciousness — the connec-
tion which is part of what I have already sought to suggest
by speaking of Science as a conative process. No treatment,
in fact, which isolates the efforts that have generated Science
from their psychological milieu can hope to do justice to its
subject, the true nature of which can only be brought out by
placing the scientific process in its proper position in a Natural
History of processes which all aim at rendering the Objective
intelligible. Only by following such a method is it possible
t6 reach a clear understanding of the relations to one another
of the various elements which a cross section of contemporary
scientific thought would exhibit.

§ 15.

Among the interpretations of the Objective which demand
comparison with the scientific, the most important from the
point of view of distribution is "animism," the system of
beliefs upon which are based those practices of " magic " which
not only are found to-day under curiously similar forms
. among all savage races, but also have preceded the existing
modes of thought among all civilised peoples. Indeed, the
researches of authors like Professor Frazerf have revealed
these ancient ideas still persisting widely beneath the modern

* Act v, scene iii.

t J. G. Frazer, The Golden Bough, 2nd ed., i, p. 74.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 49

intellectual surface, and have even seemed to justify a fear
lest the depths should some day be upturned and the results
of centriries of man's toils and genius be overwhelmed. More-
over, they have shown that magic, so far from being an
'unorganised collection of bizarre superstitions, has every claim
to the title of a logical intellectual systems-based upon funda-
mental principles, to repudiate which would be at the same time
to repudiate science.* At the base of the whole structure we
find the " scientific " Principle of Uniformity, which differs
from the fundamental principle of Logic, that " of the same
the same is always true/'f only by the addition of what we
may, perhaps, call an " existence postulate " that " the same "
for the purpose of predication actually occurs. As Dr. Frazer
points out,t the principle takes the special form of arguments
based either upon Similarity or upon Contiguity. Thus to
secure the destruction of a distant foe, you procure a waxen
effigy of him, and submit it to slow-roasting or to other ill-
treatment, in the confident expectation that the unfortunate
original will suffer analogous torments. Your hope springs,
of course, from the belief that the two cases have a " core
of identity " sufficient to make the " substitution of similars "
effective. Again, if you have succeeded in wounding your
adversary, and seek to complete your work by recovering the
spearhead and allowing it to rust away, in order that he may
simultaneously languish and die, you are assuming this time
that the intimate association between weapon and wound
has set up so much identity between two situations that
their future developments must to a large extent be tlie
same.

/ It seems highly probable that beliefs of this character

V arose as interpretations of observed facts, and it is most

unlikely that they have survived through ages without the

* Op. cit., i, pp. 61, 62.

t Cf, Bradley, Principles of Logic, p. 133.

£ Op. cit.j i, Ch. II, esp. pp. 10-18 and 56 et seq.

50 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

support of facts which have been taken to be verifications
of them ; there must, at least, have been a widespread belief
that they " worked." Formally, then, the processes are
unexceptionable, and differ from a modern investigation
apparently only in the material circumstance that now-a-days
we should not fix upon these particular " cores of identity "
in the situations contemplated as having any relevance to the
similarity between the courses of their subsequent development
alleged to be observed. Since, however, mistaken beliefs as to
the significance of certain elements of phenomena have been
common in the history of Science, if we are to find an
essential difference between Science and Magic we must look
elsewhere.

We can find the differentia we are seeking only by con-
sidering the whole primitive attitude towards the Objective,
the system of beliefs and interests by which new phenomena
were " apperceived." The primitive thinker had not reached
the clear distinctions we make between the dead world and
our living and conscious selves, and peopled the physical
environment with active individual principles whose wills had
constantly to be reckoned with. Moreover, his attitude
towards this environment was determined to a predominant
extent by considerations that touched the immediate safety
and wellbeing of himself and of his tribe. To a very large
extent it was the attitude of a being who combined with
the passions and vices of a man the terror of the child in the
presence of the unknown. Bearing these two facts in mind,
the failure to distinguish between the animate and the
inanimate which made him regard the environment as a great
community of beings, for the most part to be dreaded or
placated, and the constant pressure of the needs of defence
and preservation, which made it necessary that something should
le done, we can understand his at first sight capricious logic,
and can see the psychological force of the considerations
which led ultimately to his submission to the burden of a

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 51

rigid system of beliefs and customary acts " heavy as frost and
deep almost as life."

§ 16.

It was such a system of interpretations of the Objective
which was losing its authority at the momentous epoch which
we mark as that of the birth of Greek Philosophy. Philosophy,
the child of Wonder, began when advancing knowledge was
banishing the nymph and dryad from the world of practical
activity to the fantasy world of the poet, when no longer the
Ionian could

" Have sight of Proteus rising from the sea ;
Or hear old Triton blow his wreathed horn."

With the realisation of the inadequacy of the once sufficing
explanations of the world's happenings, there arose the need for
more satisfactory ones, while the widening and deepening of
intellectual interests that came with an age of comparative
personal and social security, brought men face to face with the
old problems of change and decay in a much more general
form. The motive of the movement, which we commonly date
from the speculations of Thales, was to seek escape from the
intellectual oppression of the world's ceaseless flux in some
abiding reality. The animistic " moment " was passed, but
men had not yet come to that realisation of the great gulf
fixed between their real selves and physical nature which is
the distinguishing mark of the modern consciousness.*:. We
find accordingly that the new effort to render the Objective
intelligible takes the form of an attempt "to give back to
Nature the life of which it had been robbed by advancing
knowledge .... simply by making it possible for that life
which had hitherto been supposed to reside in each thing, to be
transferred to the one thing of which all others were passing
forms." f Animism was replaced by Hylozoism.

* Martineau, Types of Ethical Theory, i, pp. 123, 124.
t Burnet, Early Greek Philosophers, p. 13.

E 2

52 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

Once more we have to distinguish the " secular conative
process " here initiated from Science. That the Greeks collected
material indispensable to the structure of Science is not to be
disputed, whatever estimate we adopt of the actual value of
their achievements on the whole and in detail. As a result of
recent research that estimate has undoubtedly tended to rise.*
We^an no longer accuse them of an entire neglect of physical
experiment, and. the late Professor Huxley, after a careful
. consideration of the existing records, arrived at " a very favour-
able estimate of the oldest anatomical investigations among"
them.f Burnet has, moreover, defended the hastiness with
which hypotheses were advanced upon the warrant of a very
slender bridge of facts, regarding this haste as naturally
characteristic of early undisciplined enthusiasm, and retorting
effectively that the same fault is by no means absent from
the history of modern investigation.}: -Finally, Jowett has
attributed to these " general notions " a positive value, regarding
them as " necessary to the apprehension ofjDarticular facts . . .
Before men can observe the world, they must be able to conceive

it.- §

Against these apologies it must be maintained that, with
certain exceptions that hardly affect the argument, the
"scientific" achievements of the Greek thinkers were simply
incidental to the search for the " abiding reality " which is the
predominant characteristic of the whole intellectual movement.
This which was true of Milesian Nature-philosophy, was still
more obviously true when their speculations gave place to the
" moralised " conceptual investigations in Being and Becoming
of Heraclitus and his Eleatic opponents. We must maintain
the same of Empedocles, though he " anticipated " the theory

* See Mach, Science of Mechanics, 2nd Eng. ed., App. I.
t On certain Errors attributed to Aristotle in Science and Culture,
p. 193.

| Burnet, op. cit., p. 26.

§ Introduction to the Timaeus : Dialogues, iv, p. 416.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 53

of organic evolution, though his pity par a were the direct
ancestors of the modern elements, and though his " mechani-
cal" Welfbildung, in which, besides these o-ro^ela, only the
forces of Love and Hate play their part, may not so fancifully
be compared with the object of physical science as conceived
(for example) by Helmholtz.* To be brief, not even the
elaborate systems of Democritus and Aristotle can be exenjpted
from the general statement that we are dealing here with
attempts to render the Objective intelligible which, on the
ground of an essential difference in the whole "situation,"
must be distinguished from Science.! To justify this statement
fully would obviously require so much time that I must ask
to be forgiven for stating dogmatically a contention the principle
of which you will, I hope, be inclined to admit without further
argument.

' - • ' • § 17. •

For the same reason it is impossible to do more than
illustrate the fact that my contention also holds good 6f many
modern thinkers, who have yet made contributions to the
fabric of Science of fundamental importance. In the case of
these moderns the individual systems of ideas by which Objec-
tive facts were apperceived were dominated by theological as well
as philosophical elements. Thus Descartes when, to complete
his philosophical system, he turns his attention to the actual
particulars of the behaviour of the res extensa, deduces (in an
imperfect form) the modern doctrine of the Conservation of
Momentum from considerations of the perfection of God t }N A

* Ueber die Erhaltung der Kraft, Einleitung, p. 6 : " Die Naturer-
scheinungen zuruckzufiihren auf [Materie und] unveranderliche, anzie-
hende und abstossende Kraf te."

t Cf. Plato's view that "the movements of the stars are only bad
diagrams illustrating the truths of ideal astronomy," or Aristotle's
conception of laws valid only " cir\ TO TroAv," with Galileo's conviction that
unbiassed investigation of matter will explain all apparent anomalies in
its behaviour. [Dialogues, Weston's trans., p. 3.]

\ Descartes, Principia Philosophiae, 2nd part, § 36.

54 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

little later Leibniz corrects the deficiencies of this principle,
pointing out that Descartes had neglected to observe that the
direction as well as the quantity of " force " (momentum) is
conserved. Our interest fastens on his further remark that if
Descartes had noticed the fact, " he would have fallen into my
System of Pre-established Harmony." *

§ 18.

But for the illustrations most suitable to my purpose I
must direct your attention to the writings of Keppler ; for the
student who picks his way discreetly through Frisch's monu-
mental editionf of the Omnia Opera of that heroic astronomer,
will gain as his reward a vivid idea of how profoundly the
whole " situation " in which Objective facts are actually central
is determined by the character of what I have already called
the "embroidery"; and will, moreover, catch sight of the
human spirit at the precise moment of one of its most interest-
ing metamorphoses.

Keppler begins (in the My sternum Cosmographicum, 1596)
as an enthusiastic young convert to the heliocentric doctrine
of Copernicus. He defends the new theory on the ground
of its superior simplicity, not, bien entendu, its simplicity as
a description of the facts, but its real and meritorious simplicity
as an actual creative plan.

" Amat [Natura] simplicitatem : amat unitatem. Nunquam
in ipsa quicquam otiosum aut superfluum extitit ; at saepius
una res multis ab ilia destinatur effectibus." J One form of
orbit, then, should be expected to suffice for all the planets,
instead of the deplorably diverse orbits of the Ptolemaic system.
In the spirit thus indicated he proceeds to determine the reasons
why the solar system could not but be precisely as it is. First
we learn why a combination of curves and linear distances (from

* Leibniz, Jfonodologie, §80 ; ThJodice'e, § 61.

t Keppler, Omnia Opera, ed. Frisch, Frankfort, 1858-71.

t Cap. I, p. 113 (Vol. I. of Frisch's ed.).

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 55

the Sun) should be exhibited : " Quantitatem autem Deus ideo
ante omnia existere voluit, ut esset curvi ad rectum com-
paratio."* Moreover, these curves will lie upon spherical
surfaces so as to exemplify the Trinity : " [Imago] Patris scilicet
in centre, Filii in superficie, Spiritus in aequalitate o-^ea-e&j?
inter punctum et ambitum."t

Similarly there must have been the best of reasons for the
choice of the particular dimensions of the orbits, the general
principle being, " Nefas est . . . . quicquam nisi pulcherrimum
facere eum qui esset optimus."{ So it was inevitable that the
Creator should lay the foundations of the planetary worlds in
accordance with the ideas He would gather from His contem-
plation of the Five Perfect Solid Figures. Imagine the sphere
of which the circular orbit of Saturn is a central section, to be
circumscribed about a cube, then the sphere which contains in a
similar manner the orbit of Jupiter will be inscribed within this
cube. Next, within the sphere of Jupiter let a regular tetra-
hedron be inscribed ; this will in turn circumscribe the sphere
of Mars. Thus we reach all the planets in turn, finding it
obvious that Man — finis et mundi et omnis creationis — should
have his habitation in the midst of the planetary host, three
celestial bodies guarding his path without, three (including the
Sun) within.§

So far you will agree, the course of Keppler's investigation
has exemplified my remark that, formally, non-scientific
attempts to render the Objective intelligible may not differ
from those which are 'admittedly scientific. "We have the
usual primary Objective basis and the usual secondary con-
struction— the Objective facts qualified by an " hypothesis."
But the secondary construction here exhibited (you will object)
is one that is capable of verification — i.e., of predicting new

* Cap. II, p. 122.

t Ibid.

I Ibid.

§ Cap. IV, p. 128.

56 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

Objective facts which contributed nothing to the determination
of that construction. The relative distances of the planets
from the sun in Keppler's system are open to calculation and
comparison with data of observation. Keppler's theological
.prepossessions do not prevent him from recognising this truth
i^ the clearest manner. " Transeamus modo," he says, " ad

orbiuin astronomiae et demonstrationes geometricos :
quae nisi consentiant, procul dubio omnem praecedentem operam
luserimus."* So the relative radii of the spheres imprisoned
in this complicated way between the regular solids are com-
puted, and the results compared with the estimates of Copernicus.
The concordance is practically perfect ! t

In 1600 Keppler left his chair at Gratz, and received from
Tycho Brahe that introduction to the Emperor Eudolph which
led to consequences of the first importance in the development
of Science. Brahe died in the same year, and the Imperial
mathematician inherited his splendid collection of observations
on the planet Mars. In 1609 appears the famous treatise,.
De Motibus Stellae Martis, in which he sets forth with the
delightful long-windedness of a leisurely age the results of his
patient study of these data. After the fashion of a day when
philosophers reasoned even of Ethics more yeometrico, Keppler
prefixes to his work a collection of Axiomata pliysica de motilus
stellarum. These are of the highest interest, for they betray
a complete change (since the My sternum, Cosmographicuni) in
the astronomer's attitude towards his facts, To determine
the particulars of the orbits of the planets we are no longer
invited to consider that they must move "ad majorem Dei
gloriani : motus a spatio dependet ; planetae aguntur vi naturali ;
vis motrix opus habet propagation a fonte ceu effuxu " ; are
among the startling " axioms " that meet us.
'• The body of the work is largely occupied by Keppler's

* Cap. XIII, p. 148.

t Table in Cap. XIV, p. 151.

UNIVERSITY

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 57

famous demonstration that the orbit of Mars instead of being
a circle, as the prepossession in favour of " perfection " had
hitherto compelled him to suppose, is actually an ellipse of
which the sun occupies one focus. It will interest us more to
attend to the remarkable change in his whole attitude towards
the Objective upon which I have already remarked. We find ^
the evidences of this change most prominent in the introduc-
tion' and the later chapters of the treatise. Ce n'est que le
premier pas qid write, and when Keppler has once been com-
pelled to seek the secondary construction that is to make the
primary facts intelligible in a disinterested study of those facts
themselves in their quantitative determination, he travels fast
towards a characteristically " modern " point of view. Since
the planets no longer move in circles they must resign with
these the crystal spheres in which since the days of Plato they
have been " quiring to the young-eyed cherubim." These
destroyed, what is to guide a planet's motion ? The anima
mundi remains, it is true, and Keppler, like his great con-
temporary Gilbert, finds nothing objectionable in the concep-
tion. He had, in fact, used the admitted existence of the
anima mundi as an argument against the Ptolemaic orbits,
inviting his readers to pity the condition of the distracted world-
souls who in that complicated system "ad tarn multa respicere
jubentur ut plane tarn duobus permixtis motibus invehant ! "*
Similar considerations seem to deter Keppler from assigning
to the anima mundi the perpetual solution of the mathematical
difficulties incidental to following an elliptical path round an
eccentric, sun. He looks elsewhere for a means of, at least,
lightening the world-soul's burden and finds what he wants
within the Objective itself in a new conception of the sun as
fans motus. This conception has not been reached without
external suggestion, and when we meet the phrase orbs virtutis '
tractoria we are left in no doubt as to the source of that

* Introd., p. 149 (Vol. 3 of Frisch's ed.).

58 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

suggestion. Keppler has been reading the newly-published
treatise De Magneto, by Gilbert of Colchester, the Father of
Experimental Science, and has fastened upon the fruitful
analogy between magnetic and stellar phenomena. The first
result is a " true doctrine of gravity " which points directly to
the completer doctrine of Newton. Kepeating the argument
given above in connexion with the anima mundi Keppler
asserts the impossibility " ut forma lapidis movendo corpus suum
quaerat punctum mathematicurn aut mundi medium."* On the
contrary, "gra vitas est affectio corporea mutua inter cognata
corpora ad unitionem sen conjunctionem (quo rerum ordine est
facultas maynetica) ut multo magis Terra trahat lapidem quam
lapis petit Terram."f There are, it is true, difficulties in the
application of the analogy. The investigations of a Galileo were
necessary before a Newton could see that the moon is actually
and always falling towards the earth. For Keppler the
difficulty is to account for their remaining apart : " Si Luna et
Terra non retinerentur m animali aut alia aliqua aequipollenti
. . . Terra ascenderet ad Lunam . . . Luna descenderet
ad Terrain . . . ibique jungerentur."|

The words italicised in this passage illustrate at once
Keppler's willingness to retain the conception of the anima
mundi and his growing preference for a facultas corporea to
a facultas animalis if the former can make the facts intelligible.
We may leave the consideration of the development of his
ideas at the point where he reaches a " secondary construction "
of the facts of the stellar observations suggested altogether by
such material analogies. In this conception the planetary
movements are ascribed to a two-fold "virtue" — one of the
planet and one of the sun. That "of the planet is compared
with the work of oars in rowing, that of the sun to the stream
of the river. And so we reach the all-important conclusion,
in which the soundness of this conception is based upon the

* Introd., p. 150. t Lvtrod., p. 151. % Ibid.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 59

solid experimental results of Gilbert: " Quale flumen, talis renms.
Flumen est species immateriata virtutis in Sole inagneticae.
Quin igitur et remus de magnete quippiam habeat ? Quid si
ergo corpora planetarum1 omnia sunt ingentes quidam rotundi
raagnetes ? De Terra (uno ex planetis, Copernico) non est
dubium. Probavit id Gulielmus Gilbertus."*

It will be noted that Keppler's final conception of the
planetary system is formally less satisfactory than the earlier
one — since it fails to suggest quantitative determinations by
which it could be verified. At the same time it will, I hope,
be agreed that when, at some moment between 1600 and 1609,
Keppler, wrestling with Brahe's records, forgot his pious pre-
possessions in his anxiety to understand the behaviour of Mars
for the sake of understanding it, he adopted for the first time an
attitude which was genuinely " scientific." The differentia of
Science, then, as a conative process whose aim is to render the
Objective intelligible, is the presence of no motive except the
desire to render it intelligible — particularly in its quantitative
determinations. No philosophical leanings, not even the desire
of power over Nature for which Bacon was willing to be her
minister, can be admitted beyond the " margin " of the apper-
ceptive area in which the Objective facts .are central. The
scientific attitude is essentially that of the savants who, drinking
to the next great discovery, coupled with their toast the hope
that it might never be of any use to anybody.

§ 19.

I need hardly say that Keppler does not provide us witli
the first example on record of the scientific attitude. Mach
holds that -the beginnings of Science are to be found in the
descriptive communications of the processes of the craft made
by older members of a guild to begirmers.t So Hoffding,J

* Pars Quarta, Cap. LVII, p. 387.

t Mach, Science of Mechanics, p. 4.

j Hoffding, Hist, of Modern Philosophy, i, p. 161.

60 T.HE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

with truth, says that "the appearance of a Leonardo or a
Galileo* is only comprehensible when taken in connexion
with Italian industry." But industrial pursuits, I suggest,
can never do more than supply the experience which forms the
starting point for the scientific process which follows only from
a specific attitude towards that experience which I have tried
already to characterise. Just as J. A. Symonds has shown us
that in the epoch of the Crusades and dominant Scholasticism
the Latin songs of the Wandering Students "gave clear and
ai-tistic utterance " to a " bold, fresh, natural, and pagan view
of human life " ; so, doubtless, ever and anon men of intellect
turned aside from the theologico-philosophical studies of their
day to the task of rendering intelligible Objective facts in
which they took an immediate interest and delight. Such a
one, in part, was Eoger Bacon, such a one was his master,
Peter of Maricourt,f such a one pre-eminently was Leonardo
da Vinci who, though his discoveries do not appear actually to
have affected the course of Science, left among his remarkable
manuscripts a presentment of the scientific attitude which can
hardly be improved. I conclude this section by quoting a
typical expression of his opinion :| " In dealing with a scientific
problem I first arrange several experiments, since my purpose
is ^to determine the problem in accordance with experience
and then to show why the bodies are compelled so to act.
That is the method which must be followed in all researches
upon the phenomena of Xature. It is true that Nature as it

* Cf. the opening words of Galileo's own Dialogues : " The constant
employments in your famous arsenal of Venice, and especially those
relating to what we call Mechanics, seem to me to afford, to a speculative
genius, a large field to philosophise in." (Tr. Weston.)

t To whom Gilbert of Colchester was much indebted See in Bridges'
edition of Bacon's Opus Majus, 1897-1900, vol. i, p. xxv.

J From Grothe, Leonardo da Vinci also Ingenieur und Philosophy
Berlin, 1874, p. 22. Cf. the following passage : " Le me pare che quelle
scienze sieno vane e piene di errori, le quali non sono nati dall' experienzar
madre di ogni certezza, e chi non terminano in nota experienza."
Frammenti litterari e filosophici, p. 94.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 61

were begins with reasoning and ends with experience, but
nevertheless, ivc must begin with experience, and by means of
it strive after the discovery of Truth."

"The interpreter of the wonders of Nature is experience.
. . . We must consult experience in the variety of cases and
circumstances until we can draw from them a general rule
that is contained in them. And for what purpose are these
rules good ? They lead us to further investigations of Nature
and to creations of art. They prevent us from deceiving
ourselves or others by promising results to ourselves which are
not to be obtained." .

62

CHAPTER III.

§ 20.

A " natural history " of the more sustained attempts that
humanity has made to render the Objective intelligible — that is
to give it a place in a definite apperceptive system — would lead
us, then, to the conclusion that its differentia is not, as has
been frequently supposed, a peculiar method, but simply and
solely a definite attitude of the " Self of the moment " towards
the Objective, a definite character of the system by which new
elements are " apperceived," a character only to be expressed
by saying that this system is dominated by a permanent
interest in the particulars of the Objective as such. The next
chapter in our natural history would examine in turn the
various special attempts to make the Objective intelligible which
are included in the genus " scientific." Such an examination
would, I submit, bring out the fact that it is difficult to declare
any concept essentially incapable of mediating a scientific
interpretation of the Objective to some thinker. Thus it has
already been pointed out that Keppler in his "scientific"
period did not shrink from continuing to utilise the conception
of the anima mundi. A less violent but essentially similar
example is the use of the concept of cause in the sense of
transeunt action — a notion with which some scientific thinkers
have entirely dispensed, while to others it is of cardinal import-
ance. Facts of the same order are the ' marked preference
of Weber and his Continental school for the concept of
action at a distance and the equally marked preference of the
British school for the concept of an intervening medium as a
means of rendering action at a distance intelligible. Especially
illuminating in this connection are the well-known facts that
Maxwell based his immensely important electro-magnetic

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 63

theory upon the concept of a " displacement " to which it is
impossible to assign a definite meaning,* while Lord Kelvin,
speaking on the same subject said, " As long as I cannot make
a mechanical model all the way through, I cannot understand,
and that is why I cannot get the electro-magnetic theory
of light."f An almost better illustration is afforded by
Dr. McBougall, who not only conceives his " neurin " as a fluid,
but defends his practice in an excellent notej by arguments
essentially the same as those I am advancing. Finally, it is
clear, that this doctrine of the relation of the scientific concept
to the primary facts does not exclude the concept of "end"
from the investigator's armoury of interpretative weapons and
so admits the methodological propriety of the practice of
" neo-vitalists " such as Bunge and Eindefleisch. Our doctrine,
moreover, has a normative value. It declares that a concept
which is to render given primary facts intelligible must be formed
as a reaction upon the stimulus of the presentation of those
facts in their actual determinations. While it admits, then,
the aid of any concept borrowed from any other context, it
refuses to allow Objective facts to be annexed simply in order
to widen the territories of an aggressive theory, and still less to
permit their primd facie deliverances to be ignored through a
bias in favour of any particular type of interpretation. Thus
" electricity " and " neurin " may both be legitimately conceived
as fluids, but the physicist is not to rule the concept of
" interaction" or of a " soul"§ out of court, and still less is he
to refuse to entertain evidence in favour of " telepathy. "||

The only restriction upon the secondary construction is that
its form shall be determined by the actual particulars of the

* See Merz, Op. ciL, ii, 93.

t Quoted by Ward, Naturalism and Agnosticism, i, p. 119.

| McDougall, "The Physiological Factors of the Attention Process,"
Mind, N.S., No. 43, p. 350.

§ Cf. James, Pr. of Psych., i, p. 137 ; McDougall, Physiological
Psychology, pp. 8 et seq., p. 78.

|| As at least one very distinguished scientist is reported to have done.

£4 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

primary facts. This condition limits the usefulness of such a
conception as the anima mundi or the " end " to a phase in the
development of knowledge of the facts when the particulars are
not capable of full determination.* At such a time such a concept
;as vitalism may legitimately be used "as a comfortable halting-
place where the reason may be laid to rest on a pillow of obscure
ideas " when there is " danger of premature and, therefore,
inadequate physico-chemical explanations of the phenomena of

life."t

§ 21.

It is in almost identical words that Bunge opens the
important work in which he treats of physiological chemistry
from a vitalistic as opposed to a " mechanistic " point of view.J
At the same time he avows that his adoption of this position
is due to his definite dissent from " the doctrine which
:some opponents of vitalism maintain, and which would have
us believe that in living beings there are no other factors
at work than simply the forces and matter of inorganic
nature."§ In opposition to this doctrine, Bunge denies that
physico-chemical explanations have been given of any really
vital phenomena — i.e., of any phenomena in which what we call
" activity " is displayed. Thus even in so simple a case as the
absorption of food from the alimentary canal, the physical
processes of diffusion and osmosis do not account for the
established facts. The epithelial cells exercise a selective
function, accepting globules of fat, but declining (for instance)

* Thus Bunge (Text Book of Physiological and Pathological Chemistry,
p. 420) explains the high temperature in fever " teleologically " as an
attempt of the organism to destroy the bacteria. But this explanation
fails to yield the precise temperature reached.

t Prof. Hering, quoted by McDougall, loc. cit.

I Bunge, Text Book of Physiological and Pathological Chemistry ;
translated from the 4th ed. by Starling, 1902, p. 1 : " No explanation is
offered by a mere term. I regard ' vital force ' as a convenient resting-
place where, to quote Kant, * reason can repose on a pillow of obscure
qualities.' "

§ Loc. cit.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 65

fine pigments.* Similarly, Vampyrdla Spiroyyrce, only a
minute unicellular mass of protoplasm, insists on feeding upon
the alga, Spirogyra, and rejects every substitute. t Still mom
remarkable is the " behaviour " of Arcella, a unicellular organism
protected by a concavo-convex shell which on the concave side
is pierced by a hole through which pseudopodia are pushed
out. Arcella insists that these pseudopodia shall be attached to
something solid, and to secure this end generates bubbles of gas
so as to raise one edge of its shell, or, ' if necessary, to turn
itself completely over.j

The last example challenges comparison with the behaviour
of the Slipper-animalcule, Paramecium, which has been studied
and interpreted by Dr. H. S. Jennings. § The Paramecia "feed
largely upon clotted masses of bacteria. If a number are
placed upon a glass slip, together with a small bacterial clot,
they will be sure to congregate around the clot and to feectupon
it. All apparently press in so as to reach it or get as near as
possible." Moreover, they appear generally to be " actuated by
some social impulse leading them to crowd together and shun
isolated positions." They display, again, definite likes and
dislikes towards the various liquids that may be intro-
duced into the water in which they are swimming ; " from
alkaline liquids they are repelled ; to slightly acid drops they
are attracted, unless the acidity be too pungent."

Dr. Jennings' interpretation of these performances is anti-
thetical to Bunge's explanation of the behaviour of Arcella.
It consists in an attempt to reduce the phenomena to cases of
" organic response " which are conceived as of the same type as
-the orderly reactions of inorganic matter. The Paramecia

* Op. cit., p. 3. Similarly Professor Japp claimed that the mould
Peiiicillium "selects" the right-handed variety of racemic acid, B.A.
Report, 1898, p. 817.

t P. 3.

IP. 7.

§ I quote partly from the account given by Lloyd Morgan, Animal
Behaviour, 1900, Ch. I.

F

66 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

crowd round the bacterial clot if, and only if, in the course of
their random movements they are brought in contact with it.
This mere contact produces an automatic " lessening* or
•cessation of the regular movements in all the cilia except
those of the mouth-groove and funnel." In a similar spirit
Dr. Jennings deals with "the seemingly social assemblages
of Paramecia, where there is no such substance to arrest their
progress," showing that this phenomenon is nothing more than
a special case of the " attraction " which feebly acid solutions
possess for the animalculse. Where one or two Paramecia are
gathered together, they form a centre for the production of
a feebly acid solution of carbon dioxide, which is, as with all
creatures, a product of their organic waste ; " and when other
Paramecia come, in the course of their random movements, into
such a centre they remain there and help to swell^the number
in the cluster." After a time the solution in the centre of the
group becomes so strong as to be repellent instead of attractive.
" The group spreads, and the Paramecia are less densely
packed." Finally, to meet the objection that in this attraction
and repulsion " at least there is, it may be urged, an element of
choice," Dr. Jennings once more takes pains to attribute the
phenomena to a definite " reaction " which the cilia of the
Paramecia undergo when they reach the boundary of the acid
drop.

The general conclusion which he reaches is, in fact, that
" the reactions of Paramecium are, as we have seen, comparable
in all essentials with those of an isolated muscle [of a frog] "*• —
and, therefore, that " we are riot compelled to assume conscious-
ness or intelligence in any form to explain its activities."!

§ 22.

Before leaving the consideration of these specimens of
^antithetical interpretations of the same kind of data we should

* Amer. J. of Psych., x, p. 515. t Ibid.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 67

not fail to note that they are the expression of definite diverse
methodological views held by the two naturalists. Thus, Bunge*
holds that " the essence of vitalism does not lie in being
content with a term and abandoning reflection, but in adopting
the only right path of obtaining knowledge which is possible,
in starting from what we know, the internal world, to explain
what we do not know, the external world."

On the other hand, Dr. Jenningsf proceeds in obedience to
the canon that such a " complex psychology as seems forced
upon us by the observed facts " should be " reduced to simple
factors " as far as possible, and, in the course of his study of the
problems presented by the Paramecia, makes it more than pro-
bable that, in his view, a factor is " simpler " the more closely it
approximates to the reactions of physics and chemistry. There
can be no doubt that at the least he would subscribe assent
to the cautious words in which Professor Lloyd Morgan, at the
end of his description of Dr. Jennings' experiments, points a
moral upon which he has consistently acted throughout the
whole of his own well-known and valuable researches : —
" One of the first lessons which the study of animal behaviour
in its organic aspect should impress upon our minds is that
living cells may react to stimuli in a manner which we perceive
to be subservient to a biological end, and yet react without
conscious purpose — that is to say, automatically. ... If
purpose there be, it lies deeper than its protoplasm, deeper than
the dim sentience which may be present or may be absent — we
cannot tell which. "J

§23. /V

It seems a fair critical summary of this controversy to say
that it gives us a concrete example of the principle that facts
lying within a new province of experience may be made

* Op. cit., p. 10.

t Amer. J. of Psych., x, p. 506.

1 Animal Behaviour, p. 10.

F 2

68 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

intelligible by means of concepts drawn from diverse fields
of experience already explored ; the choice of concept being
determined only partly by the actual Objective data, and for
the other part by psychological conditions.* Of course, it may
be maintained that there is really a question of facts at issue :
a purpose either does or does not lie at the back of the
phenomena. Here we are confronted by a difficulty which will
be considered at a later stage."f We may decline to discuss it
in the present connection on the ground that few thinkers,
if any, would regard the Objectivity of a purpose as capable of
proof except as a " residual phenomenon " after the application
of all possible physico-chemical explanations. It is not likely
that any biologists would maintain that this stage has actually
been reached. We may conclude then that the only considera-
tions which should interfere with the purely individual choice
of one type of interpretation over another are considerations, if
such can be arrived at, of the relative usefulness of particular
kinds of hypotheses at the particular stage of exploration of the
province of fact under discussion.^

§24.

From this point of view the opposition between " causal "
and "final" explanations appears in a new light. Both
" cause " (" Ursache ") and " end " or " purpose " (" Zweck ") are
concepts abstracted from definite experiential contexts. We
have maintained that any such concept may be used to make
Objective facts intelligible to an individual investigator : the
question as to the Objectivity of 'the cause or the purpose may
ultimately have to be decided in a given case, but at the

* It is not implied, of course, that the selection of the Objective data
for interpretation is independent of these psychological conditions.
Bunge and Dr. Jennings, starting from different points of view, inevitably
deal with data of somewhat different characters. See p. 144.

t Infra, Ch. V.

j See, for example, Bieganski, Neo- Vitalismus in der moderne Biologie
(Ann. der Natur philosophic, IV, i).

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 69

outset, at any rate, is irrelevant. Since the same question
may arise with regard to other concepts which are used to
make a certain group of facts intelligible (e.g., with regard
to the Objectivity of the " ether " or of " atoms ") it 'is clear
that causal and final explanations are to be regarded as on the
same level with an indefinite number of others. This being
the case it seems hardly worth while to distinguish these
particular types of interpretation by a special terminology
which simply keeps alive the memory of a controversy which,
from the point of view adopted here, has vanished.

Nevertheless, these traditional terms may very usefully be
retained with somewhat new implications.* Thus, without
intending any suggestion of " transeunt action " we may use
the term " causal law " as a convenient name for statements
which exhibit the dependence of the behaviour of a thing upon
external conditions. In this sense it is a causal law that two
" material particles " in one another's presence, but infinitely
remote from all other matter, would exhibit accelerations of
their velocities, directed along the straight line joining them,
inversely proportional to their masses and to the square of the
distance between them. The statement of any such a " causal
law " may be called a " causal explanation " of the behaviour of
the particles in question, being usefully distinguished by this
term from explanation by an " empirical law " which would
simply give a formula for the behaviour of a single particle
without reference to the behaviour of any other body. But,
as a matter of fact, scientific investigators always' endeavour to
bring the observed behaviour of bodies under some definite
concept or concepts. Thus, in Mechanics, it is sought to
describe the behaviour of bodies by means of the concept of
the Conservation of Energy, or of Force (Conservation of
Momentum), or of Least Action, or one of a number of

* The first formal suggestion of this use with which I am acquainted
is in an interesting article by Ostwald (jun.) and Blossfeldt, Ueber Kausale
und finale Erklaruny (Ann. der NaturpJiilosophie, III, 1, pp. Ill et seq.).

70 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

subsidiary descriptive notions such as Harmonic Vibration.
Similarly, the chemist seeks to bring other Objective happen-
ings under the concept of Equivalence, while the biologist
will employ in a similar manner the notion of Natural Selection
or of Panmixia. These attempts to bring the particulars of
experience under such special concepts may one and all be
called " final explanations " ; the purpose which makes that
term appropriate being, of course, the conscious endeavour of
the investigator to apply this special form of solution to the
problems before him. On this "view, then, all the special
interpretations of the sciences are final explanations ; while
all are to be regarded as subordinated to the general notion
of causal explanation, in so far as they aim at exhibiting the
dependence of the behaviour of things upon external conditions.
In conclusion, attention should be directed to the fact that
we are here concerned only to maintain the right of every
investigator to choose his own weapons wherewith to reduce to
intelligibility the primary facts within the province of the
Objective which he has marked out for conquest. Upon the
subject of the choice at any particular stage in any of the
special sciences each such science must be " self -normative."
This contention does not forbid us to hold, with Professor
Boyce Gibson,* that, on the whole, " the essential discovery
of modern science is that its ideal, the thorough understanding
of nature, can be reached only by subordinating the idea of end
entirely to that of law," nor yet to. agree with Lotzef in feeling
" certain of being on the right track, when [we] seek in that
which should be the ground of that which is."J

* A Philosophical Introduction to Ethics, p. 53.

t Metaphysics, ii, p. 319.

J I much regret that the extremely interesting little book on The
Interpretation of Nature", by Professor Lloyd Morgan, did not fall into
my hands until after the above discussion had been written. I may,
perhaps, be allowed to refer the reader for a fuller discussion of the origin
and import of the causal concept to my paper " On Causal Explanation '
in the Proceedings of the Aristotelian Society for 1906-7.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 71
§ '20.

I am aware that in view of the vigorous and important
attack upon " hypotheses " made by writers of such eminence
as Ostwald * my defence of them will appear reactionary. I
venture to think, however, that Ostwald fails to distinguish
between the real value and the psychological value of hypotheses.
Hypotheses, such as Maxwell's displacement, the weight of a
molecule, electrons, the carbon-tetrahedron, entropy, heat itself,
may not be verifiable and, therefore, have no real value, but
their psychological value as " leading us to further investigations
of Nature" and prompting to fresh determinations of the
Objective may be immense. Ostwald's assertion that scientific
advance has taken place in spite of, and not by means of,
hypothesis! is, at best, a half truth. It is true that hypotheses
have temporarily delayed the progress of Science in some parti-
cular field, but when they have disappeared they have generally
been devoured by their own children — -objective determinations
to which they led. To maintain that these determinations
would have been made without the hypotheses — for example
that Maxwell, without the concept of electro-magnetic
displacements in the field around a varying current would have
thought of locating at points in the field the disembodied
relations expressed by his differential equations, the manipula-
tion of which led Hertz to discoveries of the highest importance,
seems itself to be an indulgence in hypothesis of a thoroughly
unwarrantable character. The point of Ostwald's objection to
a hypothesis — a Bild% used to make the phenomena intelligible
— is that the Bild will invariably contain elements which are
not present in the original observations. There are two answers
to this objection. In%the first place it may be urged that this

* Ostwald, Vorlesungen iiber Naturphilosophie, X, esp. pp. 211-215.

t Ostwald, op. cit., p. 225.

J " Dass .... man durch die Benutzung des Bildes in die Darstellung
der Erscheinung Bestandtheile hineinbringt, die dem Bilde angehoren,
nicht aber der Erscheinung selbst," op. cit., p. 212.

72 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

property of the hypothesis is that which above all makes it
valuable. The portion of the Objective under investigation
must be the seat of other relations than those " apperceived 'r
by the concept, and it is already probable that the original
analogy will extend to the other properties of the concept
whose correspondence with properties of the Objective under
examination has not yet been established. Thus " a descriptive
theory of this kind does more than serve as a vehicle for the
clear expression of well-known results, it often renders
important services by suggesting the possibility of the existence
of new phenomena/'*

In the second place, physicists are so sensible of the aid
they receive from such a descriptive hypothesis, that they da
not discard it even when it is recognised as containing elements-
actually inconsistent with known Objective determinations.
The conception of the ether as a Motionless fluid passing among
the molecules of matter " more freely than the wind through a
grove of trees," has been none the less useful because incom-
patible with the rigidity which the facts also seem to demand.
Ultimately, of course, such incompatibility will not be tolerated,
but its very presence sets a further problem — the replacement
of the inconsistent hypotheses, both having reference to the
same province of Objective fact, by another which shall do-
justice at -once to all its elements. Such a complete corre-
spondence between the elements of the descriptive hypothesis
and of the province of the Objective is, of coivifee, the ideal
of the scientific process to which the successive concepts by
which it is sought to render the facts intelligible approach, as
Mach says, " asymptotically.''! Were it attained the " picture "
and the " object " would coincide^ and we should have " a

* Prof. J. J. Thomson, introducing his conception of the " Faraday
tube " as an alternative to Maxwell's " displacement." Recent Researches
in Electricity and Magnetism, 1893, p. 1.

t Mach, Principien der Warmelehre, 1900, p. 461.

t "Wenn Bild und Gegenstand in alien Stucken uberemstinimten, so-

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 73-

complete systematized representation/' " a complete synoptic
[ilbersiclitliches] inventory of the facts of the province " of the
Objective free from the extraneous elements that hypothesis
admittedly introduces.* When this consummation has been
reached in any department of Science, descriptive hypotheses
will still have a psychological value for the purposes of
exposition and assimilation. Meanwhile they will continue ta
play an indispensable part in the conquest of the Objective,
whether in definite form as Lord Kelvin's " mechanical model
all through," or a vague form like Maxwell's " displacement,"
being, as it were, points dc rcpbrc without which great systems
of reasoning cannot be built, just as transient ones require the
aid of shadowy visual, auditory or kinsesthetic images.

Finally it may be pointed out that it is of small con-
sequence to the progress of the special sciences whether the
investigator attaches real value to his hypothesis, or whether he
recognises that it is merely psychological. Lord Kelvin and
Principal Eiicker are quoted by Dr. Wardf as examples of
the former class, holding that in the ether and in atoms and
molecules we have realities behind the veil of phenomena,
while Maxwell in his attitude towards his earlier model of the
ether, J Wollaston, Davy, Liebig and Faraday in their attitude
towards Dalton's atoms, are given by Dr. Merz§ as examples of
the second. It seems probable that in the case of the latter
class of investigators , their attitude towards their conception is

waren sie eben dasselbe, d. h. man kann eine Erscheinung vollkommen
nur durch sich selbst abbilden." Ostwald, op. cit., p. 212.

* Macli, loc. cit.

t Ward, op. cit., i, pp. 113 and 306.

| "I do not bring it forward as a mode of connection existing in

nature It is, however, a mode of connection which is mechanically

conceivable and easily investigated .... so that I venture to say that
any one who understands tlie provisional and temporary character of this*
hypothesis, will find himself rather helped than hindered by it in his
search after the true interpretation of the phenomena." Collected Papers,
i, p. 486 ; quoted by Merz, op. cit., ii, p. 83.

§ Merz, op. cit., i, p. 418.

\

74 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

rhythmic, at one time yielding to a temporary belief in them, at
another time criticising them as from an external point of view.

§26.

These properties of the hypothesis may all be well illus-
trated by a brief study of the part played in the development
of the science of Chemistry by the concept of the atom. I have
already expressed my acceptance of the view that the real
significance of the atom in the thought of the Greek school of
atomists was its metaphysieal significance — its value in con-
nection with the solution of the problem of reconciling the
Eleatic and Heraclitian views of trie universe. The atom had
had a respectably long philosophical history before it appeared
in the speculations of Democritus, but there can be no doubt
that its origin was the ordinary hard body of well-defined shape
and size of everyday experience. In taking this concept from
its accustomed context, and using it to "explain" the whole
of the structure and behaviour of the world, Democritus
neglected those qualitative differences between things which the
6/jLoiofjbepeicu of Anaxagoras had preserved ;* but he retained
the shape and size and the definite modes of movement which
characterise the molar body as we know it, and built up his
whole system on the basis of their assumed explanatory value.
In this connection it is interesting to note that Bacon, who
preferred Democritus to the idealist philosophers of antiquity,
was well aware that an atomic theory must necessarily be, an
attempt to explain some of the phenomena of matter by means
of others. Accordingly he twits Democritus with being " sibi
impar et fere contrarius"t not merely for retaining in a
theoretical construction which is intended to explain all pro-
perties of things, some properties unexplained, but also actually
using these retained properties as the means of explanation of

* See Gompertz (Eng. trans.), Greek Thinkers, i, p. 331.
t De Principiis atque Originibus : Works ; ed. Ellis and Spedding, iii,
p. 82.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 75

the rest. Bacon's criticism is, as a matter of fact, only directed
upon the assumption of definite modes of motion for the atoms,*
which modes are admittedly drawn from observation of bodies
on the earth's surface, but the spirit of his objection is the same
as that which I have expressed to the notion that hypotheses
have a real as distinguished from a psychological value.

§27.

It is well known that Bacon's general approval of the
methods and views of the ancient atomists was part of the
positive aspect of his rejection of the Aristotelian 'scholastic
system of philosophy, and that his own method of Forms
did not show that he had found in his study of the neglected
work of the laughing philosopher the clue that would lead men
out of the labyrinth of their ignorance of the world about them.
Nor did the Lord Chancellor, who " wrote on Science like a
Lord Chancellor," perform the service of pointing out to
others the way to the Promised Land which he was himself not
qualified by temperament to enter. That function fell to the
lot of Gassendi, who, turning to the atomic philosophy as he
found it expressed in the luscious hexameters of Lucretius, " not
by accident, nor out of mere love of opposition .... embraced
with a sure glance exactly what was best suited to modern
times and to the empirical tendency of his age. Atomism, by
his means drawn again from antiquity, attained a lasting
importance, however much it was gradually modified as it
passed through the hands of later inquirers."!

The interests of Gassendi, like those of his friend, our own
Thomas Hobbes, were in problems of a philosophical as opposed
to a scientific character ; but in an age when the notion of a
division of intellectual labour had not yet arisen, and men still
essayed the task of encompassing a knowledge of the answers

* "Debuit enim motum heterogeneuui atomo tribuere, non minus
quam corpus heterogeneum et virtutem heterogeneam." Loc. cit.
t Lange, History of Materialism, Eng. trans., 1877, i, p. 255.

76 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

that their fellows have given or are giving to all the problems-
of Time and Existence, it was inevitable that -this revival of
atomism in the fields of speculation should attract the attention
of the two great men who were destined to be most prominent
in naturalising the philosophy of Gassendi and Hobbes in the
positive sciences.*

§28.

To Eobert Boyle belongs the credit (in addition to his
most notable discoveries in Physics) of applying the new
corpuscular notions to rescue the investigation of the chemical
properties of matter from the unworthy hands of the alchemists
In his dialogues entitled The Sceptical Chymist, he conducts
(through the mouth of the character Carneades) a polemic
against the " Peripatetick Doctrine," in which, carrying a step
further the revolt initiated by " Paracelsus and some few other
sooty Empiriks/'t he propounds in a perfectly clear form the
modern concepts of chemical compounds and elements. "It-
seems not absurd to conceive that at the first Production of
mixt Bodies, the Universal Matter whereof they among other
Parts of the Universe consisted, was actually divided into little
Particles of several sizes and shapes variously moved." J
" Neither is it [im] possible," continues Carneades, " that of
these minute Particles divers of the smallest and neighbouring
ones were here and there associated into minute Masses or
Clusters, and did by their Coalitions constitute great store of
such little primary Concretions or Masses as were not easily
dissipable into such Particles as composed them."§

It is clear from the discussion that follows, as well as from
the more systematic exposition given in another work|| that the

* Lange, op. tit., i, p. 300.

t I quote from the English edition, Oxford, 1680.
} P. 37.
§ P. 38.

|| Boyle, The Origines of Formes and Qualities (According to the Cor-
puscular Philosophy). I quote from the second edition, Oxford, 1667.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 77

•" primary Particles," " Minima or Prima Naturalia "* are
analogous to the " ions " of the " one Universal Matter " from
which are built up the "atoms" of the elements to which
Boyle's " primary Concretions " correspond, while the
•" molecules " of the modern chemist are anticipated by the
"grosser and more compounded Corpuscles "f that compose
such bodies as the " exotick Compounds " already mentioned.:}

Mercury (to take one of Boyle's examples) " will with divers
Metals compose an Amalgam, with divers Menstruums it seems
to be turned into a liquor, with Aquafortis it will be brought
into either a red or white Powder or precipitate, with Oyl of
Vitriol into a pale yellow one, with Sulphur it will compose a
blood-red and volatile Cinaber, with some Saline Bodies it will

ascend in form of a salt which will be dissoluble in water

And yet out of all these exotick Compounds, we may recover
the very same running Mercury that was the main ingredient
of them, and was so disguised in them."§

In this way a concept drawn from the context of meta-
physical speculation served as the instrument which enabled
Boyle to render a definite group of primary facts intelligible.
The outcome of its application in a few cases was that a definite
new problem emerged into view : the problem of showing that
of the manifold substances that compose the material universe
the majority can be regarded as compounds of a few simple
elements which may be thought of as existing side by side in
the compounds, disguised but not destroyed. When the
corpuscular hypothesis had led to the formulation of this per-
manent problem for the chemist face to face with a new
substance, it had, for the time being, accomplished its task, and
ceased to be any longer an instrument of investigation in
chemistry.

* Origines of Formes, p. 47.

• „/• • t Ibid., p. 48.

I The Sceptical Chymist, p. 41.

§ P. 40.

78 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

§29.

Meanwhile, however, a vastly more powerful intellect than
that of Boyle was preparing the atom for a wider career of
usefulness. Newton's Principia made it possible for Laplace
a hundred years later to present to Napoleon the atomic
Weltanschauung in which he had found no need of any such
hypothesis as that of a Creator. In an admirable chapter
on " The Astronomical View of Nature," Dr. Merz has drawn
a most lively and interesting picture of the way in which
Newton's astronomical views " spread into molar and molecular
physics," and has traced this enormously fertile stream of
investigation arid discovery to its sources in the notion of
attraction (to which for the first time Newton gave a quanti-
tative meaning), and in Newton's own suggestion made in the
famous "Query 31 " with which the "Optics" closes, that the
behaviour of material substances might in a multitude of cases
be interpreted by means of the notion of attractive or repulsive
forces acting between the hard, solid particles of which they
might be supposed in ultimate analysis to consist.

§ 30.

But it was not until ten years ago, when the twelve
volumes of John Dalton's lecture and laboratory notes were
discovered in the rooms of the Literary and Philosophical
Society of Manchester, where his experimental work was
carried out,* that it became more than a shrewd biographer's
surmise that the father of modern chemical theory had drawn
his inspiration from the same source. At the conclusion of
the notes on a course of twenty lectures delivered at the Eoyal
Institution^ during the winter of 1809-10, Dalton copied

* Eoscoe and Harden, JNew View of Dalton's Atomic Theory,- 1896,
p. 12.

t P. 125.

. THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 79

several extracts from Query 31, and also the enunciation of
the 23rd Proposition of Book II of the Prinripia, to which
in the 17th lecture he had made a definite reference. From
Query 31 Dalton doubtless drew the general conviction that
" the Changes of corporeal Things are to be placed only in
various Separations and new Associations and Motions of
these permanent Particles";* and, in particular, laid hold of
Newton's opinion that "God in the Beginning form'd matter
in solid, massy, hard, impenetrable, moveable Particles, of such
Sizes and Figures, and with such other Properties, and in such
Proportion to Space, as most conduced to the End for which he
form'd them."f The stress that Dalton lays upon this asser-
tion of the possibility of the existence of " particles of matter
of several sizes and figures, and in several proportions to the
space they occupy, and perhaps of different densities and
forces,"J is due to his preoccupation with the problem of
accounting for the fact that " the oxygen gas being specifically
heaviest should not form a distinct stratum of air at the bottom
of the atmosphere and the azotic gas one at the top of the
atmosphere."! Before " modern discoveries [had] ascertained
that the atmosphere contains three or more elastic fluids, of
different specific gravities,"! the difficulty had not arisen.
Newton had shown that " particles flying each other with
forces that are reciprocally proportional to the distances of
their centres, compose -an elastic fluid whose density is as the
compression "IF — that is which follows a law that Boyle had
shewn to be true of ajr; \ Moreover, it was possible to suppose
that the repulsion necessary to make the solid atoms of a single

* Newton, Opticks, p. 376.

t Opticks, p. 375.

I This passage is given by Eoscoe and Harden (p. 125) as a further
quotation from Query 31. It appears rather to be a note of DaJ ton's own
upon the passage printed above.

§ Koscoe and Harden, op. tit., p. 14.

|| P. 13.

IF Newton's Principia (Motte's trans., 1729), ii, p. 77.

30 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

gas " fly each other " could be supplied by surrounding them
with a relatively bulky elastic atmosphere of heat.* But
Dalton's problem was to understand how uniform diffusion
could occur in a mixture of atoms of different densities ; and it
proved to be impossible to bring about this result by adjusting
their atmospheres of heat, without contradicting known facts
.about the specific gravities of the gases involved.! Then it
occurred to Dal ton to make the particles ' of his oxygen,
^azote, and water vapour of different sizes (meaning by size
"the hard particles at the centre and the atmosphere of heat
taken together "). If this were done, he supposed that (as in
a mixture of shot of two or three different sizes) the particles
-of one size would not be able to form a system in equilibrium
with particles of another size, but would ignore their presence ;
-equilibrium being reached only when the different types of
particles had become equally diffused. But there was an
apparent obstacle to this view. " At the time " when Dalton
"formed the theory of mixed gases" he "had a confused idea,
as many have, that the particles of elastic fluids are all of the
same size ; that a given volume of oxygenous gas contains just
-as many particles as the same volume of hydrogenous."!

Fortunately, perhaps, for the history of chemistry, Dalton
.saw reason to reject§ a doctrine which later became equally
with his own ideas part of the fundamental basis of chemical
theory. This obstacle removed, he believed that his theory of
gaseous interdiffusion held the field, and at once " it became an
object to determine the relative sizes and weights, together with the
relative number of atoms in a given volume. . . . Other bodies
besides elastic fluids, namely, liquids and solids, were subject, to
investigation, in consequence of their combining with elastic
fluids. Thus a train of investigation was laid for determining

* Roscoe and Harden, op. cit., p. 19.

t Pp. 14, 15.

% Dalton, A New System of Chemical Philosophy, 1808, p. 188.

§ Dalton, op. cit., p. 71.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 81

the number and weight of all chemical elementary principles
which enter into any sort of combination one with another."*

From these most interesting notes it appears that the origin
of Dal ton's Atomic Theory was not an attempt to explain the
fact that combination may occur between the same two sub-
stances in more than one proportion,! but an attempt to render
intelligible a group of physical phenomena. In the course of
this attempt, Dalton turned for data of the relative weights of
his atoms to the determination of the composition of different
substances that had been made by contemporaries of repute :
being already, from his physical considerations, convinced that
chemical combination takes place between particles of different
weights. " The extension of this idea to substances in general
necessarily led him to the law of combination in multiple pro-
portions," J and thus, as the result of this renewed application
of the concept of the atom, a new definite problem was
presented to chemists which has guided their researches ever
since ; a problem which would generally be stated as that of
determining what atoms of the elements enter into the com-
position of the " compound atom " or molecule of every
compound.

§31.

In the face of these historical facts it seems somewhat
paradoxical to maintain, as Professor Divers does in his
Presidential Address to the Chemical Section of the British
Association,§ that the atomic theory propounded by Dalton
" is not founded upon the metaphysical conception of material
discontinuity, and is not explained or illuminated by it."||

* Roscoe and Harden, op. cit., p. 17.

t This common belief is due to Thomson, History of Chemistry, i, p. 80,
ii, p. 291.

J Koscoe and Harden, op. cit., p. 51.

§ B.A. Reports, 1902, pp. 557-75. Cf. the criticism of Divers' views
by Meldrum, Avogadro and Dalton, 1904.

|| P. 558.

G

82 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

But a study of this valuable and interesting address shows
'that Professor Divers' real concern is to propound and illus-
trate a doctrine of the functions of the atomic hypothesis fjiul
hypothesis which is identical with that defended in this essay.
For him, as for us, the starting point of the science of chemistry
is a definite series of Objective facts. " The facts of a chemical
nature about common salt which cause the statement to be
made that it is a chemical compound of chlorine and sodium
are such as these : — Salt can be wholly changed into sodium
and chlorine ; these substances brought together change into
salt and nothing else ; salt and sodium, each under conditions
appropriate to it, change into the same substance, called also
a sodium compound, such as sodium hydroxide: salt and
chlorine, each in its OWB way, change into the same chlorine
compound, such as hydrochloric acid ; neither sodium nor
chlorine, one apart from the other or the other's chemical
compounds, ever changes into salt . . . ." Some such
''primary facts" as these, presenting themselves in ,a few
instances to Kobert Boyle, were apperceived, that is, were
rendered intelligible to him by the concept of undissolved
"primary concretions" of ultimate hard particles of different
•kinds existing side by side in the "grosser corpuscles," and
recoverable therefrom without change. This concept, as we
saw, led to the formulation of a definite problem — the problem
of determining in the case of a given substance what other
substances can "combine" to form it: or, as Professor Divers
(following Professor Duhem in his " return to Aristotle "*)
has taught us to say, what substances "change" into it and
under what conditions. Quite similarly, the really important
result from .a critical point of view of the renewed application
v by Dalton of theaatomic concept to chemical facts, is that it has
led to the definite fqrmulation of a further question. This is
the problem of determining what weights of substances display

* Duhem, Le Mixte et la Combinaison chimique, Ch. I.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 83

a new kind of equivalence to one another, which we may
express by saying that they are "chemically equal/'* just
as by saying that two bodies have "equal mass" we imply
that they display a definite kind of dynamical equivalence
to one another.f The atom — a concept which entered into
modern science in the seventeenth century from a phrtesophic
context in which its function was to harmonize the contradic-
tions of Being and Becoming — thus comes to be " the qualita-
tive and quantitative expression of the powers possessed by
substances to change into others/'J an expression which
contains no more than an historical allusion to the view
that " matter " is composed of actual minute indivisible
particles.§ In possession of the concept of a definite Objective
relation constantly showing itself between definite Objective
facts — the facts of chemical combination — we have, indeed,
reached the point at which at least one particular grou"p of
elements of the " picture " and the " object " coincide/; |j and we
have no further use, except, possibly, for purposes of exposition
and the like, of the earlier concept which first enabled us
to bring intelligibility into this region of Objective fact&IL

t

§ 32.

Whenever in the foregoing allusion has been made to the
fully determined particulars of ax province of the Objective, it
is highly probable that the reader- will have assumed that
quantitative or at least numerical determinations were intended.
It is a commonplace that Science only moves with security

m

where she can measure. Quite recently we have seen this

* Divers, op. cit., pp. 559 and 563.
t Vide infra, p. 99 ; Divers, op. cit., p. 562.
I Divers, op. tit., p. 566.
§ P. 559.
|| Supra, p. 72.

IT Cf. an article " On Methods of Teaching the Atomic Theory," con-
tributed by the writer to The School World, February, 1906.

G 2

84 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

truth demonstrated anew in the field of Biology, where Pro-
fessor Karl Pearson* has so brilliantly illustrated old Eoger
Bacon's dictum that Mathematics is the " gateway and key
to all other Sciences " ; while, doubtless, even before Egyptian
priests began to survey the lands left dry after the inundations
of the Nile, men felt the application of number and measure to
the spatial world to be natural and obvious.

But as Mr. Kussell has shown, if A is 12 inches and B
24 inches from 0, there is really an element of convention in
the familiar assertion that B's distance from 0 is twice as great
as A's.f Those distances are definite relations which cannot
strictly be identified with the relation of one number to
another. The fuller truth is that it is possible, since the
numbers form a " continuous series," to correlate every position
on the straight line 0 B with a single number, while there is a
practical convenience in arranging the " one-one correlation "
in such a way that if the distance (i.e., the spatial relation
itself before the advent of measurement) between 0 and A is
equal to that between A and B, the difference between the
numbers assigned to 0 and B is twice the difference between
those assigned to A and 0.

By the simple device of measuring with the foot rule, we
are able to overcome the difficulty that different perceived
distances between A and B have yet the same " representative
value,"t that is, refer to the same real distance. Much the
same holds good of such conceptions as temperature .and
weight. The same body at the same time may be pronounced
by two different persons to be hot and cold, a result which is
taken to mean not that the thing is both hot and cold, but that
the felt hotness and coldness are simply different representatives
of the same objective value. If a thermometer is placed in

* Pearson, Phil. Trans., vol. 185 et seq., also The 'Grammar of
Science, 2nd fed.

t Op. cit., p. 180 ; also supra, p. 29.
J Stout, Proc. Arist. Soc., 1903, loc. cit.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 85

contact with the body it is taken for granted that the different
positions of the surface of the mercury are each correlated
with one objective condition of the body. Thus if the thermo-
meter gives the same reading in the wind as it does behind a
screen, then the air, although it feels colder in the open, must
really be in the same objective condition, have, as Boyle
vividly expresses it, the same temper, in both places.* If now
we " graduate " the stem of the thermometer upon the foot-rule
method, we shall have a series of numbers correlated with the
various " tempers " or temperatures of the body. In this case
the statement that one difference of temperature is double
.another has obviously still more of the conventional character
than we noted in the case of distances,! for we have no
method of deciding that the difference between temperatures A
and B is equal to the difference in the case of B and C com-
parable with the use of the foot-rule in spatial measurement or

•of the pendulum in time determination.

*

§ 33.

The relation of this " objective condition " to what we have
hitherto described as " elements of the Objective " obviously
•demands discussion. It will at once be noted that there is no
reason to suppose that the " real " hotness of the body is ever
presented to any of the observers ; at any rate, it is certain
that it is never presentecl with any certificate of its supremacy to
the presentations of hotness which other observers experience.
We have found ourselves, in fact, compelled to regard all such
presentations as'standing on the same level of Objectivity^ and
have been driven in consequence into a certain amount of
metaphysical system-building. It is clearly opposed to the

* This problem is discussed by Boyle in his Experimental History of
Cqld, 1665, First Discourse ; also p. 513.

"'t Kelvin's "absolute thermometric scale" seeks to avoid this
•conventionality. A brief account of the theory of this scale of tem-
peratures is given in Ch. IV, infra, § 52.

J Supra, p. 15.

86 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

spirit of the whole of our previous argument to admit at mis
point the Objectivity, in our own sense, of a concept
which is quite obviously an hypothesis imported — like other
..hypotheses which we have seen and have yet to see — into the
' undoubtedly Objective data simply in order to render them
intelligible. Yet at the same time an hypothesis which is
instinctively adopted by everyone to whom the data are
'presented is something so very much like the Objective form
of synthesis that constitutes other Objective elements into a
" thing "* that its case seems to have a right to special
consideration.

Ft would appear that justice is done to its claims by the
admission that it is a postulate which has reference, strictly, not
to truth, but to action. It is true1 that in the sense explained
in § 5 a thing may not absurdly be said to have at the same
place and the same time many not-identical hotn.esses, but for
practical — -especially for social — purposes the recognition that
this is so would be highly inconvenient. For such purposes
the richness and many coloured variety of the actual primary
facts must be replaced by an artificial simplicity ; a result
which is reached by the instinctive hypothesis of a " same
representative value " which is attributed to the diverse
Objective presentations. The innumerable hotnesses perceived
around the body are conceived to represent, or to have
reference to, a " state of the body itself " which, since it is
independent of perception, is truly the same for all. This
postulate — which we may call the postulate of " pragmatic
objectivity" — performs then, a function somewhat analogous
to that of the Kantian categories ; it makes certain kinds of
social activity possible which without it could hardly take
place.

It is possible that we have here a not unhappy example of
many cases in which a distinction may profitably be made

* Supra, p. 17.

THE AIM .AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 87

between what is " pragmatically " and what is Objectively
true; between elements which are " the same for all" because
th^y are " relevant to purpose " and not because they are
Objective ; and elements whose sameness for all and relevance
to pui-pose are co-ordinate consequences of their Objectivity.
It is, I would further surmise, to "pragmatic truth" that
the maxim " Simplex sigillum veri " properly applies. Since
Lotze uttered his protest against its erection into a universal
logical principle* the progress of science has done much to
make more and more doubtful Keppler's early opinion that
" amat Natura simplicitatem " ;f while it has made the con-
ceptual simplification of the data a more and more essential
condition of theoretic success.^

To return to the case under consideration, we may make one
or two observations which throw light upon the readiness with
which the suggestion of u a • same representative value " is
accepted. As we have seen, the commonsense view of a
" thing " must, when subjected to philosophical rectification,
include the notion of an indefinite number of hotnesses which
are bound in one nexus of Objective relations. These
hotnesses undoubtedly have an identical reference to some-
thing, namely, to the " thing " or whole composed of qualities
and relational nexus to which they belong — or, rather, to what
may be called the cross-section of this whole at the moment
of observation. It is not difficult to understand that by the
plain man, whose keen interest in the practical efficiency of an
idea is unequally yoked with a very poorly developed disposi-
tion to criticism of its philosophical adequacy, they come to be
thought of as having reference to an identical hotness, much as
the qualities of individual Englishmen are thought of as having
reference to a " typical Englishman." Finally, the acceptance
of the recurrence of the thermometer index to the same point

* Logic, ii, p. 88.

t Poincare, Science et Hypothec, p. 173.

| See Larmor, B.A. Report, 1900, pp. 617-8.

88 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

as a sign that the same state of hotness has recurred presents
little difficulty. It is simply a fact that while many hotnesses
may be simultaneously perceived around the place which a
body " occupies " in the narrower sense of the term, presenta-
tions such as the one in question do not appeai\ in such
bewildering multiplicity. Thus, the position of the index which
is believed to be actually the same for all is readily correlated
with the postulated real state of the body.

Later, when attempts are made to fix with greater accuracy
the position of the index, the old difficulty breaks out anew.
The assumed uniqueness of the determination vanishes ; for it
is found that no observer is consistent with other observers, nor
even with himself. This time, moreover, it is not possible to
appeal from the fluctuating presentations to a stable one which
can be assumed to be correlated with the Objective value which
the former fail to indicate unambiguously. We are confined to
the circle of the actual discordant observations, and, if we are
to reach this real Objective value at all, must place our hopes
entirely upon some fortunate manipulation of our data. As is
well known, the .mathematical theory of observations indicates
a method by which we may calculate the " probable error " of
the series of determinations in question — that is, a method
which enables us to determine a number which represents a
deviation from the arithmetical mean of the observations within
which it is an " even chance " that the object of our search — the
real position of the index — will fall.

It would be difficult to decide the true interpretation of this
result. It is, of course, possible that the different observers
actually have the same presentations but fail to distinguish
them with accuracy— the error (of which we do not claim to be
able to give a rational account)* following the law assumed in
the mathematical theory. On the other hand, it is, perhaps,
permissible to suggest that when the observations are simply

* Cf. § 6 (last paragraph).

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 89

records of diverse presentations that have occurred to the same
•or to different observers under identical external conditions,
their diversity cannot properly be compared with the results of
imperfect muscular or instrumental adjustments such as those
which produce (to use an illustration much favoured in this
•connection) the distribution of hits around the bull's-eye of a
target. What we have maintained to be true of the diverse
perceived hotnesses may be true of the diverse perceived
positions of an index. They may have reference not to a " real "
position of the index which the actual perceptions aim at with
only imperfect success, but to an Objective relational nexus
which contains them. The fact that these presentations form a
.series between the extreme terms of which there is (in the
terminology of § 8) only a small distance, may render the
ordinary assumption of the theory of errors "pragmatically"
valid; just as Hooke's Law is pragmatically valid for small
deformations of any solid body.

In conclusion attention should be drawn to the fact that in
the view defended in this section and in the first chapter, the
various perceptions of hotness round a body are not regarded as
"effects" of a thing-in-itself lying beyond perception. There
.are Objective relations of implication between the several
hotnesses at a given moment of time, and in this sense alone
can it be admitted that " the principle of causality underlies the
whole procedure "* by which changes of hotness at a given
place are regarded as due to changes in the thing itself — this
" thing itself" being no more than the whole body of perceived
-qualities plus the nexus of Objective relations by which they
are connected. This interpretation of the facts seems capable
of satisfying the demand of common-sense for a "thing" that
.shall be more than a mere bundle of " possibilities of sensation" ;
while at the same time it avoids the serious difficulties that ensue
when a thing-in-itself, inaccessible to observation, is made the

* Stout, Froc. Arist. Soc., N.S., IV, p. 145.

90 THE ADI AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

" cause " of the series of actually presented facts. Nevertheless
this interpretation is by no means incompatible with the belief
that things have an " inner being " — possibly " ultimately
psychical," like the inner being of another person's toothache*
— so long as this inner being is not regarded as the reality of
which the facts accessible to observation are merely appearances.

§ 34.

When a hot body is placed near colder ones it gets colder,
they get hotter. These primary facts become intelligible — are
systematised — by the thought of a transference of " some-
thing" from the one to the other. This something is heat.
Black,t who made such important conquests for Science by
means of this concept, was one of those who are able to keep
on their guard against the dangers which Ostwald sees in the
BUd. He declines to form any definite conception of the
relation of the heat to the substance which occupies the same
space, on the ground 'that no Objective facts are before hinrto
justify his doing so. But if heat is regarded as a substance at
all, the " amount " of it which reaches the cold bodies must be
thought of as equal to that which left the hot body. .The
problem is set therefore of finding " something constant " at
both ends, so to speak, of the transaction. If a steady flame is
the " source of heat," it is impossible not to suppose that the
" quantity of heat " leaving the flame per minute is always the
same. Let us place above the flame in succession different
weights of water each for the same length of time. Examina-
tion of the results shows that the product ' of the weight of
water by the rise of temperature is in each case the same.
This constant product, then, may be identified with the
" quantity of heat " of which we are in search.

This simple example will serve to illustrate the weighty

* Stout, op. cit., p. 158.

t Black, Lectures on Chemistry, 1803.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 91

remark, made so long ago as 1867 by Rankine,* that " one of
the chief objects of mathematical physics is to ascertain, by
the help of experiment and observation, what physical
quantities are ' conserved.' "

The illustration also brings out the fact that the con-
stancies established in such investigations tire of an entirely
conventional character and refer to nothing Objectively " trans-
ferred." We assumed that the two temperature changes were
different aspects of the same transaction, an assumption whose
consequences are made psychologically available by throwing it
into the form of a transfer of " heat." We correlated the
various terms of the series of temperatures and weights which
appear in this transaction with numbers. If our initial
assumption was correct, it seems now that some manipulation of
the data — here the weights and temperature changes — must
yield an equality, the particular form of this manipulation
depending upon the particular manner in which the number
series has been correlated with the series of objective states of
the body. Our success in finding the desired manipulation
implies that, in the language of Lotze/f the bodies *dp " take
note " of one another's changes of condition, " and that the
data We have manipulated, that is the original data with which
numbers were correlated, are the complete expression of that
" notice."- In short, it is the verification in a particular case of
the postulate of the rationality of the world.

§ 35.

On the other hand, failure to find- this manipulation will
always prompt the investigator to look further afield to
discover other modifications of things which must be con-
sidered as elements, hitherto unrecognised, of the transaction
in question. Thus, to pursue our illustration a little further,

* Quoted by Merz, op. tit., p. 140. See also Divers, in B.A. Report,
1902, pp. 557, et seq.

t Lotze, Metaphysics (Eng. Trans.), i, § 45, p. 118.

92 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

if a definite " amount of heat " is " given " repeatedly to a mass
of air whose volume in one experiment is kept constant and
in other experiments is made to vary in an arbitrary manner, it
will be impossible to find a manipulation of the mass and
temperature changes of the air which will yield a constant
number that can be equated to the " quantity of heat given."
In other words, the change of temperature of the air is, in
this case, not a complete " notice " of the change in the body
which acted as the " source " of the heat. Some other partial
expression or expressions of that notice must be sought, and
in this case will be found, in the u work done " by or upon
the gas during the experiment. As is well known, it was
the perception of the significance in this and other cases of
the " work done " that led Mayer, Joule, and (later) Helmholtz
to the enunciation of the essentially modern conception of
" energy."

§36.

The result of such a process as has been described in the
foregoing section is always to bring the bodies between which
the new relations have been established into some sort of
unity.' This unity will take various forms and display various
degrees of permanence, to attempt a classification of which
would be scarcely profitable. But it is interesting to note
how frequently the " notice " taken by one body of another's
behaviour is thought of in the form of a transference of
substance. Besides heat and energy, momentum, electricity,
magnetism, and other cases will suggest themselves. This
transference sometimes is thought of in a shape which implies
that one part of the system is gaining what another part is
losing. Such is the case of the transference of heat which we
have just considered. In other cases the transference takes a
"circulatory" form as in the case of the "current" of electricity.
In this case the various bodies concerned — wire, battery,
galvanometer needle, &c. — are thought of as forming one

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 93

compound " thing " which displays many of the properties of
the genuine " things " of common sense. Thus the perception
of a certain deflection of a galvanometer needle may be
regarded as the perception of a certain " current," just as, when
a certain noise falls upon a man's ear, he asserts that he
" hears " a hansom cah.*

§ 37.

We have now reached, perhaps, a point from which we
obtain a clearer view of the circumstances under which, in the
history of Science, psychical events came to be excluded from
the causal series. To suppose that they are legally banished
under the terms of Hume's famous edict against investigations
that do not " contain any experimental reasoning concerning
matter of fact and existence," is a view that no one could hold
" except to save a theory."f And if they suffer through the
condemnation pronounced against inquiries that do not
"contain any abstract reasoning concerning quantity or
number" we see that this defect is not essential to their
nature as events, for "series" prevail in the psychical as
widely as in the physical world. The difficulty is reduced to
the practical difficulty of establishing for the terms of these
series (which, as I have pointed out in the first chapter, are
regarded as being " the same for all ") an unambiguous
correlation with the terms of the equally Objective number
series which happens, like much of the physical, to be not only
the same for all, but also accessible to all. ) Were such a cor- I
relation established it would apparently be possible to
determine whether certain psychical changes and physical
changes are or are not complete expressions of the " notice "
which soul takes of body or body of soul.

* Cf. Poincare, La Valeur de la Science, p. 227.
t Bradley, Appearance and Reality ', p. 324.

94

CHAPTEli IV. ; x

§38.

The special characteristic of tlie foregoing doctrine 'is its
insistence upon the co-ordinate Objectivity of many different
'•• tyPes °^ experience and the paramount importance of respecting
this Objectivity and of regarding the particulars of experience
'in which it is prima facie present as the ultimate facts. These
may, indeed, be shown to be related to other elements of the
Objective, previously unknown, or not known to have a special
relation to those in question, but may never be thought of .as
"explained" (in the sense of'. being explained away) by the
discovery of these relations. At the. same time our doctrine
does not forbid the scientist to build upon the basis of these
primary facts secondary constructions in which elements from
other categories of experience are involved. The alien
elements thus introduced often serve the important function
of bringing to light relations between the primary facts which
could hardly have been discovered without their aid. This
truth has been illustrated sufficiently in the last chapter.
The secondary construction can, without difficulty, be conceived
to be extended so as to unify the primary facts both in the
province from which the interpretative elements are borrowed
and the province in which they are applied. In this way a
result may be achieved which, for the purposes of scientific
investigation and exposition, may be regarded as a reduction
of the facts of the latter province to facts of the former.
It is obvious that the extent to which unification of the
various provinces of scientific inquiry can thus be brought
about is identical with the range over which a concept, or a
group of concepts, drawn from a single experiential context,
can be applied to make facts intelligible.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 95

It is well known that physicists have repeatedly attempted
a reduction of the happenings of the physical universe in terms
of the concepts of mass and motion. This chapter will be
devoted to a brief analysis of the main features of these
attempts.

§39.

We have seen that "the investigators of the seventeenth
and eighteenth centuries had reached clear concepts of the
temperature of a body as its real state of hotness or coldness,
and of heat as an hypothetical substance whose transference
from one body to another renders intelligible the changes of
temperature that ocfcur when bodies are brought near to one
another. Black, to whom this concept in its. modern form is
due, has the credit of applying it to phenomena — many of
which he 'himself discovered — which before his consideration
had not received adequate theoretical explanation. The most
important of these were the phenomena attending the trans-
formation of a substance from the solid to the liquid, or from
the liquid to the gaseous condition. If (for example)* a pot
of water is heated above a steady flame its temperature will
not rise indefinitely. When the " boiling point " has been
reached the whole of the water will be converted gradually
into steam — the most interesting fact (unknown before Black's
researches) being that the temperature of the steam is no
higher than that of the water. The constant and fairly regular
rise of temperature before the boiling was attributed to a
regular transference of heat from the flame to the water. It
would be contradictory to suppose that this transference ceased
when the water began to boil, f Consequently, since the heat-
substance is not "sensible" in the form of an increased
temperature of the water or steam, it must have entered into
a special combination with the particles of the water — a

* Black, Treatise on Chemistry, 1803, i, p. 164.

96 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

combination that accounts for the change of state which the-
water undergoes : it becomes " latent heat."*

§40.

But the concept which Black had handled with such
splendid audacity was not destined to a long career of
theoretical adequacy — although for elementary didactic pur-
poses it is still indispensable.f Within fifty years of Black's
researches the experiments of Eomford and Davy had made it
clear that heat must be thought of as capable of being
generated without limit by mechanical work.

The effect of this discovery may be represented symbolically
by the statement that it was tow' impossible any longer to-
conceive heat as a substance ; critically, it implied that the
changes of temperature of bodies cannot be regarded as a closed
series of events; but that a change of temperature may, in
fact, sometimes be the way in which a body " takes note " of
a change of the " mechanical " order in another body. At this
point, therefore, it becomes necessary to submit the funda-
mental notions of mechanics to a brief critical scrutiny.

§41.

The development of these ideas presents, in many respects,
an interesting parallel with the development of the ideas of the
science of Heat.J We begin with a number of kinsesthetic and
" pressure " sensations whose object is conceived vaguely as the
" force" of the body to which they are due — the heavy stone, the
stretched string, the bent rod, the moving missile. It is recog-
nised (as in the case of hotness and coldness) that these sensa-
tions, even when they are different, may often refer to the same

* Black, op. cit., pp. 176-7.

t See a lecture by the present writer printed in the Educational Times
for May, 1905.

J I have considered it from this point of view in an article on
"'Mass' and 'Force' in School Mathematics " in School (December, 1905)
The next few paragraphs are in the main a condensation of that article.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 97

objective* " force." Thus, though the force required to stretch
a given elastic cord, or to bend a given rod to a definite extent,
will seem very different when one is fatigued and when one
is fresh, very different in the cases of a man and a child, we
admit instinctively that it must " really " be the same in every
case where the same change has been brought about by
manipulating the cord or the rod in the same manner. Such
changes in the cord or rod would readily be regarded, like the
positions of the index of the thermometer, as objective records
of the forces applied, and would need only to be correlated
with numbers by means of an arbitrary scale (such as the
divisions of a foot rule arranged suitably in connection with the
cord or the rod) to be practically useful. In whatever way
the change in the cord or the rod has been brought about it is
always possible to find a body whose weight applied in the
same manner as the force in question can produce the game
change in length or shape, so that we pass readily to the
practice" of equating forces to the weights of bodies. When
this stage is reached it is possible by a very natural convention
of a character previously discussedf to proceed beyond the
mere recording of forces to their meastwement : a force equal
to the weight of 2 Ibs. is thought of as twice a force equal to
the weight of 1 Ib.

- §42.

At this stage it is clear that a different method must be
used for estimating the "force" which is ejectively thought
of as present in a moving body ; and the method just described
loses its theoretical security even for static measurements when
it is discovered that the weight of the same body is different
at different points of the earth's surface.^ The need is now

* In the sense explained on p. 86.

t Supra, p. 28.

| Vaguer notions of the distinction between the mass and the weight
of a body arose historically from simpler experiences (cf. Maxwell, Theory
of Heat, 9th ed., p. 86).

H

98 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

felt of another concept to replace the weight of a body as
a measure of what has vaguely been thought of as the
" quantity of matter " in it. Examined critically, this need
is found to resolve itself into an instinctive demand for some
constant coefficient that shall characterize the behaviour of
the same body in all " dynamical transactions "* into which it
enters, including those in which it presents the property of
a variable weight. Investigations similar to those which
Huygens and his English contemporaries carried out upon
colliding bodiesf supply the^best materials for a clear answer
to the demand thus expressed, and lead at the same time to
a perfectly general and satisfactory concept of force. If two
inelastic bodies moving directly towards one another with
equal velocities of any given value bring one another to rest,
it is found that however their velocities are changed in amount,

* Clerk Maxwell's happy phrase.

t No. 43 of the Philosophical Transactions (January 11, 1668-9) con-
tains an enunciation of Dr. Wallis's General Laws of Motion (pp. 864-6),
and of Wren's Lex Naturae de Collisions corporum (pp. 867-8). Wallis
dealt with inelastic bodies ; Wren (like Huygens) with ideally elastic
bodies. In No. 46 (April 12, 1669) Regulce de Motu Corporum ex mutuo
impulsu, "communicated by Mr. Christian Huygens in a letter to the
Royal Society " were published, like the communications of Wallis and
Wren, " in the language of the learned " for the benefit of Continental as
well as of .English inquirers. In connection with the hypothetical course
of investigation followed in the text it is interesting to note that Wren
had' formed a concept of " velocitates corporum proprise et maxime
naturales [quse] sunt ad corpora reciproce proportionates." The property
of such velocities is that "[duo] corpora, R et S, habentia proprias
velocitates, etiam post impulsum retinent proprias ;" while a general law
(for elastic bodies) is expressed in the form, " Quantum R superat et S
deficit a propria velocitate ante impulsum, tantum ex impulsu abstrahitur
ab R et additur ipsi S, et e contra."

Huygens, on the other hand, reaches, in Regula 8 (p. 928) the
general law of the text in the form " Quantitas motus [i.e., momentum]
duoruni corporum augeri minuive potest per eorum occursum ; et semper
ibi remanet eadem quantitas versus eandum partem, ablata inde quanti-
tate motus contrarii."

It is instructive to notice, also, that Wren uses for the modern
" mass " the unexplained term " corpus," while Huygens explicitly states
of his bodies that " eorum moles oestimatur ex pondere."

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 99

so long as they remain equal the same result will follow.
Consequently these bodies may be thought of as possessing
a certain equivalence to one another which we may express
by saying that they have equal mass. Directly or indirectly
it may be shown that bodies of equal mass as thus defined
have also, at the same place of observation, equal weight.
Since we have agreed to the convention that a body whose
weight is equal to the weight of two other bodies of equal
weight shall be considered as having twice the weight of one
of them, it is natural to think of such a body as having at the
same time twice the mass of one of them. But the circum-
stances under which the term " equal mass " became applied
to the original pair of bodies do not suggest directly any sense
which the assertion " A's mass is twice B's " can bear ; so that
experiments must be instituted to discover transactions in
which it will be possible to find a useful meaning for the
assertion in question that shall be in accordance with the
assumption that mass is directly proportional to weight. This
meaning is reached when it is found that if we take two
inelastic bodies A and B of which A is n times as heavy as
B and therefore has, ex hypothesi, n times the mass, these
bodies will bring one another* to rest if they collide with
velocities inversely proportional to these masses. As the
result of such observations the " mass " of a body becomes
definitely conceived as the reciprocal of the ratio which the
velocity of the body in question must bear to that of a standard
body whose mass is taken as unity, if in inelastic collision the
two bodies are to bring one another to rest.

§43.

We have now to reach the conviction that this coefficient
characterises the behaviour of the body in other transactions
besides collisions. Before this step is taken we must note that
the equivalence which we have described as equality of mass
is only a special case of the more general equivalence pre-

H 2

100 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

sentecl by bodies so circumstanced that the product of mass
into velocity is the same for each. To this equivalence is given
the name, equality of momentum. Thus the momentum
which Descartes thought of as a measure of .the " force "
of a moving body is properly a piece of descriptive apparatus
by means of which an account may readily be given of cases
in which colliding bodies bring one another to rest. By ^,n
easy extension of the experiments to cases in which rest does not
result, it is found that this descriptive instrument has a still
wider range of usefulness. Eecognising that momentum is a
" vector quantity,"* we may bring all cases of collision under
one formula : " The total momentum of the two bodies is the
same after as it was before the impact ; momentum being
merely transferred from one body to the other."

But experience presents us with cases in which twcT BocHes
enter into transactions with one another which involve a
gradual change of momentum instead of the sensibly instan-
taneous transference which characterises impact. In certain
cases (e.g., when two bodies of different mass are drawn together
by an elastic cord stretched between them) it can be demon-
strated that the movements of the bodies under these new con-
ditions can still be brought under the old descriptive formula :
momentum is not lost during the transaction but is constantly
being transferred from one body to the other. But in this
case of gradual transference the question of the rate of
transference at a given moment is one that is bound to arise ;

* I.e., a quantity which is completely specified only when its direction
and " sense," as well as its magnitude, are given. Such quantities can be
represented' by vectors, that is by straight lines of proportional length,
drawn in the appropriate directions, the " sense " being indicated by an
arrow-head on the line. Vector quantities, must, moreover, follow the
law of vector addition ; in other words, two or more of them must be
replaceable by a single one, which is represented by the "sum" of the
vectors of the others. The "sum" of the vectors, a, j3, y, . . . ., is the
vector obtained by drawing a, £, y . . . . end to end (the arrows all
pointing the same way round) and joining the first point of a to the last
point of the last vector.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 101

and when it arises is seen instinctively to depend upon the
" pull " in the cord which is drawing the bodies together. Wk
seem to perceive, in fact, an alternative method of measuring .
the " force " in the cord ; namely, by the rate of transference of
momentum from one body to another which the cord in a given
state of elongation mediates. The soundness of this method is
capable of an indirect verification. Imagine a number of
precisely similar cords to be suspended vertically and to be
stretched, each to the same extent, by loads of different
materials attached to their lower ends. Then we are bound to
think of the elongations of the cords as being caused by equal
" forces," the equal weights of the suspended loads. If at the
same instant the loads are all set free and allowed to fall
(in vacuo}, we are bound once more to think that the circum-
stances of their motion are determined by the same " weight "
which caused them to elongate the cords to the same degree in
each case. If the forces in the stretched cords could be
measured unambiguously not only by the weights which
produce the given elongation but also by the rate of change of
momentum of a body drawn along freely by a similar cord
stretched to an equal degree ; then it would be reasonable to
expect that the different loads now supposed to be falling
freely would exhibit equal changes of momentum. But since
their weights are equal their masses are equal also. It remains
to be shewn, therefore, that the falling masses gain velocity at
the same rate irrespective of the material of which they are
composed. As is well known, there is abundant experimental
evidence that this is the case.

Making, finally, an easy abstraction from the complications
introduced by the extended character (and consequent rotatory
movements) of the bodies in our experiments, we are now
prepared to hold with much confidence that when two
"Articles" are involved in a "dynamical transaction" the
rates of change of momentum along the straight line joining
them are at any moment equal and opposite, and that this rate

102 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

of transference of momentum is the complete expression of
the " notice " which one particle is taking, at the moment, of
the positional changes of the other.

§44.

We have thus reached by a route partly historical, partly
hypothetical, the classical Newtonian position. The concept
of a special type of transaction ideally supposed to take place
between two mass-particles, is extended in the Newtonian
mechanics to account for all happenings in the material world.
But to "effect this extension it is necessary to think of the
behaviour of a given particle at a given time as an expression
of the fact that it is "taking note" of the behaviour of every
other particle in the universe. It is obvious that it cannot do
so in the simple way which alorie~has been contemplated "by us
up to the present — the particle cannot exhibit at one and the
same time a rate of change of momentum directed along
each of the straight lines joining it to the other particles of
the universe, and equal in each case to the rate of change of
momentum exhibited simultaneously by the particle at the
other end of the straight line. "The distinct and general
formulation of the principle " by which (in modern terms)
the actual ra"be of change of momentum exhibited by
the particle is regarded as the " vector sum " of these
elementary rates, is one of the " chief advances " which
we owe to Newton.* But, as Mr. Eussell has, in effect,
pointed out in his penetrating criticism of the Newtonian
doctrine,f the acceptance of the vector law involves an important
widening of our concept of the way in which one material point
may " take note " of the behaviour of others ; for it is clearly
only by a convention that we can regard this rate of change of
momentum actually presented as the sum of rates of momentum

* Mach, op. cit., p. 1 92.
t Op. cit.t Ch. LV.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 103

none of which are present in the " resultant." The apparent
further difficulty that it would be, in practice, impossible to
effect the vector summation of the " notice " taken by a particle
of all the other particles in the universe, disappears when we
adopt a formula, like Newton's Law of Gravitation, which
makes the rate of change of momentum exhibited by two
particles at a given moment a function of the distance between
them of such a character that for great distances it becomes
negligible.

§45.

It is of more importance to note with Mr. Russell,* that in
the present state of mathematical / theory, it is no longer
possible to think of a particle as possessing a " velocity "
or an "acceleration," or (consequently) a rate of change
of momentum. Beturning to our single pair of particles^-
A and B, moving towards one another along a straight line, the
Objective facts are that at the moments ti, t2, and ts, A
occupies the points PI, P2, and P3 ; while B occupies the points
pi, p2, and ps. Now suppose that? r seconds after A leaves the
point PI, it occupies a point between PI and P2 distant B from PI.
Then the fraction S/r will have a definite numerical value
which it is customary to call the average velocity of A
during the time r. It is clear that if we make this time
shorter repeatedly we shall obtain a compact series of these
fractions. Under these circumstances it will be possible, as a
rule, to specify a number, Vi, which is the limit of this series of
fractions ; that is such a number that it is impossible that any
number should lie between it and the compact series of fractions.
This number is the velocity of the mass particle at the point
PI. Thus defined, it is obvious that the velocity is nothing
that can be thought of as a state of A. If in a similar way the
velocity of the particle B at the point p\ be supposed to be deter-
mined, and to have the value v\t then the doctrine of mass which

* Op. cit., p. 473.

104 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

we have seen to be equivalent to Newton's doctrine asserts that
the ratio of the numbers Vi and v\ (if the particles be supposed to
start from rest at Piand^i as in the ideal experiment described
on p. 100) characterises the behaviour of the two particles
throughout the " transaction " ; so that if when the particles are
at any other pair of points, P and p, the velocities are V and v,
the ratio of the changes in the values of the velocities, i.e., the
ratio (V— Vi)/(^— ^i) will preserve the value which we have
already defined as the inverse ratio of the masses of the particles.
It follows from the well-known relations between the velocity of a
point, the time it is in movement and the distance through which
it passes, that the distances PiP2 and p\p% will be again in this
characteristic ratio ; though their actual values will depend upon
the " law " which gives the velocity of one of the points at different
distances from the other. It will be clear, in fact, that in
general a given pair of these second positions will imply a
definite law of this kind ; so that when we have determined
two pairs of positions of the particles, such as PI, ?2, and p\t p2,
we have the data (namely, the mass ratio and the " law of
distance " involved) which are alone necessary to determine the
subsequent movements of the particles. Generalizing this
result, we reach the conclusion that whenever a particle takes
up at a definite moment a definite new point of space its
occupation of this point may be regarded as a " taking note " of
any two previous configurations of all the points of the universe,
including itself, and, therefore, a taking note of all such con-
figurations.

§46.

In view of the fact that we have ascribed only psychological
value to the concepts of momentum and force, regarding them
merely as instruments for the " economic description " of
Objective facts, suspicion may arise that this view of the
" causal law " which governs the behaviour of a mass-particle
is conventional also, so that if another set of descriptive

> THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 105

notions were employed it might not necessarily be implied in
the resulting system of mechanics. Hertz's mechanics* is
a system in which the notion of geometrical connections
between the mass-particles (supplemented when occasion
demands by hypothetical " concealed masses ") replaces the
" forces " of the Newtonian secondary construction. For this
reason Mach regards the Hertzian mechanics as an attempt to
substitute integral laws for the differential laws that make up
the ordinary science of mechanics.f

That this does not imply that the motion of a particle is
determined by consideration of only one configuration of the
system to which it belongs instead of by two, as in the
Newtonian mechanics, follows from the consideration that
Hertz's principles lead us to the assertion that under any given
set of conditions supplied in the " integral " form of geometrical
connections the particle will take the path of the least possible
curvature. But to specify the " curvature " of a path we must
make use of a second differential coefficient, just as we must to
specify an acceleration. Thus, by an argument closely
analogous to that of the preceding section it may be made clear
that in this case also one configuration of the particles of the
system must be thought of as determined by at least two
previous configurations.

§47.

A very little consideration will show that this concept of
the " notice " which one part of the universe takes of the rest
must be applicable to other elements of the warp and woof of
Nature besides the obvious movements of matter. "We have
seen, for example, that the science of Heat is built up upon
a basis of correlation of the number series with the observed
series of states of bodies which are called temperatures. We

* Hertz, Principles of Mechanics, Eng. trans, by D. E. Jones,
t Mach, Science of Mechanics, 2nd Eng. ed.. p. 255.

106 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

have seen that under the guidance of the concept of " heat " as
a substance which can be transferred from one body to another
it has been found possible to unify the facts of temperature
change by the observation that (in an ideal case where only
two bodies, A and B, are concerned) changes in temperature of
A bear a definite and constant ratio to the like changes of B.
The analogy to the law connecting the changes of velocity of
mass particles is obvious and might be taken as a support
of the view that temperatures may be reduced to dynamical
terms ; the state which we call by the name temperature being
really a " state of motion " of the particles of a body. But as,
following Mr. Eussell and the modern mathematicians, we have
seen that velocity is not a state at all, it is clear that the
analogy must be differently conceived. The temperature of
the body A at a given moment is the " notice " it takes of two
preceding pairs of simultaneous temperatures of B and of itself,
just as the position of a particle is the " notice " it takes of two
preceding configurations of the universe to which it belongs.*

The only conclusion that it seems legitimate to draw from
the consideration oi this and other similar cases is that the law
we have been studying is the necessary condition that must be
present if it is to be possible to regarfPone kind -of changes in
bodies as expressions of the " notice " taken by them of changes
of a different order elsewhere. It seems safe to assert that no
order of phenomena which does not exhibit this law can be
brought into relation with one in which the law obtains, but it
would, perhaps, be dangerous to maintain that such orders do
not exist.

* I have ignored in this discussion the fact that the " specific heat " of
bodies is only approximately constant. The recognition of this fact would
make the argument more complicated, but would not affect it essentially.
Cy. J. J. Thomson, Applications of Dynamics to Physics and Chemistry^
1888, p. 90.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 107

§48.

Whether, as Mach would have us believe,* Newton was well
aware that his concepts of quantitas motus, vis impressa, and
the like are merely of psychological value in connection with
the investigation and transformed statement of actual facts, or
whether he attributed to them objective validity, is a matter
upon which it would be neither easy nor profitable for us to-
come to a decision. There is no doubt whatever that the latter
is true of the majority of his contemporaries and successors, at
the very least until the time of D'Alembert, if not, indeed,
until the present day. Thus allusion has already been made to
the famous controversy between the followers of Descartes
and Leibniz, upon the subject of the " correct " method of
measuring that " force " of a body in motion which we have
noted as one of the vague notions in which the science of
mechanics has its birth. From the standpoint of the develop-
ment of the science, the great value of the controversy was thatr
by concentrating attention upon the theoretical aspect of the
method by which individual problems in mechanics were
solved, it led to an elaboration of definite bodies of mechanical
doctrine with a definite nomenclature. From the critical point
of view, the controversy is most interesting as throwing light
upon the way in which the most pretentious and far-reaching
scientific doctrines are based upon a mass of vague experiences,,
of " instinctive perceptions," the precise nature of which it is
not always given to the founders of the science to recognise
for what it really is. Thus even Thomson and Taitf admit
that Newton's " axioms " " must be considered as resting on
convictions drawn from observation and experiment, not on
intuitive perception." But Newton, like Gauss, exhibited his
(edifice of thought to the world only "when the scaffolding had
been removed," and has left us no precise indication of the

* Science of Mechanics, p. 193.

t Thomson and Tait, A Treatise on Natural Philosophy, 1867, p. 178.

108 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

" observations and experiments " from which he drew the con-
viction that the product of mass and velocity was the
determining circumstance in dynamical transactions. Hence
we have found it advisable to devise a hypothetical series of
experiences by the consideration of which this conviction might
be reached. It is otherwise with the conviction ihat the " vis
viva," the product of the mass into the square of the velocity,
is the determining circumstance.

§49.

In 1669 Huygens acquainted the world, by means of the
Philosophical Transactions "of our own Eoyal Society, with
his laws of motion of colliding elastic bodies. In 1703 a
posthumous Tractatus de motu corporum ex percussione was pub-
lished,* in which these laws appeared systematically deduced
upon the basis of certain assumptions, which as a rule are of the
kind called by Mach " instinctive perceptions," and are clearly
set forth.

Of this character, for example, is the assumption that the
velocities, relative to an observer, with which two colliding
bodies separate, will not be changed if the two bodies and
the observer have an additional common velocity imposed upon
them — as they would have, for instance, if the experiment were
performed upon a uniformly moving boat.f But when Huygens
reaches the case of elastic bodies of unequal mass, and
endeavours to prove the proposition^ that if such bodies collide
with velocities inversely proportional to their masses, they will
separate with the same velocities, he finds himself able to do so
only by showing that a contrary supposition is forbidden by an
axiom of complete certainty in mechanics. " which states that
in a movement of bodies which is caused by their own weights,

* Translated as No. 138 of Ostwald's Klassiker der exakten Wissen-
tchoften. I have used this translation,
t Tractatus, p. 370.
t No. VIII. Pp. 381-6.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 109

their common centre of^gravity cannot possibly rise."* Here
we have an appeal to a principle, used again with great effect
in Huygens' famous solution of the problem of the " compound
pendulum,"! which, in the hands of the Bernouillis, developed
into the Principle of Vis Viva& which Helmholtz made his
starting point in his epoch-making work on the " Conservation
of Energy."§ The same instinctive perception that a body or
a system cannot, under the influence of its own weight, move in
such a way that the centre of gravity is raised — a conviction
which is recognised as identical with the denial of the possi-
bility of the perpetuum mobile^ — was shown by Machl" to
underlie the Principle of Virtual Work (or Velocities) which
Lagrange made the basis of his Mtcanique Analytique, and hence
of the method of " generalized co-ordinates " which plays so
important a part in modern mathematical physics.**

§ 50.

Helmholtz starts with the Newtonian concept that the
task of physical science is to reduce the phenomena of Nature
to unchanging, attracting, and repelling forces acting between
mass-points; the magnitude of the forces depending merely
upon the distances between the points. He showsff that in
such a system of mass-points and forces the vis viva will be

* Tractatus, p. 382.

t Horologium oscillatorium, 1673. The solution is described by Mach>
op. cit., pp. 174-9.

| Huygens had, in the Regulas of 1669, already formulated for
collisions of perfectly elastic bodies the rule : " Summa- productorum a
mole cujuslibet corpori duri, ducta in quadratum suae celeritatis eadem
semper est ante et post occursum eorum " (Eeg. 6, Phil. Trans., No. 46,
p. 928).

§ Helmholtz, Ueber die Erhaltung der Kraft, 1847.

|| Cf. Helmholtz, op. cit., p. 8 ; also Mach, Popular Scientific Lectures^
Eng. trans., 1894, p. 147.

TT " On the Conservation of Energy " in Popular Lectures, 1894.
** See Routh, Rigid Dynamics, i, Ch. VIII.
tt Op. cit., pp. 10-12.

110 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

conserved: meaning by this statement that whenever a
particle returns to a position which it has temporarily left)
in an otherwise constant configuration, its vis viva will regain
its old value. This somewhat narrow principle is extended in
the next section* to the case when the particle, still under
the action of central forces, has not yet returned to its
original position. In this case, by distinguishing a new
quantity (to which Eankin gave the name "potential
energy " f while calling Helmholtz's vis viva " kinetic
energy ") he was able to show that the sum (as we should
•call it) of the kinetic and potential energies was constant.
In this way Helmholtz reached the Principle of the
'Conservation of "Force" [Energy] for the case of particles
under the action of central forces. The next two sections
of the work are devoted to the consideration first of the cases
in which the principle of conservation is already implicitly
known to apply, and next to the most interesting casesj (such as
inelastic collision and friction) which have hitherto been thought
to be cases of an absolute loss of " force " [energy], It is in this
chapter that Helmholtz brings into line with his own systematic
mathematical treatment the experiments in which Joule has
shortly before obtained evidence of an " equivalence " between
work and heat ; and so, for the first time, makes it appear
possible that the hitherto autonomous province of the science
•of Heat will be annexed by the expansion of the classical
Newtonian Mechanics.

§ 51.

But the doctrine of the Conservation of Energy, although it
formulates conveniently a connection between dynamical events
and temperature changes, does not of itself effect the reduction
we are seeking. It does not, that is, enable us to describe

* Op. cit., pp. 13-19.

t See Merz, op. cit., ii, p. 139.

| See Merz, op. cit.t pp. 25-6.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. Ill

transactions in which temperature changes occur in terms of
motions and mass-coefficients only. The critical treatment to
which we have subjected the concepts, kinetic energy and heat,
has been sufficient to show that they belong to the order not of
primary facts but of secondary constructions. The substantial
shape we give them is legitimate if we remember that it
makes the results of experimental or mathematical investigation
" psychologically available." But the identification of heat as a
" form " of energy has no real value; merely extending, in fact,
the psychological usefulness of the substantial conception of
energy to render intelligible the fact that a change in the value
of the arithmetical quantity ^mv2 may at times be accom-
panied by the appearance of temperature changes under
conditions which are conventionally described by the statement
that a quantity of heat has appeared or disappeared which
bears a definite ratio to the change in the value of the former
quantity.

§52.

The correlation of temperature states with definite
quantities of mechanical work theoretically effected by Lord
Kelvin's absolute thermodynamic scale, would appear at first
sight to be, in effect, • equivalent to the desired reduction.
Lord Kelvin's argument* showed that in a reversible heat
engine, the "efficiency" (that is, the ratio of the quantity
of heat converted into work by the engine to the total
quantity which it draws from the " source ") depends upon
nothing but the actual temperatures of the source and the
" condenser." Suppose, then, two reversible engines drawing
heat from the same source A but returning it (less the
equivalents of the work done by each) to different condensers
B and C. Let the engines work until each has drawn the
same total quantity of heat from A. Then the amounts of
work done will depend merely upon the temperatures of B

* See Preston, Theory of Heat, p. 611 et seq.

112 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

and C, and may be taken therefore as measures of the
differences between these temperatures and that of A. Thus,
if the second engine does twice as much work as the
former, the difference of temperature between A and C may
conveniently be taken to be twice as great as in the case of
A and B. We may assign, then, any arbitrary number to the
temperature of A, and in this way correlate with any standard
temperatures available (such as the boiling and melting points
of pure substances), numbers which will be independent of the
properties of any particular substance. It will be convenient,
however, so to choose the number assigned to the temperature
of A that in the ideal case in which all the heat drawn from
the source is converted into work, the number assigned to the
temperature of the condenser shall be zero.*

§53.

Unfortunately, however, this correlation of temperatures
with quantities of mechanical work does not enable us to
dispense with the notion of temperature as a definite state of
a body which is the object of our sensations of hotness or
coldness. The argument which made this unambiguous corre-
lation possible is not developed from mechanical data only,
even if we ignore the difference between heat and kinetic
energy. It involves our direct experience (formulated in the
axioms of Clausius and Lord Kelvin) that hot bodies in the
presence of colder ones get cooler, never hotter, except
(ultimately) by the intervention of animate agencies. f

To effect the complete annexation and assimilation of the
science of Heat, Dynamics must, then, give some account of
the behaviour of solid and liquid bodies from which (as in
the Kinetic Theory of Gases) this irreversible property of
temperature can be deduced.

* The "absolute "zero.
t Preston, op. cit., p. 612.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 113

The most important attempts in this direction have been
reviewed by Professor G. H. Bryan in an important report
" On our Knowledge of Thermodynamics, with particular
reference to the Second Law."*

There are three conditions which must be satisfied by any
dynamical analogue of temperature. It must play the part
which temperature plays of determining the changes in the
system (when it is brought into contact with another system)
which result in " thermal equilibrium " ; it must be shown
that, with the dynamical analogue, systems which are "at the
same temperature " as a third system, are in " thermal
equilibrium " with one another ; lastly, the analogue must exhibit
the property known sometimes as Clausius' Theorem, sometimes
as the Second Law of Thermodynamics. This law states that,
for all reversible transformations which a body undergoes,
if the energy supplied in the form of heat be divided by
the absolute temperature at which it enters the body the
quotient will be a "perfect differential." This expression
implies that whenever the body undergoes a series of changes
of pressure, volume, and temperature, the integral! of the
quotient in question throughout the series defines an
unambiguous change in the body called a change of entropy,
which is independent (if the condition of reversibility is
maintained) of the route by which the body passes from its
initial to its final state. If we concern ourselves only with the
quantity of heat given and make no reference to the absolute
temperature at which it passes into the body, we do not possess
the data for determining unambiguously the final state of the
body. Hence the absolute temperature (which converts the

* B. A. Report, 1891, pp. 85 et seq.

t Suppose the series of changes to be broken up into any number of
sections. Also suppose the heat received during each section to be
divided by the absolute temperature at the beginning of that section, and
the sum of the quotients to be taken. As the number of sections is
increased the sum will approach a limit (p. 103). This limit is the
integral.

I

114 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

increment of heat into a perfect differential) may be called an
integrating divisor. The third property of (absolute) tempera-
ture may be expressed, then, by saying that it must be an
integrating divisor of the energy which is given to the system
as heat.

§64

Almost the only writer who has attempted to devise kinetic
analogues which shall satisfy at once all these conditions is
Helmholtz,* whose u monocyclic systems " can be shown to do
so if certain assumptions are allowed.

In view of the account which we have already given of the
part which Helmholtz played in the derivation of the principle
of the conservation of energy as an extension of principles
founded on the postulate that dynamical transactions can
be reduced to attracting and repelling forces between mass-
particles,f it is interesting to find that he has attempted
a dynamical interpretation of temperature by a method that
avoids the postulation of the existence of an infinitely large
number of molecules. This can be done by means of assump-
tions which express the kinetic and potential energies of the
system in terms of the amounts and variations of a number of
co-ordinates — a term which, by a usage introduced by Lagrange,
may be extended from its simple geometrical significance to
any quantity which can be used to fix the spatial configuration
of a system.^

But the whole difficulty of the present case is the faqt that
we are ignorant of the " co-ordinate " which describes the
configuration of the molecules vhen a body is at a definite
temperature. The molecules are, in fact, " concealed masses,"
"whose position still remains unknown when the co-ordinates
accessible to observation have been completely specified." §

* Bryan, loc. cit., p. 104.
t Supra, § 50.
I Bryan, loc. cit., p. 96.
§ Hertz, Mechanics, p. 223.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 115

Their configuration, therefore, is to be fixed by what Professor
J. iJ. Thomson has called " unconstrainable " co-ordinates, in
distinction from the " controllable " co-ordinates which are
subject to the direct control of an experimenter.* Finally,
by supposing the " concealed masses " to form a " cyclical
system," that is, to possess constantly recurrent motions, it
is possible to " ignore " the unconstrainable co-ordinates, and
deal only with their time-rates of change.!

(A simple example of a monocyclic system in which the
concealed motion is determined by one " unconstrainable "
co-ordinate, would be a perfectly uniform rotating wheel.
The angular position of a point on the wheel would be the
" unconstrainable " co-ordinate sufficient to fix the position
of the wheel at a given moment, while its time-rate — the
angular velocity of the wheel — would alone affect the expres-
sion for the kinetic energy of the system.)

The main features of the course of Helmholtz's somewhat
recondite argument are as follows : — An expression is first
obtained for the additional energy that may be imparted to
a simple monocyclic systemj by means of the concealed masses
only. This energy corresponds to the heat given to raise the
temperature of a body. It is next shown that the whole
kinetic energy of the concealed masses is an integrating divisor
of the expression for this additional portion of energy. Thus it
appears that the kinetic energy of the concealed masses — or
a product of this kinetic energy by any one of a certain class
of functions§ — possesses the property which Clausius' theorem
indicates as distinctive of absolute temperature. || Helmholtz

* Thomson, Applications of Dynamics to Physics and Chemistry, 1888,
p. 94.

t Thomson, Applications, p. 14 ; Hertz, Mechanics, p. 209.

J Le., a system containing only one set of cyclical movements, which
can themselves be defined by one co-ordinate.

§ Functions of the "generalized momentum" of the system.

|| Supra, p. 113.

I 2

116 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

next succeeds in showing* that one of the products just
mentioned can always be found which is an. integrating divisor
of the energy of the compound cyclical system built up by
coupling any two given monocyclic systems together. This
result is easily interpreted as the analogue of the common
temperature at which two bodies are in thermal equilibrium.!
Finally, the investigation proves that these systems possess the
third property of temperature — that if two of them can each
be coupled with a third, they can be equally well coupled with
one another. J-

In this way Helmholtz has shown that all the thermo-
dynamical properties of matter can be represented dynamically
by means of monocyclic systems which are capable of being
coupled together. But owing to the somewhat arbitrary
nature of the assumptions which the argument makes, it cannot
be said that he has proved a dynamical origin for temperature§
— even in the modified sense that he has established other
Objective facts of a dynamical order which constantly accom-
pany it. Moreover, the investigation has only attempted to
cover reversible heat phenomena, and appears to be incapable
of" including the cases of irreversible phenomena, to which
actual experience practically confines us.|| "We may conclude,
then, that even the most complete and successful attempt that
has been made to reduce temperature to a dynamical pheno-
menon, not only reaches its end by inspired assumptions instead
of by the road of inevitable deductions, but also fails to consider
the enormous bulk of the primary facts in question. It does
not, therefore, escape the condemnation which Professor Bryan

* Bryan, op. cit., p. 102.

t As a simple instance of such dynamical analogues, Helmholtz gives
the case of two revolving wheels which may be coupled together by
joining their axles if their angular velocities are equal. (Bryan, op. cit^
p. 101.)

| Bryan, op. cit., p. 103.

§ Op. cit., p. 104.

|| Op. cit., p. 108.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 117

pours on all the attempts which he passes under review, that
they seek " to prove the Second Law, about v/hich we know
something, by means of molecules, about which we know much

less." *

§ 55.

It would appear that even the partial success which has
been reached in the attempt to unify by means of dynamical
concepts alone the phenomena which lie in the old provinces of
Heat and Mechanics, is dependent, at least in the present state
of science, upon the use of methods which (in Professor J. J.
Thomson's words)! " do not require an intimate knowledge of
the system to which they are applied." If we are modest
enough to abandon the hope that we may, by the application of
dynamics, "discover the properties of such systems [as are
investigated in physics] in an altogether d priori fashion," we
may yet most usefully employ dynamical methods " to predict
their behaviour under certain circumstances after having
observed it under others."! "The way in which dynamical
considerations may enable us to connect phenomena in different
branches in physics," is illustrated by Professor Thomson in the
following manner :

"Let us suppose that we have a number of pointers on a
dial, and that behind the dial the various pointers are con-
nected by a quantity of mechanism of the nature of which we
-are entirely ignorant. Then if we move one of the pointers,
A, say, it may happen that we set another, B, in motion.

" If, now, we observe how the velocity and position of B
depend on the velocity and position of A, we can by the aid
of dynamics foretell the motion of A when the velocity and
position of B are assigned, and we can do this even though we
are ignorant of the nature of the mechanism connecting the

* Op. dt., p. 121.

t J. J. Thomson, Applications, p. 8.

| Thomson, op. cit., p. 5.

118 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

two pointers." As the generalization of this example we read
that " the observation of the motion of B when that of A is
assigned may be taken to represent the experimental investiga-
tion of some phenomenon in Physics ; while the deduction by
dynamics of the motion of A when that of B is assigned may
represent the prediction ... of a new phenomenon which is
a consequence of the one investigated experimentally."*

For the application of the method contemplated — which is
a development of Hamilton's or Lagrange's principle — it is
only necessary to suppose that the system is possessed of
a constant amount of energy which expresses itself by
various ''co-ordinates," a term, originally of a strictly
limited geometrical significance, which was extended to any
quantities which can be used to determine the position of a
body in spacej or the spatial configuration of the parts of a
systemj and in the present connection is further extended to
elements which " fix the configuration " of the system in a wider
sense, such as elastic strains, electrical and magnetic phenomena.
This constant quantity of energy may be regarded either as the
sum of the " kinetic " and " potential " energies of the system,
or (in view of the fact that potential energy is a somewhat
unsatisfactory term, corresponding directly to nothing Objec-
tive^ as the sum of the kinetic energies of the given system
(determined by the "positional co-ordinates") and of sub-
sidiary connected systems such as the ether (determined by
" kinosthenic " or " ignored " co-ordinates). It is clear that
terms in this expression for the energy of the system may be
contributed by each of the co-ordinates that fix its configura-
tion, and that the coefficients of such terms may involve any
of the other co-ordinates, and possibly more than one at the

* Thomson, op. cit., p. 6. Cf. Eayleigh, Theory of Sound, pp. 71
and 75.

t Routh, Rigid Dynamics, i, p. 57.

1 Rayleigli, op. cit., p. 67.

§ Cf. Bradley, Appearance and Reality, pp. 384 et seq.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 119

same time. For example, the part of the kinetic energy of
a small sphere which is due to its velocity is affected, not
merely by its mass but also by the electric charge which it may
bear, the effect being the same as if its mass were increased.
At the same time the presence of the charge adds an indepen-
dent term to the expression of the energy. Applying to this
expression the operations prescribed by Lagraiige's equation —
regarding the electric charge as the variable — it is easy to
deduce the value of a quantity which must bear the same
relation to the energy due to the electric charge as ordinary
force does to ordinary kinetic energy.

By an extension of its original meaning, based upon this
analogy, and parallel to the extension of the meaning of the
term " co-ordinate," we may call any such quantity a " force "
of a special " type," determined by the particular co-ordinate
with regard to which* the differentiations have been per-
formed. In this case the force will be an " electro-motive
force," which will involve the velocity of the sphere. But this
result implies that the "electrical capacity" of a moving
sphere is different from that of the same sphere at rest, a
result which before was unknown.f Thus from the known
presence of a certain term in the formula for the energy,
a term which expressed a known connection between two
phenomena — the amount of the electrical charge on a moving
body and its apparent mass — we have been able to deduce
a reciprocal effect previously unknown.

It is important to notice that in Professor Thomson's
equations the energy is always expressed as a quadratic
function of the co-ordinates or their rates of change. It
follows that the " generalized forces " which are obtained by
Lagrange's method contain either second differential coefficients
analogous to acceleration, or else quantities analogous to

* And its differential coefficient with respect to the time,
t Thomson, op. cit., pp. 31-4.

120 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

differences of temperature. In accordance, then, with the results
we reached in §§ 45 — 7, both these types of co-ordinates must be
subject to the law that the value assumed by one of them at a
given moment can be regarded as an expression of the " notice "
taken by the system or a part of the system of two previous
configurations.

§56.

The consideration of this example may suffice to illustrate
the fact that the sphere of operations of the method can only
include a given co-ordinate when it is possible, in the assumed
absence of other co-ordinates, to express the whole of the
energy of the system in terms of the former. Consequently
when Professor Thomson approaches questions in which
temperature effects are involved* it is necessary for him to
form and to justify some definite concept of the relation of the
whole energy of a body to its temperature, when the body is
merely hot, and is not the seat of an electric charge, of elastic
strain, etc."f Thomson follows Hehnholtz in assuming that
the quantity analogous to temperature is the kinetic energy
due to the molecular or unconstrainable co-ordinates. It is
unnecessary, therefore, to do more than quote Professor Bryan's
criticism, that in Professor Thomson's argument " properties of
temperature are assumed which . . . have not hitherto been
satisfactorily deduced from dynamical principles.''^

§57.

In face of mathematical difficulties that have baffled even
such giants of analysis as Helmholtz, Maxwell, and Thomson,

* Op. cit., Ch. VI

t So long as the specific heat of a body is constant the rise of tem-
perature may be (at constant volume) a measure of the energy given to it
in the form of heat. This fact alone does not enable us to express as a
function of the temperature the whole energy of the body.

+ Bryan, loc. cit., p. 121.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 121

it is not surprising that investigators have turned aside from
" the high priori road," and, accepting the Carnot-Clausius
theorem as a law of our experience which holds good of all
transactions in which temperature changes occur, have sought
to unify the phenomena of the provinces of mechanics,
chemistry, and physics into one science of " energetics " which,
contenting itself with measuring the bare faces of sensible
phenomena,* expressly declines to attempt to " explain " one
set of objective phenomena in terms of another — and so comes
to exhibit a practice that accords with the philosophical
tenets of this essay.

The essential features of this method may be summed up
by the statement that it seeks some principle of determining
equilibrium in the " configuration " of a system (taking this
term in the wide sense explained on p. 118) that shall be
equivalent to the principle of virtual velocities which is
fundamental in dynamics, yet shall not, like the latter, be
expressed in an idiom peculiar to one of the special sciences
whose particulars are to be related.

The argument which seeks this goal starts from the
•experiences summarized in Clausius' law that if a system
goes through a series of modifications of a reversible character
its entropy suffers an unique and unambiguous change which
depends only on the initial and final states of the system ;f
while if the modifications are not reversible the change of
entropy which would have occurred under the conditions of
reversibility will be exceeded by an amount which Clausius
called the " sum of the non-compensated transformations." J
If now we agree to restrict our attention to cases in which the
transformations occur at a constant temperature, this additional
quantity of entropy can be regarded as corresponding to

* James?, " Humanism and Truth," Mind, N.S., No. 52, p. 459.
t Seep. 113, supra.

| Duhem, Le Potential Thermodynamique, 1886, p. 6, from which, in
the main, this section is derived.

122 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

a definite amount of " non-compensated work." It is not
difficult to obtain a mathematical expression for this work
involving the change of entropy (under reversible conditions)
the " internal energy/' the (constant) temperature of the body
in question, and the external work done during the transforma-
tion. Finally, after assuming limitations to the character of
the external work conformable with the conditions under
which an investigation of this kind would commonly be
pursued, it becomes possible to regard the "non-compensated
work " as the difference between the values at the end and
the beginning of the operation of a special function which its
inventor, Duhem, calls "the thermodynamic potential."* In
this function we have, for a large number of important cases
in which " physical phenomena affect one another," the desired
means of specifying the state of equilibrium. For <: lorsque le
potentiel thermodynamique est minimum, le systeme est dans
un etat d'equilibre stablest

It would be unsuitable to follow in these pages the extremely
technical development and applications of this principle. It
must suffice to note that by means which are equivalent to the
use of the thermodynamic potential, but before Duhem had
published his systematic theory of the processes, Willard Gibbs
had shown how to reduce to intelligibility large areas of pre-
viously intractable chemical and physical phenomena; while
Helmholtz had explained, by thermodynamical reasoning, the
astonishing discovery that the heat developable in an electric
circuit is only a fraction of the heat equivalent of the chemical
decompositions occurring in the cell — a fact which he showed
to be a particular case of the law that only part of the heat
drawn from any source is available for transformation into
useful work. This part, to which he „ gave the name of
"free energy," is identical (in this case) with Duhem's
thermodynamic potential. Upon the work of these pioneers —

* Op. tit., p. 8.
t Op. cit., p. 9.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 123

particularly upon that of Willard Gibbs — so great a theoretical
fabric has been erected in recent years that chemistry is
rapidly becoming absorbed into the circle of the mathematical

sciences.

§58.

The results of the last two chapters may now conveniently
be summarised. Starting from the concept of Science as a
conative process, which aims at rendering Objective facts intelli-
gible to an individual consciousness, and reaches that end by
building up those primary facts into secondary constructions or
apperceptive systems, by means of ideas drawn from other
contexts of experience, we took up the positions (1) that no
concept is to be deemed essentially incapable of rendering
primary facts intelligible on the ground of the context of
experience from which it is drawn ; and (2) that in no case is
the concept or hypothesis to replace, in the sense of accounting
for the "reality" of, the Objective facts which it has been
employed to render intelligible.

With regard to the first position, we came, in effect, to the
conclusion that each science must be (to use an expression
employed somewhat similarly by Professor Bosanquet) " self-
normative " with respect to the hypotheses it ~ uses; certain
hypotheses being ancillary to progress at one stage, or in one
sub-province of the science, others at another stage, or in another
sub^provincey The only general statements of a normative
character that we could make are that the secondary con-
struction must be a genuine reaction upon the primary facts,
and must not be forced upon those facts cCb extra ; and that,
while such concepts as " end " are, or may be, appropriate at
certain stages of the science, progress will show itself in the
adoption of hypotheses which suggest and make possible " a
complete synoptic view " of the primary facts, that is of hypo-
theses which can subserve a mathematical treatment.

With regard to the second position, we examined in the
present chapter, of course very briefly, though, it may be hoped

124 THE ADI AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

not inadequately for our purpose, the most systematic of the
attempts that have been made to render the whole range of
sensible facts intelligible by means of the concepts of " mass "
and " motion " which are themselves drawn only from one
province of primary facts. While, even if this attempt had
been successful,' we should not have felt compelled to admit that
the Objectivity of the sensible facts of hotness and coldness
had been in any way affected by the discovery of other
Objective facts that were found to have a constant relation to
them, yet, as a matter of fact, we drew the conclusion that the
reduction of the secondary qualities to primary qualities, which
is one of the chief motives of this attempt, "has not been
effected, and that the failure to effect it has encouraged the
development of methods of dealing with Objective facts which
respects their Objectivity. Using one of Lotze's pregnant
phrases, we have regarded as the goal of this method the
employment of correlation of the facts with the members of
the number series as a means of analysing and recording the
performances of things, so as to detect the manner in which one
Keal " takes note " of the behaviour of others with which it has
relations. The establishing of these relations, when the primary
facts which are the expression of them lie in different provinces,
has been most powerfully aided by the concept of Energy, which
attributes, in effect, the "quelquc chose qui demeure constant " which
is the form which our conviction of relationship inevitably
takes,* to the passage of a substance (in the strict philosophical,
together with an admixture of the ordinary, sense) from one
body to another, where it may appear in a transformed shape.
In accordance with our general doctrine, while admitting the
psychological usefulness of the concept, and, indeed, insisting
upon its psychological necessity, we must refuse to allow
writers like Ostwald, against whom we have actually been
defending hypotheses, to regard it as the one reality to which

* Poincare, Science et Hypothese, 1902, p. 153.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 125

we can reduce not only "things" and their "qualities," both
primary and secondary, but even, perhaps, psychical existences
as well.* Having served its purpose of leading to the most
complete "synoptic inventory" that physical science has
reached, it must, unless its Objectivity (in the sense in which
we use that term) has been incidentally established, resign the
claim to belong to the realm of facts which it has served both to-
render intelligible and to increase, and be contented to be
recognised as merely the powerful assimilative and expository
instrument that no one will deny it to be.

* Ostwald, Naturphilosophie, esp. Ch. 18.

126

CHAPTER V,

§ 59.

The pre-critical view that in certain concepts of Science we
reach the realities which lie at the back of perceived
phenomena, is one which will always have an attraction for
the actual workers in Science. It implies, perhaps, a certain
aloofness from practical life to resist conclusions supported
by evidence upon which one would act with confidence even in
aifairs of the highest moment. From this point of view Huxley*
pours ridicule upon those who would decline to accept the
geologist's reading of the palseontological record. If they were
consistent, he argues, they would decline to draw the usual
conclusions from the oyster shells outside the fishmonger's door,
or the mutton bone in the dust-bin.

In the class of cases which Huxley adduces there are few
who would reject his conclusions ; there are few of us, again,
who would be satisfied, as Professor Karl Pearson leads us to
suppose that he would be,f to " describe and classify [our]
immediate sense-impressions and [our] stored sense-impresses
by the aid of the theory of evolution," even " had the
universe been created just as it is yesterday " ; or with a
theory of matter upon which the negative " ether-sinks " (to
which nothing perceptual appears to correspond) " would long
ago have passed out of the range of ether-squirts " (which
correspond to perceptual matter), so that we need not concern
ourselves about their fate. There are few, I repeat, who would
not be troubled with " obstinate questionings " as to the truth
as well as the " economy " of these conceptions. The scruples

* In his lecture "On the Method of Zadig," Science and Culture,
p. 139.

t Pearson, The Grammar of Science, 1st ed., pp. 418 and 319.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 127

r

of such seem to imply the conviction — conscious or unconscious
— th at the business of Science is, as I have so often insisted,
to render the Objective intelligible,, and that the Objective
thus systematised must ultimately be ^he whole Objecfive and
nothing but the Objective.* No gap in either th£v spatial Or
the time series is to be tolerated, nor can we suffer any place in
either of the series to be filled by the hypothetical masquerading
as Objective. .

But this principle, apparently so simple, and so clear,
discloses unsuspected difficulties of application when we try to
determine by its aid the precise value and import of the
concepts by means of which we seek to make accessible
Objective phenomena intelligible. Many of these concepts
assign positions in the spatial and temporal series to things
which it is either essentially or else practically impossible to
verify. " Attraction " is an example of the first class, " atoms "
of the second. What is the actual standing of such entities ?
It cannot be denied that some of the evidence is forthcoming
which, if completed, would establish their existence, and if this
evidence actually produces conviction in men of the highest
intellect supremely conversant with the facts, what more is to
be said ? The denudation which " the Kazor of Occam " would
produce would depend entirely upon the hand that wielded it.
If it were applied by Lord Kelvin, the ether, for example,
would be safe : if by Professor Karl Pearson, its fate would be
.at least doubtful. f If it were handled by Professor J. J.
Thomson, the " Faraday tubes " would disappear, while "ions"
would, I imagine, remain. The truth seems to be that while
•cases of this kind were few and isolated, men's attitude towards
them might be indeterminate — each case was judged upon its
merits. But when with the advance of Science a whole compact
system of concepts appeared claiming to represent what " goes

* See Sigwart, Logic, ii, § 61.

t Grammar of Science, 1st ed., p. 214

128 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

on behind what we see and feel " over the whole surface of the
Objective, it became inevitable that individuals should take
up a definite general attitude towards them, only to be aban-
doned exceptionally : that is, that they should adopt a more
or less explicit philosophy of Science. For those who accepted
the claims of the new concepts, " atoms," " energy," " ether,"

and the like became metaphvsical terms, the names of ultimate

» .

realities, or of an hierarchy of realities, of which what we have
described as the Objective is only the appearance. As meta-
physical entities it was inevitable that they should eventually
claim to be able to account for the whole of experience. Thus
was developed that " mechanical philosophy " which has recently
suffered such a severe cross-examination by the author of
Naturalism and Agnosticism.

§ 60.

We may, it would seeni, distinguish usefully between three
distinct types of " secondary construction " to all of which the
name hypothesis has been indifferently given. In the first
kind the data are a number of facts of experience which
form an incomplete spatio-temporal system of a familiar type.
The hypothesis here simply suggests additional elements of the
same order that would make the system complete. A detective's
hypothesis of a crime is of this kind. It interpolates between
the data or "clues " other spatio-temporal facts of the same order
as the clues, which with the latter would make a system which
from its conformity with our experience would be felt to be com-
plete. The hypothesis that the collection of fossil bones labelled
Diplodocus Carnegii in the Natural History Museum once formed
the skeleton of an enormous living reptile, is another example
of the same kind of hypothesis. It is evident that the spatio-
temporal links which such an hypothesis introduces between
the data are always of such a character that they might
at least conceivably have been verified. Thus some museum
on the surface of Mars might conceivably contain, besides a

THE AIM. AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 129

duplicate skeleton, a sufficiently authenticated photograph of
Diplodocus in the flesh, a souvenir of a Martian visit to Earth
in Jurassic times ! At any rate, there can be no doubt that
the man who feels that the evolutionary hypothesis gives a
satisfactory explanation of existing biological facts, believes in
the vast majority of cases that it does so because it supplies
the " missing links " of a spatio-temporal chain, all of which
could have been verified by a human observer if he had been
present.

§ 61.

An important sub-class of this type is formed by hypotheses
of an " ejective " character. The counsel in the court of law
who seeks to persuade the jury that the accused was actuated
by certain motives, .or had a certain intention, is employing an
hypothesis of such a kind. It is clear that it has the marks
of conformity with our experience and (in a certain sense) of
homogeneity with the facts between which it is interpolated.
On the other hand, it is essentially un verifiable. Whether such
hypotheses as " attraction " or .the concept of " vital force "
should be included here is doubtful, They can hardly be so
included unless they can be said in the given cases to be
interpolations conformable with our experience. Thus we are
undoubtedly conscious of attraction and repulsion, but are we,
therefore, entitled to " eject " them into the matter of a planet ?
Much depends here upon the general character of our convic-
tions. Thus Gilbert of Colchester could write in 1600,
* Miserable were the condition of the stars, abject the lot of
the earth, if that wonderful dignity of life be denied them,
which is conceded to worms, ants, moths, plants, and toad-
stools."* From such a standpoint there can be little doubt that
"attraction" falls fairly into the present class of hypotheses.
For the modern physicist who belongs to a generation that has
learnt to disbelieve in consciousness where there is no evidence

* De Magnete (Eng. ed., publ. by the Gilbert Club, 1900, p. 209).

K

130 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

of nervous tissue, the notion belongs just as clearly to the third
class to be considered below.

§ 62.

In the second type of hypothesis^ the elements which are
added to make the secondary constructions are not spatio-
temporal existences but relations between such existences.
Such an hypothesis was Newton's belief that the attraction of the
earth for the moon could be calculated from its attraction for a
stone on the earth's surface in accordance with the law of
inverse squares, or Joule's conviction, maintained for years in
spite of contradictory experimental results, that a definite
equivalence existed between heat and work. Hypotheses of this
kind share with the former type the characteristic of being,
at least ideally, verifiable.

§63.

In the last class we find the typical hypothesis of science as
opposed to the hypothesis of history and common sense, the
hypothesis which Ostwald has attempted to banish from
scientific method. In general, its marks are — (1) a lack of the
homogeneity between the data and the added' or interpolated
elements which characterised the first type ; (2) the unverifiable
character of the added elements; and (3) that the secondary
construction does not merely complete the data but actually
replaces them.

The first two of these marks, at least, are present in the
case of the hypothesis of heat, by means of which temperature
changes are explained. The entity which is thought of as
" flowing " from the hot body to the cooler body is not thought
of as of the same order of existence as the sensations of 'hotness
and coldness which are the actual data here ; and quite
obviously it is completely unverifiable — no one has ever
pretended to exhibit heat apart from the phenomena of hotness
and coldness which it is invoked to render intelligible.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 131

The concept of molecule as used to explain physical and
chemical phenomena appears to be in possession of all these
marks. The data are the modes of behaviour of molar bodies
which cannot be regarded as homogeneous with the modes of
behaviour of the individual molecules of which the former are
assumed to exhibit only the statistical result.* Secondly, if
physicists' calculations of the "size" of molecules and their
conclusions as to the " wave-length of light " are both to be
accepted, molecules must in all probability be un verifiable. t
Finally, in this case it is one of the expressed " objects of
Physical Science to explain natural phenomena by means of
the properties of matter in motion," J where by " explain " is
probably meant to exhibit the reality of which the phenomena
in question are only the appearances.

§ 64.

On the question of the validity of these various types of hypo-
theses our own doctrine is clear. The first and second classes
consist of hypotheses which suggest definite interpolations where
interpolations are demanded by our previous experiences. The
" other context " from which an hypothesis of this type is drawn
consists of experiences of the same class as those which the
hypothesis ideally completes. This completion, as such, is the
sole object of such hypotheses, which, when verified, become
merged in the Objective facts which they have served to make
intelligible. None of these things can be said of hypotheses of
the third class. They are interpolated where there is little or
nothing to warrant interpolation. § They are drawn from

* Cf. in particular "Ward, Naturalism and Agnosticism, i, pp. 92-111.

t I leave it to the competent to decide whether this does not make
the assumption of their existence self -contradictory.

| J. J . Thomson, Applications of Dynamics to Chemistry and Physics,
p. 15.

§ Personal opinions will always differ as to the amount of such
warrant in particular cases — e.g., in interposition of an " ether " to explain
action at a distance." Cf. Larmor, B.A. Report, 1900, p, 627.

K 2

132 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

contexts of experience which are not of the same class as
the phenomena in question. Finally, so far from becoming
merged in the Objective facts which they render intelligible,
we have seen that their fate is to disappear altogether when
they have enabled us to arrive at a " complete synoptic
inventory " of these facts.

In support of this view many arguments may be brought.
The one most relevant (from the standpoint here adopted) is that
the concept of the "realities" which are to replace the sensible
data are themselves abstracted from those data. Thus Duhem*
not only argues, in a spirit entirely consonant with the spirit
of this essay, that water is not really the hydrogen and oxygen
which disappear when it is formed, but also shows that the
atomic hypothesis upon which it is possible to conceive the
" elements " as still present in the " compounds " is derived
historically from Newton's famous Query 31.f In this passage
Newton suggests the application of the ideas that he had gained
from his study of planetary bodies to the analysis of the
behaviour of the bodies manipulated in experiments. Similar
observations occur in several of Mr. Merz' splendid chapters,
and have been repeatedly illustrated in the course of our
discussions in Chapters III and IV. More recently still it
has been pointed outj that the most thorough-going quasi-
metaphysical attempt to account for perceived physical events
is vitiated by the same circle. The most striking feature of the
electric theory of matter is that it exhibits the property of
" mass " as the consequence of the motion of " electrons." But
to reach this result properties of the electromagnetic field are
appealed to, and these properties are defined by differential
equations into which the notion of mass derived from the study
of molar bodies itself enters.

* In his Le Mixte et la Combinaison chimique, 1902, and in other
writings. See also supra, p. 82.

t Supra, p. 78.

J See a review of works on " Electrontheorie," Jiy H. A. Wilson, in
Nature for June 22, 1905.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. lo$
§ 65.

Opposed to the thinkers who adopt the view of the value of
scientific concepts which has just been repudiated, are those
who have felt themselves forced to take up one of the various
positions included under the name of the descriptive view of
Science. Most of these positions have a relation to the wider
philosophical position of Humanism,* which makes them par-
ticularly interesting at the present moment.

" The great Poincare," says Professor James,f misses
Humanism by a hair. He has demonstrated^ in a brilliant
manner the conventional character of Science, and has laid
special stress upon the manner in which one theory has
succeeded another in the same physical field. § He appears to
accept what we may perhaps call the disintegrating results of
mathematical physics, regarding perceived things and events as
really due to the superposition of a great number of similar
elementary phenomena.! Moreover, he removes from the
Objective every element — such as the secondary qualities —
which cannot be proved to be " the same for all " by the use of
language. " Pas de discours. pas d'objectiviteV'IT If, then,
perception gives us no reality and the hypotheses of Science are
only conventions, what is there that remains ? We find that
while hypothesis may succeed hypothesis — as, for example,
Maxwell's electro-magnetic theory of light succeeded Fresnel's
undulatory theory — the differential equations remain the same,
the expression of veritable relations between, real terms which
Nature hides from us eternally, though Fresnel may think of
them as movements and Maxwell as electric currents** It is

* See James, " Humanism and Truth," Mind, N.S., No. 52, p. '462.
t James, loc. cit.

J In the essays reprinted in La Science et VHypothese, and the more
recent La Valeur de la Science.

§ See, e.g., La Science et VHypothese, Ch. X.
|| Op. cit., p. 187.

IF La Valeur de la Science, p. 262.
** La Science et VHypothhe, p. 190.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

through its knowledge of these Objective relations that Science
has so much theoretical hold over the inscrutable reals, that it is
able to predict the future ; but that same knowledge has clearly a
certain " intellectual " value quite apart from its value as a
collection of recettes pratiques.

§ 66.

Even this amount of intellectual value seems to disappear in
the view of Science advocated by M. Le Boy.* For this writer
the laws of Science, when they are not conventional definitions,
are simply recettes pratiques, " not true but efficacious?' " not
concerning our knowledge so much as our actions" " rather
enabling us to capture the order of Nature, than revealing it to
us."j~ Moreover, these laws have reference to artificial facts —
faits scientiftques — created by the scientist out of the faits bruts
of perception.

M. Le Boy's scientific fact seems to correspond to a large
extent with our " secondary construction " by which the
" primary fact " is apperceived. An " atom " and an " eclipse "
are examples given. Poincare adds an " electric current " as the
scientific fact constructed from the brute fact of a galvanometer
deflection; also ttfc "corrected reading "obtained by treatment of
a number of direct readings. We may add ourselves the " rigid
bar" by which the actual elastic lever is replaced in theory.
But there is this important difference between Le Boy's con-
ception and our own: the laws of science as conceived by
him seem hardly to touch the brute facts, which, not being
scientific, are outside Science.^ This is why the law is in so
many cases merely a rule of action. In our view, on the other

* See the discussion reported at length in the Bulletin de la Societc
fran$aise de Philosophic, Mai, 1901. Le Roy's views are criticised by
Poincare in the essay reprinted in La Valeur de la Science, Ch. X. •

t Bulletin, p. 5.

% Poincare, La Valeur de la Science, p. 221. Cf. Bulletin, p. 21, where
M. Le Roy says, " C'est ce qu'on ajoute au fait brut pour constituer le fait
scientifique qui est le plus important. '

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 135

hand, the whole object of the secondary construction is to render
the primary facts intelligible, to bring out real relations between
the brute facts which constitute the scientific fact, and to lead
to the discovery of new brute facts related to those already
recognised within the system.

The same kind of inversion of the relations of primary fact
and scientific construction is shown by the illustrations given
of the dictum that laws are frequently definitions. Such a one
is the law that " phosphorus melts at 44°," which is asserted by
M. Le Eoy to be merely a definition of phosphorus. One feels
here in a peculiarly tantalising form the want of security of
the relations between ideas and the reality beyond which some
of us find in other presentments of Pragmatism. The definition
"works," substances melting at 44° are actually encountered,
but one has about their identity much the same kind of doubt
as pursued the school-boy who feared that Shakespeare's plays
were not written by Shakespeare but by another man of the
same name.

§ 67.

Mach's splendid labours in this field are too well-known to
need characterisation. For the founder and chief apostle of the
new doctrine the concepts of Science are, as with us, means to
" <*/an end, an end which is conceived as " the economic exposition
of actual facts."* It is clear that this principle of " economy "
pushes analysis further than the principle of intelligibility-
which we have been considering. It suggests, as Machf applies
it, a value for the race as well as for the individual in what we
have-thought of simply as a psychological phenomenon. This
suggestion is of the highest interest and importance, and as
such may be gladly accepted. But when the same circum-
stance is made the ground upon which Mach is claimed by
Professor James (in the article already quoted) as, a Humanist,

* Mach, Science of Mechanics, 1902 ed., p. 555.
t See op. cit., pp. 481 et seq.

136 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

it seems necessary to determine what are the exact admissions
implied by one's applause. I am prepared to admit that the
results of Science have this economical value ; prepared to admit
that by Natural Selection or in some other way Nature may
have arranged that Science shall be pursued so that this value
shall be secured to the race; but, as before, I hesitate when
asked to grant that this relevance to purpose constitutes the
essence of the results in question. And Humanism is nothing
more than an interesting genetic psychology if we do not take
it as telling us not merely the circumstances under which we
come to recognise such things as thinghood, or the conservation
of energy, but what they are prior to our recognition. My own
view of the principle of the conservation of energy I have
endeavoured to explain. It is a concept by means of which
a .definite range of given facts is made intelligible to
an individual thinker. In consequence of this circumstance
it has j an economical value. Further, it is the property
of " secondary constructions," into which such concepts
and the corresponding primary facts enter, that they lead to-
the " apperception " of new primary facts — reals or relations
between reals — this being the external characteristic which
distinguishes the scientific from other attempts to render the
primary facts intelligible. Finally, the conception is a con-
vention in that another could conceivably have been found to
render the same facts intelligible, and, if " scientific," would
liave led to the recognition of the same real relations between
the real things. The conception, in fact, plays the part which
Lotze attributes to all ideas — the part of a tool which fits the
mind and also fits reality.

If pressed to consider also the case of thinghood, I should
have first to remark that I find between concepts of this order
and the concepts of Science a distinct break. In this I differ
from Mach, who does not appear to distinguish the process by
which we supply a core to a mass of sensations, and so create
a "thing" from the process by which we make a secondary

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 137

construction out of certain data by means of the concept of a
transference of something (" energy ") that remains constant in
amount. We seem to have here the thought which Professor
James expresses in his article on Humanism and Truth* and
the writers of the essay on The Nature of the Hypothesis.^
According to this thought Eeality is not the same after our
judgment as before ; it is " increased and elevated " by the
act of judgment. The implication seems to be that scientific
judgments simply continue a process wjiich " common-sense "
judgments begin. There are aspects of the two .processes of
judgment of which this notion of continuity holds good ; we
may grant to Messrs. Ashley and Dewey that the hypothesis
is a predicate, and to Mach and Professor James that the
concepts, both of " thing " and " energy," are economical.
But, as I have already pointed out, " the secondary construc-
tions " of Science which correspond to the " reality qualified
by an ideal content" of the ordinary judgment contain no
element that is not drawn from the common-sense stratum of
consciousness. For example, if one body is cooling while
another is simultaneously growing warmer, the secondary
construction in which these primary facts are synthesised
contains besides these facts merely the thought of another
thing being transferred from one body to the other. On the
other hand, the synthesis by which we bind the various qualities
into the "thing" does not present us with anything analogous
to this. The secondary construction is of a totally different
character from the elements ; the process does not reach its
end by the j.deal addition of a new element of the same type.
Further, the hypothesis has, we have shown, merely a transient
function. Setting aside purposes of exposition and convenience
in conceptual handling, its function is to point the way to
the discovery of new facts, including relations, and then to

* P. 468.

t In Studies in Logical Theory, ed. Dewey, 1903.

138 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

efface itself. Finally, at any moment it^ at least ideally
possible by criticism of the whole construction to separate
the primary facts from the interpretative "embroidery," and
to realise that the synthesis was not strictly inevitable.
Whewell's dictum that " fact and theory have no essential
difference except in the degree of their certainty and familiarity.
Theory, when it becomes firmly established and steadily lodged
in the mind, becomes fact " * — which is approved by Professor
Deweyf — ignores this power of critical analysis. It is, in fact,
an early pragmatic pronouncement which, like most pragmatist
doctrines, is excellently true within its proper sphere of appli-
cation. From the standpoint of the practical man — the
scientific workman in the engineering workshop, the labora-
tory, or the study — the theories of his predecessors may
often and may well be his facts. No one would censure a
Marconi for accepting the Maxwell-Hertz theory of electro-
magnetic ether waves as the factual basis of an attempt to
biing England and America into wireless telegraphic com-
munication ; or a German chemist for directing his search
for the means of manufacturing a " natural " perfume in
accordance with the stereo-chemical concept of the molecule
introduced by Van t'Hoff, or the " ring " concept of Kekule.
But the circumstance that theories are often taken with more
or less naive conviction as the primary data for sequences of
actions in practical life, does not contradict the doctrine, the
truth of which is established by^such critical investigations
as those of Mach, that all , theories ard ultimately founded
upon a solid primary basis of Objective fact, which can, in
most cases, be exposed with certainty below the superstructure
which ages of man's toil and genius have erected upon it.

* Whewell, The Philosophy of -the Inductive Sciences, 1840, p. 45.
t Op. cit. p. 164.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 139
§ 68.

The admission that an actual break of continuity occurs
between the ordinary judgments of common-sense and the
judgments which involve hypotheses belonging to the class
which we distinguished in § 63 will probably be obtained
without difficulty. A less ready acquiescence may be expected
when we proceed to maintain that a break can also be dis-
tinguished between common-sense judgments and scientific
judgments which involve hypotheses of our first class. It
will be remembered that these hypotheses consist in the
interpolation or extrapolation of facts homogeneous with the
data so as to form spatio-temporal systems of primary fact
consonant with experience. The very circumstance that the
added elements present this homogeneity with the data
makes the separation between the common-sense judgment
and the scientific judgment on this plane less conspicuous than
in the former case ; nevertheless, it may be held to exist.

A trivial example may make the distinction clear. If I
say, "that man has a rolling gait," the synthesis has the
inevitable character that is the mark of the primary fact, the
common-sense judgment. If, on the other hand, I assert
(on the ground of his rolling gait) " that man is a sailor/'
my synthesis has the secondary character which is not inevit-
able. If then you ask me if such a judgment is " scientific,"
I do not think I ought to hesitate to say "Yes" simply
because the instance js trivial. The secondary construction is
undoubtedly a reaction upon certain primary facts, and it has
the property of leading to the observation of other primary
" substantive " facts, and yet other facts, relation g between
these. In these respects it seems precisely like such a judg-
ment as " this substance is copper sulphate," based upon an
experiment in chemical analysis — a judgment which would
generally be admitted as scientific. The farmer judgment,
in fact, is related to the " unconditional universal," " All men

140 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

with such and such a rolling gait are sailors," in the same way
as the latter is, to " all things that have such and such
properties are copper sulphate." Both these assert a " per-
manent connection of qualities in the Real " — that is, are the
final products of a process in which primary facts have been
unified, systematised, or made intelligible by a concept which
has not failed to lead to discoveries of fresh primary facts
without limit in its province.

To this argument one very obvious answer suggests itself.
If the judgment " that man is a sailor " is to be describecKas
a secondary construction upon a basis of primary or presented
fact, because in the mind of the observer a synthesis of this
basis with non-presented elements occurs, how can the same
term be withheld from the judgment " that man has a rolling
gait " ? Surely (it may be maintained) the application to a
certain complex of presented elements of the term m an
implies the erection on the basis of that complex of a secondary
construction into which a multitude of non-presented elements
enters. The idea man is itself, in fact, nothing but an
hypothesis whose function is to render the primary data
intelligible ; for it contains a reference to an immense body
of 'spatio-temporal experiences homogeneous with the given
material.

I venture to doubt whether this argument is really as
strong as it is generally assumed to be. Granting that, as
modern theories of perception teach, the primary data are
" complicated " by references to non-presented material, and
that tha secondary construction gives the presented object its
meaning, it seems possible, yet, to maintain that in the
common-sense judgment the essential feature is an act of
analysis or discrimination, and that the secondary non-
presented elements are confined in the main to the function
of supplying a motive for this discrimination. On this view
the term "man," when it appears in a common-sense judgment,,
connotes merely a number of presented facts — very largely

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 141

visual facts. If Madame Tussaud can frame of wax and paint
so cunning a counterfeit that the country cousin commits
himself to the judgment " there is a policeman " and proceeds
to act upon it, we have a case in which complication has
followed with, perhaps, excusable haste upon an imperfect
discrimination of the data. When the counterfeit fails to
react to a question after the manner of genuine policemen,
more careful discrimination at once takes place and reveals
the absence of some essential visual elements of the complex
to which the name " man " is assigned, or the presence of others
which are incompatible with that complex.

This account must be qualified by the admission that very
often it is impossible to tell from the form of the judgment
whether it is of the common-sense type or the scientific.
Hearing a certain peculiar cough and a certain shuffling step
in the hall I flee incontinently from X, whose long-winded
stories are the bane of the club. Here my action may follow
upon a scientific judgment. The complication, instead of
serving as the motive for a more careful inspection of the
data, leads to the formation and acceptance of a secondary
construction based upon those data. It is, on the whole, more
" economical " to act upon this construction than to wait until
so much of the Objective complex is presented as will render
discrimination on the common-sense level possible. Thus
while in a given case the judgment which follows upon the
presentation of the given primary facts may "Belong to either
of the two types distinguished, and the occasions upon which
the reaction upon the presentations takes the one form or the
other are not separated by any clear intrinsic difference, yet
the judgments themselves are so essentially different that we
are justified in denying to them the continuity which the
pragmatic theory of knowledge seems to demand. And tha
difference for which we have found it necessary to contend at
the level of scientific hypotheses of our first class is so plainly
recognisable at the level of hypotheses of the third class that

142 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

to establish it needs no arguments beyond those that have
already been considered in previous sections.

§ 69.

The doctrine of the primary facts presented in this essay is
a variety of what has recently been called the "New Realism " ;*
and I hesitate to say that I have derived it from Mr. G-. E.
Moore and Mr. Bertrand E-ussell only because I have no right
to suggest the responsibility of these philosophers for details
which they might quite possibly reject. The essence of the
doctrine is the view that a large part of the contents of our
consciousness from moment to moment consists of elements
which exhibit themselves as having a certain unique " priority "
to our conscious processes. These elements constitute what
I have described as the Objective. They fall into three well
marked genera — physical exlstents, psychical existents, and
subsistents, which share with the former the characteristics of
being regarded as " the same for all," and of having a certain
relevance to human purpose, expressed by saying that " they
have to be reckoned with."

In our consideration of the first of these genera we found
ourselves driven to defend in substance, but to modify in detail
the common-sense notion which regards the material world as
composed of Objective " things " into whose structure primary
and secondary qualities enter on equal terms; occupying at
different moments of an Objective flux of time definite points
in an Objective space. We also devoted some attention to the
fact that the elements of the Objective are very often members
of series, and gave special study to the series of numbers which
plays so important a part in the development of Science.

The aim of the scientific process as it occurs in the
individual is to render the Objective in its actual determina-
tions intelligible. This happens when primary facts enter into

* See an article by Professor J. S. Mackenzie in Mind, N.S., No. 59.

THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD. 143

an " apperceptive system." They may be apperceived by
means of any concept drawn from any other context of
experience, and if by means of this concept the actual
particulars of experience are systematised, the " end " of the
process will have been reached. But if the process has been
of the kind intended by the term scientific, it will have the
further property of leading to other determinations of the
Objective, and these further determinations are the actual
achievements of Science, and its " end," therefore, from the
universal point of view. Since primary facts present them-
selves for the most part in series, the most useful method of
determining the Objective consists in correlating terms of
these series with the members of the number series — the
property of this series being that single members of it can be
substituted for combinations of other members in accordance
with definite laws easily applied. By means of such combina-
tions it is often possible to characterise simply the relations
between things, and to ascertain what changes in terms of
relations can be regarded as complete expressions of those
relations. Such cases typify the ideal of the scientific process
which is actually exhibited in a large number of grades, which
nevertheless are sharply distinguished from the processes by
which the Objective itself is recognised — a fact which is
claimed in support of the view of the unique character of the
latter.

In virtue of this feature our doctrine presents a contrast
with the pragmatist theories of Science which regard the
primary facts and secondary constructions as merely different
stages of a homogeneous productive process to which it is
impossible to assign either a lower or an upper limit.

Finally, it may be claimed that the concept here
defended avoids the error contained in the theories of Science
given by other writers, which have been justly criticised
because they fail to represent the actual relations between
hypothesis and fact. My concept allows the hypothesis

144 THE AIM AND ACHIEVEMENTS OF SCIENTIFIC METHOD.

to determine largely what primary facts shall he apper-
ceived, and admits that the fact before the individual, i.e.,
the secondary construction, is constituted by the apperception.
At the same time the implication that the Objective in this
construction is an ideal upon which we can never actually set
the finger, is rejected ; and it is maintained that to a critical
scrutiny the Objective reveals itself in ordinary cases, though in
some cases it may not be easy to determine it without reference
to the " confirmatory tests " of sameness for all and relevance
to purpose.

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