Popular lectures on scientific subjects

POPULAR LECTURES

SCIENTIFIC SUBJECTS.

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POPULAR LECTURES

ON

SCIENTIFIC SUBJECTS

BY

HERMANN VON HELMHOLTZ

TRANSLATED BY

E. ATKINSON, PH.D., F.C.S.

FORMERLY PROFESSOR OF EXPERIMENTAL SCIENCE, STAFF COLLKOB

FIRST SERIES
WITH AS INTRODUCTION by PROFESSOR TYNDALL

LONDON

LONGMANS, GREEN, AND CO.

AND NEW YORK : 15 EAST 16th STREET

1895

All rights reserved

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TEANSLATOE'S PBEFACE

TO NEW EDITION.

IN 1891 the seventieth birthday of Hermann von Helm-
holtz, and the forty-ninth anniversary of his taking the
degree of Doctor of Medicine, was celebrated in Berlin in
a manner which, whether for its universality or import-
ance, has, perhaps, never fallen to the lot of a Man of
Science during his lifetime.

The demand for a new edition of the translation of
the two volumes of Von Helmholtz's Popular Scientific
Lectures suggests that this is an appropriate occasion for
a re-issue under conditions which make them accessible
to a larger circle of readers.

The present edition is identical with the preceding
ones ; discussing as they do, with the hand of a master,
fundamental scientific problems, these Lectures are not
likely to be soon out of date.

To the second volume has been added a remarkable
autobiographical account of the Author's scientific career
and development, which formed the subject of an address
given by Von Helmholtz in reply to the addresses of
congratulation on the occasion of his Jubilee. This is
taken, by kind permission of the publisher, M. Hirschwald,
from a collection of the addresses and speeches delivered
on that occasion.

E. ATKINSON.

March 1893.

425182

TRANSLATOR'S PREFACE.

IN bringing this Translation of Helmholtz's Popular
Scientific Lectures before the public, I have to thank Mr.
A. J. Ellis for having placed at the disposal of the
Publishers the translation of the third Lecture ; and also
Dr. Francis, the Editor of the ' Philosophical Magazine,'
for giving ine permission to use the translation of the fifth
Lecture, which originally appeared in that Journal.

In addition to the Editorial charge of the book, my
own task has been limited to the translation of two of the
Lectures. I should have hesitated to undertake the work,
had I not from the outset been able to rely upon the aid
of several gentlemen whose names are appended to the
Contents. One advantage gained from this division of
labour is, that the publication of the work has been
accelerated; but a far more important benefit has been
secured to it, in the co-operation of translators who have
brought to the execution of their task special knowledge
of their respective subjects.

E. ATKINSON.

AUTHOR'S PBEFACE.

IN COMPLIANCE with many requests, I beg to offer to the
public a series of popular Lectures which I have delivered
on various occasions. They are designed for readers who,
without being professionally occupied with the study of
Natural Science, are yet interested in the scientific results
of such studies. The difficulty, felt so strongly in printed
scientific lectures, namely, that the reader cannot see the
experiments, has in the present case been materially
lessened by the numerous illustrations which the publishers
have liberally furnished.

The first and second Lectures have already appeared in
print ; the first in a university programme, which, how-
ever, was not published. The second appeared in the
'Kieler Monatsschrift' for May, 1853, but, owing to the
restricted circulation of that journal, became but little
known; both have, accordingly, been reprinted. The
third and fourth Lectures have not previously appeared.

These Lectures, called forth as they have been by
incidental occasions, have not, of course, been composed
in accordance with a rigidly uniform plan. Each of them
has been kept perfectly independent of the others. Hence

viii AUTHOR'S PREFACE.

some amount of repetition has been unavoidable, and the
first four may perhaps seem somewhat confusedly thrown
together. If I may claim that they have any leading
thought, it would be that I have endeavoured to illustrate
the essence and the import of Natural laws, and their
relation to the mental activity of man. This seems to me
the chief interest and the chief need in Lectures before a
public whose education has been mainly literary.

I have but little to remark with reference to individual
Lectures. The set of Lectures which treat of the
Theory of Vision have been already published in the
* Preussische Jahrbiicher,' and have acquired, therefore,
more of the character of Eeview articles. As it was
possible in this second reprint to render many points
clearer by illustrations, I have introduced a number of
woodcuts, and inserted in the text the necessary explan-
ations. A few other small alterations have originated in
my having availed myself of the results of new series of
experiments.

The fifth Lecture, on the Interaction of Natural
Forces, originally published sixteen years ago, could not
be left entirely unaltered in this reprint Yet the alter-
ations have been as slight as possible, and have merely
been such as have become necessary by new experimental
facts, which partly confirm the statements originally made,
and partly modify them.

The seventh Lecture, on the Conservation of Force,
developes still further a portion of the fifth. Its main
object is to elucidate the cardinal physical ideas of work,
and of its unalterability. The applications and conse-

AUTHOR'S PREFACE. is

quences of the law of the Conservation of Force are com-
paratively more easy to grasp. They have in recent times
been treated by several persons in a vivid and interesting
manner, so that it seemed unnecessary to publish the cor-
responding part of the cycle of lectures which I delivered
on this subject; the more so as some of the more
important subjects to be discussed will, perhaps in the
immediate future, be capable of more definite treatment
than is at present possible.

On the other hand, I have invariably found that the
fundamental ideas of this subject always appear difficult
of comprehension not only to those who have not passed
through the school of mathematical mechanics ; but even
to those who attack the subject with diligence and in-
telligence, and who possess a tolerable acquaintance with
natural science. It is not to be denied that these ideas
are abstractions of a quite peculiar kind. Even such a
mind as that of Kant found difficulty in comprehending
them; as is shown by his controversy with Leibniiz.
Hence I thought it worth while to furnish in a popular
form an explanation of these ideas, by referring them to
many of the better known mechanical and physical ex-
amples ; and therefore I have only for the present given
the first Lecture of that series which is. devoted to this
object.

The last Lecture was the opening address for the
' Naturforscher-Versammlung,' in Innsbruck. It was not
delivered from a complete manuscript, but from brief
notes, and was not written out until a year after. The
present form has, therefore, no claim to be considered an

x AUTHOR'S PREFACE.

accurate reproduction of that address. I have added it to
the present collection, for in it I have treated briefly what
is more fully discussed in the other articles. Its title to
the place which it occupies lies in the fact that it attempts
to bring the views enunciated in the preceding Lectures
into a more complete and more comprehensive whole.

In conclusion, I hope that these Lectures may meet
with that forbearance which lectures always require when
they are not heard, but are read in print.

THE AUTHOR.

CONTENTS.

LECTUBB PAGE

I. ON THE KELATION OF NATURAL SCIENCE TO SCIENCE IN
GENERAL. Translated by H. W. EVE, Esq., M.A., F.C.S.,
Wellington College . . . .*• '••'.. ... 1

II. ON GOETHE'S SCIENTIFIC EESEARCHES. Translated by H. W.

EVE, Esq 29

III. ON THE PHYSIOLOGICAL CAUSES OF HARMONY or Music.

Translated by A. J. ELLIS, Esq., M.A., F.K.S. . .53

IV. ICE AND GLACIERS. Translated by Dr. ATKINSON, F.C.S.,

Professor of Experimental Science, Staff College . . .95

V. ON THE INTERACTION OF THE NATURAL FORCES. Translated

by Professor TYNDALL, LL.D., F.R.S. . . . . .137

VI. THE EECENT PROGRESS OF THE THEORY OF VISION. Translated
by Dr. PYE-SMITH, B.A., F.R.C.P., Guy's Hospital .

L The Eye as an Optical Instrument . . . .175

n. The Sensation of Sight . . . . . .202

m. The Perception of Sight . . . . . .237

VII. ON THE CONSERVATION OF FORCE. Translated by Dr. AT-
KINSON . .277

VIII. ON THE AIM AND PROGRESS OF PHYSICAL SCIENCB. Translated

by Dr. W. FLIGHT, F.C.S., British Museum . . .319

INTRODUCTION.

IN the year 1850, -when I was a student in the University of
Marburg, it was my privilege to translate for the 'Philosophical
Magazine' the celebrated memoirs of Clausius, then just pub-
lished, on the Moving Force of Heat.

In 1851, through the liberal courtesy of the late Professor
Magnus, I was enabled to pursue my scientific labours in his
laboratory in Berlin. One evening during my residence there
my friend Dr. Du Bois-Raymond put a pamphlet into my hands,
remarking that it was 'the production of the first head in Europe
since the death of Jacobi,' and that it ought to be translated
into English. Soon after my return to England I translated
the essay and published it in the 'Scientific Memoirs/ then
brought out under the joint-editorship of Huxley, Henfrey,
Francis, and myself.

This essay, which was communicated in 1847 to the Physical
Society of Berlin, has become siifficiently famous since. It was
entitled 'Die Erhaltung der Kraft,' and its author was Helmholtz,
originally Military Physician in the Prussian service, afterwards
Professor of Physiology in the Universities of Kb'nigsberg and
Heidelberg, and now Professor of Physics in the University of
Berlin.

Brought thus face to face with the great generalisation of
the Conservation of Energy, I sought, to the best of my ability,
to master it by independent thought in all its physical details.
I could not forget my indebtedness to Helmholtz and Clausius,

xiv INTRODUCTION.

or fail to see tlw probable influence of their writings on the
science of the coming time. For many years, therefore, it was
my habit to place every physical paper published by these
eminent men within the reach of purely English readers.

The ti-anslation of the lecture on the ' Wechselwirkung der
Naturkrafte,' printed in the following series, had this origin.
It appears here with the latest emendations of the author
ntroduced by Dr. Atkinson.

The evident aim of these Lectures is to give to those 'whose
education has been mainly literary,' an intelligent interest
in the researches of science. Even among such persons the
reputation of Helmholtz is so great as to render it almost super-
fluous for me to say that the intellectual nutriment here offered
is of the very first quality.

Soon after the publication of the ' Tonempfindungen ' by
Helmholtz, I endeavoured to interest the Messrs. Longman in the
work, urging that the publication of a translation of it would
be an honour to their house. They went carefully into the
question of expense, took sage counsel regarding the probable
sale, and came reluctantly to the conclusion that it would not
be remunerative. ' I then recommended the translation of these
' Populare Yortrage,' and to this the eminent publishers imme-
diately agreed.

Hence the present volume, brought out under the editorship
of Dr. Atkinson, of the Staff College, Sandhurst. The names
of the translators are, I think, a guarantee that their work will
be worthy of their original.

JOHN TYNDALL.

ROYAL INSTITUTION:
March 1873.

1 Since the date of the foregoing letter from Professor Tyndall, Messrs.
Longman & Co. have made arrangements for the translation of Helmholtz's
Tonempfindungen, by Mr. Alexander J. Ellis, F.R.S. &c.

ON THE

RELATION OF NATURAL SCIENCE1
TO GENERAL SCIENCE,

Academical Discourse delivered at Heidelberg, November 22, 1862,
Br DB. H. HELMHOLTZ, SOJIETIMK PKOBECTOB.

TO-DAY we are met, according to annual custom, in grateful
commemoration of an enlightened sovereign of this kingdom,
Charles Frederick, who, in an a-ge when the ancient fabric of
European society seemed tottering to its fall, strove, with lofty
purpose and untiring zeal, to promote the welfare of his sub-
jects, and, above all, their moral and intellectual development.
Rightly did he judge that by no means could he more effectually
realise this beneficent intention than by the revival and the
encouragement of this University. Speaking, as I do, on such
an occasion, at once in the name and in the presence of the
whole University, I have thought it well to try and take, as far

1 The German word Naturwissensctiaft has no exact equivalent in modern
English, including, as it does, both the Physical and the Natural Sciences.
Curiously enough, in the original charter of the Royal Society, the phrase
Natural Knowledge covers the same ground, but is there used in opposition
to supernatural kn OTvleclge. (Note in Buckle's Civilisation, vol. ii. p. 34] .)—
Tu.

r. B

2 Oy THE RELATION OK

as is permitted by the narrow standpoint of a single student,
a general view of the connection of the several sciences, and of
their study.

It may, indeed, he thought that, at the present day, those
relations between the different sciences which have led us to
combine them under the name Universitas Litterarum, have
become looser than ever. We see scholars and scientific men
absorbed in specialities of such vast extent, that the most
universal genius cannot hope to master more than a small
section of our present range of knowledge. For instance, the
philologists of the last three centuries found ample occupation
in the study of Greek and Latin ; at best they added to it the
knowledge of two or three European languages, acquired for
practical purposes. But now comparative philology aims at
nothing less than an acquaintance with all the languages of all
branches of the human family, in order to deduce from them
the laws by which language itself has been formed, and to this
gigantic task it has already applied itself with superhuman
industry. Even classical philology is no longer restricted to
the study of those works which, by their artistic perfection
and precision of thought, or because of the importance of their
contents, have become models of prose and poetry to all ages.
On the contrary, we have learnt, that every lost fragment of
an ancient author, every gloss of a pedantic grammarian, every
allusion of a Byzantine court-poet, every broken tombstone
found in the wilds of Hungary or Spain or Africa, may con-
tribute a fresh fact, or fresh evidence, and thus serve to increase
our knowledge of the past. And so another group of scholars are
busy with the vast scheme of collecting and cataloguing, for the
use of their successors, every available relic of classical antiquity.
Add to this, in history, the study of original documents, the
critical examination of parchments and papei-s accumulated in
the archives of states and of towns ; the combination of details
scattered up and down in memoirs, in correspondence, and iu
biographies ; the deciphering of hieroglyphics and cuneiform in-
scriptions; in natural history the more and more comprehensive
classification of minerals, plants, and animals, as wellliving aa

NATURAL SCIENCE TO GENERAL SCIENCE. 3

extinct ; and there opens out before us an expanse of knowledge the
contemplation of which may well bewilder us. In all these sciences
the range of investigation widens as fast as the means of obser-
vation improve. The zoologists of past times were content to
have described the teeth, the hair, the feet, and other external
characteristics of an animal. The anatomist, on the other hand,
confined himself to human anatomy, so far as he could make
it out by the help of the knife, the saw, and the scalpel, with
the occasional aid of injections of the vessels. Human anatomy
then passed tor an unusually extensive and difficult study. Now
we are no longer satisfied with the comparatively rough science
which bore the name of human anatomy, and which, though
without reason, was thought to be almost exhausted. We
have added to it comparative anatomy — that is, the anatomy
of all animals — and microscopic anatomy, both of them sciences
of infinitely wider range, which now absorb the interest of
students.

The four elements of the ancients and of mediaeval alchemy
have been increased to sixty-four, the last four of which are
due to a method invented in our own University, which pro-
mises still further discoveries.1- But not merely is the number
of the elements far greater, the methods of producing compli-
cated combinations of them have been so vastly improved, that
what is called organic chemistry, which embraces only com-
pounds of carbon with oxygen, hydrogen, nitrogen, and a few
other elements, has already taken iunk as an independent
science.

' As the stars of heaven for multitude ' was in ancient times
the natural expression for a number beyond our comprehension,
Pliny even thinks it almost presumption (' rem etiam Deo im-
probam ') on the part of Hipparchus to have undertaken to
count the stars and to determine their relative positions. And
yet none of the catalogues up to the seventeenth centuiy, con-
structed without the aid of telescopes, give more than from

1 That is the method of spectrum analysis, duo to Bunsen and Kirchoff, both
of Heidelberg. The elements alluded to are csesiura, rubidium, thallium, and
iridium.

4 ON THE RELATION OF

1,000 to 1,500 stars of magnitudes from the first to the fifth.
At present several observatories are engaged in continuing these
catalogues down to stars of the tenth magnitude ; so that up-
wards of 200,000 fixed stai-s are to be catalogued and their places
accurately determined. The immediate result of these obser-
vations has been the discovery of a great number of new
planets; so that, instead of the six known in 1781, there are
now seventy-five.1

The contemplation of this astounding activity in all branches
of science may well make us stand aghast at the audacity of
man, and exclaim with the Chorus in the ' Antigone ' : ' Who
can survey the whole field of knowledge 1 Who can grasp the
clues, and then thread the labyrinth 1 ' One obvious consequence
of this vast extension of the limits of science is, that every
student is forced to choose a narrower and narrower field for
his own studies, and can only keep up an imperfect acquaintance
even with allied fields of research. It almost raises a smile to
hear that in the seventeenth century Kepler was invited to
Grate as professor of mathematics and moral philosophy : and
that at Leyden, in the beginning of the eighteenth, Boerhave
occupied at the same time the chairs of botany, chemistry, and
clinical medicine, and therefore practically that of pharmacy as
well. At present we require at least four professors, or, in an
university with its full complement of teachers, seven or eight,
to represent all these branches of science. And the same is
true of other faculties.

One of my strongest motives for discussing to-day the con-
nection of the different sciences is that I am myself a student
of natural philosophy; and that it has been made of late a
reproach against natural philosophy that it has struck out a
path of its own, and has separated itself more and more widely
from the other sciences which are united by common philological
and historical studies. This opposition has, in fact, been long
apparent, and seems to me to have grown up mainly under the
influence of the Hegelian philosophy, or, at any rate, to have

« At the end of November 1864, the 82nd of the small planets, Alcmene. WM
discovered. There are now 109.

NATURAL SCIENCE TO GENERAL SCIENCE. 5

been brought out into more distinct relief by that philosophy.
Certainly, at the end of the last century, when the Kantian
philosophy reigned supreme, such a schism had never been pro-
claimed ; on the contrary, Kant's philosophy rested on exactly
the same ground as the physical sciences, as is evident from his
own scientific works, especially from his * Cosmogony,' based
upon Newton's Law of Gravitation, which afterwards, under
the name of Laplace's Nebular Hypothesis, came to be uni-
versally recognised. The sole object of Kant's 'Critical Phi-
losophy' was to test the sources and the authority of our
knowledge, and to fix a definite scope and standard for the
researches of philosophy, as compared with other sciences.
According to his teaching, a principle discovered a priori by
pure thought was a rule applicable to the method of pure
thought, and nothing further ; it could contain no real, pos tivo
knowledge. The 'Philosophy of Identity* l was bolder. It
started with the hypothesis that not only spiritual phenomena,
but even the actual world — nature, that is, and man — were the
result of an act of thought on the part of a creative mind,
similar, it was supposed, in kind to the human mind. On this
hypothesis it seemed competent for the human mind, even with-
out the guidance of external experience, to think over again the
thoughts of the Creator, and to rediscover them by its own
inner activity. Such was the view with which the ' Philosophy
of Identity' set to work to construct a priori the results of
other sciences. The process might be more or loss successful in
matters of theology, law, politics, language, art, histoiy, in short,
in all sciences the subject-matter of which really grows out of
our moral nature, and which are therefore properly classed
together under the name of moral sciences. The state, the
church, art and language, exist in order to satisfy certain moral
needs of man. Accordingly, whatever obstacles nature, or
chance, or the rivalry of other men may interpose, the efforts of
the human mind to satisfy its needs, being systematically directed
to one end, must eventually triumph over all such fortuitous

1 So called because it proclaimed the identity not onh' of subject and object,
but of contradictories, such *w existence and, non-existence.— To.

6 ON THE RELATION OF

hindrances. Under these circumstances, it would not be a
downright impossibility for a philosopher, starting from an exact
knowledge of the mind, to predict the general course of human
development under the above-named conditions, especially if
he has before his eyes a basis of observed facts, on which to
build his abstractions. Moreover, Hegel was materially assisted,
in his attempt to solve this problem, by the profound and philo-
sophical views on historical and scientific subjects with which
the writings of his immediate predecessors, both po?ts and phi-
losophers, abound. He had, for the ruost part, only to collect
and combine them, in order to produce a system calculated to
impress people by a number of acute and original observations.
He thus succeeded in gaining the enthusiastic approval of most
of the educated men of his time, and in raising extravagantly
sanguine hopes of solving the deepest enigma of human life ; all
the more sanguine doubtless, as the connection of his system
was disguised under a strangely abstract phraseology, and was
perhaps really understood by but few of his worshippers.

But even granting that Hegel was more or less successful in
constructing, a priori, the leading results of the moral sciences,
still it was no proof of the correctness of the hypothesis of
Identity, with which he started. The facts of nature would
have been the crucial test. That in the moral sciences traces of
the activity of the human intellect and of the several stages of
its development should present themselves, was a matter of
course ; but surely, if nature really reflected the result of the
thought of a creative mind, the system ought, without difficulty,
to find a place for her comparatively simple phenomena and
processes. It was at this point that Hegel's philosophy, we
venture to say, utterly broke down. His system of nature
eeemed, at least to natural philosophers, absolutely crazy. Of
all the distinguished scientific men who were his contem-
poraries, not one was found to stand up for his ideas. Accord-
ingly, Hegel himself, convinced of the importance of winning
for his philosophy in the field of physical science that recog-
nition which had been so freely accorded to it elsewhere,
launched out, with unusual vehemence and acrimony, against

NATURAL SCIENCE TO GENERAL SCIENCE. 7

the natural philosophers, and especially against Sir Isaac Newton,
ns the first and gieatest representative of physical investigation.
The philosophers accused the scientific men of narrowness; the
scientific men retorted that the philosophers were crazy. And
so it came about that men of science began to lay some stress on
the banishment of all philosophic influences from their work ;
while some of them, including men of the greatest acuteness,
went so far as to condemn philosophy altogether, not merely a=?
useless, but as mi;chievous dreaming. Thus, it must be con-
fessed, not only were the illegitimate pretensions of the Hegelian
system to subordinate to itself all other studies rejected, but no
regard was paid to the rightful claims of philosophy, that is,
the criticism of the sources of cognition, and the definition of
the functions of the intellect.

In the moral sciences the course of things was different,
though it ultimately led to almost the same result. In all
branches of those studies, in theology, politics, jurisprudence,
sesthetics, philology, there started up enthusiastic Hegelians,
who tried to reform their several departments in accordance
with the doctrines of their master, and, by the royal road of
speculation, to reach at once the promised land and gather in
the harvest, which had hitherto only bean approached by long
and laborious study. And so, for some time, a hard and fast
line was drawn between the moral and the physical sciences ;
in fact, the very name of science was often denied to the
latter.

The feud did not long subsist in its original intensity. The
physical sciences proved conspicuously, by a brilliant series of
discoveries and practical applications, that they contained a
healthy germ of extraordinary fertility ; it was impossible any
longer to withhold from them recognition and respect. And
even in other departments of science, conscientious investigators
of facts soon protested against the over-bold nights of specu-
lation. Still, it cannot be overlooked that the philosophy of
Hegel and Schilling did exercise a beneficial influence ; since their
time the attention of investigators in the moral sciences had
been constantly and more keenly directed to the scope of those

8 OX THE RELATION OF

sciences, and to their intellectual contents, and therefore the
great amount of labour bestowed on those systems has not
been entirely thrown away.

We see, then, that in proportion as the experimental inves-
tigation of facts has recovered its importance in the moral
sciences, the opposition between them and the physical sciences
has become less and less marked. Yet we must not forget
that, though this opposition was brought out in an unnecessarily
exaggerated form by the Hegelian philosophy, it has its founda-
tion in the nature of things, and must, sooner or later, make
itself felt. It depends partly on the nature of the intellectual
processes the two groups of sciences involve, partly, as their
very names imply, on the subjects of which they treat. It is
not easy for a scientific man to convey to a scholar or a jurist a
clear idea of a complicated process of nature ; he must demand
of them a certain power of abstraction from the phenomena, as
well as a certain skill in the use of geometrical and mechanical
conceptions, in which it is difficult for them to follow him. On
the other hand an artist or a theologian will perhaps find the
natural philosopher too much inclined to mechanical and
material explanations, which seem to them commonplace, and
chilling to their feeling and enthusiasm. Nor will the scholar
or the historian, who have some common ground with the
theologian and the jurist, fare better with the natural philo-
sopher. They will find him shockingly indifferent to literary
treasures, perhaps even more indifferent than he ought to be to
the history of his own science. In short, there is no denying
that, while the moral sciences deal directly with the nearest
and dearest interests of the human mind, and with the insti-
tutions it has brought into being, the natural sciences are con-
cerned with dead, indifferent matter, obviously indispensable
for the sake of its practical utility, but apparently without any
immediate bearing on the cultivation of the intellect.

It has been shown, then, that the sciences have branched
out into countless ramifications, that there has grown up
between different groups of them a real and deeply felt opposi-
tion, that) finally no single intellect can embrace the whole range

NATURAL SCIENCE TO GENERAL SCIENCE. 9

or even a considerable portion of it. Is it still reasonable to
keep them together in one place of education ? Is the union
of the four faculties to form one University a mere relic of the
Middle Ages? Many valid arguments have been adduced for
separating them. Why not dismiss the medical faculty to the
hospitals of our great towns, the scientific men to the Poly-
technic Schools, and form special seminaries for the theologians
and jurists ? Long may the German universities be preserved
from such a fate ! Then, indeed, would the connection between
the different sciences be finally broken. How essential that
connection is, not only from an university point of view, as
tending to keep alive the intellectual energy of the country, but
also on material grounds, to secure the successful application of
that energy, will be evident from a few considerations.

First, then, I would say that union of the different faculties
is necessary to maintain a healthy equilibrium among the in-
tellectual energies of students. Each study tries certain of our
intellectual faculties more than the rest, and strengthens them
accordingly by constant exercise. Bat any sort of one-sided
development is attended with danger ; it disqualifies us for
using those faculties that are less exercised, and so renders us
less capable of a general view ; above all it leads us to overvalue
ourselves. Any one who has found himself much more suc-
cessful than others in some one department of intellectual labour,
is apt to forget that there are many other things which they can
do better than he can : a mistake — I would have every student
remember — which is the worst enemy of all intellectual
activity.

How many men of ability have forgotten to practise that
criticism of themselves which is so essential to the student, and
BO hard to exercise, or have been completely crippled in their
progress, because they have thought dry, laborious drudgery
beneath them, and have devoted all their energies to the quest
of brilliant theories and wonder-working discoveries ! How
many such men have become bitter misanthropes, and put an end
to a melancholy existence, because they have failed to obtain
among their fellows that recognition which must be won by

10 ON THE RELATION OF

labour and result?, but which is ever withheld from mere self-con-
scious genius ! And the more isolated a man is, the more liable
is he to this danger ; while, on the other hand, nothing is more
inspiriting than to feel yourself forced to strain every nerve to
win the admiration of men whom you, in your turn, must admire.

In comparing the intellectual processes involved in the
pursuit of the several branches of science, we are struck by
certain generic differences, dividing one group of sciences from
another. At the same time it must not be forgotten that every
man of conspicuous ability has his own special mental constitution
which fits him for one line of thought rather than another.
Compare the work of two contemporary investigators even
in closely allied branches of science, and you will generally be
able to convince yourself that the more distinguished the men are
the more clearly does their individuality come out, and the less
qualified would either of them be to carry on the other's researches.
To-day I cau, of course, do nothing more than characterise
some of the most general of these differences.

I have already noticed the enormous mass of the materials
accumulated by science. It is obvious that the organisation
and arrangement of them must be proportionately perfect, if
we are not to be hopelessly lost in the maze of erudition.
One of the reasons why we can so far surpass our predecessors
in each individual study is that they have shown us how to
organise our knowledge.

This organisation consists, in the first place, of a mechanical
arrangement of materials, such as is to be found in our cata-
logues, lexicons, registers, indexes, digests, scientific and literary
annuals, systems of natural history, and the like. By these
appliances thus much at least is gained, that such know-
ledge as cannot be carried about in the memory is immedi-
ately accessible to anyone who wants it. With a good lexicon u
school-boy of the present day can achieve results in the inter-
pretation of the classics which an Erasmus, with the erudition
of a lifetime, could hardly attain. Works of this kind form, so
to speak, our intellectual principal with the interest of which
we trade : it is, so to speak, like capital invested in land. The

NATURAL SCIENCE TO GENERAL SCIENCE. 11

[earning buried in catalogues, lexicons, and indexes looks as
bare and uninviting as the soil of a farm ; the uninitiated cannot
Bee or appreciate the labour and capital already invested there ;
to them the work of the ploughman seems infinitely dull, weary,
and monotonous. But though the compiler of a lexicon or of
a system of natural history must be prepared to encounter
labour as weary and as obstinate as the ploughman's, yet it
need not be supposed that his work is of a low type, or that it is
by any means as dry and mechanical as it looks when we have
it before us in black and white. In this, as in any other sort of
scientific work, it is necessary to discover every fact by careful
observation, then to verify and collate them, and to separate
what is important from what is not. All this requires a
man with a thorough grasp both of the object of the
compilation and of the matter and methods of the science;
and for such a man every detail has its bearing on the
whole, and its special interest. Otherwise dictionary-making
would be the vilest drudgery imaginable.1 That the influence
of the progressive development of scientific ideas extends to
these works is obvious from the constant demand for new
lexicons, new natural histories, new digests, new catalogues of
stars, all denoting advancement in the art of methodising and
organising science.

But our knowledge is not to lie dormant in the shape of
catalogues. The very fact that we must carry it about in black
and white shows that our intellectual mastery of it is incomplete.
It is not enough to be acquainted with the facts; scientific
knowledge begins only when their laws and their causes are un-
veiled. Our materials must be worked up by a logical process;
and the first step is to connect like with like, and to elaborate a
general conception embracing them all. Such a conception, as
the name implies, takes a number of single facts together, and
stands as their representative in our mind. We call it a general
conception, or the conception of a genus, when it embraces a
number of existing objects; we call it a law when it embraces a
beries of incidents or occurrences. When, for example, I havo
1 Condendaque lexi^a mandat damnatis. — TB.

12 OX THE RELATION OF

made out that all mammals — that is, all warm-blooded, vivi-
parous animals — breathe through lungs, have two chambers in
the heart, and at least three tympana] bones, I need no longer
remember these anatomical peculiarities in the individual cases
of the monkey, the dog, the horse, and the whale; the general
rule includes a vast number of single instances, and represents
them in my memory. V/hen I enunciate the law of refraction,
not only does this law embrace all cases of rays falling at all
possible angles on a plane surface of water, and inform me of
the result, but it includes all cases of rays of any colour incident
on transparent surfaces of any form and any constitution what-
soever. This law, therefore, includes an infinite number of
cases, which it would have been absolutely impossible to carry
in one's memory. Moreover, it should be noticed that not only
does this law include the cases which we ourselves or other men
have already observed, but that we shall not hesitate to apply it
to new cases, not yet observed, with absolute confidence in the
reliability of our results. In the same way, if we were to find
a new species of mammal, not yet dissected, we are entitled to
assume, with a confidence bordering on a certainty, that it has
lungs, two chambers in the heart, and three or more tympanal
bones.

Thus, when we combine the results of experience by a pro-
cess of thought, and form conceptions, whether general concep-
tions or laws, we not only bring our knowledge into a form in
which it can be easily used and easily retained, but we actually
enlarge it, inasmuch as we feel ourselves entitled to extend the
rules and the laws we have discovered to all similar cases that
may be hereafter presented to us.

The above-mentioned examples are of a class in which the
mental process of combining a number of single cases so as to
form conceptions is unattended by farther difficulties, and can be
distinctly followed in all its stages. But in complicated cases it
is not so easy completely to separate like facts from unlike, and
to combine them into a clear well-defined conception. Assume
that we know a man to be ambitious ; we shall perhaps be able
to predict with tolerable certainty that if he has to act under

NATURAL SCIENCE TO GENERAL SCIENCE. 13

certain conditions, he will follow the dictates of his ambition,
and decide on a certain line of action. But, in the first place,
we cannot define with absolute precision what constitutes an
ambitious man, or by what standard the intensity of his ambition
is to be measured; nor, again, can we say precisely what degree
of ambition must operate in order to impress the given direction,
on the actions of the man under those particular circumstances.
Accordingly, we institute comparisons between the actions of
the man in question, as far as we have hitherto observed them,
and those of other men who in similar cases have acted as he
has done, and we draw our inference respecting his future actions
without being able to express either the major or the minor pre-
miss in a clear, sharply defined form — perhaps even without hav-
ing convinced ourselves that our anticipation rests on such an
analogy as I have described. In such cases our decision proceeds
only from a certain psychological instinct, not from conscious
reasoning, though in reality we have gone through an intellectual
process identical with that which leads us to assume that a
newly discovered mammal has lungs.

This latter kind of induction, which can never be perfectly
assimilated to forms of logical reasoning, nor pressed so far as to
establish universal laws, plays a most important part in human
life. The whole of the process by which we translate our sen-
sations into perceptions depends upon it, as appears especially
from the investigation of what are called illusions. For in-
stance, when the retina of the eye is irritated by a blow, we
imagine we see a light in our field of vision, because we have,
throughout our lives, felt irritation in the optic nerves only
when there was light in the field of vision, and have become
accustomed to identify the sensations of those nerves with the
presence of light in the field of vision. Moreover, such is the
complexity of the influences affecting the formation both of
character in general and of the mental condition at any given
moment, that this same kind of induction necessarily plays a
leading part in the investigation of psychological processes. In
fact, in ascribing to ourselves free-will, that is, full power to act
as we please without being subject to a stern inevitable law of

14 OX THE RELATION OF

causality, we deny in toto the possiblity of referring at least one
of the ways in which our mental activity expresses itself to a
rigorous law.

We might possibly, in opposition to logical induction which
reduces a question to clearly defined universal propositions, call
this kind of reasoning (esthetic induction, because it is most con-
spicuous in the higher class of works of art. It is an essential
part of an artist's talent to reproduce by words, by form, by
colour, or by music, the external indications of a character or a
state of mind, and by a kind of instinctive intuition, uncon-
trolled by any definable rule, to seize the necessary steps by
which we pass from one mood to another. If we do find that
the artist has consciously worked after general rules and abstrac-
tions, we think his work poor and commonplace, and cease to
admire. On the contrary, the works of great artists bring be-
fore us characters and moods with such a lifelikeness, with such
a wealth of individual traits and such an overwhelming con-
viction of truth, that they almost seem to be more real than the
reality itself, because all disturbing influences are eliminated.

Now if, after these reflections, we proceed to review the
different sciences, nnd to classify them according to the method
by which they must arrive at their results, we are brought face
to face with a generic difference between the natural and the
moral sciences. The natural sciences are for the most part in
a position to reduce their inductions to sharply defined general
rules and principles; the ir oral sciences, on the other hand, have,
in by far the most numerous cases, to do with conclusions
arrived at by psychological instinct. Philology, in so far as it
is concerned with the interpretation and emendation of the
texts handed down to us, must seek to feel out, as it were, the
meaning which the author intended to express, and the accessory
notions which he wished his words to suggest : and for that pur-
pose it is necessary to start with a correct insight, both into the
personality of the author, and into the genius of the language
in which he wrote. All this affords scope for aesthetic, but
not for strictly logical, induction. It is only possible to pass
judgment, if you have ready in your memory a great number of

NATURAL SCIENCE TO GENERAL SCIENCE. 15

similar facts, to be instantaneously confronted with the question
you are trying to solve. Accordingly, one of the first requisites
for studies of this class ia an accurate and ready memory.
Many celebrated historians and philologists have, in fact,
astounded their contemporaries by their extraordinary strength of
memory. Of course memory alone is insufficient without a
knack of everywhere discovering real resemblance, and without
a delicately and fully trained insight into the springs of human
action ; while this again is unattainable without a certain
warmth of sympathy and an interest in observing the working
of other men's minds. Intercourse with our follow-men in
daily life must lay the foundation of this insight, but the study
of history and art serves to make it richer and completer, for
there we see men acting under comparatively unusual conditions,
and thus come to appreciate the full scope of the energies which
lie hidden in our breasts.

None of this group of sciences, except grammar, lead us, as a
rule, to frame and enunciate general laws, valid under all circum-
stances. The laws of grammar are a product of the human
will, though they can hardly be said to have been framed de-
liberately, but rather to have grown up gradually, as they were
wanted. Accordingly, they present themselves to a learner
rather in the form of commands, that is, of laws imposed by
external authority.

With these sciences theology and jurisprudence are naturally
connected. In fact, certain branches of history and philology
serve both as stepping-stones and as handmaids to them. The
general laws of theology and jurisprudence are likewise com-
mands, laws imposed by external authority to regulate, from a
moral or juridical point of view, the actions of mankind; not
laws which, like those of nature, contain generalisations from a
vast multitude of facts. At the same time the application of a
grammatical, legal, moral, or theological rule is couched, like the
application of a law of nature to a paiticular case, in the forms of
logical inference. The rule forms the major premiss of the
syllogism, while the minor must settle whether the case in ques-
tion satisfies the conditions to which the rule is intended to

16 ON THE RELATION OF

apply. The solution of this latter problem, whether in gram-
matical analysis, where the meaning of a sentence is to be
evolved, or in the legal criticism of the credibility of the facts
alleged, of the intentions of the parties, or of the meaning of
the documents they have put into court, will, in most cases, be
again a matter of psychological insight. On the other hand, it
should not be forgotten that both the syntax of fully developed
languages and a system of jurisprudence gradually elaborated, as
ours has been, by the practice of more than 2,000 years,1 have
reached a high pitch of logical completeness and consistency; so
that, speaking generally, the cases which do not obviously fall
under some one or other of the laws actually laid down are quite
exceptional. Such exceptions there will always be, for the legis-
lation of man can never have the absolute consistency and
perfection of the laws of nature. In such cases there is no
course open but to try and guess the intention of the legislator ;
or, if needs be, to supplement it after the analogy of his decisions
in similar cases.

Grammar and jurisprudence have a certain advantage as
means of training the intellect, inasmuch as they tax pretty
equally all the intellectual powers. On this account secondary
education among modern European nations is based mainly
upon the grammatical study of foreign languages. The mother-
tongue and modern foreign languages, when acquired solely by
practice, do not call for any conscious logical exercise of thought,
though we may cultivate by means of them an appreciation for
artistic beauty of expression. The two classical languages,
Latin and Greek, have, besides their exquisite logical subtlety
and aesthetic beauty, an additional advantage, which they seem
to possess in common with most ancient and original languages
— they indicate accurately the relations of words and sentences
to each other by numerous and distinct inflexions. Languages
are, as it were, abraded by long use ; grammatical distinctions
are cut down to a minimum for the sake of brevity and rapidity

i It should be remembered that the Roman law, which has only partially
and indirectly influenced English practice, is the recognised basis of German
jurisprudence. — TB.

NATURAL SCIENCE TO GENERAL SCIENCE. 17

of expression, and are thus made less and less definite, as is
obvious from the comparison of any modern European language
with Latin; in English the process has gone further than in
any other. This seems to me to be really the reason why the
modern languages are far less fitted than the ancient for instru-
ments of education.1

As grammar is the staple of school education, legal studies
are used, and rightly, as a means of training persons of maturer
age, even when not specially required for professional purposes.

We now come to those sciences which, in respect of the kind
of intellectual labour they require, stand at the opposite end
of the series to philology and history ; namely, the natural and
physical sciences. I do not mean to say that in many branches
even of these sciences an instinctive appreciation of analogies
and a certain artistic sense have no part to play. On the
contrary, in natural history the decision which characteristics
are to be looked upon as important for classification, and which
as unimportant, what divisions of the animal and vegetable
kingdoms are more natural than others, is really left to an
instinct of this kind, acting without any strictly definable rule.
And it is a very suggestive fact that it was an artist, Goethe,
who gave the first impulse to the researches of comparative
anatomy into the analogy of corresponding organs in different
animals, and to the parallel theory of the metamorphosis of
leaves in the vegetable kingdom; and thus, in fact, really
pointed out the direction which the science has followed ever
since. But even in those departments of science where we
have to do with the least understood vital processes, it is,
speaking generally, far easier to make out general and compre-
hensive ideas and principles, and to express them in definite
language, than in cases where we must base our judgment on
the analysis of the human mind. It is only when we come to
the experimental sciences to which mathematics are applied,
and especially when we come to pure mathematics, that we

1 Those to whom German is not a foreign tongue may, perhaps, be per-
mitted to hold different views on the efficacy of modern languages in educa-
tion.—TR.

I. O

18 OX THE RELATION OF

see the peculiar characteristics of the natural and physical
sciences fully brought out.

The essential differentia of these sciences seems to me to
consist in the comparative ease with which the individual
results of observation and experiment are combined under
general laws of unexceptionable validity and of an extra-
ordinarily comprehensive character. In the moral sciences, on
the other hand, this is just the point where insuperable diffi-
culties are encountered. In mathematics the general propo-
sitions which, under the name of axioms, stand at the head of
the reasoning, are so few in number, so comprehensive, and so
immediately obvious, that no proof whatever is needed for
them. Let me remind you that the whole of algebra and
aiithmetic is developed out of the three axioms : —

' Things which are equal to the same things are equal to
one another.'

' If equals be added to equals, the wholes are equal.'

' If unequals be added to equals, the wholes are unequal.'
And the axioms of geometry and mechanics are not more
numerous. The sciences we have named are developed out ot
these few axioms by a continual process of deduction from
them in more and more complicated cases. Algebra, however,
does not confine itsslf to finding the sum of the most hetero-
geneous combinations of a finite number of magnitudes, but in
the higher analysis it teaches us to sum even infinite series,
he terms of which increase or diminish according to the most
various laws ; to solve, in fact, problems which could never be
completed by direct addition. An instance of this kind shows
us the conscious logical activity of the mind in its purest and
most perfect form. On the one hand we see the laborious nature
of the process, the extreme caution with which it is necessary
to advance, the accuracy required to determine exactly the
scope of such universal principles as have been attained, the
difficulty of forming and understanding abstract conceptions.
On the other hand, we gain confidence in the certainty, the
range, and the fertility of this kind of intellectual work.

The fertility of the method comes out moie strikingly in

NATURAL SCIENCE TO GENERAL SCIENCE. 1'J

applied mathematics, especially in mathematical physics, in-
cluding, of course, physical astronomy. From the time when
Xewton discovered, by analysing the motions of the planets on
mechanical principles, that every particle of ponderable matter
in the universe attracts every other particle with a force vary-
ing inversely as the square of the distance, astronomers have
been able, in virtue of that one law of gravitation, to calculate
with the greatest accuracy the movements of the planets to the
remotest past and the most distant future, given only the posi-
tion, velocity, and mass of each body of our system at any one
time. More than that, we recognise the operation of this la\v
in the movements of double stars, whose distances from us are
so great that their light takes years to reach us ; in some
cases, indeed, so great that all attempts to measure them have
failed.

This discovery of the law of gravitation and its consequences
is the most imposing achievement that the logical power of the
human mind has hitherto performed. I do not mean to say
that there have not been men who in power of abstraction have
equalled or even surpassed Newton and the other astronomers,
who either paved the way for his discovery, or have carried it
out to its legitimate consequences ; but there has never been
presented to the human mind such an admirable subject as
those involved and complex movements of the planets, which
hitherto had served merely as food for the astrological super-
stitions of ignorant star-gazers, and were now reduced to a single
law, capable of rendering the most exact account of the minutest
detail of their motions.

The principles of this magnificent discovery have been suc-
cessfully applied to several other physical sciences, among which
physical optics and the theory of electricity and magnetism are
especially worthy of notice. The experimental sciences have
one great advantage over the natural sciences in the investiga-
tion of general laws of nature : they can change at pleasure the
conditions under which a given result takes place, and can thus
confine themselves to a small number of characteristic instances,
in order to discover the law. Of course its validity must then
c '2

20 ON THE RELATION OF

stand the test of application to more complex cases. Accord-
ingly the physical sciences, when once the right methods have
been discovered, have made proportionately rapid progress.
Not only have they allowed us to look back into primaeval
chaos, where nebulous masses were forming themselves into
suns and planets, and becoming heated by the energy of their
contraction; not only have they permitted us to investigate
the chemical constituents of the solar atmosphere and of the
remotest fixed stars, but they have enabled us to turn the
forces of surrounding nature to our own uses and to make them
the ministers of our will.

Enough has been said to show how widely the intellectual
processes involved in this group of sciences differ, for the most
part, from those required by the moral sciences. The mathe-
matician need have no memory whatever for detached facts, the
physicist hardly any. Hypotheses based on the recollection of
similar cases may, indeed, be useful to guide one into the right
track, but they have no real value till they have led to a precise
and strictly defined law. Nature does not allow us for a moment
to doubt that we have to do with a rigid chain of cause and
effect, admitting of no exceptions. Therefore to us, as her
students, goes forth the mandate to labour on till we have dis-
covered unvarying laws ; till then we dare not rest satisfied, for
then only can our knowledge grapple victoriously with time
and space and the forces of the universe.

The iron labour of conscious logical reasoning demands great
perseverance and great caution ; it moves on but slowly, and is
rarely illuminated by brilliant flashes of genius. It knows
little of that facility with which the most varied instances come
thronging into the memory of the philologist or the historian.
Rather is it an essential condition of the methodical progress of
mathematical reasoning that the mind should remain concen-
trated on a single point, undisturbed alike by collateral ideas on
the one hand, and by wishes and hopes on the other, and moving
on steadily in the direction it has deliberately chosen. A cele-
brated logician, Mr. John Stuart Mill, expresses his conviction
that the inductive sciences have of late done more for the advance

NATURAL SCIENCE TO GENERAL SCIENCE. 21

of logical methods than the labours of philosophers properly so
called. One essential ground for such an assertion must un-
doubtedly be that in no department of knowledge can a fault
in the chain of reasoning be so easily detected by the incorrect-
ness of the results as in those sciences in which the results of
reasoning can be most directly compared with the facts of
nature.

Though I have maintained that it is in the physical sciences,
and especially in such branches of them as are treated mathe-
matically, that the solution of scientific problems has been most
successfully achieved, you will not, I trust, imagine that I wish
to depreciate other studies in comparison with them. If the
natural and physical sciences have the advantage of great per-
fection in form, it is the privilege of the moral sciences to deal
with a richer material, with questions that touch more nearly
the interests and the feelings of men, with the human mind
itself, in fact, in its motives and the different branches of its
activity. They have, indeed, the loftier and the more difficult
task, but yet they cannot afford to lose sight of the example of
their rivals, which, in form at least, have, owing to the more
ductile nature of their materials, made greater progress. Not
only have they something to learn from them in point of method,
but they may also draw encouragement from the greatness of
their results. And I do think that our age has learnt many
lessons from the physical sciences. The absolute, unconditional
reverence for facts, and the fidelity with which they are col-
lected, a certain distrustfulness of appearances, the effort to
detect in all cases relations of cause and effect, and the tendency
to assume their existence, which distinguish our century from
preceding ones, seem to me to point to such an influence.

I do not intend to go deeply into the question how fir
mathematical studies, as the representatives of conscious logical
reasoning, should take a more important place in school educa-
tion. But it is, in reality, one of the questions of the day. ID
proportion as the range of science extends, its system and or-
ganisation must be improved, and it must inevitably come about
that individual students will find themselves compelled to go

22 OX THE RELATION OF

through a stricter course of training than grammar is in a
position to supply. What strikes me in my own experience
of students who pass from our classical schools to scientific and
medical studies, is, first, a certain laxity in the application of
strictly universal laws. The grammatical rules in which they
have been exercised are for the mcst part followed by long
lists of exceptions; accordingly they are not in the habit of
relying implicitly on the certainty of a legitimate deduction
from a strictly universal law. Secondly, I find them for the
most part too much inclined to trust to authority, even in cases
where they might form an independent judgment. In fact, in
philological studies, inasmuch as it is seldom possible to take in
the whole of the premisses at a glance, and inasmuch as the de-
cision of disputed questions often depends on an aesthetic feeling
for beauty of expression, and for the genius of the language,
attainable only by long training, it must often happen that the
student is referred to authorities even by the best teachers.
Both faults are traceable to a certain indolence and vagueness
of thought, the sad effects of which are not confined to sub-
sequent scientific studies. But certainly the best remedy for
both is to be found in mathematics, where there is absolute
certainty in the reasoning, and no authority is recognised but
that of one's own intelligence.

So much for the several branches of science considered as
exercises for the intellect, and as supplementing each other in
that respect. But knowledge is not the sole object of man upon
earth. Though the sciences arouse and educate the subtlest
powers of the mind, yet a man who should study simply for the
sake of knowing, would assuredly not fulfil the purpose of his
existence. We cften see men of considerable endowments, to
whom their good or bad fortune has secured a comfortable
livelihood or good social position, without giving them, at the
same time, ambition or energy enough to make them work,
dragging out a weary, unsatisfied existence, while all the time
they fancy they are following the noblest aim of life by constantly
devoting themselves to the increase of their knowledge, and the
cultivation of their minds. Action alone gives a man a lifo

NATURAL SCIENCE TO GENERAL SCIENCE. 23

worth living ; and therefore he must aim either at the practical
application of his knowledge, or at the extension of the limits
of science itself. For to extend the limits of science is really to
work for the progress of humanity. Thus we pass to the second
link, uniting the different sciences, the connection, namely,
between the subjects of which they treat.

Know-ledge is power. Our age, more than any other, is in a
position to demonstrate the truth of this maxim. We have
taught the forces of inanimate nature to minister to the wants
vrf human life and the designs of the human intellect. The
application of steam has multiplied our physical strength a
million-fold ; weaving and spinning machines have relieved us
of labours the only merit of which consisted in a deadening
monotony. The intercourse between men, with its far-reaching
influence on material and intellectual progress, has increased to
an extent of which no one could have even dreamed within the
lifetime of the older among us. But it is not merely on the
machines by which our powers are multiplied ; not merely on
rifled cannon and armour-plated ships; not merely on accumu-
lated stores of money and the necessaries of life, that the power of
a nation rests : though these things have exercised so unmistak-
able an influence that even the proudest and most obstinate des-
j >otisms of our times have been forced to think of removing restric-
tions on industry, and of conceding to the industrious middle classes
a due voice in their councils. But political organisation, the
administration of justice, and the moral discipline of individual
citizens are no less important conditions of the preponderance
of civilised nations ; and so surely as a nation remains in-
accessible to the influences of civilisation in these respects, so
surely is it on the high road to destruction. The several con-
ditions of national prosperity act and react on each other;
where the administration of justice is uncertain, where the
interests of the majority cannot be asserted by legitimate means,
the development of the national resources, and of the power
depending upon them, is impossible ; nor, again, is it possible
to make good soldiers except out of men who have learnt under
just laws to educate the sense of honour that characterises an

24 OX THE RELATION OF

independent man, certainly not out of those who nave lived the
submissive slaves of a capricious tyrant.

Accordingly every nation is interested in the progress of know
ledge on the simple ground of self-preservation, even were there no
higher .wants of an ideal character to be satisfied; and not merely
in the development of the physical sciences, and their technical
application, but also in the progress of legal, political, and moral
sciences, and of the accessory historical and philological studies.
No nation which would be independent and influential can afford
to be left behind in the race. Nor has this escaped the notice of the
cultivated peoples of Europe. Never before was so large a part
of the public resources devoted to universities, schools, and
scientific institutions. We in Heidelberg have this year occasion
to congratulate ourselves on another rich endowment granted by
our government and our parliament.

I "svas speaking, at the beginning of my address, of the in-
creasing division of labour and the improved organisation among
scientific workers. In fact, men of science form, as it were, an
organised army labouring on behalf of the whole nation, and
generally under its direction and at its expense, to augment the
stock of such knowledge as may serve to promote industrial
enterprise, to increase wealth, to adorn life, to improve political and
social relations, and to further the moral development of indivi-
dual citizens. After the immediate practical results of their work
we forbear to inquire ; that we leave to the uninstructed. "We
are convinced that whatever contributes to the knowledge of
the forces of nature or the powers of the human mind is worth
cherishing, and may, in its own due time, bear practical fruit,
very often where we should least have expected it. Who, when
Galvani touched the muscles of a frog with different metals,
and noticed their contraction, could have dreamt that eighty
years afterwards, in virtue of the self-same process, whose
earliest manifestations attracted his attention in his anatomical
researches, all Europe would be traversed with wires, flashing
intelligence from Madrid to St. Petersburg with the speed of
lightning? In the hands of Galvani, and at first even in
Volta's, electrical currents were phenomena capable of exerting

NATURAL SCIENCE TO GENERAL SCIENCE. 25

only the feeblest forces, and could not be detected except by the
most delicate apparatus. Had they been neglected, on the
ground that the investigation of them promised DO immediate
practical result, we should now be ignorant of the most import-
ant and most interesting of the links between the various forces
of nature. When young Galileo, then a student at Pisa, noticed
one day during divine service a chandelier swinging backwards
and forwards, and convinced himself, by counting his pulse, that
the duration of the oscillations was independent of the arc
through which it moved, who could know that this discovery
would eventually put it in our power, by means of the pendulum,
to attain an accuracy in the measurement of time till then
deemed impossible, and would enable the storm-tossed seaman
in the most distant oceans to determine in what degree of longi-
tude he was sailing ?

Whoever, in the pursuit of science, seeks after immediate
practical utility, may generally rest assured that he will seek in
vain. All that science can achieve is a perfect knowledge and a
perfect understanding of the action of natural and moral forces.
Each individual student must be content to find his reward in
rejoicing over new discoveries, as over new victories of mind
over reluctant matter, or in enjoying the aesthetic beauty of a
well-ordered field of knowledge, where the connection and the
filiation of every detail is clear to the mind, and where all
denotes the presence of a ruling intellect ; he must rest satisfied
with the consciousness that he too has contributed something to
the increasing fund of knowledge on which the dominion of man
over all the forces hostile to intelligence reposes. He will,
indeed, not always be permitted to expect from his fellow-men
appreciation and reward adequate to the value of his work. It
is only too true that many a man to whom a monument has
been erected after his death would have been delighted to receive
during his lifetime a tenth part of the money spent in doing
honour to his memory. At the same time, we must acknowledge
that the value of scientific discoveries is now far more fully recog-
nised than formerly by public opinion, and that in stances of the
authors of great advance in science starving in obscurity have

20 ON THE RELATION OF

become rarer and rarer. On the contrary, the governments and
peoples of Europe have, as a rule, admitted it to be their duty
to recompense distinguished achievements in science by appro'
priate appointments or special rewards.

The sciences have then, in this respect, all one common aim,
to establish the supremacy of intelligence over the world:
while the moral sciences aim directly at making the resources of
intellectual life more abundant and more interesting, and seek
to separate the pure gold of truth from alloy, the physical
sciences are striving indirectly towards the same goal, inasmuch
as they labour to make mankind more and more independent of
the material restraints that fetter their activity. Each student
works in his own department, he chooses for himself those tasks
for which he is best fitted by his abilities and his training.
But each one must be convinced that it is only in connection
with others that he can further the great work, and that therefore
he is bound, not only to investigate, but to do his utmost to
make the resu'ts of his investigation completely and easily
accessible. If ho does this, he will derive assistance from others,
and will in his turn be able to render them his aid. The annals
of science abound in evidence how such mutual services have
been exchanged, even between departments of science apparently
most remote. Historical chronology is essentially based on
astronomical calculations of eclipses, accounts of which are pre-
Bsrved in ancient histories. Conversely, many of the important
data of astronomy — for instance, the invariabi'ity of the length
of the day, and the periods of several comets — rest upon ancient
historical notices. Of late years, physiologists, especially Briicke,
have actually undertaken to draw up a complete system of all
the vocables that can be produced by the organs of speech, and to
base upon it propositions for an universal alphabet, adapted to
all human languages. Thus physiology has entered the service
of comparative philology, and has already succeeded in account-
ing for many apparently anomalous substitutions, on the ground
that they are governed, not as hitherto supposed, by the laws of
euphony, but by similarity between the movements of the mouth
that produce them. Again, comparative philology gives us

NATURAL SCIENCE TO GENERAL SCIENCE. 27

information about the relationships, the separations, and the
migrations of tribes in pi-ehistoric times, and of the d-::gree of
civilisation which they had reached at the time when they
parted. For the names of objects to which they had already
learnt to give distinctive appellations reappear as words common
to their later languages. So thatthe study of languages actually
gives us historical data for periods respecting which no other
historical evidence exists.1 Yet again I may notice the help
which not only the sculptor, but the archaeologist, concerned
with the investigation of ancient statues, derives from anatomy.
And if I may be permitted to refer to my own most recent studies,
I would mention that it is possible, by reference to physical
r.coustics and to the physiological theory of the sensation of
hearing, to account for the elementary principles on which our
musical system is constructed, a problem essentially within the
sphere of aesthetics. In fact, it is a general principle that the
physiology of the organs of sense is most intimately connected
with psychology, inasmuch as physiology traces in our sensations
the results of mental processes which do not fall within the
sphere of consciousness, and must therefore have remained inac-
cessible to us.

I have been able to quote only some of the most striking
instances of this interdependence of different sciences, and such
as could be explained in a few words. Naturally, too, I have
tried to choose them from the most widely separated sciences.
But far wider is of course the influence which allied sciences
exert upon each other. Of that I need not speak, for each of
you knows it from his own experience.

In conclusion, I would say, let each of us think of himself,
not as a man seeking to gratify his own thirst for knowledge,
or to promote his own private advantage, or to shine by his
own abilities, but rather as a fellow-labourer in one great com-
mon work bearing upon the highest interests of humanity.
Then assuredly we shall not fail of our reward in the approval
of our own conscience and the esteem of our fallow-citizens.

1 See, for example, Mommseu's Rome, Book I. ch. ii. — TR.

28 ON THE RELATION OF NATURAL SCIENCE.

To keep up these relations between all seai-chers after truth and
all branches of knowledge, to animate them all to vigorous co-
operation towards their common end, is the great office of the
Universities. Therefore is it necessary that the four faculties
should 'ever go hand in hand, and in this conviction will we
strive, so far as in us lies, to press onward to the fulfilment of
our in-eat mission,

ON

GOETHE'S SCIENTIFIC EESEAECHES.

A Lecture delivered before the German Society of Kionigsberg, in the
Sprinff of 1853.

IT could not but be that Goethe, whose comprehensive genius
was most strikingly apparent in that sober clearness with which
he grasped and reproduced with lifelike freshness the realities
of nature and human life in their minutest details, should, by
those very qualities of his mind, be drawn towards the study of
physical science. And in that department, he was not content
with acquiring what others could teach him, but he soon at-
tempted, as so original a mind was sure to do, to strike out an in-
dependent and a very characteristic line of thought. He directed
his energies not only to the descriptive but also to the experi-
mental sciences; the chief results being his botanical and
osteological treatises on the one hand, and his theory of colour
on the other. The first germs of these researches belong for
the most part to the last decade of the eighteenth century,
though some of them were not completed nor published till
later. Since that time science has not only made great progress
but has widely extended its range. It has assumed in some
respects an entirely new aspect, it has opened out new fields of
research and undergone many changes in its theoretical views.
I. shall attempt in the following Lecture to sketch the rela-
tion of Goethe's researches to the present standpoint of science,
and to bring out the guiding idea that is common to them all.

30 oy GOETHE'S sciE?rnFic RESEARCHES.

The peculiar character of the descriptive sciences — botany,
zoology, anatomy, and the like — is a necessary result of tbe
work imposed upon them. They undertake to collect and sift
an enormous mass of facts, and, above all, to bring them into a
logical order or system. Up to this point their work is only
the dry task of a lexicographer ; their system is nothing more
than a muniment-room in which the accumulation of papers is
so arranged that any one can find what he wants at any moment.
The more intellectual part of their work and their real interest
only begins when they attempt to feel after the scattered traces
of law and order in the disjointed, heterogeneous mass, and out
of it to construct for themselves an orderly system, accessible at
a glance, in which every detail has its due place, and gains
additional interest from its connection with the whole.

In such studies, both the organising capacity and theinsighu
of our poet found a congenial sphere — the epoch was moreover
propitious to him. He found ready to his hand a sufficieiit
store of logically arranged materials in botany and comparative
anatomy, copious and systematic enough to admit cf a compre-
hensive view, and to indicate the way to some happy glimpse
of an all-pervading law ; while his contemporaries, if they mado
any efforts in this direction, wandered without a compass, or
else they were so absorbed in the dry registration of facts, that
they scarcely ventured to think of anything beyond. It was
reserved for Goethe to introduce two ideas of infinite fruit-
fulness.

The first was the conception that the differences in the
anatomy of different animals are to be looked upon as variations
from a common phase or type, induced by differences of habit,
locality, cr food. The observation which led him to this fertile
conception was by no means a striking one ; it is to be found in
a monograph on the intermaxillary bone, written as early as
1786. It was known that in most vertebrate animals (that is,
mammalia, birds, amphibia, and fishes) the upper jaw consists
of two bones, the upper jaw-bone and the intermaxillary bone.
The former always contains in the mammalia the molar and
the canine teeth, the latter the incisors. Man, who is dis-

ON GOETHE'S SCIENTIFIC RESEARCHES. 31

tinguished from all other animals by the absence of the
projecting snout, has, on the contrary, on each side only one
bone, the upper jaw-bone, containing all the teeth. This being
so, Goethe discovered in the human skull faint traces of the
sutures which in animals unite the upper and middle jaw-bones,
and concluded from it that man had originally possessed an
intermaxillary bone, which had subsequently coalesced with the
upper jaw-bone. This obscure fact opened up to him a source
of the most intense interest in the field of osteology, generally
so much decried as the driest of studies. That details of
structure should be the same in man and in animals when the
parts continue to perform similar functions had involved
nothing extraordinary. In fact, Camper had already attempted,
on this principle, to trace similarities of structure even between
man and fishes. But the persistence of this similarity, at least
in a rudimentary form, even in a case when it evidently does
not correspond to any of the icquirements of the complete
human structure, and consequently needs to be adapted to
them by the coalescence of two parts originally separate, was
what struck Goethe's far-seeing eye, and suggested to him a
far more comprehensive view than had hitherto been taken.
Further studies soon convinced him of the universality of his
newly discovered principle, so that in 1795 and 1796 he was
able to define more clearly the idea that had struck him in 1786,
and to commit it to wi iting in his ' Sketch of a General Intrp-
duction to Comparative Anatomy.' He there lays down with
the utmost confidence and precision that all differences in the
structure of animals must be looked upon as variations of a
single primitive type, induced by the coalescence, the alteration,
the increase, the diminution, or even the complete removal of
single parts of the stru.ture; the very principle, in fact, which
has become the leading idea of comparative anatomy in its
present stage. Nowhere ha? it been better or more clearly ex-
pressed than in Goathe's writings. Subsequent authorities have
made but few essential alterations in his theory. The most
important of these is, that we no longer undertake to construct
a common type for the whole animal kingdom, but are content

32 ON GOETHE'S SCIENTIFIC RESEARCHES.

with one for each of Cuvier's great divisions. The industry of
Goethe's successors has accumulated a well-sifted stock of facts,
infinitely more copious than what he could command, and has
followed up successfully into the minutest details what he could
only indicate in a general way.

The second leading conception which science owes to Goethe
enunciated the existence of an analogy between the different
parts of one and the same organic being, similar to that which
we have just pointed out as subsisting between corresponding
parts of different species. In most organisms we see a great
repetition of single parts. This is most striking in the veget-
able kingdom ; each plant has a great number of similar stem
leaves, similar petals, similar stamens, and so on. According
to Goethe's own account, the idea first occurred to him while look-
ing at a fan-palm at Padua. He was struck by the immense
variety of changes of form which the successively developed
stem-leaves exhibit, by the way in which the first simple root
leaflets are replaced by a series of more and more divided leaves,
till we come to the most complicated.

He afterwards succeeded in discovering the transformation
of stem-leaves into sepals and petals, and of sepals and petals
into stamens, nectaries, and ovaries, and thus he was led to the
doctrine of the metamorphosis of plants, which he published in
1790. Just as the anterior extremity of vertebrate animals
takes different forms, becoming in man and in apes an arm, in
other animals a paw with claws, or a forefoot with a hoof, or a
fin, or a wing, but always retains the same divisions, the same
position, and the same connection with the trunk, so the leaf
appears as a cotyledon, stem-leaf, sepal, petal, stamen, nectary,
ovary, <fec., all resembling each other to a certain extent in origin
and composition, and even capable, under certain unusual con-
ditions, of passing from one form into the other, as, for example,
may be seen by any one who looks carefully at a full-blown rose,
where some of the stamens are completely, some of them pai-tially,
changed into petals. This view of Goethe's, like the other, is
now completely adopted into science, and. enjoys the universal
assent of botanists, though of course some details are stiD

ON GOETHE'S SCIENTIFIC RESEARCHES. 33

matters of controversy, as, for instance, whether the bud is a
single leaf or a branch.

In the animal kingdom, the composition of an individual
out of several similar parts is very striking in the great sub-
kingdom of the articulata — for example, in insects and worms. The
larva of an insect, or the caterpillar of a butterfly, consists of a
number of perfectly similar segments ; only the first and last of
them differ, and that but slightly, from the others. After their
transformation into perfect insects, they furnish clear and simple
exemplifications of the view which Goethe had grasped in his
doctrine of the metamorphosis of plants, the development,
namely, of apparently very dissimilar forms from parts ' origin-
ally alike. The posterior segments retain their original simple
form ; those of the breastplate are drawn closely together, and
develop feet and wings, while those of the head develop jaws
and feelers ; so that in the perfect insect, the original segments
are recognised only in the posterior part of the body. In the
vertebrata, again, a repetition of similar parts is suggested by
the vertebral column, but has ceased to be observable in the ex-
ternal form. A fortunate glance at a broken sheep's skull,
which Goethe found by accident on the sand of the Lido at
Venice, suggested to him that the skull itself consisted of a series
of very much altered vertebrae. At first sight, no two things
can be more unlike than the broad uniform cranial cavity of the
mammalia, inclosed by smooth plates, and the narrow cylindrical
tube of the spinal marrow, composed of short, massy, jagged
bones. It was a bright idea to detect the transformation in
the skull of a mammal ; the similarity is more striking in the
amphibia and fishes. It should be added that Goethe left this
idea unpublished for a long time, apparently because he was not
quite sure how it would be received. Meantime, in 1806, the
same idea occurred to Oken, who introduced it to the scientific
world, and afterwards disputed with Goethe the priority of
discovery. In fact, Goethe had waited till 1817, when the
opinion had begun to find adherents, and then declared that he
had had it in his mind for thirty years. Up to the present day
the number and composition of the vertebrae of the skull are a

L 2>

34 ON GOETHE'S SCIENTIFIC RESEARCHES.

subject of controversy, but the principle has maintained ita
ground.

Goethe's views, however, on the existence of a common type
in the animal kingdom do not seem to have exercised any direct
influence on the progress of science. The doctrine of the meta-
morphosis of plants was introduced into botany as his distinct
and recognised property; but his views on osteology were at
first disputed by anatomists, and only subsequently attracted
attention when the science had, apparently on independent
grounds, found its way to the same discovery. He himself com-
plains that his first ideas of a common type had encountered
nothing but contradiction and scepticism at the time when
he was working them out in his own mind, and that even
men of the freshest and most original intellect, like the two
Von Humboldts, had listened to them with something like
impatience. But it is almost a matter of course that in any
natural or physical science, theoretical ideas attract the attention
of its cultivators only when they are advanced in connection
with the whole of the evidence on which they rest, and thus
justify their title to recognition. Be that as it may, Goethe is
entitled to the credit of having caught the first glimpse of the
guiding ideas to which the sciences of botany and anatomy were
tending, and by which their present form is determined.

But great as is the respect which Goethe has secured by his
achievements in the descriptive natural sciences, the denuncia-
tion heaped by all physicists on his researches in their depart-
ment, and especially on his ' theory of colour,' is at least as uncom-
pi'omising. This is not the place to plunge into the controversy
that raged on the subject, and so I shall only attempt to state
clearly the points at issue, and to explain what principle was
involved, and what is the latent significance of the dispute.

To this end it is of some importance to go back to the history
of the origin of the theory, and to its simplest form, because at
that stage of the controversy the points at issue are obvious, and
admit of easy and distinct statement, unincumbered by disputes
about the correctness of detached facts and complicated theories.

Goethe himself describes very gracefully, in the confession at

ON GOETHE'S SCIENTIFIC RESEARCHES. 35

the end of his ' Theory of Colour.' how he came to take up the
subject. Finding himself unable to grasp the aesthetic principles
involved in effects of colour, he resolved to resume the study of
the physical theory, which he had been taught at the university,
and to repeat for himself the experiments connected with it.
With that view he borrowed a prism of Hofrath Biitter, of Jena,
but was prevented by other occupations from carrying out his
plan, and kept it by him for a long time unused. The owner of
the prism, a very orderly man, after several times asking in vain,
sent a messenger with instructions to bring it back directly.
Goethe took it out of the case, and thought he would take one
more peep through it. To make certain of seeing something, he
turned it towards a long white wall, under the impression that
as there was plenty of light there he could not fail to see a
brilliant example of the resolution of light into different colours;
a supposition, by the way, which shows how little Newton's
theory of the phenomena was then present to his mind. Of
course he was disappointed. On the white wall he saw no
colours ; they only appeared where it was bounded by darker
objects. Accordingly he made the observation — which, it should
be added, is fully accounted for by Newton's theory — that
colour can only be seen through a prism where a dark object
and a bright one have the same boundary. Struck by this
observation, which was quite new to him, and convinced that it
was irreconcilable with Newton's theory, he induced the owner
of the prism to relent, and devoted himself to the question with
the utmost zeal and interest. He prepared sheets of paper with
black and white spaces, and studied the phenomenon under
every variety of condition, until he thought he had sufficiently
proved his rules. He next attempted to explain his supposed
discovery to a neighbour, who was a physicist, and was dis-
agreeably surprised to be assured by him that the experiments
were well known, and fully accounted for in Newton's theory.
Every other natural philosopher whom he consulted told him
exactly the same, including even the brilliant Lichtenberg,
whom he tried for a long time to convert, but in vain. I£e
"tudied Newton's writings, and fancied he had found some

D2

30 ON GOETHE'S SCIENTIFIC RESEARCHES.

fallacies in them which accounted for the error. Unable to con-
vince any of his acquaintances, he at last resolved to appear
before the har of public opinion, and in 1791 and 1792 published
the first and second parts of his 'Contributions to Physical
Optics.'

In that work he describes the appearances presented by white
discs on a black ground, black discs on a white ground, and
coloured discs on a black or white ground, when examined
through a prism. As to the results of the experiments, there is
no dispute whatever between him and the. physicists. He de-
scribes the phenomena he saw with great truth to nature; the
style is lively, and the arrangement such as to make a conspectus
of them easy and inviting ; in short, in thus as in all other cases
where facts are to be described, he proves himself a master. At
the same time he expresses his conviction that the facts he has
adduced are calculated to refute Newton's theory. There are
two points especially which he considers fatal to it : first, that
the centre of a broad white surface remains white when seen
through a prism; and secondly, that even a black streak on a
white ground can be entirely decomposed into colours.

Newton's theory is based on the hypothesis that there exists
light of different kinds, distinguished from one another by the
sensation of colour which they produce in the eye. Thus there
is red, orange, yellow, green, blue, and violet light, and light of
all intermediate colours. Different kinds of light, or differently
coloured lights, produce, when mixed, derived colours, which to
a certain extent resemble the original colours from which they
are derived ; to a certain extent form new tints. White is a
mixture of all the before-named coloiirs in certain definite pro-
portions. But the primitive colours can always be reproduced
by analysis from derived colours, or from white, while themselves
incapable of analysis or change. The cause of the colours of
transparent and opaque bodies is, that when white light falls
upon them they destroy some of its constituents and send to
the eye other constituents, but no longer mixed in the right
proportions to produce white light. Thus a piece of red glass
looks red because it transmits only red rays. Consequently aD

ON GOETHE'S SCIENTIFIC RESEARCHES. 37

colour is derived solely from a change in the proportions in
which light is mixed, and is, therefore, a property of light, not
of the coloured bodies, which only furnish an occasion for its
manifestation.

A prism refracts transmitted light; that is to say, deflects it
BO that it makes a certain angle with its original direction; the
rays of simple light of different colours have, according to
Newton, different refrangibilities, and therefore, after refraction
in the prism, pursue different courses and separate from each
other. Accordingly a luminous point of infinitely small dimen-
sions appears, when seen through the pi-ism, to be first displaced,
and, secondly, extended into a coloured line, the so-called pris^
matic spectrum, Avhich shows what are called the primary
colours in the order above-named. If, however, you look at a
broader luminous surface, the spectra of the points near the
middle are superposed, as may be seen from a simple geometrical
investigation, in such proportions as to give white light, except
at the edges, where certain of the colours are free. This white
surface appears displaced, as the luminous point did; but in-
stead of being coloured throughout, it has on one side a margin
of blue and violet, on the other a margin of red and yellow. A
black patch between two bright surfaces may be entirely covered
by their coloured edges; and when these spectra meet in the
middle, the red of the one and the violet of the other combine
to form purple. Thus the colours into which, at first sight, it
seems as if the black were analysed are in reality due, not to
the black strip, but to the white on each side of it.

It is evident that at the first moment Goethe did not recol-
lect Newton's theory well enough to be able to find out the
physical explanation of the facts I have just glanced at. It was
afterwards laid before him again and again, and that in a
thoroughly intelligible form, for he speaks about it several times
in terms that show he understood it quite correctly. But he is
still so dissatisfied with it that he persists in his assertion that
the facts just cited are of a nature to convince any one who
observes them of the absolute incorrectness of Newton's theory.
Neither here nor in his later controversial writings does he ever

425182

38 oy GOETEE'S SCIENTIFIC RESEARCHES.

clearly state in what he conceives the insufficiency of the ex-
planation to consist. He merely repeats again and again that
it is quite absurd. And yet I cannot see how any one, whatever
his views about colour, can deny that the theory is perfectly
consistent with itself; and that if the hypothesis from which it
starts be granted, it explains the observed facts completely and
even simply. Newton himself mentions these spurious spectra
in several passages of his optical works, without going into
any special elucidation of the point, considering, of course, that
the explanation follows at once from his hypothesis. And he
seems to have had good reason to think so ; for Goethe no sooner
began to call the attention of his scientific friends to the pheno-
mena than all with one accord, as he himself tells us, met his
difficulties with this explanation from Newton's principles, which,
though not actually in his writings, instantly suggested itself to
every one who knew them.

A reader who tries to realise attentively and thoroughly
every step in this part of the controversy is ant to experience at
this point an uncomfortable, almost a painful, feeling to see a man
of extraordinary abilities persistently declaring that there is an
obvious absurdity lurking in a few inferences appai'ently quite
clear and simple. He searches and searches, and at last unable,
with all his efforts, to find any such absurdity, or even the ap-
pearance of it, he gets into a state of mind in which his own
ideas are, so to speak, crystallisad. But it is just this obvious,
flat contradiction that makes Goethe's point of view in 1792 so
interesting and so important. At this point he has not as yet
developed any theory of his own; there is nothing under dis-
cussion but a few easily grasped facts, as to the correctness of
which both parties are agreed, and yet both hold distinctly
opposite views; neither of them even understands what his
opponent is driving at. On the one side are a number of phy-
sicists, who, by a long series of the ablest investigations, the
most elaborate calculations, and the most ingenious inventions,
have Drought optics to such perfection that it, and it alone,
nmong the physical sciences, was beginning almost to rival
astronomy in accuracy. Some of them have made the pheno-

ON GOETHE'S SCIENTIFIC RESEARCHES. 39

ir.ena the subject of direct investigation ; all of them, thanks
to the accuracy with which it is possible to calculate beforehand
the result of every variety in the construction and combination
of instruments, have had the opportunity of putting the infer-
ences deduced from Newton's views to the test of experiment,
and all, without exception, agree in accepting them. On the other
aide is a man whose remarkable mental endowments, and
AT hose singular talent for seeing through whatever obscures
reality, we have had occasion to recognise, not only in poetry, but
also in the descriptive parts of the natural sciences ; and this
man assures us with the utmost zeal that the physicists are
wrong : he is so convinced of the correctness of his own view,
that he cannot explain the contradiction except by assuming
narrowness or malice on their part, and finally declares that he
cannot help looking upon his own achievement in the theory of
colour as far more valuable than anything he has accomplished
in poetry.1

So flat a contradiction leads us to suspect that there must
be behind some deeper antagonism of principle, some difference
of organisation between his mind and theirs, to prevent them
from understanding each other. I will try to indicate in the
following pages what I conceive to be the grounds of this anta-
gonism.

Goethe, though he exercised his powers in many spheres
of intellectual activity, is nevertheless, par excellence, a poet.
Now in poetry, as in every other art, the essential thing is to
make the material of the art, be it words, or music, or colour,
the direct vehicle of an idea. In a perfect work of art, the idea
must be present and dominate the whole, almost unknown to
the poet himself, not as the result of a long intellectual process,
but as inspired by a direct intuition of the inner eye, or by an
outburst of excited feeling.

An idea thus embodied in a work of art, and dre?sei in the
garb of reality, does indeed make a vivid impression by appeal-
ing directly to the senses, but loses, of course, that universality
and that intelligibility which it would have had if presented in
1 See Eckermann's Conversation*

40 ON GOETHE'S SCIENTIFIC RESEARCHES.

the form of an abstract notion. The poet, feeling how the
charm of his works is involved in an intellectual process of this
type, seeks to apply it to other materials. Instead of trying to
arrange the phenomena of nature under definite conceptions,
independent of intuition, he sits down to contemplate them as
he would a work of art, complete in itself, and certain to yield
up its central idea, sooner or later, to a sufficiently susceptible
student. Accordingly, when he sees the skull on the Lido,
which suggests to him the vertebral theory of the cranium, he
remarks that it serves to revive his old belief, already confirmed
by experience, that Nature has no secrets from the attentive
observer. So again in his first conversation with Schiller on
the 'Metamorphosis of Plants.' To Schiller, as a follower of
Kant, the idea is the goal, ever to be sought, but ever unattain-
able, and therefore never to be exhibited as realised in a phe-
nomenon. Goethe, on the other hand, as a genuine poet,
conceives that he finds in the phenomenon the direct expression
of the idea. He himself tells us that nothing brought out
more sharply the separation between himself and Schiller.
This, too, is the secret of his affinity with the natural philosophy
of Schelling and Hegel, which likewise proceeds from the
assumption that Nature shows us by direct intuition the several
steps by which a conception is developed. Hence too the ardour
with which Hegel and his school defended Goethe's scientific
views. Moreover, this view of Nature accounts for the war
which Goethe continued to wage against complicated experi-
mental researches. Just as a genuine work of art cannot bear
retouching by a strange hand, so he would have us believe
Nature resists the interference of the experimenter who torturea
her and disturbs her ; and, in revenge, misleads the impertinent
kill-joy by a distorted image of herself.

Accordingly, in his attack upon Newton he often sneers at
spectra, tortured through a number of narrow slits and glasses,
and commends the experiments that can be made in the open air
under a bright sun, not merely as particularly easy and parti-
cularly enchanting, but also as particularly convincing ! The
poetic turn of mind is very marked even in his morphological

ON GOETHE'S SCIENTIFIC RESEARCHES. 41

researches. If we only examine what has really been accom-
plished by the help of the ideas which he contributed to science,
we shall be struck by the very singular relation which they bear
to it. !No one will refuse to be convinced if you lay before him
the series of transformations by which a leaf passes into a
etainen, an arm into a fin or a wing, a vertebra into the occipital
bone. The idea that all the parts of a flower are modified leaves
reveals a connecting law which surprises us into acquiescence.
But now try and define the leaf-like organ, determine its essential
characteristics, so as to include all the forms that we have named.
You will find yourself in a difficult}-, for all distinctive marks
vanieh, and you have nothing left, except that a leaf in the
wider sense of the term is a lateral appendage of the axis of
a plant. Try then to express the proposition 'the parts of the
flower are modified leaves ' in the language of scientific defi-
nition, and it reads, ' the parts of the flower are lateral appen-
dnges of the axis.' To see this does not require a Goethe. So
igain it has been objected, and not unjustly, to the vertebral
theory, that it must extend the notion of a vertebra so much
that nothing is left but the bare fact — a vertebra is a bone. We
are equally perplexed if we try to express in clear scientific
language what we mean by saying that such and such a part of
one animal corresponds to such and such a part of another. We
ilo not mean that their physiological use is the same, for the
name piece which in bird serves as the lower jaw, becomes
in mammals a tiny tympana! bone. Nor would the shape, the
position, or the connection of the part in question with other
parts serve to identify it in all cases. But yet it has been found
possible in most cases, by following the intermediate steps, to
determine with tolerable certainty which parts correspond to
each other. Goethe himself said this very clearly : he says, in
speaking of the vertebral thory of the sk\ill, ' Such an aper^i,
such an intuition, conception, representation, notion, idea, or
whatever you choose to call it, always retains something
esoteric and indefinable, struggle as you will against it ; as a
general principle, it may be enunciated, but cannot be proved ;
in detail it may be exhibited, but can never be put in a cut and

42 ON GOETHE'S SCIENTIFIC RESEARCHES.

dry form.' And so, or nearly so, the problem stands to this
day. The difference may be brought out still more clearly if wo
consider how physiology, which investigates the relations of vital
processes as cause and effect, would have to treat this idea of a
common type of animal structure. The science might ask, Is
it, on the one hand, a correct view, that during the geological
periods that have passed over the earth, one species has been
developed from another, so that, for example, the breast-fin of
the fish has gradually changed into an arm or a wing ? Or
again, shall we say that the different species of animals were
created equally perfect — that the points of resemblance between
them are to be ascribed to the fact that in all vertebrate animals
the first steps in development from the egg can only be effected
by Nature in one way, almost identical in all cases, and that
the later analogies of structure are determined by these features,
common to all embryos 1 Probably the majority of observers
incline to the latter view,1 for the agreement between the
embryos of different vertebrate animals, in the earlier stages, is
very striking. Thus even young mammals have occasionally
rudimentary gills on the side of the neck, like fishes. It seems,
in fact, that what are in the mature animals corresponding parts
originate in the same way during the process of development, so
that scientific men have lately begun to make use of embryology
as a sort of check on the theoretical views of comparative ana-
tomy. It is evident that by the application of the physiological
views just suggested, the idea of a common type would acquire
definiteness and meaning as a distinct scientific conception.
Goethe did much : he saw by a happy intuition that there was a
law, and he followed up the indications of it writh great shrewdness.
But what law it was he did not see ; nor did he even try to
find it out. That was not in his line. Moreover, even in the
present condition of science, a definite view on the question is
impossible ; the very form in which it should be proposed is
scarcely yet settled. And therefore we readily admit that in this
department Goethe did all that was possible at the time when he
lived. I said just now that he treated nature like a work of
1 This was writteu before the appearance of Darwin's Origin of Species.

ON GOETHE'S SCIENTIFIC RESEARCHES. 43

art. In his studies on morphology, he reminds one of a spectator
at a play, with strong artistic sympathies. His delicate instinct
makes him feel how all the details fall into their places, and
work harmoniously together, and how some common purpose
governs the whole ; and yet while this exquisite order and sym-
metry give him intense pleasure he cannot formulate the dominant
idea. That is reserved for the scientific critic of the drama,
while the artistic spectator feels perhaps, as Goethe did in the
presence of natural phenomena, an antipathy to such dissection,
fearing, though without reason, that his pleasure may be spoilt
by it.

Goethe's point of view in the Theory of Colour is much the
same. We havo seen that he rebels against the physical theory
just at the point where it gives complete and consistent expla-
nations from principles once accepted. Evidently it is not the
insufficiency of the theory to explain individual cases that is a
stumbling-block to him. He takes offence at the assumption
made for the sake of explaining the phenomena, which seem to
him so absurd, that he looks upon the interpretation as no inter-
pretation at all Above all, the idea that white light could be
composed of coloured light seems to have been quite inconceiv-
able to him ; at the very beginning of the controversy, he rails
at the disgusting Newtonian white of the natural philosophers,
an expression which seems to show that this was the assumption
that most annoyed him.

Again, in his later attacks on Newton, which were not
published till after his Theory of Colour was completed, he
rather strives to show that Newton's facts might be explained
on his own hypothesis, and that therefore Newton's hypothesis
was not fully proved, than attempts to prove that hypothesis
inconsistent with itself or with the facts. Nay, he seems to
consider the obviousness of his own hypothesis so overwhelming,
that it need only be brought forward to upset Newton's entirely.
There are only a few passages where he disputes the experiments
described by Newton. Some of them, apparently, he could not
succeed in refuting, because the result is not equally easy to
observe in all positions of the lenses used, and because he waa

44 ON GOETHE'S SCIENTIFIC RESEARCHES.

unacquainted with the geometrical relations by which the most
favourable positions of them are determined. In other experi-
ments on the separation of simple coloured light by means of
prisms alone, Goethe's objections are not quite groundless, inas-
much as the isolation of single colours cannot by this means be
so effectually carried out, that after refraction through another
prism there are no traces of other tints at the edges. A com-
plete isolation of light of one colour can only be effected by
very carefully arranged apparatus, consisting of combined
prisms and lenses, a set of experiments which Goethe postponed
to a supplement, and finally left unnoticed. When he complains
of the complication of these contrivances, we need only think
of the laborious and roundabout methods which chemists must
often adopt to obtain certain elementary bodies in a pui-e form ;
and we need not be surprised to find that it is impossible to
solve a similar problem in the case of light in the open air in a
garden, and with a single prism in one's hand.1 Goethe must,
consistently with his theory, deny in toto the possibility of
isolating pure light of one colour. Whether he ever experi-
mented with the proper apparatus to solve the problem remains
doubtful, as the supplement in which he promised to detail
these experiments was never published.

To give some idea of the passionate way in which Goethe,
usually so temperate and even courtier-like, attacks Newton, I
quote from a few pages of the controversial part of his work
the following expressions, which he applies to the propositions
of this consummate thinker in physical and astronomical
science — ' incredibly impudent' ; ' mere twaddle' ; ' ludicrous ex-
planation' ; ' admirable for school-children in a go-cart' j ' but I
see nothing will do but lying, and plenty of it.'2

1 I venture to add that 1 am acquainted -with the impossibility of decom-
posing or changing simple coloured light, the two principles which form the
basis of Newton's theory, not merely by hearsay, but from actual observation,
having been under the necessity in one of my own researches of obtaining light
of one colour in a state of the greatest possible purity. (See Poggendorff's
Annalen, vol. Ixxxvi. p. 501, on Sir D. Brewster's New Analysis of Sunlight.)

2 Something parallel to this extraordinary proceeding of Goethe's may be
found in Hobbes's attack on Wallis. — TB.

ON GOETHE'S SCIENTIFIC RESEARCHES. 45

Thus, in the theory of colour, Goethe remains faithful to
his principle, that Nature must reveal her secrets of her own
free will ; that she is but the transparent representation of the
ideal world. Accordingly, he demands, as a preliminary to the
investigation of physical phenomena, that the observed facts
shall be so arranged that one explains the other, and that thus
we may attain an insight into their connection without ever
having to trust to anything but our senses. This demand of
his looks most attractive, but is essentially wrong in principle.
For a natural phenomenon is not considered in physical science
to be fully explained until you have traced it back to the
ultimate forces which are concerned in its production and its
maintenance. Now, as we can never become cognisant of forces
qua forces, but only of their effects, we are compelled in every
explanation of natural phenomena to leave the sphere of sense,
and to pass to things which are not objects of sense, and are
denned only by abstract conceptions. When we find a stove
warm, and then observe that a fire is burning in it, we say,
though somewhat inaccurately, that the former sensation is
explained by the latter. But in reality this is equivalent to say-
ing, we are always accustomed to find heat where fire is bum-
ing ; now, a fire is burning in the stove, therefore we shall find
heat there. Accordingly we bring our single fact under a more
general, better-known fact, rest satisfied with it, and call it
falsely an explanation. Evidently, however, the generality of the
observation does not necessarily imply an insight into causes; such
an insight is only obtained when we can make out what forces
are at work in the fire, and how the effects depend upon them.

But this step into the region of abstract conceptions, which
must necessarily be taken if we wish to penetrate to the causes
of phenomena, scares the poet away. In writing a poem he
has been accustomed to look, as it were, right into the subject,
and to reproduce his intuition without formulating any of the
steps that led him to it. And his success is proportionate to
the vividness of the intuition. Such is the fashion in which he
would have Nature attacked. But the natural philosopher in-
sists on transporting him into a world of invisible atoms and

46 ON GOETHE'S SCIENTIFIC RESEARCHES.

movements, of attractive and repulsive forces, whose intricate
actions and reactions, though governed by strict laws, can
scarcely be taken in ai a glance. To him the impiessions of
sense are not an irrefragable authority ; he examines what claim
they have to be trusted ; he asks whether things which they
pronounce alike are really alike, and whether things which they
pronounce different are really different ; and often finds that he
must answer, no ! The result of such examination, as at present
understood, is that the organs of sense do indeed give us informa-
tion about external effects produced on them, but convey thorte
effects to our consciousness in a totally different form, so that
the character of a sensuous perception depends not so much on
the properties of the object perceived as on those of the organ
by which we I'eceive the information. All that the optic nerve
conveys to us, it conveys under the form of a sensation of light,
whether it be the rays of the sun, or a blow in the eye, or an
electric current passing through it. Again, the auditory nerve
translates everything into phenomena of sound, the nerves of the
skin into sensations of temperature or touch. The same electric
current whose existence is indicated by the optic nerve as a flash
of light, or by the organ of taste as an acid flavour, excites in
the nerves of the skin the sensation of burning. The same ray
of sunshine, which is called light when it falls on the eye, \re
call heat when it falls on the skin. But on the other hand, in
spite of their different effects upon our organisation, the daylight
which enters through our windows, and the heat radiated by an
iron stove, do not in reality differ more or less from each other
than the red and blue constituents of light. In fact, just as in
the Undulatory Theory the red rays are distinguished from the
blue rays only by their longer period of vibration, and their
smaller refrangibility, so the dark heat rays of the stove have a
still longer period and still smaller refrangibility than the red
lays of light, but are in every other respect exactly similar to
them. All these rays, whether luminous or non-luminous, have
heating properties, but only a certain number of them, to which
for that reason we give the name of light, can penetrate through
the transparent part of the eye to the optic nerve, and excite a

ox GOETHE'S SCIENTIFIC RESEARCHES. 47

sensation of light. Perhaps the relation between our senses and
the external world may be best enunciated as follows : our sen-
sations are for us only symbols of the objects of the external
world, and correspond to them only in some such way as written
characters or articulate words to the things they denote. They
give us, it is true, information respecting the properties of things
without us, but no better information than we give a blind man
about colour by verbal descriptions.

We see that science has arrived at an estimate of the senses
very different from that which was present to the poet's mind.
A.nd Newton's assertion that white was composed of all the
colours of the spectrum was the first gerni of the scientific view
which has subsequently been developed. For at that time there
were none of those galvanic observations which paved the way
to a knowledge of the functions of the nerves in the production
of sensations. Natural philosophers asserted that white, to the
^ye the simplest and purest of all our sensations of colour, was
compounded of less pure and complex materials. It seems to
have flashed upon the poet's mind that all his principles were
unsettled by the results of this assertion, and that is why the
hypothesis seems to him so unthinkable, so ineffably absurd.
We must look upon his theoiy of colour as a forlorn hope, as
a desperate attempt to rescue from the attacks of science the
belief in the direct truth of our sensations. And this will ac-
count for the enthusiasm with which he strives to elaborate and to
defend his theory, for the passionate irritability with which he
attacks his opponent, for the overweening importance which he
attaches to these researches in comparison with his other achieve-
ments, and for his inaccessibility to conviction or compromise.

If we now turn to Goethe's own theories on the subject,
we must, en the grounds above stated, expect to find that he
cannot, without being untrue to his own principle, give us
anything deserving to be called a scientific explanation of the
phenomena, and that is exactly what happens. He starts with
the proposition that all colours are darker than white, that they
have something of shade in them (on the physical theoiy, white
compounded of all colours must necessarily be brighter than

48 ON GOETHE'S SCIENTIFIC RESEARCHES.

any of its constituents). The direct mixture of dark and light,
of black and white, gives grey ; the colours must therefore owe
their existence to some form of the co-operation of light and
shade. Goethe imagines he has discovered it in the phenomena
presented by slightly opaque or hazy media. Such media usually jj
look blue when the light falls on them and they are seen in
front of a dark object, but yellow when a bright object is looked
at through them. Thus in the daytime the air looks blue
against the dark background of the sky, and the sun, when
viewed, as is the case at sunset, through a thick and hazy
stratum of air, appears yellow. The physical explanation of
this phenomenon, which, however, is not exhibited by all such
media, as, for instance, by plates of unpolished glass, would lead
us too far from the subject. According to Goethe, the semi-opaque
medium imparts to the light something corporeal, something of
the nature of shade, such as is requisite, he would say, for the
formation of colour. This conception alone is enough to perplex
any one who looks upon it as a physical explanation. Does he
mean to say that material particles mingle with the light and
fly away with it ? But this is Goethe's fundamental experiment,
this is the typical phenomenon under which he tries to reduce
all the phenomena of colour, especially those connected with
the prismatic spectritrn. He looks upon all transparent bodies
as slightly hazy, and assumes that the prism imparts to the
image which it shows to an observer something of its own
opacity. Here, again, it is hard to get a definite conception of
what is meant. Goethe seems to have thought that a prism
never gives perfectly defined images, but only indistinct, half-
obliterated ones, for he puts them all in the same class with the
double images which are exhibited by parallel plates of glass
and by Iceland spar. The images formed by a prism are, it
is true, indistinct in compound light, but they are perfectly
defined when simple light is used. If you examine, he says, a
bright surface on a dark ground through a prism, the image is
displaced and blurred by the prism. The anterior edge is
pushed forward over the dark background, and consequently 9
hazy light on a dark ground appears blue, while the other edge

ON GOETHE'S SCIENTIFIC RESEARCHES. 49

is covered by the image of the black surface which comes after it,
and, consequently, being a light image behind a hazy dark colour,
appears yellowish-red. But why the anterior edge appears in
front of the ground, the posterior edge behind it, and not vice
versd, he does not explain. Let us analyse this explanation,
and try to grasp clearly the conception of an optical image.
When I see a bright object reflected in a mirror, the reason is
that the light which proceeds from it is thrown back exactly as
if it came from an object of the same kind behind the mirror.
The eye of the observer receives the impression accordingly,
and therefore he imagines he really sees the object. Every one
knows there is nothing real behind the mirror to correspond to
the image — that no light can penetrate thither, but that what
is called the image is simply a geometrical point, in which the
reflected rays, if produced backwards, would intersect. And,
accordingly, no one expects the image to produce any real effect
behind the mirror. In the same way the prism shows us images
of objects which occupy a different position from the objects
themselves ; that is to say, the light which an object sends to
the prism is refracted by it, so that it appears to come from an
object lying to one side, called the image. This image, again;
is not real ; it is, as in the case of reflection, the geometrical
point in which the refracted rays intersect when produced back-
wards. And yet, according to Goethe, this image is to produce
real effects by its displacement; the displaced patch of light
makes, he says, the dark space behind it appear blue, just as an
imperfectly transparent body would, and so again the displaced
dark patch makes the bright space behind appear reddish-yellow.
That Goethe really treats the image as an actual object in the
place it appears to occupy is obvious enough, especially as he is
compelled to assume, in the course of his explanation, that the
blue and red edges of the bright space are respectively before
and behind the dark image which, like it, is displaced by the
prism. He does, in fact, remain loyal to the appearance pre-
sented to the senses, and treats a geometrical locus as if it were
a material object. Again, he does not scruple at one time to
make red and blue destroy each other, as, for example, in the

I. E

50 ON GOETHE'S SCIENTIFIC RESEARCHES.

blue edge of a red surface seen through the prism, and at
another to construct out of them a beautiful purple, as when
the blue and red edges of two neighbouring white surfaces
meet in a black ground. And when he comes to Newton's
more complicated experiments, he is driven to still more mar-
vellous expedients. As long as you treat his explanations as a
pictorial way of representing the physical processes, you may
acquiesce in them, and even frequently find them vivid and
characteristic, but as physical elucidations of the phenomena
they are absolutely irrational.

In conclusion, it must be obvious to evpry one that the
theoretical part of the Theory of Colour is not natural philo-
sophy at all ; at the same time we can, to a certain extent, see
that the poet wanted to introduce a totally different method
into the study of Nature, and more or less understand how he
rame to do so. Poetry is concerned solely with the ' beautiful
show which makes it possible to contemplate the ideal ; how
that show is produced is a matter of indifference. Even nature
is, in the poet's eyes, but the sensible expression of the spiritual.
The natural philosopher, on the other haud, tries to discover the
levers, the cords, and the pulleys which work behind the scenes,
and shift them. Of course the sight of the machinery spoils
the beautiful show, and therefore the poet would gladly talk it
out of existence, and ignoring cords and pulleys as the chimeras
of a pedant's brain, he would have us believe that the scenes
shift themselves, or are governed by the idea of the drama.
And it is just characteristic of Goethe that he, and he alone
among poets, must needs break a lance with natural philosophers.
Other poets are either so entirely carried away by the fire of
their enthusiasm that they do not trouble themselves about the
disturbing influences of the outer world, or else they rejoice
in the triumphs of mind over matter, even on that unpropitious
battlefield. But Goethe, whom no intensity of subjective feeling
could blind to the realities around him, cannot rest satisfied
until he has stamped reality itself with the image and super-
scription of poetry. This constitutes the peculiar beauty of his
poetry, and at the same time fully accounts for his resolute

ON GOETHE'S SCIENTIFIC RESEARCHES. 51

hostility to the machinery that every moment threatens to
disturb his poetic repose, and for his determination to attack
the enemy in his own camp.

But we cannot triumph over the machinery of matter
by ignoring it; we can triumph over it only by subordinating
it to the aims of our moral intelligence. We must familiarise
ourselves with its levers and pulleys, fatal though it be to poetic
contemplation, in order to be able to govern them after our own
will, and therein lies the complete justification of physical
investigation, and its vast importance for the advance of human
civilisation.

From what I have said it will be apparent that Goethe did
follow the same line of thought in all his contributions to science,
but that the problems he encountered were of diametrically
opposite characters. And, perhaps, when it is understood how
the self-same characteristic of his intellect, which in one branch
of science won for him immortal renown, entailed upon him
egregious failure in the other, it will tend to dissipate, in the
minds of many worshippers of the great poet, a lingering pre-
judice against natural philosophers, whom they suspect of being
blinded by narrow professional pride to the loftiest inspirations
of genius.

63

ON THE

PHYSIOLOGICAL CAUSES OF HAEMONT
IN MUSIC,

A Lecture delivered in Bonn during the Winter nf 1857.

LADIES AND GENTLEMEN, — In the native town of Beethoven, the
mightiest among the heroes of harmony, no subject seemed
to me better adapted for a popular audience than music itself.
Following, therefore, the direction of my researches during the
last few years, I will endeavour to explain to you what physics
and physiology have to say regarding the most cherished art of
the Rhenish land — music and musical relations. Music has
hitherto withdrawn itself from scientific treatment more than
any other art. Poetry, painting, and sculpture borrow at least
the material for their delineations from the world of experience.
They portray nature and man. Not only can their material be
critically investigated in respect to its correctness and truth
to nature, but scientific art-criticism, however much enthusiasts
may have disputed its right to do so, has actually succeeded in
making some progress in investigating the causes of that aesthetic
pleasure which it is the intention of these arts to excite. In
music, on the other hand, it seems at first sight as if those were
still in the right who reject all ' anatomisation of pleasurable
sensations.' This art, borrowing no part of its material from
the experience of our senses, not attempting to describe, and

54 ON THE PHYSIOLOGICAL CAUSES OF

only exceptionally to imitate the outer world, necessarily with-
draws from scientific consideration the chief points of attack
which other arts present, and hence seems to be as incompre-
hensible and wonderful as it is certainly powerful in its effects.
"We are, therefore, obliged, and we purpose, to confine ourselves,
in the first place, to a consideration of the material of the art,
musical sounds or sensations. It always struck me as a wonder-
ful and peculiarly interesting mystery, that in the theory of
musical sounds, in the physical and technical foundations of
music, which above all other arts seems in its action on the
mind as the most immaterial, evanescent, and tender creator of
incalculable and indescribable states of consciousness, that here
in especial the science of purest and strictest thought — mathe-
matics— .should prove pre-eminently fertile. Thorough bass is a
kind of applied mathematics. In considering musical intervals,
divisions of time, and so forth, numerical fractions, and some-
times even logarithms, play a prominent part. Mathematics
and music ! the most glaring possible opposite.? of human
thought! and yet connected, mutually sustained! It is as
if they would demonstrate the hidden consensus of all the
actions of our mind, which in the revelations of genius makes
us forefeet unconscious utterances of a mysteriously active
intelligence.

When I considered physical acoustics from a physiological
point of view, and thus more closely followed up the part which
the ear plays in the perception of musical sounds, much became
clear of which the connection had not been previously evident.
I will attempt to inspire you with some of the interest which
these questions have awakened in niy own mind, by endeavour-
ing to exhibit a few of the results of physical and physiological
acoustics.

The short space of time at my disposal obliges me to confine
my attention to one particular point; but I shall select the
most important of all, which will best show you the significance
and results of scientific investigation in this field ; I mean the
foundation of concord. It is an acknowledged fact that the
numbers of the vibrations of concordant tones bear to each

HARMONY IX MUSIC. 55

other ratios expressible by small whole numbers. But why ?
What have the ratios of small whole numbers to do with con
cord? This is an old lidd e, propounded by Pythagoras, and
hitherto unsolved. Let us see whether the means at the com-
mand of modern science will furnish the answer.

First of all, what is a musical tone? Common experience
teaches us that all sounding bodies are in a state of vibration.
This vibration can be seen and felt ; and in the case of loud
sounds we feel the trembling of the air even without touching
the sounding bodies. Physical science has ascertained that any
series of impulses which produce a vibration of the air will, if
repeated with sufficient rapidity, generate sound.

This sound becomes a musical tone, when such rapid im-
pulses recur with perfect regularity and in precisely equal times.
Irregular agitation of the air generates only noise. The pitch
of a musical tone depends on the number of impulses which
take place in a given time ; the more there are in the same time
the higher or sharper is the tone. And, as before remarked,
there is found to be a close relationship between the well-known
harmonious musical intervals and the number of the vibrations
of the air. If twice as many vibrations are performed in the
same time for one tone as for another, the first is the octave
above the second. If the numbers of vibrations in the same
time are as 2 to 3, the two tones form a fifth ; if they are as 4
to 5, the two tones form a major third.

If you observe that the numbers of the vibrations which
generate the tones of the major chord C E G c are in the ratio
of the numbers 4:5:6:8, you can deduce from these all
other relations of musical tones, by imagining a new major
chord, having the same relations of the numbers of vibrations,
to be formed upon each of the above-named tones. The num-
bers of vibrations within the limits of audible tones which
would be obtained by executing the calculation thus indicated
are extraordinarily different. Since the octave above any tone
has twice as many vibrations as the tone itself, the second octave
above will have four times, the third has eight times as many.
Our modern pianofortes have seven octaves. Their highest

fG ON THE PHYSIOLOGICAL CAUSES OF

tones, therefore, perform 128 vibrations in the time that theii
lowest tone makes one single vibration.

The deepest Cj which our pianos usually possess answers to
the sixteen-foot open pipe of the organ — musicians call it the
' contra-C ' — and makes thirty-three vibrations in one second of
time. This is very nearly the limit of audibility. You will
have observed that these tones have a dull, bad quality of sound
on the piano, and that it is difficult to determine their pitch and
the accuracy of their tuning. On the organ the contra-C is
somewhat more powerful than on the piano, but even here some
uncertainty is felt in judging of its pitch. On larger organs
there is a whole octave of tones below the contra-C, reaching to
the next lower C, with 16^ vibrations in a second. But the ear
can scarcely separate these tones from an obscure drone ; and
the deeper they are the more plainly can it distinguish the sepa-
rate impulses of the air to which they are due. Hence they
are used solely in conjunction with the next higher octaves, to
strengthen their notes, and produce an impression of greater
depth.

With the exception of the organ, all musical instruments,
however diverse the methods in which their sounds are pro-
duced, have their limit of depth at about the same point in the
scale as the piano ; not because it would be impossible to produce
slower impulses of the air of sufficient power, but because the
ear refuses its office, and hears slower impulses separately, without
gathering them up into single tones.

The often-repeated assertion of the French physicist Savart,
that he heard tones of eight vibrations in a second, upon a
peculiarly constructed instrument, seems due to an error.

Ascending the scale from the contra-C, pianofortes usually
have a compass of seven octaves, up to the so-called five-accented
c, which has 4,224 vibrations in a second. Among orchestral
instruments it is only the piccolo flute which can reach as high,
and this will give even one tone higher. The violin usually
mounts no higher than the e below, which has 2,640 vibrations
— of course we excapt the gymnastics of heaven-scaling virtuosi,
who are ever striving to excruciate their audience by some new

HARMONY IN MUSIC. 57

impossibility. Such performers may aspire to three whole
octaves lying above the five-accented c, and very painful to the
ear, for their existence has been established by Despretz, who,
by exciting small tuning-forks with a violin bow, obtained and
heard the eight-accented c, having 32,770 vibrations in a second.
Here the sensation of tone seemed to have reached its upper
limit, and the intervals were really undistinguishable in the
later octaves.

The musical pitch of a tone depends entirely on the number
of vibrations of the air in a second, and not at all upon the
mode in which they are produced. It is quite indifferent whether
they are generated by the vibrating strings of a piano or violin,
the vocal chords of the human larynx, the metal tongues of the
harmonium, the reeds of the clarionet, oboe, and bassoon, the
trembling lips of the trumpeter, or the air cut by a sharp edge
in organ pipes and flutes.

A tone of the same number of vibrations has always the
same pitch, by whichever one of these instruments it is pro-
duced. That which distinguishes the note A of a piano, for
example, from the equally high A of the violin, flute, clarionet,
or trumpet, is called the quality of the tone, and to this we shall
have to recur presently.

As an interesting example of these assertions, I beg to show you a
peculiar physical instrument for producing musical tones, called the
siren, Fig. 1, which is especially adapted to establish the properties
resulting from the ratios of the numbers of vibrations.

In order to produce tones upon this instrument, the portvents g0
and gj are connected by means of flexible tubes with a bellows. The
air enters into round brass boxes, a0 and a^ and escapes by the per-
forated covers of these boxes at c0 and c,. But the holes for the
escape of air are not perfectly free. Immediately before the covers
of both boxes there are two other perforated discs, fastened to a per-
pendicular axis k, which turns with great readiness. In the figure,
only the perforated disc can be seen at c0, and immediately below it
is the similarly perforated cover of the box. In the upper box, c]?
only the edge of the disc is visible. If then the holes of the disc are
precisely opposite to those of the cover, the ah- can escape freely.
But if the disc is made to revolve, so that some of its unperforated

HARMONY IN MUSIC. 59

portions stand before the holes of the box, the air cannot escape at
all. On turning the disc rapidly, the vent-holes of the box are alter-
nately opened and closed. During the opening, air escapes ; during
the closure, no air can pass. Hence the continuous stream of air from
the bellows is converted into a series of discontinuous puffs, which,
when they follow one another with sufficient rapidity, gather them-
selves together into a tone.

Each of the revolving discs of this instrument (_which is more
complicated in its construction than any one of the kind hitherto
'uade, and hence admits of a much greater number of combinations
of tone) has four concentric circles of holes, the lower set having
8, 10, 12, 18, and the upper set 9, 12, 15, and 16 holes respectively.
The series of holes in the covers of the boxes are precisely the same
as those in the discs, but under each of them lies a perforated ring,
which can be so arranged, by means of the stops i i i i, that the
corresponding holes of the cover can either communicate freely with
the inside of the box, or are entirely cut off from it; We are thus
enabled to use any one of the eight series of holes singly, or com-
bined two and two, or three and three together, in any arbitrary
manner.

The round boxes, h0 h0 and hj hn of which halves only are drawn
in the figure, serve by their resonance to soften the harshness of the
tone.

The holes in the boxes and discs are cut obliquely, so that when
the air enters the boxes through one or more of the series of holes,
the wind itself drives the discs round with a perpetually increasing
velocity.

On beginning to blow the instrument, we first hear separate im-
pulses of the air, escaping as puffs, as often as the holes of the disc
pass in front of those of the box. These puffs of air follow one an-
other more and more quickly, as the velocity of the revolving discs
increases, just like the puffs of steam of a locomotive on beginning to
move with the train. They next produce a whirring and whizzing,
which constantly becomes more rapid. At last we hear a dull drone,
which, as the velocity further increases, gradually gains in pitch and
strength.

Suppose that the discs have been brought to a velocity of 33 re-
volutions in a second, and that the series with 8 holes has been
opened. At each revolution of the disc all these 8 holes will pass
before each separate hole of the cover. Hence there will be 8 puffs
for each revolution of the disc, or S times 33, that is, 20-i pulls in a

60 ON THE PHYSIOLOGICAL CAUSES OF

second. This gives us the once-accented c' of our musical scale [that
is, ' middle c,' written on the leger line between the bass and treble
staves]. But on opening the series of 16 holes instead, we have twice
as many, or 16 times 33, that is, 528 vibrations in a second. We
hear exactly the octave above the first </, that is, the twice-accented
c'' [or c on the third space of the treble staff]. By opening both the
series of 8 and 16 holes at once, we have both c' and c" at once, and
can convince ourselves that we have the absolutely pure concord of
the octave. By taking 8 and 12 holes, which give numbers of vibra-
tions in the ratio of 2 to 3, we have the concord of a perfect fifth.
Similarly 12 and 16 or 9 and 12 give fourths, 12 and 15 give a major
third, and so on.

The upper box is furnished with a contrivance for slightly sharpen-
ing or flattening the tones which it produces. This box is movable
upon an axis, and connected with a toothed wheel, which is worked
by the driver attached to the handle d. By turning the handle
slowly while one of the series of holes in the upper box is in use, the
tone will be sharper or natter, according as the box moves in the
opposite direction to the disc, or in the same direction as the disc.
When the motion is in the opposite direction, the holes meet those of
the disc a little sooner than they otherwise would, the time of vibra-
tion of the tone is shortened, and the tone becomes sharper. The
contrary ensues in the other case.

Now, on blowing through 8 holes below and 16 above, we have a
perfect octave, as long as the upper box is still ; but when it is ic
motion, the pitch of the upper tone is slightly altered, and the octave
becomes false.

On blowing through 12 holes above and 18 below, the result is a
perfect fifth as long as the upper box is at rest, but if it moves the
concord is perceptibly injured.

These experiments with the siren show us, therefore : —

1. That a series of puffs following one another with sufficient
rapidity produce a musical tone.

2. That the more rapidly they follow one another, the sharper is
the tone.

3. That when the ratio of the number of vibrations is exactly 1 to
2, the result is a perfect octave ; when it is 2 to 3, a perfect fifth ;
when it is 3 to 4, a pure fourth, and so on. The slightest alteration
in these ratios destroys the purity of the concord.

You will perceive, from what has been hitherto adduced,

HARMOXY IX MUSIC. 61

Chat the human ear is affected by vibrations of the air, within
certain degrees of rapidity — viz. from about 20 to about 32,000
in a second — and that the sensation of musical tone arises from
this affection.

That the sensation thus excited is a sensation of musical
tone does not depend in any way upon the peculiar manner in.
•which the air is agitated, but solely on the peculiar powers of
sensation possessed by our ears and auditory nerves. I re-
marked, a little while ago, that when the tones are loud the
agitation of the air is perceptible to the skin. In this way
deaf mutes can perceive the motion of the air which we call
sound. But they do not hear, that is, they have no sensation of
tone in the ear. They feel the motion by the nerves of the skin,
producing that peculiar description of sensation called whirring.
The limits of the rapidity of vibration within which the ear
feels an agitation of the air to be sound, depend also wholly
upon the peculiar constitution of the ear.

When the siren is turned slowly, and hence the puffs of
air succeed each other slowly, you hear no musical sound. By
the continually increasing rapidity of its revolution, no essential
change is produced in the kind of vibration of the air. Nothing
new happens externally to the ear. The only new result is the
sensation experienced by the ear, which then for the first time
begins to be affected by the agitation of the air. Hence the
more rapid vibrations receive a new name, and are called Sound.
If you admire paradoxes, you may say that aerial vibrations do
not become sound until they fall upon a hearing ear.

I must now describe the propagation of sound through the
atmosphere. The motion of a mass of air through which a
tone passes belongs to the so-called wave-motions — a class of
motions of great importance in physics. Light, as well as
sound, is one of these motions.

The name is derived from the analogy of waves on the sur-
face of water, and these will best illustrate the peculiarity of
this description of motion.

When a point in a surface of still water is agitated — as by
throwing in a stone — the motion thus caused is propagated in

62 ON THE PHYSIOLOGICAL CAUSES OF

the form of waves, which spread in rings over the surface of
the water. The circles of waves continue to increase even after
rest has been restored at the point first affected. At the same
time the waves become continually lower, the further they are
removed from the centre of motion, and gradually disappear. On
each wave- ring we distinguish ridges or crests, and hollows or
troughs.

Crest and trough together form a wave, and we measure its
length from one crest to the next.

While the wave passes over the surface of the fluid, the
particles of the water which form it do not move on with it.
This is easily seen, by floating a chip of straw on the water.
When the waves reach the chip, they raise or depress it, but
when they have passed over it the position of the chip is not
perceptibly changed.

Now a light floating chip has no motion different from that
of the adjacent particles of water. Hence we conclude that
these particles do not follow the wave, but, after some pitching
up and down, remain in their original position. That which
really advances as a wave is, consequently, not the particles of
water themselves, but only a superficial form, which continues
to be built up by fresh particles of water. The paths of the
separate particles of water are more nearly vertical circles, in
which they revolve with a tolerably uniform velocity, as long
as the waves pass over them.

In Fig. 2 the dark wave-line ABC represents a section of the
surface of the water over which waves are running in the direction
of the arrows above a and c. The three circles a, b, and c represent
the paths of particular particles of water at the surface of the wave.
The particle which revolves in the circle b is supposed, at the time
that the surface of the water presents the form A B C, to be at its
highest point B, and the particles revolving in the circles a and
c to he simultaneously in their lowest positions.

The respective particles of water revolve in these circles in the
direction marked by the arrows. The dotted curves represent other
positions of the passing waves, at equal intervals of time, partly
before the assumption of the ABC position (as for the crests be-
tween a and b), and partly after the same (for the crests between b

HARMONY IN MUSIC. 63

and c). The positions of the crests are marked with figures. The
same figures in the three circles show where the respective revolving
particle would be, at the moment the wave assumed the corresponding
form. It will be noticed that the particles advance by equal arcs of
the circles, as the crest of the wave advances by equal distances
parallel to the water level.

In the circle b it will be further seen that the particle of water in
its positions 1, 2, 3 hastens to meet the approaching wave-crests,
] , 2, 3, rises on its left-hand side, is then carried on by the crest from
4 to 7 in the direction of its advance, afterwards halts behind it,
sinks down again on the right side, and finally reaches its original
position at 13. (In the 'Lecture itself, Fig. 2 was replaced by a
working model, in which the movable particles, connected by threads,
really revolved in circles, while connecting elastic threads represented
the surface of the water.)

All particles at the surface of the water, as you see by this draw-
ing, describe equal circles. The particles of water at different depths
move in the same way, but as the depths increase, the diameters of
their circles of revolution rapidly diminish.

In this way, then, arises the appearance of a progressive motion
along the surface of the water, while in reality the moving particles
of water do not advance with the wave, but perpetually revolve in
their small circular orbits.

To return from waves of water to waves of sound. Ima-
gine an elastic fluid like air to replace the water, and the
waves of this replaced water to be compressed by an inflexible
plate laid on their surface, the fluid being prevented from escap-
ing laterally from the pressure. Then on the waves being thus
flattened out, the ridges where the fluid had been heaped up

64 ON THE PHYSIOLOGICAL CAUSES OF

will produce much greater density than the hollows, from which
the fluid had been removed to form the ridges. Hence the
ridges are replaced by condensed strata of air, and the hollows
by rarefied strata. Now further imagine that these compressed
waves are propagated by the same law as before, and that also
the vertical circular orbits of the several particles of water
are compressed into horizontal straight lines. Then the waves
of sound will retain the peculiarity of having the particles of
air only oscillating backwards and forwards in a straight line,
while the wave itself remains merely a progressive form of
motion, continually composed of fresh particles of air. The
immediate result then would be waves of sound spr/eading out
horizontally from their origin.

But the expansion of waves of sound is not limited, like
those of water, to a horizontal surface. They can spread out in
any direction whatsoever. Suppose the circles generated by
a stone thrown into the water to extend in all directions of space,
and you will have the spherical waves of air by which sound is
propagated.

Hence we can continue to illustrate the peculiarities of the
motion of sound by the well-known visible motions of waves
of water.

The length of a wave of water, measured from crest to
crest, is extremely different. A falling drop, or a breath of air,
gently curls the surface of the water. The waves in the wake
of a steamboat toss the swimmer or skiff severely. But the
waves of a stormy ocean can find room in their hollows for the
keel of a ship of the line, and their ridges can scarcely be
overlooked from the mast-head. The waves of sound present
similar differences. The little curls of water with short lengths
of wave correspond to high tones, the giant ocean billows to
deep tones. Thus the contrabass C has a wave thirty-five feet
long, its higher octave a wave of half the length, while the
highest tones of a piano have waves of only three inches in
length.

1 The exact lengths of waves corresponding to certain notes, or symbols of
tone, depend upon the standard pitch assigned to one particular note, and

HARMONY IN MUSIC. 65

You perceive that the pitch of the tone corresponds to the
length of the wave. To this we should add that the height of
the ridges, or, transferred to air, the degree of alternate con-
densation and rarefaction, corresponds to the loudness and
intensity of the tone. But waves of the same height may
have different forms. The crest of the ridge, for example, may
be rounded off or pointed. Corresponding varieties also occur
in waves of sound of the same pitch and loudness. The so-
called timbre or quality of tone is what corresponds to the form
of the waves of water. The conception of form is transferred
from waves of water to waves of sound. Supposing waves of
water of different forms to be pressed flat as before, the surface,
having been levelled, will of course display no differences of
form, but, in the interior of the mass of water, we shall have
different distributions of pressure, and hence of density, which
exactly correspond to the differences of form in the still uncom-
pressed surface. In this sense then we can continue to speak of

FIG. 3,

the form of waves of sound, and can represent it geometrically.
"VVe make the curve rise where the pressure, and hence density,
increases, and fall where it diminishes — just as if we had a
compressed fluid beneath the curve, which would expand to
the height of the curve in order to regain its natural density.

Unfortunately, the form of waves of sound, on which de-
pends the quality of the tones produced by various sounding
bodies, can at present be assigned in only a very few cases.

Among the forms of waves of sound which we are able to
determine with more exactness is one of great importance, here
termed the simple or pure wave-form, and represented in Fig. 3.

this differs in different countries. Hence the figures of the author have
been left unreduced. They are sufficiently near to those usually adopted in
England, to occasion no difficulty to the reader in these general remarks. — TB
I. V

66

ON THE PHYSIOLOGICAL CAUSES OF

It can be seen in waves of watei only when their height is
small in comparison with their length, and they run over a
smooth surface without external disturbance, or without any
action of wind. Ridge and hollow are gently rounded, off,
equally broad and symmetrical, so that, if we inverted the carve,
the ridges would exactly fit into the hollows, and conversely.
This form of wave would be more precisely defined by saying

FIG. 4.

that the particles of water describe exactly circular orbits of
small diameters, with exactly uniform velocities. To this simple
wave-form corresponds a peculiar species of tone, which, from
reasons to be hereafter assigned, depending upon its relation to
quality, we will term a simple tone. Such tones are produced
by striking a tuning-fork and holding it before the opening of a
properly tuned resonance tube. The tone of tuneful human

HARMONY IN MUSIC.

67

voices, singing the vowel oo in too, in the middle positions of their
register, appears not to differ materially from this form of wave.
We also know the laws of the motion of strings with suffi-
cient accuracy to assign in some cases the form of motion which
they impart to the air. Thus Fig. 4 represents the forms suc-
cessively assumed by a string struck, as in the German Zither,
by a pointed style [the plectrum of the ancient lyra, or the quill
of the old harpsichord, which may be easily imitated on a
guitar]. A a represents the form assumed by the string at the
moment of percussion. Then, at equal intervals of time, follow
the forms B, C, D, E, F, G ; and then, in inverse order, F, E, D,
C, B, A, and so on in perpetual repetition. The form of motion
which such a string, by means of an attached sounding-board,
imparts to the surrounding air, probably corresponds to the
broken line in Fig. 5, where h h indicates the position of equili-
brium, and the letters a b c d e f g show the line of the wave
which is produced by the action of several forms of string
marked by the corresponding capital letters in Fig. 4. It is
easily seen how greatly this form of wave (which of course

FIG. 5.

could not occur in water) differs from that of Fig. 3 (inde-
pendently of magnitude), as the string only imparts to the air a
series of short impulses, alternately directed to opposite sides.1
The waves of air produced by the tone of a violin would, on

FIG. 6.

the same principle, be represented by Fig. 6. During each

1 It is here as*nmed that the sounding-board and air in contact with it
immediately obey the impulse given by the end of the string without exercising
a perceptible reaction on the motion of the string.

C8 OX THE PHYSIOLOGICAL CAUSES OP

period of vibration the pressure increases uniformly, and at the
end falls back suddenly to its minimum.

It is to such differences in the forms of the waves of sound
that the variety of quality in musical tones is due. "We may
even carry the analogy farther. The more uniformly rounded the
form of wave, the softer and milder is the quality of tone. The
more jerking and angular the wave-form, the more piercing the
quality. Tuning-forks, with their rounded forms of wave (Fig.
3), have an extraordinarily soft quality; and the qualities of
tone generated by the zither and violin resemble in harshness
the angularity of their wave-forms. (Figs. 5 and 6.)

Finally, I would direct your attention to an instructive
spectacle, which I have never been able to view without a cer-
tain degree of physico-scientific delight, because it displays to
the bodily eye, on the surface of water, what otherwise could
only be recognised by the mind's eye of the mathematical thinker
in a mass of air traversed in all directions by waves of sound.
I allude to the composition of many different systems of waves,
as they pass over one another, each undisturbedly pursuing its
own path. We can watch it from the parapet of any bridge
spanning a river, but it is most complete and sublime when
viewed from a cliff beside the sea. It is then rare not to see
innumerable systems of waves, of various length, propagated in
various directions. The longest come from the deep sea and dash
against the shore. Where the boiling breakers burst shorter
waves arise, and run back again towards the sea. Perhaps
a bird of prey darting after a fish gives rise to a system of
circular waves, which, rocking over the undulating surface, are
propagated with the same regularity as on the mirror of an in-
land lake. And thus, from the distant horizon, where white
lines of foam on the steel blue surface betray the coming trains
of wave, down to the sand beneath our feet, where the impres-
sion of their arcs remains, there is unfolded before our eyes a
sublime image of immeasurable power and unceasing variety,
which, as the eye at once recognises its pervading order and law,
enchains and exalts without confusing the mind.

Now, just in the same way you must conceive the air of a

HARMONY IN MUSIC. 69

concert-hall or TxJl-room traversed in every direction, and not
merely on the surface, by a variegated crowd of intersecting
wave-systems. From the mouths of the male singers proceed
waves of six to twelve feat in length ; from the lips of the song-
stresses dart shorter waves, from eighteen to thirty-six inches
long. The rustling of silken skirts excites little curls in the
air, each instrument in the orchestra emits its peculiar waves,
and all these systems expand spherically from their respective
centres, dart through each other, are reflected from the walls of
the room, and thus rush backwards and forwards, until they
succumb to the greater force of newly generated tones.

Although this spectacle is veiled from the material eye, we
have another bodily organ, the ear, specially adapted to reveal
it to us. This analyses the interdigitation of the wavas, which
in snch cases would be far more confused than the intersection
of the water undulations, separates the several tones which
compose it, and distinguishes the voices of men and women —
nay, even of individuals — the peculiar qualities of tone given
out by each instrument, the rustling of the dresses, the footfalls
of the walkers, and so on.

It is necessary to examine the circumstances with greater
minuteness. When a bird of prey dips into the sea, rings of
waves arise, which are propagated as slowly and regularly upon
the moving surface as upon a surface at rest. These rings are
cut into the curved surface of the waves in precisely the same
way as they would have been into the still surface of a lake.
The form of the external surface of the water is determined in
this, as in other more complicated cases, by taking the height
of each point to be the height of all the ridges of the waves
which coincide at this point at one time, after deducting the sum
of all similarly simultaneously coincident hollows. Such a sum of
positive magnitudes (the ridges) and negative magnitudes (the
hollows), where the latter have to be subtracted instead of being
added, is called an algebraical sum. Using this term, then, we
may say that the height of every point of the surface of t/M
water is equal to the algebraical sum of all the portions of the
waves which at that moment there concur.

70 OX THE PHYSIOLOGICAL CAUSES OF

It is the same with the waves of sound. They, too, are
added together at every point of the mass of air, as well as
in contact with the listener's ear. For them also the degree of
condensation and the velocity of the particles of air in the
passages of the organ of hearing are equal to the algebraical
sums of the separate degrees of condensation and of the velo-
cities of the waves of sound, considered apart. This single motion
of the air produced by the simultaneous action of various sound-
ing bodies, has now to be analysed by the air into the separate
paits which correspond to their separate effects. For doing this
the ear is much more unfavourably situated than the eye. The
latter surveys the whole und ulat ing surface at a glance. But the
ear can, of course, only perceive the motion of the particles of air
which impinge upon it. And yet the ear solves its problem with
the grea test exactness, certainty, and determinacy. This power
of the ear is of supreme importance for hearing. Were it not
present it would be impossible to distinguish different tones.

Some recent anatomical discoveries appear to give a clue to
the explanation of this important power of the ear.

You will all have observed the phenomena of the sym pathetic
production of tones in musical instruments, especially stringed
instruments. The string of a pianoforte when the damper is
raised begins to vibrata as soon as its proper tone is produced
in its neighbourhood with sufficient force by some other means.
When this foreign tone ceases the tone of the string will be
heard to continue some little time longer. If we put little paper
riders on the string they will be jerked off when its tone is thus
produced in the neighbourhood. This sympathetic action of
the string depends on the impact of the vibrating particles of
air against the string arid its sounding-board.

Each separate wave-crest (or condensation) of air which
passes by the string is, of course, too weak to produce a sensible
motion in it. But when a long series of wave-crests (or con-
densations) strike the string in such a manner that each succeed-
ing one increases the slight tremor which resulted from the
action of its predecessors, the effect finally becomes sensible.
It is a process of exactly the same nature as the swinging of a

HARMONY EN MUSIC. 71

heavy bell. A powerful man can scarcely move it sensibly by
a single impulse. A boy, by pulling the rope at regular intervals
corresponding to the time of its oscillations, can gradually bring
it into violent motion.

This peculiar reinforcement of vibration depends entirely
on the rhythmical application of the impulse. When the bell
has been once made to vibrate as a pendulum in a very small
arc, and the boy always pulls the rope as it falls, and at a time
that his pull augments the existing velocity of the bell, this
velocity, increasing slightly at each pull, will gradually become
considerable. But if the boy apply his power at irregular in-
tervals, sometimes increasing and sometimes diminishing the
motion of the bell, he will produce no sensible effect.

In the same way that a mere boy is thus enabled to swing
a heavy bell, the tremors of light and mobile air suffice to set
in motion the heavy and solid mass of steel contained in a
tuning-fork, provided that the tone which is excited in the air
is exactly in unison with that of the fork, because in this case
also every impact of a wave of air against the fork increases
the motions excited by the like previous blows.

This experiment is most conveniently performed on a fork,
Fig. 7, which is fastened to a sounding-board, the air being
excited by a similar fork of precisely the same pitch. If one is
struck, the other will be found after a fe\v seconds to be sound-
ing also. Then damp the first fork, by touching it for a moment
with a finger, and the second will continue the tone. The
second will then bring the first into vibration, and so on.

But if a very small piece of wax be attached to the ends of
one of the forks, whereby its pitch will be rendered scarcely
percept'bly lower than the other, the sympathetic vibration of
the second fork ceases, because the times of oscillation are no
longer the same in each. The blows which the waves of air
excited by the first inflict upon the sounding-board of the second
fork, are indeed for a time in the same direction as the motions
of the second fork, and consequently increase the latter, but
after a very short time they cease to be so, and consequently
destroy the slight motion which they had previously excited.

72 ON THE PHYSIOLOGICAL CAUSES OF

Lighter and more mobile elastic bodies, as for example
strings, can be set in motion by a much smaller number of
aerial impulses. Hence they can be set in sympathetic motion
much more easily than tuning-forks, and by means of a musical
tone which is far less accurately in unison with themselves.

Now, then, if several tones are sounded in the neighbour-
hood of a pianoforte, no string can be set in sympathetic
vibration unless it is in unison with one of those tones. For
example, depress the forte pedal (thus raising the dampers), and
put paper riders on all the strings. They will of course leap
FIG. 7.

off when their strings are put in vibration. Then let several
voices or instruments sound tones in the neighbourhood. All
those riders, and only those, will leap off which are placed upon
strings that correspond to tones of the same pitch as those
sounded. You perceive that a pianoforte is also capable of
analysing the wave confusion of the air into its elementary con-
stituents.

The process which actually goes on in our ear is probably
very like that just described. Deep in the petrous bone out of
which the internal ear is hollowed lies a peculiar organ, the
cochlea or snail shell — a cavity filled with water, and so called

HARMOJfY IN MUSIC.

73

from its resemblance to the shell of a common garden snail.
This spiral passage is divided throughout its length into three
sections, upper, middle, and lower, by two membranes stretched
in the middle of its height. The Marchese Corti discovered
some very remarkable formations in the middle section. They
FIG. 8.

Tly^-—^

consist of innumerable plates, microscopically smnll, and
arranged orderly side by side, like the keys of a piano. They are
connected at one end with the fibres of the auditory nerve, and
at the other with the stretched membrane.

Fig. 8 shows this extraordinarily complicated arrangement

74 ON TEE PHYSIOLOGICAL CAUSES OF

for a small part of the partition of the cochlea. The arches
which leave the membrane at d and are reinserted at e, reach-
ing their greatest height between m and o, are probably the
parts which are suited for vibration. They are spun round
with innumerable fibrils, among which some nerve fibres can be
recognised, coming to them through the holes near c. The
transverse fibres g, h, i, k, and the cells o, also appear to belong
to the nervous system. There are about three thousand arches
similar to d e, lying orderly beside each other, like the keys of
a piano in the whole length of the partition of the cochlea.

In the so-called vestibulum, also, where the nerves expand
upon little membranous bags swimming in water, elastic appen-
dages, similar to stiff hairs, have been lately discovered at the
ends of the nerves. The anatomical arrangement of these
appendages leaves scarcely any room, to doubt that they are set
into sympathetic vibration by the waves of sound which are
conducted through the ear. Now if we venture to conjecture
— it is at present only a conjecture, but after careful considera-
tion I am led to think it very probable — that every such
appendage is tuned to a certain tone like the strings of a piano,
then the recent experiment with a piano shows you that when
(and only when) that tone is sounded the corresponding hair-
like appendage may vibrate, and the corresponding nerve-fibre
experience a sensation, so that the presence of each single such
tone in the midst of a whole confusion of tones must be in-
dicated by the corresponding sensation.

Experience then shows us that the ear really possesses the
power of analysing waves of air into their elementary forms.

By compound motions of the air, we have hitherto meant
such as have been caused by the simultaneous vibration of
several elastic bodies. Now, since the forms of the waves of
sound of different musical instniments are different, there is
room to suppose that the kind of vibration excited in the pas-
sages of the ear by one such tone will be exactly the same as
the kind of vibration which in another case is there excited by
two or more instruments sounded together. If the ear analyses
the motion into its elements in the latter case, it cannot well

HARMONY IN MUSIC.

75

avoid doing so in the former, where the tone is due to a single
source. And this is found to be really the case.

I have previously mentioned the form of wave with gently
rounded crests and hollows, and termed it simple or pure (p. 65).
In reference to this form the French mathematician Fourier has
established a celebrated and important theorem which may be
translated from mathematical into ordinary language thus : Any
form of wave whatever can be compounded of a number of
simple waves of different lengths. The longest of these simple
waves has the same length as that of the given form of wave,
the others have lengths one half, one third, one fourth, &c., as
great.

By the different modes of uniting the crests and hollows of
these simple waves, an endless multiplicity of wave-forms may
be produced.

TIG. 9.

For example, the wave-curves A and B, Fi^. 9, represent waves
oi simple tones, 13 making twice as many vibrations as A in a second