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BOOK 060. IN8 V. 1 c. 1
INTERNATIONAL CONGRESS OF ARTS
AND SCIENCES # INTERNATIONAL CONG

3 T1S3 00057781 T

THIS FIRST EDITION DE LUXE,

printed from type, is limited to five hundred

sets, of which this is copy

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INTERNATIONAL CONGRESS

OF ARTS AND SCIENCE

. ^

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INTERNATIONAL CONGRESS

ARTS AND SCIENCE

EDITED BY

Howard J. Rogers, A.M., LL.D.

DIRECTOR OF CONGRESSES

VOLUME I

HISTORY OF THE CONGRESS

By THE Editor

SCIENTIFIC PLAN OF THE CONGRESS
By Professor Hugo Munsterberg

PHILOSOPHY AND MATHEMATICS

UNIVERSITY ALLIANCE

LONDON NEW YORK

ILLUSTRATIONS

VOLUME I

FACING
PAGE

XMA Mater Frontispiece

^ Photogravure from the statue by Daniel C. French

Dr. Howard J. Rogers 1

Photogravure from a photograph

Dr. Simon Newcomb 135

Photogravure from a photograph

Dr. Benno Erdmann 352

Photogravure from a photograph

Portraits of Dr. Charles Emile Picard, Dr. Heinrich Maschke and

Dr. E. H. Moore 452

Photogravure from a photograph

ORGANIZATION OF THE CONGRESS

PRESIDENT OF THE EXPOSITION:
HON. DAVID R. FRANCIS, A.M., LL.D.

DIRECTOR OF CONGRESSES:

HOWARD J. ROGERS, A.M., LL.D.

Universal Exposition, 1904.

ADMINISTRATIVE BOARD

NICHOLAS MURRAY BUTLER; Ph.D., LL.D.

President of Columbia University, Chairman.

WILLIAM R. HARPER, Ph.D., LL.D.

President of the University of Chicago.

R. H. JESSE, Ph.D., LL.D.

President of the University of Missouri.

HENRY S. PRITCHETT, Ph.D., LL.D.
President of the Massachusetts Institute of Technology.

HERBERT PUTNAM, Litt.D., LL.D.
Librarian of Congress.

FREDERICK J. V. SKIFF, A.M.

Director of the Field Columbian Museum.

OFFICERS OF THE CONGRESS

PRESIDENT:
SIMON NEWCOMB, Ph.D., LL.D.

Retired Professor U. S. N.

VICE-PRESIDENTS:

HUGO MUNSTERBERG, Ph.D., LL.D.

Professor of Psychology in Harvard University.

ALBION W. SMALL, Ph.D., LL.D.

Professor of Sociology in the University of Chicago.

TABLE OF CONTENTS

THE HISTORY OF THE CONGRESS

Howard J. Rogers, A.M., LL.D.

Programme

Purpose and Plan of the Congress .
Organization of the Congress .
Officers of the Congress
Speakers and Chairmen
Chronological Order of Proceedings
Programme of Social Events . • .
List of Ten-Minute Speakers

THE SCIENTIFIC PLAN OF THE CONGRESS

Hugo Munsterberg, Ph.D., LL.D,

47
50
52
53
54
77
81
82

PROCEEDINGS OF THE CONGRESS

Introductory Address.

The Evolution of the Scientific Investigator . . . . . . 135

Simon Newcomb, Ph.D., LL.D,

Division A — Normative Science.

The Sciences of the Ideal 151

Professor Josiah Roycb.

Department I — Philosophy.

Chairman's Address 171

Professor Borden P. Bowne.

Philosophy: Its Fundamental Conceptions and its Methods . .173

Professor George Holmes Howison.

The Development of Philosophy in the Nineteenth Century . . . 194

Professor George Trumbull Ladd.

Section A — Metaphysics.

The Relations between Metaphysics and the Other Sciences . . . 227
Professor Alfred Edward Taylor.

The Present Problems of Metaphysics 246

Professor Alexander Thomas Ormond,
Short Papers _ 259

Section B — Philosophy op Religion.

The Relation of the Philosophy of Religion to the Other Sciences . 263

Professor Otto Pfleiderer.

viii TABLE OF CONTENTS

Main Problems of the Philosophy of Religion: Psychology and Theory of
Knowledge in the Science of Religion ....... 275

Professor Ernst Troeltsch.

Short Papers 289

Section C — Logic.

The Relations of Logic to Other Disciplines . . . . . .296

Professor William Alexander Hammond.

The Field of Logic • . .313

Professor Frederick J. E. Woodbridge.

Section D — Methodology of Science.

On the Theory of Science ......... 333

Professor Wilhelm Ostwald.

The Content and Validity of the Causal Law ...... 353

Professor Benno Erdmann.

Section E — Ethics.

The Relations of Ethics 391

Professor William Ritchie Sorley.

Problems of Ethics 403

Professor Paul Hensel.

Section F — Esthetics.

The Relation of Esthetics to Psychology and Philosophy . . . .417
Professor Henry Rutgers Marshall.

The Fundamental Questions of Contemporary Esthetics .... 434
Professor Max Dessoir.

Special Bibliography prepared by Professor Dessoir for his Address . .447

Short Papers . 448

General Bibliography for Department of Philosophy 449

Department II — Mathematics.

The Fundamental Conceptions and Methods of Mathematics . . . 456
Professor Maxime Bocher.

The History of Mathematics in the Nineteenth Century . . . . 474
Professor James P. Pierpont.

Section A — Algebra and Analysis.

On the Development of Mathematical Analysis and its Relations to Some

Other Sciences ........... 497

Professor Emile Picard.

On Present Problems of Algebra and Analysis ...... 518

Professor Heinrich Maschke.

Short Papers • 531

TABLE OF CONTENTS ix

Section B — Geometry.

A Study of the Development of Geometric Methods 535

M. Jean Gaston Darboux.

The Present Problems of Geometry ........ 559

Dr. Edward Kasner.

Short Papers 587

Section C — Applied Mathematics.

The Relations of Applied Mathematics 591

Professor Ludwig Boltzmann.

The Principles of Mathematical Physics ....... 604

Professor Henri Poincare.

General Bibliography of the Department of Mathematics .... 623

Special Bibliography accompanying Professor Boltzmann' s Address . . 625

CONTENTS OF THE SERIES . .627

CONGRESS OF ARTS AND SCIENCE

■ ■■ - :, _.

THE HISTORY OF THE CONGRESS

BY HOWARD J. ROGERS A.M., LL.D.

The forces which bring to a common point the thousandfold energies
of a universal exposition can best promote an international congress
of ideas. Under national patronage and under the spur of interna-
tional competition the best products and the latest inventions of
man in science, in literature, and in art are grouped together in orderly-
classification. Whether the motive underlying the exhibits be the
promotion of commerce and trade, or whether it be individual
ambition, or whether it be national pride and loyalty, the resultant
is the same. The space within the boundaries of the exposition is
a forum of the nations where equal rights are guaranteed to every
representative from any quarter of the globe, and where the sover-
eignty of each nation is recognized whenever its flag floats over a
national pavilion or an exhibit area. The productive genius of every
governed people contends in peaceful rivalry for world recognition,
and the exposition becomes an international clearing-house for
practical ideas.

For the demonstration of the value of these products men thor-
oughly skilled in their development and use are sent by the various
exhibitors. The exposition by the logic of its creation thus gathers
to itself the expert representatives of every art and industry. For
at least two months in the exposition period there are present the
members of the international jury of awards, selected specially by
the different governments for their thorough knowledge, theoretical
and practical, of the departments to which they are assigned, and
selected further for their ability to impress upon others the correct-
ness of their views. The renown of a universal exposition brings, as
visitors, students and investigators bent upon the solution of prob-
lems and anxious to know the latest contributions to the facts and
the theories which underlie every phase of the world's development.

The material therefore is ready at hand with which to construct
the framework of a conference of parts, or a congress of the whole
of any subject. It was a natural and logical step to accompany the
study of the exhibits with a debate on their excellence, an analysis
of their growth, and an .argument for their future. Hence the con-
gress. The exposition and the congress are correlative terms. The
former concentres the visible products of the brain and hand of man ;
the congress is the literary embodiment of its activities.

2 THE HISTORY OF THE CONGRESS

Yet it was not till the Paris Exposition of 1889 that the idea of
a series of congresses, international in membership and universal in
scope, was fully developed. The three preceding expositions, Paris,
1878, Philadelphia, 1876, and Vienna, 1873, had held under their
auspices many conferences and congresses, and indeed the germ of
the congress idea may be said to have been the establishment of the
International Scientific Commission in connection with the Paris
Exposition of 1867; but all of these meetings were unrelated and
sometimes almost accidental in their organization, although many
were of great scientific interest and value.

The success of the series of seventy congresses in Paris in 1889
led the authorities of the World's Columbian Exposition in 1893
to establish the World's Congress Auxiliary designed "to supple-
ment the exhibit of material progress by the Exposition, by a por-
trayal of the wonderful achievements of the new age in science,
literature, education, government, jurisprudence, morals, charity,
religion, and other departments of human activity, as the most
effective means of increasing the fraternity, progress, prosperity,
and peace of mankind." The widespread interest in this series of
meetings is a matter easily within recollection, but they were in
no wise interrelated to each other, nor more than ordinarily com-
prehensive in their scope.

It remained for the Paris Exposition of 1900 to bring to a perfect
organization this type of congress development. By ministerial
decree issued two years prior to the exposition the conduct of the
department was set forth to the minutest detail. One hundred
twenty-five congresses, each with its separate secretary and organiz-
ing committee, were authorized and grouped under twelve sections
corresponding closely to the exhibit classification. The principal
delegate, M. Gariel, reported to a special commission, which was
directly responsible to the government. The department was ad-
mirably conducted and reached as high a degree of success as a highly
diversified, ably administered, but unrelated system of international
conferences could. And yet the attendance on a majority of these
congresses was disappointing, and in many there was scarcely any
one present outside the immediate circle of those concerned in its
development. If this condition could prevail in Paris, the home of
arts and letters, in the immediate centre of the great constituency
of the University and of many scientific circles and learned societies,
and within easy traveling distance of other European university
and literary centres, it was fair to presume that the usefulness of this
class of congress was decreasing. It certainly was safe to assume,
on the part of the authorities of the St. Louis Exposition of 1904,
that such a series could not be a success in that city, owing to its
geographical position and the limited number of university and

THE HISTORY OF THE CONGRESS 3

scientific circles within a reasonable traveling distance. Something
more than a repetition of the stereotyped form of conference was
admitted to be necessary in order to arouse interest among scholars
and to bring credit to the Exposition.

This was the serious problem which confronted the Exposition of
St. Louis. No exposition was ever better fitted to serve as the ground-
work of a congress of ideas than that of St. Louis. The ideal of the
Exposition, which was created in time and fixed in place to com-
memorate a great historic event, was its educational influence. Its
appeal to the citizens of the United States for support, to the Federal
Congress for appropriations, and to foreign governments for coopera-
tion, was made purely on this basis. For the first time in the history
of expositions the educational influence was made the dominant
factor and the classification and installation of exhibits made con-
tributory to that principle. The main purpose of the Exposition was
to place within reach of the investigator the objective thought of
the world, so classified as to show its relations to all similar phases
of human endeavor, and so arranged as to be practically available
for reference and study. As a part of the organic scheme a congress
plan was contemplated which should be correlative with the exhibit
features of the Exposition, and whose published proceedings should
stand as a monument to the breadth and enterprise of the Exposition
long after its buildings had disappeared and its commercial achieve-
ments grown dim in the minds of men.

DEVELOPMENT OF THE CONGRESS

The Department of Congresses, to which was to be intrusted this
difficult task, was not formed until the latter part of 1902, although
the question was for a year previous the subject of many discussions
and conferences between the President of the Exposition, Mr.
Francis; the Director of Exhibits, Mr. Skiff; the Chief of the Depart-
ment of Education, Mr. Rogers; President Nicholas Murray Butler
of Columbia University, and President William R. Harper of Chicago
University. To the disinterested and valuable advice of the two last-
named gentlemen during the entire history of the Congress the Ex-
position is under heavy obligations. During this period proposals had
been made to two men of international reputation to give all their
time for two years to the organization of a plan of congresses which
should accomplish the ultimate purpose of the Exposition authorities.
Neither one, however, could arrange to be relieved of the pressure of
his regular duties, and the entire scheme of supervision was conse-
quently changed. The plan adopted was based upon the idea of an
advisory board composed of men of high literary and scientific
standing who should consider and recommend the kind of congress
most worthy of promotion, and the details of its development.

4 THE HISTORY OF THE CONGRESS

In November, 1902, Howard J. Rogers, LL.D., was appointed
Director of Congresses, and the members of the Advisory (afterwards
termed Administrative) Board selected as follows: —

Chairman: Nicholas Murray Butler, Ph.D., LL.D., President
Columbia University.

William R. Harper, Ph.D., LL.D., President University of
Chicago.

Honorable Frederick W. Holls, A.M., LL.B., New York.

R. H. Jesse, Ph.D., LL.D., President University of Missouri.

Henry S. Pritchett, Ph.D., LL.D., President Massachusetts
Institute of Technology.

Herbert Putnam, Litt.D., LLD., Librarian of Congress.

Frederick J. V. Skiff, A.M., Director of Field Columbian Mu-
seum.

The action of the Executive Committee of the Exposition, ap-
proved by the President, was as follows : —

There shall be appointed by the President of the Exposition Company a
Director of Congresses who shall report to the President of the Exposition Com-
pany.

There shall be appointed by the President of the Exposition Company an
Advisory Board of seven persons, the chairman to be named by the President,
who shaU meet at the call of the Director of Congresses, or the Chairman of the
Advisory Board.

The expenses of the members of the Advisory Board while on business of the
Exposition shall be a charge against the funds of the Exposition Company.

The duties of the said Advisory Board shall be: to consider and make recom-
mendations to the Director of Congresses on all matters submitted to them; to
determine the number and the extent of the congresses; the emphasis to be
placed upon special features; the prominent men to be invited to participate;
the character of the programmes; and the methods for successfully carrying out
the enterprise.

There shall be set aside from the Exposition funds for the maintenance of the
congresses the sum of two hundred thousand dollars ($200,000).

The standing Committee on Congresses from the Exposition board
of directors was shortly afterwards appointed and was composed of
five of the most prominent men in St. Louis : —

Chairman: Hon. Frederick W. Lehmann, Attorney at Law.

Breckenridge Jones, Banker.

Charles W. Knapp, Editor of The St Louis Republic.

John Schroers, Manager of the Westliche Post.

A. F. Shapleigh, Merchant.

To this committee were referred for consideration by the President
all matters of policy submitted by the Director of Congresses. This
committee had jurisdiction over all congress matters, including not
only the Congress of Arts and Science, but also the many miscel-
laneous congresses and conventions, and a great part of the success

THE HISTORY OF THE CONGRESS 5

of the congresses is due to their broad-minded and liberal deter-
mination of the questions laid before them.

IDEA OF THE CONGRESS OF ARTS AND SCIENCE

It is impossible to ascribe the original idea of the Congress of
Arts and Science to any one person. It was a matter of slow growth
from the many conferences which had been held for a year by men
of many occupations, and as finally worked out bore little resemblance
to the original plans under discussion. The germ of the idea may fairly
be said to have been contained in Director Skiff's insistence to the
Executive Committee of the Exposition that the congress work
stand for something more than an unrelated series of independent
gatherings, and that some project be authorized which would at once
be distinctive and of real scientific worth. To support this view
Director Skiff brought the Executive Committee to the view of
expending $200,000, if need be, to insure the project. Starting from
this suggestion many plans were brought forward, but one which
seems to belong of right to the late Honorable Frederick W. Holls,
of New York City, contained perhaps the next recognizable step in
advance. This thought was, briefly, that a series of lectures on
scientific and literary topics by men prominent in their respective
fields be delivered at the Exposition and that the Exposition pay
the speakers for their services. This point was thoroughly discussed
by Mr. Holls and President Butler, and the next step in the evolution
of the Congress was the idea of bringing these lecturers together at
the Exposition at about the same time or all during one month. At
this stage Professor Hugo Miinsterberg, who was the guest of Mr.
Holls and an invited participant in the conference, made the import-
ant suggestion that such a series of unrelated lectures, even though
given by most eminent men, would have little or no scientific value,
but that if some relation, or underlying thought, could be intro-
duced into the addresses, then the best work could be done, which
would be of real value to the scientific world. He further stated that
only in this case would scientific leaders be likely to favor the plan
of a St. Louis congress, as they would feel attracted not so much
through the honorariums to be given for their services as through
the valuable opportunity of developing such a contribution to scien-
tific thought. Subsequently Professor Miinsterberg was asked by
Mr. Holls to formulate his ideas in a manner to be submitted to the
Exposition authorities. This was done in a communication under
date of October 20, 1902, which contained logically presented the
foundation of the plan afterwards worked out in detail. At this
juncture the Department of Congresses was organized, as has been
stated, the Director named, and the Administrative Board appointed,
and on December 27, 1902, the first meeting of the Director with
the Administrative Board took place in New York City.

6 THE HISTORY OF THE CONGRESS

A thorough canvass of the subject was made at this meeting and
as a result the following recommendations were made to the Exposi-
tion authorities : —

(1) That the sessions of this Congress be held within a period
of four weeks, beginning September 15, 1904.

(2) That the various groups of learned men who may come together
be asked to discuss their several sciences or professions with reference
to some theme of universal human interest, in order that thereby
a certain unity of interest and of action may be had. Under such a
plan the groups of men who come together would thus form sections
of a single Congress rather than separate congresses.

(3) As a subject which has universal significance, and one likely
to serve as a connecting thread for all of the discussions of the Con-
gress, the theme "The Progress of Man since the Louisiana Pur-
chase" was considered by the Administrative Board fit and suggest-
ive. It is believed that discussions by leaders of thought in the
various branches of pure and applied science, in philosophy, in politics,
and in religion, from the standpoint of man's progress in the century
which has elapsed, would be fruitful, not only in clearing the thoughts
of men not trained in science and in government, but also in preparing
the way for new advances.

(4) The Administrative Board further recommends that the Con-
gress be made up from men of thought and of action, whose work
would probably fall under the following general heads : —

a. The Natural Sciences (such as Astronomy, Biology, Mathe-
matics, etc.).

h. The Historical, Sociological, and Economic group of studies
(History, Political Economy, etc.).

c. Philosophy and Religion.

d. Medicine and Surgery.

e. Law, Politics, and Government (including development and
history of the colonies, their government, revenue and prosperity,
arbitration, etc.).

/. Applied Science (including the various branches of engineer-
ing).

(5) The Administrative Board recommends further referring to
a special committee cf seven the problem of indicating in detail the
method in which this plan can best be carried out. To this com-
mittee is assigned the duty of choosing the general divisions of the
Congress, the various branches of science and of study in these divi-
sions, and of recommending to the Administrative Board a detailed
plan of the sections in which, in their judgment, those who come to
the Congress maybe most effectively grouped, with a view not only
to bring out the central theme, but also to represent in a helpful way
and in a suggestive manner the present boundary of knowledge in the

THE HISTORY OF THE CONGRESS 7

various lines of study and investigation which the committee may
think wise to accept.

These recommendations were transmitted by the Director of
Congresses to the Committee on Congresses, approved by them, and
afterwards approved by the Executive Committee and the President.
The first four recommendations were of a preliminary character, but
the fifth contained a distinct advance in the formation of a Committee
on Plan and Scope which should be composed of eminent scientists
capable of developing the fundamental idea into a plan which should
harmonize with the scientific work in every field. The committee
selected were as follows : —

De. Simon Newcomb, Ph.D., LL.D., Retired Professor of Mathe-
matics, U. S. Navy.

Prof. Hugo Munsterberg, Ph.D., LL.D., Professor of Psycho-
logy, Harvard University.

Prof. John Bassett Moore, LL.D., ex-assistant Secretary of
State, and Professor of International Law and Diplomacy, Columbia
University.

Prof. Albion W. Small, Ph.D., Professor of Sociology, Uni-
versity of Chicago.

Dr. William H. Welch, M.D., LL.D., Professor of Pathology,
Johns Hopkins University.

Hon. Elihu Thomson, Consulting Engineer General Electric
Company.

Prof. George F. Moore, D.D., LL.D., Professor of Comparative
Religion, Harvard University.

In response to a letter from President Butler, Chairman of the
Administrative Board, giving a complete resume of the growth of
the idea of the Congress to that time, all of the members of the com-
mittee, with the exception of Mr. Thomson, met at the Hotel Man-
hattan on January 10, 1903, for a prehminary discussion. The entire
field was canvassed, using the recommendations of the Administrative
Board and the aforementioned letter of Professor Miinsterberg's to
Mr. HoUs as a basis, and an adjournment taken until January 17
for the preparation of detailed recommendations.

The Committee on Plan and Scope again met, all members being
present, at the Hotel Manhattan on January 17, and arrived at
definite conclusions, which were embodied in the report to the
Administrative Board, a meeting of which had been called at the
Hotel Manhattan for January 19, 1903. The report of the Com-
mittee on Plan and Scope is of such historic importance in the devel-
opment of the Congress that it is given as follows, although many
points were afterwards materially modified : — ■

8 THE HISTORY OF THE CONGRESS

New York, January 19, 1903.
President Nicholas Murray Butler,

Chairman Administrative Board of World's Congress at
The Louisiana Purchase Exposition:

Dear Sir, — The undersigned, appointed by your Board a committee on the
scope and plan of the proposed World's Congress, at the Louisiana Purchase
Exposition, have the honor to submit the following report: —

The authority under which the Committee acted is found in a commimication
addressed to its members by the Chairman of the Administrative Board. A
subsequent commimication to the Chairman of the Committee indicated that the
widest scope was allowed to it in preparing its plan. Under this authority the
Committee met on January 10, 1903, and again on January 17. The Committee
was,, from the beginning, unanimous in accepting the general plan of the Admin-
istrative Board, that there should be but a single congress, which, however, might
be divided and subdivided, in accord with the general plan, into divisions, depart-
ments, and sections, as its deliberations proceed.

PLANS OF THE CONGRESS

As a basis of discussion two plans were drawn up by members of the Committee
and submitted to it. The one, by Professor Miinsterberg, started from a compre-
hensive classification and review of human achievement in advancing knowledge,
the other,' by Professor Small, from an equally comprehensive review of the great
public questions involved in human progress.

Professor Miinsterberg proposed a congress having the definite task of bringing
out the unity of knowledge with a view of correlating the scattered theoretical and
practical scientific work of our day. This plan proposed that the congress should
continue through one week. The first day was to be devoted to the discussion of
the most general problem of knowledge in one comprehensive discussion and four
general divisions. On the second day the congress was to divide into several
groups and on the remaining days into yet more specialized groups, as set forth
in detail in the plan.

The plan by Professor Small proposed a congress which would exhibit not
merely the scholar's interpretation of progress in scholarship, but rather the
scholar's interpretation of progress in civilization in general. The proposal was
based on a division of human interests into six great groups: —
I. The Promotion of Health.
II. The Production of Wealth.

III. The Harmonizing of Human Relations.

IV. Discovery and Spread of Knowledge.
V. Progress in the Fine Arts.

VI. Progress in Religion.

The plan agreed with the other in beginning with a general discussion and then
subdividing the" congress into divisions and groups.

As a third plan the Chairman of the Committee suggested the idea of a congress
of publicists and representative men of all nations and of all civilized peoples,
which should discuss relations of each to all the others and throw light on the
question of promoting the unity and progress of the race.

After due consideration of these plans the Committee reached the conclusion
that the ends aimed at in the second and third plans could be attained by taking
the first plan as a basis, and including in its subdivisions, so far as was deemed
advisable, the subjects proposed in the second and third plans. They accordingly
adopted a resolution that "Mr. Miinsterberg's plan be adopted as setting forth

THE HISTORY OF THE CONGRESS 9

the general object of the Congress and definmg the scope of its work, and that
Mr. Small's plan be communicated to the General Committee as containing sug-
gestions as to details, but without recommending its adoption as a whole."

DATE OF THE CONGRESS

Your Committee is of opinion that, in view of the climatic conditions at St. Louis
during the summer and early autimm, it is desirable that the meeting of this
general Congress be held during the six days beginning on Monday, September 19,
1904, and continuing until the Saturday following. Special associations choosing
St. Louis as their meeting-place may then convene at such other dates as may be
deemed fit; but it is suggested that learned societies whose field is connected with
that of the Congress should meet during the week beginning September 26!

The sectional discussions of the Congress will then be continued by these
societies, the whole forming a continuous discussion of human progress during
the last century.

PLAN OF ADDRESSES

The Committee believe that in order to carry out the proposed plan in the most
effective way it is necessary that the addresses be prepared by the highest living
authorities in each and every branch. In the last subdivisions, each section
embraces two papers; one on the history of the subject during the last one hun-
dred years and the other on the problems of to-day.

The programme of papers suggested by the Committee as embraced in Pro-
fessor Miinsterberg's plan may be summarized as foUows : —

On the first day four papers wiU be read on the general subject, and four on
each of the four large divisions, twenty in all. On the second day those four divi-
sions wiU be divided into twenty groups, or departments, each of which will have
four papers referring to the divisions and relations of the sciences, eighty in all.
On the last four days, two papers in each of the 120 sections, 240 in aU, thus
making a total of 340 papers.

In view of the fact that the men who wiU make the addresses should not be
expected to bear all the expense of their attendance at the Congress, it seems
advisable that the authorities of the Fair should provide for the expenses neces-
sarily incurred in the journey, as weU as pay a small honorarium for the addresses.
The Committee suggest, therefore, that each American invited be offered $100 for
his traveling expenses and each European $400. In addition to this that each
receive $150 as an honorarium. Assuming that one half of those invited to deliver
addresses wiU be Americans and one half Europeans, this arrangement wiU involve
the expenditure of $136,000. This estimate will be reduced if the same person
prepares more than one address. It wiU also be reduced if more than half of the
speakers are Americans, and increased in the opposite case.

As the Committee is not advised of the amount which the management of the
Exposition may appropriate for the purpose of the Congress, it cannot, at present,
enter further into details of adjustment, but it records its opinion that the sum
suggested is the least by which the ends sought to be attained by the Congress can
be accomplished. To this must be added the expenses of administration and
publication.

All addresses paid for by the Congress should be regarded as its property, and
be printed and published together, thus constituting a comprehensive work
exhibiting the unity, progress, and present state of knowledge.

This plan does not preclude the delivery of more than one address by a single
scholar. The directors of the Exposition may sometimes find it advisable to ask
the same scholar to deliver two addresses, possibly even three.

10 THE HISTORY OF THE CONGRESS

The Committee recommends that full liberty be allowed to each section of the
Congress in arranging the general character and programme of its discussions
within the field proposed.

As an example of how the plan will work m the case of any one section, the
Committee take the case of a neurologist desiring to profit by those discussions
which relate to his branch of medicine. This falls under C of the four main
divisions as related to the physical sciences. His interest on the first day will
therefore be centred in Division C, wliere he may hear the general discussion of
the physical sciences and the relations to the other sciences. On the second day
he wiU hear four papers in Group IS on the subjects embraced in the general
science of antliropology ; one on its fundamental conceptions; one on its
methods and two on the relation of anthropology to the sciences most closely con-
nected with it. During the remaining four days he will meet with the represent-
atives of medicine and its related subjects, who will divide into sections, and
listen to four papers in each section. One paper wiU consider the progress of
that section in the last one hundred years, one paper will be devoted to the
problems of to-day, lea\'ing room for such contributions and discussions as may
seem appropriate during the remainder of the day.

COOPERATION OF LEARNED SOCIETIES INVOKED

In presenting this general plan, your Committee wishes to point out the diffi-
culty of deciding in advance what subjects should be included in every section.
Therefore, the Committee deems it of the utinost importance to secure the advice
and assistance of learned societies in this country in perfecting the details of the
proposed plan, especially the selection of speakers and the programme of work in
each section. It will facilitate the latter purpose if such societies be invited and
encouraged to hold meetings at St. Louis during the week immediately preceding,
or, preferably, the week following the General Congress. The selection of speakers
should be made as soon as possible, and, in any case, before the end of the present
academic year, in order that formal invitations may be issued and final arrange-
ments made with the speakers a year in advance of the Congress.

CONCLUDING SUGGESTIONS

With the view of securing the cooperation of the governments and leading
scholars of the principal countries of Western and Central Europe in the proposed
Congress, it seems advisable to send two commissioners to these countries for this
purpose. It seems unnecessary to extend the operations of this commission out-
side the European continent or to other than the leading countries. In other
cases arrangements can be made by correspondence.

It is the opinion of the Committee that an American of world-wide reputation
as a scholar should be selected to preside over the Congress.
AH which is respectfully submitted.

(Signed) Simon Newcomb,

Chairman ;
George F. Moore,
John B. Moore,
Hugo Munsterberg,
Albion W. Small,
William H. Welch,
Elihu Thomson,

Committee.

THE HISTORY OF THE CONGRESS 11

The Administrative Board met on January 19 to receive the report
of the Committee on Plan and Scope which was presented by Dr.
Newcomb. Professor Miinsterberg and Professor John Bassett Moore
were also present by invitation to discuss the details of the scheme.
In the afternoon the Board went into executive session, and the
following recommendations were adopted and transmitted by the
Director of Congresses to the Committee on Congresses of the Expo-
sition and to the President and Executive Committee, who duly
approved them.

To the Director of Congresses : —

The Administrative Board have the honor to make the following recommenda-
tions in reference to the Department of Congresses : —

(1) That there be held in coimection with the Universal Exposition of St. Louis
in 1904, an International Congress of Arts and Science.

(2) That the plan recommended by the Committee on Plan and Scope for a
general congress of Arts and Science, to be held during the six days beginning on
Monday, September 19, 1904, be approved and adopted, subject to such revision
in point of detail as may be advisable, preserving its fundamental principles.

(3) That Simon Newcomb, LL.D., of Washington, D. C, be named for President
of the International Congress of Arts and Science, provided for in the foregoing
resolution.

(4) That Professor Miinsterberg, of Harvard University, and Professor Albion
W. Small, of the University of Chicago, be invited to act as Vice-Presidents of
the Congress.

(5) That the Directors of the World's Fair be requested to change the name of
this Board from the "Advisory Board" to the "Administrative Board of the
International Congress of Arts and Science."

(6) That the detailed arrangements for the Congress be intrusted to a com-
mittee consisting of the President and two Vice-Presidents already named, sub-
ject to the general oversight and control of the Administrative Board, and that
the Directors of the Exposition be requested to make appropriate provision for
their compensation and necessary expenses.

(7) That it be recommended to the Directors of the World's Fair that appro-
priate provision should be made in the office of the Department of Congresses for
an executive secretary and such clerical assistance as may be needed.

(8) That the following payment be recommended to those scholars who accept
invitations to participate and do a specified piece of work, or submit a specified
contribution in the International Congress of Arts and Science: For traveling
expenses for a European scholar, $500. For traveling expenses for an American
scholar, $150.

(9) That provision be made for the publication of the proceedings of the Con-
gress in suitable form to constitute a permanent memorial of the work of the
World's Fair for the promotion of science and art, under competent editorial
supervision.

(10) That an appropriation of $200,000 be made to cover expenses of the
Department of Congresses, of which sum $130,000 be specifically appropriated for
an International Congress of Arts and Science, and the remainder to cover aU
expenses connected with the publication of the proceedings of said Interna-
tional Congress of Arts and Science, and the expenses for promotion of aU other
congresses.

12 THE HISTORY OF THE CONGRESS

In addition to the foregoing recommendations, Professor Miinster-
berg was requested at his eariiest convenience to furnish each member
with a revised plan of his classification, which would reduce as far as
possible the number of sections into which the Congress was finally
to be divided.

With the adjournment of the Board on January 19 the Congress
may be fairly said to have been launched upon its definite course,
and such changes as were thereafter made in the programme did not
in any wise affect the principle upon which the Congress was based,
but were due to the demands of time, of expediency, and in some
cases to the accidents attending the participation. The organization
of the Congress and the personnel of its officers from this time on
remained unchanged, and the history of the meeting is one of steady
and progressive development. The Committee on Plan and Scope
were discharged of their duties, with a vote of thanks for the
laborious and painstaking work which they had accomplished and
the thoroughly scientific and novel plan for an international congress
which they had recommended.

It was determined by the Administrative Board to keep the serv-
ices of three of the members of the Committee on Plan and Scope,
who should act as a scientific organizing committee and who should
also be the presiding officers of the Congress. The choice for President
of the Congress fell without debate to the dean of American scientific
circles, whose eminent services to the Government of the United
States and whose recognized position in foreign and domestic sci-
entific circles made him particularly fitted to preside over such an
international gathering of the leading scientists of the world. Dr.
Simon Newcomb, retired Professor of Mathematics, United States
Navy. Professor Hugo Miinsterberg, of Harvard University, and Pro-
fessor Albion W. Small, of the University of Chicago, were designated
as the first and second Vice-Presidents respectively.

The work of the succeeding spring, with both the Organizing Com-
mittee and the Administrative Board, was devoted to the perfecting
of the programme and the selection of foreign scientists to be invited
to participate in the Congress. The theory of the development of
the programme and its logical bases are fully and forcibly treated by
Professor Miinsterberg in the succeeding chapter, and therefore will
not be touched upon in this record of facts. As an illustration of the
growth of the programme, however, it is interesting to compare its
form, which was adopted at the next meeting of the Organizing
Committee on February 23, 1903, in New York City, with its final
form as given in the completed programme presented at St. Louis
in September, 1904 (pp. 47-49). No better illustration can be given
of the immense amount of labor and painstaking adjustment, both
to scientific and to physical conditions, and of the admirable adapt-

THE HISTORY OF THE CONGRESS

ability of the original plan to the exigencies of actual practice. At
the meeting of February 23, 1903, which was attended by all of the
members of the Organizing Committee and by President Butler of
the Administrative Board, it was determined that the number of
Departments should be sixteen, with the following designations: —

A. NORMATIVE SCIENCES

1. Philosophical Sciences. 2. Mathematical Sciences.

B. HISTORICAL

3. Political Sciences.

4. Legal Sciences.

5. Economic Sciences.

6. Philological Sciences.

SCIENCES

7. Pedagogical Sciences.

8. ^Esthetic Sciences.

9. Theological Sciences.

C. PHYSICAL SCIENCES

10. General Physical Sciences.

11. Astronomical Sciences.

12. Geological Sciences.

13. Biological Sciences.

14. Anthropological Sciences.

D. MENTAL SCIENCES
15. Psychological Sciences. 16. Sociological Sciences.

Indo-Iranian Languages.
Semitic Languages.
Classical Languages.
Modem Languages.
History of Education.
Educational Institutions.
History of Architecture.
History of Fine Arts.
History of Music.
Oriental Literature.
Classical Literature.
Modem Literature.
Architecture.
Fine Arts.
Music.

Primitive Religions.
Asiatic Religions.
Semitic Rehgions.
Christianity.
Religious Institutions.
Mechanics and Soimd.
Light and Heat.
Electricity.
Inorganic Chemistry.
Organic Chemistry.
Physical Chemistry.
Mechanical Technology.
Optical Technology.
Electrical Technology.

SECTIONS

, a Metaphysics.

6. a

b Logic.

c Ethics.

d .Esthetics.

, a Algebra.

7. a

b Geometry.

c Statistical Methods.

8. a

. a Classical Political History of

Asia.

b Classical PoUtical History of

Europe.

c Medieval Political History of

Europe.

d Modem Pohtical History of

Europe.

e Pohtical History of America.

9. a

. a History of Roman Law.

b History of Common Law.

aa Constitutional Law.

bb Criminal Law.

cc Civil Law.

10. a

dd History of International Law.

. a History of Economic Institu-

tions.

6 History of Econopaic Theories.

c Economic Law.

aa Finance.

bb Commerce and Transportation.

cc Labor.

THE HISTORY OF THE CONGRESS

SECTIONS — continued

10. dd Chemical Teclinology.

11. a Theoretical Astronomy.
b Astrophysics.

12. a Geodesy.
b Geology.

c Mineralogy.

d Physiography.

e Meteorology.
aa Surveying.
bb Metallurgy.

13. a Botany.

b Plant Physiology.

c Ecology.

d Bacteriology.

e Zoology.

/ Embryology.

g Comparative Anatomy.

h Ph5rsiology.
aa Agronomy.
bb Veterinary Medicine.

14. Anthropological Sciences:
a Human Anatomy.

b Human Physiology.
c Neurology.

d Physical Chemistry.

e Pathology.

/ Raceomatology.
aa Hygiene.
bb Contagious Diseases.

cc Internal Medicine.
dd Surgery.
ee Gynecology.
// Ophthalmology.
gg Therapeutics.
hh Dentistry.

15. Psychological Sciences:
a General Psychology.

b Experimental Psychology.
c Comparative Psychology.
d Child Psychology.
e Abnormal Psychology.

16. Sociological Sciences:
a Social Morphology.

b Social Psychology.
c Laws of Civilization.
d Laws of Language and Myths.
e Etlmology.
aa Social Technology.

It was also resolved, that the discussion of subjects falling under
the first four divisions should be held in the forenoon of each of the
four days, from Wednesday until Saturday, and those relating to
the three divisions of Practical Science in the afternoon of the same
days. The programme was thus rearranged by the addition of the
following : —

E. UTILITARIAN SCIENCES

17. Medical Sciences:
a Hygiene.

b Sanitation.

c Contagious Diseases.

d Internal Medicine.

e Psychiatry.

/ Surgery.

g Gynecology.

h Ophthalmology.

i Otology.

J Therapeutics.

k Dentistry.

18. Practical Economic Sciences:

a Extractive Productions of
Wealth.

b Transportation,
c Commerce.
d Postal Service.
e Money and Banking.
19. Technological Sciences:
a Mechanical Technology.
b Electrical Technology.
c Chemical Technology.
d Optical Technology.
e Surveying.
/ Metallurgy.
g Agronomy.
h Veterinarv Medicine.

THE HISTORY OF THE CONGRESS 15

F. REGULATIVE SCIENCES

20. Practical Political Sciences : c Criminal Law.
a Internal Practical Politics. d Civil Law.

& National Practical Politics. 22. Practical Social Sciences:

c Tariff. a Treatment of the Poor.

d Taxation. h Treatment of the Defective,

e Municipal Practical Politics. c Treatment of the Dependent.

/ Colonial Practical Politics. d Treatment of Vice and Crime.

21. Practical Legal Sciences: e Problems of Labor.

a International Law. / Problems of the Family.

6 Constitutional Law.

G. CULTURAL SCIENCES

23. Practical Educational Sciences : /Publications.

a Kindergarten and Home. 24. Practical ^Esthetic Sciences:
h Primary Education. a Architecture,

c Universities and Research — h Fine Arts.

Secondary. c Music.

d Moral Education. d Landscape Architecture.

e Esthetic Education. 25. Practical Religious Sciences:
/ Manual Training. a Religious Education.

g University. h Training for Religious Ser\'ice.

h Libraries. c Missions.

i Museums. d Religious Influence.

The programme was again thoroughly revised at the meeting of the
Organizing Committee on April 9, 1903, at Hotel Manhattan, and as
thus amended was submitted to the Administrative Board at a meet-
ing held in New York on April 11. A careful consideration of the
programme at this meeting, and a final revision made at the meeting
of the Administrative Board at the St. Louis Club April 30, 1903,
brought it practically into its final shape, with such minor changes
as were found necessary in the latter days of the Congress due to the
unexpected declinations of foreign speakers at the last moment. The
continuous and exacting work done in perfecting the programme by
each member of the Organizing Committee and by the Chairman of
the Administrative Board deserves special mention, and was pro-
ductive of the best results by its logical appeal to the scientific world.
The programme as finally worked out in orderly detail, shortened in
many departments by various exigencies, may be found on pages 47
to 49 of this volume.

PARTICIPATION AND SUPPORT

The general plan of the Congress having been determined and the
prcgramme practically perfected by M&y 1, 1903, two most import-
ant questions demanded the attention of the Administrative Board:
first, the participation in the Congress, both foreign and domestic;

16 THE HISTORY OF THE CONGRESS

second, the support of the scientific pubUc. At a meeting of the Board
held in New York City April 11, 1903, these points were given full
consideration. It was determined that the list of speakers both for-
eign and domestic should be made up on the advice of men of letters
and of scientific thought in this country, and accordingly there was
sent to the officers of the various scientific societies in the United
States, to heads of university departments and to every prominent
exponent of science and art in this country, a printed announcement
and tentative programme of the Congress, and a letter asking advice
as to the scientists best fitted in view of the object of the Congress
to prepare an address. From the hundreds of replies received in
response to this appeal were made up the original lists of invited
speakers, and only those were placed thereon who were the choice of
a fair majority of the representatives of the particular science under
selection. The Administrative Board reserved to itself the full right
to reject any of these names or to change them so as to promote the
best interests of the Congress, but in nearly every instance it would
be safe to say that the person selected was highly satisfactory to the
great majority of his fellow scientists in this country. Many changes
were unavoidably made at the last moment to meet the situation
caused by withdrawals and declinations, but the list of second choices
was so complete, and in many cases there was such a delicate balance
between the first and second choice, that there was no difficulty
in keeping the standard of the programme to its original high
plane.

It was early determined that the seven Division speakers and the
forty-eight Department speakers, which occupied the first two days
of the programme, should be Americans, and that these Division and
Department addresses should be a contribution of American scholar-
ship to the general scientific thought of the world. This decision
commended itself to the scientific public both at home and abroad,
and it was so carried out. It was further determined that the Division
and Department speakers and the foreign speakers should be selected
during the summer of 1903, and that the American participation in
the Section addresses should be determined after it was definitely
known what the foreign participation would be. In view of the
importance of the Congress, it was deemed inadvisable to attempt
to interest foreign scientific circles by correspondence, and it was
further decided to pay a special compliment to each invited speaker
by sending an invitation at the hands of special delegates. Arrange-
ments were therefore made for Dr. Newcomb and Professors Miinster-
berg and Small to proceed to Europe during the summer of 1903, and
to present in person to the scientific circles of Europe and to the
scientists specially desired to deliver addresses the complete plan
and scope of the Congress and an invitation to participate.

THE HISTORY OF THE CONGRESS 17

INVITATIONS TO FOREIGN SPEAKERS

The members of the Organizing Committee, armed with very strong
credentials from the State Department to the diplomatic service
abroad, sailed in the early summer of 1903 to present the invitation of
the Exposition to the selected scientists. Dr. Newcomb sailed May 6,
Professor Miinsterberg May 30, and Professor Small June 6. A general
interest in the project had at this time become aroused, and there
was assured a respectful hearing. Both the President of the United
States and the Emperor of Germany expressed their warm interest
in the plan, and the State Department at Washington gave to the
Congress both on this occasion and on succeeding occasions its effect-
ive aid. The Director of Congresses wishes to express his obligations
both to the late Secretary Hay and to Assistant-Secretary Loomis for
their valuable suggestions and courteous cooperation in all matters
relating to the foreign participation. Strong support was also given
the Committee and the plan of the Congress by Commissioner-General
Lewald of Germany, and Commissioner-General Lagrave of France.
Throughout the entire Congress period, both of these energetic Com-
missioners-General placed themselves actively at the disposition of
the Department in promoting the attendance of scientists from their
respective countries.

Geographically the division between the three members of the
Organizing Committee gave to Dr. Newcomb, France; to Professor
Miinsterberg, Germany, Austria, and Switzerland; and to Professor
Small, England, Russia, Italy, and a part of Austria. It was also
agreed that Dr. Newcomb should have special oversight of the
departments of Mathematics, Physics, Astronomy, Biology, and
Technology; Professor Miinsterberg, special charge of Philosophy,
Philology, Art, Education, Psychology, and Medicine; and that
Professor Small should look after Politics, Law, Economics, Theology,
Sociology, and Religion. The Committee worked independently of
each other, but met once during the summer at Munich to compare
results and to determine their closing movements.

The public and even the Exposition authorities have probably
never realized the delicacy and the extremely careful adjustment
exercised by the Organizing Committee in their summer's campaign.
Scientists are as a class sensitive, jealous of their reputations, and
loath to undertake long journeys to a distant country for congress
purposes. The amount of labor devolving upon the Committee to
find the scientists scattered over all Europe; the careful and pains-
taking presentation to each of the plan of the Congress; the appeal
to their scientific pride; the hearing of a thousand objections, and
the answering of each; the disappointments incurred; the substi-
tutions made necessary at the last moment; — all sum up a task of

18 THE HISTORY OF THE CONGRESS

the greatest difficulty and of enormous labor. The remarkable success
with which the mission was crowned stands out the more promi-
nently in view of these conditions. When the Committee returned in
the latter part of September, they had visited every important coun-
try of Europe, delivered more than one hundred fifty personal invita-
tions, and for the one hundred twenty-eight sections had secured one
hundred seventeen acceptances.

At a meeting of the Administrative Board, which met with the
Organizing Committee on October 13, 1903, a full report of the
European trip was received and ways and means considered for insur-
ing the attendance from abroad. A list of the foreign acceptances was
ordered printed at once for general distribution, and the Chairman of
the Administrative Board was requested to address a letter to each
of the foreign scientists confirming the action of the special delegates
and giving additional information as to the length of addresses, and
rules and details governing the administration of the Congress.

DEATH OF FREDERICK W. HOLLS

The. number of the Administrative Board was decreased during
the summer by the sudden death of the Hon. Frederick W. HoUs, on
July 23, 1903. Mr. Holls had been intensely interested in the develop-
ment of the Congress from its earliest days, and was very instru-
mental in determining the form in which it was finally promoted.
His great influence abroad as a member of the Hague Conference,
and his high standing in legal and literary circles in this country,
rendered him one of the most prominent members of the Board. A
resolution of regret at his untimely death was spread upon the min-
utes of the Administrative Board at the meeting in October, and it
was decided that his place upon the Board should remain unfilled.

DOMESTIC PARTICIPATION

At this same meeting of October 13, active measures were taken to
forward the American participation in the Congress. The necessity
was now very evident that our strongest men of science must be
induced to take part, in order to compare favorably with the leading
minds which Europe was sending. The Organizing Committee were
instructed to consult the American scientific societies and associations
regarding the selection of American speakers, and also in reference
to presiding officials for each section. Six weeks was considered suf-
ficient for this task, and the Committee were asked to submit to the
Administrative Board at a meeting in New York, on December 3
and 4, their recommendations for American speakers.

An immense amount of detailed labor, in the way of correspond-
ence, now devolved upon the Organizing Committee as well as upon
the Director of Congresses, and a branch office was established in

THE HISTORY OF THE CONGRESS 19

Washington equipped with clerks and stenographers under the charge
of Dr. Newcomb, who devoted the greater portion of his time for the
next six months to the many details connected with the selection
of foreign and American speakers and chairmen. The meeting of the
Administrative Board in New York in December, and a similar
meeting with the Organizing Committee held at the St. Louis Club on
December 28, were given over entirely to perfecting the personnel of
the programme. Great care was exerted in selecting the chairmen
of the departments and sections, inasmuch as they must be men of
international reputation and conceded strength. For the secretary-
ships younger men of promise and ability were selected, chiefly from
university circles. Both the chairmen and secretaries served without
compensation.

The work of the late winter was a continuance of the perfecting of
details, and at a meeting of the Administrative Board held in New
York in February, 1904, a final approval was given to the programme
and the speakers. The imminent approach of the Exposition and the
work of the college commencement season made it impossible for
further general meetings, and on June 1 the Organizing Committee
was constituted a committee with power to fill vacancies in the pro-
gramme or to amend the programme as circumstances might demand.
All suggestions with reference to details were to be made directly to
the Director of Congresses, upon whom devolved from this time for-
ward the entire executive control of the Congress.

ASSEMBLY HALLS

The highly diversified nature of the Congress and the holding of
one hundred twenty-eight section meetings in four days' time ren-
dered necessary a large number of meeting-places centrally located.
The Exposition was fortunate in having the use of the new plant of
the Washington University, nine large buildings of which had been
erected. Many of these buildings contained lecture halls and assembly
rooms, seating from one hundred fifty to fifteen hundred people.
Sixteen halls were necessary to accommodate the full number of
sections running at any one time, and of this number twelve were
available in the group of University Buildings; the other four were
found in the lecture halls of the Education Building, Mines and
Metallurgy Building, Agriculture Building, and the Transportation
Building. The opening exercises, at which the entire Congress was
assembled, was held in Festival Hall, capable of seating three
thousand people. In the assignment of halls care was taken so far as
possible to assign the larger halls to the more popular subjects, but it
often happened that a great speaker was of necessity assigned to
a smaller hall. Two of the halls also proved bad for speaking owing
to the traffic of the Intramural Railway, and there was lacking in

20 THE HISTORY OF THE CONGRESS

nearly all of the halls that academic peace and quiet which usually
surrounds gatherings of a scientific nature. This, however, was to be
expected in an exposition atmosphere, and was readily acquiesced
in by the speakers themselves, and very little objection was heard to
the halls as assigned. Every one seemed to recognize the fact that the
immediate value of the meeting lay in the commingling and fellowship,
and that the addresses, of which one could hear at most only one in six-
teen, could not be judged in the proper light until their publication.

SUPPORT OF THE SCIENTIFIC PUBLIC

A strong effort was made by the Organizing Committee to secure
the attendance of an audience which should not only in its proportions
be complimentary to the eminence of the speakers, but also be thor-
oughly appreciative of the addresses and conversant with the topic
under discussion. Letters were therefore sent to all of the prominent
scientific societies in the United States, asking that wherever possible
the meetings of the society be set for the Congress week in St. Louis,
and wherever this was not possible that the societies send special
delegates to attend the Congress, and urge their membership to make
an effort to be present. Personal letters were also sent to the leading
members of the different professions and sciences, to the faculties of
universities and colleges, urging them to attend, and pointing out the
necessity of the support of the American scientific public.

Special invitations were also sent in the name of the Organizing
Committee to the leading authorities of the various subjects under
discussion in the Congress, asking them to contribute a ten-minute
paper to any section in which they were particularly interested. The
result of this careful campaign, in addition to the general exploita-
tion which the Congress received, was such a flattering attendance of
American scientists, as to be both a compliment to the European
speakers and a benefit to scientific thought. Many societies, such as
the American Neurological Association, American Philological Asso-
ciation, American Mathematical Society, Physical and Chemical
Societies of America, American Astronomical Society, Germanic Con-
gress, American Electro-Therapeutic Association, held their annual
meetings during the week of the Congress, although the date rendered
it impossible for the majority of the associations to meet at that time.
The eighth International Geographic Congress adjourned from Wash-
ington to St. Louis to meet with the Congress of Arts and Science. In
response to the special invitations, two hundred forty-seven ten-
minute addresses were promised and one hundred two actually read.

RECEPTION OF FOREIGN GUESTS

Every effort was made by the Department of Congresses to assist
the foreign speakers in their traveling arrangements and to make

THE HISTORY OF THE CONGRESS

matters as easy and comfortable as possible. A letter of advice was
mailed to each speaker prior to his departure, carefully setting forth
the conditions of American travel, routes to be followed, reception
committees to be met, and other essential details. The official badge
of the Congress was also mailed, so that those wearing them might
be easily identified by the reception committees both in New York
and St. Louis. Nine tenths of the speakers cam.e by the way of New
York, and in order to facilitate the clearance of their baggage and to
provide for their fitting entertainment in New York, a special recep-
tion committee was formed composed of the following members : —

F. P. Keppel, Columbia University, New York City, Chairman.
Prof. Herbert V. Abbott, New York. Robert Hoguet, New York.

R. Arrowsmith, New York.
C. William Beebe, New York.
George Bendelari, New York.
Edward W. Berry, Passaic.
J. Fuller Berry, Old Forge,
Rev. H. C. Birckhead, New York.
Dr. James H. Canfield, New York.
Rev. G. A. Carstenson, New York.
Prof. H. S. Crampton, New York.
Sanford L. Cutler, New York.
Dr. Israel Davidson, New York.
William H. Davis, New York.
Prof. James C. Egbert, New York.
Dr. Haven Emerson, New York.
Prof. T. S. Fiske, New York.
J. D. Fitz-Gerald, II, Newark.
W. D. Forbes, Hoboken.
Clyde Furst, Yonkers.
William K. Gregory, New York.
George CO. Haas, New York.
Prof. W. A. Hervey, New York.
Carl Herzog, New York.

Dr. Percy Hughes, Brooklyn.
Prof. A. V. W. Jackson, New York.
Albert J. W. Kern, New York.
Prof. Charles F. Kroh, Orange.
Dr. George F. Kunz, New York.
Prof. L. A. Lousseaux, New York.
Frederic L. Luqueer, Brooklyn.
R. A. V. Minckwitz, New York.
Charles A. Nelson, New York.
Dr. Harry B. PenhoUow, New York.
Prof. E. D. Perry, New York.
John Pohhnan, New York.
Dr. Ernest Richard, New York.
Dr. K. E. Richter, New York.
Edward Russ, Hoboken.
Prof. C. L. Speranza, Oak Ridge.
Prof. Francis H. Stoddard, New York.
Dr. Anthony Spitzka, Goodground.
Harvey W. Thayer, Brooldyn.
Prof. H. A. Todd, New York.
Dr. E. M. Wahl, New York.
Prof. F. H. WUkens, New York.

To each foreign speaker was extended the courtesies of the Century
and the University clubs while remaining in New York City, Mention
should also be made of the assistance of the Treasury Department
and of the courtesy of Collector of the Port, Hon. N. N. Stranahan,
through whom special privileges of the Port were extended to
the members of the Congress. The work of the reception committee
was most satisfactorily and efficiently performed, and was highly
appreciated by the foreign guests.. Special acknowledgment is due
Mr, F. P. Keppel, of Columbia University, for his painstaking and
efficient management of the affairs of the committee in New York,
Many of the speakers proceeded singly to St. Louis, stopping at vari-
ous places, but the great majority went directly to the University of
Chicago, where they were entertained during the week preceding the
Congress by President Harper and Professor Small, of the University

THE HISTORY OF THE CONGRESS

of Chicago. The arrivals at St. Louis were made on Saturday the 17th
and Sunday the 18th of September. Many of the participants had
arrived at earUer dates, and fully twenty of the speakers were mem-
bers of the International Jury of Awards for their respective countries,
and had been in St. Louis since September 1, the beginning of the
Jury work.

A reception committee similar to that in New York was also
formed at St. Louis from the members of the University Club, and
their duties were to meet all incoming trains and conduct the members
of the Congress personally to their stopping-places, and assist them
in all matters of detail. This committee was comprised of the follow-
ing members, nearly all of the University Club, who performed
their work efficiently and enthusiastically to the great satisfaction
of the Exposition and to the thorough appreciation of the foreign
guests : —

V. M. Porter, Chairman, St. Louis.

E. H. Angert, St. Louis.

Gouverneur Calhoun, St. Louis.

W. M. Chauvenet, St. Louis.

H. G. Cleveland, St. Louis.

Mr. M. B. Clopton, St. Louis.

Walter Fischel, St. Louis.

W. L. R. Gifford, St. Louis.

E. M. Grossman, St. Louis.

L. W. Hagerman, St. Louis.

Louis La Beaume, St. Louis.

Carl H. Lagenburg,
Sears Lelimann,
G. F. Paddock,
T. G. Rutledge,
Luther Ely Smith,
J. Clarence Taussig,
C. E. L. Thomas,
W. M. Tompkins,
G. T. Weitzel,
Tyrrell Williams,

St. Louis.
St. Louis.
St. Louis,
St. Louis.
St. Louis.
St. Louis.
St. Louis.
St. Louis.
St. Louis.
St. Louis.

The itinerary of the foreign speakers after leaving St. Louis at the
end of the Congress took them on appointed trains to Washington,
where they were given an official reception by President Roosevelt
and a reception by Dr. Simon Newcomb, President of the Congress.
From here they proceeded to Harvard University, Cambridge, Mass.,
where they were given a reception by Prof. Hugo Miinsterberg,
and were entertained as guests of Harvard University. Thence the
great majority of the speakers returned to New York, where the}^
were the guests of Columbia University, and were given a farewell
dinner by the Association of Old German Students. Many of the
speakers, however, visited other portions of the country before
returning to Europe.

The foreign speakers while in St. Louis were considered the guests
of the Exposition Company, and were relieved from all care and
expense for rooms and entertainment. Those who were accompanied
by their wives and daughters were entertained by prominent St. Louis
families, and those who came singly were quartered in the dormitory
of the Washington University, which was set aside for this purpose
during the week of the Congress. The dormitory arrangement proved
a very happy circumstance, as nearly one hundred foreign and Amer-

THE HISTORY OF THE CONGRESS 23

ican scientists of the highest rank were thrown in contact, much after
the fashion of their student days, and thoroughly enjoyed the novelty
and fellowship of the plan. The dormitory contained ninety-six
rooms newly fitted up with much care and with all modern con-
veniences. Light breakfasts were served in the rooms, and special
service provided at the call of the occupants. The situation of the
dormitory also in the Exposition grounds in close proximity to the
assembly halls was highly appreciated, and although at times there
were minor matters which did not run so smoothly, the almost
unanimous expression of the guests of the Exposition was one of
delight and appreciation of the arrangements. Special mention ought
in justice to be made to those residents of St. Louis who sustained
the time-honored name of the city for hospitality and courtesy by
entertaining those foreign members of the Congress who were accom-
panied by the immediate members of their family. They were as
follows: —

Dr. C. Barck Mr. Edward Mallinckrodt

Dr. William Bartlett Mr. George D. Markham

Judge W. F. Boyle Mr. Thomas McKittrick

Mr. Robert Brookings Mr. Theodore Meier

Mrs. J. T. Davis Dr. S. J. Niccolls

Dr. Samuel Dodd Dr. W. F. Nolker

Mr. L. D. Dozier Dr. S. J. Schwab

Dr. W. E. Fischel Dr. Henry Schwartz

Mr. Louis Fusz Mr. Corwin H. Spencer

Mr. August Gehner Dr. William Taussig

Dr. M. A. Goldstein Mr. G. H. Tenbroek

Mr. Charles H. Huttig Dr. Herman Tuholske

Dr. Ernest Jonas Hon. Rolla Wells

Mr. R. McKittrick Jones Mr. Edwards Whitaker

Mr. F. W. Lehmann Mr. Charles Wuelfing

Dr. Robert Luedeking Mr. Max Wuelfing.

DETAIL OF THE CONGRESS

The immense amount of detail work which devolved upon the
Department in the matter of preparing halls for the meetings, receiv-
ing guests, providing for their comfort, issuing the programmes,
managing the detail of the receptions, banquets, invitations, etc.,
providing for registration, payment of honorariums, and furnishing
information on every conceivable topic, rendered necessary the for-
mation of a special bureau which was placed in charge of Dr. L. O.
Howard of Washington, D. C, as Executive Secretary. Dr. Howard's
long experience as Secretary of the American Association for the
Advancement of Science rendered him particularly well qualified to
assume this laborious and thankless task. By mutual arrangement
the Director of Congresses and the Executive Secretary divided
the field of labor. The Director had, in addition to the general over-

24 THE HISTORY OF THE CONGRESS

sight of the Congress, special supervision of the local reception com-
mittee, the entertainment of the guests, official banquets and enter-
tainments, and all financial details. The Executive Secretary took
entire charge of the programme, assignment of rooms in the dormi-
tory, care and supervision of the dormitory, assignment of halls for
speakers, registration books and bureau of information. Dr. Howard
arrived on September 1 to begin his duties, and remained until
September 30.

WEEK OF THE CONGRESS

The opening session of the Congress was set for Monday afternoon,
September 19, at 2.30 o'clock in Festival Hall. The main programme
of the Congress began Tuesday morning. The sessions were held in
the mornings and afternoons, the evenings being left free for social
affairs. The list of functions authorized in honor of the Congress of
Arts and Science were as follows : —

Monday evening, September 19, grand fete night in honor of the
guests of the Congress, with special musical programme about the
Grand Basin and lagoons, boat rides and lagoon fete; this function
was unfortunately somewhat marred by inclement weather. It was
the only evening free in the entire week, however, for members of
the Congress to witness the illuminations and decorative evening
effects.

Banquet given by the St. Louis Chemical Society at the Southern
Hotel to members of the chemical sections of the Congress.

Tuesday evening, September 20, general reception by the Board
of Lady Managers to the officers and speakers of the Congress and
officials of the Exposition.

Wednesday afternoon, September 21, garden fete given to the
members of the Congress at the French National Pavilion by the
Commissioner-General from France. The gardens of the miniature
Grand Trianon were never more beautiful than on this brilliant after-
noon, and the presence of the Garde Republicaine band and the entire
official representation of the Exposition, lent a color and spirit to the
affair unsurpassed during the Exposition period.

Wednesday evening, reception by the Imperial German Commis-
sioner-General to the officers and speakers of the Congress and the
officials of the Exposition, at the German State House. The magni-
ficent hospitality which characterized this building during the entire
Exposition period was fairly outdone on this occasion, and the func-
tion stands prominent as one of the brilliant successes of the Exposi-
tion period.

Thursday evening, September 22, Shaw banquet at the Bucking-
ham Club to the foreign delegates and officers of the Congress.
Through the courtesy of the trustees of Shaw's Garden and of the

THE HISTORY OF THE CONGRESS 25

officers of Washington University, the annual banquet provided for
men of science, letters, and affairs, by the will of Henry B. Shaw,
founder of the Missouri Botanical Gardens, was given during this
week as a compliment to the noted foreign scientists who were the
guests of the city of St. Louis.

Friday evening, September 23, official banquet given by the
Exposition to the speakers and officials of the Congress and the
officials of the Exposition, in the banquet hall of the Tyrolean Alps.

Saturday evening, September 24, banquet at the St. Louis Club
given by the Round Table of St. Louis, to the foreign members of the
Congress. The Round Table is a literary club which meets at banquet
six times annually for discussion of topics of interest to the literary
and scientific world.

Banquet given by the Imperial Commissioner-General from Japan
to the Japanese delegation to the Congress and to the Exposition
officials and Chiefs of Departments.

Dinner given by Commissioner-General from Great Britain to the
English members of the Congress.

OPENING OF THE CONGRESS

The assembhng of the Congress on the afternoon of September 19,
in the magnificent auditorium of Festival Hall which crowned Cascade
Hill and the Terrace of States, was marked with simple ceremonies
and impressive dignity. The great organ pealed the national hymns
of the countries participating and closed with the national anthem
of the United States. In the audience were the members of the Con-
gress representing the selected talent of the world in their field of
scientific endeavor, and about them were grouped an audience drawn
from every part of the United States to promote by their presence the
success of the Congress and to do honor to the noted personages who
were the guests of the Exposition and of the Nation. On the stage
were seated the officials of the Congress, the honorary vice-presidents
from foreign nations, and the officials of the Exposition.

At the appointed hour the Director of Congresses, Dr. Howard J.
Rogers, called the meeting to order, and outlined in a few words the
object of the Congress, welcomed the foreign delegates, and presented
the members, both foreign and American, to the President of the
Exposition, Hon. David R. Francis.

The President spoke as follows : —

What an ambitious undertaking is a universal exposition! But how worthy
it is of the highest effort! And, if successful, how far-reaching are its results,
how lasting its benefits! Who shall pass judgment on that success? On what
evidence, by what standards shall their verdicts be formed? The development
of society, the advancement of civilization, involve many problems, encounter
many and serious difficulties, and have met with deplorable reactions which
decades and centuries w^e required to repair. The proper study of mankind is

26 THE HISTORY OF THE CONGRESS

man, and any progress in science that ignores or loses sight of his welfare and
happiness, however admirable and wonderful such progress may be, disturbs the
equilibrium of society.

The tendency of the times toward centralization or unification is, from an
economic standpoint, a drifting in the right direction, but the piloting must be
done by skillful hands, under the supervision and control of far-seeing minds, who
will remember that the masses are human beings whose education and expanding
intelligence are constantly broadening and emphasizing their individuality. A
universal exposition affords to its visitors, and those who systematically study its
exhibits and its phases, an unequaled opportunity to view the general progress and
development of all countries and all races. Every line of human endeavor is here
represented.

The conventions heretofore held on these grounds and many planned to be
held — aggregating over three hundred — have been confined in their delibera-
tions to special lines of thought or activity. This international congress of arts
and sciences is the most comprehensive in its plan and scope of any ever held,
and is the first of its kind. The lines of its organization, I shall leave the Director
of Exhibits, who is also a member of the administrative board of this congress, to
explain. You who are members are already advised as to its scope, and your
almost universal and prompt acceptance of the invitations extended to you to
participate, implies an approval which we appreciate, and indicates a willingness
and a desire to cooperate in an effort to bring into intelligent and beneficial corre-
lation all branches of science, all lines of thought. You need no argument to con-
vince you of the eminent fitness of making such a congress a prominent feature
of a universal exposition in which education is the dominant feature.

The administrative board and the organizing committee have discharged their
onerous and responsible tasks with signal fidelity and ability, and the success that
has rewarded their efforts is a lasting monument to their wisdom. The manage-
ment of the Exposition tenders to them, collectively and individually, its grateful
acknowledgments. The membership in this congress represents the world's elect
in research and in thought. The participants were selected after a careful survey
of the entire field ; no limitations of national boundaries or racial affiliations
have been observed. The Universal Exposition of 1904, the city of St. Louis,
the Louisiana territory whose acquisition we are celebrating, the entire country,
and all participating in or visiting this Exposition are grateful for your coming,
and feel honored by your presence.

We are proud to welcome you to a scene where are presented the best and high-
est material products of all countries and of every civilization, participated in by
all peoples, from the most primitive to the most highly cultured — a marker in the
progress of the world, and of which the International Congress of Arts and Science
is the crowning feature.

May the atmosphere of this universal exposition, cliarged as it is mth the
restless energies of every phase of human activity and permeated by that ineffable
sentiment of universal brotherhood engendered by the intelligent sons of God, con-
gregating for the friendly rivalries of peace, inspire you with even higher thoughts
— imbue you with still broader sympathies, to the end that by your future labors
you may be still more helpful to the human race and place your fellow men under
yet deeper obligations.

Director Frederick J. V. Skiff was then introduced by the Presi-
dent as representing the Division of Exhibits, whose untiring labors
had filled the magnificent Exposition palaces surrounding the Festival
Hall with the visible products of those sciences and arts, the theory,

THE HISTORY OF THE CONGRESS 27

progress, and problems of which the Congress was assembled to
consider.

Mr. Skiff spoke as follows : —

The division of exhibits of the Universal Exposition of 1904 has looked for-
ward to this time, when the work it has performed is to be reviewed and discussed
by this distinguished body. I do not, of course, intend to convey the idea that
the international congress is to inspect or criticise the exhibitions, but I do mean
to say that the deliberations of this organization are contemporaneous with and
share the responsibility for the accomplishments of which the exhibitions made
are the visible evidences.

The great educational yield of a universal exposition comes from the intellec-
tual more than from the mechanical processes. It is the material condition of the
times. It is as weU the duty of the responsible authorities to go yet further and
record the thoughts and theories, the investigations, experiments, and observa-
tions of which these material things are the tangible results.

A congress of arts and science, whose membership is drawn from aU educational
as well as geographical zones, not only accounts for and analyzes the philosophy
of conditions, but points the way for further advance along the lines consistent
with demonstration. Its contribution to the hour is at once a history and a
prophecy.

The extent to which the deliberations and utterances of this congress may
regulate the development of society or give impulse to succeeding generations, it
is impossible to estimate, but not unreasonable to anticipate. The plans of the
congress matured in the minds of the best scholars; the classification of its pur-
pose, the scope, the selection of its distinguished participants, gave to the hopes
and ambitions of the management of the Exposition inspiration of a most exalted
degree. At first these ambitions were — not without reason — regarded as too
high. The plane upon which the congress had been inaugurated, the aim, the
broad intent, seemed beyond the merits, if not beyond the capacity, of this liitherto
not widely recognized intellectual centre. But the courage of the inception, the
loftiness of the purpose, appealed so profoundly to the toilers for truth and the
apostles of fact, that we find gathered here to-day in the heart of the new Western
continent the great minds whose impress on society has rendered possible the intel-
lectual heights to which this age has ascended and now beckon forward the stu-
dents of the world to limitless possibilities.

While international congresses of literature, science, art, and industry have been
accomplished by previous expositions, yet to classify and select the topics in sym-
pathy with the classification and installation of the exhibits material is a step
considerably in advance of the custom. The men who build an exposition must
by temperament, if not by characteristic, be educators. They must be in sym-
pathy with the welfare of humanity and its higher destiny. The exhibitions at this
Exposition are not the haphazard gatherings of convenient material, but the out-
come of a plan to illustrate the productiveness of mankind at this particular time,
carefully digested, thoroughly thought out, and conscientiously executed. The
exhibit, therefore, in each of the departments of the classification, as well as in the
groups of the different departments, are of such character, and so arranged as to
reflect the best that the world can do along departmental lines, and the best that
different peoples can do along group lines. The congresses accord with the ex-
hibits, and the exhibits give expression to the congresses.

Education has been the keynote of this Exposition. Were it not for the educa-
tional idea, the acts of government providing vast sums of money for the up-
building of this Exposition would have been impossible. This congress reflects
one idea vastly outstripping others, and that is, in the unity of thought in the

28 THE HISTORY OF THE CONGRESS

universal concert of purpose. It is the first time, I believe, that there has been an
international gathering of the authorities of all the sciences, and in that respect
the congress initiates and establishes the universal brotherhood of scholars.

A thought uncommunicated is of little value. An unrecorded achievement
is not an asset of society. The real lasting value of this congress wiU consist of the
printed record of its proceedings. The delivery of the addresses, reaching and
appealing to, as must necessarily be the case, a very limited number of people,
can be considered as only a method of reaching the lasting and perpetual good of
civilization.

In just the degree that this Exposition in its various divisions shall make a
record of accomplishments, and lead the way to further advance, this enterprise
has reached the expectations of its contributors and the hopes of its promoters.
This congress is the peak of the mountain that this Exposition has builded on
the highway of progress. From its heights we contemplate the past, record the
present, and gaze into the future.

This universal exposition is a world's university. The International Congress
of Arts and Science constitutes the faculty; the material on exhibition are the
laboratories and the museums; the students are mankind.

That in response to invitation of the splendid committee of patriotic men, to
whom all praise is due for their efforts in this crowning glory of the Exposition, so
eminent a gathering of the scholars and savants of the world has resulted, speaks
unmistakably for the fraternity of the world, for the sympathy of its citizenship,
and for the patriotism of its people.

In reply to these addresses of the officials of the Exposition, the
honorary Vice-Presidents for Great Britain, France, Germany, Rus-
sia, Austria, Italy, and Japan made brief responses in behalf of their
respective countries.

Sir William Ramsay of London spoke in the place of Hon. James
Bryce, extending England's thanks for the courtesy which had been
shown her representatives and declaring that England, particularly
in the scientific field, looked upon America as a relative and not as
a foreign country.

France was represented by Professor Jean Gaston Darboux, Per-
petual Secretary of the Academy of Sciences of Paris, who spoke as
follows : —

Mr. President, Ladies and Gentlemen, — My first word wiU be to thank
you for the honor which you have been so courteous as to pay my country in
reserving for her one of the vice-presidencies of the Congress. Since the time of
Franklin, who received at the hands of France the welcome which justice and his
own personal genius and worth demanded, most affectionate relations have not
ceased to unite the scientists of France and the scientists of America. The dis-
tinction which you have here accorded to us will contribute stiU further to render
these relations more intimate and more fraternal. In choosing me among so many
of the better fitted delegates sent by my country, you have without doubt wished
to pay special honor to the Academic des Sciences and to the Institut de France,
which I have the honor of representing in the position of Perpetual Secretary.
Permit me therefore to thank you in the name of these great societies, which are
happy to count in the number of their foreign associates and of their correspond-
ents so many of the scholars of America. In like manner as the Institut de France,
so the Congress which opens to-day seeks to unite at the same time letters, science,

THE HISTORY OF THE CONGRESS 29

and arts. We shall be happy and proud to take part in this work and contribute
to its success.

Germany was represented by Professor Wilhelm Waldeyer, of the
University of Berlin, who replied as follows: —

Mk. President, Honored Assemblage, — The esteemed invitation which has
been offered to me in this significant hour of the opening of the Congress of Arts
and Science to greet the members of this congress, and particularly my esteemed
compatriots, I have had no desire to decline. I have been for a fortnight imder
the free sky of this mighty city — so I must express myself, since enclosing walls
are unknown in the United States — and this fact, together with the hospitality
offered me in such delightful manner by the Chairman of the Committee on Con-
gresses, Mr. Frederick W. Lehmann, has almost made me a St. Louis man. There-
fore I may perhaps take it upon myseff to greet you here.

I confess that I arrived here with some misgiving — some doubts as to whether
the great task which was here imdertaken under most difficult circumstances
could be accomplished with even creditable success. These doubts entirely dis-
appeared the first time I entered the grounds of the World's Fair and obtained a
general view of the method, beautiful as well as practical, by which the treasures
gathered from the whole world were arranged and displayed. I trust you, too, will
have a like experience; and will soon recognize that a most earnest and good work
is here accomplished.

And I must remark at this time that we Germans may indeed be well satisfied
here; the unanimous and complete recognition which our cooperation in this
great work has received is almost disconcerting.

What can be said of the whole Exposition with reference to its extent and the
order in which everything is arranged, I may well say concerning the depart-
ments of science, especially interesting to us. In this hour in which the Congress
of Arts and Science is being opened, we shall not express any thanks to those who
took this part of the work upon their shoulders — a more difficult task indeed than
all the others, for here the problem is not to manage materials, but heads and
minds. And as I see here assembled a large number of German professors — I, too,
belong to the profession — of whom it is said, I know not with how much justice,
that they are hard to lead, the labors of the Directors and Presidents of the
Congress could not have been, and are not now, small. Neither shall we to-day
prophesy into what the Congress may develop. The greater number of speakers
cannot expect to have large audiences, but even to-day we can safely say this : the
imposing row of volumes in which shall be given to posterity the reviews here to
be presented concerning the present condition, and future problems of the sciences
and arts as they appear to the scientific world at the beginning of the twentieth
century, will provide a mommaental work of lasting value. This we may confi-
dently expect. The thanks which we to-day do not wish to anticipate in words, let
us show by our actions to our kind American hosts, and especially to the directors
of the World's Fair and of this Congress. With exalted mind, forgetting all little
annoyances which may and will come, let us go forward courageously to the work,
and let us do our best. Let us grasp heartily the open hand honestly extended to
us.

May this Congress of Arts and Science worthily take part in the great and
undisputed success which even to-day we must acknowledge the World's Fair
at St. Louis.

For Austria Dr. Theodore Escherich, of the University of Vienna,
responded as follows : —

30 THE HISTORY OF THE CONGRESS

In the name of the many Austrians present at the Congress I express the thanks
of my compatriots to the Committee which summoned us, for their invitation and
the hospitality so cordially extended. . . .

I congratulate the authorities upon the idea of opening this Congress. How
many world-expositions have already been held without an attempt having been
made to exhibit the spirit that has created this world of beautiful and useful
things ? It was reserved for these men to find the form in which the highest results
of human thought — Science — represented in the persons of her representatives,
could be incorporated in the compass of the World's Fair. The conception of this
International Congress of all Sciences in its originality and audacity, in its univer-
sality and comprehensive organization, is truly a child of the " young-American
spirit." . . .

After this Congress has come to a close and the collection of the lectures de-
livered, an unparalleled encyclopaedia of human knowledge, both in extent and
content, will have appeared. We may say that this Fair has become of epochal
importance, not alone for trade and manufactures, but also for science. These
proud palaces wiU long have disappeared and been forgotten when this work, a
monumentum aere perennius, shaU still testify to future generations the standard
of scientific attainment at the beginning of the twentieth century.

Short acknowledgments were then made for Russia by Dr. Oscar
Backlund, of the Astronomical Observatory at Pulkowa, Russia, and
for Japan by Prof. Nobushige Hozumi, of the Imperial University at
Tokio, Japan.

The last of the Vice-Presidents to respond to the addresses of wel-
come was Signor Attilio Brunialti, Councilor of State for Italy, who
after a few formal words in English broke into impassioned eloquence
in his native tongue, and in brilliant diction and graceful periods
expressed the deep feeling and profound joy which Italy, the mother
of arts, felt in participating in an occasion so historic and so magni-
ficent. Signor Brunialti said in part : —

I thank you, gentlemen, for the honor you have paid both to my country and
myself by electing me a Vice-President of this great scientific assembly. Would
that I could thank you in words in which vibrate the heart of Rome, the scientific
spirit of my land, and aU that it has given to the world for the progress of science,
literature, and art. You know Italy, gentlemen, you admire her, and therefore
it is for this also that my thanks are due to you. What ancient Rome has con-
tributed to the common patrimony of civilization is also refiected here in a thou-
sand ways, and a classical education, held in such honor, by a young and practical
people such as yours, excites our admiration and also our astonisliment. By giant
strides you are reviving the activity of Italy at the epoch of the Communes, when
all were animated by unwearying activity and our manufactures and arts held
the first place in Europe. I have already praised here the courageous spirit which
has suggested the meeting of this Congress — a Congress that will remain famous
in the annals of science. Many things in your country have aroused in me grow-
ing surprise, but nothing has struck me more, I assure you, than this homage to
science which is pushing aU the wealthy classes to a noble rivalry for the increase
of education and mental cultivation.

You have already large libraries and richly endowed universities, and every
kind of school, where the works of Greece and Rome are perhaps even more appre-
ciated and adapted to modern improvements than with us old classical nations.

THE HISTORY OF THE CONGRESS ' 31

Full of energy, activity, and wealth, you have before you perpetual progress, and
what, up to this, your youth has not allowed you to give to the world, you will
surely be able to give in the future. Use freely all the treasures of civilization, art,
and science that centuries have accumulated in the old world, and especially in
my beloved Italy; fructify them with your youthful initiation and with your
powerful energy. By so doing you will contribute to peace, and then we may say
with truth that we have prepared your route by the work of centuries; and like
unto those who from old age are prevented from following the bold young man
of Longfellow in his course, we will accompany you with our greetings and our
alterable affection.

By my voice, the native country of Columbus, of Galileo, of Michelangelo and
Raphael, of Macchiavelli and Volta, salutes and with open arms hails as her hope-
ful daughter young America, — thanking and blessing her for the road she has
opened to the sons of Italy, workmen and artists, to civilization, to science, and to
modem research and thought.

The Chairman of the Administrative Board, President Nicholas
Murray Butler, of Columbia University, was prevented by illness in
his family from being present at the Congress, and in place of the
address to have been delivered by him on the idea and development
of the Congress and the work of the Administrative Board, President
William R. Harper, of the University of Chicago, spoke on the same
subject as follows: —

I have been asked within a few hours by those in authority to present to you
on behalf of the Administrative Board of this International Congress a statement
concerning the origin and purpose of the congress. It is surely a source of great
disappointment to all concerned that the chairman of the board. President Butler,
is prevented from being present.

Many of us recall the fact that at the Paris Exposition of 1889 the first attempt
was made to do something systematic in the way of congresses. This attempt was
the natural outcome of the opinion which had come to exist that so splendid an
opportunity as was afforded by the coming together of leaders in every depart-
ment of activity should not be suffered to pass by unimproved. What could be
more natural in the stimulating and thought-provoking atmosphere of an exposi-
tion than the proposal to make provision for a consideration and discussion of
some of the problems so closely related to the interests represented by the exposi-
tion?

The results achieved at the Paris Exposition of 1889 were so striking as to lead
those in charge of the World's Columbian Exposition in Chicago, 1893, to organize
what was called the World's Congress Auxiliary, including a series of congresses,
in which, to use the language of the original decree, " the best workers in general
science, philosophy, literature, art, agriculture, trade, and labor were to meet to
present their experiences and results obtained in all those various lines of thought
up to the present time." Seven years later, in connection with the Paris Exposition
of 1900, there was held another similar series of international congresses. The
general idea had in this way slowly but surely gained recognition.

The authorities of the Universal Exposition at St. Louis, from the first, recog-
nized the desirability of providing for a congress which should exceed in its scope
those that had before been attempted. In the earliest days of the preparation for
this Exposition Mr. Frederick J. V. Skiff, the Director of the Field Columbian
Museum, my nearest neighbor in the city of Chicago, took occasion to present this
idea, and particularly to emphasize the specific point that something should be

32 • THE HISTORY OF THE CONGRESS

undertaken which not only might add dignity and glory to the great name of the
Exposition, but also constitute a permanent and valuable contribution to the
sum of human knowledge. After a consideration of the whole question, which
extended over many months, the committee on international congresses resolved
to establish an administrative board of seven members, to which should be com-
mitted the responsibility of suggesting a plan in detail for the attainment of the
ends desired. This Board was appointed in November, 1902, and consisted of
President Nicholas Murray Butler, of Columbia University, New York; President
R. H. Jesse, of the University of Missouri; President Henry S. Pritchett, of the
Massachusetts Institute of Technology; Dr. Herbert Putnam, Librarian of Con-
gress; Mr. Frederick J. V. Skiff, of the Field Columbian Museum, Chicago; Fred-
erick G. Holls, of New York City, and the present speaker.

This Board held several meetings for the study of the questions and problems
involved in the great undertaking. Much valuable counsel was received and con-
sidered. The Board was especially indebted, however, to Prof. Hugo Miinsterberg
of Harvard University for specific material which he placed at their disposal —
material which, with modification, served as the basis of the plans adopted by the
Board, and reconunended to the members of the Exposition.

At the same time the Administrative Board recommended the appointment of
Dr. Howard J. Rogers as the Director of Congresses, and nominated Prof. Simon
Newcomb of the United States Navy to be President of the Congress, and Pro-
fessors Hugo Miinsterberg of Harvard University and Albion W. Small of the
University of Chicago to be Vice-Presidents of the Congress; the three to consti-
tute the Organizing Committee of the Congress. This Organizing Committee was
later empowered to visit foreign countries and to extend personal invitations to men
distinguished in the arts and sciences to participate in the Congress. The recep-
tion accorded to these, our representatives, was most cordial. Of the 150 invita-
tions thus extended, 117 were accepted; and of the 117 learned savants who
accepted the invitation, 96 are here in person this afternoon to testify by their pre-
sence the interest they have felt in this great concourse of the world's leaders. I
am compelled by necessity this afternoon to omit many points of interest in rela-
tion to the origin and history of the undertaking, all of which will be published in
due time.

After many months of expectancy we have at last come together from aU the
nations of the world. But for what purpose? I do not know that to the statement
already published in the programme of the Congress anything can be added which
wiU really improve that statement. The purpose, as it has seemed to some of us,
is threefold:

In the first place, to secure such a general survey of the various fields of learn-
ing, with aU their "subdivisions and multiplication of specialties," as will at the
same time set forth their mutual relations and connections, and likewise constitute
an effort toward the unification of knowledge. This idea of unity has perhaps been
uppermost in the minds of all concerned with the work of organizing the Congress.

In the second place, to provide a platform from which might be presented the
various problems, a solution of wliich wiU be expected of the scholarship of the
future. This includes a recognition of the fundamental principles and conception
that underlie these mutual relations, and therefore serve necessarily as the basis
of all such future work. Here again the controlling idea is that of unity and law,
in other words, universal law.

In the third place, to bring together in person and spirit distinguished investi-
gators and scholars from all the countries of the world, in order that by contact of
one with another a mutual sympathy may be promoted, and a practical coopera-
tion may be effected among those whose lifework leads them far apart. Here, still
again, unity of result is sought for.

THE HISTORY OF THE CONGRESS 33

As we now take up the work of this convention, which abeady gives sure
promise of being notable among the conventions that have called together men
of different nations, let us confidently assure ourselves that the great purpose
which has throughout controlled in the different stages of its organization will be
realized; that because the Congress has been held, the nations of the earth will
find themselves drawn more closely together; that human thought will possess
a more unified organization and human life a more unified expression.

Following these addresses of welcome and of response came the
first paper of the specific programme, designed to be introductory to
the division, department, and section addresses of the week. This
address, which will be found in full in its proper place, on pages 135 to
147 of this volume, was given by Dr. Simon Newcomb, President of
the Congress and Chairman of the Organizing Committee, whose
labors for fifteen months were thus brought to a brilliant conclusion.

At the close of Dr. Newcomb 's address the assembly was dismissed
by a few words of President Francis, in which he placed at the disposi-
tion of the members of the Congress the courtesies and privileges of
the Exposition, and expressed the hope and belief that their presence
and the purpose for which they were assembled, would be the crown-
ing glory of the Universal Exposition of 1904.

On Tuesday, September 20, the seven division addresses and the
twenty-four department addresses were given, all the speakers being
Americans : Royce, in Normative Science; Wilson, in Historical
Science; Woodward, in Physical Science; Hall, in Mental Science;
Jordan, in Utilitarian Science; Lowell, in Social Regulation; and
Harris, in Social Culture, treating the main divisions of science and
their applications, each dwelling particularly on the scope of the great
field included in his address and the unification of the work therein.
The forty-eight department speakers divided the field of knowledge,
one address in each department giving the fundamental conceptions
and methods, the other the history and development of the work of
the department during the last century.

With Wednesday the international participation began, and in the
one hundred twenty-eight sections into which the departments were
divided one half of the speakers were drawn, so far as circum-
stances permitted, from foreign scientific circles. With the exception
of the last two sections, Religious Influence Personal, and Religious
Influence Social, the work of the Congress closed on Saturday after-
noon. These two sections having four speakers each were placed, one
on Sunday morning and one on Sunday afternoon, in Festival Hall,
and passes to the grounds given upon application to any one desiring
to attend. Large numbers availed themselves of the privilege, and the
closing hours of the Congress were eminently suitable and worthy of
its high success. At the end of the afternoon session in Festival Hall,
Vice-President of the Congress, Dr. Albion W. Small, reviewed in a
few words the work of the week, its meaning to science, its possible

34 THE HISTORY OF THE CONGRESS

effect upon American thought, and then formally announced the
Congress closed.

OFFICIAL BANQUET

The official banquet given by the Exposition to all participants,
members, and officials of the Congress, on Friday evening, at the
Tyrolean Alps banquet hall, proved a charming conclusion to the
labors of the week. No better place could be imagined for holding it,
within the grounds of an exposition, than the magnificently propor-
tioned music and dining hall of the "Alps." A room 160 feet by 105
feet, capable of seating fifteen hundred banqueters; the spacious,
oval, orchestral stage at the south end; the galleries and boxes along
the sides of the hall done in solid German oak; the beautiful and
impressive mural decorations, the work of the best painters of Ger-
many; the excellence of the cuisine, and the thoroughly drilled corps
of waiters, rendered the physical accessories of a banquet as nearly
perfect as possible in a function so extensive.

The banquet was the largest held during the Exposition period,
eight hundred invitations being issued and nearly seven hundred
persons present. The music was furnished by the famous Garde
Republicaine Band of France, as the Exposition orchestra was
obliged to fill its regular weekly assignment at Festival Hall, The
decorations of the hall, the lights and flowers, the musical pro-
gramme, the galleries and boxes filled with ladies representing the
official and social life of the Exposition, and the distinguished body
of the Congress, formed a picture which appealed to the admiration
and enthusiasm of every one alike. No attempt was made to assign
seats to the banqueters outside the speakers' table, and little coteries
and clusters of scientists, many of whom were making acquaintances
and intellectual alliances during this week which would endure for
a lifetime, were scattered about the hall, giving an interest and an ani-
mation to the scene quite beyond the powers of description. In one
corner were Harnack, Budde, Jean Reville, and Cuthbert Hall, chat-
ting as animatedly as though their religious theories were not as far
apart as the poles; in another, Waldeyer, Escherich, Jacobi, Allbutt,
and Eatasato formed a medical group, the counterpart of which would
be hard to find unless in another part of this same hall; still again
were Erdmann, Sorley, Ladd, Royce, and Creighton as the centre of
a group of philosophers of world renown. So in every part of the
picture which met the eye were focused the leaders of thought and
action in their respective fields. The tout ensemble of the Congress was
here brought out in its strongest effect, as, with the exception of the
opening exercises at Festival Hall at which time many had not arrived ,
it was the only time when the entire membership was together. The
banquet coming at the close of the week was also fortunate, as by this

THE HISTORY OF THE CONGRESS 35

time the acquaintances made, and the common incidents and anec-
dotes experienced, heightened the enjoyment of all.

The toastmaster of the banquet and presiding officer, Hon. David
R. Francis, was never in a happier vein than when he assumed the
gavel and proposed the health of the President of the United States
and the rulers of all nations represented at the board.

President Francis said : —

Members op the International Congress op Arts and Science :

On the fa9ade at the base of the Louisiana Monument, which is the central
feature of this Exposition picture, is a group of Livingston, Monroe, and Marbois.
It represents the signing of the treaty, which by peaceful negotiation transferred
an empire from France to the United States. Upon the inscription are the words
of Livingston, " We have lived long and accompUshed much, but this is the
crowning act of our lives."

It is that transfer of an empire which this Exposition is held to commemo-
rate. And paraphrasing the words of Livingston, permit me to say that I have
presided over many dinners, but this is the crowning act of my career.

In opening the deUberations of the International Congress of Arts and Science,
I made the statement that a Universal Exposition is an ambitious undertaking.
I stated also that the International Congress of Arts and Science is the crowning
feature of this Exposition. I did not venture the assertion then which I have the
presumption to make now, that the most difficult task in connection with this
Universal Exposition was the assembling of an International Congress of Arts
and Science. I venture to make the statement now, because I feel that I am justi-
fied in doing so by the success which up to the present has attended your delibera-
tions. Any congregation of the leaders of thought in the world is a memorable
occasion. This is the first systematic one that has ever been attempted. Whether
it proves successful or not, it will be long remembered in the history of the civilized
countries that have participated in it. If it be but the precursor of other like
assemblages it wiU stiU be long remembered, and in that event it wiU be entitled
to unspeakable credit if it accomplishes anything toward the realization of the
very laudable objects which prompted its assembhng.

The effort to unify aU human knowledge and to establish the inter-relations
thereof is a bold conception, and requires the courage that characterizes the
people who live in the western section of the United States. If it be the last effort
of the kind it will still be remembered, and this Universal Exposition, if it had
done nothing else to endear it to cultured people of this and other countries, will
not be forgotten. The savants assembled by the caU of this Exposition have pur-
sued their respective lines of thought and research, prompted by no desire other
than one to find a solution of the problem which confronts humanity. By bringing
you together and making an effort to determine and establish the relations between
all lines of human knowledge, we have certainly made an advance in the right
direction. If your researches, if the results of your studies, can be utilized by
the human race, then we who have been the instruments of that great blessing
wiU be entitled to credit secondary only to the men who are the discoverers of
the scientific knowledge whose relations we are endeavoring to establish. The
Management of the Universal Exposition of 1904 salutes the International Con-
gress of Arts and Science. We drink to the perpetuation of that organization, and
I shall caU upon its distinguished President, Professor Newcomb, to respond to
the sentiment.

Dr. Newcomb in a few words thanked the members of the Congress

36 THE HISTORY OF THE CONGRESS

for their participation, which had made possible the brilliant success
of the enterprise, portrayed its effect and the influence of its perpetua-
tion, and then extended to all the invitation from the President of
the United States to attend the reception at the White House on the
following Tuesday.

In responding to these toasts the senior Honorary Vice-President,
Hon. James Bryce, of Great Britain, spoke in matchless form and
held the attention of the vast hall closely while he portrayed in a few
words the chief glories of England in the field of science, and the
pride the English nation felt in the glorious record made by her
eldest daughter, the United States, Mr. Bryce spoke extempora-
neously, and his remarks cannot be given in full.

For Germany, Commissioner-General Lewald responded in an
eloquent address, in which, after thanking the Exposition and the
American Government for the high honor done the German nation in
selecting so large a percentage of the speakers from German scien-
tific circles, he enlarged upon the close relations which had existed
between German university thought and methods and American
thought and practice, due to the vast number of American students
who had pursued their post-graduate courses in the universities of
Germany. He dwelt upon the pride that Germany felt in this sincerest
form of tribute to German supremacy in scientific thought, and of the
satisfaction which the influence in this country of German-trained
students afforded. He described at length the great exhibit made by
German universities in the education department of the Exposition,
and pointed to it as demonstrating the supremacy of German scienti-
fic thought and accurate methods. Dr. Lewald closed with a brilliant
peroration, in which he referred to the immense service done for the
cause of science in the last fifty years of German history and to the
patronage and support of the Emperor, not only to science in general,
but to this great international gathering of scientific experts, and
drank to the continued cordial relations of Germany and America
through its university circles and scientific endeavors.

For the response from France, Prof. Gaston Darboux was dele-
gated by Commissioner-General Gerald, who was unable to be present
on account of sickness. In one of the most beautiful and polished
addresses of the evening. Professor Darboux spoke in French, of which
the following is a translation : —

Gentlemen, — Graciously invited to respond in the name of the delegates
of France who have accepted the invitation of the American Government, I con-
sider it my duty in the first place to thank this great nation for the honor which
it has paid to us, and for the welcome which it has extended to us. Those of you
who are doing me the honor to listen, know of that disagreeable feeling of isolation
which at times the traveler in the midst of a strange people experiences; — that
feeling I know only from hearsay. We have not had a moment of time to experi-
ence it. They are accustomed in Europe to portray the Americans as exclusively

THE HISTORY OF THE CONGRESS 37

occupied with business affairs. They throw in our faces the famous proverb/ Busi-
ness is Business/ and give it to us as the rule of conduct for Americans. We are
able to testify entirely to the contrary, since the inhabitants of this beautiful coun-
try are always seeking to extend to strangers a thousand courtesies. Above aU, we
have encountered no one who has not been anxious to go out of his way to give
to us, even before we had asked it, such information as it was necessary for us to
have. And what shall I say of the welcome which we have received here at the
hands of our American confreres, — Monsieur the President of the Exposition,
Monsieur the Director of Congresses and other worthy colaborers? The authori-
ties of the Exposition and the inhabitants of St. Louis have rivaled each other in
making our stay agreeable and our ways pleasant in the heart of this magnificent
Exposition, of which we shall ever preserve the most enchanting memory.

We should have wished to see in a more leisurely manner, and to make
acquaintance with the attractions without number with which the Exposition
literally swarms (men of letters and men of science love at times to disport
themselves) and to study the exhibits classified in a method so exact in the
palaces of an architecture so original and so impressive. But Monsieur Newcomb
has not permitted this. The Congress of which he is the illustrious President offers
so much in the way of attractions, — of a kind a little rigorous it is true, — and so
much of work to be accomplished, that to our very great regret we have had to
refuse many invitations which it would have been most agreeable to accept. The
Americans wiU pardon us for this, I am sure; they know better than any one else
the value of time, but they know also that human strength has some limits, espe-
cially among us poor Europeans, for I doubt whether an American ever knows
the meaning of fatigue.

Messieurs, the Congress which is about to terminate to-morrow has been truly
a very great event. It is the first time, I beUeve, that there has been seen assembled
in one grand international reunion that which our great minister, Colbert, had in
mind, and that which we have realized for the first time in our Institut de France,
— the union of letters, science, and arts. That this union shall maintain itself in
the future is the dearest wish of my heart.

Science is a unit, even as the Universe. The aspects which it presents know
neither boundaries of states nor the pohtical divisions established between peoples .
In all civilized countries they calculate with the same figures, they measure with
the same instruments, they employ the same classifications, they study the same
historic facts, economics, and morals. If there exists among the different nations
some differences in methods, these differences are slight. They are a benefit at the
same time as well as a necessity. For the doing of the immense amount of work
of research imposed on that part of humanity which thinks, it is necessary that
the subjects of study should not be identically the same, or better, if they are
identical, that the difference between the points of view from which they are con-
sidered in the different countries contribute to our better knowledge of their
nature, their results, and their applications. It is necessary then that each people
preserve their distinctive genius, their particular methods which they use to
develop the qualities they have inherited. In exactly the same way that it is
important in an orchestra that each instrument play in the most perfect manner,
and with the timbre which accords with its nature, the part which is given to it,
so in science as in music, the harmony between the players is a necessary condi-
tion, which each one ought to exert himself to realize. Let us endeavor then in
scientific research to execute in the most perfect manner that part of the task
which fate has devolved upon us, but let us endeavor also to maintain that accord
which is a necessary condition to the harmony which wiU alone be able in the
future to assure the progress of humanity.

Gentlemen, in this international reunion it would not be fitting that I dwell

38 THE HISTORY OF THE CONGRESS

upon the services which my country has been able to render to science; and on
the other hand it would be difficult for me to say to you exactly what part America
is called upon to take in this concert of civilized nations ; but I am certain that the
part wiU be worthy of the great nation which has given to itself a constitution so
hberal and which in so short a space of time has known how to conquer, and
measure in value, a territory so immense that it extends from ocean to ocean. I
lift my glass to the honor of American science; I drink to the future of that great
nation, for which we, as weU as aU other Frenchmen, hold so much of common
remembrance, so much of close and Uving sympathy, and so much of profound
admiration. I am the more happy to do this in this- most beautiful territory of
Louisiana, which France in a former age ceded freely to America.

Perhaps the treat of the evening was the response made in behalf
of the Empire of Japan by Professor Hozumi, of the Faculty of Law
of the University of Tokio.

Unfortunately this response was not preserved in full, but Professor
Hozumi dwelt with much feeling on the world-wide significance of the
Congress and the common plane upon which all nations might meet
in the pursuit of science and the manifold applications of scientific
principles. He paid a beautiful tribute to the educational system of the
United States and to the great debt which Japan owed to American
scholars and to American teachers for their aid in establishing mod-
ern educational principles and methods in the Empire of Japan. The
impetus given to scientific study in Japan by the Japanese students
trained in American universities was also earnestly dwelt upon, and
the close relations which had always existed between Japanese and
American students and instructors feelingly described. In the field
of science Japan was yet young, but she had shown herself a close
and apt pupil, and her period of initiative and original research was
at hand. In bacteriology, in medicine, in seismology, oceanography,
and other fields, Japan has made valuable contributions to science
and established the right to recognition in an international gathering
of this nature. It was with peculiar and grateful pride and pleasure
that the Japanese Government had sent its delegation to this Con-
gress of selected experts in response to the invitation of the American
Government. Near the close of his address Professor Hozumi made
a gracious and happy allusion, based upon the conflict with Russia,
in which he said that of all places where men meet, and of all places
sunned by the light of heaven, this great Congress, built on the high
plane of the brotherhood of science and the fellowship of scholars,
was the only place where a Japanese and a Russian could meet in
mutual accord, with a common purpose, and clasp hands in unity of
thought. This chivalrous and beautiful idea, given here so imper-
fectly from memory, brought the great assembly to its feet in rounds
of cheers. In closing, Professor Hozumi expressed the earnest belief
that the benefits of science from a gathering of this nature would
quickly be felt, by a closer cooperation in the application of theory

THE HISTORY OF THE CONGRESS 39

and practical principles and a simultaneous advance in all parts of the
world.

The closing response of the evening for the foreign members was
made for Italy by Signor Attilio Brunialti, whose brilliant eloquence
at many times during the week had won the admiration of the mem-
bers of the Congress. Under the inspiration of this assemblage he
fairly surpassed himself, and the following translation of his remarks
but poorly indicates the grace and brilliant diction of the original : —

I have had the good fortune to be present in this wonderful country at three
international Congresses, that of science, the peace parliament, and the geo-
graphic. I wish to record the impression they have excited in my mind, already so
favorably inchned by your never-to-be-forgotten and gracious reception. You
must, please, allow me to address you in my own language, because the Latin
tongue inspires me, because I wish to affirm more solemnlj^ my nationality, and
also, because I cannot express my feelings weU in a language not familiar to me.
My country, the land of Columbus, of GaUleo, the nation that more than all others
in Europe is an element of peace, is already in itself the synthesis of the three
Congresses. And I can caU to mind that tliis land is indebted to geography for
the fact of its being made kno-mi to the world, because the immortal Genoese
pointed it out to people fighting in the old world for a small territory, and opened
to mortals new and extensive countries destined to receive the valiant and the
audacious of the entire world and to rise like yours to immortal glory.

Thus the poet can sing, —

L' avanza, 1' avanza
Divino straniero,
Conosci la stanza
Che i fati ti diero;
Se lutti, se lagrime
Ancora rinterra
L' giovin la terra.

Thus Columbus of old could point out to men — who run down each other,
disputing even love for fear that man may become a wolf for man — the vast
and endless wastes awaiting laborers, and give to man the treasures of the fruit-
ful land. 'Tis in the name of peace that I greet modern science in all its forms,
and I say to you chemists: "Invent new means of destruction;" and to you
mechanics and shipbuilders: " Give us invulnerable men-of-war and such per-
fect caimons, that your owti progress may contribute to make war rarer in the
world." Then will men, amazed at their own destructive progress, be drawn
together by brotherly love, by the development of common knowledge and
sympathy, and by the study of geography be led to know that there is plenty of
room for every one in the world to contribute to progress and civilization.

Americans! these sentiments are graven in your country; in point of fact, it is
a proof of the harmony that reigns in this Congress between guests come from all
parts of the world, that I, an Italian, am allowed to address you in my own lan-
guage on American ground, near the Tyrolean Alps, greeted by the music of the
Republicaine French Garde, united in eternal bonds of friendship by the two
great goddesses of the modern world, — Science and Peace.

The last speaker of the evening was Hon. Frederick W. Lehmann,
Chairman of the Exposition Committee on Congresses, who in elo-
quent periods set forth the ambition of the city of St. Louis and the

40 THE HISTORY OF THE CONGRESS

Exposition of 190-1 in creating a Congress of intellect on the same high
plane that had characterized the educational ideals of the Exposition,.
and the intense satisfaction which the officials of the Congress felt in
its brilliant outcome, and the possibilities which it promised for an
unequaled contribution to scientific hterature.

At the close of these addresses the members of the Congress and
the spectators in the gallery sang, in full chorus and under the lead of
the Garde Repubhcaine Band, the various national anthems, closing
with "The Star Spangled Banner."

PUBLICATION OF THE REPORT

In accordance with the recommendation of the Administrative
Board to the Committee on Congresses, the Executive Committee
appointed Dr. Howard J. Rogers, Director of Congresses, editor of
the proceedings of the Congress of Arts and Science. The Congress
records were removed from St. Louis to Albany, Xew York, the home
of the Director, from which place the pubHcation has been prepared.
Upon collecting the papers it was found that they could be di'V'ided
logically, and with a fair degree of similarity in size, into eight volumes,
each of which should cover a definite and distinct portion of the pro-
gramme. These are as follows : —

Volume 1. History of the Congress, Scientific Plan of the Congress.

Philosophy, Mathematics.
Volume 2. Pohtical and Economic Hjstorv^, Histor}' of Law, History

of Rehgion.
Volume 3. History of Language, History of Literature, History of

Art.
Volume 4. Physics, Chemistry, Astronomy, Sciences of the Earth.
Volume 5. Biolog}-, Anthropology, Psychology, Sociology.
Volume 6. Medicine, Technology.

Volimie 7. Economics, PoUtics, Jurisprudence, Social Science.
Volume 8. Education, Rehgion.

The details and specifications of the volumes were prepared for
competitive bids and submitted to twelve of the prominent pubUsh-
ers of the country. The most advantageous bid was received from
Houghton, ^lifflin & Company of Boston, Mass., and was accepted
by the Exposition Company. The Administrative Board and the
authorities of the Exposition feel deeply pleased at the result, inas-
much as the imprint of this firm guarantees a work in fuU accord with
the high plane upon which the Congress has been conducted.

It was determined to print the entire proceedings in the English
language, inasmuch as the Congress was held in an English-speaking
countr}- and the vast majority of the papers were read in that lan-
guage. The consent of ever}"- foreign speaker was obtained for this

THE HISTORY OF THE CONGRESS 41

procedure. It was found, after collecting, that the number of addresses
to be translated was forty-four. The translators were selected by
the editor upon the advice of the members of the Administrative
Board and Organizing Committee, and great care was taken to find
persons not only thoroughly trained in the two languages and pos-
sessing a good EngUsh style, but also persons who were thoroughly
conversant with the subject on which the paper treated. Many of
the translators were suggested by the foreign speakers themselves.
As a result of this careful selection, the editor feels confident that the
original value of the papers has been in no wise detracted from, and
that both in form and content the translations are thoroughly satis-
factory.

It wall be found that some addresses are not closely related to the
scheme of the Congress. Either through some misunderstanding of the
exact purpose of the Congress, or through too close devotion to their
own particular phase of investigation, some half-dozen speakers sub-
mitted papers deaUng with special lines of work. These, while valu-
able and scholarly from their standpoint, do not accord with a series
of papers prepared with a view to general relations and historical
perspective. The exceptions are so few, however, as not seriously to
interfere -^ith the unity of the plan.

In the arrangement of the papers the order of the official pro-
gramme is followed exactly, with the exception that, under Historical
Science, Departments 3, 4, and 8, covering Histor}^ of Politics, Law,
and Religion, are combined in one volume; and Departments 5, 6,
and 7, covering History of Language, Literature, and Art, are com-
bined in the succeeding volume. In volume one, the first chapter is
devoted to the history of the Congress, T\Titten by the editor, in which
is set forth the plain narrative of the growth and development of
the Congress, as much for the benefit of similar undertakings in the
future as for the interest of those participating in this Congress. The
second chapter contains the scientific introduction, written by Prof.
Hugo Miinsterberg of Harvard University, First Vice-President of
the Congress and Member of the Organizing Committee. This is
written for the purpose of giving in detail the principles upon which
the classification was based, and the relations which the different
sections and departments held to each other.

_ Each paper is prefaced by a very short biographical note in cate-
gorical form, for the purpose of insuring the identity of the speaker
as long in the future as the volumes may exist. Appended to the ad-
dresses of each department is a short bibliography, which is essential
for a general study of the subject in question. These are in no wise
exhaustive or complete, but are rather designed to be a small, valu-
able, working reference Hbrary for students. The bibliographies have
been prepared by eminent experts in the departments of the Con-

42 THE HISTORY OF THE CONGRESS

gress, but are necessarily somewhat uneven, as some of the writers
have gone into the subject more thoroughly than others. The general
arrangement of the bibliographies is: 1. Historical books and stand-
ard works dealing with the subject. 2. General books for the whole
department. 3. Books for sections of departments.

Appended also to the addresses of each department and sections
are resumes of the ten-minute addresses delivered by invitation at
the meeting of the department or section. Many of these papers are of
high value; but inasmuch as very few of them were written in accord
with the plan of the Congress, and with the main thought to be de-
veloped by the Congress, but deal rather with some interesting and
detached phase of the subject, it has been deemed best not to print
them in full, but to indicate in brief the subject and the treatment
given it by the writer. Those which do accord with the plan of the
Congress are given more extensive treatment.

CONCLUSION

What the results of the Congress will be; what influence it may
have; was it worth the work and cost, are questions often fairly asked.

The lasting results and influences are of course problematical.
They depend upon the character and soundness of the addresses, and
whether the uniform strength of the publication will make the work
as a whole, what it undoubtedly is in parts, a source-book for the
future on the bases of scientific theory at the beginning of the twenti-
eth century, and a reliable sketch of the growth of science during the
nineteenth century. Critical study of the addresses will alone deter-
mine this, but from the favorable reception of those already pub-
lished in reviews, and from editorial acquaintance with the others,
it seems assured. That portion of the section addresses which deals
with the inter-relations of science and demonstrates both its unity
and variety of processes is new and authoritative thought, and will be
the basis of much discussion and remodeling of theories in the future.

The immediate results of the Congress are highly satisfactory,
and fully repay the work and the cost both from a scientific and an
exposition standpoint. As an acknowledgment of the prominence
of scientific methods, as a public recognition of the work of scientists,
as the means of bringing to one place the most noted assemblage of
thinkers the world has ever seen, as an opportunity for scholars to
meet and know each other better, the Congress was an unqualified
success and of enduring reputation. From the Exposition point of
view, it was equally a success; not financially, nor was there ever
a thought that it would be. Probably not more than seven thousand
persons outside of St. Louis came primarily to attend the Congress,
and their admission fees were a bagatelle; the revenue derived from
the sale of the Proceedings will not meet the cost of printing. There

THE HISTORY OF THE CONGRESS 43

has been no money value sought for in the Congress, — none received.
Its value to the Exposition lies solely in the fact that it is the final
argument to the world of the initial claims of the officials of the
Exposition that its purpose was purely educational. Coordinate with
the material exhibits, sought, classified, and installed on a rigidly
scientific classification, the Congress, which relates, illumines, and
defends the principles upon which the material portion was founded,
has triumphantly vindicated the good faith, the wisdom, and the
foresight of the Universal Exposition of 1904. This printed record of
its proceedings will be a monument not only to the spirit of Science,
but to the spirit of the Exposition, which will endure as long as the
records of man are preserved.

In conclusion, the editor wishes to express his obligations to the
many speakers and officers of the Congress, who have evinced great
interest in the publication and assisted by valuable suggestions and
advice. In particular, he acknowledges the help of President Butler
of Columbia University, Professor Miinsterberg of Harvard Uni-
versity, and Professor Small of the University of Chicago. Acknow-
ledgments are with justice and pleasure made to the Committee on
Congresses of the Exposition, and the able chairman, Hon. Frederick
W. Lehmann, for their unwavering and prompt support on aU mat-
ters of policy and detail, without which the full measure of success
could not have been achieved. To the efficient secretary of the
Department of Congresses, Mr. James Green Cotchett, an expression
of obligation is due for his indefatigable labors during the Congress
period, and for his able and painstaking work in compiling the
detailed records of this publication.

At a meeting of the Executive Committee of the Exposition on
January 3, 1905, there was unanimously voted the following resolu-
tion, recommended by the Administrative Board and approved by
the Committee on Congresses : —

Moved : that a vote of thanks and an expression of deepest obliga-
tion be tendered to Dr. Simon Newcomb, President of the Congress,
Prof. Hugo Miinsterberg, vice-president of the Congress, and Prof.
Albion W. Small, vice-president of the Congress, for their efficient,
thorough, and comprehensive work in connection with the pro-
gramme of the Congress, the selection and invitation of speakers,
and the attention to detail in its execution. That, in view of the
enormous amount of labor devolving upon these three gentlemen
for the past eighteen- months, to the exclusion of all opportunities
for literary and other work outside their college departments, an
honorarium of twenty-five hundred dollars be tendered to each of
them.

44 THE HISTORY OF THE CONGRESS

At a subsequent meeting the following resolution was also passed : —
Moved : that the Directors of the Louisiana Purchase Exposition
Company place upon the record an expression of their appreciation
of the invaluable aid so freely given by the Administrative Board
of the Congress of Arts and Science. In organization, guidance, and
results the Congress was the most notable of its kind in history.
For the important part performed wisely and zealously by the Admin-
istrative Board the Exposition Management extends this aeknow-
ledgment.

SUMMARY OF EXPENSES OF THE CONGRESS

Office expenses $7,025 82

Travel 3,847 24

Exploitation, Organizing Committee abroad . . , 8,663 16

Traveling expenses, American Speakers 31,350

Traveling expenses. Foreign Speakers 49,000

Honorariums 7,500

Banquet 3,500

Expenses for editing proceedings . 5,875

Estimated cost of printing proceedings ..... 22,000 $138,761 22

INTERNATIONAL
CONGRESS OF ARTS AND SCIENCE

UNIVERSAL EXPOSITION ST. LOUIS

SEPTEMBER 19-25 1904

PROGRAMME AND LIST OF SPEAKERS

PROGRAMME

Purpose and Plan of the Congress

Organization of the Congress

Speakers and Chairmen

Chronological Order of Proceedings

Programme of Social Events

List of Ten-minute Speakers

List of Chairmen and Principal Speakers

INDEX SUBJECTS
Division A. Normative Science

Department i. Philosophy

Sec. A. Metaphysics

B. Philosophy of Religion

C. Logic

D. Methodology of Science

E. Ethics

F. Esthetics

Department 2. Mathematics

Sec. A. Algebra and Anatysis

B. Geometry

C. Applied Mathematics

Division B. Historical Science

Department 3. Political and
Economic History

Sec. A. History of Asia

B. History of Greece and Rome

C. Mediaeval History

D. Modem History of Europe

E. History of America

F. History of Economic Institu-

tions

Department 4. History of Law

Sec. A. History of Roman Law

B. History of Common Law

C. Comparative Law

Department 5. History of
Language

Sec. A. Comparative Language

B. Semitic Language

C. Indo-Iranian Languages

D. Greek Language •

E. Latin Language

F. English Language

G. Romance Languages
H. Germanic Languages

Department 6. History of Lit-
erature

Sec. A. Indo-Iranian Literature

B. Classical Literature

C. Enghsh Literature

D. Romance Literature

E. Germanic Literature

F. Slavic Literature

G. BeUes-Lettres

Department 7. History of Art

Sec. A. Classical Art

B. Modem Architecture

C. Modem Painting

Department 8. History of Re-
ligion

Sec. A. Brahminism and Buddhism

B. Mohammedism

C. Old Testament

D. New Testament

E. History of the Christian

Church

PROGRAMME

Division C. Physical Science

Department 9. Physics

Sec. A. Physics of Matter

B. Physics of Ether

C. Physics of the Electron

Department 10. Chemistry-
Sec. A. Inorganic Chemistry

B. Organic Chemistry

C. Physical Chemistry

D. Physiological Chemistry

Department 11. Astronomy

Sec. A. Astrometry
B. Astrophysics

Department 12. Sciences of the
Earth

Sec. A. Geophysics

B. Geology

C. Palaeontology

D. Petrology and Mineralogy

E. Physiography

F. Geography

G. Oceanography
H. Cosmical Physics

Department 13. Biology

Sec. A. Phylogeny

B. Plant Morphology

C. Plant Physiology

D. Plant Pathology

E. Ecology

F. Bacteriology

G. Animal Morphology
H. Embryology

I. Comparative Anatomy
J. Human Anatomy
K. Physiology

Department 14. Anthropology

Sec. A. Somatology

B. Archaeology

C. Ethnology

Division D. Mental Science

Department 15. Psychology

Sec. A. General Psychology

B. Experimental Psychology

C. Comparative and Genetic

Psychology

D. Abnormal Psychology

Department 16. Sociology

Sec. B. Social Structure
C. Social Psychology

Division E. Utilitarian Sciences

Department 17. Medicine

Sec. A. Pubhc Health

B. Preventive Medicine

C. Pathology

D. Therapeutics and Phar-

macology

E. Internal Medicine

F. Neurology

G. Psychiatry
H. Surgery

I. G5Tiecology

J. Ophthalmology

K. Otology and Laryngology

L. Pediatrics

Department 18. Technology

Sec. A. Civil Engineering

B. Mechanical Engineering

C. Electrical Engineering

D. Mining Engineering

E. Technical Chemistry

F. Agriculture

Department 19. Economics

Sec. A. Economic Theory

B. Transportation

C. Commerce and Exchange

D. Money and Credit

E. Public Finance

F. Insurance

PROGRAMME

Division F.
Department 20. Politics

Sec. A. Political Theory

B. Diplomacy

C. National Administration

D. Colonial Administration

E. Municipal Administration

Department 21. Jurisprudence

Sec. A. International Law

B. Constitutional Law

C. Private Law

Social Regulation

Department 22. Social Science

Sec. A. The Family

B. The Rural Community

C. The Urban Commimity

D. The Indu^rial Group

E. The Dependent Group

F. The Criminal Group

Division G. Social Culture

Department 23. Education

Sec. A. Educational Theory

B. The School

C. The College

D. The University

E. The Library

Department 24. Religion

Sec. A. General Religious Educa-
tion

B. Professional Rehgious Edu-

cation

C. Religious Agencies

D. Religious Work

E. Religious Influence: Per-

sonal

F. Religious Influence: Social

PURPOSE AND PLAN OF THE CONGRESS

The idea of the Congress grows out of the thought that the sub-
division and multiphcation of specialties in science has reached a stage
at which investigators and scholars may derive both inspiration and
profit from a general survey of the various fields of learning, planned
with a view of bringing the scattered sciences into closer mutual
relations. The central purpose is the unification of knowledge, an
effort toward which seems appropriate on an occasion when the
nations bring together an exhibit of their arts and industries. An
assemblage is therefore to be convened at which leading represent-
atives of theoretical and applied sciences shall set forth those general
principles and fundamental conceptions which connect groups of
sciences, review the historical development of special sciences, show
their mutual relations and discuss their present problems.

The speakers to treat the various themes are selected in advance
from the European and American continents. The discussions will
be arranged on the following general plan : —

After the opening of the Congress on Monday afternoon, Septem-
ber 19, will follow, on Tuesday forenoon, addresses on main divisions
of science and its applications, the general theme being the unification
of each of the fields treated. These will be followed by two addresses
on each of the twenty-four great departments of knowledge. The
theme of one address in each case will be the Fundamental Concep-
tions and Methods, while the other will set forth the progress during
the last century. The preceding addresses will be delivered by Ameri-
cans, making the work of the first two days the contribution of
American scholars.

On the third day, with the opening of the sections, the international
work will begin. One hundred twenty-eight sectional meetings will
be held on the four remaining days of the Congress, at each of
which two papers will be read, the theme of one being suggested by
the relations of the special branch treated to other branches; the
other by its present problems. Three hours will be devoted to each
sectional meeting, thus enabling each hearer to attend eight such
meetings, if he so desires. The programme is so arranged that related
subjects will be treated, as far as possible, at different times. The
length of the principal addresses being limited to forty-five minutes
each, there will remain at least one hour for five or six brief communi-
cations in each section. The addresses in each department will be
collected and published in a special volume.

PURPOSE AND PLAN OF THE CONGRESS 51

It is hoped that the living influence of this meeting will be yet more
important than the formal addresses, and that the scholars whose
names are announced in the following programme of speakers and
chairmen will form only a nucleus for the gathering of thousands who
feel in sympathy with the efforts to bring unity into the world of
knowledge.

ORGANIZATION OF THE CONGRESS

PRESIDENT OF THE EXPOSITION:
HON. DAVID R. FRANCIS, A.M., LL.D.

DIRECTOR OF CONGRESSES,

HOWARD J. ROGERS, A.M., LL.D..

Universal Exposition, 1904.

ADMINISTRATIVE BOARD

NICHOLAS MURRAY BUTLER, Ph.D., LL.D.

President of Columbia University, Chairman.

WILLIAM R. HARPER, Ph.D., LL.D.
President of the University of Chicago.

R. H. JESSE, Ph.D., LL.D.
President of the University of Missouri.

HENRY S. PRITCHETT, Ph.D., LL.D.
President of the Massachusetts Institute of Technology.

HERBERT PUTNAM, Litt.D., LL.D.
Librarian of Congress.

FREDERICK J. V. SKIFF, A.M.
Director of the Field Columbian Museum.

OFFICERS OF THE CONGRESS

PRESIDENT:

SIMON NEWCOMB, Ph.D., LL.D.

Retired Processor U. S. N.

VICE-PRESIDENTS:
HUGO MtJNSTERBERG, Ph.D., LL.D.

Professor of Psychology in Harvard University.

ALBION W. SMALL, Ph.D., LL.D.

Professor of Sociology in The University of Chicago.

HONORARY VICE-PRESIDENTS:

RIGHT HONORABLE JAMES BRYCE, M.P.

Great Britain.

M. GASTON DARBOUX,

France.

PROFESSOR WILHELM WALDEYER,

Germany.

DR. OSKAR BACKLUND,

Russia.

PROFESSOR THEODORE ESCHERICH,

Austria,

SIGNOR ATTILIO BRUNIALTI,
Italy.

PROFESSOR N. HOZUMI,
Japan.

EXECUTIVE SECRETARY:
DR. L. 0. HOWARD,

Permanent Secretary American Association
for the Advancement of Science.

SPEAKERS AND CHAIRMEN

DIVISION A— NORMATIVE SCIENCE

Speakek : Professor Josiah Royce, Harvard University.
{Hall 6, September 20, 10 a. m.)

Chairman :
Speakers :

SECTION A.

Chairman:
Speakers:

Secretary :

DEPARTMENT 1 — PHILOSOPHY

(Hall 6, September 20, 11.15 a. m.)

Professor Borden P. Bowne, Boston University.
Professor George H. Howison, University of Cali-
fornia.
Professor George T. Ladd, Yale University.

METAPHYSICS. {Hall 6, September 21, 10 a. m.)

Professor A. C. Armstrong, Wesleyan University.
Professor A. E. Taylor, McGlII University, Montreal.
Professor Alexander T. Ormond, Princeton Uni-
versity.
Professor A. O. Lovejoy, Washington University.

SECTION B. PHILOSOPHY OF RELIGION. {Hall 1, September 21, 3 p. m.)

Chairman: Professor Thomas C. Hall, Union Theological Sem-
inary, N. Y.

Speakers: Professor Otto Pfleiderer, University of Berlin.
Professor Ernst Troeltsch, University of Heidel-
berg.

Secretary: Dr. W. P. Montague, Columbia University.

SECTION C. LOGIC. {Hall 6, September 22, 10 a. m.)

Chairman: Professor George M. Duncan, Yale University.
Speakers: Professor William A. Hammond, Cornell University.
Professor Frederick J. E. Woodbridge, Columbia
University.
Secretary: Dr. W. H. Sheldon, Columbia University.

SECTION D. METHODOLOGY OF SCIENCE. {Hall 6, September 22, 3 p. m.)

Chairman: Professor James E. Creighton, Cornell University.
Speakers: Professor Wilhelm Ostwald, University of Leipzig.

Professor Benno Erdmann, University of Bonn.
Secretary: Dr. R. B. Perry, Harvard University.

SECTION E.

Chairman:
Speakers :

Secretary;

ETHICS. {Hall 6, September 23, 10 a. m.)

Professor George H. Palmer, Harvard University.
Professor William R. Sorley, University of Cam-
bridge.
Professor Paul Hensel, University of Erlangen,
Professor F. C. Sharp, University of Wisconsin.

SPEAKERS AND CHAIRMEN

SECTION F. AESTHETICS. {Hall 4, September 23, 3 p. to.)
Chairman: Professor James H. Tufts, University of Chicago.
Speakers :

Secretary:

Chairman:
Speakers:

SECTION A.

Chairman :
Speakers:

Secretary :

SECTION B.

Chairman:

Speakers:

Secretary:

SECTION C.

Chairman:

Speakers:
Secretary :

Dr. Henry Rutgers Marshall, New York City.
Professor Max Dessoir, University of Berlin.
Professor Max Meyer, University of Missouri.

DEPARTMENT 2 — MATHEMATICS

{Hall 7, September 20, 11.15 a. m.)
Professor Henry S. White, Northwestern Univers-
ity.
Professor Maxime B6cher, Harvard University.
Professor James P. Pierpont, Yale University.

ALGEBRA AND ANALYSIS. {Hall 9, September 22, 10 o. to.)

Professor E. H. Moore, University of Chicago.
Professor Emile Pi card, The Sorbonne; Member

of the Institute of France.
Professor Heinrich Maschke, University of Chicago.
Professor G. A. Bliss, University of Chicago.

GEOMETRY. {Hall 9, September 24, 10 a. to.)

Professor M. W. Haskell, University of California
Darboux, Perpetual Secretary of

the

M, Gaston

Academy of Sciences, Paris.

Dr. Edward Kasner, Columbia University.

Professor Thomas J. Holgate, Northwestern Uni-
versity.

{Hall 7, September 24, 3 p. to.)
Webster, Clark University,

APPLIED MATHEMATICS.

Professor Arthur G.

Worcester, Mass.
Professor Ludwig Boltzmann, University of Vienna.
Professor Henri Poincare, The Sorbonne; Member

of the Institute of France.
Professor Henry T. Eddy, University of Minnesota.

DIVISION B — HISTORICAL SCIENCE

{Hall 3, September 20, 10 a. m.)
Speaker: President Woodrow Wilson, Princeton University.

DEPARTMENT 3 — POLITICAL AND ECONOMIC HISTORY

{Halt 4:, September 20, 11.15 a. to.)

Chairman:

Speakers: Professor William M. Sloane, Columbia University.
Professor James H. Robinson, Columbia University.

SPEAKERS AND CHAIRMEN

SECTIONS A AND B. HISTORY OF GREECE, ROME. AND ASIA. (.Hall 3,
September 21, 10 a. m.)

Chairman:

Speakers;

Secretary:

Professor Thomas D. Seymour, Yale University.

Professor John P. Mahaffy, University of Dublin,

Professor Ettore Pais, University of Naples. Direc-
tor of the National Museum of Antiquities, Naples.

Professor Henri Cordier, Ecole des Langues Viv-
antes Orientales, Paris.

Professor Edward Capps, University of Chicago.

SECTION C. MEDIAEVAL HISTORY. {Hall 6, September 21, 3 p. to.)

Chairman: Professor Charles H. Haskins, Harvard University.
Speakers : Professor Karl Lamprecht, University of Leipzig.

Professor George B. Adams, Yale University.
Secretary: Professor Earle W. Dow, University of Michigan.

SECTION D.

Chairman :

Speakers:

Secretary :

SECTION E.

Chairman :
Speakers:

Secretary
SECTION F.

MODERN HISTORY OF EUROPE. {Hall 3, September 22,

10 a. m.)
Honorable James B. Perkins, Rochester, N. Y.
Professor J. B. Bury, University of Cambridge.
Professor Charles W. Colby, McGill University,

Montreal.
Professor Ferdinand Schwill, University of Chicago.

HISTORY OF AMERICA. {Hall 1, September 24, lO a. m.)

Dr. James Schouler, Boston.

Professor Frederic J. Turner, University of Wis-
consin.
Professor Edward G. Bourne, Yale University.
Professor Evarts B. Greene, University of Illinois.

{Hall 2, Septem-

HISTORY OF ECONOMIC INSTITUTIONS.

ber 23, 3 p. m.)

Chairman: Professor Frank A. Fetter, Cornell University.
Speakers: Professor J. E. Conrad, University of Halle.

Professor Simon N. Patten, University of Penn-
sylvania.
Secretary: Dr. J. Pease Norton, Yale University.

DEPARTMENT 4 — HISTORY OF LAW
{Hall 5, September 20, 11.15 a. to.)

Chairman: Honorable David J. Brewer, Associate Justice of
the Supreme Court of the United States.

Speakers: Honorable Emlin McClain, Judge of the Supreme
Court of Iowa, Iowa City.
Professor Nathan Abbott, Leland Stanford Jr.
University.

SECTION A. HISTORY OF ROMAN LAW. {Hall 11, September 21, 3 p. to.)

Chairman:

Speakers: Mr. W. H. Buckler, Baltimore, Md.

Professor Munroe Smith, Columbia University.

SPEAKERS AND CHAIRMEN

SECTION B. HISTORY OF COMMON LAW. (,Hall 11, September 21, 10 a. m.)

Chairman: Professor John D. Lawson, University of Missouri.
Speakers : Honorable Simeon E, Baldwin, Judge of the Supreme
Court of Errors, New Haven, Conn.
Professor John H. Wigmore, Northwestern Uni-
versity.
Secretary: Professor C. H. Huberich, University of Texas.

SECTION C. COMPARATIVE LAW. {Hall 14, September 24, 3 p. m.)

Chairman: Honorable Jacob M. Dickinson, Chicago.
Speakers: Professor Nobushige Hozumi, TJniversity of Tokio.
Professor Alfred Nerincx, University of Louvain.
Secretary:

DEPARTMENT 5 — HISTORY OF LANGUAGE
{Hall 4, September 20, 2 p. m.)

Chairman: Professor George Hempl, University of Michigan,
Speakers: Professor T. R, Lounsbury, Yale University.

President Benjamin Ide Wheeler, University of
California.

SECTION A. COMPARATIVE LANGUAGE. (Hall 4, September 21, 10 a. m.)

Chairman: Professor Francis A. March, Lafayette College.
Speakers: Professor Carl D. Buck, University of Chicago.

Professor Hans Oertel, Yale University.
Secretary: Professor E. W. Fay, University of Texas, Austin,
Texas.

SECTION B. SEMITIC LANGUAGES. {Hall 4, September' 21, 3 p. m.)

Chairman: Professor G. F. Moore, Harvard University.
Speakers: Professor James A. Craig, University of Michigan.

Professor Crawford H. Toy, Harvard University.
Secretary:

SECTION C. INDO-IRANLAJ? LANGUAGES. {Hall 8, September 22, 10 a. m.)

Chairman:

Speakers: Professor Sylvain Levi, College de France, Paris.

Professor Arthur A, Macdonell, University of
Oxford.
Secretary :

SECTION D. GREEK LANGUAGE. (Hall 3, September 22, 3 p. m.) '

Chairman : Professor Martin L. D'Ooge, University of Michigan.
Speakers: Professor Herbert W. Smyth, Harvard University.
Professor Milton W. Humphreys, University of
Virginia.
Secretary: Professor J. E. Harry, University of Cincinnati.

SECTION E. LATIN LANGUAGE. (Hall 9, September 23, 10 a. m.)
Chairman: Professor Maurice Hutton, University of Toronto.
Speakers: Professor E. A. Sonnenschein, University of Bir-
mingham.
Professor William G. Hale, University of Chicago.
Secretary: Professor F. W. Shipley, Washington University.

58 SPEAKERS AND CHAIRMEN

SECTION F. ENGLISH LANGUAGE. {Hall 3, September 23, 3 p. to.)

Chairman: Professor Charles M. Gayley, University of Cal-
ifornia.

Speakers: Professor Otto Jespersen, University of Copen-
hagen.
Professor George L. Kittredge, Harvard University.

Secretary :

SECTION G. ROMANCE LANGUAGES. {Hall 5, September 24, 10 a. to.)

Chairman:

Speakers: Professor Paul Meyer, College de France, Paris.
Professor Henry A. Todd, Columbia University.
Secretary: Professor E. E. Brandon, Miami University.

SECTION H. GERMANIC LANGUAGES. {Hall 3, September 24, 3 p. m.)

Chairman: Professor Gustaf E. Karsten, Cornell University.
Speakers: Professor Eduard Sievers, University of Leipzig,

Professor Herman Collitz, Bryn Mawr College.
Secretary :

DEPARTMENT 6 — HISTORY OF LITERATURE

{Hall 6, September 20, 4.15 p. m.)

Chairman:

Speakers : Professor James A. Harrison, University of Virginia.
Professor Charles M. Gayley, University of Cali-
fornia.

SECTION A. IltoO-IRANDm LITERATURE. {Hall 8, September 24, 3 p. to.)

Chairman: Professor Maurice Bloomfield, Johns Hopkins

University.
Speaker: Professor A. V. W. Jackson, Columbia University.
Secretary :

SECTION B. CLASSICAL LITERATURE. {Hall 3, September 21, 3 p. m.)

Chairman: Professor Andrew F. West, Princeton University.
Speakers: Professor Paul Shorey, University of Chicago.

Professor John H. Wright, Harvard University.
Secretary: Professor F. G. Moore, Dartmouth College.

SECTION C. ENGLISH LITERATURE. {Hall 1, September 22, 10 a. to.)

Chairman:

Speakers: Professor Francis B. Gummere, Haverford College.

Professor John Hoops, University of Heidelberg.
Secretary :

SECTION D. ROMANCE LITERATURE. {Hall 8, September 22, 3 p. to.)

Chairman: Professor Adolphe Cohn, Columbia University.
Speakers: Professor Pio Rajna, Institute of Higher Studies,

Florence, Italy.
Professor Alc^e Fortier, Tulane University, New

Orleans.
Secretary: Dr. Comfort, Haverford College.

SPEAKERS AND CHAIRMEN

SECTION E.

Chairman:
Speakers:

Secretary :
SECTION F.

Chair]V[an:
Speakers:

Secretary :
SECTION G.

Chairman:
Speakers:

GERMANIC LITERATURE. {Hall 3, September 23, 10 a. m.)

Professor Kuno Francke, Harvard University.
Professor August Sauer, University of Prague.
Professor J. Minor, University of Vienna.
Professor D. K. Jessen, Bryn Mawr College.

Secretary:

Chairman:
Speakers :

SECTION A.

Chairman:
Speakers :

Secretary
SECTION B.

Chairman:
Speakers :

Secretary :
SECTION C.

Chairman:
Speakers:

Secretary:

SLAVIC LITERATURE. {Hall 8, September 21, 10 a. m.)

Mr. Charles R. Crane, Chicago.
Professor Leo Wiener, Harvard University.
Professor Paul Boyer, Ecole des Langues Vivantes

Orientales, Paris.
Mr. S. N. Harper, University of Chicago.

BELLES-LETTRES. {Hall 3, September 24, 10 a. m.)

Professor Robert Herrick, University of Chicago.
Professor Henry Schofield, Harvard University.
Professor Brander Matthews, Columbia Univers-
ity.

DEPARTMENT 7 — HISTORY OF ART
{Hall 8, September 20, 11.15 a. m.)

Professor Halsey C. Ives, Washington University,

St. Louis.
Professor Rufus B. Richardson, New York, N. Y.
Professor John C. Van Dyke, Rutgers College.

CLASSICAL ART. {Hall 12, September 22, 10 a. to.)

Professor Rufus B. Richardson, New York City.
Professor Adolph Furtwangler, University of

Munich.
Professor Frank B. Tarbell, University of Chicago.
: Dr. p. Baur, Yale University.

MODERN ARCHITECTURE. {Hall 7, September 22, 3 p. m.)

Mr. Charles F. McKim, New York City.
Professor C. Enlart, University of Paris.
Professor Alfred D. F. Hamlin, Columbia Uni-
versity.
Mr. Guy Lowell, Boston, Mass.

MODERN PAINTING. {Hall 4, September 24, 3 p. to.)

Professor Richard Muther, University of Breslau.
Mr. Okakura Kakuzo, Japan.

DEPARTMENT 8 — HISTORY OF RELIGION

{Hall 5, September 20, 2 p. to.)

Chairman: Rev. Wm. Eliot Griffis, Ithaca, N. Y.
Speakers: Professor George F. Moore, Harvard University.
Professor Nathaniel Schmidt, Cornell University.

SPEAKERS AND CHAIRMEN

SECTION A. BRAHMANISM AND BUDDHISM.

10 a. m.)

Chairman:
Speakers :

{Hall 8, September 23,

Professor Hermann Oldenberg, University of Kiel.
Professor Maurice Bloomfield, Johns Hopkins
University.
Secretary: Dr. Reginald C. Robbins, Harvard University.

SECTION B. MOHAMMEDISM. {Hall 8, September 23, 3 p. m.)

Chairman: Professor James R. Jewett, University of Chicago.
Speakers: Professor Ignaz Goldziher, University of Budapest.
Professor Duncan B. Macdonald, Hartford Theo-
logical Seminary.
Secretary:

SECTION C. OLD TESTAMENT. {Hall 4, September 22, 10 a. m.)

Chairman: Professor A. S. Carrier, McCormick Theological

Seminary.
Speakers: Professor James F. McCurdy, University College of

Toronto.
Professor Karl Budde, University of Marburg.
Secretary: Professor James A. Kelso, Western Theological

Seminary, Allegheny, Pa.

SECTION D. NEW TESTAMENT. {Hall 1, September 23, 10 a. m.)

Chairman: Professor Andrew C. Zenos, McCormick Theological

Seminary.
Speakers: Professor Benjamin W. Bacon, Yale University.

Professor Ernest D. Burton, University of Chicago.
Secretary: Professor Clyde W. Votaw, University of Chicago.

SECTION E. HISTORY OF THE CHRISTIAN CHURCH. {Hall 2, Sep-
tember 24, 10 a. m.)

Chairman: Dr. Eri Baker Hulbert, University of Chicago.
Speakers: Professor Adolf Harnack, University of Berlin.
Professor Jean Reville, Faculty of Protestant
Theology, Paris.
Secretary :

DIVISION C — PHYSICAL SCIENCE

{Hall 4, September 20, 10 a. m.)
Speaker: Professor Robert S. Woodward, Columbia University.

DEPARTMENT 9 — PHYSICS

{Hall 6, September 20, 2 p. m.)

Chairman: Professor Henry Crew, Northwestern University.
Speakers: Professor Edward L. Nichols, Cornell University.
Professor Carl Barus, Brown University.

SPEAKERS AND CHAIRMEN

SECTION A. PHYSICS OF MATTER. (Hall 11, September 23, 10 a. m.)

Chairman: Professor Samuel W. Stratton, Director of the
National Bureau of Standards, Washington.

Speakers: Professor Arthur L. Kimball, Amherst College.

Professor Francis E. Nipher, Washington Uni-
versity.

Secretary: Professor R. A. Milliken, University of Chicago.

SECTION B. PHYSICS OF ETHER. (Hall 11, September 23, 3 p. m.)

Chairman: Professor Henry Crew, Northwestern University.

Speaker: Professor DeWitt B. Brace, University of Ne-
braska.

Secretary: Professor Augustus Trowbridge, University of
Wisconsin.

SECTION C. PHYSICS OF THE ELECTRON. {Hall 5, September 22, 3 p. m.)

Chairman: Professor A. G. Webster, Clark University.
Speakers: Professor P. Langevin, College de France.

Professor Ernest Rutherfurd, McGill University,
Montreal.
Secretary: Professor W. J. Humphreys, University of Virginia.

. DEPARTMENT 10 — CHEMISTRY
{Hall 5, September 20, 4.15 p. m.)

Chairman: Professor James M. Crafts, Massachusetts Institute

of Technology.
Speakers: Professor John U. Nef, University of Chicago.

Professor Frank W. Clarke, Chief Chemist, U. S.

Geological Survey.

SECTION A. INORGANIC CHEMISTRY. {Hall 16, September 21, 10 a. m.)

Chairman: Professor John W. Mallet, University of Virginia.

Speakers: Professor Henri Moissan, The Sorbonne; Member
of the Institute of France.
Sir William Ramsay, K.C.B., Royal Institution,
London.

Secretary: Professor William L. Dudley, Vanderbilt Univers-
ity.

SECTION B. ORGANIC CHEMISTRY. {Hall 16, September 21, 3 p. m.)

Chairman: Professor Albert B. Prescott, University of Michi-
gan.
Speakers: Professor Julius Stieglitz, University of Chicago.
Professor William A. Noyes, National Bureau of
Standards.
Secretary :

SECTION C. PHYSICAL CHEMISTRY. {Hall 16, September 22, 10 a. m.)

Chairman: Professor Wilder D. Bancroft, Cornell University.
Speakers: Professor J. H. Van t'Hoff, University of Berlin.
Professor Arthur A. Noyes, Massachusetts Institute
of Technology.
Secretary: Mr. W. R. Whitney, Schenectady, N. Y.

SPEAKERS AND CHAIRMEN

SECTION D. PHYSIOLOGICAL CHEMISTRY. {Hall 16, September 22,
3 p. m.)

Chairman: Professor Wilbur O. At water, Wesleyan Univers-
ity.
Speakers: Professor 0. Cohnheim, University of Heidelberg.
Professor Russell H. Chittenden, Yale Univers-
ity.
Secretary: Dr. C. L. Alsberg, Harvard University.

DEPARTMENT 11— ASTRONOMY

{Hall 8, September 20, 4.15 p. m.)

Chairman: Professor George C. Comstock, Director of the
Observatory, Madison, Wisconsin.

Speakers: Professor Lewis Boss, Director of Dudley Observa-
tory.
Professor Edward C. Pickering, Director of Har-
vard Observatory.

SECTION A. ASTROMETRY. {Hall 9, September 21, 10 a. m.)

Professor Ormond Stone, University of Virginia.

Chairman:
Speakers :

Secretary:

SECTION B.

Chairman:

Speakers :

Secretary:

Dr. Oskar Backlund, Director of the Observatory,
Pulkowa, Russia.

Professor John C. Kapteyn, University of Gronin-
gen, Holland.

Professor W. S. Eichelberger, U. S. Naval Observ-
atory.

ASTROPHYSICS. {Hall 9, September 21, 3 p. m.)

Professor George E. Hale, Director of the Yerkes
Observatory.

Professor Herbert H. Turner, F.R.S., Univers-
ity of Oxford.

Professor William W. Campbell, Director of the
Lick Observatory, Mt. Hamilton, California.

Mr. W. S. Adams, Yerkes Observatory.

DEPARTMENT 12 — SCIENCES OF THE EARTH

{Hall 3, September 20, 11.15 a. m.)

Chairman: Dr. G. K. Gilbert, U. S. Geological Survey.
Speakers: Professor Thomas C. Chamberlin, University of
Chicago.
Professor William M. Davis, Harvard University.

SECTION A. GEOPHYSICS. {Hall 14, September 21, 10 a. m.)

Chairman: Professor Christopher W. Hall, University of

Minnesota.
Speaker: Dr. George F. Becker, Geologist, U. S. Geological

Survey.
Secretary: Professor E. M. Lehnerts, Minnesota State Normal

School.

SPEAKERS AND CHAIRMEN

SECTION B.

Chairman:

Speakers:

Secretary :
SECTION C.

Chairman:

Speakers :

Secretary :
SECTION D.

Chairman:

Speaker:
Secretary :

SECTION E.

Chairman:
Speakers :

Secretary :
SECTION F.

Chairman :
Speakers :

Secretary:
SECTION G.

Chairman :

Speakers :

Secretary :
SECTION H.

Chairman :
Speakers:

Secretary :

GEOLOGY. {Hall 14, September 21, 3 p. m.)

Professor T. C. Chamberlin, University of Chicago.
President Charles R. Van Hise, University of Wis-
consin.

Professor R. D. Salisbury, University of Chicago.

PALAEONTOLOGY. {Hall 11, September 22, 10 a. m.)

Professor William B. Scott, Princeton University.
Dr. a. S. Woodward, F.R.S., British Museum of

Natural History, London.
Professor Henry F. Osborn, Columbia University.
Dr. John M. Clarke, Albany, N. Y.

PETROLOGY AND MINERALOGY. {Hall 9, September 22,
3 p. m.)

Dr. Oliver C. Farrington, Field Columbian Museum,

Chicago.
Professor F. Zirkel, University of Leipzig.

PHYSIOGRAPHY. {Hall 12, September 21, 10 a. m.)

Mr. Henry Gannett, United States Geological Survey.
Professor Albrecht Penck, University of Vienna.
Professor Israel C. Russell, University of Michigan.
Dr. John M. Clarke, Albany, N. Y.

GEOGRAPHY. {Hall 11, September 22, 3 p. m.)

Professor Israel C. Russell, University of Michigan.
Dr. Hugh R. Mill, Director British Rainfall Organ-
ization, London.
Professor H. Yule Oldham, Cambridge, England.
Professor R. D. Salisbury, University of Chicago.

OCEANOGRAPHY. {Hall 8, September 21, 3 p. m.)

Rear-Admiral John R. Bartlett, United States

Navy.
Sir John Murray, K.C.B., F.R.S., Edinburgh.
Professor K. Mitsukuri, University of Tokio.

COSMICAL PHYSICS. {Hall 10, September 22, 10 a. m.)

Professor Francis E. NiPHER,Washington University.
Professor Svante Arrhenius, University of Stock-
holm, Stockholm.
Dr. Abbott L. Rotch, Blue Hill Observatory.
Dr. L. a. Bauer, Washington, D. C.

DEPARTMENT 13 — BIOLOGY

{Hall "2, September 20, 11.15 a. m.)

Chairman: Professor William G. Farlow, Harvard University.

Speakers: Professor John M. Coulter, IJniversity of Chicago.

Professor Jacques Loeb, University of California.

SPEAKERS AND CHAIRMEN

SECTION A. PHYLOGENY. {Hall 2, September 21, 3 p. to.)

Chairman:
Speakers:

Secretary:

Professor T. H. Morgan, Columbia University.
Professor Hugo de Vries, University of Amsterdam.
Professor Charles 0. Whitman, University of
Chicago.

SECTION B. PLANT MORPHOLOGY. {Hall 2, September 22, 10 a. to.)

Chairman: Professor William Trelease, Washington Univers-
ity, St. Louis.

Speakers: Professor Frederick O. Bower, University of Glas-
gow.
Professor Karl F. Goebel, University of Munich.

Secretary: Professor F. E. Lloyd, Columbia University.

SECTION C. PLANT PHYSIOLOGY. {Hall 4, September 22, 3 p. to.)

Chairman: Professor Charles R. Barnes, University of Chicago.
Speakers: Professor Julius Wiesner, University of Vienna.
Professor Benjamin M. Duggar, University of Mis-
souri.
Secretary: Professor F. C. Newcomb, University of Michigan.

SECTION D. PLANT PATHOLOGY. {Hall 7, September 23, 10 a. to.)

Chairman: Professor Chas. E. Bessey, University of Nebraska.
Speakers : Professor Joseph C. Arthur, Purdue University.

Merton B. Waite, U. S. Department of Agriculture.
Secretary: Dr. C. S. Shear, U. S. Department of Agriculture.

SECTION E. ECOLOGY. {Hall 7, September 23, 3 p. to.)

Chairman:
Speakers:

Secretary :

Professor Oskar Drude, Kon. Technische Hoch-

schule, Dresden.
Professor Benjamin Robinson, Harvard University.
Professor F. E. Clements, University of Nebraska.

SECTION F. BACTERIOLOGY. {Hall 15, September 24, 10 a. to.)

Professor Harold C. Ernst, Harvard University.

Chairman:
Speakers :

Secretary :

Professor Edwin O. Jordan, University of Chicago.
Professor Theobald Smith, Harvard University.
Dr. p. H. Hiss, Jr., Columbia University.

SECTION G.

Chairman:

Speakers :

Secretary :

ANIMAL MORPHOLOGY. {Hall 2, September 21, 10 a. to.)

Dr. Leland 0. Howard, Department of Agriculture,

Washington, D. C.
Professor Charles B. Davenport, University of

Chicago.
Professor Alfred Giard, The Sorbonne; Member

of the Institute of France.
Professor C. H. Herrick, Dennison University.

SPEAKERS AND CHAIRMEN

SECTION H. EMBRYOLOGY. {Hall 9, September 23, 3 p. m.)

Chairman: Professor Simon H. Gage, Cornell University.
Speakers: Professor Oskar Hertwig, University of Berlin,

Professor William K. Brooks, Johns Hopkins
University.
Secretary: Professor T. G. Lee, University of Minnesota.

SECTION I. COMPARATIVE ANATOMY. {Hall 2, September 24, 3 p. m.)

Chairman: Professor James P. McMurrich, University of
Michigan.

Speakers: Professor William E. Ritter, University of Cali-
fornia.
Professor Yves Delage, The Sorbonne ; Member of
the Institute of France.

Secretary: Professor Henry B. Ward, University of Nebraska.

SECTION J. HUMAN ANATOMY. {Hall 2, September 22, 3 p. m.)

Chairman: Professor George A. Piersol, University of Penn-
sylvania.
Speakers: Professor Wilhelm Waldeyer, University of Berlin.
Professor H. H. Donaldson, University of Chicago.
Secretary: Dr. R. J. Terry, Washington University.

SECTION K. PHYSIOLOGY. {Hall 4, September 23, 10 a. m.)

Chairman: Dr. S. J. Meltzer, New York.

Speakers: Professor Max Verworn, University of Gottingen.
Professor William H. Howell, Johns Hopkins Uni-
versity.
Secretary: Dr. Reid Hunt, Washington.

DEPARTMENT 14 — ANTHROPOLOGY

{Hall 8, September 20, 2 p. m.)

Chairman: Professor Frederic W. Putnam, Harvard Univers-
ity.

Speakers: Dr. WJ McGee, President American Anthropological
Association, Washington, D. C.
Professor Franz Boas, Columbia University.

SECTION A. SOMATOLOGY. {Hall 16, September 23, 3 p. m.)

Chairman: Dr. Edward C. Spitzka, New York City.
Speakers: Professor L. Manouvrier, School of Anthropology^

Paris.
Dr. George A. Dorsey, Field Columbian Museum^.

Chicago.
Secretary: Dr. E. A. Spitzka, New York City.

SECTION B. ARCHAEOLOGY. {Hall 16, September 24, 10 a. m.)

Chairman: Mr. M. H. Saville, American Museum of Natural

History," New York.
Speakers: Senor Alfredo Chavero, Inspector of the National

Museum, Mexico.
Professor Edouard Seler, University of Berlin.
Secretary: Professor William C. Mills, Ohio State University.

SPEAKERS AND CHAIRMEN

SECTION C. ETHNOLOGY. {Hall 16, September 24, 3 p. m.)

Chairman: Miss Alice C. Fletcher, President of the Washing-
ton Anthropological Society.
Speakers: Professor Frederick Starr, University of Chicago.
Professor A. C. Haddon, University of Cambridge.
Secretary: Professor F. W. Shipley, Washington University.

Speaker:

DIVISION D. — MENTAL SCIENCE

{Hall 7, September 20, 10 a. m.)

President G. Stanley Hall, Clark University, Wor-
cester, Mass.

DEPARTMENT 15 — PSYCHOLOGY

{Hall 7, September 20, 2 p. m.)

Chairman:

Speakers : Professor James McK. Cattell, Columbia University.
Professor J. Mark Baldwin, Johns Hopkins Uni-
versity.

SECTION A. GENERAL PSYCHOLOGY. {Hall 6, September 23, 3 p. m.)

Chairman: Professor Jos. Royce, Harvard University.

Speakers: Professor Harald Hoeffding, University of Copen-
hagen.
Professor James Ward, University of Cambridge,
England.

Secretary: Dr. W. H. Davis, Lehigh University.

SECTION B. EXPERIMENTAL, PSYCHOLOGY. {Hall 2, September 23,
10 a. m.)

Chairman: Professor Edward A. Pace, Catholic University of

America.
Speakers : Professor Robert MacDougal, New York University.
Professor Edward B. Titchener, Cornell University.
Secretary: Dr. R. S. Wood worth, Columbia University.

SECTION C. COMPARATIVE AND GENETIC PSYCHOLOGY. {Hall 6,
September 24, 10 a. m.)

Chairman: Professor Edmund C. Sanford, Clark University,

Worcester, Mass.
Speakers: Principal C. Lloyd Morgan, University College,
Bristol.
Professor Mary W. Calkins, Wellesley College.
Secretary: Dr. R. M. Yerkes, Harvard University.

SECTION D. ABNORMAL PSYCHOLOGY. {Hall 6, September 24, 3 p. m.)

Chairman: Dr. Edward Cowles, Waverley, Mass.

Speakers: Dr. Pierre Janet, College de France, Paris.

Dr. Morton Prince, Boston.

Secretary: Dr. Adolph Meyer, New York City.

SPEAKERS AND CHAIRMEN
DEPARTMENT 16 — SOCIOLOGY

Chaieman:

{Hall 7, September 20, 4.15 p. m.)
Professor Frank W. Blackmar, University of Kan-

sas.

Speakers: Professor Franklin H. Giddings, Columbia Uni-
versity.
Professor George E. Vincent, University of Chicago.

SECTION A. SOCIAL STRUCTURE. {Hall 15, September 21, 10 a. m.)

Chairman: Professor Frederick W. Moore, Vanderbilt Uni-
versity.
Speakers: Field Marshal Gustav Ratzenhofer, Vienna.

Professor F. Toennies, University of Kiel.

Professor Lester F. Ward, U. S. National Museum.
Secretary: Professor Jerome Dowd, University of Wisconsin.

SECTION B. SOCIAL PSYCHOLOGY. {Hall 15, September 23, 10 a. m.)

Chairman: Professor Charles A. Ell wood. University of Mis-
souri.

Speakers:

Secretary: Professor E. C. Hayes, Miami University

Professor Wm. I. Thomas, University of Chicago.
Professor Edward A. Ross, University of Nebraska.

Speaker:

DIVISION E— UTILITARIAN SCIENCES

{Hall 1, September 20, 10 a. m.)

President David Starr Jordan, Leland Stanford Jr.
University.

DEPARTMENT 17 — MEDICINE
{Hall 1, September 20, 4,15 p. m.)

Chairman: Dr. William Osler, Johns Hopkins University.
Speakers: Dr. William T. Councilman, Harvard University.
Dr. Frank Billings, University of Chicago.

SECTION A. PUBLIC HEALTH. {Hall 13, September 21, 10 a. m.)

Chairman: Dr. Walter Wyman, Surgeon-General of the U. S.

Marine Hospital Service.
Speakers: Professor William T. Sedgwick, Massachusetts

Institute of Technology.
Dr. Ernst J. Lederle, Former Commissioner of

Health, New York City.
Secretary: Dr. H. .M Bracken, St. Paul, Minn.

68 SPEAKERS AND CHAIRMEN

SECTION B. PREVENTIVE MEDICINE. {Hall 13, September 21, 3 p. m.)

Chairman: Dr. Joseph M. Mathews, President of the State Board

of Health, Louisville, Ky.
Speaker: Professor Ronald Ross, F.R.S., School of Tropical

Medicine, University College, Liverpool.
Secretary: Dr. J. N. Hurty, Indianapolis, Ind.

SECTION C. PATHOLOGY. {Hall 13, September 22, 10 a. m.)

Chairman: Professor Simon Flexner, Director of the Rocke-
feller Institute.
Speakers: Professor Ludwig Hektoen, University of Chicago.
Professor Johannes Orth, University of Berlin,
Professor Shibasaburo Kitasato, University of
Tokio.
Secretary: Dr. W. McN. Miller, University of Missouri.

SECTION D. THERAPEUTICS AND PHARMACOLOGY. {Hall 13, Sep-
tember 24, 3 p. m.)

Chairman: Dr. Hobart A. Hare, Jefferson Medical College.
Speakers: Professor Oscar Liebreich, University of Berlin.

Sir Lauder Brunton, F.R.S., London.
Secretary: Dr. H. B. Favill, Chicago, 111.

SECTION E. INTERNAL MEDICINE. {Hall 13, September 23, 3 p. m.)

Chairman: Professor Frederick C. Shattuck, Harvard Uni-
versity.

Speakers: Professor T. Clifford Allbutt, F.R.S. , University
of Cambridge.
Professor William S. Thayer, Johns Hopkins Uni-
versity.

Secretary: Dr. R. C. Cabot, Boston, Mass.

SECTION F. NEUROLOGY. {Hall 13, September 22, 3 p. m.)

Chairman: Professor Lewellyn F. Barker, University of

Chicago.
Speaker: Professor James J. Putnam, Harvard University.
Secretary :

SECTION G. PSYCHLATRY. {Hall 7, September 22, 10 a. m.)

Chairman:

Speakers: Dr. Charles L. Dana, Cornell University, New York.

Dr. Edward Cowles, Boston.
Secretary: Dr. C. G. Chaddock, St. Louis, Mo.

SECTION H. SURGERY. {Hall 13, September 23, 10 a. m.)

Chairman: Professor Carl Beck, Post-Graduate Medical School,

New York.
Speakers: Dr. Frederic S. Dennis, F.R.C.S., Cornell Medical

College, New York City.
Professor Johannes Orth, University of Berlin.
Secretary: Dr. J. F. Binnie, Kansas City, Mo.

SPEAKERS AND CHAIRMEN 69

SECTION I. GYNECOLOGY. {Hall 13, September 24, 10 a. m.)

Chairman: Professor Howard A. Kelly, Johns Hopkins Uni-
versity.

Speaker: Professor J. Clarence Webster, Rush Medical Col-
lege, Chicago.

Secretary: Dr. G. H. Noble, Atlanta, Ga.

SECTION J. OPHTHALMOLOGY. (Hall 7, September 24, 10 a. m.)

Chairman: Dr. George C. Harlan, Philadelphia, Pa.
Speakers: Dr. Edward Jackson, Denver, Col.

Dr. George M. Gould, Philadelphia, Pa.
Secretary: Dr. Wm. M. Sweet, Jefferson Medical College, Phil-
adelphia, Pa.

SECTION K. OTOLOGY AND LARYNGOLOGY. (Hall 7, September 21,
10 a. m.)

Chairman: Professor William C. Glasgow, Washington Uni-
versity, St. Louis.

Speaker: Sir Felix Semon, C.V.O., Physician Extraordinary
to His Majesty, the King, London.

Secretary: Dr. S. Spencer, Allenhurst, N. J.

SECTION L. PEDIA.TRICS. {Hall 7, September 21, 3 p. m.)

Chairman: Professor Thomas M. Rotch, Harvard University.
Speakers : Professor Theodore Escherich, University of Vienna.

Professor Abraham Jacobi, Columbia University.
Secretary: Dr. Samuel S. Adams, Washington, D. C.

DEPARTMENT 18 — TECHNOLOGY.

{Hall 3, September 20, 2 p. m.)

Chairman: Chancellor Winfield S. Chaplin, Washington Uni-
versity, St. Louis.

Speaker: Professor Henry T. Bovey, F.R.S., McGill Uni-
versity, Montreal.

SECTION A. CIVIL ENGINEERING. {Hall 10, September 21, 10 a. m.)

Chairman: Professor William H. Burr, Columbia University.

Speakers: Dr. J. A. L. Waddell, Consulting Engineer, Kansas
City.
Mr. Lewis M. Haupt, Consulting Engineer, Phila-
delphia.

Secretary :

SECTION B. MECHANICAL ENGINEERING. {Hall 10, September 23,
3 p. m.)

Chairman: Professor James E. Denton, Stevens Institute of

Technology.
Speaker: Professor Albert W. Smith, Leland Stanford Jr.

University.
Secretary: Mr. George Dinkel, Jr., Jersey City.

70 SPEAKERS AND CHAIRMEN

SECTION C. ELECTRICAL ENGINEERING. {Hall 10, September 22,
3 p. m.)

Chairman:

Speakers: Professor Arthur E. Kennelly, Harvard Univers-
ity.
Professor Michael I. Pupin, Columbia University.
Secretary: Mr. Carl Hering, Philadelphia, Pa.

SECTION D. MINING ENGINEERING. {Hall 11, September 24, 10 a. m.)

Chairman: Mr. John Hays Hammond, New York City.

Speakers: Professor Robert H. Richards, Massachusetts
Institute of Technology.
Professor Samuel B. Christy, University of Cali-
fornia.

Secretary: Dr. Joseph Struthers, New York City.

SECTION E. TECHNICAL CHEMISTRY. {Hall 16, September 23, 10 a. m.)

Chairman: Dr. H. W. Wiley, Department of Agriculture.
Speakers: Professor Charles E. Munroe, George Washington
University.
Professor William H. Walker, Massachusetts In-
stitute of Technology.
Secretary: Dr. Marcus Benjamin, U. S. National Museum.

SECTION F. AGRICULTURE. {Hall 10, September 24, 3 p. m.)

Chairman: Professor H. J. Wheeler, Kingston, R. I.
Speakers: Professor Charles W. Dabney, Jr., University of
Cincinnati.
Professor Liberty H. Bailey, Cornell University.
Secretary: Professor William Hill, University of Chicago.

DEPARTMENT 19 — ECONOMICS

{Hall 1, September 20, 11.15 a. m.)

Chairman: Professor Emory R. Johnson, University of Penn-
sylvania.
Speakers: Professor Frank A. Fetter, Cornell University.

Professor Adolph C. Miller, University of Cali-
fornia.

SECTION A. ECONOMIC THEORY. {Hall 15, September 22, 10 a. m.)

Chairman:

Speakers: Professor John B. Clark, Columbia University.

Professor Jacob H. Hollander, Johns Hopkins
University.

Professor Jesse E. Pope, University of Missouri.

Secretary
SECTION B.

Chairman :

TRANSPORTATION. {Hall 10, September 23, 10 a. m.)

Professor J. Lawrence Laughlin, University of
Chicago.
Speakers: Professor Eugene von Philippovich, University
of Vienna.
Professor William Z. Ripley, Harvard University.
Secretary: Mr. George G. Tunell. Chicago.

SPEAKERS AND CHAIRMEN 71

SECTION C. COMMERCE AND EXCHANGE. {Hall 10, September 24,
10 a. m.)

Chairman:

Speakers: Professor E. D. Jones, University of Michigan.
Professor Carl Plehn, University of California.
Secretary :

SECTION D. MONEY AND CREDIT. (Hall 5, September 24, 3 p. m.)

Chairman: Mr. B. E. Walker, Canadian Bank of Commerce,

Toronto.
Speakers: Mr. Horace White, New York City.

Professor J. Lawrence Laughlin, University of

Chicago.
Secretary: Professor John Cummings, University of Chicago.

SECTION E. PUBLIC FINANCE. {Hall 1, September 21, 10 a. m.)

Chairman :

Speakers: Professor Henry C. Adams, University of Michigan.
Professor Edwin R. A. Seligman, Columbia Uni-
versity.
Secretary:

SECTION F. INSURANCE. {Hall 10, September 21, Z p. to.)

Chairman: Dr. Emory McClintock, Actuary, Mutual Life In
surance Company, New York.

Speakers: Mr. Frederick L. Hoffman, Statistician, Prudential
Insurance Company, Newark.
Professor Balthasar H. Meyer, University of Wis-
consin.

Secretary :

DIVISION F — SOCIAL REGULATION

{Hall 2, September 20, 10 a. to.)
Speaker: Professor Abbott L. Lowell, Harvard University.

DEPARTMENT 20 — POLITICS

{Hall 2, September 20, 2 p. to.)

Chairman:

Speakers: Professor William A. Dunning, Columbia Univers-
ity.
Chancellor E. Benjamin Andrews, University of
Nebraska.

SECTIONS A AND C. POLITICAL THEORY AND NATIONAL ADMINIS-
TRATION. {Hall 15, September 22, 3 p. to.)

Chairman :

Speakers: Professor W. W. Willoughby, Johns Hopkins Uni-
versity.

72 SPEAKERS AND CHAIRMEN

Professor George G. Wilson, Brown University.
Right Hon. James Bryce, London, England.
Secretary: Dr. Charles E. Merriam, University of Chicago.

SECTION B. DIPLOMACY. (Hall 1, September 23, 3 p. m.)

Chairman:

Speakers: Honorable John W. Foster, Ex-Secretary of State.
Honorable David Jayne Hill, Minister of the United
States to Switzerland.
Secretary:

SECTION D. COLONIAL ADMINISTRATION. {Hall 4, September 24,
10 a. m.)

Chairman: Professor Harry P. Judson, University of Chicago.
Speakers : Professor Bernard J. Moses, University of California.
Professor Paul S. Reinsch, University of Wisconsin.
Secretary:

SECTION E. MUNICIPAL ADMINISTRATION. {Hall 15, September 24,
3 p. m.)

Chairman:

Speakers: Mr. Albert Shaw, Editor American Monthly Review
of Reviews.
Miss Jane Addams, Hull House, Chicago.
Secretary: Professor John A. Fairlie, University of Michigan.

DEPARTMENT 21 — JURISPRUDENCE

{Hall 3, September 20, 4.15 p. m.)

Chairman: Professor George W. Kirchwey, Columbia Uni-
versity.

Speakers: President Charles W. Needham, Columbian Uni-
versity, Washington.
Professor Joseph H. Beale, Harvard University.

SECTION A. INTERNATIONAL LAW. {Hall 14, September 22, 10 a. m.)

Chairman: Professor James B. Scott, Columbia University.
Speakers: Professor H. LaFontaine, Member of the Senate,
Brussels, Belgium.
Professor Charles Noble Gregory, University of

Iowa.
Count Albert Apponyi, Hungary.
Secretary: Dr. W. C. Dennis, Leland Stanford Jr. University.

SECTION B. CONSTITUTIONAL LAW. {Hall 14, September 24, 10 a. m.)

Chairman: Professor Henry St. George Tucker, George

Washington University, Washington.
Speakers: Signor Attilio Brunialti, Councilor of State, Rome.

Professor John W. Burgess, Columbia University.

Professor Ferdinand Larnaude, University of Paris.
Secretary :

SPEAKERS AND CHAIRMEN

SECTION C.

Chaieman :
Speakees :

PRIVATE LAW. {Hall 14, September 23, 3 p. m.)
Peofessoe James B. Ames, Dean, Harvard Law School.

Peofessoe Eenst Feeund, University of Chicago.
HoNOEABLE Edwaed B. Whitnet, New York.
Seceetaet: Dean William Deapee Lewis, University of Penn-
sylvania.

Chaieman;

Speakees :

SECTION A.

Chaieman :

Speakees:

Seceetaet:
SECTION B.

Chaieman :

Speakees:

Seceetaet :

SECTION C.

Chaieman:
Speakees :

Seceetaet :

SECTION D.

Chaieman :
Speakees:

Seceetaet :
SECTION E.

Chaieman:
Speakees :

Seceetaet :

SECTION F.

Chaieman :
Speakee :

Seceetaet :

DEPARTMENT 22 — SOCIAL SCIENCE

{Hall 1, September 20, 2 p. m.)

Me. Waltee L. Sheldon, Ethical Society, St. Louis.
Peofessoe Felix Adlee, Columbia University.
Peofessoe Geaham Tatloe, Chicago Theological

Seminary.

THE FAMILY. {Hall 5, September 21, 10 a. m.)

Peofessoe Samuel G. Smith, University of Minnesota.
De. Samuel W. Dike, Auburndale, Mass.
Peofessoe Geoege Elliott Howaed, University of
Nebraska.

THE RURAL COMMUNITY. {Hall 5, September 21, Z p. m.)

Hon. Aaeon Jones, Master of National Grange, South

Bend, Ind.
Peofessoe Max Webee, University of Heidelberg.
Peesident Kenton L. Butteefield, Rhode Island

State Agricultural College.
Peofessoe William Hill, University of Chicago.

THE URBAN COMMUNITY, {Hall 5, September 22, 10 a. m.)

Peofessoe T. Jasteow, University of Berlin.
Peofessoe Louis Wuaein, University of Geneva.

THE INDUSTRIAL GROUP. {Hall 14, September 22, 3 p. m.)

Peofessoe Weenee Sombaet, University of Breslau.
Peofessoe Richaed T. Elt, University of Wisconsin.
Peofessoe Thomas S. Adams, Madison, Wis.

THE DEPENDENT GROUP. {Hall 5, September 23, 10 a. m.)

Me. Robeet W. DeFoeest, New York City.
Peofessoe Chaeles R. Hendeeson, University of

Chicago.
De. Emil MtJNSTEEBEEG, President City Charities,

Berlin.

THE CRIMINAL GROUP. {Hall 5, September 23, 3 p. m.)

Me. Feedeeick H. Wines, Secretary State Charities
Aid Association, Upper Montclair, N. J.

SPEAKERS AND CHAIRMEN

Speaker :

DIVISION G — SOCIAL CULTURE

(Hall 5, September 20, 10 a. m.)

Honorable William T, Harris, United States Com-
missioner of Education.

DEPARTMENT 23 — EDUCATION

{Hall 2, September 20, 4.15 p. m.)

Chairman:

Speakers: President Arthur T. Hadley, Yale University.

The Right Rev. John L. Spalding, Bishop of Peoria.

SECTION A. EDUCATIONAL THEORY. {Hall 12, September 24, 3 p. m.)

Professor Charles DeGarmo, Cornell University.
Professor Wilhelm Rein, University of Jena.
Professor Elmer E. Brown, University of Califor-
nia.
Dr. G. M. Whittle, Cornell University.

Chairman :
Speakers :

Secretary:

SECTION B. THE SCHOOL. {Hall 12, September 23, 10 a. m.)

Chairman: Dr. F. Louis Soldan, Superintendent Public Schools,
St. Louis.

Speakers : Dr. Michael E. Sadler, University of Manchester.
Dr. William H. Maxwell, Superintendent Public
Schools, New York City.

Secretary: Professor A. S. Langsdorf, Washington Univers-
ity.

SECTION C. THE COLLEGE. {Hall 12, September 23, 3 p. m.)

Chairman: President W. S. Chaplin, Washington University.
Speakers: President William DeWitt Hyde, Bowdoin College.

President M. Carey Thomas, Bryn Mawr College.
Secretary: Professor H. H. Horne, Dartmouth College.

SECTION D. THE UNIVERSITY. {Hall 12, September 24, 10 a. m.)

Chairman :

Speakers: Professor C. Chabot, University of Lyons.

Professor Edward Delavan Perry, Columbia Uni-
versity.
Secretary :

SECTION E. THE LIBRARY. {Hall 12, September 22, 3 p. m.)

Chairman: Mr. Frederick M. Crunden, Librarian St. Louis

Public Library.
Speakers: Mr. William A. E. Axon, Manchester, England.

Professor Guido Biagi, Royal Librarian, Florence.
Secretary : Mr. C. P. Pettus, Washington University.

SPEAKERS AND CHAIRMEN

Chairman :
Speakers :

SECTION A.

Chairman :
Speakers :

Secretary :
SECTION B.

Chairman:

Speakers :

Secretary :

SECTION C.

Chairman:

Speakers :
Secretary:

SECTION D.

Chairman:
Speakers :

Secretary :

DEPARTMENT 24 — RELIGION

{Hall 4, September 20, 4.15 p. m.)

Bishop John H. Vincent, Chautauqua, N. Y.
President Henry C. King, Oberlin College.
Professor Francis G. Peabody, Harvard University.

GENERAL RELIGIOUS EDUCATION. {Hall 11, September
24, 3 p. m.)

Professor Edwin D. Starbuck, Earlham College,
Richmond, Ind.

Professor George A. Coe, Northwestern Univers-
ity.

Dr. Walter L. Hervey, Examiner Board of Education,
New York City.

PROFESSIONAL RELIGIOUS EDUCATION. {Hall 1, Sep-
tember 22, 3 p. m.)

President Charles Cuthbert Hall, Union Theo-
logical Seminary.

Professor Frank K. Sanders, Yale University.

Professor Herbert L. Willett, Disciples Divinity
House, Chicago, lU.

RELIGIOUS AGENCIES. {Hall 15, September 23, 3 p. m.)

President Edgar G. Mullins, Southern Baptist
Theological Seminary, LouisviUe, Ky.

Rev. Washington Gladden, Columbus, Ohio.

Rev. James M. Buckley, Editor The Christian Ad-
vocate, New York.

Dr. Ira Landrith, General Secretary Religious Edu-
cation Association, Chicago, 111.

RELIGIOUS WORK. {Hall 1, September 24, 3 p. m.)

Rt. Rev. Thomas F. Gailor, Memphis.

Rev. Floyd W. Tomkins, Church of the Holy Trinity,

Philadelphia.
Rev. Henry C. Mabie, Corresponding Secretary

American Baptist Missionary Union.

SECTION E. RELIGIOUS INFLUENCE: PERSONAL. {Festival Hall, Sep-
tember 25, 10 a. m.)

Chairman:
Speakers :

Secretary :

Chancellor J. H. Kirkland, Vanderbilt University.
Rev. Hugh Black, Edinburgh, Scotland.
Professor John E. McFadyen, Knox CoUege.
Rev. Samuel Eliot, Boston, Mass.
Rev. Edward B. Pollard, Georgetown, Ky.
Professor Clyde W. Votaw, University of Chicago.

SPEAKERS AND CHAIRMEN

SECTION F. RELIGIOUS INFLUENCE: SOCIAL. (Festival Hall, Septem-
ber 25, 3 p. m.)
Chairman: Dr. J. H. Garrison, St. Louis.
Speakers: President Joseph Swain, Swarthmore College.
Dr. Emil G. Hirsch, Chicago, 111.
Professor Edward C. Moore, Harvard University.
Dr. Josiah Strong, League for Social Service, New
York.
Secretary: Professor Clyde W. Votaw, University of Chicago.

CHRONOLOGICAL ORDER OF PROCEEDINGS

MONDAY, SEPTEMBER 19-

3 P. M. Opening exercises of the Congress. Festival Hall (Hall 17).

The Congress will be called to order by the Director of Congresses,
who will introduce the President of the Exposition.

Welcoming addresses will be delivered by the President of the
Exposition and other officials,

A reply to these addresses of welcome will be made on behalf of the
Congress by the Honorary Vice-President for Great Britain.

The Chairman of the Administrative Board will give an account of
the origin and purpose of the Congress.

The President of the Congress will then be introduced and will
deUver an introductory address, after which adjournment will follow.

TUESDAY, SEPTEMBER 20.

10.00 A. M. Meetings of the seven Divisions. The Divisional ad-
dresses will be given as follows: —

Hall 1, Utilitarian Sciences. Hall 5, Social Culture.

Hall 2, Social Regulation. Hall 6, Normative Science.

Hall 3, Historical Science. Hall 7, Mental Science.
Hall 4, Physical Science.

11.15 to 6.00 p. M. Meetings of the Departments, with addresses: —
Meeting at 11.15 a. m. Meeting at 2 p. m.

DEPARTMENTS. DEPARTMENTS.

Hall 1, Economics. Hall 1, Social Science.

Hall 2, Biology. Hall 2, PoHtics.

Hall 3, Sciences of the Earth. Hall 3, Technology.

Hall 4, Political History. Hall 4, History of Language.

Hall 5, History of Law. Hall 5, History of Rehgion.

Hall 6, Philosophy. ' Hall 6, Physics.

Hall 7, Mathematics. Hall 7, Psychology.

Hall 8, History of Art. Hall 8, Anthropology.

Adjournment at 1 p. m. Adjournment at 3.45 p. m.

CHRONOLOGICAL ORDER OF PROCEEDINGS

Meeting at 4.15 p. m.

DEPARTMENTS.

Hall 1, Medicine. Hall 5, Chemistry.

HaU 2, Education. Hall 6, History of Literature.

Hall 3, Jurisprudence. Hall 7, Sociology.

Hall 4, Religion. Hall 8, Astronomy.

Adjournment at 6. p. m.

On the four days following, the Sectional meetings will be held.
The duration of each session will be three hours. The morning ses-
sions will extend from 10 a. m. until 1 p. m.; the afternoon sessions
from 3 P. M. to 6 p. m.

The meetings of some of the religious sections will be held on
Sunday, September 25, in Festival Hall. Further announcements
concerning these Sunday Meetings will be made in Registration Hall,
in the daily press of St. Louis, and in the World's Fair Official Pro-
gramme.

WEDNESDAY, SEPTEMBER 21.

Meeting at 10 a. m.

1, Public Finance.

2, Animal Morphology.

3, History of Greece, Rome,
and Asia.

4, Comparative Language.

5, The FamUy.

6, Metaphysics.

7, Otology and Laryngo-
logy.

8, Slavic Literature.

9, Astrometry.
Hall 10, Civil Engineering.

HaU 11, History of Common Law.
Hall 12, Physiography.
Hall 13, Public Health.
Hall 14, Geophysics.
Hall 15, Social Structure.
Hall 16, Inorganic Chemistry.
Adjournment at 1 p. m.

Hall
HaU
HaU

Hall
HaU
Hall
HaU

HaU
HaU

Meeting at 3 p. m.
Hall 1, Philosophy of Religion.
HaU 2, Phylogeny.
Hall 3, Classical Literature.
Hall 4, Semitic Languages.
Hall 5, The Rural Community.
Hall 6, Medieval History.
Hall 7, Pediatrics,
Hall 8, Oceanography.
Hall 9, Astrophysics.
HaU 10, Insurance.
HaU 11, History of Roman Law.
HaU 13, Preventive Medicine.
Hall 14, Geology.
Hall 16, Organic Chemistry.
Adjournment at 6 p. m.

Immediately following the Section of Geophysics in the morning,
and the Section of Geology in the afternoon, in Room 14, the Eighth
International Geographic Congress will hold sessions in the same
room. Hall 14, Mines and Metallurgy Building.

CHRONOLOGICAL ORDER OF PROCEEDINGS 79

THURSDAY, SEPTEMBER 22.

Meeting at 10 a. m.
Hall 1, English Literature.
Hall 2, Plant Morphology.
Hall 3, Modern History of Eu-
rope.
Hall 4, Old Testament.
Hall 5, The Urban Community.
Hall 6, Logic.
HaU 7, Psychiatry.
Hall 8, Indo-Iranian Languages.
Hall 9, Algebra and Analysis.
Hall 10, Cosmical Physics.
Hall 11, Palaeontology.
Hall 12, Classical Art.
Hall 13, Pathology.
Hall 14, International Law.
Hall 15, Economic Theory.
Hall 16, Physical Chemistry.
Adjournment at 1 p. m.

Meeting at 3 p. m.

Hall 1, Professional Religious
Education.

Hall 2, Human Anatomy.

Hall 3, Greek Language.

HaU 4, Plant Physiology.

Hall 5, Physics of the Electron.

Hall 6, Methodology of Science.

Hall 7, Modern Architecture.

Hall 8, Romance Literature.

Hall 9, Petrology and Mineral-
ogy.

HaU 10, Electrical Engineering.

Hall 11, Geography.

HaU 12, The Library.

HaU 13, Neurology.

Hall 14, The Industrial Group.

Hall 15, PoHtical Theory and Na-
tional Administration.

HaU 16, Physiological Chemistry.
Adjournment at 6 p. m.

FRIDAY, SEPTEMBER 23.

Meeting at 10 a. m.
Hall 1, New Testament.
Hall 2, Experimental Psycho-
logy.
Hall 3, Germanic Literature.
Hall 4, Physiology.
Hall 5, The Dependent Group.
HaU 6, Ethics.
HaU 7, Plant Pathology.
Hall 8, Brahmanism and Buddh-
ism.
Hall 9, Latin Language.
Hall 10, Transportation.
HaU 11, Physics of Matter.
HaU 12, The School. .
Hall 13, Surgery.
Hall 15, Social Psychology.
Hall 16, Technical Chemistry.
Adjournment at 1 p. m.

Meeting at 3 p. m.
Hall 1, Diplomacy.
Hall 2, History of Economic In-
stitutions.
Hall 3, English Language.
HaU 4, Esthetics.
HaU 5, The Criminal Group.
HaU 6, General Psychology.
HaU 7, Ecology.
HaU 8, Mohammedism.
HaU 9, Embryology.
HaU 10, Mechanical Engineering.
HaU 11, Physics of Ether.
HaU 12, The CoUege.
HaU 13, Internal Medicine.
Hall 14, Private Law.
HaU 15, Religious Agencies.
Hall 16, Somatology.

Adjournment at 6 p, m.

CHRONOLOGICAL ORDER OF PROCEEDINGS

SATURDAY, SEPTEMBER 24.

Meeting at 10 a. m.

1, History of America,

2, History of the Christian
Church.

3, Belles-Lettres.

4, Colonial Administration.

5, Romance Languages.

6, Comparative and Gene-
tic Psychology.

7, Ophthalmology.

8, History of Asia.

9, Geometry.

Hall 10, Commerce and Exchange.

Hall 11, Mining Engineering.

HaU 12, The University.

Hall 13, Gynecology.

Hall 14, Constitutional Law.

Hall 15, Bacteriology.

Hall 16, Archaeology.

Adjournment at 1 p. m.

Hall
Hall

Hall
Hall
Hall
Hall

Hall
Hall
Hall

Meeting at 3 p. m.

Hall 1, Religious Work.

Hall 2, Comparative Anatomy.

Hall 3, Germanic Languages.

Hall 4, Modern Painting.

Hall 5, Money and Credit.

Hall 6, Abnormal Psychology.

Hall 7, Applied Mathematics.

Hall 8, Indo-Iranian Literature.

Hall 10, Agriculture.

Hall 11,

Hall 12, Educational Theory.

Hall 13, Therapeutics and Phar-
liiacology.

Hall 14, Comparative Law.

Hall 15, Municipal Administra-
tion.

Hall 16, Ethnology.

Adjournment at 6 p. m.

SUNDAY, SEPTEMBER 25.

Festival Hall.

Meeting at 10 a. m.
Religious Influence: Personal.

Meeting at 3 p. m.
Religious Influence: Social.

PROGRAMME OF SOCIAL EVENTS

Monday Evening, September 19. — Grand Fete night in honor
of the Congress of Arts and Science. Special illuminations about the
Grand Basin. Lagoon fete.

Banquet by the St. Louis Chemical Society, at the Southern Hotel,
to the members of the Chemical Sections.

Tuesday Evening, September 20. — General Reception by
Board of Lady Managers to the officers and speakers of the Congress
and officials of the Exposition.

Wednesday Afternoon, September 2L — Garden fete to be
given to the members of the Congress of Arts and Science, at the
French Pavilion, by the Commissioner-General from France.

Wednesday Evening, September 2L — General reception by the
German Imperial Commissioner-General to the members of the Con-
gress of Arts and Science, at the German State House.

Thursday Evening. — Shaw banquet at the Buckingham Club to
the foreign delegates.

Friday Evening, September 23. — General banquet to the
speakers and officials of the Congress of Arts and Science in the
banquet-hall of the Tyrolean Alps. 8 p. m.

Saturday Evening, September 24. — Banquet at St. Louis Club
by Round Table of St. Louis, to the foreign members of the Congress.

Banquet given by Imperial Commissioner-General from Japan to
the Japanese delegation to the Congress and Exposition officials.

Dinner given by Commissioner-General from Great Britain to the
English members of the Congress.

ALPHABETICAL LIST OF MEMBERS
WHO MADE 10-MINUTE ADDRESSES

The following list differs from the original programme, in that it
contains the names only of those who actually read addresses. It
was planned that each Section should meet for three hours. When
authors of ten-minute papers were not present, and where not enough
of these shorter papers were offered to fill out the time, the Chairmen
invited discussions from the floor until the time was filled.

Professor R. G. Aitken
James W. Alexander, Esq.
Frederick Almy
Professor S. G. Aslimore
Professor L. A. Bauer
Dr. Marcus Benjamin
Professor H. T. Blickfeldt
Professor Ernest W. Brown
Dr. Henry Dickson Bruns

Dr. F. K. Cameron

Rear- Admiral C. M. Chester,

U. S. N.
H. H. Clayton, Esq.
Professor Charles A. Coffin
Dr. George Coronilas
Professor J. E. Denton

Professor L. W. Dowling
Professor H. C. Elmer
Professor A. Emch
Professor H. R. Fanclough
Professor W. S. Ferguson

Dr. Carlos Finley
Dr. C. E. Fisk
Homer Folks, Esq.
Professor F. C. French
H. L. Gannt, Esq.

Dr. F. P. Gorham
Professor Evarts B. Greene
Stansbury Hagar, Esq.
J. D. Hague, Esq.

Lick Observatory
New York City
Buffalo, N. Y.
Union College
Carnegie Institute
National Museum
Leland Stanford Univ.
Haverford College
New Orleans

Dep't of Agriculture
United States Naval

Observatory
Blue Hill Observatory
New York City
Athens, Greece
Stevens Institute

Univ. of Wisconsin
Cornell Univ.
Univ. of Colorado
Leland Stanford Univ.
Univ. of CaHfornia

Havana
Centralia, lU.
New York City
Univ. of Nebraska
Schenectady, N. Y.

Brown Univ.
Univ. of Illinois
Brooklyn, N. Y.
New York City

Astronomy
Insurance
Social Science
Latin Language
Cosmical Physios
Technical Chemistry
Geometry
Lunar Theory
Municipal Administra-
tion
Physical Chemistry
Astronomy

Cosmical Physics

Modern Painting

Tuberculosis

Mechanical Engineer-
ing

Geometry

Latin Language

Geometry

Classical Literature

History of Greece,
Rome, and Asia

Pathology

History of America

Social Science

Philosophy of Religion

Mechanical Engineer-
ing

Bacteriology

History of America

Ethnology

Mining Engineering

MEMBERS WHO MADE 10-MINUTE SPEECHES 83

Professor G. B. Halstead
Professor A. D. F. Hamlin
Professor H. Hancock
Professor J. A. Harris
Professor M. W. Haskell
Professor J. T. Hatfield
Professor E. C. Hayes
Professor W. E. Heidel
Dr. C. L. Herrick
Dr. C. Judson Herrick
Professor W. H. Hobbs

Professor A. R. Hohlfeld
Professor H. H. Home
Dr. E. V. Huntiagton
Dr. Reid Hunt
Dr. J. N. Hurty
Professor J. J. Hutchinson
Rev. Thomas E. Judge

Professor L. Kahlenburg
Professor Albert G. Keller

Professor George Lefevre
President Henry C. King
Dr. Ira Landrith
Professor M. D. Learned
Professor A. O. Leuschner
Dr. E. P. Lyon
Dr. Duncan B. Macdonald

Professor A. MacFarlane
Professor James McMahon
Mr. Edward Mallinckrodt
Professor H. P. Manning
Professor G. A. MiUer
Dr. W. C. Mills
Professor W. S. Milner
Professor F. G. Moore
Dr. W. P. Montague
Clarence B. Moore, Esq.
Professor F. R. Moulton
Dr. J. G. Needham
Professor Alex. T. Ormond
Professor Frederic L. Paxton
Dr. Carl Pfister

Professor M. B. Porter
Dr. A. J. Reynolds
Professor S. P. Sadtler

Dr. John A. Sampson
Oswald Schreiner, Esq.

Kenyon College
Columbia Univ.
Univ. of Cincinnati
St. Louis, Mo.
Univ. of California
Northwestern Univ.
Miami Univ.
Iowa CoUege
GranvUle, Ohio
Granville, Ohio
Univ. of Wisconsin

Univ. of Wisconsin
Dartmouth CoUege
Harvard Univ.
U. S. Marine Hospital
Indianapolis, Ind.
Cornell Univ.
Catholic Review of Re-
views
Univ. of Wisconsin
Yale University

Univ. of Missouri

OberUn CoUege

Belmont CoUege

Univ. of Pennsylvania

Univ. of California

St. Louis Univ.

Hartford Theological
Seminary

Chatham, Ontario

CorneU Univ.

St. Louis, Mo.

Brown Univ.

Leland Stanford Univ.

Ohio State Univ.

Univ. of Toronto

Dartmouth CoUege

Columbia Univ.

Philadelphia

Univ. of Chicago

Lake Forest Univ.

Princeton Univ.

Univ. of Colorado

St. Mark's Hospital,
New York City

Univ. of Texas

Chicago

Philadelphia CoUege of
Pharmacy

Albany, N. Y.

U. S. Dep't of Agricul-
ture

Geometry
Esthetics
Geometry
Plant Morphology
Algebra and Analysis
Germanic Language
Social Psychology
Greek Language
Neurology
Animal Morphology
Petrology and Mineral-
ogy
Germanic Literature
Educational Theory
Algebra and Analyses
Alcohol, etc.
Public Health
Algebra and Analysis
General Religious Edu-
cation
Physical Chemistry
Municipal Administra-
tion
Comparative Anatomy
Education, The College
Religious Agencies
Germanic Literature
Astronomy
Physiology
Semitic Languages

Applied Mathematics

AppUed Mathematics

Chemistry

Geometry

Algebra and Analysis.

Archaeology

Classical Literature

Metaphysics

Archaeology.

Astronomy.

Animal Morphology

Philosophy of ReUgion

History of America

Surgery

Algebra and Analysis

Public Health

Technical Chemistry
G5maecology

Chemistry

84 MEMBERS WHO MADE 10-MINUTE SPEECHES

Rev. Frank Sewall

Professor H. C. Sheldon

Professor Frank C. Sharp
Professor J. B. Shaw
Professor W. B. Smith
Professor Marshall S. Snow
Professor Henry Snyder
Professor Edwain D. Starbuck

Professor George B. Stewart

John M. Stahl
Professor J. Stieglitz
Professor Robert Stein
Mr. Teitaro Suzuki

Col. T. W. Symonds, U. S. A.
Professor Teissier
Judge W. H. Thomas
Professor O. H. Tittmann
Professor Alfred M. Tozzer
Dr. Benjamin F. Trueblood
Professor Clyde W. Votaw
Professor John B. Watson
Professor H. L. WiUett

President Mary E. WooUey

H. Zwaarddemaker

Washington, D. C.

Boston Univ.

Univ. of Wisconsin
Milliken Univ.
Tulane Univ.
Washington Univ.
Univ. of Minnesota
Earlham College

Auburn Theological

Seminary
Quincy, lU.
Univ. of Chicago
U. S. Geological Survey
La SaUe, 111.

Washington, D. C.
Lyons, France
Montgomery, Ala.
U. S. C. and G. Survey
Peabody Museum
Univ. of Missouri
Univ. of Chicago
Univ. of Chicago
Disciples Divinity

House, Chicago
Mt. Holyoke College

Utrecht

Social Science, The
FamUy

History of the Chris-
tian Church

Ethics

Algebra and Analysis

New Testament

History of America

Social Science

General Religious Edu-
cation

Professional Religious
Education

The Rural Community

Chemistry

Comparative Language

Brahmanism and
Buddhism

Civil Engineering

Pathology

Private Law

Astronomy

Anthropology

Medieval History

New Testament

Psychology

Professional Religious
Education

Education, The Col-
lege

Otology and Laryngo-
logy.

THE SCIENTIFIC PLAN OF THE CONGRESS

BY PROF. HUGO MUNSTERBERG

THE PURPOSE OF THE CONGRESS .

1. The Centralization of the Congress

The history of the Congress has been told. It remains to set forth the
principles which controlled the work of the Congress week, and thus
scientifically to introduce the scholarly undertaking, the results of
which are to speak for themselves in the eight volumes of this pub-
lication. Yet in a certain way this scientific introduction has once
more to use the language of history. It does not deal with the ex-
ternal development of the Congress, and the story which it has to tell
is thus not one of dates and names and events. But the principles
which shaped the whole undertaking have themselves a claim to his-
torical treatment; they do not lie before us simply as the subject for a
logical disputation or as a plea for a future work. That was the situa-
tion of three years ago. At that time various ideas and opposing
principles entered into the arena of discussion; but now, since the
work is completed, the question can be only of what principles, right
or wrong, have really determined the programme. We have thus to
interpret that state of mind out of which the purposes and the scientific
arrangement of the Congress resulted; and no after-thought of to-day
would be a desirable addition. Whatever possible improvements of
the plan may suggest themselves in the retrospect can be given only
a closing word. It was certainly easy to learn from experience, but
first the experience had to be passed through. We have here to inter-
pret the view from that standpoint from which the experience of the
Congress was still a matter of the future, and of an uncertain future
indeed, full of doubts and fears, and yet full of hopes and possibilities.
The St. Louis World's Fair promised, through the vast extent of
its grounds, through the beautiful plans of the buildings, through the
eagerness of the United States, through the participation of all coun-
tries on earth, and through the gigantic outlines of the internal plans,
to become the most monumental expression of the energies with
which the twentieth century entered on its course. Commerce and
industry, art and social work, politics and education, war and peace,

86 THE SCIENTIFIC PLAN OF THE CONGRESS

country and city, Orient and Occident, were all to be focussed for
a few summer months in the ivory city of the Mississippi Valley. It
seemed most natural that science and productive scholarship should
also find its characteristic place among the factors of our modern
civilization. Of course the scientist had his word to say on almost every
square foot of the Exposition. Whether the building was devoted to
electricity or to chemistry, to anthropology or to metallurgy, to civic
administration or to medicine, to transportation or to industrial arts,
it was everywhere the work of the scientist which was to win the tri-
umph; and the Palace of Education, the first in any universal exposi-
tion, was to combine under its roof not only the school work of all
countries, but the visible record of the world's universities and tech-
nical schools as well. And yet it seemed not enough to gather the
products and records of science and to make science serve with its
tools and inventions. Modern art, too, was to reign over every hall
and to beautify every palace, and yet demanded its own unfolding in
the gallery of paintings and sculptures. In the same way it was not
enough for science to penetrate a hundred exhibitions and turn the
wheels in every hall, but it must also seek to concentrate all its ener-
gies in one spot and show the cross-section of human knowledge in
our time, and, above all, its own methods.

An exhibition of scholarship cannot be arranged for the eyes. The
great work which grows day by day in quiet libraries and laboratories,
and on a thousand university platforms, can express itself only
through words. Yet heaped up printed volumes would be dead to
a World's Fair spectator; how to make such words living was the
problem. Above all, scholarship does not really exhibit its methods,
if it does not show itself in production. It is no longer scholarship
which speaks of a truth-seeking that has been performed instead of
going on with the search for further truth. If the world's science was
to be exhibited, a form had to be sought in which the scholarly
work on the spot would serve the ideals of knowledge, would add to
the storehouse of truth, and would thus work in the service of human
progress at the same moment in which it contributed to the com-
pleteness of the exhibition.

The effort was not without precedent. Scholarly production had
been connected with earlier expositions, and the large gatherings of
scholars at the Paris Exposition were still in vivid memory. A large
number of scientific congresses of specialists had been held there, and
many hundred scholarly papers had been read. Yet the results hardly
suggested the repetition of such an experiment. Every one felt too
strongly that the outcome of such disconnected congresses of special-
ists is hardly comparable with the glorious showing which the arts
and industries have made and were to make again. In every other
department of the World's Fair the most careful preparation secured

THE CENTRALIZATION OF THE CONGRESS 87

an harmonious effect. The scholarly meetings alone failed even to aim
at harmony and unity. Not only did the congresses themselves stand
apart without any inner relation, grouped together by calendar dates
or by their alphabetical order from Anthropology to Zoology; but
in every congress, again, the papers read and the manuscripts pre-
sented were disconnected pieces without any programme or correla-
tion. Worse than that, they could not even be expected in their isolat-
edness to add anything which would not have been worked out and
communicated to the world just as well without any congress. The
speaker at such a meeting is asked to contribute anything he has at
hand, and he accepts the invitation because he has by chance a com-
pleted paper or a research ready for publication. In the best case it
would have appeared in the next number of the specialistic magazine,
in not unfrequent cases it has appeared already in the last number.
Such a congress is then only an accident and does not itself serve the
progress of knowledge.

Even that would be acceptable if at least the best scholars would
come out with their latest investigations, or, still more delightful, if
they would enter into an important discussion. But experience has
too often shown that the conditions are most favorable for the oppo-
site outcome. The leading scholars stay away partly to give beginners
the chance to be heard, partly not to be grouped with those who
habitually have the floor at such gatherings. These are either the men
whose day has gone by or those whose day has not yet come; and
both groups tyrannize alike an unwilling audience. Yet it may be said
that in scientific meetings of specialists the reading of papers is non-
essential and no harm is done even if they do not contribute anything
to the status of scholarship ; their great value lies in the personal con-
tact of fellow workers and in the discussions and informal exchange of
opinions. All that is true, and completely justifies the yearly meetings
of scholarly associations. But these advantages are much diminished
whenever such gatherings take on an international character, and
thus introduce the confusion of tongues. And hardly any one can
doubt that the turmoil of a world's fair is about the worst possible
background for such exchange of thought, which demands repose and
quietude. Yet even with the certainty of all these disadvantages the
city of Paris, with its large body of scholars, with its venerable schol-
arly traditions, and with its incomparable attractions, could overcome
every resistance, and its convenient location made it natural that in
vacation time, in an exposition summer, the scholars should gather
there, not on account of, but in spite of, the hundred congresses.
With this the city of St. Louis could make no claim to rivalry. Its
recent growth, its minimum of scholarly tradition, its great distance
from the old centres of knowledge even in the New World, the apathy
of the East and the climatic fears of Europe, all together made it clear

88 THE SCIENTIFIC PLAN OF THE CONGRESS

that a mere repetition of unrelated congresses would be not only
uselesS; but a disastrous failure. These very fears, however, them-
selves suggested the remedy.

If the scholarly work of our time was to be represented at St. Louis,
something had to be attempted which should be not simply an imita-
tion of the branch-congresses which every scientific specialty in every
country is calling every year. Scholarship was to be asked to show
itself really in process, and to produce for the World's Fair meeting
something which without it would remain undone. To invite the
scholars of the world for their leisurely enjoyment and reposeful dis-
cussion of work done elsewhere is one thing; to call them together
for work which they would not do otherwise, and which ought to be
done, is a very different thing. The first had in St. Louis all odds
against it; it seemed worth while to try the second. And it seemed
not only worth while in the interest of scholarship, it seemed, above
all, the only way to give to the scholarship of our time a chance for
the complete demonstration of its productive energies.

The plan of unrelated congresses, with chance combinations of
papers prepared at random, was therefore definitively replaced by the
plan of only one representative gath^ering, bound together by one
underlying thought, given thus the unity of one scholarly aim, whose
fulfillment is demanded by the scientific needs of our time, and is
hardly to be reached by other methods. Every arbitrary and indi-
vidual choice was then to be eliminated and every effort was to be
controlled by the one central purpose; the work thus to be organized
and prepared with the same carefulness of adjustment and elabora-
tion which was doubtless to be applied in the admirable exhibitions
of the United States Government or in the art exhibition. The open
question was, of course, what topic could fulfill these various demands
most completely; wherein lay the greatest scholarly need of our time;
what task could be least realized by the casual efforts of scholarship
at random; where was the unity of a world organization most needed?

One thought was very naturally suggested by the external circum-
stances. St. Louis had asked the nations of the world to a celebration
of the Louisiana Purchase. Historical thoughts thus gave meaning
and importance to the whole undertaking. The pride of one century's
development had stimulated the gigantic work from its inception. An
immense territory had been transformed from a half wilderness into
a land with a rich civilization, and with a central city in which eight
thousand factories are at work. No thought lay nearer than to ask
how far this century was of similar importance for the changes in the
world of thought. How have the sciences developed themselves since
the days of the Louisiana Purchase? That is a topic which with com-
plete uniformity might be asked from every special science, and which
might thus offer a certain unity of aim to scholars of all scientific de-

THE CENTRALIZATION OF THE CONGRESS 89

nominations. There was indeed no doubt that such an historical ques-
tion would have to be raised if we were to live up to the commemora-
tive idea of the whole Fair. And yet it seemed still more certain that
the retrospective problem did not justify itself as a central topic for a
World's Congress. There were sciences for which the story of the last
hundred years was merely the last chapter of a history of three thou-
sand years and other sciences whose life history did not begin until
one or two decades ago. It would thus be a very external uniformity;
the question would have a very different meaning for the various
branches of knowledge, and the treatment would be of very unequal
interest and importance. More than that, it would not abolish the
unrelated character of the endeavors; while the same topic might
be given everywhere, yet every science would remain isolated; there
would be no internal unity, and thus no inner reason for bringing
together the best workers of all spheres. And finally the mere retro-
spective attitude brings with it the depressing mood of perfunctory
activity. Certainly to look back on the advance of a century can be
most suggestive for a better understanding of the way which lies
before us; and we felt indeed that the occasion for such a back-
ward glance ought not to be missed. Yet there would be something
lifeless if the whole meeting were devoted to the consideration of work
that had been completed; a kind of necrological sentiment would
pervade the whole ceremony, while our chief aim was to serve the
progress of knowledge and thus to stimulate living interests.

This language of life spoke indeed in the programme of another
plan which seemed also to be suggested by the character of the
Exposition. The St. Louis Fair desired not merely to look backward
and to revive the historical interest in the Louisiana Purchase,
but its first aim seemed to be to bring into sharp relief the factors
which serve to-day the practical welfare and the achievements of
human society. If all the scholars of all sciences were to convene
under one flag, would it not thus seem most harmonious with the
occasion, if, as the one controlling topic, the question were proposed,
" What does your science contribute to the practical progress of man-
kind? " No one can deny that such a formulation would fit in well
with the lingering thoughts of every World's Fair visitor. Whoever
wanders through the aisles of exhibition palaces and sees amassed the
marvelous achievements of industry and commerce, and the thousand
practical arts of modern society, may indeed turn most naturally to
a gathering of scholars with the question, '■' What have you to offer
of similar import?" All. your thinking and speaking and writing, are
they merely words on words, or do you also turn the wheels of this
gigantic civilization?

Such a question would give a noble opening indeed to almost every
science. Who would say that the opportunity is confined to the man of

90 THE SCIENTIFIC PLAN OF THE CONGRESS

technical science? Does not the biologist also prepare the achievements
of modern medicine, does not the mathematician play his most impor-
tant role in our mastery over stubborn nature, do we not need lan-
guage for our social intercourse, and law and religion for our practical
social improvement? Yes, is there any science which has not directly
or indirectly something to contribute to the practical development of
the modern man and his civilization? All this is true, and yet the
perspective of this truth, too, appears at once utterly distorted if we
take the standpoint of science itself. The one end of knowledge is to
reach the truth. The belief in the absolute value of truth gives to it
meaning and significance. This value remains the controlling influ-
ence even where the problem to be solved is itself a practical one, and
the spirit of science remains thus essentially theoretical even in the
so-called applied sciences. But incomparably more intense in that
respect is the spirit of all theoretical disciplines. Philosophy and
mathematics, history and philology, chemistry and biology, astro-
nomy and geology, may be and ought to be helpful to practical
civilization everywhere; and every step forward which they take
will be an advance for man's practical life too. And yet their real
meaning never lies in their technical by-product. It is not the
scholar who peers in the direction of practical use who is most loyal
to the deepest demand of scholarship, and every relation to prac-
tical achievement is more or less accidental or even artificial for
the real life interests of productive scholarship.

But if the contrast between his real intention and his social tech-
nical successes may not appear striking to the physicist or chemist,
it would appear at least embarrassing to the scholars in many other
departments and directly bewildering to not a few. Perhaps two
thirds of the sciences to which the best thinkers of our time are faith-
fully devoted would then be grouped together and relegated to a
distant corner, their only practical technical function would be to
contribute material to the education of the cultured man. For what
else do we study Sanscrit or medieval history or epistemology? And
finally even the uniform topic of practical use would not have
brought the different sciences nearer to each other; the Congress
would still have remained a budget of disconnected records of scholar-
ship. If the practical side of the Exposition was to suggest anything,
it should then not be more than an appeal not to overlook the impor-
tance of the applied sciences which too often play the r61e of a mere
appendix to the system of knowledge. The logical one-sidedness
which considers practical needs as below the dignity of pure science
was indeed to be excluded, but to choose practical service as the one
controlling topic would be far more anti-scientific.

THE UNITY OF KNOWLEDGE 91

2. The Unity of Knowledge

There was another side of the Exposition plan which suggested a
stronger topic. The World's Fair was not only an historical memorial
work, and was not only a show of the practical tools of technical civil-
ization; its deepest aim was after all the effort to bring the energies of
our time into inner relation. The peoples of the whole globe, sepa-
rated by oceans and mountains, by language and custom, by politics
and prejudice, were here to come in contact and to be brought into
correlation by better mutual understanding of the best features of
their respective cultures. The various industries and arts, the most
antagonistic efforts of commerce and production, separated by the
rivalry of the market and by the diversity of economic interests
were here to be brought together in harmony, were to be correlated
for the eye of the spectator. It was a near-lying thought to choose
correlation as the controlling thought of a scientific World's Congress
too. That was the topic which was finally agreed upon: the inner
relation of the sciences of our day.

The fitness and the external advantages of such a scheme are
evident. First of all, the danger of disconnectedness now disappears
completely. If the sciences are to examine what binds them together,
their usual isolation must be given up for the time being and a con-
certed effort must control the day. The bringing together of scholars
of all scientific specialties is then no longer a doubtful accidental fea-
ture, but becomes a condition of the whole undertaking. More than
that, such a topic, with all that it involves, makes it a matter of course
that the call goes out to the really leading scholars of the time. To
aim at a correlation of sciences means to seek for the fundamental
principles in each territory of knowledge and to look with far-seeing
eye beyond the limits of its field; but just this excludes from the
outset those who like to be the self-appointed speakers in routine
gatherings. It excludes from the first the narrow specialist who does
not care for anything but for his latest research, and ought to exclude
not less the vague spirits who generalize about facts of which they
have no concrete substantial knowledge, as their suggestions towards
correlation would lack inner productiveness and outer authority.
Such a plan has room only for those men who stand high enough to
see the whole field and who have yet the full authority of the special-
istic investigator; they must combine the concentration on specialized
productive work with the inspiration that comes from looking over
vast regions. With such a topic the usual question does not come up
whether one or another strong man would feel attracted to take part
in the gathering, but it would be justified and necessary to confine the
active participation from the outset to those who are leaders, and
thus to guarantee from the beginning a representation of science

92 THE SCIENTIFIC PLAN OF THE CONGRESS

equal in dignity to the best efforts of the exhibiting countries in all
other departments. In this way such a plan had the advantage of
justifying through its topic the administrative desire to bring all
sciences to the same spot, and at the same time of excluding all par-
ticipants but the best scholars: with isolated gatherings or with
second-rate men, this subject would have been simply impossible.

Yet all these halfway external advantages count Uttle compared
with the significance and importance of the topic for the inner Hfe of
scientific thought of our time. We aU felt it was the one topic which
the beginning of the twentieth century demanded and which could
not be dealt with otherwise than by the combined labors of all nations
and of all sciences. The World's Fair was the one great opportunity
to make a first effort in this direction; we had no right to miss this
opportunity. Thus it was decided to have a congress with the definite
purpose of working towards the unity of human knowledge, and with
the one mission, in this time of scattered speciahzing work, of bringing
to the consciousness of the world the too-much neglected idea of the
unity of truth. To quote from our first tentative programme: " Let
the rush of the world's work stop for one moment for us to consider
what are the underlying principles, what are their relations to one
another and to the whole, what are their values and purposes; in
short, let us for once give to the world's sciences a hohday. The work-
aday functions are much better fulfilled in separation, when each
scholar works in his own laboratory or in his hbrary; but this holiday
task of bringing out the underlying unity, this synthetic work, this
demands really the cooperation of all, this demands that once at least
all sciences come together in one place at one time."

Yet if our work stands for the unity of knowledge, aims to consider
the fundamental conceptions which bind together all the specialistic
results, and seeks to inquire into the methods which are common to
various fields, all this is after all merely a symptom of the whole spirit
of our times. A reaction against the narrowness of mere fact-diggers
has set in. A mere heaping up of disconnected, unshaped facts begins
to disappoint the world; it is felt too vividly that a mere dictionary of
phenomena, of events and laws, makes our knowledge larger but not
deeper, makes our Hfe more complex but not more valuable, makes
our science more difficult but not more harmonious. Our time longs
for a new synthesis and looks towards science no longer merely with
a desire for technical prescriptions and new inventions in the interest
of comfort and exchange. It waits for knowledge to fulfill its higher
mission, it waits for science to satisfy our higher needs for a view
of the world which shall give unity to our scattered experience. The
indications of this change are visible to every one who observes the
gradual turning to philosophical discussion in the most different
fields of scientific life.

THE UNITY OF KNOWLEDGE 93

When after the first third of the nineteenth century the great
philosophic movement which found its cUmax in HegeUanism came
to disaster in consequence of its absurd neglect of hard solid facts, the
era of naturalism began its triumph with contempt for all philosophy
and for all deeper unity. IdeaHsm and philosophy were stigmatized as
the enemies of true science and natural science had its great day. The
rapid progress of physics and chemistry fascinated the world and pro-
duced modern technique; the sciences of life, physiology, biology,
medicine, followed; and the scientific method was carried over from
body to mind, and gave us at the end of the nineteenth century mod-
ern psychology and sociology. The lifeless and the living, the physical
and the mental, the individual and the social, all had been conquered
by analytical methods. But just when the cUmax was reached and all
had been analyzed and explained, the time was ripe for disillusion,
and the lack of deeper unity began to be felt with alarm in every
quarter. For seventy years there had been nowhere so much philo-
sophizing going on as suddenly sprung up among the scientists of
the last decade. The physicists and the mathematicians, the chemists
and the biologists, the geologists and the astronomers, and, on the
other side, the historians and the economists, the psychologists and
the sociologists, the jurists and the theologians — all suddenly found
themselves again in the midst of discussions on fundamental princi-
ples and methods, on general categories and conditions of knowledge,
in short, in the midst of the despised philosophy. And with those
discussions has come the demand for correlation. Everywhere have
arisen leaders who have brought unconnected sciences together and
emphasized the unity of large divisions. The time seems to have come
again when the wave of naturalism and realism is ebbing, and a new
idealistic philosophical tide is swelling, just as they have always alter-
nated in the civilization of two thousand years.

No one dreams, of course, that the great synthetic apperception, for
which our modern time seems ripe, will come through the delivery of
some hundred addresses, or the discussions of some hundred audiences.
An ultimate unity demands the gigantic thought of a single genius,
and the work of the many can, after all, be merely the preparation
for the final work of the one. And yet history shows that the one will
never come if the many have not done their share. What is needed
is to fill the sciences of our time with the growing consciousness of
belonging together, with the longing for fundamental principles, with
the conviction that the desire for correlation is not the fancy of
dreamers, but the immediate need of the leaders of thought. And in
this preparatory work the St. Louis Congress of Arts and Science
seemed indeed called for an important part when it was committed
to this topic of correlation.

To call the scholars of the world together for concerted action

94 THE SCIENTIFIC PLAN OF THE CONGRESS

towards the correlation of knowledge meant, of course, first of all, to
work out a detailed programme, and to select the best authorities
for every special part of the whole scheme. Nothing could be left to
chance methods and to casual contributions. The preparation needed
the same administrative strictness which would be demanded for an
encyclopedia, and the same scholarly thoroughness which would be
demanded for the most scientific research. A plan was to be devised
in which every possible striving for truth would find its place, and
in which every section would have its definite position in the system.
And such a ground-plan given, topics were to be assigned to every
department and sub-department, the treatment of which would bring
out the fundamental principles and the inner relations in such a way
that the papers would finally form a close-woven intellectual fabric.
There would be plenty of room for a retrospective glance at the his-
torical development of the sciences and plenty of room for emphasis
on their practical achievements; but the central place would always
belong to the effort towards unity and internal harmonization.

We thus divided human knowledge into large parts, and the parts
into divisions, and the divisions into departments, and the depart-
ments into sections. As the topic of the general divisions — we pro-
posed seven of them — it was decided to discuss the Unity of the
whole field. As topic for the departments — we had twenty-four of
them — the addresses were to discuss the fundamental Conceptions
and Methods and the Progress during the last century; and in the
sections, finally — our plan provided for one hundred and twenty-
eight of them — the topics were in every one the Relation of the
special branch to other branches, and those most important Present
Problems which are essential for the deeper principles of the special
field. In this way the ground-plan itself suggested the unity of the
practically separated sciences; and, moreover, our plan provided
from the first that this logical relation should express itself externally
in the time order of the work. We were to begin with the meetings of
the large divisions, the meetings of the departments were to foUow,
and the meetings of the sections and their ramifications would follow
the departmental gatherings.

3. The Objections to the Plan

It was evident that even the most modest success of that gigantic
imdertaking depended upon the right choice of speakers, upon the
value of the ground-plan, and upon many external conditions; thus
no one was in doubt as to the difficulty in realizing such a scheme.
Yet there were from the scholarly side itself objections to the prin-
ciples involved, objections which might hold even if those other
conditions were successfully met. The most immediate reason for

THE OBJECTIONS TO THE PLAN 95

reluctance lies in the specializing tendencies of our time. Those
who devote all their working energy as loyal sons of our analyzing
period of science to the minute detail of research come easily into the
habit of a nervous fear with regard to any wider general outlook. The
man of research sees too often how ignorance hides itself behind gen-
eralities. He knows too well how much easier it is to formulate vague
generalities than to contribute a new fact to human knowledge, and
how often untrained youngsters succeed with popular text-books
which are rightly forgotten the next day. Methodical science must
thus almost encourage this aversion to any deviation from the path
of painstaking speciahstic labor. Then, of course, it seems almost
a scientific duty to declare war against an undertaking which ex-
plicitly asks everywhere for the wide perspectives and the last prin-
ciples, and does not aim at adding at this moment to the mere treasury
of information.

But such a view is utterly one-sided, and to fight against such one-
sidedness and to overcome the speciahzing narrowness of the scat-
tered sciences was the one central idea of the plan. If there existed
no scholars who despise the philosophizing connection, there would
have hardly been any need for this whole undertaking; but to yield
to such philosophy-phobia means to declare the analytic movement
of science permanent, and to postpone a synthetic movement in-
definitely. Our time has just to emphasize, and the leaders of thought
daily emphasize it more, that a mere heaping up of information can
be merely a preparation for knowledge, and that the final aim is
a Weltanschauung, a unified view of the whole of reality. All that
our Congress had to secure was thus merely that the generalizing dis-
cussion of principles should not be left to men who generalized be-
cause they lacked the substantial knowledge which is necessary to
specialize. The thinkers we needed were those who through special-
istic work were themselves led to a point where the discussion of gen-
eral principles becomes unavoidable. Our plan was by no means
antagonistic to the patient labors of analysis; the aim was merely to
overcome its one-sidedness and to stimulate the synthesis as a neces-
sary supplement.

But the objections against a generalizing plan were not confined to
the mistaken fear that we sought to antagonize the productive work
of the specialist. They not seldom took the form of a general aver-
sion to the logical side of the ground-plan. It was often said that such
a scheme has after all interest only for the logician, for whom science
as such is an object of study, and who must thus indeed classify the
sciences and determine their logical relation. The real scientist, it
was said, does not care for such methodological operations, and should
be suspicious from the first of such philosophical high-handedness.
The scientist cannot forget how often in the history of civilization

96 THE SCIENTIFIC PLAN OF THE CONGRESS

science was the loser when it trusted its problems to the metaphy-
sical thinker who substituted his lofty speculations for the hard
work of the investigator. The true scholar will thus not only object
to generalizing " commonplaces" as against solid information, but he
will object as well to logical demarcation lines and systematization
as against the practical scientific work which does not want to be
hampered by such philosophical subtleties. Yet all these fears and
suspicions were still more mistaken.

Nothing was further from our intentions than a substitution of
metaphysics for concrete science. It was not by chance that we took
such pains to find the best specialists for every section. No one was
invited to enter into logical discussions and to consider the relations
of science merely from a dialectic point of view. The topic was every-
where the whole living manifoldness of actual relations, and the logi-
cian had nothing else to do than to prepare the programme. The
outlines of the programme demanded, of course, a certain logical
scheme. If hundreds of sciences are to take part, they have to be
grouped somehow, if a merely alphabetical order is not adopted; and
even if we were to proceed alphabetically, we should have to decide
beforehand what part of knowledge is to be recognized as a special
science. But the logical order of the ground-plan refers, of course,
merely to the simple relation of coordination, subordination, and
superordination, and the logician is satisfied with such a classification.
But the endless variety of internal relations is no longer to be dealt
with from the point of view of mere logic. We may work out the
ground-plan in such a way that we understand that logically zoology
is coordinated to botany and subordinated to mechanics and super-
ordinated to ichthyology; but this minimum of determination gives,
of course, not even a hint of that world of relations which exists from
the standpoint of the biologist between the science of zoology and
the science of botany, or between the biological and the mechanical
studies. To discuss these relations of real scientific life is the work of
the biologist and not at all of the logician.

The foregoing answers also at once an objection which might seem
more justified at the first glance. It has been said that we were under-
taking the work of bringing about a synthesis of scientific endeavors,
and that we yet had that synthesis already completed in the pro-
gramme on which the work was to be based. The scholars to be in-
vited would be bound by the programme, and would therefore have
no other possibility than to say with more words what the programme
had settled beforehand. The whole effort would then seem determined
from the start by the arbitrariness of the proposed ground-plan.
Now it cannot be denied indeed that a certain factor of arbitrariness
has to enter into a programme. We have already referred to the fact
that some one must decide beforehand what fraction of science is to be

THE OBJECTIONS TO THE PLAN 97

acknowledged as a self-dependent discipline. If a biologist were to
work out the scheme, he might decide that the whole of philosophy
was just one science; while the philosopher might claim a large num-
ber of sections for logic and ethics and philosophy of religion, and so
on. And the philosopher, on the other hand, might treat the whole of
medicine as one part in itself, while the physician might hold that even
otology has to be separated from rhinology. A certain subjectivity of
standpoint is unavoidable, and we know very well that instead of the
one hundred and twenty-ei'ght sections of our programme we might
have been satisfied with half that number or might have indulged in
double that number. And yet there was no possible plan which would
have allowed us to invite the speakers without defining beforehand
the sectional field which each was to represent. A certain courage of
opinion was then necessary, and sometimes also a certain adjustment
to external conditions.

Quite similar was the question of classification. Just as we had to
take the responsibility for the staking-out of every section, we had
also to decide in favor of a certain grouping, if we desired to organ-
ize the Congress and not simply to bring out haphazard results. The
principles which are sufficient for a mere directory would never allow
the shaping of a programme which can be the basis for synthetic work.
Even a university catalogue begins with a certain classification, and
yet no one fancies that such catalogue grouping inhibits the freedom
of the university lecturer. It is easy to say, as has been said, that the
essential trait of the scientific life of to-day is its live-and-let-live
character. Certainly it is. In the regular work in our libraries and
laboratories the year round, everything depends upon this demo-
cratic freedom in which every one goes his own way, hardly asking
what his neighbor is doing. It is that which has made the specialistic
sciences of our day as strong as they are. But it has brought about at
the same time this extreme tendency to unrelated specialization with
its discouraging lack of unity; this heaping up of information without
an outer harmonious view of the world ; and if we were really at least
once to satisfy the desire for unity, then we had not the right to yield
fully to this live-and-let-live tendency. Therefore some principle of
grouping had to be accepted, and whatever principle had been chosen,
it would certainly have been open to the criticism that it was a pro-
duct of arbitrary decision, inasmuch as other principles might have
been possible.

A classification which in itself expresses all the practical relations in
which sciences stand to each other is, of course, absolutely impossible.
A programme which should try to arrange the place of a special disci-
pline in such a way that it would become the neighbor of all those other
sciences with which it has internal relation is unthinkable. On the
other hand, only if we had tried to construct a scheme of such exagger-

98 THE SCIENTIFIC PLAN OF THE CONGRESS

ated ambitions should we have been really guilty of anticipating a
part of that which the specialistic scholars were to tell us. The Con-
gress had to leave it to the invited participants to discuss the totality
of relations which practically exist between their fields and others,
and the organizers confined themselves to that minimum of classifica-
tion which just indicates the pure logical relations, a minimum which
every editor of encyclopedic work would be asked to initiate without
awakening suspicions of interference with the ideas of his contributors.

The only justified demand which could be met was that a system
of division and classification should be proposed which should give
fair play to every existing scientific tendency. The minimum of classi-
fication was to be combined with the maximum of freedom, and to
secure that a careful consideration of principles was indeed necessary.
To bring logical order into the sciences which stand out clearly with
traditional rights is not difficult; but the chances are too great that
certain tendencies of thought might fail to find recognition or might
be suppressed by scientific prejudice. Any serious omission would
indeed have necessarily inhibited the freedom of expression. To
secure thus the greatest inner fullness of the programme, seemed in-
deed the most important task in the elaboration of the ground-plan.
The fears that we might offer empty generalization instead of schol-
arly facts, or that we might simply heap up encyclopedic information
instead of gaining wide perspectives, or that we might interfere with
the living connections of sciences by the logical demarcation lines, or
that we might disturb the scholar in his freedom by determining
beforehand his place in the classification, — all these fears and objec-
tions, which were repeatedly raised when the plan was first proposed,
seemed indeed unimportant compared with the fear that the pro-
gramme might be unable to include all scientific tendencies of the
time.

That would have been, indeed, the one fundamental mistake, as the
whole Congress work was planned in the service of the great synthetic
movement which pervades the intellectual life of to-day. The under-
taking would be useless and even hindering if it were not just the newer
and deeper tendencies that came to most complete expression in it.
Everything depended, therefore, upon the fullest possible representa-
tion of scientific endeavors in the plan. But no one can become aware
of this manifoldness and of the logical relations who does not go back to
the ultimate principles of the human search for truth. We have, there-
fore, to enter now into a full discussion of the principles which have
controlled the classification and subdivision of the whole work. The
discussion is necessarily in its essence a philosophical one, as it was
earlier made plain that philosophy must lay out the plan, while in the
realization of the plan through concrete work the scientist alone, and
not the logician, has to speak. Yet here again it may be said that

THE DEVELOPMENT OF CLASSIFICATION 99

while our discussion of principles in its essence is logical, in another
respect it is a merely historical account. The question is not what
principles of classification are to be acknowledged as valuable now
that the work of the Congress lies behind us, but what principles were
accepted and really led to the organization of the work in that form in
which it presents itself in the records of the following volumes.

THE CLASSIFICATION OF THE SCIENCES

L The Development of Classification

The problem of dividing and subdividing the whole of human know-
ledge and of thus bringing order into the manifoldness of scientific
efforts has fascinated the leading thinkers of all ages. It may often be
difficult to say how far the new principles of classification themselves
open the way for new scientific progress and how far the great forward
movements of thought in the special sciences have in turn influenced
the principles of classification. In any case every productive age has
demanded the expression of its deepest energy in a new ordering of
human science. The history of these efforts leads from Plato and
Aristotle to Bacon and Locke, to Bentham and Ampere, to Kant and
Hegel, to Comte and Spencer, to Wundt and Windelband. And yet
we can hardly speak of a real historical continuity. In a certain way
every period took up the problem anew, and the new aspects resulted
not only from the development of the sciences themselves which were
to be classified, but still more from the differences of logical interest.
Sometimes the classification referred to the material, sometimes to
the method of treatment, sometimes to the mental energies involved,
and sometimes to the ends to be reached. The reference to the mental
faculties was certainly the earliest method of bringing order into
human knowledge, for the distinction of the Platonic philosophy be-
tween dialectics, physics, and ethics pointed to the threefold charac-
ter of the mind, to reason, perception, and desire; and it was on the
threshold of the modern time, again, when Bacon divided the intel-
lectual globe into three large parts according to three fundamental
psychical faculties: memory, imagination, and reason. The memory
gives us history; the imagination, poetry; the reason, philosophy,
or the sciences. History was further divided into natural and civil
history; natural history into normal, abnormal, and artificial phe-
nomena; civil history into political, literary, and ecclesiastical history.
The field of reason was subdivided into man, nature, and God; the
domain of man gives, first, civil philosophy, parted off into inter-

100 THE SCIENTIFIC PLAN OF THE CONGRESS

course^ business, and government, and secondly, the philosophy of
humanity, divided into that of body and of soul, wherein medicine
and athletics belong to the body, logic and ethics to the soul. Nature,
on the other hand, was divided into speculative and applied science,
— the speculative containing both physics and metaphysics; the
applied, mechanics and magic. All this was full of artificial con-
structions, and yet still more marked by deep insight into the needs
of Bacon's time, and not every modification of later classifiers was
logically a step forward.

Yet modern efforts had to seek quite different methods, and the
energies which have been most effective for the ordering of knowledge
in the last decades spring unquestionably from the system of Comte
and his successors. He did not aim at a system of ramifications; his
problem was to show how the fundamental sciences depend on each
otiier. A series was to be constructed in which each member should
presuppose the foregoing. The result was a simplicity which is cer-
tainly tempting, but this simplicity was reached only by an artificial
emphasis which corresponded completely to the one-sidedness of
naturalistic thought. It was a philosophy of positivism, the back-
ground for the gigantic work of natural science and technique in the
last two thirds of the nineteenth century. Comte 's fundamental
thought is that the science of Morals, in which we study human nature
for the government of human life, is dependent on sociology. Socio-
logy, however, depends on biology; this on chemistry; this on
physics; this on astronomy; and this finally on mathematics. In this
way, all mental and moral sciences, history and philology, jurispru-
dence and theology, economics and politics, are considered as socio-
logical phenomena, as dealing with functions of the human being.
But as man is a living organism, and thus certainly falls under
biology, all the branches of knowledge from history to ethics, from
jurisprudence to aesthetics, can be nothing but subdivisions of biology.
The living organism, on the other hand, is merely one type of the
physical bodies on earth, and biology is thus itself merely a depart-
ment of physics. But as the earthly bodies are merely a part of the
cosmic totality, physics is thus a part of astronomy; and as the whole
universe is controlled by mathematical laws, mathematics must be
superordinated to all sciences.

But there followed a time which overcame this thinly disguised
example of materialism. It was a time when the categories of the
ph5^siologist lost slightly in credit and the categories of the psycho-
logist won repute. This newer movement held that it is artificial to
consider ethical and logical life, historic and legal action, literary and
religious emotions, merely as physiological functions of the living
organism. The mental life, however necessarily connected with brain
processes, has a positive reality of its own. The psychical facts repre-

THE DEVELOPMENT OF CLASSIFICATION 101

sent a world of phenomena which in its nature is absolutely different
from that of material phenomena, and, while it is true that every
ethical action and every logical thought can, from the standpoint of
the biologist, be considered as a property of matter, it is not less true
that the sciences of mental phenomena, considered impartially, form
a sphere of knowledge closed in itself, and must thus be coordinated,
not subordinated, to the knowledge of the physical world. We should
say thus: all knowledge falls into two classes, the physical sciences
and the mental sciences. In the circle of physical sciences we have the
general sciences, like physics and chemistry, the particular sciences of
special objects, like astronomy, geology, mineralogy, biology, and the
formal sciences, like mathematics. In the circle of mental sciences we
have correspondingly, as a general science, psychology, and as the
particular sciences all those special mental and moral sciences which
deal with man's inner life, like history or jurisprudence, logic or ethics,
and aU the rest. Such a classification, which had its philosophical
defenders about twenty years ago, penetrated the popular thought
as fully as the positivism of the foregoing generation, and was cer-
tainly superior to its materialistic forerunner.

Of course it was not the first time in the history of civilization that
materialism was replaced by dualism, that biologism was replaced by
psychologism; and it was also not the first time that the development
of civilization led again beyond this point: that is, led beyond the
psychologizing period. There is no doubt that our time presses
on, with all its powerful internal energies, away from this Weltan-
schauung of yesterday. The materialism was anti-philosophic, the
psychological dualism was unphilosophic. To-day the philosophical
movement has set in. The one-sidedness of the nineteenth century
creed is felt in the deeper thought all over the world : popular move-
ments and scholarly efforts alike show the signs of a coming idealism,
which has something better and deeper to say than merely that our
life is a series of causal phenomena. Our time longs for a new inter-
pretation of reality; from the depths of every science wherein for
decades philosophizing was despised, the best scholars turn again to a
discussion of fundamental conceptions and general principles. Histor-
ical thinking begins again to take the leadership which for half a cen-
tury belonged to naturalistic thinking; specialistic research demands
increasingly from day to day the readjustment toward higher unities,
and the technical progress which charmed the world becomes more
and more simply a factor in an ideal progress. The appearance of this
unifying congress itself .is merely one of a thousand symptoms of
this change appearing in our public life, and if the scientific philo-
sophy is producing to-day book upon book to prove that the world
of phenomena must be supplemented by the world of values, that
description must yield to interpretation, and that explanation must

102 THE SCIENTIFIC PLAN OF THE CONGRESS

be harmonized with appreciation: it is but echoing in technical
terms the one great emotion of our time.

This certainly does not mean that any step of the gigantic material-
istic, technical, and psychological development will be reversed, or
that progress in any one of these directions ought to cease. On the
contrary, no time was ever more ready to put its immense energies
into the service of naturalistic work; but it does mean that our time
recognizes the one-sidedness of these movements, recognizes that they
belong only to one aspect of reality, and that another aspect is pos-
sible; yes, that the other aspect is that of our immediate life, with its
purposes and its ideals, its historical relations and its logical aims.
The claim of materialism, that aU psychical facts are merely functions
of the organism, was no argument against psychology, because,
though the biological view was possible, yet the other aspect is cer-
tainly a necessary supplement. In the same way it is no argument
against the newer view that all purposes and ideals, all historical
actions and logical thoughts, can be considered as psychological phe-
nomena. Of course we can consider them as such, and we must go on
doing so in the service of the psychological and sociological sciences;
but we ought not to imagine that we have expressed and understood
the real character of our historical or moral, our logical or religious
life when we have described and explained it as a series of phenomena.
Its immediate reality expresses itself above all in the fact that it has
a meaning, that it is a purpose which we want to understand, not by
considering its causes and effects, but by interpreting its aims and
appreciating its ideals.

We should say, therefore, to-day that it is most interesting and
important for the scientist to consider human life with all its strivings
and creations from a biological, psychological, sociological point of
view; that is, to consider it as a system of causal phenomena; and
many problems worthy of the highest energies have still to be solved
in these sciences. But that which the jurist or the theologian, the
student of art or of history, of literature or of politics, of education or
of morality, is dealing with, refers to the other aspect in which inner
life is not a phenomenon but a system of purposes, not to be ex-
plained but to be interpreted, to be approached not by causal but by
teleological methods. In this case the historical sciences are no longer
sub-sections of psychological or of sociological sciences; the concep-
tion of science is no longer identical with the conception of the
science of phenomena. There exist sciences which do not deal with
the description or explanation of phenomena at all, but with the
internal relation and connection, the interpretation and appreciation
of purpose. In this way modern thought demands that sciences of
purpose be coordinated with sciences of phenomena. Only if all these
tendencies of our time are fully acknowledged can the outer frame-

THE FOUR THEORETICAL DIVISIONS 103

work of our classification offer a fair field to every scientific thought,
while a positivistic system would cripple the most promising tend-
encies of the twentieth century.

2. The Four Theoretical Divisions

We have first to determine the underlying structure of the classifi-
cation, that is, we have to seek the chief Divisions, of which our plan
shows seven; four theoretical and three practical ones. It will be a
secondary task to subdivide them later into the 24 Departments and
128 Sections. We desire to divide the whole of knowledge in a funda-
mental way, and we must therefore start with the question of prin-
ciple:— what is knowledge? This question belongs to epistemology,
and thus falls, indeed, into the domain of philosophy. The positivist
is easily inclined to substitute for the philosophical problem the
empirical question: how did that which we call knowledge grow
and develop itself in our individual mind, or in the mind of the
nations? The question becomes, then, of course, one which must be
answered by psychology, by sociology, and perhaps by biology. Such
genetic inquiries are certainly very important, and the problem of
how the processes of judging and conceiving and thinking are pro-
duced in the individual or social consciousness, and how they are to
be explained through physical and psychical causes, deserves fullest
attention. But its solution cannot even help us as regards the funda-
mental problem, what we mean by knowledge, and what the ultimate
value of knowledge may be, and why we seek it. This deeper logical
inquiry must be answered somehow before those genetic studies of the
psychological and the sociological positivists can claim any truth at
all, and thus any value, for their outcome. To explain our present
knowledge genetically from its foregoing causes means merely to con-
nect the present experience, which we know, with a past experience,
which we remember, or with earlier phenomena which we construct
on the basis of theories and hypotheses; but in any case with facts
which we value as parts of our knowledge and which thus presuppose
the acknowledgment of the value of knowledge. We cannot deter-
mine by hnking one part of knowledge with another part of know-
ledge whether we have a right to speak of knowledge at aU and to
rely on it.

We can thus not start from the childhood of man, or from the begin-
ning of humanity, or from any other object of knowledge, but we
must begin with the state which logically precedes all knowledge;
that is, with our immediate experience of real life. Here, in the naive
experience in which we do not know ourselves as objects which we
perceive, but where we feel ourselves in our subjective attitudes as
agents of will, as personalities, here we find the original reality not yet

104 THE SCIENTIFIC PLAN OF THE CONGRESS

shaped and remoulded by scientific conceptions and by the demands
of knowledge. And from this basis of primary, naive reality we must
ask ourselves what we mean by seeking knowledge, and how this
demand of ours is different from the other activities in which we work
out the meaning and the ideals of our life.

One thing is certain, we cannot go back to the old dogmatic stand-
point, whether rationalistic or sensualistic. In both cases dogmatism
took for granted that there is a real world of things which exist in
themselves independent of our subjective attitudes, and that our
knowledge has to give us a mirror picture of that self-dependent
world. Sensualism averred that we get this knowledge through our
perceptions; rationalism, that we get it by reasoning. The one as-
serted that experience gives us the data which mere abstract reason-
ing can never supply; the other asserted that our knowledge speaks
of necessity which no mere perception can find out. Our modern
time has gone through the school of philosophical criticism, and the
dogmatic ideas have lost for us their meaning. We know that the
world which we think as independent cannot be independent of the
forms of our thinking, and that no science has reference to any other
world than the world which is determined by the categories of our
apperception. There cannot be anything more real than the immedi-
ate pure experience, and if we seek the truth of knowledge, we do not
set out to discover something which is hidden behind our experience,
but we set out simply to make something out of our experience which
satisfies certain demands. Our immediate experience does not contain
an objective thing and a subjective picture of it, but they are com-
pletely one and the same piece of experience. We have the object of
our immediate knowledge not in the double form of an outer object
independent of ourselves and an idea in us, but we have it as our
object there in the practical world before science for its special pur-
poses has broken up that bit of reality into the physical material
thing and the psychical content of consciousness. And if this double-
ness does not hold for the immediate reality of pure experience, it
cannot enter through that reshaping and reconstructing and connect-
ing and interpreting of pure experience which we call our knowledge.
All that science gives to us is just such an endlessly enlarged expe-
rience, of which every particle remains objective and independent,
inasmuch as it is not in us as psychical individuals, while yet com-
pletely dependent upon the forms of our subjective experience. The
ideal of truth is thus not to gain by reason or by observation ideas
in ourselves which correspond as well as possible to absolute things,
but to reconstruct the given experience in the service of certain
purposes. Everything which completely fulfills the purposes of this
intentional reconstruction is true.

What are these purposes? One thing is clear from the first : There

THE FOUR THEORETICAL DIVISIONS 105

cannot be a purpose where there is not a will. If we come from pure
experience to knowledge by a purposive transformation, we must
acknowledge the reality of will in ourselves, or rather, we must find
ourselves as will in the midst of pure experience before we reach any
knowledge. And so it is indeed. We can abstract from all those recon-
structions which the sciences suggest to us and go back to the most
immediate naive experience; but we can never reach an experience
which does not contain the doubleness of subject and object, of will
and world. That doubleness has nothing whatever to do with the
difference of physical and psychical; both the physical thing and the
psychical idea are objects. The antithesis is not that between two
kinds of objects, since we have seen that in the immediate experience
the objects are not at all split up into the two groups of material and
mental things; it is rather the antithesis between the object in its
undifferentiated state on the one side and the subject in its will-atti-
tude on the other side. Yes, even if we speak of the subject which
stands as a unity behind the will-attitudes, we are already reconstruct-
ing the real experience in the interest of the purposes of knowledge.
In the immediate experience, we have the will-attitudes themselves,
and not a subject which wills them.

If we ask ourselves finally what is then the ultimate difference
between those two elements of our pure experience, between the object
and the will-attitude, we stand before the ultimate data: we call that
element which exists merely through a reference to its opposite, the
object, and we call that element of our experience which is complete
in itself, the attitude of the will. If we experienced hking or dislik-
ing, affirming or denying, approving or disapproving in the same way
in which we experience the red and the green, the sweet and the sour,
the rock and the tree and the moon, we should know objects only.
But we do experience them in quite a different way. The rock and
the tree do not point to anything else, but the approval has no real-
ity if it does not point to its opposition in disapproval, and the denial
has no meaning if it is not meant in relation to the affirmative. This
doubleness of our primary experience, this having of objects and of
antagonistic attitudes must be acknowledged wherever we speak of ex-
perience at all. We know no object without attitude, and no attitude
without object. The two are one state; object and attitude form
a unity which we resolve by the different way in which we experience
these two features of the one state: we find the object and we live
through the attitude. It is a different kind of awareness, the having
of the object and the taking of the attitude. In real life our will is
never an object which we simply perceive. The psychologist may treat
the will as such, but in the immediate experience of real life, we are
certain of our action by doing it and not by perceiving our doing; and
this our performing and rejecting is really our self which we posit as

106 THE SCIENTIFIC PLAN OF THE CONGRESS

absolute reality, not by knowing it, but by willing it. This corner-
stone of the Fichtean philosophy was forgotten throughout the un-
critical and unphilosophical decades of a mere naturalistic age. But
our time has finally come to give attention to it again.

Our pure experience thus contains will-attitudes and objects of will,
and the different attitudes of the will give the fundamental classes of
human activity. We can easily recognize four different types of will-
relation towards the world. Our will submits itself to the world; our
will approves the world as it is; our will approves the changes in the
world; our will transcends the world. Yet we must make at once one
more most important discrimination. We have up to this point sim-
plified our pure experience too much. It is not true that we experience
only objects and our own will-attitudes. Our will reaches out not only
to objects, but also to other subjects. In our most immediate experi-
ence, not reshaped at all by theoretical science, our will is in agree-
ment or disagreement with other wills; tries to influence them, and
receives influences and suggestions from them. The pseudo-philo-
sophy of naturalism must say of course that the will does not stand in
any direct relation to another will, but that the other persons are for
us simply material objects which we perceive, like other objects, and
into which we project mental phenomena like those which we flnd in
ourselves 'by the mere conclusion of analogy. But the complex recon-
structions of physiological psychology are therein substituted for the
primary experience. If we have to express the agreement or disagree-
ment of wills in the terms of causal science, we may indeed be obliged
to transform the real experience into such artificial constructions;
but in our immediate consciousness, and thus at the starting-point of
our theory of knowledge, we have certainly to acknowledge that we
understand the other person, accept or do not accept his suggestion,
agree or disagree with him, before we know anything of a difference
between physical and mental objects.

We cannot agree with an object. We agree directly with a will,
which does not come to us as a foreign phenomenon, but as a proposi-
tion which we accept or decline. In our immediate experience will
thus reaches will, and we are aware of the difference between our will-
attitude as merely individual and our will-attitude as act of agree-
ment with the will-attitude of other individuals. We can go still
further. The circle of other individuals whose will we express in our
own will-act may be narrow or wide, may be our friends or the nation,
and this relation clearly constitutes the historical significance of our
attitude. In the one case our act is a merely personal choice for
personal purposes without any general meaning; in the other case it
is the expression of general tendencies and historical movements. Yet
our will-decisions can have connections still wider than those with our
social community or our nation, or even with all living men of to-day.

THE FOUR THEORETICAL DIVISIONS 107

It can seek a relation to the totality of those whom we aim to acknow-
ledge as real subjects. It thus becomes independent of the chance
experience of this or that man, or this or that movement, which
appeals to us, but involves in an independent way the reference to
every one who is to be acknowledged as a subject at all. Such refer-
ence, which is no longer bound to any special group of historical in-
dividuals, thus becomes strictly over-individual. We can then dis-
criminate three stages: our merely individual wiU; secondly, our will
as bound by other historical individuals; and thirdly, our over-
individual will, which is not influenced by any special individual,
but by the general demands for the idea of a personality.

Each of those four great types of will-attitude which we insisted on
— that is, of submitting, of approving the given, of approving change,
and of transcending — can be carried out on these three stages, that
is, as individual act, as historical act, and as over-individual act.
And we may say at once that only if we submit and approve and
change and transcend in an over-individual act, do we have Truth
and Beauty and Morality and Conviction. If we approve, for instance,
a given experience in an individual will-act, we have simply personal
enjoyment and its object is simply agreeable; if we approve it in har-
mony with other individuals, we reach a higher attitude, yet one which
cannot claim absolute value, as it is dependent on historical considera-
tions and on the tastes and desires of a special group or a school or a
nation or an age. But if we approve the given object just as it is in an
over-individual will-act, then we have before us a thing of beauty,
whose value is not dependent upon our personal enjoyment as indi-
viduals, but is demanded as a joy forever, by every one whom we
acknowledge at all as a complete subject. In exactly the same way,
we may approve a change in the world from any individual point of
view: we have then to do with technical, practical achievements; or
we may approve it in agreement with others: we then enter into the
historical interests of our time. Or we may approve it, finally, in an
over-individual way, without any reference to any special person-
ality: then only is it valuable for all time, then only is it morally good.
And if our will is transcending experience in an individual way, it can
again claim no more than a subjective satisfaction furnished by any
superstition or hope. But if the transcending will is over-individual,
it reaches the absolute values of religion and metaphysics.

Exactly the same differences, finally, must occur when our will sub-
mits itself to experience. This submission may be, again, an individ-
ual decision for individual purposes; no absolute value belongs to it.
Or it may be again a yielding to the suggestions of other individuals;
or it may, finally, again be an over-individual submission, which seeks
no longer a personal interest. This submission is not to the authority
of others, and is without reference to any individual; we assume

108 THE SCIENTIFIC PLAN OF THE CONGRESS

that every one who is to share with us our world of experience has to
share this submission too. That alone is a submission to truth, and
experience, considered in so far as we submit ourselves to it over-
individually, constitutes our knowledge.

The system of knowledge is thus the system of experience with all
that is involved in it in so far as it demands submission from our over-
individual will, and the classification which we are seeking must be
thus a division and subdivision of our over-individual submissions.
But the submission itself can be of very different characters and these
various types must give the deepest logical principles of scientific
classification. To point at once to the fundamental differences: our
will acknowledges the demands of other wills and of objects. We can-
not live our life — and this is not meant in a biological sense, but,
first of all, in a teleological sense — our life becomes meaningless, if
our will does not respect the reality of will-demands and of objects of
will. Now we have seen that the will which demands our decision may
be either the individual will of other subjects or the over-individual
will, which belongs to every subject as such and is independent of any
individuality. We can say at once that in the same way we are led to
acknowledge that the object has partly an over-individual character,
that is, necessarily belongs to the world of objects of every possible
subject, and partly an individual character, as our personal object.
We have thus four large groups of experiences to which we submit
ourselves: over-individual will-acts, individual will-acts, over-indi-
vidual objects, individual objects. They constitute the first four large
divisions of our system.

The over-individual will-acts, which are as such teleologically bind-
ing for every subject and therefore norms for his will, give us the
Normative Sciences. The individual will-acts in the world of historical
manifoldness give us the Historical Sciences. The objects, in so far
as they belong to every individual, make up the physical world, and
thus give us the Physical Sciences; and finally the objects, in so far
as they belong to the individual, are the contents of consciousness,
and thus give us the Mental Sciences. We have then the demarca-
tion lines of our first four large divisions : the Normative, the Histor-
ical, the Physical, and the Mental Sciences. Yet their meaning and
method and difference must be characterized more fully. We must
understand why we have here to deal with four absolutely different
types of scientific systems, why the over-individual objects lead us
to general laws and to the determination of the future, while the study
of the individual will-acts, for instance, gives us the system of history,
which turns merely to the past and does not seek natural laws; and
why the study of the norms gives us another kind of system in which
neither a causal nor an historical, but a purely logical connection pre-
vails. Yet all these methodological differences result necessarily from

THE PHYSICAL AND THE MENTAL SCIENCES 109

the material with which these four different groups of sciences are
working.

Let us start again from the consideration of our original logical
purpose. We feel ourselves bound and limited in our will by physical
things, by psychical contents, by the demands of other subjects, and
by norms. The purpose of all our knowledge is to develop completely
all that is involved in this bondage. We want to develop in an over-
individual way all the obligations for our submission which are
necessarily included in the given objects and the given demands of
subjects. We start of course everywhere and in every direction from
the actual experience, but we expand the experience by seeking those
objects and those demands to which, as necessarily following from the
immediately given experience, we must also submit. And in thus
developing the whole system of submissions, the interpretation of
the experience itself becomes transformed: the physicist may per-
haps substitute imperceptible atoms for the physical object and the
psychologist may substitute sensations for the real idea, and the
historian may substitute combinations of influences for the real per-
sonality, and the student of norms may substitute combinations of
conflicting demands for the one complete duty; yet in every case the
substitution is logically necessary and furnishes us what we call truth
inasmuch as it is needed to develop the concrete system of our sub-
missions and thus to express our confidence in the order-lines of real-
ity. And each of these substitutions and supplementations becomes,
as material of knowledge, itself a part of the world of experience.

3. The Physical and the Mental Sciences

The physicist, we said, speaks of the world of objects in so far as
they belong to every possible subject, and are material for a merely
passive spectator. Of course the pure experience does not offer us any-
thing of that kind. We insisted that the objects of our real life are
objects of our will and of our attitudes, and are at the same time un-
differentiated into the physical things outside of us and the psychical
ideas in us. To reach the abstraction of the physicist, we have thus to
cut loose the objects from our will and to separate the over-individual
elements from the individual elements. Both transformations are
clearly demanded by our logical aims. As to the cutting loose from our
will, it means considering the object as if it existed for itself, as if it
were a mere passively given material and not a material of our per-
sonal interests. But just that is needed. We want to find out how
far we have to submit ourselves to the object. If we want to live our
life, we must adjust our attitudes to things, and, as we know our will,
we must seek to understand the other factor in the complex experi-
ence, the object of our will, and we must find out what it involves in

110 THE SCIENTIFIC PLAN OF THE CONGRESS

itself. But we do not understand the object and the submission which
it demands if we do not completely understand its relation to our
desires. Our total submission to the thing thus involves our acknow-
ledgment of aU that we have to expect from it. And although the
real experience is a unity of will and thing, we have thus the most
immediate interest in considering what we have to expect from the
thing in itself, without reference to our will. That means finding out
the effects of the given object with a subject as the passive spec-
tator. We eliminate artificially, therefore, the activity of the subject
and construct as presupposition for this circle of knowledge a nowhere
existing subject without activity, for which the thing exists merely
as a cause of the effects which it produces.

The first step towards natural science is, therefore, to dissolve
the real experience into thing and personality; that is, into object
and active subject, and to eliminate in an artificial abstraction the
activity of the subject, making the object material of merely passive
awareness, and related no longer to the will but merely to other
objects. It may be more difficult to understand the second step which
naturalism has to take before a natural science is possible. It must
dissolve the object of will into an over-individual and an individual
part and must eliminate the individual. That part of my objects
which belongs to me alone is their psychical side; that which belongs
to all of us and is the object of ever new experience is the physical
object. As a physicist, in the widest sense of the word, I have to ignore
the objects in so far as they are my ideas and have to consider the
stones and the stars, the inorganic and the organic objects, as they
are outside of me, material for every one. The logical purpose of this
second abstraction may be perhaps formulated in the following way.

We have seen that the purpose of the study of the objects is to find
out what we have to expect from them; that is, to what effects of the
given thing we have to submit ourselves in anticipation. The ideal
aim is thus to understand completely how present objects and future
objects — that is, how causes and effects — are connected. The first
stage in such knowledge of causal connections is, of course, the obser-
vation of empirical consequences. Our feeling of expectation grows
with the regularity of observed succession; yet the ideal aim can
never be fulfilled in that way. The mere observation of regularities
can help us to reduce a particular case to a frequently observed type,
but what we seek to understand is the necessity of the process. Of
course we have to formulate laws, and as soon as we acknowledge
a special law to be expressive of a necessity, the subsumption of the
particular case under the law will satisfy us even if the necessity of the
connection is not recognized in the particular case. We are satisfied
because the acknowledgment of the law involved all possible cases.
But we do not at all feel that we have furnished a real explanation if

THE PHYSICAL AND THE MENTAL SCIENCES 111

the law means to us merely a generalization of routine experiences,
and if thus no absolute validity is attached to the law. This necessity
between cause and effect must thus have its ultimate reason in our
own understanding. We must be logically obliged to connect the
objects in such a way, and wherever observation seems to contradict
that which is logically necessary, we must reshape our idea of the
object till the demands of reason are fulfilled. That is, we must sub-
stitute for the given object an abstraction which serves the purpose of
a logically necessary connection. That demand is clearly not satisfied
if we simply group the totality of such causal judgments under the
single name. Causality, and designate thus all these judgments as
results of a special disposition of the understanding. We never under-
stand why just this cause demands just this effect so long as we rely
on such vague and mystical power of our reason to link the world by
causality.

But the situation changes at once if we go still further back in the
categories of our understanding. While a mere demand for causality
never explains what cause is to be linked with what effect, the vague-
ness disappears when we understand this demand for causality itself
as the product of a more fundamental demand for identity. That an
object remains identical with itself does not need for us any further
interpretation. That is the ultimate presupposition of our thought,
and where a complete identity is found nothing demands further
explanation. All scientific effort aims at so rethinking different ex-
periences that they can be regarded as partially identical, and every
discovery of necessary connection is ultimately a demonstration of
identity. If we seek connections with the final aim to understand
them as necessary, we must conceive the world of our objects in such
a way that it is possible to consider the successive experiences as parts
of a self-identical world; that is, as parts of a world in which no sub-
stance and no energy can disappear or appear anew. To reach this end
it is obviously needed that we eliminate from the world of objects all
that cannot be conceived as identically returning in a new experience;
that is, all that belongs to the present experience only. We do elimin-
ate this by taking it up conceptually into the subject and calling it
psychical, and thus leaving to the object merely that which is con-
ceived as belonging to the world of everybody's experience, that is, of
over-individual experience. The whole history of natural science is
first of all the gigantic development of this transformation, resolution,
and reconstruction. The objects of experience are re-thought till
everything is eliminated which cannot be conceived as identical with
itself in the experiences of all iadividuals and thus as belonging to the
over-individual world. All the substitutions of atoms for the real thing,
and of energies for the real changes, are merely conceptional schemes
to satisfy this demand.

112 THE SCIENTIFIC PLAN OF THE CONGRESS

The logically primary step is thus not the separation of the physical
and the psychical things plus the secondary demand to connect the
physical things causally; the order is exactly opposite. The primary
desire is to connect the real objects and to understand them as causes
and effects. This understanding demands not only empirical observa-
tion, but insight into the necessary connection. Necessary connec-
tion, on the other hand, exists merely for identical objects and identi-
cal qualities. But in the various experiences only that is identical
which is independent of the momentary individual experiences, and
therefore we need as the ultimate aim a reconstruction of the object
into the two parts, the one perceptional, which refers to our individual
experience 5 and the other conceptional, which expresses that which
can be conceived as identical in every new experience. The ideal of
this constructed world is the mechanical universe in which every
atom moves by causal necessity because there is nothing in that
universe, no element of substance and no element of energy, which
will not remain identical in all changes of the universe which are pos-
sibly to be expected. It becomes completely determinable by antici-
pation and the system of our submissions to the object can be com-
pletely'constructed. The totality of intellectual efforts to reconstruct
such a causally connected over-individual world of objects clearly
represents a unity of its own. It is the system of physical sciences.

The physical universe is thus not the totality of our objects. It is a
substitution for our real objects, constructed by eliminating the indi-
vidual parts of our objects of experience. These individual parts are
the psychical aspects of our objective experience, and they clearly
awake our scientific interest too. The physical sciences need thus as
counterpart a division of mental sciences. Their aim must be the same.
We want to foresee the psychical results and to understand causally
the psychical experience. Yet it is clear that the plan of the mental
sciences must be quite different in principle from that of the sciences
of nature. The causal connection of the physical universe was ulti-
mately anchored in the identity of the object through various experi-
ences; while the object of experience was psychical for us just in so
far as it could never be conceived as identical in different phases of
reality. The psychical object is an ever new creation; my idea can
never be your idea. Their meaning may be identical, but the psych-
ical stuff, the content of my consciousness, can never be object for
any one else, and even in myself the idea of to-day is never the idea
of yesterday or to-morrow. But if there cannot be identity in different
psychical experiences, it is logically impossible to connect them
directly by necessity. If we yet want to master their successive
appearance, we must substitute an indirect connection for the direct
one, and must describe and explain the psychical phenomena through
reference to the physical world. It is in this way that modern psycho-

THE HISTORICAL AND THE NORMATIVE SCIENCES 113

logy has substituted elementary sensations for the real contents of
consciousness and has constructed relations between these element-
ary mental states on the basis of processes in the organism, especially
brain processes. Here, again, reality is left behind and a mere concep-
tional construction is put in its place. But this construction fulfills
its purpose and thus gives us truth; and if the basis is once given, the.
psychological sciences can build up a causal system of the conscious
processes in the individual man and in society.

4. The Historical^ and the Normative Sciences

The two divisions of the physical and mental sciences represent our
systematized submission to objects. But we saw from the first that it
is an artificial abstraction to consider in our real experience the object
alone. We saw clearly that we, as acting personalities, in our will and
in our attitudes, do not feel ourselves in relation to objects, merely, but
to will-acts ; and that these will-acts were the individual ones of other
subjects or the over-individual ones which come to us in our conscious-
ness of norms. The sciences which deal with our submissions to the
individual will-acts of others are the Historical Sciences. Their start-
ing-point is the same as that of the object sciences, the immediate
experience. But the other subjects reach our individuality from the
start in a different way from the objects. The wills of other subjects
come to us as propositions with which we have to agree or disagree ;
as suggestions, which we are to imitate or to resist; and they carry in
themselves that reference to an opposite which, as we saw, character-
izes all will-activity. The rock or the tree in our surroundings may
stimulate our reactions, but does not claim to be in itself a decision
with an alternative. But the political or legal or artistic or social or
religious will of my neighbors not only demands my agreement or
disagreement, but presents itself to me in its own meaning as a free
decision which rejects the opposite, and its whole meaning is de-
stroyed if I consider it like the tree or the rock as a mere phenom-
enon, as an object in the world of objects. Whoever has clearly
understood that politics and religion and knowledge and art and law
come to me from the first quite differently from objects, can never
doubt that their systematic connection must be most sharply sepa-
rated from all the sciences which connect impressions of objects, and
is falsified if the historical disciplines are treated simply as parts of
the sciences of phenomena — for instance, as parts of sociology, the
science of society as a psycho-physical object.

Just as natural science transcends the immediately experienced
object and works out the whole system of our necessary submissions
to the world of objects, so the historical sciences transcend the social
will-acts which approach us in our immediate experience, and again

114 THE SCIENTIFIC PLAN OF THE CONGRESS

seek to find what we are really submitting to if we accept the sugges-
tions of our social surroundings. And yet this similar demand has
most dissimilar consequences. We submit to an object and want to
find out what we are really submitting to. That cannot mean any-
thing else, as we have seen, than to seek the effects of the object and
thus to look forward to what we have to expect from the object.
On the other hand, if we want to find out what we are really sub-
mitting to if we agree with the decision of our neighbor, the only
meaning of the question can be to ask what our neighbor really is
deciding on, what is contained in his decision; and as his decision
must mean an agreement or disagreeme"ht with the will-act of another
subject, we cannot understand the suggestion which comes to us
without understanding in respect to what propositions of others it
takes a stand. Our interest is in this case thus led from those sub-
jects of will which enter into our immediate experience to other sub-
jects whose purposes stand in the relation of suggestion and demand
to the present ones. And if we try to develop the system of these
relations, we come to an endless chain of will-relations, in which one
individual will always points back in its decisions to another indi-
vidual will with which it agrees or disagrees, which it imitates or
overcomes by a new attitude of will; and the whole network of these
will-relations is the political or religious or artistic or social history
of mankind. This system of history as a system of teleologically
connected will-attitudes is elaborated from the will-propositions
which reach us in immediate experience, with the same necessity
with which the mechanical universe of natural science is worked out
from the objects of our immediate experience.

The historical system of will-connections is similar to the system of
object-connections, not only in its starting in the immediate experi-
ence, but further in its also seeking identities. Without this feature
history would not offer to our understanding real connections. We
must link the will-attitudes of men by showing the identity of the
alternatives. Just as the physical thing is substituted by a large
number of atoms which remain identical in the causal changes, in
the same way the personality is substituted by an endless manifold-
ness of decisions and becomes linked with the historical community
by the thought that each of these partial decisions refers to an alter-
native which is identical with that of other persons. And yet there
remains a most essential difference between the historical and the
causal connection. In a world of things the mere identical continu-
ity is sufficient to determine the phenomena of any given moment.
In a world of wiU the identity of alternatives cannot determine be-
forehand the actual decision ; that belongs to the free activity of the
subject. If this factor of freedom were left out, man would be made
an object and history a mere appendix of natural science. The

THE HISTORICAL AND THE NORMATIVE SCIENCES 115

connection of the historian can therefore never be a necessary one,
however much we may observe empirical regularities. If there were
no identities, our reason could not find connection in history; but if the
historical connections were necessary, like the causal ones, it would
not be history. The historian is, therefore, unable and without the
ambitien to look into the future like the naturalist; his domain is
the past.

Yet will-attitudes and will-acts can also be brought into necessary
connection; that is, we can conceive will-acts as teleologically iden-
tical with each other and exempt from the freedom of the individual.
That is clearly possible only if they are conceived as beyond the free-
dom of individual decision and related to the over-individual subject.
The question is then no longer how this special man wills and decides,
but how far a certain will-decision binds every possible individual who
performs this act if he is to share our common world of will and mean-
ing. Such an over-individual connection of will-acts is what we call
the logical connection. It shares with all other connections the depend-
ence upon the category of identity. The logical connection shows
how far one act or combination of acts involves, and thus is partially
identical with, a new combination. This logical connection has, in
common with the causal connection, necessity; and in common with
the historical connection, teleological character. Any individual will-
act of historical life may be treated for certain purposes as such a
starting-point of over-individual relations; it would then lead to that
scientific treatment which gives us an interpretation, for instance, of
law. Such interpretative sciences belong to the system of history in
the widest sense of the word.

The chief interest, however, must belong to the logical connections
of those will-acts which themselves have over-individual character.
A merely individual proposition can lead to necessary logical connec-
tion, but cannot claim that scientific importance which belongs to
the logical connection of those propositions which are necessary for
the constitution of every real experience : the science of chess cannot
stand on the same level with the science of geometry, the science of
local legal statutes not on the same level with the system of ethics.
The logical connections of the over-individual attitudes thus consti-
tute the fourth large division besides the physical, the mental, and the
historical sciences. It must thus comprise the systems of all those
propositions which are presuppositions of our common reality, in-
dependent of the free individual decision. Here belong the acts of
approval — the ethical approval of changes and achievements, as
weU as the aesthetic approval of the given world ; the acts of convic-
tion — the religious convictions of a superstructure of the world as
well as the metaphysical convictions of a substructure; and above
all, the acts of affirmation and submission, the logical as well as the

116 THE SCIENTIFIC PLAN OF THE CONGRESS

mathematical. But to be consistent we must really demand that
merely the over-individual logical connections are treated in this
division. If we deal, for instance, with the sesthetical or ethical acts as
psychological experiences, or as historical propositions, they belong
to the psychical or historical division. Only the philosophical system
of ethics or aesthetics finds its place in this division. It is difficult to
find a suitable name for this whole system of logical connections of
over-individual attitudes. Perhaps it would be most correct to call it
the Sciences of Values, inasmuch as every one of these over-individual
decisions constitutes a value in our world which our individual will
finds as an absolute datum like the objects of experience. Seen from
another point of view, these values appear as norms which bind our
practical will inasmuch as these absolute values demand of our will to
realize them, and it may thus be permitted to designate this whole
group of sciences as a Division of Normative Sciences.

Our logical explanation of the meaning of these four divisions
naturally began with the interpretation of that science which usually
takes precedence in popular thought — with the science of nature,
that is, and passed then to those groups whose methodological situa-
tion is seen rather vaguely by our positivistic age. But as soon as we
have once defined and worked out the boundary lines of each of these
four divisions, it would appear more logical to change their order and
to begin with that division whose material is those over-individual
will-acts on which all possible knowledge must depend, and then to
turn to those individual will-acts which determine the formulation
of our present-day knowledge, and then only to go to the objects of
knowledge, the over-individual and the individual ones. In short, we
must begin with the normative sciences, consider in the second place
the historical sciences, in the third place the physical sciences, and
in the fourth place the psychical sciences. There cannot be a scientific
judgment which must not find its place somewhere in one of these
four groups. And yet can we really say that these four great divisions
complete the totality of scientific efforts? The plan of our Congress
contains three important divisions besides these.

5. The Three Divisions of Practical Sciences

The three divisions which still lie" before us represent Practical
Knowledge. Have we a logical right to put them on an equal leve.l
with the four large divisions which we have considered so far? Might it
not rather be said that all that is knowledge in those practical sciences
must find its place somewhere in the theoretical field, and that every-
thing outside of it is not knowledge, but art ? It cannot be denied
indeed that the logical position of the practical sciences presents seri-
ous problems. That the function of the engineer or of the physician,

THE THREE DIVISIONS OF PRACTICAL SCIENCES 117

of the lawyer or of the minister, of the diplomat or of the teacher,
contains elements of an art cannot be doubted. They all need not
only knowledge, but a certain instinct and power and skill, and their
schooling thus demands a training and discipline through imitation
which cannot be substituted by mere learning. Yet when it comes to
the classification of sciences, it seems very doubtful whether practical
sciences have to be acknowledged as special divisions, inasmuch as
the factor of art must have been eliminated at the moment they are
presented as sciences. The auscultation of the physician certainly
demands skill and training, yet this practical activity itself does not
enter into the science of medicine as presented in medical writings.
As soon as the physician begins to deal with it scientifically, he
needs, as does any scholar, not the stethoscope, but the pen. He
must formulate judgments; and as soon as he simply describes and
analyzes and explains and interprets his stethoscopic experiences,
his statements become a system of theoretical ideas.

We can say in general that the science of medicine or of engineering,
of jurisprudence or of education, contains, as science, no element of art,
but merely theoretical judgments which, as such, can find their place
somewhere in the complete systems of the theoretical sciences. If the
physician describes a disease, its symptoms, the means of examining
them, the remedies, their therapeutical effects, and the prophylaxis,
in short, everything which the physician needs for his art, he does not
record anything which would not belong to an ideally complete de-
scription and explanation of the processes in the human body. In the
same way it can be said that if the engineer characterizes the con-
ditions under which an iron bridge will be safe, it is evident that he
cannot introduce any facts which would not find their logical place in
an ideally complete description of the properties of inorganic nature;
and finally, the same is true for the statements of the politician, the
jurist, the pedagogue, or the minister. Whatever is said about their
art is a theoretical judgment which connects facts of the ideally
complete system of theoretical science; in their case the facts of
course belong in first line to the realm of the psychological, his-
torical, and normative sciences. There never has been or can be
practical advice in the form of words which is not in principle a state-
ment of facts which belong to the absolute totality of theoretical
knowledge. Seen from this point of view, it is evident that all our
knowledge is fundamentally theoretical, and that the conception of
practical knowledge is logically unprecise.

But the opposite point of view might also be taken. It might be
said that after all every kind of knowledge is practical, and our own
deduction of the meaning of science might be said to suggest such
interpretation. We acknowledged at the outset that the so-called
theoretical knowledge is by no means a passive mirror-picture of an

118 THE SCIENTIFIC PLAN OF THE CONGRESS

independent outside world; but that in every judgment real expe-
rience is remoulded and reshaped in the service of certain purposes of
will. Here lies the true core of that growing popular philosophy
of to-day which, under the name of pragmatism, or under other titles,
mingles the purposive character of our knowledge and the evolution-
ary theories of modern biology in the vague notion that men created
knowledge because the biological struggle for existence led to such
views of the world; and that we call true that correlation of our
experiences which has approved itself through its harmony with
the phylogenetic development. Certainly we must reject such circle
philosophies. We must see clearly that the whole conception of a
biological development and of a struggle of organisms is itself only
a part of our construction of causal knowledge. We must have know-
ledge to conceive ourselves as products of a phylogenetic history, and
thus cannot deduce from it the fact, and, still less, the justification
of knowledge. Yet one element of this theory remains valuable:
knowledge is indeed a purposive activity, a reconstruction of the
world in the service of ideals of the will. We have thus from one side
the suggestion that all knowledge is merely theoretical, from the other
side the claim that all knowledge is practical activity. It seems as if
both sides might agree that it is superfluous and unjustified to make
a demarcation line through the field of knowledge and to separate
two sorts of knowledge, theoretical and practical. For both theories
demand that all knowledge be of one kind , and they disagree only as
to whether we ought to call it all theoretical or all practical.

Yet the true situation is not characterized by such an antithesis.
If we say that all knowledge is ultimately practical, we are speaking
from an epistemological point of view, inasmuch as we take it then as
a reconstruction of the world through the purposive activity of the
over-individual subject. On the other hand it is an empirical point of
view from which ultimately all knowledge, that of the physician and
engineer and lawyer, as well as that of the astronomer, appears theo-
retical. But this antithesis can, therefore, not decide the further
empirical question, whether or not in the midst of theoretical know-
ledge two kinds of sciences may be discriminated, of which the one
refers to empirical practical purposes and the other not. Such an
inquiry would have nothing to do with the epistemological problem of
pragmatism ; it would be strictly non-philosophical, just as the separa-
tion of chemistry into organic and inorganic chemistry. This empir-
ical question is indeed to be answered in the affirmative. If we ask
what causes bring about a certain effect, for the sake of a practical
purpose of ours, — for instance, the curing a patient of a disease, — no
one can state facts which are not in principle to be included in the
complete system of physical causes and effects and thus in the system
of physical sciences. And yet it may well be that the physical sciences,

THE THREE DIVISIONS OF PRACTICAL SCIENCES 119

as such, have not the slightest reason to mention the effect of that
special drug on that special pathological alteration of the tissues of
the organism. The descriptions and explanations of science are not a
mere heaping up of material, but a steady selection in the interest of
the special aim of the science. No physical science describes every
special pebble on the beach; no historical science deals with the chance
happenings in the daily life of any member of the crowd. And we
already well know the point of view from which the selection is to be
performed. We want to know in the physical and psychical sciences
whatever is involved in the object of our experience, and in the his-
torical and normative sciences whatever is involved in the demands
which reach our will. But whether we have to do with the objects or
with the demands, in both cases we have systems before us which are
determined only by the objects or demands themselves, without any
relation to our individual will and our own practical activity. Theo-
retically, of course, our will, our activity, our organism, our person-
ality is included in the complete system; and if we knew absolutely
everything of the empirical effects of the object or of the consequences
of these demands, we should find among them their relation to our
individual interests; but that relation would be but one chance
case among innumerable others, and the sciences would not have the
slightest interest in giving any attention to that particular case. Thus
if our knowledge of chemical substances were complete, we should
certainly have to know theoretically that a few grains of antipyrine
introduced into the organism have an influence on those brain centres
which regulate the temperature of the human body. Yet if the chem-
ist does not share the interest of the physician who wants to fight
a fever, he would have hardly any reason for examining this particular
relation, as it hardly throws light on the chemical constitution as
such. In this way we might say in general that the relation of the
world to us as acting individuals is in principle contained in the total
system of the relations of our world of experience, but has a strictly
accidental place there and can never be in itself a centre around which
the scientific data are clustered, and science will hardly have an inter-
est in giving any attention to its details.

This relation of the world, the phj^sical, the psychical, the histor-
ical, and the normative world, to our individual, practical purposes
can, however, indeed become the centre of scientific interest, and it is
evident that the whole inquiry receives thereupon a perfectly new
direction which demands not only a completely new grouping of facts
and relations, but also "a very different shading in elaboration. As
long as the purpose was to understand the world without relation to
our individual aims, science had to gather endless details which are
for us now quite indifferent, as they do not touch our aims; and in
other respects science was satisfied with broad generalizations and

120 THE SCIENTIFIC PLAN OF THE CONGRESS

abstractions where we have now to examine the most minute details.
In short, the shifting of the centre of gravity creates perfectly new
sciences which must be distinguished ; and if we call them again theo-
retical and practical sciences, it is clear that this difference has then
no longer anything to do with the philosophical problems from which
we started.

The term practical may be preferable to the other term which is
sometimes used : Applied Science. If we construct the antithesis of
theoretical and applied science, the underlying idea is clearly that we
have to do on the practical side with a discipline which teaches how
to apply a science which logically exists as such beforehand. Engin-
eering, for instance, is an applied science because it applies the
science of physics; but this is not reallj^ our deepest meaning here.
Our practical sciences are not meant as mere applications of theo-
retical sciences. They are logically somewhat degraded if they are
treated in such a way. Their real logical meaning comes out only if
they are acknowledged as self-dependent sciences whose material is
differentiated from that of the theoretical sciences by the different
point of view and purpose. They are methodologically perfectly inde-
pendent, and the fact that a large part or theoretically even every-
thing of their teaching overlaps the teaching of certain theoretical
sciences ought not to have any influence on their logical standing.
The practical sciences could be conceived as completely self-depend-
ent, without the existence of any so-called theoretical sciences;
that is, the relations of the world of experience to our individual
aims might be brought into complete systems without working out in
principle the system of independent experience. We might have a
science of engineering without acknowledging an independent science
of theoretical physics besides it. To be sure, such a science of engin-
eering would finally develop itself into a system which would con-
tain very much that might just as well be called theoretical physics;
yet all would be held together by the point of view of the engineer,
and that part of theoretical physics which the engineer applies might
just as well be considered as depracticalized engineering. If this
logical self-dependence of the practical science holds true even for
such technological disciplines, it is still more evident that it would
cripple the meaning and independent character of jurisprudence and
social science, or of pedagogy and theology, to treat them simply as
applied sciences, that is, as applications of theoretical science.

This point of view determines, also, of course, the classification of
the Practical Sciences. If they were really merely applied sciences
it would be most natural to group them according to the classification
of the theoretical sciences which are to be applied. We should then
have applied physical sciences, applied psychological sciences, applied
historical sciences, and applied normative sciences. Yet even from the

THE THREE DIVISIONS OF PRACTICAL SCIENCES 121

standpoint of practice, we should come at once into difficulties, and
indeed much of the superficiality of practical sciences to-day results
from the hasty tendency to consider them as applied sciences only,
and thus to be determined by the points of view of the theoretical dis-
cipline which is to be applied. Then, for instance, pedagogy becomes
simply applied psychology, and the psychological point of view is
substituted for the educational one. Pedagogy then becomes simply
a selection of those chapters in psychology which deal with the mental
functions of the child. Yet as soon as we really take the teachers'
point of view, we understand at once that it is utterly artificial to sub-
stitute the categories of the psychologist for those of immediate
practical will-relations and to consider the child in the class-room as
a causal system of pyscho-physical elements instead of a personality
which is teleologically to be interpreted, and whose aims are not to be
connected with causal effects but with over-individual attitudes. In
this way the historical relation and the normative relation have to
play at least as important a role in the pedagogical system as the
psycho-physical relation, and we might quite as well call education
applied history and applied ethics.

Almost every practical science can be shown in this way to apply
a number of theoretical sciences; it synthesizes them to a new unity.
But better, we ought to say, that it is a unity in itself from the start,
and that it only overlaps with a number of theoretical sciences. If
we want to classify the practical sciences, we have thus only the one
logical principle at our disposal : we must classify them in accordance
with the group of human individual aims which control those dif-
ferent disciplines. If all practical sciences deal with the relation of
the world of experience to our individual practical ends, the classes of
those ends are the classes of our practical sciences, whatever combina-
tions of applied theoretical sciences may enter into the group. Of
course a special classification of these aims must remain somewhat
arbitrary ; yet it may seem most natural to separate three large divi-
sions. We called them the Utilitarian Sciences, the Sciences of Social
Regulation, and the Sciences of Social Culture. Utilitarian we may
call those sciences in which our practical aim refers to the world of
things; it maybe the technical mastery of nature or the treatment
of the body, or the production, distribution, and consumption of the
means of support. The second division contains everything in which
our aim does not refer to the thing, but to the other subjects; here
naturally belong the sciences which deal with the political, legal, and
social purposes. And finally the sciences of culture refer to those aims
in which not the individual relations to things or to other subjects are
in the foreground, but the purposes of the teleological development of
the subject himself; education, art, and religion here find their place.
It is, of course, evident that the material of these sciences frequently

122 THE SCIENTIFIC PLAN OF THE CONGRESS

allows the emphasis of different aspects. For instance, education,
which aims primarily at self-development, might quite well be con-
sidered also from the point of view of social regulation; and still
more naturally could the utilitarian sciences of the economic distri-
bution of the means of support be considered from this point of
view. Yet a classification of sciences nowhere suggests by its
boundary lines that there are no relations and connections between
the different parts; on the contrary, it is just the manifoldness of
these given connections which makes it so desirable to become con-
scious of the principles involved, and thus to emphasize logical
demarcation lines, which of course must be obliterated as soon as
any material is to be treated from every possible point of view. It may
thus well be that, for instance, a certain industrial problem could be
treated in the Normative Sciences from the point of view of ethics; in
the Historical Sciences, from the point of view of the history of
economic institutions; in the Physical Sciences, from the point of
view of physics or chemistry; in the Mental Sciences, from the point
of view of sociology; in the Utilitarian Sciences, from the point of
view of medicine or of engineering, or of commerce and transporta-
tion; and finally in the Regulative Sciences, from the point of view of
political administration, or in the Social Sciences, from the standpoint
of the urban community, and so on. The more complex the relations
are, the more necessary is it to make clean distinctions between the
different logical purposes with which the scientific inquiries start.
Practical life may demand a combination of historical, sociological,
psychological, economical, social, and ethical considerations; but not
one of these sciences can contribute its best if the consciousness of
these differences is lost and the deliberate combination is replaced by
a vague mixture of the problems.

6. The Subdivisions

We have now before us the ground-plan of the scheme, the four
theoretical divisions, and the three practical divisions; every addi-
tional comment on the classification must be of secondary importance,
as it has to refer to the smaller subdivisions, which cannot change the
principles of the plan, and which have not seldom, indeed, been a re-
sult of practical considerations. If, for instance, our Division of Cul-
tural Sciences shows in the final plan merely the departments of
Education and of Religion, while the originally planned Department
of Art is left out, there was no logical reason for it, but merely the
practical ground that it seemed difficult to bring such a practical art
section to a desirable scientific level; we confine art, therefore, to
the normative aesthetic and historical points of view. Or, to choose
another illustration, if it happened that the normative sciences were

THE SUBDIVISIONS 123

finally organized without a section for the philosophy of law, this re-
sulted from the fact that the American jurists, in contrast with their
Continental European colleagues, showed a general lack of appre-
ciation for such a section. A few sections had to be left out even for
the chance reason that the leading speakers were obliged to with-
draw at a time when it was too late to ask substitutes to work up
addresses. And almost everywhere there had to be something arbi-
trary in the limitation of the special sections. Though Otology and
Laryngology were brought together into one section, they might just
as well have been placed in two ; and Rhinology , which was left out,
might have been added as a third in that company. As to this sub-
tler ramification, the plan has been changed several times during the
period of the practical preparation of the plan, and much is the result
of adjustment to questions of personalities. No one claims, thus,
any special logical value for the final formulation of the sectional
details, for which our chief aim was not to go beyond eight times
sixteen, that is 128, sections, inasmuch as it was planned to have
the meetings at eight different time-periods in sixteen different halls.
If we had fulfilled all the wishes which were expressed by specialists,
the number would have been quickly doubled.

Yet a few remarks may be devoted to the branching off within the
seven divisions, as a short discussion of some of these details may
throw additional light on the general principles of the whole plan. If
we thus begin with the Normative Sciences, we stand at once before
one feature of the plan which has been in an especially high degree
a matter of both approval and criticism : the fact that Mathematics
is grouped with Philosophy. The Division was to contain, as we have
seen, the systems of logically connected wiU-acts of the over-individ-
ual subject. That Ethics or Logic or ^Esthetics or Philosophy of
Religion deals with such over-individual attitudes cannot be doubted;
but have we a right to coordinate the mathematical sciences with
these philosophical sciences? Has Mathematics not a more natural
place among the physical sciences coordinated with and introductory
to Mechanics, Physics, and Astronomy? The mathematicians them-
selves would often be inclined to accept without hesitation this neigh-
borhood of the physical sciences. They would say that the mathe-
matical objects are independent realities whose properties we study
like those of nature, whose relations we "observe," whose existence
we "discover," and in which we are interested because they belong to
the real world. All this is true, and yet the objects of the mathema-
tician are objects made by the logical will only, and thus different
from all phenomena into which sensation enters. The mathema-
tician, of course, does not reflect on the purely logical origin of the
objects which he studies, but the system of knowledge must give to
the study of the mathematical objects its place in the group where the

124 THE SCIENTIFIC PLAN OF THE CONGRESS

functions and products of the over-individual attitudes are classified.
The mathematical object is a free creation, and a creation not only
as to the combination of elements — that would be the case with
many laboratory substances of the chemist too — but a creation as to
the elements themselves, and the value of that creation, its " mathe-
matical interest," is to be judged by ideals of thought; that is, by
logical purposes. No doubt this logical purpose is its application in
the world of objects and the mathematical concepts must thus fit the
objective world so absolutely that mathematics can be conceived as a
description of the world after abstracting not only from the will-rela-
tions, as physics does, but also from the content. Mathematics would,
then, be the phenomenalistic science of the form and order of the
world. In this way, mathematics has indeed a claim to places in both
divisions: among the physical sciences if we emphasize its applica-
bility to the world, and among the teleological sciences if we empha-
size the free creation of the objects by the logical will. But if we really
go back to epistemological principles, our system has to prefer the
latter emphasis; that is, we must coordinate mathematics with logic
and not with physics.

As to the subdivision of philosophy, it is most essential for us to
point to the negative fact that of course psychology cannot have a
place in the philosophical department, as part of the Normative Divi-
sion. There is perhaps no science whose position in the system of
knowledge offers so many methodological difficulties as psychology.
Historical tradition of course links it with philosophy; throughout a
great part of its present endeavors it is, on the other hand, linked with
physiology. Thus we find it sometimes coordinated with logic and
ethics, and sometimes, especially in the classical positivistic systems,
coordinated with the sciences of the organic functions. We have seen
why a really logical treatment has to disregard those historical and
practical relations and has to separate the psychological sciences from
the philosophical and the biological sciences. Yet even this does
not complete the list of problems which must be settled, inasmuch
as modern thinkers have frequently insisted that psychology itself
allows a twofold aspect. We can have a psychology which describes
and explains the mental life by analyzing it into its elements and by
connecting these elements through causality. But there may be
another psychology which treats inner life in that immediate unity in
which we experience it and seeks to interpret it as the free function
of personality. This latter kind of psychology has been called volun-
taristic psychology as against the phenomenalistic psychology which
seeks description and explanation. Such voluntaristic psychology
would clearly belong again to a different division. It would be a
theory of individual life as a function of will, and would thus be
introductory to the historical sciences and to the normative sciences

THE SUBDIVISIONS 125

too. Yet we left out this teleological psychology from our programme,
as such a science is as yet a programme only. Wherever an effort is
made to realize it, it becomes an odd mixture of an inconsistent phe-
nomenalistic psychology on the one side, and philosophy of history,
logic, ethics, and aesthetics on the other side. The only science which
really has a right to call itself psychology is the one which seeks to
describe and to explain inner life and treats it therefore as a system
of psychical objects, that is, as contents of consciousness, that is, as
phenomena. Psychology belongs, then, in the general division of
psychical sciences as over against physical sciences, and both deal
with objects as over against philosophy and history, which deal with
subjects of will.

The subdivision of the Historical Sciences offers no methodological
difficulty as soon as those epistemological arguments are acknow-
ledged by which we sharply distinguish history from the Physical
and Mental Sciences. If history is a system of will-relations which
is in teleological connection with the will-demands that surround us,
then political history loses its predominant role, and the history of
law and of literature, of language and of economy, of art and relig-
ion, become coordinated with political development, while the mere
anthropological aspect of man is relegated to the physical sciences.
The more complete original scheme was here again finally condensed
for practical reasons; for instance, the planned departments on the
History of Education, on the History of Science, and on the History
of Philosophy were sacrificed, and the department of Economic His-
tory was joined to that of Political History. In the same way we felt
obliged to omit in the end many important sections in the depart-
ments; we had, for instance, in the History of Language at first a sec-
tion on Slavic Languages ; yet the number of scholars interested was
too small to justify its existence beside a section on Slavic Literature.
Also the History of Music was omitted from the History of Art; and
the History of Law was planned at first with a fuller ramification.

The division of Physical Sciences naturally suggested that kind of
subdivision which the positivistic classification presents as a com-
plete system of sciences. Considering physics and chemistry as the
two fundamental sciences of general laws, we turn first to astronomy,
then from the science of the whole universe to the one planet, to the
sciences of the earth; thence to the living organisms on the earth; and
from biology to the still narrower circle of anthropology. The special
classification of physics offers a certain difficulty. To divide it in text-
book fashion into sound, light, electricity, etc., seems hardly in har-
mony with the effort to seek logical principles in the other parts of the
classification. The three groups which we finally formed, Physics of
Matter, Physics of Ether, and Physics of Electron, may appear some-
what too much influenced by the latest theories of to-day, yet it

126 THE SCIENTIFIC PLAN OF THE CONGRESS

seemed preferable to other principles. In the biological department,
criticism seems justified in view of the fact that we constructed
a special section, Human Anatomy. A strictly logical scheme might
have acknowledged that human anatomy is to-day not a separate
science, and that it has resolved itself into comparative anatomy.
Sections of Invertebrate and Vertebrate Anatomy might have been
more satisfactory. The final arrangement was a concession to the
practical interests of the physicians, who have naturally to emphasize
the anatomy of the human organism.

In the division of Mental Sciences, we have the Department of
Sociology. We were, of course, aware that the sociological interest
includes not only the psychological, but also the physiological life
of society, and that it thus has relations to the physical sciences
too. Yet these relations are logically not more fundamental than
those of the individual mental life to the functions of the indi-
vidual organism. Much of the physiological side was further to
be handed over to the Department of Anthropology, and thus we
felt justified in grouping sociology with psychology under the Men-
tal Sciences, as the psychology of the social organism. Here, too,
a larger number of sections was intended and only the two most
essential ones, Social Structure and Social Psychology, were finally
admitted.

The ramifications of the practical sciences had to follow the general
principle that their character is determined by purpose and not by
material. The difficulty was here merely in the extreme specialization
of the practical disciplines, which suggests on the whole the forming of
very small units, while our plan was to provide for fifty practical sec-
tions only. It seemed, therefore, incongruous to have the whole of
Internal Medicine or the whole of Private Law condensed into one
section. Yet as the purpose of the scheme was a theoretical and not a
practical one, even where the theory of practical sciences was in ques-
tion, we felt justified in constructing coordinated sections, even where
the practical importance was very unequal. On the other hand, some
glaring defects just here are due merely to chance circumstances.
That there were, for instance, no sections on Criminal Law or Eccle-
siastical Law in the Department of Jurisprudence, nor on Legal Pro-
cedure, resulted from the unfortunate accident that in these cases the
speakers who were to come from Europe were withheld by illness or
public duties. The absence of the Department of Art in the Division
of Social Culture, and thus of the Sections on the theory and practice
of the different arts, has been explained before. It is evident that
also in the Economical Department the practical development has
interfered with the original symmetrical arrangement of the sec-
tions. This is not true of the Religious Department, whose six
sections express the tendencies of the original plan. The fre-

THE RESULTS OF THE CONGRESS 127

quently expressed criticism that the different rehgions and their
denominations ought to have found place there shows a mis-
conception of our purpose; a Parliament of Religion did not belong
to this plan.

Ill

THE RESULTS OF THE CONGRESS

The programme of the Congress, as outlined in the previous
pages, was in this case somewhat more than a mere programme. It
not onl}^ invited to do a piece of work, but it sought to contribute to
the work itself. Yet the chief work had to be done by others, and
their part needed careful preparation. Yet very little of the prepar-
ation showed itself to the eyes of the larger public, and few were fully
aware what a complex organization was growing up and how many
persons of mark were cooperating.

It was essential to find for every address the best man. Specialists
only could suggest to the committees where to find him. It has been
told before how our invitations were brought to the foreigners first
till the desired number of foreign participants was secured, and how
the Americans followed. As could not be otherwise expected, interfer-
ences of all kinds disturbed the ideal configuration of the first list of
acceptances; substitutes had sometimes to be relied on; and yet,
when on the nineteenth of September President Francis welcomed the
Congress of Arts and Science in the gigantic Festival Hall of the St.
Louis Exposition, the Committee knew that almost four hundred
speakers had completed their manuscripts, and that it was a galaxy
which far surpassed in importance that of any previous international
congress. And the list of those who stood for the success of the work
was not confined to the official speakers. Each Department and each
Section had its own honorary President, who was also chosen by the
consent of leading specialists and whose introductory remarks were to
give additional importance to the gathering. At their side stood the
hundred and thirty Secretaries, carefully chosen from among the pro-
ductive scholars of the younger generation. And a large number of
informal, yet officially invited contributors, had announced valuable
discussions and addresses for almost every Section. Invitations to
membership finally had been sent to the universities and scholarly
societies of all countries.

That the turmoil of a world's fair is out of harmony with the
scholar's longing for repose and quietude is a natural presupposition,
which has not been disproved by the experience of St. Louis. When
Professor Newcomb, our President, spoke to the opening assembly on
the dignity of scholarship, the scholar's peaceful address was accentu-

128 THE SCIENTIFIC PLAN OF THE CONGRESS

ated by the thunder of the cannons with which Boer and British
forces were playing at war near by. The roaring of the Pike over-
powered many a quiet session, and the patient speaker had not seldom
to fight heroically with a brass band on the next lawn. The trains
were delayed, trunks were mixed up, and the sultry St. Louis weather
stirred much secret longing for the seashore and the mountains, which
most had to leave too early for that pilgrimage to the Mississippi
Valley. Yet all this could have been easily foreseen, and every one
knew that all this would soon be forgotten. These slight discomforts
were many times made up for by the overwhelming beauty of that
ivory city in which the civilization of the world was focused by the
united energy of the nations, and it seemed well worth while to cross
the ocean for the delight of that enchantment which came with every
evening's myriad illumination. And every day brought interesting
festivities. No one will forget the receptions of the foreign commis-
sioners, or the charming hospitality of the leading citizens of St. Louis,
or the enthusiastic banquet which brought one thousand speakers
and presidents and official members of the Congress together as guests
of the master mind of the Exposition, President Francis.

While the discomfort of external shortcomings was thus easily bal-
anced, it is more doubtful whether the internal shortcomings of the
work can be considered as fully compensated for. It would be impos-
sible to overlook these defects in the realization of our plans, even if it
may be acknowledged that they were unavoidable under the given
conditions. The principal difficulty has been that many speakers
have not really treated the topic for the discussion of which they were
invited. This deviation from the plan took various forms. There was
in some cases a fundamental attitude taken which did not harmonize
with those logical principles which had led to the classification; for
instance, we had sharply separated, for reasons fully stated above,
the Division of History from the Division of Mental Sciences, includ-
ing sociology; yet some papers for the Division of History clearly
indicated sympathy with the traditional positivistic view, according
to which history becomes simply a part of sociology. And sunilar
variations of the general plan occur in almost every division. But
there cannot be any objection to this secondary variety as long as the
whole framework gives the primary uniformity. Certainly no ore of
the contributors is to be blamed for it; no one was pledged to the
philosophy of the general plan, and probably few would have agreed
if any one had had the idea of demanding from every contributor an
identical background of general convictions. Such monotony would
have been even harmful, as the work would have become inexpressive
of the richness of tendencies in the scholarly life of our time. This was
not an occasion where educated clerks were to work up in a second-
hand way a report whose general trend was determined beforehand;

THE RESULTS OF THE CONGRESS 129

the work demanded original thinkers, with whom every word grows
out of a rich individual view of the totality. If every paper had been
meant merely as a detailed amplification of the logical principles
on which the whole plan was based, it would have been wiser to set
young Doctor candidates to work, who might have elaborated the
hint of the general scheme. To invite the leaders of knowledge meant
to give them complete freedom and to confine the demands of the plan
to a most general direction.

The same freedom, which every one was to have as to the general
standpoint, was intended also for all with regard to the arrangement
and limitation of the topic. All the sectional addresses were supposed
to deal either with relations or with fundamental problems of to-day.
It would have been absurd to demand that in every case the totality
of relations or of problems should be covered or even touched. The
result would have become perfunctory and insignificant. No one
intended to produce a cyclopedia. It was essential everywhere to
select that which was most characteristic of the tendencies of the age
and most promising for the science of the twentieth century. Those
problems were to be emphasized whose solution is most demanded for
the immediate progress of knowledge, and those relations had to be
selected through which new connections, new synthetic thoughts
prepare themselves to-day. That this selection had to be left to the
speaker was a matter of course.

Yet it may be said that in all these directions, with reference to the
general standpoint and with reference to problems and relations,
the Organizing Committee had somewhat prepared the choice through
the selection of the speakers themselves. As the standpoints of the
leading speakers were well known, it was not difficult to invite as far
as possible for every place a scholar whose general views would be
least out of harmony with the principles of the plan. For instance,
when we had the task before us of selecting the divisional speakers for
the Normative and for the Mental Sciences, it was only natural to
invite for the first a philosopher of idealistic type and for the latter a
philosopher of positivistic stamp, inasmuch as the whole scheme gave
to the mental sciences the same place which they would have had in
a positivistic scheme, while the normative sciences would have lost
the meaning which they had in our plan if a positivist had simply
psychologized them. In the same way we gave preference as far as
possible, for the addresses on relations, to those scholars whose pre-
vious work was concerned with new synthetic movements, and as
speakers on problem^ those were invited who were in any case
engaged in the solution of those problems which seemed central in
the present state of science. Thus it was that on the whole the ex-
pectation was justified that the most characteristic relations and the
most characteristic problems would be selected if every imdted

130 THE SCIENTIFIC PLAN OF THE CONGRESS

speaker spoke essentially on those relations and on those problems
with which his own special work was engaged.

Yet there is no doubt that this expectation was sometimes ful-
filled beyond our anticipation, in an amount of specialization which
was no longer entirely in harmony with the general character of the
undertaking. The general problem has become sometimes only the
starting-point or almost the pretext for speaking on some relation
or problem &o detailed that it can hardly stand as a representative
symbol of the whole movement in that sectional field. Especially in
the practical sciences more room was sometimes taken for particu-
lar hobbies and chance aspects than in the eyes of the originators the
occasion may have called for. Yet on the whole this was the excep-
tion. The overwhelming majority of the addresses fulfilled nobly the
high hopes of the Boards, and even in those exceptional cases where
the speaker went his own way, it was usually such an original and
stimulating expression of a strong personality that no one would care
to miss this tone in the symphony of science.

Even now of course, though the Congress days have passed, and
only typewritten manuscripts are left from all those September
meetings, it would be easy to provide, by editorial efforts, for a greater
uniformity and a smoother harmonization. Most of the authors
would have been quite willing to retouch their addresses in the
interest of greater objective uniformity and to accept the hint of an
editorial committee in elaborating more fully some points and in con-
densing or eliminating others. Much was written in the desire to bring
a certain thought for discussion before such an eminent audience,
while the speaker would be ready to substitute other features of the
subject for the permanent form of the printed volume. Yet such
editorial supervision and transformation would be not only immodest
but dangerous. We might risk gaining some external uniformity, but
only to lose much of the freshness and immediacy and brilliancy of
the first presentation. And who would dare to play the critical judge
when the international contributors are the leaders of thought ?
There was therefore not the slightest effort made to suggest revision
of the manuscripts, for which the whole responsibility must thus fall
to the particular author. The reduction to a uniform language
seemed, on the other hand, most natural, and those who had delivered
their addresses in French, German, or Italian themselves welcomed
the idea that their papers should be translated into English by com-
petent specialists. The short bibliographies, selected mostly through
the chairman of the departments, and the very full index with refer-
ences may add to the general usefulness of the eight volumes in which
the work is to be presented.

But the significance of the Congress of Arts and Science ought not
to be measured and valued only by reference to this printed result.

THE RESULTS OF THE CONGRESS 131

Its less visible side-effects seem in no way less important for scholar-
ship, and they are fourfold. There was, first, the personal contact
between the scholarly public and the leaders of thought; there was,
secondly, the first academic alliance between the United States and
Europe; there was, thirdly, the first demonstration of a world con-
gress crystallized about one problem; there was, fourthly, the unique
accentuation of the thought of unity in all human science; and each
of these four movements will be continued and reinforced by the pub-
lication of these proceedings.

The first of these four features, the contact of the scholarly public
with the best thinkers of our time, had, to be sure, its limitations. It
was not sought to create a really popular congress. Neither the level
of the addresses, nor the size of the halls, nor the number of invita-
tions sent out, nor the general conditions of a world's fair at which
the expense of living is high and the distractions thousandfold,
favored the attendance of crowds. It was planned from the first that
on the whole scholars and specialists should attend and that the army
should be made up essentially of officers. If in an astronomical section
perhaps thirty men were present, among whom practically every one
was among the best known directors of observatories or professors of
mathematics, astronomy, or physics, from all countries of the globe,
much more was gained than if three thousand had been in the audi-
ence, brought together by an interest of curiosity in moon and stars.
For the most part there must have been between a hundred and two
hundred in each of the 128 sectional meetings, and that was more
than the organizers expected. This direct influence on the inter-
ested public is now to be expanded a thousandfold by the mission
work of these volumes. The concentration of these hundreds of
addresses into a few days made it in any case impossible to listen to
more than to a small fraction; these volumes will bring at last all
speakers to coordinated effectiveness; and while one hall suffered
from bad acoustics, another from bad ventilation, and a third from
the passing of the intermural trains, here at least is an audience in
which nothing will disturb the sensitive nerves of the willing follower.

But much more emphasis is due to the second feature. The Con-
gress was an epoch-making event for the international world of
scholarship from the fact that it was the first great undertaking in
which the Old and the New Worlds stood on equal levels and in which
Europe really became acquainted with the scientific life of these
United States. The contact of scholarship between America and Eu-
rope has, indeed, grown in importance through many decades. Many
American students had studied in European and especially in German
universities and had come back to fill the professorial chairs of the
leading academic institutions. The spirit of the Graduate School and
the work towards the Doctor's degree, yes, the whole productive

132 THE SCIENTIFIC PLAN OF THE CONGRESS

scholarship of recent decades had been influenced by European ideals,
and the results were no longer ignored at the seats of learning through-
out the whole world. European scholars had here and there come as
visiting lecturers or as assimilated instructors, and a few American
scholars belonged to the leading European Academies. Yet, whoever
knew the real development of American post-graduate university life,
the rapid advance of genuine American scholarship, the incomparable
progress of the scientific institutions of the New World, of their libra-
ries and laboratories, museums and associations, was well aware that
Europe had hardly noticed and certainly not fully understood the
gigantic strides of the country which seemed a rival only on commer-
cial and industrial ground. Europe was satisfied with the traditional
ideas of America's scientific standing which reflected the situation of
thirty years ago, and did not understand that the changes of a few
lustres mean in the New World more than under the firmer traditions
of Europe. American scientific literature was still neglected; Ameri-
can universities treated in a condescending and patronizing spirit
and with hardly any awareness of the fundamental differences in the
institutions of the two sides. Those European scholars who crossed
the ocean did it with missionary, or perhaps with less unselfish, inten-
tions, and the Americans who attended European congresses were
mostly treated with the friendliness which the self-satisfied teacher
shows to a promising pupil. The time had really come when the con-
trast between the real situation and the traditional construction
became a danger for the scientific life of the time. Both sides had to
suffer from it. The Americans felt that their serious and important
achievements did not come to their fullest effectiveness through the
insistent neglect of those who by the tradition of centuries had
become the habitual guardians of scientific thought. A kind of feeling
of dependency as it usually develops in weak colonies too often
depressed the conscientious scholarship on American soil as the result
of this undue condescension. Yet the greater harm was to the other
side. Once before Europe had had the experience of surprise when
American successes presented themselves where nothing of that kind
was anticipated in the Old World. It was in the field of economic
life that Europe looked down patronizingly on America's industrial
efforts, and yet before she was fully aware how the change resulted,
suddenly the warning signal of the "American danger" was heard
everywhere. The surprise in the intellectual field will not be less.
The unpreparedness was certainly the same. Of course, there cannot
be any danger of rivalry in the scientific field, inasmuch as science
knows no competition but only cooperation. And yet it cannot be
without danger for European science if it willfully neglects and reck-
lessly ignores this eager working of the modern America. For both
sides a change in the situation was thus not only desirable, but neces-

THE RESULTS OF THE CONGRESS 133

sary; and to prepare this change, to substitute knowledge for ignor-
ance, nothing could have been more effective than this Congress of
Arts and Science.

Even if we abstract from the not inconsiderable number of those
European scholars who followed naturally in the path of the invited
guests, and if we consider merely the function of these invited par-
ticipants, the importance of the procedure is evident. More than a
hundred leading scholars from all European countries came under
conditions where academic fellowship on an equal footing was a neces-
sary part of the work. There was not the slightest premium held out
which might have attracted them had not real interacademic interest
brought them over the ocean, and no missionary spirit was appealed
to, as everything was equally divided between American and foreign
contributors. It was a real feast of international scholarship, in
which the importance and the number of foreigners stamped it as
the first significant alliance of the spirit of learning in the New and the
Old Worlds. And it was essentially for this purpose that the week of
personal intermingling in St, Louis itself was preceded and followed
by happy weeks of visits to leading universities. Almost every one
of those one hundred European scholars visited Harvard and Yale,
Chicago and Johns Hopkins, Columbia and Pennsylvania, saw the
treasures of Washington and examined the exhibitions of American
scholarship in the World's Fair itself. The change of opinion, the dis-
appearance of prejudice, the growth of confidence, the personal inter-
collegiate ties which resulted from all that, have been evident since
those days all over Europe. And it is not surprising that it is just
the most famous and most important of the visitors, famous and im-
portant through their width and depth of view, whose expression
of appreciation and admiration for the new achievements has been
loudest.

We insisted that the effectiveness of the Congress showed itself in
two other directions still : on the one side, there was at last a congress
with a unified programme, a congress which stood for a definite
thought, and which brought all its efforts to bear on the solution of
one problem. There seemed a far-reaching agreement of opinion that
this new principle of congress administration had successfully with-
stood the test of practical realization. Mere conglomerations of un-
connected meetings with casual programmes and unrelated papers
cannot claim any longer to represent the only possible form of inter-
national gatherings of scholars. More than that, their superfluous
and disheartening character will be felt in future more strongly
than before. No congress will appear fully justified whose printed
proceedings do not show a real plan in its programme. And the
consciousness of this mission of the Congress will certainly be again
reinforced by the publication of these volumes, inasmuch as it is

134 THE SCIENTIFIC PLAN OF THE CONGRESS

evident that they represent a substantial contribution to the know-
ledge of our time which would not have been made without the
special stimulating occasion of the Congress.

And, finally, whether such a congress is held again or not, the
impulse of this one cannot be lost on account of the special end to
which all its efforts have been directed: the unity of scientific know-
ledge. We had emphasized from the first that here was the centre
of our purposes in a time whose scientific specialization necessarily
involves a scattering of scholarly work and which yet in its deepest
meaning strives for a new synthesis, for a new unity, which is to give
to all this scattered labor a real dignity and significance; truly
nothing was more needed than an intense accentuation of the internal
harmony of all human knowledge. But for that it is not enough that
the masses feel instinctively the deep need of such unifying move-
ments, nor is it enough that the philosophers point with logical argu-
ments towards the new synthesis. The philosopher can only stand by
and point the way; the specialists themselves must go the way. And
here at last they have done so. Leaders of thought have interrupted
their specialistic work and have left their detailed inquiries to seek
the fundamental conceptions and methods and principles which bind
all knowledge together, and thus to work towards that unity from
which all special work derives its meaning. Whether or not their
cooperation has produced anything which is final is a question almost
insignificant compared with the fundamental fact that they cooper-
ated at all for this ideal synthetic purpose. This fact can never lose
its influence on the scholarly effort of our age, and will certainly find
its strongest reinforcement in this unified publication. It has ful-
filled its noblest purpose if it adds strength to the deepest movement
of our time, the movement towards unity of meaning in the scattered
manifoldness of scientific endeavor with which the twentieth century
has opened.

PROCEEDINGS OF THE CONGRESS

INTRODUCTORY ADDRESS

DELIVERED AT THE OPENING EXERCISES AT FESTIVAL HALL BY
PROFESSOR SIMON NEWCOMB, PRESIDENT OF THE CONGRESS

THE EVOLUTION OF THE SCIENTIFIC INVESTIGATOR

As we look at the assemblage gathered in this hall, comprising so
many names of widest renown in. every branch of learning, — we
might almost say in every field of human endeavor, — the first in-
quiry suggested must be after the object of our meeting. The answer
is, that our purpose corresponds to the eminence of the assemblage.
We aim at nothing less than a survey of the realm of knowledge, as
comprehensive as is permitted by the limitations of time and space.
The organizers of our Congress have honored me with the charge of
presenting such preliminary view of its field as may make clear the
spirit of our undertaking.

Certain tendencies characteristic of the science of our day clearly
suggest the direction of our thoughts most appropriate to the oc-
casion. Among the strongest of these is one toward laying greater
stress on questions of the beginning of things, and regarding a know-
ledge of the laws of development of any object of study as necessary
to the understanding of its present form. It may be conceded that
the principle here involved is as applicable in the broad field before
us as in a special research into the properties of the minutest or-
ganism. It therefore seems meet that we should begin by inquir-
ing what agency has brought about the remarkable development
of science to which the world of to-day bears witness. This view is re-
cognized in the plan of our proceedings, by providing for each great
department of knowledge a review of its progress during the century
that has elapsed since the great event commemorated by the scenes
outside this hall. But such reviews do not make up that general
survey of science at large which is necessary to the development of
our theme, and which must include the action of causes that had
their origin long before our time. The movement which culminated

136 INTRODUCTORY ADDRESS

in making the nineteenth century ever memorable in history is the
outcome of a long series of causes, acting through many centuries,
which are worthy of especial attention on such an occasion as this.
In setting them forth we should avoid laying stress on those visible
manifestations which, striking the eye of every beholder, are in no
danger of being overlooked, and search rather for those agencies whose
activities underlie the whole visible scene, but which are liable to be
blotted out of sight by the very brilliancy of the results to which they
have given rise. It is easy to draw attention to the wonderful qualities
of the oak; but from that very fact, it may be needful to point out
that the real wonder lies concealed in the acorn from which it grew.

Our inquiry into the logical order of the causes which have made
our civilization what it is to-day will be facilitated by bringing to
mind certain elementary considerations — ideas so familiar that
setting them forth may seem like citing a body of truisms — and
yet so frequently overlooked, not only individually, but in their
relation to each other, that the conclusion to which they lead may be
lost to sight. One of these propositions is that psj^chical rather than
material causes are those which we should regard as fundamental in
directing the development of the social organism. The human
intellect is the really active agent in every branch of endeavor, —
the primum mobile of civilization, — and all those material mani-
festations to which our attention is so often directed are to be re-
garded as secondary to this first agency. If it be true that " in the
world is nothing great but man; in man is nothing great but mind,"
then should the keynote of our discourse be the recognition of this
first and greatest of powers.

Another well-known fact is that those applications of the forces
of nature to the promotion of human welfare which have made our
age what it is, are of such comparatively recent origin that we need
go back only a single century to antedate their most important fea-
tures, and scarcely more than four centuries to find their beginning.
It follows that the subject of our inquiry should be the commence-
ment, not many centuries ago, of a certain new form of intellectual
activity.

Having gained this point of view, our next inquiry'will be into the
nature of that activity, and its relation to the stages of progress
which preceded and followed its beginning. The superficial observer,
who sees the oak but forgets the acorn, might tell us that the special
qualities which have brought out such great results are expert
scientific knowledge and rare ingenuity, directed to the application
of the powers of steam and electricity. From this point of view the
great inventors and the great captains of industry were the first
agents in bringing about the modern era. But the more careful
inquirer will see that the work of these men was possible only through

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 137

a knowledge of the laws of nature, which had been gained by men
whose work took precedence of theirs in logical order, and that
success in invention has been measured by completeness in such
knowledge. While giving all due honor to the great inventors, let
us remember that the first place is that of the great investigators,
whose forceful intellects opened the way to secrets preAaously hidden
from men. Let it be an honor and not a reproach to these men, that
they were not actuated by the love of gain, and did not keep utilita-
rian ends in view in the pursuit of their researches. If it seems that in
neglecting such ends they were leaving undone the most important
part of their work, let us remember that nature turns a forbidding
face to those who pay her court with the hope of gain, and is respons-
ive only to those suitors whose love for her is pure and undefiled.
Not only is the special genius required in the investigator not that
generally best adapted to applying the discoveries which he makes,
but the result of his having sordid ends in view would be to nar-
row the field of his efforts, and exercise a depressing effect upon his
activities. The true man of science has no such expression in
his vocabulary as "useful knowledge." His domain is as wide
as nature itself, and he best fulfills his mission when he leaves to
others the task of applying the knowledge he gives to the world.

We have here the explanation of the well-known fact that the
functions of the investigator of the laws of nature, and of the in-
ventor who applies these laws to utilitarian purposes, are rarely
united in the same person. If the one conspicuous exception which
the past century presents to this rule is not unique, we should prob-
ably have to go back to Watt to find another.

From this viewpoint it is clear that the primary agent in the
movement which has elevated man to the masterful position he now
occupies, is the scientific investigator. He it is whose work has de-
prived plague and pestilence of their terrors, alleviated human suffer-
ing, girdled the earth with the electric wire, bound the continent
with the iron way, and made neighbors of the most distant nations.
As the first agent which has made possible this meeting of his re-
presentatives, let his evolution be this day our worthy theme. As we
follow the evolution of an organism by studying the stages of its
growth, so we have to show how the work of the scientific investi-
gator is related to the ineffectual efforts of his predecessors.

In our time we think of the process of development in nature as
one going continuously forward through the combination of the
opposite processes of evolution and dissolution. The tendency of our
thought has been in the direction of banishing cataclysms to the
theological limbo, and viewing nature as a sleepless plodder, en-
dowed with infinite patience, waiting through long ages for results.
I do not contest the truth of the principle of continuity on which

138 INTRODUCTORY ADDRESS

this view is based. But it fails to make known to us the whole truth.
The building of a ship from the time that her keel is laid until she is
making her way across the ocean is a slow and gradual process; yet
there is a cataclysmic epoch opening up a new era in her history. It
is the moment when, after lying for months or years a dead, inert,
immovable mass, she is suddenly endowed with the power of motion,
and, as if imbued with life, glides into the stream, eager to begin the
career for which she was designed.

I think it is thus in the development of humanity. Long ages
may pass during which a race, to all external observation, appears to
be making no real progress. Additions may be made to learning, and
the records of history may constantly grow, but there is nothing in
its sphere of thought, or in the features of its life, that can be called
essentially new. Yet, nature may have been all along slowly working
in a way which evades our scrutiny until the result of her operations
suddenly appears in a new and revolutionary movement, carrying
the race to a higher plane of civilization.

It is not difficult to point out such epochs in human progress. The
greatest of all^ because it was the first, is one of which we find no
record either in written or geological history. It was the epoch when
our progenitors first took conscious thought of the morrow, first used
the crude weapons which nature had placed within their reach to
kill their prey, first built a fire to warm their bodies and cook their
food. I love to fancy that there was some one first man, the Adam
of evolution, who did all this, and who used the power thus acquired
to show his fellows how they might profit by his example. When
the members of the tribe or community which he gathered around
him began to conceive of life as a whole, — to include yesterday, to-
day, and to-morrow in the same mental grasp — to think how they
might apply the gifts of nature to their own uses, — a movement
was begun which should ultimately lead to civilization.

Long indeed must have been the ages required for the development
of this rudest primitive community into the civilization revealed to
us by the most ancient tablets of Egypt and Assyria. After spoken
language was developed, and after the rude representation of ideas
by visible marks drawn to resemble them had long been practiced,
some Cadmus must have invented an alphabet. When the use of
written language was thus introduced, the word of command ceased
to be confined to the range of the human voice, and it became pos-
sible for master minds to extend their influence as far as a written
message could be carried. Then were communities gathered into
provinces; provinces into kingdoms; kingdoms into the great
empires of antiquity. Then arose a stage of civilization which we
find pictured in the most ancient records, — a stage in which men
were governed by laws that were perhaps as wisely adapted to their

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 139

conditions as our laws are to ours, — in which the phenomena of
nature were rudely observed, and striking occurrences in the earth
or in the heavens recorded in the annals of the nation.

Vast was the progress of knowledge during the interval between
these empires and the century in which modern science began. Yet,
if I am right in making a distinction between the slow and regular
steps of progress, each growing naturally out of that which preceded
it, and the entrance of the mind at some fairly definite epoch into an
entirely new sphere of activity, it would appear that there was only
one such epoch during the entire interval. This was when abstract
geometrical reasoning commenced, and astronomical observations
aiming at precision were recorded, compared, and discussed. Closely
associated with it must have been the construction of the forms of
logic. The radical difference between the demonstration of a theorem
of geometry and the reasoning of every-day life which the masses of
men must have practiced from the beginning, and which few even
to-day ever get beyond, is so evident at a glance that I need not
dwell upon it. The principal feature of this advance is that, by one
of those antinomies of the human intellect of which examples are not
wanting even in our own time, the development of abstract ideas
preceded the concrete knowledge of natural phenomena. When we
reflect that in the geometry of EucHd the science of space was
brought to such logical perfection that even to-day its teachers are
not agreed as to the practicability of any great improvement upon
it, we cannot avoid the feeling that a very slight change in the
direction of the intellectual activity of the Greeks would have led to
the beginning of natural science. But it would seem that the very
purity and perfection which was aimed at in their system of geometry
stood in the way of any extension or application of its methods and
spirit to the field of nature. One example of this is worthy of atten-
tion. In modern teaching the idea of magnitude as generated by
motion is freely introduced. A line is described by a moving point;
a plane by a moving line; a solid by a moving plane. It may, at first
sight, seem singular that this conception finds no place in the Euclid-
ian system. But we may regard the omission as a mark of logical
purity and rigor. Had the real or supposed advantages of introduc-
ing motion into geometrical conceptions been suggested to Euclid,
we may suppose him to have replied that the theorems of space are
independent of time; that the idea of motion necessarily implies
time, and that, in consequence, to avail ourselves of it would be to
introduce an extraneous element into geometry.

It is quite possible that the contempt of the ancient philosophers
for the practical application of their science, which has continued in
some form to our own time, and which is not altogether unwholesome,
was a powerful factor in the same direction. The result was that,

140 INTRODUCTORY ADDRESS

in keeping geometry pure from ideas which did not belong to it, it
failed to form what might otherwise have been the basis of physical
science. Its founders missed the discovery that methods similar to
those of geometric demonstration could be extended into other and
wider fields than that of space. Thus not only the development of
applied geometry, but the reduction of other conceptions to a rigorous
mathematical form was indefinitely postponed.

Astronomy is necessarily a science of observation pure and simple,
in which experiment can have no place except as an auxiliary. The
vague accounts of striking celestial phenomena handed down by the
priests and astrologers of antiquity were followed in the time of the
Greeks by observations having, in form at least, a rude approach to
precision, though nothing like the degree of precision that the astro-
nomer of to-day would reach with the naked eye, aided by such
instruments as he could fashion from the tools at the command of
the ancients.

The rude observations commenced by the Babylonians were
continued with gradually improving instruments, — first by the
Greeks and afterward by the Arabs, — but the results failed to afford
any insight into the true relation of the earth to the heavens. What
was most remarkable in this failure is that, to take a first step forward
which would have led on to success, no more was necessary than a
course of abstract thinking vastly easier than that required for work-
ing out the problems of geometry. That space is infinite is an unex-
pressed axiom, tacitly assumed by Euclid and his successors. Com-
bining this with the most elementary consideration of the properties
of the triangle, it would be seen that a body of any given size could
be placed at such a distance in space as to appear to us like a point.
Hence a body as large as our earth, which was known to be a globe
from the time that the ancient Phoenicians navigated the Mediter-
ranean, if placed in the heavens at a sufficient distance, would look
like a star. The obvious conclusion that the stars might be bodies
like our globe, shining either by their own light or by that of the sun,
would have been a first step to the understanding of the true system
of the world.

There is historic evidence that this deduction did not wholly
escape the Greek thinkers. It is true that the critical student will
assign little weight to the current belief that the vague theory of
Pythagoras — that fire was at the centre of all things — implies a
conception of the heliocentric theory of the solar system. But the
testimony of Archimedes, confused though it is in form, leaves no
serious doubt that Aristarchus of Samos not only propounded the
view that the earth revolves both on its own axis and around the sun,
but that he correctly removed the great stumbling-block in the way
of this theory by adding that the distance of the fixed stars was

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 141

infinitely greater than the dimensions of the earth's orbit. Even the
world of philosophy was not yet ready for this conception, and, so far
from seeing the reasonableness of the explanation, we find Ptolemy
arguing against the rotation of the earth on grounds which careful
observations of the phenomena around him would have shown to be
ill-founded.

Physical science, if we can apply that term to an uncoordinated
body of facts, was successfully cultivated from the earliest times.
Something must have been known of the properties of metals, and
the art of extracting them from their ores must have been practiced,
from the time that coins and medals were first stamped. The pro-
perties of the most common compounds were discovered by alchem-
ists in their vain search for the philosopher's stone, but no actual
progress worthy of the name rewarded the practitioners of the black
art.

Perhaps the first approach to a correct method was that of Archi-
medes, who by much thinking worked out the law of the lever,
reached the conception of the centre of gravity, and demonstrated
the first principles of hydrostatics. It is remarkable that he did not
extend his researches into the phenomena of motion, whether spon-
taneous or produced by force. The stationary condition of the human
intellect is most strikingly illustrated by the fact that not until the
time of Leonardo was any substantial advance made on his discovery.
To sum up in one sentence the most characteristic feature of ancient
and medieval science, we see a notable contrast between the precision
of thought implied in the construction and demonstration of geo-
metrical theorems and the vague indefinite character of the ideas of
natural phenomena generally, a contrast which did not disappear
until the foundations of modern science began to be laid.

We should miss the most essential point of the difference between
medieval and modern learning if we looked upon it as mainly a differ-
ence either in the precision or the amount of knowledge. The devel-
opment of both of these qualities would, under any circumstances,
have been slow and gradual, but sure. We .can hardly suppose that
any one generation, or even any one century, would have seen the
complete substitution of exact for inexact ideas. Slowness of growth
is as inevitable in the case of knowledge as in that of a growing organ-
ism. The most essential point of difference is one of those seemingly
slight ones, the importance of which we are too apt to overlook. It
was like the drop of blood in the wrong place, which some one has
told us makes all the difference between a philosopher and a maniac.
It was all the difference between a living tree and a dead one, between
an inert mass and a growing organism. The transition of knowledge
from the dead to the living form must, in any complete review of the
subject, be looked upon as the really great event of modern times.

142 INTRODUCTORY ADDRESS

Before this event the intellect was bound down by a scholasticism
which regarded knowledge as a rounded whole, the parts of which
were written in books and carried in the minds of learned men. The
student was taught from the beginning of his work to look upon
authority as the foundation of his beliefs. The older the authority the
greater the weight it carried. So effective was this teaching that it
seems never to have occurred to individual men that they had all the
opportunities ever enjoyed by Aristotle of discovering truth, with the
added advantage of all his knowledge to begin with. Advanced as
was the development of formal logic, that practical logic was wanting
which could see that the last of a series of authorities, every one of
which rested on those which preceded it, could never form a surer
foundation for any doctrine than that supplied by its original pro-
pounder.

The result of this view of knowledge was that, although during the
fifteen centuries following the death of the geometer of Syracuse
great universities were founded at which generations of professors
expounded all the learning of their time, neither professor nor student
ever suspected what latent possibilities of good were concealed in the
most familiar operations of nature. Every one felt the wind blow, saw
water boil, and heard the thunder crash, but never thought of inves-
tigating the forces here at play. Up to the middle of the fifteenth
century the most acute observer could scarcely have seen the dawn
of a new era.

In view of this state of things, it must be regarded as one of the most
remarkable facts in evolutionary history that four or five men, whose
mental constitution was either typical of the new order of things or
who were powerful agents in bringing it about, were all born during
the fifteenth century, four of them at least at so nearly the same time
as to be contemporaries.

Leonardo da Vinci, whose artistic genius has charmed succeeding
generations, was also the first practical engineer of his time, and the
first man after Archimedes to make a substantial advance in develop-
ing the laws of motion. That the world was not prepared to make
use of his scientific discoveries does not detract from the significance
which must attach to the period of his birth.

Shortly after him was born the great navigator whose bold spirit
was to make known a new world, thus giving to commercial enterprise
that impetus which was so powerful an agent in bringing about a
revolution in the thoughts of men.

The birth of Columbus was soon followed by that of Copernicus,
the first after Aristarchus to demonstrate the true system of the
world. In him more than in any of his contemporaries do we see the
struggle between the old forms of thought and the new. It seems
almost pathetic and is certainly most suggestive of the general view

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 143

of knowledge taken at that time that, instead of claiming credit for
bringing to light great truths before unknown, he made a labored
attempt to show that, after all, there was nothing really new in his
system, which he claimed to date from Pythagoras and Philolaus.
In this connection it is curious that he makes no mention of Aris-
tarchus, who I think will be regarded by conservative historians as
his only demonstrated predecessor. To the hold of the older ideas
upon his mind we must attribute the fact that in constructing his
system he took great pains to make as little change as possible in
ancient conceptions.

Luther, the greatest thought-stirrer of them all, practically of the
same generation with Copernicus, Leonardo, and Columbus, does not
come in as a scientific investigator, but as the great loosener of chains
which had so fettered the intellect of men that they dared not think
otherwise than as the authorities thought.

Almost coeval with the advent of these intellects was the invention
of printing with movable type. Gutenberg was born during the first
decade of the century, and his associates and others credited with the
invention not many years afterward. If we accept the principle on
which I am basing my argument, that we should assign the first place
to the birth of those psychic agencies which started men on new lines
of thought, then surely was the fifteenth the wonderful century.

Let us not forget that, in assigning the actors then born to their
places, we are not narrating history, but studying a special phase of
evolution. It matters not for us that no university invited Leonardo
to its halls, and that his science was valued by his contemporaries
only as an adjunct to the art of engineering. The great fact still is
that he was the first of mankind to propound laws of motion. It is
not for anything in Luther's doctrines that he finds a place in our
scheme. No matter for us whether they were sound or not. What he
did toward the evolution of the scientific investigator was to show by
his example that a man might question the best-established and most
venerable authority and still live — still preserve his intellectual
integrity — still command a hearing from nations and their rulers.
It matters not for us whetJier Columbus ever knew that he had dis-
covered a new continent. His work was to teach that neither hydra,
chimera, nor abyss — neither divine injunction nor infernal machina-
tion — was in the way of men visiting every part of the globe, and
that the problem of conquering the world reduced itself to one of
sails and rigging, hull and compass. The better part of Copernicus
was to direct man to a viewpoint whence he should see that the
heavens were of like matter with the earth. All this done, the acorn
was planted from which the oak of our civilization should spring.
The mad quest for gold which followed the discovery of Columbus,
the questionings w^hich absorbed the attention of the learned, the

144 INTRODUCTORY ADDRESS

indignation excited by the seeming vagaries of a Paracelsus, the fear
and trembhng lest the strange doctrine of Copernicus should under-
mine the faith of centuries, were all helps to the germination of the
seed — stimuli to thought which urged it on to explore the new fields
opened up to its occupation. This given, all that has since followed
came out in regular order of development, and need be here con-
sidered only in those phases having a special relation to the purpose
of our present meeting.

So slow was the growth at first that the sixteenth century may
scarcely have recognized the inauguration of a new era. Torricelli
and Benedetti were of the third generation after Leonardo, and
Galileo, the first to make a substantial advance upon his theory, was
born more than a century after him. Only two or three men appeared
in a generation who, working alone, could make real progress in dis-
covery, and even these could do little in leavening the minds of their
fellow men with the new ideas.

Up to the middle of the seventeenth century an agent which all
experience since that time shows to be necessary to the most pro-
ductive intellectual activity was wanting. This was the attraction of
like minds, making suggestions to each other, criticising, comparing,
and reasoning. This element was introduced by the organization of
the Royal Society of London and the Academy of Sciences of Paris.

The members of these two bodies seem like ingenious youth sud-
denly thrown into a new world of interesting objects, the purposes and
relations of which they had to discover. The novelty of the situation
is strikingly shown in the questions which occupied the minds of the
incipient investigators. One natural result of British maritime enter-
prise was that the aspirations of the Fellows of the Royal Society
were not confined to any continent or hemisphere. Inquiries were
sent all the way to Batavia to know "whether there be a hill in
Sumatra which burneth continually, and a fountain which runneth
pure balsam." The astronomical precision with which it seemed pos-
sible that physiological operations might go on was evinced by the
inquiry whether the Indians can so prepare that stupefying herb
Datura that " they make it lie several days, months, years, according
as they will, in a man's body without doing him any harm, and at
the end kill him without missing an hour's time." Of this continent
one of the inquiries was whether there be a tree in Mexico that yields
water, wine, vinegar, milk, honey, wax, thread, and needles.

Among the problems before the Paris Academy of Sciences those
of physiology and biology took a prominent place. The distillation
of compounds had long been practiced, and the fact that the more
spirituous elements of certain substances were thus separated nat-
urally led to the question whether the essential essences of life might
not be discoverable in the same way. In order that all might par-

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 145

ticipate in the experiments, they were conducted in open session of
the Academy, thus guarding against the danger of any one member
obtaining for his exclusive personal use a possible elixir of life. A
wide range of the animal and vegetable kingdom, including cats, dogs,
and birds of various species, were thus analyzed. The practice of
dissection was introduced on a large scale. That of the cadaver of an
elephant occupied several sessions, and was of such interest that the
monarch himself was a spectator.

To the same epoch with the formation and first work of these two
bodies belongs the invention of a mathematical method which in its
importance to the advance of exact science may be classed with the
invention of the alphabet in its relation to the progress of society at
large. The use of algebraic symbols to represent quantities had its
origin before the commencement of the new era, and gradually grew
into a highly developed form during the first two centuries of that
era. But this method could represent quantities only as fixed. It is
true that the elasticity inherent in the use of such symbols permitted
of their being applied to any and every quantity; yet, in any one
application, the quantity was considered as fixed and definite. But
most of the magnitudes of nature are in a state of continual variation;
indeed, since all motion is variation, the latter is a universal charac-
teristic of all phenomena. No serious advance could be made in the
application of algebraic language to the expression of physical phe-
nomena until it could be so extended as to express variation in quan-
tities, as well as the quantities themselves. This extension, worked
out independently by Newton and Leibnitz, may be classed as the
most fruitful of conceptions in exact science. With it the way was
opened for the unimpeded and continually accelerated progress of the
last two centuries.

The feature of this period which has the closest relation to the
purpose of our coming together is the seemingly unending subdivision
of knowledge into specialties, many of which are becoming so minute
and so isolated that they seem to have no interest for any but their
few pursuers. Happily science itself has afforded a corrective for its
own tendency in this direction. The careful thinker will see that in
these seemingly diverging branches common elements and common
principles are coming more and more to light. There is an increasing
recognition of methods of research, and of deduction, which are com-
mon to large branches, or to the whole of science. We are more and
more recognizing the principle that progress in knowledge implies its
reduction to more exact forms, and the expression of its ideas in
language more or less mathematical. The problem before the organ-
izers of this Congress was, therefore, to bring the sciences together,
and seek for the unity which we believe underlies their infinite
diversity.

146 INTRODUCTORY ADDRESS

The assembling of such a body as now fills this hall was scarcely
possible in any preceding generation, and is made possible now only
through the agency of science itself. It differs from all preceding inter-
national meetings by the universality of its scope, which aims to
include the whole of knowledge. It is also unique in that none but
leaders have been sought out as members. It is unique in that so
many lands have delegated their choicest intellects to carry on its
work. They come from the country to which our republic is indebted
for a third of its territory, including the ground on which we stand;
from the land which has taught us that the most scholarly devotion to
the languages and learning of the cloistered past is compatible with
leadership in the practical application of modern science to the arts
of life; from the island whose language and literature have found
a new field and a vigorous growth in this region; from the last seat
of the holy Roman Empire; from the country which, remembering
a monarch who made an astronomical observation at the Greenwich
Observatory, has enthroned science in one of the highest places in its
government; from the peninsula so learned that we have invited one
of its scholars to come and tell us of our own language; from the land
which gave birth to Leonardo, Galileo, Torricelli, Columbus, Volta —
what an array of immortal names! — from the little republic of
glorious history which, breeding men rugged as its eternal snow-
peaks, has yet been the seat of scientific investigation since the day of
the Bernoullis; from the land whose heroic dwellers did not hesitate
to use the ocean itself to protect it against invaders, and which now
makes us marvel at the amount of erudition compressed within its
little area; from the nation across the Pacific, which, by half a cen-
tury of unequaled progress in the arts of life, has made an important
contribution to evolutionary science through demonstrating the
falsity of the theory that the most ancient races are doomed to be
left in the rear of the advancing age — in a word, from every great
centre of intellectual activity on the globe I see before me eminent
representatives of that world-advance in knowledge which we have
met to celebrate. May we not confidently hope that the discussions
of such an assemblage will prove pregnant of a future for science
which shall outshine even its brilliant past?

Gentlemen and scholars all! You do not visit our shores to find
great collections in which centuries of humanity have given expression
on canvas and in marble to their hopes, fears, and aspirations. Nor
do you expect institutions and buildings hoary with age. But as you
feel the vigor latent in the fresh air of these expansive prairies, which
has collected the products of human genius by which we are here
surrounded, and, I may add, brought us together; as you study the
institutions which we have founded for the benefit, not only of our
own people, but of humanity at large; as you meet the men who, in

EVOLUTION OF THE SCIENTIFIC INVESTIGATOR 147

the short space of one century, have transformed this valley from a
savage wilderness into what it is to-day — then may you find com-
pensation for the want of a past like yours by seeing with prophetic
eye a future world-power of which this region shall be the seat. If such
is to be the outcome of the institutions which we are now building up,
then may your present visit be a blessing both to your posterity and
ours by making that power one for good to all mankind. Your deliber-
ations will help to demonstrate to us and to the world at large that the
reign of law must supplant that of brute force in the relations of the
nations, just as it has supplanted it in the relations of individuals.
You "udll help to show that the war which science is now waging
against the sources of diseases, pain, and misery offers an even nobler
field for the exercise of heroic qualities than can that of battle. We
hope that when, after your all too fleeting sojourn in our midst, you
return to your own shores, you will long feel the influence of the new
air you have breathed in an infusion of increased vigor in pursuing
your varied labors. And if a new impetus is thus given to the great
intellectual movement of the past century, resulting not only in
promoting the unification of knowledge, but in widening its field
through new combinations of effort on the part of its votaries, the
projectors, organizers, and supporters of this Congress of Arts and
Science wiU be justified of their labors.

DIVISION A — NORMATIVE SCIENCE

Speaker : Professor Josiah Royce, Harvard University
{Hall 6, September 20, 10 a. m.)

THE SCIENCES OF THE IDEAL

BY JOSIAH ROYCE

[Josiah Royce, Professor of History of Philosophy, Harvard University, since
1892. b. Grass Valley, Nevada County, California, November 20, 1855.
A.B, University of California, 1875; Ph.D. Johns Hopkins, 1878; LL.D.
University of Aberdeen, Scotland; LL.D. Johns Hopkins. Instructor in
English Literature and Logic, University of California, 1878-82. Instruct-
or and Assistant Professor, Harvard University, 1882-92. Author of Re-
ligious Aspect of Philosophy; History of California; The Feud of Oak field
Creek; The Spirit of Modern Philosophy; Studies of Good and Evil; The
World and the Individual ; Gifford Lectures ; and numerous other works and
memoirs.]

I SHALL not attempt, in this address, either to justify or to criticise
the name, normative science, under which the doctrines which con-
stitute this division are grouped. It is enough for my purpose to
recognize at the outset that I am required, by the plans of this Con-
gress, to explain what scientific interests seem to me to be common
to the work of the philosophers and of the mathematicians. The
task is one which makes severe demands upon the indulgence of the
listener, and upon the expository powers of the speaker, but it is a
task for which the present age has well prepared the way. The spirit
which Descartes and Leibnitz illustrated seems likely soon to become,
in a new and higher sense, prominent in science. The mathematicians
are becoming more and more philosophical. The philosophers, in the
near future, will become, I believe, more and more mathematical.
It is my office to indicate, as well as the brief time and my poor powers
may permit, why this ought to be so.

To this end I shall first point out what is that most general com-
munity of interest which unites all the sciences that belong to our
division. Then I shall indicate what type of recent and special
scientific work most obviously bears upon the tasks of all of us alike.
Thirdly, I shall state some results and problems to which this type
of scientific work has given rise, and shall try to show what promise
we have of an early increase of insight regarding our common interests.

152 NORMATIVE SCIENCE

The most general community of interest which unites the various
scientific activities that belong to our division is this: We are all
concerned with what may be called ideal truth, as distinct from
physical truth. Some of us also have a strong interest in physical
truth; but none of us lack a notable and scientific concern for the
realm of ideas, viewed as ideas.

Let me explain what I mean by these terms. Whoever studies
physical truth (taking that term in its most general sense) seeks to
observe, to collate, and, in the end, to control, facts which he regards
as external to his own thought. But instead of thus looking mainly
without, it is possible for a man chiefly to take account, let us say,
of the consequences of his own hypothetical assumptions — assump-
tions which may possess but a very remote relation to the physical
world. Or again, it is possible for such a student to be mainly de-
voted to reflecting upon the formal validity of his own inferences, or
upon the meaning of his own presuppositions, or upon the value and
the interrelation of human ideals. Any such scientific work, reflective,
considerate principally of the thinker's own constructions and pur-
poses, or of the constructions and purposes of humanity in general,
is a pursuit of ideal truth. The searcher who is mainly devoted to
the inquiry into what he regards as external facts, is indeed active;
but his activity is moulded by an order of existence which he conceives
as complete apart from his activity. He is thoughtful; but a power
not himself assigns to him the problems about which he thinks. He
is guided by ideals; but his principal ideal takes the form of an ac-
ceptance of the world as it is, independently of his ideals. His deal-
ings are with nature. His aim is the conquest of a foreign realm.
But the student of what may be called, in general terms, ideal truth,
while he is devoted as his fellow, the observer of outer nature, to
the general purpose of being faithful to the verity as he finds it, is
still aware that his own way of finding, or his own creative activity
as an inventor of hypotheses, or his own powers of inference, or his
conscious ideals, constitute in the main the object into which he is
inquiring, and so form an essential aspect of the sort of verity which
he is endeavoring to discover. The guide, then, of such a student is,
in a peculiar sense, his OAvn reason. His goal is the comprehension of
his own meaning, the conscious and thoughtful conquest of himself.
His great enemy is not the mystery of outer nature, but the imper-
fection of his reflective powers. He is, indeed, as unwilling as is any
scientific worker to trust private caprices. He feels as little as does
the observer of outer facts, that he is merely noting down, as they
pass, the chance products of his arbitrary fantasy. For him, as for
any scientific student, truth is indeed objective; and the standards

THE SCIENCES OF THE IDEAL 153

to which he conforms are eternal. But his method is that of an inner
considerateness rather than of a curiosity about external phenomena.
His objective world is at the same time an essentially ideal world,
and the eternal verity in whose light he seeks to live has, throughout
his undertakings, a peculiarly intimate relation to the purposes of
his own constructive will.

One may then sum up the difference of attitude which is here in
question by saying that, while the student of outer nature is ex-
plicitly conforming his plans of action, his ideas, his ideals, to an
order of truth which he takes to be foreign to himself — the student
of the other sort of truth, here especially in question, is attempting
to understand his own plans of action, that is, to develop his ideas,
or to define his ideals, or else to do both these things.

Now it is not hard to see that this search for some sort of ideal
truth is indeed characteristic of every one of the investigations
which have been grouped together in our division of the normative
sciences. Pure mathematics shares in common with philosophy
this type of scientific interest in ideal, as distinct from physical or
phenomenal truth. There is, to be sure, a marked contrast between
the ways in which the mathematician and the philosopher approach,
select, and elaborate their respective sorts of problems. But there
is also a close relation between the two types of investigation in
question. Let us next consider both the contrast and the analogy in
some of their other most general features.

Pure mathematics is concerned with the investigation of the logical
consequences of certain exactly stateable postulates or hypotheses —
such, for instance, as the postulates upon which arithmetic and analy-
sis are founded, or such as the postulates that he at the basis of any
type of geometry. For the pure mathematician, the truth of these
hypotheses or postulates depends, not upon the fact that physical
nature contains phenomena answering to the postulates, but solely
upon the fact that the mathematician is able, with rational consist-
ency, to state these assumed first principles, and to develop their
consequences. Dedekind, in his famous essay, " Was Sind und Was
Sollen die Zahlen," called the whole numbers " freie Schopfungen des
Menschlichen Geistes; " and, in fact, we need not enter into any dis-
cussion of the psychology of our number concept in order to be able
to assert that, however we men first came by our conception of the
whole numbers, for the mathematician the theory of numerical truth
must appear simply as the logical development of the consequences
of a few fundamental j&rst principles, such as those which Dedekind
himself, or Peano, or other recent writers upon this topic, have, in
various forms, stated. A similar formal freedom marks the develop-
ment of any other theory in the realm of pure mathematics. Pure
geometry, from the modern point of view, is neither a doctrine forced

154 NORMATIVE SCIENCE

upon the human mind by the constitution of any primal form of
intuition, nor yet a branch of physical science, limited to describing
the spatial arrangement of phenomena in the external world. Pure
geometry is the theory of the consequences of certain postulates
which the geometer is at liberty consistently to make; so that there
are as many types of geometry as there are consistent systems of
postulates of that generic type of which the geometer takes account.
As is also now well known, it has long been impossible to define pure
mathematics as the science of quantity, or to limit the range of the
exactly stateable hypotheses or postulates with which the mathema-
tician deals to the world of those objects which, ideally speaking,
can be viewed as measurable. For the ideally defined measurable
objects are by no means the only ones whose properties can be stated
in the form of exact postulates or hypotheses; and the possible range
of pure mathematics, if taken in the abstract, and viewed apart from
any question as to the value of given lines of research, appears to be
identical with the whole realm of the consequences of exactly state-
able ideal hypotheses of every type.

One limitation must, however, be mentioned, to which the asser-
tion just made is, in practice, obviously subject. And this is, indeed,
a momentous limitation. The exactly stated ideal hypotheses whose
consequences the mathematician develops must possess, as is some-
times said, sufiicient intrinsic importance to be worthy of scientific
treatment. They must not be trivial hypotheses. The mathema-
tician is not, like the solver of chess problems, merely displaying
his skill in dealing with the arbitrary fictions of an ideal game. His
truth is, indeed, ideal; his world is, indeed, treated by his science as
if this world were the creation of his postulates a " freie Schopfung."
But he does not thus create for mere sport. On the contrary, he re-
ports a significant order of truth. As a fact, the ideal systems of the
pure mathematician are customarily defined with an obvious, even
though often highly abstract and remote, relation to the structure
of our ordinary empirical world. Thus the various algebras which
have been actually developed have, in the main, definite relations
to the structure of the space world of our physical experience. The
different systems of ideal geometry, even in all their ideality, still
cluster, so to speak, about the suggestions which our daily experi-
ence of space and of matter give us. Yet I suppose that no mathe-
matician would be disposed, at the present time, to accept any brief
definition of the degree of closeness or remoteness of relation to or-
dinary experience which shall serve to distinguish a trivial from
a genuinely significant branch of mathematical theory. In general, a
mathematician who is devoted to the theory of functions, or to group
theory, appears to spend little time in attempting to show why the
development of the consequences of his postulates is a significant

THE SCIENCES OF THE IDEAL 155

enterprise. The concrete mathematical interest of his inquiry sustains
him in his labors, and wins for him the sympathy of his fellows. To
the questions, " Why consider the ideal structure of just this system
of object at all? " " Why study various sorts of numbers, or the
properties of functions, or of groups, or the system of points in
projective geometry? " — the pure mathematician in general, cares
to reply only, that the topic of his special investigation appears to
him to possess sufficient mathematical interest. The freedom of his
science thus justifies his enterprise. Yet, as I just pointed out, this
freedom is never mere caprice. This ideal interest is not without a
general relation to the concerns even of common sense. In brief, as
it seems at once fair to say, the pure mathematician is working under
the influence of more or less clearly conscious philosophical motives.
He does not usually attempt to define what distinguishes a signi-
ficant from a trivial system of postulates, or what constitutes a pro-
blem worth attacking from the point of view of pure mathematics.
But he practically recognizes such a distinction between the trivial
and the significant regions of the world of ideal truth, and since
philosophy is concerned with the significance of ideas, this recogni-
tion brings the mathematician near in spirit to the philosopher.

Such, then, is the position of the pure mathematician. What, by
way of contrast, is that of the philosopher? We may reply that to
state the formal consequences of exact assumptions is one thing; to
reflect upon the mutual relations, and the whole significance of such
assumptions, does indeed involve other interests; and these other
interests are the ones which directly carry us over to the realm of
philosophy. If the theory of numbers belongs to pure mathematics,
the study of the place of the number concept in the system of
human ideas belongs to philosophy. Like the mathematician, the
philosopher deals directly with a realm of ideal truth. But to unify
our knowledge, to comprehend its sources, its meaning, and its re-
lations to the whole of human life, these aims constitute the proper
goal of the philosopher. In order, however, to accomplish his aims,
the philosopher must, indeed, take account of the results of the
special physical science; but he must also turn from the world of
outer phenomena to an ideal world. For the unity of things is never,
for us mortals,, any thing that we find given in our experience. You
cannot see the unity of knowledge; you cannot describe it as a phe-
nomenon. It is for us now, an ideal. And precisely so, the mean-
ing of things, the relation of knowledge to life, the significance of
our ideals, their bearing upon one another — these are never, for us
men, phenomenally present data. Hence the philosopher, however
much he ought, as indeed he ought, to take account of phenomena,
and of the results of the special physical sciences, is quite as deeply
interested in his own way, as the mathematician is interested in his

156 NORMATIVE SCIENCE

way, in the consideration of an ideal realm. Only, unlike the mathe-
matician, the philosopher does not first abstract from the empirical
suggestions upon which his exact ideas are actually based, and then
content himself merely with developing the logical consequences of
these ideas. On the contrary, his main interest is not in any idea or
fact in so far as it is viewed by itself, but rather in the interrelations,
in the common significance, in the unity, of all fundamental ideas,
and in their relations both to the phenomenal facts and to life! On
the whole, he, therefore, neither consents, like the student of a special
science of experience, to seek his freedom solely through conformity
to the phenomena which are to be described; nor is he content, like
the pure mathematician, to win his truth solely through the exact
definition of the formal consequences of his freely defined hypotheses.
He is making an effort to discover the sense and the unity of the
business of his own life.

It is no part of my purpose to attempt to show here how this gen-
eral philosophical interest differentiates into the various interests of
metaphysics, of the philosophy of religion, of ethics, of aesthetics,
of logic. Enough — I have tried to illustrate how, while both the
philosopher and the mathematician have an interest in the meaning
of ideas rather than in the description of external facts, still there
is a contrast which does, indeed, keep their work in large measure
asunder, namely, the contrast due to the fact that the mathematician
is directly concerned with developing the consequences of certain
freely assumed systems of postulates or hypotheses; while the philo-
sopher is interested in the significance, in the unity, and in the re-
lation to life, of all the fundamental ideals and postulates of the
human mind.

Yet not even thus do we sufficiently state how closely related
the two tasks are. For this very contrast, as we have also suggested,
is, even within its own limits, no final or perfectly sharp contrast.
There is a deep analogy between the two tasks. For the mathema-
tician, as we have just seen, is not evenly interested in developing
the consequences of any and every system of freely assumed pos-
tulates. He is no mere solver of arbitrary ideal puzzles in general.
His systems of postulates are so chosen as to be not trivial, but sig-
nificant. They are, therefore, in fact, but abstractly defined aspects
of the very system of eternal truth whose expression is the universe.
In this sense the mathematician is as genuinely interested as is the
philosopher in the significant use of his scientific freedom. On the
other hand, the philosopher, in reflecting upon the significance and
the unity of fundamental ideas, can only do so with success in case
he makes due inquiry into the logical consequences of given ideas.
And this he can accomplish only if, upon occasion, he employs the
exact methods of the mathematician, and develops his sj^stems of

THE SCIENCES OF THE IDEAL 157

ideal truth with the precision of which only mathematical research
is capable. As a fact, then, the mathematician and the philosopher
deal with ideal truth in ways which are not only contrasted, but
profoundly interconnected. The mathematician, in so far as he con-
sciously distinguishes significant from trivial problems, and ideal
systems, is a philosopher. The philosopher, in so far as he seeks
exactness of logical method, in his reflection, must meanwhile aim
to be, within his own limits, a mathematician. He, indeed, will not
in future, like Spinoza, seek to reduce philosophy to the mere develop-
ment, in mathematical form, of the consequences of certain arbitrary
hypotheses. He will distinguish between a reflection upon the unity
of the system of truth and an abstract development of this or that
selected aspect of the system. But he will see more and more that,
in so far as he undertakes to be exact, he must aim to become, in
his own way, and with due regard to his own purposes, mathemat-
ical; and thus the union of mathematical and philosophical inquiries,
in the future, will tend to become closer and closer.

So far, then, I have dwelt upon extremely general considerations
relating to the unity and the contrast of mathematical and philo-
sophical inquiries. I can well conceive, however, that the individual
worker in any one of the numerous branches of investigation which
are represented by the body of students whom I am privileged to
address, may at this point mentally interpose the objection that all
these considerations are", indeed, far too general to be of practical
interest to any of us. Of course, all we who study these so-called
normative sciences are, indeed, interested in ideas, for their own
sakes — in ideas so distinct from, although of course also somehow
related to, phenomena. Of course, some of us are rather devoted to
the development of the consequences of exactly stated ideal hypo-
theses, and others to reflecting as we can upon what certain ideas and
ideals are good for, and upon what the unity is of all ideas and ideals.
Of course, if we are wise enough to do so, we have much to learn
from one another. But, you will say, the assertion of all these things
is a commonplace. The expression of the desire for further mutual
cooperation is a pious wish. You will insist upon asking further:
" Is there just now any concrete instance in a modern type of research
which furnishes results such as are of interest to all of us? Are
we actually doing any productive work in common? Are the philo-
sophers contributing anything to human knowledge which has a
genuine bearing upon the interests of mathematical science? Are
the mathematicians contributing anything to philosophy?"

These questions are perfectly fair. Moreover, as it happens, they

158 NORMATIVE SCIENCE

can be distinctly answered in the affirmative. The present age is one
of a rapid advance in the actual unification of the fields of investi-
gation which are included within the scope of this present division.
What little time remains to me must be devoted to indicating, as
well as I can, in what sense this is true. I shall have still to deal
in very broad generalities. I shall try to make these generalities
definite enough to be not wholly unfruitful.

We have already emphasized one question which may be said to
interest, in a very direct way, both the mathematician and the
philosopher. The ideal postulates, whose consequences mathemat-
ical science undertakes to develop, must be, we have said, significant
postulates, involving ideas whose exact definition and exposition
repay the labor of scientific scrutiny. Number, space, continuity,
functional correspondence or dependence, group-structure — these
are examples of such significant ideas; the postulates or ideal
assumptions upon which the theory of such ideas depends are signi-
ficant postulates, and are not the mere conventions of an arbitrary
game. But now what constitutes the significance of an idea, or
of an abstract mathematical theory? What gives an idea a worthy
place in the whole scheme of human ideas? Is it the possibility of
finding a physical application for a mathematical theory which
for us decides what is the value of the theory? No, the theory of
functions, the theory of numbers, group theory, have a significance
which no mathematician would consent to measure in terms of the
present applicability or non-applicability of these theories in physical
science? In vain, then, does one attempt to use the test of applied
mathematics as the main criticism of the value of a theory of pure
mathematics. The value of an idea, for the sciences which con-
stitute our division, is dependent upon the place which this idea
occupies in the whole organized scheme or system of human ideas.
The idea of number, for instance, familiar as its applications are,
does not derive its main value from the fact that eggs and dollars
and star-clusters can be counted, but rather from the fact that the
idea of numbers has those relations to other fundamental ideas
which recent logical theory has made prominent — relations, for
instance, to the concept of order, to the theory of classes or collec-
tions of objects viewed in general, and to the metaphysical concept
of the self. Relations of this sort, which the discussions of the num-
ber concept by Dedekind, Cantor, Peano, and Russell have recently
brought to light — such relations, I say, constitute what truly justi-
fied Gauss in calling the theory of numbers a " divine science." As
against such deeper relations, the countless applications of the
number concept in ordinary life, and in science, are, from the truly
philosophical point of view, of comparatively small moment. What
we want, in the work of our division of the sciences, is to bring to

THE SCIENCES OF THE IDEAL 159

light the unity of truth, either, as in mathematics, by developing
systems of truth which are significant by virtue of their actual rela-
tions to this unity, or, as in philosophy, by explicitly seeking the
central idea about which all the many ideas cluster.

Now, an ancient and fundamental problem for the philosophers
is that which has been called the problem of the categories. This
problem of the categories is simply the more formal aspect of the
whole philosophical problem just defined. The philosopher aims to
comprehend the unity of the system of human ideas and ideals. Well,
then, what are the primal ideas? Upon what group of concepts do
the other concepts of human science logicall}'' depend? About what
central interests is the system of human ideals clustered? In ancient
thought Aristotle already approached this problem in one way.
Kant, in the eighteenth century, dealt with it in another. We stu-
dents of philosophy are accustomed to regret what we call the ex-
cessive formalism of Kant, to lament that Kant was so much the
slave of his own relatively superficial and accidental table of catego-
ries, and that he made the treatment of every sort of philosophical
problem turn upon his own schematism. Yet we cannot doubt that
Kant was right in maintaining that philosophy needs, for the suc-
cessful development of every one of its departments, a well-devised
and substantially complete system of categories. Our objection to
Kant's over-confidence in the virtues of his own schematism is due
to the fact that we do not now accept his table of categories as an
adequate view of the fundamental concepts. The efforts of philo-
sophers since Kant have been repeatedly devoted to the task of
replacing his scheme of categories by a more adequate one. I am
far from regarding these purely philosophical efforts made since
Kant as fruitless, but they have remained, so far, very incomplete,
and they have been held back from their due fullness of success by
the lack of a sufficiently careful survey and analysis of the processes
of thought as these have come to be embodied in the living sciences.
Such concepts as number, quantity, space, time, cause, continuity,
have been dealt with by the pure philosophers far too summarily
and superficially. A more thoroughgoing analysis has been needed.
But now, in comparatively recent times, there has developed a re-
gion of inquiry which one may call by the general name of modern
logic. To the constitution of this new region of inquiry men have
principally contributed who began as mathematicians, but who, in
the course of their work, have been led to become more and more
philosophers. Of late, however, various philosophers, who were
originally in no sense mathematicians, becoming aware of the im-
portance of the new type of research, are in their turn attempting
both to assimilate and to supplement the undertakings which were
begun from the mathematical side. As a result, the logical problem

160 NORMATIVE SCIENCE

of the categories has to-day become almost equally a problem for
the logicians of mathematics and for those students of philosophy
who take any serious interest in exactness of method in their own
branch of work. The result of this actual cooperation of men from
both sides is that, as I think, we are to-day, for the first time, in
sight of what is still, as I freely admit, a somewhat distant goal,
namely, the relatively complete rational analysis and tabulation of
the fundamental categories of human thought. That the student of
ethics is as much interested in such an investigation as is the meta-
physician, that the philosopher of religion needs a well-completed
table of categories quite as much as does the pure logician, every
competent student of such topics ought to admit. And that the
enterprise in question keenly interests the mathematicians is shown
by the prominent part which some of them have taken in the re-
searches in question. Here, then, is the type of recent scientific work
whose results most obviously bear upon the tasks of all of us alike.
A catalogue of the names of the workers in this wide field of
modern logic would be out of place here. Yet one must, indeed,
indicate what lines of research are especially in question. From the
purelj'" mathematical side, the investigations of the type to which I
now refer may be viewed (somewhat arbitrarily) as beginning with
that famous examination into one of the postulates of Euclid's
geometry which gave rise to the so-called non-Euclidean geometry.
The question here originally at issue was one of a comparatively
limited scope, namely, the question whether Euclid's parallel-line
postulate was a logical consequence of the other geometrical prin-
ciples. But the investigation rapidly develops into a general study
of the foundations of geometry — a study to which contributions
are still almost constantly appearing. Somewhat independently
of this line of inquiry there grew up, during the latter half of the
nineteenth century, that reexamination of the bases of arithmetic
and analysis which is associated with the names of Dedekind, Weier-
strass, and George Cantor. At the present time, the labors of a num-
ber of other inquirers (amongst whom we may mention the school
of Peano and Fieri in Italy, and men such as Poincare and Couturat
in France, Hilbert in Germany, Bertrand Russell and Whitehead in
England, and an energetic group of our American mathematicians
— men such as Professor Moore, Professor Halsted, Dr. Hunting-
ton, Dr. Veblen, and a considerable number of others) have been
added to the earlier researches. The result is that we have recently
come for the first time to be able to see, with some completeness,
what the assumed first principles of pure mathematics actually are.
As was to be expected, these principles are capable of more than
one formulation, according as they are approached from one side or
from another. As was also to be expected, the entire edifice of pure

THE SCIENCES OF THE IDEAL • 161

mathematics, so far as it has yet been erected, actually rests upon
a very few fundamental concepts and postulates, however you may
formulate them. What was not observed, however, by the earlier,
and especially by the philosophical, students of the categories, is
the form which these postulates tend to assume when they are
rigidly analyzed.

This form depends upon the precise definition and classification
of certain types of relations. The whole of geometry, for instance,
including metrical geometry, can be developed from a set of postu-
lates which demand the existence of points that stand in certain
ordinal relationships. The ordinal relationships can be reduced,
according as the series of points considered is open or closed, either
to the well-known relationship in which three points stand when
one is between the other two upon a right line, or else to the ordinal
relationship in which four points stand when they are separated by
pairs; and these two ordinal relationships, by means of various log-
ical devices, can be regarded as variations of a single fundamental
form. Cayley and Klein founded the logical theory of geometry here
in question. Russell, and in another way Dr. Veblen, have given
it its most recent expressions. In the same way, the theory of whole
numbers can be redut^ed to sets of principles which demand the exist-
ence of certain ideal objects in certain simple ordinal relations. Dede-
kind and Peano have worked out such ordinal theories of the num-
ber concept. In another development of the theory of the cardinal
whole numbers, which Russell and Whitehead have worked out,
ordinal concepts are introduced only secondarily, and the theory
depends upon the fundamental relation of the equivalence or non-
equivalence of collections of objects. But here also a certain simple
type of relation determines the definitions and the development of
the whole theory.

Two results follow from such a fashion of logically analyzing the
first principles of mathematical science. In the first place, as just
pointed out, we learn how jew and simple are the conceptions and pos-
tulates upon which the actual edifice of exact science rests. Pure
mathematics, we have said, is free to assume what it chooses. Yet
the assumptions whose presence as the foundation principles of the
actually existent pure mathematics an exhaustive examination thus
reveals, show by their fe^^^less that the ideal freedom of the mathe-
matician to assume and to construct what he pleases, is indeed, in
practice, a very decidedly limited freedom. The limitation is, as we
have already seen, a limitation which has to do with the essential
significance of the fundamental concepts in question. And so the
result of this analysis of the bases of the actually developed and
significant branches of mathematics, constitutes a sort of empirical
revelation of what categories the exact sciences have practically

162 NORMATIVE SCIENCE

found to be of such significance as to be worthy of exhaustive treat-
ment. Thus the instinctive sense for significant truth, which has all
along been guiding the development of mathematics, comes at least
to a clear and philosophical consciousness. And meanwhile the es-
sential categories of thought are seen in a new light.

The second result still more directly concerns a philosophical logic.
It is this: Since the few types of relations which this sort of ana-
lysis reveals as the fundamental ones in exact science are of such
importance, the logic of the present day is especially required to face
the questions : What is the nature of our concept of relations f What
are the various possible types of relations? Upon what does the
variety of these types depend? What unity lies beneath the variety?

As a fact, logic, in its modern forms, namely, first that symbolic
logic which Boole first formulated, which Mr. Charles S. Peirce and
his pupils have in this country already so highly developed, and
which Schroeder in Germany, Peano's school in Italy, and a num-
ber of recent English writers have so effectively furthered — and
secondly, the logic of scientific method, which is now so actively
pursued, in France, in Germany, and in the English-speaking coun-
tries — this whole movement in modern logic, as I hold, is rapidly
approaching new solutions of the problem of the fundamental nature
and the logic of relations. The problem is one in which we are all
equally interested. To De Morgan in England, in an earlier genera-
tion, and, in our time, to Charles Peirce in this country, very im-
portant stages in the growth of these problems are due. Russell, in
his work on the Principles of Mathematics has very lately under-
taken to sum up the results of the logic of relations, as thus far
developed, and to add his own interpretations. Yet I think that
Russell has failed to get as near to the foundations of the theory
of relations as the present state of the discussion permits. For
Russell has failed to take account of what I hold to be the most
fundamentally important generalization yet reached in the general
theory of relations. This is the generalization set forth as early as
1890, by Mr. A. B. Kempe, of London, in a pair of wonderful but
too much neglected, papers, entitled, respectively. The Theory of
Mathematical Form, and The Analogy between the Logical Theory
of Classes and the Geometrical Theory of Points. A mere hint first
as to the more precise formulation of the problem at issue, and then
later as to Kempe 's special contribution to that problem, may be in
order here, despite the impossibility of any adequate statement.

Ill

The two most obviously and universally important kinds of rela-
tions known to the exact sciences, as these sciences at present exist,
are: (1) The relations of the type of equality or equivalence; and

THE SCIENCES OF THE IDEAL 163

(2) the relations of the type of before and after, or greater and less.
The first of these two classes of relations, namely, the class repre-
sented, although by no means exhausted, by the various relations
actually called, in different branches of science by the one name
equality, this class I say, might well be named, as I myself have
proposed, the leveling relations. A collection of objects between
any two of which some one relation of this type holds, may be said
to be a collection whose members, in some defined sense or other,
are on the same level. The second of these two classes of relations,
namely, those of the type of before and after, or greater and less
■ — this class of relations, I say, consists of what are nowadays often
called the serial relations. And a collection of objects such that, if
any pair of these objects be chosen, a determinate one of this pair
stands to the other one of the same pair in some determinate rela-
tion of this second type, and in a relation which remains constant
for all the pairs that can be thus formed out of the members of this
collection — any such collection, I say, constitutes a one-dimen-
sional open series. Thus, in case of a file of men, if you choose any
pair of men belonging to the file, a determinate one of them is, in the
file, before the other. In the number series, of any two numbers,
a determinate one is greater than the other. Wherever such a state
of affairs exists, one has a series.

Now these two classes of relations, the leveling relations and the
serial relations, agree with one another, and differ from one another
in very momentous ways. They agree with one another in that both
the leveling and the serial relations are what is technically called
transitive; that is, both classes conform to what Professor James
has caUed the law of "skipped intermediaries." Thus, if A is equal
to B, and B is equal to C, it follows that A is equal to C. If A is
before B, and B is before C, then A is before C. And this property,
which enables you in your reasonings about these relations to skip
middle terms, and so to perform some operation of elimination, is
the property which is meant when one calls relations of this type
transitive. But, on the other hand, these two classes of relations
differ from each other in that the leveling relations are, while the
serial relations are not, symmetrical or reciprocal. Thus, if A is equal
to B, B is equal to A. But if X is greater than Y, then Y is not
greater than X, but less than X. So the leveling relations are sym-
metrical transitive relations. But the serial relations are transitive
relations which are not symmetrical.

All this is now well known. It is notable, however, that nearly
all the processes of our exact sciences, as at present developed,
can be said to be essentially such as lead either to the placing of sets
or classes of objects on the same level, by means of the use of sym-
metrical transitive relations, or else to the arranging of objects in

164 NORMATIVE SCIENCE

orderly rows or series, by means of the use of transitive relations
which are not symmetrical. This holds also of all the applications
of the exact sciences. Whatever else you do in science (or, for that
matter, in art), you alwaj^s lead, in the end, either to the arrang-
ing of objects, or of ideas, or of acts, or of movements, in rows or
series, or else to the placing of objects or ideas of some sort on the
same level, by virtue of some equivalence, or of some invariant
character. Thus numbers, functions, lines in geometry, give you
examples of serial relations. Equations in mathematics are classic
instances of leveling relations. So, of course, are invariants. Thus,
again, the whole modern theory of energy consists of two parts,
one of which has to do with levels of energy, in so far as the quan-
tity of energy of a closed system remains invariant through all the
transformations of the system, while the other part has to do with
the irreversible serial order of the transformations of energy them-
selves, which follow a set of unsymmetrical relations, in so far as
energy tends to fall from higher to lower levels of intensity within
the same system.

The entire conceivable universe then, and all of our present exact
science, can be viewed, if you choose, as a collection of objects or
of ideas that, whatever other types of relations may exist, are at
least largely characterized either by the leveling relations, or by
the serial relations, or by complexes of both sorts of relations. Here,
then, we are plainly dealing with very fundamental categories.
The "between" relations of geometry can of course be defined, if
you choose, in terms of transitive relations that are not symmet-
rical. There are, to be sure, some other relations present in exact
science, but the two types, the serial and leveling relations, are
especially notable.

So far the modern logicians have for some time been in substan-
tial agreement. Russell's brilliant book is a development of the
logic of mathematics very largely in terms of the two types of rela-
tions which, in my own way, I have just characterized; although
Russell gives due regard, of course, to certain other types of rela-
tions.

But hereupon the question arises, "Are these two types of rela-
tions what Russell holds them to be, namely, ultimate and irre-
ducible logical facts, unanalyzable categories — mere data for the
thinker? Or can we reduce them still further, and thus simplify
yet again our view of the categories?

Here is where Kempe's generalization begins to come into sight.
These two categories, in at least one very fundamental realm of
exact thought, can be reduced to one. There is, namel}'', a world
of ideal objects which especially interest the logician. It is the
world of a totality of possible logical classes, or again, it is the ideal

THE SCIENCES. OF THE IDEAL 165

world, equivalent in formal structure to the foregoing, but composed
of a totality of possible statements, or thirdly, it is the world, equiva-
lent once more, in formal structure, to the foregoing, but consisting
of a totality of possible acts of will, of possible decisions. When we
proceed to consider the relational structure of such a world, taken
merely in the abstract as such a structure, a relation comes into
sight which at once appears to be peculiarly general in its nature.
It is the so-caUed illative relation, the relation which obtains between
two classes when one is subsumed under tke other, or between two
statements, or two decisions, when one implies or entails the other.
This relation is transitive, but may be either symmetrical or not
symmetrical; so that, according as it is symmetrical or not, it may
be used either to establish levels or to generate series. In the order
system of the logician's world, the relational structure is thus, in
any case, a highly general and fundamental one.

But this is not all. In this the logician's world of classes, or of
statements, or of decisions, there is also another relation observable.
This is the relation of exclusion or mutual opposition. This is a
purely symmetrical or reciprocal relation. It has two forms —
obverse or contradictory opposition, that is, negation proper, and
contrary opposition. But both these forms are purely symmetrical.
And by proper devices each of them can be stated in terms of the
other, or reduced to the other. And further, as Kempe incidentally
shows, and as Mrs. Ladd Franklin has also substantiall}'- shown in
her important theory of the syllogism, it is possible to state every
proposition, or complex of propositions involving the illative relation,
in terms of this purely symmetrical relation of opposition. Hence,
so far as mere relational form is concerned, the illative relation itself
may be wholly reduced to the symmetrical relation of opposition.
This is our first result as to the relational structure of the realm of
pure logic, that is, the realm of classes, of statements, or of deci-
sions.

It follows that, in describing the logician's world of possible classes
or of possible decisions, all unsymmetrical, and so all serial, relations
can be stated solely in terms of symmetrical relations, and can be entirely
reduced to such relations. Moreover, as Kempe has also very prettily
shown, the relation of opposition, in its two forms, just mentioned,
need not be interpreted as obtaining merely between pairs of objects.
It may and does obtain between triads, tetrads, n-ads of logical en-
tities; and so all that is true of the relations of logical classes may
consequently be stated merely by ascribing certain perfectly sym-
metrical and homogeneous predicates to pairs, triads, tetrads, n-ads
of logical objects. The essential contrast between symmetrical
and unsymmetrical relations thus, in this ideal realm of the logi-
cian, simply vanishes. The categories of the logician's world of

166 NORMATIVE SCIENCE

classes, of statements, or of decisions, are marvelously simple. All
the relations present may be viewed as variations of the mere con-
ception of opposition as distinct from non-opposition.

All this holds, of course, so far, merely for the logician's world of
classes or of decisions. There, at least, all serial order can actually
be derived from wholly symmetrical relations. But Kempe now
very beautifully shows (and here lies his great and original contri-
bution to our topic) — he shows, I say, that the ordinal relations
of geometry, as well as of the number-system, can all be regarded
as indistinguishable from mere variations of those relations which,
in pure logic, one finds to he the symmetrical relations obtaining within
pairs or triads of classes or of statements. The formal identity of the
geometrical relation called "between" with a purely logical relation
which one can define as existing or as not existing amongst the mem-
bers of a given triad of logical classes, or of logical statements, is
shown by Kempe in a fashion that I cannot here attempt to expound.
But Kempe's result thus enables one, as I believe, to simplify the
theory of relations far beyond the point which Russell in his brilliant
book has reached. For Kempe's triadic relation in question can be
stated, in what he calls its obverse form, in perfectly symmetrical
terms. And he proves very exactly that the resulting logical rela-
tion is precisely identical, in all its properties, with the fundamental
ordinal relation of geometry.

Thus the order-systems of geometry and analysis appear simply
as special cases of the more general order-system of pure logic. The
whole, both of analysis and of geometrj^, can be regarded as a de-
scription of certain selected groups of entities, which are chosen,
according to special rules, from a single ideal world. This general
and inclusive ideal world consists simply of all the objects which can
stand to one another in those symmetrical relations wherein the pure lo-
gician finds various statements, or various decisions inevitably standing.
'' Let me," says in substance Kempe, " choose from the logician's
ideal world of classes or decisions, what entities I will; and I will
show you a collection of objects that are in their relational structure,
precisely identical with the points of a geometer's space of n dimen-
sions." In other words, all of the geometer's figures and relations can
be precisely pictured by the relational structure of a selected system
of classes or of statements, whose relations are wholly and explicitly
logical relations, such as opposition, and whose relations may all
be regarded, accordingly, as reducible to a single type of purely
symmetrical relation.

Thus, for all exact science, and not merely for the logician's special
realm, the contrast between symmetrical and unsymmetrical rela-
tions proves to be, after all, superficial and derived. The purely
logical categories, such as opposition, and such as hold within the

THE SCIENCES OF THE IDEAL 167

calculus of statements, are, apparently, the basal categories of all
the exact science that has yet been developed. Series and levels are
relational structures that, sharply as they are contrasted, can be
derived from a single root.

I have restated Kempe's generalization in my own way. I think
it the most promising step towards new light as to the categories
that we have made for some generations.

In the field of modern logic, I say, then, work is doing which is
rapidly tending towards the unification of the tasks of our entire
division. For this problem of the categories, in all its abstractness,
is still a common problem for all of us. Do you ask, however, what
such researches can do to furnish more special aid to the workers
in metaphysics, in the philosophy of religion, in ethics, or in aesthetics,
beyond merely helping towards the formulation of a table of cate-
gories — then I reply that we are already not without evidence that
such general researches, abstract though they may seem, are bear-
ing fruits which have much more than a merely special interest.
Apart from its most general problems, that analysis of mathemat-
ical concepts to which I have referred has in any case revealed
numerous unexpected connections between departments of thought
which had seemed to be very widely sundered. One instance of such
a connection I myself have elsewhere discussed at length, in its gen-
eral metaphysical bearings. I refer to the logical identity which
Dedekind first pointed out between the mathematical concept of
the ordinal number of series and the philosophical concept of the
formal structure of an ideally completed self. I have maintained
that this formal identity throws light upon problems which have as
genuine an interest for the student of the philosophy of religion as
for the logician of arithmetic. In the same connection it may be
remarked that, as Couturat and Russell, amongst other writers,
have very clearly and beautifully shown, the argument of the Kant-
ian mathematical antinomies needs to be explicitly and totally
revised in the light of Cantor's modern theory of infinite collections.
To pass at once to another, and a very different instance : The mod-
ern mathematical conceptions of what is called group theory have
already received very wide and significant applications, and promise
to bring into unity regions of research which, until recently, appeared
to have little or nothing to do with one another. Quite lately, how-
ever, there are signs that group theory will soon prove to be of im-
portance for the definition of some of the fundamental concepts of
that most refractory branch of philosophical inquiry, aesthetics. Dr.
Emch, in an important paper in the Monist, called attention, some
time since, to the symmetry groups to which certain aesthetically
pleasing forms belong, and endeavored to point out the empirical
relations between these groups and the aesthetic effects in question .

168 NORMATIVE SCIENCE

The grounds for such a connection between the groups in question
and the observed sesthetic effects, seemed, in the paper of Dr. Emch
to be left largely in the dark. But certain papers recently published
in the country by Miss Ethel Puffer, bearing upon the psychology
of the beautiful (although the author has approached the subject
without being in the least consciously influenced, as I understand,
by the conceptions of the mathematical group theory), still actually
lead, if I correctly grasp the writer's meaning, to the doctrine that
the sesthetic object, viewed as a psychological whole, must possess
a structure closely, if not precisely, equivalent to the ideal structure
of what the mathematician calls a group. I myself have no authority
regarding sesthetic concepts, and speak subject to correction. But
the unexpected, and in case of Miss Puffer's research, quite unin-
tended, appearance of group theory in recent sesthetic analysis is to
me an impressive instance of the use of relatively new mathematical
conceptions in philosophical regions which seem, at first sight, very
remote from mathematics.

That both the group concept and the concept of the self just sug-
gested are sure to have also a wide application in the ethics of the
future, I am myself well convinced. In fact, no branch of philosophy is
without close relations to all such studies of fundamental categories.

These are but hints and examples. They suffice, I hope, to show
that the workers in this division have deep common interests, and
will do well, in future, to study the arts of cooperation, and to regard
one another's progress with a watchful and cordial sympathy. In a
word: Our common problem is the theory of the categories. That
problem can be solved only by the cooperation of the mathema-
ticians and of the philosophers.

DEPARTMENT I — PHILOSOPHY

{Hall 6, September 20, 11.15 a. m.)

CHAiRiLVN: Professor Borden P. Bowne, Boston University.
Speakers: Professor George H. Howison, University of California.
Professor George T. Ladd, Yale University.

In opening the Department of Philosophy, the Chairman, Pro-
fessor Borden P. Bowne, LL.D., of Boston University, made an
interesting address on the Philosophical Outlook. Professor Bowne
said in part : —

I congratulate the members of the Philosophical Section on the improved out-
look in philosophy. In the generation just passed, philosophy was somewhat at
a discount. The great and rapid development of physical science and invention,
together with the profound changes in biological thought, produced for a time a
kind of chaos. New facts were showered upon us in great abundance, and we had
no adequate philosophical preparation for dealing with them. Such a condition is
always disturbing. The old mental equilibrium is overthrown and readjustment
is a slow process. Besides, the shallow sense philosophy of that time readily lent
itself to mechanical and materialistic interpretations, and for a while it seemed
as if all the higher faiths of humanity were permanently discredited. All this has
passed away. Philosophical criticism began its work and the naive dogmatism of
materialistic naturalism was soon disposed of. It quickly appeared that our trouble
was not due to the new facts, but to the superficial philosophy by which they had
been interpreted. Now that we have a better philosophy, we have come to live in
perfect peace with the facts once thought disturbing, and even to welcome them as
valuable additions to knowledge. . . .

The brief naturalistic episode was not without instruction for us. It showed
conclusively the great practical importance of philosophy. Had we had thirty
years ago the current philosophical insight, the great development of the physical
and biological sciences would have made no disturbance whatever. But being
interpreted by a crude scheme of thought, it produced somewhat of a storm.
Philosophy may not contribute much of positive value, but it certainly has an
important negative fimction in the way of suppressing pretentious dogmatism
and fictitious knowledge, which often lead men astray. It is these things which
produce conflicts of science and religion or which find in evolution the solvent of
all mysteries and the source of all knowledge.

Concerning the partition of territory between science and philosophy, there
are two distinct questions respecting the facts of experience. First, we need to
know the facts in their temporal and spatial order, and the way they hang together
in a system of law. To get this knowledge is the function of science, and in this
work science has Inalienable rights and a most important practical function. This
work cannot be done by speculation nor interfered with by authority of any kind.
It is not surprising, then, that scientists in their sense of contact with reality

172 PHILOSOPHY

should be indignant with, or feel contempt for, any who seek to limit or proscribe
their research. But supposing this work all done, there remains another question
respecting the causality and interpretation of the facts. This question belongs to
philosophy. Science describes and registers the facts with their temporal and
spatial laws; philosophy studies their causality and significance. And while the
scientist justly ignores the philosopher who interferes with his inquiries, so the
philosopher may justly reproach the scientist who fails to see that the scientific
question does not touch the philosophic one. . . .

In the field of metaphysics proper I note a strong tendency toward personal
idealism, or as it might be called, Personalism; that is, the doctrine that sub-
stantial reality can be conceived only under the personal form and that all else is
phenomenal. This is quite distinct from the traditional idealisms of mere concep-
tionism. It holds the essential fact to be a community of persons with a Supreme
Person at their head whUe the phenomenal world is only expression and means
of communication. And to this view we are led by the failure of philosophizing on
the impersonal plane, which is sure to lose itself in contradiction and impossi-
bility. Under the form of mechanical naturalism, with its tendencies to mate-
riahsm and atheism, impersonalism has once more been judged and found want-
ing. We are not Ukely to have a recurrence of this view imless there be a return
to philosophical barbarism. But impersonalism at the opposite pole in the form
of abstract categories of being, causahty, unity, identity, continuity, sufficient
reason, etc., is equally untenable. Criticism shows that these categories when
abstractly and impersonally taken cancel themselves. On the impersonal plane we
can never reach unity from plurality, or pluraMty from unity; and we can never
find change in identity, or identity in change. Continuity in time becomes mere
succession without the notion of potentiality, and this in turn is empty. Exist-
ence itself is dispersed into nothingness through the infinite divisibility of space
and time, while the law of the sufficient reason loses itself in barren tautology and
the infinite regress. The necessary logical equivalence of cause and effect in any
impersonal scheme makes all real explanation and progress impossible, and shuts
us up to an unintelligible oscillation between potentiahty and actuality, to which
there is no corresponding thought. . . .

Philosophy is still mihtant and has much work before it, but the omens are
auspicious, the problems are better understood, and we are coming to a synthesis
of the results of past generations of thinking which wiU be a very distinct progress.
Philosophy has aheady done good service, and never better than in recent times,
by destroying pretended knowledge and making room for the higher faiths of
humanity. It has also done good service in helping these faiths to better rational
form, and thus securing them against the defilements of superstition and the
cavilings of hostile critics. With aU its aberrations and shortcomings, philosophy
deserves well of humanity.

FUNDAMENTAL METHODS AND CONCEPTIONS 173

PHILOSOPHY: ITS FUNDAMENTAL CONCEPTIONS AND

ITS METHODS

BY GEORGE HOLMES HOWISON

[George Holmes Howison, Mills Professor of Intellectual and Moral Philo-
sophy and Civil Polity, University of California, b. Montgomery County,
Maryland, 1834. A.B. Marietta College, 1852 ; M.A. 1855 ; LL.D. ibid.
1883. Post-graduate, Lane Theological Seminary, University of Berlin,
and Oxford. Headmaster High School, Salem, Mass., 1862-64; Assistant
Professor of Mathematics, Washington University, St. Louis, 1864-66; Tile-
ston Professor of Political Economy, ibid. 1866-69; Professor of Logic and
the Philosophy of Science, Massachusetts Institute of Technology, 1871-79;
Lecturer on Ethics, Harvard University, 1879-80; Lecturer on Logic and
Speculative Philosophy, University of Michigan, 1883-84. Member and vice-
president St. Louis Philosophical Society; member California Historical
Society; American Historical Association; American Association for the
Advancement of Science ; National Geographic Society, etc. Author of
Treatise on Analytic Geometry, 1869; The Limits of Evolution, 1901, 2d edi-
tion, 1904; joint author and editor of The Conception of God, 1897, etc. Editor
Philosophical Publications of University of California; American Editorial
Representative Hibbert Journal, London.]

The duty has been assigned me, honored colleagues, of address-
ing you on the Fundamental Conceptions and the Methods of our
common pursuit — philosophy. In endeavoring to deal with the
subject in a way not unworthy of its depth and its extent, I have
found it impossible to bring the essential material within less com-
pass than would occupy, in reading, at least four times the period
granted by our programme. I have therefore complied with the rule
of the Congress which directs that, if a more extended writing be
left with the authorities for publication, the reading must be re-
stricted to such a portion of it as will not exceed the allotted time.
I will accordingly read to you, first, a brief summary of my entire
discussion, by way of introduction, and then an excerpt from the
larger document, which may serve for a specimen, as our scholastic
predecessors used to say, of the whole inquiry I have carried out.
The impression will, of course, be fragmentary, and I must 'ask
beforehand for your most benevolent allowances, to prevent a judg-
ment too unfavorable.

The discussion naturally falls into two main parts: the first
dealing with the Fundamental Conceptions; and the second, with
the Methods.

In the former, after presenting the conception of philosophy
itself, as the consideration of things in the light of the whole, I take up
the involved Fundamental Concepts in the following order : —
I. Whole and Part;

II. Subject and Object (Knowing and Being, Mind and Matter;
Dualism, Materialism, Idealism);

III. Reality and Appearance (Noumenon and Phenomenon);

174 PHILOSOPHY

IV. Cause and Effect (Ground and Consequence; Causal System);
V. One and Many (Number System; Monism and Pluralism);

VI. Time and Space (their relation to Number; their Origin and

Ileal Meaning) ;
VII. Unconditioned and Conditioned (Soul, World, God; their
Reinterpretation in terms of Pluralism) ;
VIII. The True, the Beautiful, the Good (their relation to the
question between Monism and Pluralism) .

These are successively dealt with as they rise one out of the other
in the process of interpreting them and applying them in the actual
creation of philosophy, as this goes on in the historic schools. The
theoretic progress of philosophy is in this way explained by them,
in its movement from natural dualism, or realism, through the
successive forms of monism, materialistic, agnostic, and idealistic,
until it reaches the issue, now coming so strongly forward within
the school of idealism, between the adherents of monism and those
of pluralism.

The importance of the Fundamental Concepts is shown to increase
as we pass along the list, till on reaching Cause and Effect, and
entering upon its full interpretation into the complete System of
Causes, we arrive at the very significant conception of the Reci-
procity OF First Causes, and through it come to the Primacy of
Final Cause, and the derivative position of the other forms of cause,
Material, Formal, Efficient. The philosophic strength of idealism,
but especially of idealistic pluralism, comes into clear light as the re-
sult of this stage of the inquiry. But it appears yet more decidedly
when One and Many, Time and Space, and their interrelations,
are subjected to analysis. So the discussion next passes to the
higher conceptions. Soul, World, God, by the pathway of the cor-
relation Unconditioned and Conditioned, and its kindred contrasts
Absolute and Relative, Necessary and Contingent, Infinite and
Finite, corroborating and reinforcing the import of idealism, and,
still more decidedly, that of its plural form. Finally, the strong
and favorable bearing of this last on the dissolution of agnosticism
and the habilitation of the ideals, the True, the Beautiful, and the
Good, in a heightened meaning, is brought out.

This carries the inquiry to the second part of it, that of the Philo-
sophical Methods. Here I recount these in a series of six: the
Dogmatic, the Skeptical, the Critical, the Pragmatic, the Genetic,
the Dialectic. These, I show, in spite of the tendency of the earlier
members in the series to over-emphasis, all have their place and
function in the development of a complete philosophy, and in fact
form an ascending series in methodic effectiveness, all that precede
the last being taken up into the comprehensive Critical Rationalism
of the last. Methodology thus passes upward, over the ascending

FUNDAMENTAL CONCEPTIONS AND METHODS 175

and widening roadways of (1) Intuition and Deduction; (2) Ex-
perience and Induction; (3) Intuition and Experience adjusted by
Critical Limits; (4) Skepticism reinforced and made gwast-affirm-
ative by Desire and Will; (5) Empiricism enlarged by substitu-
tion of cosmic and psychic history for subjective consciousness;
(6) Enlightened return to a Rationalism critically established by
the inclusion of the preceding elements, and by the sifting and the
grading of the Fundamental Concepts through their behavior when
tested by the effort to make them universal. In this way, the
methods fall into a System, the organic principle of which is this
principle of Dialectic, which proves itself alone able to establish
necessary truths; that is, truths indeed, — judgments that are seen
to exclude their opposites, because, in the attempt to substitute the
opposite, the place of it is still filled by the judgment which it aims
to dislodge.

And now, with your favoring leave, I will read the excerpt from
my larger text. ^

The task to which, in an especial sense, the cultivators of philo-
sophy are summoned by the plans of the present Congress of Arts
and Science, is certainly such as to stir an ambition to achieve it.
At the same time, it tempers eagerness by its vast difficulty, and the
apprehension lest this may prove insuperable. The task, the officers
of the Congress tell us, is no less than to promote the unification of
all human knowledge. It requires, then, the reduction of the enor-
mous detail in our present miscellany of sciences and arts, which to
a general glance, or even to a more intimate view, presents a con-
fusion of differences that seems overwhelming, to a system never-
theless clearly harmonious, — founded, that is to say, upon uni-
versal principles which control all differences by explaining them,
and which therefore, in the last resort, themselves flow lucidly from
a single supreme principle. Simply to state this meaning of the task
set us, is enough to awaken the doubt of its practicability.

This doubt, we are bound to confess, has more and more impressed
itself upon the general mind, the farther this has advanced in the
experience of scientific discovery. The very increase in the multi-
plicity and complexity of facts and their causal groupings increases
the feeling that at the root of things there is " a final inexplicability "
— total reality seems, more and more, too vast, too profound, for us
to grasp or to fathom. And yet, strangely enough, this increasing
sense of mysterious vastness has not in the least prevented the modern
mind from more and .more asserting, with a steadily increasing in-
sistence, the essential and unchangeable unity of that whole of things
which to our ordinary experience, and even to all our sciences, appears
such an endless and impenetrable complex of differences, — yes, of
contradictions. In fact, this assertion of the unity of all things, under

176 PHILOSOPHY

the favorite name of the Unity of Nature, is the pet dogma of modern
science; or, rather, to speak with right accuracy, it is the stock-in-
trade of a philosophy of science, current among many of the leaders
of modern science; for every such assertion, covering, as it tacitly
and unavoidably does, a view about the absolute whole, is an asser-
tion belonging to the province of philosophy, before whose tribunal
it must come for the assessment of its value. The presuppositions
of all the special sciences, and, above all, this presupposition of the
Unity and Uniformity of Nature, common to all of them, must thus
come back for justification and requisite definition to philosophy —
that uppermost and all-inclusive form of cognition which addresses
itself to the whole as whole. In their common assertion of the Unity
of Nature, the exponents of modern science come unawares out of
their own province into quite another and a higher; and in doing so
they show how unawares they come, by presenting in most instances
the curious spectacle of proclaiming at once their increasing belief
in the unity of things, and their increasing disbelief in its pene-
trability by our intelligence : —

In's Innere der Natur,
Dringt kein erschaffner Geist,

is their chosen poet's expression of their philosophic mood. Curious
we have the right to call this state of the scientific mind, because
it is to critical reflection so certainly self-contradictory. How can
there be a real unity belonging to what is inscrutable? — what evi-
dence of unity can there be, except in intelligible and explanatory
continuity?

But, at all events, this ver}^ mood of agnostic self-contradiction,
into which the development of the sciences casts such a multitude
of minds, brings them, — brings all of us, — as already indicated,
into that court of philosophy where alone such issues lawfully belong,
and where alone they can be adjudicated. If the unification of the
sciences can be made out to be real by making out its sole sufficient
condition, namely, that there is a genuine, and not a merely nominal,
unity in the whole of reality itself, — a unity that explains because
it is itself, not simply intelligible, but the only completely intelligible
of things, — this desirable result must be the work of philosophy.
However difficult the task may be, it is rightly put upon us who belong
to the Department listed first among the twenty-four in the pro-
gramme of this representative Congress.

I cannot but express my own satisfaction, as a member of this
Department, nor fail to extend my congratulations to you who are
my colleagues in it, that the Congress, in its programme, takes
openly the affirmative on this question of the possible unification of
knowledge. The Congress has thus declared beforehand for the

FUNDAMENTAL CONCEPTIONS AND METHODS 177

practicability of the task it sets. It has even declared for its not
distant accomplishment; indeed, not impossibly, its accomplishment
through the transactions of the Congress itself; and it indicates, by
no uncertain signs, the leading, the determining part that philosophy
must have in the achievement. In fact, the authorities of the Congress
themselves suggest a solution of their own for their problem. In their
programme we see a renewed Hierarchy of the Sciences, and at the
summit of this appears now again, after so long a period of humiliating
obscuration, the figure of Philosophy, raised anew to that supremacy,
as Queen of the Sciences, which had been hers from the days of Plato
to those of Copernicus, but which she began to lose when modern
physical and historical research entered upon its course of sudden
development, and which, until recently, she has continued more and
more to lose as the sciences have advanced in their career of discover-
ies, — ever more unexpected, more astonishing, yet more convincing
and more helpful to the welfare of mankind. May this sign of her
recovered empire not fail! If we rejoice at the token, the Congress
has made it our part to see that the title is vindicated. It is ours to
show this normative function of philosophy, this power to reign as the
unifying discipline in the entire realm of our possible knowledge; to
show it by showing that the very nature of philosophy — its ele-
mental concepts and its directing ideals^ its methods taken in their
systematic succession — is such as must result in a view of universal
reality that will supply the principle at once giving rise to all the
sciences and connecting them all into one harmonious whole.

Such, and so grave, my honored colleagues, is the duty assigned to
this hour. Sincerely can I say. Would it had fallen to stronger hands
than mine! But since to mine it has been committed, I will undertake
it in no disheartened spirit; rather, in that temper of animated hope
in which the whole Congress has been conceived and planned. And
I draw encouragement from the place, and its associations, where
we are assembled — from its historic connections not only with the
external expansion of our country, but with its growth in culture,
and especially with its growth in the cultivation of philosophy. For
your speaker, at least, can never forget that here in St. Louis, the
metropolis of the region by which our national domain was in the
Louisiana Purchase so enlarged, — here was the centre of a move-
ment in philosophic study that has proved to be of national import.
It is fitting that we all, here to-day, near to the scene itself, com-
memorate the public service done by our present National Commis-
sioner of Education and his group of enthusiastic associates, in
beginning here, in the middle years of the preceding century, those
studies of Kant and his great idealistic successors that unexpectedly
became the nucleus of a wider and more penetrating study of philo-
sophy in all parts of our country. It is with quickened memories

178 PHILOSOPHY

belonging to the spot where, more than five-and-thirty years ago, it
was my happy fortune to take some part with Dr. Harris and his
companions, that I begin the task assigned me. The undertaking
seems less hopeless when I can here recall the names and the con-
genial labors of Harris, of Davidson, of Brockmeyer, of Snider, of
Watters, of Jones, — half of them now gone from life. They " builded
better than they knew; " and, humbly as they may themselves have
estimated their ingenuous efforts to gain acquaintance with the great-
est thoughts, history will not fail to take note of what they did, as
marking one of the turning-points in the culture of our nation. The
publication of the Journal of Speculative Philosophy, granting all
the subtractions claimed by its critics on the score of defects (of
which its conductors were perhaps only too sensible) , was an influence
that told in all our circles of philosophical study, and thence in the
whole of our social as well as our academic life.

[Here I enter upon the discussion of the subject proper, beginning,
as above indicated, with the Fundamental Conceptions. Having
followed these through the contrasts Whole and Part, Subject and
Object, Reality and Appearance (or Noumenon and Phenomenon),
and developed the bearing of these on the procedure of thought from
the dualism of natural realism to materialism and thence to idealism,
with the issue now coming on, in this last, between monism and
pluralism, I strike into the contrast Cause and Effect, and, noting
its unfolding into the more comprehensive form of Ground and Con-
sequence, go on thence as follows : ]

It is plain that the contrast Ground and Consequence will enable
us to state the new issue with closer precision and pertinence than
Reality and Appearance, Noumenon and Phenomenon, can supply;
while, at the same time. Ground and Consequence exhibits Cause and
Effect as presenting a contrast that only fulfills what Noumenon and
Phenomenon foretold and strove towards; in fact, what was more
remotely, but not less surely, also indicated by Whole and Part,
Knowing and Being, Subject and Object. For in penetrating to the
coherent meaning of these conceptions, the philosophic movement,
as we saw, advanced steadily to the fuller and fuller translating of
each of them into the reality that unifies by explanation, instead of
pretending to explain by merely unifying; and this, of course, will
now be put forward explicitly, in the clarified category of Cause and
Effect, transfigured from a physical into a purely logical relation.
What idealism now says, in terms of this, is that the Cause (or, as
we now read it, the Ground) of all that exists is the Subject; is
Mind, the intelligently Self-conscious; and that all things else, the
mere objects, material things, are its Consequence, its Outcome, —

FUNDAMENTAL CONCEPTIONS AND METHODS 179

in that sense its Effect. And what the new phiraHstic ideahsm says,
is that the assemblage of individual minds — intelligence being
essentially personal and individual, and never merely universal
and collective — is the true total Cause of all, and that every mind
thus belongs to the order of First Causes; nevertheless, that part,
and the most significant part, of the nature of ever}^ mind, essential
to its personality and its reason, is its recognition of other minds in
the very act of its own self -definition. That is to say, a mind by its
spontaneous nature as intelligence, by its intrinsic rational or logical
genius, puts itself as member of a system of minds; all minds are put
by each other as Ends — completely standard and sacred Objects,
as much parts of the system of true Causes as each is, in its capacity
of Subject; and we have a noumenal Reality that is properly to be
described as the eternal Federal Republic of Spirits.

Consequently, the relation of Cause and Effect now expands and
heightens into a sj^'stem of the Reciprocity of First Causes; causes,
that is, which, while all coefficients in the existence and explanation
of that natural world of experience which forms their passive effect,
their objects of mere perception, are themselves related only in the
higher way of Final Causes — that is, Defining-Bases and Ends —
of each other, making them the logical Complements, and the Ob-
jects of conduct, all for each, and each for all. Hence, the system
of causation undergoes a signal transformation, and proves to be
organized by Final Cause as its basis and root, instead of by Efficient
Cause, or Originating Ground, as the earlier stages of thinking had
always assumed.

The causal relation between the absolute or primary realities
being purely Final, or Defining and Purposive; that is to say, the
uncoercive influence of recognition and ideality; all the other forms
of cause, as grouped by Aristotle, — Material, Formal, and Efficient,
— are seen to be the derivatives of Final Cause, as being supphed
by the action of the minds that, as absolute or underived realities,
exist only in the relation of mutual Complements and Ends. Accord-
ingly, Efficient Cause operates only from minds, as noumena, to
matter, as their phenomenon, their presented contents of experience;
or, in a secondary and derivative sense, from one phenomenon to
another, or from one group of phenomena to another group, these
playing the part of transmitters, or (as some logicians would say)
Instrumental Causes, or Means. Cause, as Material, is hence defined
as the elementary phenomenon, and the combinations of this; and
therefore, strictly taken, is merely Effect (or Outcome) of the self-
active consciousness, whose spontaneous forms of conception and
perception become the Formal Cause that organizes the sum of
phenomena into cosmic harmony or unity.

180 PHILOSOPHY

Here, accordingly, comes into view the further and in some respects
deeper conceptual pair, Many and One. The history of philosophic
thought proves that this antithesis is darkly obscure and deeply
ambiguous; for about it have centred a large part of the conflicts
of doctrine. This pair has already been used, implicitly, in exhibiting
the development of the preceding group, Cause and Effect; and
in so using it we have supplied ourselves with a partial clarification
of it, and with one possible solution of its ambiguity. We have seen,
namely, how our strong natural persuasion that philosophy guided
by the fundamental concept Cause must become the search for the
One amid the wilderness of the Many, and that this search cannot
be satisfied and ended except in an all-inclusive Unit, in which the
Many is embraced as the integral and originated parts, completely
determined, subjected, and controlled, may give way to another
and less oppressive conception of unity; a conception of it as the
harmony among many free and independent primary realities,
a harmony founded on their intelligent and reasonable mutual
recognition. This conception casts at least some clearing light upon
the long and dreary disputes over the Many and the One; for it
exposes, plainly, the main source of them. They have arisen out of
two chief ambiguities, — the ambiguity of the concept One, and the
ambiguity of the concept Cause in its supreme meaning. The normal
contrast between the One and the Many is a clear and simple con-
trast: the One is the single unit, and the Many is the repetition of
the unit, or is the collection of the several units. But if we go on to
suppose that there is a collection or sum oi all possible units, and
call this the Wliole, then, since there can 'be no second such, we call
it also "one" (or the One, by way of preeminence), overlooking the
fact that it differs from the simple one, or unit, in genere; that it is
in fact not a unit at all, not an elementary member of a series, but
the annulment of all series; that our name "one" has profoundly
changed its meaning, and now stands for the Sole, the Only. Thus,
by our forgetfulness of differences, we fall into deep water, and,
with the confused illusions of the drowning, dream of the One and
All as the single punctum originatioms of all things, the Source and
Begetter of the very imits of which it is in reality only the resultant
and tTie derivative. Or, from another point of view, and in another
mood, we riglitly enough take the One to mean the coherent, the
intelligible, the consistent, the harmonious; and putting the Many,
on the misleading hint of its contrast to the unit, in antithesis to
this One of harmony, we fall into the ?belief that the Many cannot
be harmonious, is intrinsically a cluster of repulsions or of collisions,
incapable of giving rise to accord; indeed, essentially hostile to it.
So, as accord is the aim and the essence of our reason, we are caught
in the snare of monism, pluralism having apparently become the

FUNDAMENTAL CONCEPTIONS AND METHODS 181

equivalent of chaos, and thus the bete noir of rational metaphysics.
Nay, in the opposed camp itself, some of the most ardent adherents
of pluralism, the hveliest of wit, the most exuberant in literary re-
sources, are the abjectest believers in the hopeless disjunction and
capriciousness of the plural, and hold there is a rift in the texture of
reality that no intelligence, " even though you dub it ' the Absolute,' "
can mend or reach across. Yet surely there is nothing in the Many, as
a sum of units, the least at war with the One as a system of harmony.
On the contrary, even in the pure form of the Number Series, the
Many is impossible except on the principle of harmony, — the units
can be collected and summed (that is, constitute the Many), only
if they cohere in a community of intrinsic kindred. Consequently
the whole question of the chaotic or the harmonic nature of a plural
world turns on the nature of the genus which we find characteristic
of the absolutely (le., the unreservedly) real, and which is to be taken
as the common denomination enabling us to count them and to sum
them. When minds are seen to be necessarily the primary realities,
but also necessarily federal as well as individual, the illusion about
the essential disjunction and non-coherence of the plurally real dis-
solves away, and a primordial world of manifold persons is seen
to involve no fundamental or hopeless anarchy of individualism,
irreducible in caprice, but an indwelling principle of harmony,
rather, that from the springs of individual being intends the control
and composure of all the disorders that mark the world of experien-
tial appearance, and so must tend perpetually to effect this.

The other main source of our confusions over the Many and the
One is the variety of meaning hidden in the concept Cause, and our
propensity to take its most obvious but least significant sense for
its supreme intent. Closest at hand, in experience, is our productive
causation of changes in our sense- world, and hence most obvious
is that reading of Cause which takes it as the producer of changes
and, with a deeper comprehension of it, of the inalterable linkage
between changes, whereby one follows regularly and surely upon
another. Thus what we have in philosophy agreed to call Efficient
Cause comes to be mistaken for the profoundest and the supreme form
of cause, and all the other modes of cause, the Material (or Stuff),
the Form (or Conception), and the End (or Purpose), its conse-
quent and derivative auxiliaries. Under the influence of this strong
impression, we either assume total reality to be One Whole, all-
embracing and all-producing of its manifold modes, or else view it
as a duality, consisting of One Creator and his manifold creatures.
So it has come about that metaphysics has hitherto been chiefly
a contention between pantheism and monotheism, or, as the latter
should for greater accuracy be called, monarchotheism; and, it
must be acknowledged, this struggle has been attended by a con-

182 PHILOSOPHY

tinued (though not continual) decline of this later dualistic theory
before the steadfast front and unyielding advance of the older
monism. Thus persistent has been the assumption that harmony can
only be assured by the unity given in some single productive causa-
tion : the only serious uncertainty has been about the most rational
way of conceiving the operation of this Sole Cause; and this doubt
has thus far, on the whole, declined in favor of the Elder Oriental
or monistic conception, as against the Hebraic conception of extra-
neous creation by fiat. The frankly confessed mystery of the latter,
its open appeal to miracle, places it at a fatal disadvantage with the
Elder Orientalism, when the appeal is to reason and intelligibility.
It is therefore no occasion for wonder that, especially since the rise
of the scientific doctrine of Evolution, with its postulate of a univer-
sal unity, self-varying yet self-fulfilling, even the leaders of theology
are more and more falling into the monistic line and swelling the
ever-growing ranks of pantheism. If it be asked here, And why not ?
— where is the harm of it ? — is not the whole question simply of what
is true? the answer is. The mortal harm of the destruction of personal-
ity, which lives or dies with the preservation or destruction of individual
responsibility; while the completer truth is, that there are other and
profounder (or, if you please, higher) truths than this of explanation
hy Efficient Cause. In fact, there is a higher conception of Cause
itself than this of production, or efficiency; for, of course, as we well
might say, that alone can be the supreme conception of Cause which
can subsist between absolute or unreserved realities, and such must
exclude their production or their necessitating control by others.
So that we ought long since to have realized that Final Cause, the
recognized presence to each other as unconditioned realities, or De-
fining Auxiliaries and Ends, is the sole causal relation that can hold
among primary realities; though among such it can hold, and in
fact must.

For the absolute reality of personal intelligences, at once indi-
vidual and universally recognizant of others, is called for by other
conceptions fundamental to philosophy. These other fundamental
concepts can no more be counted out or ignored than those we have
hitherto considered; and when we take them up, we shall see how
vastly more significant they are. They alone will prove supreme,
truly organizing, normative; they alone can introduce gradation in
truths, for they alone introduce the judgment of worth, of valuation;
they alone can give us counsels of perfection, for they alone rise
from those elements in our being which deal with ideals and with
veritable Ideas. So let us proceed to them.

Our path into their presence, however, is through another pair,
not so plainly antithetic as those we have thus far considered. This

FUNDAMENTAL CONCEPTIONS AND METHODS 183

pair that I now mean is Time and Space, which, though not ob-
viously antinomic, yet owes its existence, as can now be shown,
to that profoundest of concept-contrasts which we earlier considered
under the head of Subject and Object, when the Object takes on its
only adequate form of Other Subject. But in passing from the con-
trast One and Many towards its rational transformation into the
moral society of Mind and Companion Minds, we break into this
pair of Time and Space, and must make our way through it by
taking in its full meaning.

Time and Space play an enormous part in all our empirical thinking,
our actual use of thought in our sense-perceptive life. And no wonder;
for, in cooperation, they form the postulate and condition of all our
possible sensuous consciousness. Only on them as backgrounds can
thought take on the peculiar clearness of an image or a picture ; only
on the screens which they supply can we literally depict an object.
And this clarity of outline and boundary is so dear to our ordinary
consciousness, that we are prone to say there is no sufficient, no real
clearness, unless we can clarify by the bounds either of place or of
date, or of both. In this mood, we are led to deny the reality and
validity of thought altogether, when it cannot be defined in the metes
and bounds afforded by Time or by Space: that which has no date
nor place, we say, — no extent and no duration, — cannot be real;
it is but a pseudo-thought, a pretense and a delusion. Here is the
extremely plausible foundation of the philosophy known as sensa-
tionism, the refined or second-thought form of materialism, in which
it begins its euthanasia into idealism.

Without delaying here to criticise this, let us notice the part that
Time and Space play in reference to the conceptual pair we last con-
sidered, the One and the Many; for not otherwise shall we find our
way beyond them to the still more fundamental conceptions which
we are now aiming to reach. Indeed, it is through our surface-appre-
hension of the pair One and Many, as this illumines experience, that
we most naturally come at the pair Time and Space; so that these are
at first taken for mere generalizations and abstractions, the purely
nominal representatives of the actual distinctions between the mem-
bers of the Many by our sense-perception of this from that, of here
from there, of now from then. It is not till our reflective attention is
fixed on the fact that there and here, now and then, are peculiar dis-
tinctions, wholly different from other contrasts of this with that, —
which may be made in all sorts of ways, by difference of quality, or of
quantity, or of relations quite other than place and date, — it is not
till we realize this peculiar character of the Time-contrast and the
Space-contrast, that we see these singular differential qualia cannot
be derived from others, not even from the contrast One and Many,
but are independent, are themselves underived and spontaneous

184 PHILOSOPHY

utterances of our intelligent, our percipient nature. But when Kant
jEirst helped mankind to the realization of this spontaneous (or
a priori) character of this pair of perceptive conditions, or Sense-
Forms, he fell into the persuasion, and led the philosophic world into
it, that though Time and Space are not derivatives of the One and
the Many read as the numerical aspect of our perceptive experiences,
yet there is between the two pairs a connection of dependence as
intimate as that first supposed, but in exactly the opposite sense;
namely, that the One and the Many are conditioned by Time and
Space, or, when it comes to the last resort, are at any rate completely
dependent upon Time. By a series of units, this view means, we really
understand a set of items discriminated and related either as points or
as instants: in the last analysis, as instants: that is, it is impossible
to apprehend a unit, or to count and sum units, unless the unit is taken
as an instant, and the units as so many instants. Numbers, Kant
holds, are no doubt pure (or quite unsensuous) percepts, — dis-
cerned particulars, — therefore spontaneous products of the mind
a priori, but made possible only by the primary pure percept Time,
or, again, through the mediation of this, by the conjoined pure per-
cept Space; so that the numbers, in their own pure character, are
simply the instants in their series. As the instants, and therefore the
numbers, are pure percepts, — particulars discerned without the
help of sense, — so pure percepts, in a primal and comprehensive
sense, argues Kant, must their conditioning postulates Time and
Space be, to supply the "element," or "medium," that will render
such pure percepts possible.

This doctrine of Kant's is certainly plausible; indeed, it is impress-
ively so; and it has taken a vast hold in the world of science, and
has reinforced the popular belief in the unreality of thought apart
from Time and Space; an unreality which it is an essential part of
Kant's system to establish critically. But as a graver result, it has
certainly tended to discredit the belief in personal identity as an
abiding and immutable reality, enthroned over the mutations of
things in Time and Space; since all that is in these is numbered and
is mutable, and is rather many than one, yet nothing is believed real
except as it falls under them, at any rate under Time. And with this
decline of the belief in a changeless self, has declined, almost as rapidly
and extensively, the belief in immortalit3^ Or, rather, the per-
manence and the identity of the person has faded into a question
regarded as unanswerable ; though none the less does this agnostic
state of belief tend to take personality, in any responsible sense of
the word, out of the region of practical concern. With what is un-
knowable, even if existing, we can have no active traffic; 'tis for
our conduct as if it were not.

So it behooves us to search if this prevalent view about the relation

FUNDAMENTAL CONCEPTIONS AND METHODS 185

of One and Many to Time and Space is trustworthy and exact. What
place and function in philosophy must Space and Time be given? —
for they certainly have a place and function; they certainly are
among the inexpugnable conceptions with which thought has to
concern itself when it undertakes to gain a view of the whole. But
it may be easy to give them a larger place and function than belong
to them by right. Is it true, then, that the One and the Many — that
the system of Numbers, in short — are unthinkable except as in
Space and Time, or, at any rate, in Time? Or, to put the question
more exactly, as well as more gravely and more pertinently. Are
Space and Time the true prindpia individui, and is Time preemi-
nently the ultimate prindpium individuationis 9 Is there accordingly
no individuality, and no society, no associative assemblage, except
in the fleeting world of phenomena, dated and placed? Simply to ask
the question, and thus bring out the full drift of this Kantian doc-
trine, is almost to expose the absurdity of it. Such a doctrine, though
it may be wisely refusing to confound personality, true individuality,
with the mere logical singular; nay, worse, with a limited and special
illustration of the singular, the one here or the one there, the one now
or the one then ; nevertheless, by confining numerability to things
material and sensible, makes personal identity something unmeaning
or impossible, and destroys part of the foundation for the relations
of moral responsibility. Though the vital trait of the person, his
genuine individuality, doubtless lies, not in his being exactly num-
erable, but in his being aboriginal and originative; in a word, in his
self-activity, in his being a centre of autonomous social recognition;
yet exactly numerable he indeed is, and must be, not confusable with
any other, else his professed autonomy, his claim of rights and his
sense of duty, can have no significance, must vanish in the universal
confusion belonging to the indefinite. Nor, on the other hand, is it at
all true that a number has to be a point or an instant, nor that things
when numbered and counted are implicitly pinned upon points or, at
all events, upon instants. It may well enough be the fact that in our
empirical use of number we have to employ Time, or even Space, but
it is a gaping non sequitur to conclude that we therefore can count
nothing but the placed and the dated. Certainly we count whenever
we distinguish, — by whatever means, on whatever ground. To
think is, in general, at least to "distinguish the things that differ;"
but this will not avail except we keep account of the differences;
hence the One and the Many lie in the very bosom of intelligence,
and this fundamental -and spontaneous contrast can not only rive
Time and Space into expressions of it, in instants and in points, but
travels with thought from its start to its goal, and as organic factor
in mathematical science does indeed, as Plato in the Republic said,
deal with absolute being, if yet dreamwise ; so that One and Many,

186 PHILOSOPHY

and Many as the sum of the ones, makes part of the measure of that
primally real world which the world of minds alone can be. If the
contrast One and Many can pass the bounds of the merely phenome-
nal, by passing the temporal and the spatial; if it applies to universal
being, to the noumenal as well as to the phenomenal; then the abso-
lutely real world, so far as concerns this essential condition, can be
a world of genuine individuals, identifiable, free, abiding, responsible,
and there can be a real moral order; if not, then there can be no
such moral world, and the deeper thought-conceptions to which we
now approach must be regarded, at the best, as fair illusions, bare
ideals, which the serious devotee of truth must shun, except in such
moments of vacancy and leisure as he may venture to surrender,
at intervals, to purely hedonic uses. But if the One and the Many
are not dependent on Time and Space, their universal validity is
possible; and it has already been shown that they are not so de-
pendent, are not. thus restricted.

And now it remains to show their actual universality, by exhibiting
their place in the structure of the absolutely real; since nobody calls
in question their pertinence to the world of phenomena. But their
noumenal applicability follows from their essential implication with
all and every difference: no difference, no distinction, that does not
carry counting; and this is quite as true as that there can be no count-
ing without difference. The One and the Many thus root in Identity
and Difference, pass up into fuller expression in Universal and Par-
ticular, hold forward into Cause and Effect, attain their commanding
presentation in the Reciprocity of First Causes, and so keep record of
the contrast between Necessity and Contingency. In short, they are
founded in, and in their turn help (indispensably) to express, all the
categories, — Quality, Quantity, Relation, Modality. Nor do they
suffer arrest there; they hold in the ideals, the True, the Beautiful,
the Good, and in the primary Ideas, the Self, the World,- and God.
For all of these differ, however close their logical linkage may be;
and in so far as they differ, each of them is a counted unit, and so they
are many. And, most profoundly of all, One and Many take footing
in absolute reality so soon as we realize that nothing short of intelli-
gent being can be primordially real, underived, and truly causal, and
that intelligence is, by its idea, at once an /-thinking and a universal
recognizant outlook upon others that think 7.

Hence Number, so far from being the derivative of Time and Space,
founds, at the bottom, in the self-definition and social recognition of
intelligent beings, and so finds a priori a valid expression in Time and
in Space, as well as in every other primitive and spontaneous form in
which intelligence utters itself. The Pythagorean doctrine of the rank
of Number in the scale of realities is only one remove from the truth :
though the numbers are indeed not the Prime Beings, they do enter

FUNDAMENTAL CONCEPTIONS AND METHODS 187

into the essential nature of the Prime Beings; are^ so to speak, the
organ of their definite reality and identity, and for that reason go
forward into the entire defining procedure by which these intelli-
gences organize their world of experiences. And the popular impres-
sion that Time and Space are derivatives from Number, is in one
aspect the truth, rather than the doctrine of Kant is; for though they
are not mere generalizations and abstractions .from numbered dates
and durations, places and extents, they do exist as relating-principles
which minds simply put, as the conditions of perceptive experiences ;
which by the nature of intelligence they must number in order to
have and to master; while Number itself, the contrast of One and
Many, enters into the very being of minds, and therefore still holds
in Time and in Space, which are the organs, or media, not of the whole
being of the mind, but only of that region of it constituted by sensa-
tion, — the material, the disjunct, the empirical. Besides, the logical
priority of Number is implied in the fact that minds in putting Time
and Space a priori must count them as two, since they discriminate
them with complete clearness, so that it is impossible to work up
Space out of Time (as Berkeley and Stuart Mill so adroitly, but so
vainly, attempted to do), or Time out of Space (as Hegel, with so little
adroitness and such patent failure, attempted to do) . No; there Time
and Space stand, fixed and inconfusable, incapable of mutual trans-
mutation, and thus the ground of an abiding difference between the
inner or psychic sense-world and the outer or physical, between the
subjective and the (sensibly) objective. By means of them, the world
of minds discerns and bounds securely between the privacy of each
and the publicity, the life "out of doors," which is common to all;
between the cohering isolation of the individual and the communicat-
ing action of the society. Indeed, as from this attained point of view
we can now clearly see, the real ground of the difference between
Time and Space, and hence between subjective perception and the
objective existence of physical things, is in the fact that a mind, in
being such, — in its very act of self-definition, — correlates itself
with a society of minds, and so, to fulfill its nature, in so far as this
includes a world of experiences, must form its experience socially as
well as privately, and hence "^dll put forth a condition of sensuous
communication, as weU as a condition of inner sensation. Thus the
dualization of the sense-world into inner and outer, psychic and
physical, subjective and objective, rests at last on the intrinsically
social nature of conscious being; rests on the twofold structure,
logically dichotomous, of the self-defining act; and we get the explan-
ation, from the nature of intelligence as such, why the Sense-Forms
are necessarily two, and only two. It is no accident that we experi-
ence all things sensible in Time or in Space, or in both together; it is
the natural expression of our primally intelligent being, concerned

188 PHILOSOPHY

as that is, directly and only, with our self and its logically necessary
complement, the other selves; and so the natural order, in its two
discriminated but complemental portions, the inner and the outer,
is founded in that moral order which is given in the fundamental act of
our intelligence. It is this resting of Space upon our veritable Objects,
the Other Subjects, that imparts to it its externalizing quality, so
that things in it are referred to the testing of all minds, not to ours
only, and are reckoned external because measured by that which is
alone indeed other than we.

In this way we may burst the restricting limit which so much of
philosophy, and so much more of ordinary opinion, has drawn about
our mental powers in view of this contrast Time and Space, espe-
cially with reference to the One and the Many, and to the persuasion
that plural distinctions, at any rate, cannot belong in the region of
absolute reality. Ordinary opinion either inclines to support a philo-
sophy that is skeptical of either Unity or Plurality being pertinent
beyond Time and Space, and thus to hold by agnosticism, or, if it
affects affirmative metaphysics, tends to prefer monism to pluralism,
when the number-category is carried up into immutable regions: to
represent the absolutely real as One, somehow seems less contradict-
ory of the "fitness of things" than to represent it as Many; more-
over, carrying the Many into that supreme region, by implying the
belonging there of mortals such as we, seems shocking to customary
piety, and full of extravagant presumption. Still, nothing short of
this can really satisfy our deep demand for a moral order, a personal
responsibility, nay, an adequate logical fulfillment of our conception of
a self as an intelligence ; while the clarification which a rational plural-
ism supplies for such ingrained puzzles in the theory of knowledge as
that of the source and finality of the contrast Time and Space, to
mention no others, should afford a strong corroborative evidence in
its behalf. And, as already said, this view enables us to pass the
limit which Time and Space are so often supposed to put, hopelessly,
upon our concepts of the ideal grade, the springs of all our aspira-
tion. To these, then, we may now pass.

We reach them through the doorways of the Necessary vs. the
Contingent, the Unconditioned vs. the Conditioned, the Infinite vs. the
Finite, the Absolute vs. the Relative; and we recognize them as our
profoundest foundation-concepts, alone deserving, as Kant so per-
tinently said, the name of Ideas, — the Soul, the World, and God.
Associated with them are what we may call our three Forms of the
Ideal, — the True, the Beautiful, the Good. These Ideas and their
affihated ideals have the highest directive and settling function in
the organization of philosophy; they determine its schools and its
history, by forming the centre of all its controlling problems; they

FUNDAMENTAL CONCEPTIONS AND METHODS 189

prescribe its great subdivisions, breaking it up into Metaphysics,
Esthetics, and Ethics, and Metaphysics, again, into Psychology
Cosmology, and Ontology, — or Theology in the classic sense, which,
in the modern sense, becomes the Philosophy of Religion; they call
into existence, as essential preparatory and auxiliary disciplines.
Logic and the Theory of Knowledge, or Epistemology. They thus
provide the true distinctions between philosophy and the sciences of
experience, and present these sciences as the carrying out, upon
experiential details, of the methodological principles which philo-
sophy alone can supply; hence they lead us to view all the sciences
as in fact the applied branches, the completing organs of philosophy,
instead of its hostile competitors.

As for the controlling questions which they start, these are such as
follow : Are the ideals but bare ideals, serving only to cast "a light
that never was, on land or sea?" — are the Ideas only bare ideas,
without any objective being of their own, without any footing in the
real, serving only to enhance the dull facts of experience with auroral
illusions? The philosophic thinker answers affirmatively, or with
complete skeptical dubiety, or with a convinced and uplifting nega-
tive, according to his less or greater penetration into the real meaning
of these deepest concepts, and depending on his view into the nature
and thought-effect of the Necessary and the Contingent, the Uncon-
ditioned and the Conditioned, the Infinite and the Finite, the Abso-
lute and the Relative.

And what, now, are the accurate, the adequate meanings of the
three Ideas? — what does our profoundest thought intend by the
Soul, by the World, by God? We know how Kant construed them,
in consequence of the course by which he came critically (as he
supposed) upon them, — as respectively the paramount Subject of
experiences; the paramount Object of experiences, or the Causal
Unity of the possible series of sensible objects; and the complete
Totality of Conditions for experience and its objects, itself therefore
the Unconditioned. It is worth our notice, that especially by his con-
struing the idea of God in this way, thus rehabilitating the classical
and scholastic conception of God as the Sum of all Realities, he laid
the foundation for that very transfiguration of mysticism, that ideal-
istic monism, which he himself repudiated, but which his three noted
successors in their several ways so ardently accepted, and which has
since so pervaded the philosophic world. But suppose Kant's alleged
critical analysis of the three Ideas and their logical basis is in fact far
from critical, far from "exactly discriminative," — and I believe
there is the clearest warrant for declaring that it is, — then the
assumed "undeniable critical basis" for idealistic monism will be
dislodged, and it will be open to us to interpret the Ideas with accu-
racy and consistency — an interpretation which may prove to estab-

190 PHILOSOPHY

lish, not at all any monism, but a rational pluralism. And this will
also reveal to us, I think, that our prevalent construing of the Uncon-
ditioned and the Conditioned, the Necessary and the Contingent, the
Infinite and the Finite, the Absolute and the Relative, suffers from
an equal inaccuracy of analysis, and precisely for this reason gives
a plausible but in fact untrustworthy support to the monistic inter-
pretation of God, and Soul, and World; or, as Hegel and his chief
adherents prefer to name them, God, Mind, and Nature. If the
Kantian analysis stands, then it seems to follow, clearly enough, that
God is the Inclusive Unit which at once embraces Mind and Nature,
Soul and World, expresses itself in them, and imparts to them their
meaning; and the plain dictate then is, that Kant's personal pre-
judice, and the personal prejudices of others like him, in favor of
a transcendent God, must give way to that conception of the Divine,
as immanent and inclusive, which is alone consistent with its being
indeed the Totality of Conditions, — the Necessary Postulate, and
the Sufficient Reason, for both Subject and Object.

But will Kant's analysis stand? Have we not here another of his
few but fatal slips, — like his doctrine of the dependence of Number
upon Time and Space, and its consequent subjection to them? It
surely seems so. If the veritable postulate of categorical syllogizing
be, as Kant thinks it is, merely the Subject, the self as experiencer of
presented phenomena, in contrast to the Object, the causally united
sum of possible phenomena ; and if the true postulate of conditional
syllogizing is this cosmic Object, as contrasted with the correlate
Subject, then it would seem we cannot avoid certain pertinent ques-
tions. Is such a postulate Subject any fit and adequate account of the
whole Self, of the Soul? — is there not a vital difference between this
subject-self and the Self as Person? — does not Kant himself imply
so, in his doctrine of the primacy of the Practical Reason? Again: Is
not the World, as explained in Kant's analysis, and as afterwards
made by him the solution of the Cosmological Antinomies, simply the
supplemental factor necessarily correlate to the subjective aspect
of the conscious life, and reduced from its uncritical role of thing-in-
itself to the intelligible subordination required by Kant's theory of
Transcendental Idealism? — and can this be any adequate account
of the Idea that is to stand in sufficing contrast to the whole Self,
the Person? — wiiat less than the Society of Persons can meet the
World-Idea for that? Further: If with Kant we take the World to
mean no more than this object-factor in self-consciousness, must not
the Soul, the total Self, from which, according to Kant's Transcen-
dental Idealism, both Space and Time issue, supplying the basis for
the immutable contrast between the experiencing subject and the
really experienced objects, — must not this whole Self be the real
meaning of the "Totality of Conditions, itself unconditioned," which

FUNDAMENTAL CONCEPTIONS AND METHODS 191

comes into view as simply the postulate of disjunctive syllogizing?
How in the world can disjunctive syllogizing, the confessed act of
the /-thinking intelligence, really postulate anything as Totality of
Conditions, in any other sense than the total of conditions for such
syllogizing? — namely, the conditioning I that organizes and does
the reasoning? There is surely no warrant for calling this total, which
simply transcends and conditions the subject and the object of sen-
sible experiences, by any loftier name than that which Kant had
already given it in the Deduction of the Categories, when he desig-
nated it the "originally synthetic unity of apperception (self-con-
sciousness)," or " the /-thinking {das ich-denke) that must accompany
all my mental presentations," — that is to say, the whole Self, or
thinking Person, idealistically interpreted. The use of the name God
in this connection, where Kant is in fact only seeking the roots of the
three orders of the syllogism ivhen reasoning has hy supposition been
restricted to the subject-matter of experience, is assuredly without war-
rant; yes, without excuse. In fact, it is because Kant sees that the
third Idea, as reached through his analysis, is intrinsically immanent,
— resident in the self that syllogizes disjunctively, and, because so
resident, incapable of passing the bounds of possible experience, —
while he also sees that the idea of God should mean a Being tran-
scendent of every other thinker, himself a distinct individual con-
sciousness, though not an empirically limited one, — it is, I say,
precisely because he sees all this, that he pronounces the Idea, though
named with the name of God, utterly without pertinence to indicate
God's existence, and so enters upon that part of his Transcendental
Dialectic which is, in chief, directed to exposing the transcendental
illusion involved in the celebrated Ontological Proof. Consistently,
Kant in this famous analytic of the syllogism should be talking, not
of the Soul, the World, and God, but of the Subject (as uniting-
principle of its sense-perceptions) , the Object (as uniting-principle of
all possible sense-percepts), and the Self (the whole / presiding over
experience in both its aspects, as these are discriminated in Time and
Space). By what rational title — even granting for the sake of argu-
ment that they are the genuine postulates of categorical and of con-
ditional syllogizing — can this Subject and this Object, these corre-
late factors in the Self, rank as Ideas with the Idea of their condi-
tioning Whole — the Self, that in its still unaltered identity fulfills, in
Practical Reason, the high role of Person? If this no more than meets
the standard of Idea, how can they meet it? How can two somethings,
neither of which is the Totality of Conditions, and both of which are
therefore in fact conditioned, deserve the same title with that which
is intrinsically the Totality of Conditions, and, as such, uncondi-
tioned? To call the conditioned and the unconditioned alike Ideas is
a confounding of dignities that Pure Reason should not tolerate,

192 PHILOSOPHY

whether the procedure be read as a levehng down or a levehng up.
Distributing the titles conferred by Pure Reason in this democratic
fashion reminds us too much, unhappily for Kant, of the Cartesian
performances with Substance; whereby God, mind, and matter be-
came alike "substances," though only God could in truth be said to
"require nothing for his existence save himself," while mind and
matter, though absolutely dependent on God, and derivative from
him, were still to be called substances in the "modified" and Pick-
wickian sense of being underived from each other.

But if Kant's naming his third syllogistic postulate the Idea of
God is inconsequent upon his analysis; or if, when the analysis is
made consequent by taking the third Idea to mean the whole Self,
the first and second postulates sink in conceptual rank, so that they
cannot with any pertinence be called Ideas, unless we are willing to
keep the same name when its meaning must be changed in genere, —
a procedure that can only encumber philosophy instead of clearing
its way, — these difficulties do not close the account; we shall find
other curious things in this noted passage, upon which part of the
characteristic outcome of Kant's philosophizing so much depends.
Besides the misnaming of the third Idea, we have already had to
question, in view of the path by which he reaches it, the fitness of
his calling the first by the title of the Soul; and likewise, though for
other and higher reasons, of his calling the second by the name of the
World. In fact, it comes home to us that all of the Ideas are, in one
way or another, misnomers; Kant's whole procedure with them, in
fine, has already appeared inexact, inconsistent, and therefore uncrit^
ical. But now we shall become aware of certain other inconsistencies.
In coming to the Subject, as thej postulate of categorical syllogizing,
Kant, you remember, does so by the path of the relation Subject and
Predicate, arguing that the chain of categorical prosyllogisms has
for its limiting concept and logical motor the notion of an absolute
subject that cannot be a predicate; and as no subject of a judgment
can of itself give assurance of fulfilling this condition, he concludes
this motor-limit of judgment-subjects to be identical with the Subject
as thinker, upon whom, at the last, all judgments depend, and who,
therefore, and who alone, can never be a predicate merely. In similar
fashion, he finds as the motor-limit of the series of conditional
prosyllogisms, which is governed by the relation Cause and Effect,
the notion of an absolute cause — a cause, that is, incapable of being
an effect; and this, as undiscoverable in the chain of phenomenal
causes, which are all in turn effects, he concludes is a pure Idea, the
reason's native conception of a necessary linkage among all changes
in Space, or of a Cosmic Unity among physical phenomena. In both
conceptions, then, whether of the unity of the Subject or of the
World, we seem to have a case of the unconditioned, as each, surely,

FUNDAMENTAL CONCEPTIONS AND METHODS 193

is a totality of conditions: the one, for all possible syllogisms by
Subject and Predicate; the other, for all possible syllogisms from
Cause and Effect. Until it can be shown that the syllogisms of the
first sort and the syllogisms of the second are both conditioned by
the system of disjunctive syllogisms, so that the Idea alleged to be the
totality of conditions for this system becomes the conditioning prin-
ciple for both the others, there appears to be no ground for contrasting
the totality of conditions presented in it with those presented in the
others, as if it were the absolute Totality of all Conditions, while the
two others are only "relative totalities," — which would be as much
as to say they were only pseudo-totalities, both being conditioned
instead of being unconditioned. But there seems to be no evidence,
not even an indication, that disjunctive reasoning conditions cate-
gorical or conditional — that it constitutes the whole kingdom, in
which the other two orders of reasoning form dependent provinces,
or that for final validation these must appeal to the disjunctive series
and the Idea that controls it. On the contrary, any such relation
seems disproved by the fact that the three types of syllogism apply
alike in all subject-matter, psychic or physical, subjective or object-
ive, concerning the Self or concerning the World, — yes, concerning
other Selves or even concerning God; whereas, if the relation were a
fact, it would require that only disjunctive reasoning can deal with the'
Unconditioned, and that conditional must confine itself to cosmic
material, while categorical pertains only to the things of inner sense..
Such considerations cannot but shake our confidence in the inqui-
sition to which Kant has submitted the Ideas of Reason, both a&
regards what they really mean and how they are to be correlated..
At all events, the analysis of logical procedure and connection on^
which his account of them is based is full of the confusions and over-
sights that have now been pointed out, and justifies us in saying
that his case is not established. Hence we are not bound to follow-
when his three successors, or their later adherents, proceed in accept-
ance of his results, and advance into various forms of idealism, all of
the monistic type, as if the general relation between the three Ideas
had been demonstrably settled by Kant in the monist sense, despite
his not knowing this, and that all we have to do is to disregard his-
recorded protests, and render his results consistent, and our idealism
"absolute," by casting out from his doctrine the distinction between
the Theoretical and the Practical Reason, with the "primacy" of the
latter, through making an end of his assumed world of Dinge an sich^
or "things in themselves." This movement, I repeat, we are not
bound to follow: a rectification of view as to the meaning of the three
Ideas becomes possible as soon as we are freed from Kant's entangled
method of discovering and defining them; and when this rectification
is effected, we shall find that the question between monism and

194 PHILOSOPHY

rational or harmonic pluralism is at least open, to say no more. Nay,
we are not to forget that by the results of our analysis of the concepts
One and Many, Time and Space, and the real relation between them,
plural metaphysics has already won a precedence in this contest.

THE DEVELOPMENT OF PHILOSOPHY IN THE NINE-
TEENTH CENTURY

BY GEORGE TRUMBULL LADD

[George Trumbull Ladd, Professor of Philosophy, Yale University, b. Jan-
uary 19, 1842, Painesville, Ohio. B.A. Western Reserve College, 1864;
B.D. Andover Theological Seminary, 1869; D.D. Western Reserve, 1879;
M.A. Yale, 1881; LL.D. Western Reserve, 1895; LL.D. Princeton, 1896.
Decorated with the 3d Degree of the Order of the Rising Sun of Japan,
1899; Pastor, Edinburg, Ohio, 1869-71; ibid., Milwaukee, Wis., 1871-79;
Professor of Philosophy, Bowdoin College, 1879-81; ibid., Yale University,
1881 ; Lecturer, Harvard, Tokio, Bombay, etc., 1885 . Member Ameri-
can Psychological Association, American Society of Naturalists, American
Philosophical Association, American Oriental Society, Imperial Educational
Society of Japan, Connecticut Academy. Author of Elements of Physiolog-
ical Psychology ; Philosophy of Knowledge; Philosophy of Mind ; A Theory of
Reality ; and many other noted scientific works and papers.]

The histor}'- of man's critical and reflective thought upon the
more ultimate problems of nature and of his own life has, indeed,
its period of quickened progress, relative stagnation, and apparent
decline. Great thinkers are born and die, "schools of philosophy,"
so-called, arise, flourish, and become discredited; and tendencies
of various characteristics mark the national or more general Zeit-
geist of the particular centuries. And always, a certain deep under-
current, or powerful stream of the rational evolution of humanity,
flows silently onward. But these periods of philosophical develop-
ment do not correspond to those which have been marked off for
man by the rhythmic motion of the heavenly bodies, or by himself
for purposes of greater convenience in practical affairs. The pro-
posal, therefore, to treat any century of philosophical development
as though it could be taken out of, and considered apart from, this
constant unfolding of man's rational life is, of necessity, doomed to
failure. And, indeed, the nineteenth century is no exception to the
general truth.

There is, however, one important and historical fact which makes
more definite, and more feasible, the attempt to present in outline
the history of the philosophical development of the nineteenth
century. This fact is the death of Immanuel Kant, February 12,
1804. In a very unusual way this event marks the close of the

PHILOSOPHY IN THE NINETEENTH CENTURY 195

development of philosophy in the eighteenth century. In a yet
more unusual way the same event defines the beginning of the philo-
sophical development of the nineteenth century. The proposal is,
therefore, not artificial, but in accordance with the truth of history,
if we consider the problems, movements, results, and present con-
dition of this development, so far as the fulfillment of our general
purpose is concerned, in the light of the critical philosophy of Kant.
This purpose may then be further defined in the following way : to
trace the history of the evolution of critical and reflective thought
over the more ultimate problems of Nature and of human life, in
the Western World during the last hundred years, and from the
standpoint of the conclusions, both negative and positive, which
are best embodied in the works of the philosopher of Konigsberg.
This purpose we shall try to fulfill in these four divisions of our theme :
(1) A statement of the problems of philosophy as they were given over
to the nineteenth century by the Kantian Critique; (2) a brief
description of the lines of movement along which the attempts at
the improved solution of these problems have proceeded, and of the
principal influences contributing to these attempts; (3) a sum-
mary of the principal results of these movements — the items, so to
say, of progress in philosophy which may be credited to the last cen-
tury; and finally, (4) a survey of the present state of these pro-
blems as they are now to be handed down by the nineteenth to the
twentieth century. Truly an immensely difficult, if not an impos-
sible task, is involved in this purpose!

I. The problems which the critical philosophy undertook defini-
tively to solve may be divided into three classes. The first is the
epistemological problem, or the problem offered by human know-
ledge — its essential nature, its fixed limitations, if such there be,
and its ontological validity. It was this problem which Kant brought
to the front in such a manner that certain subsequent writers on
philosophy have claimed it to be, not only the primary and most
important branch of philosophical discipline, but to comprise the
sum-total of what human reflection and critical thought can suc-
cessfully compass. "We call philosophy self-knowledge," says one
of these writers. " The theory of knowledge is the true prima philo-
sophia," says another. Kant himself regarded it as the most im-
perative demand of reason to establish a science that shall "deter-
mine a priori the possibility, the principles, and the extent of all
cognitions." The burden of the epistemological problem has pressed
heavily upon the thought of the nineteenth century; the different
attitudes toward this problem, and its different alleged solutions,
have been most influential factors in determining the philosophical
discussions, divisions, schools, and permanent or transitory achieve-
ments of the centurv.

196 PHILOSOPHY

In the epistemological problem as offered by the Kantian philo-
sophy of cognition there is involved the subordinate but highly
important question as to the proper method of philosophy. Is the
method of criticism, as that method was employed in the three
Critiques of Kant, the exclusive, the sole appropriate and product-
ive way of advancing human philosophical thought? I do not
think that the experience of the nineteenth century warrants an
affirmative answer to this question of method. This experience has
certainly, however, resulted in demonstrating the need of a more
thorough, consistent, and fundamental use of the critical method
than that in which it was employed by Kant. And this improved use
of the critical method has induced a more profound study of the
psychology of cognition, and of the historical development of philo-
sophy in the branch of epistemology. More especially, however, it
has led to the reinstatement of the value-judgments, as means of
cognition, in their right relations of harmony with the judgments
of fact and of law.

The second of the greater problems which the critical philosophy
of the eighteenth handed on to the nineteenth century is the onto-
logical problem. This problem, even far more than the epistemo-
logical, has excited the intensest interest, and called for the pro-
foundest thought, of reflective minds during the last hundred years.
This problem engages in the inquiry as to what Reality is; for to
define philosophy from the ontological point of view renders it
"the rational science of reality;" or, at least, "the science of the
supreme and most important realities." In spite of the fact that
the period immediately following the conclusion of the Kantian
criticism was the age when the people were singing

"Da die Metaphysik vor Kurzem unheerbt dbging,
Werden die Dinge an sich jetzo suh hasta verkauft,"

the cultivation of the ontological problem, and the growth of sys-
tematic metaphysics in the nineteenth century, had never pre-
viously been surpassed. In spite of, or rather because of, the fact
that Kant left the ancient body of metaphysics so dismembered and
discredited, and his otvti ontological structure in such hopeless con-
fusion, all the several buildings both of Idealism and of Realism
either rose quickly or were erected upon the foundations made bare
by the critical philosophy.

But especially unsatisfactory to the thought of the first quarter
of the nineteenth century was the Kantian position with reference
to the problem in which, after all, both the few who cultivate philo-
sophy and the multitude who share in its fruits are always most
truly interested; and this is the ethico-religious problem. In the
judgment of the generation which followed him, Kant had achieved

PHILOSOPHY IN THE NINETEENTH CENTURY 197

for those who accepted his points of view, his method of philo-
sophizing, and his results, much greater success in "removing know-
ledge" than in "finding room for faith." For he seemed to have
left the positive truths of Ethics so involved in the negative posi-
tions of his critique of knowledge as greatly to endanger them; and
to have entangled the conceptions of religion with those of morality
in a manner to throw doubt upon them both.

The breach between the human cognitive faculties and the onto-
logical doctrines and conceptions on which morality and religion
had been supposed to rest firmly, the elaborately argued distrust
and skepticism which had been aimed against the ability of human
reason to reach reality, and the consequent danger which threatened
the most precious judgments of worth and the ontological value
of ethical and sesthetical sentiments, could not remain unnoticed,
or fail to promote ceaseless and earnest efforts to heal it. The hitherto
accepted solutions of the problems of cognition, of being, and of
man's ethico-religious experience, could not survive the critical
philosophy. But the solutions which the critical philosophy itself
offered could not fail to excite opposition and to stimulate further
criticism. Moreover, certain factors in human nature, certain inter-
ests in human social life, and certain needs of humanity, not fully
recognized and indeed scarcely noticed by criticism, could not
fail to revive and to enforce their ancient, perennial, and valid
claims.

In a word, Kant left the main problems of philosophy involved
in numerous contradictions. The result of his penetrating but ex-
cessive analysis was unwarrantably to contrast sense with under-
standing; to divide reason as constitutive from reason as regulative;
to divorce the moral law from our concrete experience of the results
of good and bad conduct, true morality from many of the noblest
desires and sentiments, and to set in opposition phenomena and
noumena, order and freedom, knowledge and faith, science and
religion. Now the highest aim of philosophy is reconciliation. What
wonder, then, that the beginning of the last century felt the stimu-
lus of the unreconciled condition of the problems of philosophy at
the end of the preceding century! The greatest, most stimulating
inheritance of the philosophy of the nineteenth century from the
philosophy of the eighteenth century was the "post-Kantian pro-
blems."

11. The lines of the movement of philosophical thought and the
principal contributory influences which belong to the nineteenth
century may be roughly divided into two classes; namely, (1)
those which tended in the direction of carrying to the utmost ex-
treme the negative and destructive criticism of Kant, and (2) those
which, either mainly favoring or mainly antagonizing the con-

198 PHILOSOPHY

elusions of the Kantian criticism, endeavored to place the positive
answer to all three of these great problems of philosophy upon
more comprehensive, scientifically defensible, and permanently
sure foundations. The one class so far completed the attempt to
remove the knowledge at which philosophy aims as, by the end of
the first half of the century, to have left no rational ground for
any kind of faith. The other class had not, even by the end of the
second half of the century, as yet agreed upon any one scheme for
harmonizing the various theories of knowledge, of reality, and of
the ground of morality and religion. There appeared, however, —
especially during the last two decades of the century, — certain
signs of convergence upon positions, to occupy which is favorable
for agreement upon such a scheme, and which now promise a new
constructive era for philosophy. The terminus of the destructive
movement has been reached in our present-day positivism and philo-
sophical skepticism. For this movement there would appear to be
no more beyond in the same direction. The terminus of the other
movement can only be somewhat dimly descried. It may perhaps
be predicted with a reasonable degree of confidence as some form
of ontological Idealism (if we may use such a phrase) that shall be
at once more thoroughly grounded in man's total experience, as
interpreted by modern science, and also more satisfactory to human
ethical, sesthetical, and religious ideals, than any form of system-
atic philosophy has hitherto been. But to say even this much is
perhaps unduly to anticipate.

If we attempt to fathom and estimate the force of the various
streams of influence which have shaped the history of the philo-
sophical development of the nineteenth century, I think there can
be no doubt that the profoundest and the most powerful is the one
influence which must be recognized and reckoned with in all the
centuries. This influence is humanity's undying interest in its
moral, civil, and religious ideals, and in the civil and religious in-
stitutions which give a faithful but temporary expression to these
ideals. In the long run, every fragmentary or systematic attempt
at the solution of the problem of philosophy must sustain the test
of an ability to contribute something of value to the realization of
these ideals. The test which the past century has proposed for its
own thinkers, and for its various schools of philosophy, is by far the
severest which has ever been proposed. For the most part unosten-
tatiously and in large measure silently, the thoughtful few and
the comparatively thoughtless multitude have been contributing,
either destructively or constructively, to the effort at satisfaction
for the rising spiritual life of man. And if in some vague but
impressive manner we speak of this thirst for spiritual satisfac-
tion as characteristic of any period of human history, we may say,

PHILOSOPHY IN THE NINETEENTH CENTURY 199

I believe, that it has been peculiarly characteristic and especially
powerful as an influence during the last hundred years. The opin-
ions, sentiments, and ideals which shape the development of the
institutions of the church and state, and the freer activities of the
same opinions, sentiments, and ideals, have been in this century,
as they have been in every century, the principal factors in deter-
mining the character of its philosophical development.

But a more definite and visible kind of influence has constantly
proceeded from the centres of the higher education. The univers-
ities — especially of Germany, next, perhaps of Scotland, but
also of England and the United States, and even in less degree of
France and Italy — have both fostered and shaped the evolution
of critical and reflective thought, and of its product as philosophy.
In Germany during the eighteenth century the greater universities
had been emancipating themselves from the stricter forms of polit-
ical and court favoritism and of ecclesiastical protection and con-
trol. This emancipation had already operated at the beginning of
the nineteenth century, and it continued more and more to operate
throughout this century, for participation in that free thought
whose spirit is absolutely essential to the flourishing of true philo-
sophy. All the other colleges and universities can scarcely repay
the debt which modern philosophy owes to the universities of Ger-
many. The institutions of the higher education which are moulded
after this spirit, and which have a generous share of this spirit,
have everywhere been schools of thought as weU as schools of learn-
ing and research. Without the increasing numbers and growing
encouragement of such centres for the cultivation of the discipline
of critical and reflective thinking, it is difficult to conjecture how
much the philosophical development of the nineteenth century would
have lost. Lihertas docendi and Academische Freiheit — without these
philosophy has one of its wings fatally wounded or severely clipped.

Not all the philosophy of the last century, however, was born
and developed in academical centres and under academical in-
fluences. In Germany, Great Britain, and France, the various
so-caUed "Academies" or other unacademical associations of men
of scientific interests and attainments — notably, the Berlin Acad-
emy, which has been called "the seat of an anti-scholastic popular
philosophy" — were during the first half of the nineteenth century
contributing by their conspicuous failures as well as by their less
conspicuous successes, important factors to the constructive new
thought of the latter half of the nineteenth century. In general,
although these men decried system and were themselves inade-
quately prepared to treat the problems of philosophy, whether
from the historical or the speculative and critical point of view, they
cannot be wholly neglected in estimating its development. Clever

200 PHILOSOPHY

reasoning, and witty and epigrammatic writing on scientific or
other allied subjects, cannot indeed be called philosophy in the
stricter meaning of the word. But this so-called "popular philo-
sophy " has greatly helped in a way to free thought from its too close
bondage to scholastic tradition. And even the despite of philosophy,
and sneering references to its "barrenness," which formerly charac-
terized the meetings and the writings of this class of its critics, but
which now are happily much less frequent, have been on the whole
both a valuable check and a stimulus to her devotees. He would be
too narrow and sour a disciple of scholastic metaphysics and sys-
tematic philosophy, who, because of the levity or scorning of "out-
siders," should refuse them all credit. Indeed, the lesson of the close
of the nineteenth century may well enough be the motto for the
beginning of the twentieth century : In philosophy — since to philo-
sophize is natural and inevitable for all rational beings — there really
are no outsiders.

In this connection it is most interesting to notice how men of the
type just referred to, were at the end of the eighteenth century
found, grouped around such thinkers as Mendelssohn, Lessing,
F. Nicolai, — representing a somewhat decided reaction from the
French realism to the German idealism. The work of the Academ-
icians in the criticism of Kant was carried forward by Jacobi,
who, at the time of his death, was the pensioned president of the
Academy at Munich. Some of these same critics of the Kantian
philosophy showed a rather decided preference for the "common-
sense" philosophy of the Scottish School.

But both inside and outside of the Universities and Academies
the scientific spirit and acquisitions of the nineteenth century have
most profoundly, and on the whole favorably, affected the develop-
ment of its philosophy. In the wider meaning of the word, " science, "
— the meaning, namely, in which BoiencQ^Wissenschajt, — philo-
sophy aims to be scientific; and science can never be indifferent
to philosophy. In their common aim at a rational and unitary sys-
tem of principles, which shall explain and give its due significance to
the totality of human experience, science and philosophy can never
remain long in antagonism; they ought never even temporarily to
be divided in interests, or in the spirit which leads each generously
to recognize the importance of the other. The early part of the last
century was, indeed, too much under the influence of that almost
exclusively speculative Natur-philosophie, of which Schelling and
Hegel were the most prominent exponents. On the other hand, the
conception of nature as a vast interconnected and unitar}^ system
of a rational order, unfolding itself in accordance with teleological
principles, — however manifold and obscure, — is a noble concep-
tion and not destined to pass away.

PHILOSOPHY IN THE NINETEENTH CENTURY 201

On the continent — at least in France, where it had attained
its highest development — the scientific spirit was, at the close
of the eighteenth century, on the whole opposed to systematization.
The impulse to both science and philosophy during both the eight-
eenth and the nineteenth centuries, over the entire continent of
Europe, was chiefly due to the epoch-making work of that greatest
of all titles in the modern scientific development of the Western
World, the Principia of Newton. In mathematics and the phys-
ical sciences, during the early third or half of the last century, Great
Britain also has a roll of distinguished names which compares most
favorably with that of either France or Germany. But in England,
France, and the United States, during the whole century, science has
lacked the breadth and philosophic spirit which it had in Germany
during the first three quarters of this period. During all that time
the German man of science was, as a rule, a scholar, an investi-
gator, a teacher, and a philosopher. Science and philosophy thrived
better, however, in Scotland than elsewhere outside of Germany, so
far as their relations in interdependence were concerned. Into the
Scottish universities Playfair introduced some of the continental
suggestions toward the end of the eighteenth century, so that there
was less of exclusiveness and unfriendly rivalry between science and
philosophy; and both profited thereby. In the United States, during
the first half or more of the century, so dominant were the theo-
logical and practical interests and influences that there was little
free development of either science or philosophy, — if we interpret
the one as the equivalent of Wissenschaft and understand the other
in the stricter meaning of the word.

The history of the development of the scientific spirit and of the
achievements of the particular sciences is not the theme of this
paper. To trace in detail, or even in its large outlines, the reciprocal
influence of science and philosophy during the past hundred years,
would itself require far more than the space allotted to me. It must
suffice to say that the various advances in the efforts of the par-
ticular sciences to enlarge and to define the conceptions and prin-
ciples employed to portray the Being of the World in its totality,
have somewhat steadily grown more and more completely meta-
physical, and more and more of positive importance for the recon-
struction of systematic philosophy. The latter has not simply been
disciplined by science, compelled to improve its method, and to ex-
amine all its previous claims. But philosophy has also been greatly
enriched by science with respect to its material awaiting synthesis,
and it has been not a little profited by the unsuccessful attempts of
the current scientific theories to give themselves a truly satisfactory
account of that Ultimate Reality which, to understand the better,
is no unworthy aim of their combined efforts.

202 ■ PHILOSOPHY

During the nineteenth century science has seen many important
additions to that Ideal of Nature and her processes, to form which
in a unitary and harmonizing but comprehensive way is the philo-
sophical goal of science. The gross mechanical conception of nature
which prevailed in the earlier part of the eighteenth century has long
since been abandoned, as quite inadequate to our experience with
her facts, forces, and laws. The kinetic view, which began with
Huygens, Euler, and Ampere, and which was so amplified by Lord
Kelvin and Clerk-Maxwell in England, and by Helmholtz and others
in Germany, on account of its success in explaining the phenomena
of light, of gases, etc., very naturally led to the attempt to develop
a kinetic theory, a doctrine of energetics, which should explain all
phenomena. But the conception of "that which moves," the ex-
perience of important and persistent qualitative differentiae, and
the need of assuming ends and purposes served by the movement,
are troublesome obstacles in the way of giving such a completeness
to this theory of the Being of the World. Yet again the amazing
success which the theory of evolution has shown in explaining the
phenomena with which the various biological sciences concern
themselves, has lent favor during the latter half of the century to
the vitalistic and genetic view of nature. For all our most elaborate
and advanced kinetic theories seem utterly to fail us as explanatory
when we, through the higher powers of the microscope, stand won-
dering and face to face with the evolution of a single living cell.
But from such a view of the essential Being of the World as evolu-
tion suggests to the psycho-physical theory of nature is not an
impassable gulf. And thus, under its growing wealth of knowledge,
science may be leading up to an Ideal of the Ultimate Reality, in
which philosophy will gratefully and gladly coincide. At any rate,
the modern conception of nature and the modern conception of
God are not so far apart from each other, as either of these con-
ceptions is now removed from the conceptions covered by the same
terms, some centuries gone by.

There is one of the positive sciences, however, with which the
development of philosophy during the last century has been par-
ticularly allied. This science is ps^^chology. To speak of its history
is not the theme of this paper. But it should be noted in passing
how the development of psychology has brought into connection
with the physical and biological sciences the development of philo-
sophy. This union, whether it be for better or for worse, — and,
on the whole, I believe it to be for better rather than for worse, —
has been in a very special way the result of the last century. In
tracing its details we should have to speak of the dependence of
certain branches of psychology on physiology, and upon Sir Charles
Bell's discovery of the difference between the sensory and the motor

PHILOSOPHY IN THE NINETEENTH CENTURY 203

nerves. This discovery was the contribution of the beginning of
the century to an entire line of discoveries, which have ended at the
close of the century with putting the localization of cerebral func-
tion upon a firm experimental basis. Of scarcely less importance
has been the cellular theory as applied (1838) by Matthias Schleiden,
a pupil of Fries in philosophy, to plants, and by Theodor Schwann
about the same time to animal organisms. To these must be added
the researches of Johannes Miiller (1801-1858), the great biologist,
a hstener to Hegel's lectures, whose law of specific energies brings
him into connection with psychology and, through psychology, to
philosophy. Even more true is this of Helmholtz, whose Lehre von
den Tonempfindungen (1862) and Physiologische Optik (1867) placed
him in even closer, though still mediate, relations to philosophy.
But perhaps especially Gustav Theodor Fechner (1801-1887), whose
researches in psycho-physics laid the foundations of whatever, either
as psychology or as philosophy, goes under this name; and whether
the doctrine have reference to the relation of man's mind and body,
or to the wider relations of spirit and matter.

In my judgment it cannot be affirmed that the attempts of the
latter half of the nineteenth century to develop an experimental
science of psychology in independence of philosophical criticism and
metaphysical assumption, or the claims of this science to have
thrown any wholly new light upon the statement, or upon the
solution of philosophical problems, have been largely successful.
But certain more definitely psychological questions have been to
a commendable degree better analyzed and elucidated; the new
experimental methods, where confined within their legitimate
sphere, have been amply justified; and certain gMas?!-metaphysical
views respecting the nature of the human mind, and even, if you
will, the nature of the Spirit in general — have been placed in a
more favorable and scientifically engaging attitude toward speculat-
ive philosophy. This seems to me to be especially true with respect
to two problems in which both empirical psychology and philosophy
have a common and profound interest. These are (1) the complex
synthesis of mental functions involved in every act of true cogni-
tion, together with the bearing which the psychology of cognition
has upon epistemological problems; and (2) the yet more complex
and profound analysis, from the psychological point of view, of what
it is to be a self-conscious and self-determining Will, a true Self,
together with the bearing which the psychology of selfhood has
upon all the problems "of ethics, aesthetics, and religion.

The more obvious and easily traceable influences which have
operated to incite and direct the philosophical development of the
nineteenth century are, of course, dependent upon the teachings and
writings of philosophers, and the schools of philosophy which they

204 PHILOSOPHY

have founded. To speak of these influences even in outhne would be
to write a manual of the history of philosophy during that hundred of
years, which has been of all others by far the most fruitful in material
results, whatever estimate may be put upon the separate or combined
values of the individual thinkers and their so-called schools. No
fewer than seven or eight relatively independent or partially antag-
onistic movements, which may be traced back either directly or
more indirectly to the critical philosophy, and to the form in which
the problems of philosophy were left by Kant, sprung up during the
century. In Germany chiefly, there arose the Faith-philosophy, the
Romantic School, and Rational Idealism; in France, Eclecticism and
Positivism (if, indeed, the latter can be called a philosophy) ; in Scot-
land, a naive and crude form of Realism, which served well for the
time as an antagonist of a skeptical idealism, but which itself con-
tributed to an improved form of Idealism; and in the United States,
or rather in New England, a peculiar kind of Transcendentalism of
the sentimental type. But all these movements of thought, and
others lying somewhere midway between, in a pair composed of any
two, together with a steadfast remainder of almost every sort of
Dogmatism, and all degrees and kinds of Skepticism, have been inter-
mixed and contending with one another, in all these countries. Such
has been the varied, undefinable, and yet intensely stimulating and
interesting character of the development of systematic and scholastic
philosophy, during the nineteenth century.

The early opposition to Kant in Germany was, in the main, two-
fold : — both to his peculiar extreme analysis with its philosophical
conclusions, and also to all systematic as distinguished from a more
popular and literary form of philosophizing. Toward the close of the
eighteenth century a group of men had been writing upon philo-
sophical questions in a spirit and method quite foreign to that held
in respect by the critical philosophy. It is not wholly without signi-
ficance that Lessing, whose aim had been to use common sense and
literary skill in clearing up obscure ideas and improving and illumin-
ing the life of man, died in the very year of the appearance of Kant's
Kritik der reineyi Vernunft. Of this class of men an historian dealing
with this period has said, " There is hardly one who does not quote
somewhere or other Pope's saying, 'The proper study of mankind
is man.'" To this class belong Hamann (1730-1788), the inspirer
of Herder and Jacobi. The former, who was essentially a poet and
a friend of Goethe, controverted Kant with regard to his doctrine of
reason, his antithesis between the individual and the race, and his
schism between things as empirically known and the known unity in
the Ground of their being and becoming. Herder's path to truth was
highly colored with flowers of rhetoric ; but the promise was that he
would lead men back to the heavenly city. Jacobi, too, with due

PHILOSOPHY IN THE NINETEENTH CENTURY 205

aUowance made for the injury wrought by his divorce of the two
philosophies, — that of faith and that of science, — and his excessive
estimate of the value-judgments which repose in the mist of a feeling-
faith, added something of worth by way of exposing the barrenness
of the Kantian doctrine of an unknowable "Thing-in-itself."

From men like Fr. Schlegel (1772-1829), whose valid protest against
the sharp separation of speculative philosophy from the sesthetical,
social, and ethical life, assumed the "standpoint of irony," little real
result in the discovery of truth could be expected. But Schleier-
macher (1768-1834), in spite of that mixture of unfused elements
which has made his philosophy " a rendezvous for the most diverse
systems," contributed valuable factors to the century's philosophical
development, both of a negative and of a positive character. This
thinker was peculiarly fortunate in the enrichment of the conception
of experience as warranting a justifiable confidence in the ontological
value of ethical, sesthetical, and religious sentiment and ideas; but he
was most unfortunate in reviving and perpetuating the unjustifiable
Kantian distinction between cognition and faith in the field of ex-
perience. On the whole, therefore, the Faith-philosophy and the
Romantic School can easily be said to have contributed more than
a negative and modifying influence to the development of the philo-
sophy of the nineteenth century. Its more modern revival toward
the close of the same century, and its continued hold upon certain
minds of the present day, are evidences of the positive but partial
truth which its tenets, however vaguely and unsystematically, con-
tinue to maintain in an aesthetically and practically attractive way.

The admirers of Kant strove earnestly and with varied success
to remedy the defects of his system. Among the earlier, less cele-
brated and yet important members of this group, were K. G. Rein-
hold (1758-1823), and Maimon (died, 1800). The former, hke
Descartes, in that he was educated by the Jesuits, began the attempt,
after rejecting some of the arbitrary distinctions of Kant and his
barren and self-contradictory "Thing-in-itself," to unify the critical
philosophy by reducing it to some one principle. The latter really
transcended Kant in his philosophical skepticism, and anticipated the
Hamiltonian form of the so-called principle of relativity. Fries (1773-
1843), and Hermes (1775-1831) — the latter of whom saw in empir-
ical psychology the only true propaedeutic to philosophy — should be
mentioned in this connection. In the same group was another, both
mathematician and philosopher, who strove more successfully than
others of this group to accept the critical standpoint of Kant and yet
to transcend his negative conclusions with regard to a theory of
knowledge. I refer to Bolzano (Prague, 1781-1848), who stands in the
same line of succession with Fries and Hermes, and whose works
on the Science of Religion (4 vols. 1834) and his Science of Know-

206 PHILOSOPHY

ledge (4 vols. 1837) are noteworthy contributions to epistemological
doctrine. In the latter we have developed at great length the import-
ant thought that the illative character of propositional judgments
implies an objective relation; and that in all truths the subject-idea
must be objective. In the work on religion there is found as thor-
oughly dispassionate and rational a defense of Catholic doctrine as
exists anywhere in philosophical literature. The limited influence of
these works, due in part to their bulk and their technical character, is
on the whole, I think, sincerely to be regretted.

It was, however, chiefly that remarkable series of philosophers
which may be grouped under the rubric of a "rational Idealism,"
who filled so full and made so rich the philosophical life of Germany
during the first half of the last century; whose philosophical thoughts
and systems have spread over the entire Western World, and who are
most potent influences in shaping the development of philosophy
down to the present hour. Of these we need do little more than that
we can do — mention their names. At their head, in time, stands
Fichte, who — although Kant is reported to have complained of this
disciple because he lied about him so much — really divined a truth
which seems to be hovering in the clouds above the master's head,
but which, if the critical philosophy truly meant to teach it, needed
helpful deliverance in order to appear in perfectly clear light. Fichte,
although he divined this truth, did not, however, free it from internal
confusion and self-contradiction. It is his truth, nevertheless, that in
the Self, as a self-positing and self-determining activity, must some-
how be found the Ground of all experience and of all Reality.

The important note which Schelling sounded was the demand that
philosophy should recognize "Nature" as belonging to the sphere
of Reality, and as requiring a measure of reflective thought which
should in some sort put it on equal terms with the Ego, for the con-
struction of our conception of the Being of the World. To Schelling it
seemed impossible to deduce, as Fichte had done, all the rich concrete
development of the world of things from the subjective needs and con-
stitutional forms of functioning which belong to the finite Self. And,
indeed, the doctrine which limits the origin, existence, and value of
all that is known about this sphere of experience to these needs, and
which finds the sufficient account of all experience with nature in
these forms of functioning, must always seem inadequate and even
grotesque in the sight of the natural sciences. Both Nature and Spirit,
thought Schelling, must be allowed to claim actual existence and
equally real value; while at the same time philosophy must reconcile
the seeming opposition of their claims and unite them in an har-
monious and self-explanatory way. In some common substratum,
in which, to adopt Hegel's sarcastic criticism, as in the darkness of
the night "all cows are black," — that is in the Absolute, as an

PHILOSOPHY IN THE NINETEENTH CENTURY 207

Identical Basis of Differences, — the reconciliation was to be accom-
plished.

But the constructive idealistic movement; in which Fichte and
Schelling bore so important a part, could not be satisfied with the
positions reached by either of these two philosophers. Neither the
physical and psychological sciences, nor the speculative interests of
religion, ethics, art, and social life, permitted this movement to stop
at this point. In all the subsequent developments of philosophy dur-
ing the first half or three quarters of the nineteenth century, undoubt-
edly the influence of Hegel was greatest of all individual thinkers. His
motif and plan are revealed in his letter of November 2, 1800, to
Schelling, namely, to transform what had hitherto been an ideal
into a thoroughly elaborate system. And in spite of his obvious
obscurities of thought and style, there is real ground for his claim to
be the champion of the common consciousness. It is undoubtedly in
Hegel's Phdnomenologie des Geistes (1807), that the distinctive fea-
tures of the philosophy of the first half of the last century most
clearly define themselves. The forces of reflection now abandon the
abstract analytic method and positions of the Kantian Critique, and
concentrate themselves upon the study of man's spiritual life as an
historical evolution, in a more concrete, face-to-face manner. Two
important and, in the main, valid assumptions underlie and guide
this reflective study: (1) The Ultimate Reality, or principle of all
realities, is Mind or Spirit, which is to be recognized and known in its
essence, not by analysis into its formal elements (the categories),
but as a living development; (2) those formal elements, or cate-
gories to which Kant gave validity merely as constitutional forms
of the functioning of the human understanding, represent, the rather,
the essential structure of Reality.

In spite of these true thoughts, fault was justly found by the par-
ticular sciences with both the speculative method of Hegel, which
consists in the smooth, harmonious, and systematic arrangement
of conceptions in logical or ideal relations to one another; and also
with the result, which reduces the Being of the World to terms of
thought and dialectical processes merely, and neglects or overlooks
the other aspects of racial experience. Therefore, the idealistic
movement could not remain satisfied with the Hegelian dialectic.
Especially did both the religious and the philosophical party revolt
against the important thought underlying Hegel's philosophy of
religion; namely, that "the more philosophy approximates to a
complete development, the more it exhibits the same need, the same
interest, and the same content, as religion itself." This, as they
interpreted it, meant the absorption of religion in philosophy.

Next after Hegel, among the great names of this period, stand
the names of Herbart and Schopenhauer. The former contributes

208 PHILOSOPHY

in an important way to the proper conception of the task and the
method of philosophy, and influences greatly the development of
psychology, both as a science that is pedagogic to philosophy, and as
laying the basis for pedagogical principles and practice. But Herbart
commits again the ancient fallacy, under the spell of which so much of
the Kantian criticism was bound; and which identifies contradictions
that belong to the imperfect or illusory conceptions of individual
thinkers with insoluble antinomies inherent in reason itself. In spite
of the little worth and misleading character of his view of perception,
and the quite complete inadequacy of the method by which, at a
single leap, he reaches the one all-explanatory principle of his philo-
sophy, Schopenhauer made a most important contribution to the
reflective thought of the century. It is true, as Kuno Fischer has
said, that it seems to have occurred to Schopenhauer only twenty-
five years after he had propounded his theory, that will, as it appears
in consciousness, is as truly phenomenal as is intellect. It is also true
that his' theory of knowledge and his conception of Reality, as meas-
ured by their powder to satisfy and explain our total experience, are
inflicted with irreconcilable contradictions. Neither can we accord
firm confidence or high praise to the "Way of Salvation" which
somehow Will can attain to follow by Eesthetic contemplation and
ascetic self-denial. Yet the philosophy of Schopenhauer rightly
insists upon our Idealistic construction of Reality having regard to
aspects of experience which his predecessors had quite too much
neglected; and even its spiteful and exaggerated reminders of the
facts which contradict the tendency of all Idealism to construct a
smooth, regular, and altogether pleasing conception of the Being of
the World, have been of great benefit to the development of the latter
half of the nineteenth century. »

In estimating the thoughts and the products of modern Idealism
we ought not to forget the larger multitude of thoughtful men, both
in Germany and elsewhere, who have contributed toward shaping
the course of reflection in the attempt to answer the problems which
the critical philosophy left to the nineteenth century. It is a singu-
lar comment upon the caprices of fame that, in philosophy as in sci-
ence, politics, and art, some of those who have really reasoned most
soundly and acutely, if not also effectively upon these problems, are
little known even by name in the history of the philosophical develop-
ment of the century. Among the earlier members of this group, did
space permit, we should wish to mention Berger, Solger, Steffens,
and others, who strove to reconcile the positions of a subjective ideal-
ism with a realistic but pantheistic conception of the Being of the
World. There are others, who like Weisse, I. H. Fichte, C. P.
Fischer, and Braniss, more or less bitterly or moderately and reas-
onably, opposed the method and the conclusions of the Hegelian dia-

PHILOSOPHY IN THE NINETEENTH CENTURY 209

lectic. Still another group earned for themselves the supposedly
opprobrious but decidedly vague title of "Dualists," by rejecting
what they conceived to be the pantheism of Hegel. Still others, like
Fries and Beneke and their successors, strove to parallel philosophy
with the particular sciences by grounding it in an empirical but
scientific psychology; and thus they instituted a line of closely con-
nected development, to which reference has already been made.

Hegel himself believed that he had permanently effected that
reconcihation of the orthodox creed with the cognition of Ultimate
Reality at which his dialectic aimed. In all such attempts at recon-
ciliation three great questions are chiefly concerned: (1) the Being of
God; (2) the nature of man; (3) the actual and the ideally satisfac-
tory relations between the two. But, as might have been expected,
a period of wild, irregular, and confused contention met the attempt
to establish this claim. In this conflict of more or less noisy and
popular as well as of thoughtful and scholastic philosophy, Hegelians
of various degrees of fidelity, anti-Hegelians of various degrees of
hostility, and ultra-Hegelians of various degrees of eccentricity, all
took a valiant and conspicuous part. We cannot follow its history;
but we can learn its lesson. Polemical philosophy, as distinguished
from quiet, reflective, and critical but constructive philosophy involves
a most uneconomical use of mental force. Yet out of this period of
conflict, and in a measure as its result, there came a period of improved
relations between science and philosophy and between philosophy and
theology, which was the dawn, toward the close of the nineteenth
century, of that better illumined day into the middle of which we
hope that we are proceeding.

Before leaving this idealistic movement in Germany, and else-
where as influenced largely b}^ German philosophy, one other name
deserves mention. This name is that of Lotze, who combined ele-
ments from many previous thinkers with those derived from his own
studies and thoughts, — the conceptions of mechanism as applied
to physical existences and to psychical life, with the search for some
monistic Principle that shall satisfy the eesthetical and ethical, as
well as the scientific demands of the human mind. This variety of
interests and of culture led to the result of his making important
contributions to psychology, logic, metaphysics, and aesthetics. If
we find his system of thinking — as I think we must — lacking in
certain important elements of consistency and obscured in places by
doubts as to his real meaning, this does not prevent us from assign-
ing to Lotze a position which, for versatility of interests, genial
quality of reflection and criticism, suggestiveness of thought and
charm of style, is second to no other in the history of nineteenth
century philosophical development.

In France and in England the first quarter of the last century

210 PHILOSOPHY

was far from being productive of great thinkers or great thoughts in
the sphere of philosophy. De Biran (1766-1824), in several important
respects the forerunner of modern psychology, after revolting from
his earlier complacent acceptance of the vagaries of Condillac and
Cabanis, made the discovery that the "immediate consciousness of
self-activity is the primitive and fundamental principle of human
cognition." Meantime it was only a little group of Academicians who
were being introduced, in a somewhat superficial way, to the thoughts
of the Scottish and the German idealistic Schools by Royer-Collard,
Jouffroy, Cousin, and others. A more independent and characteristic
movement was that inaugurated by Auguste Comte (1798-1857),
who, having felt the marked influence of Saint-Simon when he was
only a boy of twenty, in a letter to his friend Valat, in the year 1824,
declares: "I shall devote my whole life and all my powers to the
founding of positive philosophy." In spite of the impossibility of
harmonizing with this point of view the vague and mystical elements
which characterize the later thought of Comte, or with its carrying
into effect the not altogether intelligent recognition of the synthetic
activity of the mind (tout se reduit toujours a lier) and certain hints as
to "first principles;" and in spite of the small positive contribution
to philosophy which Comtism could claim to have made; it has in
a way represented the value of two ideas. These are (1) the necessity
for philosophy of studying the actual historical forces which have
been at work and which are displayed in the facts of history; and
(2) the determination not to go by mere unsupported speculation
beyond experience in order to discover knowable Reality. There is,
however, a kind of subtle irony in the fact that the word " Positivism "
should have come to stand so largely for negative conclusions, in the
very spheres of philosophy, morals, and religion where affirmative
conclusions are so much desired and sought.

That philosophy in Great Britain was in a nearly complete con-
dition of decadence during the first half or three quarters of the
nineteenth century was the combined testimony of writers from such
different points of view as Carlyle, Sir William Hamilton, and John
Stuart Mill. And yet these very names are also witnesses to the fact
that this decadence was not quite complete. In the first quarter of
the century Coleridge, although he had failed, on account of weakness
both of mind and of character, in his attempt to reconcile religion to
the thought of his own age, on the basis of the Kantian distinction be-
tween reason and the intellect, had sowed certain seed-thoughts which
became fertile in the soil of minds more vigorous, logical, and practi-
cal than his own. This was, perhaps, especially true in America, where
inquirers after truth were seeking for something more satisfactory
than the French skepticism of the revolutionary and following period.
Carlyle 's mocking sarcasm was also not without wholesome effect.

PHILOSOPHY IN THE NINETEENTH CENTURY 211

But it was Sir William Hamilton and John Stuart Mill whose
thoughts exercised a more powerful formative influence over the
minds of the younger men. The one was the flower of the Scottish
Realism, the other of the movement started by Bentham and the
elder Mill.

That the Scottish Realism should end by such a combination
with the skepticism of the critical philosophy as is implied in Ham-
ilton's law of the relativity of all knowledge, is one of the most
curious and interesting turns in the history of modern philosophy.
And when this law was so interpreted by Dean Mansel in its appli-
cation to the fundamental cognitions of religion as to lay the founda-
tions upon which the most imposing structure of agnosticism was
built by Herbert Spencer, surely the entire swing around the circle,
from Kant to Kant again, has been made complete. The attempt of
Hamilton failed, as every similar attempt must always fail. Neither
speculative philosophy nor religious faith is satisfied with an ab-
stract conception, about the correlate of which in Reality nothing
is known or ever can be known. But every important attempt of
this sort serves the double purpose of stimulating other efforts to
reconstruct the answer to the problem of philosophy, on a basis of
positive experience of an enlarged type; and also of acting as a real,
if only temporary practical support to certain value- judgments
which the faiths of morality, art, and religion both implicate and,
in a measure, validate.

The influence of John Stuart Mill, as it was exerted not only in
his conduct of life while a servant of the East India Company, but
also in his writings on Logic, Politics, and Philosophy, was, on the
whole, a valuable contribution to his generation. In the additions
which he made to the Utilitarianism of Bentham we have done, I
believe, all that ever can be done in defense of this principle of ethics.
And his posthumous confessions of faith in the ontological value of
certain great conceptions of religion are the more valuable because of
the nature of the man, and of the experience which is their source.
Perhaps the most permanent contribution which Mill made to the
development of philosophy proper, outside of the sphere of logic,
ethics, and politics, was his vigorous polemical criticism of Hamil-
ton's claim for the necessity of faith in an "Unconditioned" whose
conception is "only a fasciculus of negations of the Conditioned in
its opposite extremes, and bound together merely by the aid of
language and their common character of incomprehensibility."

The history of the development of philosophy in America during
the nineteenth century, as during the preceding century, has been
characterized in the main by three principal tendencies. These
may be called the theological, the social, and the eclectic. From
the beginning down to the present time the religious influence and

212 ■ PHILOSOPHY

the interest in political and social problems have been dominant.
And yet withal, the student of these problems in the atmosphere
of this country likes, in a way, to do his own thinking and to make
his own choices of the thoughts that seem to him true and best
fitted for the best form of life. In spite of the fact that the different
streams of European thought have flowed in upon us somewhat
freely, there has been comparatively little either of the adherence
to schools of European philosophy or of the attempt to develop a
national school. Doubtless the influence of English and Scottish
thinking upon the academical circles of America was greatest for
more than one hundred and fifty years after the gift in 1714 by
Governor Yale of a copy of Locke's Essay to the college which bore
his name, — and especially upon the reflections and published
works of Jonathan Edwards touching the fundamental problems
of epistemology, ethics, and religion. During the early part of this
century these views awakened antagonism from such writers as
Dana, Whedon, Hazard, Nathaniel Taylor, Jeremiah Day, Henry P.
Tappan, and other opponents of the Edwardean theology, and also
from such advocates of so-called "free-thinking," as had derived
their motifs and their views from English deistical writers like
Shaftesbury, or from the skepticism of Hume.

A more definite philosophical movement, however, which had
established itself somewhat firmly in scholastic centres by the year
1825, and which maintained itself for more than half a century,
went back to the arrival in this country of John Witherspoon, in
1768, to be the president of Princeton, bringing with him a library
of three hundred books. It was the appeal of the Scottish School to
the "plain man's consciousness" and to so-called "common sense,"
which was relied upon to controvert all forms of philosophy which
seemed to threaten the foundations of religion and of the ethics
of politics and sociology. But even during this period, which was
characterized by relatively little independent thinking in scholastic
circles, a more pronounced productivity was shown by such writers
as Francis Wayland, and others; but, perhaps, especially by Laurens
P. Hickok, whose works on psychology and cosmology deserve
especial recognition: while in psychology, as related to philosophical
problems, the principal names of this period are undoubtedly the
presidents of Yale and Princeton, — Noah Porter and James Mc-
Cosh, — both of whom (but especially the former) had their views
modified by the more scientific psychology of Europe and the pro-
founder thinking of Germany.

It was Germany's influence, however, both directly and indirectly
through Coleridge and a few other English writers, that caused a
ferment of impressions and ideas which, in their effort to work them-
selves clear, resulted in what is known as New England "Tran-

PHILOSOPHY IN THE NINETEENTH CENTURY 213

scendentalism/^ In America this movement can scarcely be called
definitely philosophical; much less can it be said to have resulted
in a system, or even in a school, of philosophy. It must also be said
to have been "inspired but not borrowed" from abroad. Its prin-
cipal, if not sole, literary survival is to be found in the works of Emer-
son. As expounded by him, it is not precisely Pantheism — certainly
not a consistent and critical development of the pantheistic theory
of the Being of the World; it is, rather, a vague, poetical, and pan-
theistical Idealism of a decidedly mystical type.

The introduction of German philosophy proper, in its nature form,
and essential being, to the few interested seriously in critical and
reflective thinking upon the ultimate problems of nature and of
human life, began with the founding of the Journal of Speculative
Philosophy, in 1867, under the direction of William T. Harris, then
Superintendent of Schools in this city.

With the work of Darwin, and his predecessors and successors,
there began a mighty movement of thought which, although it is
primarily scientific and more definitely available in biological science,
has already exercised, and is doubtless destined to exercise in the
future, an enormous influence upon philosophy. Indeed, we are
already in the midst of the preliminary confusions and contentions,
but most fruitful considerations and discoveries belonging to a
so-called philosophy of evolution.

This development has, in the sphere of systematic philosophy,
reached its highest expression in the voluminous works produced
through the latter half of the nineteenth century by Mr. Herbert
Spencer, whose recent death seems to mark the close of the period
we have under consideration. The metaphysical assumptions and
ontological value of the system of Spencer, as he wished it to be
understood and interpreted, have perhaps, though not unnaturally,
been quite too much submerged in the more obvious expressions of
its agnostic positivism. In its psychology, however, the assumption
of "some underlying substance in contrast to all changing forms,"
distinguishes it from a pure positivism in a very radical way. But
more especially in philosophy, the metaphysical postulate of a
mysterious Unity of Force that somehow manages to reveal itself,
and the law of its operations, to the developed cognition of the
nineteenth century philosopher, however much it seems to involve
the system in internal contradictions, certainly forbids that we
should identify it with the positivism of Auguste Comte. In our
judgment, however, it is in his ethical good sense and integrity of
judgment, — a good sense and integrity which commits to ethics
rather than to sociology the task of determining the highest type
of human life, — and in basing the conditions for the prevalence and
the development of the highest type of life upon ethical principles

214 PHILOSOPHY

and upon the adherence to ethical ideas, that Herbert Spencer will
be found most clearly entitled to a lasting honor.

III. The third number of our difScult tasks is to summarize the
principal results, to inventory the net profits, as it were, of the devel-
opment of philosophy during the nineteenth century. This task is
made the more difficult by the heterogeneous nature and as yet
unclassified condition of the development. With the quickening
and diversifying of all kinds and means of intercourse, there has
come the breaking-down of national schools and idiosyncrasies of
method and of thought. In philosophy, Germany, France, Great
Britain, and indeed, Italy, have come to intermingle their streams
of infiuence; and from all these countries these streams have been
flowing in upon America. In psychology, especially, as well as in all
the other sciences, but also to some degree in philosophy, returning
streams of influence from America have, during the last decade or
two, been felt in Europe itself.

It must also be admitted that the attempts at a reconstruction of
systematic philosophy which have followed the rapid disintegration
of the Hegelian system, and the enormous accumulations of new
material due to the extension of historical studies and of the par-
ticular sciences, — including especially the so-called "new psycho-
logy, " — have not as yet been fruitful of large results. In philo-
sophy, as in art, politics, and even scientific theory, the spirit and
the opportunity of the time are more favorable to the gathering of
material and to the projecting of a bewildering variety of new opin-
ions, or old opinions put forth under new names, than to that candid,
patient, and prolonged reflection and balancing of judgment which
a worthy system-building inexorably requires. The age of breaking up
the old, without assimilating the new, has not yet passed away. And
whatever is new, startling, large, even monstrous, has in many
quarters the seeming preference, in philosophy's building as in other
architecture. To the confusion which reigns even in scholastic
circles, contributions have been arriving from the outside, from
philosophers like Nietzsche, and from men great in literature like
Tolstoi. Nor has the matter been helped by the more recent extreme
developments of positivism and skepticism, which often enough,
without any consciousness of their origin and without the respect
for morality and religion which Kant always evinced, really go back
to the critical philosophy.

In spite of all this, however, the last two decades or more have
shown certain hopeful tendencies and notable achievements, look-
ing toward the reconstruction of systematic philosophy. In this
attempt to bring order out of confusion, to enable calm, prolonged,
and reflective thinking to build into its structure the riches of the
new material which the evolution of the race has secured, a place

PHILOSOPHY IN THE NINETEENTH CENTURY 215

of honor ought to be given to France, where so much has been done
of late to blend with clearness of style and independence of thought
that calm reflective and critical judgment which looks all sides of
human experience sympathetically but bravely in the face. In
psychology Ribot, and in philosophy, Fouillee, Renouvier, Secretan,
and others, deserve grateful recognition. No friend of philosophy
can, I think, fail to recognize the probable benefits to be derived
from that movement with which such names as Mach and Ostwald
in Germany are connected, and which is sounding the call to the
men of science to clear up the really distressing obscurity and con-
fusion which has so long clung to their fundamental conceptions;
and to examine anew the significance of their assumptions, with
a view to the construction of a new and improved doctrine of the
Being of the World. And if to these names we add those of the
numerous distinguished investigators of psychology as pedagogic
to philosophy, and, in philosophy, of Deussen, Eucken, von Hart-
mann, Riehl, Wundt, and others, we may well affirm that new light
will continue to break forth from that country which so powerfully
aroused the whole Western. World at the end of the eighteenth and
beginning of the nineteenth centuries. In Great Britain the name
and works of Thomas Hill Green have influenced the attempts at
a reconstruction of systematic philosophy in a manner to satisfy at
one and the same time both the facts and laws of science and the
sesthetical, ethical, and religious ideals of the age, in a very consider-
able degree. And in this attempt, both as it expresses itself in theo-
retical psychology and in the various branches of philosophical
discipline, writers like Bradley, Eraser, Flint, Hodgson, Seth, Stout,
Ward, and others, have taken a conspicuous part. Nor are there
wanting in Holland, Italy, and even in Sweden and Russia, thinkers
equally worthy of recognition, and recognized, in however limited
and unworthy fashion, in their own land. The names of those in
America who have labored most faithfully, and succeeded best, in
this enormous task of reconstructing philosophy in a systematic
way, and upon a basis of history and of modern science, I do not
need to mention; they are known, or they surely ought to be known,
to us all.

In attempting to summarize the gains of philosophy during the
last hundred years, we should remind ourselves that progress in
philosophy does not consist in the final settlement, and so in the
"solving" of any of its great problems. Indeed, the relations of
philosophy to its grounds in experience, and the nature of its method
and of its ideal, are such that its progress can never be expected
to put an end to itself. But the content of the total experience of
humanity has been greatly enriched during the last century; and
the critical and reflective thought of trained minds has been led

216 PHILOSOPHY

toward a more profound and comprehensive theory of ReaUty,
and toward a doctrine of values that shall be more available for the
improvement of man's political, social, and religious life.

In view of this truth respecting the limitations of systematic
philosophy, I think we may hold that certain negative results,
which are customarily adduced as unfavorable to the claims of
philosophical progress, are really signs of improvement during the
latter half of the nineteenth century. One is an increased spirit
of reserve and caution, and an increased modesty of claims. This
result is perhaps significant of riper wisdom and more trustworthy
maturity. Kant believed himself to have established for philosophy
a system of apodeictic conclusions, which were as completely forever
to have displaced the old dogmatism as Copernicus had displaced
the Ptolemaic astronomy. But the steady pressure of historical and
scientific studies has made it increasingly difficult for any sane
thinker to claim for any system of thinking such demonstrable val-
idity. May we not hope that the students of the particular sciences,
to whom philosophy owes so much of its enforced sanity and sane
modesty, will themselves soon share freely of the philosophic spirit
with regard to their own metaphysics and ethical and religious
standpoints, touching the Ultimate Reality? Even when the recoil
from the overweening self-satisfaction and crass complacency of the
earher part of the last century takes the form of melancholy, or of
acute sadness, or even of a mild despair of philosophy, I am not sure
that the last state of that man is not better than the first.

In connection with this improvement in spirit, we may also note an
improvement in the method of philosophy. The purely speculative
method, with its intensely interesting but indefensible disregard of
concrete facts, and of the conclusions of the particular sciences, is no
longer in favor even among the most ardent devotees and advocates
of the superiority of philosophy to those sciences. At the same time,
philosophy may quite properly continue to maintain its position of
independent critic, as well as of docile pupil, toward the particular
sciences.

In the same connection must be mentioned the hopeful fact that
the last two or three decades have shown a decided improvement in
the relations of philosophy toward the positive sciences. There are
plain signs of late that the attitude of antagonism, or of neglect,
which prevailed so largely during the second and third quarters of the
nineteenth century, is to be replaced by one of friendship and mutual
helpfulness. And, indeed, science and philosophy cannot long or
greatly flourish without reciprocal aid, if by science we mean a true
Wissenschaft and if we also mean to base philosophy upon our total
experience. For science and philosophy are really engaged upon the
same task, — to understand and to appreciate the totality of man's

PHILOSOPHY IN THE NINETEENTH CENTURY 217

experience. They, therefore, have essential and permanent relations
of dependence for material, for inspiration and correction, and for
other forms of helpfulness. While, then, their respective spheres have
been more clearly delimited during the last century, their inter-
dependence has been more forcefully exhibited. Both of them have
been developing a systematic exposition of the universe. Both
of them desire to enlarge and deepen the conception of the Being of
the World, as made known to the totality of human experience, in
its Unity of nature and significance. We cannot believe that the end
of the nineteenth century would sustain the charge which Fontenelle
made in the closing years of the seventeenth century: '' L'Acade?nie
des Sciences ne prend la nature que par petites parcelles." Science itself
now bids us regard the Universe as a dynamical Unity, teleologically
conceived, because in a process of evolution under the control of
immanent ideas. Philosophy assumes the same point of view, rather
at the beginning than at the end of defining its purpose; and so feels
a certain glad leap at its heart-strings, and an impulse to hold out
the hand to science, when it hears such an utterance as that of Poin-
care: Ce n'est pas le mechanisme le vrai, le seul hut ; c'est Vunite.

Shall we not say, then, that this double-faced but wholly true
lesson has been learned: namely, that the so-called philosophy of
nature has no sound foundation and no safeguard against vagaries
of every sort, unless it follows the lead of the positive sciences of
nature; but that the sciences themselves can never afford a full
satisfaction to the legitimate aspirations of human reason unless they,
too, contribute to the philosophy of nature — writ large and con-
ceived of as a real-ideal Unity.

That nature, as known and knowable by man, is a great artist,
and that man's sesthetical consciousness may be trusted as having
a certain ontological value, is the postulate properly derived from the
considerations advanced in the latest, and in some respects the most
satisfactory, of the three Critiques of Kant. The ideal way of looking
at natural phenomena which so delighted the mind of Goethe has now
been placed on broad and sound foundations by the fruitful indus-
tries of many workmen, — such as Karl Ernst von Baer and Charles
Darwin, — whose morphological and evolutionary conceptions of the
universe have transformed the current conceptions of cosmic pro-
cesses. But the world of physical and natural phenomena has thereby
been rendered not less, but more, of a Cosmos, an orderly totality.

In addition to these more general but somewhat vague evaluations
of the progress of philosophy during the nineteenth century, we are
certainly called upon to face the question whether, after all, any
advance has been made toward the more satisfactory solution of the
definite problems which the Kantian criticism left unsolved. To this
question I believe an affirmative answer may be given in accordance

218 PHILOSOPHY

with the facts of history. It will be remembered that the first of these
problems was the epistemological. Certainly no little improvement
has been made in the psychology of cognition. We can no longer
repeat the mistakes of Kant, either tvith respect to the uncritical
assumptions he makes regarding the origin of knowledge in the
so-called "faculties" of the human mind or regarding the analysis
of those faculties and their interdependent relations. It is not the
Scottish philosophy alone which has led to the conclusion that, in the
word of the late Professor Adamson, " What are called acts or states
of consciousness are not rightly conceived of as having for their
objects their own modes of existence as ways in which a subject is
modified." And in the larger manner both science and philosophy, in
their negations and their affirmations, and even in their points of
view, have better grounds for the faith of human reason in its power
progressively to master the knowledge of Reality than was the case
a hundred years ago. Nor has the skepticism of the same era, whether
by shallow scoffing at repeated failures, or by pious sighs over the
limitations of human reason, or by critical analysis of the cognitive
faculties "according to well-established principles," succeeded in
limiting our speculative pretensions to the sphere of possible expe-
rience,— in the Kantian meaning both of "principles" and of
"experience." But what both science and philosophy are com-
pelled to agree upon as a common underlying principle is this: The
proof of the most fundamental presuppositions, as well as of the
latest more scientifically established conclusions, of both science and
philosophy, is the assistance they afford in the satisfactory explana-
tion of the totality of racial experience.

In the evolution of the ontological problem, as compared with the
form in which it was left by the critical philosophy, the past century
has also made some notable advances. To deny this would be to dis-
credit the development of human knowledge so far as to say that we
know no more about what nature is, and man is, than was known
a hundred years ago. To say this, however, would not be to speak
truth of fact. And here we may not unnaturally grow somewhat
impatient with that metaphysical fallacy which places an impassable
gulf between Reality and Experience. No reality is, of course,,
cognizable or believable by man which does not somehow show its
presence in his total experience. But no growth of experience is pos-
sible without involving increase of knowledge representing Reality.
For Reality is no absent and dead, or statical, Ding-an-Sich. Cogni-
tion itself is a commerce of realities. And are there not plain signs
that the more thoughtful men of science are becoming less averse to
the recognition of the truth of ontological philosophy; namely, that
the deeper meaning of their own studies is grasped only when they
recognize that they are ever face to face with what they call Energy

PHILOSOPHY IN THE NINETEENTH CENTURY 219

and we call Will, and with what they call laws and we call Mind as
significant of the progressive realization of immanent ideas. This
Ultimate Reality is so profound that neither science nor philosophy
will ever sound all its depths, and so comprehensive as more than to
justify all the categories of both.

Probably, on the whole, there has been less progress made toward a
satisfactory solution of the problems offered by the value-judgments
of ethics and religion, in the form in which these problems were left
by the critical philosophy. The century has illustrated the truth of
Falckenberg's statement: "In periods which have given birth to a
skeptical philosophy, one never looks in vain for the complementary
phenomenon of mysticism." Twice during the century the so-called
"faith-philosophy," or philosophy of feeling, has been borne to the
front, to raise a bulwark against the advancing hosts of agnostics —
occasioned in the first period by the negations of the Kantian criti-
cism, and in the second by the positive conclusions of the physical
and biological sciences. This form of protesting against the neglect
or disparagement of important factors which belong to man's ses-
thetical, ethical, and religious experience, is reasonable and must be
heard. But the extravagances with which these neglected factors
have been posited and appraised, to the neglect of the more defini-
tively scientific and strictly logical, is to be deplored. The great work
before the philosophy of the present age is the reconciliation of the
historical and scientific conceptions of the Universe with the legiti-
mate sentiments and ideals of art, morality, and religion. But surely
neither rationalism nor "faith-philosophy" is justified in pouring out
the living child with the muddy water of the bath.

IV. The attempt to survey the present situation of philosophy,
and to predict its immediate future, is embarrassed by the fact that
we are all immersed in it, are a part of its spirit and present form.
But if nearness has its embarrassments, it has also its benefits. Those
who are amidst the tides of life may know better, in a way, how these
tides are tending and what is their present strength, than do those
who survey them from distant, cool, and exalted heights. "Fur
jeden einzelnen hildet der Vater und der Sohn eine greifbare Kette von
Lehensereignungen und Erfahrungen." The very intensely vital and
formative but unformed condition of systematic philosophy — its
protoplasmic character — contains promises of a new life. If we
may believe the view of Hegel that the systematizing of the thought
of any age marks the time when the peculiar living thought of that
age is passing into a period of decay, we may certainly claim for our
present age the prospect of a prolonged vitality.

The nineteenth century has left us with a vast widening of the
horizon, — outward into space, backward in time, inward toward the
secrets of life, and downward into the depths of Reality. With this

220 PHILOSOPHY

there has been an increase in the profundity of the conviction of the
spiritual unity of the race. In the consideration of all of its problems
in the immediate future and in the coming century — so far as we can
see forward into this century — philosophy will have to reckon with
certain marked characteristics of the human spirit which form at the
same time inspiring stimuli and limiting conditions of its endeavors
and achievements. Chief among these are the greater and more
firmly established principles of the positive sciences, and the pre-
valence of the historical spirit and method in the investigation of all
manner of problems. These influences have given shape to the con-
ception which, although it is as yet by no means in its final or even
in thoroughly self-consistent form, is destined powerfully to affect
our philosophical as well as our scientific theories. This conception is
that of Development. But philosophy, considered as the product of
critical and reflective thinking over the more ultimate problems of
nature and of human life, is itself a development. And it is now, more
than ever before, a development interdependently connected with all
the other great developments.

Philosophy, in order to adapt itself to the spirit of the age, must
welcome and cultivate the freest critical inquiry into its own methods
and results, and must cheerfully submit itself to the demand for
evidences which has its roots in the common and essential experience
of the race. Moreover, the growth of the spirit of democracy, which,
on the one hand, is distinctly unfavorable to any system of philosophy
whose tenets and formulas seem to have only an academic validity
or a merely esoteric value, and which, on the other hand, requires
for its satisfaction a more tenable, helpful, and universally appli-
cable theory of life and reality, cannot fail, in my judgment, to influ-
ence favorably the development of philosophy. In the union of the
speculative and the practical; in the harmonizing of the interests of
the positive sciences, with their judgments of fact and law, and the
interests of art, morality, and religion, with their value-judgments
and ideals; in the synthesis of the truths of Realism and Idealism, as
they have existed hitherto and now exist in separateness or antago-
nism; in a union that is not accomplished by a shallow eclecticism, but
by a sincere attempt to base philosophy upon the totality of human
experience; — in such a union as this must we look for the real pro-
gress of philosophy in the coming century.

Just now there seem to be two somewhat heterogeneous and not
altogether well-defined tendencies toward the reconstruction of sys-
tematic philosophy, both of which are powerful and represent real
truths conquered by ages of intellectual industry and conflict. These
two, however, need to be internally harmonized, in order to obtain a
satisfactory statement of the development of the last century. They
may be called the evolutionary and the idealistic. The one tendency

PHILOSOPHY IN THE NINETEENTH CENTURY 221

lays emphasis on mechanism, the other on spirit. Yet it is most
interesting to notice how many of the early workmen in the investi-
gation of the principle of the conservation and correlation of energy
took their point of departure from distinctly teleological and spiritual
conceptions. " I was led/' said Colding, — to take an extreme case, —
at the Natural Science Congress at Innsbruck, 1869, " to the idea of
the constancy of national forces by the religious conception of life."
And even Moleschott, in his Autobiography, posthumously published,
declares : " I myself was well aware that the whole conception might
be converted; for since all matter is a bearer of force, endowed with
force or penetrated with spirit, it would be just as correct to call it
a spiritualistic conception." On the other hand, the modern, better
instructed Idealism is much inclined, both from the psychological and
from the more purely philosophical points of view, to regard with
duly profound respect all the facts and laws of that mechanism of
Reality, which certainly is not merely the dependent construction
of the human mind functioning according to a constitution that
excludes it from Reality, but is rather the ever increasingly more
trustworthy revealer of Reality. This tendency to a union of the
claims of both Realism and Idealism is profoundly influencing the
solution of each one of these problems which the Kantian criticism
left to the philosophy of the nineteenth century. In respect of the
epistemological problem, philosophy — as I have already said —
is not likely again to repeat the mistakes either of Kant or of the
dogmatism which his criticism so effectually overthrew. It was a
wise remark of the physician Johann Benjamin Erhard, in a letter
dated May 19, 1794, a propos of Fichte: "The philosophy which
proceeds from a single fundamental principle, and pretends to deduce
everything from it, is and always will remain a piece of artificial
sophistry: only that philosophy which ascends to the highest prin-
ciple and exhibits everything else in perfect harmony with it, is the
true one." This at least ought — one would say — to have been
made clear by the century of discussion over the epistemological
problem, since Kant. You cannot deduce the Idea from the Reality,
or the Reality from the Idea. The problem of knowledge is not, as
Fichte held in the form of a fundamental assumption, an alternative
of this sort. The Idea and Reality are, the rather already there,
and to be recognized as in a living unity, in every cognitive experi-
ence. Psychology is constantly adding something toward the pro-
blem of cognition as a problem in synthesis; and is then in a way
contributing to the better scientific understanding of the philo-
sophical postulate which is the confidence of human reason in its
ability, by the harmonious use of all its powers, progressively to
reach a better and fuller knowledge of Reality.

The ontological problem will necessarily always remain the un-

222 PHILOSOPHY

solved, in the sense of the very incompletely solved problem of
philosophy. But as long as human experience develops, and as long
as philosophy bestows upon experience the earnest and candid
efforts of reflecting minds, the solution of the ontological problem
will be approached, but never fully reached. That Being of the
World which Kant, in the negative and critical part of his work,
left as an X, unknown and unknowable, the last century has filled
with a new and far richer content than it ever had before. Especially
has this century changed the conception of the Unity of the Uni-
verse in such manner that it can never return again to its ancient
form. On the one hand, this Unity cannot be made comprehensible
in terms of any one scientific or philosophical principle or law.
Science and philosophy are both moving farther and farther away
from the hope of comprehending the variety and infinite manifold-
ness of the Absolute in terms of any one side or aspect of man's
complex experience. But, on the other hand, the confidence in this
essential Unity is not diminished, but is the rather confirmed. As
humanity itself develops, as the Selfhood of man grows in the
experience of the world which is its own environment, and of the
world within which it is its own true Self, humanity may reasonably
hope to win an increased, and increasingly valid, cognition of the
Being of the World as the Absolute Self.

Closely connected, and in a way essentially identical with the
ontological problem, is that of the origin, validity, and rational
value of the ideas of humanity. May it not be said that the nine-
teenth century transfers to the twentieth an increased interest in
and a heightened appreciation of the so-called practical problems
ef philosophy. Science and philosophy certainly ought to combine
— and are they not ready to combine? — in the effort to secure
a more nearly satisfactory understanding and solution of the pro-
blems afforded by the sesthetical, ethical, and religious sentiments
and ideals of the race. To philosophy this combination means that
it shall be more fruitful than ever before in promoting the uplift and
betterment of mankind. The fulfillment of the practical mission of
philosophy involves the application of its conceptions and prin-
ciples to education, politics, morals, as a matter of law and of cus-
tom, and to religion as matter both of rational faith and of the con-
duct of life.

How, then, can this brief and imperfect sketch of the outline of the
development of philosophy in the nineteenth century better come to
a close than by words of encouragement and of exhortation as well.
There are, in my judgment, the plainest signs that the somewhat
too destructive and even nihilistic tendencies of the second and
third quarters of the nineteenth century have reached their limit;
that the strife of science and philosophy, and of both with religion,

PHILOSOPHY IN THE NINETEENTH CENTURY 223

is lessening, and is being rapidly displaced by the spirit of mutual
fairness and reciprocal helpfulness; and that reasonable hopes of
a new and a splendid era of reconstruction in philosophy may be
entertained. For I cannot agree with the dictum of a recent writer
on the subject, that " the sciences are coming less and less to admit
of a synthesis, and not at all of a synthetic philosopher."

On the contrary, I hold that, with an increased confidence in the
capacity of human reason to discover and validate the most secret
and profound, as well as the most comprehensive, of truths, philo-
sophy may well put aside some of its shyness and hesitancy, and may
resume more of that audacity of imagination, sustained by ontological
convictions, which characterized its work during the first half of the
nineteenth century. And if the latter half of the twentieth century
does for the constructions of the first half of the same century, what
the latter half of the nineteenth century did for the first half of that
century, this new criticism will only be to illustrate the way in which
the human spirit makes every form of its progress.

Therefore, a summons of all helpers, in critical but fraternal spirit,
to this work of reconstruction, for which two generations of enormous
advance in the positive sciences has gathered new material, and for
the better accomplishment of which both the successes and the
failures of the philosophy of the nineteenth century have prepared
the men of the twentieth century, is the winsome and imperative
voice of the hour.

SECTION A — METAPHYSICS

{Hall 6, September 21, 10 a. m.)

Chairman: Professor A. C. Armstrong, Wesleyan University.
Speakers: Professor A. E. Taylor, McGill University, Montreal.

Professor Alexander T. Ormond, Princeton University.
Secretary: Professor A. O. Lovejoy, Washington University.

The Chairman of the Section, Professor A. C. Armstrong, of Wes-
leyan University, in opening the meeting referred to the contin-
ued vitahty of metaphysics as shown by its repeated revivals after
the many destructive attacks upon it in the later modern times:
he congratulated the Section on the fact that the principal speakers
were scholars who had made notable contributions to metaphysical
theory.

THE RELATIONS BETWEEN METAPHYSICS AND THE
OTHER SCIENCES

BY PEOFESSOR ALFRED EDWARD TAYLOR

[Alfred Edward Taylor, Frothingham Professor of Philosophy, McGill Uni-
versity, Montreal, Canada, b. Oundle, England, December 22, 1869. M.A.
Oxford. Fellow, Merton College, Oxford, 1891-98, 1902- ; Lecturer in
Greek and Philosophy, Owens College, Manchester, 1896-1903; Assistant
Examiner to University of Wales, 1899-1903; Green Moral Philosophy Prize-
man, Oxford, 1899; Frothingham Professor of Philosophy, McGill Uni-
versity, 1903- ; Member Philosophical Society, Owens College, American
Philosophical Association. Author of The Problem of Conduct; Elements of
Metaphysics.]

When we seek to determine the place of metaphysics in the gen-
eral scheme of human knowledge, we are at once confronted by an
initial difficulty of some magnitude. There seems, in fact, to be no
one universally accepted definition of our study, and even no very
general consensus among its votaries as to the problems with which
the metaphysician ought to concern himself. This difficulty, serious
as it is, does not, however, justify the suspicion that our science is,
like alchemy or astrology, an illusion, and its high-sounding title
a mere "idol of the market-place," one of those nomina rerum quae
non sunt against which the Chancellor Bacon has so eloquently
warned mankind. If it is hard to determine precisely the scope of

228 METAPHYSICS

metaphysics, it is no less difficult to do the same thing for the un-
doubtedly legitimate sciences of logic and mathematics. And in all
three cases the absence of definition merely shows that we are deal-
ing with branches of knowledge which are, so to say, still in the
making. It is not until the first principles of science are already
firmly laid beyond the possibility of cavil that we must look for
general agreement as to its boundary lines, though excellent work
may be done, long before this point has been reached, in the estab-
lishment of individual principles and deduction of consequences
from them. To revert to the parallel cases I have just cited, many
mathematical principles of the highest importance are formulated in
the Elements of Euclid, and many logical principles in the Organon
of Aristotle; yet it is only in our own time that it has become possible
to offer a general definition either of logic or of mathematics, and
even now it would probably be true to say that the majority of
logicians and mathematicians trouble themselves very little about
the precise definition of their respective studies.

The state of our science then compels me to begin this address
with a more or less arbitrary, because provisional, definition of the
term metaphysics, for which I claim no more than that it may serve
to indicate with approximate accuracy the class of problems which
I shall have in view in my subsequent use of the word. By meta-
physics, then, I propose to understand the inquiry which used
formerly to be known as ontology, that is, the investigation into the
general character which belongs to real Being as such, the science, in
Aristotelian phraseology, of ovra ■§ ovra. Or, if the term " real " be
objected against as ambiguous, I would suggest as an alternative
account the statement that metaphysics is the inquiry into the general
character by which the content of true assertions is distinguished
from that of jalse assertions. The two definitions here offered will,
I think, be found equivalent when it is borne in mind that what the
second of them speaks of is exclusively the content which is asserted
as true in a true proposition, not the process of true assertion, which,
like all other processes in the highest cerebral centres, falls under
the consideration of the vastly different sciences of psychology and
cerebral physiology. Of the two equivalent forms of statement, the
former has perhaps the advantage of making it most clear that it
is ultimately upon the objective distinction between the reality and
the unreality of that which is asserted for truth, and not upon any
psychological peculiarity in the process of assertion itself that the
distinction between true and untrue rests, while the second may be
useful in guarding against misconceptions that might be suggested
by too narrow an interpretation of the term '' reality," such as, e. g.,
the identification of the " real" with what is revealed by sensuous
perception.

METAPHYSICS AND THE OTHER SCIENCES 229

From the acceptance of such a definition two important conse-
quences would follow. (1) The first is that metaphysics is at once
sharply discriminated from any study of the psychical process of
knowledge, if indeed, there can be any such study distinct from the
psychology of conception and belief, which is clearly not itself the
science we have in view. For the psychological laws of the formation
of concepts and beliefs are exemplified equally in the discovery and
propagation of truth and of error. And thus it is in vain to look to
them for any explanation of the difference between the two. Nor
does the otherwise promising extension of Darwinian conceptions
of the "struggle for existence" and the ''survival of the fittest"
to the field of opinions and convictions appear to affect this con-
clusion. Such considerations may indeed assist us to understand
how true convictions in virtue of their " usefulness" gradually come
to be established and extended, but they require to presume the
truth of these convictions as an antecedent condition of their " use-
fulness" and consequent establishment. I should infer, then, that
it is a mistake in principle to seek to replace ontology by a " theory
of knowledge," and should even be inclined to question the very
possibility of such a theory as distinct from metaphysics on the one
hand and empirical psychology on the other. (2) The second con-
sequence is of even greater importance. The inquiry into the gen-
eral character by which the contents of true assertions are discrim-
inated from the contents of false assertions must be carefully dis-
tinguished from any investigation into the truth or falsehood of
special assertions. To ask how in the end truth differs from falsehood
is to raise an entirely different problem from that created by asking
whether a given statement is to be regarded as true or false. The dis-
tinction becomes particularly important when we have to deal with
what Locke would call assertions of "real existence," i. e., assertions
as to the occurrence of particular events in the temporal order. All
such assertions depend, in part at least, upon the admission of what
we may style "empirical" evidence, the immediate unanalyzed
witness of simple apprehension to the occurrence of an alleged
matter of fact. Thus it would follow from our proposed conception
of metaphysics that metaphysics is in principle incapable either of
establishing or refuting any assertion as to the details of our immedi-
ate experience of empirical fact, though it may have important bear-
ings upon any theory of the general nature of true Being which we
may seek to found upon our alleged experiences. In a word, if our
conception be the corre'ct one, the functions of a science of meta-
physics in respect of our knowledge of the temporal sequence of
events psychical and physical must be purely critical, never con-
structive, — a point to which I shall presently have to recur.

One more general reflection, and we may pass to the consideration

230 METAPHYSICS

of the relation of metaphysics to the various already organzied
branches of human knowledge more in detail. The admission that
there is, or may be, such a study as we have described, seems of itself
to involve the recognition that definite knowledge about the character
of what really " is, " is attainable, and thus to commit us to a position
of sharp opposition both to consistent and thorough-going agnos-
ticism and also to the latent agnosticism of Kantian and neo-Kant-
ian "critical philosophy." In recognizing ontology as a legitimate
investigation, we revert in principle to the "dogmatist" position
common, e. g., to Plato, to Spinoza and to Leibniz, that there is genu-
ine truth which can be known, and that this genuine truth is not
confined to statements about the process of knowing itself. In
fact, the "critical" view that the only certain truth is truth about
the process of knowing seems to be inherently self-contradictory.
For the knowledge that such a proposition as, e. g., " I know only
the laws of my own apprehending activity, " is true, would itself be
knowledge not about the process of knowing but about the content
known. Thus metaphysics, conceived as the science of the general
character which distinguishes truth from falsehood, presupposes
throughout all knowledge the presence of what we may call a " tran-
scendent object," that is, a content which is never identical with
the process by which it is apprehended, though it may no doubt be
maintained that the two, the process and its content, if distinct, are
yet not ultimately separable. That they are in point of fact not
ultimately separable would seem to be the doctrine which, under
various forms of statement, is common to and characteristic of all the
"idealistic" systems of metaphysics. So much then in defense of a
metaphysical point of view which seems to be closely akin to that
of Mr. Bradley and of Professor Royce, to mention only two names
of contemporary philosophers, and which might, I think, for the
purpose of putting it in sharp opposition to the " neo-Kantian "
view, not unfairly be called, if it is held to need a name, "neo-
Leibnizian."

In passing on to discuss in brief the nature of the boundary lines
which divide metaphysics from other branches of study, it seems
necessary to start with a clear distinction between the "pure" or
"formal" and the "applied" or "empirical" sciences, the more so
as in the loose current employment of language the name " science "
is frequently given exclusively to the latter. In every-day life, when
we are told that a certain person is a "man of science," or as the
detestable jargon of our time likes to say, a "scientist, " we expect to
find that he is, e.g., a geologist, a chemist, a biologist, or an electrician.
We should be a little surprised to find on inquiry that our " man of
science" was a pure mathematician, and probably more than a little
to learn that he was a formal logician. The distinction between the

METAPHYSICS AND THE OTHER SCIENCES 231

pure and the empirical sciences may be roughly indicated by saying
that the latter class comprises all those sciences which yield infor-
mation about the particular details of the temporal order of events
phj^sical and psychical, whereas the pure sciences deal solely with the
general characteristics either of all truths, or of all truths of some
well-defined class. More exactly we may say that the marks by
which an empirical is distinguished from a pure science are two.
(1) The empirical sciences one and all imply the presence among
their premises of empirical propositions, that is, propositions which
assert the actual occurrence of some temporal fact, and depend upon
the witness of immediate apprehension, either in the form of sense-
perception or in that of what is commonly called self -consciousness.
In the vague language made current by Kant, they involve an appeal
to some form of unanalyzed "intuition." The pure sciences, on the
other hand, contain no empirical propositions either among their pre-
mises or their conclusions. The principles which form their premises
are self-evidently true propositions, containing no reference to the
actual occurrence of any event in the temporal order, and thus in-
volving no appeal to any form of "intuition." And the conclusions
established in a pure science are all rigidly logical deductions from
such self-evident premises. That the universality of this distinction
is still often overlooked even by professed writers on scientific method
seems explicable by two simple considerations. On the one hand, it
is easy to overlook the important distinction between a principle
which is self-evident, that is, which cannot be denied without explicit
falsehood, and a proposition affirmed on the warrant of the senses,
because, though its denial cannot be seen to be obviously false,
the senses appear on each fresh appeal to substantiate the asser-
tion. Thus the Euclidean postulate about parallels was long falsely
supposed to possess exactly the same kind of self-evidence as
the dictum de omni and the principle of identity which are part
of the foundations of all logic. And further Kanf, writing under
. the influence of this very confusion, has given wide popularity to
the view that the best known of the pure sciences, that of mathe-
matics, depends upon the admission of empirical premises in the
form of an appeal to intuition of the kind just described. Fortunately
the recent developments of arithmetic at the hands of such men
as Weierstrass, Cantor, and Dedekind seem to have definitely refuted
the Kantian view as far as general arithmetic, the pure science of
number, is concerned, by proving that one and all of its propositions
are analytic in the strict sense of the word, that is, that they are
capable of rigid deduction from self-evident premises, so that, in
what regards arithmetic, we may say with Schroder that the famous
Kantian question "how are sjmthetic judgments a priori possible?"
is now known to be meaningless. As regards geometry, the case ap-

232 METAPHYSICS

pears to a non-mathematician like myself more doubtful. Those
who hold with Schroder that geometry essentially involves, as Kant
thought it did, an appeal to principles not self-evident and depend-
ent upon an appeal to sensuous "intuition," are logically bound
to conclude with him that geometry is an " empirical," or as W. K.
Clifford called it, a "physical" science, different in no way from
mechanics except in the relative paucity of the empirical premises
presupposed, and to class it with the applied sciences. On the other
hand, if Mr. Bertrand Russell should be successful in his promised
demonstration that all the principles of geometry are deducible from
a few premises which include nothing of the nature of an appeal to
sensuous diagrams, geometry too would take its place among the
pure sciences, but only on condition of our recognizing that its
truths, like those of arithmetic, are one and all, as Leibniz held,
strictly analytical. Thus we obtain as a first distinction between the
pure and the empirical sciences the principle that the propositions
of the former class are all analytical, those of the latter all synthetic.
It is not the least of the services which France is now rendering to
the study of philosophy that we are at last being placed by the
labors of M. Couturat in a position to appreciate at their full worth
the views of the first and greatest of German philosophers on this
distinction, and to understand how marvelously they have been
confirmed by the subsequent history of mathematics and of logic.

(2) A consequence of this distinction is that only the pure or
formal sciences can be matter of rigid logical demonstration. Since
the empirical or applied sciences one and all contain empirical pre-
mises, i. e., premises which we admit as true only because they have
always appeared to be confirmed by the appeal to " intuition,"
and not because the denial of them can be shown to lead to false-
hood, the conclusions to which they conduct us must one and all
depend, in part at least, upon induction from actual observation of
particular temporal sequences. This is as much as to say that all
propositions in the applied sciences involve somewhere in the course
of the reasoning by which they are established the appeal to the
calculus of Probabilities, which is our one method of eliciting general
results from the statistics supplied by observation or experiment.
That this is the case with the more concrete among such applied
sciences has long been universally acknowledged. That it is no less
true of sciences of such wide range as mechanics may be said, I
think, to have been definitely established in our own day by the
work of such eminent physicists as Kirchhoff and Mach. In fact,
the recent developments of the science of pure number, to which
reference has been made in a preceding paragraph, combined with
the creation of the "descriptive " theory of mechanics, may fairly
be said to have finally vindicated the distinction drawn by Leibniz

METAPHYSICS AND THE OTHER SCIENCES 233

long ago between the truths of reason and the truths of empirical
fact, a distinction which the Kantian trend of philosophical specu-
lation tended during the greater part of the nineteenth century to
obscure, while it was absolutely ignored by the empiricist opponents
of metaphysics both in England and in Germany. The philosoph-
ical consequences of a revival of the distinction are, I conceive, of
far-reaching importance. On the one side, recognition of the em-
pirical and contingent character of all general propositions estab-
lished by induction appears absolutely fatal to the current mechan-
istic conception of the universe As a realm of purposeless sequences
unequivocally determined by unalterable "laws of nature," a result
which has in recent years been admirably illustrated for the Eng-
lish-speaking world by Professor Ward's weU-known Gifford lectures
on "Naturalism and Agnosticism." Laws of physical nature, on the
empiristic view of applied science, can mean no more than observed
regularities, obtained by the application of the doctrine of chances,
— regularities which we are indeed justified in accepting with con-
fidence as the basis for calculation of the future course of temporal
sequence, but which we have no logical warrant for treating as ulti-
mate truths about the final constitution of things. Thus, for exam-
ple, take the common assumption that our physical environment
is composed of a multitude of particles each in every respect the
exact counterpart of every other. Reflection upon the nature of
the evidence by which this conclusion, if supported at all, has to
be supported, should convince us that at most all that the state-
ment ought to mean is that individual differences between the ele-
mentary constituents of the physical world need not be allowed
for in devising practical formulae for the intelligent anticipation of
events. When the proposition is put forward as an absolute truth
and treated as a reason for denying the ultimate spirituality of the
world, we are well within our rights in declining the consequence
on the logical ground that conclusions from an empirical premise
must in their o-rti nature be themselves empirical and contingent.

On the other hand, the extreme empiricism which treats all know-
ledge whatsoever as merely relative to the total psychical state
of the knower, and therefore in the end problematic, must, I appre-
hend, go down before any serious investigation into the nature of
the analytic truths of arithmetic, a consequence which seems to be
of some relevance in connection with the philosophic view popularly
known as Pragmatism. Thus I should look to the coming regeneration
of metaphysics, of which there are so many signs at the moment, on
the one hand, for emphatic insistence on the right, e. g., of physics
and biology and psychology to be treated as purely empirical
sciences, and as such freed from the last vestiges of any domination
by metaphysical presuppositions and foregone conclusions, and on

234 METAPHYSICS

the other, for an equally salutary purgation of formal studies like
logic and arithmetic from the taint of corruption by the irrelevant
intrusion of considerations of empirical psychology.

We cannot too persistently bear in mind that there is, correspond-
ing to the logical distinction between the analytic and the synthetic
proposition, a deep and broad general difference between the wants
of our nature ministered to by the formal and the applied sciences
respectively. The formal sciences, incapable of adding anything to
our detailed knowledge of the course of events, as we have seen,
enlighten us solely as to the general laws of interconnection by which
all conceivable systems of true assertions are permeated and bound
together. In a different connection it would be interesting to de-
velop further the reflection that the necessity of appealing to such
formal principles in all reasoning about empirical matters of fact
contains the explanation of the famous Platonic assertion that the
''Idea of Good" or supreme principle of organization and order in
the universe, is itself not an existent, but something en i-n-eKava t^9
ovcTLa?, "transcending even existence," and the very similar declara-
tion of Hegel that the question whether "God" — in the sense of
such a supreme principle — exists is frivolous, inasmuch as existence
(Dasein) is a category entirely inadequate to express the Divine
nature. For my present purpose it is enough to remark that the
need to which the formal sciences minister is the demand for that
purely speculative satisfaction which arises from insight into the
order of interconnection between the various truths w^hich compose
the totality of true knowledge. Hence it seems a mistake to say, as
some theorists have done, that were we born with a complete know-
ledge of the course of temporal sequences throughout the universe,
and a faultless memory, we should have no need of logic or meta-
physics, or in fact of inference. For even a mind already in possession
of all true propositions concerning the course of events, would still
lack one of the requisites for complete intellectual satisfaction
unless it were also aware, not only of the individual truths, but of
the order of their interdependence. What Aristotle said long ago
with reference to a particular instance may be equally said univers-
ally of all our empirical knowledge; ''even if we stood on the
moon and saw the earth intercepting the light of the sun, we should
still have to ask for the reason why." The purposes ministered to
by the empirical sciences, on the other hand, always include some re-
ference to the actual manipulation in advance by human agency of
the stream of events. We study mechanics, for instance, not merely
that we may perceive the interdependence of truths, but that we
may learn how to maintain a system of bodies in equilibrium, or how
to move masses in a given direction with a given momentum. Hence
it is true of applied science, though untrue of science as a whole, that

METAPHYSICS AND THE OTHER SCIENCES 235

it would become useless if the whole past and future course of events
were from the first familiar to us. And, incidentally it may be ob-
served, it is for the same reason untrue of inference, though true of
inductive inference, that it is essentially a passage from the known
to the unknown.

In dealing with the relation of metaphysics to the formal sciences
generally, the great difficulty w^hich confronts us is that of determin-
ing exactly the boundaries which separate one from another. Among
such pure sciences we have by universal admission to include at
least two, pure formal logic and pure mathematics, as distinguished
from the special applications of logic and mathematics to an empiri-
cal material. Whether we ought also to recognize ethics and aesthet-
ics, in the sense of the general determination of the nature of the
good and the beautiful, as non-empirical sciences, seems to be a more
difficult question. It seems clear, for instance, that ethical discus-
sions, such as bulk so largely in our contemporary literature, as to what
is the right course of conduct under various conditions, are concerned
throughout with an empirical material, namely, the existing pecu-
liarities of human nature as we find it, and must therefore be regarded
as capable only of an empirical and therefore problematic solution.
Accordingly I was at one time myself tempted to regard ethics as
a purely empirical science, and even published a lengthy treatise
in defense of that point of view and in opposition to the whole
Kantian conception of the possibility of a constructive Metaphysik
der Sitten. It seems, however, possible to hold that in the question
"What do we mean by good?" as distinguished from the question
" What in particular is it right to do? " there is no more of a reference
to the empirical facts of human psycholog}'' than in the question
"What do we mean by truth?" and that there must therefore be
a non-empirical answer to the problem. The same would of course
hold equally true of the question "What is beauty?" If there are,
however, such a pure science of ethics and again of aesthetics, it
must at least be allowed that for the most part these sciences are
still undiscovered, and that the ethical and sesthetical results hitherto
established are in the main of an empirical nature, and this must
be my excuse for confining the remarks of the next two paragraphs
to the two great pure sciences of which the general principles may
be taken to be now in large measure known.

That metaphysics and logic should sometimes have been absolutely
identified, as for instance by Hegel, will not surprise us when we
consider how hard it becomes on the view here defended to draw any
hard and fast boundary fine between them. For metaphysics, accord-
ing to this conception of its scope, deals with the formulation of the
self-evident principles implied, in there being such a thing as truth
and the deductions which these principles warrant us in drawing.

236 METAPHYSICS

Thus it might be fairly said to be the supreme science of order, and
it would not be hard to show that all the special questions commonly
included in its range, as to the nature of space, time, causation, con-
tinuity, and so forth, are all branches of the general question, how
many types of order among concepts are there, and what is their
nature. A completed metaphysics would thus appear as the realiza-
tion of Plato's splendid conception of dialectic as the ultimate reduc-
tion of the contents of knowledge to order by their continuous de-
duction from a supreme principle (or, we may add, principles) . Now
such a view seems to make it almost impossible to draw any ulti-
mate distinction between logic and metaphysics. For logic is strictly
the science of the mutual implication of propositions, as we see as
soon as we carefully exclude from it all psychological accretions. In
the question what are the conditions under which one proposition
or group of propositions imply another, we exhaust the whole scope
of logic pure and proper, as distinguished from its various empirical
applications. This is the important point which is so commonly
forgotten when logic is defined as being in some way a study of " psy-
chical processes," or when the reference to the presence of "minds"
in which propositions exist, is intended into logical science. We can-
not too strongly insist that for logic the question so constantly raised
in a multitude of text-books, what processes actually take place when
we pass from the assertion of the premises to the assertion of the
conclusion, is an irrelevant one, and that the only logical problem
raised by inference is whether the assertion of the premises as true
warrants the further assertion of the conclusion, supposing it to be
made. (At the risk of a little digression I cannot help pointing out that
the confusion between a logical and a psychological problem is com-
mitted whenever we attempt, as is so often done, to make the self-
evidence of a principle identical with our psychological inability to
believe the contradictory. From the strictly logical point of view,
all that is to be said about the two sides of such an ultimate contra-
diction is that the one is true and the other is false. Whether it is
or is not possible, as a matter of psychical fact for me to affirm with
equal conviction, both sides of a contradiction, knowing that I am
doing so, is a question of empirical psychology which is possibly
insoluble, and at any rate seems not to have received from the
psychologists the attention it deserves. But the logician, so far as
I can see, has no interest as a logician in its solution. For him it
would still be the case even though all mankind should actually and
consciously affirm both sides of a given contradiction, that one of the
affirmations would be true, and the other untrue.) Logic thus seems
to become either the whole or an integral part of the science of order,
and there remain only two possible ways of distinguishing it from
metaphysics. It might be suggested that logical order, the order of

METAPHYSICS AND THE OTHER SCIENCES 237

implication between truths, is only one species of a wider genus,
order in general by the side, for example, of spatial, temporal, and
numerical order, and thus that logic is one subordinate branch of
the wider science of metaphysics. Such a view, of course, implies
that there are a plurality of ultimately independent forms of order
irreducible to a single type. Whether this is the case, I must confess
myself at present incompetent to decide, though the signal success
with which the principles of number have already been deduced
from the fundamental definitions and axioms of symbolic logic, and
number itself defined, as by Mr. Russell, in terms of the purely logical
concept of class-relation, seems to afford some presumption to the
contrary. Or it may be held that the difference is purely one of the
degree of completeness with which the inquiry into order is pursued.
Thus the ordinary symbolic logic of what Schroder has called the
"identical calculus," or "calculus of domains," consists of a series
of deductions from the fundamental concepts of class and number,
identical equality, totality or the "logical 1," zero or the null-class,
and the three principles of identity, subsumption, and negation. The
moment you cease to accept these data in their totality as the given
material for your science, and to inquire into their mutual coherence,
by asking for instance whether any one of them could be denied,
and yet a body of consistent results deduced from the rest, your
inquiry, it might be said, becomes metaphysics. So, again, the dis-
cussion of the well-known contradictions which arise when we try to
apply these principles in their entirety and without modification to
classes of classes instead of classes of individuals, or of the problem
raised by Peano and Russell, whether the assertions "Socrates is
a man" and "the Greeks are men" affirm the same or a different
relation between their subject and predicate (which seems indeed to
be the same question differently stated), would generally be allowed
to be metaphysical. And the same thing seems to be equally true
of the introduction of time-relations into the interpretation of our
symbols for predication employed by Boole in his treatment of
hypotheticals, and subsequently adopted by his successors as the
foundation of the "calculus of equivalent statements."

However we may decide such questions, we seem at least driven
by their existence to the recognition of two important conclusions.
(1) The relation between logical and metaphysical problems is so close
that you cannot in consistency deny the possibility of a science of
metaphysics unless you are prepared with the absolute skeptic to
go the length of denying the possibility of logic also, and reducing
the first principles of inference to the level of formulae which have
happened hitherto to prove useful but are, for all we know, just as
likely to fail us in future application as not. (Any appeal to the
doctrine of chances would be out of place here, as that doctrine is

238 METAPHYSICS

itself based on the very principles at stake.) (2) The existence of
fundamental problems of this kind which remained almost or wholly
unsuspected until revealed in our own time by the creation of a science
of symbolic logic should console us if ever we are tempted to suspect
that metaphysics is at any rate a science in which all the main con-
structive work has already been accomplished by the great thinkers
of the past. To me it appears, on the contrary, that the recent enor-
mous developments in the purely formal sciences of logic and mathe-
matics, with the host of fundamental problems they open up, give
promise of an approaching era of fresh speculative construction
which bids fair to be no less rich in results than any of the great
"golden" periods in the past history of our science. Indeed, but
that I would avoid the slightest suspicion of a desire to advertise
personal friends, I fancy I might even venture to name some of those
to whom we may reasonably look for the work to be done.

Of the relation of metaphysics to pure mathematics it would be
impertinent for any but a trained mathematician to say very much.
I must therefore be content to point out that the same difficulty
in drawing boundary lines meets us here as in the case of logic. Not
so long ago this difficulty might have been ignored, as it still is by too
many writers on the philosophy of science. Until recently mathematics
would have been thought to be adequately defined as the science of
numerical and quantitative relations, and adequatel}^ distinguished
from metaphysics by the non-quantitative and non-numerical char-
acter of the latter, though it would probably have been admitted that
the problem of the definition of quantity and number themselves is
a metaphysical one. But in the present state of our knowledge such
an account seems doubly unsatisfactory. On the one hand, we have
to recognize the existence of branches of mathematics, such as the
so-called descriptive geometry, which are neither quantitative nor
numerical, and, on the other, quantity as distinct from number appears
to play no part in mathematical science, while number itself, thanks
to the labors of such men as Cantor and Dedekind, seems, as I have
said before, to be known now to be only a special type of order in
a series. Thus there appears to be ground for regarding serial order
as the fundamental category of mathematics, and we are thro-^oi back
once more upon the difficult task of deciding how many ultimately
irreducible types of order there may be before we can undertake any
precise discrimination between mathematical and metaphysical
science. Ho^^ever we may regard the problem, it is at least certain
that the recent researches of mathematicians into the meaning of
such concepts as continuity and infinity have, besides opening up new
metaphysical problems, done much to transfigure the familiar ones,
as all readers of Professor Royce must be aware. For instance I
imagine all of us here present, even the youngest, were brought up on

METAPHYSICS AND THE OTHER SCIENCES 239

the Aristotelian doctrine that there is and can be no such thing as an
actually existing infinite collection, but which of us would care to
defend that time-honored position to-day? Similarly with continuity
all of us were probably once on a time instructed that whereas " quan-
tity" is continuous, number is essentially "discrete," and is indeed
the typical instance of what we mean by the non-continuous. To-day
we know that it is in the number-series that we have our one certain
and familiar instance of a perfect continuum. Still a third illustration
of the transforming light which is thrown upon old standing meta-
physical puzzles by the increasing formal development of mathe-
matics may be found in the difficulties attendant upon the conception
of the "infinitely little," once regarded as the logical foundation of
the so-called Differential Calculus. With the demonstration, which
may be found in Mr. Russell's important work, that "infinitesimal,"
unlike "infinite," is a purely relative term, and that there are no
infinitesimal real numbers, the supposed logical significance of the
concept seems simply to disappear. Instances of this kind could easily
be multiplied almost indefinitely, but those already cited should be
sufficient to show how important are the metaphysical results which
may be anticipated from contemporary mathematical research, and
how grave a mistake it would be to regard existing metaphysical con-
struction, e. g., that of the Hegelian system, as adequate in principle
to the present state of our organized knowledge. In fact, all the mate-
rials for a new Kategorienlehre, which may be to the knowledge of our
day what Hegel's Logic was to that of eighty years ago, appear to lie
ready to hand when it may please Providence to send us the meta-
physician who knows how to avail himself of them. The proof, given
since this address was delivered, by E. Zermelo, that every assem-
blage can be well ordered, is an even more startling illustration of
the remarks in the text.

It remains to say something of the relation of metaphysical specu-
lation to the various sciences which make use of empirical premises.
On this topic I may be allowed to be all the more brief, as I have quite
recently expressed my views at fair length in an extended treatise
(Elements of Metaphysics, Bks. 3 and 4), and have nothing of conse-
quence to add to what has been there said. The empirical sciences,
as previously defined, appear to fall into two main classes, distin-
guished by a difference which corresponds to that often taken in the
past as the criterion by which science is to be separated from philo-
sophy. We may study the facts of temporal sequence either with a
view to the actual control of future sequences or with a view to
detecting under the sequence some coherent purpose. It is in the
former way that we deal with facts in mechanics, for instance, or in
chemistry, in the latter that we treat them when we study history for
the purpose of gaining insight into national aims and character. We

240 METAPHYSICS

may, if we please, with Professor Royce, distinguish the two attitudes
toward fact as the attitude respectively of description and of appre-
ciation or evaluation. Now as regards the descriptive sciences, the
position to which, as I believe, metaphysicians are more and more
tending is that here metaphysics has, strictly speaking, no right at all
to interfere. Just because of the absence from metaphysics itself of all
empirical premises, it can be no business of the metaphysician to
determine what the course of events will be or to prescribe to the
sciences what methods and hypotheses they shall employ in the work
of such determination. Within these sciences any and every hypothe-
sis is sufficiently justified, whatever its nature, so long as it enables
us more efficiently than any other to perform the actual task of calcu-
lation and prediction. And it was owing to neglect of this caution
that the Naturphilosophie of the early nineteenth century speedily fell
into a disrepute fully merited by its ignorant presumption. As regards
the physical sciences, the metaphysician has indeed by this time
probably learned his lesson. We are not likely to-day to repeat the
mistake of supposing that it is for us as metaphysicians to dictate
what shall be the physicist's or chemist's definition of matter or mass
or elementary substance or energy, or how he shall formulate the
laws of motion or of chemical composition. Here, at any rate, we can
see that the metaphysician's work is done when his analysis has made
it clear that we are dealing with no self-evident truths such as the
laws of number, but with inductive, and therefore problematic and
provisional results of empirical assumptions as to the course of facts,
assumptions made not because of their inherent necessity, but because
of their practical utility for the special task of calculation. It is only
when such empirical assumptions are treated as self-evident axioms,
in fact when mechanical science gives itself out as a mechanistic
philosophy, that the metaphysician obtains a right to speak, and then
only for the purpose of showing by analysis that the presence of the
empirical postulates which is characteristic of the natural sciences of
itself excludes their erection into a philosophy of first principles.

What is important in this connection is that we should recognize
quite clearly that psychology stands in this respect on precisely the
same logical footing as physics or chemistry. It is tempting to sup-
pose that in psychology, at any rate, we are dealing throughout with
absolute certainties, realities which "consciousness" apprehends just
as they are without any of that artificial selection and construction
which, as we are beginning to see, is imposed upon the study of physi-
cal nature by the limitations of our purpose of submitting the course
of events to calculation and manipulation. And it is a natural conse-
quence of this point of view to infer that since psychology deals
directly with reahties, it must be taken as the foundation of the meta-
physical constructions which aim at understanding the general char-

METAPHYSICS AND THE OTHER SCIENCES 241

acter of the real as such. The consequence, indeed, disappears at once
if the views maintained in this address as to the intimate relation of
metaphysics and logic, and the radical expulsion from logic of all
discussion of mental processes as such, be admitted. But it is still
important to note that the premises from which the conclusion in
question was drawn are themselves false. We must never allow our-
selves to forget that, as the ever-increasing domination of psychology
by the highly artificial methods of observation and experiment intro-
duced by Fechner and Wundt is daily making more apparent,
psychology itself, like physics, deals not directly with the concrete
realities of individual experience, but with an abstract selected from
that experience, or rather a set of artificial symbols only partially
corresponding with the realities symbolized, and devised for the spe-
cial object of submitting the realm of mental sequences to mathemat-
ical calculation. We might, in fact, have based this inference upon
the single reflection that every psychological "law" is obtained, like
physical laws, by the statistical method of elimination of individual
peculiarities, and the taking of an average from an extended series
of measurements. For this very reason, no psychological law can
possibly describe the unique realities of individual experience. We
have in psychology, as in the physical sciences, the duty of suspecting
exact correspondence between the single case and the general "law"
to be of itself proof of error somewhere in the course of our computa-
tion. These views, which I suppose I learned in the first instance from
Mr. F. H. Bradley's paper called A Defence of Phenomenalism in
Psychology , may now, I think, be taken as finally established beyond
doubt by the exhaustive analysis of Professor Miinsterberg's Grund-
zuge der Psychologic. They possess the double advantage of freeing
the psychologist once for all from any interference by the meta-
physician in the prosecution of his proper study, and delivering
metaphysics from the danger of having assumptions whose sole justi-
fication lies in their utility for the purpose of statistical computation
thrust upon it as self-evident principles. For their full discussion I
may perhaps be allowed to refer to the first three chapters of the
concluding book of my Elements of Metaphysics.

When we turn to the sciences which aim at the appreciation or
evaluation of empirical fact, the case seems rather different. It may
fairly be regarded as incumbent on the metaphysician to consider
how far the general conception he has formed of the character of
reality can be substantiated and filled in by our empirical knowledge
of the actual course of temporal sequence. And thus the way seems
to lie open to the construction of what may fairly be called a Philo-
sophy of Nature and History. For instance, a metaphysician who has
rightly or wrongly convinced himself that the universe can only be
coherently conceived as a society of souls or wills may reasonably go

242 METAPHYSICS

on to ask what views seem best in accord with our knowledge of
human character and animal intelligence as to the varying degrees of
organized intelligence manifested by the members of such a hierarchy
of souls, and the nature and amount of mutual intercourse between
them. And again, he may fairly ask what general way of conceiving
what we loosely call the inanimate world would at once be true to
fundamental metaphysical principles and free from disagreement
with the actual state of our physical hypotheses. Only he will need to
bear in mind that since conclusions on these points involve appeal
to the present results of the inductive sciences, and thus to purely
empirical postulates, any views he may adopt must of necessity share
in the problematic and provisional character of the empirical sciences
themselves, and can have no claim to be regarded as definitely de-
monstrated in respect of their details. I will here only indicate very
briefly two lines of inquiry to which these reflections appear appli-
cable. The growth of evolutionary science, with the new light it has
thrown upon the processes by M^hich useful variations may be estab-
lished without the need for presupposing conscious preexisting design,
naturally gives rise to the question whether such unconscious factors
are of themselves sufficient to account for the actual course of devel-
opment so far as it can be traced, or whether the actual history of the
world offers instances of results which, so far as we can see, can only
have issued from deliberate design. And thus we seem justified in
regarding the problem of the presence of ends in Nature as an intel-
'ligible and legitimate one for the philosophy of the future. I would
only suggest that such an inquiry must be prosecuted throughout by
the same empirical methods, and with the same consciousness of the
provisional character of any conclusions we may reach which would
be recognized as in place if we were called on to decide whether some
peculiar characteristic of an animal group or some singular social
practice in a recently discovered tribe does or does not indicate
definite purpose on the part of breeders or legislators.

The same remarks, in my opinion, apply to the familiar problems
of Natural Theology relative to the existence and activity of such
non-human intelligences as are commonly understood by the names
" God " or " gods." Hume and Kant, as it seems to me, have definitely
shown between them that the old-fashioned attempts to demonstrate
from self-evident principles the existence of a supreme personal intel-
ligence as a condition of the very being of truth all involve unavoid-
able logical paralogisms. I should myself, indeed, be prepared to go
further, and to say that the conception of a single personality as the
ground of truth and reality can be demonstrated to involve contra-
diction, but this I know is a question upon which some philosophers
for whom I entertain the profoundest respect hold a contrary opinion.
The more modest question, however, whether the actual course of

METAPHYSICS AND THE OTHER SCIENCES 243

human history affords probable ground for believing in the activity
of one or more non-human personalities as agents in the development
of our species I cannot but think a perfectly proper subject for
empirical investigation, if only it be borne in mind that any conclusion
upon such a point is inevitably affected by the provisional character
of our information as to empirical facts themselves , and can claim in
consequence nothing more than a certain grade of probability. With
this proviso, I cannot but regard the question as to the existence of
a God or of gods as one upon which we may reasonably hope for
greater certainty as our knowledge of the empirical facts of the
world's history increases. And I should be inclined only to object to
any attempt to foreclose examination by forcing a conclusion either
in the theistic or in the atheistic sense on alleged grounds of a priori
metaphysics. In a word, I would maintain not only with Kant that
the " physico-theological " argument is specially deserving of our
regard, but with Boole that it is with it that Natural Theology
must stand or fall.

NOTE ON EXTENSION AND INTENSION OF TERMS

Among the numerous difficulties which beset the teaching of the
elements of formal logic to beginners, one of the earliest is that of
deciding whether all names shall be considered to have meaning both
in extension and intension. As we all know, the problem arises in
connection with two classes of names, (1) proper names of individ-
uals, (2) abstract terms. I should like to indicate what seems to me
the true solution of the difficulty, though I do not remember to have
seen it advocated anywhere in just the form I should prefer.

(1) As to proper names. It seems clear that those who regard the
true proper name as a meaningless label are nearer the truth than
those who assert with Jevons that a proper name has for its intension
all the predicates which can be truly ascribed to the object named.
As has often been observed, it is a sufficient proof that, for example,
John does not mean " a human being of the male sex," to note that he
who names his daughter, his dog, or his canoe John, makes no false
assertion, though he may commit a solecism. So far the followers of
Mill seem to have a satisfactory answer to Jevons, when they say, for
example, that he confuses the intension of a term with its accidental or
acquired associations. (So, again, we can see that Socrates cannot
mean "the wisest of the Greek philosophers," by considering that I
may perfectly well understand the statement "there goes Socrates"
without being aware that Socrates is wise or a Greek or a philosopher.)
And if we objected that no proper name actually in use is ever with-
out some associations which in part determine its meaning by restrict-
ing its applicability, it would be a valid rejoinder that in. pure logic
we have to consider not the actual usages of language, but those that

244 METAPHYSICS

would prevail in an ideal language purged of all elements of irre-
levancy. In such an ideal scientific language, it might be said, the
proper name would be reduced to the level of a mere mark serviceable
for identification, but conveying no implication whatever as to the
special nature of the thing identified. Thus it would be indifferent
what mark we attach to any particular individual, just as in mathe-
matics it is indifferent what alphabetical symbol we appropriate to
stand for a given class or number. I think, however, that even in such
an ideal scientific language the proper name would have a certain
intension. In the first place, the use of proper name seems to inform
us that the thing named is not unique, is not the only member of
a class. To a monotheist, for instance, the name "God" is no true
proper name, nor can he consistently give a proper name to his
Deity. It is only where one member of a class has to be distinguished
from others that the bestowal of a proper name has a meaning.
And, further, to give a thing a proper name seems to imply that the
thing is itself not a class. In logic we have, of course, occasion to form
the concept of classes which have other classes for their individual
members. But the classes which compose such classes of classes could
not themselves be identified by means of proper names. Thus the
employment of a proper name seems to indicate that the thing
named is not the only member of its class, and further that it is not
itself a class of individuals. Beyond this it seems to be a mere question
of linguistic convention what information the use of a proper name
shall convey. Hence it ought to be said, not that the proper name has
no intension, but that it represents a limiting case in which intension
is at a minimum.

(2) As to abstract terms. Ought we to say, with so many English
formal logicians, that an abstract term is always singular and non-
intensional? The case for asserting that such terms are all singular,
I own, seems unanswerable. For it is clear that if the name of an
attribute or relation is equally the name of another attribute or rela-
tion, it is ambiguous and thus not properly one term at all. To say, for
example, that whiteness means two or more distinct qualities seems
to amount to saying that it has no one definite meaning. Of course, it
is true that milk is white, paper is white, and snow is white, and yet
the color-tones of the three are distinct. But what we assert here is,
not that there are different whitenesses, but only that there are differ-
ent degrees of approximation to a single ideal standard or type of
whiteness. It is just because the whiteness we have in view is one and
not many that we can intelligibly assert, for example, that newly
fallen snow is whiter than any paper. All the instances produced by
Mill to show that abstract terms may be general seem to me either to
involve confusion between difference of kind and difference in degree
of approximation to type, or else to depend upon treating as abstract

METAPHYSI€S AND THE OTHER SCIENCES 245

a term which is really concrete. Thus when we say red, blue, green,
are different kinds of color, surely what we mean is different kinds of
colored surface. Qua colored, they are not different; I mean just as
much and no more when I say "a red thing is colored," or "has
color," as when I say " a green thing is colored." If Mill were right, the
proposition "red is a color" ought to mean exactly the same as "red
is red." Or, to put it in another way, it would become impossible to
form in thought any concept of a single class of colored things.

But need we infer because abstract terms are singular that there-
fore they have no intension and are mere meaningless marks? Com-
monly as this inference is made, it seems to me clearly mistaken. It
seems, in fact, to rest upon the vague and ill-defined principle that
an attribute can have no attributes of its own. That it is false is
shown, I think, by the simple reflection that scientific definitions
are one and all statements as to the meaning of abstract names of
attributes and relations. For example, the definition of a circle is
a statement as to the meaning of circularity, the legal definition of
responsible persons a statement as to the meaning of the abstraction
"responsibility," and so on. (We only evade the point if we argue
that abstract terms when used as the subjects of propositions are
really being employed concretely. For "cruelty is odious," for
instance, does not merely mean that cruel acts are odious acts,
but that they are odious because they are cruel.) In fact, the doc-
trine that abstract terms have no intension would seem, if thought
out, to lead to the view that there are only classes of individuals, but
no classes of classes. Thus to say "cruel acts are odious because
cruel " implies, not only that I can form the concept of a class of cruel
acts, but also that of classes of odious acts of which the class of cruel
acts in its turn is a member. And to admit as much as this is to admit
that the class of cruel acts, considered as a member of the class of odious
acts, shares the common predicate of odiousness with the other classes
of acts composing the higher class. Hence the true account of abstract
terms seems to me to be that we have in them another limiting case,
a case in which the extension and the intension are coincident. Inci-
dentally, by illustrating the ambiguity of the principle that attributes
have no attributes of their own, our discussion seems to indicate the
advantage of taking the purely extensional view as opposed to the
predicative view of the import of propositions as the basis of an ele-
mentary treatment of logical doctrine.

THE PRESENT PROBLEMS OF METAPHYSICS '

BY ALEXANDER T. ORMOND

[Alexander Thomas Ormond, McCosh Professor of Philosophy, Princeton
University, since 1897. b. 1847, Punxsutawney, Pennsylvania. Mental
Science Fellow, Princeton, 1877-78; Post-grad. Bonn and" Berhn, 1884^85;
Ph.D. Princeton, 1880; A.B. ibid. 1877; LL.D. Miami, 1899. Professor of
Philosophy and History, University of Minnesota, 1880-83; Professor of
Mental Science and Logic, Princeton University, 1883-97. Member Ameri-
can Philosophical Association, American Psychological Association.]

THE PRELIMINARY QUESTION

The living problems of any science arise out of two sources: (1) out
of what men may think of it, in view of its nature and claims, and
(2) the problems that at any period are vital to it, and in the solution
of which it realizes the purpose of its existence. Now if we distinguish
the body of the sciences which deal with aspects of the world's phenom-
ena— and here I would include both the psychic and the physical —
from metaphysics, which professes to go behind the phenomenon and
determine the world in terms of its inner, and, therefore, ultimate real-
ity, it may be truly said of the body of the sciences that they are in a
position to disregard in a great measure questions that arise out of the
first source, inasmuch as the data from which they make their de-
parture are obvious to common observation. Our world is all around
us, and its phenomena either press upon us or are patent to our
observation. Lying thus within the field of observation, it does
not occur to the average mind to question either the legitimacy or
the possibility of that effort of reflection which is devoted to their
investigation and interpretation. Metaphysics, however, enjoys no
such immunity as this, but its claims are liable to be met with skep-
ticism or denial at the outset, and this is due partly to the nature of
its initial claims, and partly to the fact that its real data are less open
to observation than are those of the sciences. I say partly to the
nature of the initial claims of metaphysics, for it is characteristic of
metaphysics that it refuses to regard the distinction between phe-
nomena and ground or inner nature, on which the sciences rest, as
final, and is committed from the outset to the claim that the real is
in its inner nature one and to be interpreted in the light of, or in
terms of, its inner unity; whereas, science has so indoctrinated the
modern mind with the supposition that only the outer movements
of things are open to knowledge, while their inner and real nature
must forever remain inaccessible to our powers; I say that the mod-

THE PRESENT PROBLEMS OF METAPHYSICS 247

ern mind has been so imbued with this pretension as to have almost
completely forgotten the fact that the distinction of phenomenon
and ground is one of science's own making. Neither the plain man
nor the cultured man, if he happens not to be tinctured with science,
finds his world a duality. The things he deals with are the realities,
and it is only when his naive realism begins to break down before
the complex demands of his growing life, that the thought occurs to
him that his world may be more complex than he has dreamed. It is
clear, then, that the distinction of our world into phenomena and
ground, on which science so largely rests, is a first product of reflec-
tion, and not a fact of observation at all.

If this be the case, it may be possible and even necessary for
reflection at some stage to transcend this distinction. At least, there
can be no reason except an arbitrary one for taking this first step of
reflection to be a finality. And there would be the same justification
for a second step that would transcend this dualism, as for the initial
step out of which the distinction arose; provided, it should be found
that the initial distinction does not supply an adequate basis for a
rational interpretation of the world that can be taken as final. Now,
it is precisely because the dualistic distinction of the sciences does fail
in this regard, that a further demand for a reflective transformation
of the data arises. Let us bear in mind that the data of the sciences
are not the simple facts of observation, but rather those facts trans-
formed by an act of reflection by virtue of which they become phe-
nomena distinguished from a more fundamental nature on which
they depend and which itself is not open to observation. The real
data of science are found only when the world of observation has been
thus transformed by an act of reflection. If then at some stage in our
effort to interpret our world it should become clear that the sciences
of phenomena, whatever value their results may possess, are not giv-
ing us an interpretation in terms that can be taken as final, and that in
order to ground such an interpretation a further transformation of our
data becomes necessary, I do not see why any of the sciences should
feel that they have cause to demur. In truth, it is out of Just such a
situation as this that the metaphysical interpretation arises (as I
propose very briefly here to show) , a situation that supplies a genuine
demand in the light of which the effort of metaphysics to understand
its world seems to possess as high a claim to legitimacy as that of the
sciences of phenomena. Let us take our stand with the plain man or
the child, within the world of unmodified observation. The things
of observation, in this world, are the realities, and at first we may
suppose have undergone little reflective transformation. The first re-
fiective effort to change this world in any way will, no doubt, be an
effort to number or count the things that present themselves to observa-
tion, and out of this effort will arise the transformation of the world

248 METAPHYSICS

that results from considering it under the concepts and categories
of number. In short, to mathematical reflection of this simple sort,
the things of observation will resolve themselves into a plurality of
countable things, which the numbering reflection becoming explicit
in its ordinal and cardinal moments will translate into a system that
will be regarded as a whole made up of the sum of its parts. The very
first step, then, in the reflective transformation of things resolves
them into a dual system, the world conceived as a cardinal whole that
is made up of its ordinal parts, and exactly equal to them. This
mathematical conception is moreover purely quantitative; involving
the exact and stable equivalence of its parts or units and that of the
sum of the parts with the whole. Now it is with this purely quantita-
tive transformation that mathematics and the mathematical sciences
begin. We may ask, then, why should *there be any other than mathe-
matical science,^ and what ground can non-mathematical science point
to as substantiating its claims? I confess I can see no other final
reason than this, that mathematical science does not meet the whole
demand we feel obliged to make on our world. If mathematics were
asked to vindicate itself, it no doubt would do so by claiming that
things present quantitative aspects on which it founds its procedure.
In like manner non-mathematical, or, as we may call it, physical or
natural science, will seek to substantiate its claims by pointing to
certain ultra-quantitative or qualitative aspects of things. It is true
that, so far as things are merely numerable, they are purely quantita-
tive ; but mathematics abstracts from the content and character of its
units and aggregates, which may and do change, so that a relation
of stable equivalence is not maintained among them. In fact, the
basis of these sciences is found in the tendency of things to be always
changing and becoming different from what they were before. The
problem of these sciences is how to ground a rational scheme of know-
ledge in connection with a fickle world like that of qualitative change.
It is here that reflection finds its problem, and noticing that the tend-
ency of this world of change is for a to pass into b and thus to lose
its own identity, the act of reflection that rationalizes the situation is
one that connects a and b by relating them to a common ground x of
which they stand as successive manifestations or symbols. X thus
supplies the thread of identity that binds the two changes a and b into
a relation to which the name causation may be applied. And just as
quantitative equivalence is the principle of relationship among the
parts of the simple mathematical world, so here in the world of the
dynamic or natural sciences, the principle of relation is natural
causation. 2 We find, then, that the non-mathematical sciences rest on

1 I do not raise the question of qualitative mathematics at all. It is clear that
the first mathematical reflection will be quantitative.

2 By natural causation I mean such a relationship between a and 6 in a phenom-
enal system as enables a through its connection with its ground to determine b.

THE PRESENT PROBLEMS OF METAPHYSICS 249

a basis that is constituted by a second act of reflection ; one that
translates our world into a system of phenomena causally inter-related
and connected with their underlying grounds.

We have now reached a point where it will be possible in a few
sentences to indicate the rise of the metaphysical reflection and the
ground on which it rests. If we consider both the mathematical and
the physical ways of looking at things, we will find that they possess
this feature in common, — they are purely external, having nothing
to say respecting the inner and, therefore, real nature of the things
with which they deal. Or, if we concede the latest claims of some of
the physical speculators and agree that the aim of physics is an
ultimate physical explanation of reality, it will still be true that the
whole standpoint of this explanation will be external. Let me explain
briefly what I mean substantially by the term external as I use it here.
Every interpretation of a world is a function of some knowing con-
sciousness, and consequently of some knowing self. This is too obvious
to need proof. A system will be external to such a knower just to the
extent that the knower finds it dominated and determined by cate-
gories that are different from those of its own determination. A world
physically interpreted is one that is brought completely under the
rubrics of physics and mathematics; whose movements yield them-
selves completely, therefore, to a mechanical calculus that gives rise
to purely descriptive formulae; or to the control of a dynamic prin-
ciple; that of natural causation, by virtue of which everything is
determined without thought of its own, by the impulse of another,
which impulse itself is not directly traceable to any thought or pur-
pose. Now, the occasion for the metaphysical reflection arises when
this situation that brings us face to face with, nay, makes us part
and parcel of, an alien system of things, becomes intolerable, and the
knower begins to demand a closer kinship with his world. The knower
finds the categories of his own central and characteristic activity in
experience. Here he is conscious of being an agent going out in forms
of activity for the realization of his world. The determining categories
of the activity he is most fully conscious of, are interest, idea, previ-
sion, purpose, and that selective activity which goes to its termina-
tion in some achieved end. The metaphysical interpretation arises out
of the demand that the world shall be brought into bonds of kinship
with the knower. And this is effected by generalizing the categories
of consciousness and applying them as principles of interpretation to
the world. The act of reflection on which the metaphysical interpre-
tation proceeds is one', then, in which the world of science is further
transformed by bringing the inner nature of things out of its isolation
and translating the world-movements into process the terms of which
are no longer phenomena and hidden ground, but rather inception and
realization, or, more specifically. Idea and Reality. And the point to

250 METAPHYSICS

be noted here is the fact that these metaphysical categories are led
up to positivity by an act of reflection that has for its guiding aim an
interpretation of the world that will be more ultimately satisfactory
to the knower than that of the physical or natural sciences; while
negatively, it is led up to by the refusal of the knowing consciousness
to rest in a world alien to its own nature and in which it is subordin-
ated to the physical and made a mere epiphenomenon.

QUESTIONS OF POINT OF VIEW, PRINCIPLE AND METHOD OF
METAPHYSICS

It is clear from what has been said that the metaphysical inter-
pretation proceeds on a presupposition radically different from that
of mathematical and physical science. The presumption of these
sciences is that the world is physical, that the physical categories
supply the norms of reality, and that consciousness and the psychic,
in general, are subordinate and phenomenal to the physical. On the
contrary, metaphysics arises out of a revolt from these presumptions
toward the opposite presumption, namely, that consciousness itself
is the great reality, and that the norms of an ultimate interpretation of
things are to be sought in its categories. This is the great transfor-
mation that conditions the possibility and value of all metaphysics.
It is the Copernican revolution which the mind must pass through,
a revolution in which matter and the physical world yields the
primacy to mind; a revolution in which consciousness becomes cen-
tral, its categories and analogies supplying the principles of final
world-interpretation. Let us consider then, in the light of this great
Copernican revolution, the questions of the point of view, principle,
and method of metaphysics. And here the utmost brevity must be
observed. If consciousness be the great reality, then its own central
activity, that effort by which it realizes its world, will determine for
us the point of view or departure of which we are in quest. This will
be inner rather than outer ; it will be motived by interest, will shape
itself into interest-directed effort. This effort will be cognitive; dom-
inated by an idea which will be an anticipation of the goal of the
effort. It will, therefore, become directive, selective, and will stand
as the end or aim of the completed effort. The whole movement will
thus take the form, genetically, of a developing purpose informed by
an idea, or teleologically , of a purpose going on to its fulfillment in some
aim which is also its motive. Now, metaphysics determines its point
of view in the following reasoning: if in consciousness we find the
type of the inner nature of things, then the point of view for the inter-
pretation of this inner nature will be to seek by generalizing the
standpoint of consciously determined effort and asserting that this

THE PRESENT PROBLEMS OF METAPHYSICS 251

is the true point of view from which the meaning of the world is to be
sought.

Having determined the metaphysical point of view, the next ques-
tion of vital importance is that of its principle. And we may cut mat-
ters short here by saying at once that the principle we are seeking is
that of sufficient reason, and we may say that a reason will be suffi-
cient when it adequately expresses the world- view or concept luider
which an investigation is being prosecuted. Let us suppose that this
world-view is that of simple mathematics, the principle of sufficient
reason here will be that of quantitative equivalence of parts; or, from
the standpoint of the whole, that of infinite divisibility. Whereas, if we
take the world of the ultra-mathematical science, which is determined
by the notion of phenomena depending on underlying ground, we will
find that the sufficient reason in this sphere takes the form of adequate
cause or condition. The determining condition or causes of any phys-
ical phenomenon supply, from that point of view, the ratio sufiidens
of its existence. We have seen that the sufficiency of a reason in the
above cases has been determined in view of that notion which defines
the kind of world the investigation is dealing with. Let us apply this
insight to the problem of the principle of metaphysics, and we will
soon conclude that no reason can be metaphysically sufficient that
does not satisfy the requirements of a world conceived under the
notion of inception and realization ; or, more specifically, idea and
reality. In short, the reason of metaphysics will refuse to regard its
world as a mechanism that is devoid of thought and intention; that
lacks, in short, the motives of internal determination and movement,
and will in all cases insist that an explanation or interpretation can
be metaphysically adequate only when its ultimate reference is to an
idea that is in the process of purposive fulffilment. Such an explana-
tion we call teleological or rational, rather than merely mechanical,
and such a principle is alone adequate to embody the ratio suffidens
of metaphysics.

Having determined the point of view and principle of meta-
physics, the question of metaphysical method will be divested of some
of its greatest difficulties. It will be clear to any one who reflects that
the very first problem in regard to the method of metaphysics will
be that of its starting-point and the kind of results it is to look for.
And little can be accomplished here until it has been settled that con-
sciousness is to have the primacy, and that its prerogative, is to supply
both standpoint and principle of the investigation. We have gone
a long way toward mastering our method when we have settled these
points: (1) that the metaphysical world is a world of consciousness;

(2) that the conscious form of effort rather than the mechanical is
the species of activity or movement with which we have to deal; and,

(3) that the world it is seeking to interpret is ultimately one of idea

252 METAPHYSICS

and reality in which the processes take the purposive form. In view of
this, the important steps of method (and we use the term method here
in the most fundamental sense) will be (1) the question of the form of
metaphysical activity or agency as contrasted with that of the phys-
ical sciences. This may be brought out in the contrast of the two
terms finality and mere efficiency, in which by mere efficiency is
meant an agency that is presumed to be thoughtless and purposeless,
and consequently without foresight. All this is embodied in the term
force or physical energy, and less explicitly in that of natural causa-
tion. Contrasted with this, finality is a term that involves the for-
ward impulse of idea, prevision, and purpose. Anything that is cap-
able of any sort of foretaste has in it a principle of prevision, selection,
choice, and purpose. The impulse that motives and runs it, that also
stands out as the end of its fulfillment, is a foretaste, an Ahnung, an
anticipation, and the whole process or movement, as well as every
part of it, will take on this character. (2) The second question of
method will be that of the nature of this category of which finality
is the form. What is its content, pure idea or pure will, or a synthesis
that includes both? We have here the three alternatives of pure
rationalism, voluntarism, and a doctrine hard to characterize in a
single word; that rests on a synthesis of the norms of both rational-
ism and voluntarism. Without debating these alternatives, I propose
here briefly to characterize the synthetic concept as supplying what
I conceive to be the most satisfactory doctrine. The principle of pure
rationalism is one of insight but is lacking in practical energy,
whereas, that of voluntarism supplies practical energy, but is lacking
in insight. Pure voluntarism is blind, whSie pure rationalism is power-
less. But the synthesis of idea and will, provided we go a step further
(as I think we must) and presuppose also a germ of feeling as interest,
supplies both insight and energy. So that the spring out of which our
world is to arise may be described as either the idea informed with
purposive energy, or purpose or will informed and guided by the idea. It
makes no difference which form of conception we use. In either case
if we include feeling as interest we are able to conceive movements
originating in some species of apprehension, taking the dynamic
form of purpose, and motived and selected, so to speak, by interest;
and in describing such activity we are simply describing these normal
movements of consciousness with which our experience makes us
most familiar. (3) The third question of method involves the relation
or correlation of the metaphysical interpretation with that of the
natural or physical science. Two points are fundamental here. In the
first place, it must be borne in mind that it is the same world with
which the plain man, the man of science, and the metaphysician are
concerned. We cannot partition off the external world to the plain
man, the atoms and ethers to the man of science, leaving the meta-

THE PRESENT PROBLEMS OF METAPHYSICS 253

physician in exclusive and solitary possession of the world of con-
sciousness. It is the same world for all. The metaphysician cannot
shift the physical world, with its oceans and icebergs, its vast plane-
tary systems and milky ways, on to the shoulders of the physicist.
This is the metaphysician's own recalcitrant world, which will doubt-
less task all his resources to explain. In the second place, though it
is the same world that is clamoring for interpretation, it is a world
that passes through successive transformations, in order to adapt itself
to progressive modes of interpretation. The plain man is called to pass
through a species of Copernican revolution that subordinates the phe-
nomenon to its ground, before he can become a man of science. In
turn, the man of science must go through the Copernican process, and
learn to subordinate his atoms and ethers to consciousness before he
can become a metaphysician. And it is this transformation that marks
one of the most fundamental steps in the method of metaphysics.
The world must experience this transformation, and it must become
habitual to the thinker to subordinate the physical to the mental
before the metaphysical point of view can be other than foreign to
him. If, then, it be the same content with which the sciences and
metaphysics are called on to deal, it is clear that we have on our
hands another problem on the answer to which the fate of meta-
physics vitally depends; the question of the correlation of its method
with that of the sciences so that it may stand vindicated as the final
interpretation of things.

Ill

QUESTION OF THE CORRELATION OF METAPHYSICS AVITH THE SCIENCES

We have reached two conclusions that are vital here: (1) that the
meta]3hysical way of looking at the world involves a transformation
of the world of physical science; (2) that it is the same world that lies
open to both science and metaphysics. Out of this arises the pro-
blem of the correlation oi the two views; the two interpretations of
the world. If science be right in conceiving the world under such
categories as quantity and natural causation; if science be right
in seeking a mechanical explanation of phenomena (that is, one that
excludes prevision, purpose, and aim); and if metaphysics be right
in refusing to accept this explanation as final and in insisting that
the principle of ultimate interpretation is teleological, that it falls
under the categories of prevision, purpose, and aim; then it is clear
that the problem of correlation is on our hands. In dealing with this
problem, it will be convenient to separate it into two questions: (1)
that of the fact; (2) that of its rationale. The fact of the correlation
is a thing of common experience. We have but to consider the way
in which this Congress of Science has been brought about in order to

254 METAPHYSICS

have an exhibition of the method of correlation. Originating first in
the sphere of thought and purpose, the design has been actuaUzed
through the operation of mechanical agencies which it has some-
how contributed to liberate. On the scale of individual experience
we have the classic instance of the arm moving through space in
obedience to a hidden will. There can be no question as to the fact
and the great difficulty of metaphysics does not arise in the task of
generalizing the fact and conceiving the world as a system of thought-
purposes working out into forms of the actual through mechanical
agencies. This generalization somehow lies at the foundation of all
metaphysical faith, and, this being the case, the real task here, aside
from the profounder question of the rationale, is that of exhibiting
the actual points of correlation; those points in the various stages
of the sciences from physics to ethics and religion, at which the
last category or result of science is found to hold as its immediate
implication some first term of the more ultimate construction of
metaphysics. The working out of this task is of the utmost import-
ance, inasmuch as it makes clear to both the man of science and the
metaphysician the intrinsic necessity of the correlation. It is a task
analogous to the Kantian deduction of the categories.

QUESTIONS OF THE ULTIMATE NATURE OP EEALITT

We come, then, to the question of the rationale of this correlation,
and it is clear here that we are dealing with a phase of the problem
of the ultimate nature of reality. For the question of the correlation
now is how it is possible that our thoughts should affect things so
that they move in response; how mind influences body or the re-
verse, how, when we will, the arm moves through space. And with-
out going into details of discussion here, let us say at once, that
whatever the situation may be for any science, — and it maybe that
some form of dualism is a necessary presupposition of science, —
for metaphysics it is clear that no dualism of substances or orders
can be regarded as final. The life of metaphysics depends on finding
the one for the many; the one that when found will also ground the
many. If, then, the phenomenon of mind and body presents the
appearance of a correspondence of two different and, so far as can
be determined, mutually exclusive agencies, the problem of meta-
physics is the reduction of these agencies to one species. Here we
come upon the issue between materialism and immaterialism. But
inasmuch as the notion of metaphysics itself seems to exclude ma-
terialism, the vital alternative is that of immaterialism. Again, if
psycho-physics presents as its basal category a parallelism between
two orders of phenomena, psychic and physical, it is the business of

THE PRESENT PROBLEMS OF METAPHYSICS 255

metaphysics to seek the explanation of this dualism in some more
ultimate and unitary conception. Now, since the very notion of
metaphysics again excludes the physical alternative from the cate-
gory of finality, we are left with the psychic term as the one that,
by virtue of the fact that it embodies a form of conscious activ-
ity, promises to be most fruitful for metaphysics. From one point •
of view, then, we have reduced our world to immaterialism; from
another, to some form or analogue of the psychic. Now it is not
necessary here to carry the inquiry further in this direction. For
what metaphysics is interested in, specially, is the fact that the
world must be reduced to one kind of being and one type of agency.
If this be done, it is clear that the dualism of body and mind and
the parallel orders of psycho-physics cannot be regarded as final, but
must take their places as phenomena that are relative and reducible
to a more fundamental unity. The metaphysician will say that the
arm moves through space in response to the will, and that every-
where the correlation between mechanical and teleological agency
takes place because in the last analysis there is only one type of agency;
an agency that finds its initiative in interest, thought, purpose,
design, and thus works out its results in the fields of space and
mechanical activities.

Furthermore, on the question to which these considerations lead
up; that of the ultimate interpretation we are to put on the reality
of the world, the issue is not so indeterminate as it might seem from
some points of view. Takhig it that the very notion of metaphysics
excludes the material and the physical as ultimate types of the real,
we are left with the notions of the immaterial and the psychic; and
while the former is indefinite, it is a fact that in the psychic and
especially in the form of it which man realizes in his own experience,
he finds an intelligible type and the only one that is available to him
for the definition of the immaterial. He has his choice, then, either
to regard the world as absolutely opaque, showing nothing but its
phenomenal dress which ceases to have any meaning; or to apply
to the world's inner nature the intelligible types and analogies of
his own form of being. That this is the alternative that is embodied
in the existence of metaphysics is clearly demonstrated by the fact
that the metaphysical interpretation embodies itself in the cate-
gories of reason, design, purpose, and aim. Whatever difficulties we
may encounter, then, in the use and application of the psychic analogy
in determining the nature of the real, it is clear that its employment
is inevitable and indispensable. Let us, then, employ the term ra-
tional to that characterization of the nature of things which to meta-
ph^'sics is thus inevitable and indispensable. The world must in the
last analysis be rational in its constitution, and its agencies and forms
of being must be construed as rational in their type.

256 METAPHYSICS

And here we come upon the last question in this field, that of the
ultimate being of the world. We have already concluded that the
real is in the last analysis rational. But we have not answered the
question whether there shall be one rational or many. Now it has
become clear that with metaphysics unity is a cardinal interest;
that, therefore, the world must be one in thought, purpose, aim.
And it is on this insight that the metaphysical doctrine of the ab-
solutfe rests. There must be one being whose thought and purpose are
all-inclusive, in order that the world may be one and that it may
have meaning as a whole. But the world presents itself as a plurality
of finite existents which our metaphysics requires us to reduce in the
last analysis to the psychic type. What of this plurality of psychic
existents? It is on this basis that metaphysics constructs its doctrine
of individuality. Allowing for latitude of opinion here, the trend
of metaphysical reflection sets strongly toward a doctrine of realitj^
that grounds the world in an Absolute whose all-comprehending
thought and purpose utters or realizes itself in the plurality of finite
individuals that constitutes the world; the degree of reality that
shall be ascribed to the plurality of individuals being a point in
debate, giving rise to the contemporary form of the issue between
idealism and realism. Allowing for minor differences, however,
there is among metaphysicians a fair degree of assent to the doctrine
that in order to be completely rational the world of individual plural-
ity must be regarded as implying an Absolute, which, whether it is
to be conceived as an individual or not, is the author and bearer of
the thought and design of the world as a whole.

QUESTIONS OF METAPHYSICAL KNOWLEDGE AND ULTIMATE CRITEEIA

OF TRUTH

We have only time to speak very briefly, in conclusion, of two
vital problems in metaphysics: (1) that of the nature and limits of
metaphysical knowledge; (2) that of the ultimate criteria of truth. In
regard to the question of knowledge, we may either identify thought
with reality, or we may regard thought as wholly inadequate to repre-
sent the real; in one case we will be gnostic, in the other agnostic.
Now whatever may be urged in favor of the gnostic alternative, it
remains true that our thought, in order to follow along intelligible
lines, must be guided by the categories and analogies of our own
experience. This fixes a limit, so that the thought of man is never in
a position to grasp the real completely. Again, whatever may be
urged in behalf of the agnostic alternative, it is to be borne in mind
that our experience does supply us with intelligible types and cate-
gories ; and that under the impulse of the infinite and absolute, or

THE PRESENT PROBLEMS OF METAPHYSICS 257

the transcendent, to which our thought responds (to put it no
stronger), a dialectical activity arises; on the one hand, the appli-
cation of the experience-analogies to determine the real; on the
other, the incessant removal of limits by the impulse of transcend-
ence (as we may call it). Thus arises a movement of approxima-
tion which while it never completely compasses its goal, yet proceeds
along intelligent lines; constitutes the mind's effort to know; and
results in an approximating series of intelligible and relatively ade-
quate conceptions. Metaphysically, we are ever approximating to
ultimate knowledge; though it can never be said that we have at-
tained it. The type of metaphysical knowledge cannot be character-
ized, therefore, as either gnostic or agnostic.

As to the question of ultimate criteria, it is clear that we are here
touching one of the living issues of our present-day thought. Shall
the judgment of truth, on which certitude must found, exclude
practical considerations of value, or shall the consideration of value
have weight in the balance of certitude ? On this issue we have at
the opposite extremes (1) the pnre rationalist who insists on the
rigid exclusion from the epistemological scale of every consideration
except that of pure logic. The truth of a thing, he urges, is always
a purely logical consideration. On the other hand, we have (2) the
pure pragmatist, who insists on the "vnll to believe" as a legitimate
datum or factor in the determination of certitude. The pragmatic
platform has two planks: (1) the ontological — we select our world
that we call real at the behest of our interests; (2) the ethical — in
such a world practical interest has the right of way in determining
what we are to accept as true as well as what we are to choose as
good. It is my purpose in thus outlining the extremes of doctrine
to close with a suggestion or two toward less ultra-conclusions. It
is a sufficient criticism on the pure rationalist's position to point out
the fact that his separation of practical and theoretic interests is a
pure fiction that is never realized anywhere. The motives of science
and the motives of practice are so blended that interest in the con-
clusion always enters as a factor in the process. A conclusion reached
by the pure rationalist's method would be one that would only
interest the pure rationalist in so far as he could divest himself of all
motives except the bare love of fact for its own sake. The pure
pragmatist is, I think, still more vulnerable. He must, to start with,
be a pure subjective idealist, otherwise he would find his world at
many points recalcitrant to his ontology. Furthermore, the mere
will to believe is arbitrary and involves the suppression of reason. In
order that the will to believe may work real conviction, the point
believed must at least amount to a postulate of the practical reason;
it must become somehow evident that the refusal to believe would
create a situation that would be theoretically unsound or irrational;

258 METAPHYSICS

as, for instance, if we assume that the immortality of the soul is a
real postulate of practical reason, it must be so because the negative
of it would involve the irrationality of our world; and therefore a
degree of theoretic imperfection or confusion. Personally I believe
the lines here converge in such a way that the ideal of truth will
always be found to have practical value; and conversely, as to prac-
tical ideals, that a sound practical postulate will have weight in the
theoretic scales. And it is doubtless true, as Professor Royce urges
in his presidential address on The Eternal and The Practical, that
all judgments must find their final warrant at the Court of the
Eternal where, so far as we can see, the theoretical and practical
coalesce into one.

At the close of the work of this Section and upon the invitation of
Dr. Armstrong, a number of distinguished members in attendance
joined freely in the discussion, to the great pleasure of the many
specialists who were present. Among those participating were
Professor Boltzmann of Vienna, Professor Hoeffding of Copenhagen,
Professor Calkins of Wellesley, and Professor French of the Uni-
versity of Nebraska, to whom replies were made by the principal
speakers, Messrs. Taylor and Ormond.

SHORT PAPERS

A short paper was contributed to the work of the Section by Professor W. P.
Montague of Columbia University, on the " Physical Reality of Secondary Quali-
ties." The speaker said that from the beginning of modern philosophy there has
existed a strong tendency among aU schools of thought — monists of the idealistic
or materialistic types, as well as outspoken dualists — to treat the distinction
between primary and secondary qualities as coincident, so far as it goes, with
the distinction between physical and psychical. Colors, sounds, odors, etc., are
regarded as purely subjective or mental in their nature, and as having no true
membership in the physical order; while correlativeiy all special forms and
relations have been in their turn extruded from the field of the psychical. Let it
be noted that introspection offers little or nothing in support of this view. There
is nothing, for example, about the color red that would make it appear more dis-
tinctively psychical or subjective than a figure or a motion. The perception of
a square or a triangle is not a square or triangular perception; but neither is the
perception of red or blue a red or blue perception. Now with the affective or
emotional contents of experience the case is quite different.

A feeling of pain is a painful feeling, a consciousness of anger is an angry con-
sciousness. Pains are more and less painful, according as we are more and less
aware of them. With feehngs and volitions esse is indeed percipi. Colors and
other secondary qualities, however, do not seem thus to increase or diminish
in their reality concomitantly with our perceptions of them. Red is red, neither
more nor less, regardless of the amount to which we attend to it. And yet it
remains true that, notwithstanding this seeming objectivity, the secondary qual-
ities have long been contrasted with the primary, and classed along with the
affective and volitional states as purely subjective facts. It has always seemed
curious that a view so important as this in its consequences, and so radically at
variance, not only with Pre-Cartesian philosophy, but also with our instinctive
beliefs, should have won its way to the position of an accepted dogma; and the
purpose of this paper was first to examine the grounds upon which this belief
rests, and second to show that the problem of the independent reality of the
physical world and the problem of the relation of physical and psychical appear
in a clearer and more hopeful light when disentangled from the quite different
problem of the relation of primary and secondary qualities.

There were two reasons why the older or Pre-Cartesian view of this question
should give place to the modern doctrine. First, because of the rediscovery of
the idea of mechanism, without which predictive science had been virtually im-
possible. The second reason for reducing the secondary qualities to a merely
subjective status lay in the fact that they are much more dependent than the
primary qualities upon the bodily organism of the one who perceives them.
In closing Professor Montague said: —

"I wish in closing to point out two consequences of the view which I have
been opposing. First, the present paradoxical status of the eternal world; second,
the equally paradoxical status of the relation of that world to the world of mind.
Berkeley was the first thinker clearly to perceive the unsubstantial nature of a
world made up solely of primary qualities. Indeed, in the last analysis, a world
of primary qualities, and nothing else, is a world of relations without terms, a
geometrical fiction, the objective (or, for that matter, the subjective) existence

260 METAPHYSICS

of which the ideaUst would be right in denying. In Biology we have abandoned
obscurantist methods, and no longer attribute the distinctive vital functions of
growth and reproduction to a vital force or vital substance, but solely to the
peculiar configuration of the material elements of a ceU. Why may we not in
psychology with equal propriety attribute the distinctively psychical functions
of subjectivity or consciousness, not to the action of a hj^per-psychical soul-sub-
stance, nor to the presence of a transcendental ego, but simply to that peculiar
configuration of. sensorv elements which constitutes a what we call psychosis? "

SECTION B — PHILOSOPHY OF RELIGION

SECTION B
PHILOSOPHY OF RELIGION

{Hall 1, September 21, 3 p. m.)

Chairman: Professor Thomas C. Hall, Union Theological Seminary, N. Y.
Speakers: Professor Otto Pfleiderer, University of Berlin.

Professor Ernst Troeltsch, University of Heidelberg.
Secretary: Dr. W. P. Montague, Columbia University.

THE RELATION OF THE PHILOSOPHY OF RELIGION TO
THE OTHER SCIENCES

BY PROFESSOR OTTO PFLEIDERER

[D. Otto Pfleiderer, Professor of Theology, University of Berlin since 1875.
b. September 1, 1839, Stetten, Wiirtemberg. Grad. Tiibingen, 1857-61.
Post-grad. ibid. 1864-68. City Professor, Heilbronn, 1868-69; Superin-
tendent, Jena, 1869-70; Professor of Theology, Jena, 1870-75. Author of
Religion and its Essential Characteristics; Religious Philosophy upon His-
torical Foundation; and many other works and papers on Theology.]

In order to answer this question, we need to consider a prelimi-
nary question, namely, whether religion can be regarded as the
object of scientific knowledge in the same manner as other processes
of the intellectual life of the race, such as law, history, and art. It
is well known that this question has not always received an affirm-
ative answer, and indeed it can never be answered in the affirmative
so long as the position is maintained that the only religion is that of
the Christian Church, whose doctrines and teachings rest upon an
immediate divine revelation, and that these must be accepted by
men in blind belief. Under the position of an authoritative ecclesias-
tical faith there can indeed exist a theoretical consideration of the
doctrines of faith, as it was the case with the scholastic theology
of the Middle Ages, which with great earnestness sought to harmon-
ize faith and knowledge; nevertheless, no one of the present day
would give to the scholastic theology the name of science with the
modern meaning of the term science. The scholastic theology used
great formal acuteness and skill in the work of defining and defend-
ing ecclesiastical traditions, still there was lacking that which for
us is the essential condition of scientific knowledge, the free examin-
ation of tradition according to the laws of human thought and the

264 PHILOSOPHY OF RELIGION

analogy of the general experience of humanity. The great hindrance
to the progress of the knowledge of religion was the accepted posi-
tion that the truth of the ecclesiastical doctrines was beyond human
reason and outside of human examination, since their truth rested
upon an immediate divine revelation. Whether this supernatural
authority was ascribed to the Church or the Bible makes very little
difference, for in either case the assumption of such an authority
is a hindrance to the free examination of that which claims to be the
divine revealed truth.

But is this assumption really justifiable in the nature of the case?
Do the doctrines of the Church rest upon a supernatural divine
revelation? So soon as this question was really earnestly considered,
and the thinking mind could not always avoid the consideration,
then there was revealed the 'inadequacy of the assumption. Two
ways of examination led to a common critical result, the philosophical
analysis of the religious consciousness and the historical comparison
of various religions. The first to enter upon these ways and at the
same time to become the founder of the modern science of religion
was the keen Scotch thinker David Hume. Truly the thought of
Hume was still a one-sided, disorganizing skepticism; even as his
theory of knowledge disturbed the truth of all our previous common-
sense opinions and conceptions, so also his philosophy of religion
sought to demonstrate that all religion cannot be proved and is full
of doubt, and that the origin of religion was neither to be found in
divine revelation nor in the reason of man, but in the passions of
the heart and in the illusions of imagination. As unsatisfactory as
this result was, nevertheless it gave an important , advance to the
rational study of religion in two directions, in that of religion being
an experience of the inner life of the soul and in that of religion
being a fact of human history.

Kant added the positive criticism of reason to the negative skep-
ticism of Hume; that is, Kant showed that the human intellect
moved independently in the formation of theoretical and practical
judgments, and that the various materials of thought, desire, and
feelings were regulated by the intellect according to innate original
ideas of the true and good and beautiful. Thus as a natural result
there came the conception that the doctrines of belief arose not as
complete truths, given by divine revelation, but, like every other
form of conscious knowledge, these came to us through the activity
of our own mind, and that therefore these doctrines cannot be re-
garded as of absolute authority for all time, but that we are to seek
to understand their origin in historical and psychical motives. So
far as one looked at the ceremonial forms of positive religion, these
motives indeed were found according to Kant in irrational concep-
tions, but as far as the essence of religion was concerned they were

RELIGION AND THE OTHER SCIENCES 265

rather found to be rooted in the moral nature of man. This is the
consciousness of obhgation of the practical reason or of the con-
science, which raises man to a faith in the moral government of the
world, in immortality and God. With the reduction of religion
from all external forms, doctrines, and ceremonies and the finding
of the real essence of religion in the human mind and spirit, the way
was opened to a knowledge of religion free from all external authority.
Those philosophers who came after Kant followed essentially this
course, though here and there they may separate in their opinions
according to their thought of the psychological function of religion.
When Kant had emphasized the close connection between religion
and the moral obligation, then came Schleiermacher, who empha-
sized the feeling of our dependence upon the Eternal, and who sought
to find the explanation of all religious thoughts and conceptions
in the many relations of the feeling to religious experience. Hegel
on the other hand sought the truth of religion in the thought of the
absolute spirit as found in the finite spirit. Thus Hegel made reli-.
gion a sort of popular philosophy.

At present all agree that all sides of the soul-life have part in
religion; now one side may be the more prominent, now another,
according to the peculiarity of certain religions or the individual
temperaments. The philosophy of religion has, in common with
scientific psychology, the question of the relation of feeling to the
intellect and the will, and as yet there may be many views of this
question. Altogether the philosophy of religion is looking for im-
portant solutions to many of its problems from the realm of the
present scientific psychology. Experiences, such as religious con-
versions, appear under this point of view as ethical changes in which
the aim of a personal life is changed from a carnal and selfish end to
that of a spiritual and altruistic purpose. These are extraordinary
and seemingly supernatural processes; nevertheless in them there
can still be found a certain development of the soul-life according
to law. Modern psychology especially has thrown light upon the
abnormal conditions of consciousness which have so often been made
manifest in the religious experience of all times. That which religious
history records concerning inspiration, visions, ecstasy, and revelation,
we now classify with the well-known appearances of hypnotism,
the induction of conceptions and motives of the will through foreign
suggestion or through self-suggestion, of the division of conscious-
ness in different egos, and in the union of several consciousnesses
into one common mediumistic fusion of thought and will. The explan-
ation of these experiences may not yet be satisfactory, but never-
theless we do not doubt the possibility of a future explanation from
the general laws controlling the life of the soul. The fact that we can
through psychological experiments produce such abnormal conditions

266 PHILOSOPHY OF RELIGION

of consciousness justifies us in taking the position, that certain
psychical laws are at the foundation of these conditions which in
their kind are as natural and regular in their functions as the physical
laws which we observe in physical experiments. These solutions
which modem psychology so far has given, and hopes stiU further
to give, are of great importance to the philosophy of reHgion. They
are an indorsement of the general principle which one hundred years
ago had been advanced by critical speculation, namely, that in all
experiences of the religious life the same principles which control
the human mind in all other intellectual and emotional fields shall
hold sway. Nothing therefore should hinder us in scientific research
from following the well-defined Tnaxims of thought, and unreservedly
applying the same methods of scientific analysis in theology as is
done generally in the other sciences.

The claim of the Church to infallibility and divine inspiration of
its dogmas is weakened under this view of the work of the philosophy
of reHgion. Prophetical inspiration and ecstasy, which usually were
thought to be supernatural revelations, are now declared by the
present psychology to come under the category of other analogous
experiences, such as the action of mental powers which, under definite
conditions of individual gifts and on historical occasions, have
manifested themselves in extraordinary forms of consciousness.
However, these enthusiastic forms of prophetical consciousness
cannot be accepted for a higher form of knowledge or even as of
divine origin and as an infallible proclamation of the truth; on the
contrary, these forms are to be judged as pathological appearances,
which may be more harmful than beneficent for the ethical value
of the prophetical intuition. At least, it has come to pass that aU
forms of revelation must come under the examination of a psycho-
logical analysis and of an analogical judgment. Hence their tradi-
tional nimbus of imique. supernatural, and absolute authority is for
all time destroyed.

We are carried to the same r^ult by the comparative study of the
history of rehgions. The study shows us that the Christian Church,
with its dogma of the divine inspiration of the Bible, does not stand
alone; that before and after Christianity other rehgions made
exaetiy the same claims for their sacred scriptures. By the pious
Brahman the Veda is regarded as infallible and eternal; he beUeves
the hymns of the old seers were not composed by the seers them-
selves, but were taken from an original copy in heaven. The Buddhist
sees in the sayings of his sacred book '■' Dhammapadam '" the exact
inheritance of the infaUible words of his omniscient teacher Buddha.
For the confessor of Ahuramazda the Zenda vesta contains the
scriptiiral revelation of the good spirit unto the prophet Zarathustra;
accordins to the rabbis the laws revealed unto Mos^ on Mount Sinai

RELIGION AND THE OTHER SCIENCES 267

were even before the creation of the world the object of the observ^a-
tion of God; for the faithful Mohammedan the Koran is the copy
of an ever-present original in heaven, the contents of which were
dictated word for word to Mohammed by the angel Gabriel. Whoever
ponders the similar claims of all these rehgions for the infallibility of
their sacred books, to him it becomes difficult to hold the dogma
of the Christian Church concerning the inspiration and infaUibihty of
the Bible as alone true and the similar dogmas of other rehgions
as being false. Rather he will accept the xievr that in all these ex-
amples there are found the same motives of the rehgions mind, 'that
here is given an expression to the same need common to aU seeking
for an absolute and abiding basis for their faith.

The study of the comparison of rehgions has discovered in religions
other than that of Christianity many very striking parallels to many
narratives and teachings of the Bible. It may be weU to recall very
briefly some of the important points. Owing to the fact that the
Assyrian cuneiform writings have now been deciphered, there has
been found a stor}' of the creation which has many characteristics
in common with those of the Bible. There is found a story of a flood,
which in its very details can be regarded as the forerunner of the
story of the flood in the Bible. There have been found Assyrian
penitential psalms, which, in consciousness of gmlt and in earnest-
ness of prayer for forgiveness, can well be compared with many
psalms of the Bible. Recentl}' the Code of the AssjTian King Ham-
murabi, who reigned two thousand three hundred years before
Christ, has been discovered. The similarity of this Code with many
of the early Mosaic Laws has called general attention to this fact. In
the Persian rehgion there are found teachings of the Kingdom of God,
of the good spirits who surround the throne of God, of the Spirit
hostile to God and of an army of his demons, of the judgment of each
soul after death, of a heaven with eternal hght and of the dark
abyss of hell, of the future struggle of the multitudes of good and bad
spirits and the ^•ictory over the bad through a divine hero and
saviour, of the general resurrection of the dead, of the awful destruc-
tion of the world and the creation of a new and better world, —
teachings which are also found in the later Jewish theology and apo-
calypse, so that the acceptance of a dependence of Jewish upon
corresponding Persian teaching can hardly be avoided. Also Grecian
influence is obser^'ed in later Jewish hterature, in proverbs, in the
wisdom of Solomon and the Son of Sirach; especially in the Alex-
andrian Jewish theology are found Platonic thoughts of an eternal,
ideal world, of the heavenly home of the soul, and the Stoic concep-
tion of a world-ruling di\ine Logos.

It is from this source that the Logos to which Philo had already
ascribed the meaning of the Son of God and the Bringer of a di^-ine

268 PHILOSOPHY OF RELIGION

revelation crossed over into Christian theology and became the
foundation of the dogma of the Church concerning the person of
Christ. Of still greater importance than even all this was the opening
of the Indian and especially the Buddhistic religious writings. In
these we have, five hundred years before Christianity, the revelation
of redemptive religion, resting upon the ethical foundation of the
abnegation of self and the withdrawal from the world. In the centre
of this religion is Gautama Buddha, the ideal teacher of redeeming
truth, whose human life was adorned by the faith of his followers
with a crown of wonderful legends; from an abode in heaven, out of
mercy to the world, he descended into the world, conceived and
born of a virgin mother, greeted and entertained by heavenly spirits,
recognized beforehand by a pious seer as the future redeemer of the
world; as a youth he manifested a wisdom beyond that of his teachers.
Then after the reception of an illuminating revelation, he victoriously
overcomes the temptation of the devil, who would cause him to be-
come faithless to his call to redemption. Then he begins to preach
of the coming of the Kingdom of Justice, and sends forth his dis-
ciples, two by two, as messengers of his gospel to all people. Although
he declares that it is not his calling to perform miracles, neverthe-
less the legends indeed tell how many sick were healed, how with the
contents of a small basket hundreds were fed, how possessed of all
knowledge he reveals hidden things; how overcoming the limitations
of space and time, swaying in the air, being transfigured in a heavenly
light, he reveals himself to his disciples just before his death. And
at last, in the faith of his followers, having passed from the position
of a human teacher to that of an eternal heavenly spirit and lord
of the world, he is exalted as the object of prayer and reverence, to
many millions of the human race in Southern and Eastern Asia.

It is hardly possible that the knowledge of this parallel from India
to the New Testament, and of the Babylonian and Persian parallel
to the Old Testament, can be without influence upon the religious
thought of Christian people. Although we may be ever so much
convinced concerning the essential superiority of our religion over
all other religions, nevertheless the dogmatic contrast between abso-
lute truth on the one side and complete falsity on the other can no
more be maintained. In place of this view there must enter the view
of a relative grade of differences between the higher and lower stages
of development. No longer can we see in other religions only mis-
takes and fiction, but under the husk of their legends many precious
kernels of truth must be seen, expressions of inner religious feelings
and of noble ethical sentiments. One should therefore accept the
position not to object to the same discrimination between husk and
kernel in the matter of one's own religion, and to recognize in its
inherited traditions and dogmas legendary elements, the explanation

RELIGION AND THE OTHER SCIENCES 269

of which is to be found in psychical motives and in historical sur-
roundings, even as they are found in the corresponding parts of
religions other than the Christian religion. Therefore the historical
comparison of religions takes us away from an absolute dogmatic
positivism to a relative evolutionary manner of study, placing all
religions without exception under the laws of time progression and
under the causal connection of the law of cause and effect. The
isolation of religion therefore is no more. It is regarded as being
a part of other human historical affairs, and must yield to the test of a
thorough unhindered research. The value of the Christian religion
can never suffer in the view of a reasonable man, when it is not ac-
cepted in blind faith, but as the result of discriminating comparison.
As the evolutionary philosophy of religion uses the method of
science without exception in the case of all historical religions, so
also it does not shrink from taking up the question of the beginning
of religion, but believes that here also is found the key in the ana-
lytical, critical, and comparative method. And here is found the
assistance of the comparative study of languages, ethnology, and
paleontology.

The celebrated Sanscrit scholar. Max Miiller, sought in the com-
parative study of mythology to prove the etymological relation of
many of the Grecian gods and heroes with those of the mythology
of India and to trace the common origin of all these mythical beings
and legends in the personification of the movements of the heavenly
bodies, the thunder and lightning, the tempest and the rain. All
mythical belief in gods of the Indo-Germanic peoples seems to have
arisen out of a poetical view and dramatic personification of the
powers of nature. Suggestive as this hypothesis is, it is not by any
means sufficient to give us a complete explanation of the subject.
In fact, others have shown that primitive religion does not altogether
consist in mythical conceptions, but mainly in reverential actions,
sacrifices, sacraments, vows, and other similar cults, which have
very little to do with the atmospherical powers of nature, but rather
with the social life of primitive people. And when once the sight
was clearly directed to the social meaning of the religious rites, it was
then observed that even the earliest legends concerning the gods
were connected far more closely with the habits and customs of
early society than with the facts of nature. Tylor's celebrated book
concerning "Primitive Civilization" is written from this standpoint,
an epoch-making book, showing the original close connection of
religion with the entire civilization of humanity, with the views of
life and death, the social customs, the forms of law, their strivings in
art and science; a book with a large amount of information, brought
together from observation on all sides. In this channel are found all
the researches which to-day are classified under the name of Folk-

270 PHILOSOPHY OF RELIGION

lore; seeking to gather the still existing characteristic customs and
forms, legends, stories, and sayings, in order to compose these and to
discover the survivals of earliest religion, poetry, and civilization of
humanity. The gain of this study pursued with so great diligence is
not to be underrated. These studies show that all that, which at one
time existed as faith in the spirit of humanity, possessed within its
very nature the strongest power of continuance, so that in new and
strange conditions and in other forms it continued to remain. Under
all changes and progress of history there is still found an unbroken
connection of constant development.

As important, however, as the possession of a general knowledge
of historical forms of development is to the philosophy of religion,
nevertheless the possession of this knowledge is not wholly a fulfill-
ment of the purpose of the philosophy of religion. To understand a
development means not merely to know how one thing follows as the
result of the other, but also to understand the law which lies at the
foundation of all empirical changes and at the same time controls
the end of the development. If this principle holds good in the
understanding of the development in the processes of nature, much
more does the principle hold good in understanding the processes of
intellectual development of humanity, which have for us not only
a theoretical, but at the same time an eminently practical interest.
The philosopher of religion sees in religious history not merely the
coming together of similar forms, but an advance from the lowest
stage of childlike ignorance to an ever purer and richer realization
of the idea of religion, a divinely ordained progress for the education
of humanity from the slavery of nature to the freedom of the spirit.
The question now arises : where do we find the principle and law of
this ever-rising development? Where do we find the measure of
judgment for the relative value of religious appearances? It is clear
that the general principle of the complete development cannot be
found in a single fact which is only one of the many manifestations
of the general principle, and it is just as clear that the absolute
norm of judgment is not found in a single fact always relative,
presenting to us the object of judgment and therefore being impos-
sible to stand as the norm of judgment. Therefore the principle of
religious development and the norm of its judgment can only be
found in the inner being of the spirit of humanity, namely, in the
necessary striving of the mind into an harmonious arrangement of
all our conceptions, or the idea of the truth, and into the complete
order of all our purposes, or the idea of the good. These ideas unite
in the highest unity, in the Idea of God. Therefore the consciousness
of God is the revelation of the original innate longing of reason after
complete unity as a principle of universal harmony and consistence in
all our thinking and willing. Hence, in the first place, arises the result

RELIGION AND THE OTHER SCIENCES 271

that the development of the consciousness of God in the history of
religion is always dependent upon the existing conditions of the two
united sides, the theoretical perception of the truth and the moral
standard of life. In the second place the result arises that the judg-
ment of the value of all appearances in the history of religion depends
as to whether and how far these appearances agree with the idea of
the true and the good, and correspond with the demands of reason
and conscience. That science which is engaged with the idea of the
good we name Ethics; that which is engaged with the last principles
of the perception of truth, using the expression of Aristotle, we
may name Metaphysics, or following Plato — Dialectic. Recognizing
then in the idea of God the synthesis of the idea of the true and the
good, the philosophy of religion is closely related with both. Ethics
and Metaphysics.

At present the relation of religion to morality is an object of much
controversy. There are many who hold that morality without religion
is not only possible but also very desirable; since they are of the
opinion that moral strength is weakened, the will is without freedom,
and its motives corrupted on account of religious conceptions. On
the other hand, the Church, considering the experience of history,
finds that religion has ever proved itself to be the strongest and most
necessary aid to morality. In this contest the philosophy of rehgion
occupies the position of a judge who is called upon to adjust the rela-
tive rights of the parties. The philosophy of religion brings to light
the historical fact that from the very beginnings of human civilization,
social life and morality were closely connected with religious con-
ceptions and usages, and indeed always so interchangeable in their
influence that the position of social civilization on the one side cor-
responded with the position of religious civilization on the other,
just as the water-level in two communicating pipes. Therefore it
follows that it is unjust and not historical to blame religion on ac-
count of the defects of a national and temporal morality; for these
defects of morality, with the corresponding errors of religion, find a
common ground in a low stage of development of the entire civiliza-
tion of the people of the time and age. Further, it becomes the task
of the philosophy of religion to examine whether this correspondence
of religion and morality, recognized in history, is also found in the
very nature of morality and religion. This question in the main is
answered without doubt in the affirmative, for it is clear that the
religious feeling of dependence upon one all-ruling power is well
adapted not only to make keen the moral consciousness of obligation
and to deepen the feeling of responsibility, but also to endow moral
courage with power and to strengthen the hope of the solution of
moral purposes. The clearer religious faith comprehends the rela-
tion of man to God, so much the more will that faith prove itself as

272 PHILOSOPHY OF RELIGION

a strong motive and a great incentive of the moral life. Such a con-
ception will not make the moral will unfree but truly free, not in the
sense of a selfish choice, but in the sense of a love that serves, knowing
itself as an instrument of the divine will, who binds us all into a
social organism, the kingdom of God. And, on the other hand, the
more ideal the moral view of life, the higher and greater its aims,
the more it recognizes its great task to care for the welfare not only
of the individual but of all, to cooperate in the welfare and develop-
ment of all forms of society, the more earnestly the moral mind will
need a sincere faith that this is God's world, that above all the
changes of time an eternal will is on the throne, whose all-wise guid-
ance causes everything to be for the best unto those who love him.
A like middle position of arbitration falls to the philosophy of
religion in the matter of the relation of religion to science. The
first demand of science is freedom of thought, according to its own
logical laws, and its fundamental assumption is the possibility of
the knowledge of the world on the basis of the unchangeable laws
of all existence and events. With this fundamental demand science
places itself in opposition to the formal character of ecclesiastical
doctrine so far as the doctrine claims infallible authority resting
upon a divine revelation. And the fundamental assumption of the
regular law of the course of the world is in opposition to the contents
of ecclesiastical doctrine concerning the miraculous interposition
in the course of nature and of history. To the superficial observer
there appears therefore to exist an irreconcilable conflict between
science and religion. Here is the work of the philosophy of religion,
to take away the appearance of an irreconcilable opposition between
science and religion, in that the philosophy of religion teaches first
of all to distinguish between the essence of religion and the ecclesias-
tical doctrines of a certain religion, and to comprehend the historical
origin of these doctrines in the forms of thought of past times. To
this purpose the method of psychological analysis and of historical
comparison mentioned above is of service. When, then, by this
critical process religion is traced to its real essence in the emotional
consciousness of God, to which the dogmatic doctrines stand as
secondary products and varied symbols, then it remains to show
that between the essence of religion and that which science demands
and presupposes, there exists not conflict but harmony. When the
idea of God is recognized as the synthesis of the ideas of the true
and the good, so then must all truth as sought by science, even as the
highest good, which the system of ethics places as the purpose of all
action — these must be recognized as the revelation of God in his
eternal reason and goodness. The laws of our rational thinking
then cannot be in conflict with divine revelation in history, and the
laws of the natural order of the world can no more stand in conflict

RELIGION AND THE OTHER SCIENCES 273

with the world-governing Omnipotence; but both, the laws of our
thinking and those of the real world, reveal themselves as the har-
monious revelations of the creative reason of God, which, according
to Plato's fitting word, is the efficient ground of being as well as of
knowing. It is therefore not merely a demand of religious belief that
there is real truth in our God-consciousness, that there should be
an activity and revelation of God himself in the human mind; it is
also in the same manner a demand of science considering its last
principles, that the world, in order to be known by us as a rational,
regulated order, must have for its principle an eternal creative
reason. Long ago the old master of thinking, Aristotle, recognized
this fact clearly, when he said that order in the world without a prin-
ciple of order could be as little thinkable as the order of an army
without a commanding general.

But while it is true that science, as the ground of the possibility
of its knowledge of the truth, must presuppose the same general
principle of intellectual knowledge which religion has as the object
of its practical belief, then by principle the apprehension is excluded
that any possible progress on the part of science in its knowledge
of the world can ever destroy religion. We are rather the more
justified in the hope that all true knowledge of science will be' a help
to religion, and will serve as the means of purifying religion from the
dross of superstition.

Truly it can easily be shown that a divine government of the
world breaking through, and now and then suspending the regular
order of nature through miraculous intervention, would not be more
majestic, but far more limited and human, than such a government
which reveals itself as everywhere and always the same in and
through its own ordained laws in the world. And again, that a
revelation prescribing secret and incomprehensible doctrines and
rites, demanding from humanity a blind faith, would far less be in
harmony with the guiding wisdom and love of God, and far less
could work for the intellectual liberty and perfection of humanity,
than such a revelation which is working in and through the reason
and conscience of humanity, and is realizing its purpose in the pro-
gressive development of our intellectual and moral capacities and
powers. When therefore science raises critical misgivings against
the supernatural and irrational doctrines of positive religion, then
the real and rightly understood interests of religion are not harmed
but rather advanced; for this criticism serves religion in helping
it to become free from the unintellectual inheritance of its early
days, in helping religion to consider its true intellectual and moral
essence, and to bring to a full display all the blessed powers which
are concealed within its nature, to press through the narrow walls of
an ecclesiasticism out into the full life of humanity, and to work as

274 PHILOSOPHY OF RELIGION

leaven for the ennoblement of humanity. Not in conflict with science
and moral culture, but only in harmony with these, can religion come
nearer to the attainment of its ideal, which consists in the worship of
God in spirit and in truth. Even though they may not be conscious
of their purpose, but nevertheless in fact all honest work of science
and all the endeavors of social and ethical humanity have part in
the attainment of this ideal.

It is the work of the philosophy of religion to make clear that all
work of the thinking and striving spirit of humanity, in its deepest
meaning, is a work in the kingdom of God, as service to God, who is
truth and goodness. It is the work of the philosophy of religion
to explain various misunderstandings, to bring together opposing
sides, and so to prepare the way for a more harmonious cooperation
of all, and for an always hopeful progress of all on the road to the
high aims of a humanity fraternally united in the divine spirit.

MAIN PROBLEMS OF THE PHILOSOPHY OF RELIGION:
PSYCHOLOGY AND THEORY OF KNOWLEDGE IN
THE SCIENCE OF RELIGION

BY PROFESSOR ERNST TROELTSCH

{Translated from the German by Dr. J. H. Woods, Harvard University.)

[Ernst Troeltscli, Professor of Systematic Theology, University of Heidelberg,
since 1894. b. February 17, 1865, Augsburg, Bavaria. Doctor of Theology.
Professor University of Bonn, 1892-94. Author of John Gerhard and Mel-
anchthon; Richard Rubbe; The Scientific Attitude and its Demands on
Theology; The Absoluteness of Christianity, and of the History of Religion;
Political Ethics and Christianity; The Historic Element in Kant's Religious
Philosophy .^

The philosophy of religion of to-day is philosophy of religion so far
only, and in such a sense, as this word means science of religion or
philosophy with reference to religion. The science of religion of
former days was first dogmatic theology, deriving its dogmas from
the Bible and from Church tradition, expounding them apologetic-
ally with the metaphysical speculation of the later period of anti-
quity, and regarding the non-Christian religions as sinful derange-
ments and obscure fragments of the primitive revelation. This
lasted sixteen centuries, and is confined to-day to strictly ecclesias-
tical circles. Next, science of religion became natural theology,
which proved the existence of God by the nature of thought and by
the constitution of reality, and also the immortality of the soul by
the concept of the soul and by moral demands, thus constructing
natural or rational dogmas and putting these dogmas into more
or less friendly relations with traditional Christianity. This lasted
about two centuries, and is to-day of the not strictly ecclesiastical
or pietistic circles, which still wish to hold fast to religion. Both
kinds of science of religion exist no longer for the strict science.
The first was, in reality, supernaturalistic dogmatics, the second
was, in reality, a substitution of philosophy for religion. The first
was demolished by the criticism of miracles in the eighteenth century,
the second by the criticism of knowledge in the nineteenth century,
which, in its turn, rests upon Hume and Kant.

The science of religion of to-day keeps in touch with that which
without doubt factually exists and is an object of actual experience,
the subjective religious' consciousness. The distrust of ecclesiastical
and rationalistic dogmas has made, in the thought of the present,
every other treatment impossible. So the spirit of empiricism has
here as at other points completely prevailed. But empiricism in this
field means psychological analysis. This analysis is pursued by the

276 PHILOSOPHY OF RELIGION

present to the widest extent : on the one side by anthropologists and
archaeologists, who investigate the life of the soul in primitive peoples
and thus indicate the particular function and condition of religion
in these states; on the other side, by the modern experimental
psychologists and psychological empiricists, who, by self-observa-
tion, and especially by the collection of observations by others and of
personal testimony, study religion, and then, from the point of view
of the concepts of experimental psychology, examine the main
phenomena thus found.

Now, such an empirical psychology of religion has been constructed
with considerable success. In this German literature, it is true, has
cooperated to a slight degree only. The German theologians have
held to the older statements of the psychology of Kant, of Schleier-
macher, of Hegel, and of Fries, alone, which, in principle, were on
the right path, but which combined the purely psychological with
metaphysical and epistemological problems to such a degree that it
was impossible to reach a really unprejudiced attitude. German
psychologists remain, furthermore, under the spell of psycho-physio-
logy and of quantitative statements of measure, and have, conse-
quently, not liked to advance into this field, which is inaccessible
to such statements. More productive than the German psychology
for this subject is the French, which has attacked the complex facts
far more courageously. Here, however, under the predominance of
positivism, there prevails, on the whole, the tendency to regard
religion, in its essence, anthropologically or medically and patho-
logically in connection with bodily conditions. This is the confusion
of conditions and origins with the essence of the thing itself, which
can be determined only by the thing, and is, by no means, bound
exclusively to these conditions. Notwithstanding, the works of
Marillier, Murisier, and Flournoy have considerably aided the
problem. More impartially than all of these, the English and Ameri-
can psychology has investigated our subject. Here we have a master-
piece in the Gifford Lectures of William James, which collects into
a single reservoir similar investigations such as have been carried on
by Coe and Starbuck. There is here no tendency to a mechanism of
consciousness, or to the dogma of the causal and necessary structure
of consciousness. And to just this is due the freshness and impartial-
ity of the analyses which James gives out of his enviable knowledge of
characteristic cases. James rightly emphasizes the endlessly different
intensity of religious experiences, and the great number of points
of view and of judgments which thereby results. He also rightly
emphasizes the connection of this different intensity with irreducible
typical constitutions of the soul's life, with the optimistic and the
melancholy disposition; hence there arise constantly, even within
the same religion, essentially different types of religiousness. Limit-

PSYCHOLOGY IN THE SCIENCE OF RELIGION 277

ing himself, then, to the most intense experiences, he decides that
the characteristic of religious states is the sense of presence of the
divine, which one might perhaps describe in other terms, but which
still continues the specifically divine, with the opposed emotional
effects of a solemn sense of contrast and of enthusiastic exaltation.
He pictures these senses of presence, and illustrates them by vision-
ary and hallucinatory representations of the abstract. With this are
connected impulsive and inhibitive conditions for the appearance of
these senses of presence and of reality, descriptions of the effects
upon the emotional life and action, and, above all, the analysis of
the event usually called conversion, in which the religious experi-
ence out of subconscious antecedents becomes, in various ways, the
centre of the soul's life. All this is description, but it is based upon
a mass of examples and explained by general psychological cate-
gories which, by the occurrence of the religious event only, receive
a thoroughly specific coloring. It is a description after the manner
of Kirchhoff's mechanics; permanent and similar types, and, like-
wise, similar conditions for their relations to the rest of the soul's
life are sought out everywhere, without maintaining to have proven
at the same time, in this way, an intellectual necessity for the con-
nection. But the characteristic peculiarity of religious phenomena
is thus conceived as in -no other previous analysis.

All this is still, however, nothing more than psychologic. For the
science of religion it accomplishes nothing more than the psycho-
logical determination of the peculiarity of the phenomenon, of its
environment, its relations and consequences. It is evident that the
phenomenon occurs in an indefinite number of varieties; and the
chosen point of departure, in unusual and excessive cases, frequently
diffuses over religion itself the character of the bizarre and abnor-
mal. Consequently nothing whatever is said about the amount of
truth or of reality in these cases. This, by the very principles of
such a psychology, is iihpossible. It analyzes, produces types and
categories, points out comparatively constant connections and inter-
actions. But this cannot be the last word for the science of religion.
It demands, above all, empirical knowledge of the phenomenon; but
it demands this only in order, on the basis of this knowledge, to be
able to answer the question of the amount of truth. But this leads
to an entirely different problem, that of the theory of knowledge,
which has its own conditions of solution. It is impossible to stop
at a merely empirical psychology. The question is not merely of
given facts, but of the amount of knowledge in these facts. But pure
empiricism will not succeed in answering this question. The question
with regard to the amount of truth is always a question of validity.
The question with regard to validity can, however, be decided only
by logical and by general, conceptual investigations. Thus we pass

278 PHILOSOPHY OF RELIGION

over from the ground of empiricism to that of rationahsm, and the
question is, what the theory of knowledge or rationahsm signifies
for the science of reUgion.

Such a synthesis of the rational and irrational, of the psychological
and the theory of knowledge, is the main problem raised by the
teaching of Kant, and the significance of Kant is that he clearly and
once for all raised the problem in this way. He had the same strong
mind for the empirical and actual as for the rational and conceptual
elements of human knowledge, and constructed science as a balance
between the two. (He destroyed forever the a priori speculative
rationalism of the necessary ideas of thought, and the analytical
deductions from them, which undertakes to call reality out of the
necessity of thought as such. He restricted regressive rationalism
to metaphysical hypotheses and probabilities, the evidence for which
rests upon the inevitability of the logical operations which leads to
them, which, however, apply general concepts without reference to
experience, and therefore become empty, and thus afford no real
knowledge.) On the other hand, he proclaimed the formal, imman-
ent rationalism of experience, in attempting to unite Hume's
truth with the truth of Leibnitz and of Plato. In this way he suc-
ceeded in grasping the great problem of thought by the root, and
in putting attempts at solutions on the right basis. So it is not a
mere national custom of German philosophizing, if we take our
bearings, for the most part, from this greatest of German thinkers,
but it is, absolutely, the most fruitful and keenest way of putting the
problem. It is true, the solutions which Kant made, and which are
closely connected with the classical mechanics of that time, with
the undeveloped condition of the psychology of that time, and with
the incompleteness of historical thinking then just beginning, have
been, meantime, more than once given up again. A simple return to
him is therefore impossible. But the problem was put by him in
a fundamental way, and his solutions need nothing more than modi-
fication and complstion.

Now all this is especially true in the case of the science of religion.
Here also Kant took the same course, which seemed to me right for
the theoretical knowledge of the natural sciences and for anthro-
pology. In practical philosophy also, to which he rightly counts
philosophy of religion, he seeks laws of the practical reason analogous
to the laws of theoretical reason, axioms of the ethical, sesthetic,
and religious consciousness which are already contained a priori
in the elementary appearances in these fields, and, in application
to concrete reality, produce just these activities of the reason. Here
also one should grasp reason only as contained in life itself, the
a priori law itself already effective in the diversity of the appearances
should make one's self clear-sighted and so competent for a criticism

PSYCHOLOGY IN THE SCIENCE OF RELIGION 279

of the stream of the soul's appearances. Seizing upon itself in the
practical reality, the practical reason criticises the psychological
complex, rejects as illusion and error that which cannot be com-
prehended in an a 'priori law, selects that part of the same which
needs basis and centre and requires only clearness with regard to
itself, clears the way for revelations of a life consciousness of its own
legality and becomes capable of the development of critically purified
experience.

If this is, in principle, valid, the Kantian thought, in the further
detail, is maintained in principle only and as a whole. The elabora-
tion itself will have to be quite different from that of his own. Even
by Kant himself, on this very point, the synthesis of empiricism and
rationalism is far from being elaborated with the necessarj'^ rigor and
consistency. And to-day we have a quite differently developed
psychology of religion, in contrast with which that presupposed by
Kant is bare and thin. Finally, there remain in the whole method of
the critical system unsolved problems; by failure to solve these, or
by too hasty solution, science of religion, especially, is affected.

To make clear the present condition of the problem, one ought,
above all, to indicate the modifications to which the Kantian theory
of religion must submit, — must submit, especially, by reason of a
more delicate psychology, such as we have, with remarkable rich-
ness, in James and the American psychologists connected with him.
There are jour points with regard to this question.

The first is the question of the relation of psychology and theory
of knowledge in the very establishment of the laws of the theory of
knowledge. Are not the search for and discovery of the laws of the
theory of knowledge themselves possible only by way of psychological
ascertainment of facts, itself then a psychological undertaking and
consequently dependent upon all its conditions? It is the much dis-
cussed question of the circle which itself lies at the outset of the
critical system. The answer to this is that this circle lies in the very
being of all knowledge, and must therefore be resolutely committed.
It signifies nothing more than the presupposition of all thought, the
trust in a reason which establishes itself only by making use of
itself. The unmistakable elements of the logical assert themselves
as logical in distinction from the psychological, and from this point
on reason must be trusted in all its confusions and entanglements to
recognize itself within the psychological. It is the courage of thought,
as Hegel says, which may presuppose that the self-knowledge of rea-
son may trust itself, presuppose that reason is contained within the
psychological; or it is the ethical and teleological presupposition of
all thought, as Lotze saj^s, which believes in knowledge and the
validity of its laws for the sake of a connected meaning for reality,
and which, therefore, trusts to recognize itself out of the psycholog-

280 PHILOSOPHY OF RELIGION

ical mass. The establishment, therefore, of the laws of the theory
of knowledge is not itself a psychological analysis, but a knowledge
of self by the logical by virtue of which it extricates itself out of the
psychological mass. Theory of knowledge, like every rationalism,
includes, it is true, very real presuppositions with regard to the sig-
nificant, rational, and teleologically connective character of reality,
and without this presupposition it is untenable; in it lies its root.
It is insight of former daj'S, the importance of which, however, must
constantly be emphasized anew, that discusses the vaUdity of the
rational as opposed to the merely empirical. But still more im-
portant than this thesis are several inferences which are given
with it.

The establishment of the laws of consciousness, in which we
produce experience, is a selection of the laws out of experience itself,
a knowledge of itself by the reason contained in the very experience
by way of the analysis which extracts it. It is then an endless task,
completed by constantly renewed attacks, and always only approxi-
mately solvable. The complete separation of the merel}'' psychological
and actual and of the logical and necessary will never be completely
accomplished, but wall always be open to doubt; one can only
attempt always to limit more vigorously the field of what is doubtful.
And with this something further is connected.

The inexhaustible production of life becomes constantly, in the
latent amount of reason, richer than the analysis discerns, or, in
other words, the laws which are brought into the light of logic will
always be less the amount of reason not brought into consciousness,
and conscious logic will always be obliged to correct itself and enrich
itself out of the unartificial logical operations arising in contact with
the object. So a finished system of a priori principles, but this sys-
tem will always be in growth, wdll be obliged unceasingly to correct
itself, and to contain open spaces.

Finally, and above all, in case of this separation, there remains
within the psychologically conditioned appearance, a residuum,
which is either not conceived, but is later reduced to law and thereby
a conceived phenomenon, or which never can be so, and is therefore
illusion and error. If the psychological and the theoretical for know-
ledge are to be separated, then that can occur, not merely to show
that both must always be together, and form real experience only
when together, but there must also be a rejection of that which is
merely psychological and not rational since it is illusion and error.
The distinction between the apparent and the real was the point
of departure which made the whole theory necessary, and, accord-
ingly, the merely psychological must remain appearance and error
side by side with that which is psychological and, at the same time,
theoretical for knowledge. There always remains in consciousness

PSYCHOLOGY IX THE SCIKSXE OF RELIGION 281

a residuum of the inconceivable, that is. inconceivable since it is
illusion and error. This amounts to saying that reality is never
fully rational, but is engaged in a struggle between the rational
and anti-rationaL The anti-rational or irrational, in the sense of
psychological illusion and error, belongs also to the real, and strives
against the rational. The true and rational reality to be attained
by thought is always in conjunction vrith the untrue reahty. the
psychological, that containing illusion and error.

All this signifies that the rationalism of the theory of knowledge
must be conditional, partly owing to the corrective and enriching
fecundation by primitive and naive thought, partly owing to never
quite separable admixture of illusion and error. So. long ago. the
system of categorical forms, as Kant constructed it for theoretical
and practical reason, began to change, and can never again acquire
the rigidity which Kant's rationalism intended to give it forever-
more. And thus the critical system's rational reality of law produc-ed
bv reason always contains below itself and beside itself the merely
psychological reahty of the factual, to which also illusion and error
belong. — a reahty which can never be rationalized, but only set
aside. This . too . is also true for me philosophy of rehgion : the rational
reduction of the psychological facts of rehgion to the general laws of
consciousness which prevail among them is a task constantly to be
resumed anew by the study of reahty. and foUo^va the movements
of primitive rehgion in order to find there first the rational bass:
the reduction is. however, always approximate, can comprehend
the main points only, and must leave much open, the rational ground
for which is not or not yet evident: finally it has unceasingly to
reckon with the irrational as illusion and error, which auaches to the
rational, and yet is not explainable by it. Hie two reahties. which
the critical system must recognize at its very foundation, continue
in strife with each other, and this strife as the strife of divine truth
with human illusion is for the science of religion of still r!>?re im-
portance.

The second correction of the Kantian reaching is only a further
consequence from this state of things. K the attitude of psychology
and theory- of knowledge requires a strict separation, it requires it
only for the purpose of more correct relation. The la^vs of the theory
of knowledge are separated from the merely psychological actuahty,
but still can be produced only out of it. Thus, sis a matter of fact,
psychological analysis is always the jwresupposition for the correct
conception of all these laws. Psychology is the entrance gate to
theory of knowledge. This is true for theoretical logic as well as for
the practical logic of the moral, the aesthetical. and the religious.
But just at this point the present, on the basis of its psychological
investigation, presses far beyond the original form. o£ the Kantian

282 PHILOSOPHY OF RELIGION

teaching. This is not the place to describe this, more closely, with
reference to the first of the subjects just mentioned. But it is im-
portant to insist that this is especially true with respect to the
Kantian doctrine of religion. The Kantian doctrine of religion is
founded on the moral and religious psychology of Deism, which had
made the connection, frequent in experience, of moral feelings with
religious emotion the sole basis of the philosophy of religion, and
had, in the manner of the psychology of the eighteenth century,
immediately changed this connection into intellectual reflections,
in accord with which the moral law demands its originator and
guarantee. Kant accepted this psychology of religion without proof
and built upon it his main law of the religious consciousness, in
accordance with which a synthetic judgment a priori is operative
in religion (arising in the moral experience of freedom), which
requires that the world be regarded as subject to the purposes of
freedom. It is, however, extremely one-sided, to give religion its
place just between the elements, and a rather violent translation of
the religious constitution into reflection. The error of this psycho-
logy of religion had been discovered and corrected already by Schleier-
macher. But Schleiermacher, for his part too, also failed to deny
himself an altogether too sudden metaphysical interpretation of the
religious a priori which he had demonstrated, since he not only
described the a priori judgment of things, from the point of view of
absolute dependence upon God, as a vague feeling, but raised this
feeling, by reason of the supposed lack of difference, in it, between
thought and will, reason and being, to a world-principle, and inter-
preted the idea of God contained in this feeling in the terms of his
Spinozism, the lack of difference between God and Nature within
the Absolute. A real theory of knowledge of religion must keep
itself much more independent of all metaphysical presuppositions
and inferences, and must admit that the essence of the religious
a priori is extorted from a thoroughly impartial psychological
analysis. And this is always the place where works, such as those
of James, come into play. Religion as a special category or form of
psychical constitution, the result of a more or less vague presence
of the divine in the soul, the feeling of presence and reality with
reference to the superhuman or infinite, that is without any doubt
a much more correct point of departure for the analysis of the rational
a priori of religion, and it remains to make this new psychology
fruitful for the theory of knowledge of religion. That will be one of
the chief tasks of the future.

The third change relates to the distinction of the empirical and
intelligible Ego, which Kant connected closely, almost indissolubly
with his main epistemological thought of the formal rationalisms
immanent in experience. Kant rationalized the whole outer and

PSYCHOLOGY IN THE SCIENCE OF RELIGION 283

inner experience, by means of a 'priori laws, into a totality, conform-
ing to law, appearing in intuitive forms of space and time, causally
and necessarily rigidly connected. The freedom autonomously
determining itself out of the logical idea, and contrasting itself with
the psychological stream, produces out of the confused psycholican
reality this scientific formation of the true reality. The product of
thought, however, swallows its own maker. For the same acts of
freedom, which autonomously produced the formation of the reality
of law, remain themselves in the temporal sequence of psychical
events, and, therefore, themselves, with that formation, lapse into
the sequence which is under mechanical law. The intelligible Ego
creates the world of law, and finds itself therein, with its activity, as
empirical Ego, that is, as product of the great world-mechanism and
of its causal sequence. It is an intolerable, violent contradiction,
and it is no solution of this contradiction to refer the empirical Ego
to appearance, and the intelligible Ego to actuality existing in itself,
if the operations of the intelligible Ego, also a constituent part of
what takes place in the soul, occur in time and so relapse irrecover-
ably into phenomenality and its mechanism. All the ingenuity
of modern interpretation of Kant has not succeeded in making this
circle more tolerable, all shifting of one and the same thing to differ-
ent points of view has only enriched scientific terminology with
masterpieces of parenthetical caution, but not removed the objection
that two different points of view do not, as a matter of fact, exist
side by side, but conflict within the same object.

This circle is especially intolerable for the psychology of religion
and its application to the theory of knowledge. The psychology of
religion certainly shows us that the deeper feeling of all religion is
not a product of the mechanical sequence, but an effect of the super-
sensuous itself as it is felt there; it believes that it arises in the
intelligible Ego by way of some kind of connection with the super-
sensuous world. This, however, becomes completely impossible for
the Kantian theory of the empirical Ego, and all distinctions of a
double point of view in no wise change the fact that these points of
view are mutually absolutely exclusive. Here we have the results
of psychology which the expression of religious emotion confirms, in
that religion can be causally reduced to nothing else, totally opposed
to the consequences of such a theory of knowledge. Kant had him-
self often enough practicall}^ felt this, and spoke then of freedom as
an experience of communion with the supersensuous as a possible
but unprovable affair, while all that, in case of a strict adherence
to the phenomenality of time and of the theory of the empirical
Ego, which is a consequence of it, is completely impossible. No-
thing can be of any assistance here except a decisive renunciation
of those epistemological positions which contradict the results of

284 PHILOSOPHY OF RELIGION

psychology, and which are themselves only doctrinaire consequences
from other positions. Nothing else is possible but the modification
of the phenomenality of time, in such a way that by no means
everj^thing which belongs to time belongs also as a matter of
course to phenomenality, but that the autonomous rational acts
which occur in the time series of consciousness possess their own
intelligible time-form. At the same time the concept of causality
closely connected with the concept of time is to be modified so
that there should be not only an immanent and phenomenal causal
connection, but also a regular interaction between phenomenal and
intelligible, psychological and rational, conscious reality. At the
same time the conclusion is also given up, that the Ego submits
unconditionally and directly to phenomenality and to causal neces-
sity, while the same Ego, once more, in the same way, as a whole,
from another point of view, is subordinate to freedom and auto-
nomy, that is, self-constitutive through ideas. The two Egos must
lie not side by side, but in and over one another. It must be
possible that, within the phenomenal Ego by a creative act of
the intelligible Ego in it, the personality should be formed and
developed as a realization of the autonomous reason, so that the
intelligible issues from the phenomenal, the rational from the psy-
chological, the former elaborates and shapes the latter, and between
both a relation of regular interaction, but not of causal constraint,
takes place. This rather deep, incisive modification is, in its turn, an
approach of the Kantian teaching to empiricism, but still at the
same time, in the destruction and subordination of the phenomenal
and intelligible world, in the emphasis upon the single personality
issuing from the act of reason, an adherence to rationalism. But
since the distinction and the interrelation between the rational and
the empirical forms the point of departure for the critical system,
and this point of departure requires at the same time the moulding
and shaping of the empirical by the rational and the rejection of the
psychological appearance; a mere parallelism is altogether impossi-
ble, but an interrelation , is included, and a task set for the effort. and
labor which constantly makes the rational penetrate the empirical.
At the very outset we have the exclusion of the parallelism and the
assertion of the interrelation. The interrelation, by its very nature,
asserts the interruption of the causal necessity and the penetration
of autonomous reason in this sequence, without being itself produced
by this sequence, although it can be stimulated and helped or inhib-
ited and weakened by it. Thus, in such a case as this, the irrational
is recognized by the side of and in the rational. In this case the irra-
tional of the event without causal compulsion by some antecedent,
or of the self-determination by the autonomous idea alone, is the irra-
tional of freedom. It is the irrational of the creative procedure

PSYCHOLOGY IN THE SCIENCE OF RELIGION 285

which constitutes the idea out of itself and produces the consequences
of the reason out of the constituted idea. But this irrational plays
everywhere in the whole life of the soul an essential part, and is not
less than decisive in the case of religion, which must be quite differ-
ent from what it is if it did not have the right to maintain that
which it declares to be true of itself, namely, that it is an act of
freedom and a gift of grace, an effect of the supersensuous permeating
the natural phenomenal life of the soul and an act of free devotion
the natural motivation.

The fourth problem arises, when we examine the rational law of
the religious nature or of the having of religion which lies in the
being and organization of the reason. The having of religion may be
demonstrated as a law of the normal consciousness from the immanent
feeling of necessity and obhgation which properly belongs to religion,
and from its organic place in the economy of consciousness, which
receives its concentration and its relation to an objective world-
reason only from religion. But precisely because religion is reduced
to this, it is clear that this is only a reduction which abstracts from
the empirical actuality just as the categories of pure reason do. This
abstraction, then, should under no circumstances itself be regarded
as the real religion. It is only the rational a priori of the psychical
appearances, but not the replacement of appearances by the truth
free from confusion. The psychical reality in which alone the truth
is effective should never be forgotten out of regard for the truth.
This is, however, the fact in the Kantian theory of religion in two
directions.

It is always noticeable that the a priori of the practical reason is
treated by Kant quite differently from the theoretical. In case of
the latter the main idea of the synthesis, immanent in experience, of
rationalism and empiricism, is retained, and the a priori of the pure
forms of intuition and of the pure categories is nothing without the
contents of concrete reality which become shaped in it. It may be
very difficult actually to grasp the cooperation of the a priori and
the empirical in the single case, and Kant's theory of the categories
may have to be entirely reshaped and approximated to a priori
hypotheses requiring verification, but the principle itself is always
the disposition of the real and genuine problem of all knowledge. In
case of the practical a priori Kant did, it is true, firmly emphasize
the formal character of the ethical, sesthetical, and religious law,
but, in doing this, does not lose quite out of sight the psychical
reality. They appear not as empty forms which attain to their
reality only when filled with the concrete ethical tasks, the artistic
creations, and the religious states, but as abstract truths of reason,
which have to take the place of the intricacies of usual consciousness.
A-t this point one has always been right in feeling a relapse on the

286 PHILOSOPHY OF RELIGION

part of Kant into the abstract, analytical, conceptual, rationalism,
and for this very reason Kant's statements about these things are
of great sublimity and rigor of principle, but scanty in content. It
is more important in case also of this a priori of the practical reason
to keep in mind that it is a purely formal a priori and in reality
must constantly be in relation with the psychical content, in order
to give this content the firm core of the real and the principle of
the critical regulation of self. So the a priori of morals is not to
be represented abstractly merely by itself, but it is to be con-
ceived in its relation to all the tasks which we feel as obligatory, and
it extends itself from that point outwards over the total expanse of
the activity of reason. Likewise the a priori of art is not to be
denoted in the abstract idea of the unity of freedom and necessity,
but to be shown in the whole expanse which is present to the soul as
artistic form or conception. Thus, in especial degree, religion is not
to be reduced to the belief of reason in a moral world-order, and
simply contrasted with all supposed religion of any other kind, but
the religious a priori should only serve in order to establish the
essential in the empirical appearance, but without stripping off this
appearance altogether, and from this point of the essential to correct
the intricacies and narrowness, the errors and false combinations of
the psychical situation. Kant, by his original thought of the a priori,
was urged in different ways to such a view, and construed epistemo-
logically the empirical psychological religion as imaginary illustra-
tions of the a priori. But that is occasional only and does not
dominate Kant's real view of religion. This is and still remains only a
translation of the usual moral and theological rationalism from the
formula of Locke and Wolff into the formula of the critical philosophy.
The same revision occurs in quite a different direction. If religion
is an a priori of reason, it is, once for all, established together with
reason, and all religion is everywhere and always religious in the same
proposition as it is in any way realized. Schleiermacher expressly
stated this in his development of the Kantian theory, and, in so far
as the practical reason is always penetrated with freedom, and con-
sequently religion itself is established with the act of moral freedom,
this was also asserted by Kant himself. Such an assertion, however,
contradicts every psychological observation whatsoever. It is true
such observation can prove that religious emotions adjust them-
selves easily to all activities of reason, but it must sharply distin-
guish what is nothing more than the religiousness of vague feeling
of supersensual regulations, which usually are joined with art and
morals, from real and characteristic religiousness, in which, each
single time, a purely personal relation of presence to the super-
sensuous takes place. But this whole problem signifies nothing else
than the actualizing of the religious a priori, which actualizing

PSYCHOLOGY IN THE SCIENCE OF RELIGION 287

always occurs in quite specific and, in spite of all difference, essen-
tially similar psychical experiences and states. This problem of the
actualizing of the religious a 'priori and of its connection with con-
crete individual psychical phenomena, Kant completely overlooked
in his abstract concept of religion, or rather, deliberately ignored,
because, as he wrote to Jacobi, he saw aU the dangers of mysticism
lurking in it. This fear was justified; for, as a matter of fact, all the
specific occurrences of mysticism, from conversion, prayer, and con-
templation to enthusiasm, vision, and ecstasy, do lurk in it. But
without this mysticism there is no real religion, and the psychology
of religion shows most clearly how the real pulse of religion beats in
the mystical experiences. A religion without it is only a preliminary
step, or a reverberation of real and actual religion. Moreover, the
states are easily conceived in a theory of knowledge, if one sees in
them the actualizing of the religious a 'priori, the production of
actual religion in the fusion of the rational law with the concrete
individual psychical fact. The mysticism recognized as essential by
the psychology of religion must find its place in the theory of know-
ledge, and it finds it as the psychological actualizing of the religious
a priori, in which alone that interlacing of the necessary, the rational,
the conformable to law, and the factual occurs, which characterizes
real religion. The dangers of such a mysticism, which are recognized
a thousandfold in experience, cannot be dispelled altogether by the
displacement of mysticism, for that would mean to displace religion
itself. It would be the same, if one should try to avoid the dangers
of illusion and error, by keeping to the pure categories alone, and
ceasing to employ them in the actual thinking of experience. Rather,
they can be dispelled only in that the actualizing of the rational
a priori is recognized in the mystical occurrences, and thus the
intricacies and one-sidedness of the mere psychological stream of
religiousness be avoided. The psychological reality of religion must
always remember the rational substance of religion, and always bring
religion as central in the system of consciousness into fruitful and
adjusted contact with the total life of the reason. Thus the psycho-
logical reality corrects and purifies itself out of its own a priori, with-
out, however, destroying itself; or rather, the actual religion in the
psychical category of the mystical occurrences will subside to a more
or less degree. Thus we have the irrational prevailing here in its third
form, which like the two others was contained in the very outset of
the critical system, in the form of the once-occurring, factual, and
individual, which, of course, has a rational basis or a rational element
in itself, but is besides a pure fact and reality. Just this is the
excellence of the rationalism immanent in experience (the critical
system) , that it makes room for this feature beside the general and
conceptual rationality. It did not make room for it to the extent

288 PHILOSOPHY OF RELIGION

really required, and it especially left no space for it in its abstract
philosophy of religion. This space must again be opened by the
theory of the actualizing of the religious a priori, and there again
lies another improvement of the critical system under the influence of
modern psychology.

If we summarize all this, we have a quantity of concessions by the
formal epistemological rationalism to the irrationality of the psycho-
logical facts and a repeated breaking down of the over-rigorous
Kantian rationalism. Contrariwise, however, the pure psychological
investigation is also compelled to withdraw from the unlimited
quantity and the absolute irrationality of the multifarious (and of
the confusion of appearance and truth) to a rational criterium,
which can be found in the rational a priori of the reason only, and in
the organic position of this a priori in the system of consciousness in
general. By this rationalism alone may the true validity of religion
be founded, and by this alone the uncultivated psychical life may
be critically regulated. Religion will be conceived in its concrete
vitality and not mutilated; it will constantly be brought out of the
jumble of its distortions, blendings, one-sidedness, narrowness, and
exuberance back again to its original content, and to its organic
relations to the totality of the life of reason, to the scientific moral
and artistic accomplishments. That is everything that science can
do for it, but is not this service great enough and indispensable
enough to justify the work of such a science? We do not stop with
nothing more than "varieties of religious experience" which is the
result of James's method; but neither do we stop with nothing more
than a rational idea of religion, which overpowers experience, as was
still so in the case of Kant. But we must learn how intimately to
combine the empirical and psychological with the critical and norma-
tive. The ideas of Hume and of Leibnitz must once more be brought
into relation with the continuations of Kant's work, and the com-
bination of the Anglo-Saxon sense for reality with the German
spirit of speculation is still the task for the new century as well as
for the century past.

SHORT PAPERS

A short paper was contributed to this Section by Professor Alexander T.
Ormond, of Princeton University, on "Some Roots and Factors of Religion."
The speaker said that religion, Uke everything else human, has its rise in man's
experience. It has also doubtless had a history that wiU present the outlines of
a development, if but the course of that development can be traced. " But ux the
case of religion our theory of development wiU be largely qualified by our judg-
ment as to its origin; while, regarding origin itself, we have to depend on hypo-
theses constructed from our more or less imperfect acquaintance with the races,
and especially the savage races, of the present. The primitive pre-rehgious man
is a construction from present data, and wiU always remain more or less hypo-
thetical. This wiU partially explain, and at the same time partially excuse, what
we will agree is the unsatisfactory character of the anthropological theories as
accounts of the origin of rehgion. But there are other reasons for this partial
failure that are less excusable. One of these is the rather singular failure of the
leading anthropologists, in dealing with the origin of religion, to distinguish
between fundamental and merely tributary causes. For instance, if we suppose
that man has m some way come into possession of a germ of religiousness, many
things wiU become genuine tributaries to its development that when urged as
explanations of the germ itself would be obviously futile. There must be a cause
for the pretty general failure to note this distinction which is vital to rehgious
theory, and I am convinced that the prLacipal cause is a certain lack of psycho-
logical insight and of philosopliical grasp in deahng with the problem of the first
data and primary roots of religion in man's nature.

"In the first place, it is needful in deahng with the rehgion of the hypothetical
man that we should have some idea of what constitutes religion in the actual
man. Now, back of aU the outward maliifestations of religion, wiU stand the
religious consciousness of the man and the cormnimity, and it wiU be this that will
determine the idea of religion in its most essential form. The developed idea
of religion, therefore, arising out of this germinal impression, would take the form
of a sense (we may now caU it concept) of relatedness to some being akin to man
himself, and yet transcending him in some real though imdetermined respects.
Anything short of this would, I think, leave rehgion in some respects unaccounted
for; while anything more would perhaps exclude some genuine manifestations of
religion.

" If the idea of religion arises out of an impression, then it will not be possible
to deny to it an inteUectual root. I make this statement with some diffidence,
because if I do not misinterpret them, some recent psychologists have practically
denied the intehectual root in their doctrine that religion can have no orig-
inal intellectual content. If I am not further misled, however, these writers
would admit that a content is achieved bj^ the symbohc use of experience. This
is perhaps all I need argue for here; since our epistemology is teaching us
that the distinction between symbolism and perception is only that between the
direct and the indirect; while here it is clear that its use in developing the signi-
ficance of the religious impression would have all the directness and, therefore, aU
the cogency of an immediate inference.

" Let us now restore the- intellectual and emotional elements of religion to their
place in a sjmthesis; we wiU then have a concrete religious experience out of
which may be analyzed at least two fundamental factors. The first of these is
what we may call the personal factor in religion. We are treading in the foot-

290 PHILOSOPHY OF RELIGION

steps of the anthropologists when we find among the most undeveloped savages
a tendency to personify the objects of their worship. When it comes to the ques-
tion of determining the role that tliis personalizing tendency has actually
played in the development of religion, the anthropologists divide into two
camps, one of these, led by Max Miiller, regarding it as a symbolic interpretation
put upon the impression of some great natural or cosmic object or phenomenon;
while others, including Herbert Spencer and Mr. Tylor, prefer to seek the originals
of religion in ancestral dream-images and ghostly apparitions. These writers
thus start with completely anthropomorphic terms, and their problem is to
de-anthropomorphize the elements to the extent necessary to constitute them data
of religion. The second factor standing over against the personal, as its opposite,
is that of transcendence. By transcendence I mean that deifying, infinitating
process that is ever working contra to the anthropomorphic influence in the
sphere of religious conceptions. The School of Spencer regard this as the only
legitimate tendency in religion. We do not argue this point here, but agree that
it is as legitimate and real a factor as that of personality. The root of this factor,
if our diagnosis of the idea of religion be correct, is to be sought in the original
impression of religion, and it no doubt has its origin in man's feeling-reaction
from that impression. We have pointed to submission as one of the religious
emotions. Now submission rests on some deeper feeling-attitude, which some
have translated into the feeling or sense of dependence. This, however, is not
adequate, since men have the sense of social dependence on finite beings, and we
have it with reference to the floor we are standing on. Rather, it seems to me,
we must translate it into the stronger and more unconditional feeling of help-
lessness. One real ground of our religious consciousness is the sense or feeling of
helplessness toward God; the sense that we have no standing in being as against
the Deity. This radical feeling utters itself in every note of the religious scale,
from the lowest superstitious terror to the highest mystical self-annihilation.

" These two factors, the forces of personalization and transcendence, are in-
separable. They constitute the terms of a dialectic within the religious con-
sciousness by virtue of which in one phase our religious conceptions are becoming
ever more adequate and satisfying, while from another point of view their in-
sufficiency grows more and more apparent. And, on the broader field of religious
history, they embody themselves in a law of tendency, which Spencer has only
half -expressed, by virtue of which the objects of religion are on one hand becoming
ever more intelligible; on the other, ever more transcendent of our conceptions."

A short paper was read by Professor F. C. French, Professor of Pliilosophy in
the University of Nebraska, on "The Bearing of Certain Aspects of the Newer
Psychology on the Philosophy of Religion." The speaker said in part:

" The relation of science to religion has received, to be sure, much study, but
to most minds hitherto this has meant the relation of only the physical sciences to
religion. The older psychology was largely speculative and metaphysical in
character. There were, of course, some who employed the empirical method in
psychology, but they were so far from comprehending the full scope of mental
phenomena that, at best, their work gave the promise of a science rather than
a science itself.

It is not the fact that the newer psychology takes account of the physiological
conditions of mental life; it is not the fact that the subject is now pursued in
laboratories with instruments of precision, that gives it its fuU standing as a
science : it is much more the fact that the psychology of to-day has found a place
in the natural system of mental things for those strange and relativeh'^ unusual
phenomena of consciousness which to the scientifically minded seemed totally
unreal and to the superstitious manifestations of the supernatural. . . .

SHORT PAPERS 291

" In showing that the abnormal can be explained in terms of the normal,
psychology does now for the phenomena of mind what the physical sciences
have long done for the phenomena of nature. . . .

" Psychology as a science postulates the reign of natural law in the subjective
sphere just as rigorously as physics postulates the reign of law in the objective
sphere. . . .

"It is not in the unusual and the abnormal that the reflective mind is to see
God. It is not through gaps in nature that we are to get glimpses of the super-
natural. Rather is it in the very nature of nature, rational, harmonious, law-
conforming, subject to scientific interpretation, that we have the best evidence
that the world is made mind- wise, that it is the work of an intelligent mind, that
there is a rational spirit at the core of the universe.

" For science the transcendent does not enter into the perceptual realm external
or internal. It is, indeed, hard for the religious mind to admit this fact in all
its fullness. Until it does, however, religion must always stand more or less in
fear of science. Once give up the perceptual, in all its bearings, to science, and
religion will find that it has lost a weak support only to gain a stronger one.
Ultimately, I believe, we shall find that the full acceptance of science in the mental
domain as well as in the physical will strengthen the rational grounds of theistic
behef."

SECTION C — LOGIC

{Hall 6, September 22, 10 a. m.)

Chairman: Professor George M. Duncan, Yale University.
Speakers: Professor William A. Hammond, Cornell University.

Professor Frederick J. E. Woodbridge, Columbia University.
Secretary: Dr. W. H. Sheldon, Columbia University.

The Chairman of this Section, Professor George M. Duncan, Pro-
fessor of Logic and Mathematics at Yale University, in introducing
the speakers spoke briefly of the scope and importance of the sub-
ject assigned to the Section; expressed, on behalf of those in attend-
ance, regret at the inability of Professor Wilhelm Windelband to
be present and take part in the work of the Section, as had been
expected; congratulated the Section on the papers to be presented
and the speakers who were to present them; and announced the
final programme of the Section.

THE RELATIONS OF LOGIC TO OTHER DISCIPLINES

BY PROFESSOR WILLIAM A. HAMMOND

[William Alexander Hammond, Assistant Professor of Ancient and Medieval
Philosophy and Esthetics, Cornell University, b. May 20, 1861, New Ath-
ens, Ohio. A.B. Harvard, 1885; Ph.D. Leipzig, 1891. Lecturer on Classics,
King's College, Windsor, N. S., 1885-88; Secretary of the University Fac-
ulty, Cornell; Member American Psychological Association, American
Philosophical Association. Author of The Characters of Theophrastus,
translated with Introduction ; Aristotle's Psychology, translated with Intro-
duction.]

In 1787; in the preface to the second edition of the Kr. d. r. V., Kant
wrote the following words: "That logic, from the earliest times,
has followed that secure method " (namely, the secure method of a
science witnessed by the unanimity of its workers and the stability
of its results) " may be seen from the fact that since Aristotle it has
not had to retrace a single step, unless we choose to consider as
improvements the removal of some unnecessary subtleties, or the
clearer definition of its matter, both of which refer to the elegance
rather than to the solidity of the science. It is remarkable, also, that
to the present day, it has not been able to make one step in advance,
so that to all appearances it may be considered as completed and
perfect. If some modern philosophers thought to enlarge it, by
introducing psychological chapters on the different faculties of
knowledge (faculty of imagination, wit, etc.), or metaphysical chapters
on the origin of knowledge or different degrees of certainty accord-
ing to the difference of objects (idealism, skepticism, etc.), or, lastly,
anthropological chapters on prejudices, their causes and remedies,
this could only arise from their ignorance of the peculiar nature of
logical science. We do not enlarge, but we only disfigure the sciences,
if we allow their respective limits to be confounded ; and the limits
of logic are definitely fixed by the fact that it is a science which has
nothing to do but fully to exhibit and strictly to prove the formal
rules of all thought (whether it be a priori or empirical, whatever be
its origin or its object, and whateA^er be the impediments, accidental
or natural, which it has to encounter in the human mind). " — [Trans-
lated by Max Miiller.] Scarcely more than half a century after the
publication of this statement of Kant's, John Stuart Mill (Intro-
duction to System of Logic) wrote: "There is as great diversity
among authors in the modes which they have adopted of defining
logic, as in their treatment of the details of it. This is what
might naturally be expected on any subject on which writers have
availed themselves of the same language as a means of delivering
different ideas. . . . This diversity is not so much an evil to be

RELATIONS OF LOGIC TO OTHER DISCIPLINES 297

complained of, as an inevitable, and in some degree a proper result
of the imperfect state of those sciences " (that is, of logic, jurispru-
dence, and ethics). "It is not to be expected that there should be
agreement about the definition of an3^thing, until there is agree-
ment about the thing itself." This remarkable disparity of opinion
is due partly to the changes in the treatment of logic from Kant to
Mill, and partly to the fact that both statements are extreme. That
the science of logic was "completed and perfect" in the time of
Kant could only with any degree of accuracy be said of the treat-
ment of syllogistic proof or the deductive logic of Aristotle. That
the diversity was so great as pictured by Mill is not historically
exact, but could be said only of the new epistemological and psycho-
logical treatment of logic and not of the traditional formal logic.
The confusion in logic is no doubt largely due to disagreement in
the delimitation of its proper territory and to the consequent variety
of opinions as to its relations to other disciplines. The rise of induct-
ive logic, coincident with the rise and growth of physical science
and empiricism, forced the consideration of the question as to the
relation of formal thought to reality, and the consequent entangle-
ment of logic in a triple alliance of logic, psychology, and meta-
physics. How logic can maintain friendly relations with both of
these and yet avoid endangering its territorial integrity has not been
made clear by logicians or psA^chologists or metaphysicians, and
that, too, in spite of persistent attempts justly to settle the issue as
to their respective spheres of influence. Until modern logic definitely
settles the question of its aims and legitimate problems, it is difficult
to see how any agreement can be reached as to its relation to the
other disciplines. The situation as it confronts one in the discus-
sion of the relations of logic to allied subjects may be analyzed as
follows :

1. The relation of logic as science to logic as art.

2. The relation of logic to psychology.

3. The relation of logic to metaphysics.

The development of nineteenth century logic has made an answer to
the last two of the foregoing problems exceedingly difficult. Indeed,
one may say that the evolution of modern epistemology has had a
centrifugal influence on logic, and instead of growth towards unity
of conception we have a chaos of diverse and discordant theories.
The apple of discord has been the theory of knowledge. A score of
years ago when Adamson wrote his admirable article in the Ency-
clopcedia Britannica (article ''Logic," 1882), he found the conditions
much the same as I now find them. " Looking to the chaotic state of
logical text-books at the present time, one would be inclined to say
that there does not exist anywhere a recognized currently received
body of speculations to which the title logic can be unambiguously

298 LOGIC

assigned, and that we must therefore resign the hope of attaining
by any empirical consideration of the received doctrine a precise
determination of the nature and Umits of logical theory." I do not,
however, take quite so despondent a view of the logical chaos as
the late Professor Adamson; rather, I believe with Professor Stratton
(Psy. Rev. vol. iii) that something is to be gained for unity and
consistency by more exact delimitation of the subject-matter of
the philosophical disciplines and their interrelations, which pre-
cision, if secured, would assist in bringing into clear relief the real
problems of the several departments of inquiry, and facihtate the
proper classification of the disciplines themselves.

The attempt to delimit the spheres of the disciplines, to state their
interrelations and classify them, was made early in the history of
philosophy, at the very beginning of the development of logic as
a science by Aristotle. In Plato's philosophy, logic is not separated
from epistemology and metaphysics. The key to his metaphysics is
given essentially in his theory of the reality of the concept, which
offers an interesting analogy to the position of logic in modern
idealism. Before Plato there was no formulation of logical theory,
and in his dialogues it is only contained in solution. The nearest
approach to any formulation is to be found in an applied logic set
forth in the precepts and rules of the rhetoricians and sophists.
Properly speaking, Aristotle made the first attempt to define the
subject of logic and to determine its relations to the other sciences.
In a certain sense logic for Aristotle is not a science at all. For
science is concerned with some ens, some branch of reality, while
logic is concerned with the methodology of knowing, with the
formal processes of thought whereby an ens or a reality is ascertained
and appropriated to knowledge. In the sense of a method whereby
all scientific knowledge is secured, logic is a propaedeutic to the
sciences. In the idealism of the Eleatics and Plato, thought and being
are ultimately identical, and the laws of thought are the laws of
being. In Aristotle's conception, while the processes of thought
furnish a knowledge of reality or being, their formal operation con-
stitutes the technique of investigation, and their systematic explana-
tion and description constitute logic. Logic and metaphysics are dis-
tinguished as the science of being and the doctrine of the thought-
processes whereby being is known. Logic is the doctrine of the
organon of science, and when applied is the organon of science. The
logic of Aristotle is not a purely formal logic. He is not interested in
the merely schematic character of the thought-processes, but in
their function as mediators of apodictic truth. He begins with the
assumption that in the conjunction and disjunction of correctly
formed judgments the conjunction or disjunction of reality is mir-
rored. Aristotle does not here examine into the powers of the mind

RELATIONS OF LOGIC TO OTHER DISCIPLINES 299

as a whole; that is done, though fragmentarily, in the De Anima and
Parva Naturalia, where the mental powers are regarded as phases of
the processes of nature without reference to normation; but in his logic
he inquires only into those forms and laws of thinking which mediate
proof. Scientific proof, in his conception, is furnished in the form of
the syllogism, whose component elements are terms and propositions.
In the little tract On Interpretation (i. e. on the judgment as inter-
preter of thought), if it is genuine, the proposition is considered in
its logical bearing. The treatise on the Categories, which discusses
the nature of the most general terms, forms a connecting link be-
tween logic and metaphysics. The categories are the most general
concepts or universal modes under which we have knowledge of
the world. They are not simply logical relations; they are existential
forms, being not only the modes under which thought regards being,
but the modes under which being exists. Aristotle's theory of the
methodology of science is intimately connected with his view of
knowledge. Scientific knowledge in his opinion refers to the essence
of things; for example, to those universal aspects of reality which
are given in particulars, but which remain self-identical amidst the
variation and passing of particulars. The universal, however, is
knoAvn only through and after particulars. There is no such thing
as innate knowledge or Platonic reminiscence. Knowledge, if not
entirely empirical, has its basis in empirical reality. Causes are
known only through effects. The universals have no existence apart
from things, although they exist realiter in things. Empirical know-
ledge of particulars must, therefore, precede in time the conceptual
or scientific knowledge of universals. In the evolution of scientific
knowledge in the individual mind, the body of particulars or of
sense-experience is to its conceptual transformation as potentiality
is to actuality, matter to form, the completed end of the former
being realized in the latter. Only in the sense of this power to trans-
form and conceptualize, does the mind have knowledge within itself.
The genetic content is experiential; the developed concept, judg-
ment, or inference is in form noetic. Knowledge is, therefore, not
a mere "precipitate of experience," nor is Aristotle a complete
empiricist. The conceptual form of knowledge is not immediately
given in things experienced, but is a product of noetic discrimination
and combination. Of a sensible object as such there is no concept;
the object of a concept is the generic essence of a thing; and the
concept itself is the thought of this generic essence. The individual
is generalized; every concept does or can embrace several individuals.
It is an " aggregate of distinguishing marks, " and is expressed in a
definition. The concept as such is neither true nor false. Truth first
arises in the form of a judgment or proposition, wherein a subject
is coupled with a predicate, and something is said about something.

300 LOGIC

A Judgment is true when the thought (whose inward process is the
judgment and the expression in vocal symbols is the proposition)
regards as conjoined or divided that which is conjoined or divided
in actuality; in other words, when the thought is congruous with
the real. While Aristotle does not ignore induction as a scientific
method, (how could he when he regards the self-subsistent individual
as the only real?) yet he says that, as a method, it labors under
the defect of being only proximate; a complete induction from all
particulars is not possible, and therefore cannot furnish demonstra-
tion. Only the deductive process proceeding syllogistically from
the universal (or essential truth) to the particular is scientifically
cogent or apodictic. Consequently Aristotle developed the science
of logic mainly as a syllogistic technique or instrument of demon-
stration. From this brief sketch of Aristotle's logical views it will
be seen that the epistemological and metaphysical relations of
logic which involve its greatest difficulty and cause the greatest
diversity in its modern exponents, were present in undeveloped
form to the mind of the first logician. It would require a mighty
optimism to suppose that this difficulty and diversity, which has
increased rather than diminished in the progress of historical philo-
sophy, should suddenly be made to vanish by some magic of re-
statement of subject-matter, or theoretical delimitation of the
discipline. As Fichte said of philosophy, " The sort of a philosophy
that a man has, depends on the kind of man he is; " so one might
almost say of logic, "The sort of logic that a man has, depends on
the kind of philosopher he is." If the blight of discord is ever re-
moved from epistemology, we may expect agreement as to the rela-
tions of logic to metaphysics. Meanwhile logic has the great body
of scientific results deposited in the physical sciences on which to
build and test, with some assurance, its doctrine of methodology;
and as philosophy moves forward persistently to the final solution
of its problems, logic may justly expect to be a beneficiary in its
established theories.

After Aristotle's death logic lapsed into a formalism more and
more removed from any vital connection with reality and oblivious
to the profound epistemological and methodological questions that
Aristotle had at least raised. In the Middle Ages it became a highly
developed exercise in inference applied to the traditional dogmas of
theology and science as premises, with mainly apologetic or polemi-
cal functions. Its chief importance is found in its application to the
problem of realism and nominalism, the question as to the nature of
universals. At the height of scholasticism realism gained its victory
by syllogistically showing the congruity of its premises with certain
fundamental dogmas of the Church, especially with the dogma of the
unity and reality of the Godhead. The heretical conclusion involved

RELATIONS OF LOGIC TO OTHER DISCIPLINES 301

in nominalism is equivalent (the accepted dogma of the Church be-
ing axiomatic) to reductio ad ahsurdum. A use of logic such as this,
tending to conserve rather than to increase the body of knowledge,
was bound to meet with attack on the awakening of post-renaissance
interest in the physical world, and the acquirement of a body of truth
to which the scholastic formal logic had no relation. The anti-scholas-
tic movement in logic was inaugurated by Francis Bacon, who
sought in his Novum Organum to give science a real content through
the application of induction to experience and the discovery of
universal truths from particular instances. The syllogism is rejected
as a scientific instrument, because it does not lead to principles, but
proceeds only from principles, and is therefore not useful for dis-
covery. It permits at most only refinements on knowledge already
possessed, but cannot be regarded as creative or productive. The
Baconian theory of induction regarded the accumulation of facts
and the derivation of general principles and laws from them as the
true and fruitful method of science. In England this empirical view
of logic has been altogether dominant, and the most illustrious Eng-
lish exponents of logical theory, Herschel, Whewell, and Mill,
have stood on that ground. Since the introduction of German
idealism in the last half century a new logic has grown up whose
chief business is with the theory of knowledge.

Kant's departure in logic is based on an epistemological examin-
ation of the nature of judgment, and on the answer to his own
question, "How are synthetic judgments a priori possible?" The
a priori elements in knowledge make knowledge of the real nature of
things impossible. Human knowledge extends to the phenomenal
world, which is seen under the a priori forms of the understanding.
Logic for Kant is the science of the formal and necessary laws of
thought, apart from any reference to objects. Pure or universal
logic aims to understand the forms of thought without regard to meta-
physical or psychological relations, and this position of Kant is the
historical beginning of the subjective formal logic.

In the metaphysical logic of Hegel, which rests on a panlogistic
basis, being and thought, form and content, are identical. Logical
necessity is the measure and criterion of objective reality. The body
of reality is developed through the dialectic self-movement of the
idea. In such an idealistic monism, formal and real logic are by the
metaphysical postulate coincident.

Schleiermacher in his dialectic regards logic from the standpoint
of epistemological realism, in which the real deliverances of the
senses are conceptually transformed by the spontaneous activity
of reason. This spirit of reahsm is similar to that of Aristotle, in which
the one-sided a priori view of knowledge is controverted. Space and
time are forms of the existence of things, and not merely a priori

302 LOGIC

forms of knowing. Logic he divides into dialectic and technical
logic. The former regards the idea of knowledge as such; the formal
or technical regards knowledge in the process of becoming or the
idea of knowledge in motion. The forms of this process are induction
and deduction. The Hegelian theory of the generation of knowledge
out of the processes of pure thought is emphatically rejected,

Lotze, who is undoubtedly one of the most influential and fruitful
writers on logic in the last century, attempts to bring logic into
closer relations with contemporary science, and is an antagonist of
one-sided formal logics. For him logic falls into the three parts of
(1) pure logic or the logic of thought; (2) applied logic or the logic
of investigation; (3) the logic of knowledge or methodology; and this
classification of the matter and problems of logic has had an im-
portant influence on subsequent treatises on the discipline. His
logic is formal, as he describes it himself, in the sense of setting forth
the modes of the operation of thought and its logical structure; it is
real in the sense that these forms are dependent on the nature of
things and not something independently given in the mind. While
he aims to maintain the distinct separation of logic and metaphysics,
he says (in the discussion of the relations between formal and real
logical meaning) the question of meaning naturally raises a meta-
physical problem: " Ich thue besser der Metaphysik die weitere
Erorterung dieses wichtigen Punktes zu iiberlassen." {Log. 2d ed.
p. 571.) How could it be otherwise when his whole view of the rela-
tions and validity of knowledge is inseparable from his realism or
teleological idealism, as he himself characterizes his own standpoint?

Drobisch, a follower of Herbart, is one of the most thoroughgoing
formalists in modern logical theory. He attempts to maintain strictly
the distinction between thought and knowledge. Logic is the science
of thought. He holds that there may be formal truth, for example,
logically valid truth, which is materially false. Logic, in other words,
is purely formal; material truth is matter for metaphysics or science.
Drobisch holds, therefore, that the falsity of the judgment expressed
in the premise from which a formally correct syllogism may be deduced,
is not subject-matter for logic. The sphere of logic is limited to the
region of inference and forms of procedure, his view of the nature
and function of logic being determined largely by the bias of his
mathematical standpoint. The congruity of thought with itself,
judgments, conclusions, analyses, etc., is the sole logical truth, as
against Trendelenburg, who took the Aristotelian position that log-
ical truth is the "agreement of thought with the object of thought."

Sigwart looks at logic mainly from the standpoint of the tech-
nology of science, in which, however, he discovers the implications
of a teleological metaphysic. Between the processes of conscious-
ness and external changes he finds a causal relation and not parallel-

RELATIONS OF LOGIC TO OTHER DISCIPLINES 303

ism. Inasmuch as thought sometimes misses its aim, as is shown
by the fact that error and dispute exist, there is need of a discipline
whose purpose is to show us how to attain and estabhsh truth and
avoid error. This is the practical aim of logic, as distinguished from
the psychological treatment of thought, where the distinction between
true and false has no more place than the distinction between good
and bad. Logic presupposes the impulse to discover truth, and it
therefore sets forth the criteria of true thinking, and endeavors
to describe those normative operations whose aim is validity of
judgment. Consequently logic falls into the two parts of (1) critical,
(2) technical, the former having meaning only in reference to the
latter; the main value of logic is to be sought in its function as art.
''Methodology, therefore, which is generally made to take a subor-
dinate place, should be regarded as the special, final, and chief aim of
our science." (Logic, vol. i, p. 21, Eng. Tr.) As an art, logic under-
takes to determine under what conditions and prescriptions judgments
are valid, but does not undertake to pass upon the validity of the con-
tent of given judgments. Its prescriptions have regard only to formal
correctness and not to the material truth of results. Logic is, there-
fore, a formal discipline. Its business is with the due procedure of
thought, and it attempts to show no more than how we may advance
in the reasoning process in such way that each step is valid and
necessary. If logic were to tell us what to think or give us the con-
tent of thought, it would be commensurate with the whole of science.
Sigwart, however, does not mean by formal thought independence of
content, for it is not possible to disregard the particular manner in
which the materials and content of thought are delivered through
sensation and formed into ideas. Further, logic having for its chief
business the methodology of science, the development of knowledge
from empirical data, it ought to include a theory of knowledge, but
it should not so far depart from its subjective limits as to include
within its province the discussion of metaphysical implications or
a theory of being. For this reason, Sigwart relegates to a postscript
his discussion of teleology, but he gives an elaborate treatment of
epistemology extending through vol. i and develops his account of
methodology in vol. ii. The question regarding the relation between
necessity, the element in which logical thought moves, and freedom,
the postulate of the will, carries one beyond the confines of logic and
is, in his opinion, the profoundest problem of metaphysics, whose
function is to deal with the ultimate relation between "subject
and object, the world' and the individual, and this is not only basal
for logic and all science, but is the crown and end of them all."

Wundt's psychological and methodological treatment of logic
stands midway between the purely formal treatises on the one hand ,
and the metaphysical treatises on the other hand. The general

304 LOGIC

standpoint of Wundt is similar to that of Sigwart, in that he dis-
covers the function of logic in the exposition of the formation and
methods of scientific knowledge; for example, in epistemology and
methodology. Logic must conform to the conditions under which
scientific inquiry is actually carried on; the forms of thought,
therefore, cannot be separate from or indifferent to the content of
knowledge; for it is a fundamental principle of science that its
particular methods are determined by the nature of its particular
subject-matter. Scientific logic must reject the theory that identifies
thought and being (Hegel) and the theory of parallelism between
thought and reality (Schleiermacher, Trendelenburg, and Ueberweg),
in which the ultimate identity of the two is only concealed. Both
of these theories base logic on a metaphysics, which makes it nec-
essary to construe the real in terms of thought, and logic, so di-
vorced from empirical reality, is powerless to explain the methods of
scientific procedure. One cannot, however, avoid the acceptance of
thought as a competent organ for the interpretation of reality, unless
one abandons all question of validity and accepts agnosticism or
skepticism. This interpretative power of thought or congruity with
reality is translated by metaphysical logic into identity. Metaphysical
logic concerns itself fundamentally with the content of knowledge, not
with its evidential or formal logical aspects, but with being and the
laws of being. It is the business of metaphysics to construct its
notions and theories of reality out of the deliverances of the special
sciences and inferences derived therefrom. The aim of metaphysics
is the development of a world-view free from internal contradictions,
a view that shall unite all particular and plural knowledges into a
whole. Logic stands in more intimate relation to the special sciences,
for here the relations are reciprocal and immediate; for example,
from actual scientific procedure logic abstracts its general laws and
results, and these in turn it delivers to the sciences as their formu-
lated methodology. In the history of science the winning of know-
ledge precedes the formulation of the rules employed, that is, pre-
cedes any scientific methodology. Logic, as methodology, is not an
a priori construction, but has its genesis in the growth of science
itself and in the discovery of those tests and criteria of truth which
are found to possess an actual heuristic or evidential value. It is
not practicable to separate epistemology and logic, for such con-
cepts as causality, analogy, validity, etc., are fundamental in logical
method, and yet they belong to the territory of epistemology, are
epistemological in nature, as one may indeed say of all the general
law^s of thought. A formal logic that is merely propaedeutic, a logic that
aims to free itself from the quarrels of epistemology, is scientifically
useless. Its norms are valueless, in so far as they can only teach the
arrangement of knowledge already possessed, and teach nothing as to

RELATIONS OF LOGIC TO OTHER DISCIPLINES 305

how to secure it or test its real validity. While formal logic aims to
put itself outside of philosophy, metaphysical logic would usurp
the place of philosoph3^ Formal logic is inadequate, because it
neither shows how the laws of thought originate, why they are
valid, nor in what sense they are applicable to concrete investigation.
Wundt, therefore, develops a logic which one may call epistemo-
logical methodological, and which stands between the extremes of
formal logic and metaphysical logic. The laws of logic must be
derived from the processes of psychic experience and the procedure
of the sciences. "Logic therefore needs," as he says, ''epistemology
for its foundation and the doctrine of methods for its completion."

Lipps takes the view outright that logic is a branch of psychology;
Husserl in his latest book goes to the other extreme of a purely
formal and technical logic, and devotes almost his entire first volume
to the complete sundering of psj^chology and logic.

Bradley bases his logic on the theory of the judgment. The logical
judgment is entirely different from the psychological. The logical
judgment is a qualification of reality by means of an idea. The
predicate is an adjective or attribute which in the judgment is
ascribed to reality. The aim of truth is to qualify reality by general
notions. But inasmuch as reality is individual and self-existent,
whereas truth is universal, truth and reality are not coincident.
Bradley's metaphysical solution of the disparity between thought
and reality is put forward in his theory of the unitary Absolute,
whose concrete content is the totality of experience. But as thought
is not the whole of experience, judgments cannot compass the whole
of reality. Bosanquet objects to this, and maintains that reality must
not be regarded as an ideal construction. The real world is the world to
which our concepts and judgments refer. In the former we have a
world of isolated individuals of definite content; in the latter, we have
a world of definitely systematized and organized content. Under the
title of the Morphology of Knowledge Bosanquet considers the evo-
lution of judgment and inference in their varied forms. " Logic starts
from the individual mind, as that within which we have the actual
facts of intelligence, which we are attempting to interpret into a sys-
tem " {Logic, vol. I, p. 247). The real world for every individual is his
world. "The work of intellectually constituting that totality which
we call the real world is the work of knowledge. The work of analyz-
ing the process of this constitution or determination is the work of
logic, which might be described ... as the reflection of knowledge
upon itself " (Logic, vol. i, p. 3). "The relation of logic to truth con-
sists in examining the characteristics by which the various phases
of the one intellectual function are fitted for their place in the
intellectual totality which constitutes knowledge " (ibid.). The real
world is the intelligible world ; reality is something to which we attain

306 LOGIC

by a constructive process. We have here a type of logic which is
essentially a metaphysic. Indeed, Bosanquet says in the course of his
first volume : " I entertain no doubt that in content logic is one with
metaphysics, and differs, if at all, simply in mode of treatment — in
tracing the evolution of knowledge in the light of its value and import,
instead of attempting to summarize its value and import apart from
the details of its evolution " {Logic, vol. i, 247).

Dewey (Studies in Logical Theory, p. 5) describes the essential
function of logic as the inquiry into the relations of thought as such
to reality as such. Although such an inquiry may involve the investi-
gation of psychological processes and of the concrete methods of
science and verification, a description and analysis of the forms of
thought, conception, judgment, and inference, yet its concern with
these is subordinate to its main concern, namely, the relation of
"thought at large to reality at large." Logic is not reflection on
thought, either on its nature as such or on its forms, but on its relations
to the real. In Dewey's philosophy, logical theory is a description of
thought as a mode of adaptation to its own conditions, and validity
is judged in terms of the efficiency of thought in the solution of its
own problems and difficulties. The problem of logic is more than
epistemological. Wherever there is striving there are obstacles ; and
wherever there is thinking there is a " material-in-question." Dewey's
logic is a theory of reflective experience regarded functionally, or
a pragmatic view of the discipline. This logic of experience aims to
evaluate the signiflcance of social research, psychology, fine and in-
dustrial art, and religious aspiration in the form of scientific statement,
and to accomplish for social values in general what the physical
sciences have done for the ph5^sical world. In Dewey's teleological
pragmatic logic the judgment is essentially instrumental, the whole
of thinking is functional, and the meaning of things is identical
with valid meaning (Studies in Logical Theory, cf. pp. 48, 82, 128).
The real world is not a self-existent world outside of knowledge, but
simply the totality of experience; and experience is a complex of
strains, tensions, checks, and attitudes. The function of logic is the
redintegration of this experience. " Thinking is adaptation to an end
through the adjustment of particular objective contents " (ihid.
p. 81). Logic here becomes a large part, if not the whole, of a meta-
physics of experience ; its nature and function are entirely determined
by the theory of reality.

In this brief and fragmentary resume are exhibited certain charac-
teristic movements in the development of logical theory, the construc-
tion put upon its subject-matter and its relation to other disciplines.
The resume has had in view only the making of the diversity of
opinion on these questions historically salient. There are three
distinct types of logic noticed here: (1) formal, whose concern is

RELATIONS OF LOGIC TO OTHER DISCIPLINES 307

merely with the structural aspect of inferential thought, and its
validity in terms of internal congruity; (2) metaphysical logic whose
concern is with the functional aspect of thought, its validity in
terms of objective reference, and its relation to reality; (3) epi-
stemological and methodological logic, whose concern is with the
genesis, nature, and laws of logical thinking as forms of scientific
knowledge, and with their technological application to the sciences
as methodology. I am not at present concerned with a criticism
of these various viewpoints, excepting in so far as they affect the
problem of the interrelationship of logic and the allied disciplines.

For my present purpose I reject the extreme metaphysical and
formal positions, and assume that logic is a discipline whose busi-
ness is to describe and systematize the formal processes of inferential
thought and to apply them as practical principles to the body of
real knowledge.

I wish now to take up seriatim the several questions touching
the various relations of logic enumerated above, and first of all the
question of the relation of logic as science to logic as art.

I. Logic as science and logic as art.

It seems true that the founder of logic, Aristotle, regarded logic
not as a science, but rather as propaedeutic to science, and not as an
end in itself, but rather technically and heuristically as an instrument.
In other words, logic was conceived by him rather in its application
or as an art, than as a science, and so it continued to be regarded
until the close of the Middle Ages, being characterized indeed as the
ars artium; for even the logica docens of the Scholastics was merely
the formulation of that body of precepts which are of practical serv-
ice in the syllogistic arrangement of premises, and the Port Royal
Logic aims to furnish I'art de penser. This technical aspect of the
science has clung to it down to the present day, and is no doubt
a legitimate description of a part of its function. But no one would
now say that logic is an art; rather it is a body of theory which
may be technically applied. Mill, in his examination of Sir William
Hamilton's Philosophy (p. 391), says of logic that it "is the art of
thinking, which means of correct thinking, and the science of the
conditions of correct thinking," and indeed, he goes so far as to say
(System of Logic, Introd. § 7) : " The extension of logic as a science
is determined by its necessities as an art." Strictly speaking, logic
as a science is purely theoretical, for the function of science as such
is merely to know. It is an organized system of knowledge, namely,
an organized system of the principles and conditions of correct
thinking. But because correct thinking is an art, it does not follow
that a knowledge of the methods and conditions of correct thinking

308 LOGIC

is art, which would be a glaring case of /Aera/iJao-ts ets dXXo yei/os. The
art-bearings of the science are given in the normative character of its
subject-matter. As a science logic is descriptive and explanatory, that
is, it describes and formulates the norms of valid thought, although
as science it is not normative, save in the sense that the principles
formulated in it may be normatively or regulatively applied, in
which case they become precepts. What is principle in science
becomes precept in application, and it is only when technically
applied that principles assume a mandatory character. Validity is not
created by logic. Logic merely investigates and states the conditions
and criteria of validity, being in this reference a science of evidence.
In the very fact, however, that logic is normative in the sense of
describing and explaining the norms of correct thinking, its practical
or applied character is given. Its principles as known are science;
its principles as applied are art. There is, therefore, no reason to
sunder these two things or to call logic an art merely or a science
merely ; for it is both when regarded from different viewpoints,
although one must insist on the fact that the rules for practical
guidance are, so far as the science is concerned, quite ah extra. Logic,
ethics, and aesthetics are all commonly (and rightly) called norm-
ative disciplines: they are all concerned with values and standards;
logic with validity and evidence, or values for cognition; ethics
with motives and moral quality in conduct, or values for volition;
aesthetics with the standards of beauty, or values for appreciation
and feeling. Yet none of them is or can be merely normative, or
indeed as science normative at all; if that were so, they would not
be bodies of organized knowledge, but bodies of rules. They might
be well-arranged codes of legislation on conduct, fine art, and evi-
dence, but not sciences. Strictly regarded, it is the descriptive and
explanatory aspect of logic that constitutes its scientific character,
while it is the specific normative aspect that constitutes its logical
character. Values, whether ethical or logical, without an examina-
tion and formulation of their ground, relations, origin, and intercon-
nection, would be merely rules of thumb, popular phrases, or pastoral
precepts. The actual methodology of the sciences or applied logic
is logic as art.

II. Relation of logic to psychology.

The differentiation of logic and psychology in such way as to be
of practical value in the discussion of the disciplines has always been
a difficult matter. John Stuart Mill was disposed to merge logic in
psychology, and Hobhouse, his latest notable apologete, draws no
fixed distinction between psychology and logic, merely saying that
they have different centres of interest, and that their provinces

RELATIONS OF LOGIC TO OTHER DISCIPLINES 309

overlap. Lipps, in his Grundziige der Logik (p. 2), goes the length
of saying that "Logic is a psychological discipline, as certainly as
knowledge occurs only in the Psyche, and thought, which is developed
in knowledge, is a psychical event." Now, if we were to take such
extreme ground as this, then ethics, aesthetics, and pure mathe-
matics would become at once branches of psychology and not coor-
dinate disciplines with it, for volitions, the feelings of appreciation,
and the reasoning of pure mathematics are psychical events. Such
a theory plainly carries us too far and would involve us in confusion.
That the demarcation between the two disciplines is not a chasmic
cleavage, but a line, and that, too, an historically shifting line, is
apparent from the foregoing historical resume.

The four main phases of logical theory include: (1) the concept
(although some logicians begin with the judgment as temporally
prior in the evolution of language), (2) judgment, (3) inference, (4) the
methodology of the sciences. The entire concern of logic is, indeed,
with psychical processes, but with psychical processes regarded from
a specific standpoint, a standpoint different from that of psychology.
In the first place psychology in a certain sense is much wider than
logic, being concerned with the whole of psychosis as such, including
the feelings and will and the entire structure of cognition, whereas
logic is concerned with the particular cognitive processes enumer-
ated above (concept, judgment, inference), and that, too, merely
from the point of view of validity and the grounds of validity. In
another sense psychology is narrower than logic, being concerned
purely with the description and explanation of a particular field of
phenomena, whereas logic is concerned with the procedure of all the
sciences and is practically related to them as their formulated
method. The compass and aims of the two disciplines are different;
for while psychology is in different references both wider and nar-
rower than logic, it is also different in the problems it sets itself,
its aim being to describe and explain the phenomena of mind in the
spirit of empirical science, whereas the aim of logic is only to explain
and establish the laws of evidence and standards of validity. Logic
is, therefore, selective and particular in the treatment of mental
phenomena, whereas psychology is. universal, that is, it covers
the entire range of mental processes as a phenomenalistic science;
logic dealing with definite elements as a normative science. By this
it is not meant that the territory of judgment and inference should
be delivered from the psychologist into the care of the logician;
through such a division of labor both disciplines would suffer. The
two disciplines handle to some extent the same subjects, so far as
names are concerned; but the essence of the logical problem is not
touched by psychology, and should not be mixed up with it, to the
confusion and detriment of both disciplines. The field of psychology.

310 LOGIC

as we have said, is the whole of psychical phenomena; the aim of
individual psychology in the investigation of its field is: (1) to give
a genetic account of cognition, feeling, and will, or whatever be the
elements into which consciousness is analyzed; (2) to explain their
interconnections causally; (3) as ^a chemistry of mental life to
analyze its complexes into their simplest elements; (4) to explain the
totality structurally (or functionally) out of the elements; (5) to
carry on its investigation and set forth its results as a purely empir-
ical science; (6) psychology makes no attempt to evaluate the
processes of mind either in terms of false and true, or good and bad.
From this description of the field and function of psychology, based
on the expressions of its modern exponents, it will be found impossible
to shelter logic under it as a subordinate discipline. If one were to
enlarge the scope of psychology to mean rational psychology, in the
sense which Professor Howison advocates {Psychological Review,
vol. Ill, p. 652), such a subordination might.be possible, but it would
entail the loss of all that the new psychology has gained by the
sharper delimitation of its sphere and problems, and would carry us
back to the position of Mill, who appears to identify psychology
with philosophy at large and with metaphysics.

In contradistinction to the aims of psychology as described in
the foregoing, the sphere and problems of logic may be summarily
characterized as follows: (1) All concepts and judgments are psycho-
logical complexes and processes and may be genetically and struc-
turally described ; that is the business of psychology. They also have
a meaning value, or objective reference, that is, they may be correct
or incorrect, congruous or incongruous with reality. The meaning,
aspect of thought, or its content as truth is the business of logic.
This subject-matter is got by regarding a single aspect in the
total psychological complex. (2) Its aim is not to describe factual
thought or the whole of thought, or the natural processes of thought,
but only certain ideals of thinking, namely, the norms of correct
thinking. Its object is not a datum, but an ideal. (3) While psycho-
logy is concerned with the natural history of reasoning, logic is
concerned with the warrants of inferential reasoning. In the term-
inology of Hamilton it is the nomology of discursive thought. To
use an often employed analogy, psychology is the physics of thought,
logic an ethics of thought. (4) Logic implies an epistemology or
theory of cognition in so far as epistemology discusses the concept
and judgment and their relations to the real world, and here is to be
found its closest connection with psychology. A purely formal logic,
which is concerned merely with the internal order of knowledge and
does not undertake to show how the laws of thought originate, why
they hold good as the measures of evidence, or in what way they are
applicable to concrete reality, would be as barren as scholasticism.

RELATIONS OF LOGIC TO OTHER DISCIPLINES 311

(5) While logic thus goes back to epistemology for its bases and for
the theoretical determination of the interrelation of knowledge and
truth, it goes forward in its application to the practical service of the
sciences as their methodology, A part of its subject-matter is therefore
the actual procedure of the sciences, which it attempts to organize
into systematic statements as principles and formulae. This body of
rules given implicitly or explicitly in the workings and structure of
the special sciences, consisting in classification, analysis, experiment,
induction, deduction, nomenclature, etc., logic regards as a concrete
deposit of inferential experience. It abstracts these principles from
the content and method of the sciences, describes and explains them,
erects them into a systematic methodology, and so creates the
practical branch of real logic. Formal logic, therefore, according to
the foregoing account, would embrace the questions of the internal
congruity and self-consistency of thought and the schematic arrange-
ment of judgments to insure formally valid conclusions; real logic
would embrace the epistemological questions of how knowledge is
related to reality, and how it is built up out of experience, on the
one hand, and the methodological procedure of science, on the other.
The importance of mathematical logic seems to be mainly in the
facilitation of logical expression through symbols. It is rather with
the machinery of the science than with its content and real problem
that the logical algorithm or calculus is concerned. In these con-
densed paragraphs sufficient has been said, I think, to show that logic
and psychology should be regarded as coordinate disciplines; for their
aims and subject-matter differ too widely to subordinate the former
under the latter mthout confusion to both.

I wish now to add a brief note on the relation of logic to another
discipline.

III. Relation of logic to metaphysics.

As currently expounded, logic either abuts immediately on the
territory of metaphysics at certain points or is entirely absorbed in it
as an integral part of the metaphysical subject-matter. I regard the
former view as not only the more tenable theoretically, but as
practically advantageous for working purposes, and necessary for
an intelligible classification of the philosophical disciplines. The
business of metaphysics, as I understand it, is with the nature of
reality; logic is concerned with the nature of validity, or with the
relations of the elements of thought within themselves (self-consist-
ency) and with the relations of thought to its object (real truth), but
not with the nature of the objective world or reality as such. Further,
metaphysics is concerned with the unification of the totality of
knowledge in the form of a scientific cosmology; logic is concerned

312 LOGIC

merely with the inferential and methodological processes whereby
this result is reached. The former is a science of content; the latter is
a science of procedure and relations. Now, inasmuch as procedure
and relations apply to some reality and differ with different forms of
reality, logic necessitates in its implications a theory of being, but
such implications are in no wise to be identified with its subject-
matter or with its own proper problems. Their consideration falls
within the sphere of metaphysics or a broadly conceived epistemo-
logy, whose business it is to solve the ultimate questions of subject
and object, thought and thing, mind and matter, that are implied
and pointed to rather than formulated by logic. Inasmuch as the
logical judgment says something about something, the scientific
impulse drives us to investigate what the latter something ultimately
is; but this is not necessary for logic, nor is it one of logic's legitimate
problems, any more than it is the proper business of the physicist to
investigate the mental implications of his scientific judgments and
hypotheses or the ultimate nature of the theorizing and perceiving
mind, or of causality to his world of matter and motion, although a
general scientific interest may drive him to seek a solution of these
ultimate metaphysical problems. Scientifically the end of logic and
of every discipline is in itself; it is a territorial unity, and its govern-
ment is administered with a unitary aim. Logic is purely a science
of evidential values, not a science of content (in the meaning of
particular reality, as in the special sciences, or of ultimate reality,
as in metaphysics) ; its sole aim and purpose, as I conceive it, is to
formulate the laws and grounds of evidence, the principles of method,
and the conditions and forms of inferential thinking. When it has
done this, it has, as a single science, done its whole work. When one
looks at the present tendencies of logical theory, one is inclined to
believe that the discipline is in danger of becoming an " Allerleiwis-
senschaft/' whose vast undefined territory is the land of " Weiss-
nichtwo." The strict delimitation of the field and problems of science
is demanded in the interest of a serviceable division of scientific labor
and in the interest of an intelligible classification of the accumulated
products of research.

THE FIELD OF LOGIC

BY FREDERICK J. E. WOODBRIDGE

[Frederick J. E. Woodbridge, Johnsonian Professor of Philosophy in Columbia
University, New York, N. Y., since 1902. b. Windsor, Ontario, Canada,
March 26, 1867. A.B. Amherst College, 1889; Union Theological Seminary,
1892; A.M. 1898, LL.D. 1903, Amherst College. Post-grad. Berlin Univers-
ity. Instructor in Philosophy, University of Minnesota, 1894-95; Professor
of Philosophy and head of department, 1895-1902. Member of American
Association for the Advancement of Science, American Philosophical Associ-
ation, American Pyschological Association. Editor of the Journal of Philo-
sophy, Psychology and Scientific Methods.]

Current tendencies in logical theory make a determination of the
field of logic fundamental to any statement of the general problems
of the science. In view of this fact, I propose in this paper to attempt
such a determination by a general discussion of the relation of logic
to mathematics, psychology, and biology, especially noting in con-
nection with biology the tendency known as pragmatism. In con-
clusion, I shall indicate what the resulting general problems appear
to be.

There may appear, at first, little to distinguish mathematics in its
most abstract, formal, and symbolic type from logic. Indeed, math-
ematics as the universal method of all knowledge has been the ideal
of many philosophers, and its right to be such has been claimed of
late with renewed force. The recent notable advances in the science
have done much to make this claim plausible. A logician, a non-
mathematical one, might be tempted to say that, in so far as mathe-
matics is the method of thought in general, it has ceased to be
mathematics; but, I suppose, one ought not to quarrel too much
with a definition, but should let mathematics mean knowledge
simply, if the mathematicians wish it. I shall not, therefore, enter
the controversy regarding the proper limits of mathematical inquiry.
I wish to note, however, a tendency in the identification of logic and
mathematics which seems to me to be inconsistent with the real
significance of knowledge. I refer to the exaltation of the freedom
of thought in the construction of conceptions, definitions, and hypo-
theses.

The assertion that mathematics is a "pure" science is often taken
to mean that it is in no way dependent on experience in the construc-
tion of its basal concepts. The space with which geometry deals
may be Euclidean or not, as we please; it may be the real space of

314 LOGIC

experience or not; the properties of it and the conclusions reached
about it may hold in the real world or they may not; for the mind is
free to construct its conception and definition of space in accordance
with its own aims. Whether geometry is to be ultimately a science
of this type must be left, I suppose, for the mathematicians to decide.
A logician may suggest, however, that the propriety of calling all
these conceptions ''space" is not as clear as it ought to be. Still
further, there seems to underlie all arbitrary spaces, as their founda-
tion, a good deal of the solid material of empirical knowledge, gained
by human beings through contact with an environing world, the
environing character of which seems to be quite independent of
the freedom of their thought. However that may be, it is evident,
I think, that the generalization of the principle involved in this idea
of the freedom of thought in framing its conception of space, would,
if extended to logic, give us a science of knowledge which would
have no necessary relation to the real things of experience, although
these are the things with which all concrete knowledge is most
evidently concerned. It would inform us about the conclusions
which necessarily follow from accepted conceptions, but it could
not inform us in any way about the real truth of these conclusions.
It would, thus, always leave a gap between our knowledge and its
objects which logic itself would be quite impotent to close. Truth
would thus become an entirely extra-logical matter. So far as the
science of knowledge is concerned, it would be an accident if knowledge
fitted the world to which it refers. Such a conception of the science
of knowledge is not the property of a few mathematicians exclusively,
although they have, perhaps, done more than others to give it its
present revived vitality. It is the classic doctrine that logic is the
science of thought as thought, meaning thereby thought in inde-
pendence of any specific object whatever.

In regard to this doctrine, I would not even admit that such a
science of knowledge is possible. You cannot, by a process of general-
ization or free construction, rid thought of connection with objects;
and there is no such thing as a general content or as content-in-
general. Generalization simply reduces the richness of content and,
consequently, of implication. It deals with concrete subject-matter
as much and as directly as if the content were individual and ^p'^'cial-
ized. "Things equal to the same thing are equal to each other," is a
truth, not about thought, but about things. The conclusions about
a fourth dimension follow, not from the fact that we have thought
of one, but from the conception about it which we have framed.
Neither generalization nor free construction can reveal the operations
of thought in transcendental independence.

It may be urged, however, that nothing of this sort was ever
claimed. The bondage of thought to content must be admitted, but

THE FIELD OF LOGIC 315

generalization and free construction, just because they give us the
power to vary conditions as we please, give us thinking in a relative
independence of content, and thus show us how thought operates
irrespective of, although not independent of, its content. The bino-
mial theorem operates irrespective of the values substituted for its
symbols. But I can find no gain in this restatement of the position.
It is true, in a sense, that we may determine the way thought operates
irrespective of any specific content by the processes of generalization
and free construction; but it is important to know in what sense.
Can we claim that such irrespective operation means that we have
discovered certain logical constants, which now stand out as the
distinctive tools of thought? Or does it rather mean that this process
of varying the content of thought as we please reveals certain real
constants, certain ultimate characters of reality, which no amount of
generalization or free construction can possibly alter? The second
alternative seems to me to be the correct one. Whether it is or not
may be left here undecided. What I wish to emphasize is the fact
that the decision is one of the things of vital interest for logic, and
properly belongs in that science. Clearly, we can never know the
significance of ultimate constants for our thinking until we know
what their real character is. To determine that character we must
most certainly pass out of the realm of generalization and free con-
struction; logic must become other than simply mathematical or
symbolic.

There is another sense in which the determination of the operations
of thought irrespective of its specific content is interpreted in con-
nection wath the exaltation of generalization and free construction.
Knowledge, it is said, is solely a matter of implication, and logic,
therefore, is the science of implication simply. If this is so, it would
appear possible to develop the whole doctrine of implication by the
use of symbols, and thus free the doctrine from dependence on the
question as to how far these symbols are themselves related to the
real things of the world. If, for instance, a implies h, then, if a is
true, 6 is true, and this quite irrespective of the real truth of a or h.
It is to be urged, however, in opposition to this view, that knowledge
is concerned ultimately only with the real truth of a and h, and
that the implication is of no significance whatever apart from this
truth. There is no virtue in the mere implication. Still further, the
supposition that there can be a doctrine of implication, simply,
seems to be based on a misconception. For even so-called formal
implication gets its significance only on the supposed truth of the
terms with which it deals. We suppose that a does imply 6, and that
a is true. In other words, we can state this law of implication only
as we first have valid instances of it given in specific, concrete cases.
The law is a generalization and nothing more. The formal statement

316 LOGIC

gives only an apparent freedom from experience. Moreover, there is
no reason for saying that a implies h unless it does so either really or
by supposition. If a really implies b, then the implication is clearly
not a matter of thinking it; and to suppose the implication is to feign
a reality, the implications of which are equally free from the processes
by which they are thought. Ultimately, therefore, logic must take
account of real implications. We cannot avoid this through the use
of a symbolism which virtually implies them. Implication can have
a logical character only because it has first a metaphysical one.

The supposition underlying the conception of logic I have been
examining is, itself, open to doubt and seriously questioned. That
supposition was the so-called freedom of thought. The argument
has already shown that there is certainly a very definite limit to this
freedom, even when logic is conceived in a very abstract and formal
way. The processes of knowledge are bound up with their- contents,
and have their character largely determined thereby. When, more-
over, we view knowledge in its genesis, when we take into considera-
tion the contributions which psychology and biology have made to
our general view of what knowledge is, we seem forced to conclude
that the conceptions which we frame are very far from being our own
free creations. They have, on the contrary, been laboriously worked
out through the same processes of successful adaptation which have
resulted in other products. Knowledge has grown up in connection
with the unfolding processes of reality, and has, by no means, freely
played over its surface. That is why even the most abstract of all
mathematics is yet grounded in the evolution of human experience.

In the remaining parts of this paper, I shall discuss further the
claims of psychology and biology. The conclusion I would draw
here is that the field of logic cannot be restricted to a realm where
the operations of thought are supposed to move freely, independent
or irrespective of their contents and the objects of a real world;
and that mathematics, instead of giving us any support for the
supposition that it can, carries us, by the processes of symbolization
and formal implication, to recognize that logic must ultimately find
its field where implications are real, independent of the processes
by which they are thought, and irrespective of the conceptions we
choose to frame.

The processes involved in the acquisition and systematization of
knowledge may, undoubtedly, be regarded as mental processes and
fall thus within the province of psychology. It may be claimed,
therefore, that every logical process is also a ps3^chological one. The
important question is, however, is it nothing more? Do its logical
and psychological characters simply coincide? Or, to put the ques-

THE FIELD OF LOGIC 317

tion in still another form, as a psychological process simply, does
it also serve as a logical one? .The answers to these questions can be
determined only by first noting what psychology can say about it
as a mental process.

In the first place, psychology can analyze it, and so determine
its elements and their connections. It can thus distinguish it from all
other mental processes by pointing out its unique elements or their
unique and characteristic connection. No one will deny that a
judgment is different from an emotion, or that an act of reasoning is
different from a volition; and no one will claim that these differences
are entirely beyond the psychologist's power to ascertain accurately
and precisely. Still further, it appears possible for him to determine
with the same accuracy and precision the distinction in content and
connection between processes which are true and those which are
false. For, as mental processes, it is natural to suppose that they
contain distinct differences of character which are ascertainable.
The states of mind called belief, certainty, conyiction, correctness,
truth, are thus, doubtless, all distinguishable as mental states. It
may be admitted, therefore, that there can be a thoroughgoing
psychology of logical processes.

Yet it is quite evident to me that the characterization of a mental
process as logical is not a psychological characterization. In fact,
I think it may be claimed that the characterization of any mental
process in a specific way, say as an emotion, is extra-psychological.
Judgments and inferences are, in short, not judgments and inferences
because they admit of psychological analysis and explanation, any
more than space is space because the perception of it can be worked
out by genetic psychology. In other words, knowledge is first know-
ledge, and only later a set of processes for psychological analysis.
That is why, as it seems to me, all psychological logicians, from Locke
to our own day, have signally failed in dealing with the problem of
knowledge. The attempt to construct knowledge out of mental
states, the relations between ideas, and the relation of ideas to
things, has been, as I read the history, decidedly without profit.
Confusion and divergent opinion have resulted instead of agreement
and confidence. On precisely the same psychological foundation,
we have such divergent views of knowledge as idealism, phenomenal-
ism, and agnosticism, with many other strange mixtures of logic,
psychology, and metaphysics. The lesson of these perplexing theories
seems to be that logic, as logic, must be divorced from psychology.

It is also of importance to note, in this connection, that the deter-
mination of a process as mental and as thus falling within the domain
of psychology strictly, has by no means been worked out to the
general satisfaction of psychologists themselves. Recent literature
abounds in elaborate discussion of the distinction between what is

318 LOGIC

a mental fact and what not, with a prevailing tendency to draw the
remarkable conclusion that all facts are somehow mental or experi-
enced facts. The situation would be worse for psychology than it is,
if that vigorous science had not learned from other sciences the valu-
able knack of isolating concrete problems and attacking them
directly, without the burden of previous logical or metaphysical
speculation. Thus knowledge, which is the peculiar province of logic,
is increased, while we wait for the acceptable definition of a mental
fact. But definitions, be it remembered, are themselves logical
matters. Indeed, some psychologists have gone so far as to claim
that the distinction of a fact as mental is a purely logical distinc-
tion. This is significant as indicating that the time has not yet come
for the identification of logic and psychology.

In refreshingly sharp contrast to the vagueness and uncertainty
which beset the definition of a mental fact are the palpable concrete-
ness and definiteness of knowledge itself. Every science, even history
and philosophy, are instances of it. What constitutes a knowledge
ought to be as definite and precise a question as could be asked.
That logic has made no more progress than it has in the answer to it
appears to be due to, the fact that it has not sufficiently grasped the
significance of its own simplicity. Knowledge has been the important
business of thinking man, and he ought to be able to tell what he does
in order to know, as readily as he tells what he does in order to build
a house. And that is why the Aristotelian logic has held its own so
long. In that logic, " the master of them that know" simply rehearsed
the way he had systematized his own stores of knowledge. Naturally
we, so far as we have followed his methods, have had practically
nothing to add. In our efforts to improve on him, we have too often
left the right way and followed the impossible method inaugurated
by Locke. Had we examined with greater persistence our own
methods of making science, we should have profited more. The
introduction of psychology, instead of helping the situation, only
confuses it.

Let it be granted, however, in spite of the vagueness of what is
meant by a mental fact, that logical processes are also mental pro-
cesses. This fact has, as I have already suggested, an important
bearing on their genesis, and sets very definite limits to the freedom
of thought in creating. It is not, however, as mental processes that
they have the value of knowledge. A mental process which is know-
ledge purports to be connected with something other than itself,
something which may not be a mental process at all. This connection
should be investigated, but the investigation of it belongs, not to
psychology, but to logic.

I am well aware that this conclusion runs counter to some meta-
physical doctrines, and especially to idealism in all its forms, with the

THE FIELD OF LOGIC 319

epistemologies based thereon. It is, of course, impossible here to
defend my position by an elaborate analysis of these metaphysical
systems. But I will say this. I am in entire agreement with idealism
in its claim that questions of knowledge and of the nature of reality
cannot ultimately be separated, because we can know reality only
as we know it. But the general question as to how we know reality
can still be raised. By this I do not mean the question, how is it
possible for us to have knowledge at all, or how it is possible for reality
to be known at all, but how, as a matter of fact, we actually do know
it? That we really do know it, I would most emphatically claim.
Still further, I would claim that what we know about it is determined,
not by the fact that we can know in general, but by the way reality,
as distinct from our knowledge, has determined. These ways appear
to me to be ascertainable, and form, thus, undoubtedly, a section
of metaphysics. But the metaphysics will naturally be realistic rather
than idealistic. ^ ,

III

Just as logical processes may be regarded as, at the same time,
psychological processes, so they may be regarded, with equal right,
as vital processes, coming thus under the categories of evolution.
The tendency so to regard them is very marked at the present day,
especially in France and in this country. In France, the movement
has perhaps received the clearer definition. In America the union of
logic and biology is complicated — and at times even lost sight of —
by emphasis on the idea of evolution generally. It is not my intention
to trace the history of this movement, but I should like to call atten-
tion to its historic motive in order to get it in a clear light.

That the theory of evolution, even Darwinism itself, has radically
transformed our historical, scientific, and philosophical methods, is
quite evident. Add to this the influence of the Hegelian philosophy,
with its own doctrine of development, and one finds the causes of
the rather striking unanimity which is discoverable in many ways
between Hegelian idealists, on the one hand, and philosophers of
evolution of Spencer's type, on the other. Although two men would,
perhaps, not appear more radically different at first sight than Hegel
and Spencer, I am inclined to believe that we shall come to recognize
more and more in them an identity of philosophical conception. The
pragmatism of the day is a striking confirmation of this opinion, for
it is often the expression of Hegelian ideas in Darwinian and Spencer-
ian terminology. The claims of idealism and of evolutionary science
and philosophy have thus sought reconciliation. Logic has been,
naturally, the last of the sciences to yield to evolutionary and genetic
treatment. It could not escape long, especially when the idea of
evolution had been so successful in its handling of ethics. If morality

320 LOGIC

can be brought under the categories of evolution, why not thinking
also? In answer to that question we have the theory that thinking
is an adaptation, judgment is instrumental. But I would not leave
the impression that this is true of pragmatism alone, or that it has
been developed only through pragmatic tendencies. It is naturally
the result also of the extension of biological philosophy. In the
biological conception of logic, we have, then, an interesting coinci-
dence in the results of tendencies differing widely in their genesis.

It would be hazardous to deny, without any qualifications, the
importance of genetic considerations. Indeed, the fact that evolution
in the hands of a thinker like Huxley, for instance, should make con-
sciousness and thinking apparently useless epiphenomena in a devel-
oping world, has seemed like a most contradictory evolutionary
philosophy. It was difficult to make consciousness a real function in
development so long as it was regarded as only cognitive in character.
Evolutionary philosophy, coupled with physics, had built up a sort of
closed system with which consciousness could not interfere, but which
it could know, and know with all the assurance of a traditional logic.
If, however, we were to be consistent evolutionists, we could not abide
by such a remarkable result. The whole process of thinking must be
brought within evolution, so that knowledge, even the knowledge of
the evolutionary hypothesis itself, must appear as an instance of
adaptation. In order to do this, however, consciousness must not be
conceived as only cognitive. Judgment, the core of logical processes,
must be regarded as an instrument and as a mode of adaptation.

The desire for completeness and consistency in an evolutionary
philosophy is not the only thing which makes the denial of genetic
considerations hazardous. Strictly biological considerations furnish
reasons of equal weight for caution. For instance, one will hardly
deny that the whole sensory apparatus is a striking instance of
adaptation. Our perceptions of the world would thus appear to be
determined by this adaptation, to be instances of adjustment. They
might conceivably have been different, and in the case of many other
creatures, the perceptions of the world are undoubtedly different.
All our logical processes, referring ultimately as they do to our per-
ceptions, would thus appear finally to depend on the adaptation
exhibited in the development of our sensory apparatus. So-called
laws of thought would seem to be but abstract statements or formu-
lations of the results of this adjustment. It would be absurd to sup-
pose that a man thinks in a sense radically different from that in
which he digests, or a flower blossoms, or that two and two are four
in a sense radically different from that in which a flower has a given
number of petals. Thinking, like digesting and blossoming, is an
effect, a product, possibly a structure.

I am not at all interested in denying the force of these considera-

THE FIELD OF LOGIC 321

tions. They have, to my mind, the greatest importance, and due
weight has, as yet, not been given to them. To one at all committed
to a unitary and evolutionary view of the world, it must indeed seem
strange if thinking itself should not be the result of evolution, or that,
in thinking, parts of the world had not become adjusted in a new
way. But while I am ready to admit this, I am by no means ready to
admit some of the conclusions for logic and metaphysics which are
often drawn from the admission. Just because thought, as a product
of evolution, is functional and judgment instrumental, it by no means
follows that logic is but a branch of biology, or that knowledge of the
world is but a temporary adjustment, which, as knowledge, might
have been radically different. In these conclusions, often drawn with
Protagorean assurance, two considerations of crucial importance
seem to be overlooked, first, that adaptation is itself metaphysical in
character, and secondly, that while knowledge may be functional and
judgment instrumental, the character of the functioning has the
character of knowledge, which sets it off sharply from all other
functions.

It seems strange to me that the admission that knowledge is as
matter of adaptation, and thus a relative matter, should, in these
days, be regarded as in any way destroying the claims of knowledge
to metaphysical certainty. Yet, somehow, the opinion widely prevails
that the doctrine of relativity necessarily involves the surrender of
auA'thing like absolute truth. '' All our knowledge is relative, and^
therefore, only partial, incomplete, and but practically trustworthy/'
is a statement repeatedly made. The fact that, if our development
had been different, our knowledge would have been different, is
taken to involve the conclusion that our knowledge cannot possibly
disclose the real constitution of things, that it is essentially condi-
tional, that it is only a mental device for getting results, that any
other system of knowledge which would get results equally well
would be equally true; in short, that there can be no such thing, as
metaphysical or epistemological truth. These conclusions do indeed
seem strange, and especially strange on the basis of evolution. For
while the evolutionary process might, conceivably, have been dif-
ferent, its results are, in any case, the results of the process. They
are not arbitrary. We might have digested without stomachs, but
the fact that we use stomachs in this important process ought not to
free us from metaphysical respect for the organ. As M. Rey suggests,
in the Revue Philosoj)hique for June, 1904, a creature without the
sense of smell would have no geometry, but- that does not make
geometry essentially hypothetical, a mere mental construction; for
we have geometry because of the working out of nature's laws.
Indeed, instead of issuing in a relativistic metaphysics of knowledge,
the doctrine of relativity should issue in the recognition of the finality

322 LOGIC

of knowledge in every case of ascertainably complete adaptation. In
other words, adaptation is itself metaphysical in character. Adjust-
ment is always adjustment between things, and yields only what it
does yield. The things or elements get into the state which is their
adjustment, and this adjustment purports to be their actual and
unequivocal ordering in relation to one another. Different conditions
might have produced a different ordering, but, again, this ordering
would be equally actual and unequivocal, equally the one ordering to
issue from them. To suppose or admit that the course of events might
have been and might be different is not at all to suppose or admit
that it was or is different; it is, rather, to suppose and admit that we
have real knowledge of what that course really was and is. This seems
to be very obvious.

Yet the evolutionist often thinks that he is not a metaphysician,
even when he brings all his conceptions systematically under the
conception of evolution. This must be due to some temporary lack of
clearness. If evolution is not a metaphysical doctrine when extended
to apply to all science, all morality, all logic, in short, all things, then
it is quite meaningless for evolutionists to pronounce a metaphysical
sentence on logical processes. But if evolution is a metaphysics, then
its sentence is metaphysical, and in every case of adjustment or
adaptation we have a revelation of the nature of reality in a definite
and unequivocal form. This conclusion applies to logical processes as
well as to others. The recognition that they are vital processes can,
therefore, have little significance for these processes in their distinct-
ive character as logical. They are like all other vital processes in
that they are vital and subject to evolution. They are unlike all
others in that thought is unlike digestion or breathing. To regard
logical processes as vital processes does not in any way, therefore,
invalidate them as logical processes or make it superfluous to consider
their claim to give us real knowledge of a real world. Indeed, it makes
such a consideration more necessary and important.

A second consideration overlooked by the Protagorean tendencies
of the day is that judgment, even if it is instrumental, purports to
give us knowledge, that is, it claims to reveal what is independent of
the judging process. Perhaps I ought not to say that this considera-
tion is overlooked, but rather that it is denied significance. It is even
denied to be essential to judgment. It is claimed that, instead of
revealing anything independent of the judging process, judgment is
just the adjustment and no more. It is a reorganization of experience,
an attempt at control. All this looks to me like a misstatement of the
facts. Judgment claims to be no such thing. It does not function as
such a thing. When I make any judgment, even the simplest, I may
make it as the result of tension, because of a demand for reorganiza-
tion, in order to secure control of experience; but the judgment

THE FIELD OF LOGIC 323

means for me something quite different. It means decidedly and
unequivocally that in reality, apart from the judging process, things
exist and operate just as the judgment declares. If it is claimed that
this meaning is illusory, I eagerly desire to know on what solid ground
its illusoriness can be established. AATien the conclusion was reached
that gravitation varies directly as the mass and inversely as the
square of the distance, it was doubtless reached in an evolutionary
and pragmatic way; but it claimed to disclose a fact which prevailed
before the conclusion was reached, and in spite of the conclusion.
Knowledge has been born of the travail of living, but it has been
born as knowledge.

When the knowledge character of judgment is insisted on, it seems
almost incredible that any one would think of denying or overlooking
it. Indeed, current discussions are far from clear on the subject.
Pragmatists are constantly denying that they hold the conclusions
that their critics almost unanimously draw. There is, therefore, a
good deal of confusion of thought yet to be dispelled. Yet there
seems to be current a pronounced determination to banish the epi-
stemological problem from logic. This is, to my mind, suspicious, even
when epistemologj'" is defined in a way which most epistemologists
would not approve. It is suspicious just because we must always
ask eventually that most epistemological and metaphysical question :
" Is knowledge true? " To answer, it is true when it functions in a way
to satisfy the needs which generated its activity, is, no doubt, correct,
but it is by no means adequate. The same answer can be made to
the inquiry after the efficiency of any vital process whatever, and is,
therefore, not distinctive. We have still to inquire into the specific
character of the needs which originate judgments and of the conse-
quent satisfaction. Just here is where the uniqueness of the logical
problem is disclosed. With conscious beings, the success of the things
they do has become increasingly dependent on their ability to discover
what takes place in independence of the knowing process. That is the
need which generates judgment. The satisfaction is, of course, the
attainment of the discovery. Now to make the judgment itself and
not the consequent action the instrumental factor seems to me to
misstate the facts of the case. Nothing is clearer than that there
is no necessity for knowledge to issue in adjustment. And it is clear
to me that increased control of experience, while resulting from
knowledge, does not give to it its character. Omniscience could idly
view the transformations of reality and yet remain omniscient.
Knowledge works, but it is not, therefore, knowledge.

These considerations have peculiar force when applied to that
branch of knowledge which is knowledge itself. Is the biological
account of knowledge correct? That question we must evidently
ask, especially when we are urged to accept the account. Can we,

324 LOGIC

to put the question in its most general form, accept as an adequate
account of the logical process a theory which is bound up with some
other specific department of human knowledge? It seems to me that
we cannot. Here we must be epistemologists and metaphysicians,
or give up the problem entirely. This by no means involves the
attempt to conceive pure thought set over against pure reality — the
kind of epistemology and metaphysics justly ridiculed by the prag-
matist — for knowledge, as already stated, is given to us in concrete
instances. How knowledge in general is possible is, therefore, as use-
less and meaningless a question as how reality in general is possible.
The knowledge is given as a fact of life, and what we have to deter-
mine is not its non-logical antecedents or its practical consequences,
but its constitution as knowledge and its validity. It may be admitted
that the question of validity is settled pragmatically. No knowledge
is true unless it yields results which can be verified, unless it can issue
in increased control of experience. But I insist again that that fact
is not sufficient for an account of what knowledge claims to be. It
claims to issue in control because it is true in independence of the
control. And it is just this assurance that is needed to distinguish
knowledge from what is not knowledge. It is the necessity of exhibit-
ing this assurance which makes it impossible to subordinate logical
problems, and forces us at last to questions of epistemology and
metaphysics.

As I am interested here primarily in determining the field of logic,
it is somewhat outside my province to consider the details of logical
theory. Yet the point just raised is of so much, importance in con-
nection with the main question that I venture the following general
considerations. This is, perhaps, the more necessary because the
pragmatic doctrine finds in the concession made regarding the test
of validity one of its strongest defenses.

Of course a judgment is not true simply because it is a judgment.
It may be false. The only way to settle its validity is to discover
whether experience actually provides what the judgment promises,
that is, whether the conclusions drawn from it really enable us to
control experience. No mere speculation will yield the desired result,
no matter with how much formal validity the conclusions may be
drawn. That merely formal validity is not the essential thing, I
have pointed out in discussing the relation of logic to mathematics.
The test of truth is pragmatic. It is apparent, therefore, that the
formal validity does not determine the actual validity. What is
this but the statement that the process of judgment is not itself the
determining factor in its real validity? It is, in short, only valid
judgments that can really give us control of experience. The impli-
cations taken up in the judgment must, therefore, be real implica-
tions which, as such, have nothing to do with the judging process,

THE FIELD OF LOGIC 325

and which, most certainly, are not brought about by it. And what
is this but the claim that judgment as such is never instrumental ?
In other words, a judgment which effected its own content would
only by the merest accident function as valid knowledge. We have
valid knowledge, then, only when the implications of the judgment
are found to be independent of the judging process. We have know-
ledge only at the risk of error. The pragmatic test of validity, instead
of proving the instrumental character of judgment, would thus
appear to prove just the reverse.

Valid knowledge has, therefore, for its content a system of real,
not judged or hypothetical implications. The central problem of
logic which results from this fact is not how a knowledge of real
implications is then possible, but what are the ascertainable types
of real implications. But, it may be urged, we need some criterion to
determine what a real implication is. I venture to reply that we
need none, if by such is meant anything else than the facts with
which we are dealing. I need no other criterion than the circle to
determine whether its diameters are really equal. And, in general,
I need no other criterion than the facts dealt with to determine
whether they really imply what I judge them to imply. Logic appears
to me to be really as simple as this. Yet there can be profound pro-
blems involved in the working out of this simple procedure. There is
the problem already stated of the most general types of real impli-
cation, or, in other words, the time-honored doctrine of categories.
Whether there are categories or basal types of existence seems to me
to be ascertainable. When ascertained, it is also possible to discover
the types of inference or implication which they afford. This is by
no means the whole of logic, but it appears to me to be its central
problem.

These considerations will, I hope, throw light on the statement
that while knowledge works, it is not therefore knowledge. It works
because its content existed before its discovery by the knowledge
process, and because its content was not effected or brought about
by that process. Judgment was the instrument of its discovery, not
the instrument which fashioned it. While, therefore, willing to admit
that logical processes are vital processes, I am not willing to admit
that the problem of logic is radically changed thereby in its formu-
lation or solution, for the vital processes in question have the unique
character of knowledge, the content of which is what it claims to be,
a system of real implications which existed prior to its discovery.

In the psychological and biological tendencies in logic, there is,
however, I think, a distinct gain for logical theory. The insistence
that logical processes are both mental and vital has done much to
take them out of the transcendental aloofness from reality in which
they have often been placed, especially since Kant. So long as

326 LOGIC

thought and object were so separated that they could never be
brought together, and so long as logical processes were conceived
wholly in terms of ideas set over against objects, there was no hope
of escape from the realm of pure hypothesis and conjecture. Locke's
axiom that "the mind, in all its thoughts and reasonings, hath no
other immediate object but its own ideas," an axiom which Kant
did so much to sanctify, and which has been the basal principle of
the greater part of modern logic and metaphysics, is most certainly
subversive of logical theory. The transition from ideas to anything
else is rendered impossible by it. Now it is just this axiom which the
biological tendencies in logic have done so much to destroy. They
have insisted, with the greatest right, that logical processes are not
set over against their content as idea against object, as appearance
against reality, but are processes of reality itself. Just as reality
can and does function in a physical or a physiological way, so also
it functions in a logical way. The state we call knowledge becomes,
thus, as much a part of the system of things as the state we call
chemical combination. The problem how thought can know anything
becomes, therefore, as irrelevant as the problem how elements can
combine at all. The recognition of this is a great gain, and the
promise of it most fruitful for both logic and metaphysics.

But, as I have tried to point out, all this surrendering of pure
thought as opposed to pure reality, does not at all necessitate our
regarding judgment as a process which makes reality different
from what it was before. Of course there is one difference, namely,
the logical one; for reality prior to logical processes is unknown. As
a result of these processes it becomes known. These processes are,
therefore, responsible for a known as distinct from an unknown
reality. But what is the transformation which reality undergoes in
becoming known? When it becomes known that water seeks its own
level, what change has taken place in the water? It would appear
that we must answer, none. The water which seeks its own level has
not been transformed into ideas or even into a human experience.
It appears to remain, as water, precisely what it was before. The
transformation which takes place, takes place in the one who knows,
a transformation from ignorance to knowledge. Psychology and bio-
logy can afford us the natural history of this transformation, but
they cannot inform us in the least as to why it should have its
specific character. That is given and not deduced. The attempts to
deduce it have, without exception, been futile. That is why we are
forced to take it as ultimate in the same way we take as ultimate
the specific character of any definite transformation. To my mind,
there is needed a fuller and more cordial recognition of this fact. The
conditions under which we, as individuals, know are certainly dis-
coverable, just as much as the conditions under which we breathe

THE FIELD OF LOGIC 327

or digest. And what happens to things when we know them is also
as discoverable as what happens to them when we breathe them or
digest them.

But here the ideahst may interpose that we can never know what
happens to things when we know them, because we can never know
them before they become known. I suppose I ought to wrestle with
this objection. It is an obvious one, but, to my mind, it is without
force. The objection, if pursued, can carry us only in a circle. The
problem of knowledge is still on our hands, and every logician of
whatever school, the offerer of this objection also, has, nevertheless,
attempted to show what the transformation is that thought works,
for all admit that it works some. Are we, therefore, engaged in a
hopeless task? Or have we failed to grasp the significance of our
problem? I think the latter. We fail to recognize that, in one way
or other, we do solve the problem, and that our attempts to solve it
show quite clearly that the objection under consideration is without
force. Take, for instance, any concrete case of knowledge, the water
seeking its own level, again. Follow the process of knowledge to the
fullest extent, we never find a single problem which is not solvable
by reference to the concrete things with which we are dealing, nor
a single solution which is not forced upon us by these things rather
than by the fact that we deal with them. The transformation wrought
is thus discovered, in the progress of knowledge itself, to be wrought
solely in the inquiring individual, and wrought by repeated contact
with the things with which he deals. In other words, all knowledge
discloses the fact that its content is not created by itself, but by the
things with which it is concerned.

It is quite possible, therefore, that knowledge should be what
we call transcendent and yet not involve us in a transcendental
logic. That we should be able to know without altering the things we
know is no more and no less remarkable and mysterious than that
we should be able to digest by altering the things we digest. In
other words, the fact that digestion alters the things is no reason
that knowledge should alter them, even if we admit that logical
processes are vital and subject to evolution. Indeed, if evolution
teaches us anything on this point, it is that knowledge processes are
real just as they exist, as real as growth and digestion, and must
have their character described in accordance with what they are. The
recognition that knowledge can be transcendent and yet its processes
vital seems to throw light on the difficulty evolution has encountered
in accounting for consciousness and knowledge. All the reactions
of the individual seem to be expressible in terms of chemistry and
physics without calling in consciousness as an operating factor. What
is this but the recognition of its transcendence, especially when the
conditions of conscious activity are quite likely expressible in chem-

328 LOGIC

ical and physical terms? While, therefore, biological considerations
result in the great gain of giving concrete reality to the processes of
knowledge, the gain is lost, if knowledge itself is denied the tran-
scendence which it so evidently discloses.

The argument advanced in this discussion has had the aim of
emphasizing the fact that in knowledge we have actually given, as
content, reality as it is in independence of the act of knowing, that the
real world is self-existent, independent of the judgments we make
about it. This fact has been emphasized in order to confine the
field of logic to the field of knowledge as thus understood. In the
course of the argument, I have occasionally indicated what some
of the resulting problems of logic are. These I wish now to state in
a somewhat more systematic way.

The basal problem of logic becomes, undoubtedly, the metaphysics
of knowledge, the determination of the nature of knowledge and its
relation to reality. It is quite evident that this is just the problem
which the current tendencies criticised have sought, not to solve,
but to avoid or set aside. Their motives for so doing have been
mainly the difficulties which have arisen from the Kantian philo-
sophy in its development into transcendentalism, and the desire
to extend the category of evolution to embrace the whole of reality,
knowledge included. I confess to feeling the force of these motives
as strongly as any advocate of the criticised opinions. But I do not
see my way clear to satisfying them by denying or explaining away
the evident character of knowledge itself. It appears far better
to admit that a metaphysics of knowledge is as yet hopeless, rather
than so to transform knowledge as to get rid of the problem; for we
must ultimately ask after the truth of the transformation. But I
am far from believing that a metaphysics of knowledge is hopeless.
The biological tendencies themselves seem to furnish us with much
material for at least the beginnings of one. Reality known is to be set
over against reality unknown or independent of knowledge, not as
image to original, idea to thing, phenomena to noumena, appearance
to reality; but reality as known is a new stage in the development of
reality itself. It is not an external mind which knows reality by
means of its own ideas, but reality itself becomes known through
its own expanding and readjusting processes. So far I am in entire
agreement with the tendencies I have criticised. But what change is
effected by this expansion and readjustment? I can find no other
answer than this simple one : the change to knowledge. And by this
I mean to assert unequivocally that the addition of knowledge to
a reality hitherto without it is simply an addition to it and not a
transformation of it. Such a view may appear to make knowledge

THE FIELD OF LOGIC 329

a wholly useless addition, but I see no inherent necessity in such a
conclusion. Nor do I see any inherent necessity of supposing that
knowledge must be a useful addition. Yet I would not be so foolish
as to deny the usefulness of knowledge. We have, of course, the
most palpable evidences of its use. As we examine them, I think we
find, without exception, that knowledge is useful just in proportion
as we find that reality is not transformed by being known. If it really
were transformed in that process, could anything else than confusion
result from the multitude of knowing individuals?

To me, therefore, the metaphysics of the situation resolves itself
into the realistic position that a developing reality develops, under
ascertainable conditions, into a known realitj'- without undergoing
any other transformation, and that this new stage marks an advance
in the efficiency of reality in its adaptations. My confidence steadily
grows that this whole process can be scientifically worked out. It is
impossible here to justify my confidence in detail, and I must leave
the matter with the following suggestion. The point from which
knowledge starts and to which it ultimately returns is always some
portion of reality where there is consciousness, the things, namely,
which, we are wont to say, are in consciousness. These things are not
ideas representing other things outside of consciousness, but real
things, which, by being in consciousness, have the capacity of repre-
senting each other, of standing for or implying each other. Know-
ledge is not the creation of these implications, but their successful
systematization. It will be found, I think, that this general state-
ment is true of every concrete case of knowledge which we possess.
Its detailed working out would be a metaphysics of knowledge, an
epistemology.

Since knowledge is the successful systematization of the implica-
tions which are disclosed in things by virtue of consciousness, a
second logical problem of fundamental importance is the determina-
tion of the most general types of implication with the categories
which underlie them. The execution of this problem would naturally
involve, as subsidiary, the greater part of formal and symbolic logic.
Indeed, vital doctrines of the syllogism, of definition, of formal
inference, of the calculus of classes and propositions, of the logic of
relations, appear to be bound up ultimately with a doctrine of cate-
gories; for it is only a recognition of basal types of existence with
their implications that can save these doctrines from mere formal-
ism. These types of existence or categories are not to be regarded
as free creations or as the contributions of the mind to experience.
There is no deduction of them possible. They must be discovered
in the actual progress of knowledge itself, and I see no reason to
suppose that their number is necessarily fixed, or that we should
necessarily be in possession of all of them. It is requisite, however,

330 LOGIC

that in every case categories should be incapable of reduction to
each other,

A doctrine of categories seems to me to be of the greatest import-
ance in the systematization of knowledge, for no problem of relation
is even stateable correctly before the type of existence to which its
terms belong has been first determined. I submit one illustration
to reinforce this general statement, namely, the relation of mind to
body. If mind and body belong to the same type of existence, we
have one set of problems on our hands; but if they do not, we have
an entirely different set. Yet volumes of discussion written on this
subject have abounded in confusion, simply because they have
regarded mind and body as belonging to radically different types of
existence and yet related in terms of the type to which one of them
belongs. The doctrine of parallelism is, perhaps, the epitome of this
confusion.

The doctrine of categories will involve not only the greater part
of formal and symbolic logic, but will undoubtedly carry the logician
into the doctrine of method. Here it is to be hoped that recent
tendencies will result in effectively breaking down the artificial dis-
tinctions which have prevailed between deduction and induction.
Differences in method do not result from differences in points of de-
parture, or between the universal and the particular, but from the
categories, again, which give the method direction and aim, and
result in different types of synthesis. In this direction, the logician
may hope for an approximately correct classification of the various
departments pf knowledge. Such a classification is, perhaps, the
ideal of logical theory.

SECTION D — METHODOLOGY OF SCIENCE

{Hall 6, September 22, 3 p. m.)

Chairman: Professor James E. Creighton, Cornell University.
Speakers: Professor Wilhelm Ostwald, University of Leipzig.
Professor Benno Erdmann, University of Bonn.
Secretary: Dr. R. B. Perry, Harvard University.

ON THE THEORY OF SCIENCE

BY WILHELM OSTWALD

{Translated from the German by Dr. R. M. Yerkes, Harvard University)

[Wilhelm Ostwald, Professor of Physical Chemistry, University of Leipzig,
since 1887. b. September 2, 1853, Riga, Russia. Grad. Candidate Chemistry,
1877; Master Chemistry, 1878; Doctor Chemistry, Dorpat. Dr. Hon.
Halle and Cambridge; Privy Councilor; Assistant, Dorpat, 1875-81;
Regular Professor, Riga, 1881-87. Member various learned and scientific
societies. Author of Manual of General Chemistry; Electro Chemistry; Foun-
dation of Inorganic Chemistry; Lectures on Philosophy of Nature; Artist's
Letters; Essays and Lectures; and many other noted works and papers on
Chemistry and Philosophy.]

One of the few points on which the philosophy of to-day is united is
the knowledge that the only thing completely certain and undoubted
for each one is the content of his own consciousness; and here the
certainty is to be ascribed not to the content of consciousness in
general, but only to the momentary content.

This momentary content we divide into two large groups, which
we refer to the inner and outer world. If we call any kind of content
of consciousness an experience, then we ascribe to the outer world
such experiences as arise without the activity of our will and cannot
be called forth by its activity alone. Such experiences never arise
without the activit}'' of certain parts of our body, which we call
sense organs. In other words, the outer world is that which reaches
our consciousness through the senses.

On the other hand, we ascribe to our inner world all experiences
which arise without the immediate assistance of a sense organ.
Here, first of all, belong all experiences which we call remembering
and thinking. An exact and complete differentiation of the two
territories is not intended here, for our purpose does not demand
that this task be undertaken. For this purpose the general orienta-
tion in which every one recognizes familiar facts of his consciousness
is sufficient.

Each experience has the characteristic of uniqueness. None of us
doubts that the expression of the poet " Everything is only repeated
in life" is really just the opposite of the truth, and that in fact no-

334 METHODOLOGY OF SCIENCE

thing is repeated in life. But to express such a judgment we must.
be in position to compare different experiences with each other, and
this possibihty rests upon a fundamental phenomenon of our con-
sciousness, memory. Memory alone enables us to put various ex-
periences in relation to each other, so that the question as to their
likeness or difference can be asked.

We find the simpler relations here in the inner experiences. A
certain thought, such as twice two is four, I can bring up in my
consciousness as often as I wish, and in addition to the content of
the thought I experience the further consciousness that I have
already had this thought before, that it is familiar to me.

A similar but somewhat more complex phenomenon appears in
the experiences in which the outer world takes part. After I have
eaten an apple, I can repeat the experience in two ways. First, as
an inner experience, I can remember that I have eaten the apple
and by an effort of my will I can re-create in myself, although with.
diminished strength and intensity, a part of the former experience
— the part which belonged to my inner world. Another part, the
sense impression which belonged to that experience, I cannot re-create
by an effort of my will, but I must again eat an apple in order to
have a similar experience of this sort. This is a complete repetition
of the experience to which the external world also contributes.
Such a repetition does not depend altogether on my own powers,
for it is necessary that I have an apple, that is, that certain condi-
tions which are independent of me and belong to the outer world
be fulfilled.

Whether the outer world takes part in the repetition of an experi-
ence or not has no influence upon the possibility of the content of
consciousness which we call memory. From this it follows that this
content depends upon the inner experience alone, and that we
remember an external event only by means of its inner constituents.
The mere repetition of corresponding sense impressions is not suffi-
cient for this, for we can see the same person repeatedly without
recognizing him, if the inner accompanying phenomena were so
insignificant, as a result of lack of interest, that their repetition
does not produce the content of consciousness known as memory.
If we see him quite frequently, the frequent repetition of the exter-
nal impression finally causes the memory of the corresponding inner
experience. .

From this it results that for the " memory "-reaction a certain
intensity of the inner experience is necessary. This threshold can be
attained either at once or by continued repetition. The repetitions
are the more effective the more rapidly they follow each other.
From this we may conclude that the memory- value of an experience,
or its capacity for calling forth the " memory "-reaction by repetition,

ON THE THEORY OF SCIENCE 335

decreases with the lapse of time. Further, we must consider the
fact mentioned above, that an experience is never exactly repeated,
and that therefore the "memory ^'-reaction occurs even where there
is only resemblance or partial agreement in place of complete agree-
ment. Here, too, there are diiferent degrees; memory takes place
more easily the more perfectly the two experiences agree, and vice
versa.

If we look at these phenomena from the physiological side, we
may say we have two kinds of apparatus or organs, one of which
does not depend upon our will, whereas the other does. The former
are the sense organs, the latter constitutes the organ of thought.
Only the activities of the latter constitute our experiences or the
content of our consciousness.

The activities of the former may call forth the corresponding pro-
cesses of the latter, but this is not always necessary. Our sense organs
can be influenced without our "noticing'' it, that is, without the
thinking apparatus being involved. An especially important reaction
of the thinking apparatus is memory, that is, the consciousness that
an experience which we have just had possesses more or less agreement
with former experiences. With reference to the organ of thought,
it is the expression of the general physiological fact that every process
influences the organ in such a way that it has a different relation to
the repetition of this process, from the first time, and moreover that
the repetition is rendered easier. This influence decreases with time.

It is chiefly upon these phenomena that experience rests. Experi-
ence results from the fact that all events consist of a complete series of
simultaneous and successive components. When a connection between
some of those parts has become familiar to us by the repetition of
similar occurrences (for instance, the succession of day and night), we
do not feel such an occurrence as something completely new, but as
something partially familiar, and the single parts or phases of it do
not surprise us, but rather we anticipate their coming or expect
them. From expectation to prediction is only a short step, and so
experience enables us to prophesy the future from the past and pre-
sent.

Now this is also the road to science; for science is nothing but
systematized experience, that is, experience reduced to its simplest
and clearest forms. Its purposes to predict from a part of a phe-
nomenon which is known another part which is not yet known.
Here it may be a question of spatial as well as of temporal phenom-
ena. Thus the scientij&c zodlogist knows how to "determine," that
is, to tell, from the skull of an animal, the nature of the other parts
of the animal to which the skull belongs; likewise the astronomer
is able to indicate the future situation of a planet from a few obser-
vations of its present situation; and the more exact the first obser-

336 METHODOLOGY OF SCIENCE

vations were, the more distant the future for which he can predict.
All such scientific predictions are limited, therefore, with reference to
their number and their accuracy. If the skull shown to the zoologist
is that of a chicken, then he will probably be able to indicate the
general characteristics of chickens, and also perhaps whether the
chicken had a top-knot or not; but not its color, and only uncertainly
its age and its size. Both facts, the possibility of prediction and its
limitation in content and amount, are an expression for the two
fundamental facts, that among our experiences there is similarity,
but not complete agreement.

The foregoing considerations deserve to be discussed and extended
in several directions. First, the objection will be made that a chicken
or a planet is not an experience; we call them rather by the most
general name of thing. But our knowledge of the chicken begins
with the experiencing of certain visual impressions, to which are
added, perhaps, certain impressions of hearing and touch. The
sight impressions (to discuss these first) by no means completely
agree. We see the chicken large or small, according to the distance;
and according to its position and movement its outline is very differ-
ent. As we have seen, however, these differences are continually
grading into one other and do not reach beyond certain limits; we
neglect to observe them and rest contented with the fact that certain
other peculiarities (legs, wings, eyes, bill, comb, etc.) remain and do
not change. The constant properties we group together as a thing,
and the changing ones we call the states of this thing. Among. the
changing properties, we distinguish further those which depend
upon us (for example, the distance) and those upon which we have
no immediate influence (for instance, the position or motion): the
first is called the subjective changeable part of our experience, while
the second is called the objective mutability of the thing.

This omission of both the subjectively and objectively changeable
portion of the experience in connection with the retention of the
constant portion and the gathering together of the latter into a
unity is one of the most important operations which we perform
with our experiences. We call it the process of abstraction, and its
product, the permanent unity, we call a concept. Plainly this pro-
cedure contains arbitrary as well as necessary factors. Arbitrary or
accidental is the circumstance that quite different phases of a given
experience come to consciousness according to our attention, the
amount of practice we have had, indeed according to our whole
intellectual nature. We may overlook constant factors and attend
to changeable ones. The objective factors-, however, become neces-
sary as soon as we have noticed them; after we have seen that the
chicken is black, it is not in our power to see it red. Accordingly, in
general, our knowledge of that which agrees must be less than it

ON THE THEORY OF SCIENCE 337

actually could be, since we have not been able to observe ever}^
agreement, and our concept is always poorer in constituents at any
given time than it might be. To seek out such elements of concepts
as have been overlooked, and to prove that they are necessary factors
of the corresponding experiences, is one of the never-ending tasks
of science. The other case, namely, that elements have been received
in the concept which do not prove to be constant, also happens, and
leads to another task. One can then leave that element out of the
concept, if further experiences show that the other elements are
found in them, or one can form a new concept which contains the
former elements, leaving out those that have been recognized as
unessential. For a long time the white color belonged to the concept
SAvan. When the Dutch black swans became known, it was possible
either to drop the element white from the concept swan (as actually
happened), or to make a new concept for the bird which is similar
to the swan but black. Which choice is made in a given case is largely
arbitrary, and is determined by considerations of expediency.

Into the formation of concepts, therefore, two factors are operat-
ive, an objective empirical factor, and a subjective or purposive
factor. The fitness of a concept is seen in relation to its purpose,
which we shall now consider.

The purpose of a concept is its use for prediction. The old logic
set up the syllogism as the type of thought-activity, and its simplest
example is the well-known

All men are mortal,

Caius is a man, ; ;

Therefore Caius is mortal.

In general, the scheme runs s,,,-. '

To the concept M belongs the element B,
C belongs under the concept M,
Therefore the element B is found in C.

One can say that this method of reasoning is in regular use even:
to this day. It must be added, however, that this use is of a quite
different nature from that of the ancients. Whereas formerly the-
setting up of the first proposition or the major premise was con-
sidered the most important thing, and the establishment of the
second proposition or minor premise was thought to be a rather
trifling matter, now the relation is reversed. The major premise con-
tains the description of a concept, the minor makes the assertion
that a certain thing belongs under this concept. What right exists;
for such an assertion? The most palpable reply would be, since-
all the elements of the concept M (including B) are found in C, G
belongs under the concept M. Such a conclusion would indeed be
binding, but at the same time quite worthless, for it only repeats the

338 METHODOLOGY OF SCIENCE

minor premise. Actually the method of reasoning is essentially
different, for the minor premise is not obtained by showing that all
the elements of the concept M are found in C, but only some of them.
The conclusion is not necessary, but only probable, and the whole
process of reasoning runs : Certain elements are frequently found to-
gether, therefore they are united in the concept M. Certain of these
elements are recognized in the thing C, therefore probably the other
elements of the concept M will be found in C.

The old logic, also, was familiar with this kind of conclusion. It
was branded, however, as the worst of all, by the name of incomplete
induction, since the absolute certainty demanded of the syllogism
did not belong to its results. One must admit, however, that the whole
of modern science makes use of no other form of reasoning than
incomplete induction, for it alone admits of a prediction, that is, an
indication of relations which have not been immediately observed.

How does science get along with the defective certainty of this
process of reasoning? The answer is, that the probability of the
conclusion can run through all degrees from mere conjecture to the
maximum probability, which is practically indistinguishable from
certainty. The probability is the greater the more frequently an
incomplete induction of this kind has proven correct in later experi-
ence. Accordingly we have at our command a number of expressions
which in their simplest and most general form have the appearance :
If an element A is met within a thing, then the element B is also
found in it (in spatial or temporal relationship).

If the relation is temporal, this general statement is known by
some such name as the law of causality. If it is spatial, one talks of
the idea (in the Platonic sense) , or the type of the thing, of substance,
etc.

From the considerations here presented we get an easy answer
to many questions which are frequently discussed in very different
senses. First, the question concerning the general validity of the
law of causality. All attempts to prove such a validity have failed,
and there has remained only the indication that without this law
we should feel an unbearable uncertainty in reference to the world.
From this, however, we see very plainly that here it is merely a
question of expediency. From the continuous flux of our experiences
we hunt out those groups which can always be found again, in order
to be able to conclude that if the element A is given, the element B
will be present. We do not find this relationship as "given," but
we put it into our experiences, in that we consider the parts which
correspond to the relationship as belonging together.

The very same thing may be said of spatial complexes. Such factors
as are always, or at any rate often, found together are taken by us as
"belonging together," and out of them a concept is formed which

ON THE THEORY OF SCIENCE 339

embraces these factors. A question as to the why has here, as with
the temporal complexes, no definite meaning. There are countless
things that happen together once to which we pay no attention
because they happen only once or but seldom. The knowledge
of the fact that such a single concurrence exists amounts to nothing,
since from the presence of one factor it does not lead to a conclusion
as to the presence of another, and therefore does not make possible
prediction. Of all the possible, and even actual combinations, only
those interest us which are repeated, and this arbitrary but expedient
selection produces the impression that the world consists only of
combinations that can be repeated ; that, in other words, the law of
causality or of the type is a general one. However general or limited
application these laws have, is more a question of our skill in finding
the constant combinations among those that are present than a ques-
tion of objective natural fact.

Thus we see the development and pursuit of all sciences going on in
such a way that on the one hand more and more constant combina-
tions are discovered, and on the other hand more inclusive relations
of this kind are found out, by means of which elements are united
with each other which before no one had even tried to bring together.
So sciences are increasing both in the sense of an increasing complica-
tion and in an increasing unification.

If we consider from this standpoint the development and procedure
of the various sciences, we find a rational division of the sum total of
science in the question as to the scope and multiplicity of the com-
binations or groups treated of in them. These two properties are in
a certain sense antithetical. The simpler a complex is, that is, the
fewer elements brought together in it, the more frequently it is met
with, and vice versa. One can therefore arrange all the sciences in
such a way that one begins with the least multiplicity and the greatest
scope, and ends with the greatest multiplicity and the least scope.
The first science will be the most general, and will therefore contain
the most general and therefore the most barren concepts; the last
will contain the most specific and therefore the richest.

What are these limiting concepts? The most general is the concept
of thing, that is, any piece of experience, seized arbitrarily from the
flux of our experiences, which can be repeated. The most specific
and richest is the concept of human intercourse. Between the science
of things and the science of human intercourse, all the other sciences
are found arranged in regular gradation. If one follows out the
scheme the following outline results:

1. Theory of order. ~]

2. Theory of numbers, or arithmetic. ! ^.^ ,,

„ ^1 . ,. V Mathematics. .

6. iheory of time.

4, Theory of space, or geometry. J

340 METHODOLOGY OF SCIENCE

5. Mechanics. ^

6. Physics. V Energetics,

7. Chemistry, j

8. Physiology. ]

9. Psychology. V Biology.
10. Sociology. J

This table is arbitrary in so far as the grades assumed can be
increased or diminished according to need. For example, mechanics
and physics could be taken together; or between physics and chem-
istry, physical chemistry could be inserted. Likewise between
physiology and psychology, anthropology might find a place; or the
first five sciences might be united under mathematics. How one
makes these divisions is entirely a practical question, which will be
answered at any time in accordance with the purposes of division;
and dispute concerning the matter is almost useless.

I should like, however, to call attention to the three great groups
of mathematics, energetics, and biology (in the wider sense). They
represent the decisive regulative thought which humanity has
evolved, contributed up to this time, toward the scientific mastery of
its experiences. Arrangement is the fundamental thought of mathe-
matics. From mechanics to chemistry the concept of energy is the
most important; and for the last three sciences it is the concept of
life. Mathematics, energetics, and biology, therefore, embrace the
totality of the sciences.

Before we enter upon the closer consideration of these sciences, it
will be well to anticipate another objection which can be raised on the
basis of the following fact. Besides the sciences named (and those
which lie between them) there are many others, as geology, history,
medicine, philology, which we find difficulty in arranging in the above
scheme, which must, however, be taken into consideration in some
way or other. They are often characterized by the fact that they
stand in relation with several of the sciences named, but even more
by the following circumstance. Their task is not, as is true of the
pure sciences above named, the discovery of general relationships,
but they relate rather to existing complex objects whose origin,
scope, extent, etc., in short, whose temporal and spatial relationships
they have to discover or to "explain." For this purpose they make
use of relations which are placed at their disposal by the first-named
pure sciences. These sciences, therefore, had better be called applied
sciences. However, in this connection we should not think only or
even chiefly of technical applications; rather the expression is used
to indicate that the reciprocal relations of the parts of an object are
to be called to mind by the application of the general rules found in
pure science.

While in such a task the abstraction process of pure science is

ON THE THEORY OF SCIENCE 341

not applicable (for the omission of certain parts and the concentra-
tion upon others which is characteristic of these is excluded by the
nature of the task), yet in a given case usually the necessity of bringing
in various pure sciences for the purpose of explanation is evident.

Astronomy is one of these applied sciences. Primarily it rests upon
mechanics, and in its instrumental portion, upon optics; in its
present development on the spectroscopic side, however, it borrows
considerably of chemistry. In like manner history is applied sociology
and psychology. Medicine makes use of all the sciences before men-
tioned, up to psychology, etc.

It is important to get clearly in mind the nature of these sciences,
since, on account of their compound nature, they resist arrangement
amongst the pure sciences, while, on account of their practical
significance, they still demand consideration. The latter fact gives
them also a sort of arbitrary or accidental character, since their
development is largely conditioned by the special needs of the time.
Their number, speaking in general, is very large, since each pure
science may be turned into an applied science in various ways; and
since in addition we have combinations of two, three, or more sciences.
Moreover, the method of procedure in the applied sciences is funda-
mentally different from that in the pure sciences. In the first it is
a question of the greatest possible analysis of a single given complex
into its scientifically comprehensible parts; while pure science, on
the other hand, considers many complexes together in order to
separate out from them their common element, but expressly dis-
claims the complete analysis of a single complex.

In scientific work, as it appears in practice, pure and applied
science are by no means sharply separated. On the one hand the
auxiliaries of investigations, such as apparatus, books, etc., demand
of the pure investigator knowledge and application in applied science;
and, on the other hand, the applied scientist is frequently unable to
accomplish his task unless he himself becomes for the time being
a pure investigator and ascertains or discovers the missing general
relationships which he needs for his task. A separation and differentia-
tion of the two forms of science was necessary, however, since the
method and the aim of each present essential differences.

In order to consider the method of procedure of pure science more
carefully, let us turn back to the table on pages 339, 340, and attend to
the single sciences separately. The theory of arrangement was men-
tioned first, although this place is usually assigned to mathematics.
However, mathematics has to do with the concepts of number and
magnitude as fundamentals, while the theory of arrangement does
not make use of these. Here the fundamental concept is rather the
thing or object of which nothing more is demanded or considered
than that it is a fragment of our experience which can be isolated and

342 METHODOLOGY OF SCIENCE

will remain so. It must not be an arbitrary combination; such a
thing would have only momentary duration, and the task of science,
to learn the unknown from the given, could not find appUcation.
Rather must this element have such a nature that it can be charac-
terized and recognized again, that is, it must already have a concept-
ual nature. Therefore only parts of our experience which can be
repeated (which alone can be objects of science) can be characterized
as things or objects. But in saying this we have said all that was
demanded of them. In other respects they may be just as different
as is conceivable.

If the question is asked. What can be said scientifically about
indefinite things of this sort? it is especially the relations of arrange-
ment and association which yield an answer. If we call any definite
combination of such things a group, we can arrange such a group
in different ways, that is, we can determine for each thing the relation
in which it is to stand to the neighboring thing. From every such
arrangement result not only the relationships indicated, but a great
number of new ones, and it appears that when the first relationships
are given the others always follow in like manner. This, however,
is the type of the scientific proposition or natural law (page 335).
From the presence of certain relations of arrangement we can deduce
the presence of others which we have not yet demonstrated.

To illustrate this fact by an example, let us think of the things
arranged in a simple row, while we choose one thing as a first member
and associate another with it as following it; with the latter another
is associated, etc. Thereby the position of each thing in the row is
determined only in relation to the immediately preceding thing.
Nevertheless, the position of every member in the whole row, and
therefore its relation to every other member, is determined by this.
This is seen in a number of special laws. If w^e differentiate former
and latter members we can formulate the proposition, among others,
if B is a later member with reference to A, and C with reference to
B, then C is also a later member wdth reference to A.

The correctness and validity of this proposition seems to us beyond
all doubt. But this is only a result of the fact that we are able to
demonstrate it very easily in countless single cases, and have so
demonstrated it. We know only cases which correspond to the
proposition, and have never experienced a contradictory case. To call
such a proposition, however, a necessity of thinking, does not appear
to me correct. For the expression necessity of thinking can only rest
upon the fact that every time the proposition is thought, that is, every
time one remembers its demonstration, its confirmation always arises.
But every sort of false proposition is also thinkable. An undeniable
proof of this is the fact that so much which is false is actually thought.
But to base the proof for the correctness of a proposition upon the

ON THE THEORY OF SCIENCE 343

impossibility of thinking its opposite is an impossible undertaking,
because every sort of nonsense can be thought : where the proof was
thought to have been given, there has always been a confusion of
thought and intuition, proof or inspection.

With this one proposition of course the theory of order is not
exhausted, for here it is not a question of the development of this
theory, but of an example of the nature of the problems of science.
Of the further questions we shall briefly discuss the problem of
association.

If we have two groups A and B given, one can associate with every
member of A one of B; that is, we determine that certain operations
which can be carried on with the members of A are also to be carried
on with those of B. Now we can begin by simply carrying out the
association, member for member. Then we shall have one of three
results: A will be exhausted while there are still members of B left,
or B will be exhausted first, or finally A and B will be exhausted at
the same time. In the first case we call A poorer than B; in the second
B poorer than A; in the third both quantities are alike.

Here for the first time we come upon the scientific concept of
equality, which calls for discussion. There can be no question of a
complete identity of the two groups which have been denominated
equal, for we have made the assumption that the members of both
groups can be of any nature whatever. They can then be as different
as possible, considered singly, but they are alike as groups. However
I may arrange the members of A, I can make a similar arrangement
of the members of B, since every member of A has one of B associated
with it; and with reference to the property of arrangement there is no
difference to be observed between A and B. If, however, A is poorer
or richer than B, this possibility ceases, for then one of the groups
has members to which none of the members in the other group cor-
responds; so that the operations carried out with these members
cannot be carried out with those of the other group.

Equality in the scientific sense, therefore, means equivalence,
or the possibility of substitution in quite definite operations or for
quite definite relations. Beyond this the things which are called
like may show any differences whatever. The general scientific
process of abstraction is again easily seen in this special case.

On the basis of the definitions just given, we can establish further
propositions. If group A equals B, and B equals C, then A also
equals C. The proof of this is that we can relate every member
of A to a correspanding member of B and by hypothesis no
member will be left. Then C is arranged with reference to B, and
here also no member is left. By this process every member of A,
through the connecting link of a member of B, is associated with
a member of C; and this association i-s preserved even if we cut out

344 METHODOLOGY OF SCIENCE

the group B. Therefore A and C are equal. The same process of
reasoning can be carried out for any number of groups.

Likewise it can be demonstrated that if A is poorer than B and B
poorer than C, then A is also poorer than C. For in the association
of B with A some members of B are left over by hypothesis, and
likewise some members of C are left over if one associates C with B.
Therefore in the association of C with A, not only those members are
left over which could not be associated with B, but also those mem-
bers of C which extend beyond B. This proposition can be extended
to any number of groups, and permits the arrangement of a number
of different groups in a simple series by beginning with the poorest
and choosing each following so that it is richer than the preceding
but poorer than the following. From the proposition just established,
it follows that every group is so arranged with reference to all other
groups that it is richer than all the preceding and poorer than all the
following.^

In this derivation of scientific proposition or laws of the simplest
kinds, the process of derivation and the nature of the result becomes
particularly clear. We arrive at such a proposition by performing
an operation and expressing the result of it. This expression enables
us to avoid the repetition of the operation in the future, since in
accordance with the law we can indicate the result immediately.
Thus an abbreviation and therefore a facihtation of the problem is
attained which is the more considerable the larger the number of
operations saved.

If we have a number of equal groups, we know by the process of
association that all of the operations with reference to arrangement
which we can perform with one of them can be performed with all the
others. It is sufficient, therefore, to determine the properties of
arrangement of one of these groups in order to know forthwith the
properties of all the others. This is an extremely important pro-
position, which is continually employed for the most various purposes.
All speaking, writing, and reading rests upon the association of
thoughts with sounds and symbols, and by arranging the signs in
accordance with our thoughts we bring it to pass that our hearers
or readers think like thoughts in like order. In a similar fashion we
make use of various systems of formulae in the different sciences,
especially in the simpler sciences; and these formulae we correlate
with phenomena and use in place of the phenomena themselves,
and can therefore derive from them certain characteristics of phe-
nomena without being compelled to use the latter. The force of this
process appears very strikingly in astronomy where, by the use of
definite formulae associated with the different heavenly bodies, we

1 Equal groups cannot be distinguished here, and therefore represent only a
group.

ON THE THEORY OF SCIENCE 345

can foretell the future positions of these bodies with a high degree of
approximation.

From the theory of order we come to the theory of number or
arithmetic by the systematic arrangement or development of an
operation just indicated (page 343). We can arrange any number of
groups in such a way that a richer always follows a poorer. But the
complex obtained in this manner is always accidental with reference
to the number and the richness of its members. A regular and com-
plete structure of all possible groups is evidently obtained only if
we start from a group of one member or from a simple thing, and by
the addition of one member at a time make further groups out of
those that we have. Thus we obtain different groups arranged ac-
cording to an increasing richness, and since we have advanced one
member at a time, that is, made the smallest step which is possible,
w^e are certain that we have left out no possible group, which is poorer
than the richest to which the operation has been carried.

This whole process is familiar; it gives the series of the positive
whole numbers, that is, the cardinal numbers. It is to be noted that
the concept of quantity has not yet been considered; what we have
gained is the concept of number. The single things or members in
this number are quite arbitrary, and especially they do not need to
be alike in any manner. Every number forms a group-type, and
arithmetic or the science of numbers has the task of investigating
the properties of these different types with reference to their division
and combination. If this is done in general form, without attention
to the special amount of the number, the corresponding science is
called algebra. On the other hand, by the application of formal rules
of formation, the number system has had one extension after another
beyond the territory of its original validity. Thus counting back-
ward led to zero and to the negative numbers; the inversion of
involution to the imaginary numbers. For the group-type of the
positive whole numbers is the simplest but by no means the only
possible one, and for the purpose of representing other manifolds
than those which are met with in experience, these new types have
proved themselves very useful.

At the same time the number series gives us an extremely useful
type of arrangement. In the process of arising it is already ordered,
and w^e make use of it for the purpose of arranging other groups.
Thus, we are accustomed to furnish the pages in a book, the seats in a
theatre, and countless other groups which we wish to make use of in
any kind of order with the signs of the number series, and thereby
we make the tacit assumption that the use of that corresponding
group shall take place in the same order as the natural numbers
follow each other. The ordinal numbers arising therefrom do not
represent quantities, nor do they represent the only possible type

346 METHODOLOGY OF SCIENCE

of arrangement, but they are again the simplest of all. We come
to the concept of magnitude only in the theory of time and space.
The theory of time has not been developed as a special science; on
the contrary, what we have to say about time first appears in me-
chanics. Meantime we can present the fundamental concepts, which
arise in this connection, with reference to such well-known charac-
teristics of time that the lack of a special science of time is no dis-
advantage.

The first and most important characteristic of time (and of space,
too) is that it is a continuous manifold; that is, every portion of
time chosen can be divided at any place whatever. In the number
series this is not the case; it can be divided only between the single
numbers. The series one to ten has only nine places of division and
no more. A minute, or a second, on the other hand, has an unlimited
number of places of division. In other words, there is nothing in the
lapse of any time which hinders us from separating or distinguishing
in thought at any given instant the time which has elapsed till then
from the following time. It is just the same with space, except that
time is a simple manifold and space a threefold, continuous manifold.

Nevertheless, when we measure them, we are accustomed to indicate
times and spaces with numbers. If we first examine, for example, the
process of measuring a length, it consists in our applying to the dis-
tance to be measured a length conceived as unchangeable, the unit
of measure, until we have passed over the distance. The number of
these applications gives us the measure or magnitude of the distance.
The result is that by the indication of arbitrarily chosen points upon
the continuous distance, we place upon it an artificial discontinuity
which enables us to associate it with the discontinuous number series.

A still further assumption, however, belongs to the concept of
measuring, namely, that the parts of the distance cut off by the unit
used as a measure be equal, and it is taken for granted that this
requirement will be fulfilled to whatever place the unit of measure
is shifted. As may be seen, this is a definition of equality carried
further than the former, for one cannot actually replace a part of
the distance by another in order to convince one's self that it has
not changed. Just as little can one assert or prove that the unit of
measure in changing its place in space remains of the same length;
we can only say that such distances as are determined by the unit of
measure in different places are declared or defined as equal. Actually,
for our eye, the unit of measure becomes smaller in perspective the
farther away from it we find ourselves.

From this example we see again the great contribution which
arbitrariness or free choice has made to all our structure of science.
We could develop a geometry in which distances which seem sub-
jectively equal to our eye are called equal, and upon this assumption

ON THE THEORY OF SCIENCE 347

we would be able to develop a self-consistent system or science. Such
a geometry, however, would have an extremely complex and imprac-
tical structure for objective purposes (as, for example, land meas-
urement), and so we strive to develop a science as free as possible
from subjective factors. Historically, we have before us a process of
this sort in the astronomy of Ptolemy and that of Copernicus. The
former corresponded to the subjective appearances in the assumption
that all heavenly bodies revolved around the earth, but proved to be
very complicated when confronted with the task of mastering these
movements with figures. The latter gave up the subjective stand-
point of the observer, who looked upon himself as the centre, and
attained a tremendous simplification by placing the centre of revo-
lution in the sun.

A few words are to be said here about the application of arithmetic
and algebra to geometry. It is well known that under definite
assumptions (coordinates), geometrical figures can be represented
by means of algebraic formulse, so that the geometrical properties
of the figure can be deduced from the arithmetical properties of the
formulse, and vice versa. The question must be asked how such a
close and univocal relationship is possible between things of such
different nature. The answer is, that here is an especially clear case
of association. The manifold of numbers is much greater than that of
surface or space, for while the latter are determined by two or three in-
dependent measurements, one can have any number of independent
number series working together. Therefore the manifold of numbers
is arbitrarily limited to two or three independent series, and in so
far determines their mutual relations (by means of the laws of cosine)
that there results a manifold, corresponding to the spatial, which can
be completely associated with the spatial manifold. Then we have
two manifolds of the same manifold character, and all characteristics
of arrangement and size of the one find their likeness in the other.

This again characterizes an extremely important scientific pro-
cedure which consists, namely, in constructing a formal manifold for
the content of experience of a certain field, to which one attributes
the same manifold character which the former possesses. Every
science reaches by this means a sort of formal language of correspond-
ing completeness, which depends upon how accurately the manifold
character of the object is recognized and how judiciously the formulse
have been chosen. While in arithmetic and algebra this task has been
performed fairly well (though by no means absolutely perfectly), the
chemical formulse, for instance, express only a relatively small part
of the manifold to be represented; and in biology as far as sociology,
scarcely the first attempts have been made in the accomplishment of
this task.

Language especially serves as such a universal manifold to repre-

348 METHODOLOGY OF SCIENCE

sent the manifolds of experience. As a result of its development
from a time of less culture, it has by no means sufficient regularity
and completeness to accomplish its purpose adequately and con-
veniently. Rather, it is just as unsystematic as the events in the
lives of single peoples have been, and the necessity of expressing
the endlessly different particulars of daily life has only allowed it to
develop so that the correspondence between word and concept is
kept rather indefinite and changeable, according to need within
somewhat wide limits. Thus all work in those sciences which must
make vital use of these means, as especially psychology and sociology,
or philosophy in general, is made extremely difficult by the ceaseless
struggle with the indefiniteness and ambiguity of language. An
improvement of this condition can be effected only by introducing
signs in place of words for the representation of concepts, as the
progress of science allows it, and equipping these signs with the
manifold which from experience belongs to the concept.

An intermediate position in this respect is taken by the sciences
which were indicated above as parts of energetics. In this realm
there is added to the concepts order, number, size, space, and time,
a new concept, that of energy, which finds application to every
single phenomenon in this whole field, just as do those more general
concepts. This is due to the fact that a certain quantity, which
is known to us most familiarly as mechanical work, on account of
its qualitative transformability and quantitative constancy, can
be shown to be a constituent of every physical phenomenon, that
is, every phenomenon which belongs to the field of mechanics,
physics, and chemistry. In other words, one can perfectly character-
ize every physical event by indicating what amounts and kinds of
energy have been present in it and into what energies they have
been transformed. Accordingly, it is logical to designate the so-
called physical phenomena as energetical.

That such a conception is possible is now generally admitted.
On the other hand, its expediency is frequently questioned, and there
is at present so much the more reason for this because a thorough
presentation of the physical sciences in the energetical sense has not
yet been made. If one applies to this question the criterion of the
scientific system given above, the completeness of the correspondence
between the representing manifold and that to be represented, there
is no doubt that all previous systematizations in the form of hypo-
theses which have been tried in these sciences are defective in this
respect. Formerly, for the purpose of representing experiences,
manifolds whose character corresponded to the character of the
manifold to be represented only in certain salient points without
consideration of any rigid agreement, indeed, even without definite
question as to such an agreement, have been employed.

ON THE THEORY OF SCIENCE 349

The energetical conception admits of that definiteness of represen-
tation which the condition of science demands and renders possible.
For each special manifold character of the field a special kind of
energy presents itself: science has long distinguished mechanical,
electric, thermal, chemical, etc., energies. All of these different
kinds hold together by the law of transformation with the mainten-
ance of the quantitative amount, and in so far are united. On the
other hand, it has been possible to fix upon the corresponding ener-
getical expression for every empirically discovered manifold. As a
future system of united energetics, we have then a table of possible
manifolds of which energy is capable. In this we must keep in mind
the fact that, in accordance with the law of the conservation, energy
is a necessarily positive quantity which also is furnished with the
property of unlimited possibility of addition; therefore, every par-
ticular kind of energy must have this character.

The very small manifold which seems to lack this condition is
much widened by the fact that every kind of energy can be separ-
ated into two factors, which are only subject to the limitation that
their product, the energy, fulfills the conditions mentioned while
they themselves are much freer. For example, one factor of a kind of
energy can become negative as well as positive; it is only necessarj'-
that at the same time the other factor should become negative,
viz., positive.

Thus it seems possible to make a table of all possible forms of
energy, by attributing all thinkable manifold characteristics to the
factors of the energy and then combining them by pairs and cutting
out those products which do not fulfill the above-mentioned con-
ditions. For a number of years I have tried from time to time to
carry out this programme, but I have not yet got far enough to
justify publication of the results obtained.

If we turn to the biological sciences, in them the phenomenon of life
appears to us as new. If we stick to the observed facts, keeping our--
selves free from aU hypotheses, we observe as the general characteris-
tics of the phenomena of life the continuous stream of energy which
courses through a relatively constant structure. Change of substance
is only a part, although a very important part, of this stream. Espe-
cially in plants we can observe at first hand the great importance of
energy in its most incorporeal form, the sun's rays. Along with this,
self-preservation and development and reproduction, the begetting
of offspring of like nature, are characteristic. All of these properties
must be present in order that an organism may come into existence ;
they must also be present if the reflecting man is to be able by
repeated experience to form a concept of any definite organism,
whether of a lion or of a mushroom. Other organisms are met with
which do not fulfill these conditions; on account of their rarity, how-

350 METHODOLOGY OF SCIENCE

ever, they do not lead to a species concept, but are excluded from
scientific consideration (except for special purposes) as deformities or
monsters.

While organisms usually work with kinds of energy which we know
well from the inorganic world, organs are found in the higher forms
which without doubt cause or assist transfers of energy, but we
cannot yet say definitely what particular kind of energy is active in
them. These organs are called nerves, and their function is regularly
that, after certain forms of energy have acted upon one end of them,
they should act at the other end and release the energies stored up
there which then act in their special manner. That energetical
transformations also take place in the nerve during the process of
nervous transmission can be looked upon as demonstrated. We
shall thus be justified in speaking of a nerve energy, while leaving it
undecided whether there is here an energy of a particular kind, or
perhaps chemical energy, or finally a combination of several energies.

While these processes can be shown objectively by the stimulation
of the nerve and its corresponding releasing reaction in the end
apparatus (for instance, a muscle), we find in ourselves, connected
with certain nervous processes, a phenomenon of a new sort which
we call self-consciousness. From the agreement of our reactions
with those of other people we conclude with scientific probability
that they also have self-consciousness; and we are justified in making
the same conclusion with regard to some higher animals. How far
down something similar to this is present cannot be determined by
the means at hand, since the analogy of organization and of behavior
diminishes very quickly; but the line is probably not very long, in
view of the great leap from man to animal. Moreover, there are many
reasons for the view that the gray cortical substance in the brain,
with its characteristic pyramidal cell, is the anatomical substratum
of this kind of nervous activity.

The study of the processes of self-consciousness constitutes the chief
task of psychology. To this science belong those fields which are gener-
ally allotted to philosophy, especially logic and epistemology, while aes-
thetics, and still more ethics, are to be reckoned with the social sciences.

The latter have to do with living beings in so far as they can be
united in groups with common functions. Here in place of the indi-
vidual mind appears a collective mind, which owing to the adjust-
ment of the differences of the members of society shows simpler
conditions than that. From this comes especially the task of the
historical sciences. The happenings in the world accessible to us are
conditioned partly by physical, partly by psychological factors, and
both show a temporal' mutability in one direction. Thus arises on
the one hand a history of heaven and earth, on the other hand a
history of organisms up to man.

ON THE THEORY OF SCIENCE 351

All history has primarily the task of fixing past events through the
effects which have remained from them. Where such are not access-
ible, only analogy is left, a very doubtful means for gaining a concep-
tion of those events. But it must be kept in mind that an event which
has left no evident traces has no sort of interest for us, for our interest
is directly proportional to the amount of change which that event has
caused in what we have before us. The task of historical science is
just as little exhausted, however, with the fixing of former events
as, for instance, the task of physics with the establishment of a single
fact, as the temperature of a given place at a given time. Rather the
individual facts must serve to bring out the general characteristics of
the collective mind, and the much-discussed historical laws are laws
of collective psychology. Just as physical and chemical laws are
deduced in order with their help to predict the course of future phys-
ical events (to be called forth either experimentally or technically), so
should the historical laws contribute to the formation and control of
social and political development. We see that the great statesmen of
all time have eagerly studied history for this purpose, and from that
we derive the assurance that there are historical laws in spite of the
objections of numerous scholars.

After this brief survey, if we look back over the road we have come,
we observe the following general facts. In every case the development
of a science consists in the formation of concepts by certain abstrac-
tions from experience, and setting of these concepts in relation with
each other so that a systematical control of certain sides of our
experience is made possible. These relations, according to their gener-
ality and reliability, are called rules or laws. A law is the more
important the more it definitely expresses concerning the greatest
possible number of things, and the more accurately, therefore, it en-
ables us to predict the future. Every law rests upon an incomplete in-
duction, and is therefore subject to modification by experience. From
this there results a double process in the development of science.

First, the actual conditions are investigated to find out whether, be-
sides those already known, new rules or laws, that is, constant relations
between individual peculiarities, cannot be discovered between them.
This is the inductive process, and the induction is always an incom-
plete one on account of the limitlessness of all possible experience.

Immediately the relationship found inductively is applied to cases
which have not yet been investigated. Especially such cases are
investigated as result from a combination of several inductive laws.
If these are perfectly 'certain, and the combination is also properly
made, the result has claim to unconditional validity. This is the
lim-it which all sciences are striving to reach. It has almost been
reached in the simpler sciences : in mathematics and in certain parts
of mechanics. This is called the deductive process.

352 METHODOLOGY OF SCIENCE

In the actual working of every science the two methods of investiga-
tion are continually changing. The best means of finding new success-
ful inductions is in the making of a deduction on a very insufficient
basis, perhaps, and subsequently testing it in experience. Sometimes
the elements of his deductions do not come into the investigator's
consciousness; in such cases we speak of scientific instinct. On the
other hand we have much evidence from great mathematicians that
they were accustomed to find their general laws by the method of
induction, by trying and considering single cases; and that the
deductive derivation from other known laws is an independent
operation which sometimes does not succeed until much later. Indeed
there is to-day a number of mathematical propositions which have
not yet reached the second stage and therefore have at present a
purely inductive empirical character. The proportion of such laws in
science increases very quickly with the rise in the scale (page 339).

Another peculiarity which may be mentioned here is that in the
scale all previous sciences have the character of applied sciences
(page 341) with reference to those which follow, since they are every-
where necessary in the technique of the latter, yet do not serve to
increase their own field but are merely auxiliaries to the latter.

If we ask finally what influence upon the shaping of the future such
investigations as those which have been sketched in outline above
can have, the following can be said. Up till now it has been considered
a completely uncontrollable event whether and where a great and
influential man of science has developed. It is obvious that such a
man is among the most costly treasures which a people (and, indeed,
humanity) can possess. The conscious and regular breeding of such
rarities has not been considered possible. While this is still the case
for the very exceptional genius, we see in the countries of the older
civilization, especially in Germany at present, a system of education
in vogue in the universities by which a regular harvest of young
scientific men is gained who not only have a mastery of knowledge
handed down, but also of the technique of discovery. Thereby the
growth of science is made certain and regular, and its pursuit is
raised to a higher plane. These results were formerly attained chiefly
by empirically and oftentimes by accidental processes. It is a task of
scientific theory to make this activity also regular and systematic, so
that success is no more dependent solely upon a special capacity for
the founding of a "school" but can also be attained by less original
minds. By the mastery of methods the way to considerably higher
performances than he could otherwise attain will be open for the
exceptionally gifted.

THE CONTENT AND VALIDITY OF THE CAUSAL LAW

BY BENNO ERDMANN

(Translated from the German by Professor Walter T. Marvin, Western Reserve

University)

[Benno Erdmann, Professor of Philosophy, University of Bonn, since 1898.
b. October 5, 1851, Glogau in Schlesien, Germany. Ph.D.; Privy Councilor.
Academical Lecturer, Berlin, 1876- ; Special Professor, Kiel, 1878-79;
Regular Prof essor, i6iU 1879-84; ibid. Breslau, 1884-90; ibid. Halle, 1890-
98. Member various scientific and learned societies. Author of The Axioms
of Geometry; Kant's Criticism; Logic; Psychological Researches on Reading
(together with Prof. Ramon Dodge) ; The Psychology of the Child and the
School; Historical Researches on Kant's Prolegomena, and many other works
and papers in Philosophy.]

We have learned to regard the real, which we endeavor to appre-
hend scientifically in universally valid judgments, as a whole that is
connected continuously in time and in space and by causation, and
that is accordingly continuously self-evolving. This continuity of
connection has the following result, namely, every attempt to classify
•the sum total of the sciences on the basis of the difference of their
objects leads merely to representative types, that is, to species which
glide into one another. We find no gaps by means of which we can
separate sharply physics and chemistry, botany and zoology, political
and economic history and the histories of art and religion, or, again,
history, philology, and the study of the prehistoric.

As are the objects, so also are the methods of science. They are
separable one from another only through a division into represent-
ative types; for the variety of these methods is dependent upon the
variety of the objects of our knowledge, and is, at the same time,
determined by the difference between the manifold forms of our
thought, itself a part of the real, with its elements also gliding into
one another.^

The threads which join the general methodology of scientific
thought with neighboring fields of knowledge run in two main direc-
tions. In the one direction they make up a closely packed cable,
whereas in the other their course diverges into all the dimensions
of scientific thought. That is to say, first, methodology has its roots
in logic, in the narrower sense, namely, in the science of the element-
ary forms of our thought which enter into the make-up of all scien-
tific methods. .Secondly, methodology has its source in the methods
themselves which actually, and therefore technically, develop in the

^ Cf. the author's "Theorie der Typeneinteilungen," Phihsophische Monat-
shefte, vol. xxx, Berlin, 1894.

354 METHODOLOGY OF SCIENCE

various fields of our knowledge out of the problems peculiar to those
fields.

It is the office of scientific thought to interpret validly the objects
that are presented to us in outer and inner perception, and that
can be derived from both these sources. We accomplish this inter-
pretation entirely through judgments and combinations of judgments
of manifold sorts. The concepts, which the older logic regarded as
the true elementary forms of our thinking, are only certain selected
types of judgment, such stereotyped judgments as those which
make up definitions and classifications, and which appear independ-
ent and fundamental because their subject-matter, that is, their
intension or extension, is connected through the act of naming with
certain words. Scientific methods, then, are the ways and means
by which our thought can accomplish and set forth, in accordance
with its ideal, this universally valid interpretation.

There belongs, accordingly, to methodology a list of problems
which we can divide, to be sure only in abstracto, into three separ-
ate groups. First, methodology has to analyze the methods which
have been technically developed in the different fields of knowledge
into the elementary forms of our thinking from which they have
been built up. Next to this work of analyzing, there comes a second
task which may be called a normative one; for it follows that we'
must set forth and deduce systematically from their sources the
nature of these manifold elements, their resulting connection, and
their vahdity. To these two offices must be added a third that we
may call a potiori a synthetic one ; for finally we must reconstruct out
of the elements of our thinking, as revealed by analysis, the methods
belonging to the different fields of knowledge and also determine
their different scope and validity.

The beginning of another conception of the office of methodology
can be found in those thoughts which have become significant,
especially in Leibnitz's fragments and drafts of a calculus ratiocinator
or a specieuse generate. The foregoing discussion has set aside all
hope that these beginnings and their recent development may give,
of the possibility of constructing the manifold possible methods a
priori, that is, before or independent of experience. However, it
remains entirely undecided, as it should in this our preliminary
account of the office of general methodology, whether or not all
methods of our scientific thought will prove to be ultimately but
branches of one and the same universal method, a thought contained
in the undertakings just referred to. Although modern empiricism,
affiliated as it is with natural science, tends to answer this question
in the affirmative even more definitely and dogmatically than any
type of the older rationalism, still the question is one that can be
decided only in the course of methodological research.

CONTENT AND VALIDITY OF THE CAUSAL LAW 355

The conception of a methodology of scientific thought can be
said to be almost as old as scientific thought itself; for it is already
contained essentially, though undifferentiated, in the Socratic
challenge of knowledge. None the less, the history of methodology,
as the history of every other science, went through the course of
which Kant has given a classical description. " No one attempts
to construct a science unless he can base it on some idea; but in the
elaboration of it the schema, nay, even the definition which he gives
in the beginning of his science, corresponds very seldom to his idea,
which, like a germ, lies hidden in the reason, and all the parts of
which are still enveloped and hardly distinguishable even under
microscopical observation." ^

We are indebted to the Greek, and especially to the Platonic-
Aristotelian philosophy for important contributions to the under-
standing of the deductive method of mathematical thought. It
was precisely this trend of philosophic endeavor which, though
furnishing for the most part the foundation of methodological
doctrine well on into the seventeenth century, offered no means
of differentiating the methods that are authoritative for our know-
ledge of facts. What Socrates was perhaps the first to call "induc-
tion," is essentially different, as regards its source and aim, from the
inductive methods that direct our research in natural and mental
science. For it is into these two fields that we have to divide the
totality of the sciences of facts, the material sciences, let us call
them, in opposition to the formal or mathematical sciences, - — that
is, if we are to do justice to the difference between sense and self-
perception, or "outer" and "inner" perception.

Two closely connected forces especially led astray the methodo-
logical opinions regarding the material sciences till the end of the
eighteenth century, and in part until the beginning of the nineteenth
century. We refer, in the first place, to that direction of thought
which gives us the right to characterize the Platonic-Aristotelian
philosophy as a '' concept philosophy;" namely, the circumstance
that Aristotelian logic caused the "concept" to be set before the
"judgment." In short, we refer to that tendency in thought which
directs the attention not to the permanent in the world's occurrences,
the uniform connections of events, but rather to the seemingly per-
manent in the things, their essential attributes or essences. Thus
the concept philosophy, as a result of its tendency to hypostasize,
finds in the abstract general concepts of things, the ideas, the eternal
absolute reality that constitutes the foundation of things and is
contained in them beside the accidental and changing properties.^

1 Kant, Kr. d. r. Y ., 2d ed., p. 862.

^ According to Plato, it is true, the ideas are separated from the sensible things;
they must be thought in a conceptual place, for the space of sense-perception is to

356 METHODOLOGY OF SCIENCE

Here we have at once the second force which inspired the ancient
methodology. These ideas, Hke the fundamentally real, constitute
that which ultimately alone acts in all the coming into existence
and the going out of existence of the manifold things. In the Aris-
totelian theory of causation, this thought is made a principle; and
we formulate only what is contained in it, when we say that, accord-
ing to it, the efficient and at the same time final causes can be
deduced through mere analysis from the essential content of the
effects; that, in fact, the possible effects of every cause can be de-
duced from the content of its definition. The conceptual determina-
tion of the causal relation, and with it in principle the sum total of
the methods in the material sciences, becomes a logical, analytical,
and deductive one. These sciences remain entirely independent of
the particular content of experience as this broadens, and so do also
the methods under discussion.

As a consequence, every essential difference between mathemat-
ical thought and the science of causes is done away with in favor
of a rationalistic construction of the methods of material science.
Accordingly, throughout the seventeenth century, the ideal of all
scientific method becomes, not the inductive method that founded
the new epoch of the science of to-day, but the deductive math-
ematical method applied to natural scientific research. The flourish
of trumpets with which Francis Bacon hailed the onslaught of the
inductive methods in the natural science of the time, helped in no
way; for he failed to remodel the traditional, Aristotelian-Scholastic
conception of cause, and, accordingly, failed to understand both
the problem of induction and the meaning of the inductive methods
of the da3^^ Descartes, Hobbes, Spinoza, and related thinkers
develop their mathesis universalis after the pattern of geometrical
thinking. Leibnitz tries to adapt his specieuse generate to the thought
of mathematical analysis. The old methodological conviction gains
its clear-cut expression in Spinoza's doctrine: " Aliquid efficitur ah
aliqua re" means " aliquid sequitur ex ejus definitione."

The logically straight path is seldom the one taken in the course
of the history of thought. The new formulation and solution of
problems influence us first through their evident significance and
consequences, not through the traditional presuppositions upon
which they are founded. Thus, in the middle of the seventeenth
century, when insight into the precise difference between mental
and physical events gave rise to pressing need for its definite formu-
lation, no question arose concerning the dogmatic presupposition

be understood as non-being, matter. The things revealed to sense, however,
occupy a middle position between being and non-being, so that they partake of
the idpas. In this sense, the statement made above holds also of the older view
of the concept philosophy.

'■ Cf. the articles on Francis Bacon by Chr. Sigwart in the Preussische Jahr-
hiXcher, xii, 1863, and xiii, 1864.

CONTENT AND VALIDITY OF THE CAUSAL LAW 357

of a purely logical (analytisch) relationship between cause and effect;
but, on the contrary, this presupposition was then for the first time
brought clearly before consciousness. It was necessary to take the
roundabout way through occasionalism and the preestablished
harmony, including the latter 's retreat to the omnipotence of God,
before it was possible to raise the question of the validity of the
presupposition that the connection between cause and effect is
analytic and rational.

Among the leading thinkers of the period this problem was re-
cognized as the cardinal problem of contemporaneous philosophy. It
is further evidence how thoroughly established this problem must
have been among the more deeply conceived problems of the time
in the middle of the eighteenth century, that Hume and Kant were
forced to face it, led on, seemingly independently of each other,
and surely from quite different presuppositions and along entirely
different ways. The historical evolution of that which from the
beginning has seemed to philosophy the solving of her true problem
has come to pass in a way not essentially different from that of the
historical evolution in all other departments of human knowledge.
Thus, in the last third of the seventeenth century, Newton and
Leibnitz succeeded in setting forth the elements of the infinitesimal
calculus; and, in the fifth decade of the nineteenth century, Robert
Mayer, Helmholtz, and perhaps Joule, formulated the law of the
conservation of energy. In one essential respect Hume and Kant
are agreed in the solution of the new, and hence contemporaneously
misunderstood, problem. Both realized that the connection be-
tween the various causes and effects is not a rational analytic, but
an empirical synthetic one. However, the difference in their presup-
positions as well as method caused this common result to make its
appearance in very different light and surroundings. In Hume's
empiricism the connection between cause and effect appears as the
mere empirical result of association; whereas in Kant's rationalism
this general relation between cause and effect becomes the funda-
mental condition of all possible experience, and is, as a conse-
quence, independent of all experience. It rests, as a means of
connecting our ideas, upon an inborn uniformity of our thought.

Thus the way was opened for a fundamental separation of the
inductive material scientific from the deductive mathematical
method. For Hume mathematics becomes the science of the rela-
tions of ideas, as opposed to the sciences of facts. For Kant philo-
sophical knowledge is the knowledge of the reason arising from
concepts, whereas the mathematical is that arising from the con-
struction of concepts. The former, therefore, studies the particular
only in the universal; the latter, the universal in the particular,
nay, rather in the individual.

358 METHODOLOGY OF SCIENCE

Both solutions of the new problem which in the eighteenth cen-
tury supplant the old and seemingly self-evident presupposition,
appear accordingly embedded in the opposition between the ration-
alistic and empiristic interpretation of the origin and validity of our
knowledge, the same opposition that from antiquity runs through
the historical development of philosophy in ever new digressions.

Even to-day the question regarding the meaning and the validity
of the causal connection stands between these contrary directions
of epistemological research; and the ways leading to its answer
separate more sharply than ever before. It is therefore more press-
ing in our day than it was in earlier times to find a basis upon which
we may build further epistemologicallj'^ and therefore methodologic-
ally. The purpose of the present paper is to seek such a basis for the
different methods employed in the sciences of facts.

As has already been said, the contents of our consciousness, which
are given us immediately in outer and inner perception, constitute
the raw material of the sciences of facts. From these various facts
of perception we derive the judgments through which we predict,
guide, and shape our future perception in the course of possible
experience. These judgments exist in the form of reproductive
ideational processes, which, if logically explicit, become inductive
inferences in the broader sense. These inferences may be said to be
of two sorts, though fundamentally only two sides of one and the
same process of thought; they are in part analogical inferences and
in part inductive inferences in the narrower sense. The former infers
from the particular in a present perception, which in previous per-
ceptions was uniformly connected with other particular contents of
perception, to a particular that resembles those other contents of per-
ception. In short, they are inferences from a particular to a particular.
After the manner of such inferences we logically formulate, for
example, the reproductive processes, whose conclusions run: "This
man whom I see before me, is attentive, feels pain, will die;" "this
meteor will prove to have a chemical composition similar to known
meteors, and also to have corresponding changes on its surface as
the result of its rapid passage through our atmosphere." The induct-
ive inferences in the narrower sense argue, on the contrary, from
the perceptions of a series of uniform phenomena to a universal,
which includes the given and likewise all possible cases, in which
a member of the particular content of the earlier perceptions is
presupposed as given. In short, they are conclusions from a partic-
ular to a universal that is more extensive than the sum of the given
particulars. For example: "All men have minds, will die;" "all
meteoric stones will prove to have this chemical composition and
those changes of surface."

CONTENT AND VALIDITY OF THE CAUSAL LAW 359

There is no controversy regarding the inner similarity of both
these types of inference or regarding their outward structure; or,
again, regarding their outward difference from the deductive in-
ferences, which proceed not from a particular to a particular or
general, but from a general to a particular.

There is, however, difference of opinion regarding their inner
structure and their inner relation to the deductive inferences. Both
questions depend upon the decision regarding the meaning and
validity of the causal relation. The contending parties are recruited
essentially from the positions of traditional empiricism and ration-
alism and from their modern offshoots.

We maintain first of all:

L The presupposition of all inductive inferences, from now on
to be taken in their more general sense, is, that the contents of
perception are given to us uniformly in repeated perceptions, that is,
in uniform components and uniform relations.

2. The condition of the validity of the inductive inferences lies
in the thoughts that the same causes will be present in the unobserved
realities as in the observed ones, and that these same causes ivill bring
forth the same effects.

3. The conclusions of all inductive inferences have, logically
speaking, purely problematic validity, that is, their contradictory
opposite remains equally thinkable. They are, accurately expressed,
merely hypotheses , whose validity needs verification through future
experience.

The first-mentioned presupposition of inductive inference must not
be misunderstood. The paradox that nothing really repeats itself,
that each stage in nature's process comes but once, is just as much
and just as little justified as the assertion, everything has already
existed. It does not deny the fact that we can discriminate in the
contents of our perceptions the uniformities of their components
and relations, in short, that similar elements are present in these
ever new complexes. This fact makes it possible that our manifold
perceptions combine to make up one continuous experience. Even
our paradox presupposes that the different contents of our percep-
tions are comparable with one another, and reveal accordingly
some sort of common nature. All this is not only a matter of course
for empiricism, which founds the whole constitution of our know-
ledge upon habits, but must also be granted by every rationalistic
interpretation of the structure of knowledge. Every one that is
well informed know^ that what we ordinarily refer to as facts already
includes a theory regarding them. Kant judges in this matter pre-
cisely as Hume did before him and Stuart Mill after him. " If cin-
nabar were sometimes red and sometimes black, sometimes light and
sometimes heavy, if a man could be changed now into this, now into

360 METHODOLOGY OF SCIENCE

another animal shape, if on the longest day the fields were some-
times covered with fruit, sometimes with ice and snow, the faculty
of my empirical imagination would never be in a position, when
representing red color, to think of heavy cinnabar." ^

The assumption that in recurring perceptions similar elements
of content, as well as of relation, are given, is a necessary condition
of the possibility of experience itself, and accordingly of all those
processes of thought which lead us, under the guidance of previous
perceptions, from the contents of one given perception to the con-
tents of possible perceptions.

A tradition from Hume down has accustomed us to associate the
relation of cause and effect not so much with the uniformity of co-
existence as with the uniformity of sequence. Let us for the present
keep to this tradition. Its first corollary is that the relation of cause
and effect is to be sought in the uninterrupted flow and connection
of events and changes. The cause becomes the uniformly preceding
event, the constant antecedens, the effect the uniformlj^ following, the
constant consequens , in the course of the changes that are presented
to consciousness as a result of foregoing changes in our sensorium.

According to this tradition that we have taken as our point of
departure, the uniformity of the sequence of events is a necessary
presupposition of the relation between cause and effect. This uni-
formity is given us as an element of our experience; for we actually
find uniform successions in the course of the changing contents of
perception. Further, as all our perceptions are in the first instance
sense-perceptions, we may call them the sensory presupposition of
the possibility of the causal relation.

In this presupposition, however, there is much more involved than
the name just chosen would indicate. The uniformity of sequence
lies, as we saw, not in the contents of perception as such, which are
immediately given to us. It arises rather through the fact that, in
the course of repeated perceptions, we apprehend through abstraction
.the uniformities of their temporal relation. Moreover, there lie in the
repeated perceptions not only uniformities of sequence, but also
uniformities of the qualitative content of the successive events
themselves, and these uniformities also must be apprehended through
abstraction. Thus these uniform contents of perception make up
series of the following form:

ai -> 61
tto — > bo

1 Kant, Kr. d. r. V., 1st ed., pp. 100 f.

CONTENT AND VALIDITY OF THE CAUSAL LAW 361

The presupposition of the possibiUty of the causal relations in-
cludes, therefore, more than mere perceptive elements. It involves
the relation of different, if you will, of peculiar contents of percep-
tion, by virtue of which we recognize ac^—^h^ . . . a,j — > h^ as events
that resemble one another and the event a^ -^ h^ qualitatively as well
as in their sequence. There are accordingly involved in our presup-
position reproductive elements which indicate the action of memory.
In order that I may in the act of perceiving a^ —^ 63 apprehend the
uniformity of this present content with that of Qo -^ &2 ^^^ a^ ^ 6j,
these earlier perceptions must in some way, perhaps through mem-
ory,^ be revived with the present perception.

In this reproduction there is still a further element, which can
be separated, to be sure only in abstracto, from the one just pointed
out. The present revived content, even if it is given in memory as
an independent mental state, is essentially different from the original
perception. It differs in all the modifications in which the memory
of lightning and thunder could differ from the perception of their
successive occurrence, or, again, the memory of a pain and the re-
sulting disturbance of attention could differ from the corresponding
original experience. However, as memory, the revived experience
presents itself as a picture of that which has been previously per-
ceived. Especially is this the case in memory properly so called,
where the peculiar space and time relations individualize the revived
experience. If we give to this identifying element in the associative
process a logical expression, we shall have to say that there is in-
volved in revival, and especially in memory, an awareness that the
present ideas recall the same content that was previously given us
in perception. To be sure, the revival of the content of previous
perceptions does not have to produce ideas, let alone memories.
Rapid, transitory, or habitual revivals, stimulated- by associative
processes, can remain unconscious, that is, they need not appear as
ideas or states of consciousness. Stimulation takes place, but con-
sciousness does not arise, provided we mean by the term " conscious-
ness" the genus of our thoughts, feelings, and volitions. None the
less it must not be forgotten that this awareness of the essential
identity of the present revived content with that of the previous
perception can be brought about in every such case of reproduction.
How all this takes place is not our present problem.

We can apply to this second element in the reproductive process,
which we have found to be essential to the causal relation, a Kantian

' It is not our present concern to ascertain how this actually happens. The
psychological presuppositions of the present paper are contained in the theory of
reproduction that I have worked out in connection with the psychology of speech
in the articles on "Die psychologischen Grundlagen der Beziehungen zwischen
Sprechen und Denken," Archiv fiir systematische Philosophie, 11, iii, und vii;
cf. note 1, page 151.

362 METHODOLOGY OF SCIENCE

term, "Recognition." This term, however, is to be taken only in the
sense called for by the foregoing statements; for the rationalistic
presuppositions and consequences which mark Kant's "Synthesis
of Recognition " are far removed from the present line of thought.

We may, then, sum up our results as follows: In the presuppo-
sition of a uniform sequence of events, which we have accepted
from tradition as the necessary condition of the possibility of the
causal relation, there lies the thought that the contents of perception
given us through repeated sense stimulation are related to one
another through a reproductive recognition.

The assumption of such reproductive recognition is not justified
merely in the cases so far considered. It is already necessary in the
course of the individual perceptions a and b, and hence in the appre-
hension of an occurrence. It makes the sequence itself in which a
and b are joined possible; for in order to apprehend b as following
upon a, in case the perception of a has not persisted in its original
form, a must be as far revived and recognized upon b's entrance into
the field of perception as it has itself passed out of that field. Other-
wise, instead of b following upon a and being related to a, there
would be only the relationless change from a to b. This holds gen-
erally and not merely in the cases where the perception of a has
disappeared before that of b begins, for example, in the case of light-
ning and thunder, or where it has in part disappeared, for example,
in the throwing of a stone.

We have represented a as an event or change, in order that uniform
sequences of events may alone come into consideration as the pre-
supposition of the causal relation. But every event has its course in
time, and is accordingly divisible into many, ultimately into infimtely
many, shorter events. Now if b comes only an infinitely short interval
later than a, and by hypothesis it must come later than a, then a
corresponding part of a must have disappeared by the time b appears.
But the infinitesimal part of a perception is just as much out of all
consideration as would be an infinitely long perception; all which only
goes to show that we have to substitute intervals of finite length in
place of this purely conceptual analysis of a continuous time inter-
val. This leaves the foregoing discussion as it stands. If b follows a
after a perceptible finite interval, then the flow or development
of a by the time of 6's appearance must have covered a course cor-
responding to that interval; and all this is true even though the
earlier stages of a remain unchanged throughout the interval pre-
ceding ?>'s appearance. The present instant of flow is distinct from
the one that has passed, even though it takes place in precisely the
same way. The former, not the latter, gives the basis of relation which
is here required, and therefore the former must be reproduced and
recognized. This thought also is included in the foregoing summary

CONTENT AND VALIDITY OF THE CAUSAL LAW 363

of what critical analysis shows to be involved in the presupposition
of a uniform sequence.

In all this we have already abandoned the field of mere perception
which gave us the point of departure for our analysis of uniform
sequence. We may call the changing course of perception only in the
narrower meaning the sensory presupposition of the causal relation.
In order that these changing contents of perception may be known
as like one another, as following one another, and as following one
another uniformly, they must be related to one another through a
recognitive reproduction.

Our critical analysis of uniform sequence is, however, not yet
complete. To relate to one another the contents of two ideas always
requires a process at once of identifying and of differentiating, which
makes these contents members of the relation, and which accordingly
presupposes that our attention has been directed to each of the two
members as weU as to the relation itself — in the present case, to the
sequence. Here we come to another essential point. We should apply
the name "thought" to every ideational process in which attention
is directed to the elements of the mental content and which leads us
to identify with one another, or to differentiate from one another, the
members of this content.^ The act of relating, which knows two
events as similar, as following one another, indeed, as following one
another uniformly, is therefore so far from being a sensation that it
must be claimed to be an act of thinking. The uniformity of sequence
of a and b is therefore an act of relating on the part of our thought,
so far as this becomes possible solely through the fact that we at one
and the same time identify with one another and differentiate from
one another a as cause and b as effect. We say " at one and the same
time," because the terms identifying and differentiating are corre-
latives which denote two different and opposing sides of one and the
same ideational process viewed logically. Accordingly, there is here
no need of emphasizing that the act of relating, which enables us to
think a as cause and b as effect, is an act of thought also, because it
presupposes on our part an act of naming which raises it to being
a component of our formulated and discursive thought. We therefore
think a as cause and b as effect in that we apprehend the former as
uniform antecedens and the latter as uniform consequens.

Have we not the right, after the foregoing analysis, to interpret
the uniform sequence of events solely as the necessary presupposi-
tion of the causal relation? Is it not at the same time the adequate
presupposition? Yes, is it not the causal relation itself? As we
know, empiricism since Hume has answered the last question in the

^ Cf. the author's "Umrisse zur Psychologie des Denkens," in Philosophische
Abhandlungen Chr. Sigwart . . . gewidmet, Tiibingen, 1900.

364 METHODOLOGY OF SCIENCE

affirmative, and rationalism since Kant has answered it in the nega-
tive.

We, too, have seemingly followed in our discussion the course of
empiricism. At least, I find nothing in that discussion which a con-
sistent empiricist might not be willing to concede; that is, if he is
ready to set aside the psychological investigation of the actual pro-
cesses which we here presuppose and make room for a critical anal3^sis
of the content of the relation of cause and effect.^ However, the

^ The difference between the two pomts of view can be made clearer by an illus-
tration. The case that we shall analyze is the dread of coming into contact Tvnth
fire. The psychological analysis of this case has to make clear the mental content
of the dread and its causes. Such dread becomes possible only when we are aware
of the burning that results from contact with fire. We could have learned to be
aware of this either immediately through our own experience, or mediately
through the communication of others' experience. In both cases it is a matter of
one or repeated experiences. In all cases the effects of earlier experiences equal
association and recall, which, in turn, result in recognition. The recognition
explaining the case imder discussion arises thus. The present stimuli of visual
perception arouse the retained impressions of previous visual perceptions of fire
and give rise to the present perception (apperception) by fusing with them. By
a process of interweaving, associations are joined to this perception. The apper-
ceptively revived elements which lie at the basis of the content of the perception
are interwoven by association with memorj^ elements that retain the additional
contents of previous perceptions of fire, viz., the burning, or, again, are interwoven
with the memory elements of the communications regarding such burning. By
means of this interweaving, the stimulation of the apperceptive element transmits
itself to the remaining elements of the association complex. The character of the
association is different under different conditions. If it be founded only upon one
experience, then there can arise a memory or a recall, in the wider sense, of the
foregoing content of the perception and feeling at the tune of the burning, or,
again, there can arise a revival wherein the stimulated elements of retention remain
unconscious. Again, the words of the mother tongue that denote the previous
mental content, and which hkewise belong to the association complex (the apper-
ceiving mass, in the wider sense), can be excited in one of these three forms and
in addition as abstract verbal ideas. Each one of these forms of verbal discharge
can lead to the innervations of the muscles involved in speech, which bring about
some sort of oral expression of judgment. Each of these verbal reproductions can
be connected with each of the foregoing sensory {sachlichen) revivals. Secondly,
if the association be founded upon repeated perceptions on the part of the person
himself, then all the afore-mentioned possibilities of reproduction become more
complicated, and, in addition, the mental revivals contain, more or less, only the
common elements of the previous perceptions, i. e., reappear in the form of
abstract ideas or their corresponding unconscious modifications. In the third
case the association is founded upon a communication of others' experience. For
the sake of simplicity, let this case be confined to the following instance. The
communication consisted in the assertion: "AH fire will burn upon contact."
Moreover, this judgment was expressed upon occasion of imminent danger of
burning. There can then arise, as is perhaps evident, all the possibilities men-
tioned in the second case, only that here there will be a stronger tendency toward
verbal reproduction and the sensory reproduction will be less fixed.

In the first two cases there was connected with the perception of the burning
an intense feeling of pain. In the third the idea of such pain added itself to the
visual perception of the moment. The associated elements of the earlier mental
contents belong likewise to the apperceiving mass excited at the moment, in fact
to that part of it excited by means of association processes, or, as we can again say,
depending upon the point from which we take our view, the associative or apper-
ceptive completion of the content of present perception. If these pain elements
are revived as memories, i. e., as elements in consciousness, they give rise to a new
disagreeable feeling, which is referred to the possible coming sensation of burning.
If the mental modifications corresponding to these pain elements remain uncon-
scious, as is often possible, there arises none the less the same result as regards our
feeling, only with less intensity. This feeling tone we call the dread.

CONTENT AND VALIDITY OF THE CAUSAL LAW 365

decision of the question, whether or not empiricism can determine
exhaustively the content that we think in the causal relation, depends
upon other considerations than those which we have until now been
called upon to undertake. We have so far only made clear what
every critical analysis of the causal relation has to concede to empiri-
cism. In reality the empiristic hypothesis is inadequate. To be sure,

As a result of the sum total of the revivals actual and possible, there is finally
produced, according to the particular circumstances, either a motor reaction or an
inhibitant of such reaction. Both innervations can take place involimtarily or
voluntarily.

The critical analysis of the fact that we dread contact with fire, even has another
purpose and accordingly proceeds on other lines. It must make clear under what
presuppositions the foresight that Ues at the basis of such dread is valid for future
experience. It must then formulate the actual process of revival that constitutes
the foundation of this feeling as a series of judgments, from which the meaning and
interconnection of the several judgments will become clear. Thus the critical
analysis must give a logical presentation of the apperceptive and associative
processes of revival.

For this purpose the three cases of the psychological analysis reduce themselves
to two: viz., first, to the case in which an immediate experience forms the basis,
and secondly, to that in which a variety of similar mediately or immediately
communicated experiences form such basis.

In the first of these logically differentiated cases, the transformation into the
speech of formulated thought leads to the following inference from analogy :

Fire A burned.

Fire B is similar to fire A.

Fire B -nail burn.

In the second case there arises a syllogism of some such form as:

All fire causes burning upon contact.
This present phenomenon is fire.

This present phenomenon wiU cause burning upon contact.

Both premises of this syllogism are inductive inferences, whose implicit meaning
becomes clear when we formulate as follows :

All heretofore investigated instances of fire have burned, therefore all fire

bums.
The present phenomenon manifests some properties of fire, wiU consequently

have all the properties thereof.

The present phenomenon will, in case of contact, cause burning.

The first syllogism goes from the particular to the particular. The second proves
itself to be (contrary to the analysis of Stuart Mill) an inference that leads from
the general to the particular. For the conclusion is the particular of the second
parts of the major and minor premises; and these second parts of the premises are
inferred from their first parts in the two possible ways of inductive inference. The
latter do not contain the case referred to in the conclusion, but set forth the con-
ditions of carrying a result of previous experience over to a new case with inductive
probability, in other words, the conditions of making past experience a means of
foreseeing future experience. It would be superfluous to give here the symbols
of the two forms of inductive inference.

We remain within the bounds of logical analysis, if we state under what condi-
tions conclusions follow necessarily from their premises, viz., the conclusions
of arguments from analogy and of syllogisms in the narrower sense, as well as
those of the foregoing inductive arguments. For the inference from analogy and
the two forms of inductive inference, these conditions are the presuppositions
already set forth in the text of the present paper, that in the as yet unobserved
portion of reality the like causes will be found and they will give rise to like effects.
For the syllogism they are the thought that the predicate of a predicate is the
(mediate) predicate of the subject. Only the further analysis of these presupposi-
tions, which is undertaken in the text, leads to critical considerations in the
narrower sense.

366 METHODOLOGY OF SCIENCE

the proof of this inadequacy is not to be taken from the obvious
argument which Reid raised against the empiricism of Hume, and
which compelled Stuart Mill in his criticism of that attack ^ to abandon
his empiristic position at this point. No doubt the conclusion to which
we also have come for the time being, goes much too far, the conclu-
sion that the cause is nothing but the uniform antecedens and the
effect merely the uniform consequens. Were it true, as we have
hitherto assumed, that every uniformly preceding event is to be
regarded as cause and every uniformly following event as effect, then
day must be looked upon as cause of night and night as cause of day.

Empiricism can, however, meet this objection without giving up
its position; in fact, it can employ the objection as an argument in its
favor; for this objection affects only the manifestly imperfect formu-
lation of the doctrine, not the essential arguments.

It should have been pointed out again and again in the foregoing
exposition that only in the first indiscriminating view of things maj^
we regard the events given us in perception as the basis of our concepts
of cause and effect. All these events are intricately mixed, those that
are given in self perception as well as those given in sense perception.
The events of both groups flow along continuously. Consequently,
as regards time, they permit a division into parts, which division
proceeds, not indeed for our perception, but for our scientific thought,
in short, conceptually, into infinity. The events of sense perception
permit also conceptually of infinite division in their spatial relations.

It is sufficient for our present purpose, if we turn our attention to
the question of divisibility in time. This fact of divisibility shows
that the events of our perception, which alone we have until now
brought under consideration, must be regarded as systems of events.
We are therefore called upon to apportion the causal relations among
the members of these systems. Only for the indiscriminating view
of our practical Weltanschauung is the perceived event a the cause
of the perceived event h. The more exact analysis of our theoretical
apprehension of the world compels us to dissect the events a and h
into the parts a^, a^, a , — h^, b^, h , and, where occasion calls for it, to
continue the same process in turn for these and further components.
We have accordingly to relate those parts to one another as causes
and effects which, from the present standpoint of analysis, follow one
another uniformly and immediately, viz., follow one another so that
from this standpoint no other intervening event must be presupposed.
In this way we come to have a well-ordered experience. The disposi-
tions to such experience which reveal themselves within the field of
practical thought taught man long before the beginning of scientific
methods not to connect causally day and night with one another, but
the rising and setting of the sun with day and night. The theoretical

' A System of Logic, Ratiocinative and Inductive, bk. in, ch. v, § 6.

CONTENT AND VALIDITY OF THE CAUSAL LAW 367

analysis, indeed, goes farther. It teaches that in what is here summed
up as rising of the sun and yonder as day, there he again intricate
elements requiring special attention, in our own day extending per-
haps to the lines of thought contained in the electro-dynamic theory
of light and of electrons. Still the ways of thought remain the same
on all the levels of penetrating analysis. We have throughout to relate
to one another as cause and effect those events which, in a well-
ordered experience, must be regarded as following one another imme-
diately. The cause is then the immediate uniform antecedens , the
effect the immediate uniform consequens. Otherwise stated, the per-
ceived events that we are accustomed, from the standpoint of the
practical Weltanschauung, to regard as causes and effects, e. g., light-
ning and thunder, from the theoretical apprehension of the world
prove to be infinitely involved collections of events, whose elements
must be related to one another as causes and effects in as far as they
can be regarded as following one another immediately. No exception
is formed by expressions of our rough way of viewing and describing
which lead us without hesitation to regard as cause one out of the very
many causes of an event, and this, too, not necessarily the immedia>te
uniformly preceding event. All this lies rather in the nature of such
a hasty view.

The present limitation of uniform sequence to cases of immediate
sequence sets aside, then, the objection from which we started, in that
it adopts as its own the essential point in question.

Moreover, the way that leads us to this necessary limitation goes
farther: it leads to a strengthening of the empiristic position. It
brings us to a point where we see that the most advanced analysis
of intricate systems of events immediately given to us in perception
as real nowhere reveals more than the simple fact of uniform sequence.
Again where we come to regard the intervals between the events that
follow one another immediately as very short, there the uniformity of
the time relation makes, it would seem, the events for us merely
causes and effects; and as often as we have occasion to proceed to
the smaller time differences of a higher order, the same process repeats
itself; for we dissect the events that make up our point of departure
into ever more complex systems of component events, and the
coarser relations of uniform sequence into ever finer immediate ones.
Nowhere, seemingly, do we get beyond the field of events in uniform
sequence, which finally have their foundation in the facts of perception
from which they are drawn. Thus there follows from this conceptual
refinement of the point of departure only the truth that nothing
connects the events as causes and effects except the immediate
uniformity of sequence.

None the less, we have to think the empiristic doctrine to the bot-
tom, if we desire to determine whether or not the hypothesis which

368 METHODOLOGY OF SCIENCE

it offers is really sufficient to enable us to deduce the causal relation.
For this purpose let us remind ourselves that the question at issue
is, whether or not this relation is merely a temporal connection of
events that are given to us in perception or that can be derived from
the data of perception.

Besides, let us grant that this relation is as thoroughly valid for
the content of our experience as empiricism has always, and ration-
alism nearly always, maintained. We presuppose, therefore, as
granted, that every event is to be regarded as cause, and hence, in
the opposite time relation, as effect, mental events that are given
to us in self perception no less than the physical whose source is our
sense perception. In other words, we assume that the totality of
events in our possible experience presents a closed system of causal
series, that is, that every member within each of the contemporary
series is connected with the subsequent ones, as well as with the
subsequent members of all the other series, backward and forward
as cause and effect; and therefore, finally, that every member of
every series stands in causal relationship with every member of
e'\iery other series. We do not then, for the present purpose, burden
ourselves with the hypothesis which was touched upon above, that
this connection is to be thought of as a continuous one, namely, that
other members can be inserted ad infinitum between any two mem-
bers of the series.

We maintain at the same time that there is no justification for
separating from one another the concepts, causality and interaction.
This separation is only to be justified through the metaphysical
hypothesis that reality consists in a multitude of independently
existing substances inherently subject to change, and that their
mutual interconnection is conditioned by a common dependence
upon a first infinite cause. ^ Every connection between cause and
effect is mutual, if we assume with Newton that to every action
there is an equal opposing reaction.

In that we bring the totality of knowable' reality, as far as it is
analyzable into events, under the causal relation, we may regard
the statement that every event requires us to seek among uniformly
preceding events for the sufficient causes of its own reality, namely,
the general causal law, as the principle of all material sciences. For
all individual instances of conformity to law which we can discover
in the course of experience are from this point of view only special
cases of the general universal conformity to law which we have just
formulated.

^ This doctrine began in the theological evolution of the Christian concept of
God. It was first fundamentally formulated by Leibnitz. It is retained in Kant's
doctrine of the harmonia generaUter stabilita and the latter's consequences for the
critical doctrine of the mundus intelligibilis. Hence it permeates the metaphysical
doctrines of the systems of the nineteenth century in various ways.

CONTENT AND VALIDITY OF THE CAUSAL LAW 369

For the empiristic interpretation, the (general) causal law is only
the highest genus of the individual cases of empirically synthetic
relations of uniform sequence. Starting from these presuppositions,
it cannot be other than a generalization from experience, that is, a
carrying over of observed relations of uniform, or, as we may now
also say, constant sequence to those which have not been or cannot
be objects of observation, as well as to those which we expect to ap-
pear in the future. Psychologically regarded, it is merely the most
general ex]3ression of an expectation, conditioned through associative
reproduction, of uniform sequence. It is, therefore, — to bring
Hume's doctrine to a conclusion that the father of modern empiricism
himself did not draw, — a species of temporal contiguity.

The general validity which we ascribe to the causal law is ac-
cordingly a merely empirical one. It can never attain apodeictic
or even assertorical validity, but purely that type of problematic
validity which we may call "real" in contradistinction to the other
type of problematic validity attained in judgments of objective as
well as of subjective and hypothetical possibility.^ No possible pro-
gress of experience can win for the empiristically interpreted causal
law any other than this real problematic validity; for experience
can never become complete a parte post, nor has it ever been com-
plete a parte ante. The causal law is valid assertorically only in so
far as it sums up, purely in the way of an inventory, the preceding
experiences. We call such assumptions, drawn from well-ordered
experience and of inductive origin, "hypotheses," whether they rest
upon generalizing inductive inferences in the narrower sense, or upon
specializing inferences from analogy. They, and at the same time
the empiristically interpreted causal law, are not hypotheses in the
sense in which Newton rightly rejected all formation of hypotheses,^
but are such as are necessarily part of all methods in the sciences of
facts in so far as the paths of research lead out beyond the content
given immediately in perception to objects of only possible experience.

The assertion of Stuart Mill, in opposition to this conclusion,
that the cause must be thought of as the "invariable antecedent"
and, correspondingly, the effect as the "invariable consequent,"^
does all honor to the genius of the thinker; but it agrees by no means
with the empiristic presuppositions which serve as the basis for his
conclusions. For, starting from these presuppositions, the "invari-
able sequence" can only mean one that is uniform and constant

1 Cf. the author's Logik, bd. i, § 61. _

■^ " Rationem vero harura gravitatis propriefcatum ex phaenomenis nondum potui
deducere, et hypotheses no'n fingo. Quicquid enim ex phaenomenis non deducitur,
hypothesis vocanda est ; et hypotheses seu metaphysicae, seu physicae, seu qualita-
tum occultarum, seu mechanicae, in philosophia experimentah locum non habent.
In hac philosophia propositiones deducuntur ex phaenomenis, et redduntur gener-
ales per indudionem." Newton, at the end of his chief work.

^ Logic, bk. iii, ch. v, § 2.

370 METHODOLOGY OF SCIENCE

according to past experience, and that we henceforth carry over
to not yet observed events as far as these prove in conformity with
it, and in this way verify the anticipation contained in our general
assertion. The same holds of the assertion through which Mill en-
deavors to meet the above-mentioned objection of E,eid, namely, that
the unchanging sequence must at the same time be demonstrably
an " unconditional " one. The language in which experience speaks to
us knows the term "the unconditioned" as little as the term "the
unchangeable," even though this have, as Mill explains, the mean-
ing that the effect "will be, whatever supposition we may make in
regard to all other things," or that the sequence will "be subject to
no other than negative conditions." For in these determinations there
does not lie exclusively, according to Mill, a probable prediction of
the future. "It is necessary to our using the word cause, that we
should believe not only that the antecedent always has been fol-
lowed by the consequent, but that as long as the present constitution
of things endures, it always will be so." Likewise, Mill, the man of
research, not the empiristic logician, asserts that there belongs to
the causal law, besides this generality referring to all possible events
of uniform sequence, also an "undoubted assurance;" although he
could have here referred to a casual remark of Hume.^ Such an
undoubted assurance, "that for every event . . . there is a law to
be found, if we only know where to find it," evidently does not know
of a knowledge referred exclusively to experience.

Hence, if the causal law is, as empiricism to be consistent must
maintain, only a general hypothesis which is necessarily subject to
verification as experience progresses, then it is not impossible that in
the course of experience events will appear that are not preceded or
followed uniformly by others, and that accordingly cannot be re-
garded as causes or effects. According to this interpretation of the
causal law, such exceptional events, whether in individual or in
repeated cases of perception, must be just as possible as those which
in the course of preceding experience have proved themselves to be
members of series of constant sequence. On the basis of previous
experience, we should only have the right to say that such exceptional
cases are less probable; and we might from the same ground expect
that, if they could be surely determined, they would only have to be
regarded as exceptions to the rule and not, possibly, as signs of a
misunderstood universal non-uniformity of occurrence. No one
wants to maintain an empirical necessity, that is, a statement that
so comprehends a present experienoe or an hypothesis developed

* Logic, hk. ni, ch. v, § 6, and end of §2. Hume says in a note to section vi of his
Enquiry concerning Human Understanding : " We ought to divide arguments into
demonstrations, proofs, and probabilities. By proofs meaning sucli arguments from
experience as leave no room for doubt or opposition." The note stands in evident
contrast to the well-known remarks at the beginning of section iv, pt. i.

CONTENT AND VALIDITY OF THE CAUSAL LAW 371

on the basis of present experience that its contradictory is rationally-
impossible. An event preceded by no other immediately and uni-
formly as cause would, according to traditional usage, arise out of
nothing. An event that was followed immediately and constantly
by no other would accordingly be an event that remained without
effect, and, did it pass away, it must disappear into nothing. The
old thought, well known in its scholastic formulation, ex nihilo nihil
fit, in nihilum nihil potest reverti, is only another expression for the
causal law as we have interpreted it above. The contradictories
to each of the clauses of the thought just formulated, that some-
thing can arise out of nothing and pass into nothing, remain there-
fore, as a consequence of empiricism, an improbable thought, to be
sure, but none the less a thought to which a real possibility must be
ascribed.

It was in all probability this that Stuart Mill wished to convey
in the much-debated passage: " I am convinced that any one accus-
tomed to abstraction and analysis, who will fairly exert his faculties
for the purpose, will, when his imagination has once learnt to enter-
tain the notion, find no difficulty in conceiving that in some one, for
instance, of the many firmaments into which sidereal astronomy
now divides the universe, events may succeed one another at random
without any fixed law; nor can anything in our experience, or in our
mental nature, constitute a sufficient, or indeed any, reason for
believing that this is nowhere the case." For Mill immediately calls
our attention to the following: "Were we to suppose (what it is
perfectly possible to imagine) that the present order of the universe
were brought to an end, and that a chaos succeeded in which there
was no fixed succession of events, and the past gave no assurance of
the future; if a human being were miraculously kept alive to witness
this change, he surely would soon cease to believe in any uniformity,
the uniformity itself no longer existing." ^

We can throw light from another side upon the thought that lie?
in this outcome of the empiristic interpretation of the causal law.
If we still desire to give the name "effect" to an event that is pre-
ceded uniformly by no other, and that we therefore have to regard
as arising out of nothing, then we must say that it is the effect of
itself, that is, its cause lies in its own reality, in short, that it is
causa sui. Therefore the assumption that a causa sui has just as
much real possibility as have the causes of our experience which are
followed uniformly by another event, is a necessary consequence of
the empiristic view of causation. This much only remains sure, there
is nothing contained in our previous experience that in any way
assures us of the validity of this possible theory.

The empiristic doctrine of causation requires, however, still fur-
' Logic, bk. iii, ch. xxi, § 1.

372 METHODOLOGY OF SCIENCE

ther conclusions. Our scientific, no less than our practical thought
has always been accustomed to regard the relation between cause
and effect not as a matter of mere sequence, not therefore as a mere
formal temporal one. Rather it has always, in both forms of our
thought, stood for a real relation, that is, for a relation of dynamic
dependence of effect upon cause. Accordingly, the effect arises out
of the cause, is engendered through it, or brought forth by it.

The historical development of this dynamic conception of cause
is well known. The old anthropopathic interpretation, which inter-
polates anthropomorphic and yet superhuman intervention between
the events that follow one another uniformly, has maintained itself
on into the modern metaphysical hypotheses. It remains standing
wherever God is assumed as the first cause for the interaction be-
tween parts of reality. It is made obscure, but not eliminated, when,
in other conceptions of the world, impersonal nature, fate, neces-
sity, the absolute identity, or an abstraction related to these, ap-
pears in the place of God. On the other hand, it comes out clearly
wherever these two tendencies of thought unite themselves in an
anthropopathic pantheism. That is, it rests only upon a differ-
ence in strength between the governing religious and scientific in-
terests, whether or not the All-One which unfolds itself in the
interconnection and content of reality is thought of more as the im-
manent God, or more as substance. Finally, we do not change our
position, if the absolute, self-active being (in all these theories a first
cause is presupposed as causa sui) is degraded to a non-intellectual
will.

However, the dynamic interpretation of cause has not remained
confined to the field of these general speculations, just because it
commanded that field so early. There is a second branch, likewise
early evolved from the stem of the anthropopathic interpretation,
the doctrine that the causal relations of dependence are effected
through "forces." These forces adhere to, or dwell in, the ultimate
physical elements which are thought of as masses. Again, as spiritual
forces they belong to the "soul," which in turn is thought of as a
substance. In the modern contrast between attractive and repulsive
forces, there lies a remnant of the Empedoklean opposition between
Love and Hate. In the various old and new hylozoistie tendencies,
the concepts of force and its correlate, mass, are eclectically united.
In consistent materialism as well as spiritualism, and in the abstract
dynamism of energetics, the one member is robbed of its independence
or even rejected in favor of the other.^

' Alongside of these d^mamic theories, there are to be found mechanical ones
that arose just as early and from the same source, Aaz., the practical Weltan-
schauung. It is not part of our purpose to discuss them. Their first scientific
expression is to be found in the doctrine of effluences and pores in Empedokles
and in Atomism.

CONTENT AND VALIDITY OF THE CAUSAL LAW 373

It is evident in what light all these dynamic conceptions appear,
when looked at from the standpoint of consistent extreme empiricism.
These "forces," to consider here only this one of the dynamic hypo-
theses, help to explain nothing. The physical forces, or those which
give rise to movement, are evidently not given to us as contents of
sense perception, and at the most they can be deduced as non-sen-
suous foundations, not as contents of possible sense perception. The
often and variously expressed belief that self perception reveals to
us here what our senses leave hidden has proved itself to be in all its
forms a delusion. The forces whose existence we assume have then
an intuitable content only in so far as they get it through the uniform-
ities present in repeated perceptions, which uniformities are to be
"explained" through them. But right here their assumption proves
itself to be not only superfluous but even misleading; for it makes us
believe that we have offered an explanation, whereas in reality we
have simply duplicated the given by means of a fiction, quite after the
fashion of the Platonic doctrine of ideas. This endeavor to give the
formal temporal relations between events, which we interpret as
causes and effects, a dynamic real substructure, shows itself thus to
be worthless in its contributions to our thought. The same holds
true of every other dynamic hypothesis. The critique called forth
by these contributions establishes therefore only the validity of the
empiristic interpretation.

If, however, we have once come so far, we may not hold ourselves
back from the final step. Empiricism has long ago taken this step,
and the most consistent among its modern German representatives
has aroused anew the impulses that make it necessary. Indeed, if
we start from the empiristic presuppositions, we must recognize that
there lies not only in the assumption of forces, but even in the habit
of speaking of causes and effects, "a clear trace of fetishism." We
are not then surprised when the statement is made: The natural
science of the future, and accordingly science in general, will, it is
to be hoped, set aside these concepts also on account of their formal
obscurity. For, so it is explained, repetitions of like cases in which
a is always connected mth h, namely, in which like results are found
under like circumstances, in short, the essence of the connection of
cause and effect, exists only in the abstraction that is necessary to
enable us to repicture the facts. In nature itself there are no causes
and effects. Die Natur ist nur einmal da.

It is, again, Stuart Mill, the man of research, not the empiricist, that
opposes this conclusion, and indeed opposes it in the form that
Auguste Comte had given it in connection with thoughts that can
be read into Hume's doctrine. Comte's "objection to the word
cause is a mere matter of nomenclature, in which, as a matter of
nomenclature, I consider him to be entirely wrong. . . . By reject-

374 METHODOLOGY OF SCIENCE

ing this form of expression, M. Comte leaves himself without any
term for marking a distinction which, however incorrectly expressed,
is not only real, but is one of the fundamental distinctions in science." ^
For my own part, the right seems to be on the side of Comte
and his recent followers in showing the old nomenclature to be worn
out, if viewed from the standpoint of empiricism. If the relation
between cause and effect consists alone in the uniformity of sequence
which is hypothetically warranted by experience, then it can be
only misleading to employ words for the members of this purely
formal relation that necessarily have a strong tang of real dynamic
dependence. In fact, they give the connection in question a peculiarity
that, according to consistent empiricism, it does not possess. The
question at issue in the empiristically interpreted causal relation is
a formal functional one, which is not essentially different, as Ernst
Mach incidentally acknowledges, from the interdependence of the
sides and angles of a triangle.

Here two extremes meet. Spinoza, the most consistent of the dog-
matic rationalists, finds himself compelled in his formulation of the
analytic interpretation of the causal relation handed down to him
to transform it into a mathematical one. Mach, the most consistent
of recent German empiricists, finds himself compelled to recognize
that the empirically synthetic relation between cause and effect
includes no other form of dependence than that which is present
in the functional mathematical relations. (In Germany empiricism
steeped in natural science has supplanted the naive materialism
saturated with natural science.) That the mathematical relations
must likewise be subjected to a purely empirical interpretation,
which even Hume denied them, is a matter of course.

However, this agreement of two opposing views is no proof that
empiricism is on the right road. The empiristic conclusions to which
we have given our attention do not succeed in defining adequately
the specific nature of the causal relation; on the contrary, they
compel us to deny such a relation. Thus they cast aside the concept
that we have endeavored to define, that is, the judgment in which
we have to comprehend whatever is peculiar to the causal connection.
But one does not untie a knot by denying that it exists.

It follows from this self-destruction of the empiristic causal hypo-
thesis that an additional element of thought must be contained in the
relation of cause and effect besides the elements of reproductive
recognition and those of identification and discrimination, all of
which are involved in the abstract comprehension of uniform se-
quence. The characteristics of the causal connection revealed by our
previous analysis constitute the necessary and perhaps adequate
conditions for combining the several factual perceptions into the
* Logic, bk. iii, ch. v, § 6.

CONTENT AND VALIDITY OF THE CAUSAL LAW 875

abstract registering idea of uniform sequence. We may, therefore,
expect to find that the element sought for lies in the tendency to
extend the demand for causal connections over the entire field of
possible experience; and perhaps we may at the same time arrive
at the condition which led Hume and Mill to recognize the complete
universality of the causal law in spite of the exclusively empirical
content that they had ascribed to it. In this further analysis also
we have to draw from the nature of our thought itself the means of
guiding our investigation.

In the first place, all thought has a formal necessity which reveals
itself in the general causal law no less than in every individual
thought process, that is, in every valid judgment. The meaning of
this formal necessity of thought is easily determined. If we presup-
pose, for example, that I recognize a surface which lies before me
as green, then the perception judgment, "This surface is green,"
that is, the apprehension of the present perceptive content in the
fundamental form of discursive thought, repeats with predicative
necessity that which is presented to me in the content of perception.
The necessity of thought contained in this perception judgment, as
mutatis mutandis in every affirmative judgment meeting the logical
conditions, is recognizable through the fact that the contradictory
judgment, "This surface is not green," is impossible for our thought
under the presupposition of the given content of perception and of
our nomenclature. It contradicts itself. I can express the contradict-
ory proposition, for instance, in order to deceive; but I cannot really
pass the judgment that is contained in it. It lies in the very nature
of our thought that the predicate of an assertive judgment can con-
tain only whatever belongs as an element of some sort (characteristic,
attribute, state, relation) to the subject content in the wider sense.
The same formal necessity of thought, to give a further instance, is
present in the thought process of mediate syllogistic predication.
The conclusion follows necessarily from the premises, for example,
the judgment, "All bodies are divisible," from the propositions,
" All bodies are extended," and, " Whatever is extended is divisible."

These elementary remarks are not superfluous; for they make
clear that the casually expressed assertion of modern natural scien-
tific empiricism, declaring in effect that there is no such thing as
necessity of thought, goes altogether too far. Such necessity can
have an admissible meaning only in so far as it denotes that in
predicting or recounting the content of possible experience every hypo-
thesis is possible for thought. Of course it is, but that is not
the subject under discussion.

The recognition of the formal necessity of thought that must be
presupposed helps us to define our present question; for it needs no
proof that this formal necessity of thought, being valid for every

376 METHODOLOGY OF SCIENCE

affirmative judgment, is valid also for each particular induction,
and again for the general causal law. If in the course of our per-
ceptions we meet uniform sequences, then the judgment, "These
sequences are uniform," comprehends the common content of many
judgments with formal necessity of thought. Empiricism, too, does
not seriously doubt that the hypothesis of a general functional, even
though only temporal, relation between cause and effect is deduced
as an expectation of possible experience Tvith necessity from our real
experience. It questions only the doctrine that the relation between
the events regarded as cause and effect has an}- other than a purely
empirical import. The reahty of an event that is preceded and fol-
lowed uniformly by no other remains for this view, as we have seen,
a possibiht}' of thought.

In opposition to empiricism, we now formulate the thesis to be
estabhshed: Wherever two events a and h are known to foUow one
another uniformly and immediately, there we must require ^ith
formal necessity that some element in the preceding a be thought of
as fundamental, which will determine sufficiently 6's appearance or
make that appearance necessary. The necessity of the relation
between the events regarded as cause and effect is, therefore, the
question at issue.

We must keep in mind from the very start that less is asserted in
this formulation than we are apt to read into it. It states merely
that something in a must be thought of as fundamental, which makes
h necessary. On the other hand, it saj's nothing as to what this
fundamental something is, or how it is constituted. It leaves entirely
undecided whether or not this something that our thought must
necessarily postulate is a possible content of perception or can be-
come such, accordingly whether or not it can become an object of
our knowledge, or whether or not it lies beyond the bounds of all
our possible experience and hence all our possible knowledge. It
contains nothing whatsoever that teUs us how the determination
of h takes place through a. The word "fundamental" is intended
to express all this absence of determination.

Thus we hope to show a necessity of thought pecuhar to the rela-
tion between cause and effect. This is the same as saying that our
proof wiU estabhsh the logical impossibility of the contradictory
assertion; for the logical impossibility of the contradictory assertion
is the only criterion of logical necessit}'. Thus the proof that we seek
can be given only indirectly. In the course of this proof, we can
disregard the immediacy of the constant sequence and confine our
attention to the uniformit}' of the sequence, not only for the sake of
brcAity, but also because, as we have seen, we have the right to
speak of near and remote causes. We may then proceed as follows.

If there is not something fundamental in a constant antecedent

CONTENT AND VALIDITY OF THE CAUSAL LAW 377

event a, which determines necessarily the constant subsequent
appearance of one and the same h, — that is, if there is nothing
fundamental which makes this appearance necessary, — then we
must assume that also c or d . . . , in short, any event you will,
we dare not say "foUows upon," but appears after a in irregular
alternation with h. This assumption, however, is impossible for our
thought, because it is in contradiction with our experience, on the
basis of which our causal thought has been developed. Therefore
the assumption of a something that is fundamental in a, and that
determines sufficiently and necessarilj" the appearance of b, is a
necessity for our thought.

The assertion of this logical impossibihty (Denkunnidglichkeit)
will at once appear thoroughly paradoxical. The reader, merely
recaUing the results of the empiristic interpretation given above,
will immediately say: "The assumption that a h does not follow
constantly upon an a, but that sometimes b, sometimes c, some-
times d . . . irregularly appears, is in contradiction only with aU
our previous experience, but it is not on this account a logical im-
possibiht3\ It is merely improbable." The reader -otU appeal espe-
cially to the discussion of Stuart ]\IiU, already quoted, in which ]\Iill
pictures in concreto such an improbable logical impossibiht}^, and
therefore at the same time establishes it in fact. Again, the reader
may bring forward the words in which Helmholtz introduces intel-
lectual beings of only two dimensions. " By the much misused
expression, 'to be able to imagine to one's self,' or, 'to think how
something happens,' I understand (and I do not see how anybody
can understand anything else thereb}^ without robbing the expression
of aU meaning) that one can picture to one's self the series of sense
impressions which one would have if such a thing actuall}' took
place in an indi\ddual case."^

Nevertheless, pertinent as are these and similar objections, they
are not able to stand the test. We ask: "Is in fact a world, or even
a portion of our world, possible for thought that display's through an
absolutely irregular alternation of events a chaos in the full sense;
or is the attempt to picture such a chaos only a mere play of words
to which not even our imagination, not to mention our thought, can
give a possible meaning? "

Perhaps we shall reach a conclusion by the easiest way, if we
subject ]\IiU's description to a test. If we reduce it to the several
propositions it contains, we get the following: (1) Every one is able
to picture to himself in hi's imagination a reahty in which events
follow one another without rule, that is, so that after an event a
now b appears, now c, etc., in complete irregularity. (2) The idea of

* Vortrcge und Reden, bd. ii, "Uber den Ursprung und die Bedeutung der
geometrischen Axiome."

378 METHODOLOGY OF SCIENCE

such a chaos accordingly contradicts neither the nature of our mind
nor our experience. (3) Neither the former nor the latter gives us
sufficient reason to believe that such an irregular alternation does
not actually exist somewhere in the observable world. (4) If such
a chaos should be presented to us as fact, that is, if we were in a
position to outlive such an alternation, then the belief in the uniform-
ity of time relations would soon cease.

Every one would subscribe to the last of these four theses, im-
mediately upon such a chaos being admitted to be a possibility of
thought; that is, he would unless he shared the rationalistic con-
viction that our thought constitutes an activity absolutely inde-
pendent of all experience. We must simply accept this conclusion
on the ground of the previous discussion and of a point still to be
brought forward.

If we grant this conclusion, however, then it follows, on the
ground of our previous demonstration of the reproductive and
recognitive, as well as thought elements involved in the uniform
sequence, that the irregularity in the appearance of the events,
assumed in such a chaos, can bring about an absolutely relationless
alternation of impressions for the subject that we should presuppose
to be doing the perceiving. If we still wish to call it perception, it
would remain only a perception in which no component of its con-
tent could be related to the others, a perception, therefore, in which
not even the synthesis of the several perception contents could be
apprehended as such. That is, every combination of the different
perception contents, by which they become components of one and
the same perception, presupposes, as we have seen, those repro-
ductive and recognitive acts in revival which are possible only where
uniformities of succession (and of coexistence) exist. Again, every
act of attention involved in identifying and discriminating, which
likewise we have seen to be possible only if we presuppose uniform-
ities in the given contents of perception, must necessarily disappear
when we presuppose the chaotic content; and yet they remain
essential to the very idea of such a chaos. A relationless chaos is
after all nothing else than a system of relations thought of without
relations! That the same contradiction obtains also in the mere
mental picturing of a manifold of chaotic impressions needs no
discussion; for the productive imagination as well as the reproduct-
ive is no less dependent than is our perceptive knowledge upon the
reproductive recognition and upon the processes of identifying and
discriminating.

Thus the mental image of a chaos could be formed only through
an extended process of ideation, which itself presupposes as active
in it all that must be denied through the very nature of the image.
A relationless knowledge, a relationless abstraction, a relationless

CONTENT AND VALIDITY OF THE CAUSAL LAW 379

reproduction or recognition, a relationless identification or discrim-
ination, in short, a relationless thought, are, as phrases, one and
all mere contradictions. We cannot picture "through our relating
thought," to use Helmholtz's expression, nor even in our imagination,
the sense impressions that we should have if our thought were re-
lationless, that is, were nullified in its very components and presup-
positions. In the case of Helmholtz's two dimensional beings, the
question at issue was not regarding the setting aside of the conditions
of our thought and the substituting conditions contradictory to
them, but regarding the setting aside of a part of the content of our
sense intuition, meanwhile retaining the conditions and forms
peculiar to our thought. In this case, therefore, we have a permissible
fiction, whereas in Mill's chaos we have an unthinkable thought.

Again, the sense impressions that must be presupposed in an
inherently relationless chaos have no possible relation to the world
of our perception, whose components are universally related to
each other through the uniformities of their coexistences and se-
quences. Accordingly, the remark with which Helmholtz concludes
the passage above quoted holds, mutatis mutandis, here also. " If there
is no sense impression known that stands in relation to an event
which has never been observed (by us), as would be the case for us
were there a motion toward a fourth dimension, and for those two
dimensional beings were there a motion toward our third dimension;
then it foUows that such an ' idea ' is impossible, as much so as that
a man completely blind from childhood should be able to ' imagine '
the colors, if we could give him too a conceptual description of them."

Hence the first of the theses in which we summed up Stuart
Mill's assumptions must be rejected. With it go also the second and
third. In this case we need not answer the question: In how far
do these theses correspond to Mill's own statements regarding the
absolute surety and universality of the causal law?

We have now found what we sought, in order to establish as a
valid assertion the seeming paradox in the proof of the necessity
that we ascribe to the relation between cause and effect. We have
proved that the assumption of a completely irregular and therefore
relationless alternation of impressions contradicts not only our
experience, but even the conditions of our thought; for these pre-
suppose the uniformities of the impressions, and consequently our
ability to relate them, all which was eliminated from our hypothetical
chaos. Hence we have also established that a necessary relation is
implied in the thought of a constant sequence of events, which
makes the uniformly following b really dependent upon the uniformly
preceding a.

From still another side, we can make clear the necessity asserted

380 METHODOLOGY OF SCIENCE

in the relation of cause and effect. We found that the connection
between each dej&nite cause and its effect is an empirically synthetic
one and has as its warrant merely experience. We saw further that
the necessity inherent in the causal connection contains merely the
demand that there shall be something fundamental in the constantly
preceding a which makes the appearance of b necessary; not, however,
that it informs us what this efficacy really is, and hence also not that
it informs us how this efficacy brings about its effect. Finally, we
had to urge that every induction, the most general no less than the
most particular, depends upon the presupposition that the same
causes will be given in the reality not yet observed as in that already
observed. This expectation is warranted by no necessity of thought,
not even by that involved in the relation of cause and effect; for
this relation begins for future experience only when the presup-
position that the same causes will be found in it is assumed as ful-
filled.^ This expectation is then dependent solely upon previous
experience, whose servants we are, whose lords we can never be.
Therefore, every induction is an hypothesis requiring the verification
of a broader experience, since, in its work of widening and completing
our knowledge, it leads us beyond the given experience to a possible
one. In this respect we can call all inductive thought empirical,
that is, thought that begins with experience, is directed to experience,
and in its results is referred to experience. The office of this progress-
ing empirical thought is accordingly to form hypotheses from which
the data of perception can be regressively deduced, and by means of
which they can be exhibited as cases of known relations of our well-
ordered experience, and thus can be explained.

The way of forming hypotheses can be divided logically into
different sections which can readily be made clear by an example.
The police magistrate finds a human corpse under circumstances
that eliminate the possibihty of accident, natural death, or suicide;
in short, that indicate an act of violence on the part of another man.
The general hypothesis that he has here to do with a crime against
life forms the guide of his investigation. The result of the circum-
stantial evidence, which we presuppose as necessary, furnishes then
a special hypothesis as following from the general hypothesis.

It is clear that this division holds for all cases of forming hypo-
theses. A general hypothesis serves every special hypothesis as a
heuristic principle. In the former we comprehend the causal explan-
ation indicated immediately by the facts revealed to our perception

^ The only empiricism which can maintain that the same causes would, in con-
formity with the causal law, be given in the unobserved reality, is one which puts
all events that can be regarded as causes in the immediately given content of
perception as its members. Such a view is not to be fovmd in Mill; and it stands
so completely in the way of all further analysis required of us by every perception
of events that no attention has been paid in the text to this extreme of extremes.

CONTENT AND VALIDITY OF THE CAUSAL LAW 381

in the special case. It contains, as we might also express it, the
genus to the specific limitations of the more exact investigation.
But each of these general hypotheses is a modification of the most
general form of building hypotheses, which we have already come
to know as the condition of the validity of all inductive inferences,
that is, as the condition for the necessity of their deduction, and,
consequently, as the condition for the thought that like causes will
be given in the reality not yet observed as in that already observed.
We have further noticed that in this most general form of building
hypotheses there lie two distinct and different valid assumptions:
beside the empirical statement that like causes will be given, which
gives the inductive conclusion the hypothetical form, there stands
the judgment that like causes bring forth like effects, a corollary of
the causal law. The real dependence of the effect upon the cause,
presupposed by this second proposition and the underlying causal
law, is not, as was the other assumption, an hypothesis, but a neces-
sary requirement or postulate of our thought. Its necessity arises out
of our thought, because our experience reveals uniformity in the
sequence of events. From this point of view, therefore, the causal
law appears as a postulate of our thought, grounded upon the uni-
formity in the sequence of events. It underlies every special case of
constructing hypotheses as well as the expectation that like causes
will be given in the reality not yet observed.

Mill's logic of induction contains the same fault as that already
present in Hume's psychological theory of cause. Hume makes
merely the causal law itself responsible for our inductive inferences,
and accordingly (as Mill likewise wrongly assumes) for our inferences
in general. But we recognize how rightly Mill came to assert, in
contradiction to his empiristic presuppositions, that the causal law
offers "an undoubted assurance of an invariable, universal, and
unconditional," that is, necessary, sequence of events, from which
no seeming irregularity of occurrence and no gap in our experience
can lead us astray, as long as experience offers uniformities of se-
quence.

Rationalism is thus in the right, when it regards the necessary
connection as an essential characteristic of the relation between
cause and effect, that is, recognizes in it a relation of real dependence.
At this point Kant and Schopenhauer have had a profounder insight
than Hume and Stuart Mill. Especially am I glad to be in agreement
with Lotze on a point which he reached by a different route and
from essentially different presuppositions. Lotze distinguishes in
pure logic between postulates, hypotheses, and fictions. He does
not refer the term "postulate" exclusively to the causal law which
governs our entire empirical thought in its formation of hypotheses,
but gives the term a wider meaning. " Postulates " are only corollaries

382 METHODOLOGY OF SCIENCE

from the inductive fundamental form of all hypothesis construction,
and correspond essentially to what we have called general or heuristic
hypotheses. His determination of the validity of these postulates,
however, implies the position to be assigned to the causal law and
therefore not to those heuristic hypotheses. " The postulate is not an
assumption that we can make or refrain from making, or, again, in
whose place we can substitute another. It is rather an (absolutely)
necessary assumption without which the content of the view at
issue would contradict the laws of our thought." ^

Still the decision that we have reached is not on this account in
favor of rationalism, as this is represented for instance by Kant and
his successors down to our own time, and professed by Lotze in the
passage quoted, when he speaks of an absolute necessity for thought.
We found that the causal law requires a necessary connection be-
tween events given us in constant sequence. It is not, however,
on that account a law of our thought or of a "pure understanding"
which would be absolutely independent of all experience. When we
take into consideration the evolution of the organic world of which
we are members, then we must say that our intellect, that is, our
ideation and with it our sense perception, has evolved in us in ac-
cordance with the influences to which we have been subjected. The
common elements in the different contents of perception which have
arisen out of other psychical elements, seemingly first in the brute
world, are not only an occasion, but also an efficient cause, for the
evolution of our processes of reproduction, in which our memory
and imagination as well as our knowledge and thought, psycholog-
ically considered, come to pass. The causal law, which the critical
analysis of the material-scientific methods shows to be a funda-
mental condition of empirical thought, in its requirement that the
events stand as causes and effects in necessary connection, or real
dependence, comprehends these uniform contents of perception
only in the way peculiar to our thought.

Doubtless our thought gives a connection to experience through
this its requirement which experience of itself could not offer. The
necessary connection of effect with cause, or the real dependence of
the former upon the latter, is not a component of possible percep-
tion. This requirement of our thought does not, however, become
thereby independent of the perceptive elements in the presupposi-
tions involved in the uniformity of sequence. The a priori in the
sense of "innate ideas," denoting either these themselves or an ab-
solutely a priori conformity to law that underlies them, for instance,
our "spontaneity," presupposes in principle that our "soul" is an
independently existing substance in the traditional metaphysical
sense down to the time of Locke. Kant's rationalistic successors,
^Logic, 1874, buch ii, kap. viii.

CONTENT AND VALIDITY OF THE CAUSAL LAW 383

for the most part, lost sight of the fact that Kant had retained these
old metaphysical assumptions in his interpretation of the tran-
scendental conditions of empirical interaction and in his cosmo-
logical doctrine of freedom. The common root of the sensibility and
of the understanding as the higher faculty of knowledge remains for
Kant the substantial force of the soul, which expresses itself (just as
in Leibnitz) as vis passiva and vis activa. The modern doctrine of
evolution has entirely removed the foundation from this rationalism
which had been undermined ever since Locke's criticism of the tra-
ditional concept of substance.

To refer again briefly to a second point in which the foregoing
results differ from the Kantian rationalism as well as from empiricism
since Hume: The postulate of a necessary connection between
cause and effect, as we have seen, in no way implies the consequence
that the several inductions lose the character of hypotheses. This
does not follow merely from the fact that all inductions besides the
causal law include the hypothetical thought that the same causes
will be given in the reality not yet observed as appear in that already
observed. The hypothetical character of all inductive inferences is
rather revealed through the circumstance that in the causal postulate
absolutely nothing is contained regarding what the efficacy in the
causes is, and how this efficacy arises.

Only such consequences of the foregoing interpretation of the
causal law and of its position as one of the bases of all scientific con-
struction of hypotheses may be pointed out, in conclusion, as will
help to make easier the understanding of the interpretation itself.

The requirement of a necessary connection, or dependence, is
added by our thought to the reproductive and recognitive presup-
positions that are contained in the uniformity of the sequence of
events. If this necessary connection be taken objectively, then
it reveals as its correlate the requirement of a real dependence of
effect upon cause. We come not only upon often and variously
used rationalistic thoughts, but also upon old and unchangeable
components of all empirical scientific thought, when we give the
name "force " to the efficacy that underlies causes. The old postu-
late of a dynamic intermediary between the events that follow one
another constantly retains for us, therefore, its proper meaning.
We admit without hesitation that the word "force" suggests fetish-
ism more than do the words "cause" and "effect;" but we do not
see how this can to any degree be used as a counter-argument. All
words that were coined in the olden time to express thoughts of the
practical Weltanschauung have an archaic tang. Likewise all of our
science and the greater part of our nomenclature have arisen out of
the sphere of thought contained in the practical Weltanschauung,

384 METHODOLOGY OF SCIENCE

which centred early in fetishism and related thoughts. If, then, we
try to free our scientific terminology from such words, we must
seek refuge in the Utopia of a lingua universalis, in short, we must
endeavor to speak a language which would make science a secret
of the few. Or will any one seriously maintain that a thought which
belongs to an ancient sphere of mental life must be false for the
very reason that it is ancient?

In any case, it is fitting that we define more closely the sense
in which we are to regard forces as the dynamic intermediaries of
uniform occurrence. Force cannot be given as a content of perception
either through our senses or through our consciousness of self;
in the case of the former, not in our kinesthetic sensations, in the
case of the latter, not in our consciousness of volition. Volition
would not include a consciousness of force, even though we were
justified in regarding it as a simple primitive psychosis, and were not
compelled rather to regard it as an intricate collection of feelings
and sensations as far as these elementary forms of consciousness are
connected in thought with the phenomena of reaction. Again,
forces cannot be taken as objects that are derived as possible percep-
tions or after the analogy of possible perceptions. The postulate of
our thought through which these forces are derived from the facts
of the uniform sequence of events, reveals them as limiting notions
(Grenzhegriffe) , as specializations of the necessary connection be-
tween cause and effect, or of the real dependence of the former upon
the latter; for the manner of their causal intermediation is in no way
given, rather they can be thought of only as underlying our percep-
tions. They are then in fact qualitates occultae ; but they are such
only because the concept of quality is taken from the contents of
our sense and self perception, which of course do not contain the
necessary connection required by our thought. Whoever, therefore,
requires from the introduction of forces new contents of percep-
tion, for instance, new and fuller mechanical pictures, expects the
impossible.

The contempt with which the assumption of forces meets, on
the part of those who make this demand, is accordingly easily
understood, and still more easily is it understood, if one takes into
consideration what confusion of concepts has arisen through the use
of the term "force" and what obstacles the assumption of forces has
put in the way of the material sciences. It must be frankly admitted
that this concept delayed for centuries both in the natural and moral
sciences the necessary analysis of the complicated phenomena
forming our data. Under the influence of the "concept philosophy "
it caused, over and over again, the setting aside of the problems
of this analytical empirical thought as soon as their solution had
been begun. This misuse cannot but make suspicious from the very

CONTENT AND VALIDITY OF THE CAUSAL LAW 385

start every new form of maintaining that forces underlie causa-
tion.

However, misuse proves as little here against a proper use as it
does in other cases. Moreover, the scruples that we found arising
from the standpoint of empiricism against the assumption of forces
are not to the point. In assuming a dynamic intermediary between
cause and effect, we are not doubling the problems whose solution is
incumbent upon the sciences of facts, and still less is it true that our
assumption must lead to a logical circle. That is, a comparison
with the ideas of the old concept philosophy, which even in the
Aristotelian doctrine contain such a duplication, is not to the point.
Those ideas are hypostasized abstractions which are taken from the
uniformly coexisting characteristics of objects. Forces, on the other
hand, are the imperceivable relations of dependence which we must
presuppose between events that follow one another uniformly, if the
uniformity of this sequence is to become for us either thinkable or
conceivable. The problems of material scientific research are not
doubled by this presupposition of a real dynamic dependence, be-
cause it introduces an element not contained in the data of percep-
tion which give these problems their point of departure. This pre-
supposition does not renew the thought of an analytic rational
connection between cause and effect which the concept philosophy
involves; on the contrary, it remains true to the principle made
practical by Hume and Kant, that the real connection between
causes and their effects is determinable only through experience,
that is, empirically and synthetically through the actual indication
of the events of uniform sequence. How these forces are constituted
and work, we cannot know, since our knowledge is confined to the
material of perception from which as a basis presentation has de-
veloped into thought. The insight that we have won from the limit-
ing notion of force helps us rather to avoid the misuse which has
been made of the concept of force. A fatal circle first arises, when we
use the unknowable forces and not the knowable events for the
purpose of explanation, that is, when we cut off short the empirical
analysis which leads ad infinitum. To explain does not mean to
deduce the known from the unknown, but the particular from the
general. It was therefore no arbitrary judgment, but an impulse
conditioned by the very nature of our experience and of our thought,
that made man early regard the causal connection as a dynamic
one, even though his conception was of course indistinct and mixed
with confusing additions.

The concept of force remains indispensable also for natural scien-
tific thought. It is involved with the causal law in every attempt to
form an hypothesis, and accordingly it is already present in every
description of facts which goes by means of memory or abstraction

386 METHODOLOGY OF SCIENCE

beyond the immediately given content of present perception. In
introducing it we have in mind, moreover, that the foundations of
every possible interpretation of nature possess a dynamic character,
just because all empirical thought, in this field as well, is subordinate
to the causal law. This must be admitted by any one who assumes
as indispensable aids of natural science the mechanical figures
through which we reduce the events of sense perception to the mo-
tion of mass particles, that is, through which we associate these
events with the elements of our visual and tactual perception. All
formulations of the concept of mass, even when they are made so
formal as in the definition given by Heinrich Hertz, indicate dynamic
interpretations. Whether the impelling forces are to be thought of
in particular as forces acting at a distance or as forces acting through
collision depends upon the answer to the question whether we have
to assume the dynamic mass particles as filling space discontinuously
or continuously. The dynamic basis of our interpretation of nature
will be seen at once by any one who is of the opinion that we can make
the connection of events intelligible without the aid of mechanical
figures", for instance, in terms of energy.

Thus it results that we interpret the events following one another
immediately and uniformly as causes and effects, by presupposing
as fundamental to them forces that are the necessary means of their
uniformity of connection. What w^e call "laws" are the judgments
in which we formulate these causal connections.

A second and a third consequence need only be mentioned here.
The hypothesis that interprets the mutual connection of psychical
and physical vital phenomena as a causal one is as old as it is natural.
It is natural, because even simple observations assure us that the
mental content of perception follows uniformly the instigating
physical stimulus and the muscular movement the instigating
mental content which we apprehend as will. We know, however,
that the physical events which, in raising the biological problem, we
have to set beside the psychical, do not take place in the periphery of
our nervous system and in our muscles, but in the central nervous
system. But we must assume, in accordance with all the psycho-
physiological data which at the present time are at our disposal, that
these events in our central nervous system do not follow the cor-
responding psychical events, but that both series have their course
simultaneously. We have here, therefore, instead of the real relation
of dependence involved in constant sequence, a real dependence of
the simultaneity or correlative series of events. This would not, of
course, as should be at once remarked, tell as such against a causal
connection between the two separate causal series. But the contested
paralielistic interpretation of this dependence is made far more
probable through other grounds. These are in part corollaries of the

CONTENT AND VALIDITY OF THE CAUSAL LAW 387

law of the conservation of energy, rightly interpreted, and in part
epistemological considerations. Still it is not advisable to burden
methodological study, for instance, the theory of induction, with.
these remote problems; and on that account it is better for our
present investigation to subordinate the psychological interdepend-
ences to the causal ones in the narrower sense.

The final consequence, too, that forces itself upon our attention
is close at hand in the preceding discussion. The tradition prevailing
since Hume, together T\dth its inherent opposition to the inter-
pretation of causal connection given by the concept philosophy,
permitted us to make the uniform sequences of events the basis of
our discussion. In so doing, however, our attention had to be called
repeatedly to one reservation. In fact, onl}^ a moment ago, in allud-
ing to the psychological interdependences, we had to emphasize
the uniform sequence. Elsewhere the arguments depended upon
the uniformity that characterizes this sequence; and rightly, for the
reduction of the causal relation to the fundamental relation of the
sequence of events is mereh" a convenient one and not the only pos-
sible one. As soon as we regard the causal connection, along -v^dth
the opposed and equal reaction, as an interconnection, then cause
and effect become, as a matter of principle, simultaneous. The sep-
aration of interaction from causation is not justifiable.

In other ways also we can so transform every causal relation
that cause and effect must be regarded as simultaneous. Every
stage, for instance, of the warming of a stone by the heat of the
sun, or of the treaty conferences of two states, presents an effect
that is simultaneous with, the totality of the acting causes. The
analysis of a cause that was at first grasped as a whole into the
multiplicity of its constituent causes and the comprehension of
the constituent causes into a whole, which then presents itself as
the effect, is a necessarj^ condition of such a type of investigation.
This conception, which is present already in Hobbes, but especially
in Herbart's '^nethod of relations," deserves preference always
where the purpose in view is not the shortest possible argumentation
but the most exact analysis.

If we turn our attention to this way of viewing the problem, —
not, however, in the form of Herbart's speculative method, — we
shall find that the results which we have gained will in no respect be
altered. We do, however, get a view bej^ond. From it we can find
the way to subordinate not onh* the uniform sequence of events,
but also the persistent characteristics and states with their mutual
relations, under the extended causal law. In so doing, we do not
fall back again into the intellectual world of the concept philosophy.
We come only to regard the persisting coexistences — in the physical
field, the bodies, in the psychical, the subjects of consciousness — as

388 METHODOLOGY OF SCIENCE

systems or modes of activity. The thoughts to which such a doctrine
leads are accordingly not new or unheard of. The substances have
always been regarded as sources of modes of activity. We have here
merely new modifications of thoughts that have been variously de-
veloped, not only from the side of empiricism, but also from that
of rationalism. They carry with them methodologically the implica-
tion that it is possible to grasp the totality of reality, as far as it
reveals uniformities, as a causally connected whole, as a cosmos.
They give the research of the special sciences the conceptual bases for
the wider prospects that the sciences of facts have through hard
labor won for themselves. The subject of consciousness is unitary as
far as the processes of memory extend, but it is not simple. On the
contrary, it is most intricately put together out of psychical com-
plexes, themselves intricate and out of their relations; all of which
impress upon us, psychologically and, in their mechanical correlates,
physiologically, an ever-recurring need for further empirical analysis.
Among the mechanical images of physical reality that form the
foundation of our interpretation of nature, there can finally be but
one that meets all the requirements of a general hypothesis of the
continuity of kinetic connections. With this must be universally
coordinated the persistent properties or sensible modes of action
belonging to bodies. The mechanical constitution of the compound
bodies, no matter at what stage of combination and formation, must
be derivable from the mechanical constitution of the elements of this
combination. Thus our causal thought compels us to trace back
the persistent coexistences of the so-called elements to combin-
ations whose analysis, as yet hardly begun, leads us on likewise to
indefinitely manifold problems. Epistemologically we come finally
to a universal phenomenological dynamism as the fundamental
basis of all theoretical interpretation of the world, at least funda-
mental for our scientific thought, and we are here concerned with
no other.

SECTION E — ETHICS

{Hall 6, September 23, 10 a. m.)

Chairman: Professor George H. Palmer, Harvard University.
Speakers: Professor William R. Sorley, University of Cambridge.
Professor Paul Hensel, University of Erlangen.
Secretary: Professor F. C. Sharp, University of Wisconsin.

THE RELATIONS OF ETHICS

BY WILLIAM RITCHIE SORLEY

[William Ritchie Sorley, Knightbridge Professor of Moral Philosophy in the
University of Cambridge; Fellow of the British Academy . b. Sellcirk, Scot-
land, 1855. M.A. Edinburgh; Litt.D. Cambridge; Hon. LL.D. Edinburgh.
Post-Graduate, Shaw Fellow, Edinburgh University, 1878; Fellow, Trinity
College, Cambridge, 1883; Lecturer, Local Lectures Syndicate and for the
Moral Science Board, Cambridge, 1882-86; Deputy for the Professor of
Philosophy, University College, London, 1886-87; Prof essor of Philosophy,
University College, Cardiff, 1888-94; Regius Professor Moral Philosophy,
Aberdeen, 1894-1900. Author of Ethics of Naturalism, 1885 (new ed. 1904);
Mining Royalties, 1889; Recent Tendencies in Ethics, 1904; Edition of
Adam^on Development of Modern Philosophy, 1903.]

There are many departments of inquiry whose scope is so well
defined by the consensus of experts that one may proceed, almost
without preliminary, to mark off the boundaries of one science from
other departments, to investigate the relations in which it stands
to them, and to exhibit the place which each occupies in the whole
scheme of human knowledge. In other departments opinion differs
not only regarding special problems and results, but concerning the
whole nature of the science and its relation to connected subjects.
The study of ethics still belongs to this latter group. In it there is no
consensus of experts. Competent scholars hold diametrically opposed
views as to its scope. They differ not merely in the answers they
give to ethical questions, but in their views as to what the fundamen-
tal question of ethics is. And this opposition of opinion as to its
nature is connected with a difference of view regarding the relation
of ethics to the sciences. By many investigators it is set in line
with the sciences of biology, psychology, and sociology; and its
problems are formulated and discussed by the application of the same
historical method as those sciences employ. On the other hand, it is
maintained that ethics implies and requires a concept so different
from the concepts used by the historical and natural sciences as to
give its problem an altogether distinct character and to indicate

392 ETHICS

for it a far more significant position in the whole scheme of human
thought.

The question of the relation of ethics to the sciences implies a view
of the nature of ethics itself and, in particular, of the fundamental
concept used in ethical judgments. If the nature of this concept and
its relation to the concepts employed in other branches of inquiry
can be determined, the relations of ethics will become clear of them-
selves. The problem of this paper will receive its most adequate
solution — so far as the time at my disposal permits — by an in-
dependent inquiry into the nature of the ethical concept in relation
to the concepts used in other sciences.

The immediate judgments of .experience fall into two broadly
contrasted classes, which may be described in brief as judgments
of fact and judgments of worth. The former are the foundations
on which the whole edifice of science (as the term is commonly used)
is built. Science has no other object than to understand the relations
of facts as exhibited in historical sequence, in causal interconnection,
or in the logical interdependence which may be discovered amongst
their various aspects. In its beginnings it may have arisen as an aid
to the attainment of practical purposes: it is still everywhere yoked
to the chariot of man's desires and aims. But it has for long
vindicated an independent position for itself. It may be turned to
what uses you will; but its essential spirit stands aloof from these
uses. It has one interest only, — to know what happens and how.
Otherwise it is indifferent to all purposes alike. It studies with
equal mind the slow growth of a plant or the swift destruction
wrought by the torpedo, the reign of a Caligula or of a Victoria; it
takes no side, but observes and describes all " just as if the question
were of lines, planes, and solids." Mathematical method does not
limit its range, but it typifies its attitude of indifference to every
interest save one, — that of knowing the what and how of things.

We can conceive an intelligence of this nature, a pure intelligence,
or mere intelligence, to whose understanding all the relations of
things are evident, with the prophetic power of the Laplacian Demon
and the gift of tongues to make its knowledge clear, and yet unable to
distinguish between good and evil or to see beauty or ugliness in
nature. We can conceive such an intelligence; but it is an unreality,
a mere abstraction from the scientific aspect of human intelligence.
Pure intelligence of this sort does not exist in man, and we have no
grounds for asserting its existence anywhere. In the experience
which forms the basis of mental life, judgments of reality are every-
where combined with and colored by judgments of worth. And the
latter are as insistent as the former, and make up as large a part of
our experience. If we go back to the original judgments of experi-
ence, we find that they are not only of the form " it is here or there,"

THE RELATIONS OF ETHICS 393

"it is of this nature or that," "it has such and such effects;" just
as a large part of our experience is of another order which may be
expressed in judgments of the form "it is good or evil/' "it is fair or
foul."

Nor does the way in which scientific judgments are elaborated
give any rationale of the distinction between good and evil. If we
ask of science "What is good?" it can give no relevant answer to the
question. Strictly speaking, it does not understand the meaning of
the question at all. The ball has gone out of bounds; and science can-
not touch it until it has been thrown back into the field. It can say
what is, and what will happen, and it can describe the methods or
laws by which things come to pass; that is all; it has only one law
for the just and the unjust.

But science is very resourceful, and is able to deal with judgments
of worth from its own point of view. For these judgments also are
facts of individual experience: they are formed by human minds
under certain conditions, betray certain relations to the judgments
of fact with which they are associated, and are connected with an
environment of social institutions and physical conditions of life:
they have a history therefore. And in these respects they become
part of the material for science: and a description of them can be
given by psychological and historical methods.

The general nature and results of the application of these methods
to ethics are too well known to need further comment, too well estab-
lished to require defense. But these results may be exaggerated and
have been exaggerated. When all has been said and done that the
historical method can say and do, the question "What is good?"
is found to remain exactly where it was. We may have learned much
as to the way in which certain kinds of conduct in certain circum-
stances promote certain ends, and as to the gradual changes which
men's ideas about good and evil, virtue and vice, have passed through;
but we have not touched the fundamental question which ethics has
to face — the question of the nature of worth or goodness or duty.
And yet it is this question only which gives significance to the
problems on which historical evolution has been able to throw light.
Moral ideas and moral institutions have all along been effective
factors in human development, as well as the subject of development
themselves. And the secret of their power has lain in this that men
have believed in those ideas as expressing a moral imperative or a
moral end, and that they have looked upon moral institutions as
embodiments of something which has worth for man or a moral
claim upon his devotion. These ideas and institutions would have
had no power apart from this belief in their validity.

But was this belief true? Were the ideas or institutions valid?
This question the man of science, as sociologist or historian, does not

394 ETHICS

answer and has no means of answering. He can show their adapta-
tion or want of adaptation to certain ends, but he can say nothing
about the vaUdity of these ends themselves. It is implied in their
efficiency that these ends were conceived as having moral value or
moral authority. But to what ends does this moral value or authority
truly belong? and what is its significance? — these are questions
which the positive sciences (such as psychology and sociology) can-
not touch and which must be answered by other methods than those
which they employ.

The moral concept is expressed in various ways and by a variety
of terms, — right, duty, merit, virtue, goodness, worth. And these
different terms indicate different aspects opened up by a single new
point of view. Thus " right '^ seems to imply correspondence with a
standard or rule, which standard or rule is some moral law or ideal
of goodness; and "merit" indicates performance of the right,
perhaps in victory over some conflicting desire; and "virtue" means
a trait of character in which performance of this sort has become
habitual. The term "worth" has conveniences which have led to
its having considerable vogue in ethical treatises since the time of
Herbart; it lends itself easily to psychological manipulation; but
it does not seem to refer to a concept fundamentally distinct from
goodness. But between "goodness" and "duty" there seems to be
this difference at any rate, that the latter term refers definitely to
something to be done by a voluntary agent, whereas, in calling some-
thing "good," we may have no thought of action at all, but only
see and name a quality.

There lies here therefore a difference which is not a mere difference
of expression.

On the one hand it may be held that good is a quality which be-
longs to certain things and has no special and immediate reference
to volition: that we say this or that is good as we say that some-
thing else is heavy or green or positively electrified. No relation to
human life at all may be implied in the one form of judgment any
more than in the other. That relation will only follow by way of
application to circumstances. Just as a piece of lead may serve as
a letter-weight because it is heavy, so certain actions may come to
be our duty because they lead to the realization of something which
is objectively good in quality.

According to the other view goodness has reference in its primary
meaning to free self-conscious agency. The good is that which
ought to be brought into existence: goodness is a quality of things,
but only in a derivative regard because these things are produced
by a good will. It is objective, too, inasmuch as it unites the individual
will with a law or ideal which has a claim upon the will; but it does
not in its primary meaning indicate something out of relation to the

THE RELATIONS OF ETHICS 395

will: if there were no will there would be no law; apart from con-
scious agency good and evil would disappear.

The question thus raised is one of real and fundamental import-
ance. "Ethics" by its very name may seem to have primary refer-
ence to conduct; and that is the view which most moralists have,
in one way or another, adopted. But the other view which gives to
the concept "good" an independence of all relation to volition is not
always definitely excluded, even by these moralists; by others it
has been definitely maintained: it seems implied in Plato's idealism,
at one stage of its development; and quite recently a doctrine of
the principles of ethics has been worked out which is based on its
explicit recognition.^

If we would attempt to decide between these two conflicting
views of the ethical concept, we must, in the first place, imitate the
procedure of science and examine the facts on which the concept
is based. To get to the meaning of such scientific concepts as "mass,"
"energy," or the like, we begin by a consideration of the facts which
the concepts are introduced to describe. These facts are in the last
resort the objects of sense perception. No examination of these
sense percepts will, as we have seen,, yield the content of the ethical
concept; good and evil are not given in sense perception — they are
themselves an estimate of, or way of regarding, the immediate
material of experience. Moral experience is thus in a manner reflex,
as so many of the English moralists have called it. Its attitude to
things is not merely receptive; and the concepts to which it gives
rise have not mere understanding in view. Objects are perceived as
they occur; and experience of them is the groundwork of science.
There is also, at the same time, an attitude of approbation or dis-
approbation; this attitude is the special characteristic of moral
experience; and from moral experience the ethical concept is formed.

This reflex experience, or reflex attitude to experience, is exhibited
in different ways. There is, to begin with, the appreciation of beauty
in its various kinds and degrees and the corresponding depreciation
of ugliness or deformity. These give rise to the concepts and judg-
ments of aesthetics. They are closely related to moral approbation
and disapprobation, so closely that there has always been a tendency
amongst a school of moralists to strain the facts by identifying them.
A certain looseness in our use of terms favors this tendency. For
we do often use good of a work of art or even scene in nature when
we mean beautiful. But if we reflect on and compare our mental
attitudes in commending, say, a sunset and self-sacrifice, it seems
to me that there can be no doubt that the two attitudes are different.
Both objects may be admired; but both are not, in the same sense,
approved. It is hard to express this difference otherwise than by

1 Prindpia Ethica, by G. E. Moore (1903).

398 ETHICS

saying that the moral attitude is present in the one and absent in the
other. But the difference is brought out by the fact that our sesthet-
ical and moral attitudes towards the same experience may diverge
from one another. We may admire the beauty of that which we
condemn as immoral. De Quincey saw a fine art in certain cases
of murder; the finish and perfection of wickedness may often stir
a certain artistic admiration, especially if we lull the moral sense to
sleep. And, on the other hand, moral approval is often tempered
by a certain aesthetic depreciation of those noble characters who do
good awkwardly, without the ease and grace of a gentleman. John
Knox and Mary Queen of Scots (if I may assume for the moment
an historical judgment which may need qualification) will each have
his or her admirers according as the moral or sesthetic attitude
preponderates — the harsh tones of the one appealing to the law
of truth and goodness, the other an embodiment of the beauty and
gaiety of life, "without a moral sense, however feeble."

Nor is sesthetic appreciation the only other reflex attitude which
has a place in our experience side by side with the moral. Judgments
about matters of fact and relations of ideas are discriminated as
true or false; an ideal of truth is formed; and conditions of its
realization are laid down. Here again we have a concept and class
of judgments analogous to our sesthetical and ethical concepts and
judgments, but not the same as them, and not likely to be confused
with them.

Beside these may be put a whole class of judgments of worth
which may be described as judgments of utiUty. We estimate and
approve or disapprove various facts of experience according to their
tendency to promote or interfere with certain ends or objects of
desire. That moral judgments are to be identified with a special
class of these judgments of utility is a thesis too well known to
require discussion here, and too important to admit of discussion in
a few words. But it may be pointed out that it is only in a very
special and restricted sense of the term "utility" that judgments
of utility have ever been identified with moral judgments. The
" jimmy " is useful to the burglar, as his instruments are useful to
the surgeon; and they are in both cases appreciated by the same
kind of reflective judgment. Judgments of utility are all of them,
properly speaking, judgments about means to ends; and the ends
may and do differ; while it is only by a forced interpretation that aU
these ends are sometimes and somehow made to resolve themselves
into pleasure.

It is enough, however, for my present purpose to recognize the
prima fade distinction of moral judgments or judgments of goodness
from other judgments of worth, such as those of utility, of beauty,
and of truth (in the sense in which these last also are judgments of

THE RELATIONS OF ETHICS 397

worth). Had the question of the origin and history of the moral
judgment been before us, a great deal more might have been neces-
sary. For our present purpose what has been already said may be
sufficient: it was required in order to enable us to approach the
consideration of the question already raised concerning the applica-
tion and meaning of the moral concept.

The question is, Does our moral experience support the assignment
of the predicate "good" or "bad" to things regarded as quite inde-
pendent of volition or consciousness? At first sight it may seem
easy to answer the question in the affirmative. We do talk of sun-
shine and gentle rain and fertile land as good, and of tornadoes and
disease and death as bad. But I think that when we do so, in nine
cases out of ten, our "good" or "bad" is not a moral good or
bad; they are predicates of utility or sometimes sesthetic predicates,
not moral predicates; and we recognize this in recognizing their
relativity: the fertile land is called good because its fertility makes
it useful to man's primary needs; but the barren and rocky moun-
tain may be better in the eyes of the tourist, though the farmer
would call it bad land. There is an appreciation, a judgment of
worth in the most general sense, in such experiences; but they are
in most cases without the special feature of moral approbation or
disapprobation.

There remains, however, the tenth case in which the moral predi-
cate does seem to be applied to the unconscious. One may instance
J. S. Mill's passionate impeachment of the course of nature, in which
"habitual injustice" and "nearly all the things which men are
hanged or imprisoned for doing to one another" are spoken of as
"nature's every-day performances;"^ and a similar indictment
was brought by Professor Huxley, twenty years after the publica-
tion of Mill's essay, against the cosmic process for its encourage-
ment of selfishness and ferocity.^ These are only examples. Litera-
ture is full of similar reflections on the indiscriminate slaughter
wrought by the earthquake or the hurricane, and on the sight of the
wicked flourishing or of the righteous begging his bread; and these
reflections find an echo in the experience of most men.

But the nature of this experience calls for remark.

In the first place, if we look more closely at the arguments of Mill
or Huxley, we see that both are cases of criticism of a philosophical
theory. Mill was refuting a view which he held (and rightly held)
to have influence still on popular thought, though it might have
ceased to be a living ethical theory — the doctrine that the standard
of right and wrong was to be found in nature; it was in keeping
with his purpose, therefore, to speak of the operations of nature as

1 J. S. Mill, Three Essays on Religion, pp. 35, 38.

^ T. H. Huxley, Evolution and Ethics (Romanes Lecture).

398 ETHICS

if they were properly the subject of moral praise or blame. In the
same way, when Huxley wrote, the old doctrine which Mill regarded
as philosophically extinct and only surviving as a popular error had
been revived by the impetus which the theory of evolution had
given to every branch of study; and Huxley was criticising the evo-
lutionist ethics of Spencer and others who looked for moral guidance
to the course of evolution. He, therefore, was led to speak of the
cosmic process as a possible subject of moral predicates, not neces-
sarily because he thought that application appropriate, but in order
to demonstrate the hollowness of the ethics of evolution by showing
that if the moral predicate could be applied at all, then the appro-
priate adjective would be not "good" but "bad."

Perhaps there is more than this in Huxley; and Mill's expressions
often betray a direct and genuine moral condemnation of the methods
of nature as methods of wickedness; and, still more clearly, this
immediate moral disapproval may be found in expressions of common
experience as yet uncolored by philosophy. But if we examine these
we find that, while there is no reference to philosophical theories
about nature, the things approved or condemned are yet looked upon
as implying consciousness. In the lower stages of development this
implication is simply animistic; at a later period it becomes theo-
logical. But throughout experience moral judgments upon nature are
not passed upon mere nature. Its forces are regarded as expressing a
purpose or mind; and it is this that is condemned or approved. The
primitive man and the child do not merely condemn the misdoings of
inanimate objects; they wreak their vengeance upon them or punish
them : and this is a consequence of their animistic interpretation of
natural forces. Gradually, in the mental growth of the child, this ani-
mistic interpretation of things gives place to an understanding of the
natural laws of their working; and at the same time and by the same
degrees, the child ceases to inflict punishment upon the chair that
has fallen on him or to condemn its misdemeanor. Here the moral
judgment is displaced by the causal judgment; and the reason of its
displacement is the disappearance of mind or purpose from amongst
the phenomena. When the child comes to understand that the
chair falls by "laws of nature" which are not the expression of will,
like the acts done by himself or his companions, he ceases to disap-
prove or to resent, though he does not cease to feel pain or to im-
prove the circumstances by setting the chair firmly on the floor.
The recognition of natural causation as all that there is in the case
leaves no room for the moral attitude. So true is this that the same
result is sometimes thought to be a consequence of the scientific
understanding even of what is called moral causation, "tout com-
prendre c'est tout pardonner " — as if knowledge of motive and cir-
cumstances were sufficient to dispense with praise or blame.

THE RELATIONS OF ETHICS 399

Moral judgments of a more mature kind on the constitution and
course of nature form the material for optimistic and pessimistic
views of the world — at least, when these views rise above the asser-
tion of a preponderance of pleasure or of pain in life. But, so far as
I can see, in such moral judgments nature is never looked upon as
consisting of dead mechanical sequences. It is because it is looked
upon as the expression of a living will or as in some way — perhaps
very vaguely conceived — animated by purpose or consciousness, that
we regard it as morally good or evil. Apart from some such theological
conception, it does not seem to me that the nature of things calls out
the attitude of moral approval or disapproval. Things are estimated
as useful for this or that end, they are seen and appreciated as
beautiful or the reverse, without any reference to them as due to an
inspiring or originating mind; and in one or other of these references
the terms "good" or "bad" may be used. But when we use the
term good in its specifically moral signification, we do not apply
it to the inanimate, except in a derivate way, on account of the
relation in which these inanimate things stand to the moral ends
and character of conscious beings.

So far, therefore, as the evidence of moral experience goes, it
does not support the view that the "good" is a quality which be-
longs to things out of relation to self-conscious activity. And, in so
far, the peculiarity of the moral experience would seem to be better
brought out by the conception "ought" than by the conception
"good."

But here a difficulty arises at once. For how can we say that any-
thing ought to be done or to be except on the assumption that it is
antecedently good? Is not such antecedent and independent good-
ness necessary in order to justify the assertion that any one ought
to produce it?

The question undoubtedly points to a difficulty; and if that diffi-
culty can be solved it may help to bring out the true significance of
the moral concept. The judgment which assigns the duty of an indi-
vidual — according to which I or any one ought to adopt a certain
course of action — involves a special application of the moral con-
cept. It binds the individual to a certain objective rule or end. The
individual's desires as mere facts of experience may point in an
altogether different direction; the purpose or volition contemplated
and approved by the moral judgment has in view the union of indi-
vidual striving with an end which is objective and, as objective, uni-
versal. This union involves an adaptation of two things which may
fall asunder, and which in every case of evil volition do fall asunder.
And the adaptation may be regarded from either side: on the side
of the individual, application to his individuality is implied; the
duty of one man is not just the same as the duty of any other; he

400 ETHICS

has his own special place and calling. But he is connected with
a larger purpose which in his consciousness becomes both an ideal
and a law, while its application is not limited to his individuality or
his circumstances.

All this is implied in the moral judgment. It is not limited to one
individual consciousness or volition. But it does not follow that the
predicate "good," in the ethical meaning of the term, is or can be
applied out of relation to consciousness altogether. At the earliest
stages of moral development we find it applied unhesitatingly
wherever conscious activity is supposed to be present — to anything
that is regarded as the embodiment of spirit; and it is applied to the
universe as a whole when the universe is thought of as the product
of mind. " Good " is not even limited to an actual existent; it neither
implies nor denies actual existence. "Such and such, if it existed,
would be a good" is as legitimate though not so primitive an expres-
sion of the moral judgment as "this existent is good." But it does
imply a relation to existence. It does not even seem possible to
distinguish except verbally between "good" and "ought to be."
And this "ought" seems to imply a reference to a purpose through
which the idea is to be realized.

This conception "ought to be" is not the same as the concept
"ought to be done by me." The latter is an application of the more
general concept to a special individual in special circumstances;
and this is the common meaning of the concept duty. The former
is the more general concept of "goodness." It may be called object-
ive, because it does not refer to any individual state of mind; it is
universal because independent of the judgments and desires of the
individual; and when the goodness is not due to its tendency towards
some further end, it may also be called absolute.

The point of the whole argument can thus be made clear if we
bear in mind the familiar distinction between "good in itself" and
"good for me now." That the latter has always a relation to con-
sciousness is obvious: it is something to be done or experienced by
me. But there must be some ground why anything is to be or ought
to be done or experienced by me at any time. Present individual
activity must rest upon or be connected with some wider or objective
basis. What is good for me points to and depends upon something
which is not merely relatively good, but good in itself or absolutely.
Yet it does not follow that this good in itself is necessarily absolute
in the sense of having significance apart altogether from conscious-
ness. Its absoluteness consists in independence of individual con-
sciousness or feeling, not in independence of consciousness altogether.
It is objective rather than absolute in the literal sense of the term.
The good in itself, like the relative good, is one aspect which can only
belong to a consciousness — to purpose. The moral judgment on

THE RELATIONS OF ETHICS 401

things — either on the universe as a whole, or on anything in the
universe which is not regarded as due to the will of man — is only-
Justified if we regard these things as in some way expressing con-
sciousness; either as directly due to it, or as aiding it, or as in con-
flict with it. From any other point of view, to speak of things as good
or evil (unless in some non-ethical sense of these terms) seems out
of place, and is unsupported by the mode of application which be-
longs to the immediate judgments of the moral consciousness. If
the moral concept has significance beyond the range of the feelings
and desires of men, it is because the objects to which it applies are
the expression of mind.

This is not put forward as a vindication of a spiritual idealism.
It is only a small contribution towards the meaning of "good." A
comprehensive idealism may not be the only view of reality with
which the conclusions reached so far will harmonize. But it is the
view with which they harmonize most simply. The conception of a
purpose to which all the events of the world are related is a form in
which the essential feature of idealism may be expressed; the view
of this purpose as good makes the idealism at the same time a moral
interpretation of reality, and allows of our classing each distinguish-
able event as good or evil according as it tends to the furtherance or
hindrance of that purpose.

This doctrine of the significance and application of the ethical
concept would enable us to reach a definite view of the nature of
ethics and of the way in which it is related to the sciences and to
metaphysics. The ethical concept is based upon the primary facts
of the moral consciousness, just as scientific concepts have as their
basis the facts of direct experience. The primary facts of the moral
consciousness are themselves of the nature of judgment — they are
approbations or disapprobations. But all facts of experience involve
judgments, though these judgments may be only of the form "it is
here" or "it is of this or that nature." Again, the primary ethical
facts or judgments cannot be assumed to be of unquestionable val-
idity: we may approve what "is not worthy of approval, or disaj>-
prove what ought to have been approved. Our moral judgments
claim validity; and their claim is of the nature of an assertion, not
that one simply feels in such and such a way, but that something
ought or ought not to be. They imply an objective standard. But
the objective standard, when more clearly understood, may modify
or even reverse them. Our primary ethical judgments — all our
ethical judgments, indeed — stand in need of revision and criti-
cism; and they receive this revision and criticism in the course of
the elaboration of the ethical concept and of its application to the
worlds of fact and possibility. In the same way it may be contended
that the direct judgments of experience upon which science is based

402 ETHICS

need criticism and correction; though their variation may be less in
amount than the variation of moral judgments. The color-blind
man identifies red with green, and his judgment on this point has to
be reversed; the hypersensitive subject often confuses images with
percepts; exact observation needs a highly trained capacity. The
correction and criticism which is needed come from objective stand-
ards; and these are the result of the comparison of many experiences
and the work of many minds.

It is no otherwise in the case of ethics. Criticism brings to light
inconsistencies in the primary judgments of approbation and disap-
probation as well as in the later developments of the moral judgment.
And these inconsistencies must be dealt with in a way similar to that
in which we deal with inconsistencies in the judgments of perception
and of science. The objective standard is not itself given once for
all; it has to be formed by accumulation and comparison of moral
experiences. Like the experiences on which science is based, these
have to be made as far as possible harmonious, and analysis has to
be employed to bring out the element of identity which often lurks
behind apparent contradiction. They have also to be made as com-
prehensive as possible, so that they may be capable of application to
all relevant facts, and that the scattered details of the moral con-
sciousness may be welded into an harmonious system. In these
general respects the criticism of ethical concepts proceeds upon the
same lines as the criticism of scientific concepts. The difference lies
in the concepts themselves, for ethics involves a point of view to
which science must always remain a stranger.

BIBLIOGRAPHICAL NOTE

The relations of ethics are discussed in ahnost every ethical treatise; special
reference may be made to the writers who have worked out the theory of worth
or value, especially von Ehrenfels, System der Wert-theorie (1897) ; Meinong, Psy-
chologisch-Ethische Untersuchungen zur Werth-theorie (1894), and an article in
Archiv fiXr Syst. Phil. 1895; Krueger, Begriff der Absolut Wertvollen (18p8); also
to articles by Standinger and by Natorp in Archiv fur Syst. Phil. (1896); by
Wentscher, Archiv fur Syst. Phil. 1899; by Westermarck, Mind, 1900; and by
Belot, Revue de M&ta'physiqae et de Morale, 1905. — W. R. S.

PROBLEMS OF ETHICS

BY PAUL HENSEL
(Translated from the German by Professor J. H. Woods, Harvard University)

[Paul Hensel, Professor of Systematic Philosophy, University of Erlangen,
since 1902. b. May 17, 1860, Great Barten, East Prussia. Ph.D. Freiburg,
Baden, 1885. Privat-docent, Strassburg, 1888-95; Special Professor, Strass-
burg, 1895-98. Author of The Ethical Basis and Ethical Transactions; Car-
lyle; The Principal Problems of Ethics.]

Since the appearance of the three chief works of Kant a certain
rhythm in the treatment of philosophical problems, first of all in
Germany, but also, in less degree, in other civilized countries, is un-
mistakable. After an intense occupation with theoretical problems
a flood of ethical discussion usually follows ; and this then is usually
resolved into a renewed revision of sesthetical problems. If I am
not deceived, we are now at the period of transition from the second
to the third epoch; so much the more favorable is the time to re-
view the present condition of ethical problems. In the first place,
then, it seems rather remarkable that recent ethical discussion, so
intensely carried on, has resulted in a definite victory for neither one
school nor the other. One thing alone, however, may with some
accuracy be said, that the school of utilitarianism of the older inter-
pretation by Bentham, which earlier prevailed almost alone in
England with a fairly strong representation in France and Germany,
seems to be withdrawn from the field. Not as if there were no men
to-day who in other times would have sworn by Bentham's flag,
rather we are here facing a fact that a theory which formerly ap-
peared in independence, now may be deemed a special case of a
more inclusive theory, which with the help of its wider horizon can
remove a w^hole series of difficulties, which apparently raised insolv-
able problems for the special theory. Utilitarianism, since it had
started with the examination of the individual, could not, even in the
master-hand of Bentham, transfer itself without remainder into the
greatest happiness of the greatest number; the interest paid on
the sacrifice offered to fellow men, again and again seemed dubitable
and probable; again and again the best calculation seemed to con-
sist in egoism pure and simple. The impossibility of an exact calcula-
tion of consequences in pleasure and in pain was likewise repeatedly
emphasized by opponents; the suggestion that we do not count the
shrewd calculator so good as the man who acts impulsively was also
not lacking: all these were difficulties, which, on the ground of the
older utilitarianism, could be evaded but not quite entirely put out
of the world.

404 ETHICS

It is then easily understood that the further combinations into
which evolution was able to advance ethical questions have resulted
in the cessation of utilitarianism as an independent system. Around
the huge system of thought of Herbert Spencer one of the great camps
of ethical workers is collected. It is not correct to count Herbert
Spencer as systematizer of Darwin's thoughts; his main thoughts
were finished, before a line of Darwin had appeared. But it is correct
that the wonderful inductions of Darwin were precisely that which
Spencer's system needed in order to begin its triumphal march
through the civilized world. Here the case is the reverse of that of
Copernicus and Giordano Bruno: the systematizer precedes the
man of special research. It is superfluous on American soil to give
a description of Spencer's thoughts; they have become parts of the
general consciousness. So it may suffice to emphasize a few character-
istic features, to which my remarks shall be attached, since, other-
wise, in view of the richness of the system, there might easily be
other sides of it in the mind of my hearers than those to which I
have here to attach importance.

The- characteristic feature of the system of Spencer is its unity and
compactness. Just as every picture has a definite point from which
it should be seen, so also the system of Spencer is a view of the world
from a quite definite point of view, — that of evolution. Systems
of evolution had already occurred in philosophy, — I mention the
vast performance of Hegel only, — but that which gives Spencer's
system its characteristic significance is that here evolution is con-
ceived not as logical, but as biological; while in the case of Hegel
nature is the vestibule of the realm of purpose, and therein alone
has its significance, Spencer takes nature as his point of departure,
and the realm of human activity represents itself to him merely as
the finest conformation of natural events. Here the whole evolution
from the nebula in world-space to the most delicate relations between
man and man are comprehended in one grand conception. The same
amount of force which then existed in world-space exists still to-day,
only in infinitely more differentiated form. The new which is pro-
duced is nothing else than the transformed old, but transformed in
an essential relation, in the direction towards constantly increasing
complexity of relations in which single things and centres of force
stand to each other.

If it be asked what this principle is which is the ground for this
differentiation, a glance at the behavior of organisms informs us. In
them we can most clearly recognize effects which result, with the
necessity of laws of nature, from increasing differentiation. The
undifferentiated individual is powerless in the presence of every
change of his environment. Banished to its accidental place, the
plant must wait for what happens to it. Only within a narrow limit

PROBLEMS OF ETHICS 405

can it maintain its existence. Better equipped we find the animal,
especially when it has gathered into social groups, either for pro-
tection against carnivora or for the breeding of progeny in common.
The young steer has an infinitely better prospect to maintain itself,
to grow up, than the single egg in the spawn of the sturgeon.

So it is, before all else, the fact of social combination which attracts
to itself the attention of the revolutionary ethicist. His ethics is
social ethics. The analysis of the historical development of mankind
forms the standard, in which the social combinations have resulted,
and in which greater and world-inclusive formations have replaced
those earlier, smaller, and smallest, usually engaged in war with
each other. It is a long way from the time when hospes was equivalent
to hostis to international expositions, and the single stages of this
way reflect themselves in the moral behavior of the individuals.
The old question, which in so many ways agitated the English
ethics of the seventeenth and eighteenth centuries, the question,
whether man should be regarded as an originally egoistic being, or
whether equally original, benevolent instincts must be ascribed to
him, is transferred by evolutionists of to-day beyond the realm of
man to that of his animal ancestors and, in this case, in favor of the
originality of egoism. But long before man appeared as an inde-
pendent species the effects of the life of the horde must have shown
themselves in him, since those communities only in which the single
members were bound to each other by sympathy had any prospect
of survival. It is therefore possible to speak of animal ethics. The
interesting attempts which Darwin had made in this field were taken
by Spencer, as a whole, into his system. It must, however, be con-
ceded that we must observe the full development of this process,
first of all, in man, and the tendency then consists in a constant
decrease of egoistic, as compared with altruistic, actions. How it
was possible that the individual was ever willing to renounce the
amounts of pleasure, which he could obtain, in favor of others,
Spencer skillfully tried to explain by the introduction of the egoistic-
altruistic feelings. These give the impulse to actions which are useful
to the community, but which give to the doer honor and distinction,
and thus, from egoistic motives, make actions which promote the
welfare of the community commendable. But those actions which
damage the community are visited with punishment of all kinds.
The theory of sanctions in Bentham and Mill here passes over into
the more extensive system of evolution. For modern theory of
evolution, by the broader biological foundation of its system, suc-
ceeds in explaining why even, in the case of those who cannot over-
look the consequences of such actions as are injurious to their own
person, these consequences are still ignored. The fact of the con-
science, for the consistent Benthamite a negligible quantity, forms

406 ETHICS

the keystone of Spencer's ethics, and affords the chance of making
the theory of heredity applicable in a new field of ethical speculation.
It is, as a matter of fact, impossible for the single individual to
calculate, by Bentham's receipt, all the consequences of pleasure or
of pain which result from the actions for his own welfare. The
individual need not, however, undertake this calculation at all. He
does not begin at the beginning of making his experiences in this
world; he enjoys the heaped-up treasure of experiences which, before
him, long-forgotten generations of ancestors had made; and the
sum of these experiences he calls his conscience. This voice of the
conscience restrains the individual from anti-social actions, which,
in accordance with experience, must lead to an injury to his own
person; in accordance, of course, with the experience not of single
ancestors but of the whole line. Here, again, a selective process in the
struggle for existence is being completed. Men with no conscience at
all or with an only imperfectly developed conscience have to contend
with disadvantages similar to those in whom the corporal adjustment
to the modern conditions of civilization have proved defective; they
are exterminated by seclusion in prison or by execution, as the others
by diseases which their bodies cannot resist. The criminal of to-day
might perhaps have been, in primitive times, a respected member of
his horde, perhaps, even a great chief. To-day he can be regarded
only as an atavistic survivor, who fits into our conditions as little
as a living ichthyosaurus into this lecture-hall. Again, it is to be
hoped, it is even definitely to be predicted, that many who to-day
are quite irreproachable in moral respects, in later times will no
longer succeed in satisfying the requirements in the form of their
grandson or great-grandson. For the progress is a biological necessity;
and he who cannot attach himself to its ways is submerged.

It is small disparagement for this vast construction of the connec-
tion between the moral life of the individual and the total evolution
of the associations of men, of organisms, of the whole, that, now espe-
cially in English ethics, a bitter strife has broken forth, which we may
regard as the one-sided elaboration of the individualistic parts of
Spencer's ethics on the one side, of the social on the other side.
While the orthodox disciples of Spencer insist that such progress
only can be kept iv. aim which must assure to the individual, to the
fit the most unrestricted possible amount of free movement, while
the whole rigor of the process of selection must fall upon the unad-
justed and the unfit, the socialist tendencies of our time tend to
advocate a reversal of this harsh result and to advocate both the
united struggle of human society, by suppressing over-energetic
individuals, and the preservation of the economically weak. Though
it would be interesting to trace this division "to its final grounds, I
must limit myself to note the fact that the socialist movement

PROBLEMS OF ETHICS 407

seems here also to be in advance, — at least, so far as European
movements of thought are concerned; and that they are in the
condition to compensate for their departure from the teachings of
the master by an appeal to the main thoughts of his system, con-
cerns me just here. Doubtless socialistic thought is on the whole
in advance when compared with liberal and individualistic thought.
And, under these circumstances, the inference for every disciple of
Darwin's theory of evolution is simple; that here again is a case
of survival of the fittest; that socialistic ideals represent a higher
form of adjustment; that just by the fact of their victory the ne-
cessity and justification of this victory is placed beyond doubt. It
helped little that the venerable thinker himself in the last years of
his rich and active life descended into the arena of the contest and
warned his beloved England against the dangers of this socialistic
tendency. It was inconsistent that he tried to brand these thoughts
as a retrograde movement, as a step backward, since his own system
with its powerful optimism affords no possibility for victorious
retrograde movements. Even imperfection and evil has for Spencer
only the significance of an imperfect progress; and the thought
that imperfection could even win the victory over the perfect, that
must be warned against it, could only be nonsense in connection
with his system. For him, as for Hegel, the final formula, obtained
it is true by a very different way, is the thesis: The actual is
rational.

But just this reference to Hegel's system makes clear to us the
opposition which Herbert Spencer's system found in Germany,
first of all, but also in -wide circles in England and in America. If
it could be objected against Hegel that the activity of the individual,
in contrast to the might of the developing process of the logical idea,
is reduced to insignificance, this consideration returns with doubled
force in contrast to the concept of the thought of development, which
is found in the modern theory of evolution of Spencer. For here
it is not teleological necessities which prevail, but causal. To have
proved evolution by the laws of nature is precisely his system's title
to fame. The question must then be raised whether an obligation
to any definite practical action can be deduced from the proof of the
necessity of any event. If the development is necessary, it will be
completed whether I cooperate with it or not. If it needs my coopera-
tion, it need not be regarded as a law of nature. It is exactly the
same difficulty which beset the Stoics, when they tried to harmon-
ize the determinism of world events with the demands which their
ethics put upon the moral resolves of the individual. It is absurd
to will any necessary event of the laws of nature; I can suspend my
action so that I count upon the occurrence of such an incident, but
I cannot make this incident the object of my will. I can decide that

408 ETHICS

I will observe an eclipse of the moon, but I cannot will the occurrence

of this eclipse of the moon, or not will it.

If we reduce the difficulty to the simplest formula, it would be as
follows: the theory of evolution did not distinguish between two
completely different kinds of attitudes on the part of human mental
activity; between the knowledge of the necessity of what exists and
its judgment by standards of value. But it is precisely with the
latter that ethics has to do. It is, like logic and aesthetics, a science
of values; the interest in the question how something has come to be,
is quite different from the interest in determining its value. Every-
thing has come to be, the valueless as well as the valued, with the
same necessity; that is a self-evident presupposition of all explana-
tory science. The bungling drawing of a school-boy and the Sistine
Madonna, the hallucinations of a lunatic and the thought of a
Herbert Spencer, a demonic crime and a deed of the purest ethical
fulfillment of duty, are, in the same sense, necessary; but with the
knowledge of this necessity we have not come a single step nearer
to the task of their valuation.

The difference between these two kinds of attitudes has perhaps
never been more clearly sketched than in Fichte's book On the
Calling of Alan. If we assume that I have a fully adequate scientific
knowledge of the course of nature, I might discern that this grain
of sand which the storm has set in motion could not drift a hair's
breadth farther, unless the whole previous course of nature had been
quite different; what then would be gained for my own moral action?
The answer must be: Nothing. More than that, if this point of view
were the only possible for man, then this action would have no
longer, as a moral action, any significance, and could have none;
since as a part of the world event alike in value to all other parts it
would remain like in value, and it would be meaningless to select and
emphasize out of this continuum of facts and environments, alike in
value, single elements as especially valuable and significant. The
man who could not resign himself to this knowledge, who could
not be satisfied to continue, in cool content, at the point of view of
the silent contemplation of causes, must fall into conflicts similar
to those which Carlyle so vividly described in Sartor Resartus. We
must then, in order to an understanding for this new problem,
provisionally disregard, above all else, whatever the theory of evo-
lution has accomplished by way of scientific explanation, and reserve
for a later investigation the ethical valuation of this sequence of
development. The question which is now to occupy us is directed,
first of all, to the subject of our moral valuation. What do we call
good or bad?

This is the main question of all normative ethics in general, and its
answer by Kant will always remain a brilliant feat in this field. He

PROBLEMS OF ETHICS 409

proved, in the first place, that this predicate can be properly applied
to no action whatever, that we can speak of a good action in figur-
ative language only, when we believe that we can make from this
action an inference with regard to something else, — the disposition
of the actor; and that the same action which we do not hesitate
to describe as good, on the supposition of the correctness of this
inference, loses directly this character as soon as doubt of the cor-
rectness of the inference arises. This disposition, which we distin-
guish in this way, which forms the substrate of our moral valuation,
we call the good will, and the Magna Charta of the Kantian ethics
consists in the celebrated thesis: Nothing can possibly be good
except a good will. This reasoning appears to be as self-evident
as its result is important.

The whole ethical process is removed within the soul. While the
theory of evolution and, still more, utilitarianism could still hope
to obtain, with the character of the work, at the same time an ex-
pression with regard to the ethical value of the action; while, in this
combination of ideas, the ethical goodness of the disposition could
be judged by the usefulness or value to civilization of the performance
done, so that both these systems would have essentially the character
of an ethics of results, we have in Kant and his successors, most
decidedly, an ethics of dispositions. It has rightly been pointed
out that this ethics could grow only upon Protestant soil, that here
the same contradiction prevails which Luther once summed up in the
words: "Good works do not make the good man, but the good man
creates good works." All the excellences, but all the weaknesses
also, of Protestantism, cling to Kant's ethics.

First, let us follow the further stages of Kant's thought. How
must a good will be constituted, so that we may count it as ethically
good? All our acts happen in order to fulfill a purpose. The character
of the action depends upon the character of the purpose, which the
actor proposes for himself, which he affirms with his will, which he
makes his own. But if the purpose be no longer willed, then all the
actions cease, which hitherto had had to be accomplished for its
fulfillment. All those purposes, which under the circumstances
cannot be willed, cannot therefore produce that lasting constitution
of the will which we understand under the term the good will. But
among the different motivations of the will, there are some which
for the observer become separated. They have not a character such
that they could, under any circumstances, cease 'to motivate the
will; they are necessary and universal determinations of the will.
The imperative which they contain and with which they demand
action has not the hypothetical form: "If thou wilt obtain this or
that, you must; " but the absolute: " Thou shalt." It is a categorical
imperative, to which the will is here subordinated, which determines

410 ETHICS

my actions; and such a categorical imperative we term duty. Only
the dutiful will is good. It is clear that this determination shows
an exact analogy to the other norms of judgment in the logical and
the sesthetical field. The principle of contradiction states nothing
at all with regard to the single thoughts, it only asserts that our think-
ing can then alone make a claim upon a logical valuation while it
fills the condition which the principle of contradiction states. Like-
wise, the impulses of our wills can be morally valued only when they
refer to an absolute "Thou shalt;" if this is not the case, they are
excluded from the range of valuation, just as the play of our fancy,
which does not recognize the principle of contradiction, is excluded
from the realm of the norm of scientific thinking.

Here again the normal action of ethics is represented as a selective
process. While the evolutionist ethicist can estimate every single
content of human consciousness with reference to the point whether
it is preservative of the species or not, and thus give it ethical value,
the realm of the Kantian ethics is much more confined. Only those
impulses of the will occur with conscious subordination under the
command of duty, or in conscious opposition to it fall within the
realm of moral valuation. All others — and their name is legion —
must be termed unmoral. Not as if they become thereby actually
valueless; they may stand as high as you please in the intellectual,
aesthetic, or religious scale of values. But to bring them under just
the moral norms of judgment would be an attempt at an unappli-
able object. This is the point, perhaps, where the Kantian ethics
gives the hardest shock to the healthy human understanding. It
will always seem a paradox that we have a moral act when a man
with strong desires for theft, after a severe inner struggle, does not
put a silver spoon into his pocket, while the man who omits all this
quite as a matter of course may have no claim upon moral desert.
And yet each one of us would feel it as an insult, if he should be
praised for such omission. The solution of this difficulty lies in the
distinction of the value of the single resolve and that of the whole
moral personality. The man who is still led into temptation by silver
spoons stands morally upon the same plane upon which the scholar
stands who struggles with extreme mental effort to calculate a simple
example in multiplication. In the case of the more advanced person
our moral approval is not aroused because he no longer needs, in
this simple case, to appeal to the law of duty, but because we be-
lieve that we may conclude that his moral personality is attacking
other more difficult problems with full force, and that he is here in
himself feeling the full weight of the contest. If we were deceived
in this, if it prove true that he, content with what had been attained,
had withdrawn to the position of the ethical capitalist, our ethical
interest in him would likewise cease, just as our intellectual interest

PROBLEMS OF ETHICS 411

ceases in the scholar for whom there are no more problems in his
science. From this point of view the result is necessary that the
category of duties, to speak with Hegel, is absolutely infinite; and
in this perhaps lies the considerable difference between modern and
ancient ethics. For ancient ethics the ideal of the wise man was
a distinctly finitely determined amount. However difficult it might
be to fulfill the conditions for it, it could still be fulfilled in a human
life; and a further advance beyond this fulfilled ideal would have
been to the Greeks an absurdity: it is the "nothing too much"
transferred to the ethical point of view. It is otherwise in modern
ethics, and with this is connected the change in that the concept of
the infinite has become a concept of value. It is as Carlyle says:
" Fulfill the next duty which presents itself to thee, and when thou
hast fulfilled it, wait for ten, twenty, a hundred to be fulfilled." But
we recognize the degree of ethical development which a man has
attained by noting that it is no longer duty to him.

If the limits of the moral valuation have been much restricted
by the introduction of the concept of unmoral actions, it has been
extended in the other direction by the insight that now every action
which happens in fulfillment of a command of duty is to be valued
as the result of a moral disposition. We come thus to the problem
which, since the time of the ancient sophists, has not ceased to occupy
minds, and which may most simply be termed the anthropological
problem. What in the world is there that is not by individuals and
by people deemed to be moral! With what strange contents the
formal " Thou shalt " of morality is filled ! In face of these contradic-
tions, is there any sense at all in speaking of ethical commands? All
skeptical attacks upon ethics find in such considerations their strong-
est support; and here again the answer is easy when we reflect upon
the analogy with science, art, and religion. Aristotle and Democritus,
Hegel and Hobbes,have taught very differently, and yet all have been
busy with science. Raphael and Menzel are surely to be valued as
artists; Mahomet and Buddha were both religious geniuses of the first
magnitude. Why should it be different in the field of ethics? What
other men have held to be moral, how they have acted, this can be
valuable to me, in order for me to become clear with regard to my
own moral determination, just as the artist sees the works of other
masters, just as the scientific man must know the theorems of others.
But all this cannot be the standard for the formation of my own life.
I am, once for all, placed in this world, to be active there; I am
responsible to myself .for what I wish to accomplish with this life.
And so it can, it is true, be an encouragement to me that other men
have felt in themselves the same motive to moral activity; I can
give them my hand as striving for the same with me through the
separating centuries and across the estranging seas. But their way

412 ETHICS

of solving the great problems of life cannot be the standard for me
save in the sense that I receive them into my will, recognize them as
valid for my own life.

So, then, the whole weight of the distinction, the whole moral
process, is transferred to the individual. He is the point of depart-
ure and the goal of the struggle for a content in life. Is this now
egoism? This much-discussed question also suffers, as I believe, by
a defect in the statement of the problem. If it is intended that that
action is meant by egoism, the motive for which is one's own welfare
or happiness, by altruism, however, the action which aims at the
happiness of others, it is quite clear that these two contrasts have as
little meaning for the ethics of disposition as the complementary
contrast of beautiful and ugly. Moral action is completely indifferent
with regard to these contrasts. Moral actions can be characterized
as altruistic as well as egoistic, and the same is the case for unmoral
or bad actions. By knowing that distinct advantages have resulted
to the doer from an action, or that "the greatest happiness of the
greatest number" has resulted from it, I have not gained one step
for the moral valuation of this action. I should surely act immorally
if I omitted an action acknowledged as moral by me because it
would involve pain for others and thus would have an anti-altruistic
character. Whence this confusion of the altruistic with the moral
arose is easy to see. Long before the child could himself act morally,
it must be accustomed to feel that its beloved self cannot be the sole
standard for its action; and to the end that it keep peace and content
with its brothers and playmates, it is properly accustomed to regard
in its action the welfare of the human beings about it. That is a
preparatory step to moral action; but, strictly speaking, it can be
counted as moral by those only who are determined not to recognize
the limits between psychological motivation and normative deter-
mination.

It would be an interesting task to trace the relations into which
the autonomous moral individual enters with the great moral
institutions which dominate the community and have combined in
usage, society, and state, and which Hegel described in a happy
expression as " objective morality." Here it is no longer the regard
for the weal or woe of fellow men which strives to gain influence
over my action; here the ethical will of past generations of my own
ancestors accosts and asks me whether I can bring my action into
harmony with that which they willed and for which they strove.
It is a slight disadvantage to the ethically directed man that, in
order to protect these moral institutions from injury, an arsenal
of punishments, of social influences, of boycotts, and of whatever
finer or coarser means of compulsion there may be, are set up. This
arsenal is necessary to sustain the social structure which alone

PROBLEMS OF ETHICS 413

affords the chance for moral action; and he who calculates with
pleasure and pain, who tries to arrange his life as happily as possible,
will be restrained by shrewd calculation from injuring the prevailing
moral institutions. The moral man has nothing to do with such
considerations. When he affirms the objective morality, he does so
because he recognizes his moral will as identical with that of previous
generations which have made these forms. But the time can come
when he discovers that a moral life within these forms is no longer
possible for him, when with deep regret he sees the bond of continuity
break which knit him in affection with the past, when he must
resolve to enter new untrodden paths, just as Copernicus was forced
to resolve to substitute a new knowledge for those which had satisfied
centuries. Such a man will endure calmly and patiently the con-
sequences which result from such a course; he will not expect to
be justified, through the purity of his intentions, in the eyes of his
fellows, if he undertakes to lay hands on the institutions which the
moral consciousness of his contemporaries recognizes as valid. But
he will also know that these same institutions owe what sacredness
they possess to the blood of previous martyrs, that these shadows
of a past can only then speak to a living generation when they have
tasted the sacred blood of sacrifice.

So then we see two great movements in our time struggling about
the ethical questions. The one has on its side the whole apparatus
of scientific conceptions, the presupposition of necessary events
without exceptions, the knowledge that the single individual is an
infinitely small element in a necessary sequence of development. It
can explain everything, deduce everything from its conditions. At
one point only its power breaks down : it cannot make the individual
comprehend why he should raise a finger to keep in motion this
machine which goes of itself.

And, opposed to this, is the other movement, which rests upon the
one fact that the point of view of its opponent, the scientific, is also
a relation of reality to values, and that man alone introduces these
values into reality, measures and tests it by these values. According
to this movement, every new human life has the question put to it,
what it can accomplish with these values, whether it is capable of
making something out of reality, out of itself, which has in itself a
value such as to raise it above the flux of appearances as the bearer
of these values. Everything previous as well as everything subsequent
vanishes before these thoughts that it is now day, that the night is
soon coming when no man can work^that at the day's end the day's
work must be done. But what each recognizes as his day's work, he
must himself find within himself. This decision is his destiny.

I cannot better close than with the words of the man whose life
had little joy, but who grappled with these questions in the soUtude

414 ETHICS

of Craigenputtock, in the supreme solitude of the human wilderness
of London, with a seriousness which still to-day proves to be soul-
wooing and soul- winning : " Centuries have passed that thou might-
est be born, and centuries are waiting in dumb expectation of what
thou wilt accomplish with this life, now that it has begun." And
what this hfe can offer Carlyle, by combining the thoughts of Fichte
and of Goethe, has united in the call:

" Work and despair not."

SECTION F — .ESTHETICS

{Hall 4, September 23, 3 p. m.)

Chairal^n: Professor James H. Tufts, University of Chicago.
Speakers: Dr. Henry Rutgers Marshall, New York City.
Professor Max Dessoir, University of Berlin.
Secretary: Professor Max Meyer, University of Missouri.

THE RELATION OF ESTHETICS TO PSYCHOLOGY AND

PHILOSOPHY

BY HENRY RUTGEES MARSHALL

[Henry Rutgers Marshall, Practicing Architect, President of the New York
Chapter, American Institute of Architects, Member of Art Commission,
Citv of New York. b. July 22, 1852, New York City. B.A. Columbia Uni-
versity, 1873; U.K. ibid. 1875; L.H.D. Rutgers College, 1903. Member
American Psychological Association, Society of American Naturalists, Fel-
low American Institute of Architects, Honorary Member National Society of
Mural Painters, Member American Philosophical Association. Author of
Pain, Pleasure, and Esthetics; ^^sthetic Principles ; Instinct and Reason.]

If conventional divisions of time are to serve as means by which we
may mark the movement of thought as it develops, we may well
say that the nineteenth century saw a real awakening in relation
to aesthetics among those who concern themselves with accurate
thinking, — a coming to consciousness, as it were, of the importance
to the philosophy of life of the existence of beauty in the world, and
of the sense of beauty in man.

And with this awakening came a marked breadth of inquiry; an
attempt to throw the light given by psychological analysis upon the
broad field of aesthetics, and an effort to grasp the relations within
the realm in which beauty holds sway to philosophy as a whole.

That the questions thus presented to us have been answered, I
imagine few, if any, would claim; rather may we say that the nine-
teenth century set the problems which it concerns the aesthetician
of the twentieth century to solve; and this without underestimating
the value of the work of the masters in aesthetics who lived and
vsTote in the century so lately closed, some of whom are fortunately
with us still.

Of these present problems M. Dessoir will treat in his address to
follow mine; in the regretted absence of Professor Lipps the privilege
has been granted to me to consider with you briefly the relations
of aesthetics to psychology, and to philosophy, which must in the

418 ^ESTHETICS

end determine the nature of the problems to be studied by the aesthe-
tician, and the import of the solutions of these problems which they
present for our consideration.

I. The Relation of Esthetics to Psychology

We live in what may well be called the era of psychological develop-
ment, an era marked by the recognition of the truth that no philo-
sophical view of life can be adequate which does not take full account
of the experience of the individual human spirit which interprets this
life. And so quite naturally for ourselves, and in all probability
quite in accord with the habit of thought of the immediate future,
we begin our study by the consideration of the relation of aesthetics
to psychology.

In turning for light to psychology, the sesthetician finds himself of
course asking what is the nature of the states of mind related to his
inquiry; and here at once he finds himself confronted with a distinc-
tion which must be made if a correct aesthetic doctrine is to become
established. He notes that there is a sharp difference between (1)
the mental attitude of an artist who produces works of beauty; and
(2) the mental attitude of a man at the moment when he appreciates
beauty in his experience.^ The failure to note this distinction has in
my view led to much confusion of thought among the sestheticians
of the past, and to the defense of dogmas which otherwise would
not have been maintained.

That this distinction is an important one becomes clear in the
fact that the sense of beauty is aroused in us by objects in nature
which bear no relation to what men call fine art. The mental state
of the appreciator of beauty has therefore a breadth which does not
belong to the mental state which accompanies, or leads to, the pro-
duction of works of beauty by the artist.

And yet it should not surprise us that this distinction has so
often been overlooked; for the theorists first follow the trend of
thought of the uncritical man, and this uncritical man does not
naturally make the distinction referred to.

For, on the one hand, even the least talented of men has some
little tendency to give part of his strength to artistic creation in one
form or another; the creative artist is guided by what is a truly racial
instinct, which under favorable conditions will appear in any man
who is not defective: each of us thus in the appreciation of beauty
throws himself to some degree into the attitude of the creative artist.

And, on the other hand, the artist, when not in creative mood, falls
back into the ranks of men who keenly appreciate beauty but who

1 Cf . my MstheHc Principles, chap, i, " The Observer's Standpoint," and
chap. Ill, "The Artist's Standpoint."

THE RELATIONS OF ESTHETICS 419

are not productive artists; he thus alternately creates and appre-
ciates, and with difficulty separates his diverse moods.

We may well consider these two distinguishable mental attitudes
separately.

In asking what is the nature of the experience which we call the
sense of beauty, we are stating what may well be held to be the most
important problem in aesthetics that is presented to the psychologist.

Man is practical before he deals with theory, and his first theo-
retical questionings are aroused by practical demands in connection
with his failures to reach the goal toward which he strives. The de-
velopment of modern aesthetic theory has in the main quite naively
followed this course, and we may properly consider first the psycho-
logical inquiries which seem to have the most direct bearing upon
practical questions.

The artist asks why his efforts so often fail, and he is led to inquire
what are the qualities in his work which he so often misses, but now
and again gains -^dth the resulting attainment of beauty.

It is thus that we naturally find the sesthetician appealing to the
psychologist, asking him what special types of impression yield
beauty, what special characteristics of our mental states involve the
fullest aesthetic experience.

The psychologist is naturally first led to consider certain striking
relations found within the beautiful object which impresses us, and
to inquire into the nature of the psj^chic functioning which is in-
volved with the impressions thus given. He thus comes to consider
the relations of the lineal parts of pleasing plane-surface figures; and
the study of these relations has given to us such investigations as
the notable ones of Fechner in respect to the "Golden Section,"
which have been supplemented by the more rigid tests of Dr. Witmer
and Doctors Haines and Davies in our own day. In similar manner
the basis of the beauty found in symmetry and in order, and the
problems related to rhythm, have been closely studied, especially
in late years by Lipps; and the fundamental principles of tonal
relation, and of melodic succession, by Helmholtz, Stumpf, and
later writers.

But all these studies of the striking characteristics found in the
object are for the psychologist necessarily involved with the study
of the distinctly subjective accompaniments in the sense of beauty
aroused by the objective forms thus brought to our attention, and
he is led to dwell upon the active part the mind takes in connection
with aesthetic appreciation. We see this tendency in Berenson's
emphasis, and perhaps on the whole over-emphasis, of the import-
ance of the interpretation of works of art, in the group of what I
would call the arts of sight, in terms of the tactile sensibilities. But

420 ^ESTHETICS

we see it much more markedly in the important studies of Lipps,
who shows us how far our appreciation of beauty in nature, and in
artistic products, is due to the sympathetic introjection of ourselves
as it were into the object, — to what he calls Einfilhlung.

But, broad as he shows the applicability of this principle to be, it
is clear that we have not in it the solution of the fundamental aesthetic
problem with which the psychologist must deal when appealed to by
the sesthetician. For no one would claim that all of this sympathetic
introjection — this Einfilhlung — is aesthetic : the aesthetic Einfilhl-
ung is of a special type. Nor to my mind does it seem clearly
shown that there are no sources of beauty which do not involve this
introjection, as would be the case if we had reached in this principle
the solution of the fundamental sesthetico-psychologic problem. For
instance, the sense of beauty experienced when I look at some one
bright star in the deep blue of the heaven seems to me to be inex-
plicable in terms of such introjection.

All this work, however, brings help to the practical artist and to
the critic. They do not acknowledge it fully to-day, but year by year,
more and more will the influence of the results of these studies be
felt as they gain the attention of thinking men.

Nevertheless, we cannot but face the fact that the practical benefit
to be gained from them is of a negative sort. There is no royal road
to the attainment of beauty; but the psychologist is able to point
out, by the methods here considered, the inner nature of certain
sources of beauty ; thus teaching the artist how he may avoid ugliness ,
and even indicating to him the main direction in which he may best
travel toward the attainment of his goal.

But, after all, the relations thus discovered in the beautiful object,
and the related special analyses of mental functioning which are
involved with our appreciation of beauty, tell us of but relatively
isolated bits of the broad realm of beauty. The objects which arouse
within us the sense of beauty are most diverse, and equally diverse
are the modes of mental functioning connected with the appreciation
of their beauty. *

And this has led to the formulation of such principles as that of
the "unity of manifoldness " of which Fechner makes so much, and
that of the monarchischen Unterordnung which Lipps has more lately
enunciated.

It is indeed of great interest to inquire why it is that the processes
which lead to the recognition of these principles are so clearly defined
in many cases where the sense of beauty is aroused. But very evi-
dently these general principles, important though they be in them-

' Nothing has shown this more clearly than the investigations of Haines and
Davies in reference to the Golden Section of which we have spoken above. See
Psychological Review, xi, 415.

THE RELATIONS OF ^ESTHETICS 421

selves, are not ones upon which we can afford to rest: for clearly
they apply in very many cases where beauty does not claim sway.

Our whole mental life exemplifies the unification of the manifold,
and monarchic subordination, whether the processes be aesthetic or
not. It does not suffice us to show, what is thus shown, that the
aesthetic states conform with conditions of our mental life that
have a broad significance, although it is of great importance to
demonstrate the fact: for our mental functioning in the apprecia-
tion of beauty appears thus as in truth an important type, but
for aU that but a special and peculiar type of the functioning which
we thus bring into prominence.

The problem then remains, what is the special nature of this
functioning which yields to us the sense of beauty?

And here in my view we have the problem which is of prime
importance to aesthetics to-day, and which psychology alone can
answer; namely, what is the characteristic that differentiates the
sense of beauty from all other of our mental states? Until this
question is answered, all else must seem of secondary importance
from the standpoint of theoretical psychology, however important
other forms of inquiry may be from a practical point of view.

When the psychologist turns his attention to this problem, he
at once perceives that he is unable to limit his inquiry to the experi-
ence of the technically trained artist, or even to that of the man of
culture who gives close attention to aesthetic appreciation.

Beauty is experienced by all men. But beauty is very clearly of
varied types, and the sense of beauty is evidently called out by
impressions of most varied nature; but the fields of what is considered
beautiful by different people so far overlap that we can rest assured
that we all refer to an experience of the same characteristic mental
state when we proclaim the existence of beauty; for when we by
general agreement use a special term as descriptive of an objective
impression, we do so because this impression excites in us certain
more or less specific mental states; and when different people use
the same term in reference to objects of diverse nature, we are wont
to assume, and are in general correct in assuming, that these objects
affect these different people in approximately the same way.

It seems probable, therefore, that if the child, who has learned how
to apply words from his elders, speaks of having a beautiful time at
his birthday party; and if the grown man speaks of a beautiful day;
and if the pathologist speaks of his preparation of morbid tissue as
beautiful; and if the artist or connoisseur speaks of the beauty of
a picture, a statue, a work of architecture, a poem, a symphony;
then the word beauty must be used to describe a certain special mental
state which is aroused in different people by very diverse objective
impressions.

422 AESTHETICS

This view is strengthened when we consider that the apphcation
of the term by individuals changes as they develop naturally or by
processes of education; and that the standards of beauty alter in
like manner in a race from generation to generation as it advances
in its development.

We must then look for the essence of beauty in some quality of
our mental states which is called up by different objective impres-
sions in different people, and under diverse conditions by different
objects at different times in the same individual.

Search for such a quality has led not a few psychologists to look
to pleasure as the quality of our mental states which is most likely
to meet our demand. It is true that the consideration of pleasure
as of the essence of the sense of beauty has not often been seriously
carried out ; apparently because so many of what we speak of as our
most vivid pleasures appear as non-sesthetic; and because pleasure
is recognized to be markedly evanescent, while beauty is thought of
as at least relatively permanent.

It is true, also, that there is a hesitancy in using the word pleasure
in this connection; many writers preferring the less definite word
"feeling" in English, and "gefiihl" in German. But by a large
number of psychologists the words pleasure and feeling are used as
synonyms; and those who, with me, agree that what we loosely call
feeling is broader than mere pleasure, must note that it is the pleas-
urable aspect alone of what is called "feeling" that is essentially
related to our experience of the sense of beauty.

All of us agree, in any event, that the sense of beauty is highly
pleasant; and, in fact, most of our aestheticians have come to assume
tacitly in their writings that the field of aesthetics must be treated
as a field of pleasure-getting; and this whether or not they attempt
to indicate the relation of pleasure-getting to the sense of beauty.

The suggestion that pleasure of a certain type is of the essence of
beauty seems the more likely to prove to be satisfactory when we
consider that pleasure is universally acknowledged to be the con-
tradictory opposite of pain; and that we have in ugliness, which is
always unpleasant, a contradictory opposite of beauty.^

Clearly then it behooves the psychologist to give to the sesthetician
an account of the nature of pleasure which shall be compatible with
the pleasurable nature of the sense of beauty; and which shall either
explain the nature of this sense of beauty in terms of pleasure, or
explain the nature of pleasure in a manner which shall throw light
upon the nature of this sense of beauty to which pleasure is so indis-
solubly attached.

' It is of course agreed that beauty and ugliness may be held together in a
complex impression: but in such cases the beauty and the ugliness are inherent
in diverse elements of the complex.

THE RELATIONS OF ESTHETICS 423

The aesthetician thus demands urgently of the psychologist an
analysis of the nature of pleasure; and an analysis of this so-
called "feeling," which shall show the relation between the two
experiences.

Concerning the latter problem I hope some day to have something
to say.

Those of you who happen to be familiar with my published works
will realize that my efforts in this field in the past have been given
largely to the study of the former problem. My own view may be
succinctly stated thus.

While all aesthetic experiences are pleasant, very evidently much
that we call pleasant is not aesthetic. We must look then for some
special differentiation of aesthetic pleasure, and this I find in its
relative permanency.

This view is led up to by a preliminary study of the psychological
nature of pleasure.

Pleasure I find to be one phase of a general quality — Pleasure-
Pain — which, under proper conditions, may inhere in any emphasis
within the field of attention, or, to use more common language, may
belong to any element of attention.

Now pleasure, as we have said, is notably evanescent, but this
does not preclude the existence of pleasurable states of attention
which are relatively permanent. This permanency may be given by
the shifting of attention from one pleasurable element to another;
by the summation of very moderate pleasures, etc., etc.

Any pleasant psychic element may become an element of an
aesthetic complex : and any psychic complex which displays a relative
permanency of pleasure is in that fact aesthetic. Our aesthetic states
are those in which many pleasant elements are combined to produce
a relative permanency of pleasure.

Our "non-aesthetic pleasures," so called, are those states which
have been experienced in the past as vividly pleasant, and to which
the name pleasure has become indissolubly attached: but they are
states which do not produce a relatively permanent pleasure in
revival; and correctly speaking, are not pleasures at the moment
when they are described as such, and at the same time as non-
aesthetic.

I am glad to feel that this view of mine is not discrepant from that
of Dr. Santayana, as given in quite different terms in his book en-
titled The Sense of Beauty. For what is relatively permanent has
the quality which I call realness; and that in experience which has
realness we tend to objectify. Hence it is quite natural to find Dr.
Santayana defining beauty as objectified pleasure.

You will not blame me I believe for thinking that my own defini-
tion cuts down closer to the root of the matter than Dr. Santayana 's.

424 AESTHETICS

But if this theory of mine is found wanting, the j£sthetician will
not cease to call upon the psychologist for some other which shall
meet the demands of introspection; and which shall accord with our
experiences of the sense of beauty, which in all their wealth of impres-
sion the sesthetician offers to the psychologist as data for the labor-
ious study asked of him.

Before leaving this subject I may perhaps be allowed to call
attention to the fact that the theoretical view, which places the essence
of the sense of beauty in pleasure-getting, if it prove to be true, is
not without such practical applications as are so properly demanded
in our time. For if this view is correct, it teaches' to the critic a lesson
of sympathetic tolerance; for he learns from it that the sources from
which the sense of beauty are derived differ very markedly in people
of diverse types: and it warns him also against the danger of an
artificial limitation of his own a)sthetic sense, which will surely
result unless he carefully avoids the narrowing of his interests.

It teaches further that there is no validity in the distinction
between fine art and aesthetics on the one hand, and beauty on the
other, on the ground, commonly accepted by the highly trained
artist and connoisseur, that a work of art may deal with what is not
beautiful.

For it appears that while the sense of beauty is the same for each
of us, the objects which call it out are in some measure different for
each.

Now it happens naturally that the objects which arouse the sense
of beauty in a large proportion of men of culture get the word beauty
firmly attached to them in common speech.

But under the view here maintained, it must be that the highly
trained artist. or critic will pass beyond these commoner men, and
find his sense of beauty aroused by objects and objective relations
quite different from those which arouse the sense of beauty in the
commoner man; so that often he may deal with the beauty of
elements in connection with which beauty is unknown to the com-
moner man, and even with elements which arouse a sense of ugliness
in the commoner man; while on the other hand the objects which
the commoner man signalizes as most beautiful, and which are cur-
rently so called, may not arouse in the trained artist or critic the
sense of beauty which is now aroused in him by effects of broader
nature, and of less common experience.

The critic and the skilled artist thus often find their aesthetic sense
aroused no longer by the objects to which the word beauty has by com-
mon consent come to be attached ; although with the commoner man he
still uses the word beauty as descriptive of the object which arouses
the aesthetic thrill in the mass of normally educated men. He may

THE RELATIONS OF ESTHETICS 425

even find his aesthetic sense aroused by what the common man calls
ugly; although it is for himself really beautiful. And he comes thus
quite improperly to think of the highest art as in a measure inde-
pendent of what he calls "mere beauty." What he has a right to
say, however, is merely this, that the highest art deals with sources
of beauty which are not appreciated by even the generally well-
cultivated man.

I have dwelt, perhaps, too long on the psychological problems
presented when the psychologist attempts to describe to the sesthet-
ician the nature of the experience of one who appreciates beauty;
and have left perhaps too little time for the consideration of the
problems presented when he is asked to consider the nature of the
experience of the artist who creates.

The man who finds strongly developed within him the creative
tendency, is wont, when he turns to theory, to lay emphasis upon
expression as of the essence of beauty.

It is, of course, to be granted that the process of Einfilhlung, —
of introjection, — above referred to, leads us to find a source of
beauty in the vague imagination of ourselves as doing what others
have done; and we may take great aesthetic delight in reading,
through his work, the mind of the man who has created the object
of beauty for us. But evidently, when we lay stress upon this intro-
jection, we are dealing with the appreciation of beauty, and not with
the force which leads to its production.

Just as clearly is it impossible to hold that expression is of the
essence of the making of beauty. For expressiveness is involved in
all of man's creative activity, much of which has no relation what-
ever to the aesthetic. The expression of the character of the genius
of the inventor of a cotton loom, or of the successful leader of an
army in a bloody battle, excites our interest and wonder; but the
expression of his character as read in the result accomplished does
not constitute it a work of beauty.

I speak of this point at this length because in my opinion views
of the nature of that here objected to could not have been upheld
by such men as Bosanquet and Veron had they kept clear the dis-
tinction referred to above between the experience of one who ap-
preciates beauty, and the experience of the creative artist; and
especially because the teaching of the doctrine thus combated is
wont to lead the artist whose cry is " Art for Art's sake" to excessive
self-satisfaction, and to lack of restraint which leads to failure.^

* In order to avoid misunderstanding, I may say here that notwithstanding
these remarks I am in full sympathy with the artist who thus expresses himself,
as will presently appear clear.

426 ESTHETICS

The strong hold which this theory has in many minds has its value,
however, in the emphasis of the fact that aesthetic creation is due
to impulses which are born of innate instincts expressing them-
selves in the production of works of beauty. And if this be so, we see
how true it must be that each of us must have in him some measure
of this instinct; and that the appearance of its appropriate impulses
should not mislead us, and induce us to devote our lives to the
worship of the Muses, unless we become convinced that no other work
can adequately express the best that is in us.

But the true artist is not troubled by such questionings. He finds
himself carried away by what is a true passion; by what is instinct-
ive and not ratiocinative.

The fact that the artist is thus impelled by what may well be called
the "art instinct" is one he could only have learned from the psy-
chologist, or when in introspective mood he became a psychologist
himself; and it carries with it corollaries of great value, which the
psychologist alone can elucidate.

It teaches the artist, for instance, that his success must be deter-
mined by the measure of this instinct that is developed within him;
that he must allow himself to be led by this instinct; that his best
work will be his "spontaneous" work. This, of course, is very far
from saying that he cannot gain by training; but it does mean that
he must learn to treat this training as his tool; that he must not
trust overmuch to his ratiocinative work, the result of which must
be assimilated by, and become part of, his impulsive nature, if he is
to be a master.

An artist is one in whom is highly developed the instinct which
leads him to create objects that arouse the sense of beauty. The
expression of this instinct marks his appropriate functioning. He
may incidentally do many useful things, and produce results apart
from his special aptitude; but as an artist his work is solely and
completely bound up in the production of works of beauty.

We naturally ask here what may be the function in life of the
expressions of such an instinct as we have been studying, and this
leads us to consider a point of more than psychological interest,
and turns our thought to our second division.

II. The Relation of ^Esthetics to Philosophy

For while the science of psychology must guide, it can never dom-
inate the thought of the philosopher who strives to gain a broad view
of the world of experience; and, as will appear below, the sesthetician
calls upon the philosopher for aid which the psychologist as such
cannot give.

. THE RELATIONS OF ESTHETICS 427

In approacMng this subject we may take at the start what we may
call the broadly philosophical view, and may consider the question
raised immediately above, where we ask what may be the function
in life of the art instinct, and what the significance of the aesthetic
production to which its expression leads.

We, in our day, are still strongly influenced by the awakening of
interest in the problems of organic development with which Darwin's
name is identified, and thus naturally look upon this problem from
a genetic point of view; from which, to my mind, artistic expression
appears, as I have elsewhere argued at length, as one of nature's
means to enforce social consolidation. But it is possible that we
are led, by the present-day interest above spoken of, to over-
emphasize the importance of the processes of the unfolding of our
capacities, and it is not improbable that those who follow us, less
blinded by the brilliancy of the achievement of the evolutionists,
may be able to look deeper than we can into the essence of the
teleological problem thus raised.

That art is worthy for art's sake is the conviction of a large body
of artists, who labor in their chosen work often with a truly martyr-
like self-abnegation; and as an artist I find myself heartily in sym-
pathy with this attitude. But aesthetics looks to philosophy for
some account of this artistic reXos, which shall harmonize the artist's
effort with that of mankind in general, from whom the artist all too
often feels himself cut off by an impassable gulf.

The study of aesthetics by the philosopher from the genetic stand-
point has, however, already brought to our attention some facts
which are both significant and helpful.

It has shown us how slow and hesitant have been the steps in the
development of aesthetic accomplishment and appreciation in the
past, and how dependent these steps have been upon economic con-
ditions. This on the one hand arouses in us a demand for a fuller
study of the relations of the artistic to the other activities of men;
and on the other hand is a source of encouragement to critic and
artist alike, each of whom in every age is apt to over-emphasize the
artistic failures of his time, and to minimize the importance of its
artistic accomplishment.

This genetic study has a further value in the guidance of our
critical judgment, in that it shows us that the artistic tendencies
of our time are but steps in what is a continuous process of develop-
ment. It shows us arts which have differentiated in the past, and
teaches us to look for further artistic differentiations of the arts in
the future; thus leading us to critical conclusions of no little im-
portance. This consideration seems to me to be of sufficient interest
to warrant our dwelling upon it a little at length.

428 ESTHETICS

The arts of greatest importance in our time may well be divided
into the arts of hearing (that is, literature, poetry, music), and the
arts of sight (that is, architecture, sculpture, painting, and the
graphic arts).

These diverse groups of arts were differentiated long before any
age of which we have a shadow of record. But many animals display
what seem to be rudimentary art instincts, in which rhythmical move-
ment (which is to be classed as an art of sight) and tonal accompani-
ment are invariably combined — as they are also in the dance and
song of the savage; and this fact would seem to indicate that in the
earliest times of man's rise from savagery the differentiation between
the arts of sight and the arts of hearing was at least very incom-
plete.

But leaving such surmises, we may consider the arts of sight and
the arts of hearing in themselves. We see them still in a measure
bound together; for many an artist, for instance, devotes his life
to the making of paintings which "tell a story," and many a poet
to the production of "word-pictures."

In general, however, it may be said that the arts of hearing and
the arts of sight express themselves in totally different languages,
so to speak, and they have thus differentiated because each can give
a special form of aesthetic delight.

Turning to the consideration of each great group, we note that
the arts of sight have become clearly differentiated on lines which
enable us to group them broadly as the graphic arts, painting,
sculpture, and architecture. Each of these latter has become im-
portant in itself, and has separated itself from the others, just so
far as it has shown that it can arouse the sense of beauty in a man-
ner which its kindred arts of sight cannot approach. It is true that
all the arts of sight hold together more closely than do the arts of
sight, as such, with the arts of hearing, as such. But it is equally
clear that the bond between the several arts of sight was closer
in earlier times than it is to-day, in the fact that modeled paint-
ing, and colored sculpture, were common media of artistic expres-
sion among the ancients, the latter being still conventional even so
late as in the times of the greatest development of art among the
Greeks.

But the modern has learned that in painting and graphics the
artist can gain a special source of beauty of color and line which he
is able to gain with less distinctness when he models the surface upon
which he works : and the experience of the ages has gradually taught
the sculptor once for all that he in his own special medium is able
to gain a special source of beauty of pure form which no other arts
can reach, and that this special type of beauty cannot be brought
into as great emphasis when he colors his modeled forms.

THE RELATIONS OF ESTHETICS 429

In my view we may well state, as a valid critical principle, that,
other things being equal, in any art the artist does best who presents
in his chosen medium a source of beauty which cannot be as well
presented by any other art. That this principle is appreciated and
widely accepted (although implicitly rather than explicitly) is
indicated by the unrationalized objection of the cultivated critic in
our day to colored sculpture or to modeled painting, and in a more
special direction to the use of body-color in aquarelle work. The
objection in all cases is apparently to the fact that the artist fails to
bring into prominence that type of beauty which his medium can
present as no other medium can.

Personally I have no objection to raise to a recombination of the
arts of sight, provided a fuller sense of beauty can thereby be
reached. But it is clear that this recombination becomes more and
more difficult as the ages of development pass; and I believe the
principle of critical judgment above enunciated is valid, based as
it is upon the inner sense of cultivated men.

Better than attempts to recombine the already differentiated
arts of sight are attempts to use them in conjunction, so that our
shiftings of attention from one type of beauty to another may carry
with them more permanent and fuller sesthetic effects ; and such
attempts we see common to-day in the conjunction of architecture
and of sculpture and of painting, in our private and public galleries,
in which are collected together works of the arts of sight.

Now if we turn to the consideration of the arts of hearing, we find
a correspondence which leads to certain suggestions of no little
importance to the critical analyst in our day.

The arts of hearing have become differentiated on lines which
enable us to group them broadly as rhetoric, poetry and literature,
and music. Each has become important in itself, and has gradually
separated itself from the others; — and this just so far as it has
shown that it can arouse in men, in a special and peculiar manner,
the sense of beauty.

It is true, as with the arts of sight, that the special arts of hearing
still hold well together.

But in relatively very modern times music, having discovered a
written language of its own, has differentiated very distinctly from
the other arts of hearing. Men have discovered that 'pure music
can arouse in a special manner the sense of beauty, and can bring to
us a form of sesthetic delight which no other art can as well give.

Poetry has long been written which is not to be sung; and it has
gained much in freedom of development in that fact.

Music in our modern times is composed by the greatest masters
for its own intrinsic worth, and not as of old as a mere accompani-

430 ESTHETICS

merit of the spoken word of the poet; the existence of the works
of Bach, to mention no others, tells of the value of this differentiation.

And here I think we may apply with justice the principle of criticism
above presented. The poet and the musician each do their best work,
other things being equal, when they emphasize the forms of beauty
which their several arts alone can give. We have here in my view
a rational ground for the repulsion many of us feel for the so-called
"programme music" of our day.

Music and literature of the highest types nowadays present
sources of beauty of very diverse character, and any effort to make
one subsidiary to the other is likely to lessen the aesthetic worth of
each, and of the combination.

Here again I may say that I have no objection to raise to a recom-
bination of the arts of hearing, provided a fuller sense of beauty
can thereby be reached. But this recombination becomes year by
3^ear more difficult as the several arts become more clearly differen-
tiated, and must in my view soon reach its limit.

The opera of to-day attempts such a recombination, but does so
either to the detriment of the musical or of the literary constituent;
that is clear when we consider the musical ineptitude of such operas
as deal with a finely developed drama, and the literary crudeness of
the plot-interest in Wagner's very best works. Such a consideration
makes very clear to us how much each of the great divisions of the
arts of hearing has gained by their differentiation, and by their inde-
pendent development.

Here as with the arts of sight we may, in my view, hope for
better eesthetic results from the development of each of the differ-
entiated arts in conjunction; rather from the persistent attempt to
recombine them, with the almost certain result that the aesthetic
value of each will be reduced.

But aesthetics demands more of philosophy than an account of the
genesis of art, with all the valuable lessons that this involves. It de-
mands, rightly, that it be given a place of honor in any system which
claims to give us a rationalized scheme of the universe of experience.

The aesthetician tells the philosopher that he 'cannot but ask
himself what significance aesthetic facts have for his pluralism, or
for his monism. He claims that this question is too often overlooked
entirely or too lightly considered; but that it must be satisfactorily
answered if the system-maker is to find acceptance of his view.
And in the attempt to answer this and kindred questions, the aesthet-
ician is not without hope that no inconsiderable light may be thrown
by the philosopher upon the solution of the problems of aesthetics
itself.

THE RELATIONS OF .ESTHETICS 431

Nor are the problems of aesthetics without relation to pure meta-
physic. The existence of aesthetic standards must be considered by
the metaphysician, and these standards, with those of logic and ethics,
must be treated by him as data for the study of ontological
problems.

But beyond this, aesthetics cries out for special aid from the
ontologist. What, he asks, is the significance of our standards of
aesthetic appreciation? What the inner nature of that which we call
the real of beauty? What its relation with the real of goodness
and the real of truth?

From a practical standpoint this last-mentioned question is of
special import at this time. For the world of art has for centuries
been torn asunder by the contention of the aesthetic realists and their
opponents.

That, in its real essence, beauty is truth, and truth beauty, is
a claim which has often been, and is still heard; and it is a claim
which must finally be adjudicated by the metaphysician who deals
wth the nature of the real.

The practical importance of the solution of this problem is brought
home forcibly to those who, like myself, seem to see marked aesthetic
deterioration in the work of those artists who have been led to listen
to the claims of aesthetic realism; who learn to strive for the expres-
sion of truth, thinking thus certainly to gain beauty.

That many great artists have announced themselves as aesthetic
realists shows how powerfully the claims of the doctrine appeal to
them. But one who studies the artistic work of Leonardo, for in-
stance, cannot but believe that he was a great artist notwithstanding
his theoretical belief, and cannot but believe that all others of his
way of thinking, so far as they are artists, are such because in them
genius has overridden their dogmatic thought.

It is clearly not without significance that the world of values is
by common consent held to be covered by the categories of the True,
the Good, and the Beautiful. This common consent seems surely
to imply that each of the three is independent of the other two,
although aU are bound together in one group. And if this is true, then
the claim of the aesthetic realist can surely not be correct.

But this claim will not be overthrown by any reference to such
a generalization as that above mentioned. The claim of the aesthetic
realist is based upon what he feels to be clear evidence founded upon
experience; and he cannot be answered unless we are able to show
him what is the basis foT his ready conviction that truth and beauty
are one and identical; and what is the true relation between the
True, the Good, and the Beautiful. And these problems, which are
in our day of vital importance to the artist, the philosopher alone
can answer.

432 ESTHETICS

In my view some aid in the solution of this problem may be gained
from the examination of the meaning of our terms. From this study
I feel convinced that we must hold that when we speak of the True,
and the Good, and the Beautiful, as mutually exclusive as above,
we use the term "true" in a narrow sense. On the other hand, the
True is often used in a broader sense, as equivalent to the Real.

This being so we may say

That the Beautiful is the Real as discovered in the world of im-
pression; the relatively permanent pleasure which gives us the sense
of beauty being the most stable characteristic of those parts of the
field of impression which interest us we may also assent

That the Good is the Real as discovered in the world of expression,
that is, of impulse, which is due to the inhibited capacity for expres-
sion, and the reaction of the self in its efforts to break down the
inhibition. And in the same way we may conclude

That the True (using the term in the narrow sense) is the Real
as discovered in the realm of experience exclusive of impression or
expression.

The Real
or

The True
(in the broad sense
of the term)

The Real of Impression — The Beautiful

The Real of Expression — The Good

The Real in realms — The True
exclusive of a and ft (in the narrower

sense of the term)

That the Beautiful is part of the Real, that is, is always the
True, using the term true in the broader sense, is not questioned: and
that, in my view, is the theoretical truth recognized by the aesthetic
realists. But in practice the aesthetic realist maintains that the
beautiful is always the true, using the term true in the narrow sense,
and in this, in my view, lies his error.

And if the relation of the beautiful to the true demands the
attention of the philosopher, equally so does the relation of the
beautiful to the good. As I look upon it, all of the true (using the
term as above explained in the narrow sense) and all of the good,
so far as either involve relatively permanent pleasure of impression,
are possible elements of beauty. But, on the other hand, it seems clear
that neither the true (still using the term in the nari'ower sense) , nor
the good, is necessarily pleasing, but may be unpleasant, and there-
fore either of them may be an element of ugliness, and as such must
lose all possibility of becoming an element in the beautiful.

One further word, in closing, upon the closely allied question as
to the nature of worth- values. There is a worth-value involved
in the Good, and a worth-value involved in the True, and a worth-

THE RELATIONS OF ESTHETICS 433

value involved in the Beautiful: and each of these worth-values
in itself seems to be involved with pleasure-getting. Now if this is
the case, then, under the theory I uphold, any worth-value should be
a possible aesthetic element, and this I think it will be granted is
true. But the distinctions between these worth-values are on differ-
ent planes, as it were. In the case of the worth-value of the Good,
we appreciate the worth-pleasure within the realm of the Real of
Expression, that is, of impulse. In the case of the worth-value of the
True (in the narrow sense) , we appreciate the worth-pleasure within
the realm of the Real in other fields than that of expression or that
of impression. In the case of the worth-value of the Beautiful, we
appreciate the worth-pleasure within the realm of the Real of Im-
pression; that is, we appreciate, with pleasure, the significance for
life of the existence of relatively permanent pleasure in and for
itself.

THE FUNDAMENTAL QUESTIONS OF CONTEMPORARY

AESTHETICS

BY MAX DESSOIR
{Translated from the German by Miss Ethel D. Puffer, Cambridge, Mass.)

[Max Dessoir, Professor of Philosophy, University of Berlin, since 1897. b.
1867, Berlin, Germany. Ph.D. Berlin, 1889; M.D. Wtirzburg, 1892. Pri-
vat-docent, University of Berlin, 1892-97. Member German Psychological
Society, Society for Psychical Research, London. Author of The Double Ego;
History of the New German Psychology ; Philosophical Reader; Msihetik u.All-
gemeine Kunstwissenschaft ; and many other works and papers an philosophy.]

In the development which our science has undergone, from its
inception up to the present day, one thought has held a central
place, — that aesthetic enjoyment and production, beauty and art,
are inseparably allied. The subject-matter of this science is held to
be, though varied, of a unitary character. Art is considered as the
representation of the beautiful, which comes to pass out of an ses-
thetic state or condition, and is experienced in a similar attitude; the
science which deals with these two psychical states, with the beau-
tiful and its modifications, and with art in its varieties, is, inasmuch
as it constitutes a unity, designated by the single name of aesthetics.

The critical thought of the present day is, however, beginning to
question whether the beautiful, the aesthetic, and art stand to one
another in a relation that can be termed almost an identity. The
undivided sway of the beautiful has already been assailed. Since
art includes the tragic and the comic, the graceful and the sublime,
and even the ugly, and since aesthetic pleasure can attach itself to
all these categories, it is clear that by "the beautiful" something
narrower must be meant than the artistically and aesthetically
valuable. Yet beauty might still constitute the end and aim and
central point of art, and it might be that the other categories but
denote the way to beauty — beauty in a state of becoming, as it
were.

But even this view, which sees in beauty the real content of art,
and the central object of aesthetic experiences, is open to serious
question. It is confronted with the fact, above all, that the beauty
enjoyed in life and that enjoyed in art are not the same. The artist's
copy of the beauty of nature takes on a quite new character. Solid
objects in space become in painting flat pictures, the existent is in
poetry transformed into matter of speech; and in every realm is a

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 435

like metamorphosis. The subjective impression might indeed be sup-
posed to remain the same, in spite of objective differentiations. But
even that is not the case. Living human beauty — an acknowledged
passport for its possessor — speaks to all our senses; it often stirs
sex-feeling in however delicate and scarce conscious a way; it
involuntarily influences our actions. On the other hand, there hangs
about the marble statue of a naked human being an atmosphere
of coolness in which we do not consider whether we are looking
upon man or woman: even the most beauteous body is enjoyed as
sexless shape, like the beauty of a landscape or a melody. To
the aesthetic impression of the forest belongs its aromatic fragrance,
to the impression of tropical vegetation its glowing heat, while
from the enjoyment of art the sensations of the lower senses are
barred. In return for what is lost, as it were, art-enjoyment involves
pleasure in the personality of the artist, and in his power to over-
come difficulties, and in the same way many other elements of pleas-
ure, which are never produced by natural beauty. Accordingly,
what we call beautiful in art must be distinguished from what goes
by that name in life, both as regards the object and the subjective
impression.

Another point, too, appears from our examples. Assuming that
we may call the pure, pleasurable contemplation of actual things
and events aesthetic, — and what reason against it could be adduced
from common usage ? — it is thus clear that the circle of the aesthetic
is wider than the field of art. Our admiring and adoring self-abandon-
ment to nature-beauties bears all the marks of the aesthetic attitude,
and needs for all that no connection with art. Further: in all in-
tellectual and social spheres a part of the productive energy expresses
itself in aesthetic forms; these products, which are not works of art,
are yet aesthetically enjoyed. As numberless facts of daily experience
show us that taste can develop and become effective independently
of art, we must then concede to the sphere of the aesthetic a wider
circumference than that of art.

This is not to maintain that the circle of art is a narrow section of
a large field. On the contrary, the aesthetic principle does not by
any means exhaust the content and purpose of that realm of human
production which taken together we call "art." Every true work of
art is extraordinarily complex in its motives and its effects; it arises
not alone from the free play of aesthetic impulse, and aims at more
than pure beauty — at more than aesthetic pleasure. The desires
and energies in which art is grounded are in no way fulfilled by
the serene satisfaction which is the traditional criterion of the aes-
thetic impression, as of the aesthetic object. In reality the arts
have a function in intellectual and social life, through which they are
closely bound up with all our knowing and willing.

436 ESTHETICS

It is, therefore, the duty of a general science of art to take account
of the broad facts of art in all its relations. Esthetics is not capable
of this task, if it is to have a determined, self-complete, and clearly
bounded content. We may no longer obliterate the differences
between the two disciplines, but must rather so sharply separate
them by ever finer distinctions that the really existent connections
become clear. The first step thereto has been taken by Hugo Spitzer.
The relation of earlier to current views is comparable to that between
materialism and positivism. While materialism ventured on a pretty
crude resolution of the spiritual into the corporeal, positivism set
up a hierarchy of forces of nature, whose order was determined
by the relation of dependence. Thus mechanical forces, physico-
chemical processes, the biological and the social-historical groups
of facts, are not traced back each to the preceding by an inner con-
nection, but are so linked that the higher orders appear as dependent
on the lower. In the same way is it now sought to link art methodo-
logically with the aesthetic. Perhaps even more closely, indeed, since
already aesthetics and the science of art often play into each other's
hands., like the tunnel-workers who pierce a mountain from opposite
points, to meet at its centre.

Often it so happens, but not invariably. In many cases research is
carried to an end, quite irrespectively of what is going on in other
quarters. The field is too great, and the interests are too various.
Artists recount their experiences in the process of creation, con-
noisseurs enlighten us as to the technique of the special arts; socio-
logists investigate the social function, ethnologists the origin, of art;
psychologists explore the aesthetic impression, partly by experiment,
partly through conceptual analysis; philosophers expound aesthetic
methods and principles; the historians of literature, music, and
pictorial art have collected a vast deal of material — and' the sum
total of these scientific inquiries constitutes the most substantial
though not the greatest part of the published discussions, which,
written from every possible point of view, abound in newspapers and
magazines. " There is left, then, for the serious student, naught but
to resolve to fix a central point somewhere, and thence to find out
a way to deal with all the rest as outlying territory " (Goethe).

Only by the mutual setting of bounds can a united effect be pos-
sible from the busy whirl of efforts. Contradictory and heterogeneous
facts are still very numerous. He who should undertake to construct
thereof a clear intelligible unity of concepts, would destroy the
energy which now proves itself in the encounters, crossing of swords,
and lively controversies of scholars, and would mutilate the fullness
of experience which now expresses itself in the manifold special
researches. System and method signify for us: to be free from one
system and one method.

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 437

If we are to consider how we answer to-day the questions put for
scientific consideration as to the facts of aesthetic life and of art, first
of all we must examine the now prevailing theories of aesthetics.
They fall in general into sesthetic objectivism and subjectivism.
By the first collective name we denote the aggregate of all theories
which find the characteristic of their field of inquiry essentially in
the quality and conformation of the object, not in the attitude of
the enjoying subject. This quality of the aesthetically valuable is
most easily characterized by setting it off against reality. Of such
theories, which explain "the beautiful" and art from their relation
to what is given in nature, naturalism stands for the identity of real-
ity and art, while the various types of idealism set forth art as more
than reality, and vice versa, formalism, illusionism, sensualism make
it less than reality.

Inasmuch as naturalism is still defended only by a handful of
artists who write, it would appear almost superfluous to consider it.
But the refutations of it which are still appearing indicate that it
must have some life. And in fact it still exists, partly as a present-
day phenomenon in literature and art, partly as the permanent
conviction of many artists. The naturalistic style testifies to revolt
against forms and notions which are dying out; it therefore only
attains a pure aesthetic interest through the theoretic ground which
is furnished to it. And this rests above all on the testimony of the
artists, who are never weary of assuring us that they immediately
reproduce what is given in perception. Some philosophical concep-
tions also play therein a certain role. The adherents of the doctrine
that only the sense-world is real come as a matter of course to the
demand that art shall hold itself strictly to the given. And what
optimist, who explains the real world as the best of all possible
worlds, can, without a logical weakening, admit a play of imagination
different from the reality.

-(Esthetic idealism, too, is borne on general philosophical premises.
However various these are, in this they all agree, that the world is
not exhausted by appearances, but has an ideal content and import,
which finds in the aesthetic and in the field of art its expression to
sense. Even H. Taine sets to art the task of showing the " dominant
character" of things. The beautiful is therefore something higher
than the chance reality, — the typical as over against the anomalous
natural objects or events. It can then be objectively determined
with reference to its typical and generic quality and in its various
kinds.

Somewhat different is the case of formalism, which to-day scarcely
anywhere sets up to be a complete system of aesthetics, but points

438 ESTHETICS

the way for many special investigations. It seeks the sesthetically
effective in the form, that is, in the relation of parts, which has
in principle nothing to do with the content of the object. Every
clearly perceptible unity in manifoldness is pleasing. As this ar-
rangement is independent of the material, the aesthetic represents
only a part of reality.

In contrast thereto, illusionism sets the world of art as a whole over
against the whole of reality. Art, we are taught, presents neither a
new aspect of the given nor hidden truth, nor piire form; it is, on the
contrary, a world of appearance only, and is to be enjoyed without
regard to connections in life or any consequences. While we other-
wise consider objects as to how they serve our interests and as to their
place in the actual connection of all things, in the esthetic experi-
ence this twofold relation is disregarded. Neither what things do
for us, nor what they do for each other, comes in question. Their
reality disappears, and the beautiful semblance comes to its own.
Konrad Lange has given to this theory — especially in the line of
a subjective side, to be later mentioned — its modern form.

Of the nearly-related sensualism, the connoisseur Fiedler and the
sculptor Hildebrand are the recent exponents; Rutgers Marshall
and certain French scholars also lean that way. It is the arts which
fix the transitory element of the sense-image, hold fast the fleeting,
make immortal the perishable, and lend stability and permanence
to all pleasure that is bound up with perception. What does painting
accomplish? Arisen, as it has, out of the demands of the eye, its
sole task is to gain for the undefined form- and color-impressions
of reality a complete and stable existence. The same thing is true
of the other arts, for their respective sense-impressions.

To sum up: If the transformation of reality is acknowledged as
a fundamental principle of art, it is also to be granted that this takes
place in two directions : — art is something at once more and less
than nature. Inasmuch as art pushes on to the vraie verite, and at
the same time disregards all that is not of the nature of semblance
or image, we take from it ideas whose quality enthralls and stimu-
lates us quite independently of their meaning. Art shows us the
hidden essence of the world and of life and at the same time the
outsides of things created for our pleasure; that is, the objects'
pure psychical value in the field of sense. It involves a lifting above
nature, and at the same time the rounding out and fulfillment of
sense. Through making of the object an image, it frees us from our
surrounding, yet leaves us at rest in it.

We turn now to sesthetic subjectivism. Under this name we com-
prehend the essence of those theories which seek to solve the riddle
of the beautiful by a general characterization of the sesthetic atti-
tude. Many of these are near akin to the objectivistic theories; some,

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 439

however, like the Einfiihlung-theoTy, take an independent place.
For the former, therefore, a mere indication will suffice. The prin-
ciple of "semblance" or illusion, for instance, takes very easily a
subjectivistic turn. The question then runs: Wherein consists the
peculiarity of the conscious processes which are set up by the
semblance? The answer as given by Meinong and Witasek starts
from the fact that the totality of psychical processes falls into two
divisions. Every process in one division has its counterpart in the
other. To perception corresponds imagination, to judgment assump-
tion, to real emotion ideal emotion, to earnest desire fancied desire.
The sesthetic emotions attached to assumptions, the semblance-emo-
tions, that is, are held to be scarcely distinguished, so far as feeling
goes, from other emotions, at most, perhaps, by less intensity. The
chief difference lies rather in the premise or basis of emotion; and
this is but a mere assumption or fiction.

A critical treatment of the foregoing cannot be given here; nor
of that view which explains the psychical condition in receiving an
sesthetic impression as a conscious self-deception, a continued and
intentional confusion of reality and semblance. The sesthetic pleas-
ure, according to this, is a free and conscious hovering between
reality and unreality; or, otherwise expressed, the never successful
seeking for fusion of original and copy. The enjoyment of a good
graphic representation of a globe would then depend on the specta-
tor's now thinking he sees a real globe, now being sure he views a flat
drawing.

While this theory has found but small acceptance, comparatively
many modern sestheticians admit the doctrine of Einfithlung. Its
leading exponent, Theodor Lipps, sees the decisive characteristic
of sesthetic enjoyment in the fusion of an alien experience with one's
own: as soon as something objectively given furnishes us the pos-
sibility of freely living ourselves into it, we feel sesthetic pleasure.
In the example of the Doric column, rearing itself and gathering
itself up to our view, Lipps has sought to show how given space-
forms are interpreted first dynamically, then anthropomorphically.
We read into the geometrical figure not only the expression of energy,
but also free purposiveness. In so far as we look at it in the light of
our own activity, and sympathize with it accordingly, in so far do
we feel it as beautiful.

Could we enter upon a critical discussion at this point, it would
appear that the Einfuhlung-iheory, like its fellows, is open to well-
founded objections. The belief in an all-explaining formula is a
delusion. In truth, every one of the enumerated principles is rela-
tively justified. And as they all have points of similarity with one
another, it is not hard for the past-master of terminology and the
technique of concepts to epitomize the common element in a single

440 AESTHETICS

phrase or thesis. Still, nothing is gained by such a general formula
in presence of the richness of the reality; and just as little — as an
exhaustive treatment would have to prove — by the concise ex-
position of a single method for our science.

The specially approved method of procedure at the present day
is that of psychological description and explanation. It seems,
indeed, very natural to see in psychical processes the real subject-
matter of Eesthetics, and in psychology, accordingly, the science to
which it is subordinate. Some philosophers, however, — among
whom I may instance Jonas Cohn, — wish to make of aesthetics a
science of values, and demand that on the basis of this pretension
the mutually contradictory judgments of taste and types of art be
tried and tested. They will have no mere descriptive and explan-
atory eesthetics, but a normative, precept-giving science. In truth,
the opposition of the schools is complete at every point; in the
writings of Volkelt and Groos we have the proof of it.

Ill

The special research in the narrower field of aesthetics is at present
almost entirely of the psychological type. Our survey can touch
upon only the salient points.

The aim of the extended and highly detailed study consists in
fixating by means of psychological analysis the course of develop-
ment, the effective elements, and the various sub-species of the
aesthetic experience. Certain philosophers seek a point of departure
for this undertaking in the aesthetic object. Thus Volkelt 's system
of aesthetics finds, for the chief elements of the aesthetic enjoyment,
corresponding features in the object; in the special field of poetry
Dilthey has undertaken 'an analysis along the same lines. For the
most part, however, such dissection is limited to the subjective side.
In Wundt's psychology, for instance, the aesthetic state of mind is
shown to be built up of sense-feelings, feelings from perceptions, in-
tellectual and emotional excitements; the most important, that is
to say, the pivotal feelings, which are bound up with space- and
time-relations, become in turn the condition and support of the
higher emotions, because they lead over from the field of sense to
that of the logical and emotional.

If we limit ourselves to the psychological, we must first ask in what
order the elements of the aesthetic impression are wont to follow
each other. The phases of this development, however, are as yet not
completely studied, although they are of great significance for the
differences in enjoyment. The second problem concerns the con-
stitution (taken as timeless) of the experience. All formulas which
attempt to fix in two words the totality of the impression fail com-

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 441

pletely, — so extraordinarily various and manifold are the factors
which enter here. What these are and how they are bound together
is the question which is for the moment occupying the scholars with
a leaning toward psychology.

The aesthetic impression is an emotion. According to the well-
known sensualistic theory of the emotions, it must therefore, in so
far as it is more than mere perception or idea, be composed of organic
sensations. G. Sergi and Karl Lange see, in fact, the peculiar mark
of the aesthetic experience in the general sensations which appear
with changes in the circulation, breathing, etc. Unprejudiced ob-
servation must satisfy every one that much in all this is true. On the
other hand, it is to be recalled that we do not reckon the organic
sensations to the objective qualities of aesthetic things, and that we
cannot explain in this way every artistic enjoyment. — In regard to
the sensations of taste, smell, and touch, it is generally granted that
they play a certain role, even if but as reproduced ideas and only
corresponding to natural beauty. Among the most important are
the attitudes and imitative movements, finely investigated by Karl
Gtoos. — To this must be added the sensuous pleasantness of visual
and auditory perceptions. Yet attempts to construct the aesthetic
enjoyment in its entirety out of such pleasure-factors have so far
failed. The undertaking is already wrecked by the fact that elements
displeasing to sense are demonstrably present, not only as negligible
admixtures, but also as necessary factors. The relations of similarity
between the contents of a sense-field, and the spatial and temporal
connections between them, are in any case incomparably more
important; we devote to them, therefore, a closer consideration.
Finally, alongside all these ideas and the emotions immediately
attaching to them, there must be arrayed the great multitude of as-
sociated ideas and connecting judgments. While scientific interest
in the associations is now greatly diminished, explanations of the
part played by the element of really active thought are many. A
universally satisfactory theory is still to appear, for the reason,
above all, that here the higher principles referred to in the second
section enter into the problems.

Elementary aesthetics, therefore, willingly turns aside from the
shore of the very complex emotions, of association, Einfiihlung and
illusion in aesthetic experience, in order to become independent of
general philosophical fundamental conceptions. Its own field lies
in the general province of the perception-feelings determined im-
mediately by the object: more exactly, of the feelings which are
induced partly by the relations of similarity, partly by the outer
connections of the content, partly by the linking of inner and outer
reference. The qualitative relation of tones and colors arouses the
so-called feelings of harmony; the arrangement in space and time

442 ESTHETICS .

awakes the so-called proportion-f eelings ; and from the cooperation
of these two arise the so-called aesthetic complication-feelings.

As to the pleasurable tone- and color-combinations, the first are
better known than the second, but even their theoretical interpre-
tation is not well settled. More diligent and successful at the present
time is the research into the proportion-feelings. So far as these
bear upon space-relations, they attach either to the outlines or to the
structure of the forms. The bounding-lines are then pleasing, one
theory holds, when they correspond to the easiest eye-movements, and
in general meet our desire for easy, effortless orientation. Another
doctrine, already referred to, explains their aesthetic value from a
co5peration of general bodily feelings, especially sensations of
breathing and equilibrium. Accurate experiments have not succeeded
in finding a real conformity to law in either the first or the second
direction. In the matter of the structure of forms, symmetry in
the horizontal position, and the proportion of the golden section
in the vertical position, receive especial attention. All those space-
shapes may be called symmetrical, whose halves are of equal value
{Esthetically. How these must be constituted, has been studied
from the simplest examples by Miinsterberg and his pupils. The
explanation of the pleasing quality rests on the fact that the spec-
tator feels the contents of the two halves — lines or colors — as light
or heavy, according to the energy expended in, the necessary eye-
movements. In the vertical position a proportion pleases (as does also
equality) which is only approximately that of the golden section.
The numerical proportion is, therefore, not the ground of pleasure,
for otherwise those forms which are thus divided would have to be
the absolutely beautiful ones, and the more a division varies from
the exact fraction, the more would it sacrifice in beauty. The ground
of pleasure is rather descried in the fact that in the case of the pleas-
ing divisions the two parts stand out as distinct and clearly character-
ized, while yet unified effect is secured through the larger division.

The temporal ordering of an aesthetic character is that of rhythm.
Concerning the aesthetic object as such — that is, concerning the
metrical forms in music and poetry, the views are still widely at
variance; this is true to a startling degree of poetr}^, because here
the element, that is to say, the word, is made up of accented and
unaccented syllables, and because the tendency of the logical con-
nections of the content to create unities cannot be done away with.
This state of confusion is so much the more to be regretted as it is
just to the art-forms that the most vivid rhythmical feelings attach.
The psychological investigations of Neumann, Bolton, and others have
nevertheless much advanced our scientific understanding of this
subject. A new point of view has taken its rise from Souriau and
Bilcher: the connection of the art-rhythm with work and other

THE FUNDAMENTAL QUESTIONS OF ESTHETICS 443

aspects of life. But the collections of data do not yet render it pos-
sible to settle the question in what manner the rhythm of work,
which runs on automatically, and is controlled by the idea of an end,
goes over into aesthetic rhythm.

The aesthetic complication-feelings are bound up Tvith the products
of the fusion of rhythm and harmony, form and color, rhythm and
form (in the dance). So long as all elements of association are
neglected, three characteristics remain to be noted: an increasing
valuation of the absolute quantity, the building-up of definite
form-qualities (Gestaltqualitdten) , and a reconciliation or harmony
of differences, wherein the quantitative element is wont to be the
unifying, the qualitative element the separating factor. I need not,
however, go any further into investigations so subtle, and even now
merely in their beginnings.

This entire fabric of experience, from M'hich but a few threads
have been drawn out to view, can now take on various shadings.
These we refer to as the aesthetic moods, or by a less psychological
name, as the aesthetic categories. The ideally beautiful and the
sublime, the tragic and the ugly, the comic and the graceful, are
the best known among them. Modern science has shown most
interest in the study of the comic and the tragic. According to Lipps
the specific emotion of the comic arises in the disappointing of a
psychical preparation for a strong impression, by the appearance of
a weak one. The pleasurable character of the experience would be
explained by the fact that the surplus of psychical impulse, like every
excess of inner energy, is felt as agreeable. The tragic mood is under-
stood no longer as arising in fear and pity, but in pathos and wonder.
Its objective correlate should not be forced to the standard of a nar-
row ethics. The demand for guilt and expiation is being given up
by progressive thinkers in aesthetics; but the constituents of tragedy
remain fast bound to the realm of harshness, cruelty, and dissonance.

From a period more or less remote there have existed poetics,
musical theory, and the science of art. To examine the presupposi-
tions methods and aims of these disciplines from the epistemological
point of view, and to sum up and compare their most important results ,
is the task of a general science of art; this has besides, in the pro-
blems of artistic creation and the origin of art, and of the classification
of the arts and their social function, certain fields of inquiry that
would otherT\ase have no definite place. They are worked, indeed,
with remarkable diligence and productiveness. Most to be regretted,
on the other hand, is that so little energy is applied to laying the
epistemological foundation.

The theory of the development of art deals with it both in its

444 ESTHETICS

individual and its generally human aspect. Concerning the genesis of
the child's understanding of art and impulse to produce it, we learn
most from the studies of his drawings at an early age. Here are to be
noted well-established results of observation, even though as yet
they are few in number. On the other hand, the unfolding of primi-
tive feeling (and of the aesthetic sensibility in general) during the
historical period can be only approximately reconstructed. The
case is somewhat more favorable for our information in regard to
the beginnings of art, especially since it has been systematically
assembled by Ernst Grosse and Yrjo Hirn. If the conditions of the
most primitive of the races now living in a state of nature can be
taken as identical with those at the beginnings of civilization, the
entire vast material of ethnology can be made use of. We gather
therefrom how close-linked with the useful and the necessary beauty
is, and see clearly that primitive art is thoroughly penetrated by
the purpose of a common enjoyment, and is effective in a social
way; but beyond such general principles one can go only with
hesitation, inasmuch as it seems scarcely possible to us, creatures of
civilization, to fix the boundaries of what is really art there.

There are three conjectures as to objective origin of art. It may
be that the separate arts have developed through variation from one
embryonic state. Or the main arts may have been separate from the
very first, having arisen independently of each other. Finally, there
are middle views, like that of Spencer, according to which poetry,
music, and the dance on the one hand, and writing, painting, and
sculpture on the other, have a common root ; Mobius recognizes
three primitive arts, to which the others are to be traced back. The
solution of this question would be especially important, could one
hope to find Darwin's maxim for all setiological investigations valid
for our field also — that is, the dictum: What is of like origin is of
like character.

As psychological conditions, from which the artistic activity is
likely first to have arisen, the following functions have been suggested
and maintained, — the plajMnstinet, imitation, the need for expres-
sion and communication, the sense for order and arrangement, the
impulse to attract others and the opposed impulse to startle others.
Each of these theories of conditions must clearly connect itself with
one or the other of the just-named three theories of art's origin; for
had music, taken in our sense and independently, existed as the orig-
inal art, one could hardly regard imitation as the psychological root
of art. All in all, art and the play-instinct seem most closely linked;
that is also true, moreover, of its development with the child.

I come now to the fundamental problems of artistic creation. It
is they which present the most obstinate difficulties to a thorough
and exact investigation, for experiment and the questionnaire —

THE FUNDAMENTAL QUESTIONS OF AESTHETICS 445

which aims at least at objectivity — are but crude means to the end in
view. At the present day, as eariier, there is no lack of very refined,
penetrating, — nay, brilliant analyses. They have a very superior
value; but this has no special significance for the present status of
the science of aesthetics, and for this reason our survey may omit
much which yet has an interest for individuals.

The influence of heredity and environment on the artist's talent
offers rich material for research. It is conceded, though, that how
the most material and the most spiritual of influences, inherited
disposition and fortune, the chances of descent and of intercourse
with one's fellows, — how all this is fused into a unified personality,
can be established only in individual cases by the biographer. A
second very productive source of material in this field has appeared
in Lombroso's teaching. The days of the most violent controversies
lie behind us. It is the general view that genius and madness are near
allied in their expression, that greatness often breaks forth in ques-
tionable forms; yet the majority perceive an essential difference ; the
genius points onward, the mind diseased harks back; the one has
purposive significance, the other not. After these more introductory
inquiries, the real work begins. It has to show in what points every
gift for art coincides with generally disseminated abilities, and just
where the specific power sets in, which the inartistic person lacks.
Take, for example, the memory. We retain this or that fact without,
in principle, any selection; the remembrance of the artist, on the
contrary, is dissociative — it favors what is needful for its own ends.
The memory of the painter battens on forms and colors, the conscious-
ness of the musician is filled with melodies, the fancy of the poet lives
in verbal images. Also there is, especially with the poet, a peculiar
understanding for human experience. In truth, the fanciful products
of the imagination are but the starting-point for the soul-know-
ledge of the poet. Without going into details we may say that by
such penetrating and delimiting analyses the superficial theory of
inspiration is refuted. Out of date, too, is the notion that the artist
creates by putting things together; on the contrary his fancy has
the whole before the parts, it gives to the world an organism, within
which the members gradually emerge. Finally, the old theory is no
longer held, according to which the work of art is already complete
in the inner man, and afterwards merely brought to light. More
definite explanation is given by the doctrine of the way in which the
artistic creation runs its course, which Eduard v, Hartmann has
skillfully portrayed. •

The distinction, differentiation, and comparison of the special
arts offers opportunity and material for numberless special studies.
Music is here the least fully represented, since it is only exceptionally
that art-philosophers feel a drawing to it. So much the more, how-

446 ESTHETICS

ever, are they inclined to the study of poetry. They are even begin-
ning to make use, for poetics, of the studies in the modern psychology
of language, since it is acknowledged that language is the essential
element, and thus more than the mere form of expression, of the
poetic art. Th. A. Meyer has thrown an apple of discord into the
question whether the poet's words must, in order to arouse pleasure,
also awake an image. As a matter of fact, the aesthetic value does not
depend on the chance-aroused sense-images, but on the language
itself and the images which belong to it alone; for the most part the
understanding of the words alone is enough to give the reader pleas-
ure in the poetic treatment. In the general theory of the visu-
ally representative arts there are two opposed doctrines. The one
emphasizes the common element, and believes to have found it in
the so-called Fernhild, or distant image; the other seeks salvation
in complete separations — as, for instance, of the so-called Griff el-
kunst, or graphic art, from painting. Only the future can decide
between them.

The existence of the total field of art as an essential factor of hu-
man endeavors involves difficulties which must be removed partly
in the philosophical consideration, partly in law and governmental
practice. The last factor must also be taken account of in theory;
for so long as we do not live in an ideal world, the state will claim
regulation of all activities expressing themselves in it, and so also
Of art. In first line it is concerned for art's relation to morality.
Secondly, the social problems arise: does art bind men together,
or part them? does it reconcile or intensify oppositions? is it demo-
cratic or aristocratic? is it a necessity or a luxury? does it further or
reject patriotic, ethical, pedagogical ends? The artistic education of
youth and the race has become a burning question. Huskin and
Morris have developed from art-critics to critics of the social order,
and Tolstoi has contracted the democratic point of view to the
most extreme degree. With the desire to transform art from the
privilege of the few to the possession of all is, finally, bound up the
wish that art shall emerge from another seclusion — that it shall not
be throned in museums and libraries, in theatres and concert-halls,
but shall mingle with our daily domestic life, and direct and color
every act of the scholar as of the peasant.

A satisfactory decision can be reached only by him who keeps in
view that art presents something extremely complex, and by no
means mere aesthetic form; that, on the other hand, the aesthetic
life is not banished to the sacred circle of the independent arts.
With this conclusion we return to the first words of our reflec-
tions herein presented.

SPECIAL BIBLIOGRAPHY PREPARED BY PROFESSOR
DESSOIR FOR HIS ADDRESS

Thaddeus L. Bolton, Rhytlim. Americ. Joum. of Psychol. 1894. vi, 145-238.

Karl Biicher, Arbeit und Rhythmus. 3 Aufi. Leipzig, 1902.

Jonas Cohn, Allgemeine ^sthetik, Leipzig, 1901.

Wilhelm Dilthey, Die Einbildungshraft des Dichters. Bausterne zu einer Poetik.

In den ZeUer gewidmeten Pliilos. Aufsatzen, Leipzig, 1887.
Konrad Fiedler, Schriften tiber Kunst, Leipzig, 1896.
Karl Groos, Der sesthetische Genuss. Giessen, 1902.
Ernst Grosse, Die Anfange der Kunst, Freiburg und Leipzig, 1894.
Eduard von Hartmann, ^sthetik, Bd. ii, Leipzig, 1887.
Adolf Hildebrand, Das Problem der Form in der bildenden Kunst. 3 Aufl.

Strassburg, 1901
Yrjo Hirn, The Origins of Art, London, 1900. Deutsch, Leipzig, 1904.
Karl Lange, Sirmesgenusse und Kunstgenuss, Wiesbaden, 1903.
Konrad Lange, Das Wesen der Kunst, 2 Bde., Berlin, 1901.
Theodor Lipps, Raumsesthetik, Leipzig, 1897. — Komik und Humor, Hamburg

imd Leipzig, 1898. — Grundlegung der ^sthetik, Hamburg und Leipzig, 1903.
Cesare Lombroso, L'uomo di genio in rapporto alia psichiatria. Torino, 1889

und ofter. Deutsch, Hamburg, 1890.
H. Rutgers Marshall, Esthetic Principles, New York, 1895.
A. Meinong, Ueber Annahmen, Leipzig, 1902.
Th. A. Meyer, Das Stilgesetz der Poesie, Leipzig, 1901.
P. J. Mobius, Ueber Kunst und Kiinstler, Leipzig, 1901.
William Mori'is, Hopes and Fears for Art, London, 1882. Deutsch, Bd. i. Die

niederen Kiinste. ii. Die Kunst des Volkes. Leipzig, 1891.
Hugo Mtinsterberg, Harvard Psychological Studies. Bd. i, Lancaster, Pa. 1903.
Ernst Nemnann, Untersuchungen fur Psychologic und ^sthetik des Rhythmus

Philos. Studien, herausg. von W. Wundt, 1894, Bd. x.
John Ruskin, Ausgewahlte Werke. Deutsch, Leipzig, 1900,
G. Sergi, Dolore e Piacere, Milano, 1897.
Paul Souriau, L'esthetique du mouvement, Paris, 1889.
Herbert Spencer, Principles of Psychology, Bd. ii, London, 1855 und ofter.

Deutsch, Leipzig, 1875, ff.
Hugo Spitzer, Hermann Hettners Kunstphilosophische Anfange, Graz, 1903.
H. Taine, Philosophie de I'Art. 2 Bde. 7 Aufl. Paris, 1895.
Leo Tolstoj, Was ist Kunst? Deutsch, Berlin, 1892.
Johannes Volkelt,- ^sthetische Zeitfragen, Muenchen, 1895. Deutch, Leipzig,

1902-03.— ^sthetik des Tragischen, Munchen, 1897.— System der^sthetik,

Bd. I, Munchen, 1905.
Stephan Witasek, Grundztige der allgemeinen ^sthetik, Leipzig.
Wilheim Wundt, Grundziige der physiologsichen Psychologie, 1904. Bd. iii. 5

Aufl. Leipzig, 1903.

SHORT PAPERS

A short paper was contributed by Professor A. D. F. Hamlin, of Columbia
University, on the "Sources of Savage Conventional Patterns." The speaker
said that, in the exhibit of the Department of the Interior, two glass cases displayed
side by side the handiwork of the American Indian of one hundred years ago and
of to-day. In the Fine Arts palace the blankets and basketry of the Navahoes
were shown beside the leather work and other handicrafts of white Americans.
In both instances the contrast between the savage and the civilized work em-
phasizes the fact that civilization tends to stifle or destroy the decorative instinct.
The savage art is spontaneous, instructive, unpremeditated. The work of the
civilized artist is thoughtful, carefully elaborated, intellectual. Among these
peoples both the crafts and the patterns are traditional, and there is little or no
ambition to innovate. The forms and combinations we admire in their work are
the result of long-continued processes of evolution and elimination in which, as in
the world of Uving organisms, the fittest have survived. The structure of savage
patterns is almost always extremely simple. There are three theories advanced to
account for them; that they were invented out of hand; that they were evolved
out of the technical processes, tools, and materials of primitive industry; that
they are descended from fetish or animistic representations of natural forms.
The first is the common view of laymen; the second was first expressed (though
chiefly with reference to civilized art) by Semper; and the third is widely enter-
tained by anthropologists.

The savage instinct for decoration has probably developed from primitive
animism — from that fear of the powers of nature, and that confounding of the
animate and inanimate world which is universally recognized as a primitive
trait. But once awakened in even the sHghtest degree, it has found exercise in
the operations of primitive industry, and given existence to a long series of repeti-
tive forms produced in weaving, basketry, string-lashing, and car\ing. The two
classes of patterns thus originated — those derived from the imitation of nature
under fetish ideas, and those derived from teclinical processes — have invariably
converged, overlapping at last in many forms of decorative art, so that the real
origin of a given pattern may be dual. Myths have invariably arisen to explain
the origin of the technical patterns, which have received magic significance and
names, in accordance with savage tendency to assign magical powers to all visible
or at least to all valued objects: all savage art is talismanic. One ought to be
cautious about dogmatizing as to origins in dealing with savage art, because both
the phenomenon of what I call convergence in ornament evolution, and that of
the myths, poetic faculty, and habit among savages, tend to confuse and obscure
the real origin of the patterns with which they deal. And finally, for the artist
as distinguished from the archaeologist and the theorist, the real lesson of savage
art is not in its origins, but in its products; in the strength, simplicity, admirable
distribution, and high decorative effects of poor and despised peoples. Savage
aU-over patterns and Greek carved ornament and decorative sculpture represent
the opposed poles of decorative design, and both are of fundamental value as
objects of study for the designer.

BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

PREPARED THROUGH THE COURTESY OF DR. RALPH BARTON PERRY,
- OF HARVARD UNIVERSITY

HISTORY OF PHILOSOPHY

BouiLLiEB, F., Philosophie Cartesienne.
Burnet, J., Early Greek Philosophy.
Erdmann, J. E., Geschichte der Philosophie.
EucKEN, R. , Lebensanschauungen der grossen Denker.
Fairbanks, A., The First Philosophers of Greece.
Falkenberg, R. , Geschichte der neueren Philosophie.
Fischer, K., Geschichte der neueren Philosophie.
Gomperz, Th., Greek Thinkers.
Hoffding, H., Geschichte der neueren Philosophie.
Levy-Bruhl, Histoire de la philosophie moderne.
Royce, J., Spirit of Modern Philosophy.
SiDGwiCK, H., History of Ethics.
Turner, W., History of Philosophy.
Ueberweg, F. , Geschichte der Philosophie.
Weber, A., Histoire de la philosophie europeenne.
Windelband, W., Geschichte der Philosophie.

Geschichte der alten Philosophie.
Zeller, E., Geschichte der griechischen Philosophie.

PHILOSOPHICAL CLASSICS

Abelard, Dialectic.

Anselm, Monologium.
Aristotle, Metaphysics.
De Anima.
Physics.

Nicomachean Ethics.
Bacon, F., Novum Organum.

Berkeley, G., The Principles of Human Knowledge.
Bruno, G., Dialogi, De la Causa Principio et Uno, etc.
Burnet, J., Early Greek Philosophy; fragments of Heraclitus, Parmenides,

Anaxagoras, etc.
Descartes, R., Discours de la Methode.

Meditationes de Prima Philosophia.
Duns Scotus, Opus Oxoniense.
FiCHTE, J. G., Wissenschaftslehre.
Hegel, G. W. F., Wissenschaft der Logik.

Encyklopadie.
Hobbes, T., Leviathan.-
Hume, D., Enquiry Concerning the Human Understanding.

Enquiry Concerning the Principles of Morals.
Kant, I., Kritik der reinen Vermmft.

Kritik der praktischen Vernunft.
Kritik der Urteilskraft.

450 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

Leibniz, G. W., Monadologie.

Theodicee.
Locke, J., An Essay Concerning Human Understanding.
LoTZE, R. H., Metaphysik.
Lucretius, De Rerum Natura.
Plato, Republic. Phaedo. Theaetetus. SsTnposium. Phaedrus. Protagoras

(and other dialogues).
Plotinus, Enneades.
St. Augustine, De Civitate Dei.
ScHELLiNG, Philosophic der Natur.

Schopenhauer, A., Die Welt als Wille und Vorstellung.
Spencer, H., Synthetic Philosophy.
Spinoza, B., Ethica.
Thomas Aquinas, Summa Theologiae.

INTRODUCTION TO PHILOSOPHY

Baldwin, J. M., Dictionary of Philosophy.

Hibben, J. G., Problems of Philosophy.

KuLPE, 0., Einleitung in die Philosophic.

Marvin, W. T., Introduction to Philosophy.

Paulsen, F., Einleitung in die Philosophic.

Perry, R. B., Approach to Philosophy.

SiDGWicK, H., Philosophy, its Scope and Relations.

Stuckenberg, J. H. W., Introduction to the Study of Philosophy.

Watson, J., Outline of Philosophy.

WiNDELBAND, W., Praludien.

METAPHYSICS

AvENARius, R., Kritik der reinen Erfahrung.

Bergson, H., Matiere et memoire.

Bradley, F. H., Appearance and Reality.

Deussen, p.. Elements of Metaphysics.

Eucken, R., Der Kampf um einen geistigen Lebensinhalt.

Fullerton, G. S., System of Metaphysics.

Hodgson, S., Metaphysics of Experience.

HowisoN, G. H., The Limits of Evolution.

James, W., The Will to Believe.

LiEBMANN, Analysis der Wirklichkeit.

Ormond, a. T., Foundations of Knowledge.

Petzoldt, J. , Philosophic der reinen Erfahrung.

Renouvier, C, Les Dilemmes de la m^taphysique pure.

Rickert, H., Der Gegenstand der Erkeuntniss.

RiEHL, A., Philosophische Kriticismus.

RoYCE, J., The World and the Individual.

Schiller, F. C. S., Humanism.

Seth, A., Man and the Cosmos.

Sturt, H. (editor) , Personal Idealism.

Taylor, A. E., Elements of Metaphysics.

VoLKELT, J., Erfahrung imd Denken.

WiNDELBAND, W., Praludicn.

WuNDT, W., System der Philosophie.

BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY 451

PHILOSOPHY OF RELIGION

BoussET, W., Das Wesen der Religion, dargestellt in ihrer Geschichte.

Caird, E., The Evolution of Religion.

DoRNER, A., Religionsphilosophie.

EucKEN, R., Der Wahrheitsgehalt der Religion.

Everett, C. C, The Psychological Elements of Religious Faith.

Hartmann, von, E., Religionsphilosophie.

HoFFDiNG, H., Religionsphilosophie.

James, W., Varieties of Religious Experience.

Martineau, J., A Study of Religion, its Sources and Contents.

MtJLLER, M., Einleitung in die vergleichende Religionswissenschaft.

Pfleiderer, O., Religionspliilosophie auf geschichtelichen Grundlage.

Rauenhopf, Religionsphilosophie.

RoYCE, J., The ReUgious Aspect of Philosophy.

Sabatier, a., Religionsphilosophie auf psychologischen und geschichtUchen

Grundlage.
Saussaye, Lehrbuch der Religionsgeschichte.
Seydel, R., Religionsphilosophie.
Teichmuller, G., Religionsphilosophie.
Tiele, C. p., Grundziige der Religionswissenschaft.

LOGIC

Bradley, F. H., The Principles of Logic.

Bosanquet, B., Logic.

Cohen, H., Die Logik der reinen Erkenntniss.

Dewey, J., Studies in Logical Theory.

Erdmann, B., Logik.

Hibben, J. G., Logic.

HoBHOUSE, L. T., Theory of Knowledge.

HussERL, Logische Untersuchungen.

Lotze, R. H., Gnmdziige der Logik.

ScHUPPE, W., Erkenntnisstheoretische Logik.

SiGWART, C, Logik.

Wundt, W., Logik.

METHODOLOGY OF SCIENCE

Cantor, G., Grundlagen einer allgemeinen Mannigfaltigkeitslehre.

Dedekind, R., Was sind und was sollen die Zahlen? .

Hertz, H., Die Principien der Mechanik.

Jevons, W. S., Principles of Science.

Mach, E., Die Analyse der Empfindung.

Munsterberg, H., Grtmdziige der Psychologie.

Natorp, p., Einleitimg in die Psychologie.

OsTWALD, W., Vorlesungen iiber Naturphilosophie.

Pearson, K., Grammar of Science.

Poincare, H., La Science et I'Hypoth^se.

Rickert, H. , Die Grenzen der naturwissenschaftlichen Begriffsbildung. .

RoYCE, J., The World and the Individual, Second Series.

Russell, B., The Principles of Mathematics.

Ward, J., Naturalism and Agnosticism.

Windelband, W., Geschichte und Naturwissenschaft.

452 BIBLIOGRAPHY: DEPARTMENT OF PHILOSOPHY

ETHICS

Alexander, S., Moral Order and Progress.

Bradley, F. H., Ethical Studies.

Cohen, H. , Ethik des reinen Willens.

GiZYCKi, G., Grundziige der Moral.

Green, T. H. , Prolegomena to Ethics.

GuYAU, M. J., Esquisse d'une morale sans obligation ni sanction.

Ladd, G. T., Philosophy of Conduct.

Martineau, J., Types of Ethical Theory.

Mezes, S. E., Ethics, Descriptive and Explanatory.

Moore, G. E., Principia Etbica.

Palmer, G. H., The Nature of Goodness.

Paulsen, F., System der Ethik.

RoYCE, J., Studies of Good and Evil.

Seth, J., Principles of Ethics.

Sidgwick, H., Methods of Ethics.

SiMMEL, G., Einleitimg in die Moralwissenschaft.

SoRLEY, W. E.., Ethics of Naturalism.

Spencer, H., Principles of Ethics.

Stephen, L., Science of Ethics.

Taylor, A. E., The Problem of Conduct.

Wundt, W., Ethik.

ESTHETICS

Cohn, Allgemeine iRsthetik.

GuYAU, M. J., Les Problemes de I'esthetique contemporaine.

HiRN, Yrjo, The Origins of Art.

Lange, K., Das Wesen der Kunst.

Lipps, T., ^sthetik.

Puffer, E., Psychology of Beauty.

SouRiAU, P., La Beauts Rationelle.

Volkelt, J., System der jEsthetik.

Witasek, S., Grundziige der allgemeinen sesthetik.

DEPARTMENT II — MATHEMATICS

{Hall 7, September 20, 11.15 a. m.)

Chairman: Peofessor Henry S. White, Northwestern University.
Speaxers: Professor Maxime Bocher, Harvard University.
Professor James P. Pierpont, Yale University.

The Chairman of the Department of Mathematics was Professor
Henry S. White, of Northwestern University. In opening the pro-
ceedings Professor White said:

" Influenced by patriotism and by pride in material progress, cities
and whole nations meet and celebrate the building of bridges, the
opening of long railways, the tunneling of difficult mountain passes,
the acquisition of new territories, or commemorate with festivity the
discovery of a continent. These things are real and significant to us
all.

" In the realm of ideas also there are events of no less moment,
discoveries and conquests that greatly enlarge the empire of human
reason. In the lapse of a century there may be many such notable
achievements, even in the domain of a single science.

" Mathematics is a science continually expanding; and its growth,
unlike some political and industrial events, is attended by universal
acclamation. We are wont to-day, as devotees of this noble and
useful science, to pass in review the newest phases of her expansion,
— I say newest, for in retrospect a century is but brief, — and to
rejoice in the deeds of the past. At the same time, also, we turn
an eye of aspiration and resolution towards the mountains, rivers,
deserts, and the obstructing seas that are to test the mathematicians
of the future."

THE FUNDAMENTAL CONCEPTIONS AND METHODS OF

MATHEMATICS

BY PROFESSOR MAXIME BQCHER

[Maxime Bocher, Professor of Mathematics, Harvard University, b. August 28,
1867, Boston, Mass. A.B. Harvard, 1888; Ph.D. Gottingen, 1891. In-
structor, Assistant Professor and Professor, Harvard University, 1891-.
Fellow of the American Academy. Author of Ueber die Reihenentwickel-
ungen der Potentialtheorie; and various papers on mathematics.]

I. Old and New Definitions of Mathematics

I AM going to ask you to spend a few minutes with me in consider-
ing the question: what is mathematics? In doing this I do not propose
to lay down dogmatically a precise definition; but rather, after hav-
ing pointed out the inadequacy of traditional views, to determine
what characteristics are common to the most varied parts of mathe-
matics but are not shared by other sciences, and to show how this
opens the way to two or three definitions of mathematics, any one of
which is fairly satisfactory. Although this is, after all, merely a dis-
cussion of the meaning to be attached to a name, I do not think that
it is unfruitful, since its aim is to bring unity into the fundamental
conceptions of the science with which we are concerned. If any of
you, however, should regard such a discussion of the meaning of words
as devoid of any deeper significance, I will ask you to regard this
question as merely a bond by means of which I have found it con-
venient to unite what I have to say on the fundamental conceptions
and methods of what, with or without definition, we all of us agree
to call mathematics.

The old idea that mathematics is the science of quantity, or that
it is the science of space and number, or indeed that it can be charac-
terized by any enumeration of several more or less heterogeneous
objects of study, has pretty well passed away among those mathe-
maticians who have given any thought to the question of what
mathematics really is. Such definitions, which might have been
intelligently defended at the beginning of the nineteenth century,
became obviously inadequate as subjects like projective geometry,
the algebra of logic, and the theory of abstract groups were de-
veloped; for none of these has any necessary relation to quantity
(at least in any ordinary understanding of that word), and the last
two have no relation to space. It is true that such examples have
had little effect on the more or less orthodox followers of Kant,
who regard mathematics as concerned with those conceptions which

CONCEPTIONS AND METHODS OF MATHEMATICS 457

are obtained by direct intuition of time and space without the aid of
empirical observation. This view seems to have been held by such
eminent mathematicians as Hamilton and DeMorgan; and it is a
very difficult position to refute, resting as it does on a purely meta-
physical foundation which regards it as certain that we can evolve
out of our inner consciousness the properties of time and space.
According to this view the idea of quantity is to be deduced from
these intuitions; but one of the facts most vividly brought home to
pure mathematicians during the last half-century is the fatal weak-
ness of intuition when taken as the logical source of our knowledge
of number and quantity.^

The objects of mathematical study, even when we confine our
attention to what is ordinarily regarded as pure mathematics are,
then, of the most varied description; so that, in order to reach a
satisfactory conclusion as to what really characterizes mathematics,
one of two methods is open to us. On the one hand we may seek
some hidden resemblance in the various objects of mathematical
investigation, and having found an aspect common to them all we
may fix on this as the one true object of mathematical study. Or,
on the other hand, we may abandon the attempt to characterize
mathematics by means of its objects of study, and seek in its methods
its distinguishing characteristic. Finally, there is the possibility of
our combining these two points of view. The first of these methods is
that of Kempe, the second will lead us to the definition of Benjamin
Peirce, while the third has recently been elaborated at great length
by Russell. Other mathematicians have naturally followed out more
or less consistently the same ideas, but I shall nevertheless take the
liberty of using the names Kempe, Peirce, and Russell as convenient
designations for these three points of view. These different methods
of approaching the question lead finally to results which, without
being identical, still stand in the most intimate relation to one an-
other, as we shall now see. Let us begin with the second method.

II. Peirce' s Definition

More than a third of a century ago Benjamin Peirce wrote: ^
Mathematics is the science which draws necessary conclusions. Accord-
ing to this view there is a mathematical element involved in every
inquiry in which exact reasoning is used. Thus, for instance,^ a
jury listening to the attempt of the counsel for the prisoner to prove
an alibi in a criminal case might reason as follows: "If the witnesses

' I refer here to such facts as that there exist continuous functions without
derivatives, whereas the direct untutored intuition of space would lead anj^ one
to believe that every continuous curve has tangents.

^ Linear Associative Algebra. Lithographed 1870. Reprinted in the American
Journal of Mathematics, vol. iv.

3 This illustration was suggested by the remarks by J. Richard, Sur la philoso-
phie des mathmeatiques. Paris, Gauthier-Villars, 1903^ p. 50.

458 MATHEMATICS

are telling the truth when they say that the prisoner was in St. Louis
at the moment the crime was committed in Chicago, and if it is
true that a person cannot be in two places at the same time, it follows
that the prisoner was not in Chicago when the crime was committed."
This, according to Peirce, is a bit of mathematics; while the further
reasoning by which the jury would decide whether or not to believe
the witnesses, and the reasoning (if they thought any necessary)
by which they would satisfy themselves that a person cannot be
in two places at once, would be inductive reasoning, which can give
merely a high degree of probability to the conclusion, but never
certainty. This mathematical element may be, as the example
just given shows, so slight as not to be worth noticing from a prac-
tical point of view. This is almost always the case in the transac-
tions of daily life and in the observational sciences. If, however, we
turn to such subjects as chemistry and mineralogy, we find the
mathematical element of considerable importance, though still
subordinate. In physics and astronomy its importance is much
greater. Finally in geometry, to mention only one other science, the
mathematical element predominates to such an extent that this
science has been commonly rated a branch of pure mathematics,
whereas, according to Peirce, it is as much a branch of applied
mathematics as is, for instance, mathematical physics.

It is clear from what has just been said that, from Peirce's point
of view, mathematics does not necessarily concern itself with quanti-
tative relations, and that any subject becomes capable of mathe-
matical treatment as soon as it has secured data from which import-
ant consequences can be drawn by exact reasoning. Thus, for
example, even though psychologists be right when they assure us
that sensations and the other objects with which they have to deal
cannot be measured, we need still not necessarily despair of one day
seeing a mathematical psychology, just as we already have a math-
ematical logic.

I have said enough, I think, to show what relation Peirce's con-
ception of mathematics has to the applications. Let us then turn
to the definition itself and examine it a little more closely. You
have doubtless already noticed that the phrase, " the science which
draws necessary conclusions, " contains a word which is very much
in need of elucidation. What is a necessary conclusion? Some of
you will perhaps think that the conception here involved is one
about which, in a concrete case at least, there can be no practical
diversity of opinion among men with well-trained minds; and in
fact when I spoke a few minutes ago about the reasoning of the
jurymen when listening to the lawyer trying to prove an alibi, I
assumed tacitly that this is so. If this really were the case, no further
discussion would be necessary, for it is not my purpose to enter into

CONCEPTIONS AND METHODS OF MATHEMATICS 459

any purely philosophical speculations. But unfortunately we can-
not dismiss the matter in this way; for it has happened not infre-
quently that the most eminent men, including mathematicians,
have differed as to whether a given piece of reasoning was exact or
not; and, what is worse, modes of reasoning which seem absolutely
conclusive to one generation no longer satisfy the next, as is shown
by the way in which the greatest mathematicians of the eighteenth
century used geometric intuition as a means of drawing what they
regarded as necessary conclusions.^

1 do not wish here to raise the question whether there is such a
thing as absolute logical rigor, or whether this whole conception of
logical rigor is a purely psychological one bound to change with
changes in the human mind. I content myself with expressing the
belief, which I will try to justify a little more fully in a moment,
that as we never have found an immutable standard of logical rigor
in the past, so we are not hkely to find it in the future. However
this may be, so much we can say with tolerable confidence, as past
experience shows, that no reasoning which claims to be exact can
make any use of intuition, but that it must proceed from definitely
and completely stated premises according to certain principles of
formal logic. It is right here that modern mathematicians break
sharply with the tradition of a 'priori synthetic judgments (that is,
conclusions drawn from intuition) which, according to Kant, form an
essential part of mathematical reasoning.

If then we agree that " necessary conclusions " must, in the present
state of human knowledge, mean conclusions drawn according to
certain logical principles from definitely and completely stated
premises, we must face the question as to what these principles
shall be. Here, fortunately, the mathematical logicians from Boole
down to C. S. Peirce, Schroder, and Peano have prepared the field
so well that of late years Peano and his followers ^ have been able
to make a rather short list of logical conceptions and principles upon
which it would seem that all exact reasoning depends.^ We must
remember, however, when we are tempted to put implicit confidence
in certain fundamental logical principles, that, owing to their extreme
generality and abstractness, no very great weight can be attached
to the mere fact that these principles appeal to us as obviously

^ All writers on elementary geometry from Euclid down almost to the close
of the nineteenth century use intuition freely, though usually unconsciously, in
obtaining results which they are unable to deduce from their axioms. The first
few demonstrations of Euclid are criticised from this point of view by Russell in
his Principles of Mathematics, vol. i, 404-407. Gauss's first proof (1799) that
every algebraic equation has a root gives a striking example of the use of intuition
in what was intended as an absolutely rigorous proof by one of the greatest and at
the same time most critical mathematical minds the world has ever seen.

^ And, independently, Frege.

2 It is not intended to assert that a single list has been fixed upon. Different
writers naturally use different lists.

460 MATHEMATICS

true; for, as I have said, other modes of reasoning which are now
universally recognized as faulty have appealed in just this way to
the greatest minds of the past. Such confidence as we feel must,
I think, come from the fact that those modes of reasoning which
we trust have withstood the test of use in an immense number of
cases and in very many fields. This is the severest test to which any
theory can be put, and if it does not break down under it we may
feel the greatest confidence that, at least in cognate fields, it will
prove serviceable. But we can never be sure. The accepted modes
of exact reasoning may any day lead to a contradiction which would
show that what we regard as universally applicable principles are
in reality applicable only under certain restrictions.^

To show that the danger which I here point out is not a purely
fanciful one, it is sufiicient to refer to a very recent example. Inde-
pendently of one another, Frege and Russell have built up the theory
of arithmetic from its logical foundations. Each starts with a definite
list of apparently self-evident logical principles, and builds up a
seemingly flawless theory. Then Russell discovers that his logical
principles when applied to a very general kind of logical class lead
to an absurdity; and both Frege and Russell have to admit that
something is wrong with the foundations which looked so secure.
Now there is no doubt that these logical foundations will be somehow
recast to meet this difficulty, and that they will then be stronger
than ever before. ^ But who shall say that the same thing will not
happen again?

It is commonly considered that mathematics owes its certainty
to its reliance on the immutable principles of formal logic. This,
as we have seen, is only half the truth imperfectly expressed. The
other half would be that the principles of formal logic owe such
degree of permanence as they have largely to the fact that they
have been tempered by long and varied use by mathematicians.
"A vicious circle!" you will perhaps say. I should rather describe
it as an example of the process known to mathematicians as the
method of successive approximations. Let us hope that in this
case it is really a convergent process, as it has every appearance of
being.

But to return to Peirce's definition. From what are these neces-

* If the view which I here maintain is correct, it follows that if the teim " abso-
lute logical rigor" has a meaning, and if we should some time arrive at this abso-
lute standard, the only indication we should ever have of the fact would be that
for a long period, several thousand years let us say, the logical principles in ques-
tion had stood the test of use. But this state of affairs might equally well mean
that during that time the human mind had degenerated, at least with regard to
some of its functions. Consider, for instance, the twenty centuries following Euclid
when, without doubt, the high tide of exact thinking attained during Euclid's gen-
eration had receded.

^ Cf. Poincard's view* in La Science et I'Hypothese, p. 179, according to which
a theory never renders a greater service to science than when it breaks down.

CONCEPTIONS AND METHODS OF MATHEMATICS 461

sary conclusions to be drawn? The answer clearly implied is, from
any premises sufficiently precise to make it possible to draw neces-
sary conclusions from them. In geometry, for instance, we have a
large number of intuitions and fixed beliefs concerning the nature
of space: it is homogeneous and isotropic, infinite in extent in every
direction, etc.; but none of these ideas, however clearly defined
they may at first sight seem to be, gives any hold for exact reasoning.
This was clearly perceived by Euclid, who therefore proceeded to
lay down a list of axioms and postulates, that is, specific facts which
he assumes to be true, and from which it was his object to deduce all
geometric propositions. That his success here was not complete
is now well known, for he frequently assumes unconsciously further
data which he derives from intuition; but his attempt was a monu-
mental one,

III. The Abstract Nature of Mathematics

Now a further self-evident point, but one to which attention seems
to have been drawn only during the last few years, is this : since we
are to make no use of intuition, but only of a certain number of
explicitly stated premises, it is not necessary that we should have
any idea what the nature of the objects and relations involved in
these premises is.^ I will try to make this clear by a simple example.
In plane geometry we have to consider, among other things, points and
straight lines. A point may have a peculiar relation to a straight
line which we express by the words, the point lies on the line. Now
one of the fundamental facts of plane geometry is that two points
determine a line, that is, if two points are given, there exists one and
only one line on which both points lie. All the facts that I have just
stated correspond to clear intuitions. Let us, however, eliminate our
intuition of what is meant by a point, a line, a point lying on a line.
A slight change of language will make it easy for us to do this. In-
stead of points and lines, let us speak of two different kinds of objects,
say A-objects and J5-objects; and instead of saying that a point
lies on a line we will simply say that an A-object bears a certain
relation R to a, jB-object. Then the fact that two points determine
a line will be expressed by saying: If any two ^-objects are given,
there exists one and only one 5-object to which they both bear the
relation R. This statement, while it does not force on us any specific
intuitions, will serve as a basis for mathematical reasoning ^ just as
well as the more familiar statement where the terms points and lines

^ This was essentially Kempe's point of view in the papers to be referred to
presently. In the geometric example which follows it was clearly brought out
by H. Wiener: Jahresbericht d. deuischen Mathematiker-Vereinigung, vol. i (1891),
p. 45.

^ In conjunction, of course, with further postulates with which we need not
here concern ourselves.

462 MATHEMATICS

are used. But more than this. Our ^-objects, our B-objects, and our
relation R may be given an interpretation, if we choose, very different
from that we had at first intended.

We may, for instance, regard the A-objects as the straight hnes in
a plane, the jB-objects as the points in the same plane (either finite
or at infinity), and when an A-object stands in the relation jR to a
5-object, this may be taken to mean that the line passes through the
point. Our statement would then become : Any two lines being given,
there exists one and only one point through which they both pass.
Or we may regard the A-objects as the men in a certain community,
the 5-objects as the women, and the relation of an A-object to a
5-object as friendship. Then our statement would be: In this com-
munity any two men have one, and only one, woman friend in com-
mon.

These examples are, I think, sufficient to show what is meant
when I say that we are not concerned in mathematics with the
nature of the objects and relations involved in our premises, except
in so far as their nature is exhibited in the premises themselves.
Accordingly mathematicians of a critical turn of mind, during the
last few years, have adopted more and more a purely nominalistic
attitude towards the objects and relations involved in mathematical
investigation. This is, of course, not the crude mixture of nominalism
and empiricism of the philosopher Hobbes, whose claim to mathe-
matical fame, it may be said in passing, is that of a circle-squarer.^
The nominalism of the present-day mathematician consists in treating
the objects of his investigation and the relations between them as
mere symbols. He then states his propositions, in effect, in the fol-
lowing form: If there exist any objects in the physical or mental
world with relations among themselves which satisfy the conditions
which I have laid down for my symbols, then such and such facts
will be true concerning them.

It will be seen that, according to Peirce's view, the mathematician
as such is in no wise concerned with the source of his premises or with
their harmony or lack of harmony with any part of the external
world. He does not even assert that any objects really exist which
correspond to his symbols. Mathematics may therefore be truly
said to be the most abstract of all sciences, since it does not deal
directly with reality.^

ThiS; then, is Peirce's definition of mathematics. Its advantages
in the direction of unifying our conception of mathematics and of
assigning to it a definite place among the other sciences are clear.

^ Hobbes practically obtains as the ratio of a circumference to its diameter
the value VlO. Cf. for instance Molesworth's edition of Hobbes's English Works,
vol. VII, p. 431.

^ Cf. the very interesting remarks along this line of C. S. Peirce in The Monist,
vol. VII, pp. 23-24.

CONCEPTIONS AND METHODS OF MATHEMATICS 463

What are its disadvantages? I can see only two. First that, as has
been already remarked, the idea of drawing necessary conclusions
is a slightly vague and shifting one. Secondly, that it lays exclusive
stress on the rigorous logical element in mathematics and ignores
the intuitional and other non-rigorous tendencies which form an
important element in the great bulk of mathematical work concern-
ing which I shall speak in greater detail later.

IV. Geometry an Experimental Science

Some of you will also regard it as an objection that there are
subjects which have almost universally been regarded as branches
of mathematics but are excluded by this definition. A striking
example of this is geometry, I mean the science of the actual space
we hve in; for though geometry is, according to Peirce's definition,
preeminently a mathematical science, it is not exclusively so. Until
a system of axioms is established mathematics cannot begin its work.
Moreover, the actual perception of spatial relations, not merel}^
in simple cases but in the appreciation of complicated theorems, is
an essential element in geometry which has no relation to mathe-
matics as Peirce understands the term. The same is true, to a con-
siderable extent, of such subjects as mechanical drawing and model-
making, which involve, besides small amounts of physics and math-
ematics, mainly non-mathematical geometry. Moreover, although the
mathematical method is the traditional one for arriving at the truth
concerning geometric facts, it is not the only one. Direct appeal to
the intuition is often a short and fairly safe cut to geometric results;
and on the other hand experiments may be used in geometry, just
as they are used every day in physics, to test the truth of a proposi-
tion or to determine the value of some geometric magnitude.^

We must, then, admit, if we hold to Peirce's view, that there is
an independent science of geometry just as there is an independent
science of physics, and that either of these may be treated by math-
ematical methods. Thus geometry becomes the simplest of the
natural sciences, and its axioms are of the nature of physical laws,
to be tested by experience and to be regarded as true only within
the limits of error of observation. This view, while it has not yet
gained universal recognition, should, I believe, prevail, and geo-
metry be recognized as a science independent of mathematics, just
as psychology is gradually being recognized as an independent
science and not as a branch of philosophy.

The view here set 'forth, according to which geometry is an ex-
perimental science like physics or chemistry, has been held ever

1 I am thinking of measurements and observations made on accurately con-
structed drawings and models. A famous example is Galileo's determination of
the area of a cycloid by cutting out a cycloid from a metallic sheet and weighing it.

464 MATHEMATICS

since Gauss's time by almost all the leading mathematicians who
have been conversant with non-Euclidean geometry.^ Recently,
however, Poincare has thrown the weight of his great authority
against this view,^ claiming that the experiments by which it is
sought to test the truth of geometric axioms are really not geometrical
experiments at all but physical ones, and that any failure of these
experiments to agree with the ordinary geometrical axioms could
be explained by the inaccuracy of the physical laws ordinarily as-
sumed. There is undoubtedly an important element of truth here.
Every experiment depends for its results not merely on the law Vv^e
wish to test, but also on other laws which for the moment we assume
to be true. But, if we prefer, we may, in many cases, assume as
true the law we were before testing and our experiment will then
serve to test some of the remaining laws. If, then,, we choose to stick
to the ordinary Euclidean axioms of geometry in spite of what any
future experiments may possibly show, we can do so, but at the cost,
perhaps, of our present simple physical laws, not merely in one
branch of physics but in several. Poincare's view ^ is that it will
always be expedient to preserve simple geometric laws at all costs,
an opinion for which I fail to see sufficient reason.

V, Kemipe's Definition

Let us now turn from Peirce's method of defining mathematics to
Kempe's, which, however, I shall present to you in a somewhat
modified form.* The point of view adopted here is to try to define
mathematics, as other sciences are defined, by describing the objects
with which it deals. The diversity of the objects with which mathe-
matics is ordinarily supposed to deal being so great, the first step
must be to divest them of what is unessential for the mathematical
treatment, and to try in this way to discover their common and
characteristic element.

The first point on which Kempe insists is that the objects of mathe-
matical discussion, whether they be the points and fines of geometry,
the numbers real or complex of algebra or analysis, the elements of
groups or anything else, are always individuals, infinite in number
perhaps, but still distinct individuals. In a particular mathematical
investigation we may, and usually do, have several different kinds of
individuals; as for instance, in elementary plane geometry, points,
straight lines, and circles. Furthermore, we have to deal with certain
relations of these objects to one another. For instance, in the example

^ Gauss, Riemann, Helmholtz are the names which will carry perhaps the
greatest weight.

^ Cf. La Science et VHypothese. Paris, 1903.

^ L. c, chapter v. In particular, p. 93.

< Kempe has set forth his ideas in rather popular form in the Proceedings of
the London Mathematical Society, vol. xxvi (1894), p. 5; and in Nature, vol. xliii
1890), p. 156, where references to his more technical writings wiU be found.

CONCEPTIONS AND METHODS OF MATHEMATICS 465

just cited; a given point may or may not lie on a given line; a given
line may or may not touch a given circle; three or more points may
or may not be coUinear, etc. This example shows how in a single
mathematical problem a large number of relations may be involved,
relations some of which connect two objects, others three, etc.
Moreover these relations may connect like or they may connect
unlike objects; and finally the order in which the objects are taken
is not by any means immaterial in general, as is shown by the relation
between three points which states that the third is coUinear v/ith and
lies between the first two.

But even this is not all; for, besides these objects and relations
of various kinds, we often have operations by which objects can be
combined to yield another object, as, for instance, addition or multi-
plication of numbers. Here the objects combined and the resulting
object are all of the same kind, but this is by no means necessary.
We may, for instance, consider the operation of combining two
points and getting the perpendicular bisector of the line connecting
them; or we may combine a point and a line and get the perpen-
dicular dropped from the point on the line.

These few examples show how diverse the relations and operations,
as vvcll as the objects of mathematics, seem at first sight to be. Out
of this apparent diversity it is not difficult to obtain a very great
uniformity by simply restating the facts in a little different language.
We shall find it convenient to indicate that the objects a, b, c, . . . ,
taken in the order named, satisfy a relation R by simply writing
R(a. 6, c, . . . ), where it should be understood that among the
objects a, h. e, . . . the same object may occur a number of times.
On the other hand, if two objects a and h are combined to yield
a third object c, we may write a o b=c,^ where the symbol o is
characteristic of the special operation with which we are concerned.

Lot us fii'st notice that the equation aoh=c denotes merely
that the three objects a, h, c bear a certain relation to one another,
say R{a, b, c). In other words the idea of an operation or law of
combination between the objects we deal with, however convenient
and useful it ma}^ be as a matter of notation, is essentially merely
a way of expressing the fact that the objects combined bear a certain
relation to the object resulting from their combination. Accordingly,
in a purely abstract discussion like the present, where questions of
practical convenience are not involved, we need not consider such
rules of combination.^

' I speak here merely of dyadic operations, — i. e., of operations by which
two objects are combined to yield a third, — these being by far the most import-
ant as well as the simplest. What is said, however, obviously applies to opera-
tions by which any number of objects are combined.

^ Even from the point of view of the technical mathematician it may some-
times be desirable to adopt the point of view of a relation rather than that of an
operation. This is seen, for instance, in laying down a system of postulates for the

466 MATHEMATICS

Furthermore, it is easy to see that when we speak of objects of
different kinds, as, for instance, the points and hnes of geometry, we
are introducing a notion which can very readily be expressed in our
relational notation. For this purpose we need merely to introduce
a further relation which is satisfied by two or more objects when and
only when they are of the same "kind/'

Let us turn finally to the relations themselves. It is customary
to distinguish here between dyadic relations, triadic relations, etc.,
according as the relation in question connects two objects, three
objects, etc. There are, however, relations which may connect any
number of objects, as, for instance, the relation of collinearity which
may hold between any number of points. Any relation holds for
certain ordered groups of objects but not for others, and it is in no
way necessary for us to fix our attention on the fact, if it be true,
that the number of objects in all the groups for which a particular
relation holds is the same. This is the point of view we shall adopt,
and we shall relegate the property that a relation is dyadic, triadic,
etc., to the background along with the various other properties
relations may have,^ all of which must be taken account of in the
proper place.

We are thus concerned in any mathematical investigation, from
our present point of view, with just two conceptions: first a set, or
as the logicians say, a class of objects a, b, c, . . .; and secondly a
class of relations R, S, T, . . . . We may suppose these objects
divested of any qualitative, quantitative, spatial, or other attributes
which they may have had, and regard them merely as satisfying or not
satisfying the relations in question, where, again, we are wholly
indifferent to the nature which these relations originally had. And
now we are in a position to state what I conceive to be really the
essential point in Kempe's definition of mathematics; although I
have omitted one of the points on which he insists most strongly,^
by saying:

If we have a certain class of objects and a certain class of relations,
and if the only questions which we investigate are whether ordered
groups of these objects do or do not satisfy the relations, the results
of the investigation are called mathematics.

theory of abstract groups (cf., for example, Huntington, Bulletin of the Ameri-
can Mathematical Society, June, 1902), where the postulate:
If a and b belong to the class, a o b belongs to the class,
which in this form looks indecomposable, immediately breaks up, when stated in
the relational form, into the following two:

1. If a and b belong to the class, there exists an element c of the class such that
R(a, b, c).

2. If a, b, c, d belong to the class, and if R{a, b, c) and R{a, b, d), then c = d.

^ For instance, the property of symmetry. A relation is said to be symmetrical
if it holds or fails to hold independently of the order in which the objects are taken.

^ Namely, that the only relation that need be considered is that of being "in-
distinguishable," i. e., a S3anmetrical and transitive relation between two groups
of objects.

CONCEPTIONS AND METHODS OF MATHEMATICS 467

It is convenient to have a term to designate a class of objects
associated with a class of relations between these objects. Such an
aggregate we will speak of as a mathematical system. If now we have
two different mathematical systems, and if a one-to-one correspond-
ence can be set up between the two classes of objects, and also
between the two classes of relations in such a way that whenever
a certain ordered set of objects of the first system satisfies a relation
of that system, the set consisting of the corresponding objects of the
second system satisfies the corresponding relation of that system,
and vice versa, then it is clear that the two systems are, from our
present point of view, mathematically equivalent, however different
the nature of the objects and relations may be in the two cases. ^ To
use a technical term, the two systems are simply isomorphic.^

It will be noticed that in the definition of mathematics just given
nothing is said as to the method by which we are to ascertain whether
or not a given relation holds between the objects of a given set. The
method used may be a purely empirical one, or it may be partly or
wholly deductive. Thus, to take a very simple case, suppose our class
of objects to consist of a large number of points in a plane and sup-
pose the only relation between them with which we are concerned
is that of collinearity. Then, if the points are given us by being
marked in ink on a piece of white paper, we can begin by taking three
pins, sticking them into the paper at three of the points; then, by
sighting along them, we can determine whether or not these points
are collinear. We can do the same with other groups of three
points, then with all groups of four points, etc. The same result
can be obtained with much less labor if we make use of certain
simple properties which the relation of collinearity satisfies, pro-
perties which are expressed by such propositions as:

R(a, h, c) implies RQ), a, c),

R{a, h, c, d) implies R{a, h, c),

R(a, b, c) and R(a, b, d) together imply R(a, b, c, d), etc.

By means of a small number of propositions of this sort it is easy
to show that no empirical observations as to the collinearity of
groups of more than three points need be made, and that it may
not be necessary to examine even all groups of three points. Having

^ The point of view here brought out, including the term isomorphism, was
first developed in a special case, — the theory of groups.

^ Inasmuch as the relations in a mathematical system are themselves objects,
we may, if we choose, take our class of objects so as to include these relations as
well as what we called objects before, some of which, we may remark in passing,
may themselves be relations. Looked at from this point of view, we need pne
additional relation which is now the only one which we explicitly eaU a relation.
If we denote this relation by inclosing the objects which satisfy it in parentheses,
then if the relation denoted before by R{a, b) is satisfied, we should now write
{R, a, b), whereas we should not have {a, R, b) (S, R, a, b), etc. Thus we see that
any mathematical system may be regarded as consisting of a class of objects and
a single relation between them.

468 MATHEMATICS

made this relatively small number of observations, the remaining
results would be obtained deductively. Finally, we may suppose
the points given by their coordinates, in which case the complete
answer to our question may be obtained by the purely deductive
method of analytic geometry.

According to the modified form of Kempe's definition which I
have just stated, mathematics is not necessarily a deductive science.
This view, while not in accord with the prevailing ideas of mathe-
maticians, undoubtedly has its advantages as well as its dangers.
The non-deductive processes, of which I shall have more to say
presently, play too important a part in the life of mathematics to
be ignored, and the definition just given has the merit of not exclud-
ing them. It would seem, however, that the definition in the form
just given is too broad. It would include, for instance, the deter-
mination by experimental methods of what pairs of chemical com-
pounds of the known elements react on one another when mixed
under given conditions.

VL Axioms and Postulates. Existence Theorems

If, however, we restrict ourselves to exact or deductive mathe-
matics, it will be seen that Kempe's definition becomes coextensive
with Peirce's. Here, in order to have a starting-point for deductive
reasoning, we must assume a certain number of facts or primitive
propositions concerning any mathematical system we wish to study,
of which all other propositions will be necessary consequences.^
We touch here on a subject whose origin goes back to Euclid and
which has of late years received great development, primarily at
the hands of Italian mathematicians.^

It is important for us to notice at this point that not merely these
primitive propositions but all the propositions of mathematics may
be divided into two great classes. On the one hand, we have pro-
positions which state that certain specified objects satisfy certain
specified relations. On the other hand are the existence theorems,
which state that there exist objects satisfying, along with certain
specified objects, certain specified relations.^ These two classes of
propositions are well known to logicians and are designated by them

^ These primitive propositions may be spoken of as axioms or postulates, ac-
cording to the point of view we wish to take concerning their source, the word
axiom, which has been much misused of late, indicating an intuitional or empirical
source.

^ Peano, Fieri, Padoa, Burali-Forti. We may mention here also Hilbert, who,
apparently without knowing of the important work of his Italian predecessors,
has also done valuable work along these lines.

^ Or we might conceivably have existence theorems which state that there
exist relations which are satisfied by certain specified objects; or these two kinds
of existence theorems might be combined. If we take the point of view explained
in the second footnote on p. 467, all existence theorems will be of the type men-
tioned in the text.

CONCEPTIONS AND METHODS OF MATHEMATICS 469

universal and particular propositions respectively.^ It is only during
the last fifty years or so that mathematicians have become conscious
of the fundamental importance in their science of existence theorems,
which until then they had frequently assumed tacitly as they needed
them, without always being conscious of what they were doing.

It is sometimes held by non-mathematicians that if mathematics
were really a purely deductive science, it could not have gained
anything like the extent which it has without losing itself in trivial-
ities and becoming, as Poincare puts it, a vast tautology.^ This
view would doubtless be correct if all primitive propositions were
universal propositions. One of the most characteristic features of
mathematical reasoning, however, is the use which it makes of aux-
iliary elements. I refer to the auxiliary points and lines in proofs
by elementary geometry, the quantities formed by combining in
various ways the numbers which enter into the theorems to be
proved in algebra, etc. Without the use of such auxiliary elements
mathematicians would be incapable of advancing a step; and
whenever we make use of such an element in a proof, we are in reality
using an existence theorem.' These existence theorems need not,
to be sure, be among the primitive propositions; but if not, they must
be deduced from primitive propositions some of which are existence
theorems, for it is clear that an existence theorem cannot be deduced
from universal propositions alone.* Thus it may fairly be said that
existence theorems form the vital principle of mathematics, but these
in turn, it must be remembered, would be impotent without the
material basis of universal propositions to work upon.

VII. RusseWs Definition

We have so far arrived at the view that exact mathematics is
the study by deductive methods of what we have called a mathe-
matical system, that is, a class of objects and a class of relations
between them. If we elaborate this position in two directions we
shall reach the standpoint of Russell.^

In the first place Russell makes precise the term deductive method

* "All men are mortals" is a standard example of a universal proposition;
while as an illustration of a particular proposition is often given: "Some men are
Greeks." That this is really an existence theorem is seen more clearly when we
state it in the form: "There exists at least one man who is a Greek."

^ Cf. La Science et VHypothese, p. 10.

' Even when in algebra we consider the sum of two numbers a + 6, we are using
the existence theorem which says that, any two numbers a and h being given,
there exists a number c which stands to them in the relation which we indicate in
ordinary language by saying that c is the sum of a and b.

* The power which resides in the method of mathematical induction, so called,
comes from the fact that this method depends on an existence theorem. It is,
however, not the only fertile principle in mathematics as Poincare would have
us believe (cf. La Science et VHypothese). In fact there are great branches of
mathematics, like elementary geometry, in which it takes little or no part.

^ The Principles of Mathematics, Cambridge, England, 1903.

470 MATHEMATICS

by laying down explicitly a list of logical conceptions and prin-
ciples which alone are to be used; and, secondly, he insists,^ on the
contrary, that no mathematical system, to use again the technical
term introduced above, be studied in pure mathematics whose exist-
ence cannot be established solely from the logical principles on which
all mathematics is based. Inasmuch as the development of mathemat-
ics during the last fifty years has shown that the existence of most,
if not all the mathematical systems which have proved to be im-
portant can be deduced when once the existence of positive integers
is granted, the point about which interest must centre here is the
proof, which Russell attempts, of the existence of this latter sys-
tem.2 This proof will necessarily require that, among the logical
principles assumed, existence theorems be found. Such theorems
do not seem to be explicitly stated by Russell, the existence theorems
which make their appearance further on being evolved out of some-
what vague philosophical reasoning. There are also other reasons,
into which I cannot enter here, why I am not able to regard the
attempt made in this direction by Russell as completely successful.^
Nevertheless, in view of the fact that the system of finite positive
integers is necessary in almost all branches of mathematics (we
cannot speak of a triangle or a hexagon without having the numbers
three and six at our disposal), it seems extremely desirable that the
system of logical principles which we lay at the foundation of all
mathematics be assumed, if possible, broad enough so that the
existence of positive integers — at least finite integers — follows from
it; and there seems little doubt that this can be done in a satisfactory
manner. When this has been done we shall perhaps be able to regard,
with Russell, pure mathematics as consisting exclusively of deduc-
tions "by logical principles from logical principles."

VIII. The Non-Deductive Elements in Mathematics

I fear that many of you will think that what I have been saying
is of an extremely one-sided character, for I have insisted merely on
the rigidly deductive form of reasoning used and the purely abstract
character of the objects considered in mathematics. These, to the
great majority of mathematicians, are only the dry bones of the
science. Or, to change the simile, it may perhaps be said that instead
of inviting you to a feast I have merely shown you the empty dishes

' In the formal definition of mathematics at the beginning of the book this is
not stated or in any way implied; and yet it comes out so clearly throughout
the book that this is a point of view which the author regards as essential, that
I have not hesitated to include it as a part of his definition.

^ Cf. also Burali-Forti, Congres internationale de philosophie. Paris, vol. iii,
p. 280.

^ PLUssell's unequivocal repudiation of nominalism in mathematics seems to
me a serious if not an insurmountable barrier to progress.

CONCEPTIONS AND METHODS OF MATHEMATICS 471

and explained how the feast would be served if only the dishes were
filled,^ I fully agree with this opinion, and can only plead in excuse
that my subject was the fundamental conceptions and methods of
mathematics, not the infinite variety of detail and application
which give our science its real vitality. In fact I should like to
subscribe most heartily to the view that in mathematics, as else-
where, the discussion of such fundamental matters derives its interest
mainly from the importance of the theory of which they are the
so-called foundations.^ I like to look at mathematics almost more
as an art than as a science; for the activity of the mathematician,
constantly creating as he is, guided though not controlled by the
external world of the senses, bears a resemblance, not fanciful I
believe but real, to the activity of an artist, of a painter let us say.
Rigorous deductive reasoning on the part of the mathematician
may be likened here to technical skill in drawing on the part of the
painter. Just as no one can become a good painter without a certain
amount of this skill, so no one can become a mathematician without
the power to reason accurately up to a certain point. Yet these
qualities, fundamental though they are, do not make a painter or
a mathematician worthy of the name, nor indeed are they the most
important factors in the case. Other qualities of a far more subtle
sort, chief among which in both cases is imagination, go to the
making of the good artist or good mathematician. I must content
myself merely by recalling to you this somewhat vague and difficult
though interesting field of speculation which arises when we attempt
to attach value to mathematical work, a field which is familiar
enough to us all in the analogous case of artistic or literary criticism.
We are in the habit of speaking of logical rigor and the considera-
tion of axioms and postulates as the foundations on which the superb
structure of modern mathematics rests; and it is often a matter of
wonder how such a great edifice can rest securely on such a small
foundation. Moreover, these foundations have not always seemed so
secure as they do at present. During the first half of the nineteenth
century certain mathematicians of a critical turn of mind — Cauchy,
Abel, Weierstrass, to mention the greatest of them — perceived to
their dismay that these foundations were not sound, and some of the
best efforts of their lives were devoted to strengthening and improv-
ing them. And yet I doubt whether the great results of mathematics

_ • Notice that just as the empty dishes could be filled by a great variety of
viands, so the empty symbols of mathematics can be given meanings of the most
varied sorts.

^ Cf . the following remark by Study, JahresbericM der deutachcn Mathematiker^
Vereinigung, vol. xi (1902), p. 313:

" So wertvoll auch Untersuchungen iiber die systematische Stellung der math-
ematischen Grundbegriffe sind . . . wertvoller ist doch noch der materielle Inhalt
der einzelnen Disciplinen, um dessentwillen allein ja derartige Untersuchungen
tiberhaupt Zweck haben, , . ,"

472 MATHEMATICS

seemed less certain to any of them because of the weakness they
perceived in the foundations on which these results are built up.
The fact is that what we call mathematical rigor is merely one of
the foundation stones of the science; an important and essential
one surely, yet not the only thing upon which we can rely. A science
which has developed along such broad lines as mathematics, with
such numerous relations of its parts both to one another and to other
sciences, could not long contain serious error without detection.
This explains how, again and again, it has come about, that the
most important mathematical developments have taken place by
methods which cannot be wholly justified by our present canons of
mathematical rigor, the logical "foundation" having been supplied
only long after the superstructure had been raised. A discussion
and analysis of the non-deductive methods which the creative
mathematician really uses would be both interesting and instructive.
Here I must content myself with the enumeration of a few of them.

First and foremost there is the use of intuition, whether geometrical,
mechanical, or physical. The great service which this method has
rendered and is still rendering to mathematics both pure and applied
is so well known that a mere mention is sufficient.

Then there is the method of experiment; not merely the physical
experiments of the laboratory or the geometrical experiments I
had occasion to speak of a few minutes ago, but also arithmetical
experiments, numerous examples of which are found in the theory
of numbers and in analysis. The mathematicians of the past fre-
quently used this method in their printed works. That this is now
seldom done must not be taken to indicate that the method itself is
not used as much as ever.

Closely allied to this method of experiment is the method of
analogy, which assumes that something true of a considerable num-
ber of cases will probably be true in analogous cases. This is, of
course, nothing but the ordinary method of induction. But in mathe-
matics induction may be employed not merely in connection with
the experimental method, but also to extend results won by deduct-
ive methods to other analogous cases. This use of induction has
often been unconscious and sometimes overbold, as, for instance,
when the operations of ordinary algebra were extended without
scruple to infinite series.

Finally there is what may perhaps be called the method of optim-
ism, which leads us either willfully or instinctively to shut our eyes
to the possibility of evil. Thus the optimist who treats a problem in
algebra or analytic geometry will say, if he stops to reflect on what
he is doing: ''1 know that I have no right to divide by zero; but
there are so many other values which the expression by which I am
dividing might have that I will assume that the Evil One has not

CONCEPTIONS AND METHODS OF MATHEMATICS 473

thrown a zero in my denominator this time." This method, if a pro-
ceeding often unconscious can be called a method, has been of great
service in the rapid development of many branches of mathematics^
though it may well be doubted whether in a subject as highly devel-
oped as is ordinary algebra it has not now survived its usefulness.^
While no one of these methods can in any way compare with
that of rigorous deductive reasoning as a method upon which to
base mathematical results, it would be merely shutting one's eyes
to the facts to deny them their place in the life of the mathematical
world, not merely of the past but of to-day. There is now, and there
always will be room in the world for good mathematicians of every
grade of logical precision. It is almost equally important that the
small band whose chief interest lies in accuracy and rigor should
not make the mistake of despising the broader though less accurate
work of the great mass of their colleagues; as that the latter should
not attempt to shake themselves wholly free from the restraint the
former would put upon them. The union of these two tendencies
in the same individuals, as it was found, for instance, in Gauss and
Cauchy, seems the only sure way of avoiding complete estrangement
between mathematicians of these two types.

^ Cf. the very suggestive remarks by Study, Jahresbericht d. Deutschen Math-
ematiker-Vereinigung, vol. xi (1902), p. 100, footnote, in which it is pointed out
how rigor, in cases of this sort, ma}' not merely serve to increase the correctness of
the result, but actually to suggest new fields for mathematical investigation.

THE HISTORY OF MATHEMATICS IN THE NINETEENTH

CENTURY

BY PROFESSOE JAMES P. PIERPONT OF YALE UNIVERSITY

The extraordinary development of mathematics in the last century-
is quite unparalleled in the long history of this most ancient of
sciences. Not only have those branches of mathematics which were
taken over from the eighteenth century steadily grown, but entirely
new ones have sprung up in almost bewildering profusion, and
many of these have promptly assumed proportions of vast extent.

As it is obviously impossible to trace in the short time allotted to
me the history of mathematics in the nineteenth century even in
merest outline, I shall restrict myself to the consideration of some
of its leading theories.

Theory of Functions of a Complex Variable

Without doubt one of the most characteristic features of mathe-
matics in the last century is the systematic and universal use of the
complex variable. Most of its great theories received invaluable aid
from it, and many owe their very existence to it. What would the
theory of differential equations or elliptic functions be to-day without
it, and is it probable that Poncelet, Steiner, Chasles, and von Staudt
would have developed synthetic geometry with such elegance and
perfection without its powerful stimulus?

The necessities of elementary algebra kept complex numbers
persistently before the eyes of every mathematician. In the eight-
eenth century the more daring, as Euler and Lagrange, used them
sparingly; in general one avoided them when possible. Three events,
however, early in the nineteenth century changed the attitude of
mathematicians toward this mysterious guest. In 1813 Argand
published his geometric interpretation of complex numbers. In
1824 came the discovery by Abel of the imaginary period of the
elliptic function. Finally Gauss in his second memoir on biquadratic
residues (1832) proclaims them a legitimate and necessary element
of analysis.

The theory of function of a complex variable may be said to have
had its birth when Cauchy discovered his integral theorem

ff(x)dx=0

published in 1825. In a long series of publications beginning with
the Cours d' Analyse (1821), Cauchy gradually developed his theory
of functions and applied it to problems of the most diverse nature;

MATHEMATICS IN THE NINETEENTH CENTURY 475

for example, existence theorems for implicit functions and the solu-
tions of certain differential equations, the development of functions
in infinite series and products, and the periods of integrals of one
and many valued functions.

Meanwhile Germany is not idle; Weierstrass and Riemann de-
velop Cauchy's theory along two distinct and original paths. Weier-
strass starts with an explicit analytical expression, a power series,
and defines his function as the totality of its analytical continua-
tions. No appeal is made to geometric intuition, his entire theory,
is strictly arithmetical. Riemann growing up under Gauss and
Dirichlet not only relies largely on geometric intuition, but he also
does not hesitate to impress mathematical physics into his service.
Two noteworthy features of his theory are the many leaved surfaces
named after him, and the extensive use of conformal representation.

The history of functions as first developed is largely a theory of
algebraic functions and their integrals. A general theory of func-
tions is only slowly evolved. For a long time the methods of Cauchy,
Riemann, and Weierstrass were cultivated along distinct lines by
their respective pupils. The schools of Cauchy and Riemann were
the first to coalesce. The entire rigor which has recently been im-
parted to their methods has removed all reason for founding, as
Weierstrass and his school have urged, the theory of functions on
a single algorithm, namely, the power series. We may therefore say
that at the close of the century there is only one theory of functions
in which the ideas of its three great creators are harmoniously united.

Let us note briefly some of its lines of advance. Weierstrass early
observed that an analytic expression might represent different
analytic functions in different regions. Associated with this is the
phenomenon of natural boundaries. The question therefore arose.
What is the most general domain of definition of an analytic function?
Runge has shown that any connected region may serve this purpose.
An important line of investigation relates to the analytic expression
of a function by means of infinite series, products, and fractions.
Here may be mentioned Weierstrass 's discovery of prime factors;
the theorems of Mittag-Leffler and Hilbert; Poincare's uniform-
ization of algebraic and analytic functions by means of a third
variable, and the work of Stieljes, Fade, and Van Vleck on infinite
fractions. Since an analytic function is determined by a single
power series, which in general has a finite circle of convergence, two
problems present themselves : determine, first, the singular points of
the analytic function so defined, and, second, an analytic expression
valid for its whole domain of definition. The celebrated memoir of
Hadamard inaugurated a long series of investigations on the first
problem; while Mittag-Leffler's star theorem is the most important
result yet obtained relating to the second.

476 MATHEMATICS

Another line of investigation relates to the work of Poincare,
Borel, Fade, et al., on divergent series. It is, indeed, a strange vicissi-
tude of our science that these series which early in the century-
were supposed to be banished once and for all from rigorous mathe-
matics should at its close be knocking at the door for readmission.

Let us finallj^ note an important series of memoirs on integral
transcendental functions, beginning with Weierstrass, Laguerre, and
Poincare.

Algebraic Functions and their Integrals

A branch of the theory of functions has been developed to such
an extent that it may be regarded as an independent theory; we
mean the theory of algebraic functions and their integrals. The
brilliant discoveries of Abel and Jacobi in the elliptic functions from
1824 to 1829 prepared the way for a similar treatment of the hyper-
elliptic case. Here a difficulty of gravest nature was met. The cor-
responding integrals have 2p linearly independent periods; but as
Jacobi had shown, a one valued function having more than two
periods admits a period as small as we choose. It therefore looked
as if the elliptic functions admitted no further generalization.
Guided by Abel's theorem, Jacobi at last discovered the solution to
the difficulty (1832) ; to get functions analogous to the elliptic func-
tions we must consider functions not of one but of p independent
variables, namely, the p independent integrals of the first species.
The great problem now before mathematicians, known as Jacobi's
Problem of Inversion, was to extend this apercu to the case of any
algebraic configuration and develop the consequences. The first to
take up this immense task were Weierstrass and Riemann, whose
results belong to the most brilliant achievements of the century.
Among the important notions hereby introduced we note the fol-
lowing: the birational transformation, rank of an algebraic con-
figuration, class invariants, prime functions, the theta and multiply
periodic functions in several variables. Of great importance is
Riemann 's method of proving existence theorems, as also his repre-
sentation of algebraic functions by means of integrals of the second
species.

A new direction was given to research in this field by Clebsch, who
considered the fundamental algebraic configuration as defining a
curve. His aim was to bring about a union of Riemann 's ideas and
the theory of algebraic curves for their mutual benefit. Clebsch's
labors were continued by Brill and Nother; in their work the tran-*
scendental methods of Riemann are placed quite in the background.
More recently Klein and his school have sought to unite the tran-
scendental methods of Riemann with the geometric direction begun
by Clebsch, making systematic use of homogeneous coordinates and

MATHEMATICS IN THE NINETEENTH CENTURY 477

the invariant theory. Noteworthy, also, is his use of normal curves
in (p — 1) way space, to represent the given algebraic configuration.
Dedekind and Weber, Hensel and Landsberg, have made use of the
ideal theory with marked success. Many of the difficulties of the
older theory, e. g., the resolution of singularities of the algebraic
configuration, are treated with a truly remarkable ease and generality.
In the theory of multiply periodic functions and the general 0
functions we mention, besides Weierstrass, the researches of Prym,
Krazer, Frobenius, Poincare, and Wirtinger.

Automorphic Functions

Closely connected with the elliptic functions is a class of functions
which has come into great prominence in the last quarter of a cen-
tury, namely, the elliptic modular and automorphic functions. Let
us consider first the modular functions of which the modulus k and
the absolute invariant J are the simplest types.

The transformation theory of Jacobi gave algebraic relations be-
tween such functions in endless number. Hermite, Fuchs, Dedekind,
and Schwarz are forerunners, but the theory of modular functions as
it stands to-day is principally due to Klein and his school. Its goal
is briefly stated thus : Determine all sub-groups of the linear group

aX-\-3
yx + d

where a, (3, y, d are integers and ao— j5;' = l; determine for each
such group associate modular functions and investigate their rela-
tion to one another and especially to J. Important features in this
theory are the congruence groups of (1); the fundamental polygon
belonging to a given sub-group, and its use as substitute for a Rie-
mann surface; the principle of reflection over a circle, the modular
forms.

The theory of automorphic functions is due to Klein and Poincare.
It is a generalization of the modular functions; the coefficients in
(1) being any real or imaginary numbers, with non-vanishing de-
terminant, such that the group is discontinuous. Both authors have
recourse to non-Euclidean geometry to interpret the substitutions (1).
Their manner of showing the existence of functions belonging to
a given group is quite different. Poincare by a brilliant stroke of
genius actually writes down their arithmetic expressions in terms
of his celebrated 0 series. Klein employs the existence methods of
Riemann. The relation of automorphic functions to differential
equations is studied by Poincare in detail. In particular, he shows that
both variables of a linear differential equation with algebraic coeffi-
cients can be expressed uniformly by their means.

478 MATHEMATICS

Differential Equations

Let us turn now to another great field of mathematical activity,
the theory of differential equations. The introduction of the theory
of functions has completely revolutionized this subject. At the
beginning of the nineteenth century many important results had
indeed been established, particularly by Euler and Lagrange; but
the methods employed were artificial, and broad comprehensive
principles were lacking. By various devices one tried to express
the solution in terms of the elementary functions and quadratures
— a vain attempt; for as we know now, the goal they strove so
laboriously to reach was in general unattainable.

A new epoch began with Cauchy, who by means of his new theory
of functions first rigorously established the existence of the solution
of certain classes of equations in the vicinity of regular points. He
also showed that many of the properties of the elliptic functions
might be deduced directly from their differential equations. Ere
long, the problem of integrating a differential equation changed
its base. Instead of seeking to express its solution in terms of the
elementary functions and quadratures, one asked what is the nature
of the functions defined by a given equation. To answer this ques-
tion we must first know what are the singular points of the integral
function and how does it behave in their vicinity. The number of
memoirs on this fundamental and often difficult question is enormous;
but this is not strange if we consider the great variety of interesting
and important classes of equations which have to be studied.

One of the first to open up this new path was Fuchs, whose classic
memoirs (1866-68) gave the theory of linear differential equations
its birth. These equations enjoy a property which renders them
particularly accessible, namely, the absence of movable singular
points. They may, however, possess points of indetermination, to
use Fuchs's terminology, and little progress has been made in this
case. Noteworthy in this connection is the introduction by v. Koch
of infinite determinants, first considered by our distinguished coun-
tryman Hill; also the use of divergent series — that invention of
the Devil, as Abel called them — by Poincare. A particular class
of linear differential equations of great importance is the hyper-
geometric equation; the results obtained by Gauss, Kummer,
Riemann, and Schwarz relating to this equation have had the great-
est influence on the development of the general theory. The vast
extent and importance of the theory of linear differential equations
may be estimated when we recall that within its borders it embraces
not only almost all the elementary functions, but also the modular
and automorphic functions.

Too important to pass over in silence is the subject of algebraic

MATHEMATICS IN THE NINETEENTH CENTURY 479

differential equations with uniform solutions. The brilliant researches
of Poinleve deserve especial mention.

Another field of great importance, especially in mathematical
physics, relates to the determination of the solution of differential
equations with assigned boundary conditions. The literature of this
subject is enormous; we may therefore be pardoned if mention is
made only of the investigation of our countrymen Bocher, Van
Vleck, and Porter.

Since 1870 the theory of differential equations has been greatly
advanced by Lie's theory of groups. Assuming that an equation or a
system of equations admits one or more infinitesimal transformations,
Lie has shown how they may be employed to simplify the problem
of integration. In many cases they give us exact information how
to conduct the solution and upon what system of auxiliary equations
the solution depends. One of the most striking illustrations of this
is the theory of ordinary linear differential equations which Picard
and Vessiot have developed, analogous to Galois's theory for algebraic
equations. An interesting result of this theory is a criterion for the
solution of such equations by quadratures. As an application, we
find that Ricatti's equation cannot be solved by quadratures. The
attempts to effect such a solution of this celebrated equation in the
century before were therefore necessarily in vain.

A characteristic feature of Lie's theories is the prominence given
to the geometrical aspects of the questions involved. Lie thinks in
geometrical images, the analytical formulation comes afterwards.
Already Morge had shown how much might be gained in geometrizing
the problem of integration. Lie has gone much farther in this direc-
tion. Besides employing all the geometrical notions of his predeces-
sors extended to 7i-way space, he has introduced a variety of new
conceptions, chief of which are his surface element and contact
transformations.

He has also used with great effect Pliicker's line geometry, and his
own sphere geometry in the study of certain types of partial differential
equations of the first and second orders which are of great geometrical
interest, for example, equations whose characteristic curves are lines
of curvature, geodesies, etc. Let us close by remarking that Lie's
theories not only afford new and valuable points of view for attack-
ing old problems, but also give rise to a host of new ones of great
interest and importance.

Groups

We turn now to the second dominant idea of the century, the
group concept.

Groups first became objects of study in algebra when Lagrange
(1770), Ruffini (1799), and Abel (1826) employed substitution groups

480 MATHEMATICS

with great advantage in their work on the quintic. The enormous
importance of groups in algebra was, however, first made clear by
Galois, whose theory of the solution of algebraic equations is one
of the great achievements of the century. Its influence has stretched
far beyond the narrow bounds of algebra.

With an arbitrary but fixed domain of rationality, Galois observed
that every algebraic equation has attached to it a certain group of
substitutions. The nature of the auxiliary equations required to
solve the given equation is completely revealed by an inspection of
this group.

Galois's theory showed the importance of determining the sub-
groups of a given substitution group, and this problem was studied
by Cauchy, Serret, Matthieu, Kirkmann, and others. The publica-
tion of Jordan's great treatise in 1870 is a noteworthy event. It
collects and unifies the results of his predecessors and contains an
immense amount of new matter.

A new direction was given to the theory of groups by the introduc-
tion by Cayley of abstract groups (1854, 1878). The work of Sylow,
Hdlder and Frobenius, Burnside and Miller, deserve especial notice.

Another line of research relates to the determination of the finite
groups in the linear group of any number of variables. These groups
are important in the theory of linear differential equations with
algebraic solutions, in the study of certain geometrical problems
as the points of inflection of a cubic, the twenty-seven lines on a
surface of the third order, in crystallography, etc. They also enter
prominently into Klein's Formen-problem. An especially important
class of finite linear groups are the congruence groups first considered
by Galois. Among the laborers in the field of linear groups, we note
Jordan, Klein, Moore, Maschke, Dickson, Frobenius, and Wiman.

Up to the present we have considered only groups of finite order.
About 1870 entirely new ideas coming from geometry and differential
equations give the theory of groups an unexpected development.
Foremost in this field are Lie and Klein.

Lie discovers and gradually perfects his theory of continuous
transformation groups and shows their relations to many different
branches of mathematics. In 1872 Klein publishes his Erlanger
Programme and in 1877 begins his investigations on elliptic modular
functions, in which infinite discontinuous groups are of primary im-
portance, as we have already seen. In the now famous Programme,
Klein asks what is the principle which underlies and unifies the
heterogeneous geometrical methods then in vogue, as, for example,
the geometry of the ancients, whose figures are rigid and invariable;
the modern projective geometry, whose figures are in ceaseless
flux passing from one form to another; the geometries of Plucker
and Lie, in which the elements of space are no longer points, but line

MATHEMATICS IN THE NINETEENTH CENTURY 481

spheres, or other configurations at pleasure, the geometry of birational
transformation, the analysis situs, etc., etc. Klein finds this answer:
In each geometry we have a system of objects and a group which
transforms these objects one into another. We seek the invariants
of this group. In each case it is the abstract group and not the con-
crete objects which is essential. The fundamental role of a group in
geometrical research is thus made obvious. Its importance is the
solution of algebraic equation, in the theory of differential equations
in the automorphic functions we have already seen. The immense
theory of algebraic invariants developed by Cayley and Sylvester,
Aronhold, Clebsch, Gordan, Hermite, Brioschi, and a host of zealous
workers in the middle of the century, also finds its place in the far
more general invariant theory of Lie's theory of groups. The same is
true of the theory of surfaces, so far as it rests on the theory of differ-
ential forms. In the theory of numbers, groups have many important
applications, for example, in the composition of quadratic forms and
the cyclotomic bodies. Finally, let us note the relation between hyper-
complex numbers and continuous groups discovered by Poincare.

In r^sum^, we may thus saj'^ that the group concept, hardly not-
iceable at the beginning of the century, has at its close become one
of the fundamental and most fruitful notions in the whole range of
our science.

Infinite Aggregates

Leaving the subject of groups, we consider now briefly another
fundamental concept, namely, infinite aggregates. In the most
diverse mathematical investigations we are confronted with such
aggregates. In geometry the conceptions of curves, surface, region,
frontier, etc., when examined carefully, lead us to a rich variety of
aggregates. In analysis they also appear, for example, the domain
of definition of an analytic function, the points where a function of
a real variable ceases to be continuous or to have a differential coeffi-
cient, the points where a series of functions ceases to be uniformly
convergent, etc.

To say an aggregate (not necessarily a point aggregate) is infinite
is often an important step; but often again only the first step. To
penetrate farther into the problem may require us to state how
infinite. This requires us to make distinctions in infinite aggregates,
to discover fruitful principles of classification, and to investigate the
properties of such classes.

The honor of having done this belongs to George Cantor. The
theory of aggregates is for the most part his creation; it has en-
riched mathematical science with fundamental and far-reaching
notions and results.

The theory falls into two parts; a theory of aggregates in general,

482 MATHEMATICS

and a theory of point aggregates. In the theory of point aggregates
the notion of limiting points gives rise to important classes of aggre-
gates as discrete, dense, everywhere dense, complete, perfect, con-
nected, etc., which are so important in the function theory.

In the general theory two notions are especially important,
namely, the one to one correspondence of the elements of two ag-
gregates, and well-ordered aggregates. The first leads to cardinal
numbers and the idea of enumerable aggregates, the second to trans-
finite or ordinal numbers.

Two striking results of Cantor's theory are these: the algebraic
and therefore the rational numbers, although everywhere dense, are
enumerable; and secondly, one-way and n-way space have the
same cardinal number.

Cantor's theory has already found many applications, especially
in the function theory, where it is to-day an indispensable instrument
of research.

Functions of Real Variables — The Critical Movement

One of the most conspicuous and distinctive features of mathe-
matical thought in the nineteenth century is its critical spirit. Be-
ginning with the calculus, it soon permeates all analysis, and toward
the close of the century it overhauls and recasts the foundation of
geometry and aspires to further conquests in mechanics and in the
immense domains of mathematical physics.

Ushered in with Lagrange and Gauss just at the close of the
eighteenth century, the critical movement receives its first decisive
impulse from the teachings of Cauchy, who in particular introduces
our modern definition of limit and makes it the foundation of the
calculus. We must also mention in this connection Abel, Bolzano,
and Dirichlet. Especially Abel adopted the reform ideas of Cauchy
with enthusiasm, and made important contributions in infinite series.

The figure, however, which towers above all others in this move-
ment, whose name has become an epithet of rigor, is Weierstrass.
Beginning at the very foundations, he creates an arithmetic of real
and complex numbers, assuming the theory of positive integers to be
given. The necessity of this is manifest when we recall that until
then the simplest properties of radicals and logarithms were utterly
devoid of a rigorous foundation; so, for example,

V2 \/5=\/l0 log 2+log 5=log 10

Characteristic of the pre-Weierstrassean era is the loose way in
which geometrical and other intuitional ideas were employed in
the demonstration of analytical theorems. Even Gauss is open to
this criticism. The mathematical world received a great shock
when Weierstrass showed them an example of a continuous function

MATHEMATICS IN THE NINETEENTH CENTURY 483

without a derivative, and Hankel and Cantor, by means of their
principle of condensation of singularities, could construct analytic
expressions for functions having in any interval however small an
infinity of points of oscillation, an infinity of points in which the
differential coefficient is altogether indeterminate, or an infinity of
points of discontinuity. Another rude surprise was Cantor's dis-
covery of the one to one correspondence between the points of a
unit segment and a unit square, followed up by Peano's example
of a space-filling curve.

These examples and many others made it very clear that the
ideas of a curve, a surface region, motion, etc., instead of being clear
and simple, were extremely vague and complex. Until these notions
had been cleared up, their admission in the demonstration of an
analytical theorem was therefore not to be tolerated. On a purely
arithmetical basis, with no appeal to our intuition, Weierstrass
develops his stately theory of functions which culminates in the
theory of Abelian and multiply periodic functions.

But the notion of rigor is relative and depends on what we are
willing to admit either tacitly or explicitly. As we observed, Gauss,
whose rigor was the admiration of his contemporaries, freely ad-
mitted geometrical notions. This Weierstrass would criticise. On
the other hand, Weierstrass has made a grave oversight: he no-
where shows that his definitions relative to the number he introduces
do not involve mutual contradictions. If he replied that such con-
tradictions would involve contradictions in the theory of positive
integers, one might ask what assurance have we that such contradic-
tions may not actually exist. A flourishing young school of mathe-
matical logic has recently grown up under the influence of Peano.
They have investigated with marked success the foundations of
analysis and geometry, and in particular have attempted to show
the non-contradictoriness of the axioms of our number-system by
making them depend on the axioms of logic, which axioms we must
admit, to reason at all.

The critical spirit, which in the first half of the century was to
be found in the writings of only a few of the foremost mathematicians,
has in the last quarter of the century become almost universal, at
least in analysis. A searching examination of the foundation of
arithmetic and the calculus has brought to light the insufficiency of
much of the reasoning formerly considered as conclusive. It became
necessary to build up these subjects anew. The theory of irrational
numbers invented by Weierstrass has been supplanted by the more
flexible theories of Dedekind and Cantor. Stolz has given us a sys-
tematic and rigorous treatment of arithmetic. The calculus has
been completely overhauled and arithmetized by Thomae, Hamack,
Peano, Stolz, Jordan, and Vallee-Poussin.

484 MATHEMATICS

Leaving the calculus, let us notice briefly the theory of functions
of real variables. The line of demarcation between these two sub-
jects is extremely arbitrary. We might properly place in the latter
all those finer and deeper questions relating to the number-system;
the study of our curve, surface, and other geometrical notions, the
peculiarities that functions present with reference to discontinuity,
oscillation, differentiation, and integration; as well as a very exten-
sive class of investigations whose object is the greatest possible
extension of the processes, concepts, and results of the calculus.
Among the many not yet mentioned who have made important
contributions to this subject we note: Fourier, Riemann, Stokes,
Dini, Tannery, Pringsheim, Arzela, Osgood, Broden, Ascoli, Borel,
Baire, Kopke, Holder, Volterra, and Lebesgue.

Closely related with the differential calculus is the calculus of
variations; in the former the variables are given infinitesimal varia-
tions, in the latter the functions. Developed in a purely formal
manner by Jacobi, Hamilton, Clebsch, and others in the first part
of the century, a new epoch began with Weierstrass, who, having
subjected the labors of his predecessors to an annihilating criticism,
placed the theory on a new and secure foundation and so opened the
path for further research by Schwarz, A. Mayer, Scheffers, v, Esche-
rich, Kneser, Osgood, Bolza, Kobb, Zermelo, and others. At the
very close of the century Hilbert has given the theory a fresh im-
pulse by the introduction of new and powerful methods, which
enable us in certain cases to neglect the second variation and sim-
plifies the consideration of the first. As application he gives the
first direct and yet simple demonstration of Dirichlet's celebrated
Principle.

Theory of Numbers — Algebraic Bodies

The theory of numbers as left by Fermat, Euler, and Legendre
was for the most part concerned with the solution of Diophantine
equations, that is, given an equation f(x, y,z, . . . ) =0 whose
coefficients are integers, find all rational, and especially all integral
solutions. In this problem Lagrange had shown the importance
of considering the theory of forms. A new era begins with the ap-
pearance of Gauss's Disquisitiones arithmeticae in 1801. This great
work is remarkable for three things: (1) The notion of divisibility
in the form of congruences is shown to be an instrument of wonder-
ful power; (2) the Diophantine problem is thrown in the back-
ground and the theory of forms is given a dominant role; (3) the
introduction of algebraic numbers, namely, the roots of unity.

The theory of formes has been further developed along the lines
of the Disquisitiones by Dirichlet, Eisenstein, Hermite, H. Smith, and
Minkowski,

MATHEMATICS IN THE NINETEENTH CENTURY 485

Another part of the theory of numbers also goes back to Gauss,
namely, algebraic numerical bodies. The Law of Reciprocity of
Quadratic Residues, one of the gems of the higher arithmetic, was
first rigorously proved by Gauss. His attempts to extend this
theorem to cubic and biquadratic residues showed that the elegant
simplicity which prevailed in quadratic residues was altogether
missing in these higher residues, until one passed from the domain
of real integers to the domain formed of the third and fourth roots of
unit3^ In these domains, as Gauss remarked, algebraic integers have
essentially the same properties as ordinary integers. Further explor-
ation in this new and promising field by Jacobi, Eisenstein, and
others soon brought to light the fact that already in the domain
formed of the twenty-third roots of unity the laws of divisibility were
altogether different from those of ordinary integers; in particular,
a number could be expressed as the product of prime factors in more
than one way. Further progress in this direction was therefore
apparently impossible.

It is Kummer's immortal achievement to make further progress
possible by the invention of his ideals. These he applied to Fermat's
celebrated Last Theorem and the Law of Reciprocity of Higher
Residues.

The next step in this direction was taken by Dedekind and Kro-
necker, who developed the ideal theory for any algebraic domain.
So arose the theory of algebraic numerical bodies, which has come
into such prominence in the last decades of the century through
the researches of Hensel, Hurwitz, Minkowski, Weber, and, above
all, Hilbert.

Kronecker has gone farther, and in his classic Grundzuge he has
shown that similar ideas and methods enable us to develop a theory
of algebraic bodies in any number of variables. The notion of divis-
ibility so important in the preceding theories is generalized by Kro-
necker still farther in the shape of his system of moduli.

Another noteworthy field of research opened up by Kronecker is
the relation between quadratic forms with negative determinant
and complex multiplication of elliptic functions. H. Smith, Gierster,
Hurwitz, and especially Weber have made important contributions.
A method of great power in certain investigations has been created
by Minkowski, which he called the Geometrie der Zahlen. Introduc-
ing a generalization of the distance function, he is led to the concep-
tion of a fundamental body (Aichkorper) . Minkowski shows that
every fundamental body is nowhere concave, and conversely to
each such body belongs a distance function. A theorem of great
importance is now the following: The minimum value which each
distance function has at the lattice points is not greater than a certain
number depending on the function chosen.

486 MATHEMATICS

We wish finally to mention a line of investigation which makes
use of the infinitesimal calculus and even the theory of functions.
Here belong the brilliant researches of Dirichlet relating to the num-
ber of classes of binary forms for a given determinant, the number
of primes in a given arithmetic progression; and Riemann's remark-
able memoir on the number of primes in a given interval.

In this analytical side of the theory of numbers we notice also the
researches of Mertens, Weber, and Hadamard.

Projective Geometry

The tendencies of the eighteenth century were predominantly
analytical. Mathematicians were absorbed for the most part in
developing the wonderful instrument of the calculus with its countless
applications. Geometry made relatively little progress. A new era
begins with Monge. His numerous and valuable contributions to
analytical descriptive and differential geometry, and especially his
brilliant and inspiring lectures at the Ecole Polytechnique (1795,
1809), put fresh life into geometry and prepared it for a new and
glorious development in the nineteenth century.

When one passes in review the great achievements which have
made the nineteenth century memorable in the annals of our science,
certainly projective geometry will occupy a foremost place. Pascal,
De la Hire, Monge, and Carnot are forerunners, but Poncelet, a pupil
of Monge, is its real creator. The appearance of his Traite des pro-
prietes projectives des figures, in 1822, gives modern geometry its
birth. In it we find the line at infinity, the introduction of imagin-
aries, the circular points at infinity, polar reciprocation, a discus-
sion of homology, the systematic use of projection, section, and
anharmonic ratio.

While the countrymen of Poncelet, especially Chasles, do not fail
to make numerous and valuable contributions to the new geometry,
the next great steps in advance are made on German soil. In 1827
Mobius publishes the Barycentrische Calcul; Pliicker's Analytisch-
geometrische Entioickelungen appears in 1828-31 and Steiner's Ent-
ivickelung der Abhdngigkeit geometrischer Gestalten von einander in
1832. In the ten years which embrace the publication of these
immortal works of Poncelet, Pliicker, and Steiner, geometry has
made more real progress than in the two thousand years which had
elapsed since the time of Appolonius. The ideas which had been
slowly taking shape since the time of Descartes suddenly crystallized
and almost overwhelmed geometry with an abundance of new ideas
and principles.

To Mobius we owe the introduction of homogeneous coordinates,
and the far-reaching conception of geometric transformation, includ-
ing collineation and duality as special cases. To Pliicker we owe the

MATHEMATICS IN THE NINETEENTH CENTURY 487

use of the abbreviate notation which permits us to study the proper-
ties of geometric figures without the intei'vention of the coordinates,
the introduction of Une and plane coordinates, and the notion of
generahzed space elements. Steiner, who has been called the greatest
geometer since Appolonius, besides enriching geometry in countless
ways, was the first to employ systematically the method of generating
geometrical figures by means of projective pencils.

Other noteworthy works belonging to this period are Pliicker's
System der analytischen Geometrie (1835), and Chasles's classic Apercu
(1837).

Already at this stage we notice a bifurcation in geometrical
methods. Steiner and Chasles become eloquent champions of the
synthetic school of geometry, while Pliicker, and later Hesse and
Cayley, are leaders in the analytical movement. The astonishing
fruitfulness and beauty of synthetic methods threatened for a short
time to drive the analytic school out of existence. The tendency
of the synthetic school was to banish more and more metrical methods.
In effecting this the anharmonic ratio became constantly more promi-
nent. To define this fundamental ratio mthout reference to measure-
ment, and so free projective geometry from the galling bondage
of metric relations, was thus a problem of fundamental importance.
The glory of this achievement, which has, as we shall see, a far
wider significance, belongs to v. Staudt. Another equally important
contribution of v. Staudt to synthetic geometry is his theory of
imaginaries. Poncelet, Steiner, Chasles operate with imaginary
elements as if they were real. Their only justification is recourse to
the so-called principles of continuity or to some other equally vague
principle. V. Staudt gives this theory a rigorous foundation, defining
the imaginary points, lines, and planes by means of involutions
without ordinal elements.

The next great advance made is the advent of the theory of alge-
braic invariants. Since projective geometry is the study of those
properties of geometric figures which remain unaltered by projective
transformations, and since the theory of invariants is the study of
those forms which remain unaltered (except possibly for a numerical
factor) by the group of linear substitutions, these two subjects are
inseparably related and in many respects only different aspects of the
same thing. It is no wonder, then, that geometers speedily applied
the new theory of invariants to geometrical problems. Among the
pioneers in this direction were Cayley, Salmon, Aronhold, Hesse,
and especially Clebsch.

Finally we must mention the introduction of the line as a space
element. Forerunners are Grassmann (1844) and Cayley (1859), but
Pliicker in his memoirs of 1865, and his work Neue Geometrie des
Raumes (1868-69), was the first to show its great value by studying

488 MATHEMATICS

complexes of the first and second order and calling attention to
their application to mechanics and optics.

The most important advance over Pliicker has been made by-
Klein, who takes as coordinates six-line complexes in involution.
Klein also observed that line geometry may be regarded as a point
geometry on a quadric in five-way space. Other laborers in this
field are Clebsch, Reye, Segre, Sturm, and Konigs.

Differential Geometry

During the first quarter of the century this important branch of
geometry was cultivated chiefly by the French. Monge and his
school study with great success the generation of surfaces in vari-
ous wayS; the properties of envelopes, evolutes, lines of curvature,
asymptotic lines, skew curves, orthogonal systems, and especially the
relation between the surface theory and partial differential equations.

The appearance of Gauss's Disquisitiones generates circa super-
ficies curvas, in 1828, marks a new epoch. Its wealth of new ideas
has furnished material for countless memoirs, and given geometry
a new direction. We find here the parametric representation of a
surface, the introduction of curvilinear coordinates, the notion of
spherical image, the Gaussian measure of curvature, and a study of
geodesies. But by far the most important contributions that Gauss
makes in this work is the consideration of a surface as a flexible,
inextensible film or membrane, and the importance given quadratic
differential forms.

We consider now some of the lines along which differential geometry
has advanced. The most important is perhaps the theory of differen-
tial quadratic forms with their associate invariants and parameters.
We mention here Lame, Beltrami, Menardi, Codazzi, Christoffel,
and Weingarten.

An especially beautiful application of this theory is the immense
subject of applicability and deformation of surfaces, in which Mind-
ing, Bauer, Beltrami, Weingarten, and Voss have made important
contributions.

Intimately related with the theory of applicability of two surfaces
is the theory of surfaces of constant curvature which play so import-
ant a part in non-Euclidean geometry. We mention here the work
of Minding, Beltrami, Dini, Backlund, and Lie.

The theory of rectilinear congruences has also been the subject
of important researches from the standpoint of differential geometry.
First studied by Monge as a sj^stem of normals to a surface and then
in connection with optics by Malus, Dupin, and Hamilton, the gen-
eral theory has since been developed by Kummer, Ribaucour,
Guichard, Darboux, Voss, and Weingarten. An important applica-
tion of this theory is the infinitesimal deformation of a surface.

MATHEMATICS IN THE NINETEENTH CENTURY 489

Minimum surfaces have been studied by Monge, Bonnet, and
Enneper. The subject owes its present extensive development prin-
cipally to Weierstrass, Riemann, Schwarz, and Lie. In it we find
harmoniously united the theory of surfaces, the theory of functions,
the calculus of variations, the theory of groups, and mathematical
physics.

Another extensive division of differential geometry is the theory of
orthogonal systems, of such importance in physics. We note espe-
cially the investigations of Dupin, jacobi, Darboux, Combescure,
and Bianchi.

Other Branches of Geometry

Under this head we group a number of subjects too important
to pass over in silence, 3^et which cannot be considered at length for
lack of time.

In the first place is the immense subject of algebraic curves and
surfaces. To develop adequately all the important and elegant
properties of curves and surfaces of the second order alone would
require a bulky volume. In this line of ideas would follow curves
and surfaces of higher order and class. Their theory is far less
complete, but this lack it amply makes good by offering an almost
bewildering variety of configurations to classify and explore. No
single geometer has contributed more to this subject than Cayley.

A theory of great importance is the geometry on a curve or sur-
face inaugurated by Clebsch in 1863.

Expressing the coordinates of a plane cubic by means of elliptic
functions and employing their addition theorems, he deduced with
hardly any calculation Steiner's theorem relating to the inscribed
polygons and various theorems concerning conies touching the curve.
Encouraged by such successes, Clebsch proposed to make use of
Riemann's theory of Abelian functions in the study of algebraic
curves of any order. The most important result was a new classifica-
tion of such curves. Instead of the linear transformation, Clebsch
in harmony with Riemann 's ideas employs the birational transforma-
tion as a principle of classification. From this standpoint we ask
what are the properties of algebraic curves which remain invariant
for such transformation.

Brill and Nother follow Clebsch. Their method is, however, alge-
braical, and rests on their celebrated Residual theorem which in
their hands takes the place of Abel's theorem. We mention further
the investigation of Castelnuovo, Weber, Krause, and Segre. An
important division of this subject is the theory of correspondences.
First studied by Chasles for curves of deficiency 0 in 1864, Cayley,
and, immediately after. Brill extended the theory to the case of any
p. The most important advance made in later years has been made

490 MATHEMATICS

by Hurwitz, who considers the totahty of possible correspondences
on an algebraic curve, making use of the corresponding integrals of
the first species.

Alongside the geometry on a curve is the vastly more difficult and
complicated geometry on a surface, or more generally, on any algebraic
spread in n-way space. Starting from a remark of Clebsch (1868),
Nother made the first great step in his famous memoir of 1868-
74. Further progress has been due to the French and Italian mathe-
maticians. Picard, Poincare, and Humbert make use of transcend-
ental methods, in which figure prominently double integrals which
remain finite on the surface and single integrals of total differentials.
On the other hand, Enriques and Castelnuovo have attacked the
subject from a more algebraic-geometric standpoint by means of
linear systems of algebraic curves on the surface.

The first invariants of a surface were discovered by Clebsch and
Nother; still others have been found by Castelnuovo and Enriques
in connection with irregular surfaces.

Leaving this subject, let us consider briefly the geometry of n
dimensions. A characteristic of nineteenth-century mathematics
is the generality of its methods and results. When such has been
impossible with the elements in hand, fresh ones have been invented;
witness the introduction of imaginarj^ numbers in algebra and the
function theory, the ideals of Kummer in the theory of numbers,
the line and plane at infinity in projective geometry. The benefit
that analysis derived from geometry was too great not to tempt
mathematicians to free the latter from the narrow limits of three
dimensions, and so give it the generality that the former has long
enjoyed. The first pioneer in this abstract field was Grassmann (1844) ;
we must, however, consider Cayley as the real founder of n-dimen-
sional geometry (1869). Notable contributions have been made by
the Italian school, Veronese, Segre, etc.

Non-Euclidean Geometry

Each century takes over as a heritage from its predecessor a
number of problems whose solution previous generations of mathe-
maticians have arduously but vainly sought. It is a signal achieve-
ment of the nineteenth century to have triumphed over some of the
most celebrated of these problems.

The most ancient of them is the Quadrature of the Circle, which
already appears in our oldest mathematical document, the Papyrus
Rhind, B.C. 2000. Its impossibility was finally shown by Lindemann
(1882).

Another famous problem relates to the solution of the quintic,
which had engaged the attention of mathematicians since the middle
of the sixteenth century. The impossibility of expressing its roots by

MATHEMATICS IN THE NINETEENTH CENTURY 491

radicals was finally shown by the youthful Abel (1824) , while Hermite
and Kroneker (1858) showed how they might be expressed by the
elliptic modular functions, and Klein (1875) by means of the icosa-
hedral irrationality.

But of all problems which have come down from the past, by far
the most celebrated and important relates to Euclid's parallel
axiom. Its solution has profoundly affected our views of space,
and given rise to questions even deeper and more far-reaching which
embrace the entire foundation of geometry and our space conception.
Let us pass in rapid review the principal events of this great move-
ment. Wallis in the seventeenth, Seccheri, Lambert, and Legendre
in the eighteenth, are the first to make any noteworthy progress
before the nineteenth century. The really profound investigations
of Seccheri and Lambert, strangely enough, were entirely over-
looked by later writers and have only recently come to light.

In the nineteenth century non-Euclidean geometry develops along
four directions, which roughly follow each other chronologically.
Let us consider them in order.

The naive-synthetic direction. — The methods employed are similar to
those of Euclid. His axioms are assumed with the exception of the
parallel axiom; the resulting geometry is what is now called hyper-
bolic or Lobatschewski's geometry. Its principal properties are de-
duced, in particular its trigonometry, which is shown to be that of a
sphere with imaginary radius as Lambert had divined. As a specific
result of these investigations the long-debated question relating to
the independence of the parallel axiom was finally settled. The great
names in this group are Lobatschewski, Bolyai, and Gauss. The first
publications of Lobatschewski are his Exposition succinct des prin-
cipes de la geometrie (1829) , and the Geometrische Untersuchungen, in
1840. Bolyai's Appendix was published in 1832, As to the extent
of Gauss's investigations, we can only judge from scattered remarks
in private letters and his reviews of books relating to the parallel
axioms. His dread of the Geschrei der Bootier, that is, the followers
of Kant, prevented him from publishing his extensive speculations.

The metric-differential direction. — This is inaugurated by three great
memoirs by Riemann, Helmholtz, and Beltrami, all published in the
same year, 1868.

Beltrami, making use of results of Gauss and Minding relating to
the applicability of two surfaces, shows that the hyperbolic geometry
of a plane may be interpreted on a surface of constant negative
curvature, the pseudosphere. By means of this discovery the purely
logical and hypothetical system of Lobatschewski and Bolyai takes
on a form as concrete and tangible as the geometry of a plane.

The work of Riemann is as original as profound. He considers
space as an n-dimensional continuous numerical multiplicity, which

492 MATHEMATICS

is distinguished from the infinity of other such multiplicities by
certain well-defined characters. Chief of them are (1) the quadratic
differential expression which defines the length of an elementary arc,
and (2) a property relative to the displacements of this multiplicity
about a point. There are an infinity of space multiplicities which
satisfy Rieniann's axioms. By extending Gauss's definition of a
curvature A;, of a surface at a point to curvature of space at a point,
by considering the geodesic surfaces passing through that point,
Riemann finds that all these spaces fall into three classes according
as k is equal to, greater, or less than 0. For n=3 and A; = 0 we have
Euclidean space; when A;<0 we have the space found by Gauss,
Lobatschewski, and Bolyai; when A;>0 we have the space first
considered in the long-forgotten writings of Seccheri and Lambert,
in which the right line is finite.

Helmholtz, like Riemann, considers space as a numerical multiplic-
ity. To characterize it further, Helmholtz makes use of the notions
of rigid bodies and free mobility. His work has been revised and ma-
terially extended by Lie from the standpoint of the theory of groups.

In the present category also belong important papers by New-
comb and Killing.

The projective direction. — We have already noticed the efforts of
the synthetic school to express metric properties by means of project-
ive relations. In this the circular points at infinity were especially
serviceable. An immense step in this direction was taken by Laguerre,
who showed, in 1853, that all angles might be expressed as an anhar-
monic ratio with reference to these points, that is, with reference to
a certain fixed conic. The next advance is made by Cayley in his
famous sixth memoir on quantics, in 1859. Taking any fixed conic
(or quadric, for space) which he calls the absolute, Cayley introduces
two expressions depending on the anharmonic ratio with reference
to the absolute. When this degenerates into the circular points
at infinity, these expressions go over into the ordinary expressions
for the distance between two points and the angle between two
lines. Thus all metric relations may be considered as projective
relations with respect to the absolute. Cayley does not seem to be
aware of the relation of his work to non-Euclidean geometry. This
was discovered by Klein, in 1871. In fact, according to the nature of
the absolute, three geometries are possible; these are precisely the
three already mentioned. Klein has made many important contri-
butions to non-Euclidean geometry. We mention his modification
of V. Staudt's definition of anharmonic ratio so as to be independ-
ent of the parallel axiom, his discovery of the two forms of Rie-
mann's space, and finally his contributions to a class of geometries
first noticed by Clifford and which are characterized by the fact that
only certain of its motions affect space as a whole.

MATHEMATICS IN THE NINETEENTH CENTURY 493

As a result of all these investigations, both in the projective as
also in the metric differential direction, we are led irresistibly to the
same conclusion, namely: The facts of experience can be explained
by all three geometries when the constant k is taken small enough.
It is, therefore, merely a question of convenience whether we adopt
the parabolic, hyperbolic, or elliptic geometry.

The critical synthetic direction represents a return to the old sya.-
thetic methods of Euclid, Lobatschewski, and Bolyai, with the added
feature of a refined and exacting logic. Its principal object is no
longer a study of non-Euclidean but of Euclidean geometry. Its
aim is to establish a system of axioms for our ordinary space which
is complete, compatible, and irreducible. The fundamental terms
point, line, plane, between, congruent, etc., are introduced as ab-
stract marks whose properties are determined by inter-relations in
the form of axioms. Geometric intuition has no place in this order
of ideas which regards geometry as a mere division of pure logic.
The efforts of this school have already been crowned with eminent
success, and much may be expected from it in the future. Its leaders
are Peano, Veronese, Fieri, Padoa, Burali-Forti, and Levi-Civitta, in
Italy, Pasch and Hilbert in Germany, and Moore in America.

Closing at this point our hasty and imperfect survey of mathe-
matics in the last century, let us endeavor to sum up its main charac-
teristics. What strikes us at once is its colossal proportions and rapid
growth in nearly all directions, the great variety of its branches, the
generality and complexity of its methods; an inexhaustible creative
imagination, the fearless introduction and employment of ideal
elements, and an appreciation for a refined and logical development
of all its parts.

We who stand on the threshold of a new century can look back on
an era of unparalleled progress. Looking into the future, an equally
bright prospect greets our eyes; on all sides fruitful fields of re-
search invite our labor and promise easy and rich returns.

Surely this is the golden age of mathematics.

SECTION A— ALGEBRA AND ANALYSIS

SECTION A — ALGEBRA AND ANALYSIS

{HaU 9, September 22, 10 a. m.)

Chairman: Professor E. H. Moore, University of Chicago.
Speakers: Professor Charles Emile Picakd, The Sorbonne; Member of the
Institute of France.
Professor Heinrich Maschke, University of Chicago.
Secretary: Professor A. G. Bliss, University of Chicago.

ON THE DEVELOPMENT OF MATHEMATICAL ANALYSIS
AND ITS RELATIONS TO SOME OTHER SCIENCES

BY CHARLES EMILE PICARD

{Translated from the French by Professor George Bruce Halsted, Kenyan College)

[Charles Emile Picard, Professor of Higher Algebra and Higher Analysis, Uni-
versity of Paris; also Professor of General Mechanics, TEcole Centrale des
Arts et Manufactures, Paris, b. Paris, France, July 24, 1856. LL.D. Clark
University, Glasgow University, University of Christiania. Member of In-
stitute of France; Academy of Science, Berlin, St. Petersburg, Bologna,
Boston, Turin, Copenhagen, Washington, and many others; Mathematical
Society of London. Former President of Mathematical Society of France,
Mathematical Societies of London and Kharkow, and many other math-
ematical societies. Author and editor of Memoirs, Traits and Discussions
of Mathematics; Theory of Algebraic Functions of Two Variables.]

It is one of the objects of a congress such as this which now
brings us together, to show the bonds between the diverse parts of
science taken in its most extended acceptation. So the organizers
of this meeting have insisted that the relations between different
sections should be put in evidence.

To undertake a study of this sort, somewhat indeterminate in
character, it is necessary to forget that all is in all; in what con-
cerns algebra and analysis, a Pythagorean would be dismayed at the
extent of his task, remembering the celebrated formula of the school:
" Things are numbers." From this point of view my subject would
be inexhaustible.

But I, for the best of reasons, will make no such pretensions.

In casting merely a glance over the development of our science
through the ages, and particularly in the last century, I hope to be
able to characterize sufficiently the role of mathematical analysis in
its relations to certain other sciences.

It would appear natural to commence by speaking of the concept
itself of whole number; but this subject is not alone of logical order,

498 ALGEBRA AND ANALYSIS

it is also of order historic and psychologic, and would draw us away
into too many discussions.

Since the concept of number has been sifted, in it have been found
unfathomable depths; thus, it is a question still pending to know,
between the two forms, the cardinal number and the ordinal number,
under which the idea of number presents itself, which of the two is
anterior to the other, that is to say, whether the idea of number
properly so called is anterior to that of order, or if it is the inverse.

It seems that the geometer-logician neglects too much in these
questions psychology and the lessons uncivilized races give us; it
would seem to result from these studies that the priority is with the
cardinal number.

It may also be there is no general response to the question, the
response varying according to races and according to mentalities.

I have sometimes thought, on this subject, of the distinction be-
tween auditives and visuals, auditives favoring the ordinal theory,
visuals the cardinal.

But I will not linger on this ground full of snares; I fear that our
modern school of logicians with difficulty comes to agreement with
the ethnologists and biologists; these latter in questions of origin
are always dominated by the evolution idea, and, for more than one
of them, logic is only the resume of ancestral experience. Mathe-
maticians are even reproached with postulating in principle that
there is a human mind in some way exterior to things, and that it
has its logic. We must, however, submit to this, on pain of con-
structing nothing. We need this point of departure, and certainly,
supposing it to have evolved during the course of prehistoric time,
this logic of the human mind was perfectly fixed at the time of the
oldest geometric schools, those of Greece; their works appear to
have been its first code, as is expressed by the story of Plato writing
over the door of his school, "Let no one not a geometer enter
here."

Long before the bizarre word algebra was derived from the Arabic,
expressing, it would seem, the operation by which equalities are
reduced to a certain canonic form, the Greeks had made algebra
without knowing it; relations more intimate could not be imagined
than those binding together their algebra and their geometry, or
rather, one would be embarrassed to classify, if there were occasion,
their geometric algebra, in which they reason not on numbers but on
magnitudes.

Among the Greeks also we find a geometric arithmetic, and one of
the most interesting phases of its development is the conflict which,
among the Pythagoreans, arose in this subject between number and
magnitude, apropos of irrationals.

Though the Greeks cultivated the abstract study of numbers, called

DEVELOPMENT OF MATHEMATICAL ANALYSIS 499

by them arithmetic, their speculative spirit showed httle taste for
practical calculation, which they called logistic.

In remote antiquity, the Egyptians and the Chaldeans, and later
the Hindus and the Arabs, carried far the science of calculation.

They were led on by practical needs; logistic preceded arithmetic,
as land-surveying and geodesy opened the way to geometry; in the
same way trigonometry developed under the influence of the in-
creasing needs of astronomy.

The history of science at its beginnings shows a close relation
between pure and applied mathematics; this we shall meet again
constantly in the course of this study.

We have remained up to this point in the domain which ordinary
language calls elementary algebra and arithmetic.

In fact, from the time that the incommensurability of certain
magnitudes had been recognized, the infinite had made its appearance,
and, from the time of the sophisms of Zeno on the impossibility of
motion, the summation of geometric progressions must have been
considered.

The procedures of exhaustion which are found in Eudoxus and in
Euclid appertain already to the integral calculus, and Archimedes
calculates definite integrals.

Mechanics also appeared in his treatise on the quadrature of the
parabola, since he first finds the surface of the segment bounded by
an arc of a parabola and its chord w4th the help of the theorem of
moments; this is the first example of the relations between me-
chanics and analysis, which since have not ceased developing..

The infinitesimal method of the Greek geometers for the measure
of volumes raised questions whose interest is even to-day not ex-
hausted.

In plane geometry, tw^o polygons of the same area are either
equivalent or equivalent-by-completion, that is to say, can be de-
composed into a finite number of triangles congruent in pairs, or
may be regarded as differences of polygons susceptible of such a
partition.

It is not the same for the geometry of space, and we have lately
learned that stereometry cannot, like planimetry, get on without
recourse to procedures of exhaustion or of limit, which require the
axiom of continuity or the Archimedes assumption.

Without insisting further, this hasty glance at antiquity shows
how completely then were amalgamated algebra, arithmetic, geo-
metry, and the first endeavors at integral calculus and mechanics, to
the point of its being impossible to recall separately their history.

In the Middle Ages and the Renaissance, the geometric algebra of
the ancients separated from geometry. Little by little algebra
properly so called arrived at independence, with its symbolism and

500 ALGEBRA AND ANALYSIS

its notation more and more perfected; thus was created this lan-
guage so admirably clear, which brings about for thought a veritable
economy and renders further progress possible.

This is also the moment when distinct divisions are organized.

Trigonometry, which, in antiquity, had been only an auxiliary of
astronomy, is developed independently; toward the same time the
logarithm appears, and essential elements are thus put in evidence.

In the seventeenth century, the analytic geometry of Descartes,
distinct from what I have just called the geometric algebra of the
Greeks by the general and systematic ideas which are at its base,
and the new-born dynamic were the origin of the greatest progress of
analysis.

When Galileo, starting from the hypothesis that the velocity of
heavy bodies in their fall is proportional to the time, from this
deduced the law of the distances passed over, to verify it afterward
by experiment, he took up again the road upon which Archimedes
had formerly entered and on which would follow after him Cavalieri,
Fermat, and others still, even to Newton and Leibnitz. The integral
calculus of the Greek geometers was born again in the kinematic of
the great Florentine physicist.

As to the calculus of derivatives or of differentials, it was founded
with precision apropos of the drawing of tangents.

In reality, the origin of the notion of derivative is in the confused
sense of the mobility of things and of the rapidity more or less great
with which phenomena happen; this is well expressed by the words
fluents and fluxions, which Newton used, and which one might
suppose borrowed from old Heraclitus.

The points of view taken by the founders of the science of motion,
Galileo, Huygens, and Newton, had an enormous influence on the
orientation of mathematical analysis.

It was with Galileo an intuition of genius to discover that, in
natural phenomena, the determining circumstances of the motion
produce accelerations: this must have conducted to the statement
of the principle that the rapidity with which the dynamic state of
a system changes depends in a determinate manner on its static state
alone. In a more general way we reach the postulate that the in-
finitesimal changes, of whatever nature they may be, occurring in
a system of bodies, depend uniquely on the actual state of this
system.

In what degree are the exceptions apparent or real? This is a ques-
tion which was raised only later and which I put aside for the
moment.

From the principles enunciated becomes clear a point of capital

DEVELOPMENT OF MATHEMATICAL ANALYSIS 501

importance for the analyst: Phenomena are ruled by differential
equations which can be formed when observation and experiment
have made known for each category of phenomena certain physical
laws.

We understand the unlimited hopes conceived from these results.
As Bertrand says in the preface of his treatise, "The early successes
were at first such that one might suppose all the difficulties of science
surmounted in advance, and believe that the geometers, without
being longer distracted by the elaboration of pure mathematics,
could turn their meditations exclusively toward the study of the
natural laws."

This was to admit gratuitously that the problems of analysis, to
which one was led, would not present very grave difficulties.

Despite the disillusions the future was to bring, this capital point
remained, that the problems had taken a precise form, and that a
classification could be established in the difficulties to be surmounted.

There was, therefore, an immense advance, one of the greatest
ever made by the human mind. We understand also why the theory
of differential equations acquired a considerable importance.

I have anticipated somewhat, in presenting things under a form
so analytic. Geometry was intermingled in all this progress. Huy-
gens, for example, followed always by preference the ancients, and
his Horologium oscillatorium rests at the same time on infinitesi-
mal geometry and mechanics; in the same way, in the Principia
of Newton, the methods followed are synthetic.

It is, above all, with Leibnitz that science takes the paths which
were to lead to what we call mathematical analysis; it is he who,
for the first time, in the latter years of the seventeenth century,
pronounces the word function.

By his systematic spirit, by the numerous problems he treated,
even as his disciples James and John Bernoulli, he established in a
final way the power of the doctrines to the edification of which had
successively contributed a long series of thinkers from the distant
times of Eudoxus and of Archimedes.

The eighteenth century showed the extreme fecundity of the new
methods. That was a strange time, the era of mathematical duels
w^here geometers hurled defiance, combats not always without
acrimony, when Leibnitzians and Newtonians encountered in the
lists.

From the purely analytic point of view, the classification and study
of simple functions is particularly interesting; the function idea, on
which analysis rests, is thus developed little by little.

The celebrated works of Euler hold then a considerable place.
However, the numerous problems which present themselves to the
mathematicians leave no time for a scrutiny of principles; the

502 ALGEBRA AND ANALYSIS

foundations themselves of the doctrine are elucidated slowly, and
the mot attributed to d'Alembert, "AUez en avant et la foi vous
viendra, " is very characteristic of this epoch.

Of all the problems started at the end of the seventeenth century
or during the first half of the eighteenth, it will suffice for me to recall
those isoperimetric problems which gave birth to the calculus of
variations.

I prefer to insist on the interpenetration still more intimate
between analysis and mechanics when, after the inductive period of
the first age of dynamics, the deductive period was reached where one
strove to give a final form to the principles. The mathematical and
formal development played then the essential role, and the analytic
language was indispensable to the greatest extension of these prin-
ciples.

There are moments in the history of the sciences and, perhaps, of
society, when the spirit is sustained and carried forward by the words
and the symbols it has created, and when generalizations present
themselves with the least effort. Such was particularly the role of
analysis in the formal development of mechanics.

Allow me a remark just here. It is often said an equation contains
only what one has put into it. It is easy to answer, first, that the
new form under which one finds the things constitutes often of itself
an important discovery.

But sometimes there is more; analysis, by the simple play of
its symbols, may suggest generalizations far surpassing the primitive
outline. Is it not so with the principle of virtual velocities, of which
the first idea comes from the simplest mechanisms; the analytic
form which translates it will suggest extensions leading far from the
point of departure.

In the same sense, it is not just to say analysis has created nothing,
since these more general conceptions are its work. Still another
example is furnished us by Lagrange's system of equations; here
calculus transformations have given the type of differential equations
to which one tends to carry back to-day the notion of mechanical
explanation.

There are in science few examples comparable to this, of the
importance of the form of an analytic relation and of the power of
generalization of which it may be capable.

It is very clear that, in each case, the generalizations suggested
should be made precise by an appeal to observation and experiment,
then it is still the calculus which searches out distant consequences
for checks, but this is an order of ideas which I need not broach here.
Under the impulse of the problems set by geometry, mechanics,
and physics, we see develop or take birth almost all the great divisions
of analysis. First were met equations with a single independent vari-

DEVELOPMENT OF MATHEMATICAL ANALYSIS 503

able. Soon appear partml differential equations, with vibrating cords,
the mechanics of fluids and the infinitesimal geometry of surfaces.

This was a wholly new analytic w^orld; the origin itself of the
problems treated was an aid which from the first steps permits no
wandering, and in the hands of Monge geometry rendered useful
services to the new-born theories.

But of all the applications of analysis, none had then more renown
than the problems of celestial mechanics set by the knowledge of the
law of gravitation and to which the greatest geometers gave their
names.

Theory never had a more beautiful triumph; perhaps one might
add that it was too complete, because it was at this moment above
all that were conceived for natural philosophy the hopes at least
premature of which I spoke above.

In all this period, especially in the second half of the eighteenth
century, what strikes us with admiration and is also somewhat
confusing, is the extreme importance of the applications realized,
while the pure theory appeared still so iU assured. One perceives it
when certain questions are raised like the degree of arbitrariness in
the integral of vibrating cords, which gives place to an interminable
and inconclusive discussion.

Lagrange appreciated these insufficiencies when he published his
theory of analytic functions, where he strove to give a precise foun-
dation to analysis.

One cannot too much admire the marvelous presentiment he had
of the role which the functions, which with him we call analytic,
were to play; but we may confess that we stand astonished before
the demonstration he believed to have given of the possibility of the
development of a function in Taylor's series.

The exigencies in questions of pure analysis were less at this
epoch. Confiding in intuition, one was content with certain probabil-
ities, and agreed implicitly about certain hypotheses that it seemed
useless to formulate in an explicit way; in reality, one had con-
fidence in the ideas which so many times had shown themselves
fecund, which is very nearly the mot of d'Alembert.

The demand for rigor in mathematics has had its successive
approximations, and in this regard our sciences have not the absolute
character so many people attribute to them.

Ill

We have now reached the first years of the nineteenth century.
As we have explained, the great majority of the analytic researches
had, in the eighteenth century, for occasion a problem of geometry,
and especially of mechanics and of physics, and we have scarcely
found the logical and aesthetic preoccupations which are to give a

504 ALGEBRA AND ANALYSIS

physiognomy so different to so many mathematical works, above all
in the latter two thirds of the nineteenth century.

Not to anticipate, however, after so many examples of the in-
fluences of physics on the developments of analysis, we meet still a
new one, and one of the most memorable, in Fourier's theory of heat.
He commences by forming the partial differential equations which
govern temperature.

What are for a partial differential equation the conditions at the
limits permitting the determination of a solution?

For Fourier, the conditions are suggested by the physical problem,
and the methods that he followed have served as models to the
physicist-geometers of the first half of the last century.

One of these consists in forming a series with certain simple solu-
tions. Fourier thus obtained the first types of developments more
general than the trigonometric developments, as in the problem of
the cooling of a sphere, where he applies his theory to the terrestrial
globe, and investigates the law which governs the variations of
temperature in the ground, trying to go even as far as numerical
applications.

In the face of so many beautiful results, we understand the enthu-
siasm of Fourier which scintillates from every line of his preliminary
discourse. Speaking of mathematical analysis, he says, "There could
not be a language more universal, more simple, more exempt from
errors and from obscurities, that is to say, more worthy to express
the invariable relations of natural things. Considered under this
point of view, it is as extended as nature herself; it defines all sen-
sible relations, measures times, spaces, forces, temperatures. This
difficult science forms slowly, but it retains all the principles once
acquired. It grows and strengthens without cease in the midst of
so many errors of the human mind."

The eulogy is magnificent, but permeating it we see the tendency
which makes all analysis uniquely an auxiliary, however incom-
parable, of the natural sciences, a tendency, in conformity, as we
have seen, with the development of science during the preceding two
centuries; but we reach just here an epoch where new tendencies
appear.

Poisson having in a report on the Fundamenta recalled the re-
proach made by Fourier to Abel and Jacobi of not having occupied
themselves preferably with the movement of heat, Jacobi wrote to
Legendre: "It is true that Monsieur Fourier held the view that
the principal aim of mathematics was public utility, and the ex-
planation of natural phenomena; but a philosopher such as he
should have known that the unique aim of science is the honor of
the human spirit, and that from this point of view a question about
numbers is as important as a question about the system of the

DEVELOPMENT OF MATHEMATICAL ANALYSIS 505

world." This was without doubt also the opinion of the grand geo-
meter of Goettingen, who called mathematics the queen of the sciences,
and arithmetic the queen of mathematics.

It would be ridiculous to oppose one to the other these two
tendencies; the harmony of our science is in their synthesis.

The time was about to arrive when one would feel the need of
inspecting the foundations of the edifice, and of making the inventory
of accumulated wealth, using more of the critical spirit. Mathematical
thought was about to gather more force by retiring into itself; the
problems were exhausted for a time, and it is not well for all seekers
to stay on the same road. Moreover, difficulties and paradoxes
remaining unexplained made necessary the progress of pure theory.

The path on which this should move was traced in its large outlines,
and there it could move with independence without necessarily losing
contact with the problems set by geometry, mechanics, and physics.

At the same time more interest was to attach to the philosophic
and artistic side of mathematics, confiding in a sort of preestab-
lished harmony between our logical and aesthetic satisfactions and the
necessities of future applications.

Let us recall rapidly certain points in the history of the revision
of principles where Gauss, Cauchy, and Abel likewise were laborers
of the first hour. Precise definitions of continuous functions, and their
most immediate properties, simple rules on the convergence of series,
were formulated; and soon was established, under very general
conditions, the possibility of trigonometric developments, legiti-
matizing thus the boldness of Fourier.

Certain geometric intuitions relative to areas and to arcs give
place to rigorous demonstration. The geometers of the eighteenth
century had necessarily sought to give account of the degree of the
generality of the solution of ordinary differential equations. Their
likeness to equations of finite differences led easily to the result; but
the demonstration so conducted must not be pressed very close.

Lagrange, in his lessons on the calculus of functions, had intro-
duced greater precision, and starting from Taylor's series, he saw
that the equation of order m leaves indeterminate the function,
and its w — 1 first derivatives for the initial value of the variable;
we are not surprised that Lagrange did not set himself the question
of convergence.

In twenty or thirty years the exigencies in the rigor of proofs had
grown. One knew that the two preceding modes of demonstration
are susceptible of all the precision necessary.

For the first, there was need of no new principle; for the second
it was necessary that the theory should develop in a new way. Up
to this point, the functions and the variables had remained real.
The consideration of complex variables comes to extend the field of

506 ALGEBRA AND ANALYSIS

analysis. The functions of a complex variable with unique derivative
are necessarily developable in Taylor's series; we come back thus
to the mode of development of which the author of the theory of
analytic functions had understood the interest, but of which the
importance could not be put fully in evidence in limiting one's self
to real variables. They also owe the grand role that they have not
ceased to play to the facility with which we can manage them, and
to their convenience in calculation.

The general theorems of the theory of analytic functions permitted
to reply with precision to questions remaining up to that time un-
decided, such as the degree of generality of the integrals of differential
equations. It became possible to push even to the end the demon-
stration sketched by Lagrange for an ordinary differential equa-
tion. For a partial differential equation or a system of such equations,
precise theorems were established. It is not that on this latter point
the results obtained, however important they may be, resolve
completely the diverse questions that may be set; because in mathe-
matical physics, and often in geometry, the conditions at the limits
are susceptible of forms so varied that the problem called Cauchy's
appears often under very severe form. I will shortly return to this
capital point.

Without restricting ourselves to the historic order, we will follow
the development of mathematical physics during the last century,
in so far as it interests analysis.

The problems of calorific equilibrium lead to the equation already
encountered by Laplace in the study of attraction. Few equations
have been the object of so many works as this celebrated equation.
The conditions at the limits may be of divers forms. The simplest
case is that of the calorific equilibrium of a body of which we main-
tain the elements of the surface at given temperatures; from the
physical point of view, it may be regarded as evident that the tem-
perature, continuous within the interior since no source of heat is
there, is determined when it is given at the surface.

A more general case is that where, the state remaining permanent,
there might be radiation toward the outside with an emissive power
varying on the surface in accordance with a given law; in particular
the temperature may be given on one portion, while there is radiation
on another portion.

These questions, which are not yet resolved in their greatest gen-
erality, have greatly contributed to the orientation of the theory of
partial differential equations. They have called attention to types of
determinations of integrals, which would not have presented them-
selves in remaining at a point of view purely abstract.

DEVELOPMENT OF MATHEMATICAL ANALYSIS 507

Laplace's equation had been met already in hydrodynamics and
in the study of attraction inversely as the square of the distance.
This latter theory has led to putting in evidence the most essential
elements, such as the potentials of simple strata and of double
strata. Analytic combinations of the highest importance were there
met, which since have been notably generalized, such as Green's
formula.

The fundamental problems of static electricity belong to the
same order of ideas, and that was surely a beautiful triumph for
theory, the discovery of the celebrated theorem on electric phe-
nomena in the interior of hollow conductors, which later Faraday
rediscovered experimentally, without having known of Green's
memoir.

All this magnificent ensemble has remained the type of the theories
already old of mathematical physics, which seem to us almost to
have attained perfection, and which exercise stiU so happy an in-
fluence on the progress of pure analysis in suggesting to it the most
beautiful problems. The theory of functions offers us another mem-
orable affiliation.

There the analytic transformations which come into play are not
distinct from those we have met in the permanent movement of
heat. Certain fundamental problems of the theory of functions of
a complex variable lost then their abstract enunciation to take a
physical form, such as that of the distribution of temperature on
a closed surface of any connection and not radiating, in calorific
equilibrium with two sources of heat which necessarily correspond
to flows equal and of contrary signs. Transposing, we face a ques-
tion relative to Abelian integrals of the third species in the theory of
algebraic curves.

The examples which precede, where we have envisaged only the
equations of heat and of attraction, show that the influence of
physical theories has been exercised not only on the general nature
of the problems to be solved, but even in the details of the analytic
transformations. Thus is currently designated in recent memoirs on
partial differential equations under the name of Green's formula,
a formula inspired by the primitive formula of the English physicist.
The theory of dynamic electricity and that of magnetism, with
Ampere and Gauss, have been the origin of important progress; the
study of curvilinear integrals and that of the integrals of surfaces
have taken thence all their developments, and formulas, such as
that of Stokes which might also be called Ampere's formula, have
appeared for the first time in memoirs on physics. The equations
of the propagation of electricity, to which are attached the names of
Ohm and Kirchoff, while presenting a great analogy with those of
heat, offer often conditions at the Hmits a little different; we know

508 ALGEBRA AND ANALYSIS

all that telegraphy by cables owes to the profound discussion of a
Fourier's equation carried over into electricity.

The equations long ago written of hydrodynamics, the equations
of the theory of electricity, those of Maxwell and of Hertz in electro-
magnetism, have offered problems analogous to those recalled above,
but under conditions still more varied. Many unsurmounted diffi-
culties are there met with; but how many beautiful results we owe
to the study of particular cases, whose number one would wish to
see increase. To be noted also as interesting at once to analysis and
physics are the profound differences which the propagation may
present according to the phenomena studied. With equations such
as those of sound, we have propagation by waves; with the equa-
tion of heat, each variation is felt instantly at every distance, but
very little at a very great distance, and we cannot then speak of
velocity of propagation.

In other cases of which Kirchoff's equation relative to the propa-
gation of electricity with induction and capacity offers the simplest
type, there is a wave front with a velocity determined but with a
remainder behind which does not vanish.

These diverse circumstances reveal very different properties of
integrals; their study has been delved into only in a few particular
cases, and it raises questions into which enter the most profound
notions of modern analysis.

I will enter into certain analytic details especially interesting for
mathematical physics.

The question of the generality of the solution of a partial differential
equation has presented some apparent paradoxes. For the same
equation, the number of arbitrary functions figuring in the general
integral was not always the same, following the form of the integral
envisaged. Thus Fourier, studying the equation of heat in an indefin-
ite medium, considers as evident that a solution will be determined
if its value for ^ = 0 is given, that is to say one arbitrary function of
the three coordinates x, y, z; from the point of view of Cauchy, we
may consider, on the contrary, that in the general solution there are
two arbitrary functions of the three variables. In reality, the ques-
tion, as it has long been stated, has not a precise signification.

In the first place, when it is a question only of analytic functions,
any finite number of functions of any number of independent vari-
ables presents, from the arithmetical point of view, no greater gen-
erality than a single function of a single variable, since in the one
case and in the other the ensemble of coefficients of the development
forms an enumerable series. But there is something more. In reality,
beyond the conditions which are translated by given functions, an

DEVELOPMENT OF MATHEMATICAL ANALYSIS 509

integral is subjected to conditions of continuity, or is to become in-
finite in a determined manner for certain elements; one may so be
led to regard as equivalent to an arbitrary function the condition
of continuity in a given space, and then we clearly see how badly
formulated is the question of giving the number of the arbitrary
functions. It is at times a delicate matter to demonstrate that con-
ditions determine in a unique manner a solution, when we do not
wish to be contented with probabilities; it is then necessary to make
precise the manner in which the function and certain of its deriva-
tives conduct themselves.

Thus in Fourier's problem relative to an indefinite medium cer-
tain hypotheses must be made about the function and its first
derivatives at infinity, if we wish to establish that the solution is
unique. x

Formulas analogous to Green's render great services, but the
demonstrations one deduces from them are not always entirely
rigorous, implicitly supposing fulfilled for the limits conditions
which are not, a 'priori at least, necessary. This is, after so many
others, a new example of the evolution of exigencies in the rigor of
proofs.

We remark, moreover, that the new study, rendered necessary,
has often led to a better account of the nature of integrals.

True rigor is fecund, thus distinguishing itself from another purely
formal and tedious, which spreads a shadow over the problems it
touches.

The difficulties in the demonstration of the unity of a solution
may be very different according as it is question of equations of
which all the integrals are or are not analytic. This is an important
point, and shows that even when we wish to put them aside, it is
necessary sometimes to consider non-analytic functions.

Thus we cannot affirm that Cauchy's problem determines in a
unique manner one solution, the data of the problem being general,
that is to say not being characteristic.

This is surely the case, if we envisage only analytic integrals,
but with non-analytic integrals there may be contacts of order
infinite. And theory here does not outstrip applications; on the
contrary, as the following example shows:

Does the celebrated theorem of Lagrange on the potentials of
velocity in a perfect fiuid hold good in a viscid fluid? Examples have
been given where the- coordinates of different points of a viscous
fluid starting from rest are not expressible as analytic functions of
the time starting from the initial instant of the motion, and where
the nul rotations as well as all their derivatives with respect to the
time at this instant are, however, not identically nul; Lagrange's
theorem, therefore, does not hold true.

510 ALGEBRA AND ANALYSIS

These considerations sufficiently show the interest it may have
to be assured that all the integrals of a system .of partial differential
equations continuous as well as all their derivatives up to a deter-
mined order in a certain field of real variables are analytic functions;
it is understood, we suppose, there are in the equations only analytic
elements. We have for linear equations precise theorems, all the
integrals being analytic, if the characteristics are imaginary, and
very general propositions have also been obtained in other cases.
. The conditions at the limits that one is led to assume are very
different according as it is question of an equation of which the
integrals are or are not analytic. A type of the first case is given
by the problem generalized by Dirichlet; conditions of continuity
there play an essential part, and, in general, the solution cannot
be prolonged fronv the two sides of the continuum which serves as
support to the data; it is no longer the same in the second case,
where the disposition of this support in relation to the characteris-
tics plays the principal role, and where the field of existence of the
solution presents itself under wholly different conditions. •

All these notions, difficult to make precise in ordinary language
and fundamental for mathematical physics, are not of less interest
for infinitesimal geometry.

It will suffice to recall that all the surfaces of constant positive
curvature are analytic, while there exist surfaces of constant nega-
tive curvature not analytic.

From antiquity has been felt the confused sentiment of a certain
economy in natural phenomena; one of the first precise examples
is furnished by Format's principle relative to the economy of time
in the transmission of light.

Then we came to recognize that the general equations of mechanics
correspond to a problem of minimum, or more exactly of variation,
and thus we obtained the principle of virtual velocities, then Ham-
ilton's principle, and that of least action. A great number of problems
appeared then as corresponding to minima of certain definite in-
tegrals.

This was a very important advance, because the existence of
a minimum could in many cases be regarded as evident, and con-
sequently the demonstration of the existence of the solution was
effected.

This reasoning has rendered immense services; the greatest geo-
meters. Gauss in the problem of the distribution of an attracting
mass corresponding to a given potential, Riemann in his theory of
Abelian functions, have been satisfied with it. To-day our attention
has been called to the dangers of this sort of demonstration; it is
possible for the minima to be simply limits and not to be actually
attained by veritable functions possessing the necessary properties

DEVELOPMENT OF MATHEMATICAL ANALYSIS 511

of continuity. We are, therefore, no longer content with the prob-
abilities offered by the reasoning long classic.

Whether we proceed indirectly or whether we seek to give a rigor-
ous proof of the existence of a function corresponding to the mini-
mum, the route is long and arduous.

Further, not the less will it be always useful to connect a ques-
tion of mechanics or of mathematical physics with a problem of
minimum; in this first of all is a source of fecund analytic trans-
formations, and besides in the very calculations of the investigation
of variations useful indications may appear, relative to the condi-
tions at the limits; a beautiful example of it was given by Kirchoff
in the delicate investigation of the conditions at the limits of the
equilibrium of flexure of plates.

I have been led to expand particularly on partial differential
equations.

Examples chosen in rational mechanics and in celestial mechanics
would readily show the role which ordinary differential equations
play in the progress of these sciences whose history, as we have seen,
has been so narrowly bound to that of analysis.

When the hope of integrating with simple functions was lost, one
strove to find developments permitting to follow a phenomenon as long
as possible, or at least to obtain information of its qualitative bearing.

For practice, the methods of approximation form an extremely
important part of mathematics, and it is thus that the highest parts
of theoretic arithmetic find themselves connected with the applied
sciences. As to series, the demonstrations themselves of the exist-
ence of integrals furnish them from the very first; thus Cauchy's
first method gives developments convergent as long as the integrals
and the differential coefficients remain continuous.

When any circumstance permits our foreseeing that such is always
the case, we obtain developments always convergent. In the pro-
blem of n bodies, we can in this way obtain developments valid so
long as there are no shocks.

If the bodies, instead of attracting, repel each other, this contin-
gency need not be feared and we should obtain developments valid
indefinitely; unhappily, as Fresnel said one day to Laplace, nature
is not concerned about analytic difficulties and the celestial bodies
attract instead of repelling each other.

One would even be tempted at times to go further than the great
physicist and say that nature has sown difficulties in the paths of
the analysts.

Thus, to take another example, we can generally decide, given a
system of differential equations of the first order, whether the gen-

512 ALGEBRA AND ANALYSIS

eral solution is stable or not about a point, and to find developments
in series valid for stable solutions it is only necessary that certain
inequalities be verified.

But if we apply these results to the equations of dynamics to dis-
cuss stability, we find ourselves exactly in the particular case which
is unfavorable. Nay, in general, here it is not possible to decide on
the stability; in the case of a function of forces having a maximum,
reasoning classic, but indirect, establishes the stability which cannot
be deduced from any development valid for every value of the time.

Do not lament these difficulties; they will be the source of future
progress.

Such are also the difficulties which still present to us, in spite of
so many works, the equations of celestial mechanics; the astro-
nomers have almost drawn from them, since Newton, by means of
series practically convergent and approximations happily con-
ducted, all that is necessary for the foretelling of the motions of the
heavenly bodies.

Th-e analysts would ask more, but they no longer hope to attain
the integration by means of simple functions or developments al-
ways convergent.

What admirable recent researches have best taught them is the
immense difficulty of the problem; a new way has, however, been
opened by the study of particular solutions, such as the periodic
solutions and the asymptotic solutions which have already been
utilized. It is not perhaps so much because of the needs of practice
as in order not to avow itself vanquished, that analysis will never
resign itself to abandon, without a decisive victory, a subject where
it has met so many brilliant triumphs; and again, what more beau-
tiful field could the theories new-born or rejuvenated of the modern
doctrine of functions find, to essay their forces, than this classic
problem of n bodies?

It is a joy for the analyst to encounter in applications equations
that he can integrate with known functions, with transcendents
already classed.

Such encounters are unhapily rare; the problem of the pendulum,
the classic cases of the motion of a solid body around a fixed point,
are examples where the elliptic functions have permitted us to effect
the integration.

It would also be extremely interesting to encounter a question
of mechanics which might be the origin of the discovery of a new
transcendent possessing some remarkable property; I should be
embarrassed to give an example of it unless in carrying back to the
pendulum the debut of the theory of elliptic functions.

The interpenetration between theory and applications is here
much less than in the questions of mathematical physics. Thus

DEVELOPMENT OF MATHEMATICAL ANALYSIS 513

is explained that, since forty years, the works on ordinary differ-
ential equations attached to analytic functions have had in great
part a theoretic character altogether abstract.

The pure theory has notably taken the advance; we have had
occasion to say that it was well it should be so, but evidently there
is here a question of measure, and we may hope to see the old pro-
blems profit by the progress accomplished.

It would not be over-difficult to give some examples, and I will re-
call only those linear differential equations, where figure arbitrary
parameters whose singular values are roots of entire transcendent
functions ; which in particular makes the successive harmonics of
a vibrating membrane correspond to the poles of a meromorphic
function.

It happens also that the theory may be an element of classifica-
tion in leading to seek conditions for which the solution falls under
a determined type, as for example that the integral may be uniform.
There have been and there yet will be many interesting discoveries
in this way, the case of the motion of a solid heavy body treated
by Madame de Kovalevski and where the Abelian functions were
utilized is a remarkable example.

VII

In studying the reciprocal relations of analysis with mechanics
and mathematical physics, we have on our way more than once
encountered the infinitesimal geometry, which has proposed so
many celebrated problems; in many difficult questions, the happy
combination of calculus and synthetic reasonings has realized con-
siderable progress, as is shown by the theories of applicable surfaces
and systems triply orthogonal.

It is another part of geometry which plays a grand role in certain
analytic researches, I mean the geometry of situation or analysis
situs. We know that Riemann made from this point of view a com-
plete study of the continuum of two dimensions, on which rests his
theory of algebraic functions and their integrals.

When this number of dimensions augments, the questions of
analysis situs become necessarily complicated; the geometric intui-
tion ceases, and the study becomes purely analytic, the mind being
guided solely by analogies which may be misleading and need to be
discussed very closely. The theory of algebraic functions of two
variables, which transports us into a space of four dimensions,
without getting from -analysis situs an aid so fruitful as does the
theory of functions of one variable, owes it, however, useful orient-
ations.

There is also another order of questions where the geometry of
situation intervenes; in the study of curves traced on a surface and

514 ALGEBRA AND ANALYSIS

defined by differential equations, the connection of this surface plays
an important role; this happens for geodesic lines.

The notion of connexity, moreover, presented itself long ago in
analysis, when the study of electric currents and magnetism led
to non-uniform potentials; in a more general manner certain multi-
form integrals of some partial , differential equations are met in
difficult theories, such as that of diffraction, and varied researches
must continue in this direction.

From a different point of view, I must yet recall the relations of
algebraic analysis with geometry, which manifest themselves so
elegantly in the theory of groups of finite order.

A regular polyhedron, say an icosahedron, is on the one hand the
solid that all the world knows; it is also, for the analyst, a group of
finite order, corresponding to the divers ways of making the poly-
hedron coincide with itself.

The investigation of all the types of groups of motion of finite
order interests not alone the geometers, but also the crystallo-
graphers; it goes back essentially to the study of groups of ternary
linear- substitutions of determinant +1, and leads to the thirty-
two systems of symmetry of the crystal lographers for the particular
complex.

The grouping in systems of polyhedra corresponding so as to fill
space exhausts all the possibilities in the investigation of the struc-
ture of crystals.

Since the epoch when the notion of group was introduced into
algebra by Galois, it has taken, in divers ways, considerable devel-
opment, so that to-day it is met in all parts of mathematics. In the
applications, it appears to us above all as an admirable instrument
of classification. Whether it is a question of substitution groups
or of Sophus Lie's transformation groups, whether it is a question
of algebraic equations or of differential equations, this comprehen-
sive doctrine permits explanation of the degree of difficulty of the
problems treated and teaches how to utilize the special circumstances
which present themselves; thus it should interest as well mechanics
and mathematical physics as pure analysis.

The degree of development of mechanics and physics has per-
mitted giving to almost all their theories a mathematical form;
certain hypotheses, the knowledge of elementary laws, have led
to differential relations which constitute the last form under which
these theories settle down, at least for a time. These latter have
seen little by little their field enlarge with the principles of thermo-
dynamics; to-day chemistry tends to take in its turn a mathemat-
ical form.

I will take as witness of it only the celebrated memoir of Gibbs
on the equilibrium of chemical systems, so analytic in character,

DEVELOPMENT OF MATHEMATICAL ANALYSIS 515

and where it needed some effort on the part of the chemists to
recognize, under their algebraic mantle, laws of high importance.

It seems that chemistry has to-day gotten out of the premathe-
matic period, by which every science begins, and that a day must
come when will be systematized grand theories, analogous to those
of our present mathematical physics, but far more vast, and com-
prising the ensemble of physicochemic phenomena.

It would be premature to ask if analysis will find in their develop-
ments the source of new progress; we do not even know before-
hand what analytic types one might find.

I haA^e constantly spoken of differential equations ruling phe-
nomena; will this always be the final form which condenses a theory?
Of this I know nothing certain, but we should, however, remember
that many hypotheses have been made of more or less experimental
nature ; among them, one is what has been called the principle of
non-heredity, which postulates that the future of a system depends
only on its present state and its state at an instant infinitely near,
or, more briefly, that accelerations depend only on positions and
velocities.

We know that in certain cases this hypothesis is not admissible,
at least with the magnitudes directly envisaged; one has sometimes
misemployed on this subject the memory of matter, which recalls
its past, and has spoken in affected terms of the life of a morsel of
steel. Different attempts have been made to give a theor}^ of these
phenomena, where a distant past seems to interfere; of them I need
not speak here. An analyst may think that in cases so complex it
is necessary to abandon the form of differential equations, and resign
one's self to envisage functional equations, where figure definite
integrals which will be the witness of a sort of heredity.

To see the interest which is attached at this moment to functional
equations, one might believe in a presentiment of the future needs
of science.

VIII

After having spoken of non-heredity, I scarcely dare touch the
question of the applications of analysis to biology.

It will be some time, no doubt, before one forms the functional
equations of biologic phenomena; the attempts so far made are
in a very modest order of ideas; yet efforts are being made to get
out of the purely qualitative field, to introduce quantitative meas-
ures. In the question of the variation of certain characteristics,
mensuration has been engaged in, and statistic measures which are
translated by curves of frequency. The modifications of these curves
with successive generations, their decompositions into distinct curves,
may give the measure of the stability of species or of the rapidity

516 ALGEBRA AND ANALYSIS

of mutations, and we know what interest attaches itself to these
questions in recent botanic researches. In all this so great is the
number of parameters that one questions whether the infinitesimal
method itself could be of any service. Some laws of a simple arith-
metic character like those of Mendel come occasionally to give
renewed confidence in the old aphorism which I cited in the begin-
ning, that all things are explained by numbers; but, in spite of
legitimate hopes, it is clear that, in its totality, biology is still far
from entering upon a period truly mathematical.

It is not so, according to certain economists, with potential econ-
omy. After Cournot, the Lausanne school made an effort extremely
interesting to introduce mathematical analysis into political econ-
omy.

Under certain hypotheses, which fit at least limiting cases, we
find in learned treatises an equation between the quantities of
merchandise and their prices, which recalls the equation of virtual
velocities in mechanics: this is the equation of economic equilib-
rium. A function of quantities plays in this theory an essential role
recalling that of the potential function. Moreover, the best author-
ized representatives of the school insist on the analogy of economic
phenomena with mechanical phenomena. "As rational mechanics,"
says one of them, "■ considers material points, pure economy con-
siders the homo oeconomicus."

Naturally, we find there also the analogues of Lagrange's equa-
tions, indispensable matrix of all mechanics.

While admiring these bold works, we fear lest the authors have
neglected certain hidden masses, as Helmholtz and Hertz would
have said. But although that may happen, there is in these doctrines
a curious application of mathematics, which, at least, in well-circum-
scribed cases, has already rendered great services.

I have terminated, messieurs, this summary history of some of
the applications of analysis, with the reflections which it has at
moments suggested to me. It is far from being complete; thus I have
omitted to speak of the calculus of probabilities, which demands
so much subtlety of mind, and of which Pascal refused to explain the
niceties to the Chevalier de Mere because he was not a geometer.

Its practical utility is of the first rank, its theoretic interest has
always been great; it is further augmented to-day, thanks to the
importance taken by the researches that Maxwell called statistical
and which tend to envisage mechanics under a wholly new light.

I hope, however, to have shown, in this sketch, the origin and
the reason of the bonds so profound which unite analysis to geometry
and physics, more generally to every science bearing on quantities
numerically measurable.

DEVELOPMENT OF MATHEMATICAL ANALYSIS 517

The reciprocal influence of analysis and physical theories has been
in this regard particularly instructive.

What does the future hold?

Problems more difficult, corresponding to an approximation of
higher order, will introduce complications which we can only vaguely
forecast, in speaking, as I have just done, of functional equations
replacing systematically our actual differential equations, or further
of integrations of equations infinite in number with an infinity of
unknowTi functions. But even though that happens, mathematical
analysis will always remain that language which, according to the
mot of Fourier, has no symbols to express confused notions, a lan-
guage endowed with an admirable power of transformation and
capable of condensing in its formulas an immense number of results.

ON PRESENT PROBLEMS OF ALGEBRA AND ANALYSIS

BY HEINEICH MASCHKE

[Heinrich Maschke, Associate Professor of Mathematics, University of
Chicago, b. Breslau, Germany, October 24, 1853. A.B. Magdalenen Gym-
nasium, Breslau, 1872; Ph.D. Gottingen, 1880. Post-graduate Heidelberg,
Breslau, Berlin, and Gottingen. Professor Mathematics Lvisenstadt. Gym-
nasium, Berlin, 1880-90; Electric Engineer at Weston Electric Company,
Newark, New Jersey, 1890-92; Assistant Professor of Mathematics, Uni-
versity of Chicago, 1892-96.]

As set forth by the Committee directing the affairs of this Interna-
tional Congress, the address which I have the distinguished privilege
of delivering to-day shall be on "Present Problems in Algebra and
Analysis," — but it is not provided by the Committee how many
of these problems shall be treated.

The different branches of algebra and analysis which have been
investigated are so numerous that it would be quite impossible to
give an approximately exhaustive representation even only of the
most important problems, within the limits of the time allowed to
me. I, therefore, have confined myself to the minimum admissible
number, namely one, or rather one group of problems.

Of this one problem, however, this Section of Algebra and Analysis
has the right to expect that it is neither purely algebraic nor purely
analytic, but one which touches both fields; and at least in this
respect I hope that my selection has been fortunate.

I purpose to speak to-day on the Theory of Invariants of Quad-
ratic Differential Quantics. Invariants suggest at once algebra,
differential quantics: analysis. At the same time the subject also
leads into geometry, — it contains, for instance, a great part of
differential geometry and of geometry of hyperspace. But is there,
indeed, any algebraic or analytic problem which does not allow
geometrical interpretation in some way or other? And when it comes
to geometry of hyperspace, — it is then only geometrical language
that we are using, — what we are actually considering are analytic
or algebraic forms. Moreover, rigorous definitions and discussions
of geometrical propositions of an invariant character in particular
can only be given by tracing them back to their analytic origin.

In the following exposition I shall first speak on the various in-
variant expressions of differential quadratics as they occur in geo-
metry of two and more dimensions, and then take up the purely
analytic representation in the second part of the paper.

This corresponds also to the historical development of the sub-

PROBLEMS OF ALGEBRA AND ANALYSIS 519

ject: geometry has here as well as in many other branches of mathe-
matics indicated the problems which in their later development
turned out to be of paramount interest in pure analysis.

A few preliminary remarks concerning the nomenclature of the
different types of invariant expressions will be necessary.

To a given differential quadratic form

• A= ^ aikdxidxk,iO'ki=aik)

where the a^^'s are functions of the n independent variables Xi, x^, . . .
Xn, we apply a general point transformation of the variables x,

Xi=Xi(y^,y„ . . . Vn).
We observe that the differentials dx are then transformed into
linear expressions of the differentials dy with the Jacobian of the
x's with respect to the y's as the substitution-determinant which
we shall call r.

By this transformation A goes into

A'=Ia'ikdyidyk.

Let now <? be a function

(a) of the coefficients aik and their first, second, . . . derivatives,

(b) of U, V, . . . and their derivatives, where U, V, . . . are any
arbitrary functions of Xj, 3:2, . . . x„.

If then 0 remains the same whether formed for the new or for
the old quantities, that is, if

0(a\,, ^, . . . , t/^i^, ...,r,...) =0(a,,, ^-^,...,U,^JL,
dyk dyX dxX dx),

...y, ...) _

we say that 0 is an invariant (in the wider sense) of A.

If 0 contains only the a^^'s and their derivatives, we call it an
invariant proper, and its order the order of the highest derivative
occurring in it. If 0 contains also one or more arbitrary functions
U,V, ... we call it a differential parameter, the definition of order
being the same as before.
* If more than one differential quadratic is given it is easily under-
stood what is meant by simultaneous invariants and simultaneous
differential parameters.

In strict analogy with the algebraic theory of invariants we call
covariants expressions 0 of the above invariantive nature, provided
that we also allow the differentials dx to enter into 0.

The first and the most important example of a differential quad-
ratic quantic is the square of the arc-element on a surface

ds^ =Edu^ +2Fdudv + Gdv\
It was Gauss who made (1827), in his Disquisitiones generales
circa superficies curvas, this expression the fundamental object of

520

ALGEBRA AND ANALYSIS

investigation. He also gave, in what has been called after him the

Gaussian Curvature

dE .

K = (E,F,G,—->- ' •),
ou

the first example of an invariant. Gauss defines this curvature
geometrically and finds for it the analytic expression

LN-M^

EG-F^

which is a simultaneous invariant of two differential quantics,

namely, of ds^ and of— =Ldu'^+2Mdudv-]-Ndv^.

This shows that K is independent of the M,'y-system on the
surface. And now Gauss expresses K in terms of E, F, G and the
first and second derivatives of these quantities alone. A direct
demonstration that K is an invariant proper of the differential
quantic ds^ alone, — that is, without passing through the second

ds^ . ,

differential quantic — , — is of course desirable.^ Each one of the
P

general methods of treating the theory of invariants, which will be

discussed in the latter part of this paper, furnishes such a direct

proof. In particular, the aspect of the formula for K, on p. 528,

deduced by the symbolic method, shows immediately the invariant

character of K.

Differential parameters were introduced into differential geometry

by Beltrami in 1863. These are the well-known expressions

J^Cp:

\dvj du dv \duj

EG-F^

F{^,4>) =

dv dv \du dv dv duj du du

EG-F^

J2(P =

Veg-f^

du dv
\/E~G^P

dip dip

\/EG-F^

where ip and 0 are the arbitrary functions which take the place of
U , V in our general definition of differential parameters. Beltrami
adopted the name " differential parameters " and also the notation

I Cf . on this subject the interesting paper by Knoblauch : " Der Gauss'sche Satz
vom Kriimmungsmass," Sitzungsberichte der Berliner Mathem. Gesellschaft. April
27, 1904.

PROBLEMS OF ALGEBRA AND ANALYSIS 521

J from Lame, who, in his Lecons sur les coordonnees curvilignes,
defined in 1859 his differential parameters

d^(p d'^cp d^<p

for the three-dimensional case where the arc-element is of the form

ds^^dx^ +dy^ ■\-dz'^.

Lame recognized the fundamental importance of these quantities
and made a systematical use of them on account of their invariance
with respect to any point-transformation preserving the form ds"^.

The general theory of invariants defines the differential parameters
J, and Jj for the case of n variables. From these general expressions
Beltrami's differential parameters are directly obtained for n = 2,
Lame's quantities (Ji)^ and Jj for the special form of ds'^ in the case
n=3.

The number of differential parameters is of course infinite, but

Darboux in his Lecons sur la theorie generale des surfaces has proved

that all of them are expressible by means of ij, J.^, f and the evident

differential parameter

dip dij) dip dip

du dv dv du
6(0, (p) = ==_ —

(by forming, for instance, Ji(J^(p) etc.) — an important theorem
which has later been extended by Staeckel to an analogous theorem
for the case of n variables.

The expression Jicp occurs already in Gauss's Disquisitiones .
By taking as parameter curves a singly infinite system of geodesies
and its orthogonal trajectories he transforms the arc-element into
the form

ds^ =dr^ -\-w?d(p'^
and shows that r satisfies the differential equation

A,r = \.

An important differential parameter is the geodesic curvature.
Its expression was thrown by Bonnet into a form which is easily
recognized as a differential parameter (of the second order). Its
numerator =0 represents the differential equation of geodesic lines
in an invariant form.

Since a transformation of the two independent variables u, v which
preserves the same value of ds"^ can also be considered as a transfor-
mation of two surfaces which are applicable to each other, it follows
that all invariants of ds^ are also invariants of a surface with respect
to the process of bending. From this reason these invariants have

522 ALGEBRA AND ANALYSIS

been called by Weingarten and Knoblauch, who were among the first
writers emphasizing and developing to a certain extent the invari-
antive side of differential geometry, in the case of invariants proper,
" Biegungsinvarianten," in the case of differential parameters, " Bie-
gungscovarianten," and this notation has been more or less generally
adopted. The notation " Biegungscovarianten " does not agree with
the definition of a covariant given above, but a differential para-
meter of ds^ can easily be modified into a covariantive form by
replacing according to the differential equation of the curve

U{u.,v) = const.

oU dU
the derivatives — and -— by /idv and — fidu.
OU ov

A surface is completely defined, apart from its location in space,
when in addition to the quadratic form ds^ also

ds^

— =^Ldu^+2Mdudv+Ndv''
P

is given, where p denotes the radius of curvature along ds, — a the-
orem which was proved (1867) by Bonnet.

With these two differential quantics given, we can now at once
form simultaneous invariants and differential parameters. The six
coefficients, E, F, G, L, M, N are, however, not independent; they
are related by three partial differential equations, — the Gaussian
relation and the two Codazzi-Mainardi equations. These three
relations are expressible in an invariantive form. The Gaussian re-
lation is

LN-M^ ^^ ^^ dE

while the two Codazzi formulas are given by the identical vanishing
of one simultaneous linear covariant.

As examples of simultaneous differential parameters and covariants
I mention the expressions which, when set equal to zero, represent
the differential equations of conjugate lines, asymptotic lines, and
lines of curvature. The differential equation of lines of curvature, for
instance, if written in terms of du, dv represents a linear simultaneous
covariant; if written as a partial differential equation derived from

U(u,v) = const.
it represents a simultaneous differential parameter involving the
arbitrary function U. The differential equation of conjugate lines,
if written in two sets of differentials du, dv and du, dv represents
a bilinear simultaneous covariant; if written as a partial differential
equation it represents a differential parameter involving two arbi-
trary functions U and V.

The theory of invariants of the above two differential quadratics.

PROBLEMS OF ALGEBRA AND ANALYSIS 523

together with the condition of the vanishing of one simultaneous
invariant proper and one simultaneous covariant, dominates then,
in a certain sense, the whole of differential geometry.

Passing now to the case of n variables we may consider the differ-
ential quadratic form

^aikdxidxk^ds^

as the square of the arc in a hyperspace of n dimensions.

The fundamental role which the Gaussian curvature plays in the
case n=2 is here represented by an invariant expresson of ds^ which
— in a certain sense — might be regarded as a generalization of the
Gaussian curvature, namely, the Riemann curvature of the hyper-
space. Riemann's investigations on this subject are found in his
paper, Ueber die Hypothesen, welche der Geometrie zu Grunde liegen,
and in the mathematical supplement to it Commentatio mathematica,
etc. in the prize-problem of the Parisian Academy, 1861.

The geometrical definition of the Riemann curvature is briefly the
following: Starting from any point P with the coordinates Xi we
consider two linear directions defined by the increments dxi and dxi.
If we remain in the vicinity of P these two directions define a plane
of two dimensions and the determinants

dxidxk — dxkdxi
may be considered as the coordinates of this plane. If now we
draw geodesic lines from the point P whose initial arc-elements
lie all in this plane, then these geodesies define a surface of two di-
mensions and the Gaussian curvature of this geodesic surface at the
point P is the Riemann curvature. The analytic expression for it is

R=-h

I iikrs) {dx^dXg — dx^dx^) (dxj^dx^. — dx^oxk)
^ lia^tfii^^ - a^^aj (dx^dx, - dx^dx^) {dxj^dx^ - dx^dx,^,

where the sum is to be taken over all values of i, k, r, s from 1 to n
with the exception of those for which i =k or r = s.

The coefficients {ikrs) are certain quantities depending on the
coefficients a^-, their first and second derivatives; they occur in the
literature mostly under the name of the " Christoffel quadruple index
symbols." A better, certainly shorter, notation would be the one
used by Ricci, namely, " Riemann symbols."

The Riemann curvature R is an invariant expression, and as its
form shows it is a covariant of two sets of differentials. For n=2
it is identical with the Gaussian curvature. For greater numbers
n the value of R depends, at a given point, on the plane-direction
at that point and in general varies with the plane. If it should be
constant for all plane-directions through one point, and if this is
so for all the points, then R is, as Schur has shown, altogether con-
stant that is, for every point.

524 ALGEBRA AND ANALYSIS

Spaces of constant Riemann curvature have been the object of
numerous interesting investigations^ but these are more or less of
a specific geometric character.

If in particular R is zero, then all the Riemann symbols vanish
and it can easily be shown that ds"^ can be transformed into the
sum of n squares

ds^=^dy\
1=1

The converse is true. In this case the hyperspace of n dimensions

is called a flat or also Euclidean space.

In every case the quadratic

can be transformed into

^aikdxidxk

iA'=l

8 = 1

where r has the maximum value '-^^^^|=^- We might say then that the
given hyperspace of n dimensions is always contained in an
Euclidean space of n+r dimensions, where r is one of the numbers,

f. 1 n{n—l)

The number r is evidently characteristic for the hyperspace the
square of the arc-element of which is the given quadratic. This
number r has been called by Ricci the class of the given differential
quadratic quantic. It is evident that this class is an invariant num-
ber, and the condition that a given differential quadratic be of class
r must certainly be an invariantive condition. For r = 0 we have
just seen that the condition is R=0. For higher values of r no at-
tempt has yet been made, so far as I know, to establish this invari-
antive condition though this problem is certainly one of fundamental
interest.

Beltrami, in his paper, Teoria generale dei parametri differenziali,
has extended the definition of his differential parameters to the
case of n variables. The definition, for instance, of the first differ-
ential parameters is

"t,A=l dXi dXk

where Aik denotes the minor of the element aik in the determinant

Wik] =0"

Beltrami shows that by means of the geodesies emanating from one
point and of the hypersurfaces orthogonal to them he can choose
his parameters such that ds"^ is transformed into

n-l

ds'^=dr'^-\- ^ hijcdyidyk,
t,t=i

where r satisfies the equation Aj- = 1, and that thus Gauss's theorems

PROBLEMS OF ALGEBRA AND ANALYSIS 525

on geodesic polar co5rdinates for n = 2 admit a perfect analogon
in hyperspace. Also in hyperspace then the determination of systems
of geodesies amounts to the integration of the partial differential
equation

This leads now to the application of differential quadratics to
analytic mechanics. If we write down the expression of the vis
viva of a (holonomous) material system in terms of generalized
coordinates q^, q^, • • -qn

dt dt

we have at once in

2Tdt''=ds''
a differential quadratic before us.

If no external forces act on the system, then a geodesic line of ds^
represents at once, as also Beltrami has shown, a path of the sys-
tem. Thus the mechanical problem is practically reduced to the
integration of the equation A^(p = \.

In the case of the existence of external forces having a potential
U , the above differential quantic has to be replaced by

I{U -\-h)aikdqidqk
and the mechanical problem is equivalent to the integration of the
equation

Ji(p = U+h
where Ji(p is the differential parameter of the quadratic form de-
noted before by ds^.

A detailed exposition of the above-mentioned researches of Bel-
trami, as well as this application to mechanics, is given in the second
volume of Darboux's Lecons sur la theorie des surfaces.

Passing now to the second part of my address, the purely ana-
lytic theory of invariants of differential quadratics, I have first
to discuss that paper which forms the foundation of almost all
later literature on the subject: Christoffel's article in Crelle's Jour-
nal, vol. Lxx (1870), "Ueber die Transformation der homogenen
Differentialausdriicke des zweiten Grades."

Christoffel puts his problem in this form: Given two differential
quadratics

A = I aijtdxidxk and A' ^la' ikdyidyk,

what are the necessary and sufficient conditions for the equivalence
of the two quadratics, that is, for the existence of a transformation
of one quantic into the other; and if these conditions are established
how can the required transformation be determined? (I should men-
tion that Lame in his already quoted work, Lecons sur les co'dr-

526 ALGEBRA AND ANALYSIS

donnees curvilignes, treats and solves the analogous problem for the
case A=dx^-\-dy^+dz'^).

Since the differentials dx are substituted linearly in terms of the
dy there exists one and only one algebraic condition for the trans-
formation, namely,

\a\k\=r^\aik\.
This condition would be sufficient if the coefficients anc and the
elements of the determinant r were constants. In our case, how-
ever, other conditions must be satisfied, namely, the conditions of
integrability in order that the expressions for the dx's are com-
plete differentials. This is the way in which Christoffel introduces
his problem to the reader.

^ The difficulty lies in the fact that the integrability conditions
lead at once to a great number of partial differential equations of
an apparently highly complex character. But Christoffel succeeds
in substituting for all these partial differential equations a purely
algebraic problem: The equivalence of two finite systems of alge-
braic forms in the sense of the algebraic theory of invariants. If
this equivalence is satisfied, — which is merely a question of algebra,

— no further discussion of the integrability conditions is required;
they are all taken care of by the equivalence of the two systems.

For the following it will be necessary to sketch briefly the char-
acter of these forms.

The first is the quadratic form A itself. The next form is a quad-
rilinear covariant G^ in four sets of differentials dx^, dx^, dx^, dx^,
the coefficients of which are precisely the quantities {i k r s ) — the
"Christoffel quadruple index symbols" or the "Riemann symbols"

— which occur in the expression for the Riemann curvature:

G^=I(ikrs) d'x^d'^Xj^d^x^d^Xg.

It is highly interesting to observe how the quantities { i k r s )
have entered into the theory from two so apparently different stand-
points. Christoffel found these expressions quite independently.
Though Riemann 's paper was written in 1861, that is, before Chris-
toffel's article which appeared in 1870, it w^as only published in 1876,
ten years after Riemann 's death, by Weber-Dedekind.

For the deduction of the following forms G^, G^, — . . . — these
forms are covariants linear in resp. 5,6, . . . sets of differentials —
Christoffel uses a certain reduction process. The coefficients (Aikr s)
for instance of G^ are obtained from (i k r s) first by differentiating
(i k r s) with respect to x^ and then by the addition of a sum of 5n
terms which are linear in the different symbols (i k r s ) with co-
efficients depending on the so-called Christoffel triple index sym-
bols of the second kind — expressions involving the quantities
aik and their first derivatives.

PROBLEMS OF ALGEBRA AND ANALYSIS 527

Continuing in this way Christoffel obtains a well-defined set of
CO variants 0^,0^,. . . , and this is his final result: the necessary
and sufficient condition for the equivalence of the two differential
quadratics is the algebraic equivalence — in the sense of the alge-
braic theory of invariants — of the forms ^, (t^, (tj, . . . G^,andA'',
G'^, G\ . . . G'^, where /x is a certain finite number.

In several papers covering the period from about 1884 up to the
present time Ricci has worked out in a systematical way the funda-
mental principles of Christoffel's investigation, and has applied his
theory to many problems in analysis, geometry, mechanics, and
mathematical physics. He recognized in particular the importance
of Christoffel's deduction of the co variants G^j^^ from G,^. He found
that this process of deduction can be applied with a proper modifi-
cation to any functions of the x's and the a^/t's and that whenever
invariantive relations with respect to the fundamental differential
quadratic A come into question, this process is always of vital im-
portance. He calls this process covariantive differentiation with
respect to the fundamental quadratic A. On the systematical use
of this covariantive differentiation Ricci based a calculus which he
called Calcolo differenziale assoluto.

A collection of all his various investigations is given in two places :

(1) In a paper published, together with Levi-Civitta in the Math,
Annalen, vol. liv.

(2) In his Lezioni sulla teoria delle superficie, Verona, Padua, 1898.
In the introduction of these autographed lectures he presents a

complete exposition of his absolute differential calculus. Charac-
teristic is the way in which he treats in his Lezioni the differential
geometry. He divides it into two parts:

(1) Properties of surfaces depending on the one differential quad-
ratic ds^.

(2) Properties of surfaces depending on the two quadratics

ds^
ds^ and — .

We are here chiefly interested in his applications to the theory of

differential invariants. This is the result in his language: In order

to obtain all invariants proper and differential parameters of order /i,

it is sufficient to determine the algebraic invariants of the system

of the following forms:

(1) The fundamental differential quantic A.

(2) The covariantive derivatives of the arbitrary functions

U, V, ... up to the order /<.

(3) (for fi>l) the quadrilinear covariant G^ and its covariantive
derivatives up to the order « — 2.

Another treatment of the invariant theory of differential quan-

528 ALGEBRA AND ANALYSIS

tics was given by myself. I applied a symbolic method to the theory
which consists chiefly in identifying the fundamental quadratic

Ittikdxidxk
with the square of a linear expression

by setting jijk=<^ik- This is strictly analogous to the introduction of
symbols in the algebraic theory. The difference, of course, comes
in at once when we have to consider also the derivatives of a^^.

A systematic development leads to expressions and formulas
which with respect to simplicity and shortness are as superior to
the formulas of the ordinary notation as the formulas of the so-
called symbolic notation in the algebraic theory are superior to the
non-symbolic expressions.

As examples I give the most important invariant expressions
for the case n=2.

Let us introduce the abbreviation

(Pi Q2 -P2 Qi ) -(PQ), where Pk=^ ^tc;

vaiia22 — a 12 oxk

let further f, (f, (l> . . . be symbols of A, so that

and let C/, y ... be arbitrary functions oi x^, x^.
Then we have

Quy^A. u,

{jU){jV)=f{UV),

{Km) =^.u,

(^^) {(f'P) (/^)) =2K (Gaussian curvature),
Q(p) (<pU) {(fU)U) : (Ji U)^ =Geodesic curvature of curve U = const.
To give also some examples of simultaneous invariant expressions
let /, ^, . . . be as before symbols of

Edv?+2Fdudv+Gdv''
and F, 0 . . . symbols of

Ldu^+2Mdudv+Ndv\
Then:

(F(Py=2K,
(/P)^=mean curvature.
The differential equations

of asymptotic curves U = c are (FUy=0,
of conjugate curves U=c, V=c: (FU)(FV)=0,
of lines of curvature U = c: (jF) (fU) (FU) = 0.
The equation (f(p)(a>F){(f(p)U) =0 gives the two Cadazzi formulas
by setting the coefficients of Ui and U2 separately equal to zero.

In these examples the invariant expressions always appear as
products of factors of the type (PR). The general theorem holds
that any product of factors of this type represents always an in-

PROBLEMS OF ALGEBRA AND ANALYSIS 529

variant expression provided that the symbols/, ip, . . . , F, 0, . . .
occur in such a connection as to permit actual meaning.

The symbolic representation of invariant expressions suggested
by the case n=2 can with6ut essential difficulty be extended to the
general case of n variables. In this treatment of the subject all the
essential quantities entering into the theory present themselves quite
naturally; they lie, so. to say, on the surface; so, for instance, all the
Christoffel symbols of the different kinds including the Riemann
symbols and in particular also the process of covariantive differ-
entiation.

The results of my investigation are chiefly laid down in the paper
"A symbolic treatment of the theory of invariants of quadratic
differential quantics of n variables," Transactions of the American
Mathematical Society, vol. iv.

A third method of investigation of our theory of invariants is
based on Lie's theory of continuous groups. The general point
transformation by which A is transformed into A' defines a so-
called " infinite " continuous group. In order to obtain the invari-
ants of A, this group must first be "extended" in Lie's sense to
include the coefficients aik of A and also the arbitrary functions
involved in the differential parameters.

Lie himself developed a short outline of the determination of
invariants in the second volume of the Mathematische Annalen for
the case n=2, and indicated in particular how the Gaussian curv-
ature and the parameter Ji(p could be found. The general plan of
investigation was taken up in the sixteenth volume of the Acta
Mathematica by Zorowski, who studied the case n = 2 in detail, adding
the complete computation of the Gaussian curvature and the most
important differential parameters.

An extension of Lie's methods to the general case of n variables
as far as the actual determination of invariants is concerned has,
so far as I know, not yet been made; only the problem of deter-
mining the number of functionally independent invariants of a given
order has been taken up. It seems that Lie's method is especially
well adapted to this particular problem. In a paper in the Atti del
Reale Instituto Veneto (1897), Levi-Civitta found a lower limit for the
number of invariants of a given order. The actual number was
determined by Haskins in the Transactions of the American Mathe-
matical Society, vol, iii, for the case of invariants proper (including
also simultaneous invariants) and in vol. v, of differential parameters.

I am at the end of my paper. I have attempted to show, in a
compendious way, what has been done in this attractive field of
reseai'ch which is so closely connected with various interesting parts

530 ALGEBRA AND ANALYSIS

of pure and applied mathematics. The number of problems that re-
main to be solved are numerous. Excepting the lowest cases as to
the number of variables and the order of the invariants, not much
more than the mere existence of the invariants is known, so that
we have hardly the right to speak of a theory of these invariants.

When it comes to the question which of the different methods
will be best adapted to a further systematical study of the subject,
it seems probable that a combination of two or more of them will
be the most promising one. But here, as always, it is the man, not
the method, that solves the problem,

SHORT PAPERS

The Section of Algebra and Analysis attracted wide interest and caused many
supplementary papers on various topics to be submitted. It is impossible to give
a resume of these, as their analytical nature demands that they be printed in full
or not at all.

The first paper was presented by Professor G. A. Miller, of Leland Stanford Jr.
University, on the " Bearing of Several Recent Theorems on Group Theory."

The second paper was read by Professor James Birney Shaw, of MiUiken
University, on " Linear Associative Algebra."

The third paper was presented by Professor M. W. Haskell, of the University
of CaUfornia, on "The Reduction of any Collineation to a Product of Perspect-
ive CoUineations."

The fourth paper was presented by Professor M. B. Porter, of the University
of Texas, " On Functions defined by an Infinite Series of Analytic Functions
of a Complex Variable."

The fifth paper was presented by Professor Edward V. Huntington, of Harvard
University, on "A Set of Postulates for Real Algebra comprising Postulates for
a One Dimensional Continuum and for the Theory of Groups."

The sixth paper was presented by Professor J. I. Hutchinson, of Cornell Uni-
versity, on " Uniformizing of Algebraic Functions."

The seventh paper was read by Professor E. R. Hedrick, of the University of
Missouri, on " Generalization of the Analytic Functions of a Complex Variable."

SECTION B — GEOMETRY

{Hall 9, September 24, 10 a. m.)

Chairman: Professor M. W. Haskell, University of California.
Speakers: M. Jean Gaston Darboux, Perpetual Secretary of the Academy of
Sciences, Paris.
Dr. Edward Kasner, Columbia University.
Secretary: Professor Thomas J. Holgate, Northwestern University.

A STUDY OF THE DEVELOPMENT OF GEOMETRIC

METHODS

BY M. JEAN GASTON DAEBOUX

{Translated from the French by Professor George Bruce Hoisted, Kenyon College)

[Jean Gaston Darboux, Perpetual Secretary Academy of Sciences, Paris; Doyen
Honorary, Professor of Higher Geometry of the Faculty of Sciences, Paris.
b. August 13, 1842, Nimes, France. Dr.Sc, LL.D., University of Cambridge,
University of Christiania, University of Heidelberg, et al. Professor of
Special Mathematics, Lycee Louis le Grand, 1867-73; Master of Confer-
ences in Superior Normal Schools, Paris, 1873-81; Professor Suppliant of
Rational Mechanics and Higher Geometry, The Sorbonne, 1873-81; since
1881, Professor Titulaire of the Faculty of Sciences, and Doyen of the Fac-
ulty of Sciences since 1889; also Professor in Higher Normal School for
Schools of Science ; Member of Bureau des Longitudes; President of the
First General Assembly of the International Association of Academies; and
Honorary Vice-President for France of the Congress of Arts and Science;
Member of Institute of France, Royal Society of London; Academies of
Berlin, St. Petersburg, Rome, Amsterdam, Munich, Stockholm; American
Philosophical Society, et al. Author of many publications and addresses
on Mathematics, and editor of the Bulletin of Science of Mathematics.]

To appreciate the progress geometry has made during the cen-
tury just ended, it is of advantage to cast a rapid glance over the
state of mathematical science at the beginning of the nineteenth
century.

We know that, in the last period of his life, Lagrange, fatigued by
the researches in analysis and mechanics, which assured him, however,
an immortal glory, neglected mathematics for chemistry (which,
according to him, was easy as algebra), for physics, for philosophic
speculations.

This mood of Lagraiige we almost always find at certain moments
of the life of the greatest savants. The new ideas which came to
them in the fecund period of youth and which they introduced into
the common domain have given them all they could have expected;
they have fulfilled their task and feel the need of turning their

536 GEOMETRY

mental activity towards wholly new subjects. This need, as we
recognize, manifested itself with particular force at the epoch of La-
grange. At this moment, in fact, the programme of researches opened
to geometers by the discovery of the infinitesimal calculus appeared
very nearly finished up. Some differential equations more or less
complicated to integrate, some chapters to add to the integral
calculus, and one seemed about to touch the very outmost bounds
of science.

Laplace had achieved the explanation of the system of the world
and laid the foundations of molecular physics. New ways opened
before the experimental sciences and prepared the astonishing
development they received in the course of the century just ended.
Ampere, Poisson, Fourier, and Cauchy himself, the creator of the
theory of imaginaries, were occupied above all in studying the appli-
cation of the analytic methods to molecular physics, and seemed to
believe that outside this new domain, which they hastened to cover,
the outlines of theory and science were finally fixed.

Modern geometry, a glory we must claim for it, came, after the
end of the eighteenth century, to contribute in large measure to the
renewing of all mathematical science, by offering to research a way
new and fertile, and above all in showing us, by brilliant successes,
that general methods are not everything in science, and that even
in thfe simplest subject there is much for an ingenious and inventive
mind to do.

The beautiful geometric demonstrations of Huygens, of Newton,
and of Clairaut were forgotten or neglected. The fine ideas introduced
by Desargues and Pascal had remained without development and
appeared to have fallen on sterile ground.

Carnot, by his Essai sur les transversales and his Geometrie de
position, above all Monge, by the creation of descriptive geometry
and by his beautiful theories on the generation of surfaces, came .to
renew a chain which seemed broken. Thanks to them, the conceptions
of the inventors of analytic geometr}', Descartes and Fermat, retook
alongside the infinitesimal calculus of Leibnitz and Newton the place
they had lost, yet should never have ceased to occupy. With his
geometry, said Lagrange, speaking of Monge, this demon of a man
will make himself immortal.

And, in fact, not only has descriptive geometry made it possible
to coordinate and perfect the procedures employed in all the arts
where precision of form is a condition of success and of excellence for
the work and its products; but it appeared as the graphic translation
of a geometry, general and purely rational, of which numerous and
important researches have demonstrated the happy fertility.

Moreover, beside the Geometrie descriptive we must not forget
to place that other masterpiece, the Application de Vanalyse a la

DEVELOPMENT OF GEOMETRIC METHODS 537

geometrie ; nor should we forget that to Monge are due the notion
of hnes of curvature and the elegant integration of the differential
equation of these lines for the case of the ellipsoid, which, it is said,
Lagrange envied him. To be stressed is this character of unity of the
work of Monge.

The renewer of modern geometry has shown us from the beginning,
what his successors have perhaps forgotten, that the alliance of
geometry and analysis is useful and fruitful, that this alliance is
perhaps for each a condition of success.

In the school of Monge were formed many geometers: Hachette,
Brianchon, Chappuis, Binet, Lancret, Dupin, Malus, Gaultier de
Tours, Poncelet, Chasles, et al. Among these Poncelet takes first
rank. Neglecting, in the works of Monge, everything pertaining to
the analysis of Descartes or concerning infinitesimal geometry, he
devoted himself exclusively to developing the germs contained in
the purely geometric researches of his illustrious predecessor.

Made prisoner by the Russians in 1813 at the passage of the Dnieper
and incarcerated at Saratoff, Poncelet employed the leisure captivity
left him in the demonstration of the principles which he has developed
in the Traite des proprietes projectives des figures, issued in 1822,.
and in the great memoirs on reciprocal polars and on harmonic
means, which go back nearly to the same epoch. So we may say the
modern geometry was born at Saratoff.

Renewing the chain broken since Pascal and Desargues, Poncelet
introduced at the same time homology and reciprocal polars, putting
thus in evidence, from the beginning, the fruitful ideas on which the
science has evolved during fifty years.

Presented in opposition to analytic geometry, the methods of Ponce-
let were not favorably received by the French analysts. But such
were their importance and their novelty, that without delay they
aroused, from divers sides, the most profound researches.

Poncelet had been alone in discovering the principles; on the
contrary, many geometers appeared almost simultaneously to study
them on all sides and to deduce from them the essential results which
they implicitly contained.

At this epoch, Gergonne was brilliantly editing a periodical which
has to-day for the history of geometry an inestimable value. The
Afinales de Mathematiques, published at Nimes from 1810 to 1831^
was during more than fifteen years the only journal in the entire
world devoted exclusively to mathematical researches.

Gergonne, who, in many regards, was a model editor for a scienti-
fic journal, had the defects of his qualities; he collaborated, often

538 GEOMETRY

against their will, with the authors of the memoirs sent him, rewrote
them, and sometimes made them say more or less than they would
have wished. Be that as it may, he was greatly struck by the origin-
ality and range of Poncelet's discoveries.

In geometry some simple methods of transformation of figures
were already known; homology even had been employed in the plane,
but without extending it to space, as did Poncelet, and especially
without recognizing its power and fruitfulness. Moreover, all these
transformations were punctual ; that is to say, they made correspond
a point to a point.

In introducing polar reciprocals, Poncelet was in the highest
degree creative, because he gave the first example of a transformation
in which to a point corresponded something other than a point.

Every method of transformation enables us to multiply the num-
ber of theorems, but that of polar reciprocals had the advantage of
making correspond to a proposition another proposition of wholly
different aspect. This was a fact essentially new. To put it in evi-
dence, Gergonne invented the system, which since has had so much
success, of memoirs printed in double columns with correlative
propositions in juxtaposition; and he had the idea of substituting
for Poncelet's demonstrations, which required an intermediary
curve or surface of the second degree, the famous "principle of
duality," of which the signification, a little vague at first, was suffi-
ciently cleared up by the discussions which took place on this subject
between Gergonne, Poncelet, and Pluecker.

Bobillier, Chasles, Steiner, Lame, Sturm, and many others whose
names escape me, were, at the same time as Pluecker and Poncelet,
assiduous collaborators of the Annales de Mathematiques. Gergonne,
having become rector of the Academy of Montpellier, was forced to
suspend in 1831 the publication of his journal. But the success it had
obtained, the taste for research it had contributed to develop, had
commenced to bear their fruit. Quetelet had established in Belgium
the Correspondance mathematique et physique. Crelle, from 1826,
brought out at Berlin the first sheets of his celebrated journal, where
he published the memoirs of Abel, of Jacobi, of Steiner.

A great number of separate works began also to appear, wherein
the principles of modern geometry were powerfully expounded and
developed.

First came in 1827 the Barycentrische Calcul of Moebius, a work
truly original, remarkable for the profundity of its conceptions, the
elegance and the rigor of its exposition; then in 1828 the Analytisch-
geometrische Entioickelungen of Pluecker, of which the second part
appeared in 1831, and which was soon followed by the System der
analytischen Geometrie of the same author, published at Berlin in
1835.

DEVELOPMENT OF GEOMETRIC METHODS 539

In 1832 Steiner brought out at Berlin his great work: Systemat-
ische Entmckelung der Ahhaengigkeit der geometrischen Gestalten von
einander, and, the following year, Die geometrischen Konstruktionen
ausgefuehrt mittels der geraden Linie und eines festen Kreises, where
was confirmed by the most elegant examples a proposition of Pon-
celet's relative to the employment of a single circle for the geometric
constructions.

Finally, in 1830, Chasles sent to the Academy of Brussels, which
happily inspired had offered a prize for a study of the principles of
modern geometry, his celebrated Apercu historique sur Vorigine et
le developpement des methodes en geometrie, followed by Memoirs
sur deux principes generaux de la science : la dualite et Vhomographie,
which was published only in 1837.

Time would fail us to give a worthy appreciation of these beautiful
works and to apportion the share of each. Moreover, to what would
such a study conduct us, but to a new verification of the general laws
of the development of science ? When the times are ripe, when the
fundamental principles have been recognized and enunciated, nothing
stops the march of ideas ; the same discoveries, or discoveries almost
equivalent, appear at nearly the same instant, and in places the most
diverse. Without undertaking a discussion of this sort, which, besides,
might appear useless or become irritating, it is, however, of import-
ance to bring out a fundamental difference between the tendencies
of the great geometers, who, about 1830, gave to geometry a scope
before unknown.

Ill

Some, like Chasles and Steiner, who consecrated their entire lives
to research in pure geometry, opposed what they called synthesis to
analysis, and, adopting in the ensemble if not in detail the tendencies
of Poncelet, proposed to constitute an independent doctrine, rival of
Descartes's analysis.

Poncelet could not content himself with the insufficient resources
furnished by the method of projections; to attain imaginaries he
created that famous principle of continuity which gave birth to such
long discussions between him and Cauchy.

Suitably enunciated, this principle is excellent and can render
great service. Poncelet was wrong in refusing to present it as a simple
consequence of analysis; and Cauchy, on the other hand, was not
willing to recognize that his own objections, applicable without
doubt to certain transcendent figures, were without force in the
applications made by the author of the Traits des proprietes pro-
jectives.

Whatever be the opinion of such a discussion, it showed at least
in the clearest manner that the geometric system of Poncelet rested

540 GEOMETRY

on an analytic foundation, and besides we know, by the untoward
publication of the manuscripts of Saratoff, that by the aid of
Descartes 's analysis were established the principles which serve as
foundation for the Traite des proprietes projectives.

Younger than Poncelet, who besides abandoned geometry for
mechanics where his works had a preponderant influence, Chasles,
for whom was created in 1847 a chair of Geometrie superieure in the
Faculty of Science of Paris, endeavored to constitute a geometric
doctrine entirely independent and autonomous. He has expounded
it in two works of high importance, the Traite de geometrie supe-
rieure, which dates from 1852, and the Traite des sections coniques,
unhappily unfinished and of which the first part alone appeared in
1865.

In the preface of the first of these works he indicates very clearly
the three fundamental points which permit the new doctrine to share
the advantages of analysis and which to him appear to mark an
advance in the cultivation of the science. These are: (1) The intro-
duction of the principle of signs, which simplifies at once the enuncia-
tions and the demonstrations, and gives to Carnot's analysis of trans-
versals all the scope of which it is susceptible; (2) the introduction of
imaginaries, which supplies the place of the principle of continuity
and furnishes demonstrations as general as those of analytic geo-
metry; (3) the simultaneous demonstration of propositions which are,
correlative, that is to say, which correspond in virtue of the principle
of duality.

Chasles studies indeed in his work homography and correlation;
but he avoids systematically in his exposition the employment of
transformations of figures, which, he thinks, cannot take the place of
direct demonstrations since they mask the origin and the true nature
of the properties obtained by their means.

There is truth in this judgment, but the advance itself of the science
permits us to declare it too severe. If it happens often that, em-
ployed without discernment, transformations multiply uselessly the
number of theorems, it must be recognized that they often aid us to
better understand the nature of the propositions even to which they
have been applied. Is it not the employment of Poncelet's projection
which has led to the so fruitful distinction between projective proper-
ties and metric properties, which has taught us also the high import-
ance of that cross-ratio whose essential property is found already
in Pappus, and of which the fundamental role has begun to appear
after fifteen centuries only in the researches of modern geometry?

The introduction of the principle of signs was not so new as Chasles
supposed at the time he wrote his Traite de Geometrie superieure.

Moebius, in his Barycentrische Calcul, had already given issue to
a desideratum of Carnot, and employed the signs in a way the largest

DEVELOPMENT OF GEOMETRIC METHODS 541

and most precise, defining for the first time the sign of a segment
and even that of an area.

Later he succeeded in extending the use of signs to lengths not
laid off on the same straight line and to angles not formed about the
same point.

Besides Grassmann, whose mind has so much analogy to that of
Moebius, had necessarily employed the principle of signs in the defini-
tions which serve as basis for his methods, so original, of studying
the properties of space.

The second characteristic which Chasles assigns to his system of
geometry is the employment of imaginaries. Here, his method was
really new, and he illustrates it by examples of high interest. One will
always admire the beautiful theories he has left us on homofocal
surfaces of the second degree, where all the known properties and
others new, as varied as elegant, flow from the general principle that
they are inscribed in the same developable circumscribed to the
circle at infinity.

But Chasles introduced imaginaries only by their symmetric func-
tions, and consequently would not have been able to define the cross-
ratio of four elements when these ceased to be real in whole or in
part. If Chasles had been able to establish the notion of the cross-
ratio of imaginary elements, a formula he gives in the Geometrie
superieure (p. 118 of the new edition) would have immediately
furnished him that beautiful definition of angle as logarithm of a
cross-ratio which enabled Laguerre, our regretted confrere, to give
the complete solution, sought so long, of the problem of the trans-
formation of relations which contain at the same time angles and
segments in homography and correlation.

Like Chasles, Steiner, the great and profound geometer, followed
the way of pure geometry; but he has neglected to give us a complete
exposition of the methods upon which he depended. However, they
may be characterized by saying that they rest upon the introduction
of those elementary geometric forms which Desargues had already
considered, on the development he was able to give to Bobillier's
theory of polars, and finally on the construction of curves and sur-
faces of higher degrees by the aid of sheaves or nets of curves of
lower orders. In default of recent researches, analysis would suffice
to show that the field thus embraced has just the extent of that into
which the analysis of Descartes introduces us without effort.

While Chasles, Steiner, and, later, as we shall see, von Staudt, were
intent on constituting a rival doctrine to analysis and set in some
sort altar against altar, Gergonne, Bobillier, Sturm, and above all
Pluecker, perfected the geometry of Descartes and constituted an

542 GEOMETRY

analytic system in a manner adequate to the discoveries of the
geometers. It is to Bobillier and to Pluecker that we owe the method
called abridged notation. Bobillier consecrated to it some pages truly
new in the last volumes of the Annates of Gergonne.

Pluecker commenced to develop it in his first work, soon followed
by a series of works where are established in a fully conscious manner
the foundations of the modern analytic geometry. It is to him that
we owe tangential coordinates, trilinear coordinates, employed with
homogeneous equations, and finally the employment of canonical
forms whose validity was recognized by the method, so deceptive
sometimes, but so fruitful, called the enumeration of constants.

All these happy acquisitions infused new blood into Descartes's
analysis and put it in condition to give their full signification to the
conceptions of which the geometry called synthetic had been unable
to make itself completely mistress.

Pluecker, to whom it is without doubt just to adjoin Bobillier,
carried off by a premature death, should be regarded as the veritable
initiator of those methods of modern analysis where the employment
of homogeneous coordinates permits treating simultaneously and,
so to say, without the reader perceiving it, together with one figure
all those deducible from it by homography and correlation.

Parting from this moment, a period opens brilliant for geometric
researches of every nature.

The analysts interpret all their results and are occupied in trans-
lating them by constructions.

The geometers are intent on discovering in every question some
general principle, usually undemonstrable without the aid of ana-
lysis, in order to make flow from it without effort a crowd of particu-
lar consequences, solidly bound to one another and to the principle
whence they are derived. Otto Hesse, brilliant disciple of Jacobi,
develops in an admirable manner that method of homogeneous
coordinates to which Pluecker perhaps had not attached its full
value. Boole discovers in the polars of Bobillier the first notion of
a CO variant ; the theory of forms is created by the labors of Cayley ,
Sylvester, Hermite, Brioschi. Later Aronhold, Clebsch and Gordan,
and other geometers still living, gave to it its final notation, estab-
lished the fundamental theorem relative to the limitation of the
number of covariant forms and so gave it all its amplitude.

The theory of surfaces of the second order, built up principally
by the school of Monge, was enriched by a multitude of elegant
properties, established principally by O. Hesse, who found later in
Paul Serret a worthy emulator and continuer.

DEVELOPMENT OF GEOMETRIC METHODS 543

The properties of the polars of algebraic curves are developed by
Pluecker and above all by Steiner. The study, already old, of curves
of the third order is rejuvenated and enriched by a crowd of new
elements. Steiner, the first, studies by pure geometry the double
tangents of curves of the fourth order, and Hesse, after him, appUes
the methods of algebra to this beautiful question, as well as to that
of points of inflection of curves of the third order.

The notion of class introduced by Gergonne, the study of a para-
dox in part elucidated by Poncelet and relative to the respective
degrees of two curves reciprocal polars one of the other, give birth
to the researches of Pluecker relative to the singularities called ordi-
nary of algebraic plane curves. The celebrated formulas to which
Pluecker is thus conducted are later extended by Cayley and by
other geometers to algebraic skew curves, by Cayley again and by
Salmon to algebraic surfaces.

The singularities of higher order are in their turn taken up by
the geometers; contrary to an opinion then very widespread, Hal-
phen demonstrates that each of these singularities cannot be con-
sidered as equivalent to a certain group of ordinary singularities, and
his researches close for a time this difficult and important question.

Analysis and geometry, Steiner, Cayley, Salmon, Cremona, meet in
the study of surfaces of the third order, and, in conformity with
the anticipations of Steiner, this theory becomes as simple and as
easy as that of surfaces of the second order.

The algebraic ruled surfaces, so important for applications, are
studied by Chasles, by Cayley, of whom we find the influence and the
mark in all mathematical researches, by Cremona, Salmon, La Gour-
nerie; so they will be later by Pluecker in a work to which we must
return.

The study of the general surface of the fourth order would seem
to be still too difficult; but that of the particular surfaces of this order
with multiple points or multiple lines is commenced, by Pluecker for
the surface of waves, by Steiner, Kummer, Cayley, Moutard, Laguerre,
Cremona, and many other investigators.

As for the theory of algebraic skew curves, grown rich in its ele-
mentary parts, it receives finally, by the labors of Halphen and of
Noether, whom it is impossible for us here to separate, the most
notable extensions.

A new theory with a great future is bom by the labors of Chasles,
of Clebsch, and of Cremona; it concerns the study of all the algebraic
curves which can be traced on a determined surface.

Homography and correlation, those two methods of transformation
which have been the distant origin of all the preceding researches,
receive from them in their turn an unexpected extension; they are
not the only methods which make a single element correspond to a

544 GEOMETRY

single element, as might have shown a particular transformation
briefly indicated by Poncelet in the Traite des proprietes projectives.

Pluecker defines the transformation by reciprocal radii vectores or
inversion, of which Sir W. Thomson and Liouville hasten to show all
the importance, as well for mathematical physics as for geometry.

A contemporary of Moebius and Pluecker, Magnus believed he had
found the most general transformation which makes a point corre-
spond to a point, but the researches of Cremona show us that the
transformation of Magnus is only the first term of a series of bira-
tional transformations which the great Italian geometer teaches us to
determine methodically, at least for the figures of plane geometry.

The Cremona transformations long retained a great interest,
though later researches have shown us that they reduce always to
a series of successive applications of the transformation of Magnus.

All the works we have enumerated, others to which we shall return
later, find their origin and, in some sort, their first motive in the con-
ceptions of modern geometry; but the moment has come to indicate
rapidly another source of great advances for geometric studies.
Legendre's theory of elliptic functions, too much neglected by the
French geometers, is developed and extended by Abel and Jacobi.
With these great geometers, soon followed by Riemann and Weier-
strass, the theory of Abelian functions which, later, algebra would
try to follow solely with its own resources, brought to the geometry
of curves and surfaces a contribution whose importance will continue
to grow.

Already, Jacobi had employed the analysis of elliptic functions
in the demonstration of Poncelet 's celebrated theorems on inscribed
and circumscribed polygons, inaugurating thus a chapter since en-
riched by a multitude of elegant results; he had obtained also, by
methods pertaining to geometry, the integration of Abelian equa-
tions.

But it was Clebsch who first showed in a long series of works all
the importance of the notion of deficiency (Geschlecht, genre) of a
curve, due to Abel and Riemann, in developing a crowd of results
and elegant solutions that the employment of Abelian integrals would
seem, so simple was it, to connect with their veritable point of
departure.

The study of points of inflection of curves of the third order, that
of double tangents of curves of the fourth order, and, in general, the
theory of osculation on Avhich the ancients and the moderns had so
often practiced, were connected with the beautiful problem of the
division of elliptic functions and Abelian functions.

In one of his memoirs, Clebsch had studied the curves which are

DEVELOPMENT OF GEOMETRIC METHODS 545

rational or of deficiency zero; this led him, toward the end of his
too short life, to envisage what may be called also rational surfaces,
those which can be simply represented by a plane. This was a vast
field for research, opened already for the elementary cases by Chasles,
and in which Clebsch was followed by Cremona and many other
savants. It was on this occasion that Cremona, generalizing his re-
searches on plane geometry, made known not indeed the totality of
birational transformations of space, but certain of the most interest-
ing among these transformations.

The extension of the notion of deficiency to algebraic surfaces is
already commenced; already also works of high value have shown
that the theory of integrals, simple or multiple, of algebraic differ-
entials will find, in the study of surfaces as in that of curves, an ample
field of important applications; but it is not proper for the reporter
on geometry to dilate on this subject .

VII

While thus were constituted the mixed methods whose principal
applications we have just indicated, the pure geometers were not
inactive. Poinsot, the creator of the theory of couples, developed,
bj"- a method purely geometric, "that, where one never for a mo-
ment loses from view the object of the research," the theory of the
rotation of a solid body that the researches of d'Alembert, Euler, and
Lagrange seemed to have exhausted; Chasles made a precious con-
tribution to kinematic by his beautiful theorems on the displacement
of a solid body, which have since been extended by other elegant
methods to the case where the motion has divers degrees of freedom.
He made known those beautiful propositions on attraction in gen-
eral, which figure without disadvantage beside those of Green and
Gauss. Chasles and Steiner met in the study of the attraction of
ellipsoids and showed thus once more that geometry has its desig-
nated place in the highest questions of the integral calculus.

Steiner did not disdain at the same time to occupy himself with
the elementary parts of geometry. His researches on the contacts of
circles and conies, on isoperimetric problems, on parallel surfaces, on
the centre of gravity of curvature, excited the admiration of all by
their simplicity and their depth.

Chasles introduced his principle of correspondence between two
variable objects which has given birth to so many applications; but
here analysis retook its. place to study the principle in its essence,
make it precise and generalize it.

It was the same concerning the famous theory of characteristics
and the numerous researches of de Jonquieres, Chasles, Cremona,
and still others, which gave the foundations of a new branch of the
science, Enumerative Geometry.

546 GEOMETRY

During many years, the celebrated postulate of Chasles was ad-
mitted without any objection: a crowd of geometers believed they
had established it in a manner irrefutable.

But, as Zeuthen then said, it is very difficult to recognize whether,
in demonstrations of this sort, there does not exist always some weak
point that their author has not perceived; and, in fact, Halphen,
after fruitless efforts, crowned finally all these researches by clearly
indicating in what cases the postulate of Chasles may be admitted
and in what cases it must be rejected.

VIII

Such are the principal works which restored geometric synthesis
to honor and assured to it, in the course of the last century, the place
belonging to it in mathematical research. Numerous and illustrious
workers took part in this great geometric movement, but we must
recognize that its chiefs and leaders were Chasles and Steiner. So
brilliant were their marvelous discoveries that they threw into the
shade, at least momentarily, the publications of other modest geo-
meters, less preoccupied perhaps in finding brilliant applications,
fitted to evoke love for geometry than to establish this science itself
on an absolutely solid foundation. Their works have received per-
haps a recompense more tardy, but their influence grows each day;
it will assuredly increase still more. To pass them over in silence
would be without doubt to neglect one of the principal factors which
will enter into future researches. We allude at this moment above
all to von Staudt. His geometric works were published in two books
of great interest: the Geometrie der Lage, issued in 1847, and the
Beitrage zur Geometrie der Lage, published in 1856, that is to say,
four years after the Geometrie swperieure. Chasles, as we have seen,
had devoted himself to constituting a body of doctrine independent
of Descartes's analysis and had not completely succeeded. We have
already indicated one of the criticisms that can be made upon this
system: the imaginary elements are there defined only by their sym-
metric functions, which necessarily exclude them from a multitude
of researches. On the other hand, the constant employment of cross-
ratio, of transversals, and of involution, which requires frequent
analytic transformations, gives to the Geometrie swperieure a char-
acter almost exclusively metric which removes it notably from the
methods of Poncelet. Returning to these methods, von Staudt
devoted himself to constituting a geometry freed from all metric
relation and resting exclusively on relations of situation.

This is the spirit in which was conceived his first work, the Geo-
metrie der Lage of 1847. The author there takes as point of departure
the harmonic properties of the complete quadrilateral and those
of homologic triangles, demonstrated uniquely by considerations

DEVELOPMENT OF GEOMETRIC METHODS 547

of geometry of three dimensions, analogous to those of which the
school of Monge made such frequent use.

In this first part of his work, von Staudt neglected entirely im-
aginary elements. It is only in the Beitrage, his second work, that
he succeeds, by a very original extension of the method of Chasles,
in defining geometrically an isolated imaginary element and dis-
tinguishing it from its conjugate.

This extension, although rigorous, is difficult and very abstract.
It may be defined in substance as follows: Two conjugate imaginary
points may always be considered as the double points of an involu-
tion on a real straight; and just as one passes from an imaginary to
its conjugate by changing i into — i, so one may distinguish the two
imaginary points by making correspond to each of them one of the
two different senses which may be attributed to the straight. In this
there is something a little artificial; the development of the theory
erected on such foundations is necessarily complicated. By methods
purely projective, von Staudt establishes a calculus of cross-ratios of
the most general imaginary elements. Like all geometry, the pro-
jective geometry employs the notion of order and order engenders
number; we are not astonished therefore that von Staudt has been
able to constitute his calculus; but we must admire the ingenuity
displayed in attaining it. In spite of the efforts of distinguished
geometers who have essayed to simplify its exposition, we fear that
this part of the geometry of von Staudt, like the geometry otherwise
so interesting of the profound thinker Grassmann, cannot prevail
against the analytical methods which have won to-day favor almost
universal. Life is short; geometers know and also practice the
principle of least action. Despite these fears, which should discour-
age no one, it seems to us that under the first form given it by von
Staudt, projective geometry must become the necessary companion
of descriptive geometry, that it is called to renovate this geometry
in its spirit, its procedures, and its applications.

This has already been comprehended in many countries, and
notably in Italy, where the great geometer Cremona did not disdain
to write for the schools an elementary treatise on projective geometry.

In the preceding articles, we have essayed to follow and bring out
clearly the most remote consequences of the methods of Monge and
Poncelet. In creating tangential coordinates and homogeneous coor-
dinates, Pluecker seemed to have exhausted all that the method of
projections and that of reciprocal polars give to analysis.

It remained for him, toward the end of his life, to return to his
first researches to give them an extension enlarging to an unexpected
degree the domain of geometry.

548 GEOMETRY

Preceded by innumerable researches on systems of straight lines,
due to Poinsot, Moebius, Chasles, Dupin, Malus. Hamilton, Krummer,
Transon, above all to Cayley, who first introduced the notion of the
coordinates of the straight, researches originating perhaps in statics
and kinematics, perhaps in geometrical optics, Pluecker's geometry of
the straight line will always be regarded as the part of his work Avhere
are met the newest and most interesting ideas.

Pluecker first set up a methodic study of the straight line, which
already is important, but that is nothing beside what he discov
ered. It is sometimes said that the principle of duality shows that
the plane as well as the point may be considered as a space element.
That is true; but in adding the straight line to the plane and point
as possible space element, Pluecker was led to recognize that any
curve, any surface, may also be considered as space element, and so
was born a new geometry which already has inspired a great number
of works, which will raise up still more in the future.

A beautiful discovery, of which we shall speak further on, has
already connected the geometry of spheres with that of straight lines
and permits the introduction of the notion of coordinates of a sphere.

The theory of systems of circles is already commenced; it will
be developed without doubt when one wishes to study the representa-
tion, which we owe to Laguerre, of an imaginary point in space by an
oriented circle.

But before expounding the development of these new ideas which
have vivified the infinitesimal methods of Monge, it is necessary to go
back to take up the history of branches of geometry that we have
neglected until now.

Among the works of the school of Monge, we have hitherto con-
fined ourselves to the consideration of those connected with -finite
geometry; but certain of the disciples of Monge devoted themselves
above all to developing the new notions of infinitesimal geometry
applied by their master to curves of double curvature, to lines of curv-
ature, to the generation of surfaces, notions expounded at least in
part in the Application de V Analyse a la Geometrie. Among these
we must cite Lancret, author of beautiful works on skew curves, and
above all Charles Dupin, the only one perhaps who followed all the
paths opened by Monge.

Among other works, we owe to Dupin two volumes Monge would
not have hesitated to sign: Les Developpements de Geometrie pure,
issued in 1813, and Les Applications de Geometrie et de Mecanique,
dating from 1822.

There we find the notion of indicatrix, which was to renovate,
after Euler and Meunier, all the theory of curvature, that of conjugate

DEVELOPMENT OF GEOMETRIC METHODS 549

tangents, of asymptotic lines which have taken so important a place
in recent researches. Nor should we forget the determination of the
surface of which all the lines of curvature are circles, nor above all
the memoir on triple systems of orthogonal surfaces where is found,
together with the discovery of the triple system formed by surfaces
of the second degree, the celebrated theorem to which the name of
Dupin wiU remain attached.

Under the influence of these works and of the renaissance of syn-
thetic methods, the geometry of infinitesimals retook in all researches
the place Lagrange had wished to take away from it forever.

Singular thing, the geometric methods thus restored were to receive
the most vivid impulse in consequence of the publication of a memoir
which, at least at first blush, would appear connected with the purest
analysis; we mean the celebrated paper of Gauss, Disquisitiones
generates circa superficies curvas, which was presented in 1827 to the
Gottingen Society, and whose appearance marked, one may say,
a decisive date in the history of infinitesimal geometry.

From this moment, the infinitesimal method took in France a free
scope before unknown.

Frenet, Bertrand, Molins, J. A. Serret, Bouquet, Puiseux, Ossian
Bonnet, Paul Serret, develop the theory of skew curves. Liouville,
Chasles, Minding, join them to pursue the methodic study of the
memoir of Gauss.

The integration made by Jacobi of the differential equation of the
geodesic lines of the ellipsoid started a great number of researches.
At the same time the problems studied in the Application de V Analyse
of Monge were greatly developed.

The determination of all the surfaces having their lines of curvature
plane or spheric completed in the happiest manner certain partial
results already obtained by Monge.

At this moment, one of the most penetrating of geometers, ac-
cording to the judgment of Jacobi, Gabriel Lame, who, like Charles
Sturm, had commenced with pure geometry and had already made to
this science contributions the most interesting by a little book pub-
lished in 1817 and by memoirs inserted in the Annates of Gergonne,
utilized the results obtained by Dupin and Binet on the system of
confocal surfaces of the second degree, and, rising to the idea of
curvilinear coordinates in space, became the creator of a wholly new
theory destined to receive in mathematical physics the most varied
applications.

Here again, in this infinitesimal branch of geometry are found the
two tendencies we have pointed out a propos of the geometry of finite
quantities.

550 GEOMETRY

Some, among whom must be placed J. Bertrand and O. Bonnet,
wish to constitute an independent method resting directly on the
employment of infinitesimals. The grand Traite de Calcul differ entiel,
of Bertrand, contains many chapters on the theory of curves and
of surfaces, which are, in some sort, the illustration of this con-
ception.

Others follow the usual analytic ways, being only intent to clearly
recognize and put in evidence the elements which figure in the first
plan. Thus did Lame in introducing his theory of differential 'para-
meters. Thus did Beltrami in extending with great ingenuity the
employment of these differential invariants to the case of two inde-
pendent variables, that is to say, to the study of surfaces.

It seems that to-day is accepted a mixed method whose origin is
found in the works of Ribaucour, under the name perimorphie. The
rectangular axes of analytic geometry are retained, but made mobile
and attached as seems best to the system to be studied. Thus dis-
appear most of the objections which have been made to the method
of coordinates. The advantages of what is sometimes called intrinsic
geometry are united to those resulting from the use of the regular
analysis. Besides, this analysis is by no means abandoned; the com-
plications of calculation which it almost always carries with it, in its
applications to the study of surfaces and rectilinear coordinates, usu-
ally disappear if one employs the notion on the invariants and the
covariants of quadratic powers of differentials which we owe to the
researches of Lipschitz and Christoffel, inspired by Riemann's studies
on the non-Euclidean geometry.

XII

The results of so many labors were not long in coming. The notion
of geodesic curvature which Gauss already possessed, but without
having published it, was given by Bonnet and Liouville; the theory
of surfaces of which the radii of curvature are functions one of the
other, inaugurated in Germany by two propositions which would
figure without disadvantage in the memoir of Gauss, was enriched
by Ribaucour, Halphen, S. Lie, and others, with a multitude of propo-
sitions, some concerning these surfaces envisaged in a general man-
ner; others applying to particular cases where the relation between
the radii of curvature takes a form particularly simple; to minimal
surfaces for example, and also to surfaces of constant curvature,
positive or negative.

The minimal surfaces were the object of works which make of
their study the most attractive chapter of infinitesimal geometry.
The integration of their partial differential equation constitutes one
of the most beautiful discoveries of Monge; but because of the im-
perfection of the theory of imaginaries, the great geometer could not

DEVELOPMENT OF GEOMETRIC METHODS 551

get from its formulas any mode of generation of these surfaces, nor
even any particular surface. We will not here retrace the detailed
history which we have presented in our Lecons sur la theorie des
surfaces ; but it is proper to recall the fundamental researches of
Bonnet which have given us, in particular, the notion of surfaces
associated with a given surface, the formulas of Weierstrass which
establish a close bond between the minimal surfaces and the functions
of a complex variable, the researches of Lie by which it was estab-
lished that just the formulas of Monge can to-day serve as founda-
tion for a fruitful study of minimal surfaces.

In seeking to determine the minimal surfaces of smallest classes
or degrees, we were led to the notion of double minimal surfaces
which is dependent on analysis situs.

Three problems of unequal importance have been studied in this
theory.

The first, relative to the determination of minimal surfaces in-
scribed along a given contour in a developable equally given, was
solved by celebrated formulas which have led to a great number of
propositions. For example, every straight traced on such a surface
is an axis of symmetry.

The second, set by S. Lie, concerns the determination of all the
algebraic minimal surfaces inscribed in an algebraic developable,
without the curve of contact being given. It also has been entirely
elucidated.

The third and the most difficult is what the physicists solve experi-
mentally, by plunging a closed contour into a solution of glycerine.
It concerns the determination of the minimal surface passing through
a given contour.

The solution of this problem evidently surpasses the resources of
geometry. Thanks to the resources of the highest analysis, it has
been solved for particular contours in the celebrated memoir of
Riemann and in the profound researches which have followed or
accompanied this memoir.

For the most general contour, its study has been brilliantly begun;
it will be continued by our successors.

After the minimal surfaces, the surfaces of constant curvature at-
tracted the attention of geometers. An ingenious remark of Bonnet
connects with each other the surfaces of which one or the other of the
two curvatures, mean curvature or total curvature, is constant.

Bour announced that the partial differential equation of surfaces
of constant curvature could be completely integrated. This result
has not been secured; it would seem even very doubtful if we con-
sider a research where S. Lie has essayed in vain to apply a general
method of integration of partial differential equations to the particu-
lar equation of surfaces of constant curvature.

552 GEOMETRY

But, if it is impossible to determine in finite terms all these sur-
faces, it has at least been possible to obtain certain of them, char-
acterized by special properties, such as that of having their lines of
curvature plane or spheric; and it has been shown, by employing a
method which succeeds in many other problems, that from every sur-
face of constant curvature may be derived an infinity of other surfaces
of the same nature, by employing operations clearly defined which
require only quadratures.

The theory of the deformation of surfaces in the sense of Gauss
has been also much enriched. We owe to Minding and to Bour the
detailed study of that special deformation of ruled surfaces which
leaves the generators rectilineal. If we have not been able, as has
been said, to determine the surfaces applicable on the sphere, other
surfaces of the second degree have been attacked with more success,
and, in particular, the paraboloid of revolution.

The systematic study of the deformation of general surfaces of the
second degree is already entered upon; it is one of those which will
give shortly the most important results.

The theory of infinitesimal deformation constitutes to-day one of
the most finished chapters of geometry. It is the first somewhat
extended application of a general method which seems to have a great
future.

Being given a system of differential or partial differential equations,
suitable to determine a certain number of unknowns, it is advantage-
ous to associate with it a system of equations which we have called
auxiliary system, and which determines the systems of solutions
infinitely near any given system of solutions. The auxiliary system
being necessarily linear, its employment in all researches gives
precious light on the properties of the proposed system and on the
possibility of obtaining its integration.

The theory of lines of curvature and of asymptotic lines has been
notably extended. Not only have been determined these two series
of lines for particular surfaces such as the tetrahedral surfaces of
Lame; but also, in developing Moutard's results relative to a par-
ticular class of linear partial differential equations of the second
order, it proved possible to generalize all that had been obtained for
surfaces with lines of curvature plane or spheric, in determining com-
pletely all the classes of surfaces for which could be solved the pro-
blem of spheric representation.

Just so has been solved the correlative problem relative to asymp-
totic lines in making known all the surfaces of which the infinitesimal
deformation can be determined in finite terms. Here is a vast field
for research whose exploration is scarcely begun.

The infinitesimal study of rectilinear congruences, already com-
menced long ago by Dupin, Bertrand, Hamilton, Kummer, has come

DEVELOPMENT OF GEOMETRIC METHODS 553

to intermingle in all these researches. Ribaucour, who has taken in
it a preponderant part, studied particular classes of rectilinear con-
gruences and, in particular, the congruences called isotropes, which
intervene in the happiest way in the study of minimal surfaces.

The triply orthogonal systems which Lame used in mathematical
physics have become the object of systematic researches. Cayley
was the first to form the partial differential equation of the third
order on which the general solution of this problem was made to
depend.

The system of homofocal surfaces of the second degree has been
generalized and has given birth to that theory of general cyclides in
which may be employed at the same time the resources of metric
geometry, of projective geometry, and of infinitesimal geometry.
Many other orthogonal systems have been made known. Among
these it is proper to signalize the cyclic systems of Ribaucour, for
which one of the three families admits circles as orthogonal trajecto-
ries and the more general systems for which these orthogonal trajec-
tories are simply plane curves.

The systematic employment of imaginaries, which we must be
careful not to exclude from geometry, has permitted the connection
of all these determinations with the study of the finite deformation
of a particular surface.

Among the methods which have permitted the establishment of
all these results, it is proper to note the systematic employment of
linear partial differential equations of the second order and of systems
formed of such equations. The most recent researches show that this
employment is destined to renovate most of the theories.

Infinitesimal geometry could not neglect the study of the two
fundamental problems set it by the calculus of variations.

The problem of the shortest path on a surface was the object of
masterly studies by Jacobi and by Ossian Bonnet. The study of
geodesic lines has been followed up; we have learned to determine
them for new surfaces. The theory of ensembles has come to permit
the following of these lines in their course on a given surface.

The solution of a problem relative to the representation of two
surfaces one on the other has greatly increased the interest of dis-
coveries of Jacobi and of Liouville relative to a particular class of
surfaces of which the geodesic lines could be determined. The results
concerning this particular case led to the examination of a new ques-
tion: to investigate all the problems of the calculus of variations of
which the solution is given by curves satisfying a given differential
equation.

Finally, the methods of Jacobi have been extended to space of
three dimensions and applied to the solution of a question which
presented the greatest difficulties: the study of properties of mini-

554 GEOMETRY

mum appertaining to the minimal surface passing through a given

contour.

XIII

Among the inventors who have contributed to the development of
infinitesimal geometry, Sophus Lie distinguishes himself by many
capital discoveries which place him in the first rank.

He was not one of those who show from infancy the most char-
acteristic aptitudes, and at the moment of quitting the University of
Christiania in 1865, he still hesitated between philology and mathe-
matics.

It was the works of Pluecker which gave him for the first time
full consciousness of his true calling.

He published in 1869 a first work on the interpretation of imagin-
aries in geometry, and from 1870 he was in possession of the directing
ideas of his whole career. I had at this time the pleasure of seeing
him often, of entertaining him at Paris, where he had come with his
friend F. Klein.

A -course by M. Sylow followed by Lie had revealed to him all the
importance of the theory of substitutions; the two friends studied
this theory in the great treatise of C. Jordan; they were fully con-
scious of the important role it was called on to play in so many
branches of mathematical science where it had not yet been applied.

They have both had the good fortune to contribute by their works
to impress upon mathematical studies the direction which to them
appeared the best.

In 1870, Sophus Lie presented to the Academy of Sciences of Paris
a discovery extremely interesting. Nothing bears less resemblance
to a sphere than a straight line, and yet Lie had imagined a singular
transformation which made a sphere correspond to a straight line,
and permitted, consequently, the connecting of every proposition
relative to straight lines with a proposition relating to spheres, and
vice versa.

In this so curious method of transformation, each property relative
to the lines of curvature of a surface furnishes a proposition relative
to the asymptotic lines of the surface attained.

The name of Lie will remain attached to these deep-lying relations
which join to one another the straight line and the sphere, those two
essential and fundamental elements of geometric research. He de-
veloped them in a memoir full of new ideas which appeared in 1872.

The works which followed this brilliant debut of Lie fully con-
firmed the hopes it had aroused. Pluecker's conception relative to
the generation of space by straight lines, by curves or surfaces
arbitrarily chosen, opens to the theory of algebraic forms a field
which has not yet been explored, which Clebsch scarcely began to
recognize and settle the boundaries of. But, from the side of infini-

DEVELOPMENT OF GEOMETRIC METHODS 555

tesimal geometry, this conception has been given its full value by
Sophus Lie. The great Norwegian geometer was able to find in it
first the notion of congruences and complexes of curves, and after-
ward that of contact transformations of which he had found, for the
case of the plane, the first germ in Pluecker. The study of these
transformations led him to perfect, at the same time with M. Mayer,
the methods of integration which Jacobi had instituted for partial
differential equations of the first order; but above all it threw the
most brilliant light on the most difficult and the most obscure parts
of the theories relative to partial differential equations of higher
order. It permitted Lie, in particular, to indicate all the cases in
which the method of characteristics of Monge is fully applicable to
equations of the second order with two independent variables.

In continuing the study of these special transformations, Lie was
led to construct progressively his masterly theory of continuous
groups of transformations and to put in evidence the very important
role that the notion of group plays in geometry. Among the essential
elements of his researches, it is proper to signalize the infinitesimal
transformations, of which the idea belongs exclusively to him.

Three great books published under his direction by able and de-
voted collaborators contain the essential part of his works and their
applications to the theory of integration, to that of complex units and
to the non-Euclidean geometry.

XIV

By an indirect way I have arrived at that non-Euclidean geometry
the study of which takes in the researches of geometers a place which
grows greater each day.

If I were the only one to talk with you about geometry, I should
take pleasure in recalling to you all that has been done on this sub-
ject since Euclid or at least from Legendre to our days.

Envisaged successively by the greatest geometers of the last cen-
tury, the question has progressively enlarged.

It commenced with the celebrated postulatum relative to parallels;
it ends with the totality of geometric axioms.

The Elements of Euclid, which have withstood the action of so
many centuries, will have at least the honor before ending of arous-
ing a long series of works admirably enchained which will contrib-
ute, in the most effective way, to the progress of mathematics, at the
same time that they furnish to the philosophers the most precise and
the most solid points of departure for the study of the origin and of
the formation of our cognitions.

I am assured in advance that my distinguished collaborator will
not forget, among the problems of the present time, this one, which is
perhaps the most important, and with which he has occupied himself

556 GEOMETRY

with so much success; and I leave to him the task of developing it
with all the amplitude which it assuredly merits.

I have just spoken of the elements of geometry. They have received
in the last hundred years extensions which must not be forgotten.
The theory of polyhedrons has been enriched by the beautiful dis-
coveries of Poinsot on the star polyhedrons and those of Moebius
on polyhedrons with a single face. The methods of transformation
have enlarged the exposition. We may say to-day that the first book
contains the theory of translation and of symmetry, that the second
amounts to the theory of rotation and of displacement, that the
third rest on homothety and inversion. But it must be recognized
that it is due to analysis that the Elements have been enriched by
their most beautiful propositions.

It is to the highest analysis that we owe the inscription of regular
polygons of seventeen sides and analogous polygons. To it we owe
the demonstrations, so long sought, of the impossibility of the quad-
rature of the circle, of the impossibility of certain geometric con-
structions with the aid of the ruler and the compasses; and to it finally
we owe the first rigorous demonstrations of the properties of maxi-
mum and of minimum of the sphere. It will belong to geometry to
enter upon this ground where analysis has preceded it.

What will be the elements of geometry in the course of the cen-
tury which has just commenced? Will there be a single elementary
book of geometry? It is perhaps America, with its schools free from
all programme and from all tradition, which will give us the best solu-
tion of this important and difficult question.

Von Staudt has sometimes been called the Euclid of the nine-
teenth century; I would prefer to call him the Euclid of projective
geometry ; but is projective geometry, interesting though it may be,
destined to furnish the unique foundation of the future elements?

The moment has come to close this over-long recital, and yet there
is a crowd of interesting researches that I have been, so to say, forced
to neglect.

I would have loved to talk with you about those geometries of
any number of dimensions of which the notion goes back to the first
days of algebra, but of which the systematic study was commenced
only sixty years ago by Cayley and by Cauchy. This kind of researches
has found favor in your country and I need not recall that our illus-
trious president, after having shown himself the worthy successor
of Laplace and Le Verrier, in a space which he considers with us as
being endowed with three dimensions, has not disdained to publish,
in the American Journal, considerations of great interest on the
geometries of n dimensions.

DEVELOPMENT OF GEOMETRIC METHODS 557

A single objection can be made to studies of this sort, and was
already formulated by Poisson: the absence of all real foundation, of
all substratum permitting the presentation, under aspects visible and
in some sort palpable, of the results obtained.

The extension of the methods of descriptive geometry, and above
all the employment of Pluecker's conceptions on the generation of
space, will contribute to take away from this objection much of its
force.

I would have liked to speak to you also of the method of equi-
pollences, of which we find the germ in the posthumous works of
Gauss, of Hamilton's quaternions, of Grassmann's methods, and in
general of systems of complex units, of the analysis situs, so inti-
mately connected mth the theory of functions, of the geometry
called kinematic, of the theory of abaci, of geometrography, of the
applications of geometry to natural philosophy or to the arts. But
1 fear, if I branched out beyond measure, some analyst, as has hap-
pened before, would accuse geometry of wishing to monopolize
everything.

My admiration for analysis, grown so fruitful and so powerful in
our time, would not permit me to conceive such a thought. But if
some reproach of this sort could be formulated to-day, it is not to
geometry, it is to analysis it would be proper, I believe, to address it.
The circle in which the mathematical studies appeared to be inclosed
at the beginning of the nineteenth century has been broken on all
sides.

The old problems present themselves to us under a new form, new
problems offer themselves, whose study occupies legions of workers.

The number of those who cultivate pure geometry has become
prodigiously restricted. Therein is a danger against which it is im-
portant to provide. We must not forget that, if analysis has acquired
means of investigation which it lacked heretofore, it owes them in
great part to the conceptions introduced by the geometers. Geometry
must not remain in some sort entombed in its triumph. It is in its
school we have learned; our successors must learn never to be blindly
proud of methods too general, to envisage the questions in themselves
and to find, in the conditions particular to each problem, perhaps
a direct way towards a solution, perhaps the means of applying in
an appropriate manner the general procedures which every science
should gather.

As Chasles said at the beginning of the Apercu historique, "The
doctrines of pure geometry offer often, and in a multitude of ques-
tions, that simple and natural way which, penetrating to the very
source of the truths, lays bare the mysterious chain which binds them
to each other and makes us know them individually in the way most
luminous and most complete."

558 GEOMETRY

Cultivate therefore geometry, which has its own advantages, with-
out wishing, on all points, to make it equal to its rival.

For the rest, if we were tempted to neglect it, it would soon find in
the apphcations of mathematics, as it did once before, means to rise
up again and develop itself anew. It is like the giant Antaus who
recovered his strength in touching the earth.

THE PRESENT PROBLEMS OF GEOMETRY

BY DR. EDWARD KASNER

[Edward Kasner, Instructor in Mathematics, Columbia University, b. New York
City, 1877. B.S. CoUege of the City of New York, 1896; A.M. Columbia
University, 1899; Pli.D. ifeid. 1899. Post-graduate, Fellow in Mathematics,
Columbia University, 1897-99 ; Student, University of Gottingen, 1899—
1900; Tutor in Mathematics, Columbia University, 1900-05; Instructor,
1905; Member American Mathematical Society; Fellow American Associa-
tion for Advancement of Science. Associate editor. Transactions American
Mathematical Society.]

In spite of the richness and power of recent geometry, it is notice-
able that the geometer himself has become more modest. It was the
ambition of Descartes and Leibnitz to discover universal methods,
applicable to all conceivable questions; later, the Ausdehnungslehre
of Grassmann and the quaternion theory of Hamilton were believed
by their devotees to be ultimate geometric analyses; and Chasles
attributed to the principles of duality and homography the same
role in the domain of pure space as that of the law of gravitation
in celestial mechanics. To-day, the mathematician admits the ex-
istence and the necessity of many theories, many geometries, each
appealing to certain interests, each to be developed by the most
appropriate methods; and he realizes that, no matter how large his
conceptions and how powerful his methods, they will be replaced
before long by others larger and more powerful.

Aside from the conceivability of other spaces with just as self-
consistent properties as those of the so-called ordinary space, such
diverse theories arise, in the first place, on account of the variety
of objects demanding consideration, — curves, surfaces, congruences
and complexes, correspondences, fields of differential elements, and
so on in endless profusion. The totality of configurations is indeed
not thinkable in the sense of an ordinary assemblage, since the total-
ity itself would have to be admitted as a configuration, that is, an
element of the assemblage. ■

However, more essential in most respects than the diversity in
the material treated is the diversity in the points of view from which
it may be regarded. Even the simplest figure, a triangle or a circle,
has an infinity of properties — indeed, recalling the unity of the
physical world, the complete study of a single figure would involve
its relations to all other figures and thus not be distinguishable from
the whole of geometry. For the past three decades the ruling thought
in this connection has been the principle (associated with the names
of Klein and Lie) that the properties which are deemed of interest
in the various geometric theories may be classified according to the

560 GEOMETRY

groups of transformations which leave those properties unchanged.
Thus almost all discussions on algebraic curves are connected with
the group of displacements (more properly the so-called principal
group), or the group of projective transformations, or the group of
birational transformations; and the distinction between such theories
is more fundamental than the distinction between the theories of
curves, of surfaces, and of complexes.

Historically, the advance has been, in general, from small to larger
groups of transformations. The change thus produced may be likened
to the varying appearance of a painting, at first viewed closely in all
its details, then at a distance in its significant features. The analogy
also suggests the desirability of viewing an object from several stand-
points, of studying geometric configurations with respect to various
groups. It is indeed true, though in a necessarily somewhat vague
sense, that the more essential properties are those invariant under
the more extensive groups ; and it is to be expected that such groups
will play a predominating role in the not far distant future.

The domain of geometry occupies a position, as indicated in the
programme of the Congress, intermediate between the domain of
analysis on the one hand and of mathematical physics on the other;
and in its development it continually encroaches upon these adjacent
fields. The concepts of transformation and invariant, the algebraic
curve, the space of n dimensions, owe their origin primarily to the
suggestions of analysis; while the null-system, the theory of vector
fields, the questions connected with the applicability and deforma-
tion of surfaces, have their source in mechanics. It is true that some
mathematicians regard the discussion of point sets, for example,
as belonging exclusively to the theory of functions, and others look
upon the composition of displacements as a part of mechanics.
While such considerations show the difficulty, if not impossibility,
of drawing strict limits about any science, it is to be observed that
the consequent lack of definiteness, deplored though it be by the
formalist, is more than compensated by the fact that such overlap-
ping is actually the principal means by which the different realms
of knowledge are bound together.

If a mathematician of the past, an Archimedes or even a Descartes,
could view the field of geometry in its present condition, the first
feature to impress him would be its lack of concreteness. There are
whole classes of geometric theories which proceed, not merely with-
out models and diagrams, but without the slightest (apparent) use
of the spatial intuition. In the main this is due, of course, to the
power of the analytic instruments of investigation as compared
with the purely geometric. The formulas move in advance of thought,
while the intuition often lags behind; in the oft-quoted words of
d'Alembert, " L'algebre est genereuse, elle donne sou vent plus qu'on

PRESENT PROBLEMS OF GEOMETRY 561

hii demande." As the field of research widens, as we proceed from
the simple and definite to the more refined and general, we naturally
cease to picture our processes and even our results. It is often neces-
sary to close our eyes and go forward blindly if we wish to advance
at all. But admitting the inevitableness of such a change in the
spirit of any science, one may still question the attitude of the geo-
meter who rests content with his blindness, who does not at least
strive to intensify and enlarge the intuition. Has not such an inten-
sification and enlargement been the main contribution of geometry
to the race, its very roison d'etre as a separate part of mathematics,
and is there any ground for regarding this service as completed?

From the point of view here referred to, a problem is not to be
regarded as completely solved until we are in position to construct
a model of the solution, or at least to conceive of such a construction.
This requires the interpretation, not merely of the results of a geo-
metric investigation, but also, as far as possible, of the intermediate
processes — an attitude illustrated most strikingly in the works of
Lie. This duty of the geometer, to make the ground won by means
of analysis really geometric, and as far as possible concretely intui-
tive, is the source of many problems of to-da}^, a few of which will
be referred to in the course of this address.

The tendency to generalization, so characteristic of modern geo-
metry, is counteracted in many cases by this desire for the concrete,
in others by the desire for the exact, the rigorous (not to be con-
fused with the rigid). The great mathematicians have acted on the
principle "Devinez avant de demontrer," and it is certainly true
that almost all important discoveries are made in this fashion. But
while the demonstration comes after the discovery, it cannot there-
fore be disregarded. The spirit of rigor, which tended at first to the
arithmetization of all mathematics and now tends to its exhibition
in terms of pure logic, has always been more prominent in analysis
than in geometry. Absolute rigor may be unattainable, but it can-
not be denied that much remains to be done by the geometers, judg-
ing even by elementary standards. We need refer only to the loose
proofs based upon the invaluable but insufficient enumeration of
constants, the so-called principle of the conservation of number, and
the discussions which confine themselves to the "general case."
Examples abound in every field of geometry. The theorem announced
by Chasles concerning the number of conies satisfying five arbitrary
conditions was proved by such masters as Clebsch and Halphen be-
fore examples invalidating the result were devised. Picard recently
called attention to the need of a new proof of Noether's theorem that
upon the general algebraic surface of degree greater than three every
algebraic curve is a complete intersection T\dth another algebraic
surface. The considerations given by Noether render the result

562 GEOMETRY

highly probable, but do not constitute a complete proof; while the
exact meaning of the term general can be determined only from
the context.

The reaction against such loose methods is represented by Study *
in algebraic geometry, and Hilbert in differential geometry. The
tendency of a considerable portion of recent work is towards the
exhaustive treatment of definite questions, including the considera-
tion of the special or degenerate cases ordinarily passed over as
unimportant. Another aspect of the same tendency is the discussion
of converses of familiar problems, with the object of obtaining con-
ditions at once necessary and sufficient, that is, completely character-
istic results.^

Another set of problems is suggested by the relation of geometry
to physics. It is the duty of the geometer to abstract from the physical
sciences those domains which may be expressed in terms of pure
space, to study the geometric foundations (or, as some would put it,
the skeletons) of the various branches of mechanics and physics.
Most of the actual advance, it is true, has hitherto come from the
physicists themselves, but undoubtedly the time has arrived for
more systematic discussions by the mathematicians. In addition to
the importance which is due to possible applications of such work,
it is to be noticed that we meet, in this way, configurations as inter-
esting and remarkable as those created by the geometer's imagina-
tion. Even in this field, one is tempted to remark, truth is stranger
than fiction.

We have now considered, briefly and inadequately, some of the
leading ideals and influences which are at work towards both the
"wddening and the deepening of geometry in general; and turn to our
proper topic, a survey of the leading problems or groups of problems
in certain selected (but it is hoped representative) fields of contem-
poraneous investigation.

Foundations

The most striking development of geometry during the past decade
relates to the critical revision of its foundations, more precisely, its
logical foundations. There are, of course, other points of view, for

^ " [Es ist eine] tief eingewurzelte Gewohnheit vieler Geometer, Satze zu formu-
lieren, die 'im allgemeinen ' gelten sollen. d. h. einen klaren Sinn iiberhaupt nicht
haben, zudem noch haufig als allgemein giiltig hingestellt oder mangelhaft be-
grundet werden. [Dies Verfahren mrd], trotz etwanigen Verweisungen auf Trager
sehr beriihmter Nam en, spateren Gesclilechtern sicher als ganz unzulassig erschei-
nen, scheint aber in unserem 'kritischen' Zeitalter von vielen als eine berechtigte
Eigenttimlichkeit der Geometrie betrachtet zu werden . . ." Jakr. Deut. Math.-
Ver., vol. XI (1902), p. 100.

^ As an example may be mentioned the theorem of Malus and Dupin, known
for almost a century, that the rays emanating from a point are converted, by any
refraction, into a normal congruence. Quite recently, Levi-Civitta succeeded in
sho-^nng that this property is characteristic; that is, any normal congruence may
be refracted into a bundle.

PRESENT PROBLEMS OF GEOMETRY 563

example, the physical, the physiological, the psychological, the meta-
physical, but the interest of mathematicians has been confined to the
purely logical aspect. The main results in this direction are due to
Peano and his co-workers; but the whole field was first brought
prominently to the attention of the mathematical world by the
appearance, five years ago, of Hilbert's elegant Festschrift.

The central problem is to lay down a system of primitive (unde-
fined) concepts or symbols and primitive (unproved) propositions
or postulates, from which the whole body of geometry (that is, the
geometry considered) shall follow by purely deductive processes.
No appeal to intuition is then necessary. " We might put the axioms
into a reasoning apparatus like the logical machine of Stanley Jevons,
and see all geometry come out of it" (Poincare). Such a system of
concepts and postulates may be obtained in a great (indeed end-
less) variety of ways: the main question, at present, concerns the
comparison of various systems, and the possibility of imposing lim-
itations so as to obtain a unique and perhaps simplest basis.

The first requirement of a system is that it shall be consistent.
The postulates must be compatible with one another. No one has yet
deduced contradictory results from the axioms of Euclid, but what
is our guarantee that this will not happen in the future? The only
method of answering this question which has suggested itself is the
exhibition of some object (whose existence is admitted) which fulfills
the conditions imposed by the postulates. Hilbert succeeded in con-
structing such an ideal object out of numbers; but remarks that the
difficulty is merely transferred to the field of arithmetic. The most
far-reaching result is the definition of number in terms of logical
classes as given by Pieri and Russell; but no general agreement is
yet to be expected in these discussions. Will the ultimate conclu-
sion be the impossibility of a direct proof of compatibility?

More accessible is the question concerning the independence of
postulates (and the analogous question of the irreducibility of con-
cepts). Most of the work of the last few years has been concentrated
on this point. In Hilbert's original system the various groups of
axioms (relating respectively to combination, order, parallels, con-
gruence, and continuity) are shown to be independent, but the dis-
cussion is not carried out completely for the individual axioms. In
Dr. Veblen's recently published system of twelve postulates, each
is proved independent of the remaining eleven.^ This marks an ad-
vance, but, of course, it does not terminate the problem. In what
respect does a group o'f propositions differ from what is termed a
single proposition? Is it possible to define the notion of an absolutely
simple postulate? The statement that any two points determine a
straight line involves an infinity of statements, and its fulfillment for
' Trans. Amer. Math. Soc, vol. v (1904).

564 GEOMETRY

certain pairs of points may necessitate its fulfillment for all pairs.
If in Euclid's system the postulate of parallels is replaced by the
postulate concerning the sum of the angles of a triangle, a well-known
example of such a reduction is obtained; for it is sufficient to as-
sume the new postulate for a single triangle, the general result being
then deducible. As other examples we may mention Peano's reduc-
tion of the Euclidean definition of the plane; and the definition of
a collineation which demands, instead of the conversion of all straight
lines into straight lines, the existence of four simply infinite systems
of such straight lines/

These examples illustrate the difficulty, if not the impossibility,
of formulating a really fundamental, that is, absolute standard of
independence and irreducibility. It is probable that the guiding
ideas will be obtained in the discussion of simpler deductive theories,
in particular, the systems for numbers and groups.

Two features are especially prominent in the actual develop-
ment of the body of geometry from its fundamental system. First,
the consideration of what may be termed the collateral geometries,
which arise by replacing one of the original postulates by its opposite,
or otherwise varying the system. Such theories serve to show the
limitation of that point of view which restricts the term general
geometry (pangeometry) to the Euclidean and non-Euclidean geo-
metries. The variety of possible abstract geometries is, of course,
inexhaustible; this is the central fact brought to light by the ex-
hibition of such systems as the non-Archimedean and the non-
arguesian. In the second place, much valuable work is being done in
discussing the various methods by which the same theorem may
be deduced from the postulates, the ideal being to use as few of the
postulates as possible. Here again the question of simplicity (simplest
proof), though it baffles analysis, forces itself upon the attention.

Among the minor problems in this field, it is sufficient to consider
that concerning the relation of the theory of volume to the axiom of
continuity. This axiom need not be used in establishing the theory
of areas of polj^gons; but after Dehn and others had proved the exist-
ence of polyhedra having the same volume though not decomposable
into mutually congruent parts (even after the addition of congruent
polyhedra), it was stated by Hilbert, and deemed evident generally,
that reference to continuity could not be avoided in three dimensions.
In a recent announcement ^ of Vahlen's forthcoming Abstrakte
Geometrie this conclusion is declared unsound. It seems probable,
however, that the difference is merely one concerning the interpreta-
tion to be given to the term continuity.

' Together with certain continuity assumptions. Cf. Bull. Amer. Math. Soc,
vol. IX (1903), p. 545.

2 Jahr. Deut. Math.-Ver., vol. xiii (1904), p. 395,

PRESENT PROBLEMS OF GEOMETRY 565

The work on logical foundations has been confined almost entirely
to the Euclidean and projective geometries. It is desirable, however,
that other geometric theories should be treated in a similar deductive
fashion. In particular, it is to be hoped that we shall soon have
a really systematic foundation for the so-called inversion geometry,
dealing with properties invariant under circular transformations.
This theory is of interest, not only for its own sake and for its appli-
cations in function theory, but also because its study serves to free
the mind from what is apt to become, without some check, slavery to
the projective point of view.

The Curve Concept — Analysis Situs

Although curves and surfaces have constituted the almost exclu-
sive material of the geometric investigation of the thirty centu-
ries of which we have record, it can hardly be claimed that the con-
cepts themselves have received their final analysis. Certain vague
notions are suggested by the naive intuition. It is the duty of mathe-
maticians to create perfectly precise concepts which agree more or
less closely with such intuitions, and at the same time, by the reac-
tion of the concepts, to refine the intuition. The problem, evidently, is
not at all determinate. It would be of interest to trace the evolution
which has actually produced several distinct curve concepts defining
more or less extensive classes of curves, agreeing in little beyond the
possession of an infinite number of points.

The more familiar special concepts or classes of curves are defined
in terms of the corresponding equation y=f(x) or function f(x).
Such are, for example: (1) algebraic curves; (2) analytic curves;
(3) graphs of functions possessing derivatives of all orders; (4) the
curves considered in the usual discussions of infinitesimal geometry,
in which the existence of first and second derivatives is assumed;
(5) the so-called regular curves Math a continuously turning tangent
(except for a finite number of corners); (6) the so-called ordinary
curves possessing a tangent and having only a finite number of
oscillations (maxima and minima) in any finite interval; (7) curves
with tangents; (8) the graphs of continuous functions.

How far are such distinctions accessible to the intuition? Of
course there are limitations. For over two centuries, from Descartes
to the publication of Weierstrass's classic example, the intuition of
mathematicians declared the classes (7) and (8) to be identical. Still
later it was found that such extraordinary (pathological or crinkly)
curves may present themselves in class (7). However, even here
partially successful attempts to connect with intuition have been
made by Wiener, Hilbert, Schoenflies, Moore, and others.

Let us consider a simpler extension in the field of ordinary curves.
If the function jT (a:) is continuous except for a certain value of x

566 GEOMETRY

where there is an ordinary discontinuity, this is indicated by a break
in the graph; \if is continuous, but the derivative y has such a dis-
continuity, this shows itself by a sharp turn in the curve; if the
discontinuity is only in the second derivative, there is a sudden
change in the radius of curvature, which is, however, relatively
difficult to observe from the figure; finally, if the third derivative
is discontinuous, the effect upon the curve is no longer apparent.
Does this mean that it is impossible to picture it? Does it not rather
indicate a limitation in the usual geometric training which goes
only as far as relations expressible in terms of tangency and curva-
ture? For the interpretation of the third derivative it is necessary
to consider say the osculating parabola at each point of the curve :
in the case referred to, as we pass over the critical point, the
tangent line and osculating circle change continuously, but there is
a sudden change in the osculating parabola. If in fact our intuition
were trained to picture osculating algebraic curves of all orders, it
would detect a discontinuity in a derivative of any order. A partial
equivalent would be the ability to picture the successive evolutes
of a given curve; a complete equivalent would be the picturing of
the successive slope curves y=f'{x), y=f"{x), etc. All this requires,
evidently, only an increase in the intensity of our intuition, not a
change in its nature.

This, however, would not apply to all questions. There are func-
tions which, while possessing derivatives of all orders (then neces-
sarily continuous), are not analytic (that is, not expressible by power
series). What is it that distinguishes the analytic curves among this
larger class? Is it possible to put the distinction in a form capable
of assimilation by an idealized intuition? In short, what is the
reall}^ geometric definition of an analytic curve ? *

Much recent work in function theory has had for its point of de-
parture a more general basis than the theory of curves, namely, the
theory of sets or assemblages of points, with special reference to
the notions of derived set and the various contents or areas. The
geometry of point sets must indeed be regarded as one of the most
important and promising in the whole field of mathematics. It
receives its distinctive character, as compared with the general
abstract theory of assemblages (Mengenlehre), from the fact that it
operates not with all one-to-one correspondences, but with the
group of analysis situs, the group of continuous one-to-one corre-
spondences. From the point of view of the larger group, there is no
distinction between a one-dimensional and a two- or many-dimen-
sional continuum (Cantor). This is still the case if the correspondence

^ One method of attack would be the interpretation of Pringsheim's condi-
tions; this requires not merely the individual derivative curves, but the limit of
the system.

PRESENT PROBLEMS OF GEOMETRY 567

is continuous but not one-to-one (Peano, 1890). In the domain of
continuous one-to-one correspondence, however, spaces of different
dimensions are not equivalent (Jiirgens, 1899).

An important class of curves, much more general than those
referred to above, consists of those point sets which are equivalent
(in the sense of analysis situs) to the straight line or segment of a
straight line. This is Hurwitz's simple and elegant geometric form-
ulation of the concept originally treated analytically by Jordan,
the most fundamental curve concept of to-day. The closed Jordan
curves are defined in analogous fashion as equivalent to the peri-
meter of a square (or the circumference of a circle).

A curve of this kind divides the remaining points of the plane into
two simply connected continua, an inside and an outside. The
necessity for proof of this seemingly obvious result is seen from the
fact that the Jordan class includes such extraordinary types as the
curve with positive content constructed recently by Osgood.^ Such
a separation of the plane may, however, be thought about by other
than Jordan curves: the concept of the boundary of a connected
region gives perhaps the most extensive class of point sets which
deserve to be called curve. Schoenflies proposes a definition for the
idea of a simple closed curve which makes it appear as the natural
extension, in a certain sense, of the polygon; a perfect set of points
P which separates the plane into an exterior region E and an interior
region / such that any E point can be connected with any I point
b}^ a path (Polygonstrecke) having only one point in common with
P. This is in effect a converse of Jordan's theorem, and shows
precisely how the Jordan curve is distinguished from other types
of boundaries of connected regions.

These discussions are mentioned here simply as aspects of a really
fundamental problem: th,e revision of the concepts and results of
that division of geometry which has been variously termed analysis
situs, theory of connection, topology, geometry of situation — a
revision to be carried out in the light of the theory of assemblages.^

Algebraic Surfaces and Birational Transformations

After the demonstration of the power of the methods based upon
projective transformation, — the chief contribution due to the
geometers of the first half of the nineteenth century, — attempts
were made to introduce other types of one-to-one correspondence or
transformation into algebraic geometry; in particular the inversion
of William Thomson and Liouville, and the quadratic transformation
of Magnus. The general theory of such Cremona transformations
was inaugurated by the Italian geometer in his memoir Sulle tras-

1 Trans. Amer. Math. Soc, vol. iv (1903), p. 107.

^ Cf. Schoenflies, Math. Annalen, vols, lviii, lix (1903, 1904).

568 GEOMETRY

formazioni geometriche delle figure piane, piiblished in 1863. Within
afewyearS; Clifford, Noether, and Rosanes, working independently,
established the remarkable result that every Cremona transforma-
tion in a plane can be decomposed into a succession of quadratic
transformations, thus bringing to light the fact that there are at
bottom only two types of algebraic one-to-one correspondence, the
homographic and the quadratic.^

The development of a corresponding theory in space has been one
of the chief aims of the geometers of Italy, Germany, and England
for the last thirty years, but the essential question of decomposition
still remains unanswered. Is it possible to reduce the general Cremona
transformation of space to a finite number of fundamental types ?

In its application to the study of the properties of algebraic
curves and surfaces, the theory of the Cremona transformation
is usually merged in the more general theory of the birational trans-
formation. By means of the latter, a correspondence is established
which is one-to-one for the points of the particular figure considered
and the transformed figure, but not for all the points of space. In
the plane theory an important result is that a curve with the most
complicated singularities can, by means of Cremona transformations;,
be converted into a curve whose only singularities are multiple
points with distinct tangents (Noether); furthermore, by means of
birational transformations, the singularities may be reduced to the
very simplest type, ordinary double points (Bertini). The known
theory of space curves is also, in this aspect, quite complete. The
analogous problem of the reduction of higher singularities of a sur-
face has been considered by Noether, Del Pezzo, Segre, Kobb, and
others, but no ultimate conclusion has yet been obtained.

One principal source of difficulty is that, while in case of two
birationally equivalent curves the correspondence is one-to-one
without exception, on the other hand, in the case of two surfaces,
there may be isolated points which correspond to curves, and just
such irregular phenomena escape the ordinary methods. Again,
not onl}^ singular points require consideration, as is the case in the
plane theory, but also singular lines, and the points may be isolated
or superimposed on the lines. Most success is to be expected from
further application of the method of projection from a higher space
due to Clifford and Veronese. In this direction the most important
result hitherto obtained is the theorem, of Picard and Simart, that
any algebraic surface (in ordinary space) can be regarded as the
projection of a surface free from singularities situated in five-dimen-
sional space.

_ * Segre recently called attention to a case where the usual methods of discus-
sion fail to apply; the proof has been completed by Castelnuovo. Cf, Atii di
Torino, vol. xxxvi (1901).

PRESENT PROBLEMS OF GEOMETRY 569

A question which awaits solution even in the case of the plane
is that relating to the invariants of the group of Cremona trans-
formations proper. The genus and the moduli of a curve are unaltered
by all birational transformations, but the problem arises: Are there
properties of curves which remain unchanged by Cremona, although
not by other birational transformations ? From the fact that
birationally equivalent curves need not be equivalent under the
Cremona group, it would seem that such invariants — Cremona
invariants proper — do exist, but no actual examples have yet been
obtained. The problem may be restated in the form: What are the
necessary and sufficient conditions which must be fulfilled by two
curves if they are to be equivalent with respect to Cremona trans-
formations? Equality of genera and moduli, as already remarked, is
necessary but not sufficient.

The invariant theory of birational transformations has for its
principal object the study of the linear systems of point groups
on a given algebraic curve, that is, the point groups cut out by
linear systems of curves. Its foundations were implicitly laid by
Riemann in his discussion of the equivalent theory of algebraic func-
tions on a Riemann surface, though the actual application to curves
is due to Clebsch. Most of the later work has proceeded along
the algebraic-geometric lines developed by Brill and Noether, the
promising purely geometric treatment inaugurated by Segre being
rather neglected.

The extension of this type of geometry to space, that is, the de-
velopment of a systematic geometry on a fundamental algebraic
surface (especially as regards the linear systems of curves situated
thereon), is one of the main tasks of recent mathematics. The
geometric treatment is given in the memoirs of Enriques and Castel-
nuovo, while the corresponding functional aspect is the subject of
the treatise of Picard and Simart on algebraic functions of two
variables, at present in course of publication.

The most interesting feature of the investigations belonging in
this field is the often unexpected light which they throw on the
inter-relations of distinct fields of mathematics, and the advantage
derived from such relations. For example, Picard (as he himself
relates on presenting the second volume of his treatise to the Paris
Academy a few months ago) ^ for a long time was unable to prove
directly that the integrals of algebraic total differentials can be
reduced, in general, to algebraic-logarithmic combinations, until
finally a method for deciding the matter was suggested by a theorem
on surfaces which Noether had stated some twenty years earlier.
Again, in the enumeration of the double integrals of the second
species, Picard arrived at a certain result, which was soon noticed
^ Comptes Rendus, February 1, 1904.

570 GEOMETRY

to be essentially equivalent to one obtained by Castelnuovo in his
investigations on linear systems; and thus there was established
a connection between the so-called numerical and linear genera of a
surface, and the number of distinct double integrals.^

A closely related set of investigations, originating with Clebsch's
theorems on intersections and Liouville's on confocal quadrics, may
be termed the "geometry of Abel's theorem." As later applications
we can merely mention Humbert's memoirs on certain metric pro-
perties of curves, and Lie's determination of surfaces of translation.

Investigations in analysis have often suggested the introduc-
tion of new types of configurations into geometry. The field of alge-
braic surfaces is especially fruitful in this respect. Thus, while in the
case of curves (excluding the rational) there always exist integrals
everywhere finite, this holds for only a restricted class of surfaces;
their determination depends on the solution of a partial differential
equation which has been discussed in a few special cases.

In addition to such relations between analysis and geometry,
important relations arise between various fields of geometry. Just
as an algebraic function of one variable is pictured by either a plane
curve or a Riemann surface (according as the independent and de-
pendent variables are taken to be real or complex), so an algebraic
function of two independent variables may be represented by either
a surface in ordinary space or a Riemannian four-dimensional mani-
fold in space of five dimensions. In the case of one variable, the
single invariant number (deficiency or genus p) which arises is
capable of definition in terms of the characteristics of the curve or
the connectivity of the Riemann surface. In passing to two variables,
however, it is necessary to consider several arithmetical invariants
— just how many is an unsettled question. For the algebraic surface
we have, for instance, the geometric genus of Clebsch, the numerical
genus of Cayley, and the so-called second genus, each of which maj^
be regarded as a generalization, from a certain point of view, of the
single genus of a curve; all are invariant with respect to birational
transformation.

The other geometric interpretation, by means of a Riemannian
manifold, has rendered necessary the study of the analysis situs of
higher spaces. The connection of such a manifold is no longer ex-
pressed by a single number as in the case of an ordinary surface, but
by a set of two or more, the so-called numbers of Betti and Riemann.
The detailed theory of these connectivities, difiicult and delicate
because it must be derived with little aid from the intuition, has been
made the subject of an extensive series of memoirs by Poincare.

From the point of view of analysis, the chief interest in these
investigations is the fact that the connectivities are related to the
^ Comptes Rendus, February 22, 1904.

PRESENT PROBLEMS OF GEOMETRY 571

number of integrals of certain types. The chief problem for the
geometer, however, is the discovery of the precise relations between
the connectivities of the Riemann manifold and the various genera
of the algebraic surface. That relations do exist between such di-
verse geometries — the one operating with all continuous, the other
with the algebraic, one-to-one correspondence — is one of the most
striking results of recent mathematics.

Geometry of Multiple Forms

For some time after its origin, the linear invariant theory of
Boole, Cayley, and Sylvester confined itself to forms containing a
single set of variables. The needs of both analysis and geometry,
however, have emphasized the importance and the necessity of
further development of the theory of forms containing two or more
sets of variables (of the same or different type), so-called multiple
forms.

In the plane we have both point coordinates (x) and line coor-
dinates (u). A form in x corresponds to a point curve (locus), a
form in w to a line curve (envelope), and a form involving both x
and u to & connex. The latter was introduced into geometry, some
thirty years ago, by Clebsch, the suggestion coming from the fact
that, even in the study of a simple form in x, covariants in x and u
present themselves, so that it seemed desirable to deal with such
forms ab initio.

Passing to space, we meet three simple elements, the point (x),
the plane (u), and the line (p). Forms in a single set of variables
represent, respectively, a surface as point locus, a surface as plane
envelope, and a complex of lines. The compound elements composed
of two simple elements are the point-plane, the point-line, and the
plane-line. The first type, leading to point-plane connexes, has been
studied extensively during the past few years; the second to a more
limited degree; the third is merely the dual of the second. To com-
plete the series, the case of the point-line-plane as element, or forms
involving x, u, and p, requires investigation.

In the corresponding w-dimensional theory it is necessary to take
account of n simple elements and the various compound elements
formed by their combinations.

The importance of such work is twofold: First, on account of
connection with the algebra of invariants. A fundamental theorem
of Clebsch states that, in the investigation of complete systems of
comitants, it is sufficient to consider forms involving not more than
one set of variables of each type : if in the given forms the types are
involved in any manner, it is possible to find an equivalent reduced
system of the kind described. On the other hand, it is impossible
to reduce the system further, so that the introduction of the n types

572 GEOMETRY

of variables is necessary for the algebraically complete discussion.
Geometry must accordingly extend itself to accommodate the
configurations defined by the new elements.

Second, on account of connection with the theory of differential
equations. The ordinary plane connex in x, u, assigns to each point
of the plane a certain number of directions (represented by the
tangents drawn to the corresponding curve), and thus gives rise to
an (algebraic) differential equation of the first order in two variables;
the point-plane connex in space, associating with each point a single
infinity of incident planes, defines a partial differential equation
of the first order; the point-line connex yields a Monge equation.
The point-line-plane case has not yet been interpreted from this
point of view.

One special problem in this field deserves mention, on account of its
many applications. This is the study of the system composed of a
quadric form in any number of variables and a bilinear form in con-
tragredient variables, that is, a quadric manifold and an arbitrary
(not merely automorphic) collineation in n-space. For n = 6, for
example, this corresponds to the general linear transformation of
line or sphere coordinates.

In addition to forms containing variables of different types, the
forms involving several sets of variables of the same type require
consideration. Forms in two sets of line coordinates present them-
selves in connection with the pfaffian problem of differential systems.
The main interest attaches, however, to forms in sets of point coor-
dinates, since it is these which occur in the theory of contact trans-
formations and of multiple correspondences. For example, while
the ordinary homography on a line is represented by a bilinear form
in binary variables, the trilinear form in similar variables gives rise
to a new geometric variety, the so-called homography of the second
class (associating with any two points a unique third point), which
has applications to the generation of cubic surfaces and to the con-
structions at the basis of photogrammetry. The theory of multilinear
forms in general deserves more attention than it has yet received.

Other important problems, connected with the geometric phases of
linear invariant theory, can merely be mentioned: (1) The general
geometric interpretation of what appears algebraically as the sim-
plest projective relation, namely, apolarity. (2) The invariant dis-
cussion of the simpler discontinuous varieties, for example, the poly-
gon considered as w-point or as n-line.^ (3) The establishment of a
system of forms corresponding to the general space curve. (4) The
study of the properties and the groups of the configurations cor-

^ Cf. F. Morley "On the geometry whose element is the 3-point of a plane,"
Trans. Amer. Math. Soc, vol. v (1904). E. Study in his Geometrie der Dynamen
develops a new foundation for kinematics by employing as element the Soma or
trirectangular trihedron.

PRESENT PROBLEMS OF GEOMETRY 573

responding in hyperspace to the simpler systems of invariants. (5)
Complete systems of orthogonal or metric invariants for the simpler
curves. ^

Transcendental Curves

To reduce to systematic order the chaos of non-algebraic curves
has been the aspiration of many a mathematician; bat, despite all
efforts, we have no theory comparable with that of algebraic curves.
The very vagueness and apparent hopelessness of the question is
apt to repel the modern mathematician, to cause him to return to
the more familiar field. The resulting concentration has led to the
powerful methods, already referred to, for studying algebraic varie-
ties. In the transcendental domain, on the other hand, we have a
multitude of interesting but particular geometric forms, — some
suggested by mechanics and physics, others derived from their relation
to algebraic curves, or by the interpretation of analytic results —
a few thousands of which have been considered of sufficient importance
to deserve specific names. ^ The problem at issue is then a practical
one (comparable with corresponding discussions in natural history) :
to formulate a principle of classification which will apply, not to all
possible curves, but to as many as possible of the usual important
transcendental curves.

The most fruitful suggestion hitherto applied has come from
the consideration of differential equations: almost all the important
transcendental curves satisfy algebraic differential equations, and
these in the great majority of cases are of the first order. Hence the
need of a systematic discussion of the curves defined by any algebraic
equation Fix, y, y') =0, the so-called panalgehraic curves of Loria. If
F is of degree n in y' and of degree v in x, y, the curve is said to belong
to a system with the characteristics (n, v) , and we thus have an im-
portant basis for classification. Closely related is the theory of the
Clebsch connex; this figure, it is true, is considered as belonging to
algebraic geometry, but it defines (by means of its principal coinci-
dence) a system of usually transcendental panalgebraic curves.

Both points of view appear to characterize certain systems of
curves rather than individual curves. The following interpretation
may serve as a simple geometric definition of the curves considered.

With any plane curve C we may associate a space curve in this
way: at each point of C erect a perpendicular to the plane whose
length represents the slope of the curve at that point; the locus of
the end points of these' perpendiculars is the associated space curve

^ Here would belong in particular the theory of algebraic curves based on link-
ages. Little advance has been made beyond the existence theorems of Kempe
and Koenigs. An important unsolved problem is the determination of the link-
age with minimum number of pieces by which a given curve can be described.

^ Cf. Loria, Spezielle Kurven, Leipzig, 1902.

574 GEOMETRY

C. Not every space curve is obtained in this way, but only those
whose tangents belong to a certain linear complex. If C is algebraic
so is C, and then an infinite number of algebraic surfaces may be
passed through the latter. If C is transcendental, so is C , and
usually no algebraic surface can be passed through it. Sometimes,
however; one such algebraic surface F exists. (If there were two,
C and C would be algebraic.) It is precisely in this case that the
curve C is panalgebraic in the sense of Loria's theory. That such a
curve belongs to a definite system is seen from the fact that while the
surface F is unique, it contains a singly infinite number of curves
whose tangents belong to the linear complex mentioned, and the
orthogonal projections of these curves constitute the required sj^stern.
The principal problems in this field which require treatment are:
first, the exhaustive discussion of the simplest systems, correspond-
ing to small values of the characteristics n and v ; second, the study of
the general case in connection with (1) algebraic differential equa-
tions, (2) connexes, and (3) algebraic surfaces and linear complexes.

Natural or Intrinsic Geometry

In spite of the immediate triumph of the Cartesian system at the
time of its introduction into mathematics, rebellion against what
may be termed the tyranny of extraneous coordinates, first expressed
in the Characteristica geometrica of Leibnitz, has been an ever-present
though often subdued influence in the development of geometry.
Why should the properties of a curve be expressed in terms of x's
and ?/'s which are defined not by the curve itself, but by its relation
to certain arbitrary elements of reference? The same curve in differ-
ent positions may have unlike equations, so that it is not a simple
matter to decide whether given equations represent really distinct or
merely congruent curves. The idea of the so-called natural or in-
trinsic coordinates had its birth during the early years of the nine-
teenth century, but it is only the systematic treatment of recent
years which has created a new field of geometry.

For a plane curve there is at each point the arc s measured from
some fixed point on the curve, and the radius of curvature p; these
intrinsic co5rdinates are connected by a relation p=f{s) which is
precisely characteristic of the curve, that is, the curves corresponding
to the equation differ only in position. There is, however, still
something arbitrary in the point taken as origin. This is eliminated
by taking as coordinates p and its derivative 8 taken with respect
to the arc; so that the final intrinsic equation is of the form 8 =F(p).
There is no difficulty in extending the method to space curves. The
two natural equations necessary are here T = (f)(p), 8=v//(p), where
p and T are the radii of first and second curvature and 8 is the arc
derivative of p.

PRESENT PROBLEMS OF GEOMETRY 575

The application to surfaces is not so evident. Thus, in Cesaro's
standard work, while the discussion of curves is consistently in-
trinsic, this is true to only a slight extent in the treatment of surfaces.
The natural geometry of surfaces is in fact only in process of forma-
tion. Bianchi proposes as intrinsic the familiar representation by
means of the two fundamental quadratic differential forms; but,
although it is true that the surfaces corresponding to a given pair
of forms are necessarily congruent, there is the disadvantage, arising
from the presence of arbitrary parameters, that the same surface
may be represented by distinct pairs of forms. One way of over-
coming this difficulty is to introduce the common feature of all pairs
corresponding to a surface, that is, the invariants of the forms: in
this direction we may cite Ricci's principle of covariant differentia-
tion and Maschke's recent application of symbolic methods.

The basis of natural geometry is, essentially, the theory of differ-
ential invariants. Under the group of motions, a given configuration
assumes go ^ positions, where r is in general 6, but may be smaller
in certain cases. The r parameters which thus enter in the analytic
representation may be eliminated by the formation of differential
equations. The aim of natural geometry is to express these differ-
ential equations in terms of the simplest geometric elements of the
given configuration.

The beginning of such a discussion of surfaces was given by Sophus
Lie in 1896 and his work has been somewhat simplified by Scheffers.
As natural coordinates we may take the principal radii of curvature
Ri, Ri, at a point of the surface, and their derivatives

taken in the principal directions. For a given surface (excluding
the Weingarten class) the radii are independent, and there are four
relations of the form

^22=/22(^l^ ■^2)-

Conversely, these equations are not satisfied by any surfaces except
those congruent or symmetric to the given surface.

It is to be noticed that four equations thus appear to be necessary
to define a surface, although two are sufficient for a twisted curve.
If a single equation in the above-mentioned natural coordinates is
considered, it is not, as in the case of ordinary coordinates, charac-
teristic: surfaces not congruent or symmetric to the given surface
would satisfy the equation. The apparent inconsistency which arises
is removed, however, by the fact that the four natural equations are

dR^

^ ^^1

, dR2

. dR:

^,0= —

o„ = —

0,,= — ■

ds.

'' ds.

" ds.

576 GEOMETRY

dependent.* It is just this that makes the subject difficult as com-
pared with the theory of curves, in which the defining equations are
entirely arbitrary. The questions demanding treatment fall under
these two headings: first, the derivation of the natural equations
of the familiar types of surfaces, and second, the study of the new
types that correspond to equations of simple form. The natural
geometry of the Weingarten class of surfaces requires a distinct basis.

The fact that intrinsic coordinates are, at bottom, differential
invariants with respect to the group of motions, suggests the exten-
sion of the same idea to the other groups. Thus in the projective
geometry of arbitrary (algebraic or transcendental) curves, coor-
dinates are required which, unlike the distances and angles ordin-
arily used, are invariant under projection. These might, for exam-
ple, be introduced as follows. At each point of the general curve C,
there is a unique osculating cubic and a unique osculating W (self-
projective) curve. Connected with each of these osculating curves
is an absolute projective invariant defined as an anharmonic ratio.
These ratios may then be taken as natural projective co5rdinates
y and o), and the natural equation on the curve is of the form
y=J'((jj), The principal advantage of such a representation is that
the necessary and sufficient condition for the equivalence of two
curves under projective transformations is simply the identity of the
corresponding equations.

Returning to the theory of surfaces, natural coordinates may
be introduced so as to fit into the so-called geometry of a flexible
but inextensible surface, originated by Gauss, in which the criterion
of equivalence is applicability, or, according to the more accurate
phraseology of Voss, isometry. Intrinsic coordinates must then be
invariant with respect to bending (Biegungsinvariante) . This pro-
perty is fulfilled, for example, by the Gaussian curvature k and the
differential parameters connected with it X=A (k, k), fjL=A(K, X),
v=A{'K, X), all capable of simple geometric interpretation. The
intrinsic equations are then of the form ix=(f)(K, X), v=(f)(K, X).

A pair of equations of this kind thus represent, not so much a
single surface S, as the totality of all surfaces applicable on S (or
into which S may be bent) — a totality which is termed a complete
group G, since no additional surfaces are obtained when the same
process is applied to any member of the totality. The discussion of
such groups is ordinarily based on the first fundamental form (repre-
senting the squared element of length), since this is the same for
isometric surfaces; though of course it changes on the introduction
of new parameters.

The simplest example of a complete isometric group is the group

^ The three relations connecting the functions /n, /12, hu /22 have been worked
out recently by S. Heller, Math. Annalen, vol. lviii (1904).

PRESENT PROBLEMS OF GEOMETRY 577

typified by the plane, consisting of all the developable surfaces. In
this case the equations of the group may be obtained explicitly, in
terms of eliminations, differentiations, and quadratures. This is,
however, quite exceptional; thus, even in the case of the surfaces
applicable on the unit sphere (surfaces of constant Gaussian curv-
ature + 1), the differential equation of the group has not been
integrated explicitly. In fact, until the year 1866, not a single case
analogous to that of the developable surfaces was discovered. Wein-
garten, by means of his theory of evolutes, then succeeded in deter-
mining the complete group of the catenoid and of the paraboloid
of revolution, and, some twenty years later, a fourth group defined
in terms of minimal surfaces.

During the past decade, the French geometers have concentrated
their efforts in this field mainly on the arbitrary paraboloid (and to
some extent on the arbitrary quadric). The difficulties even in this
extremely restricted and apparently simple case are great, and are
only gradually being conquered by the use of almost the whole
wealth of modern analysis and the invention of new methods which
undoubtedly have wider fields of application. The results obtained
exhibit, for example, connections with the theories of surfaces of
constant curvature, isometric surfaces, Backlund transformations,
and motions with two degrees of freedom. The principal workers
are Darboux, Goursat, Bianchi, Thybaut, Cosserat, Servant, Gui-
chard, and Raffy.

Geometry im Grossen

The questions we have just been considering, in common with
almost all the developments of general or infinitesimal geometry,
deal with the properties of the figure studied im Kleinen, that is,
in the sufficiently small neighborhood of a given point. Algebraic
geometry, on the other hand, deals with curves and surfaces in their
entirety. This distinction, however, is not inherent in the subject-
matter, but is rather a subjective one due to the limitations of our
analysis: our results being obtained by the use of power series are
valid only in the region of convergence. The properties of a curve
or surface (assumed analytic) considered as a whole are represented
not by means of function elements, but by means of the entire func-
tions obtained say by analytic continuation.

Only the merest traces of such a transcendental geometry im
Grossen are in existence, but the interest of many investigators is
undoubtedly tending in this direction. The difficulty of the problems
which arise (in spite of their simple and natural character) and the
delicacy of method necessary in their treatment may be compared
to the corresponding problems and methods of celestial mechanics.
The calculation of the ephemeris of a planet for a limited time is

578 GEOMETRY

a problem im Kleinen, while the discovery of periodic orbits and the
theory of the stability of the solar system are typical problems im
Grossen.

The principal problems in this field of geometry are connected
with closed curves and surfaces. Of special importance are the inves-
tigations relating to the closed geodesic lines which can be drawn
on a given surface, since these are apt to lead to the invention of
methods applicable to the wider field of dynamics. Geodesies may
in fact be defined dynamically as trajectories of a particle constrained
to the surface and acted upon either by no force or by a force due to
a force function U whose first differential parameter is expressible
in terms of U. The few general theorems known in this connection
are due in the main to Hadamard (Journal de Mathematiques, 1897,
1898). Thus, on a closed surface whose curvature is everywhere
positive, a point describing a geodesic must cross any existing closed
geodesic an infinite number of times, so that, in particular, two
closed geodesies necessarily intersect.^ On a surface of negative
curvature, under certain restrictions, there exist closed geodesies
of various topological types, as well as geodesies which approach
these asymptotically.

As regards surfaces all of whose geodesies are closed, the investi-
gations have been confined entirely to the case of surfaces of revo-
lution, the method employed being that suggested by Darboux in
the Cours de Mecanique of Despeyrons. Last year Zoll ^ succeeded
in determining such a surface (beyond the obvious sphere) which
differs from the other known solutions in not having any singularities.
Analogous problems in connection with closed lines of curvature
and asymptotic lines will probably soon secure the consideration
they deserve.

A problem of different type is the determination of applicability
criteria valid for entire surfaces. The ordinary conditions (in terms
of differential parameters) assert, for example, the applicability of
any surface of constant positive curvature upon a sphere; but the
bending is actually possible only for a sufficiently small portion of the
surface. A spherical surface as a whole cannot be applied on any
other surface, that is, cannot be bent without extension or tearing.
This result is analogous to the theorem known to Euclid, although
first proved by Cauchy, that a closed convex polyhedral surface is
necessarily rigid. Lagrange, Minding, and Jellet stated the result for
all closed convex surfaces, but the complete discussion is due to
H. Liebmann.^ The theory of the deformation of concave surfaces

^ In a paper read before the St. Louis meeting of the American Mathematical
Society, Poincar^ stated reasons which make very probable the existence of at
least three closed geodesies on a surface of this kind.

^ Math. Annalen, vol. lvii (1903).

^ Gottingen Nachrichten, 1899; Math. Annalen, vols, liii, liv.

PRESENT PROBLEMS OF GEOMETRY 579

is far more complicated, and awaits solution even in the case of
polyhedral surfaces.

Beltrami's visualization of Lobachevsky's geometry by pictur-
ing the straight lines of the Lobachevsky plane as geodesies on
a surface of constant negative curvature is well known. However,
since the known surfaces of this kind, like the pseudosphere, have
singular Hues, this method really depicts only part of the plane. In
fact Hilbert (Transactions of the American Mathematical Society
for 1900), by very refined considerations, has shown that an analytic
surface of constant negative curvature which is everywhere regular
does not exist, so that the entire Lobachevsky plane cannot be
depicted by any analytic surface.^ There remains undecided the
possibility of a complete representation by means of a non-analytic
surface. The partial differential equation of the surfaces of negative
constant curvature is of the hyperbolic type and hence does admit
non-analytic solutions.^ (This is not true for surfaces of positive
curvature, since the equation is then of elliptic type.) The discussion
of non-analytic curves and surfaces will perhaps be one of the really
new features of future geometry, but it is not yet possible to indicate
the precise direction of such a development.^

Other theories belonging essentially to geometry im Grossen
are the questions of analysis situs, or topology, to which reference has
been made on several occasions, and the properties of the very
general convex surfaces introduced by Minkowski in connection
with his Geometrie der Zahlen.

Systems of Curves — Differential Equations

Although projective geometry has for its domain the investigation
of all properties unaltered by collineation, attention has been con-
fined almost exclusively to the algebraic configuration, so that pro-
jective is often confused with algebraic geometry. To the more
general projective geometry belong, for example, the ideas of oscu-
lating conic of an arbitrary curve and the asymptotic lines of an
arbitrary surface, and Mehmke's theorem which asserts that when
two surfaces touch each other, the ratio of their Gaussian curvatures
at the point of contact is an (absolute) projective invariant. The
field for investigation in this direction is of course very extensive,
but we may mention as a problem of special importance the deriva-

^ The entire projective plane, on the other hand, can be so depicted on a sur-
face devised by W. Boy {Inaugural Dissertation, Gottingen, 1901).

^ According to Bernstein (Math. Annalen, vol. lix, 1904, p. 72), the proof given
by Lutkemeyer (Inaugural Dissertation, Gottingen, 1902) is not valid, though
the conclusion is correct.

^ Lebesgue (Comptes Rendus, 1900, and These, 1902) has examined the theory
of surfaces applicable on a plane without assuming the existence of derivatives
for the defining functions, and thereby obtains an example of a non-ruled develop-
able.

580 GEOMETRY

tion of the conditions for the projective equivalence of surfaces in
terms of their fundamental quadratic forms.

Coordinate with what has just been stated, that general configura-
tions may be studied from the projective point of view, is the fact
that algebraic configurations may be studied in relation to general
transformation theory. One may object that, with respect to the
group of all (analytic) point transformations, the algebraic con-
figurations do not form a hody,^ that is, are not converted into
algebraic configurations; but such a body is obtained by adjoining
to the algebraic all those transcendental configurations which are
equivalent to algebraic. As this appears to have been overlooked,
it seems desirable to give a few concrete instances, of interest in
showing the effect of looking at familiar objects from a new and
more general point of view.

As a first example, consider the idea of a linear system of plane
curves. In algebraic geometry, a linear system is understood to be
one represented by an equation of the form

where the X's are parameters and the i^'s are polynomials in x,y. On
the other hand, in general (infinitesimal) geometry, a system is defined
to be linear when it can be reduced (by the introduction of new
parameters) to the same form where the F's are arbitrary functions.
The first definition is invariant under the projective group; the sec-
ond, under the group of all point transformations. If now we apply
the second definition to algebraic curves, the result does not coincide
with that given by the first definition. Thus, every one-parameter
system is linear in the general sense, while only pencils of curves are
linear in the projective sense. The first case of real importance is,
however, the two-parameter system, since here each point of view
gives restricted, though not identical, types. An example in point
is furnished by the vertical parabolas tangent to a fixed line, the
equation of the system being y = (ax+hy. From the algebraic or
projective point of view, this is a quadratic system since the para-
meters are involved to the second degree; but the system is linear
from the general point of view since its equation may be written
ax-\-h — \/y=0. This suggests the problem: Determine the systems
of algebraic curves which are linear in the general sense.

As a second example, consider, from both points of view, the
equivalence of pencils of straight lines in the plane. By means of
collineations any two pencils may be converted into any other two;

^ The most extensive group for which the algebraic configurations form a body
consists of all algebraic transformations. It is rather remarkable that even this
theory has received no development.

* Halphen, Laguerre, Forsj4;h. This theory has been extended to simultaneous
equations and applied geometrically by E. J. Wilczynski {Trans. Amer. Math.
Soc, 1901-1904).

PRESENT PROBLEMS OF GEOMETRY 581

but if three pencils are given, it is necessary to distinguish the case
where the three base points are in a straight line from the case where
they are not so situated. We thus have two protectively distinct
cases, which may be represented canonically by: (1) a;=const.,
2/= const., x+ 2/= const., and (2) x= const., t/ = const., y/z =const.
The first type may, however, be converted into the second by the
transcendental transformation x^=^,y^=^,so that, in the general
group of point transformations, all sets of three pencils are equivalent.
The discussion for four or more pencils yields the rather surprising
result that the projective classification remains valid for the larger
group.

Dropping these special considerations on algebraic systems, let us
pass to the theory of arbitrary systems of curves, or, what is equiva-
lent, the geometry of differential equations. While belonging to the
cycle of theories due primarily to Sophus Lie, it has received little
development in the purely geometric direction. Most attention has
been devoted to special classes of differential equations with respect
to special groups of transformations. Thus there is an extensive
theory of the homogeneous linear equations with respect to the
group x^=$(x), yi=yTj(x) which leaves the entire class invariant.^
A special theosy which deserves development is that of equations of
the first order with respect to the infinite group of conformal trans-
formations.

As regards the general group of all point transformations, all
equations of the first order are equivalent, so that the first case of
interest is the theory of the two-parameter systems. The invariants
of the differential equation of second order have been discussed
most completely in the prize essay of A. Tresse (submitted to the
Jablonowski Gesellschaft in 1896), with application to the equiva-
lence problem. A specially important class, treated earlier by Lie
and R. Liouville, consists of the equations of cubic type

y''=Ay''+By'' + Cy'+D,
where the coefficients are functions of x, y. It includes, in particular,
the general linear system and all systems capable of representing
the geodesies of any surface. While the analytical conditions which
characterize these subclasses are known, little advance has been
made in their geometric interpretation.

Perhaps the simplest configuration belonging to the field considered,
that is, having properties invariant under all point transformations,
is that composed of three simply infinite systems of curves, which
may be represented analytically by an equation of third degree in
y' with one-valued functions of x, y for coefficients. In the case of
equations of the fourth and higher degree in y' , certain invariants

^ The elementary (metric) theory of curve systems has been too much neglected ;
it may be compared in interest and extent with the usual theory of surfaces.

582 GEOMETRY

may be found immediately from the fact that when x and y undergo
an arbitrary transformation, the derivative y' undergoes a fractional
linear transformation (of special type). The invariants found from
this algebraic principle are, however, in a sense, trivial, and the real
problem remains almost untouched: to determine the essential
invariants due to the differential relations coimecting the coefficients
in the linear transformation of the derivative.

General Theory of Transformations

Closely connected with the geometry of differential equations
that we have been considering is the geometry of point transform-
ations. In the former theory the transformations enter only as
instruments, in the latter these instruments are made the subject-
matter of the investigation. The distinction is parallel to that which
occurs in projective geometry between the theory of projective
properties of curves and surfaces and the properties of collineations.
(It may be remarked, however, that although a transformation is
generally regarded as dynamic and a configuration as static, the
distinction is not at all essential. Thus a point transformation or
correspondence between the points of a plane may be viewed as
simply a double infinity of point pairs; on the other hand, a curve
in the plane may be regarded as the equivalent of a correspondence
between the points of two straight lines. ^)

We consider first two problems concerning the general (analytic)
point transformation which are of interest and importance from the
theoretic standpoint. The one relates to the discussion of the char-
acter of such a transformation in the neighborhood of a given point.
Transon's theorem states that the effect of any analytic transform-
ation upon an infinitesimal region is the same as that of a pro-
jective transformation. This is true, however, only in general; it
ceases to hold when the derivatives of the defining functions vanish
at the point considered. What is the character of the transformation
in the neighborhood of such singular points ?

A more fundamental problem relates to the theory of equiva-
lence. Consider a transformation T which puts in correspondence
the points P and Q of a plane. Let the entire plane be subjected to
a transformation S which converts P into P' and Q into Q\ We thus
obtain a new transformation T' in which P' and Q' are corresponding
points. This is termed the transform of 3" by means of /S,the relation
being expressed symbolically by T' =S~'^TS. The question then arises
whether all transformations are equivalent, that is, can any one be
converted into any other in the manner defined. The answer de-
pends on certain functional equations which also arise in connection

^ Geometry on a straight line, in its entirety, is as rich as geometry in a plane
or in space of any number of dimensions.

PRESENT PROBLEMS OF GEOMETRY 583

with the question whether an arbitrary transformation belongs to
a continuous group. The problem deserves treatment not merely for
the analytic transformations, but also for the algebraic and for
the continuous transformations.^

Aside from such fundamental questions, further development
is desirable both in the study of the general properties (associated
curve systems and contact relations) of an arbitrary transforma-
tion, and in the introduction of new special types of transformation,
for instance, those which may be regarded as natural extensions of
familiar types.

The main problems in the theory of point transformation are
connected with certain fields of application which we now pass in
review.

1. Cartography. A map may be regarded, abstractly, as the point
by point representation of one surface upon another, the case of
especial practical importance being, of course, the representation of
a spherical or spheroidal surface upon the plane. As it is impossible
to map any but the developable surfaces without distortion upon a
plane, the chief types of available representation are characterized
by the invariance of certain elements, as angles or areas, or the
simple depiction of certain curves, as of geodesies by straight lines.
Most attention has been devoted to the conformal type, but the
question proposed by Gauss remains unsolved: what is the best
conformal representation of a given surface on the plane, that is,
the one accompanied by the minimum distortion? The answer, of
course, depends on the criterion adopted for measuring the degree
of distortion, and it is in this direction that progress is to be
expected.

2. Mathematical theory of elasticity. As a geometric foundation
for the mechanics of continua, it is necessary to study the most
general deformation of space, defined say by putting Xi, yi, Zi equal
to arbitrary functions of x, y, z. The most elegant analytical repre-
sentation, as given for instance in the memoir of E. and F. Cosserat
(Annales de Toulouse, volume 10), is obtained by introducing the
elements of length ds and ds^ before and after deformation, and the
related quadratic diifferential form dsf — ds^=2e^dx^+2e^dy^+2e3dz^
+2y^ dydz-\-2y^ dxdz+2y^dxdy. The theory is thus seen to be ana-
logous to though of course more complicated than the usual theory of
surfaces. The six functions of x, y, z which appear as coefficients
in this form are termed the components of the deformation. Their

' This problem is not to be confused with the similar (but simpler) question
connected with Lie's division of (analytic) groups into demokratisch and aristo-
kratisch. In those of the first kind all the infinitesimal transformations are
equivalent, in those of the second there exist non-equivalent infinitesimal trans-
formations. Lie shows that all finite groups are aristokratisch, while the groups
of all (analytic) point and contact transformations are demokratisch. Cf. Leip-
ziger Berichte, vol. xlvii (1895), p. 271.

584 GEOMETRY

importance is due to the fact that they vanish only when the trans-
formation is a rigid displacement, so that two deformations have
the same components when, and only when, they differ by a dis-
placement. The case where the components are constants leads to
the homogeneous deformation (or afiine transformation of the geo-
meters), the type considered almost exclusively in the usual dis-
cussions of elasticity. It would seem desirable to study in detail
the next case which presents itself, namely, that in which the com-
ponents are linear functions of x, y, z.

In the general deformation, the six components are not inde-
pendent, but are connected by nine differential equations analogous
to those of Codazzi. The fact that a transformation is defined by
three independent functions indicates, however, that there should be
only three distinct relations between the components. This means
that the nine equations of condition which occur in the standard
theory are themselves interdependent; but their relations (analogous
to syzygies among syzygies in the algebra of forms) do not appear
to have been worked out.

3. Vector -fields. From its beginning in the Faraday-Maxwell
theory of electricity until the present day, the course which the
discussion of vector fields has followed has been guided almost
entirely by external considerations, namely, the physical applications.
While this is advantageous in many respects, it cannot be denied
that it has led to lack of symmetry and generality. The time seems
to be ripe for a more systematic mathematical development. The
vector field deserves to be introduced as a standard form into geo-
metry.

Abstractly, such a field is equivalent to a point transformation of
space, since each is represented by three scalar relations in six variables.
Instead of taking these variables as the coordinates of corresponding
points, it is more convenient to consider three as the coordinates
X, y, z oi a. particle and t'he other three as components u, v, w of its
velocity; we thus picture the set of functional relations by means
of the steady motion of a hypothetical space-filling fluid. This image
should be of service even in abstract analysis ; for its role is analogous
to that of the curve in dealing with a single relation between two
variables. The streaming of a material fluid is, of course, not suffi-
ciently general for such a purpose, since, in virtue of the equation of
continuity, it images only a particular class of vector fields.

In addition to the ordinary vector fields, physics makes use of
so-called hypervector fields, which, geometrically, lead to configur-
ations consisting of a triply infinite system of quadric surfaces, one
for each point of space. In the special case of interest in hydro-
dynamics (irrotational motion), the configuration simplifies in that
the quadrics are ellipsoids about the corresponding points as centres.

PRESENT PROBLEMS OF GEOMETRY 585

This is equivalent to the tensor field which arises in studying the
moments of inertia of an arbitrary distribution of mass. The more
general case actually arises in Maxwell's theory of magnetism.

4. As a final domain of application we mention the class of ques-
tions which have received systematic treatment, under the title of
nomography, only during the past few years. This subject deals with
the methods of representing graphically, in a plane, functional
relations containing any number of variables. Thus a function of
two independent variables, z=f{x, y), may be represented by the
system of plane curves f(x, y) = c, each marked with the correspond-
ing value of the parameter. This " parametered " system is then
a cartesian graphical table, which is the simplest type of abacus or
nomogram.

By means of any point transformation, one nomogram is con-
verted into another which may serve to represent the same functional
relation. The importance of this process of conversion (the so-called
anamorphosis of Lalanne and Massau) depends on the fact that it
may replace a complicated table by a simpler. The problems which
arise (for example, the determination of aU relations between three
variables which can be represented by a nomogram composed of
three systems of straight lines^) are of both practical and theoretical
interest. The literature is scattered through the French, ItaUan,
and German technological journals, but a systematic presentation
of the main results is to be found in the Traite de Nomographie
of d'Ocagne (Paris, 1899).

We return to the abstract theory of transformations. The type
of transformation we have been considering, converting point into
point, is only a special case of more general types. The most im-
portant extension hitherto made depends upon the introduction of
differential elements. Thus the lineal element or directed point
(x, y, y') leads to transformations which in general convert a point
into a system of elements; when the latter form a curve, every curve
is converted into a curve and the result is termed a contact trans-
formation. Backlund has shown that no extension results from the
elements of second or higher order: osculation transformations are
necessarily contact transformations. The discussion of elements of
infinitely high order, defined by an infinite set of coordinates {x, y,
y' 1 y" , • • •); i^is-y perhaps lead to a real extension. The question may
be put in this form: Are there transformations (in addition to or-
dinary contact transformations) which convert analytic curves into
analytic curves in such a way that contact is an invariant relation?
The idea of curve transformation in general will probably be worked

^ The case of three systems of circles has also been discussed. See d'Ocagne,
Journal de I'Ecole Polytechnique, 1902.

586 GEOMETRY

out in the near future : what is the most general mode of setting up
a correspondence which associates with every Jordan curve another
Jordan curve? Such discussions are aspects of geometry with an
infinite number of dimensions.

After a review of the kind given in this paper, one is tempted to
ask: What is it which influences the mathematician in selecting
certain (out of an infinite number of equally conceivable) problems
for investigations? It is true, of course, that his subject is ideal,
self-created, and that "Das Wesen der Mathematik liegt in ihrer
Freiheit." Georg Cantor would indeed replace the term pure mathe-
matics by free mathematics. This freedom, however, is not entirely
caprice. The investigators of each age have always felt it their
duty to deal with the unsolved questions and to generalize the re-
sults and conceptions inherited from the past, to correlate with
other fields of contemporaneous thought, to keep in contact, as far
as possible, with the whole body of truth. This is not all, however.
The influence of aesthetic considerations, though less subject to
analysis, has been, and still is, of at least equal importance in guiding
the course of mathematical development.

SHORT PAPERS

The Section of Geometry was very fully attended and productive of extended
discussion and a number of supplementary papers. For the same reason as in the
Section of Algebra and Analysis it is impossible to give a satisfactory resume of
the short papers on this subject owing to their close technical reasoning.

The first paper was presented by Professor Harris Hancock, of the University
of Cincinnati, on "Algebraic Minimal Surfaces."

The second paper was presented by Professor H. T. Bhchfeldt, of Leland Stan-
ford Jr. University, on the subject "Concerning some Geometrical Properties
of Surfaces of Revolution."

The third paper was presented by Professor George Bruce Halsted, of Kenyon
College, on " Non-Euclidean Spherics."

The fourth paper was presented by Professor Arnold Emch, of the University
of Colorado, on "The Configuration of the Points of Inflection of a Plane
Cubic and their Harmonic Polars."

The fifth paper was presented by Professor H. P. Manning, of Brown University,
on " Representation of Complex Variables in Space of Four Dimensions."

The sixth paper was read by Professor G. A. Bliss, of the University of Missouri,
on "Concerning Calcidus of Variations."

The seventh paper was presented by Professor L. W. Dowling, of the University
of Wisconsin, on "Certain Universal Curves."

SECTION C— APPLIED MATHEMATICS

{Hall 7, September 24, 3 p. m.)

Chairman : Professor Arthur G. Webster, Clark University, Worcester,
Mass.

Speakers: Professor Ludwig Boltzmann, University of Vienna.

Professor Henri Poincare, The Sorbonne; Member of the Insti-
tute of France.

Secretary: Professor Henry T. Eddy, University of Minnesota.

THE RELATIONS OF APPLIED MATHEMATICS

BY LUDWIG BOLTZMANN
{Translated from the German by Professor S. Epsteen, University of Chicago)

[Ludwig Boltzmann, Professor of Physics, University of Vienna, since 1902.
b. Vienna, Austria, 1840. Studied, Vienna, Heidelberg, and Berlin. Professor
of Physico-Mathematics, University of Gratz, 1869-73; Professor of Mathe-
matics, University of Vienna, 1873-76; Professor of Experimental Physics,
University of Gratz, 1876-90; Professor of Theoretical Physics, University
of Munich, 1891-95; ibid. University of Vienna, 1895-1900; Professor of
Physics, University of Leipzig, 1900-02. Author of Vorlesungen iiber Max-
well's Theorie der Elekt rizitat und des Lichts; Vorlesungen iiber Kinetische
Gastheorie; Vorlesungen iiber die Prinzipe der Mechanik.]

My present lecture has been put under the heading of applied
mathematics, while my activity as a teacher and investigator be-
longs to the science of physics. The immense gap which divides
the latter science into two distinct camps has almost nowhere been
so sharply emphasized as in the division of the lecture material
of this scientific congress, which covers such an enormous range of
subjects that one may designate it as a flood, or, to preserve local
coloring, as a Niagara of scientific lectures. I speak of the division
of physics into theoretical and experimental. Although I have
been assigned, as representative of theoretical physics, to "A. —
Normative Science," experimental physics appears much later under
'' C. — Physical Science." Between them lie history, science of lan-
guage, literature, art, and science of religion. Over all this, however,
the theoretical physicist must extend his hand to the experimental
physicist. We shall therefore not be able to avoid entirely the ques-
tion of the justification of dividing physics into two parts and, in
particular, into theoretical and experimental.

Let us listen first of all to an investigator of a time when natural
science had not yet grown beyond its first beginnings, to Emmanuel
Kant. Kant requires of each science that it should be developed

592 APPLIED MATHEMATICS

logically from unified principles and firmly established theories.
Natural science seems to him a primary science only in so far as
it rests on a mathematical basis. Thus, he does not reckon the chem-
istry of his day among the sciences, because it rests merely upon
an empirical basis and lacks a unified, regulative principle.

From this point of view theoretical physics is preferred to ex-
perimental physics, and occupies, in a sense, a higher rank. Experi-
mental physics was merely to gather the material, but it remained
for the theoretical physics to form the structure.

But the succession in the order of rank becomes reversed when
we take into account the acquisitions of the last decades as well as
the progress which is to be expected in the immediate future. The
chain of experimental discoveries of the last century received a
fitting completion with the discovery of the Rontgen rays. Con-
nected with these there appear in the present century a multitude
of new rays, with the most enigmatical properties, which have the
profoundest effects upon our conceptions of nature. The more
enigmatical these newly discovered facts are, and the more they
seem at first to contradict our present conceptions, the greater the
successes which they promise for the future. But this is not the occa-
sion for the discussion of these experimental triumphs. I must leave
to the representatives of experimental physics at this Congress the
prolific problem of portraying all of the fruits which have hitherto
been gathered in this domain, one might almost say, daily, and
those which are to be expected.

The representative of theoretical physics scarcely finds himself in
an equally fortunate position. Great activity does indeed prevail
in this domain. One could almost say that it is in process of revolu-
tion. Only how much less tangible are the results here attained in
comparison with those in experimental physics! It appears here
that in a certain sense experimentation deserves precedence over
all theory. An immediate fact is at once comprehensible. Its fruits
may become evident in the shortest time, such as the various appli-
cations of the Rontgen rays and the utilization of the Hertz waves
in wireless telegraphy. The battle which the theories have to fight
is, however, an infinitely wearisome one; indeed, it seems as if cer-
tain disputed questions which existed from the beginning wiU live
as long as the science.

Every firmly established fact remains forever unchangeable; at
most, it may be generalized, completed, additions may be made,
but it cannot be completely upset. Thus it is explained why the
development of experimental physics is continuously progressive,
never making a sudden jump, and never visited by great tremblings
and revolutions. It occurs only in rare instances that something
which was regarded as a fact turns out afterwards to have been an

RELATIONS OF APPLIED MATHEMATICS 593

error, and in such cases the explanations of the errors follow soon,
and they are not of great influence on the structure of the science as
a whole.

It is, indeed, strongly emphasized that every established and
logically recognized truth must remain incontrovertible. Although
this cannot be doubted, experience teaches that the structure of our
theories is by no means composed entirely of such incontrovertibly
established truths. They are composed rather of many arbitrary
pictures of the connections between phenomena, of so-called hypo-
theses.

Without some departure, however slight, from direct observation,
a theory or even an intelligibly connected practical description for
predicting the facts of nature cannot exist. This is equally true of
the old theories whose foundations have become questionable, and
of the most modern ones, which are resigning themselves to a great
illusion if they regard themselves as free from hypotheses.

The hypotheses may perhaps be indefinite, or may be in the shape
of mathematical formulae, or the thought may be equivalent to the
latter, but expressed in words. In the latter cases the agreement
with given data may be checked step by step; a complete revolu-
tion of that previously constructed is indeed not absolutely impos-
sible, as, for example, if the law of the conservation of energy should
turn out to be incorrect. But such a revolution will be exceedingly
rare and highly improbable.

Such an indefinite, slightly specialized theory might serve as a
guiding thread for experiments whose purpose is a detailed develop-
ment of knowledge previously acquired and which is proceeding in
barren channels ; beyond this its usefulness does not reach.

In contradistinction to these are the hypotheses which give the
imagination room for play and by boldly going beyond the material
at hand afford continual inspiration for new experiments, and are
thus pathfinders for the most unexpected discoveries. Such a theory
will indeed be subject to change, a very complicated mass of inform-
ation will be brought together and will then be replaced by a new
and more comprehensive theory in which the old one will be the pic-
ture of a limited type of phenomena. Examples of this are the theory
of emission in regard to the description of the phenomena of catoptrics
and dioptrics, the hypothesis of an elastic ether in the representation
of the phenomena of interference and refraction of light, and the
notion of the electric fluid in the description of the phenomena of
electrostatics.

Moreover the theories which proudly designate themselves as free
from hypotheses are not exempt from great revolutions; thus, no one
will doubt that the so-called theory of energy will have completely
to alter its form if it desires to remain effective.

594 APPLIED MATHEMATICS

The accusation has been made that physical hypotheses have
sometimes proved injurious and have delayed the progress of the
science. This accusation is based chiefly upon the role which the
hypothesis of the electric fluid has played in the development of the
theory of electricity. This hypothesis was brought to a high stage
of perfection by Wilhelm Weber, and the general recognition which
his works found in Germany did indeed stand in the road of the
theory of Maxwell. In a similar manner Newton's emanation theory
stood in the way of the theory of undulations. But such incon-
veniences can scarcely be entirely avoided in the future. It will al-
ways be the tendency to complete as far as possible the prevailing
view, and to make it self-sufficient whenever such a theory is self-
consistent and does not in any way lead to a contradiction, whether
it consist of mechanical models, of geometrical pictures, or of mathe-
matical formulas. It will always be possible that a new theory will
arise which has not yet been tested by experiment and which will
represent a much larger field of phenomena. In such cases the older
theory will count the largest following until this field of phenomena
is brought into the range of experiment, and decisive tests demon-
strate the superiority of the newer one. It is certainly useful, if the
theory of Weber be always held up as a warning example, that one
should bear in mind the essential progressiveness of the intellect.
The services of Weber are not decreased by this, however; Maxwell
himself speaks of his theory with the greatest wonder. Indeed, this
instance cannot be taken into consideration against the usefulness of
hypotheses, since Maxwell's theory contained as much of the hypo-
thetical as any other. And this was eliminated only after it became
generally known through Hertz, Poynting, and others.

The accusation has also been raised against hypotheses in physics
that the creation and development ' of mathematical methods for
the computation of the hypothetical molecular motions has been
useless and even harmful. This accusation I cannot recognize as
substantiated. Were it so, the theme selected for my present lecture
would be an unfortunate one; and this fact may excuse me for
having lingered on this much-discussed subject and for having sought
to justify the use of hypotheses in physics.

I have not chosen for the thesis of my present lecture the entire
development of physical theory. Several years ago I treated this
subject at the German Naturforscherversammlung in Munich, and
although some new developments have taken place since then, I
should have to repeat myself a great deal. Moreover, one who has
committed himself to one faction is not in a position to judge the
other factions in a completely objective manner. I do not refer to a
criticism of its value; my lecture shall not criticise, but shall judge.
I am also convinced of the value of the views of my opponents and

RELATIONS OF APPLIED MATHEMATICS 595

only arise to repel them when they attempt to belittle mine. But
one can scarcely give as complete an account according to subject-
matter, and an exposition of the inter-relations of all ideas in the
views of another, as in his own,

I shall therefore select as the goal of my lecture to-day not merely
the kinetic theory of molecules, but, moreover, a highly specialized
branch of it. Far from denying that it contains hypotheses, I must
rather characterize it as a bold advance beyond the facts of observa-
tion. And I nevertheless do not consider it unworthy of this occa-
sion; this much faith do I have in hypotheses which present certain
peculiarities of observation in a new light or which bring forth rela-
tions among them which cannot be reached by other methods. We
must indeed be mindful of the fact that hypotheses require and are
capable of continuous development, and are only then to be aban-
doned when all the relations which they represent can be better
understood in some other manner.

To the above-mentioned problems, which are as old as the science
and still unsolved, belongs the one if matter is continuous, or if it
is to be considered as made up of discrete parts, of very many, but
not in the mathematical sense infinite, individuals. This is one of
the difficult questions which form the boundary of philosophy and
physics.

Even some decades ago, scientists felt very shy of going deeply
into the discussion of such questions. The one before us is too real
to be entirely avoided; but one cannot discuss it without touching on
some profounder still, such as upon the nature of the law of causation,
of matter, of force, and so forth. The latter are the ones of which it
was said that they did not trouble the scientist, that they belonged
entirely to philosophy. To-day the situation is different, there is
evident a tendency among scientists to consider philosophic questions,
and properly so. One of the first rules of science is never to trust
blindly to the instrument with which one works, but to test it in
all directions. How, then, are we to trust blindly to inherited and
historically developed conceptions, particularly when there are
instances known where they led us into error ? But in the examina-
tion of even the simplest elements, where is the boundary between
science and philosophy at which we should pause ?

I hope that none of the philosophers present will take offense or
perceive an accusation, if I say boldly that by assigning this question
to philosophy the resulting success has been rather meagre. Philo-
sophy has done noticeably little toward the explanation of these
questions, and from her own one-sided point of view she can do so just
as little as natural science can from hers. If real progress is possible,
it is only to be expected by cooperation of both of these sciences.
May I therefore be pardoned if I touch slightly upon these questions

596 APPLIED MATHEMATICS

although not a specialist; their connection with the aim of my lec-
ture is very intimate.

Let us consult the famous thinker already quoted, Emmanuel
Kant, on the question if matter is continuous, or if it is composed
of atoms. He treats of this in his Antimonies. Of all the questions
there raised, he shows that both the pro and con can be logically
demonstrated. It can be shown rigorously that there is no limit to
the divisibility of matter while an infinite divisibility contradicts the
laws of logic. Kant shows likewise that a beginning and end of time,
a boundary where space ceases, are as inconceivable as absolutely
endless duration, absolutely endless extension.

This is by no means the sole instance where philosophical thought
becomes tangled in contradictions; indeed, one finds them at every
step. The ordinary things of philosophy are sources of insolvable
riddles; to explain our perceptions it invents the concept of matter
and then finds that it is altogether unsuited to possess perception
itself or to generate perception in a spirit. With consummate acumen
it constructs the concept of space, or of time, and finds that it is
absolutely impossible that things should exist in this space, that
events should occur during this time. It finds insurmountable
difficulties in the relation of cause to effect, of body and soul, in
the possibility of consciousness, in short, everywhere and in every-
thing. Indeed, it finally finds it inexplicable and self-contradictory
that anything can exist at all, that something originated and is cap-
able of continuing, that everything can be classified according to
our categories, nor that there is a quite perfect classification. Such
a classification will always be a variable one and adapted to the
requirements of the moment. Also the breaking up of physics into
theoretical and experimental is merely a consequence of the preval-
ent division of methods and will not last forever.

My present thesis is quite different from the one that certain
questions are beyond the boundary of human comprehension. For
according to the latter, there is a deficiency, an incompleteness in the
human intelligence, while I consider the existence of these questions,
these problems, as an illusion. By superficial consideration it seems
astonishing, after this illusion is recognized, that the impulse to
answer those questions does not cease. Habit of thought is much too
powerful to release us.

It is here as with the ordinary illusion which continues operative
after its cause is recognized. In consequence of this is the feeling of
uncertainty, of want of satisfaction which the scientist feels when he
philosophizes. These illusions will yield but very slowly and gradually,
and I consider it as one of the chief problems of philosophy to set
forth clearly the uselessness of reaching beyond the limits of our
habits of thought and to strive, in the choice and combination of

RELATIONS OF APPLIED MATHEMATICS 597

concepts and words, to give the most useful expression of facts in a
manner which is independent of our inherited habits. Then all these
complications and contradictions must vanish. It must be made
clear what is stone in the structure of our thoughts and what is
mortar, and the oppressive sentiment, that the simplest things are
the most inexpUcable and the most trivial are the most mysterious,
becomes mere imagination-change.

To call upon logic seems to me as if one were to put on for a trip
into the mountains a long flowing robe, which always wrapped
itself about the feet so that one fell at the first steps while on the level.
The source of this kind of logic is the immoderate trust in the so-
called laws of thought. It is certain that we could not gather experi-
ence did we not have certain forms of connecting phenomena, that is
to say, of thought, innate. If we wish to call these laws of thought,
they are indeed a priori to the extent that they accompany every
experience in our soul, or if we prefer, in our brain. Only nothing
seems to me less reasonable than the conclusion from the reasoning
in this sense to certainty, to infallibility. These laws of thought
have been developed according to the same laws of evolution as
the optical apparatus of the eye, the acoustic apparatus of the ear,
and the pumping arrangements of the heart. In the course of human
development everything useless was eliminated, and thus a unity
and finish arose which might be mistaken for infallibility. Thus the
perfection of the eye, of the ear, of the arrangement of the heart
excite our admiration, without the absolute perfection of these
organs being emphasized, however. Just so little should the laws of
thought be regarded as absolutely infallible. They are the very ones
which have developed with regard to seizing that which is most
necessary and practically useful in the maintenance of life. With
these, the results of experimental investigation show more relation
than the examination of the mechanism of thought. We should,
therefore, not be surprised that the customary forms of thought
for the abstract are not entirely suited to practical applications
in far removed problems of philosophy, and that they have not
become applicable since the days of Thales. Therefore the simplest
things seem to be the most puzzling to the philosopher. And he
finds everywhere contradictions; these are nothing more, however,
than useless, incorrect facsimiles of that which is given us through
our thoughts. In facts there can be no contradictions. As soon as
contradictions seem unavoidable we must test, extend, and seek
to modify that which we call laws of thought, but which are only
inherited, customary representations, preserved for aeons, for the
description of practical needs. Just as to the inherited discoveries
of the cylinder, the carriage, the plow, numerous artificial ones have
been consciously added, so must we improve, artificially and con-

598 APPLIED MATHEMATICS

sciously, our likewise inherited concepts. Our problem cannot be
to quote facts before the judgment seat of our laws of thought, but
to fit our mental representations and concepts to the facts. Since
we attempt to express with clearness such complicated processes
merely by words, written, spoken, or inwardly thought, it might
also be said that we should combine the words in such wise as to
give the most appropriate expression of the facts, that the relations
indicated by our words should be most adequate for the relations
among the actualities. When the problem is enunciated in this
fashion, its appropriate solution may still offer the greatest difficulties,
but one knows then the end in view and will not stumble on self-
made difficulties.

Much that is useless in the usage and in the bearing of the nature
of life is brought forth by a method of treatment which, being
useful in most cases, becomes through habit a second nature, until
one cannot set it aside when it becomes inapplicable somewhere.
I say that the adaptability goes beyond the point aimed at. This
happens frequently in the commonplaces of thought, and becomes
the source of apparent contradictions between the laws of thought
and the world, as well as between the laws of thought themselves.

Thus, the regularity of the phenomena of nature is the funda-
mental condition for all cognition; thus comes the habit of inquiring
of everything the cause, the non-resisting compulsion, and we
inquire also concerning the cause, why everything must have a cause.
In fact people strove for a long time to determine if cause and effect
is a necessary bond or merely an accidental sequence, and if it did
or did not have a unique meaning to say that a certain particular
phenomenon was connected with, and a necessary consequence of,
a definite group of other phenomena.

Similarly, something is said to be useful, valuable, if it satisfies the
needs of the individual or of humanity; but we go beyond the mark
if we ask concerning the value of life itself, if such it seem to have,
because it has no purpose outside of itself. The same happens when
we strive vainly to explain the simplest concepts, out of which all
others are built, by means of simpler ones still, to explain the simplest
fundamental laws.

We should not attempt to deduce nature from our concepts, but
should adapt the latter to the former. We should not believe our
inherited rules of thought to be conditions preceding our more com-
plicated experiences, for they are not so for the simplest essentials.
They arose slowly in connection with simple experiences and passed,
by heredity, to the more highly organized being. Thus is explained
how synthetic judgments arise which were formed by our ancestors
and were born in us, and are in this sense a 'priori. Their great
power is also seen in this way, but not their infallibility.

RELATIONS OF APPLIED MATHEMATICS 599

In saying that such judgments as " everything is red or is not red "
are results of experience, I do not mean that every person checks this
empty truth by experience, but that he learns that his parents called
everything either red or not red and that he preserves this nomen-
clature.

It might seem as if we had gone somewhat deeply into philosophical
questions, but I believe that the views we have reached could not
have been attained in a shorter and simpler manner. For we have
reached an impartial judgment how the question of the atomistic
structure of matter is to be viewed. We shall not invoke the law of
thought that there is no limit to the divisibility of matter. This law
is of no more value than if a naive person were to say that no matter
where he went upon the earth the plumb-line directions seemed
always to be parallel and therefore there were no antipodes.

On the one hand we shall start from facts only, and on the other
we shall take nothing into consideration except the effort to attain
to the most adequate expression of these facts.

Regarding the first point, the numerous facts of the theory of
heat, of chemistry, of crystallography, show that bodies which are
apparently continuous do not by any means fill the entire volume
indistinguishably and uniformly with matter. Indeed, it appears
that the space which they occupy is filled with innumerably many
individuals, molecules, and atoms, which are extraordinarily small,
but not infinitely small in the mathematical sense. Their sizes can
be computed in different manners and always with the same result.

The fruitfulness of this line of thought has been verified in the
most recent time. All the phenomena which are observed with the
cathode rays, the Becquerel rays, etc., indicate that we are dealing
with diminutive, moving particles, electrons. After a vigorous
battle, this view vanquished completely the opposing explanation of
these phenomena by the theory of undulations. Not only did the
former theory give a better explanation of the previously known
facts, it inspired new experiments and permitted the prediction of
unknown phenomena, and thus it developed into an atomistic theory
of electricity. If it continue to develop with the same success as
in past years, if phenomena, such as the one observed by Ramsay
on the transmutation of radium into helium, do not remain isolated,
this theory promises deductions concerning the nature and structure
of atoms as yet undreamed of. Computation shows that electrons are
much smaller than the atoms of ponderable matter; and the hypo-
thesis that the atoms are built up of many elements, as well as
various interesting views on the character and structure of this com-
position, is to-day on every tongue. The word atom should not
lead us into error, it comes from a past time; no physicist ascribes
indivisibility to the atoms.

600 APPLIED MATHEMATICS

It is not my intention to confine the thought merely to the above
facts and their resulting consequences; these are not sufficient to
carry through the question as to the finite or infinite divisibility
of matter. If we are going to think of the atoms of chemistry as
made up of electrons, what would hinder us from considering the
electrons as particles filled with rarefied,, continuous matter? We
shall adhere faithfully to the previously developed philosophical
principles and shall examine in the most unhampered manner the
concepts themselves in order to express them in a consistent and
most useful form.

It appears now, that we are unable to define the infinite in any other
way except as the limit of continually increasing magnitudes, at
least no one has hitherto been able to set up any other intelligible
conception of the infinite. Should we desire a verbal picture of the
continuum, we must first think of a large finite number of particles
which are endowed with certain properties and study the totality
of these particles. Certain properties of this totality may approach
a definite limit as the number of particles is increased, and their
size decreased. It can be asserted, concerning these properties, that
they belong to the continuum, and it is my opinion that this is the
only self-consistent definition of a continuum which is endowed
with certain properties.

The question if matter is composed of atoms or is continuous
becomes then the question if the observed properties are accurately
satisfied by the assumption of an exceedingly great number of
such particles or, by increasing number, their limit. We have not
indeed answered the old philosophical question, but we are cured of
the effort to answer it in a senseless and hopeless manner. The
thought-process, that we must investigate the properties of a finite
totality and then let the number of members of this totality increase
greatly, remains the same in both cases. It is nothing other than
the abbreviated expression in algebraic symbols of exactly the same
thought when, as often happens, differential equations are made
the basis of a mathematical-physical theory.

The members of the totality which we select as the picture of the
material body cannot be thought of as absolutely at rest, for there
would then be no motion of any kind, nor can the members be thought
of as relatively at rest in one and the same body, for in this case it
would be impossible to account for the fluids. No effort has been
made to subject them to anything more than to the general laws
of mechanics. In order to explain nature we shall therefore select
a totality of an exceedingly large number of very minute funda-
mental individuals which are constantly in motion, and which are
subject to the laws of mechanics. But an objection is raised that
will be an appropriate introduction to the final considerations of

RELATIONS OF APPLIED MATHEMATICS 601

this lecture. The fundamental equations of mechanics do not alter
their form in the slightest way when the algebraic sign of the time is
changed. All pure mechanical events can therefore occur equally
well in one sense as in its opposite, that is, in the sense of increasing
time or of diminishing time. We remark, however, that in ordinary
Hfe future and past do not coincide as completely as the directions
right and left, but that the two are distinctly different.

This becomes still more definite by means of the second law of the
mechanical theory of heat, which asserts that when an arbitrary
system of bodies is left to itself, uninfluenced by other bodies, the
sense in which changes of condition occur can be assigned. A certain
f miction of the condition of all the bodies, the entropy, can be
determined, which is such that every change that occurs must be in
the sense of carrying with it an increase of this function; thus,
with increasing time the entropy increases. This law is indeed an
abstraction, just as the principle of Galileo; for it is impossible, in
strict rigor, to isolate a system of bodies from all others. But since
it has given correct results hitherto, in connection with all the other
laws, we assume it to be correct, just as in the case of the principle of
Galileo.

It follows from this law that every closed system of bodies must
tend toward a definite final condition for which the entropy is a
maximum. The outcome of this law, that the universe must come
to a final state in which nothing more can occur, has caused aston-
ishment; but this outcome is only comprehensible on the assump-
tion that the universe is finite and subject to the second law of the
mechanical theory of heat. If one regards the universe as infinite,
the above-mentioned difficulties of thought arise again if one does
not consider the infinite as a mere limit of the finite. Since there is
nothing analogous to the second law in the differential equations
of mechanics, it follows that it can be represented mechanically only
by the initial conditions. In order to find the assumptions suit-
able for this purpose, we must reflect that, to explain the appar-
ent continuity of bodies, we had to assume that every family
of atoms, or more generally, of mechanical individuals, existed in
incredibly many different initial positions. In order to treat this
assumption mathematically, a new science was founded whose pro-
blem is, not the study of the motion of a single mechanical system,
but of the properties of complexes of very many mechanical systems
which begin with a great variety of initial conditions. The task of
systematizing this science, of compiling it into a large book, and of
giving it a characteristic name, was executed by one of the greatest
American scholars, and in regard to abstract thinking, purely theo-
retic investigation, perhaps the greatest, Willard Gibbs, the recently
deceased professor at Yale University. He called this science statis-

602 APPLIED MATHEMATICS

tical mechanics, and it falls naturally into two parts. The first in-
vestigates the conditions under which the outwardly visible proper-
ties of a complex of very many mechanical individuals is not in any
wise altered; this first part I shall call statistical statics. The sec-
ond part investigates the gradual changes of these outwardly visible
properties when those conditions are not fulfilled; it may be called
statistical dynamics. At this point we may allude to the broad view
which is opened by applying this science to the statistics of ani-
mated beings, of human society, of sociology, etc., and not merely
upon mechanical particles. A development of the details of this
science would only be possible in a series of lectures and by means
of mathematical formulas. Apart from mathematical difficulties it is
not free from difficulties of principle. It is based upon the theory
of probabilities. The latter is as exact as any other branch of mathe-
matics if the concept of equal probabilities, which cannot be de-
duced from the other fundamental notions, is assumed. It is here
as in the method of least squares which is only free from objection
when certain definite assumptions are made concerning the equal
probability of elementary errors. The existence of this fundamental
difficulty explains why the simplest result of statistical statics, the
proof of Maxwell's speed law among the molecules of a gas, is still
being disputed.

The theorems of statistical mechanics are rigorous consequences
of the assumptions and will always remain valid, just as all well-
founded mathematical theorems. But its application to the events
of nature is the prototype of a physical hypothesis. Starting from
the simplest fundamental assumption of the equal probabilities, we
find that aggregates of very many individuals behave quite ana-
logously as experience shows of the material world. Progressive or
visible rotary motion must always go over into invisible motion of
the minutest particles, into heat, as Helmholtz characteristically
says: ordered motion tends always to go over into not ordered
motion; the mixture of different substances as well as of different
temperatures, the points of greater or less intense molecular
motion, must always tend toward homogeneity. That this mixture
was not complete from the start, that the universe began in such
an improbable state, belongs to the fundamental hypotheses of the
entire theory; and it may be said that the reason for this is as little
known as the reason why the universe is just so and not otherwise.
But we may take a different point of view. Conditions of great mix-
ture and great differences in temperature are not absolutely impos-
sible according to the theory but are very highly improbable. If the
universe be considered as large enough there will be, according to the
laws of probability, here and there places of the size of fixed stars,
of altogether improbable distributions. The development of such

RELATIONS OF APPLIED MATHEMATICS 603

a spot would be one-sided both in its structure and subsequent dis-
solution. Were there thinking beings at such a spot their impressions
of time would be the same as ours, although the course of events in
the universe as a whole would not be one-sided. The above-developed
theory does indeed go boldly beyond our experience, but it has the
merit which every such theory should have of showing us the facts
of experience in an entirely new light and of inspiring us to new
thought and reflection. In contradistinction to the first fundamental
law, the second one is merely based on probability, as Gibbs pointed
out in the '70's of the last century.

I have not avoided philosophical questions, in the firm hope that
cooperation between philosophy and natural science wiU give new
sustenance to both; indeed, that only in this manner a consistent
argument can be carried through. I agree with Schiller when he
says to the scientists and philosophers of his day, " Let there be strife
between you, and the union wiU come speedily;" I believe that the
time for this union has now arrived.

THE PRINCIPLES OF MATHEMATICAL PHYSICS

BY JULES HENRI POINCARE

(Translated from the French by George Bruce Halsted, Kenyan College)

[Jules Henri Poincare, Professor University of Paris, and the Polytechnic
School; Member of Bureau of Longitude, b. Nancy, April 29, 1854. D.Sc.
August 3, 1879; D.Sc. Cambridge and Oxford, 1879; Charge of the Course
of the Faculty of Sciences at Caen; Master of Conference of the Faculty of
Sciences of Paris, 1881; Professor of the same Faculty, 1886; Member of the
Institute of France, 1887; Corresponding Member of the National Academy
of Washington; Philosophical Society of Philadelphia; the Academies of
Berlin, London, St. Petersburg, Vienna, Rome, Munich, Gottingen, Bologna,
Turin, Naples, Venice, Amsterdam, Copenhagen, Stockholm, etc. Written
books and numerous articles for reviews and periodicals.]

What is the actual state of mathematical physics? What are the
problems it is led to set itself? What is its future? Is its orientation
on the point of modifying itself?

Will the aim and the methods of this science appear in ten years
to our immediate successors in the same light as to ourselves; or,
on the contrary, are we about to witness a profound transformation?
Such are the questions we are forced to raise in entering to-day upon
our investigation.

If it is easy to propound them, to answer is difficult.

If we feel ourselves tempted to risk a prognostication, we have,
to resist this temptation, only to think of all the stupidities the
most eminent savants of a hundred years ago would have uttered,
if one had asked them what the science of the nineteenth century
would be. They would have believed themselves bold in their pre-
dictions, yet after the event how very timid we should have found
them.

Mathematical physics, we know, was born of celestial mechanics,
which engendered it at the end of the eighteenth century, at the
moment when the latter was attaining its complete development.
During its first years especially, the infant resembled in a striking
way its mother.

The astronomic universe is formed of masses, very great without
doubt, but separated by intervals so immense that they appear to
us only as material points. These points attract each other in the
inverse ratio of the square of the distances, and this attraction is
the sole force which influences their movements. But if our senses
were sufficiently subtle to show us all the details of the bodies which
the physicist studies, the spectacle we should there discover would
scarcely differ from what the astronomer contemplates. There also
we should see material points, separated one from another by inter-

PRINCIPLES OF MATHEMATICAL PHYSICS 605

vals enormous in relation to their dimensions, and describing orbits
following regular laws.

These infinitesimal stars are the atoms. Like the stars properly
so called, they attract or repel each other, and this attraction or this
repulsion directed following the straight line which joins them, de-
pends only on the distance. The law according to which this force
varies as function of the distance is perhaps not the law of Newton,
but it is an analogous law; in j)lace of the exponent — 2, we have
probably a different exponent, and it is from this change of exponent
that springs aU the diversity of physical phenomena, the variety of
qualities and of sensations, all the world colored and sonorous which
surrounds us, — in a word, all nature.

Such is the primitive conception in all its purity. It only remains
to seek in the different cases what value should be given to this
exponent in order to explain all the facts. It is on this model that
Laplace, for example, constructed his beautiful theory of capillarity;
he regards it only as a particular case of attraction, or as he says
of universal gravitation, and no one is astonished to find it in the
middle of one of the five volumes of the Mecanique celeste.

More recently Briot believed he had penetrated the final secret
of optics in demonstrating that the atoms of ether attract each other
in the inverse ratio of the sixth power of the distance; and does not
Maxwell himself say somewhere that the atoms of gases repel each
other in the inverse ratio of the fifth power of the distance? We have
the exponent — 6, or — 5 in place of the exponent — 2, but it is
always an exponent.

Among the theories of this period, one alone is an exception, that
of Fourier; in it are indeed atoms, acting at a distance one upon the
other; they mutually transmit heat, but the}^ do not attract, they
never budge. From this point of view, the theory of Fourier must
have appeared to the eyes of his contemporaries, even to Fourier
himself, as imperfect and provisional.

This conception was not without grandeur; it was seductive, and
many among us have not finally renounced it; we know that we
shall attain the ultimate elements of things only by patiently disen-
tangling the complicated skein that our senses give us; that it is
necessary to advance step by step, neglecting no intermediary; that
our fathers were wrong in wishing to skip stations; but we believe
that when we shall have arrived at these ultimate elements, there
again wiU be found the majestic simplicity of celestial mechanics.

Neither has this conception been useless; it has rendered us an
inestimable service, since it has contributed to make precise in us
the fundamental notion of the physical law.

I will explain myself; how did the ancients understand law? It
was for them an internal harmony, static, so to say, and immutable;

606 APPLIED MATHEMATICS '

or it was like a model that nature constrained herself to imitate. A
law for us is not that at all; it is a constant relation between the
phenomenon of to-day and that of to-morrow; in a word, it is a
differential equation.

The ideal form of physical law is the law of Newton which first
covered it; and then how has one to adapt this form to physics?
by copying as much as possible this law of Newton, that is, in imi-
tating celestial mechanics.

Nevertheless, a day arrived when the conception of central forces
no longer appeared sufficient, and this is the first of those crises of
which I just now spoke.

Then investigators gave up trying to penetrate into the detail
of the structure of the universe, to isolate the pieces of this vast
mechanism, to analyze one by one the forces which put them in
motion, and were content to take as guides certain general prin-
ciples which have precisely for their object the sparing us this minute
study.

How so? Suppose that we have before us any machine; the ini-
tial wheel-work and the final wheel-work alone are visible, but the
transmission, the intermediary wheels by which the movement is
communicated from one to the other are hidden in the interior
and escape our view; we do not know whether the communication
is made by gearing or by belts, by connecting-rods or by other dis-
positives.

Do we say that it is impossible for us to understand anything about
this machine so long as we are not permitted to take it to pieces?
You know well we do not, and that the principle of the conservation
of energy suffices to determine for us the most interesting point. We
easily ascertain that the final wheel turns ten times less quickly than
the initial wheel, since these two wheels are visible; we are able
thence to conclude that a couple applied to the one will be balanced
by a couple ten times greater applied to the other. For that there
is no need to penetrate the mechanism of this equilibrium and to
know how the forces compensate each other in the interior of the
machine; it suffices to be assured that this compensation cannot fail
to occur.

Well, in regard to the universe, the principle of the conservation
of energy is able to render us the same service. This is also a machine,
much more complicated than all those of industry, and of which
almost all the parts are profoundly hidden from us; but in observing
the movement of those that we can see, we are able, by aid of this
principle, to draw conclusions which remain true whatever may be
the details of the invisible mechanism which animates them.

The principle of the conservation of energy, or the principle of
Mayer, is certainly the most important, but it is not the only one;

PRINCIPLES OF MATHEMATICAL PHYSICS 607

there are others from which we are able to draw the same advantage.
These are:

The principle of Carnot, or the principle of the degradation of
energy.

The principle of Newton, or the principle of the equality of action
and reaction.

The principle of relativity, according to which the laws of phys-
ical phenomena should be the same, whether for an observer
fixed, or for an observer carried along in a uniform move-
ment of translation; so that we have not and could not
have any means of discerning whether or not we are carried
along in such a motion.

The principle of the conservation of mass, or principle of
Lavoisier.
I would add the principle of least action.

The application of these five or six general principles to the differ-
ent physical phenomena is sufficient for our learning of them what
we could reasonably hope to know of them.

The most remarkable example of this new mathematical physics
is, beyond contradiction, Maxwell's electro-magnetic theory of light.

We know nothing of the ether, how its molecules are disposed,
whether they attract or repel each other; but we know that this
medium transmits at the same time the optical perturbations and
the electrical perturbations; we know that this transmission should
be made conformably to the general principles of mechanics, and
that suffices us for the establishment of the equations of the electro-
magnetic field.

These principles are results of experiments boldly generalized;
but they seem to derive from their generality itself an eminent
degree of certitude.

In fact the more general they are, the more frequently one has
the occasion to check them, and the verifications, in multiplying
themselves, in taking forms the most varied and the most unex-
pected, finish by no longer leaving place for doubt.

Such is the second phase of the history of mathematical physics,
and we have not yet emerged from it.

Do we say that the first has been useless? that during fifty yearg
science went the wrong way, and that there is nothing left but to
forget so many accumulated efforts as vicious conceptions condemned
in advance to non-success?

Not the least in the world ; the second phase could not have come
into existence without the first?

The hypothesis of central forces contained all the principles; it
involved them as necessary consequences; it involved both the con-

608 APPLIED MATHEMATICS

servation of energy and that of masses, and the equahty of action
and reaction; and the law of least action, which would appear, it
is true, not as experimental verities, but as theorems, and of which
the enunciation would have at the same time a something more pre-
cise and less general than under their actual form.

It is the mathematical physics of our fathers which has familiar-
ized us little by little with these divers principles; which has taught
us to recognize them under the different vestments in which they
disguise themselves. One has to compare them to the data of ex-
perience, to find how it was necessary to modify their enunciation
so as to adapt them to these data; and by these processes they
have been enlarged and consolidated.

So we have been led to regard them as experimental verities;
the conception of central forces became then a useless support, or
rather an embarrassment, since it made the principles partake of its
hypothetical character.

The frames have not therefore broken, because they were elastic;
but they have enlarged; our fathers, who established them, did not
work in vain, and we recognize in the science of to-day the general
traits of the sketch which they traced.

Are we about to enter now upon the eve of a second crisis? Are
these principles on which we have built all about to crumble away
in their turn? For some time, this may well have been asked.

In hearing me speak thus, you think without doubt of radium,
that grand revolutionist of the present time, and in fact I will come
back to it presently; but there is something else.

It is not alone the conservation of energy which is in question;
all the other principles are equally in danger, as we shall see in pass-
ing them successively in review.

Let us commence with the principle of Carnot. This is the only
one which does not present itself as an immediate consequence of
the hypothesis of central forces; more than that, it seems, if not
directly to contradict that hypothesis, at least not to be reconciled
with it without a certain effort.

If physical phenomena were due exclusively to the movements
of atoms whose mutual attraction depended only on the distance,
it seems that all these phenomena should be reversible; if all the
initial velocities were reversed, these atoms, always subjected to
the same forces, ought to go over their trajectories in the contrary
sense, just as the earth would describe in the retrograde sense this
same elliptic orbit which it describes in the direct sense, if the initial
conditions of its movement had been reversed. On this account, if
a physical phenomenon is possible, the inverse phenomenon should
be equally so, and one should be able to reascend the course of
time.

PRINCIPLES OF MATHEMATICAL PHYSICS 609

But it is not so in nature, and this is precisely what the principle
of Carnot teaches us; heat can pass from the warm body to the cold
body; it is impossible afterwards to make it reascend the inverse
way and reestablish differences of temperature which have been
effaced.

Motion can be wholly dissipated and transformed into heat by
friction; the contrary transformation can never be made except in
a partial manner.

We have striven to reconcile this apparent contradiction. If the
world tends toward uniformity, this is not because its ultimate parts,
at first unlike, tend to become less and less different, it is because,
shifting at hazard, they end by blending. For an eye which should
distinguish all the elements, the variety would remain always as
great, each grain of this dust preserves its originality and does not
model itself on its neighbors; but as the blend becomes more and
more intimate, our gross senses perceive no more than the uniform-
ity. Behold why, for example, temperatures tend to a level, without
the possibility of turning backwards.

A drop of wine falls into a glass of water; whatever may be the
law of the internal movements of the liquid, we soon see it colored
to a uniform rosy tint, and from this moment, however well we
may shake the vase, the wine and the water do not seem capable of
further separation. Observe what would be the type of the reversible
physical phenomenon: to hide a grain of barley in a cup of wheat
is easy; afterwards to find it again and get it out is practically im-
possible.

All this Maxwell and Boltzmann have explained; the one who has
seen it most clearly, in a book too little read because it is a little
difficult to read, is Gibbs, in his Elementary Principles of Statistical
Mechanics.

For those who take this point of view, the principle of Carnot is
only an imperfect principle, a sort of concession to the infirmity of
our senses; it is because our eyes are too gross that we do not dis-
tinguish the elements of the blend; it is because our hands are too
gross that we cannot force them to separate; the imaginary demon
of Maxwell, who is able to sort the molecules one by one, could well
constrain the world to return backward. Can it return of itself? That
is not impossible; that is only infinitely improbable.

The chances are that we should long await the concourse of cir-
cumstances which would permit a retrogradation, but soon or late
they would be realized, after years whose number it would take
millions of figures to write.

These reservations, however, all remained theoretic and were not
very disquieting, and the principle of Carnot retained all its practical
value.

610 APPLIED MATHEMATICS

But here the scene changes.

The biologist, armed with his microscope, long ago noticed in his
preparations disorderly movements of little particles in suspension:
this is the Brownian movement; he first thought this was a vital
phenomenon, but he soon saw that the inanimate bodies danced with
no less ardor than the others; then he turned the matter over to the
physicists. Unhappily, the physicists remained long uninterested in
this question; the light is focused to illuminate the microscopic pre-
paration, thought they; with light goes heat; hence inequalities of
temperature and interior currents produce the movements in the
liquid of which we speak.

M. Gouy, however, looked more closely, and he saw, or thought
he saw, that this explanation is untenable, that the movements
become more brisk as the particles are smaller, but that they are not
influenced by the mode of illumination.

If, then, these movements never cease, or rather are reborn with-
out ceasing, without borrowing anything from an external source
of energy, what ought we to believe? To be sure, we should not
renounce our belief in the conservation of energy, but we see under
our eyes now motion transformed into heat by friction, now heat
changed inversely into motion, and that without loss since the move-
ment lasts forever. This is the contrary of the principle of Carnot.

If this be so, to see the world return backward, we no longer
have need of the infinitely subtle eye of Maxwell's demon; our
microscope suffices us. Bodies too large, those, for example, which
are a tenth of a millimeter, are hit from all sides by moving atoms,
but they do not budge, because these shocks are very numerous and
the law of chance makes them compensate each other: but the
smaller particles receive too few shocks for this compensation to
take place with certainty and are incessantly knocked about. And
thus already one of our principles is in peril.

We come to the principle of relativity : this not only is confirmed
by daily experience, not only is it a necessary consequence of the
hypothesis of central forces, but it is imposed in an irresistible way
upon our good sense, and yet it also is battered.

Consider two electrified bodies; though they seem to us at rest,
they are both carried along by the motion of the earth; an electric
charge in motion, Rowland has taught us, is equivalent to a current;
these two charged bodies are, therefore, equivalent to two parallel
currents of the same sense and these two currents should attract
each other. In measuring this attraction, we measure the velocity
of the earth; not its velocity in relation to the sun or the fixed stars,
but its absolute velocity.

I know it will be said that it is not its absolute velocity that
is measured, but its velocity in relation to the ether. How unsatis-

PRINCIPLES OF MATHEMATICAL PHYSICS 611

factory that is! Is it not evident that from a principle so under-
stood we could no longer get anything? It could no longer tell us
anything just because it would no longer fear any contradiction.

If we succeed in measuring anything, we should always be free
to say that this is not the absolute velocity in relation to the ether,
it might always be the velocity in relation to some new unknown
fluid with which we might fill space.

Indeed, experience has taken on itself to ruin this interpretation
of the principle of relativity; all attempts to measure the velocity
of the earth in relation to the ether have led to negative results.
This time experimental physics has been more faithful to the prin-
ciple than mathematical physics; the theorists, to put in accord
their other general views, would not have spared it; but experiment
has been stubborn in confirming it.

The means have been varied in a thousand ways and finally
Michelson has pushed precision to its last limits; nothing has come
of it. It is precisely to explain this obstinacy that the mathematicians
are forced to-day to employ all their ingenuity.

Their task was not easy, and if Lorentz has gotten through it,
it is only by accumulating hypotheses.

The most ingenious idea has been that of local time.

Imagine two observers who wish to adjust their watches by
optical signals; they exchange signals, but as they know that the
transmission of light is not instantaneous, they take care to cross
them.

When the station B perceives the signal from the station A, its
clock should not mark the same hour as that of the station A at the
moment of sending the signal, but this hour augmented by a con-
stant representing the duration of the transmission. Suppose, for
example, that the station A sends its signal when its clock marks
the hour 0, and that the station B perceives it when its clock marks
the hour t. The clocks are adjusted if the slowness equal to t repre-
sents the duration of the transmission, and to verify it the station B
sends in its turn a signal when its clock marks 0; then the station A
should perceive it when its clock marks t. The time-pieces are then
adjusted. And in fact, they mark the same hour at the same phys-
ical instant, but on one condition, namely, that the two stations are
fixed. In the contrary case the duration of the transmission mil not
be the same in the two senses, since the station A, for example,
moves forward to meet the optical perturbation emanating from B,
while the station B flies away before the perturbation emanating
from A. The watches adjusted in that manner do not mark, there-
fore, the true time; they mark what one may caU the local time, so
that one of them goes slow on the other. It matters little, since we
have no means of perceiving it. All the phenomena which happen

612 APPLIED MATHEMATICS

at A, for example, will be late, but all will be equally so, and the
observer who ascertains them will not perceive it, since his watch is
slow; so, as the principle of relativity would have it, he will have no
means of knowing whether he is at rest or in absolute motion.

Unhappily, that does not suffice, and complementary hypotheses
are necessary; it is necessary to admit that bodies in motion undergo
a uniform contraction in the sense of the motion. One of the dia-
meters of the earth, for example, is shrunk by 200 000 000 ^^ conse-
quence of the motion of our planet, while the other diameter retains
its normal length. Thus, the last little differences find themselves
compensated. And then there still is the hypothesis about forces.
Forces, whatever be their origin, gravity as well as elasticity, would
be reduced in a certain proportion in a world animated by a uniform
translation; or, rather, this would happen for the components perpen-
dicular to the translation; the components parallel would not change.

Resume, then, our example of two electrified bodies; these bodies
repel each other, but at the same time if all is carried along in a
uniform translation, they are equivalent to two parallel currents of
the same sense which attract each other. This electro-dynamic
attraction diminishes, therefore, the electro-static repulsion, and the
total repulsion is more feeble than if the two bodies were at rest.
But since to measure this repulsion we must balance it by another
force, and all these other forces are reduced in the same proportion,
we perceive nothing.

Thus, all is arranged, but are all the doubts dissipated?

What would happen if one could communicate by non-luminous
signals whose velocity of propagation differed from that of light?
If, after having adjusted the watches by the optical procedure, one
wished to verify the adjustment by the aid of these new signals,
then would appear divergences which would render evident the com-
mon translation of the two stations. And are such signals incon-
ceivable, if we admit with Laplace that universal gravitation is
transmitted a million times more rapidly than light?

Thus, the principle of relativity has been valiantly defended in
these latter times, but the very energy of the defense proves how
serious was the attack.

Let us speak now of the principle of Newton, on the equality of
action and reaction.

This is intimately bound up with the preceding, and it seemB
indeed that the fall of the one would involve that of the other.
Thus we should not be astonished to find here the same difficulties.

Electrical phenomena, we think, are due to the displacements of
little charged particles, called electrons, immersed in the medium
that we call ether. The movements of these electrons produce per-
turbations in the neighboring ether; these perturbations propagate

PRINCIPLES OF MATHEMATICAL PHYSICS 613

themselves in every direction with the velocity of light, and in turn
other electrons, originally at rest, are made to vibrate when the
perturbation reaches the parts of the ether which touch them.

The electrons, therefore, act upon one another, but this action is
not direct, it is accomplished through the ether as intermediary.

Under these conditions can there be compensation between action
and reaction, at least for an observer who should take account
only of the movements of matter, that is to say, of the electrons, and
who should be ignorant of those of the ether that he could not see?
Evidently not. Even if the compensation should be exact, it could
not be simultaneous. The perturbation is propagated with a finite
velocity; it, therefore, reaches the second electron only when the
first has long ago entered upon its rest.

This second electron, therefore, will undergo, after a delay, the
action of the first, but certainly it will not react on this, since around
this first electron nothing any longer budges.

The analysis of the facts permits us to be still more precise. Imagine
for example, a Hertzian generator, like those employed in wireless
telegraphy; it sends out energy in every direction; but we can
provide it with a parabolic mirror, as Hertz did with his smallest
generators, so as to send all the energy produced in a single direction.

What happens, then, according to the theory? It is that the
apparatus recoils as if it were a gun and as if the energy it has
projected were a bullet; and that is contrary to the principle of
Newton, since our projectile here has no mass, it is not matter, it
is energy.

It is still the same, moreover, with a beacon light provided with
a reflector, since light is nothing but a perturbation of the electro-
magnetic field. This beacon light should recoil as if the light it
sends out were a projectile. What is the force that this recoil should
produce? It is what one has called the Maxwell-Bartholdi pressure.
It is very minute, and it has been difficult to put it into evidence
even with the most sensitive radiometers; but it suffices that it exists.

If all the energy issuing from our generator falls on a receiver,
this will act as if it had received a mechanical shock, which will
represent in a sense the compensation of the recoil of the generator;
the reaction will be equal to the action, but it will not be simulta-
neous; the receiver will move on but not at the moment when the
generator recoils. If the energy propagates itself indefinitely with-
out encountering a receiver, the compensation will never be made.

Do we say that the space which separates the generator from
the receiver and which the perturbation must pass over in going
from the one to the other is not void, that it is full not only of ether,
but of air; or even in the interplanetary spaces of some fluid subtle
but still ponderable; that this matter undergoes the shock like the

614 APPLIED MATHEMATICS

receiver at the moment when the energy reaches it, and recoils in its
turn when the perturbation quits it? That would save the principle
of Newton, but that is not true.

If energy in its diffusion remained always attached to some ma-
terial substratum, then matter in motion would carry along light
with it, and Fizeau has demonstrated that it does nothing of the
sort, at least for air. This is what Michelson and Morley have since
confirmed.

One may suppose also that the movements of matter, properly
so called, are exactly compensated by those of the ether; but that
would lead us to the same reflections as just now. The principle so
extended would explain everything, since whatever might be the
visible movements, we should always have the power of imagining
hypothetical movements which compensated them.

But if it is able to explain everything, this is because it does
not permit us to foresee anything; it does not enable us to decide
between different possible hypotheses, since it explains everything
beforehand. It therefore becomes useless.

And then the suppositions that it would be necessary to make
on the movements of the ether are not very satisfactory.

If the electric charges double, it would be natural to imagine
that the velocities of the divers atoms of ether double also, and for
the compensation, it would be necessary that the mean velocity of
the ether quadruple.

This is why I have long thought that these consequences of
theory, contrary to the principle of Newton, would end some day
by being abandoned, and yet the recent experiments on the move-
ments of the electrons issuing from radium seem rather to confirm
them.

I arrive at the principle of Lavoisier on the conservation of masses :
in truth this is one not to be touched without unsettling all mechanics.

And now certain persons believe that it seems true to us only
because we consider in mechanics merely moderate velocities, but
that it would cease to be true for bodies animated by velocities com-
parable to that of light. These velocities, it is now believed, have
been realized; the cathode rays or those of radium may be formed
of very minute particles or of electrons which are displaced with
velocities smaller no doubt than that of light, but which might be its
one tenth or one third.

These rays can be deflected, whether by an electric field, or by
a magnetic field, and we are able by comparing these deflections, to
measure at the same time the velocity of the electrons and their mass
(or rather the relation of their mass to their charge). But when
it was seen that these velocities approached that of light, it was
decided that a correction was necessary.

PRINCIPLES OF MATHEMATICAL PHYSICS 615

These molecules, being electrified, could not be displaced with-
out agitating the ether; to put them in motion it is necessary to
overcome a double inertia, that of the molecule itself and that of the
ether. The total or apparent mass that one measures is composed,
therefore, of two parts: the real or mechanical mass of the mole-
cule and the electro-dynamic mass representing the inertia of the
ether.

The calculations of Abraham and the experiments of Kaufmann
have then shown that the mechanical mass, properly so called, is
null, and that the mass of the electrons, or, at least, of the negative
electrons, is of exclusively electro-dynamic origin. This forces us to
change the definition of mass; we cannot any longer distinguish
mechanical mass and electro-dynamic mass, since then the first would
vanish;' there is no mass other than electro-dynamic inertia. But
in this case the mass can no longer be constant, it augments with the
velocity, and it even depends on the direction, and a body animated
by a notable velocity will not oppose the same inertia to the forces
which tend to defiect it from its route, as to those which tend to
accelerate or to retard its progress.

There is still a resource; the ultimate elements of bodies are
electrons, some charged negatively, the others charged positively.
The negative electrons have no mass, this is understood; but the
positive electrons, from the little we know of them, seem much
greater. Perhaps they have, besides their electro-dynamic mass,
a true mechanical mass. The veritable mass of a body would, then,
be the sum of the mechanical masses of its positive electrons, the
negative electrons not counting; mass so defined could still be con-
stant.

Alas, this resource also evades us. Recall what we have said
of the principle of relativity and of the efforts made to save it. And
it is not merely a principle which it is a question of saving, such
are the indubitable results of the experiments of Michelson.

Lorentz has been obliged to suppose that all the forces, what-
ever be their origin, were affected with a coefficient in a medium
animated by a uniform translation; this is not sufficient; it is still
necessary, says he, that the masses of all the particles he influenced
by a translation to the same degree as the electro-magnetic fnasses
of the electrons.

So the mechanical masses will vary in accordance with the same
laws as the electro-dynamic masses; they cannot, therefore, be con-
stant.

Need I point out that the fall of the principle of Lavoisier in-
volves that of the principle of Newton? This latter signifies that
the centre of gravity of an isolated system moves in a straight line;
but if there is no longer a constant mass, there is no longer a centre

616 APPLIED MATHEMATICS

of gravity, we no longer know even what this is. This is why I
said above that the experiments on the cathode rays appeared to
justify the doubts of Lorentz on the subject of the principle of
Newton.

From all these results, if they are confirmed, would arise an
entirely new mechanics, which would be, above all, characterized by
this fact, that no velocity could surpass that of light, any more than
any temperature could fall below the zero absolute, because bodies
would oppose an increasing inertia to the causes, which would tend
to accelerate their motion; and this inertia would become infinite
when one approached the velocity of light.

Nor for an observer carried along himself in a translation he
did not suspect could any apparent velocity surpass that of light;
there would then be a contradiction, if we recall that this observer
would not use the same clocks as a fixed observer, but, indeed, clocks
marking "local time."

Here we are then facing a question I content myself with stating.
If there is no longer any mass, what becomes of the law of Newton?

Mass has two aspects, it is at the same time a coefficient of iner-
tia and an attracting mass entering as factor into Newtonian attrac-
tion. If the coefficient of inertia is not constant, can the attracting
mass be? That is the question.

At least, the principle of the conservation of energy yet remains
to us, and this seems more solid. Shall I recall to you how it was
in its turn thrown into discredit? This event has made more noise
than the preceding and it is in all the records.

From the first works of Becquerel, and, above all, when the
Curies had discovered radium, one saw that every radio-active body
was an inexhaustible source of radiations. Its activity would seem
to subsist without alteration throughout the months and the years.
This was already a strain on the principles; these radiations were in
fact energy, and from the same morsel of radium this issued and for-
ever issued. But these quantities of energy were too slight to be
measured; at least one believed so and was not much disquieted.

The scene changed when Curie bethought himself to put radium
into a calorimeter; it was seen then that the quantity of heat in-
cessantly created was very notable.

The explanations proposed were numerous; but in so far as no
one of them has prevailed over the others, we cannot be sure there
is a good one among them.

Sir William Ramsay has striven to show that radium is in process
of transformation, that it contains a store of energy enormous but
not inexhaustible.

The transformation of radium, then, would produce a million
times more of heat than all known transformations; radium would

PRINCIPLES OF MATHEMATICAL PHYSICS 617

wear itself out in 1250 years; you see that we are at least certain
to be settled on this point some hundreds of years from now. While
waiting our doubts remain.

In the midst of so many ruins what remains standing? The prin-
ciple of least action has hitherto remained intact, and Larmor appears
to believe that it will long survive the others; in reality, it is still
more vague and more general.

In presence of this general ruin of the principles, what attitude
will mathematical physics take?

And first, before too much perplexity, it is proper to ask if all this
is really true. All these apparent contradictions to the principles are
encountered only among infinitesimals; the microscope is necessary
to see the Brownian movement; electrons are very light; radium is
very rare, and no one has ever seen more than some milligrams of
it at a time.

And, then, it may be asked if, beside the infinitesimal seen, there
be not another infinitesimal unseen counterpoise to the first.

So, there is an interlocutory question, and, as it seems, only
experiment can solve it. We have, therefore, only to hand over the
matter to the experimenters, and, while waiting for them to deter-
mine the question finally, not to preoccupy ourselves with these dis-
quieting problems, but quietly continue our work, as if the princi-
ples were still uncontested. We have much to do without leaving
the domain where they may be applied in all security; we have
enough to employ our activity during this period of doubts.

And as to these doubts, is it indeed true that we can do nothing
to disembarrass science of them? It may be said, it is not alone
experimental physics that has given birth to them; mathematical
physics has well contributed. It is the experimenters who have seen
radium throw out energy, but it is the theorists who have put in
evidence all the difficulties raised by the propagation of light across
a medium in motion; but for these it is probable we should not have
become conscious of them. Well, then, if they have done their best
to put us into this embarrassment, it is proper also that they help us
to get out of it.

They must subject to critical examination all these new views
I have just outlined before you, and abandon the principles only
after having made a loyal effort to save them.

What can they do in this sense? That is what I will try to ex-
plain.

Among the most interesting problems of mathematical physics,
it is proper to give a special place to those relating to the kinetic
theory of gases. Much has already been done in this direction, but
much still remains to be done. This theory is an eternal paradox.
We have reversibility in the premises and irreversibility in the con-

618 APPLIED MATHEMATICS

elusions; and between the two an abyss. Statistic considerations,
the law of great numbers, do they suffice to fill it? Many points
still remain obscure to which it is necessary to return, and doubtless
many times. In clearing them up, we shall understand better the
sense of the principle of Carnot and its place in the ensemble of
dynamics, and we shall be better armed to interpret properly the
curious experiment of Gouy, of which I spoke above.

Should we not also endeavor to obtain a more satisfactory theory
of the electro-dynamics of bodies in motion? It is there especially,
as I have sufficiently shown above, that difficulties accumulate.
Evidently we must heap up hypotheses, we cannot satisfy all the
principles at once; heretofore, one has succeeded in safeguarding
some only on condition of sacrificing the others; but all hope of
obtaining better results is not yet lost. Let us take, therefore, the
theory of Lorentz, turn it in all senses, modify it little by little, and
perhaps everything will arrange itself.

Thus in place of supposing that bodies in motion undergo a con-
traction in the sense of the motion, and that this contraction is the
same whatever be the nature of these bodies and the forces to which
they are otherwise submitted, could we not make an hypothesis
more simple and more natural?

We might imagine, for example, that it is the ether which is
modified when it is in relative motion in reference to the material
medium which it penetrates, that when it is thus modified, it no
longer transmits perturbations with the same velocity in every direc-
tion. It might transmit more rapidly those which are propagated
parallel to the medium, whether in the same sense or in the opposite
sense, and less rapidly those which are propagated perpendicularly.
The wave surfaces would no longer be spheres, but ellipsoids, and we
could dispense with that extraordinary contraction of all bodies.

I cite that only as an example, since the modifications one niight
essay would be evidently susceptible of infinite variation.

It is possible also that the astronomer may some day furnish us data
on this point; he it was in the main who raised the question in
making us acquainted with the phenomenon of the aberration of light.
If we make crudely the theory of aberration, we reach a very curious
result. The apparent positions of the stars differ from their real
positions because of the motion of the earth, and as this motion is
variable, these apparent positions vary. The real position we cannot
know, but we can observe the variations of the apparent position.
The observations of the aberration show us, therefore, not the
movement of the earth, but the variations of this movement; they
cannot, therefore, give us information about the absolute motion
of the earth. At least this is true in first approximation, but it
would be no longer the same if we could appreciate the thousandths

PRINCIPLES OF MATHEMATICAL PHYSICS 619

of a second. Then it would be seen that the amplitude of the oscil-
lation depends not alone on the variation of the motion, variation
which is well known, since it is the motion of our globe on its elliptic
orbit, but on the mean value of this motion; so that the constant of
aberration would not be altogether the same for all the stars, and the
differences would tell us the absolute motion of the earth in space.

This, then, would be, under another form, the ruin of the prin-
ciple of relativity. We are far, it is true, from appreciating the
thousandths of a second, but after all, say some, the total absolute
velocity of the earth may be much greater than its relative velocity
with respect to the sun. If, for example, it were 300 kilometers per
second in place of 30, this would suffice to make the phenomena
observable.

I believe that in reasoning thus we admit a too simple theory
of aberration. Michelson has shown hs, I have told you, that the
physical procedures are powerless to put in evidence absolute mo-
tion; I am persuaded that the same will be true of the astronomic
procedures, however far one pushes precision.

However that may be, the data astronomy will furnish us in
this regard will some day be precious to the physicist. While wait-
ing, I believe the theorists, recalling the experience of Michelson,
may anticipate a negative result, and that they would accomplish
a useful work in constructing a theory of aberration which would
explain this in advance.

But let us come back to the earth. There also we may aid the
experimenters. We can, for example, prepare the ground by study-
ing profoundly the dynamics of electrons; not, be it understood,
in starting from a single hypothesis, but in multiplying hypotheses
as much as possible. It will be, then, for the physicists to utilize
our work in seeking the crucial experiment to decide between these
different hypotheses.

This dynamics of electrons can be approached from many sides,
but among the ways leading thither is one which has been somewhat
neglected, and yet this is one of those which promise us most of sur-
prises. It is the movements of the electrons which produce the line
of the emission spectra; this is proved by the phenomenon of Zee-
mann; in an incandescent body, what vibrates is sensitive to the
magnet, therefore electrified. This is a very important first point,
but no one has gone farther; why are the lines of the spectrum
distributed in accordance with a regular law?

These laws have been studied by the experimenters in their least
details; they are very precise and relatively simple. The first study
of these distributions recalled the harmonics encountered in acous-
tics; but the difference is great. Not only the numbers of vibrations
are not the successive multiples of one number, but we do not

620 APPLIED MATHEMATICS

even find anything analogous to the roots of those transcendental
equations to which so many problems of mathematical physics con-
duct us: that of the vibrations of an elastic body of any form, that
of the Hertzian oscillations in a generator of any form, the problem
of Fourier for the coohng of a solid body.

The laws are simpler, but they are of whoUy other nature, and to
cite only one of these differences, for the harmonics of high order
the number of vibrations tends toward a finite Umit, instead of
increasing indefinitely.

That has not yet been accounted for, and I believe that there we
have one of the most important secrets of nature. Lindemann has
made a praiseworthy attempt, but, to my mind, without success;
this attempt should be renewed. Thus we shall penetrate, so to say,
into the inmost recess of matter. And from the particular point of
\'iew which we to-day occupy, when we know why the vibrations
of incandescent bodies differ from ordinary elastic vibrations, w^hy
the electrons do not behave themselves like the matter which is familiar
to us. we shall better comprehend the dynamics of electrons and
it will be perhaps more easy for us to reconcile it with the princi-
ples.

Suppose, now, that all these efforts fail, and after all I do not
beheve they wiU, what must be done? Will it be necessary to seek
to mend the broken principles in giving what we French call a coup
de pouce 9 That is evidently always possible, and I retract nothing
I have formerly said.

Have you not written, you might say if you wished to seek a
quarrel with me, have you not written that the principles, though of
experimental origin, are now unassailable by experiment because
they have become conventions? And now you have just told us the
most recent conquests of experiment put these principles in danger.
Well, formerly I was right and to-day I am not wrong.

Formerly I was right, and what is now happening is a new proof
of it. Take, for example, the calorimeter experiment of Curie on
radium. Is it possible to reconcile that with the principle of the
conservation of energy?

It has been attempted in many ways; but there is among them
one I should like you to notice.

It has been conjectured that radium was only an intermediary,
that it only stored radiations of unknown nature which flashed
through space in every direction, traversing aU bodies, save radium,
without being altered by this passage and without exercising any
action upon them. Radium alone took from them a little of their
energy and afterward gave it out to us in divers forms.

What an advantageous explanation, and how convenient! First,
it is unverifiable and thus irrefutable. Then again it will serve to

PRINCIPLES OF MATHEMATICAL PHYSICS 621

account for any derogation whatever to the principle of Mayer; it
responds in advance not only to the objection of Curie, but to all
the objections that future experimenters might accumulate. This
new and unknown energy would serve for everything. This is just
what I have said, and we are thereby shown that our principle is
unassailable by experiment.

And after all, what have we gained by this cowp de pouce ?
The principle is intact, but thenceforth of what use is it?
It permitted us to foresee that in such or such circumstance we
could count on such a total quantity of energy; it limited us; but
now where there is put at our disposition this indefinite provision of
new energy, we are limited by nothing; and as I have written else-
where, if a principle ceases to be fecund, experiment, without con-
tradicting it directly, will be likely to condemn it.

This, therefore, is not what would have to be done, it would be
necessary to rebuild anew.

If we were cornered down to this necessity, we should moreover
console ourselves. It would not be necessary to conclude that science
can weave only a Penelope's web, that it can build only ephemeral
constructions, which it is soon forced to demolish from top to bot-
tom with its own hands.

As I have said, we have already passed through a like crisis. I
have shown you that in the second mathematical physics, that of
the principles, we find traces of the first, that of the central forces;
it will be just the same if we must learn a third.

When an animal exuviates, and breaks its too narrow carapace to
make itself a fresh one, we easily recognize under the new envelope
the essential traits of the organism which have existed.

We cannot foresee in what way we are about to expand; perhaps
it is the kinetic theory of gases which is about to undergo develop-
ment and serve as model to the others. Then, the facts which first
appeared to us as simple, thereafter will be merely results of a very
great number of elementary facts which only the laws of chance
make cooperate for a common end. Physical law will then take an
entirely new aspect; it will no longer be solely a differential equation,
it will take the character of a statistical law.

Perhaps, likewise, we should construct a whole new mechanics,
of which we only succeed in catching a glimpse, where inertia increas-
ing with the velocity, the velocity of light would become an impass-
able limit.

The ordinary mechanics, more simple, would remain a first approx-
imation, since it would be true for velocities not too great, so that we
should still find the old dynamics under the new.

We should not have to regret having believed in the principles,
and even, since velocities too great for the old formulas would always

622 APPLIED MATHEMATICS

be only exceptional, the surest way in practice would be still to act
as if we continued to believe in them. They are so useful, it would be
:necessary to keep a place for them. To determine to exclude them
altogether would be todepriveone'sself of a precious weapon. I hasten
to say in conclusion we are not yet there, and as yet nothing proves
that the principles will not come forth from the combat victorious
and intact.

SHORT PAPERS

Three short papers were read in the Section of Applied Mathematics, the first
by Professor Henry T. Eddy, of the University of Minnesota, on " The Electro-
magnetic Theory and the Velocity of Light."

The second paper was presented by Professor Alexander Macfarlane, of Chat-
ham, Ontario, "On the Exponential Notation in Vector-analysis."

The third paper was presented by Professor James McMahon, of Cornell Uni-
versity, " On the Use of N-fold Riemann Spaces in Applied Mathematics."

WORKS OF REFERENCE

(prepared through the eOURTESY OF PROFESSOR GEORGE BRUCE HAI-STED,
OP KENTON COLLEGE, AND PROFESSOR LUDWIG BOLTZMAJSTN, OP THE UNIVERSITY

OP Vienna)

Allman, G. J., Greek Geometry from Thales to Euclid. Dublin, Hodges, 1889.
BuRNSiDE, W. S., Theory of Groups. Cambridge University Press, 1897.
Cajori, F., The Modem Theory of Equations. New York, The Macmillan Co.,
1904.
The Teaching and History of Mathematics in the United States.

Washington, 1890, Bureau of Education.
A History of Elementary Mathematics. New York, 1896, MacmiUan.
Cantor, Von M., Vorlesungen iiber Geschichte der Mathematik. In 3 Banden.

Leipzig, Teubner, 1894-1901.
Cakhart, D., Surveying. Boston, 1888, Giim & Co.
Carr, G. S., S3mopsis of Elementary Results in Pure Mathematics. London^

1886, Hodgson.
Chrtstal, G., Algebra, 2 ed. 2 vols. London, 1900, Black.
Cox, HoMERSHAM, Principles of Arithmetic, Cambridge, 1885, Deighton.
Darboux, Gaston, Lecons sur les Systemes Orthogonaux et les coordonnees-
curvilignes. Paris, 1898, Gauthier-Villars.
Lecons sur la Theorie Generale des Surfaces et les Applications G6o-
metrique du Calcul Infinitesimal. 4 vols. 1887-1896. Paris, Gau-
thier-Villars.
Frost, P., Treatise on Curve Tracing. London, 1872, Macmillan.
Gibson, G. A., An Introduction to the Calculus Based on Graphic Methods.

London, 1904, Macmillan.
Hagen, J. G., Synopsis der Hoheren Mathematik. Berlin, 1891-1901, Dames.
Halsted, G. B., Projective Geometry. New York, Wiley & Sons, 1905.

Lobachevski's Geometrical Researches on the Theory of Parallels,.

4 ed. Gambler, Ohio, The Neomon, 1905.
Bolyai's Science Absolute of Space. Gambler, Ohio, The Neomon,.

1905.
The Elements of Geometry. 6 ed. New York, Wiley & Sons, 1903
Rational Geometry. New York, Wiley & Sons, 1904.
Mensuration. 4 ed. Boston, Ginn & Co., 1903.

Synthetic Geometry. The Lemoine-Brocard Geometry. 3 ed. New
York, 1899, Wiley & Sons.
Harkness and Morley, Introduction to the Theory of Analytic Functions,

London, 1898, Macmillan.
Hilbert, D., Grundlagen der Geometric. 2 ed. Leipzig, 1903, Teubner.
Jessop, C. M., a Treatise on the Line Complex. Cambridge University Press,

1903.
Jordan, M. Camillb, Cours d' Analyse de I'Ecole Poly technique. 2 ed. 3 vols,

Paris, 1903, Gauthier-Villars.
Langley, E. M., Computation. London, 1895, Longmans.
Levett and Davidson, Plane Trigonometry. London, Macmillan, 1892.
Love, A. E. H., Theoretical Mechanics. Cambridge University Press, 1897.
Mach, E., The Science of Mechanics, a critical and historical account of its
development. Translated by T. J. McCormack. 2 ed. Chicago, 1902, The Open
Court Pub. Co.

624 BIBLIOGRAPHY

Mellor, J. W., Higher Mathematics for Students of Chemistry and Physics.

London, 1902, Longmans.
Morgan, R. B., Elementary Graphs. London, 1903, Blackie.
Mueller, Felix, Vocabulaire Mathematique, Francais-Allemand et AUemand-

Francais, contenant les termes technique employes dans les mathematiques

pures et appliques. Leipzig, 1900, Teubner, vi, 316.
Emil Picard and Georges Simart, Theorie des Fonctions Algebriques de

deux Variables Independants. Paris, Gauthier-Villars.
Picard, Eaiil, Traits d' Analyse. 4 vols. Tome i, 2 ed. 1901. Tome ii, 1893.

Tome III, 1896. Paris, Gauthier-Villars.
Poincare, H., La Valeur de la Science. Paris, E. Flammarion, 1905.
La Science et I'Hypothese. Paris, E. Flammarion.
Les Methodes Nouvelles de la Mecanique celeste. 3 vols. Paris,

Gauthier-Villars, 1893.
Calcul des Probabilites. Paris, Carr6 et Naud, 1896.
Russell, Bertrand, The Principles of Mathematics. Cambridge University

Press, 1903.
Salmon, G., Analytic Geometry of Three Dimensions. 4 ed. Dublin, 1882,
Simpkins.
Treatise on the Higher Plane Curves. 3 ed. Dublin, 1879, Hodges.
Treatise on the Conic Sections. 6 ed. London, 1879, Longmans.
-Lessons Introductory to the Modern Higher Algebra. 4 ed. Dublin,
1885, Simpkins.
Scott, R. F., Theory of Determinants. 2 ed. Revised by G. B. Mathews.

Cambridge University Press, 1905.
Simon, Dr. Max, Euklid und die sechs Planimetrischen Bucher. Leipzig,

Teubner, 1901.
Todhunter, a History of the Theory of Probability. Cambridge, 1865, Macmillan.
Todhunter and Leathen, Spherical Trigonometry. London, Macmillan, 1901.
Willson, F. N., Descriptive Geometry and Mechanical Drawing. New York,

Macmillan, 1904.
Whitehead, A. N., Universal Algebra. Cambridge University Press, 1898.
Withers, J. W., Euchd's Parallel Postulate. Chicago, The Open Court Pub.

Co., 1905.
Whittaker, E. T., Modern Analysis. Cambridge University Press, 1902.
WoLFFiNG, Ernst, Mathematische Biicherschatz, Systematische Verzerchniss

der Wichtigsten Deutschen und Aiislandischen Lehrbiicher und Monographien

des 19. Jahrhunderts auf dem Gebiete der Mathematischen Wissenschaften.

Leipzig, Teubner, 1903.
Index Du Repertoire Bibliographique des Sciences Mathematiques. 2 ed. 1898.

Paris, Gauthiers-Villars.
Repertoriam der Hoheren Mathematik. i, Theil: Analysis, ii, Theil: Geometric.

Leipzig, 1902, Teubner.
Encylopedie des Sciences Mathematiques pures et appliques. Edition francaise,

redig^e et pubhee d'apres I'edition allemande sous la direction de Jules Molk.

Paris, Gauthier-Villars, 1904.

SPECIAL WORKS OF REFERENCE

(accompanying PARTICUIiARLT PROFESSOR BOLTZMANN'S ADDRBSs)

BoLTZMANN, LuDWiG, Studien iiber das Gleichgewicht der leb. Kraft zwischen

Bewegten Materiellen Punkten. Wien, Sitz. Ber. ii, 58, p. 517, 1868.
Losung eines Mechanischen Problems. Wien, Sitz. Ber. ii, 58 p. 1035,

1868.
Uber das Warmegleichgewicht zwischen Mehratomigen Gasmolekulen.

Wien, Sitz. Ber. n, 63 p. 397, 1871.
Einige allgemeine Satze iiber Warmegleichgewicht. Wien, Sitz. Ber.

II, 63, p. 679, 1871.
Weitere Studien iiber das Warmegleichgewicht unter Gasmal. Wien,

Sitz. Ber, ii, 66, p. 275, 1872.
Uber das Warmegleichgewicht von Gasen, anf welche Aussere Kraf te

wirken. Wien, Sitz. Ber. ii, 72, p. 427, 1875.
Uber die Anfstellung mid Integration der Gelichmigen, welche die

Molecularbewegung in Gasen bestimmen. Wien, Sitz. Ber. ii, 74,

p. 503, 1876.
Bemerkmigen iiber einige Probleme der Mechanischen Warmetheorie.

Wien, Sitz. Ber. ii, 75, p. 62, 1877.
Uber die Natur der Gasmolekule. Wien, Sitz. Ber. ii, 74, p. 553,

1876.
Uber die Beiziehung zwischen dem Hauptsatze der Mechanischen

Warmetheorie u. der Wahrscheinlichkeitsrechnung. Wien, Sitz.

Ber. II, 76, October, 1877.
Weitere Bemerkungen iiber einige Probleme der mechanischen War-
metheorie. Wien, Sitz. Ber. ii, 78, p. 7, 1878.
Uber das Arbeitsquantum welches bei chemischen Verbindungem

gewonnen werden kann. Wien, Sitz. Ber. ii, 88, p. 861, 1883.
Uber die Eigenschaften monocyklischer imd anderer damit verwandter

dynamik. Joum. f. r. u. d. Math. 100, p. 201, 1885.
Uber das Maxwellsche Vertheilungsgesetz der Geschwindigkeiteni.

Wied. Ann. 55, p. 223, 1895.
Uber eine Abhanlung Zerme las. Wied. Ann. 60, p. 392, 1897; SI,

p. 773, 1896.
Uber die Sogenannte H-Curoe. Math. Ann. 50, p. 325, 1898.
Vorlesungen iiber Gastheorie, i, 1896, ii, 1898, bei Barth, Leipzig,

besonders ii, Abschn. iii, und vii; and French translation appensdo

in. to vol. II.
On the EquiUbrium of Vis Viva. Phil. Mag. v, p. 153, 1893.
Encyklopadie der Math. Wissenschaften, vol. iv. D. Mechanik der

aus zahlreichen diskreten TheUen bestehenden Systeme. 27 Eia>-

greiben der Wahrscheinlichkeitsrechn. Teubner, 1905.
BuRBURT, Samuel H. On Jeans's Theory of Gases. Phil. Mag. vi, 5, p. 134,

6, p. 529, 1903.
On the Variation of Entropy by W. Gibbs. Phil. Mag. vi, 6, p.. 251,

1903.
Cf. also Phil. Mag. January, 1904, October, 1890, etc.

626 SPECIAL BIBLIOGRAPHY

CuLVERWELii, Edward P., Lord Kelvin's Test Case on the Maxwell-Boltzmann

Law. Nat. 46, p. 76, 1892.
GiBBS, WiLLAKD, Elementary Principles of Statistical Mechanics. Scribner Sons,

1903.
Jeans, J. H., The Kinetic Theory of Gases Developed from a New Standpoint.
PhU. Mag. VI, 5, p. 597, 6, p. 720, 1903.
On the Vibrations set up in Molecules by Collisions. Phil. Mag. vi,

6, p. 279, 1903.
The Dynamic Theory of Gases, Cambridge University Press, 1904.
LiENARD, Notes sur la Th^orie Cinetique des gaz. Joum. de Physique, iv, 2, p.

677, 1903.
Maxwell, James CilAJRK, Illustrations of the Dynamical Theory of Gases.
Phil. Mag. IV, 19, p. 19, 1860; 20, p. 21, 1860.
Scientific Paper, i, p. 379.
Dynamical Theory of Gases. Phil. Mag. rv, ser. vol. 35, p. 729; Scient.

pap. II, p. 26.
On Boltzmann's Theorem. Cambr. Phil. Trans. 12, part 3, p. 547,

1879; Scient. pap. ii, 713.
On Stresses in Rarefied Gases. Phil. Trans. Roy. Soc. 1879, i, p. 231;
Scient. pap. ii, p. 681.
Ratleigh, Lord, On Maxwell's Investigations respecting Boltzmann's Theorem.
Phil. Mag. V, 33, p. 356, 1892.
Dynamical Problems in Illustration of the Theory of Gases. Phil.

Mag. V, 32, p. 424, 1891.
The Law of Partition of Kinetic Energy. Phil. Mag. v, 49, p. 98,
1900.
Waals, Jun. Van der. Die statistische Naturanschauung. Rieckes Physikal.

Zeitschrift 4., p. 508, 1903.
Zermelo, Uber die Mechanische Erklarung irreversibler Vorgange. Wied. Ann.
57, p. 485; 59, p. 793, 1896. Cf. also Poincar^'s Thermodynamique.

CONTENTS OF THE SERIES

Volume I. History of the Congress; The Scientific Plan of the Congress; Intro-
ductory Address; Department of Philosophy (6 sections); Department
of Mathematics (3. sections).

Volume II. Department of Political and Economic History (6 sections); De-
partment of History of Law (3 sections) ; Department of History of Religion
(5 sections).

Volume III. Department of History of Language (8 sections); Department of
History of Literature (7 sections) ; Department of History of Art (3 sections.)

Volume IV. Department of Physics (3 sections); Department of Chemistry
(4 sections); Department of Astronomy (2 sections); Department of
Sciences of the Earth (8 sections).

Volume V. Department of Biology (11 sections); Department of Anthropolog}^
(3 sections) ; Department of Psychology (4 sections) ; Department of Socio-
logy (2 sections). ,

Volume VI. Department of Medicine (12 sections) ; Department of Technology'
(6 sections).

Volume VII. Department of Economics (6 sections) ; Department of Politics
(5 sections); Department of Jurisprudence (3 sections); Department of
Social Science (6 sections).

Volume VIII. Department of Education (5 sections) ; Department of Religion
(6 sections).

PRINTED BY H. O. HOUGHTON & CO.

CAMBRIDGE, MASS.

U.S. A.

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