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<pre>

The Project Gutenberg EBook of Popular scientific lectures, by Ernst Mach

This eBook is for the use of anyone anywhere at no cost and with
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Title: Popular scientific lectures

Author: Ernst Mach

Translator: Thomas Joseph McCormack

Release Date: April 22, 2012 [EBook #39508]

Language: English

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</pre>

<p><span class="pagenum"><a name="Page_i" id="Page_i">[Pg i]</a></span></p>

<p><span class="pagenum"><a name="Page_ii" id="Page_ii">[Pg ii]</a></span></p>
<h1>POPULAR SCIENTIFIC LECTURES.</h1>


<hr/>

<h2><a name="BY_THE_SAME_AUTHOR" id="BY_THE_SAME_AUTHOR">BY THE SAME AUTHOR.</a></h2>


<blockquote><p><span class="smcap">The Science of Mechanics.</span> Translated from the
Second German Edition by T. J. McCormack.
250 Cuts and Illustrations. 534 Pages. Half
Morocco, Gilt Top. Price, $2.50.</p>

<p><span class="smcap">Contributions to the Analysis of the Sensations.</span>
Translated by C. M. Williams. With Notes and
New Additions by the Author. 200 Pages. 36
Cuts. Price, $1.00.</p>

<p><span class="smcap">Popular Scientific Lectures.</span> Translated by T.
J. McCormack. Third Revised and Enlarged
Edition. 411 Pages. 59 Cuts. Cloth, $1.50;
Paper, 50 cents.</p></blockquote>

<p class="center">THE OPEN COURT PUBLISHING CO.,<br />
<span class="small">324 DEARBORN ST., CHICAGO.</span>
</p><p><span class="pagenum"><a name="Page_iii" id="Page_iii">[Pg iii]</a></span></p>

<hr/>


<p class="center big bold">POPULAR<br />
SCIENTIFIC LECTURES</p>

<p class="center"><span class="small">BY</span><br/>
ERNST MACH</p>

<p class="center small">FORMERLY PROFESSOR OF PHYSICS IN THE UNIVERSITY OF PRAGUE, NOW
PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE
SCIENCE IN THE UNIVERSITY OF VIENNA</p>

<p class="center"><span class="small">TRANSLATED<br/>
BY<br/></span>
THOMAS J. McCORMACK</p>

<p class="center">THIRD EDITION, REVISED AND ENLARGED</p>
<hr/>
<p class="center small">WITH FIFTY-NINE CUTS AND DIAGRAMS</p>
<hr/>
<p class="center big">CHICAGO<br/>
THE OPEN COURT PUBLISHING COMPANY</p>

<p class="center small">FOR SALE BY<br/>
<span class="smcap">Kegan Paul, Trench, Truebner &amp; Co.</span>, LONDON<br/>
1898
</p><p><span class="pagenum"><a name="Page_iv" id="Page_iv">[Pg iv]</a></span></p>


<hr/>

<p class="center">COPYRIGHT</p>

<p class="center"><span class="smcap">By The Open Court Publishing Co.</span></p>

<div class="center">
<table border="0" cellpadding="0" cellspacing="0" summary="copyright dates">
<tr><td align="left">Pages 1-258&nbsp;</td><td align="left">&nbsp;in 1894.</td></tr>
<tr><td align="left">Pages 338-374&nbsp;</td><td align="left">&nbsp;in 1894.</td></tr>
<tr><td align="left">Pages 259-281&nbsp;</td><td align="left">&nbsp;in 1896.</td></tr>
<tr><td align="left">Pages 282-308&nbsp;</td><td align="left">&nbsp;in 1897.</td></tr>
<tr><td align="left">Pages 309-337&nbsp;</td><td align="left">&nbsp;in 1898.</td></tr>
</table></div>

<hr class="chap" />
<p><span class="pagenum"><a name="Page_v" id="Page_v">[Pg v]</a></span></p>




<h2><a name="AUTHORS_PREFACE_TO_THE_FIRST" id="AUTHORS_PREFACE_TO_THE_FIRST">AUTHOR'S PREFACE TO THE FIRST
EDITION.</a></h2>


<p>Popular lectures, owing to the knowledge they presuppose,
and the time they occupy, can afford only a <i>modicum</i>
of instruction. They must select for this purpose easy subjects,
and restrict themselves to the exposition of the simplest and the
most essential points. Nevertheless, by an appropriate choice of
the matter, the <i>charm</i> and the <i>poetry</i> of research can be conveyed
by them. It is only necessary to set forth the attractive and the
alluring features of a problem, and to show what broad domains
of fact can be illuminated by the light radiating from the solution
of a single and ofttimes unobtrusive point.</p>

<p>Furthermore, such lectures can exercise a favorable influence
by showing the substantial sameness of scientific and every-day
thought. The public, in this way, loses its shyness towards scientific
questions, and acquires an interest in scientific work which is
a great help to the inquirer. The latter, in his turn, is brought to
understand that his work is a small part only of the universal process
of life, and that the results of his labors must redound to the
benefit not only of himself and a few of his associates, but to that
of the collective whole.</p>

<p>I sincerely hope that these lectures, in the present excellent
translation, will be productive of good in the direction indicated.</p>

<p class="right">
<span class="smcap">E. Mach.</span><br />
</p>

<p><span class="smcap">Prague</span>, December, 1894.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_vii" id="Page_vii">[Pg vii]</a><br /><a name="Page_vi" id="Page_vi">[Pg vi]</a></span></p>




<h2><a name="TRANSLATORS_NOTE_TO_THE" id="TRANSLATORS_NOTE_TO_THE">TRANSLATOR'S NOTE TO THE
THIRD EDITION.</a></h2>


<p>The present third edition of this work has been enlarged by
the addition of a new lecture, "On Some Phenomena Attending
the Flight of Projectiles." The additions to the second
consisted of the following four lectures and articles: Professor
Mach's Vienna Inaugural Lecture, "The Part Played by Accident
in Invention and Discovery," the lecture on "Sensations of Orientation,"
recently delivered and summing up the results of an important
psychological investigation, and two historical articles (see
Appendix) on Acoustics and Sight.</p>

<p>The lectures extend over a long period, from 1864 to 1898,
and differ greatly in style, contents, and purpose. They were first
published in collected form in English; afterwards two German
editions were called for.</p>

<p>As the dates of the first five lectures are not given in the footnotes
they are here appended. The first lecture, "On the Forms
of Liquids," was delivered in 1868 and published with that "On
Symmetry" in 1872 (Prague). The second and third lectures, on
acoustics, were first published in 1865 (Graz); the fourth and fifth,
on optics, in 1867 (Graz). They belong to the earliest period of
Professor Mach's scientific activity, and with the lectures on electrostatics
and education will more than realise the hope expressed in
the author's Preface.</p>

<p>The eighth, ninth, tenth, eleventh, and twelfth lectures are of<span class="pagenum"><a name="Page_viii" id="Page_viii">[Pg viii]</a></span>
a more philosophical character and deal principally with the methods
and nature of scientific inquiry. In the ideas summarised in
them will be found one of the most important contributions to the
theory of knowledge made in the last quarter of a century. Significant
hints in psychological method, and exemplary specimen-researches
in psychology and physics, are also presented; while in
physics many ideas find their first discussion that afterwards, under
other names and other authorship, became rallying-cries in this
department of inquiry.</p>

<p>All the proofs of this translation have been read by Professor
Mach himself.</p>

<p class="right">
<span class="smcap">T. J. McCormack.</span><br />
</p>

<p><span class="smcap">La Salle, Ill.</span>, May, 1898.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_ix" id="Page_ix">[Pg ix]</a></span></p>




<h2><a name="TABLE_OF_CONTENTS" id="TABLE_OF_CONTENTS">TABLE OF CONTENTS.</a></h2>


<ul class="toc">
<li>&nbsp; &nbsp;<span class="label smcap">page</span></li>
<li><a href="#THE_FORMS_OF_LIQUIDS">The Forms of Liquids</a>&nbsp; &nbsp;<span class="label">1</span></li>
<li><a href="#THE_FIBRES_OF_CORTI">The Fibres of Corti</a>&nbsp; &nbsp;<span class="label">17</span></li>
<li><a href="#ON_THE_CAUSES_OF_HARMONY">On the Causes of Harmony</a>&nbsp; &nbsp;<span class="label">32</span></li>
<li><a href="#THE_VELOCITY_OF_LIGHT">The Velocity of Light</a>&nbsp; &nbsp;<span class="label">48</span></li>
<li><a href="#WHY_HAS_MAN_TWO_EYES">Why Has Man Two Eyes?</a>&nbsp; &nbsp;<span class="label">66</span></li>
<li><a href="#ON_SYMMETRY">On Symmetry</a>&nbsp; &nbsp;<span class="label">89</span></li>
<li><a href="#ON_THE_FUNDAMENTAL_CONCEPTS">On the Fundamental Concepts of Electrostatics</a>&nbsp; &nbsp;<span class="label">107</span></li>
<li><a href="#ON_THE_PRINCIPLE_OF_THE_CONSERVATION">On the Principle of the Conservation of Energy</a>&nbsp; &nbsp;<span class="label">137</span></li>
<li><a href="#THE_ECONOMICAL_NATURE_OF">On the Economical Nature of Physical Inquiry</a>&nbsp; &nbsp;<span class="label">186</span></li>
<li><a href="#ON_TRANSFORMATION_AND_ADAPTATION">On Transformation and Adaptation in Scientific Thought</a>&nbsp; &nbsp;<span class="label">214</span></li>
<li><a href="#ON_THE_PRINCIPLE_OF_COMPARISON">On the Principle of Comparison in Physics</a>&nbsp; &nbsp;<span class="label">236</span></li>
<li><a href="#THE_PART_PLAYED_BY_ACCIDENT_IN">On the Part Played by Accident in Invention and Discovery</a>&nbsp; &nbsp;<span class="label">259</span></li>
<li><a href="#ON_SENSATIONS_OF_ORIENTATION93">On Sensations of Orientation</a>&nbsp; &nbsp;<span class="label">282</span></li>
<li><a href="#ON_SOME_PHENOMENA_ATTENDING">On Some Phenomena Attending the Flight of Projectiles</a>&nbsp; &nbsp;<span class="label">309</span></li>
<li><a href="#ON_INSTRUCTION_IN_THE_CLASSICS">On Instruction in the Classics and the Mathematico-Physical Sciences</a>&nbsp; &nbsp;<span class="label">338</span></li>
<li><a href="#APPENDIX">Appendixes.</a>
<ul class="tocsub"><li><a href="#A_CONTRIBUTION_TO_THE_HISTORY_OF_ACOUSTICS">A Contribution to the History of Acoustics</a>&nbsp; &nbsp;<span class="label">375</span></li>
<li><a href="#REMARKS_ON_THE_THEORY_OF_SPATIAL_VISION">Remarks on the Theory of Spatial Vision</a>&nbsp; &nbsp;<span class="label">386</span></li></ul></li>
<li><a href="#INDEX">Index</a>&nbsp; &nbsp;<span class="label">393</span></li>
</ul>
<p><span class="pagenum"><a name="Page_x" id="Page_x">[Pg x]</a><br /><a name="Page_1" id="Page_1">[Pg 1]</a></span></p>


<hr class="chap" />

<h2><a name="THE_FORMS_OF_LIQUIDS" id="THE_FORMS_OF_LIQUIDS">THE FORMS OF LIQUIDS.</a></h2>


<p>What thinkest thou, dear Euthyphron, that the
holy is, and the just, and the good? Is the holy
holy because the gods love it, or are the gods holy because
they love the holy? By such easy questions did
the wise Socrates make the market-place of Athens unsafe
and relieve presumptuous young statesmen of the
burden of imaginary knowledge, by showing them how
confused, unclear, and self-contradictory their ideas
were.</p>

<p>You know the fate of the importunate questioner.
So called good society avoided him on the promenade.
Only the ignorant accompanied him. And finally he
drank the cup of hemlock&mdash;a lot which we ofttimes
wish would fall to modern critics of his stamp.</p>

<p>What we have learned from Socrates, however,&mdash;our
inheritance from him,&mdash;is scientific criticism.
Every one who busies himself with science recognises
how unsettled and indefinite the notions are which he
has brought with him from common life, and how, on
a minute examination of things, old differences are<span class="pagenum"><a name="Page_2" id="Page_2">[Pg 2]</a></span>
effaced and new ones introduced. The history of science
is full of examples of this constant change, development,
and clarification of ideas.</p>

<p>But we will not linger by this general consideration
of the fluctuating character of ideas, which becomes a
source of real uncomfortableness, when we reflect that
it applies to almost every notion of life. Rather shall
we observe by the study of a physical example how
much a thing changes when it is closely examined, and
how it assumes, when thus considered, increasing definiteness
of form.</p>

<p>The majority of you think, perhaps, you know
quite well the distinction between a liquid and a solid.
And precisely persons who have never busied themselves
with physics will consider this question one of
the easiest that can be put. But the physicist knows
that it is one of the most difficult. I shall mention
here only the experiments of Tresca, which show that
solids subjected to high pressures behave exactly as
liquids do; for example, may be made to flow out in
the form of jets from orifices in the bottoms of vessels.
The supposed difference of kind between liquids and
solids is thus shown to be a mere difference of degree.</p>

<p>The common inference that because the earth is
oblate in form, it was originally fluid, is an error, in
the light of these facts. True, a rotating sphere, a few
inches in diameter will assume an oblate form only
if it is very soft, for example, is composed of freshly
kneaded clay or some viscous stuff. But the earth,<span class="pagenum"><a name="Page_3" id="Page_3">[Pg 3]</a></span>
even if it consisted of the rigidest stone, could not
help being crushed by its tremendous weight, and must
perforce behave as a fluid. Even our mountains could
not extend beyond a certain height without crumbling.
The earth <i>may</i> once have been fluid, but this by no
means follows from its oblateness.</p>

<p>The particles of a liquid are displaced on the application
of the slightest pressure; a liquid conforms
exactly to the shapes of the vessels in which it is contained;
it possesses no form of its own, as you have
all learned in the schools. Accommodating itself in
the most trifling respects to the conditions of the vessel
in which it is placed, and showing, even on its surface,
where one would suppose it had the freest play, nothing
but a polished, smiling, expressionless countenance,
it is the courtier <i>par excellence</i> of the natural bodies.</p>

<p>Liquids have no form of their own! No, not for the
superficial observer. But persons who have observed
that a raindrop is round and never angular, will not be
disposed to accept this dogma so unconditionally.</p>

<p>It is fair to suppose that every man, even the weakest,
would possess a character, if it were not too difficult
in this world to keep it. So, too, we must suppose
that liquids would possess forms of their own, if
the pressure of the circumstances permitted it,&mdash;if
they were not crushed by their own weights.</p>

<p>An astronomer once calculated that human beings
could not exist on the sun, apart from its great heat,
because they would be crushed to pieces there by their<span class="pagenum"><a name="Page_4" id="Page_4">[Pg 4]</a></span>
own weight. The greater mass of this body would
also make the weight of the human body there much
greater. But on the moon, because here we should
be much lighter, we could jump as high as the church-steeples
without any difficulty, with the same muscular
power which we now possess. Statues and "plaster"
casts of syrup are undoubtedly things of fancy, even
on the moon, but maple-syrup would flow so slowly
there that we could easily build a maple-syrup man on
the moon, for the fun of the thing, just as our children
here build snow-men.</p>

<p>Accordingly, if liquids have no form of their own
with us on earth, they have, perhaps, a form of their
own on the moon, or on some smaller and lighter heavenly
body. The problem, then, simply is to get rid of
the effects of gravity; and, this done, we shall be able
to find out what the peculiar forms of liquids are.</p>

<p>The problem was solved by Plateau of Ghent, whose
method was to immerse the liquid in another of the
same specific gravity.<a name="FNanchor_1_1" id="FNanchor_1_1"></a><a href="#Footnote_1_1" class="fnanchor">[1]</a> He employed for his experiments
oil and a mixture of alcohol and water. By
Archimedes's well-known principle, the oil in this mixture
loses its entire weight. It no longer sinks beneath
its weight; its formative forces, be they ever so
weak, are now in full play.</p>

<p>As a fact, we now see, to our surprise, that the oil,
instead of spreading out into a layer, or lying in a<span class="pagenum"><a name="Page_5" id="Page_5">[Pg 5]</a></span>
formless mass, assumes the shape of a beautiful and
perfect sphere, freely suspended in the mixture, as
the moon is in space. We can construct in this way a
sphere of oil several inches in diameter.</p>

<p>If, now, we affix a thin plate to a
wire and insert the plate in the oil
sphere, we can, by twisting the wire
between our fingers, set the whole ball
in rotation. Doing this, the ball assumes
an oblate shape, and we can, if
we are skilful enough, separate by such
rotation a ring from the ball, like that
which surrounds Saturn. This ring is
finally rent asunder, and, breaking up
into a number of smaller balls, exhibits
to us a kind of model of the origin of
the planetary system according to the
hypothesis of Kant and Laplace.</p>

<div class="figright" style="width: 150px;">
<img src="images/i_015.jpg" width="150" height="491" alt="" />
<span class="caption">Fig. 1.</span>
</div>

<p>Still more curious are the phenomena
exhibited when the formative
forces of the liquid are partly disturbed
by putting in contact with the liquid's
surface some rigid body. If we immerse,
for example, the wire framework of a cube in our
mass of oil, the oil will everywhere stick to the wire
framework. If the quantity of oil is exactly sufficient
we shall obtain an oil cube with perfectly smooth walls.
If there is too much or too little oil, the walls of the
cube will bulge out or cave in. In this manner we<span class="pagenum"><a name="Page_6" id="Page_6">[Pg 6]</a></span>
can produce all kinds of geometrical figures of oil, for
example, a three-sided pyramid, a cylinder (by bringing
the oil between two wire rings), and so on. Interesting
is the change of form that occurs when we
gradually suck out the oil by means of a glass tube
from the cube or pyramid. The wire holds the oil
fast. The figure grows smaller and smaller, until it is
at last quite thin. Ultimately it consists simply of a
number of thin, smooth plates of oil, which extend
from the edges of the cube to the centre, where they
meet in a small drop. The same is true of the pyramid.</p>

<div class="figcenter" style="width: 300px;">
<img src="images/i_016.jpg" width="300" height="204" alt="" />
<span class="caption">Fig. 2.</span>
</div>

<p>The idea now suggests itself that liquid figures as
thin as this, and possessing, therefore, so slight a
weight, cannot be crushed or deformed by their weight;
just as a small, soft ball of clay is not affected in this
respect by its weight. This being the case, we no
longer need our mixture of alcohol and water for the
production of figures, but can construct them in the<span class="pagenum"><a name="Page_7" id="Page_7">[Pg 7]</a></span>
open air. And Plateau, in fact, found that these thin
figures, or at least very similar ones, could be produced
in the air, by dipping the wire nets described
in a solution of soap and water and quickly drawing
them out again. The experiment is not difficult. The
figure is formed of itself. The preceding drawing
represents to the eye the forms obtained with cubical
and pyramidal nets. In the cube, thin, smooth films
of soap-suds proceed from the edges to a small, quadratic
film in the centre. In the pyramid, a film proceeds
from each edge to the centre.</p>

<p>These figures are so beautiful that they hardly admit
of appropriate description. Their great regularity
and geometrical exactness evokes surprise from all who
see them for the first time. Unfortunately, they are of
only short duration. They burst, on the drying of the
solution in the air, but only after exhibiting to us the
most brilliant play of colors, such as is often seen in
soap-bubbles. Partly their beauty of form and partly
our desire to examine them more minutely induces us
to conceive of methods of endowing them with permanent
form. This is very simply done.<a name="FNanchor_2_2" id="FNanchor_2_2"></a><a href="#Footnote_2_2" class="fnanchor">[2]</a> Instead of
dipping the wire nets in solutions of soap, we dip them
in pure melted colophonium (resin). When drawn
out the figure at once forms and solidifies by contact
with the air.</p>

<p>It is to be remarked that also solid fluid-figures can<span class="pagenum"><a name="Page_8" id="Page_8">[Pg 8]</a></span>
be constructed in the open air, if their weight be light
enough, or the wire nets of very small dimensions. If
we make, for example, of very fine wire a cubical net
whose sides measure about one-eighth of an inch in
length, we need simply to dip this net in water to obtain
a small solid cube of water. With a piece of blotting
paper the superfluous water may be easily removed
and the sides of the cube made smooth.</p>

<p>Yet another simple method may be devised for observing
these figures. A drop of water on a greased
glass plate will not run if it is small enough, but will
be flattened by its weight, which presses it against
its support. The smaller the drop the less the flattening.
The smaller the drop the nearer it approaches
the form of a sphere. On the other hand, a drop suspended
from a stick is elongated by its weight. The
undermost parts of a drop of water on a support are
pressed against the support, and the upper parts are
pressed against the lower parts because the latter cannot
yield. But when a drop falls freely downward
all its parts move equally fast; no part is impeded by
another; no part presses against another. A freely
falling drop, accordingly, is not affected by its weight;
it acts as if it were weightless; it assumes a spherical
form.</p>

<p>A moment's glance at the soap-film figures produced
by our various wire models, reveals to us a great
multiplicity of form. But great as this multiplicity is,<span class="pagenum"><a name="Page_9" id="Page_9">[Pg 9]</a></span>
the common features of the figures also are easily discernible.</p>

<div class="poem"><div class="stanza">
<span class="i0">"All forms of Nature are allied, though none is the same as the other;<br /></span>
<span class="i0">Thus, their common chorus points to a hidden law."<br /></span>
</div></div>

<p>This hidden law Plateau discovered. It may be
expressed, somewhat prosily, as follows:</p>

<p>1) If several plane liquid films meet in a figure
they are always three in number, and, taken in pairs,
form, each with another, nearly equal angles.</p>

<p>2) If several liquid edges meet in a figure they are
always four in number, and, taken in pairs, form, each
with another, nearly equal angles.</p>

<p>This is a strange law, and its reason is not evident.
But we might apply this criticism to almost all laws.
It is not always that the motives of a law-maker are
discernible in the form of the law he constructs. But
our law admits of analysis into very simple elements
or reasons. If we closely examine the paragraphs
which state it, we shall find that their meaning is simply
this, that the surface of the liquid assumes the shape
of smallest area that is possible under the circumstances.</p>

<p>If, therefore, some extraordinarily intelligent tailor,
possessing a knowledge of all the artifices of the higher
mathematics, should set himself the task of so covering
the wire frame of a cube with cloth that every piece
of cloth should be connected with the wire and joined
with the remaining cloth, and should seek to accomplish
this feat with the greatest saving of material, he<span class="pagenum"><a name="Page_10" id="Page_10">[Pg 10]</a></span>
would construct no other figure than that which is here
formed on the wire frame in our solution of soap and
water. Nature acts in the construction of liquid figures
on the principle of a covetous tailor, and gives no
thought in her work to the fashions. But, strange to
say, in this work, the most beautiful fashions are
of themselves produced.</p>

<p>The two paragraphs which state our law apply primarily
only to soap-film figures, and are not applicable,
of course, to solid oil-figures. But the principle that
the superficial area of the liquid shall be the least
possible under the circumstances, is applicable to all
fluid figures. He who understands not only the letter
but also the reason of the law will not be at a loss
when confronted with cases to which the letter does
not accurately apply. And this is the case with the
principle of least superficial area. It is a sure guide
for us even in cases in which the above-stated paragraphs
are not applicable.</p>

<p>Our first task will now be, to show by a palpable
illustration the mode of formation of liquid figures by
the principle of least superficial area. The oil on the
wire pyramid in our mixture of alcohol and water, being
unable to leave the wire edges, clings to them, and
the given mass of oil strives so to shape itself that its
surface shall have the least possible area. Suppose
we attempt to imitate this phenomenon. We take a
wire pyramid, draw over it a stout film of rubber, and
in place of the wire handle insert a small tube leading<span class="pagenum"><a name="Page_11" id="Page_11">[Pg 11]</a></span>
into the interior of the space enclosed by the rubber
(Fig. 3). Through this tube we can blow in or suck
out air. The quantity of air in the enclosure represents
the quantity of oil. The stretched rubber film,
which, clinging to the wire edges,
does its utmost to contract, represents
the surface of the oil endeavoring
to decrease its area. By
blowing in, and drawing out the air,
now, we actually obtain all the oil
pyramidal figures, from those bulged
out to those hollowed in. Finally, when
all the air is pumped or sucked out, the
soap-film figure is exhibited. The rubber
films strike together, assume the form of planes,
and meet at four sharp edges in the centre of the
pyramid.</p>

<div class="figleft" style="width: 150px;">
<img src="images/i_021.jpg" width="150" height="256" alt="" />
<span class="caption">Fig. 3.</span>
</div>

<div class="figright" style="width: 300px;">
<img src="images/i_021-1.jpg" width="300" height="173" alt="" />
<span class="caption">Fig. 4.</span>
</div>

<p>The tendency of soap-films to assume smaller forms
may be directly demonstrated by a method of Van der
Mensbrugghe. If we dip a square wire frame to which<span class="pagenum"><a name="Page_12" id="Page_12">[Pg 12]</a></span>
a handle is attached into a solution of soap and water,
we shall obtain on the frame a beautiful, plane film of
soap-suds. (Fig. 4.) On this we lay a thread having its
two ends tied together. If, now, we puncture the part
enclosed by the thread, we shall obtain a soap-film
having a circular hole in it, whose circumference is
the thread. The remainder of the film decreasing in
area as much as it can, the hole assumes the largest
area that it can. But the figure of largest area, with
a given periphery, is the circle.</p>

<div class="figcenter" style="width: 400px;">
<img src="images/i_022.jpg" width="400" height="182" alt="" />
<span class="caption">Fig. 5.</span>
</div>

<p>Similarly, by the principle of least superficial area,
a freely suspended mass of oil assumes the shape of a
sphere. The sphere is the form of least surface for a
given content. This is evident. The more we put
into a travelling-bag, the nearer its shape approaches
the spherical form.</p>

<p>The connexion of the two above-mentioned paragraphs
with the principle of least superficial area may
be shown by a yet simpler example. Picture to yourselves
four fixed pulleys, <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, and two movable<span class="pagenum"><a name="Page_13" id="Page_13">[Pg 13]</a></span>
rings <i>f</i>, <i>g</i> (Fig. 5); about the pulleys and through the
rings imagine a smooth cord passed, fastened at one
extremity to a nail <i>e</i>, and loaded at the other with a
weight <i>h</i>. Now this weight always tends to sink, or,
what is the same thing, always tends to make the portion
of the string <i>e h</i> as long as possible, and consequently
the remainder of the string, wound round the
pulleys, as short as possible. The strings must remain
connected with the pulleys, and on account of the rings
also with each other. The conditions of the case, accordingly,
are similar to those of the liquid figures discussed.
The result also is a similar one. When, as
in the right hand figure of the cut, four pairs of strings
meet, a different configuration must be established.
The consequence of the endeavor of the string to
shorten itself is that the rings separate from each other,
and that now at all points only three pairs of strings
meet, every two at equal angles of one hundred and
twenty degrees. As a fact, by this arrangement the
greatest possible shortening of the string is attained;
as can be easily proved by geometry.</p>

<p>This will help us to some extent to understand the
creation of beautiful and complicated figures by the
simple tendency of liquids to assume surfaces of least
superficial area. But the question arises, <i>Why</i> do
liquids seek surfaces of least superficial area?</p>

<p>The particles of a liquid cling together. Drops
brought into contact coalesce. We can say, liquid
particles attract each other. If so, they seek to come<span class="pagenum"><a name="Page_14" id="Page_14">[Pg 14]</a></span>
as close as they can to each other. The particles at
the surface will endeavor to penetrate as far as they
can into the interior. This process will not stop, cannot
stop, until the surface has become as small as under
the circumstances it possibly can become, until as
few particles as possible remain at the surface, until
as many particles as possible have penetrated into the
interior, until the forces of attraction have no more
work to perform.<a name="FNanchor_3_3" id="FNanchor_3_3"></a><a href="#Footnote_3_3" class="fnanchor">[3]</a></p>

<p>The root of the principle of least surface is to be
sought, accordingly, in another and much simpler
principle, which may be illustrated by some such analogy
as this. We can <i>conceive</i> of the natural forces of
attraction and repulsion as purposes or intentions of
nature. As a matter of fact, that interior pressure
which we feel before an act and which we call an intention
or purpose, is not, in a final analysis, so essentially
different from the pressure of a stone on its support,
or the pressure of a magnet on another, that it is
necessarily unallowable to use for both the same term&mdash;at
least for well-defined purposes.<a name="FNanchor_4_4" id="FNanchor_4_4"></a><a href="#Footnote_4_4" class="fnanchor">[4]</a> It is the purpose
of nature, accordingly, to bring the iron nearer
the magnet, the stone nearer the centre of the earth,
and so forth. If such a purpose can be realised, it is
carried out. But where she cannot realise her purposes,
nature does nothing. In this respect she acts
exactly as a good man of business does.</p>
<p><span class="pagenum"><a name="Page_15" id="Page_15">[Pg 15]</a></span></p>
<p>It is a constant purpose of nature to bring weights
lower. We can raise a weight by causing another,
larger weight to sink; that is, by satisfying another,
more powerful, purpose of nature. If we fancy we
are making nature serve our purposes in this, it will
be found, upon closer examination, that the contrary
is true, and that nature has employed us to attain her
purposes.</p>

<p>Equilibrium, rest, exists only, but then always, when
nature is brought to a halt in her purposes, when the
forces of nature are as fully satisfied as, under the
circumstances, they can be. Thus, for example, heavy
bodies are in equilibrium, when their so-called centre
of gravity lies as low as it possibly can, or when as
much weight as the circumstances admit of has sunk
as low as it can.</p>

<p>The idea forcibly suggests itself that perhaps this
principle also holds good in other realms. Equilibrium
exists also in the state when the purposes of the parties
are as fully satisfied as for the time being they can
be, or, as we may say, jestingly, in the language of
physics, when the social potential is a maximum.<a name="FNanchor_5_5" id="FNanchor_5_5"></a><a href="#Footnote_5_5" class="fnanchor">[5]</a></p>

<p>You see, our miserly mercantile principle is replete
with consequences.<a name="FNanchor_6_6" id="FNanchor_6_6"></a><a href="#Footnote_6_6" class="fnanchor">[6]</a> The result of sober research, it
has become as fruitful for physics as the dry questions
of Socrates for science generally. If the principle
seems to lack in ideality, the more ideal are the fruits
which it bears.</p>
<p><span class="pagenum"><a name="Page_16" id="Page_16">[Pg 16]</a></span></p>
<p>But why, tell me, should science be ashamed of
such a principle? Is science<a name="FNanchor_7_7" id="FNanchor_7_7"></a><a href="#Footnote_7_7" class="fnanchor">[7]</a> itself anything more
than&mdash;a business? Is not its task to acquire with the
least possible work, in the least possible time, with the
least possible thought, the greatest possible part of
eternal truth?</p>
<p><span class="pagenum"><a name="Page_17" id="Page_17">[Pg 17]</a></span></p>



<h2><a name="THE_FIBRES_OF_CORTI" id="THE_FIBRES_OF_CORTI">THE FIBRES OF CORTI.</a></h2>


<p>Whoever has roamed through a beautiful country
knows that the tourist's delights increase
with his progress. How pretty that wooded dell must
look from yonder hill! Whither does that clear brook
flow, that hides itself in yonder sedge? If I only
knew how the landscape looked behind that mountain!
Thus even the child thinks in his first rambles. It is
also true of the natural philosopher.</p>

<p>The first questions are forced upon the attention of
the inquirer by practical considerations; the subsequent
ones are not. An irresistible attraction draws
him to these; a nobler interest which far transcends the
mere needs of life. Let us look at a special case.</p>

<p>For a long time the structure of the organ of hearing
has actively engaged the attention of anatomists.
A considerable number of brilliant discoveries has been
brought to light by their labors, and a splendid array
of facts and truths established. But with these facts
a host of new enigmas has been presented.</p>

<p>Whilst in the theory of the organisation and functions<span class="pagenum"><a name="Page_18" id="Page_18">[Pg 18]</a></span>
of the eye comparative clearness has been attained;
whilst, hand in hand with this, ophthalmology
has reached a degree of perfection which the preceding
century could hardly have dreamed of, and by the
help of the ophthalmoscope the observing physician
penetrates into the profoundest recesses of the eye,
the theory of the ear is still much shrouded in mysterious
darkness, full of attraction for the investigator.</p>

<p>Look at this model of the ear. Even at that familiar
part by whose extent we measure the quantity of
people's intelligence, even at the external ear, the
problems begin. You see here a succession of helixes
or spiral windings, at times very pretty, whose significance
we cannot accurately state, yet for which there
must certainly be some reason.</p>

<div class="figright" style="width: 300px;">
<img src="images/i_028.jpg" width="300" height="205" alt="" />
<span class="caption">Fig. 6.</span>
</div>

<p>The shell or concha of the ear, <i>a</i> in the annexed
diagram, conducts the sound into the curved auditory
passage <i>b</i>, which is terminated by a thin membrane,
the so-called tympanic membrane, <i>e</i>. This membrane
is set in motion by the sound, and in its turn sets in
motion a series of little bones of very peculiar formation,
<i>c</i>. At the end of all is the labyrinth
<i>d</i>. The labyrinth consists of a group of
cavities filled with a liquid, in which the
innumerable fibres of the nerve of hearing
are imbedded. By the vibration of the chain of
bones <i>c</i>, the liquid of the labyrinth is shaken, and the
auditory nerve excited. Here the process of hearing<span class="pagenum"><a name="Page_19" id="Page_19">[Pg 19]</a></span>
begins. So much is certain. But the details of the
process are one and all unanswered questions.</p>

<p>To these old puzzles, the Marchese Corti, as late
as 1851, added a new enigma. And, strange to say,
it is this last enigma, which, perhaps, has first received
its correct solution. This will be the subject of our
remarks to-day.</p>

<p>Corti found in the cochlea, or snail-shell of the
labyrinth, a large number of microscopic fibres placed
side by side in geometrically graduated order. According
to Kölliker their number is three thousand. They
were also the subject of investigation at the hands of
Max Schultze and Deiters.</p>

<p>A description of the details of this organ would
only weary you, besides not rendering the matter much
clearer. I prefer, therefore, to state briefly what in
the opinion of prominent investigators like Helmholtz
and Fechner is the peculiar function of Corti's fibres.
The cochlea, it seems, contains a large number of
elastic fibres of graduated lengths (Fig. 7), to which
the branches of the auditory nerve are
attached. These fibres, called the fibres,
pillars, or rods of Corti, being of unequal
length, must also be of unequal elasticity,
and, consequently, pitched to different
notes. The cochlea, therefore, is a species of pianoforte.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_029.jpg" width="300" height="290" alt="" />
<span class="caption">Fig. 7.</span>
</div>

<p>What, now, may be the office of this structure,
which is found in no other organ of sense? May it<span class="pagenum"><a name="Page_20" id="Page_20">[Pg 20]</a></span>
not be connected with some special property of the
ear? It is quite probable; for the ear possesses a very
similar power. You know that it is possible to follow
the individual voices of a symphony. Indeed, the
feat is possible even in a fugue of Bach, where it is certainly
no inconsiderable achievement. The ear can
pick out the single constituent tonal parts, not only of a
harmony, but of the wildest clash of music imaginable.
The musical ear analyses every agglomeration of tones.</p>

<p>The eye does not possess this ability. Who, for
example, could tell from the mere sight of white, without
a previous experimental knowledge of the fact,
that white is composed of a mixture of other colors?
Could it be, now, that these two facts, the property of
the ear just mentioned, and the structure discovered
by Corti, are really connected? It is very probable.
The enigma is solved if we assume that every note of
definite pitch has its special string in this pianoforte
of Corti, and, therefore, its special branch of the auditory
nerve attached to that string. But before I can
make this point perfectly plain to you, I must ask
you to follow me a few steps into the dry domain of
physics.</p>

<p>Look at this pendulum. Forced from its position
of equilibrium by an impulse, it begins to swing with a
definite time of oscillation, dependent upon its length.
Longer pendulums swing more slowly, shorter ones
more quickly. We will suppose our pendulum to execute
one to-and-fro movement in a second.</p><p><span class="pagenum"><a name="Page_21" id="Page_21">[Pg 21]</a></span></p>

<p>This pendulum, now, can be thrown into violent
vibration in two ways; either by a <i>single</i> heavy impulse,
or by a <i>number</i> of properly communicated slight
impulses. For example, we impart to the pendulum,
while at rest in its position of equilibrium, a very slight
impulse. It will execute a very small vibration. As
it passes a third time its position of equilibrium, a
second having elapsed, we impart to it again a slight
shock, in the same direction with the first. Again after
the lapse of a second, on its fifth passage through the
position of equilibrium, we strike it again in the same
manner; and so continue. You see, by this process
the shocks imparted augment continually the motion
of the pendulum. After each slight impulse, the pendulum
reaches out a little further in its swing, and
finally acquires a considerable motion.<a name="FNanchor_8_8" id="FNanchor_8_8"></a><a href="#Footnote_8_8" class="fnanchor">[8]</a></p>

<p>But this is not the case under all circumstances.
It is possible only when the impulses imparted synchronise
with the swings of the pendulum. If we
should communicate the second impulse at the end of
half a second and in the same direction with the first
impulse, its effects would counteract the motion of the
pendulum. It is easily seen that our little impulses
help the motion of the pendulum more and more, according
as their time accords with the time of the
pendulum. If we strike the pendulum in any other
time than in that of its vibration, in some instances, it
is true, we shall augment its vibration, but in others<span class="pagenum"><a name="Page_22" id="Page_22">[Pg 22]</a></span>
again, we shall obstruct it. Our impulses will be less
effective the more the motion of our own hand departs
from the motion of the pendulum.</p>

<p>What is true of the pendulum holds true of every
vibrating body. A tuning-fork when it sounds, also
vibrates. It vibrates more rapidly when its sound is
higher; more slowly when it is deeper. The standard
<i>A</i> of our musical scale is produced by about four hundred
and fifty vibrations in a second.</p>

<p>I place by the side of each other on this table two
tuning-forks, exactly alike, resting on resonant cases.
I strike the first one a sharp blow, so that it emits a
loud note, and immediately grasp it again with my
hand to quench its note. Nevertheless, you still hear
the note distinctly sounded, and by feeling it you may
convince yourselves that the other fork which was not
struck now vibrates.</p>

<p>I now attach a small bit of wax to one of the forks.
It is thrown thus out of tune; its note is made a little
deeper. I now repeat the same experiment with the
two forks, now of unequal pitch, by striking one of
them and again grasping it with my hand; but in the
present case the note ceases the very instant I touch
the fork.</p>

<p>What has happened here in these two experiments?
Simply this. The vibrating fork imparts to the air and
to the table four hundred and fifty shocks a second,
which are carried over to the other fork. If the other
fork is pitched to the same note, that is to say, if it<span class="pagenum"><a name="Page_23" id="Page_23">[Pg 23]</a></span>
vibrates when struck in the same time with the first,
then the shocks first emitted, no matter how slight they
may be, are sufficient to throw the second fork into rapid
sympathetic vibration. But when the time of vibration
of the two forks is slightly different, this does not
take place. We may strike as many forks as we will, the
fork tuned to <i>A</i> is perfectly indifferent to their notes;
is deaf, in fact, to all except its own; and if you strike
three, or four, or five, or any number whatsoever, of
forks all at the same time, so as to make the shocks
which come from them ever so great, the <i>A</i> fork will
not join in with their vibrations unless another fork <i>A</i>
is found in the collection struck. It picks out, in other
words, from all the notes sounded, that which accords
with it.</p>

<p>The same is true of all bodies which can yield
notes. Tumblers resound when a piano is played, on
the striking of certain notes, and so do window panes.
Nor is the phenomenon without analogy in other provinces.
Take a dog that answers to the name "Nero."
He lies under your table. You speak of Domitian,
Vespasian, and Marcus Aurelius Antoninus, you call
upon all the names of the Roman Emperors that occur
to you, but the dog does not stir, although a slight
tremor of his ear tells you of a faint response of his
consciousness. But the moment you call "Nero" he
jumps joyfully towards you. The tuning-fork is like
your dog. It answers to the name <i>A</i>.</p>

<p>You smile, ladies. You shake your heads. The<span class="pagenum"><a name="Page_24" id="Page_24">[Pg 24]</a></span>
simile does not catch your fancy. But I have another,
which is very near to you: and for punishment you shall
hear it. You, too, are like tuning-forks. Many are the
hearts that throb with ardor for you, of which you take
no notice, but are cold. Yet what does it profit you!
Soon the heart will come that beats in just the proper
rhythm, and then your knell, too, has struck. Then
your heart, too, will beat in unison, whether you will
or no.</p>

<p>The law of sympathetic vibration, here propounded
for sounding bodies, suffers some modification for
bodies incompetent to yield notes. Bodies of this
kind vibrate to almost every note. A high silk hat,
we know, will not sound; but if you will hold your
hat in your hand when attending your next concert you
will not only hear the pieces played, but also feel them
with your fingers. It is exactly so with men. People
who are themselves able to give tone to their surroundings,
bother little about the prattle of others. But the
person without character tarries everywhere: in the
temperance hall, and at the bar of the public-house&mdash;everywhere
where a committee is formed. The high
silk hat is among bells what the weakling is among
men of conviction.</p>

<p>A sonorous body, therefore, always sounds when
its special note, either alone or in company with others,
is struck. We may now go a step further. What will
be the behaviour of a group of sonorous bodies which
in the pitch of their notes form a scale? Let us picture<span class="pagenum"><a name="Page_25" id="Page_25">[Pg 25]</a></span>
to ourselves, for example (Fig. 8), a series of rods
or strings pitched to the notes <i>c d e f g</i>.... On a
musical instrument the accord <i>c e g</i> is struck. Every
one of the rods of Fig. 8 will see if its special note is
contained in the accord, and if it finds
it, it will respond. The rod <i>c</i> will give
at once the note <i>c</i>, the rod <i>e</i> the note <i>e</i>,
the rod <i>g</i> the note <i>g</i>. All the other
rods will remain at rest, will not sound.</p>

<div class="figright" style="width: 300px;">
<img src="images/i_035.jpg" width="300" height="337" alt="" />
<span class="caption">Fig. 8.</span>
</div>

<p>We need not look about us long
for such an instrument. Every piano
is an instrument of this kind, with which the experiment
mentioned may be executed with splendid success.
Two pianos stand here by the side of each other,
both tuned alike. We will employ the first for exciting
the notes, while we will allow the second to respond;
after having first pressed upon the loud pedal,
so as to render all the strings capable of motion.</p>

<p>Every harmony struck with vigor on the first piano
is distinctly repeated on the second. To prove that
it is the same strings that are sounded in both pianos,
we repeat the experiment in a slightly changed form.
We let go the loud pedal of the second piano and
pressing on the keys <i>c e g</i> of that instrument vigorously
strike the harmony <i>c e g</i> on the first piano. The harmony
<i>c e g</i> is now also sounded on the second piano.
But if we press only on one key <i>g</i> of one piano, while
we strike <i>c e g</i> on the other, only <i>g</i> will be sounded on<span class="pagenum"><a name="Page_26" id="Page_26">[Pg 26]</a></span>
the second. It is thus always the like strings of the
two pianos that excite each other.</p>

<p>The piano can reproduce any sound that is composed
of its musical notes. It will reproduce, for example,
very distinctly, a vowel sound that is sung into
it. And in truth physics has proved that the vowels
may be regarded as composed of simple musical
notes.</p>

<p>You see that by the exciting of definite tones in the
air quite definite motions are set up with mechanical
necessity in the piano. The idea might be made use
of for the performance of some pretty pieces of wizardry.
Imagine a box in which is a stretched string
of definite pitch. This is thrown into motion as often
as its note is sung or whistled. Now it would not be
a very difficult task for a skilful mechanic to so construct
the box that the vibrating cord would close a
galvanic circuit and open the lock. And it would not
be a much more difficult task to construct a box which
would open at the whistling of a certain melody. Sesame!
and the bolts fall. Truly, we should have here
a veritable puzzle-lock. Still another fragment rescued
from that old kingdom of fables, of which our day
has realised so much, that world of fairy-stories to
which the latest contributions are Casselli's telegraph,
by which one can write at a distance in one's own hand,
and Prof. Elisha Gray's telautograph. What would
the good old Herodotus have said to these things who
even in Egypt shook his head at much that he saw?<span class="pagenum"><a name="Page_27" id="Page_27">[Pg 27]</a></span>
&#7952;&#956;&#959;&#7985; &#956;&#7953;&#957;e &#959;&#973; &#960;&#953;&#963;&#964;&#945;, just as simple-heartedly as then,
when he heard of the circumnavigation of Africa.</p>

<p>A new puzzle-lock! But why invent one? Are
not we human beings ourselves puzzle-locks? Think
of the stupendous groups of thoughts, feelings, and
emotions that can be aroused in us by a word! Are
there not moments in all our lives when a mere name
drives the blood to our hearts? Who that has attended
a large mass-meeting has not experienced what
tremendous quantities of energy and motion can be
evolved by the innocent words, "Liberty, Equality,
Fraternity."</p>

<p>But let us return to the subject proper of our discourse.
Let us look again at our piano, or what will
do just as well, at some other contrivance of the same
character. What does this instrument do? Plainly,
it decomposes, it analyses every agglomeration of
sounds set up in the air into its individual component
parts, each tone being taken up by a different string;
it performs a real spectral analysis of sound. A person
completely deaf, with the help of a piano, simply by
touching the strings or examining their vibrations with
a microscope, might investigate the sonorous motion of
the air, and pick out the separate tones excited in it.</p>

<p>The ear has the same capacity as this piano. The
ear performs for the mind what the piano performs for
a person who is deaf. The mind without the ear is
deaf. But a deaf person, with the piano, does hear
after a fashion, though much less vividly, and more<span class="pagenum"><a name="Page_28" id="Page_28">[Pg 28]</a></span>
clumsily, than with the ear. The ear, thus, also decomposes
sound into its component tonal parts. I shall
now not be deceived, I think, if I assume that you
already have a presentiment of what the function of
Corti's fibres is. We can make the matter very plain to
ourselves. We will use the one piano for exciting the
sounds, and we shall imagine the second one in the
ear of the observer in the place of Corti's fibres, which
is a model of such an instrument. To every string of
the piano in the ear we will suppose a special fibre of
the auditory nerve attached, so that this fibre and this
alone, is irritated when the string is thrown into vibration.
If we strike now an accord on the external
piano, for every tone of that accord a definite string of
the internal piano will sound and as many different
nervous fibres will be irritated as there are notes in
the accord. The simultaneous sense-impressions due
to different notes can thus be preserved unmingled and
be separated by the attention. It is the same as with
the five fingers of the hand. With each finger I can
touch something different. Now the ear has three thousand
such fingers, and each one is designed for the
touching of a different tone.<a name="FNanchor_9_9" id="FNanchor_9_9"></a><a href="#Footnote_9_9" class="fnanchor">[9]</a> Our ear is a puzzle-lock
of the kind mentioned. It opens at the magic melody
of a sound. But it is a stupendously ingenious lock.
Not only one tone, but every tone makes it open; but
each one differently. To each tone it replies with a
different sensation.</p>
<p><span class="pagenum"><a name="Page_29" id="Page_29">[Pg 29]</a></span></p>
<p>More than once it has happened in the history of
science that a phenomenon predicted by theory, has
not been brought within the range of actual observation
until long afterwards. Leverrier predicted the
existence and the place of the planet Neptune, but it
was not until sometime later that Galle actually found
the planet at the predicted spot. Hamilton unfolded
theoretically the phenomenon of the so-called conical
refraction of light, but it was reserved for Lloyd some
time subsequently to observe the fact. The fortunes
of Helmholtz's theory of Corti's fibres have been somewhat
similar. This theory, too, received its substantial
confirmation from the subsequent observations of
V. Hensen. On the free surface of the bodies of Crustacea,
connected with the auditory nerves, rows of little
hairy filaments of varying lengths and thicknesses
are found, which to some extent are the analogues of
Corti's fibres. Hensen saw these hairs vibrate when
sounds were excited, and when different notes were
struck different hairs were set in vibration.</p>

<p>I have compared the work of the physical inquirer
to the journey of the tourist. When the tourist ascends
a new hill he obtains of the whole district a
different view. When the inquirer has found the solution
of one enigma, the solution of a host of others
falls into his hands.</p>

<p>Surely you have often felt the strange impression experienced<span class="pagenum"><a name="Page_30" id="Page_30">[Pg 30]</a></span>
when in singing through the scale the octave
is reached, and nearly the same sensation is produced
as by the fundamental tone. The phenomenon finds its
explanation in the view here laid down of the ear. And
not only this phenomenon but all the laws of the theory
of harmony may be grasped and verified from this
point of view with a clearness before undreamt of.
Unfortunately, I must content myself to-day with the
simple indication of these beautiful prospects. Their
consideration would lead us too far aside into the fields
of other sciences.</p>

<p>The searcher of nature, too, must restrain himself
in his path. He also is drawn along from one beauty
to another as the tourist from dale to dale, and as circumstances
generally draw men from one condition of
life into others. It is not he so much that makes the
quests, as that the quests are made of him. Yet let
him profit by his time, and let not his glance rove aimlessly
hither and thither. For soon the evening sun
will shine, and ere he has caught a full glimpse of the
wonders close by, a mighty hand will seize him and
lead him away into a different world of puzzles.</p>

<p>Respected hearers, science once stood in an entirely
different relation to poetry. The old Hindu
mathematicians wrote their theorems in verses, and
lotus-flowers, roses, and lilies, beautiful sceneries,
lakes, and mountains figured in their problems.</p>

<p>"Thou goest forth on this lake in a boat. A lily
juts forth, one palm above the water. A breeze bends<span class="pagenum"><a name="Page_31" id="Page_31">[Pg 31]</a></span>
it downwards, and it vanishes two palms from its previous
spot beneath the surface. Quick, mathematician,
tell me how deep is the lake!"</p>

<p>Thus spoke an ancient Hindu scholar. This poetry,
and rightly, has disappeared from science, but from
its dry leaves another poetry is wafted aloft which cannot
be described to him who has never felt it. Whoever
will fully enjoy this poetry must put his hand to
the plough, must himself investigate. Therefore,
enough of this! I shall reckon myself fortunate if you
do not repent of this brief excursion into the flowered
dale of physiology, and if you take with yourselves the
belief that we can say of science what we say of poetry,</p>

<div class="poem"><div class="stanza">
<span class="i0">"Who the song would understand,<br /></span>
<span class="i0">Needs must seek the song's own land;<br /></span>
<span class="i0">Who the minstrel understand<br /></span>
<span class="i0">Needs must seek the minstrel's land."<br /></span>
<span class="pagenum"><a name="Page_32" id="Page_32">[Pg 32]</a></span></div></div>




<h2><a name="ON_THE_CAUSES_OF_HARMONY" id="ON_THE_CAUSES_OF_HARMONY">ON THE CAUSES OF HARMONY.</a></h2>


<p>We are to speak to-day of a theme which is perhaps
of somewhat more general interest&mdash;<i>the causes of
the harmony of musical sounds</i>. The first and simplest
experiences relative to harmony are very ancient. Not
so the explanation of its laws. These were first supplied
by the investigators of a recent epoch. Allow me
an historical retrospect.</p>

<p>Pythagoras (586 B. C.) knew that the note yielded
by a string of steady tension was converted into its
octave when the length of the string was reduced one-half,
and into its fifth when reduced two-thirds; and
that then the first fundamental tone was consonant
with the two others. He knew generally that the same
string under fixed tension gives consonant tones when
successively divided into lengths that are in the proportions
of the simplest natural numbers; that is, in
the proportions of 1:2, 2:3, 3:4, 4:5.</p>

<p>Pythagoras failed to reveal the causes of these laws.
What have consonant tones to do with the simple natural
numbers? That is the question we should ask<span class="pagenum"><a name="Page_33" id="Page_33">[Pg 33]</a></span>
to-day. But this circumstance must have appeared
less strange than inexplicable to Pythagoras. This
philosopher sought for the causes of harmony in the
occult, miraculous powers of numbers. His procedure
was largely the cause of the upgrowth of a numerical
mysticism, of which the traces may still be detected in
our oneirocritical books and among some scientists, to
whom marvels are more attractive than lucidity.</p>

<p>Euclid (300 B. C.) gives a definition of consonance
and dissonance that could hardly be improved upon,
in point of verbal accuracy. The consonance (&#963;&#965;&#956;&#966;&#969;&#957;&#8055;&#945;)
of two tones, he says, is the mixture, the
blending (&#954;&#961;&#8118;&#963;&#953;&#962;) of those two tones; dissonance
(&#948;&#953;&#945;&#966;&#969;&#957;&#8055;&#945;), on the other hand, is the incapacity of
the tones to blend (&#7936;&#956;&#953;&#958;&#8055;&#945;), whereby they are made
harsh for the ear. The person who knows the correct
explanation of the phenomenon hears it, so to speak,
reverberated in these words of Euclid. Still, Euclid
did not know the true cause of harmony. He had unwittingly
come very near to the truth, but without
really grasping it.</p>

<p>Leibnitz (1646-1716 A. D.) resumed the question
which his predecessors had left unsolved. He, of
course, knew that musical notes were produced by vibrations,
that twice as many vibrations corresponded
to the octave as to the fundamental tone, etc. A passionate
lover of mathematics, he sought for the cause
of harmony in the secret computation and comparison
of the simple numbers of vibrations and in the secret<span class="pagenum"><a name="Page_34" id="Page_34">[Pg 34]</a></span>
satisfaction of the soul at this occupation. But how,
we ask, if one does not know that musical notes are
vibrations? The computation and the satisfaction at
the computation must indeed be pretty secret if it is
unknown. What queer ideas philosophers have! Could
anything more wearisome be imagined than computation
as a principle of æsthetics? Yes, you are not
utterly wrong in your conjecture, yet you may be sure
that Leibnitz's theory is not wholly nonsense, although
it is difficult to make out precisely what he meant by
his secret computation.</p>

<p>The great Euler (1707-1783) sought the cause of
harmony, almost as Leibnitz did, in the pleasure which
the soul derives from the contemplation of order in the
numbers of the vibrations.<a name="FNanchor_10_10" id="FNanchor_10_10"></a><a href="#Footnote_10_10" class="fnanchor">[10]</a></p>

<p>Rameau and D'Alembert (1717-1783) approached
nearer to the truth. They knew that in every sound
available in music besides the fundamental note also
the twelfth and the next higher third could be heard;
and further that the resemblance between a fundamental
tone and its octave was always strongly marked.
Accordingly, the combination of the octave, fifth, third,
etc., with the fundamental tone appeared to them "natural."
They possessed, we must admit, the correct
point of view; but with the simple naturalness of a
phenomenon no inquirer can rest content; for it is precisely
this naturalness for which he seeks his explanations.</p>
<p><span class="pagenum"><a name="Page_35" id="Page_35">[Pg 35]</a></span></p>
<p>Rameau's remark dragged along through the whole
modern period, but without leading to the full discovery
of the truth. Marx places it at the head of his
theory of composition, but makes no further application
of it. Also Goethe and Zelter in their correspondence
were, so to speak, on the brink of the truth.
Zelter knew of Rameau's view. Finally, you will be
appalled at the difficulty of the problem, when I tell
you that till very recent times even professors of physics
were dumb when asked what were the causes of
harmony.</p>

<p>Not till quite recently did Helmholtz find the solution
of the question. But to make this solution clear
to you I must first speak of some experimental principles
of physics and psychology.</p>

<p>1) In every process of perception, in every observation,
the attention plays a highly important part.
We need not look about us long for proofs of this.
You receive, for example, a letter written in a very
poor hand. Do your best, you cannot make it out.
You put together now these, now those lines, yet you
cannot construct from them a single intelligible character.
Not until you direct your attention to groups
of lines which really belong together, is the reading of
the letter possible. Manuscripts, the letters of which
are formed of minute figures and scrolls, can only be
read at a considerable distance, where the attention is<span class="pagenum"><a name="Page_36" id="Page_36">[Pg 36]</a></span>
no longer diverted from the significant outlines to the
details. A beautiful example of this class is furnished
by the famous iconographs of Giuseppe Arcimboldo in
the basement of the Belvedere gallery at Vienna. These
are symbolic representations of water, fire, etc.: human
heads composed of aquatic animals and of combustibles.
At a short distance one sees only the details,
at a greater distance only the whole figure. Yet
a point can be easily found at which, by a simple voluntary
movement of the attention, there is no difficulty
in seeing now the whole figure and now the smaller
forms of which it is composed. A picture is often seen
representing the tomb of Napoleon. The tomb is surrounded
by dark trees between which the bright heavens
are visible as background. One can look a long time
at this picture without noticing anything except the
trees, but suddenly, on the attention being accidentally
directed to the bright background, one sees
the figure of Napoleon between the trees. This case
shows us very distinctly the important part which attention
plays. The same sensuous object can, solely
by the interposition of attention, give rise to wholly
different perceptions.</p>

<p>If I strike a harmony, or chord, on this piano, by
a mere effort of attention you can fix every tone of
that harmony. You then hear most distinctly the
fixed tone, and all the rest appear as a mere addition,
altering only the quality, or acoustic color, of the primary
tone. The effect of the same harmony is essentially<span class="pagenum"><a name="Page_37" id="Page_37">[Pg 37]</a></span>
modified if we direct our attention to different
tones.</p>

<p>Strike in succession two harmonies, for example,
the two represented in the annexed diagram, and first
fix by the attention the upper note <i>e</i>, afterwards the
base <i>e</i>-<i>a</i>; in the two cases you will hear the same
sequence of harmonies differently.
In the first case, you have the impression
as if the fixed tone remained
unchanged and simply altered
its <i>timbre</i>; in the second case,
the whole acoustic agglomeration
seems to fall sensibly in depth.
There is an art of composition to guide the attention
of the hearer. But there is also an art of hearing,
which is not the gift of every person.</p>

<div class="figright" style="width: 200px;">
<img src="images/i_047.jpg" width="200" height="178" alt="" />
<span class="caption">Fig. 9.</span>
</div>

<p>The piano-player knows the remarkable effects obtained
when one of the keys of a chord that is struck
is let loose. Bar 1 played on the piano sounds almost
like bar 2. The note which lies next to the key let
loose resounds after its release as if it were freshly
struck. The attention no longer occupied with the
upper note is by that very fact insensibly led to the
upper note.</p>

<div class="figcenter" style="width: 500px;">
<img src="images/i_047-1.jpg" width="500" height="175" alt="" />
<span class="caption">Fig. 10.</span>
</div>
<p><span class="pagenum"><a name="Page_38" id="Page_38">[Pg 38]</a></span></p>

<p>Any tolerably cultivated musical ear can perform
the resolution of a harmony into its component parts.
By much practice we can go even further. Then,
every musical sound heretofore regarded as simple
can be resolved into a subordinate succession
of musical tones. For example,
if I strike on the piano the note 1, (annexed
diagram,) we shall hear, if we
make the requisite effort of attention,
besides the loud fundamental note the
feebler, higher overtones, or harmonics,
2 ... 7, that is, the octave, the twelfth, the double
octave, and the third, the fifth, and the seventh of
the double octave.</p>

<div class="figright" style="width: 200px;">
<img src="images/i_048.jpg" width="200" height="237" alt="" />
<span class="caption">Fig. 11.</span>
</div>

<p>The same is true of every musically available
sound. Each yields, with varying degrees of intensity,
besides its fundamental note, also the octave, the
twelfth, the double octave, etc. The phenomenon is
observable with special facility on the open and closed
flue-pipes of organs. According, now, as certain overtones
are more or less distinctly emphasised in a
sound, the <i>timbre</i> of the sound changes&mdash;that peculiar
quality of the sound by which we distinguish the music
of the piano from that of the violin, the clarinet, etc.</p>

<p>On the piano these overtones can be very easily
rendered audible. If I strike, for example, sharply
note 1 of the foregoing series, whilst I simply press
down upon, one after another, the keys 2, 3, ... 7,
the notes 2, 3, ... 7 will continue to sound after the<span class="pagenum"><a name="Page_39" id="Page_39">[Pg 39]</a></span>
striking of 1, because the strings corresponding to
these notes, now freed from their dampers, are thrown
into sympathetic vibration.</p>

<p>As you know, this sympathetic vibration of the like-pitched
strings with the overtones is really not to be
conceived as sympathy, but rather as lifeless mechanical
necessity. We must not think of this sympathetic
vibration as an ingenious journalist pictured it, who
tells a gruesome story of Beethoven's F minor sonata,
Op. 2, that I cannot withhold from you. "At the
last London Industrial Exhibition nineteen virtuosos
played the F minor sonata on the same piano. When
the twentieth stepped up to the instrument to play by
way of variation the same production, to the terror of
all present the piano began to render the sonata of its
own accord. The Archbishop of Canterbury, who
happened to be present, was set to work and forthwith
expelled the F minor devil."</p>

<p>Although, now, the overtones or harmonics which
we have discussed are heard only upon a special effort
of the attention, nevertheless they play a highly important
part in the formation of musical <i>timbre</i>, as also
in the production of the consonance and dissonance of
sounds. This may strike you as singular. How can
a thing which is heard only under exceptional circumstances
be of importance generally for audition?</p>

<p>But consider some familiar incidents of your every-day
life. Think of how many things you see which
you do not notice, which never strike your attention<span class="pagenum"><a name="Page_40" id="Page_40">[Pg 40]</a></span>
until they are missing. A friend calls upon you; you
cannot understand why he looks so changed. Not
until you make a close examination do you discover
that his hair has been cut. It is not difficult to tell
the publisher of a work from its letter-press, and yet
no one can state precisely the points by which this
style of type is so strikingly different from that style.
I have often recognised a book which I was in search
of from a simple piece of unprinted white paper that
peeped out from underneath the heap of books covering
it, and yet I had never carefully examined the
paper, nor could I have stated its difference from other
papers.</p>

<p>What we must remember, therefore, is that every
sound that is musically available yields, besides its
fundamental note, its octave, its twelfth, its double
octave, etc., as overtones or harmonics, and that these
are important for the agreeable combination of several
musical sounds.</p>

<p>2) One other fact still remains to be dealt with.
Look at this tuning-fork. It yields, when struck, a perfectly
smooth tone. But if you strike in company with
it a second fork which is of slightly different pitch, and
which alone also gives a perfectly smooth tone, you
will hear, if you set both forks on the table, or hold
both before your ear, a uniform tone no longer, but a
number of shocks of tones. The rapidity of the shocks
increases with the difference of the pitch of the forks.
These shocks, which become very disagreeable for the<span class="pagenum"><a name="Page_41" id="Page_41">[Pg 41]</a></span>
ear when they amount to thirty-three in a second, are
called "beats."</p>

<p>Always, when one of two like musical sounds is
thrown out of unison with the other, beats arise. Their
number increases with the divergence from unison, and
simultaneously they grow more unpleasant. Their
roughness reaches its maximum at about thirty-three
beats in a second. On a still further departure from
unison, and a consequent increase of the number of
beats, the unpleasant effect is diminished, so that tones
which are widely apart in pitch no longer produce
offensive beats.</p>

<p>To give yourselves a clear idea of the production
of beats, take two metronomes and set them almost
alike. You can, for that matter, set the two exactly
alike. You need not fear that they will strike alike.
The metronomes usually for sale in the shops are poor
enough to yield, when set alike, appreciably unequal
strokes. Set, now, these two metronomes, which strike
at unequal intervals, in motion; you will readily see
that their strokes alternately coincide and conflict with
each other. The alternation is quicker the greater the
difference of time of the two metronomes.</p>

<p>If metronomes are not to be had, the experiment
may be performed with two watches.</p>

<p>Beats arise in the same way. The rhythmical
shocks of two sounding bodies, of unequal pitch, sometimes
coincide, sometimes interfere, whereby they alternately<span class="pagenum"><a name="Page_42" id="Page_42">[Pg 42]</a></span>
augment and enfeeble each other's effects.
Hence the shock-like, unpleasant swelling of the tone.</p>

<p>Now that we have made ourselves acquainted with
overtones and beats, we may proceed to the answer of
our main question, Why do certain relations of pitch
produce pleasant sounds, consonances, others unpleasant
sounds, dissonances? It will be readily seen that
all the unpleasant effects of simultaneous sound-combinations
are the result of beats produced by those
combinations. Beats are the only sin, the sole evil of
music. Consonance is the coalescence of sounds without
appreciable beats.</p>

<div class="figright" style="width: 182px;">
<img src="images/i_052.jpg" width="182" height="800" alt="" />
<span class="caption">Fig. 12.</span>
</div>

<p>To make this perfectly clear to you I have constructed
the model which you see in Fig. 12. It represents
a claviatur. At its top a movable strip of wood
<i>aa</i> with the marks 1, 2 ... 6 is placed. By setting
this strip in any position, for example, in that where the
mark 1 is over the note <i>c</i> of the claviatur, the marks
2, 3 ... 6, as you see, stand over the overtones of <i>c</i>.
The same happens when the strip is placed in any
other position. A second, exactly similar strip, <i>bb</i>,
possesses the same properties. Thus, together, the
two strips, in any two positions, point out by their<span class="pagenum"><a name="Page_43" id="Page_43">[Pg 43]</a></span>
marks all the tones brought into play upon the simultaneous
sounding of the notes indicated by the marks 1.</p>

<p>The two strips, placed over the same fundamental
note, show that also all the overtones of those notes
coincide. The first note is simply intensified by the
other. The single overtones of a sound lie too far apart
to permit appreciable beats. The second sound supplies
nothing new, consequently, also, no new beats.
Unison is the most perfect consonance.</p>

<p>Moving one of the two strips along the other is
equivalent to a departure from unison. All the overtones
of the one sound now fall alongside those of the
other; beats are at once produced; the combination
of the tones becomes unpleasant: we obtain a dissonance.
If we move the strip further and further along,
we shall find that as a general rule the overtones always
fall alongside each other, that is, always produce
beats and dissonances. Only in a few quite definite
positions do the overtones partially coincide. Such
positions, therefore, signify higher degrees of euphony&mdash;they
point out <i>the consonant intervals</i>.</p>

<p>These consonant intervals can be readily found experimentally
by cutting Fig. 12 out of paper and moving
<i>bb</i> lengthwise along <i>aa</i>. The most perfect consonances
are the octave and the twelfth, since in these two cases
the overtones of the one sound coincide absolutely
with those of the other. In the octave, for example,
1<i>b</i> falls on 2<i>a</i>, 2<i>b</i> on 4<i>a</i>, 3<i>b</i> on 6<i>a</i>. Consonances,
therefore, are simultaneous sound-combinations not<span class="pagenum"><a name="Page_44" id="Page_44">[Pg 44]</a></span>
accompanied by disagreeable beats. This, by the way,
is, expressed in English, what Euclid said in Greek.</p>

<p>Only such sounds are consonant as possess in common
some portion of their partial tones. Plainly we
must recognise between such sounds, also when struck
one after another, a certain affinity. For the second
sound, by virtue of the common overtones, will produce
partly the same sensation as the first. The octave is
the most striking exemplification of this. When we
reach the octave in the ascent of the scale we actually
fancy we hear the fundamental tone repeated. The
foundations of harmony, therefore, are the foundations
of melody.</p>

<p>Consonance is the coalescence of sounds without
appreciable beats! This principle is competent to introduce
wonderful order and logic into the doctrines
of the fundamental bass. The compendiums of the
theory of harmony which (Heaven be witness!) have
stood hitherto little behind the cook-books in subtlety
of logic, are rendered extraordinarily clear and simple.
And what is more, all that the great masters, such as
Palestrina, Mozart, Beethoven, unconsciously got
right, and of which heretofore no text-book could render
just account, receives from the preceding principle
its perfect verification.</p>

<p>But the beauty of the theory is, that it bears upon
its face the stamp of truth. It is no phantom of the
brain. Every musician can hear for himself the beats
which the overtones of his musical sounds produce.<span class="pagenum"><a name="Page_45" id="Page_45">[Pg 45]</a></span>
Every musician can satisfy himself that for any given
case the number and the harshness of the beats can
be calculated beforehand, and that they occur in exactly
the measure that theory determines.</p>

<p>This is the answer which Helmholtz gave to the
question of Pythagoras, so far as it can be explained
with the means now at my command. A long period
of time lies between the raising and the solving of this
question. More than once were eminent inquirers
nearer to the answer than they dreamed of.</p>

<p>The inquirer seeks the truth. I do not know if the
truth seeks the inquirer. But were that so, then the
history of science would vividly remind us of that
classical rendezvous, so often immortalised by painters
and poets. A high garden wall. At the right a
youth, at the left a maiden. The youth sighs, the
maiden sighs! Both wait. Neither dreams how near
the other is.</p>

<p>I like this simile. Truth suffers herself to be
courted, but she has evidently no desire to be won.
She flirts at times disgracefully. Above all, she is determined
to be merited, and has naught but contempt
for the man who will win her too quickly. And if,
forsooth, one breaks his head in his efforts of conquest,
what matter is it, another will come, and truth is always
young. At times, indeed, it really seems as if
she were well disposed towards her admirer, but that
admitted&mdash;never! Only when Truth is in exceptionally
good spirits does she bestow upon her wooer a glance<span class="pagenum"><a name="Page_46" id="Page_46">[Pg 46]</a></span>
of encouragement. For, thinks Truth, if I do not do
something, in the end the fellow will not seek me at all.</p>

<p>This one fragment of truth, then, we have, and it
shall never escape us. But when I reflect what it has
cost in labor and in the lives of thinking men, how it
painfully groped its way through centuries, a half-matured
thought, before it became complete; when I
reflect that it is the toil of more than two thousand
years that speaks out of this unobtrusive model of
mine, then, without dissimulation, I almost repent me
of the jest I have made.</p>

<p>And think of how much we still lack! When, several
thousand years hence, boots, top-hats, hoops, pianos,
and bass-viols are dug out of the earth, out of the
newest alluvium as fossils of the nineteenth century;
when the scientists of that time shall pursue their
studies both upon these wonderful structures and upon
our modern Broadways, as we to-day make studies of
the implements of the stone age and of the prehistoric
lake-dwellings&mdash;then, too, perhaps, people will be unable
to comprehend how we could come so near to
many great truths without grasping them. And thus
it is for all time the unsolved dissonance, for all time
the troublesome seventh, that everywhere resounds in
our ears; we feel, perhaps, that it will find its solution,
but we shall never live to see the day of the pure
triple accord, nor shall our remotest descendants.</p>

<p>Ladies, if it is the sweet purpose of your life to
sow confusion, it is the purpose of mine to be clear;<span class="pagenum"><a name="Page_47" id="Page_47">[Pg 47]</a></span>
and so I must confess to you a slight transgression
that I have been guilty of. On one point I have told
you an untruth. But you will pardon me this falsehood,
if in full repentance I make it good. The model
represented in Fig. 12 does not tell the whole truth, for
it is based upon the so-called "even temperament"
system of tuning. The overtones, however, of musical
sounds are not tempered, but purely tuned. By means
of this slight inexactness the model is made considerably
simpler. In this form it is fully adequate for
ordinary purposes, and no one who makes use of it in
his studies need be in fear of appreciable error.</p>

<p>If you should demand of me, however, the full
truth, I could give you that only by the help of a mathematical
formula. I should have to take the chalk into
my hands and&mdash;think of it!&mdash;reckon in your presence.
This you might take amiss. Nor shall it happen.
I have resolved to do no more reckoning for to-day.
I shall reckon now only upon your forbearance, and
this you will surely not gainsay me when you reflect
that I have made only a limited use of my privilege to
weary you. I could have taken up much more of
your time, and may, therefore, justly close with Lessing's
epigram:</p>

<div class="poem"><div class="stanza">
<span class="i0">"If thou hast found in all these pages naught that's worth the thanks,<br /></span>
<span class="i0">At least have gratitude for what I've spared thee."<br /></span>
<span class="pagenum"><a name="Page_48" id="Page_48">[Pg 48]</a></span></div></div>




<h2><a name="THE_VELOCITY_OF_LIGHT" id="THE_VELOCITY_OF_LIGHT">THE VELOCITY OF LIGHT.</a></h2>


<p>When a criminal judge has a right crafty knave
before him, one well versed in the arts of prevarication,
his main object is to wring a confession from
the culprit by a few skilful questions. In almost a similar
position the natural philosopher seems to be placed
with respect to nature. True, his functions here are
more those of the spy than the judge; but his object
remains pretty much the same. Her hidden motives
and laws of action is what nature must be made to
confess. Whether a confession will be extracted depends
upon the shrewdness of the inquirer. Not without
reason, therefore, did Lord Bacon call the experimental
method a questioning of nature. The
art consists in so putting our questions that they
may not remain unanswered without a breach of etiquette.</p>

<p>Look, too, at the countless tools, engines, and instruments
of torture with which man conducts his
inquisitions of nature, and which mock the poet's
words:</p><p><span class="pagenum"><a name="Page_49" id="Page_49">[Pg 49]</a></span></p>

<div class="poem"><div class="stanza">
<span class="i0">"Mysterious even in open day,<br /></span>
<span class="i0">Nature retains her veil, despite our clamors;<br /></span>
<span class="i0">That which she doth not willingly display<br /></span>
<span class="i0">Cannot be wrenched from her with levers, screws, and hammers."<br /></span>
</div></div>

<p>Look at these instruments and you will see that the
comparison with torture also is admissible.<a name="FNanchor_11_11" id="FNanchor_11_11"></a><a href="#Footnote_11_11" class="fnanchor">[11]</a></p>

<p>This view of nature, as of something designedly
concealed from man, that can be unveiled only by
force or dishonesty, chimed in better with the conceptions
of the ancients than with modern notions. A
Grecian philosopher once said, in offering his opinion
of the natural science of his time, that it could only be
displeasing to the gods to see men endeavoring to spy
out what the gods were not minded to reveal to them.<a name="FNanchor_12_12" id="FNanchor_12_12"></a><a href="#Footnote_12_12" class="fnanchor">[12]</a>
Of course all the contemporaries of the speaker were
not of his opinion.</p>

<p>Traces of this view may still be found to-day, but
upon the whole we are now not so narrow-minded.
We believe no longer that nature designedly hides
herself. We know now from the history of science
that our questions are sometimes meaningless, and
that, therefore, no answer can be forthcoming. Soon
we shall see how man, with all his thoughts and quests,
is only a fragment of nature's life.</p>

<p><span class="pagenum"><a name="Page_50" id="Page_50">[Pg 50]</a></span></p><p>Picture, then, as your fancy dictates, the tools of
the physicist as instruments of torture or as engines of
endearment, at all events a chapter from the history of
those implements will be of interest to you, and it will
not be unpleasant to learn what were the peculiar difficulties
that led to the invention of such strange apparatus.</p>

<p>Galileo (born at Pisa in 1564, died at Arcetri in
1642) was the first who asked what was the velocity
of light, that is, what time it would take for a light
struck at one place to become visible at another, a
certain distance away.<a name="FNanchor_13_13" id="FNanchor_13_13"></a><a href="#Footnote_13_13" class="fnanchor">[13]</a></p>

<p>The method which Galileo devised was as simple
as it was natural. Two practised observers, with
muffled lanterns, were to take up positions in a dark
night at a considerable distance
from each other, one at
<i>A</i> and one at <i>B</i>. At a moment
previously fixed upon, <i>A</i> was
instructed to unmask his lantern; while as soon as <i>B</i>
saw the light of <i>A</i>'s lantern he was to unmask his.
Now it is clear that the time which <i>A</i> counted from
the uncovering of his lantern until he caught sight of
the light of <i>B</i>'s would be the time which it would take
light to travel from <i>A</i> to <i>B</i> and from <i>B</i> back to <i>A</i>.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_060.jpg" width="300" height="93" alt="" />
<span class="caption">Fig. 13.</span>
</div>

<p>The experiment was not executed, nor could it, in
the nature of the case, have been a success. As we<span class="pagenum"><a name="Page_51" id="Page_51">[Pg 51]</a></span>
now know, light travels too rapidly to be thus noted.
The time elapsing between the arrival of the light at
<i>B</i> and its perception by the observer, with that between
the decision to uncover and the uncovering of
the lantern, is, as we now know, incomparably greater
than the time which it takes light to travel the greatest
earthly distances. The great velocity of light will be
made apparent, if we reflect that a flash of lightning
in the night illuminates instantaneously a very extensive
region, whilst the single reflected claps of thunder
arrive at the observer's ear very gradually and in appreciable
succession.</p>

<p>During his life, then, the efforts of Galileo to determine
the velocity of light remained uncrowned with
success. But the subsequent history of the measurement
of the velocity of light is intimately associated
with his name, for with the telescope which he constructed
he discovered the four satellites of Jupiter,
and these furnished the next occasion for the determination
of the velocity of light.</p>

<p>The terrestrial spaces were too small for Galileo's
experiment. The measurement was first executed
when the spaces of the planetary system were employed.
Olaf Römer, (born at Aarhuus in 1644, died
at Copenhagen in 1710) accomplished the feat (1675-1676),
while watching with Cassini at the observatory
of Paris the revolutions of Jupiter's moons.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_062.jpg" width="600" height="373" alt="" />
<span class="caption">Fig. 14.</span>
</div>

<p>Let <i>AB</i> (Fig. 14) be Jupiter's orbit. Let <i>S</i> stand
for the sun, <i>E</i> for the earth, <i>J</i> for Jupiter, and <i>T</i> for<span class="pagenum"><a name="Page_52" id="Page_52">[Pg 52]</a></span>
Jupiter's first satellite. When the earth is at <i>E</i><sub>1</sub> we
see the satellite enter regularly into Jupiter's shadow,
and by watching the time between two successive
eclipses, can calculate its time of revolution. The
time which Römer noted was forty-two hours, twenty-eight
minutes, and thirty-five seconds. Now, as the
earth passes along in its orbit towards E<sub>2</sub>, the revolutions
of the satellite grow apparently longer and longer:
the eclipses take place later and later. The greatest
retardation of the eclipse, which occurs when the earth
is at <i>E</i><sub>2</sub>, amounts to sixteen minutes and twenty-six
seconds. As the earth passes back again to <i>E</i><sub>1</sub>, the
revolutions grow apparently shorter, and they occur
in exactly the time that they first did when the earth
arrives at <i>E</i><sub>1</sub>. It is to be remarked that Jupiter changes
only very slightly its position during one revolution of
the earth. Römer guessed at once that these periodical
changes of the time of revolution of Jupiter's satellite<span class="pagenum"><a name="Page_53" id="Page_53">[Pg 53]</a></span>
were not actual, but apparent changes, which were
in some way connected with the velocity of light.</p>

<p>Let us make this matter clear to ourselves by a simile.
We receive regularly by the post, news of the
political status at our capital. However far away we
may be from the capital, we hear the news of every
event, later it is true, but of all equally late. The
events reach us in the same succession of time as that
in which they took place. But if we are travelling
away from the capital, every successive post will have
a greater distance to pass over, and the events will
reach us more slowly than they took place. The reverse
will be the case if we are approaching the capital.</p>

<p>At rest, we hear a piece of music played in the
same <i>tempo</i> at all distances. But the <i>tempo</i> will be
seemingly accelerated if we are carried
rapidly towards the band, or retarded if
we are carried rapidly away from it.<a name="FNanchor_14_14" id="FNanchor_14_14"></a><a href="#Footnote_14_14" class="fnanchor">[14]</a></p>

<div class="figleft" style="width: 150px;">
<img src="images/i_063.jpg" width="150" height="143" alt="" />
<span class="caption">Fig. 15.</span>
</div>

<p>Picture to yourself a cross, say the
sails of a wind-mill (Fig. 15), in uniform
rotation about its centre. Clearly, the rotation of the
cross will appear to you more slowly executed if you
are carried very rapidly away from it. For the post
which in this case conveys to you the light and brings
to you the news of the successive positions of the cross
will have to travel in each successive instant over a
longer path.</p>
<p><span class="pagenum"><a name="Page_54" id="Page_54">[Pg 54]</a></span></p>
<p>Now this must also be the case with the rotation
(the revolution) of the satellite of Jupiter. The greatest
retardation of the eclipse (16-1/2 minutes), due to
the passage of the earth from <i>E</i><sub>1</sub> to <i>E</i><sub>2</sub>, or to its removal
from Jupiter by a distance equal to the diameter
of the orbit of the earth, plainly corresponds to the
time which it takes light to traverse a distance equal to
the diameter of the earth's orbit. The velocity of light,
that is, the distance described by light in a second, as
determined by this calculation, is 311,000 kilometres,<a name="FNanchor_15_15" id="FNanchor_15_15"></a><a href="#Footnote_15_15" class="fnanchor">[15]</a>
or 193,000 miles. A subsequent correction of the diameter
of the earth's orbit, gives, by the same method,
the velocity of light as approximately 186,000 miles a
second.</p>

<p>The method is exactly that of Galileo; only better
conditions are selected. Instead of a short terrestrial
distance we have the diameter of the earth's orbit,
three hundred and seven million kilometres; in place
of the uncovered and covered lanterns we have the
satellite of Jupiter, which alternately appears and disappears.
Galileo, therefore, although he could not
carry out himself the proposed measurement, found
the lantern by which it was ultimately executed.</p>

<p>Physicists did not long remain satisfied with this
beautiful discovery. They sought after easier methods
of measuring the velocity of light, such as might
be performed on the earth. This was possible after the
difficulties of the problem were clearly exposed. A
measurement of the kind referred to was executed in
1849 by Fizeau (born at Paris in 1819).</p>
<p><span class="pagenum"><a name="Page_55" id="Page_55">[Pg 55]</a></span></p>
<p>I shall endeavor to make the principle of Fizeau's
apparatus clear to you. Let <i>s</i> (Fig. 16) be a disk free
to rotate about its centre, and perforated at its rim
with a series of holes. Let <i>l</i> be a luminous point
casting its light on an unsilvered glass, <i>a</i>, inclined at
an angle of forty-five degrees to the axis of the disk.
The ray of light, reflected at this point, passes through
one of the holes of the disk and falls at right angles
upon a mirror <i>b</i>, erected at a point about five miles
distant. From the mirror <i>b</i> the light is again reflected,
passes once more through the hole in <i>s</i>, and, penetrating
the glass plate, finally strikes the eye, <i>o</i>, of the observer.
The eye, <i>o</i>, thus, sees the image of the luminous
point <i>l</i> through the glass plate and the hole of
the disk in the mirror <i>b</i>.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_065.jpg" width="600" height="226" alt="" />
<span class="caption">Fig. 16.</span>
</div>

<p>If, now, the disk be set in rotation, the unpierced
spaces between the apertures will alternately take the
place of the apertures, and the eye o will now see the
image of the luminous point in <i>b</i> only at interrupted
intervals. On increasing the rapidity of the rotation,<span class="pagenum"><a name="Page_56" id="Page_56">[Pg 56]</a></span>
however, the interruptions for the eye become again
unnoticeable, and the eye sees the mirror <i>b</i> uniformly
illuminated.</p>

<p>But all this holds true only for relatively small velocities
of the disk, when the light sent through an
aperture in <i>s</i> to <i>b</i> on its return strikes the aperture at
almost the same place and passes through it a second
time. Conceive, now, the speed of the disk to be so increased
that the light on its return finds before it an
unpierced space instead of an aperture, it will then no
longer be able to reach the eye. We then see the
mirror <i>b</i> only when no light is emitted from it, but
only when light is sent to it; it is covered when light
comes from it. In this case, accordingly, the mirror
will always appear dark.</p>

<p>If the velocity of rotation at this point were still
further increased, the light sent through one aperture
could not, of course, on its return pass through the
same aperture but might strike the next and reach
the eye by that. Hence, by constantly increasing the
velocity of the rotation, the mirror <i>b</i> may be made to
appear alternately bright and dark. Plainly, now, if
we know the number of apertures of the disk, the number
of rotations per second, and the distance <i>sb</i>, we
can calculate the velocity of light. The result agrees
with that obtained by Römer.</p>

<p>The experiment is not quite as simple as my exposition
might lead you to believe. Care must be
taken that the light shall travel back and forth over<span class="pagenum"><a name="Page_57" id="Page_57">[Pg 57]</a></span>
the miles of distance <i>sb</i> and <i>bs</i> undispersed. This
difficulty is obviated by means of telescopes.</p>

<p>If we examine Fizeau's apparatus closely, we shall
recognise in it an old acquaintance: the arrangement
of Galileo's experiment. The luminous point <i>l</i> is the
lantern <i>A</i>, while the rotation of the perforated disk performs
mechanically the uncovering and covering of the
lantern. Instead of the unskilful observer <i>B</i> we have
the mirror <i>b</i>, which is unfailingly illuminated the instant
the light arrives from <i>s</i>. The disk <i>s</i>, by alternately
transmitting and intercepting the reflected light, assists
the observer <i>o</i>. Galileo's experiment is here executed,
so to speak, countless times in a second, yet the total
result admits of actual observation. If I might be
pardoned the use of a phrase of Darwin's in this field,
I should say that Fizeau's apparatus was the descendant
of Galileo's lantern.</p>

<p>A still more refined and delicate method for the
measurement of the velocity of light was employed by
Foucault, but a description of it here would lead us
too far from our subject.</p>

<p>The measurement of the velocity of sound is easily
executed by the method of Galileo. It was unnecessary,
therefore, for physicists to rack their brains further
about the matter; but the idea which with light
grew out of necessity was applied also in this field.
Koenig of Paris constructs an apparatus for the measurement
of the velocity of sound which is closely allied
to the method of Fizeau.</p><p><span class="pagenum"><a name="Page_58" id="Page_58">[Pg 58]</a></span></p>

<p>The apparatus is very simple. It consists of two
electrical clock-works which strike simultaneously,
with perfect precision, tenths of seconds. If we place
the two clock-works directly side by side, we hear
their strokes simultaneously, wherever we stand. But
if we take our stand by the side of one of the works
and place the other at some distance from us, in general
a coincidence of the strokes will now not be heard.
The companion strokes of the remote clock-work arrive,
as sound, later. The first stroke of the remote
work is heard, for example, immediately after the first
of the adjacent work, and so on. But by increasing
the distance we may produce again a coincidence of the
strokes. For example, the first stroke of the remote
work coincides with the second of the near work, the
second of the remote work with the third of the near
work, and so on. If, now, the works strike tenths of
seconds and the distance between them is increased
until the first coincidence is noted, plainly that distance
is travelled over by the sound in a tenth of a
second.</p>

<p>We meet frequently the phenomenon here presented,
that a thought which centuries of slow and
painful endeavor are necessary to produce, when once
developed, fairly thrives. It spreads and runs everywhere,
even entering minds in which it could never
have arisen. It simply cannot be eradicated.</p>

<p>The determination of the velocity of light is not the
only case in which the direct perception of the senses<span class="pagenum"><a name="Page_59" id="Page_59">[Pg 59]</a></span>
is too slow and clumsy for use. The usual method
of studying events too fleet for direct observation consists
in putting into reciprocal action with them other
events already known, the velocities of all of which
are capable of comparison. The result is
usually unmistakable, and susceptible of
direct inference respecting the character of
the event which is unknown. The velocity
of electricity cannot be determined by direct
observation. But it was ascertained
by Wheatstone, simply by the expedient of
watching an electric spark in a mirror rotating with
tremendous known velocity.</p>

<div class="figleft" style="width: 150px;">
<img src="images/i_069.jpg" width="150" height="238" alt="" />
<span class="caption">Fig. 17.</span>
</div>

<div class="figright" style="width: 200px;">
<img src="images/i_069-1.jpg" width="200" height="189" alt="" />
<span class="caption">Fig. 18.</span>
</div>

<p>If we wave a staff irregularly hither and thither,
simple observation cannot determine how quickly it
moves at each point of its course. But let us look at
the staff through holes in the rim of a
rapidly rotating disk (Fig. 17). We
shall then see the moving staff only
in certain positions, namely, when a
hole passes in front of the eye. The
single pictures of the staff remain for a
time impressed upon the eye; we think we see several
staffs, having some such disposition as that represented
in Fig. 18. If, now, the holes of the disk are equally
far apart, and the disk is rotated with uniform velocity,
we see clearly that the staff has moved slowly
from <i>a</i> to <i>b</i>, more quickly from <i>b</i> to <i>c</i>, still more quickly
from <i>c</i> to <i>d</i>, and with its greatest velocity from <i>d</i> to <i>e</i>.</p><p><span class="pagenum"><a name="Page_60" id="Page_60">[Pg 60]</a></span></p>

<p>A jet of water flowing from an orifice in the bottom
of a vessel has the appearance of perfect quiet and
uniformity, but if we illuminate it for a second, in a
dark room, by means of an electric flash we shall see
that the jet is composed of separate drops. By their
quick descent the images of the drops are
obliterated and the jet appears uniform.
Let us look at the jet through the rotating
disk. The disk is supposed to be rotated so
rapidly that while the second aperture
passes into the place of the first, drop 1
falls into the place of 2, 2 into the place of 3, and so on.
We see drops then always in the same places. The
jet appears to be at rest. If we turn the disk a trifle
more slowly, then while the second aperture passes
into the place of the first, drop 1 will have fallen somewhat
lower than 2, 2 somewhat lower than 3, etc.
Through every successive aperture we shall see drops
in successively lower positions. The jet will appear to
be flowing slowly downwards.</p>

<div class="figright" style="width: 150px;">
<img src="images/i_070.jpg" width="150" height="184" alt="" />
<span class="caption">Fig. 19.</span>
</div>

<p>Now let us turn the disk more rapidly. Then while
the second aperture is passing into the place of the
first, drop 1 will not quite have reached the place of 2,
but will be found slightly above 2, 2 slightly above 3,
etc. Through the successive apertures we shall see
the drops at successively higher places. It will now
look as if the jet were flowing upwards, as if the drops
were rising from the lower vessel into the higher.</p>

<p>You see, physics grows gradually more and more<span class="pagenum"><a name="Page_61" id="Page_61">[Pg 61]</a></span>
terrible. The physicist will soon have it in his power
to play the part of the famous lobster chained to the
bottom of the Lake of Mohrin, whose direful mission,
if ever liberated, the poet Kopisch humorously describes
as that of a reversal of all the events of the
world; the rafters of houses become trees again, cows
calves, honey flowers, chickens eggs, and the poet's
own poem flows back into his inkstand.</p>
<p><span class="pagenum"><a name="Page_62" id="Page_62">[Pg 62]</a></span></p><p><span class="pagenum"><a name="Page_63" id="Page_63">[Pg 63]</a></span></p><p><span class="pagenum"><a name="Page_64" id="Page_64">[Pg 64]</a></span></p>
<hr class="tb" />

<p>You will now allow me the privilege of a few general
remarks. You have seen that the same principle
often lies at the basis of large classes of apparatus
designed for different purposes. Frequently it is some
very unobtrusive idea which is productive of so much
fruit and of such extensive transformations in physical
technics. It is not otherwise here than in practical
life.</p>

<p>The wheel of a waggon appears to us a very simple
and insignificant creation. But its inventor was certainly
a man of genius. The round trunk of a tree
perhaps first accidentally led to the observation of the
ease with which a load can be moved on a roller.
Now, the step from a simple supporting roller to a
fixed roller, or wheel, appears a very easy one. At
least it appears very easy to us who are accustomed
from childhood up to the action of the wheel. But if
we put ourselves vividly into the position of a man
who never saw a wheel, but had to invent one, we shall
begin to have some idea of its difficulties. Indeed, it
is even doubtful whether a single man could have accomplished
this feat, whether perhaps centuries were
not necessary to form the first wheel from the primitive
roller.<a name="FNanchor_16_16" id="FNanchor_16_16"></a><a href="#Footnote_16_16" class="fnanchor">[16]</a></p>

<p>History does not name the progressive minds who
constructed the first wheel; their time lies far back of
the historic period. No scientific academy crowned
their efforts, no society of engineers elected them
honorary members. They still live only in the stupendous
results which they called forth. Take from
us the wheel, and little will remain of the arts and industries
of modern life. All disappears. From the
spinning-wheel to the spinning-mill, from the turning-lathe
to the rolling-mill, from the wheelbarrow to the
railway train, all vanishes.</p>

<p>In science the wheel is equally important. Whirling
machines, as the simplest means of obtaining quick
motions with inconsiderable changes of place, play a
part in all branches of physics. You know Wheatstone's
rotating mirror, Fizeau's wheel, Plateau's perforated
rotating disks, etc. Almost the same principle
lies at the basis of all these apparatus. They differ
from one another no more than the pen-knife differs,
in the purposes it serves, from the knife of the anatomist
or the knife of the vine-dresser. Almost the same
might be said of the screw.</p>

<p>It will now perhaps be clear to you that new
thoughts do not spring up suddenly. Thoughts need
their time to ripen, grow, and develop in, like every
natural product; for man, with his thoughts, is also a
part of nature.</p>

<p>Slowly, gradually, and laboriously one thought is
transformed into a different thought, as in all likelihood
one animal species is gradually transformed into new
species. Many ideas arise simultaneously. They fight
the battle for existence not otherwise than do the
Ichthyosaurus, the Brahman, and the horse.</p>

<p>A few remain to spread rapidly over all fields of
knowledge, to be redeveloped, to be again split up, to
begin again the struggle from the start. As many
animal species long since conquered, the relicts of
ages past, still live in remote regions where their enemies
cannot reach them, so also we find conquered
ideas still living on in the minds of many men. Whoever
will look carefully into his own soul will acknowledge
that thoughts battle as obstinately for existence
as animals. Who will gainsay that many vanquished
modes of thought still haunt obscure crannies of his
brain, too faint-hearted to step out into the clear light
of reason? What inquirer does not know that the
hardest battle, in the transformation of his ideas, is
fought with himself.</p>

<p>Similar phenomena meet the natural inquirer in all
paths and in the most trifling matters. The true inquirer
seeks the truth everywhere, in his country-walks
and on the streets of the great city. If he is
not too learned, he will observe that certain things,
like ladies' hats, are constantly subject to change. I
have not pursued special studies on this subject, but
as long as I can remember, one form has always
gradually changed into another. First, they wore hats
with long projecting rims, within which, scarcely accessible
with a telescope, lay concealed the face of the
beautiful wearer. The rim grew smaller and smaller;
the bonnet shrank to the irony of a hat. Now a tremendous
superstructure is beginning to grow up in its
place, and the gods only know what its limits will be.
It is not otherwise with ladies' hats than with butterflies,
whose multiplicity of form often simply comes
from a slight excrescence on the wing of one species
developing in a cognate species to a tremendous fold.
Nature, too, has its fashions, but they last thousands
of years. I could elucidate this idea by many additional
examples; for instance, by the history of the
evolution of the coat, if I were not fearful that my
gossip might prove irksome to you.</p>
<p><span class="pagenum"><a name="Page_65" id="Page_65">[Pg 65]</a></span></p><p><span class="pagenum"><a name="Page_66" id="Page_66">[Pg 66]</a></span></p>
<hr class="tb" />

<p>We have now wandered through an odd corner of
the history of science. What have we learned? The
solution of a small, I might almost say insignificant,
problem&mdash;the measurement of the velocity of light.
And more than two centuries have worked at its solution!
Three of the most eminent natural philosophers,
Galileo, an Italian, Römer, a Dane, and Fizeau, a
Frenchman, have fairly shared its labors. And so it
is with countless other questions. When we contemplate
thus the many blossoms of thought that must
wither and fall before one shall bloom, then shall we
first truly appreciate Christ's weighty but little consolatory
words: "Many be called but few are chosen."</p>

<p>Such is the testimony of every page of history.
But is history right? Are really only those chosen
whom she names? Have those lived and battled in
vain, who have won no prize?</p>

<p>I doubt it. And so will every one who has felt the
pangs of sleepless nights spent in thought, at first fruitless,
but in the end successful. No thought in such
struggles was thought in vain; each one, even the most
insignificant, nay, even the erroneous thought, that
which apparently was the least productive, served to
prepare the way for those that afterwards bore fruit.
And as in the thought of the individual naught is in
vain, so, also, it is in that of humanity.</p>

<p>Galileo wished to measure the velocity of light.
He had to close his eyes before his wish was realised.
But he at least found the lantern by which his successor
could accomplish the task.</p>

<p>And so I may maintain that we all, so far as inclination
goes, are working at the civilisation of the future.
If only we all strive for the right, then are we <i>all</i>
called and <i>all</i> chosen!</p>




<h2><a name="WHY_HAS_MAN_TWO_EYES" id="WHY_HAS_MAN_TWO_EYES">WHY HAS MAN TWO EYES?</a></h2>


<p>Why has man two eyes? That the pretty symmetry
of his face may not be disturbed, the
artist answers. That his second eye may furnish a
substitute for his first if that be lost, says the far-sighted
economist. That we may weep with two eyes
at the sins of the world, replies the religious enthusiast.</p>

<p>Odd opinions! Yet if you should approach a modern
scientist with this question you might consider
yourself fortunate if you escaped with less than a rebuff.
"Pardon me, madam, or my dear sir," he would
say, with stern expression, "man fulfils no purpose in
the possession of his eyes; nature is not a person, and
consequently not so vulgar as to pursue purposes of
any kind."</p>

<p>Still an unsatisfactory answer! I once knew a professor
who would shut with horror the mouths of his
pupils if they put to him such an unscientific question.</p>

<p>But ask a more tolerant person, ask me. I, I candidly
confess, do not know exactly why man has two<span class="pagenum"><a name="Page_67" id="Page_67">[Pg 67]</a></span>
eyes, but the reason partly is, I think, that I may see
you here before me to-night and talk with you upon
this delightful subject.</p>

<p>Again you smile incredulously. Now this is one of
those questions that a hundred wise men together
could not answer. You have heard, so far, only five of
these wise men. You will certainly want to be spared
the opinions of the other ninety-five. To the first you
will reply that we should look just as pretty if we were
born with only one eye, like the Cyclops; to the second
we should be much better off, according to his
principle, if we had four or eight eyes, and that in this
respect we are vastly inferior to spiders; to the third,
that you are not just in the mood to weep; to the
fourth, that the unqualified interdiction of the question
excites rather than satisfies your curiosity; while of
me you will dispose by saying that my pleasure is not
as intense as I think, and certainly not great enough
to justify the existence of a double eye in man since
the fall of Adam.</p>

<p>But since you are not satisfied with my brief and
obvious answer, you have only yourselves to blame
for the consequences. You must now listen to a longer
and more learned explanation, such as it is in my
power to give.</p>

<p>As the church of science, however, debars the question
"Why?" let us put the matter in a purely orthodox
way: Man has two eyes, what <i>more</i> can he see with
two than with one?</p><p><span class="pagenum"><a name="Page_68" id="Page_68">[Pg 68]</a></span></p>

<p>I will invite you to take a walk with me? We see
before us a wood. What is it that makes this real
wood contrast so favorably with a painted wood, no
matter how perfect the painting may be? What makes
the one so much more lovely than the other? Is it the
vividness of the coloring, the distribution of the lights
and the shadows? I think
not. On the contrary, it
seems to me that in this
respect painting can accomplish
very much.</p>

<p>The cunning hand of
the painter can conjure up
with a few strokes of his
brush forms of wonderful
plasticity. By the help of
other means even more
can be attained. Photographs
of reliefs are so
plastic that we often imagine
we can actually lay
hold of the elevations and
depressions.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_078.jpg" width="300" height="496" alt="" />
<span class="caption">Fig. 20.</span>
</div>

<p>But one thing the painter never can give with the
vividness that nature does&mdash;the difference of near and
far. In the real woods you see plainly that you can
lay hold of some trees, but that others are inaccessibly
far. The picture of the painter is rigid. The picture
of the real woods changes on the slightest movement.<span class="pagenum"><a name="Page_69" id="Page_69">[Pg 69]</a></span>
Now this branch is hidden behind that; now that behind
this. The trees are alternately visible and invisible.</p>

<p>Let us look at this matter a little more closely.
For convenience sake we shall remain upon the highway,
I, II. (Fig. 20.) To the right and the left lies the
forest. Standing at I, we see, let us say, three trees
(1, 2, 3) in a line, so that the two remote ones are
covered by the nearest. Moving further along, this
changes. At II we shall not have to look round so far
to see the remotest tree 3 as to see the nearer tree 2,
nor so far to see this as to see 1. <i>Hence, as we move
onward, objects that are near to us seem to lag behind as
compared with objects that are remote from us, the lagging
increasing with the proximity of the objects.</i> Very remote
objects, towards which we must always look in the
same direction as we proceed, appear to travel along
with us.</p>

<p>If we should see, therefore, jutting above the brow
of yonder hill the tops of two trees whose distance
from us we were in doubt about, we should have in
our hands a very easy means of deciding the question.
We should take a few steps forward, say to the right,
and the tree-top which receded most to the left would
be the one nearer to us. In truth, from the amount
of the recession a geometer could actually determine
the distance of the trees from us without ever going
near them. It is simply the scientific development of<span class="pagenum"><a name="Page_70" id="Page_70">[Pg 70]</a></span>
this perception that enables us to measure the distances
of the stars.</p>

<p><i>Hence, from change of view in forward motion the
distances of objects in our field of vision can be measured.</i></p>

<p>Rigorously, however, even forward motion is not
necessary. For every observer is composed really of
<i>two</i> observers. Man has <i>two</i> eyes. The right eye is
a short step ahead of the left eye in the right-hand direction.
Hence, the two eyes receive <i>different</i> pictures
of the same woods. The right eye will see the
near trees displaced to the left, and the left eye will
see them displaced to the right, the displacement being
greater, the greater the proximity. This difference is
sufficient for forming ideas of distance.</p>

<p>We may now readily convince ourselves of the following
facts:</p>

<p>1. With one eye, the other being shut, you have a
very uncertain judgment of distances. You will find
it, for example, no easy task, with one eye shut, to
thrust a stick through a ring hung up before you; you
will miss the ring in almost every instance.</p>

<p>2. You see the same object differently with the
right eye from what you do with the left.</p>

<p>Place a lamp-shade on the table in front of you
with its broad opening turned downwards, and look
at it from above. (Fig. 21.) You will see with your
right eye the image 2, with your left eye the image 1.
Again, place the shade with its wide opening turned
upwards; you will receive with your right eye the image<span class="pagenum"><a name="Page_71" id="Page_71">[Pg 71]</a></span>
4, with your left eye the image 3. Euclid mentions
phenomena of this character.</p>

<p>3. Finally, you know that it is easy to judge of
distances with both eyes. Accordingly your judgment
must spring in some way from a co-operation of the
two eyes. In the preceding example the openings in
the different images received by the two eyes seem
displaced with respect to one another, and this displacement
is sufficient for the inference that the one
opening is nearer than the other.</p>

<div class="figcenter" style="width: 450px;">
<img src="images/i_081.jpg" width="450" height="446" alt="" />
<span class="caption">Fig. 21.</span>
</div>

<p>I have no doubt that you, ladies, have frequently
received delicate compliments upon your eyes, but I<span class="pagenum"><a name="Page_72" id="Page_72">[Pg 72]</a></span>
feel sure that no one has ever told you, and I know not
whether it will flatter you, that you have in your eyes,
be they blue or black, little geometricians. You say
you know nothing of them? Well, for that matter,
neither do I. But the facts are as I tell you.</p>

<p>You understand little of geometry? I shall accept
that confession. Yet with the help of your two eyes
you judge of distances? Surely that is a geometrical
problem. And what is more, you know the solution
of this problem: for you estimate distances correctly.
If, then, <i>you</i> do not solve the problem, the little geometricians
in your eyes must do it clandestinely and whisper
the solution to you. I doubt not they are fleet little
fellows.</p>

<p>What amazes me most here is, that you know nothing
about these little geometricians. But perhaps they
also know nothing about you. Perhaps they are models
of punctuality, routine clerks who bother about
nothing but their fixed work. In that case we may
be able to deceive the gentlemen.</p>

<p>If we present to our right eye an image which looks
exactly like the lamp-shade for the right eye, and to
our left eye an image which looks exactly like a lamp-shade
for the left eye, we shall imagine that we see
the whole lamp-shade bodily before us.</p>

<p>You know the experiment. If you are practised in
squinting, you can perform it directly with the figure,
looking with your right eye at the right image, and
with your left eye at the left image. In this way the<span class="pagenum"><a name="Page_73" id="Page_73">[Pg 73]</a></span>
experiment was first performed by Elliott. Improved
and perfected, its form is Wheatstone's stereoscope,
made so popular and useful by Brewster.</p>

<p>By taking two photographs of the same object from
two different points, corresponding to the two eyes, a
very clear three-dimensional picture of distant places
or buildings can be produced by the stereoscope.</p>

<p>But the stereoscope accomplishes still more than
this. It can visualise things for us which we never see
with equal clearness in real objects. You know that
if you move much while your photograph is being
taken, your picture will come out like that of a Hindu
deity, with several heads or several arms, which, at
the spaces where they overlap, show forth with equal
distinctness, so that we seem to see the one picture
<i>through</i> the other. If a person moves quickly away
from the camera before the impression is completed,
the objects behind him will also be imprinted upon
the photograph; the person will look transparent.
Photographic ghosts are made in this way.</p>

<p>Some very useful applications may be made of this
discovery. For example, if we photograph a machine
stereoscopically, successively removing during the
operation the single parts (where of course the impression
suffers interruptions), we obtain a transparent
view, endowed with all the marks of spatial solidity,
in which is distinctly visualised the interaction of parts
normally concealed. I have employed this method for<span class="pagenum"><a name="Page_74" id="Page_74">[Pg 74]</a></span>
obtaining transparent stereoscopic views of anatomical
structures.</p>

<p>You see, photography is making stupendous advances,
and there is great danger that in time some
malicious artist will photograph his innocent patrons
with solid views of their most secret thoughts and
emotions. How tranquil politics will then be! What
rich harvests our detective force will reap!</p>
<p><span class="pagenum"><a name="Page_75" id="Page_75">[Pg 75]</a></span><span class="pagenum"><a name="Page_76" id="Page_76">[Pg 76]</a></span><span class="pagenum"><a name="Page_77" id="Page_77">[Pg 77]</a></span><span class="pagenum"><a name="Page_78" id="Page_78">[Pg 78]</a></span><span class="pagenum"><a name="Page_79" id="Page_79">[Pg 79]</a></span><span class="pagenum"><a name="Page_80" id="Page_80">[Pg 80]</a></span><span class="pagenum"><a name="Page_81" id="Page_81">[Pg 81]</a></span><span class="pagenum"><a name="Page_82" id="Page_82">[Pg 82]</a></span><span class="pagenum"><a name="Page_83" id="Page_83">[Pg 83]</a></span><span class="pagenum"><a name="Page_84" id="Page_84">[Pg 84]</a></span><span class="pagenum"><a name="Page_85" id="Page_85">[Pg 85]</a></span><span class="pagenum"><a name="Page_86" id="Page_86">[Pg 86]</a></span></p>
<hr class="tb" />

<p>By the joint action of the two eyes, therefore, we
arrive at our judgments of distances, as also of the
forms of bodies.</p>

<p>Permit me to mention here a few additional facts
connected with this subject, which will assist us in the
comprehension of certain phenomena in the history of
civilisation.</p>

<p>You have often heard, and know from personal experience,
that remote objects appear perspectively
dwarfed. In fact, it is easy to satisfy yourself that
you can cover the image of a man a few feet away
from you simply by holding up your finger a short distance
in front of your eye. Still, as a general rule,
you do not notice this shrinkage of objects. On the
contrary, you imagine you see a man at the end of a
large hall, as large as you see him near by you. For
your eye, in its measurement of the distances, makes
remote objects correspondingly larger. The eye, so to
speak, is aware of this perspective contraction and is
not deceived by it, although its possessor is unconscious
of the fact. All persons who have attempted to draw
from nature have vividly felt the difficulty which this
superior dexterity of the eye causes the perspective
conception. Not until one's judgment of distances is
made uncertain, by their size, or from lack of points
of reference, or from being too quickly changed, is the
perspective rendered very prominent.</p>

<p>On sweeping round a curve on a rapidly moving
railway train, where a wide prospect is suddenly
opened up, the men upon distant hills appear like
dolls.<a name="FNanchor_17_17" id="FNanchor_17_17"></a><a href="#Footnote_17_17" class="fnanchor">[17]</a> You have at the moment, here, no known
references for the measurement of distances. The
stones at the entrance of a tunnel grow visibly larger
as we ride towards it; they shrink visibly in size as we
ride from it.</p>

<p>Usually both eyes work together. As certain views
are frequently repeated, and lead always to substantially
the same judgments of distances, the eyes in
time must acquire a special skill in geometrical constructions.
In the end, undoubtedly, this skill is so
increased that a single eye alone is often tempted to
exercise that office.</p>

<p>Permit me to elucidate this point by an example.
Is any sight more familiar to you than that of a vista
down a long street? Who has not looked with hopeful
eyes time and again into a street and measured its
depth. I will take you now into an art-gallery where
I will suppose you to see a picture representing a vista
into a street. The artist has not spared his rulers to
get his perspective perfect. The geometrician in your
left eye thinks, "Ah ha! I have computed that case a
hundred times or more. I know it by heart. It is a
vista into a street," he continues; "where the houses
are lower is the remote end." The geometrician in
the right eye, too much at his ease to question his
possibly peevish comrade in the matter, answers the
same. But the sense of duty of these punctual little
fellows is at once rearoused. They set to work at their
calculations and immediately find that all the points
of the picture are equally distant from them, that is,
lie all upon a plane surface.</p>

<p>What opinion will you now accept, the first or the
second? If you accept the first you will see distinctly
the vista. If you accept the second you will see nothing
but a painted sheet of distorted images.</p>

<p>It seems to you a trifling matter to look at a picture
and understand its perspective. Yet centuries
elapsed before humanity came fully to appreciate this
trifle, and even the majority of you first learned it from
education.</p>

<p>I can remember very distinctly that at three years
of age all perspective drawings appeared to me as
gross caricatures of objects. I could not understand
why artists made tables so broad at one end and so
narrow at the other. Real tables seemed to me just
as broad at one end as at the other, because my eye
made and interpreted its calculations without my intervention.
But that the picture of the table on the
plane surface was not to be conceived as a plane painted
surface but stood for a table and so was to be imaged
with all the attributes of extension was a joke that I
did not understand. But I have the consolation that
whole nations have not understood it.</p>

<p>Ingenuous people there are who take the mock
murders of the stage for real murders, the dissembled
actions of the players for real actions, and who can
scarcely restrain themselves, when the characters of the
play are sorely pressed, from running in deep indignation
to their assistance. Others, again, can never forget
that the beautiful landscapes of the stage are
painted, that Richard III. is only the actor, Mr. Booth,
whom they have met time and again at the clubs.</p>

<p>Both points of view are equally mistaken. To look
at a drama or a picture properly one must understand
that both are <i>shows</i>, simply <i>denoting</i> something real.
A certain preponderance of the intellectual life over
the sensuous life is requisite for such an achievement,
where the intellectual elements are safe from destruction
by the direct sensuous impressions. A certain
liberty in choosing one's point of view is necessary, a
sort of humor, I might say, which is strongly wanting
in children and in childlike peoples.</p>

<p>Let us look at a few historical facts. I shall not
take you as far back as the stone age, although we
possess sketches from this epoch which show very original
ideas of perspective. But let us begin our sight-seeing
in the tombs and ruined temples of ancient
Egypt, where the numberless reliefs and gorgeous colorings
have defied the ravages of thousands of years.</p>

<p>A rich and motley life is here opened to us. We
find the Egyptians represented in all conditions of life.
What at once strikes our attention in these pictures
is the delicacy of their technical execution. The contours
are extremely exact and distinct. But on the
other hand only a few bright colors are found, unblended
and without trace of transition. Shadows are
totally wanting. The paint is laid on the surfaces in
equal thicknesses.</p>

<p>Shocking for the modern eye is the perspective.
All the figures are equally large, with the exception of
the king, whose form is unduly exaggerated. Near and
far appear equally large. Perspective contraction is
nowhere employed. A pond with water-fowl is represented
flat, as if its surface were vertical.</p>

<p>Human figures are portrayed as they are never
seen, the legs from the side, the face in profile. The
breast lies in its full breadth across the plane of representation.
The heads of cattle appear in profile,
while the horns lie in the plane of the drawing. The
principle which the Egyptians followed might be best
expressed by saying that their figures are pressed in
the plane of the drawing as plants are pressed in a
herbarium.</p>

<p>The matter is simply explained. If the Egyptians
were accustomed to looking at things ingenuously
with both eyes at once, the construction of perspective
pictures in space could not be familiar to them.
They saw all arms, all legs on real men in their natural
lengths. The figures pressed into the planes resembled
more closely, of course, in their eyes the
originals than perspective pictures could.</p>

<p>This will be better understood if we reflect that
painting was developed from relief. The minor dissimilarities
between the pressed figures and the originals
must gradually have compelled men to the adoption
of perspective drawing. But physiologically the
painting of the Egyptians is just as much justified as
the drawings of our children are.</p>

<p>A slight advance beyond the Egyptians is shown
by the Assyrians. The reliefs rescued from the ruined
mounds of Nimrod at Mossul are, upon the whole,
similar to the Egyptian reliefs. They were made known
to us principally by Layard.</p>

<p>Painting enters on a new phase among the Chinese.
This people have a marked feeling for perspective
and correct shading, yet without being very logical
in the application of their principles. Here, too,
it seems, they took the first step but did not go far.
In harmony with this immobility is their constitution,
in which the muzzle and the bamboo-rod play significant
functions. In accord with it, too, is their
language, which like the language of children has not
yet developed into a grammar, or, rather, according
to the modern conception, has not yet degenerated
into a grammar. It is the same also with their music
which is satisfied with the five-toned scale.</p>

<p>The mural paintings at Herculaneum and Pompeii
are distinguished by grace of representation, as also
by a pronounced sense for perspective and correct illumination,
yet they are not at all scrupulous in construction.
Here still we find abbreviations avoided.
But to offset this defect, the members of the body are
brought into unnatural positions, in which they appear
in their full lengths. Abridgements are more frequently
observed in clothed than in unclothed figures.</p>

<p>A satisfactory explanation of these phenomena first
occurred to me on the making of a few simple experiments
which show how differently one may see the
same object, after some mastery of one's senses has
been attained, simply by the arbitrary
movement of the attention.</p>

<div class="figcenter" style="width: 150px;">
<img src="images/i_090.jpg" width="150" height="191" alt="" />
<span class="caption">Fig. 22.</span>
</div>

<p>Look at the annexed drawing (Fig. 22).
It represents a folded sheet of paper with
either its depressed or its elevated side
turned towards you, as you wish. You can
conceive the drawing in either sense, and
in either case it will appear to you differently.</p>

<p>If, now, you have a real folded sheet of paper on
the table before you, with its sharp edges turned towards
you, you can, on looking at it with one eye, see
the sheet alternately elevated, as it really is, or depressed.
Here, however, a remarkable phenomenon
is presented. When you see the sheet properly, neither
illumination nor form presents anything conspicuous.
When you see it bent back you see it perspectively
distorted. Light and shadow appear much brighter
or darker, or as if overlaid thickly with bright colors.
Light and shadow now appear devoid of all cause.
They no longer harmonise with the body's form, and
are thus rendered much more prominent.</p>

<p>In common life we employ the perspective and
illumination of objects to determine their forms and
position. Hence we do not notice the lights, the
shadows, and the distortions. They first powerfully
enter consciousness when we employ a different construction
from the usual spatial one. In looking at
the planar image of a camera obscura we are amazed
at the plenitude of the light and the profundity of the
shadows, both of which we do not notice in real objects.</p>

<p>In my earliest youth the shadows and lights on pictures
appeared to me as spots void of meaning. When
I began to draw I regarded shading as a mere custom
of artists. I once drew the portrait of our pastor, a
friend of the family, and shaded, from no necessity,
but simply from having seen something similar in
other pictures, the whole half of his face black. I was
subjected for this to a severe criticism on the part of
my mother, and my deeply offended artist's pride is
probably the reason that these facts remained so
strongly impressed upon my memory.</p>

<p>You see, then, that many strange things, not only
in the life of individuals, but also in that of humanity,
and in the history of general civilisation, may be explained
from the simple fact that man has two eyes.</p>

<p>Change man's eye and you change his conception
of the world. We have observed the truth of this fact
among our nearest kin, the Egyptians, the Chinese,
and the lake-dwellers; how must it be among some of
our remoter relatives,&mdash;with monkeys and other animals?
Nature must appear totally different to animals
equipped with substantially different eyes from those
of men, as, for example, to insects. But for the present
science must forego the pleasure of portraying this
appearance, as we know very little as yet of the mode
of operation of these organs.</p>

<p>It is an enigma even how nature appears to animals
closely related to man; as to birds, who see
scarcely anything with two eyes at once, but since
their eyes are placed on opposite sides of their heads,
have a separate field of vision for each.<a name="FNanchor_18_18" id="FNanchor_18_18"></a><a href="#Footnote_18_18" class="fnanchor">[18]</a></p>

<p>The soul of man is pent up in the prison-house of
his head; it looks at nature through its two windows,
the eyes. It would also fain know how nature looks
through other windows. A desire apparently never to
be fulfilled. But our love for nature is inventive, and
here, too, much has been accomplished.</p>

<p>Placing before me an angular mirror, consisting of
two plane mirrors slightly inclined to each other, I see
my face twice reflected. In the right-hand mirror I
obtain a view of the right side, and in the left-hand
mirror a view of the left
side, of my face. Also
I shall see the face of a
person standing in front
of me, more to the right with my right eye, more to
the left with my left. But in order to obtain such
widely different views of a face as those shown in the
angular mirror, my two eyes would have to be set much
further apart from each other than they actually are.</p>

<div class="figright" style="width: 220px;">
<img src="images/i_093.jpg" width="220" height="54" alt="" />
<span class="caption">Fig. 23.</span>
</div>

<p>Squinting with my right eye at the image in the
right-hand mirror, with my left eye at the image in
the left-hand mirror, my vision will be the vision of a
giant having an enormous head with his two eyes set
far apart. This, also, is the impression which my own
face makes upon me. I see it now, single and solid.
Fixing my gaze, the relief from second to second is
magnified, the eyebrows start forth prominently from
above the eyes, the nose seems to grow a foot in
length, my mustache shoots forth like a fountain from
my lip, the teeth seem to retreat immeasurably. But
by far the most horrible aspect of the phenomenon is
the nose.</p>

<p>Interesting in this connexion is the telestereoscope
of Helmholtz. In the telestereoscope we view a landscape
by looking with our right eye (Fig. 24) through
the mirror <i>a</i> into the mirror <i>A</i>, and with our left eye
through the mirror <i>b</i> into the mirror <i>B</i>. The mirrors
<i>A</i> and <i>B</i> stand far apart.
Again we see with the
widely separated eyes
of a giant. Everything
appears dwarfed and
near us. The distant
mountains look like
moss-covered stones at our feet. Between, you see the
reduced model of a city, a veritable Liliput. You
are tempted almost to stroke with your hand the soft
forest and city, did you not fear that you might prick
your fingers on the sharp, needle-shaped steeples, or
that they might crackle and break off.</p>

<div class="figleft" style="width: 350px;">
<img src="images/i_094.jpg" width="350" height="201" alt="" />
<span class="caption">Fig. 24.</span>
</div>

<p>Liliput is no fable. We need only Swift's eyes,
the telestereoscope, to see it.</p>

<p>Picture to yourself the reverse case. Let us suppose
ourselves so small that we could take long walks
in a forest of moss, and that our eyes were correspondingly
near each other. The moss-fibres would appear
like trees. On them we should see strange, unshapely
monsters creeping about. Branches of the oak-tree,
at whose base our moss-forest lay, would seem to us
dark, immovable, myriad-branched clouds, painted
high on the vault of heaven; just as the inhabitants
of Saturn, forsooth, might see their enormous ring.
On the tree-trunks of our mossy woodland we should
find colossal globes several feet in diameter, brilliantly
transparent, swayed by the winds with slow, peculiar
motions. We should approach inquisitively and should
find that these globes, in which here and there animals
were gaily sporting, were liquid globes, in fact
that they were water. A short, incautious step, the
slightest contact, and woe betide us, our arm is irresistibly
drawn by an invisible power into the interior of
the sphere and held there unrelentingly fast! A drop
of dew has engulfed in its capillary maw a manikin,
in revenge for the thousands of drops that its big human
counterparts have quaffed at breakfast. Thou
shouldst have known, thou pygmy natural scientist,
that with thy present puny bulk thou shouldst not joke
with capillarity!</p>

<p>My terror at the accident brings me back to my
senses. I see I have turned idyllic. You must pardon
me. A patch of greensward, a moss or heather forest
with its tiny inhabitants have incomparably more
charms for me than many a bit of literature with its
apotheosis of human character. If I had the gift of
writing novels I should certainly not make John and
Mary my characters. Nor should I transfer my loving
pair to the Nile, nor to the age of the old Egyptian
Pharaohs, although perhaps I should choose that time
in preference to the present. For I must candidly
confess that I hate the rubbish of history, interesting
though it may be as a mere phenomenon, because we
cannot simply observe it but must also <i>feel</i> it, because
it comes to us mostly with supercilious arrogance,
mostly unvanquished. The hero of my novel would be
a cockchafer, venturing forth in his fifth year for the
first time with his newly grown wings into the light,
free air. Truly it could do no harm if man would thus
throw off his inherited and acquired narrowness of
mind by making himself acquainted with the world-view
of allied creatures. He could not help gaining
incomparably more in this way than the inhabitant of
a small town would in circumnavigating the globe and
getting acquainted with the views of strange peoples.</p>
<p><span class="pagenum"><a name="Page_87" id="Page_87">[Pg 87]</a></span></p><p><span class="pagenum"><a name="Page_88" id="Page_88">[Pg 88]</a></span></p><p><span class="pagenum"><a name="Page_89" id="Page_89">[Pg 89]</a></span></p>
<hr class="tb" />

<p>I have now conducted you, by many paths and by-ways,
rapidly over hedge and ditch, to show you what
wide vistas we may reach in every field by the rigorous
pursuit of a single scientific fact. A close examination
of the two eyes of man has conducted us not
only into the dim recesses of humanity's childhood,
but has also carried us far beyond the bourne of human
life.</p>

<p>It has surely often struck you as strange that the
sciences are divided into two great groups; that the
so-called humanistic sciences, belonging to the so-called
"higher education," are placed in almost a hostile
attitude to the natural sciences.</p>

<p>I must confess I do not overmuch believe in this
partition of the sciences. I believe that this view will
appear as childlike and ingenuous to a matured age
as the want of perspective in the old paintings of Egypt
does to us. Can it really be that "higher culture" is to
be gotten only from a few old pots and palimpsests,
which are at best mere scraps of nature, or that more
is to be learned from them alone than from all the rest
of nature? I believe that both these sciences are simply
parts of the same science, which have begun at
different ends. If these two ends still act towards
each other as the Montagues and Capulets, if their retainers
still indulge in lively tilts, I believe that after
all they are not in earnest. On the one side there is
surely a Romeo, and on the other a Juliet, who, some
day, it is hoped, will unite the two houses with a less
tragic sequel than that of the play.</p>

<p>Philology began with the unqualified reverence and
apotheosis of the Greeks. Now it has begun to draw
other languages, other peoples and their histories, into
its sphere; it has, through the mediation of comparative
linguistics, already struck up, though as yet somewhat
cautiously, a friendship with physiology.</p>

<p>Physical science began in the witch's kitchen. It
now embraces the organic and inorganic worlds, and
with the physiology of articulation and the theory of
the senses, has even pushed its researches, at times
impertinently, into the province of mental phenomena.</p>

<p>In short, we come to the understanding of much
within us solely by directing our glance without, and
<i>vice versa</i>. Every object belongs to both sciences.
You, ladies, are very interesting and difficult problems
for the psychologist, but you are also extremely pretty
phenomena of nature. Church and State are objects
of the historian's research, but not less phenomena of
nature, and in part, indeed, very curious phenomena.
If the historical sciences have inaugurated wide extensions
of view by presenting to us the thoughts of
new and strange peoples, the physical sciences in a
certain sense do this in a still greater degree. In
making man disappear in the All, in annihilating him,
so to speak, they force him to take an unprejudiced
position without himself, and to form his judgments by
a different standard from that of the petty human.</p>

<p>But if you should ask me now why man has two
eyes, I should answer:</p>

<p>That he may look at nature justly and accurately;
that he may come to understand that he himself, with
all his views, correct and incorrect, with all his <i>haute
politique</i>, is simply an evanescent shred of nature;
that, to speak with Mephistopheles, he is a part of the
part, and that it is absolutely unjustified,</p>

<div class="poem"><div class="stanza">
<span class="i0">"For man, the microcosmic fool, to see<br /></span>
<span class="i0">Himself a whole so frequently."<br /></span>
</div></div>




<h2><a name="ON_SYMMETRY" id="ON_SYMMETRY">ON SYMMETRY.</a></h2><p><a name="FNanchor_19_19" id="FNanchor_19_19"></a><a href="#Footnote_19_19" class="fnanchor">[19]</a></p>


<p>An ancient philosopher once remarked that people
who cudgelled their brains about the nature of
the moon reminded him of men who discussed the
laws and institutions of a distant city of which they
had heard no more than the name. The true philosopher,
he said, should turn his glance within, should
study himself and his notions of right and wrong; only
thence could he derive real profit.</p>

<p>This ancient formula for happiness might be restated
in the familiar words of the Psalm:</p>

<div class="poem"><div class="stanza">
<span class="i0">"Dwell in the land, and verily thou shalt be fed."<br /></span>
</div></div>

<p>To-day, if he could rise from the dead and walk
about among us, this philosopher would marvel much
at the different turn which matters have taken.</p>
<p><span class="pagenum"><a name="Page_90" id="Page_90">[Pg 90]</a></span></p>
<p>The motions of the moon and the other heavenly
bodies are accurately known. Our knowledge of the
motions of our own body is by far not so complete.
The mountains and natural divisions of the moon have
been accurately outlined on maps, but physiologists
are just beginning to find their way in the geography
of the brain. The chemical constitution of many fixed
stars has already been investigated. The chemical
processes of the animal body are questions of much
greater difficulty and complexity. We have our <i>Mécanique
céleste</i>. But a <i>Mécanique sociale</i> or a <i>Mécanique
morale</i> of equal trustworthiness remains to be written.</p>

<p>Our philosopher would indeed admit that we have
made great progress. But we have not followed his
advice. The patient has recovered, but he took for his
recovery exactly the opposite of what the doctor prescribed.</p>

<p>Humanity is now returned, much wiser, from its
journey in celestial space, against which it was so
solemnly warned. Men, after having become acquainted
with the great and simple facts of the world without,
are now beginning to examine critically the world
within. It sounds absurd, but it is true, that only after
we have thought about the moon are we able to take
up ourselves. It was necessary that we should acquire
simple and clear ideas in a less complicated domain,
before we entered the more intricate one of psychology,
and with these ideas astronomy principally furnished
us.</p><p><span class="pagenum"><a name="Page_91" id="Page_91">[Pg 91]</a></span></p>

<p>To attempt any description of that stupendous
movement, which, originally springing out of the physical
sciences, went beyond the domain of physics and is
now occupied with the problems of psychology, would
be presumptuous in this place. I shall only attempt
here, to illustrate to you by a few simple examples the
methods by which the province of psychology can be
reached from the facts of the physical world&mdash;especially
the adjacent province of sense-perception. And I wish
it to be remembered that my brief attempt is not to be
taken as a measure of the present state of such scientific
questions.</p>
<p><span class="pagenum"><a name="Page_92" id="Page_92">[Pg 92]</a></span></p><p><span class="pagenum"><a name="Page_93" id="Page_93">[Pg 93]</a></span></p><p><span class="pagenum"><a name="Page_94" id="Page_94">[Pg 94]</a></span></p><p><span class="pagenum"><a name="Page_95" id="Page_95">[Pg 95]</a></span></p><p><span class="pagenum"><a name="Page_96" id="Page_96">[Pg 96]</a></span></p><p><span class="pagenum"><a name="Page_97" id="Page_97">[Pg 97]</a></span></p><p><span class="pagenum"><a name="Page_98" id="Page_98">[Pg 98]</a></span></p><p><span class="pagenum"><a name="Page_99" id="Page_99">[Pg 99]</a></span></p><p><span class="pagenum"><a name="Page_100" id="Page_100">[Pg 100]</a></span></p><p><span class="pagenum"><a name="Page_101" id="Page_101">[Pg 101]</a></span></p><p><span class="pagenum"><a name="Page_102" id="Page_102">[Pg 102]</a></span></p><p><span class="pagenum"><a name="Page_103" id="Page_103">[Pg 103]</a></span></p><p><span class="pagenum"><a name="Page_104" id="Page_104">[Pg 104]</a></span></p>
<hr class="tb" />

<p>It is a well-known fact that some objects please us,
while others do not. Generally speaking, anything
that is constructed according to fixed and logically
followed rules, is a product of tolerable beauty. We see
thus nature herself, who always acts according to fixed
rules, constantly producing such pretty things. Every
day the physicist is confronted in his workshop with
the most beautiful vibration-figures, tone-figures, phenomena
of polarisation, and forms of diffraction.</p>

<p>A rule always presupposes a repetition. Repetitions,
therefore, will probably be found to play some
important part in the production of agreeable effects.
Of course, the nature of agreeable effects is not exhausted
by this. Furthermore, the repetition of a
physical event becomes the source of agreeable effects
only when it is connected with a repetition of sensations.</p>

<p>An excellent example that repetition of sensations
is a source of agreeable effects is furnished by the
copy-book of every schoolboy, which is usually a treasure-house
of such things, and only in need of an Abbé
Domenech to become celebrated. Any figure, no matter
how crude or poor, if several times repeated, with
the repetitions placed in line, will produce a tolerable
frieze.</p>

<div class="figright" style="width: 300px;">
<img src="images/i_102.jpg" width="300" height="185" alt="" />
<span class="caption">Fig. 25.</span>
</div>

<p>Also the pleasant effect of symmetry is due to the
repetition of sensations. Let us abandon ourselves a
moment to this thought, yet not imagine when we have
developed it, that we have fully exhausted the nature
of the agreeable, much less of the beautiful.</p>

<p>First, let us get a clear conception of what symmetry
is. And in preference to a definition let us take
a living picture. You know that the reflexion of an
object in a mirror has a great likeness to the object itself.
All its proportions and outlines are the same.
Yet there is a difference between the object and its reflexion
in the mirror, which you will readily observe.</p>

<p>Hold your right hand before a mirror, and you will
see in the mirror a left hand. Your right glove will
produce its mate in the glass. For you could never
use the reflexion of your right glove, if it were present
to you as a real thing, for covering your right hand,
but only for covering your left. Similarly, your right
ear will give as its reflexion a left ear; and you will at
once perceive that the left half of your body could very
easily be substituted for the reflexion of your right half.
Now just as in the place of a missing right ear a left ear
cannot be put, unless the lobule of the ear be turned upwards,
or the opening into the concha backwards, so,
despite all similarity of form, the reflexion of an object
can never take the place of the object itself.<a name="FNanchor_20_20" id="FNanchor_20_20"></a><a href="#Footnote_20_20" class="fnanchor">[20]</a></p>

<p>The reason of this difference between the object
and its reflexion is simple. The reflexion appears as
far behind the mirror as the object is in front of it. The
parts of the object, accordingly, which are nearest the
mirror will also be nearest the mirror in the reflexion.
Consequently, the succession of the parts in the reflexion
will be reversed, as may best be seen in the reflexion
of the face of a watch or of a manuscript.</p>

<p>It will also be readily seen, that if a point of the object
be joined with its reflexion in the image, the line
of junction will cut the mirror at right angles and be
bisected by it. This holds true of all corresponding
points of object and image.</p>

<p>If, now, we can divide an object by a plane into
two halves so that each half, as seen in the reflecting
plane of division, is a reproduction of the other half,
such an object is termed symmetrical, and the plane
of division is called the plane of symmetry.</p>

<p>If the plane of symmetry is vertical, we can say
that the body is vertically symmetrical. An example
of vertical symmetry is a Gothic cathedral.</p>

<p>If the plane of symmetry is horizontal, we can say
that the object is horizontally symmetrical. A landscape
on the shores of a lake with its reflexion in the
water, is a system of horizontal symmetry.</p>

<p>Exactly here is a noticeable difference. The vertical
symmetry of a Gothic cathedral strikes us at once,
whereas we can travel up and down the whole length
of the Rhine or the Hudson without becoming aware
of the symmetry between objects and their reflexions
in the water. Vertical symmetry pleases us, whilst
horizontal symmetry is indifferent, and is noticed only
by the experienced eye.</p>

<p>Whence arises this difference? I say from the fact
that vertical symmetry produces a repetition of the
same sensation, while horizontal symmetry does not.
I shall now show that this is so.</p>

<p>Let us look at the following letters:</p>

<p class="center">d b
q p
</p>

<p>It is a fact known to all mothers and teachers, that
children in their first attempts to read and write, constantly
confound d and b, and q and p, but never d
and q, or b and p. Now d and b and q and p are the
two halves of a <i>vertically</i> symmetrical figure, while d
and q, and b and p are two halves of a <i>horizontally</i> symmetrical
figure. The first two are confounded; but
confusion is only possible of things that excite in us
the same or similar sensations.</p>

<p>Figures of two flower-girls are frequently seen on
the decorations of gardens and of drawing-rooms, one
of whom carries a flower-basket in her right hand and
the other a flower-basket in her left. All know how
apt we are, unless we are very careful, to confound these
figures with one another.</p>

<p>While turning a thing round from right to left is
scarcely noticed, the eye is not at all indifferent to the
turning of a thing upside down. A human face which
has been turned upside down is scarcely recognisable
as a face, and makes an impression which is altogether
strange. The reason of this is not to be sought in the
unwontedness of the sight, for it is just as difficult to
recognise an arabesque that has been inverted, where
there can be no question of a habit. This curious fact
is the foundation of the familiar jokes played with the
portraits of unpopular personages, which are so drawn
that in the upright position of the page an exact picture
of the person is presented, but on being inverted
some popular animal is shown.</p>

<p>It is a fact, then, that the two halves of a vertically
symmetrical figure are easily confounded and that they
therefore probably produce very nearly the same sensations.
The question, accordingly, arises, <i>why</i> do the
two halves of a vertically symmetrical figure produce
the same or similar sensations? The answer is: Because
our apparatus of vision, which consists of our
eyes and of the accompanying muscular apparatus is
itself vertically symmetrical.<a name="FNanchor_21_21" id="FNanchor_21_21"></a><a href="#Footnote_21_21" class="fnanchor">[21]</a></p>

<p>Whatever external resemblances one eye may have
with another they are still not alike. The right eye of
a man cannot take the place of a left eye any more
than a left ear or left hand can take the place of a
right one. By artificial means, we can change the part
which each of our eyes plays. (Wheatstone's pseudoscope.)
But we then find ourselves in an entirely new
and strange world. What is convex appears concave;
what is concave, convex. What is distant appears
near, and what is near appears far.</p>

<p>The left eye is the reflexion of the right. And the
light-feeling retina of the left eye is a reflexion of the
light-feeling retina of the right, in all its functions.</p>

<p>The lense of the eye, like a magic lantern, casts
images of objects on the retina. And you may picture
to yourself the light-feeling retina of the eye, with its
countless nerves, as a hand with innumerable fingers,
adapted to feeling light. The ends of the visual nerves,
like our fingers, are endowed with varying degrees of
sensitiveness. The two retinæ act like a right and a
left hand; the sensation of touch and the sensation of
light in the two instances are similar.</p>

<p>Examine the right-hand portion of this letter T:
namely, T. Instead of the two retinæ on which this
image falls, imagine feeling the object, my two hands.
The &#9484;, grasped with the right hand, gives a different
sensation from that which it gives when grasped with
the left. But if we turn our character about from right
to left, thus: &#9488;, it will give the same sensation in the
left hand that it gave before in the right. The sensation
is repeated.</p>

<p>If we take a whole T, the right half will produce in
the right hand the same sensation that the left half
produces in the left, and <i>vice versa</i>.</p>

<p>The symmetrical figure gives the same sensation
twice.</p>

<p>If we turn the T over thus: &#9500;, or invert the half
T thus: L, so long as we do not change the position
of our hands we can make no use of the foregoing reasoning.</p>

<p>The retinæ, in fact, are exactly like our two hands.
They, too, have their thumbs and index fingers, though
they are thousands in number; and we may say the
thumbs are on the side of the eye near the nose,
and the remaining fingers on the side away from the
nose.</p>

<p>With this I hope to have made perfectly clear that
the pleasing effect of symmetry is chiefly due to the
repetition of sensations, and that the effect in question
takes place in symmetrical figures, only where
there is a repetition of sensation. The pleasing effect
of regular figures, the preference which straight lines,
especially vertical and horizontal straight lines, enjoy,
is founded on a similar reason. A straight line,
both in a horizontal and in a vertical position, can cast
on the two retinæ the same image, which falls moreover
on symmetrically corresponding spots. This also,
it would appear, is the reason of our psychological
preference of straight to curved lines, and not their
property of being the shortest distance between two
points. The straight line is felt, to put the matter
briefly, as symmetrical to itself, which is the case also
with the plane. Curved lines are felt as deviations
from straight lines, that is, as deviations from symmetry.<a name="FNanchor_22_22" id="FNanchor_22_22"></a><a href="#Footnote_22_22" class="fnanchor">[22]</a>
The presence of a sense for symmetry in people
possessing only one eye from birth, is indeed a riddle.
Of course, the sense of symmetry, although primarily
acquired by means of the eyes, cannot be wholly limited
to the visual organs. It must also be deeply
rooted in other parts of the organism by ages of practice
and can thus not be eliminated forthwith by the
loss of one eye. Also, when an eye is lost, the symmetrical
muscular apparatus is left, as is also the
symmetrical apparatus of innervation.</p>

<p>It appears, however, unquestionable that the phenomena
mentioned have, in the main, their origin in
the peculiar structure of our eyes. It will therefore
be seen at once that our notions of what is beautiful
and ugly would undergo a change if our eyes were different.
Also, if this view is correct, the theory of the
so-called eternally beautiful is somewhat mistaken. It
can scarcely be doubted that our culture, or form of
civilisation, which stamps upon the human body its
unmistakable traces, should not also modify our conceptions
of the beautiful. Was not formerly the development
of all musical beauty restricted to the narrow
limits of a five-toned scale?</p>

<p>The fact that a repetition of sensations is productive
of pleasant effects is not restricted to the realm of
the visible. To-day, both the musician and the physicist
know that the harmonic or the melodic addition
of one tone to another affects us agreeably only when
the added tone reproduces a part of the sensation
which the first one excited. When I add an octave
to a fundamental tone, I hear in the octave a part of
what was heard in the fundamental tone. (Helmholtz.)
But it is not my purpose to develop this idea
fully here.<a name="FNanchor_23_23" id="FNanchor_23_23"></a><a href="#Footnote_23_23" class="fnanchor">[23]</a> We shall only ask to-day, whether there
is anything similar to the symmetry of figures in the
province of sounds.</p>

<p>Look at the reflexion of your piano in the mirror.</p>

<p>You will at once remark that you have never seen
such a piano in the actual world, for it has its high
keys to the left and its low ones to the right. Such
pianos are not manufactured.</p>

<p>If you could sit down at such a piano and play in
your usual manner, plainly every step which you
imagined you were performing in the upward scale
would be executed as a corresponding step in the
downward scale. The effect would be not a little surprising.</p>

<p>For the practised musician who is always accustomed
to hearing certain sounds produced when certain
keys are struck, it is quite an anomalous spectacle
to watch a player in the glass and to observe that he
always does the opposite of what we hear.</p>

<p>But still more remarkable would be the effect of
attempting to strike a harmony on such a piano. For
a melody it is not indifferent whether we execute a
step in an upward or a downward scale. But for a
harmony, so great a difference is not produced by reversal.
I always retain the same consonance whether
I add to a fundamental note an upper or a lower third.
Only the order of the intervals of the harmony is reversed.
In point of fact, when we execute a movement
in a major key on our reflected piano, we hear a
sound in a minor key, and <i>vice versa</i>.</p>

<p>It now remains to execute the experiments indicated.
Instead of playing upon the piano in the mirror,
which is impossible, or of having a piano of this
kind built, which would be somewhat expensive, we
may perform our experiments in a simpler manner, as
follows:</p>

<p>1) We play on our own piano in our usual manner,
look into the mirror, and then repeat on our real piano
what we see in the mirror. In this way we transform
all steps upwards into corresponding steps downwards.
We play a movement, and then another movement,
which, with respect to the key-board, is symmetrical
to the first.</p>

<p>2) We place a mirror beneath the music in which
the notes are reflected as in a body of water, and play
according to the notes in the mirror. In this way also,
all steps upwards are changed into corresponding,
equal steps downwards.</p>

<p>3) We turn the music upside down and read the
notes from right to left and from below upwards. In
doing this, we must regard all sharps as flats and all
flats as sharps, because they correspond to half lines
and spaces. Besides, in this use of the music we can
only employ the bass clef, as only in this clef are the
notes not changed by symmetrical reversal.</p>

<p>You can judge of the effect of these experiments
from the examples which appear in the annexed musical
cut. (Page 102.) The movement which appears in
the upper lines is symmetrically reversed in the lower.</p>

<p>The effect of the experiments may be briefly formulated.
The melody is rendered unrecognisable. The
harmony suffers a transposition from a major into a
minor key and <i>vice versa</i>. The study of these pretty
effects, which have long been familiar to physicists
and musicians, was revived some years ago by Von
Oettingen.<a name="FNanchor_24_24" id="FNanchor_24_24"></a><a href="#Footnote_24_24" class="fnanchor">[24]</a></p>

<div class="figcenter" style="width: 525px;">
<img src="images/i_112.jpg" width="525" height="800" alt="" />
<span class="caption">Fig. 26.</span><br />
<a href="music/112a.mid">Listen to 1.</a><br />
<a href="music/112b.mid">Listen to 2.</a><br />
<a href="music/112c.mid">Listen to 3.</a><br />
<a href="music/112d.mid">Listen to 4.</a><br />
<a href="music/112e.mid">Listen to 5.</a><br />
<a href="music/112f.mid">Listen to 6.</a><br />
<a href="music/112g.mid">Listen to 7.</a><br />
<a href="music/112h.mid">Listen to 8.</a><br />
</div>

<p>(See pages 101 and 103.)]</p>

<p>Now, although in all the preceding examples I have
transposed steps upward into equal and similar steps
downward, that is, as we may justly say, have played
for every movement the movement which is symmetrical
to it, yet the ear notices either little or nothing of
symmetry. The transposition from a major to a minor
key is the sole indication of symmetry remaining. The
symmetry is there for the mind, but is wanting for
sensation. No symmetry exists for the ear, because a
reversal of musical sounds conditions no repetition of
sensations. If we had an ear for height and an ear
for depth, just as we have an eye for the right and an
eye for the left, we should also find that symmetrical
sound-structures existed for our auditory organs. The
contrast of major and minor for the ear corresponds to
inversion for the eye, which is also only symmetry for
the mind, but not for sensation.</p>

<p>By way of supplement to what I have said, I will
add a brief remark for my mathematical readers.</p>

<p>Our musical notation is essentially a graphical representation
of a piece of music in the form of curves,
where the time is the abscissæ, and the logarithms of
the number of vibrations the ordinates. The deviations
of musical notation from this principle are only
such as facilitate interpretation, or are due to historical
accidents.</p>

<p>If, now, it be further observed that the sensation
of pitch also is proportional to the logarithm of the
number of vibrations, and that the intervals between
the notes correspond to the differences of the logarithms
of the numbers of vibrations, the justification
will be found in these facts of calling the harmonies
and melodies which appear in the mirror, symmetrical
to the original ones.</p>
<p><span class="pagenum"><a name="Page_105" id="Page_105">[Pg 105]</a></span></p><p><span class="pagenum"><a name="Page_106" id="Page_106">[Pg 106]</a></span></p><p><span class="pagenum"><a name="Page_107" id="Page_107">[Pg 107]</a></span></p>
<hr class="tb" />

<p>I simply wish to bring home to your minds by these
fragmentary remarks that the progress of the physical
sciences has been of great help to those branches of
psychology that have not scorned to consider the results
of physical research. On the other hand, psychology
is beginning to return, as it were, in a spirit
of thankfulness, the powerful stimulus which it received
from physics.</p>

<p>The theories of physics which reduce all phenomena
to the motion and equilibrium of smallest particles,
the so-called molecular theories, have been
gravely threatened by the progress of the theory of the
senses and of space, and we may say that their days
are numbered.</p>

<p>I have shown elsewhere<a name="FNanchor_25_25" id="FNanchor_25_25"></a><a href="#Footnote_25_25" class="fnanchor">[25]</a> that the musical scale is
simply a species of space&mdash;a space, however, of only
one dimension, and that, a one-sided one. If, now, a
person who could only hear, should attempt to develop
a conception of the world in this, his linear space, he
would become involved in many difficulties, as his space
would be incompetent to comprehend the many sides
of the relations of reality. But is it any more justifiable
for us, to attempt to force the whole world into the
space of our eye, in aspects in which it is not accessible
to the eye? Yet this is the dilemma of all molecular
theories.</p>

<p>We possess, however, a sense, which, with respect
to the scope of the relations which it can comprehend,
is richer than any other. It is our reason. This stands
above the senses. It alone is competent to found a
permanent and sufficient view of the world. The
mechanical conception of the world has performed
wonders since Galileo's time. But it must now yield
to a broader view of things. A further development of
this idea is beyond the limits of my present purpose.</p>

<p>One more point and I have done. The advice of
our philosopher to restrict ourselves to what is near
at hand and useful in our researches, which finds a
kind of exemplification in the present cry of inquirers
for limitation and division of labor, must not be too
slavishly followed. In the seclusion of our closets, we
often rack our brains in vain to fulfil a work, the
means of accomplishing which lies before our very
doors. If the inquirer must be perforce a shoemaker,
tapping constantly at his last, it may perhaps be permitted
him to be a shoemaker of the type of Hans
Sachs, who did not deem it beneath him to take a
look now and then at his neighbor's work and to
comment on the latter's doings.</p>

<p>Let this be my apology, therefore, if I have forsaken
for a moment to-day the last of my specialty.</p>




<h2><a name="ON_THE_FUNDAMENTAL_CONCEPTS" id="ON_THE_FUNDAMENTAL_CONCEPTS">ON THE FUNDAMENTAL CONCEPTS
OF ELECTROSTATICS.</a><a name="FNanchor_26_26" id="FNanchor_26_26"></a><a href="#Footnote_26_26" class="fnanchor">[26]</a></h2>


<p>The task has been assigned me to develop before
you in a popular manner the fundamental quantitative
concepts of electrostatics&mdash;"quantity of electricity,"
"potential," "capacity," and so forth. It
would not be difficult, even within the brief limits of
an hour, to delight the eye with hosts of beautiful experiments
and to fill the imagination with numerous
and varied conceptions. But we should, in such a
case, be still far from a lucid and easy grasp of the
phenomena. The means would still fail us for reproducing
the facts accurately in thought&mdash;a procedure
which for the theoretical and practical man is of equal
importance. These means are the <i>metrical concepts</i> of
electricity.</p>

<p>As long as the pursuit of the facts of a given province
of phenomena is in the hands of a few isolated
investigators, as long as every experiment can be easily
repeated, the fixing of the collected facts by provisional<span class="pagenum"><a name="Page_108" id="Page_108">[Pg 108]</a></span>
description is ordinarily sufficient. But the case is
different when the whole world must make use of the
results reached by many, as happens when the science
acquires broader foundations and scope, and
particularly so when it begins to supply intellectual
nourishment to an important branch of the practical
arts, and to draw from that province in return stupendous
empirical results. Then the facts must be so
described that individuals in all places and at all times
can, from a few easily obtained elements, put the facts
accurately together in thought, and reproduce them
from the description. This is done with the help of
the metrical concepts and the international measures.</p>

<p>The work which was begun in this direction in the
period of the purely scientific development of the science,
especially by Coulomb (1784), Gauss (1833), and
Weber (1846), was powerfully stimulated by the requirements
of the great technical undertakings manifested
since the laying of the first transatlantic cable,
and brought to a brilliant conclusion by the labors of
the British Association, 1861, and of the Paris Congress,
1881, chiefly through the exertions of Sir William
Thomson.</p>

<p>It is plain, that in the time allotted to me I cannot
conduct you over all the long and tortuous paths which
the science has actually pursued, that it will not be
possible at every step to remind you of all the little
precautions for the avoidance of error which the early
steps have taught us. On the contrary, I must make<span class="pagenum"><a name="Page_109" id="Page_109">[Pg 109]</a></span>
shift with the simplest and rudest tools. I shall conduct
you by the shortest paths from the facts to the
ideas, in doing which, of course, it will not be possible
to anticipate all the stray and chance ideas which may
and must arise from prospects into the by-paths which
we leave untrodden.</p>
<p><span class="pagenum"><a name="Page_110" id="Page_110">[Pg 110]</a></span></p><p><span class="pagenum"><a name="Page_111" id="Page_111">[Pg 111]</a></span></p><p><span class="pagenum"><a name="Page_112" id="Page_112">[Pg 112]</a></span></p><p><span class="pagenum"><a name="Page_113" id="Page_113">[Pg 113]</a></span></p><p><span class="pagenum"><a name="Page_114" id="Page_114">[Pg 114]</a></span></p><p><span class="pagenum"><a name="Page_115" id="Page_115">[Pg 115]</a></span></p><p><span class="pagenum"><a name="Page_116" id="Page_116">[Pg 116]</a></span></p><p><span class="pagenum"><a name="Page_117" id="Page_117">[Pg 117]</a></span></p><p><span class="pagenum"><a name="Page_118" id="Page_118">[Pg 118]</a></span></p><p><span class="pagenum"><a name="Page_119" id="Page_119">[Pg 119]</a></span></p><p><span class="pagenum"><a name="Page_120" id="Page_120">[Pg 120]</a></span></p><p><span class="pagenum"><a name="Page_121" id="Page_121">[Pg 121]</a></span></p><p><span class="pagenum"><a name="Page_122" id="Page_122">[Pg 122]</a></span></p><p><span class="pagenum"><a name="Page_123" id="Page_123">[Pg 123]</a></span></p><p><span class="pagenum"><a name="Page_124" id="Page_124">[Pg 124]</a></span></p><p><span class="pagenum"><a name="Page_125" id="Page_125">[Pg 125]</a></span></p><p><span class="pagenum"><a name="Page_126" id="Page_126">[Pg 126]</a></span></p><p><span class="pagenum"><a name="Page_127" id="Page_127">[Pg 127]</a></span></p>
<hr class="tb" />

<p>Here are two small, light bodies (Fig. 27) of equal
size, freely suspended, which we "electrify" either
by friction with a third body or by contact with a body
already electrified. At once a repulsive force is set
up which drives the two bodies away from each other
in opposition to the action of gravity. This force could
accomplish anew the same mechanical work which
was expended to produce it.<a name="FNanchor_27_27" id="FNanchor_27_27"></a><a href="#Footnote_27_27" class="fnanchor">[27]</a></p>

<div class="figleft" style="width: 150px;">
<img src="images/i_119-1.jpg" width="150" height="253" alt="" />
<span class="caption">Fig. 27.</span>
</div>

<div class="figright" style="width: 250px;">
<img src="images/i_119-2.jpg" width="250" height="170" alt="" />
<span class="caption">Fig. 28.</span>
</div>

<p>Coulomb, now, by means of delicate experiments
with the torsion-balance, satisfied himself that if the
bodies in question, say at a distance of two centimetres,
repelled each other with the same force with
which a milligramme-weight strives to fall to the
ground, at half that distance, or at one centimetre,
they would repel each other with the force of four
milligrammes, and at double that distance, or at four
centimetres, they would repel each other with the force
of only one-fourth of a milligramme. He found that
the electrical force acts inversely as the square of the
distance.</p>

<p>Let us imagine, now, that we possessed some
means of measuring electrical repulsion by weights,
a means which would be supplied, for example, by our
electrical pendulums; then we could make the following
observation.</p>

<p>The body <i>A</i> (Fig. 28) is repelled by the body <i>K</i> at
a distance of two centimetres with a force of one milligramme.
If we touch <i>A</i>, now, with an equal body <i>B</i>,
the half of this force of repulsion will pass to the body
<i>B</i>; both <i>A</i> and <i>B</i>, now, at a distance of two centimetres
from <i>K</i>, are repelled only with the force of one-half
a milligramme. But both together are repelled
still with the force of one milligramme. Hence, <i>the
divisibility of electrical force</i> among bodies in contact <i>is
a fact</i>. It is a useful, but by no means a necessary
supplement to this fact, to imagine an electrical fluid
present in the body <i>A</i>, with the quantity of which the
electrical force varies, and half of which flows over to
<i>B</i>. For, in the place of the new physical picture,
thus, an old, familiar one is substituted, which moves
spontaneously in its wonted courses.</p>

<p>Adhering to this idea, we define the <i>unit</i> of electrical
quantity, according to the now almost universally
adopted centimetre-gramme-second (C. G. S.) system,
as that quantity which at a distance of one centimetre
repels an equal quantity with unit of force, that
is, with a force which in one second would impart to
a mass of one gramme a velocity-increment of a centimetre.
As a gramme mass acquires through the action
of gravity a velocity-increment of about 981 centimetres
in a second, accordingly, a gramme is attracted
to the earth with 981, or, in round numbers, 1000 units
of force of the centimetre-gramme-second system,
while a milligramme-weight would strive to fall to the
earth with approximately the unit force of this system.</p>

<p>We may easily obtain by this means a clear idea of
what the unit quantity of electricity is. Two small
bodies, <i>K</i>, weighing each a gramme, are hung up by
vertical threads, five metres in length and almost
weightless, so as to touch each other. If the two bodies
be equally electrified and move apart upon electrification
to a distance of one centimetre, their charge is approximately
equivalent to the electrostatic unit of electric
quantity, for the repulsion then holds in equilibrium
a gravitational force-component of approximately
one milligramme, which strives to bring the bodies together.</p>

<p>Vertically beneath a small sphere suspended from
the equilibrated beam of a balance a second sphere is
placed at a distance of a centimetre. If both be equally
electrified the sphere suspended from the balance will
be rendered apparently lighter by the repulsion. If by
adding a weight of one milligramme equilibrium is
restored, each of the spheres contains in round numbers
the electrostatic unit of electrical quantity.</p>

<p>In view of the fact that the same electrical bodies
exert at different distances different forces upon one
another, exception might be taken to the measure of
quantity here developed. What kind of a quantity is
that which now weighs more, and now weighs less, so
to speak? But this apparent deviation from the
method of determination commonly used in practical
life, that by weight, is, closely considered, an agreement.
On a high mountain a heavy mass also is less
powerfully attracted to the earth than at the level of
the sea, and if it is permitted us in our determinations
to neglect the consideration of level, it is only because
the comparison of a body with fixed conventional
weights is invariably effected at the same level. In
fact, if we were to make one of the two weights equilibrated
on our balance approach sensibly to the centre
of the earth, by suspending it from a very long thread,
as Prof. von Jolly of Munich suggested, we should
make the gravity of that weight, its heaviness, proportionately
greater.</p>

<p>Let us picture to ourselves, now, two different
electrical fluids, a positive and a negative fluid, of such
nature that the particles of the one attract the particles
of the other according to the law of the inverse squares,
but the particles of the same fluid repel each other by
the same law; in non-electrical bodies let us imagine
the two fluids uniformly distributed in equal quantities,
in electric bodies one of the two in excess; in
conductors, further, let us imagine the fluids mobile,
in non-conductors immobile; having formed such pictures,
we possess the conception which Coulomb developed
and to which he gave mathematical precision.
We have only to give this conception free play in our
minds and we shall see as in a clear picture the fluid
particles, say of a positively charged conductor, receding
from one another as far as they can, all making
for the surface of the conductor and there seeking out
the prominent parts and points until the greatest possible
amount of work has been performed. On increasing
the size of the surface, we see a dispersion,
on decreasing its size we see a condensation of the particles.
In a second, non-electrified conductor brought
into the vicinity of the first, we see the two fluids immediately
separate, the positive collecting itself on the
remote and the negative on the adjacent side of its
surface. In the fact that this conception reproduces,
lucidly and spontaneously, all the data which arduous
research only slowly and gradually discovered, is contained
its advantage and scientific value. With this,
too, its value is exhausted. We must not seek in nature
for the two hypothetical fluids which we have
added as simple mental adjuncts, if we would not go
astray. Coulomb's view may be replaced by a totally
different one, for example, by that of Faraday, and the
most proper course is always, after the general survey
is obtained, to go back to the actual facts, to the electrical
forces.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_124-1.jpg" width="300" height="310" alt="" />
<span class="caption">Fig. 29.</span>
</div>

<div class="figright" style="width: 300px;">
<img src="images/i_124-2.jpg" width="300" height="276" alt="" />
<span class="caption">Fig. 30.</span>
</div>

<p>We will now make ourselves familiar with the concept
of electrical quantity, and with the method of
measuring or estimating it. Imagine a common Leyden
jar (Fig. 29), the inner and outer coatings of which
are connected together by means of two common metallic
knobs placed about a centimetre apart. If the
inside coating be charged with the quantity of electricity
+<i>q</i>, on the outer coating a distribution of the
electricities will take place. A positive quantity almost
equal<a name="FNanchor_28_28" id="FNanchor_28_28"></a><a href="#Footnote_28_28" class="fnanchor">[28]</a> to the quantity +<i>q</i> flows off to the earth, while
a corresponding quantity-<i>q</i> is still left on the outer
coating. The knobs of the jar receive their portion of
these quantities and when the quantity <i>q</i> is sufficiently
great a rupture of the insulating air between the knobs,
accompanied by the self-discharge of the jar, takes
place. For any given distance and size of the knobs,
a charge of a definite electric quantity <i>q</i> is always necessary
for the spontaneous discharge of the jar.</p>

<p>Let us insulate, now, the outer coating of a Lane's
unit jar <i>L</i>, the jar just described, and put in connexion
with it the inner coating of a jar <i>F</i> exteriorly connected
with the earth (Fig. 30). Every time that <i>L</i> is
charged with +<i>q</i>, a like quantity +<i>q</i> is collected on
the inner coating of <i>F</i>, and the spontaneous discharge
of the jar <i>L</i>, which is now
again empty, takes place. The
number of the discharges of
the jar <i>L</i> furnishes us, thus,
with a measure of the quantity
collected in the jar <i>F</i>, and
if after 1, 2, 3, ... spontaneous
discharges of <i>L</i> the jar <i>F</i> is
discharged, it is evident that the charge of <i>F</i> has been
proportionately augmented.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_125.jpg" width="300" height="255" alt="" />
<span class="caption">Fig. 31.</span>
</div>

<p>Let us supply now, to effect the spontaneous discharge,
the jar <i>F</i> with knobs of the same size and
at the same distance apart as those of the jar <i>L</i> (Fig.
31). If we find, then, that five discharges of the unit
jar take place before one spontaneous discharge of the
jar <i>F</i> occurs, plainly the jar <i>F</i>, for equal distances between
the knobs of the two jars, equal striking distances,
is able to hold five times the quantity of electricity
that <i>L</i> can, that is, has five times the <i>capacity</i>
of <i>L</i>.<a name="FNanchor_29_29" id="FNanchor_29_29"></a><a href="#Footnote_29_29" class="fnanchor">[29]</a></p>

<div class="figright" style="width: 400px;">
<img src="images/i_126.jpg" width="400" height="345" alt="" />
<span class="caption">Fig. 32.</span>
</div>

<p>We will now replace the unit jar <i>L</i>, with which we
measure electricity, so to speak, <i>into</i> the jar <i>F</i>, by a
Franklin's pane, consisting of two parallel flat metal
plates (Fig. 32), separated only by air. If here, for
example, thirty spontaneous discharges of the pane are
sufficient to fill the jar, ten discharges will be found
sufficient if the air-space between the two plates be
filled with a cake of sulphur. Hence, the capacity
of a Franklin's pane of sulphur is about three times
greater than that of one of the same shape and size
made of air, or, as it is the custom to say, the specific
inductive capacity of sulphur (that of air being taken
as the unit) is about 3.<a name="FNanchor_30_30" id="FNanchor_30_30"></a><a href="#Footnote_30_30" class="fnanchor">[30]</a> We are here arrived at a
very simple fact, which clearly shows us the significance
of the number called <i>dielectric constant</i>, or <i>specific
inductive capacity</i>, the knowledge of which is so important
for the theory of submarine cables.</p>

<p>Let us consider a jar <i>A</i>, which is charged with a
certain quantity of electricity. We can discharge the
jar directly. But we can also discharge the jar <i>A</i>
(Fig. 33) partly into a jar <i>B</i>, by connecting the two
outer coatings with each other. In this operation a
portion of the quantity of electricity passes, accompanied
by sparks, into the jar <i>B</i>, and we now find both
jars charged.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_127-1.jpg" width="300" height="283" alt="" />
<span class="caption">Fig. 33.</span>
</div>

<div class="figright" style="width: 300px;">
<img src="images/i_127-s.jpg" width="300" height="214" alt="" />
<span class="caption">Fig. 34.</span>
</div>

<p>It may be shown as follows that the conception of
a constant quantity of electricity can be regarded as
the expression of a pure fact. Picture to yourself any
sort of electrical conductor (Fig. 34); cut it up into a
large number of small pieces, and place these pieces by
means of an insulated rod at a distance of one centimetre
from an electrical body which acts with unit of
force on an equal and like-constituted body at the
same distance. Take the sum of the forces which
this last body exerts on the single pieces of the conductor.
The sum of these forces will be the quantity
of electricity on the whole conductor. It remains the
same, whether we change the form and the size of the
conductor, or whether we bring it near or move it
away from a second electrical conductor, so long as we
keep it insulated, that is, do not discharge it.</p>

<p>A basis of reality for the notion of electric quantity
seems also to present itself from another quarter.
If a current, that is, in the usual view, a definite
quantity of electricity per second, is sent through a
column of acidulated water; in the direction of the
positive stream, hydrogen, but in the opposite direction,
oxygen is liberated at the extremities of the column.
For a given quantity of electricity a given quantity
of oxygen appears. You may picture the column
of water as a column of hydrogen and a column of
oxygen, fitted into each other, and may say the electric
current is a chemical current and <i>vice versa</i>. Although
this notion is more difficult to adhere to in the field of
statical electricity and with non-decomposable conductors,
its further development is by no means hopeless.</p>

<p>The concept quantity of electricity, thus, is not so
aerial as might appear, but is able to conduct us with
certainty through a multitude of varied phenomena,
and is suggested to us by the facts in almost palpable
form. We can collect electrical force in a body, measure
it out with one body
into another, carry it
over from one body into
another, just as we can
collect a liquid in a vessel,
measure it out with
one vessel into another,
or pour it from one into
another.</p>

<p>For the analysis of
mechanical phenomena,
a metrical notion, derived
from experience,
and bearing the designation <i>work</i>, has proved itself
useful. A machine can be set in motion only when
the forces acting on it can perform work.</p>

<div class="figleft" style="width: 350px;">
<img src="images/i_129.jpg" width="350" height="431" alt="" />
<span class="caption">Fig. 35.</span>
</div>

<p>Let us consider, for example, a wheel and axle
(Fig. 35) having the radii 1 and 2 metres, loaded respectively
with the weights 2 and 1 kilogrammes. On
turning the wheel and axle, the 1 kilogramme-weight,
let us say, sinks two metres, while the 2 kilogramme-weight
rises one metre. On both sides the product</p>

<p class="center">KGR. M. KGR. M.</p>

<p class="center"> 1 × 2 = 2 × 1.
</p>

<p>is equal. So long as this is so, the wheel and axle will
not move of itself. But if we take such loads, or so
change the radii of the wheels, that this product (kgr.
× metre) on displacement is in excess on one side,
that side will sink. As we see, this product is characteristic
for mechanical events, and for this reason has
been invested with a special name, <i>work</i>.</p>

<p>In all mechanical processes, and as all physical
processes present a mechanical side, in all physical
processes, work plays a determinative part. Electrical
forces, also, produce only changes in which work is performed.
To the extent that forces come into play in
electrical phenomena, electrical phenomena, be they
what they may, extend into the domain of mechanics
and are subject to the laws which hold in this domain.
The universally adopted measure of work,
now, is the product of the force into the distance
through which it acts, and in the C. G. S. system, the
unit of work is the action through one centimetre of
a force which would impart in one second to a
gramme-mass a velocity-increment of one centimetre,
that is, in round numbers, the action through a centimetre
of a pressure equal to the weight of a milligramme.
From a positively charged body, electricity,
yielding to the force of repulsion and performing work,
flows off to the earth, providing conducting connexions
exist. To a negatively charged body, on the other
hand, the earth under the same circumstances gives
off positive electricity. The electrical work possible
in the interaction of a body with the earth, characterises
the electrical condition of that body. We will call
the work which must be expended on the unit quantity
of positive electricity to raise it from the earth to the
body <i>K</i> the <i>potential</i> of the body <i>K</i>.<a name="FNanchor_31_31" id="FNanchor_31_31"></a><a href="#Footnote_31_31" class="fnanchor">[31]</a></p>

<p>We ascribe to the body <i>K</i> in the C. G. S. system
the potential +1, if we must expend the unit of work
to raise the positive electrostatic unit of electric quantity
from the earth to that body; the potential -1, if
we gain in this procedure the unit of work; the potential
0, if no work at all is performed in the operation.</p>

<p>The different parts of one and the same electrical
conductor in electrical equilibrium have the same potential,
for otherwise the electricity would perform
work and move about upon the conductor, and equilibrium
would not have existed. Different conductors of
equal potential, put in connexion with one another, do
not exchange electricity any more than bodies of equal
temperature in contact exchange heat, or in connected
vessels, in which the same pressures exist, liquids
flow from one vessel to the other. Exchange of electricity
takes place only between conductors of different
potentials, but in conductors of given form and position
a definite difference of potential is necessary for
a spark, which pierces the insulating air, to pass
between them.</p>

<p>On being connected, every two conductors assume
at once the same potential. With this the means
is given of determining the potential of a conductor
through the agency of a second conductor expressly
adapted to the purpose called an electrometer, just as
we determine the temperature of a body with a thermometer.
The values of the potentials of bodies obtained
in this way simplify vastly our analysis of their
electrical behavior, as will be evident from what has
been said.</p>

<p>Think of a positively charged conductor. Double
all the electrical forces exerted by this conductor on a
point charged with unit quantity, that is, double the
quantity at each point, or what is the same thing,
double the total charge. Plainly, equilibrium still subsists.
But carry, now, the positive electrostatic unit
towards the conductor. Everywhere we shall have to
overcome double the force of repulsion we did before,
everywhere we shall have to expend double the work.
By doubling the charge of the conductor a double potential
has been produced. Charge and potential go
hand in hand, are proportional. Consequently, calling
the total quantity of electricity of a conductor <i>Q</i>
and its potential <i>V</i>, we can write: <i>Q</i> = <i>CV</i>, where <i>C</i>
stands for a constant, the import of which will be understood
simply from noting that <i>C</i> = <i>Q</i>/<i>V</i>.<a name="FNanchor_32_32" id="FNanchor_32_32"></a><a href="#Footnote_32_32" class="fnanchor">[32]</a> But the
division of a number representing the units of quantity
of a conductor by the number representing its
units of potential tells us the quantity which falls to
the share of the unit of potential. Now the number
<i>C</i> here we call the capacity of a conductor, and have
substituted, thus, in the place of the old relative determination
of capacity, an absolute determination.<a name="FNanchor_33_33" id="FNanchor_33_33"></a><a href="#Footnote_33_33" class="fnanchor">[33]</a></p>

<p>In simple cases the connexion between charge, potential,
and capacity is easily ascertained. Our conductor,
let us say, is a sphere of radius <i>r</i>, suspended
free in a large body of air. There being no other conductors
in the vicinity, the charge <i>q</i> will then distribute
itself uniformly upon the surface of the sphere, and
simple geometrical considerations yield for its potential
the expression <i>V</i> = <i>q</i>/<i>r</i>. Hence, <i>q</i>/<i>V</i> = <i>r</i>; that is,
the capacity of a sphere is measured by its radius, and
in the C. G. S. system in centimetres.<a name="FNanchor_34_34" id="FNanchor_34_34"></a><a href="#Footnote_34_34" class="fnanchor">[34]</a> It is clear
also, since a potential is a quantity divided by a length,
that a quantity divided by a potential must be a length.</p>

<p>Imagine (Fig. 36) a jar composed of two concentric
conductive spherical shells of the radii <i>r</i> and <i>r</i><sub>1</sub>,
having only air between them. Connecting the outside
sphere with the earth, and charging the inside
sphere by means of a thin, insulated wire passing
through the first, with the quantity <i>Q</i>, we shall have
<i>V</i> = (<i>r</i><sub>1</sub>-<i>r</i>)/(<i>r</i><sub>1</sub><i>r</i>)<i>Q</i>, and for the capacity in this case
(<i>r</i><sub>1</sub><i>r</i>)/(<i>r</i><sub>1</sub>-<i>r</i>), or, to take
a specific example, if <i>r</i> = 16
and <i>r</i><sub>1</sub> = 19, a capacity of
about 100 centimetres.</p>

<div class="figright" style="width: 350px;">
<img src="images/i_134.jpg" width="350" height="352" alt="" />
<span class="caption">Fig. 36.</span>
</div>

<p>We shall now use these
simple cases for illustrating
the principle by which
capacity and potential are
determined. First, it is
clear that we can use the
jar composed of concentric spheres with its known capacity
as our unit jar and by means of this ascertain,
in the manner above laid down, the capacity of any
given jar <i>F</i>. We find, for example, that 37 discharges
of this unit jar of the capacity 100, just charges the
jar investigated at the same striking distance, that is,
at the same potential. Hence, the capacity of the jar
investigated is 3700 centimetres. The large battery
of the Prague physical laboratory, which consists of
sixteen such jars, all of nearly equal size, has a capacity,
therefore, of something like 50,000 centimetres,
or the capacity of a sphere, a kilometre in diameter,
freely suspended in atmospheric space. This remark
distinctly shows us the great superiority which Leyden
jars possess for the storage of electricity as compared
with common conductors. In fact, as Faraday pointed
out, jars differ from simple conductors mainly by their
great capacity.</p>

<div class="figcenter" style="width: 450px;">
<img src="images/i_135.jpg" width="450" height="356" alt="" />
<span class="caption">Fig. 37.</span>
</div>

<p>For determining potential, imagine the inner coating
of a jar <i>F</i>, the outer coating of which communicates
with the ground, connected by a long, thin wire
with a conductive sphere <i>K</i> placed free in a large atmospheric
space, compared with whose dimensions
the radius of the sphere vanishes. (Fig. 37.) The
jar and the sphere assume at once the same potential.
But on the surface of the sphere, if that be sufficiently
far removed from all other conductors, a uniform layer
of electricity will be found. If the sphere, having the
radius <i>r</i>, contains the charge <i>q</i>, its potential is <i>V</i> = <i>q</i>/<i>r</i>.
If the upper half of the sphere be severed from the
lower half and equilibrated on a balance with one of
whose beams it is connected by silk threads, the upper
half will be repelled from the lower half with the force
<i>P</i> = <i>q</i><sup>2</sup>/8<i>r</i><sup>2</sup> = 1/8<i>V</i><sup>2</sup>. This repulsion <i>P</i> may be counter-balanced
by additional weights placed on the beam-end,
and so ascertained. The potential is then <i>V</i> =
&#8730;(8<i>P</i>).<a name="FNanchor_35_35" id="FNanchor_35_35"></a><a href="#Footnote_35_35" class="fnanchor">[35]</a></p>

<p>That the potential is proportional to the square
root of the force is not difficult to see. A doubling or
trebling of the potential means that the charge of all
the parts is doubled or trebled; hence their combined
power of repulsion quadrupled or nonupled.</p>

<p>Let us consider a special case. I wish to produce
the potential 40 on the sphere. What additional weight
must I give to the half sphere in grammes that the
force of repulsion shall maintain the balance in exact
equilibrium? As a gramme weight is approximately
equivalent to 1000 units of force, we have only the
following simple example to work out: 40×40 = 8×
1000.<i>x</i>, where <i>x</i> stands for the number of grammes.
In round numbers we get <i>x</i> = 0.2 gramme. I charge
the jar. The balance is deflected; I have reached,
or rather passed, the potential 40, and you see when I
discharge the jar the associated spark.<a name="FNanchor_36_36" id="FNanchor_36_36"></a><a href="#Footnote_36_36" class="fnanchor">[36]</a></p>

<p>The striking distance between the knobs of a machine
increases with the difference of the potential,
although not proportionately to that difference. The
striking distance increases faster than the potential
difference. For a distance between the knobs of one
centimetre on this machine the difference of potential
is 110. It can easily be increased tenfold. Of the
tremendous differences of potential which occur in
nature some idea may be obtained from the fact that
the striking distances of lightning in thunder-storms
is counted by miles. The differences of potential in
galvanic batteries are considerably smaller than those
of our machine, for it takes fully one hundred elements
to give a spark of microscopic striking distance.</p>
<p><span class="pagenum"><a name="Page_128" id="Page_128">[Pg 128]</a></span></p><p><span class="pagenum"><a name="Page_129" id="Page_129">[Pg 129]</a></span></p><p><span class="pagenum"><a name="Page_130" id="Page_130">[Pg 130]</a></span></p><p><span class="pagenum"><a name="Page_131" id="Page_131">[Pg 131]</a></span></p><p><span class="pagenum"><a name="Page_132" id="Page_132">[Pg 132]</a></span></p><p><span class="pagenum"><a name="Page_133" id="Page_133">[Pg 133]</a></span></p><p><span class="pagenum"><a name="Page_134" id="Page_134">[Pg 134]</a></span></p>
<hr class="tb" />

<p>We shall now employ the ideas reached to shed
some light upon another important relation between
electrical and mechanical phenomena. We shall investigate
what is the potential <i>energy</i>, or the <i>store of
work</i>, contained in a charged conductor, for example,
in a jar.</p>

<p>If we bring a quantity of electricity up to a conductor,
or, to speak less pictorially, if we generate by
work electrical force in a conductor, this force is able
to produce anew the work by which it was generated.
How great, now, is the energy or capacity for work of
a conductor of known charge <i>Q</i> and known potential
<i>V</i>?</p>

<p>Imagine the given charge <i>Q</i> divided into very small
parts <i>q</i>, <i>q</i><sub>1</sub>, <i>q</i><sub>2</sub> ..., and these little parts successively
carried up to the conductor. The first very small
quantity <i>q</i> is brought up without any appreciable work
and produces by its presence a small potential <i>V</i><sub>'</sub>. To
bring up the second quantity, accordingly, we must do
the work <i>q</i><sub>'</sub><i>V</i><sub>'</sub>, and similarly for the quantities which
follow the work <i>q</i><sub>''</sub><i>V</i><sub>''</sub>, <i>q</i><sub>'''</sub><i>V</i><sub>'''</sub>, and so forth. Now,
as the potential rises proportionately to the quantities
added until the value <i>V</i> is reached, we have, agreeably
to the graphical representation of Fig. 38, for the
total work performed,</p>

<p class="center"><i>W</i> = 1/2<i>QV</i>,
</p>

<p>which corresponds to the total energy of the charged
conductor. Using the equation <i>Q</i> = <i>CV</i>, where <i>C</i>
stands for capacity, we also have,</p>

<p class="center"><i>W</i> = 1/2<i>CV</i><sup>2</sup>, or <i>W</i> = <i>Q</i><sup>2</sup>/2<i>C</i>.
</p>

<p>It will be helpful, perhaps, to elucidate this idea
by an analogy from the province of mechanics. If we
pump a quantity of liquid, <i>Q</i>, gradually into a cylindrical
vessel (Fig. 39), the level of the liquid in the
vessel will gradually rise. The more we have pumped
in, the greater the pressure we must overcome, or the
higher the level to which we must lift the liquid. The
stored-up work is rendered again available when the
heavy liquid <i>Q</i>, which reaches up to the level <i>h</i>, flows
out. This work <i>W</i> corresponds to the fall of the whole
liquid weight <i>Q</i>, through the distance <i>h</i>/2 or through
the altitude of its centre of gravity. We have</p>

<p class="center"><i>W</i> = 1/2<i>Qh</i>.
</p>

<p>Further, since <i>Q</i> = <i>Kh</i>, or since the weight of the
liquid and the height <i>h</i> are proportional, we get also</p>

<p class="center"><i>W</i> = 1/2<i>Kh</i><sup>2</sup> and <i>W</i> = <i>Q</i><sup>2</sup>/2<i>K</i>.
</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_139-1.jpg" width="300" height="208" alt="" />
<span class="caption">Fig. 38.</span>
</div>

<div class="figright" style="width: 300px;">
<img src="images/i_139-2.jpg" width="300" height="283" alt="" />
<span class="caption">Fig. 39.</span>
</div>

<p>As a special case let us consider our jar. Its capacity
is <i>C</i> = 3700, its potential <i>V</i> = 110; accordingly,
its quantity <i>Q</i> = <i>CV</i> = 407,000 electrostatic units and
its energy <i>W</i> = 1/2<i>QV</i> = 22,385,000 C. G. S. units of
work.</p>

<p>The unit of work of the C. G. S. system is not readily
appreciable by the senses, nor does it well admit of
representation, as we are accustomed to work with
weights. Let us adopt, therefore, as our unit of work
the gramme-centimetre, or the gravitational pressure
of a gramme-weight through the distance of a centimetre,
which in round numbers is 1000 times greater
than the unit assumed above; in this case, our numerical
result will be approximately 1000 times smaller.
Again, if we pass, as more familiar in practice, to the
kilogramme-metre as our unit of work, our unit, the
distance being increased a hundred fold, and the weight
a thousand fold, will be 100,000 times larger. The
numerical result expressing the work done is in this
case 100,000 times less, being in round numbers 0.22
kilogramme-metre. We can obtain a clear idea of the
work done here by letting a kilogramme-weight fall 22
centimetres.</p>

<p>This amount of work, accordingly, is performed on
the charging of the jar, and on its discharge appears
again, according to the circumstances, partly as sound,
partly as a mechanical disruption of insulators, partly
as light and heat, and so forth.</p>

<p>The large battery of the Prague physical laboratory,
with its sixteen jars charged to equal potentials,
furnishes, although the effect of the discharge is imposing,
a total amount of work of only three kilogramme-metres.</p>

<p>In the development of the ideas above laid down
we are not restricted to the method there pursued; in
fact, that method was selected only as one especially
fitted to familiarise us with the phenomena. On the
contrary, the connexion of the physical processes is so
multifarious that we can come at the same event from
very different directions. Particularly are electrical
phenomena connected with all other physical events;
and so intimate is this connexion that we might justly
call the study of electricity the theory of the general
connexion of physical processes.</p>

<p>With respect to the principle of the conservation
of energy which unites electrical with mechanical phenomena,
I should like to point out briefly two ways of
following up the study of this connexion.</p>

<p>A few years ago Professor Rosetti, taking an influence-machine,
which he set in motion by means of
weights alternately in the electrical and non-electrical
condition with the same velocities, determined the
mechanical work expended in the two cases and was
thus enabled, after deducting the work of friction, to
ascertain the mechanical work consumed in the development
of the electricity.</p>

<p>I myself have made this experiment in a modified,
and, as I think, more advantageous form. Instead
of determining the work of friction by special trial, I
arranged my apparatus so that it was eliminated of itself
in the measurement and could consequently be
neglected. The so-called fixed disk of the machine, the
axis of which is placed vertically, is suspended somewhat
like a chandelier by three vertical threads of
equal lengths <i>l</i> at a distance <i>r</i> from the axis. Only
when the machine is excited does this fixed disk, which
represents a Prony's brake, receive, through its reciprocal
action with the rotating disk, a deflexion <i>&#945;</i> and a
moment of torsion which is expressed by <i>D</i> = <i>(Pr<sup>2</sup>/l)&#945;</i>,
where <i>P</i> is the weight of the disk.<a name="FNanchor_37_37" id="FNanchor_37_37"></a><a href="#Footnote_37_37" class="fnanchor">[37]</a> The angle <i>&#945;</i> is
determined by a mirror set in the disk. The work expended
in <i>n</i> rotations is given by <i>2n&#960;D</i>.</p>

<p>If we close the machine, as Rosetti did, we obtain
a continuous current which has all the properties of a
very weak galvanic current; for example, it produces a
deflexion in a multiplier which we interpose, and so
forth. We can directly ascertain, now, the mechanical
work expended in the maintenance of this current.</p>

<p>If we charge a jar by means of a machine, the energy
of the jar employed in the production of sparks,
in the disruption of the insulators, etc., corresponds
to a part only of the mechanical work expended, a
second part of it being consumed in the arc which
forms the circuit.<a name="FNanchor_38_38" id="FNanchor_38_38"></a><a href="#Footnote_38_38" class="fnanchor">[38]</a> This machine, with the interposed
jar, affords in miniature a picture of the transference
of force, or more properly of work. And in fact nearly
the same laws hold here for the economical coefficient
as obtain for large dynamo-machines.</p>

<p>Another means of investigating electrical energy is
by its transformation into heat. A long time ago
(1838), before the mechanical theory of heat had attained
its present popularity, Riess performed experiments
in this field with the help of his electrical
air-thermometer or thermo-electrometer.</p>

<div class="figleft" style="width: 200px;">
<img src="images/i_143.jpg" width="200" height="320" alt="" />
<span class="caption">Fig. 40.</span>
</div>

<p>If the discharge be conducted
through a fine wire
passing through the globe of
the air-thermometer, a development
of heat is observed
proportional to the expression
above-discussed <i>W</i> = 1/2<i>QV</i>.
Although the total energy has
not yet been transformed
into measurable heat by this
means, in as much as a portion
is left behind in the spark in the air outside the thermometer,
still everything tends to show that the total
heat developed in all parts of the conductor and along
all the paths of discharge is the equivalent of the work
1/2<i>QV</i>.</p>

<p>It is not important here whether the electrical energy
is transformed all at once or partly, by degrees.
For example, if of two equal jars one is charged with
the quantity <i>Q</i> at the potential <i>V</i> the energy present
is 1/2<i>QV</i>. If the first jar be discharged into the second,
<i>V</i>, since the capacity is now doubled, falls to <i>V</i>/2.
Accordingly, the energy 1/4<i>QV</i> remains, while 1/4<i>QV</i> is
transformed in the spark of discharge into heat. The
remainder, however, is equally distributed between
the two jars so that each on discharge is still able to
transform 1/8<i>QV</i> into heat.</p>
<p><span class="pagenum"><a name="Page_135" id="Page_135">[Pg 135]</a></span></p><p><span class="pagenum"><a name="Page_136" id="Page_136">[Pg 136]</a></span></p><p><span class="pagenum"><a name="Page_137" id="Page_137">[Pg 137]</a></span></p>
<hr class="tb" />

<p>We have here discussed electricity in the limited
phenomenal form in which it was known to the inquirers
before Volta, and which has been called, perhaps
not very felicitously, "statical electricity." It is
evident, however, that the nature of electricity is everywhere
one and the same; that a substantial difference
between statical and galvanic electricity does not exist.
Only the quantitative circumstances in the two provinces
are so widely different that totally new aspects
of phenomena may appear in the second, for example,
magnetic effects, which in the first remained unnoticed,
whilst, <i>vice versa</i>, in the second field statical attractions
and repulsions are scarcely appreciable. As a fact,
we can easily show the magnetic effect of the current
of discharge of an influence-machine on the galvanoscope
although we could hardly have made the original
discovery of the magnetic effects with this current.
The statical distant action of the wire poles of
a galvanic element also would hardly have been noticed
had not the phenomenon been known from a
different quarter in a striking form.</p>

<p>If we wished to characterise the two fields in their
chief and most general features, we should say that in
the first, high potentials and small quantities come
into play, in the second small potentials and large
quantities. A jar which is discharging and a galvanic
element deport themselves somewhat like an air-gun
and the bellows of an organ. The first gives forth
suddenly under a very high pressure a small quantity
of air; the latter liberates gradually under a very slight
pressure a large quantity of air.</p>

<p>In point of principle, too, nothing prevents our retaining
the electrostatical units in the domain of galvanic
electricity and in measuring, for example, the
strength of a current by the number of electrostatic
units which flow per second through its cross-section.
But this would be in a double aspect impractical. In
the first place, we should totally neglect the magnetic
facilities for measurement so conveniently offered by
the current, and substitute for this easy means a method
which can be applied only with difficulty and is not
capable of great exactness. In the second place our
units would be much too small, and we should find
ourselves in the predicament of the astronomer who
attempted to measure celestial distances in metres instead
of in radii of the earth and the earth's orbit; for
the current which by the magnetic C. G. S. standard
represents the unit, would require a flow of some
30,000,000,000 electrostatic units per second through
its cross-section. Accordingly, different units must
be adopted here. The development of this point, however,
lies beyond my present task.</p>




<h2><a name="ON_THE_PRINCIPLE_OF_THE_CONSERVATION" id="ON_THE_PRINCIPLE_OF_THE_CONSERVATION">ON THE PRINCIPLE OF THE CONSERVATION
OF ENERGY.</a><a name="FNanchor_39_39" id="FNanchor_39_39"></a><a href="#Footnote_39_39" class="fnanchor">[39]</a></h2>


<p>In a popular lecture, distinguished for its charming
simplicity and clearness, which Joule delivered in
the year 1847,<a name="FNanchor_40_40" id="FNanchor_40_40"></a><a href="#Footnote_40_40" class="fnanchor">[40]</a> that famous physicist declares that the
living force which a heavy body has acquired by its
descent through a certain height and which it carries
with it in the form of the velocity with which it is impressed,
is the <i>equivalent</i> of the attraction of gravity
through the space fallen through, and that it would be
"absurd" to assume that this living force could be destroyed
without some restitution of that equivalent.
He then adds: "You will therefore be surprised to
hear that until very <i>recently</i> the universal opinion has
been that living force could be absolutely and irrevocably
destroyed at any one's option." Let us add
that to-day, after forty-seven years, the <i>law of the conservation
of energy</i>, wherever civilisation exists, is accepted
as a fully established truth and receives the
widest applications in all domains of natural science.</p>

<p><span class="pagenum"><a name="Page_138" id="Page_138">[Pg 138]</a></span></p><p>The fate of all momentous discoveries is similar.
On their first appearance they are regarded by the
majority of men as errors. J. R. Mayer's work on the
principle of energy (1842) was rejected by the first
physical journal of Germany; Helmholtz's treatise
(1847) met with no better success; and even Joule, to
judge from an intimation of Playfair, seems to have
encountered difficulties with his first publication (1843).
Gradually, however, people are led to see that the new
view was long prepared for and ready for enunciation,
only that a few favored minds had perceived it much
earlier than the rest, and in this way the opposition of
the majority is overcome. With proofs of the fruitfulness
of the new view, with its success, confidence
in it increases. The majority of the men who employ
it cannot enter into a deep-going analysis of it; for
them, its success is its proof. It can thus happen that
a view which has led to the greatest discoveries, like
Black's theory of caloric, in a subsequent period in a
province where it does not apply may actually become
an obstacle to progress by its blinding our eyes to facts
which do not fit in with our favorite conceptions. If
a theory is to be protected from this dubious rôle, the
grounds and motives of its evolution and existence
must be examined from time to time with the utmost
care.</p>

<p>The most multifarious physical changes, thermal,<span class="pagenum"><a name="Page_139" id="Page_139">[Pg 139]</a></span>
electrical, chemical, and so forth, can be brought
about by mechanical work. When such alterations
are reversed they yield anew the mechanical work in
exactly the quantity which was required for the production
of the part reversed. This is the <i>principle of
the conservation of energy</i>; "energy" being the term
which has gradually come into use for that "indestructible
something" of which the measure is mechanical
<i>work</i>.</p>

<p>How did we acquire this idea? What are the
sources from which we have drawn it? This question
is not only of interest in itself, but also for the important
reason above touched upon. The opinions which
are held concerning the foundations of the law of energy
still diverge very widely from one another. Many
trace the principle to the impossibility of a perpetual
motion, which they regard either as sufficiently proved
by experience, or as self-evident. In the province of
pure mechanics the impossibility of a perpetual motion,
or the continuous production of <i>work</i> without
some <i>permanent</i> alteration, is easily demonstrated. Accordingly,
if we start from the theory that all physical
processes are purely <i>mechanical</i> processes, motions of
molecules and atoms, we embrace also, by this <i>mechanical</i>
conception of physics, the impossibility of a
perpetual motion in the <i>whole</i> physical domain. At
present this view probably counts the most adherents.
Other inquirers, however, are for accepting only a
purely <i>experimental</i> establishment of the law of energy.</p><p><span class="pagenum"><a name="Page_140" id="Page_140">[Pg 140]</a></span></p>

<p>It will appear, from the discussion to follow, that
<i>all</i> the factors mentioned have co-operated in the development
of the view in question; but that in addition
to them a logical and purely formal factor, hitherto
little considered, has also played a very important part.</p>


<h3>I. THE PRINCIPLE OF THE EXCLUDED PERPETUAL
MOTION.</h3>

<p>The law of energy in its modern form is not identical
with the principle of the excluded perpetual motion,
but it is very closely
related to it. The latter
principle, however, is by
no means new, for in the
province of mechanics it
has controlled for centuries
the thoughts and investigations
of the greatest thinkers. Let us convince
ourselves of this by the study of a few historical
examples.</p>

<div class="figright" style="width: 300px;">
<img src="images/i_150.jpg" width="300" height="272" alt="" />
<span class="caption">Fig. 41.</span>
</div>

<p>S. Stevinus, in his famous work <i>Hypomnemata mathematica</i>,
Tom. IV, <i>De statica</i>, (Leyden, 1605, p. 34),
treats of the equilibrium of bodies on inclined planes.</p>

<p>Over a triangular prism <i>ABC</i>, one side of which,
<i>AC</i>, is horizontal, an endless cord or chain is slung,
to which at equal distances apart fourteen balls of
equal weight are attached, as represented in cross-section
in Figure 41. Since we can imagine the lower<span class="pagenum"><a name="Page_141" id="Page_141">[Pg 141]</a></span>
symmetrical part of the cord <i>ABC</i> taken away, Stevinus
concludes that the four balls on <i>AB</i> hold in equilibrium
the two balls on <i>BC</i>. For if the equilibrium were
for a moment disturbed, it could never subsist: the
cord would keep moving round forever in the same direction,&mdash;we
should have a perpetual motion. He says:</p>

<blockquote><p>"But if this took place, our row or ring of balls would come
once more into their original position, and from the same cause the
eight globes to the left would again be heavier than the six to the
right, and therefore those eight would sink a second time and these
six rise, and all the globes would keep up, of themselves, <i>a continuous
and unending motion, which is false</i>."<a name="FNanchor_41_41" id="FNanchor_41_41"></a><a href="#Footnote_41_41" class="fnanchor">[41]</a></p></blockquote>

<p>Stevinus, now, easily derives from this principle
the laws of equilibrium on the inclined plane and numerous
other fruitful consequences.</p>

<p>In the chapter "Hydrostatics" of
the same work, page 114, Stevinus sets
up the following principle: "Aquam
datam, datum sibi intra aquam locum
servare,"&mdash;a given mass of water preserves
within water its given place.</p>

<div class="figright" style="width: 150px;">
<img src="images/i_151.jpg" width="150" height="172" alt="" />
<span class="caption">Fig. 42.</span>
</div>

<p>This principle is demonstrated as follows (see Fig.
42):</p>

<blockquote><p>"For, assuming it to be possible by natural means, let us suppose
that A does not preserve the place assigned to it, but sinks
down to D. This being posited, the water which succeeds A will,
for the same reason, also flow down to <i>D</i>; <i>A</i> will be forced out of
its place in <i>D</i>; and thus this body of water, for the conditions in it
are everywhere the same, <i>will set up a perpetual motion, which is
absurd</i>."<a name="FNanchor_42_42" id="FNanchor_42_42"></a><a href="#Footnote_42_42" class="fnanchor">[42]</a></p></blockquote><p><span class="pagenum"><a name="Page_142" id="Page_142">[Pg 142]</a></span></p>

<p>From this all the principles of hydrostatics are deduced.
On this occasion Stevinus also first develops
the thought so fruitful for modern analytical mechanics
that the equilibrium of a system is not destroyed by
the addition of rigid connexions. As we know, the
principle of the conservation of the centre of gravity
is now sometimes deduced from D'Alembert's principle
with the help of that remark. If we were to reproduce
Stevinus's demonstration to-day, we should have
to change it slightly. We find no difficulty in imagining
the cord on the prism possessed of unending uniform
motion if all hindrances are thought away, but
we should protest against the assumption of an accelerated
motion or even against that of a uniform motion,
if the resistances were not removed. Moreover,
for greater precision of proof, the string of balls might
be replaced by a heavy homogeneous cord of infinite
flexibility. But all this does not affect in the least the
historical value of Stevinus's thoughts. It is a fact,
Stevinus deduces apparently much simpler truths from
the principle of an impossible perpetual motion.</p>
<p><span class="pagenum"><a name="Page_143" id="Page_143">[Pg 143]</a></span></p>
<p>In the process of thought which conducted Galileo
to his discoveries at the end of the sixteenth century,
the following principle plays an important part, that
a body in virtue of the velocity acquired in its descent
can rise exactly as high as it fell. This principle,
which appears frequently and with much clearness in
Galileo's thought, is simply another form of the principle
of excluded perpetual motion, as we shall see it
is also in Huygens.</p>

<p>Galileo, as we know, arrived at the law of uniformly
accelerated motion by <i>a priori</i> considerations, as that
law which was the "simplest and most natural," after
having first assumed a different law which he was compelled
to reject. To verify his law he executed experiments
with falling bodies on inclined planes, measuring
the times of descent by the weights of the water
which flowed out of a small orifice in a large vessel.
In this experiment he assumes as a fundamental principle,
that the velocity acquired in descent down an
inclined plane always corresponds to the vertical height
descended through, a conclusion which for him is the
immediate outcome of the fact that a body which has
fallen down one inclined plane can, with the velocity it
has acquired, rise on another plane of any inclination
only to the same vertical height. This principle of
the height of ascent also led him, as it seems, to the
law of inertia. Let us hear his own masterful words
in the <i>Dialogo terzo</i> (<i>Opere</i>, Padova, 1744, Tom. III).
On page 96 we read:</p><p><span class="pagenum"><a name="Page_144" id="Page_144">[Pg 144]</a></span></p>

<blockquote><p>"I take it for granted that the velocities acquired by a body
in descent down planes of different inclinations are equal if the
heights of those planes are equal."<a name="FNanchor_43_43" id="FNanchor_43_43"></a><a href="#Footnote_43_43" class="fnanchor">[43]</a></p></blockquote>

<p>Then he makes Salviati say in the dialogue:<a name="FNanchor_44_44" id="FNanchor_44_44"></a><a href="#Footnote_44_44" class="fnanchor">[44]</a></p>

<blockquote><p>"What you say seems very probable, but I wish to go further
and by an experiment so to increase the probability of it that it shall
amount almost to absolute demonstration. Suppose this sheet of
paper to be a vertical wall, and from a nail driven in it a ball of lead
weighing two or three ounces to hang by a very fine thread <i>AB</i> four
or five feet long. (Fig. 43.) On the wall mark a horizontal line <i>DC</i>
perpendicular to the vertical <i>AB</i>, which latter ought to hang about
two inches from the wall. If now the thread <i>AB</i> with the ball
attached take the position <i>AC</i> and the ball be let go, you will see
the ball first descend through the arc <i>CB</i> and passing beyond
<i>B</i> rise through the arc
<i>BD</i> almost to the level
of the line <i>CD</i>, being
prevented from reaching
it exactly by the resistance
of the air and
of the thread. From
this we may truly conclude
that its impetus at
the point <i>B</i>, acquired by
its descent through the
arc <i>CB</i>, is sufficient to
urge it through a similar arc <i>BD</i> to the same height. Having
performed this experiment and repeated it several times, let us
drive in the wall, in the projection of the vertical <i>AB</i>, as at <i>E</i> or
at <i>F</i>, a nail five or six inches long, so that the thread <i>AC</i>, carrying
as before the ball through the arc <i>CB</i>, at the moment it reaches
the position <i>AB</i>, shall strike the nail <i>E</i>, and the ball be thus compelled
to move up the arc <i>BG</i> described about <i>E</i> as centre.
Then we shall see what the same impetus will here accomplish,
acquired now as before at the same point <i>B</i>, which then drove the
same moving body through the arc <i>BD</i> to the height of the horizontal
<i>CD</i>. Now gentlemen, you will be pleased to see the ball
rise to the horizontal line at the point <i>G</i>, and the same thing also
happen if the nail be placed lower as at <i>F</i>, in which case the ball
would describe the arc <i>BJ</i>, always terminating its ascent precisely
at the line <i>CD</i>. If the nail be placed so low that the length of
thread below it does not reach to the height of <i>CD</i> (which would
happen if <i>F</i> were nearer <i>B</i> than to the intersection of <i>AB</i> with the
horizontal <i>CD</i>), then the thread will wind itself about the nail.
This experiment leaves no room for doubt as to the truth of the
supposition. For as the two arcs <i>CB</i>, <i>DB</i> are equal and similarly
situated, the momentum acquired in the descent of the arc <i>CB</i> is
the same as that acquired in the descent of the arc <i>DB</i>; but the
momentum acquired at <i>B</i> by the descent through the arc <i>CB</i> is capable
of driving up the same moving body through the arc <i>BD</i>;
hence also the momentum acquired in the descent <i>DB</i> is equal to
that which drives the same moving body through the same arc
from <i>B</i> to <i>D</i>, so that in general every momentum acquired in the
descent of an arc is equal to that which causes the same moving
body to ascend through the same arc; but all the momenta which
cause the ascent of all the arcs <i>BD</i>, <i>BG</i>, <i>BJ</i>, are equal since they
are made by the same momentum acquired in the descent <i>CB</i>, as
the experiment shows: therefore all the momenta acquired in the
descent of the arcs <i>DB</i>, <i>GB</i>, <i>JB</i> are equal."</p></blockquote>

<p><span class="pagenum"><a name="Page_145" id="Page_145">[Pg 145]</a></span></p>
<div class="figcenter" style="width: 300px;">
<img src="images/i_155.jpg" width="300" height="246" alt="" />
<span class="caption">Fig. 43.</span>
</div>
<p><span class="pagenum"><a name="Page_146" id="Page_146">[Pg 146]</a></span></p>

<p>The remark relative to the pendulum may be applied
to the inclined plane and leads to the law of inertia.
We read on page 124:<a name="FNanchor_45_45" id="FNanchor_45_45"></a><a href="#Footnote_45_45" class="fnanchor">[45]</a></p>
<p><span class="pagenum"><a name="Page_147" id="Page_147">[Pg 147]</a></span></p>
<blockquote><p>"It is plain now that a movable body, starting from rest at <i>A</i>
and descending down the inclined plane <i>AB</i>, acquires a velocity
proportional to the increment of its time: the velocity possessed
at <i>B</i> is the greatest of the velocities acquired, and by its nature
immutably impressed, provided all causes of new acceleration or
retardation are taken away: I say acceleration, having in view its
possible further progress along the plane extended; retardation, in
view of the possibility of its being reversed and made to mount the
ascending plane <i>BC</i>. But in the horizontal plane <i>GH</i> its equable
motion, according to its velocity as acquired in the descent from <i>A</i>
to <i>B</i>, will be continued <i>ad infinitum</i>." (Fig. 44.)</p></blockquote>

<div class="figcenter" style="width: 400px;">
<img src="images/i_157.jpg" width="400" height="156" alt="" />
<span class="caption">Fig. 44.</span>
</div>

<p>Huygens, upon whose shoulders the mantel of Galileo
fell, forms a sharper conception of the law of inertia
and generalises the principle respecting the heights of
ascent which was so fruitful in Galileo's hands. He
employs the latter principle in the solution of the problem
of the centre of oscillation and is perfectly clear in
the statement that the principle respecting the heights
of ascent is identical with the principle of the excluded
perpetual motion.</p>

<p>The following important passages then occur (Hugenii,
<i>Horologium oscillatorium, pars secunda</i>). <i>Hypotheses</i>:</p>

<p><span class="pagenum"><a name="Page_148" id="Page_148">[Pg 148]</a></span></p><blockquote><p>"If gravity did not exist, nor the atmosphere obstruct the motions
of bodies, a body would keep up forever the motion once impressed
upon it, with equable velocity, in a straight line."<a name="FNanchor_46_46" id="FNanchor_46_46"></a><a href="#Footnote_46_46" class="fnanchor">[46]</a></p></blockquote>

<p>In part four of the <i>Horologium de centro oscillationis</i>
we read:</p>

<blockquote><p>"If any number of weights be set in motion by the force of
gravity, the common centre of gravity of the weights as a whole
cannot possibly rise higher than the place which it occupied when
the motion began.</p>

<p>"That this hypothesis of ours may arouse no scruples, we
will state that it simply imports, what no one has ever denied, that
heavy bodies do not move <i>upwards</i>.&mdash;And truly if the devisers of
the new machines who make such futile attempts to construct a
perpetual motion would acquaint themselves with this principle,
they could easily be brought to see their errors and to understand
that the thing is utterly impossible by mechanical means."<a name="FNanchor_47_47" id="FNanchor_47_47"></a><a href="#Footnote_47_47" class="fnanchor">[47]</a></p></blockquote>

<p>There is possibly a Jesuitical mental reservation
contained in the words "mechanical means." One
might be led to believe from them that Huygens held
a non-mechanical perpetual motion for possible.</p>

<p>The generalisation of Galileo's principle is still
more clearly put in Prop. IV of the same chapter:</p>

<blockquote><p>"If a pendulum, composed of several weights, set in motion
from rest, complete any part of its full oscillation, and from that
point onwards, the individual weights, with their common connexions
dissolved, change their acquired velocities upwards and ascend
as far as they can, the common centre of gravity of all will be carried
up to the same altitude with that which it occupied before the
beginning of the oscillation."<a name="FNanchor_48_48" id="FNanchor_48_48"></a><a href="#Footnote_48_48" class="fnanchor">[48]</a></p></blockquote>

<p><span class="pagenum"><a name="Page_149" id="Page_149">[Pg 149]</a></span></p><p>On this last principle now, which is a generalisation,
applied to a system of masses, of one of Galileo's
ideas respecting a single mass and which from Huygens's
explanation we recognise as the principle of excluded
perpetual motion, Huygens grounds his theory
of the centre of oscillation. Lagrange characterises
this principle as precarious and is rejoiced at James
Bernoulli's successful attempt, in 1681, to reduce the
theory of the centre of oscillation to the laws of the
lever, which appeared to him clearer. All the great
inquirers of the seventeenth and eighteenth centuries
broke a lance on this problem, and it led ultimately,
in conjunction with the principle of virtual velocities,
to the principle enunciated by D'Alembert in 1743 in
his <i>Traité de dynamique</i>, though previously employed
in a somewhat different form by Euler and Hermann.</p>

<p>Furthermore, the Huygenian principle respecting
the heights of ascent became the foundation of the
"law of the conservation of living force," as that was
enunciated by John and Daniel Bernoulli and employed<span class="pagenum"><a name="Page_150" id="Page_150">[Pg 150]</a></span>
with such signal success by the latter in his
<i>Hydrodynamics</i>. The theorems of the Bernoullis differ
in form only from Lagrange's expression in the <i>Analytical
Mechanics</i>.</p>

<p>The manner in which Torricelli reached his famous
law of efflux for liquids leads again to our principle.
Torricelli assumed that the liquid which flows out of
the basal orifice of a vessel cannot by its velocity of
efflux ascend to a greater height than its level in the
vessel.</p>

<p>Let us next consider a point which belongs to pure
mechanics, the history of the principle of <i>virtual motions</i>
or <i>virtual velocities</i>. This principle was not first
enunciated, as is usually stated, and as Lagrange also
asserts, by Galileo, but earlier, by Stevinus. In his
<i>Trochleostatica</i> of the above-cited work, page 72, he
says:</p>

<blockquote><p>"Observe that this axiom of statics holds good here:</p>

<p>"As the space of the body acting is to the space of the body
acted upon, so is the power of the body acted upon to the power of
the body acting."<a name="FNanchor_49_49" id="FNanchor_49_49"></a><a href="#Footnote_49_49" class="fnanchor">[49]</a></p></blockquote>

<p>Galileo, as we know, recognised the truth of the
principle in the consideration of the simple machines,
and also deduced the laws of the equilibrium of liquids
from it.</p>

<p>Torricelli carries the principle back to the properties
of the centre of gravity. The condition controlling<span class="pagenum"><a name="Page_151" id="Page_151">[Pg 151]</a></span>
equilibrium in a simple machine, in which power
and load are represented by weights, is that the common
centre of gravity of the weights shall not sink.
Conversely, if the centre of gravity cannot sink equilibrium
obtains, because heavy bodies of themselves
do not move upwards. In this form the principle of
virtual velocities is identical with Huygens's principle
of the impossibility of a perpetual motion.</p>

<p>John Bernoulli, in 1717, first perceived the universal
import of the principle of virtual movements for all
systems; a discovery stated in a letter to Varignon.
Finally, Lagrange gives a general demonstration of
the principle and founds upon it his whole <i>Analytical
Mechanics</i>. But this general demonstration is based
after all upon Huygens and Torricelli's remarks. Lagrange,
as is known, conceives simple pulleys arranged
in the directions of the forces of the system, passes a
cord through these pulleys, and appends to its free
extremity a weight which is a common measure of all
the forces of the system. With no difficulty, now, the
number of elements of each pulley may be so chosen
that the forces in question shall be replaced by them.
It is then clear that if the weight at the extremity cannot
sink, equilibrium subsists, because heavy bodies
cannot of themselves move upwards. If we do not go
so far, but wish to abide by Torricelli's idea, we may
conceive every individual force of the system replaced
by a special weight suspended from a cord passing
over a pulley in the direction of the force and attached<span class="pagenum"><a name="Page_152" id="Page_152">[Pg 152]</a></span>
at its point of application. Equilibrium subsists then
when the common centre of gravity of all the weights
together cannot sink. The fundamental supposition
of this demonstration is plainly the impossibility of a
perpetual motion.</p>

<p>Lagrange tried in every way to supply a proof free
from extraneous elements and fully satisfactory, but
without complete success. Nor were his successors
more fortunate.</p>

<p>The whole of mechanics, thus, is based upon an
idea, which, though unequivocal, is yet unwonted and
not coequal with the other principles and axioms of
mechanics. Every student of mechanics, at some stage
of his progress, feels the uncomfortableness of this
state of affairs; every one wishes it removed; but seldom
is the difficulty stated in words. Accordingly, the
zealous pupil of the science is highly rejoiced when he
reads in a master like Poinsot (<i>Théorie générale de
l'équilibre et du mouvement des systèmes</i>) the following
passage, in which that author is giving his opinion of
the <i>Analytical Mechanics</i>:</p>

<blockquote><p>"In the meantime, because our attention in that work was first
wholly engrossed with the consideration of its beautiful development
of mechanics, which seemed to spring complete from a single
formula, we naturally believed that the science was completed or
that it only remained to seek the demonstration of the principle of
virtual velocities. But that quest brought back all the difficulties
that we had overcome by the principle itself. That law so general,
wherein are mingled the vague and unfamiliar ideas of infinitely
small movements and of perturbations of equilibrium, only grew<span class="pagenum"><a name="Page_153" id="Page_153">[Pg 153]</a></span>
obscure upon examination; and the work of Lagrange supplying
nothing clearer than the march of analysis, we saw plainly that the
clouds had only appeared lifted from the course of mechanics because
they had, so to speak, been gathered at the very origin of that
science.</p>

<p>"At bottom, a general demonstration of the principle of virtual
velocities would be equivalent to the establishment of the whole
of mechanics upon a different basis: for the demonstration of a
law which embraces a whole science is neither more nor less than
the reduction of that science to another law just as general, but
evident, or at least more simple than the first, and which, consequently,
would render that useless."<a name="FNanchor_50_50" id="FNanchor_50_50"></a><a href="#Footnote_50_50" class="fnanchor">[50]</a></p></blockquote>

<p>According to Poinsot, therefore, a proof of the
principle of virtual movements is tantamount to a total
rehabilitation of mechanics.</p>

<p>Another circumstance of discomfort to the mathematician
is, that in the historical form in which mechanics
at present exists, dynamics is founded on
statics, whereas it is desirable that in a science which
pretends to deductive completeness the more special
statical theorems should be deducible from the more
general dynamical principles.</p>
<p><span class="pagenum"><a name="Page_154" id="Page_154">[Pg 154]</a></span></p>
<p>In fact, a great master, Gauss, gave expression to
this desire in his presentment of the principle of least
constraint (Crelle's <i>Journal für reine und angewandte
Mathematik</i>, Vol. IV, p. 233) in the following words:
"Proper as it is that in the gradual development of a
science, and in the instruction of individuals, the easy
should precede the difficult, the simple the complex,
the special the general, yet the mind, when once it has
reached a higher point of view, demands the contrary
course, in which all statics shall appear simply as a
special case of mechanics." Gauss's own principle,
now, possesses all the requisites of universality, but
its difficulty is that it is not immediately intelligible
and that Gauss deduced it with the help of D'Alembert's
principle, a procedure which left matters where
they were before.</p>

<p>Whence, now, is derived this strange part which
the principle of virtual motion plays in mechanics?
For the present I shall only make this reply. It would
be difficult for me to tell the difference of impression
which Lagrange's proof of the principle made on me
when I first took it up as a student and when I subsequently
resumed it after having made historical researches.
It first appeared to me insipid, chiefly on
account of the pulleys and the cords which did not fit
in with the mathematical view, and whose action I
would much rather have discovered from the principle<span class="pagenum"><a name="Page_155" id="Page_155">[Pg 155]</a></span>
itself than have taken for granted. But now that I
have studied the history of the science I cannot imagine
a more beautiful demonstration.</p>

<p>In fact, through all mechanics it is this self-same
principle of excluded perpetual motion which accomplishes
almost all, which displeased Lagrange, but
which he still had to employ, at least tacitly, in his own
demonstration. If we give this principle its proper
place and setting, the paradox is explained.</p>

<p>The principle of excluded perpetual motion is thus
no new discovery; it has been the guiding idea, for
three hundred years, of all the great inquirers. But
the principle cannot properly be <i>based</i> upon mechanical
perceptions. For long before the development of
mechanics the conviction of its truth existed and even
contributed to that development. Its power of conviction,
therefore, must have more universal and
deeper roots. We shall revert to this point.</p>


<h3>II. MECHANICAL PHYSICS.</h3>

<p>It cannot be denied that an unmistakable tendency
has prevailed, from Democritus to the present day, to
explain <i>all</i> physical events <i>mechanically</i>. Not to mention
earlier obscure expressions of that tendency we
read in Huygens the following:<a name="FNanchor_51_51" id="FNanchor_51_51"></a><a href="#Footnote_51_51" class="fnanchor">[51]</a></p>

<blockquote><p>"There can be no doubt that light consists of the <i>motion</i> of a
certain substance. For if we examine its production, we find that
here on earth it is principally fire and flame which engender it, both
of which contain beyond doubt bodies which are in rapid movement,
since they dissolve and destroy many other bodies more solid
than they: while if we regard its effects, we see that when light is
accumulated, say by concave mirrors, it has the property of combustion
just as fire has, that is to say, it disunites the parts of
bodies, which is assuredly a proof of <i>motion</i>, at least in the <i>true
philosophy</i>, in which the causes of all natural effects are conceived
as <i>mechanical</i> causes. Which in my judgment must be accomplished
or all hope of ever understanding physics renounced."<a name="FNanchor_52_52" id="FNanchor_52_52"></a><a href="#Footnote_52_52" class="fnanchor">[52]</a></p></blockquote><p><span class="pagenum"><a name="Page_156" id="Page_156">[Pg 156]</a></span></p>

<p>S. Carnot,<a name="FNanchor_53_53" id="FNanchor_53_53"></a><a href="#Footnote_53_53" class="fnanchor">[53]</a> in introducing the principle of excluded
perpetual motion into the theory of heat, makes the
following apology:</p>

<blockquote><p>"It will be objected here, perhaps, that a perpetual motion
proved impossible for <i>purely mechanical actions</i>, is perhaps not so
when the influence of <i>heat</i> or of electricity is employed. But can
phenomena of heat or electricity be thought of as due to anything
else than to <i>certain motions of bodies</i>, and as such must they not be
subject to the general laws of mechanics?"<a name="FNanchor_54_54" id="FNanchor_54_54"></a><a href="#Footnote_54_54" class="fnanchor">[54]</a></p></blockquote>

<p><span class="pagenum"><a name="Page_157" id="Page_157">[Pg 157]</a></span></p><p>These examples, which might be multiplied by
quotations from recent literature indefinitely, show
that a tendency to explain all things mechanically
actually exists. This tendency is also intelligible.
Mechanical events as simple motions in space and
time best admit of observation and pursuit by the help
of our highly organised senses. We reproduce mechanical
processes almost without effort in our imagination.
Pressure as a circumstance that produces motion
is very familiar to us from daily experience. All
changes which the individual personally produces in
his environment, or humanity brings about by means
of the arts in the world, are effected through the instrumentality
of <i>motions</i>. Almost of necessity, therefore,
motion appears to us as the most important
physical factor. Moreover, mechanical properties may
be discovered in all physical events. The sounding
bell trembles, the heated body expands, the electrified
body attracts other bodies. Why, therefore, should
we not attempt to grasp all events under their mechanical
aspect, since that is so easily apprehended and
most accessible to observation and measurement? In
fact, no objection <i>is</i> to be made to the attempt to elucidate
the properties of physical events by mechanical
<i>analogies</i>.</p>

<p>But modern physics has proceeded <i>very far</i> in this
direction. The point of view which Wundt represents
in his excellent treatise <i>On the Physical Axioms</i> is probably<span class="pagenum"><a name="Page_158" id="Page_158">[Pg 158]</a></span>
shared by the majority of physicists. The axioms
of physics which Wundt sets up are as follows:</p>

<p>1. All natural causes are motional causes.</p>

<p>2. Every motional cause lies outside the object
moved.</p>

<p>3. All motional causes act in the direction of the
straight line of junction, and so forth.</p>

<p>4. The effect of every cause persists.</p>

<p>5. Every effect involves an equal countereffect.</p>

<p>6. Every effect is equivalent to its cause.</p>

<p>These principles might be studied properly enough
as fundamental principles of mechanics. But when
they are set up as axioms of physics, their enunciation
is simply tantamount to a negation of all events except
motion.</p>

<p>According to Wundt, all changes of nature are
mere changes of place. All causes are motional causes
(page 26). Any discussion of the philosophical grounds
on which Wundt supports his theory would lead us
deep into the speculations of the Eleatics and the
Herbartians. Change of place, Wundt holds, is the
<i>only</i> change of a thing in which a thing remains identical
with itself. If a thing changed <i>qualitatively</i>, we
should be obliged to imagine that something was annihilated
and something else created in its place, which
is not to be reconciled with our idea of the identity of
the object observed and of the indestructibility of
matter. But we have only to remember that the Eleatics
encountered difficulties of exactly the same sort<span class="pagenum"><a name="Page_159" id="Page_159">[Pg 159]</a></span>
in motion. Can we not also imagine that a thing is
destroyed in <i>one</i> place and in <i>another</i> an exactly similar
thing created? After all, do we really know <i>more</i>
why a body leaves one place and appears in another,
than why a <i>cold</i> body grows <i>warm</i>? Granted that we
had a perfect knowledge of the mechanical processes
of nature, could we and should we, for that reason,
<i>put out of the world</i> all other processes that we do not
understand? On this principle it would really be the
simplest course to deny the existence of the whole
world. This is the point at which the Eleatics ultimately
arrived, and the school of Herbart stopped
little short of the same goal.</p>

<p>Physics treated in this sense supplies us simply
with a diagram of the world, in which we do not know
reality again. It happens, in fact, to men who give
themselves up to this view for many years, that the
world of sense from which they start as a province of
the greatest familiarity, suddenly becomes, in their
eyes, the supreme "world-riddle."</p>

<p>Intelligible as it is, therefore, that the efforts of
thinkers have always been bent upon the "reduction
of all physical processes to the motions of atoms," it
must yet be affirmed that this is a chimerical ideal.
This ideal has often played an effective part in popular
lectures, but in the workshop of the serious inquirer
it has discharged scarcely the least function.
What has really been achieved in mechanical physics
is either the <i>elucidation</i> of physical processes by more<span class="pagenum"><a name="Page_160" id="Page_160">[Pg 160]</a></span>
familiar <i>mechanical analogies</i>, (for example, the theories
of light and of electricity,) or the exact <i>quantitative</i>
ascertainment of the connexion of mechanical processes
with other physical processes, for example, the
results of thermodynamics.</p>


<h3>III. THE PRINCIPLE OF ENERGY IN PHYSICS.</h3>

<p>We can know only from <i>experience</i> that mechanical
processes produce other physical transformations, or
<i>vice versa</i>. The attention was first directed to the connexion
of mechanical processes, especially the performance
of work, with changes of thermal conditions
by the invention of the steam-engine, and by its great
technical importance. Technical interests and the
need of scientific lucidity meeting in the mind of S.
Carnot led to the remarkable development from which
thermodynamics flowed. It is simply <i>an accident of
history</i> that the development in question was not connected
with the practical applications of <i>electricity</i>.</p>

<p>In the determination of the maximum quantity of
<i>work</i> that, generally, a heat-machine, or, to take a
special case, a steam-engine, can perform with the
expenditure of a <i>given</i> amount of heat of combustion,
Carnot is guided by mechanical analogies. A body can
do work on being heated, by expanding under pressure.
But to do this the body must receive heat from a <i>hotter</i>
body. Heat, therefore, to do work, must pass from a
hotter body to a colder body, just as water must fall
from a higher level to a lower level to put a mill-wheel<span class="pagenum"><a name="Page_161" id="Page_161">[Pg 161]</a></span>
in motion. Differences of temperature, accordingly,
represent forces able to do work exactly as do differences
of height in heavy bodies. Carnot pictures to
himself an ideal process in which no heat flows away
unused, that is, without doing work. With a given expenditure
of heat, accordingly, this process furnishes
the maximum of work. An analogue of the process
would be a mill-wheel which scooping its water out of
a higher level would slowly carry it to a lower level
without the loss of a drop. A peculiar property of the
process is, that with the expenditure of the same work
the water can be raised again exactly to its original
level. This property of <i>reversibility</i> is also shared by
the process of Carnot. His process also can be reversed
by the expenditure of the same amount of work,
and the heat again brought back to its original temperature
level.</p>

<p>Suppose, now, we had <i>two</i> different reversible processes
<i>A</i>, <i>B</i>, such that in <i>A</i> a quantity of heat, <i>Q</i>,
flowing off from the temperature <i>t</i><sub>1</sub> to the lower temperature
<i>t</i><sub>2</sub> should perform the work <i>W</i>, but in <i>B</i> under
the same circumstances it should perform a greater
quantity of work <i>W</i> + <i>W'</i>; then, we could join <i>B</i> in
the sense assigned and <i>A</i> in the reverse sense into a
<i>single</i> process. Here <i>A</i> would reverse the transformation
of heat produced by <i>B</i> and would leave a surplus
of work <i>W'</i>, produced, so to speak, from nothing.
The combination would present a perpetual motion.</p>

<p>With the feeling, now, that it makes little difference<span class="pagenum"><a name="Page_162" id="Page_162">[Pg 162]</a></span>
whether the mechanical laws are broken directly
or indirectly (by processes of heat), and convinced of
the existence of a <i>universal</i> law-ruled connexion of nature,
Carnot here excludes for the first time from the
province of <i>general</i> physics the possibility of a perpetual
motion. <i>But it follows, then, that the quantity
of work W, produced by the passage of a quantity of heat
Q from a temperature t<sub>1</sub> to a temperature t<sub>2</sub>, is independent
of the nature of the substances as also of the character
of the process, so far as that is unaccompanied by
loss, but is wholly dependent upon the temperature t<sub>1</sub>, t<sub>2</sub>.</i></p>

<p>This important principle has been fully confirmed
by the special researches of Carnot himself (1824), of
Clapeyron (1834), and of Sir William Thomson (1849),
now Lord Kelvin. The principle was reached <i>without
any assumption whatever</i> concerning the nature of heat,
simply by the exclusion of a perpetual motion. Carnot,
it is true, was an adherent of the theory of Black, according
to which the sum-total of the quantity of heat
in the world is constant, but so far as his investigations
have been hitherto considered the decision on
this point is of no consequence. Carnot's principle
led to the most remarkable results. W. Thomson
(1848) founded upon it the ingenious idea of an "absolute"
scale of temperature. James Thomson (1849)
conceived a Carnot process to take place with water
freezing under pressure and, therefore, performing
work. He discovered, thus, that the freezing point is
lowered 0·0075° Celsius by every additional atmosphere<span class="pagenum"><a name="Page_163" id="Page_163">[Pg 163]</a></span>
of pressure. This is mentioned merely as an
example.</p>

<p>About twenty years after the publication of Carnot's
book a further advance was made by J. R. Mayer
and J. P. Joule. Mayer, while engaged as a physician
in the service of the Dutch, observed, during a
process of bleeding in Java, an unusual redness of the
venous blood. In agreement with Liebig's theory of
animal heat he connected this fact with the diminished
loss of heat in warmer climates, and with the diminished
expenditure of organic combustibles. The total
expenditure of heat of a man at rest must be equal to
the total heat of combustion. But since <i>all</i> organic actions,
even the mechanical actions, must be set down
to the credit of the heat of combustion, some connexion
must exist between mechanical work and expenditure
of heat.</p>

<p>Joule started from quite similar convictions concerning
the galvanic battery. A heat of association
equivalent to the consumption of the zinc can be made
to appear in the galvanic cell. If a current is set up,
a part of this heat appears in the conductor of the
current. The interposition of an apparatus for the
decomposition of water causes a part of this heat to
disappear, which on the burning of the explosive gas
formed, is reproduced. If the current runs an electromotor,
a portion of the heat again disappears, which,
on the consumption of the work by friction, again
makes its appearance. Accordingly, both the heat<span class="pagenum"><a name="Page_164" id="Page_164">[Pg 164]</a></span>
produced and the work produced, appeared to Joule
also as connected with the consumption of material.
The thought was therefore present, both to Mayer and
to Joule, of regarding heat and work as equivalent
quantities, so connected with each other that what is
lost in one form universally appears in another. The
result of this was a <i>substantial</i> conception of heat and
of work, and <i>ultimately a substantial conception of energy</i>.
Here every physical change of condition is regarded
as energy, the destruction of which generates
work or equivalent heat. An electric charge, for example,
is energy.</p>

<p>In 1842 Mayer had calculated from the physical
constants then universally accepted that by the disappearance
of one kilogramme-calorie 365 kilogramme-metres
of work could be performed, and <i>vice versa</i>.
Joule, on the other hand, by a long series of delicate
and varied experiments beginning in 1843 ultimately
determined the mechanical equivalent of the kilogramme-calorie,
more exactly, as 425 kilogramme-metres.</p>

<p>If we estimate every change of physical condition
by the <i>mechanical work</i> which can be performed upon
the <i>disappearance</i> of that condition, and call this measure
<i>energy</i>, then we can measure all physical changes
of condition, no matter how different they may be,
with the same common measure, and say: <i>the sum-total
of all energy remains constant</i>. This is the form that
the principle of excluded perpetual motion received at<span class="pagenum"><a name="Page_165" id="Page_165">[Pg 165]</a></span>
the hands of Mayer, Joule, Helmholtz, and W. Thomson
in its extension to the whole domain of physics.</p>

<p>After it had been proved that heat must <i>disappear</i>
if mechanical work was to be done at its expense,
Carnot's principle could no longer be regarded as a
complete expression of the facts. Its improved form
was first given, in 1850, by Clausius, whom Thomson
followed in 1851. It runs thus: "If a quantity of heat
<i>Q'</i> is transformed into work in a reversible process,
<i>another</i> quantity of heat <i>Q</i> of the absolute<a name="FNanchor_55_55" id="FNanchor_55_55"></a><a href="#Footnote_55_55" class="fnanchor">[55]</a> temperature
<i>T<sub>1</sub></i> is lowered to the absolute temperature <i>T<sub>2</sub></i>."
Here <i>Q'</i> is dependent only on <i>Q</i>, <i>T<sub>1</sub></i>, <i>T<sub>2</sub></i>, but is independent
of the substances used and of the character of
the process, so far as that is unaccompanied by loss.
Owing to this last fact, it is sufficient to find the relation
which obtains for some one well-known physical
substance, say a gas, and some definite simple process.
The relation found will be the one that holds
generally. We get, thus,</p>

<p class="center"><i>Q'/(Q' + Q)</i> = <i>(T<sub>1</sub>-T<sub>2</sub>)/T<sub>1</sub></i> (1)
</p>

<p>that is, the quotient of the available heat <i>Q'</i> transformed
into work divided by the sum of the transformed
and transferred heats (the total sum used), the
so-called <i>economical coefficient</i> of the process, is,</p>

<p><i>(T<sub>1</sub>-T<sub>2</sub>)/T<sub>1</sub></i>.</p>
<p><span class="pagenum"><a name="Page_166" id="Page_166">[Pg 166]</a></span></p>

<h3>IV. THE CONCEPTIONS OF HEAT.</h3>

<p>When a cold body is put in contact with a warm
body it is observed that the first body is warmed and
that the second body is cooled. We may say that the
first body is warmed <i>at the expense of</i> the second body.
This suggests the notion of a thing, or heat-substance,
which passes from the one body to the other. If two
masses of water <i>m</i>, <i>m'</i>, of unequal temperatures, be
put together, it will be found, upon the rapid equalisation
of the temperatures, that the respective changes
of temperatures <i>u</i> and <i>u'</i> are inversely proportional to
the masses and of opposite signs, so that the algebraical
sum of the products is,</p>

<p class="center"><i>mu</i> + <i>m'u'</i> = 0.
</p>

<p>Black called the products <i>mu</i>, <i>m'u'</i>, which are decisive
for our knowledge of the process, <i>quantities of heat</i>.
We may form a very clear <i>picture</i> of these products
by conceiving them with Black as measures of the
quantities of some substance. But the essential thing
is not this picture but the <i>constancy</i> of the sum of these
products in simple processes of conduction. If a quantity
of heat disappears at one point, an equally large
quantity will make its appearance at some other point.
The retention of this idea leads to the discovery of
specific heat. Black, finally, perceives that also something
else may appear for a vanished quantity of heat,
namely: the fusion or vaporisation of a definite quantity<span class="pagenum"><a name="Page_167" id="Page_167">[Pg 167]</a></span>
of matter. He adheres here still to this favorite
view, though with some freedom, and considers the
vanished quantity of heat as still present, but as <i>latent</i>.</p>

<p>The generally accepted notion of a caloric, or heat-stuff,
was strongly shaken by the work of Mayer and
Joule. If the quantity of heat can be increased and
diminished, people said, heat cannot be a substance,
but must be a <i>motion</i>. The subordinate part of this
statement has become much more popular than all the
rest of the doctrine of energy. But we may convince
ourselves that the motional conception of heat is now
as unessential as was formerly its conception as a substance.
Both ideas were favored or impeded solely
by accidental historical circumstances. It does not
follow that heat is not a substance from the fact that
a mechanical equivalent exists for quantity of heat.
We will make this clear by the following question
which bright students have sometimes put to me. Is
there a mechanical equivalent of electricity as there is
a mechanical equivalent of heat? Yes, and no. There
is no mechanical equivalent of <i>quantity</i> of electricity
as there is an equivalent of <i>quantity</i> of heat, because
the same quantity of electricity has a very different
capacity for work, according to the circumstances in
which it is placed; but there <i>is</i> a mechanical equivalent
of electrical energy.</p>

<p>Let us ask another question. Is there a mechanical
equivalent of water? No, there is no mechanical
equivalent of quantity of water, but there is a mechanical<span class="pagenum"><a name="Page_168" id="Page_168">[Pg 168]</a></span>
equivalent of weight of water multiplied by
its distance of descent.</p>

<p>When a Leyden jar is discharged and work thereby
performed, we do not picture to ourselves that the
quantity of electricity disappears as work is done, but
we simply assume that the electricities come into different
positions, equal quantities of positive and negative
electricity being united with one another.</p>

<p>What, now, is the reason of this difference of view
in our treatment of heat and of electricity? The reason
is purely historical, wholly conventional, and, what
is still more important, is wholly indifferent. I may
be allowed to establish this assertion.</p>

<p>In 1785 Coulomb constructed his torsion balance,
by which he was enabled to measure the repulsion of
electrified bodies. Suppose we have two small balls,
<i>A</i>, <i>B</i>, which over their whole extent are similarly
electrified. These two balls will exert on one another,
at a certain distance <i>r</i> of their centres, a certain repulsion
<i>p</i>. We bring into contact with <i>B</i> now a ball
<i>C</i>, suffer both to be equally electrified, and then measure
the repulsion of <i>B</i> from <i>A</i> and of <i>C</i> from <i>A</i> at the
same distance <i>r</i>. The sum of these repulsions is again
<i>p</i>. Accordingly something has remained constant.
If we ascribe this effect to a substance, then we infer
naturally its constancy. But the essential point of the
exposition is the divisibility of the electric force <i>p</i> and
not the simile of substance.</p>

<p>In 1838 Riess constructed his electrical air-thermometer<span class="pagenum"><a name="Page_169" id="Page_169">[Pg 169]</a></span>
(the thermoelectrometer). This gives a measure
of the quantity of heat produced by the discharge of
jars. This quantity of heat is not proportional to the
quantity of electricity contained in the jar by Coulomb's
measure, but if <i>Q</i> be this quantity and <i>C</i> be the
capacity, is proportional to <i>Q</i><sup>2</sup>/2<i>C</i>, or, more simply
still, to the energy of the charged jar. If, now, we
discharge the jar completely through the thermometer,
we obtain a certain quantity of heat, <i>W</i>. But if
we make the discharge through the thermometer into
a second jar, we obtain a quantity less than <i>W</i>. But we
may obtain the remainder by completely discharging
both jars through the air-thermometer, when it will
again be proportional to the energy of the two jars. On
the first, incomplete discharge, accordingly, a part of
the electricity's capacity for work was lost.</p>

<p>When the charge of a jar produces heat its energy
is changed and its value by Riess's thermometer is decreased.
But by Coulomb's measure the quantity remains
unaltered.</p>

<p>Now let us imagine that Riess's thermometer had
been invented before Coulomb's torsion balance, which
is not a difficult feat, since both inventions are independent
of each other; what would be more natural than
that the "quantity" of electricity contained in a jar
should be measured by the heat produced in the thermometer?
But then, this so-called quantity of electricity
would decrease on the production of heat or on
the performance of work, whereas it now remains unchanged;<span class="pagenum"><a name="Page_170" id="Page_170">[Pg 170]</a></span>
in that case, therefore, electricity would not
be a <i>substance</i> but a <i>motion</i>, whereas now it is still a
substance. The reason, therefore, why we have other
notions of electricity than we have of heat, is purely
historical, accidental, and conventional.</p>

<p>This is also the case with other physical things.
Water does not disappear when work is done. Why?
Because we measure quantity of water with scales, just
as we do electricity. But suppose the capacity of
water for work were called quantity, and had to be
measured, therefore, by a mill instead of by scales;
then this quantity also would disappear as it performed
the work. It may, now, be easily conceived
that many substances are not so easily got at as water.
In that case we should be unable to carry out the one
kind of measurement with the scales whilst many other
modes of measurement would still be left us.</p>

<p>In the case of heat, now, the historically established
measure of "quantity" is accidentally the work-value
of the heat. Accordingly, its quantity disappears when
work is done. But that heat is not a substance follows
from this as little as does the opposite conclusion that
it is a substance. In Black's case the quantity of heat
remains constant because the heat passes into no <i>other</i>
form of energy.</p>

<p>If any one to-day should still wish to think of heat
as a substance, we might allow that person this liberty
with little ado. He would only have to assume that
that which we call quantity of heat was the energy of<span class="pagenum"><a name="Page_171" id="Page_171">[Pg 171]</a></span>
a substance whose quantity remained unaltered, but
whose energy changed. In point of fact we might
much better say, in analogy with the other terms of
physics, energy of heat, instead of quantity of heat.</p>

<p>When we wonder, therefore, at the discovery that
heat is motion, we wonder at something that was never
discovered. It is perfectly indifferent and possesses
not the slightest scientific value, whether we think of
heat as a substance or not. The fact is, heat behaves
in some connexions like a substance, in others not.
Heat is latent in steam as oxygen is latent in water.</p>


<h3>V. THE CONFORMITY IN THE DEPORTMENT OF THE
ENERGIES.</h3>

<p>The foregoing reflexions will gain in lucidity from
a consideration of the conformity which obtains in the
behavior of all energies, a point to which I called attention
long ago.<a name="FNanchor_56_56" id="FNanchor_56_56"></a><a href="#Footnote_56_56" class="fnanchor">[56]</a></p>

<p>A weight <i>P</i> at a height <i>H</i><sub>1</sub> represents an energy
<i>W</i><sub>1</sub> = <i>PH</i><sub>1</sub>. If we suffer the weight to sink to a lower
height <i>H</i><sub>2</sub>, during which work is done, and the work
done is employed in the production of living force,
heat, or an electric charge, in short, is transformed,
then the energy <i>W</i><sub>2</sub> = <i>PH</i><sub>2</sub> is still <i>left</i>. The equation
subsists</p>

<p><span class="pagenum"><a name="Page_172" id="Page_172">[Pg 172]</a></span></p><p><i>W</i><sub>1</sub>/<i>H</i><sub>1</sub> = <i>W</i><sub>2</sub>/<i>H</i><sub>2</sub>, (2)
or, denoting the <i>transformed</i> energy by <i>W</i>' = <i>W</i><sub>1</sub>-<i>W</i><sub>2</sub>
and the <i>transferred</i> energy, that transported to the
lower level, by <i>W</i> = <i>W</i><sub>2</sub>,</p>

<p class="center"><i>W</i>'/(<i>W</i>' + <i>W</i>) = (<i>H</i><sub>1</sub>-<i>H</i><sub>2</sub>)/<i>H</i><sub>1</sub>, (3)
</p>

<p>an equation in all respects analogous to equation (1)
at page 165. The property in question, therefore, is
by no means peculiar to heat. Equation (2) gives the
relation between the energy taken from the higher
level and that deposited on the lower level (the energy
left behind); it says that these <i>energies</i> are proportional
to the <i>heights of the levels</i>. An equation analogous
to equation (2) may be set up for <i>every</i> form of
energy; hence the equation which corresponds to
equation (3), and so to equation (1), may be regarded
as valid for every form. For electricity, for example,
<i>H</i><sub>1</sub>, <i>H</i><sub>2</sub> signify the potentials.</p>

<p>When we observe for the first time the agreement
here indicated in the transformative law of the energies,
it appears surprising and unexpected, for we do
not perceive at once its reason. But to him who pursues
the comparative historical method that reason
will not long remain a secret.</p>

<p>Since Galileo, mechanical work, though long under
a different name, has been a <i>fundamental concept</i> of
mechanics, as also a very important notion in the applied
sciences. The transformation of work into living<span class="pagenum"><a name="Page_173" id="Page_173">[Pg 173]</a></span>
force, and of living force into work, suggests directly
the notion of energy&mdash;the idea having been first
fruitfully employed by Huygens, although Thomas
Young first called it by the <i>name</i> of "energy." Let
us add to this the constancy of weight (really the constancy
of mass) and we shall see that with respect to
mechanical energy it is involved in the very definition
of the term that the capacity for work or the potential
energy of a weight is proportional to the height of the
level at which it is, in the geometrical sense, and that
it decreases on the lowering of the weight, on transformation,
proportionally to the height of the level.
The zero level here is wholly arbitrary. With this,
equation (2) is given, from which all the other forms
follow.</p>

<p>When we reflect on the tremendous start which
mechanics had over the other branches of physics, it
is not to be wondered at that the attempt was always
made to apply the notions of that science wherever
this was possible. Thus the notion of mass, for example,
was imitated by Coulomb in the notion of
quantity of electricity. In the further development
of the theory of electricity, the notion of work was
likewise immediately introduced in the theory of potential,
and heights of electrical level were measured
by the work of unit of quantity raised to that level.
But with this the preceding equation with all its consequences
is given for electrical energy. The case with
the other energies was similar.</p><p><span class="pagenum"><a name="Page_174" id="Page_174">[Pg 174]</a></span></p>

<p><i>Thermal</i> energy, however, appears as a special
case. Only by the peculiar experiments mentioned
could it be discovered that heat is an energy. But the
measure of this energy by Black's quantity of heat is
the outcome of fortuitous circumstances. In the first
place, the accidental slight variability of the capacity
for heat <i>c</i> with the temperature, and the accidental
slight deviation of the usual thermometrical scales
from the scale derived from <i>the tensions of gases</i>, brings
it about that the notion "quantity of heat" can be set
up and that the quantity of heat <i>ct</i> corresponding to a
difference of temperature <i>t</i> is nearly proportional to
the energy of the heat. It is a quite accidental historical
circumstance that Amontons hit upon the idea
of measuring temperature by the tension of a gas. It
is certain in this that he did not think of the work of
the heat.<a name="FNanchor_57_57" id="FNanchor_57_57"></a><a href="#Footnote_57_57" class="fnanchor">[57]</a> But the numbers standing for temperature,
thus, are made proportional to the tensions of
gases, that is, to the work done by gases, with otherwise
equal changes of volume. It thus happens that
<i>temperature heights</i> and <i>level heights of work</i> are proportional
to one another.</p>

<p>If properties of the thermal condition varying
greatly from the tensions of gases had been chosen,
this relation would have assumed very complicated
forms, and the agreement between heat and the other
energies above considered would not subsist. It is<span class="pagenum"><a name="Page_175" id="Page_175">[Pg 175]</a></span>
very instructive to reflect upon this point. A <i>natural
law</i>, therefore, is not implied in the conformity of the
behavior of the energies, but this conformity is rather
conditioned by the uniformity of our modes of conception
and is also partly a matter of good fortune.</p>


<h3>VI. THE DIFFERENCES OF THE ENERGIES AND THE
LIMITS OF THE PRINCIPLE OF ENERGY.</h3>

<p>Of every quantity of heat <i>Q</i> which does work in a
reversible process (one unaccompanied by loss) between
the absolute temperatures <i>T</i><sub>1</sub>, <i>T</i><sub>2</sub>, only the portion</p>

<p>(<i>T</i><sub>1</sub>-<i>T</i><sub>2</sub>)/<i>T</i><sub>1</sub></p>

<p>is transformed into work, while the remainder is transferred
to the lower temperature-level <i>T</i><sub>2</sub>. This transferred
portion can, upon the reversal of the process,
with the same expenditure of work, again be brought
back to the level <i>T</i><sub>1</sub>. But if the process is not reversible,
then more heat than in the foregoing case flows
to the lower level, and the surplus can no longer be
brought back to the higher level <i>T</i><sub>2</sub> without some <i>special</i>
expenditure. W. Thomson (1852), accordingly,
drew attention to the fact, that in all non-reversible,
that is, in all real thermal processes, quantities of heat
are lost for mechanical work, and that accordingly a
dissipation or waste of mechanical energy is taking
place. In all cases, heat is only partially transformed
into work, but frequently work is wholly transformed<span class="pagenum"><a name="Page_176" id="Page_176">[Pg 176]</a></span>
into heat. Hence, a tendency exists towards a diminution
of the <i>mechanical</i> energy and towards an increase
of the <i>thermal</i> energy of the world.</p>

<p>For a simple, closed cyclical process, accompanied
by no loss, in which the quantity of heat <i>Q_</i>{1} is taken
from the level <i>T_</i>{1}, and the quantity <i>Q_</i>{2} is deposited
upon the level <i>T_</i>{2}, the following relation, agreeably to
equation (2), exists,</p>

<p class="center">-(<i>Q</i><sub>1</sub>/<i>T</i><sub>1</sub>) + (<i>Q</i><sub>2</sub>/<i>T</i><sub>2</sub>) = 0.
</p>

<p>Similarly, for any number of compound reversible
cycles Clausius finds the algebraical sum</p>

<p class="center">&#931;<i>Q</i>/<i>T</i> = 0,
</p>

<p>and supposing the temperature to change continuously,</p>

<p class="center">&#8747;<i>dQ</i>/<i>T</i> = 0 (4)
</p>

<p>Here the elements of the quantities of heat deducted
from a given level are reckoned negative, and the elements
imparted to it, positive. If the process is not
reversible, then expression (4), which Clausius calls
<i>entropy</i>, increases. In actual practice this is always
the case, and Clausius finds himself led to the statement:</p>

<p>1. That the energy of the world remains constant.</p>

<p>2. That the entropy of the world tends toward a
maximum.</p>

<p>Once we have noted the above-indicated conformity
in the behavior of different energies, the <i>peculiarity</i><span class="pagenum"><a name="Page_177" id="Page_177">[Pg 177]</a></span>
of thermal energy here mentioned must strike us.
Whence is this peculiarity derived, for, generally every
energy passes only partly into another form, which is
also true of thermal energy? The explanation will be
found in the following.</p>

<p>Every transformation of a special kind of energy <i>A</i>
is accompanied with a fall of potential of that particular
kind of energy, including heat. But whilst for the
other kinds of energy a transformation and therefore a
loss of energy on the part of the kind sinking in potential
is connected with the fall of the potential, with
heat the case is different. Heat can suffer a fall of
potential without sustaining a loss of energy, at least
according to the customary mode of estimation. If a
weight sinks, it must create perforce kinetic energy,
or heat, or some other form of energy. Also, an electrical
charge cannot suffer a fall of potential without
loss of energy, i. e., without transformation. But heat
can pass with a fall of temperature to a body of greater
capacity and the same thermal energy still be preserved,
so long as we regard <i>every quantity</i> of heat as
energy. This it is that gives to heat, besides its
property of energy, in many cases the character of a
material <i>substance</i>, or quantity.</p>

<p>If we look at the matter in an unprejudiced light,
we must ask if there is any scientific sense or purpose
in still considering as energy a quantity of heat that
can no longer be transformed into mechanical work,
(for example, the heat of a closed equably warmed<span class="pagenum"><a name="Page_178" id="Page_178">[Pg 178]</a></span>
material system). The principle of energy certainly
plays in this case a wholly superfluous rôle, which is
assigned to it only from habit.<a name="FNanchor_58_58" id="FNanchor_58_58"></a><a href="#Footnote_58_58" class="fnanchor">[58]</a> To maintain the principle
of energy in the face of a knowledge of the dissipation
or waste of mechanical energy, in the face of
the increase of entropy is equivalent almost to the
liberty which Black took when he regarded the heat
of liquefaction as still present but latent.<a name="FNanchor_59_59" id="FNanchor_59_59"></a><a href="#Footnote_59_59" class="fnanchor">[59]</a> It is to be
remarked further, that the expressions "energy of the
world" and "entropy of the world" are slightly permeated
with scholasticism. Energy and entropy are
<i>metrical</i> notions. What meaning can there be in applying
these notions to a case in which they are not
applicable, in which their values are not determinable?</p>

<p>If we could really determine the entropy of the
world it would represent a true, absolute measure of
time. In this way is best seen the utter tautology of
a statement that the entropy of the world increases
with the time. Time, and the fact that certain changes
take place only in a definite sense, are one and the
same thing.</p>

<p><span class="pagenum"><a name="Page_179" id="Page_179">[Pg 179]</a></span></p>
<h3>VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.</h3>

<p>We are now prepared to answer the question, What
are the sources of the principle of energy? All knowledge
of nature is derived in the last instance from experience.
In this sense they are right who look upon
the principle of energy as a result of experience.</p>

<p>Experience teaches that the sense-elements &#945;&#946;&#947;&#948;...
into which the world may be decomposed, are subject
to change. It tells us further, that certain of these
elements are <i>connected</i> with other elements, so that they
appear and disappear together; or, that the appearance
of the elements of one class is connected with the
disappearance of the elements of the other class. We
will avoid here the notions of cause and effect because
of their obscurity and equivocalness. The result
of experience may be expressed as follows: <i>The
sensuous elements of the world (&#945;&#946;&#947;&#948;...) show themselves
to be interdependent.</i> This interdependence is
best represented by some such conception as is in
geometry that of the mutual dependence of the sides
and angles of a triangle, only much more varied and
complex.</p>

<p>As an example, we may take a mass of gas enclosed
in a cylinder and possessed of a definite volume (&#945;),
which we change by a pressure (&#946;) on the piston, at
the same time feeling the cylinder with our hand and<span class="pagenum"><a name="Page_180" id="Page_180">[Pg 180]</a></span>
receiving a sensation of heat (&#947;). Increase of pressure
diminishes the volume and increases the sensation
of heat.</p>

<p>The various facts of experience are not in all respects
alike. Their common sensuous elements are
placed in relief by a process of abstraction and thus
impressed upon the memory. In this way the expression
is obtained of the features of <i>agreement</i> of extensive
groups of facts. The simplest sentence which we can
utter is, by the very nature of language, an abstraction
of this kind. But account must also be taken of the
<i>differences</i> of related facts. Facts may be so nearly related
as to contain the same kind of a &#945;&#946;&#947;..., but the
relation be such that the &#945;&#946;&#947;... of the one differ
from the &#945;&#946;&#947;... of the other only by the number of
equal parts into which they can be divided. Such
being the case, if rules can be given for deducing <i>from
one another</i> the numbers which are the measures of
these &#945;&#946;&#947;..., then we possess in such rules the <i>most
general</i> expression of a group of facts, as also that expression
which corresponds to all its differences. This
is the goal of quantitative investigation.</p>

<p>If this goal be reached what we have found is that
between the &#945;&#946;&#947;... of a group of facts, or better, between
the numbers which are their measures, a number
of equations exists. The simple fact of change
brings it about that the number of these equations
must be smaller than the number of the &#945;&#946;&#947;.... If
the former be smaller by one than the latter, then one<span class="pagenum"><a name="Page_181" id="Page_181">[Pg 181]</a></span>
portion of the &#945;&#946;&#947;... is <i>uniquely</i> determined by the
other portion.</p>

<p>The quest of relations of this last kind is the most
important function of special experimental research,
because we are enabled by it to complete in thought
facts that are only partly given. It is self-evident that
only experience can ascertain that between the &#945;&#946;&#947;...
relations exist and of what kind they are. Further,
only experience can tell that the relations that exist
between the &#945;&#946;&#947;... are such that changes of them
can be reversed. If this were not the fact all occasion
for the enunciation of the principle of energy, as is
easily seen, would be wanting. In experience, therefore,
is buried the ultimate well-spring of all knowledge
of nature, and consequently, in this sense, also
the ultimate source of the principle of energy.</p>

<p>But this does not exclude the fact that the principle
of energy has also a logical root, as will now be
shown. Let us assume on the basis of experience that
one group of sensuous elements &#945;&#946;&#947;... determines
<i>uniquely</i> another group &#955;&#956;&#957;.... Experience further
teaches that changes of &#945;&#946;&#947;... can be <i>reversed</i>. It
is then a logical consequence of this observation, that
every time that &#945;&#946;&#947;... assume the same values this
is also the case with &#955;&#956;&#957;.... Or, that purely <i>periodical</i>
changes of &#945;&#946;&#947;... can produce no <i>permanent</i>
changes of &#955;&#956;&#957;.... If the group &#955;&#956;&#957;... is a mechanical
group, then a perpetual motion is excluded.</p><p><span class="pagenum"><a name="Page_182" id="Page_182">[Pg 182]</a></span></p>

<p>It will be said that this is a vicious circle, which
we will grant. But psychologically, the situation is
essentially different, whether I think simply of the
unique determination and reversibility of events, or
whether I exclude a perpetual motion. The attention
takes in the two cases different directions and diffuses
light over different sides of the question, which logically
of course are necessarily connected.</p>

<p>Surely that firm, logical setting of the thoughts noticeable
in the great inquirers, Stevinus, Galileo, and
the rest, which, consciously or instinctively, was supported
by a fine feeling for the slightest contradictions,
has no other purpose than to limit the bounds of
thought and so exempt it from the possibility of error.
In this, therefore, the logical root of the principle of
excluded perpetual motion is given, namely, in that
universal conviction which existed even before the development
of mechanics and co-operated in that development.</p>

<p>It is perfectly natural that the principle of excluded
perpetual motion should have been first developed in
the simple domain of pure mechanics. Towards the
transference of that principle into the domain of general
physics the idea contributed much that all physical
phenomena are mechanical phenomena. But the
foregoing discussion shows how little essential this
notion is. The issue really involved is the recognition
of a general interconnexion of nature. This once established,
we see with Carnot that it is indifferent<span class="pagenum"><a name="Page_183" id="Page_183">[Pg 183]</a></span>
whether the mechanical laws are broken directly or
circuitously.</p>

<p>The principle of the excluded perpetual motion is
very closely related to the modern principle of energy,
but it is not identical with it, for the latter is to be
deduced from the former only by means of a definite
<i>formal conception</i>. As may be seen from the preceding
exposition, the perpetual motion can be excluded without
our employing or possessing the notion of <i>work</i>.
The modern principle of energy results primarily from
a <i>substantial</i> conception of work and of every change
of physical condition which by being reversed produces
work. The strong need of such a conception,
which is by no means necessary, but in a formal sense
is very convenient and lucid, is exhibited in the case
of J. R. Mayer and Joule. It was before remarked
that this conception was suggested to both inquirers
by the observation that both the production of heat
and the production of mechanical work were connected
with an expenditure of substance. Mayer says: "Ex
nihilo nil fit," and in another place, "The creation or
destruction of a force (work) lies without the province
of human activity." In Joule we find this passage:
"It is manifestly <i>absurd</i> to suppose that the powers
with which God has endowed matter can be destroyed."</p>

<p>Some writers have observed in such statements the
attempt at a <i>metaphysical</i> establishment of the doctrine
of energy. But we see in them simply the formal need
of a simple, clear, and living grasp of the facts, which<span class="pagenum"><a name="Page_184" id="Page_184">[Pg 184]</a></span>
receives its development in practical and technical life,
and which we carry over, as best we can, into the
province of science. As a fact, Mayer writes to Griesinger:
"If, finally, you ask me how I became involved
in the whole affair, my answer is simply this: Engaged
during a sea voyage almost exclusively with the study
of physiology, I discovered the new theory for the
sufficient reason that I <i>vividly felt the need of it</i>."</p>

<p>The substantial conception of work (energy) is by
no means a necessary one. And it is far from true that
the problem is solved with the recognition of the need
of such a conception. Rather let us see how Mayer
gradually endeavored to satisfy that need. He first
regards quantity of motion, or momentum, <i>mv</i>, as the
equivalent of work, and did not light, until later, on
the notion of living force (<i>mv<sup>2</sup>/2</i>). In the province
of electricity he was unable to assign the expression
which is the equivalent of work. This was done later
by Helmholtz. The formal need, therefore, is <i>first</i>
present, and our conception of nature is subsequently
gradually <i>adapted</i> to it.</p>

<p>The laying bare of the experimental, logical, and
formal root of the present principle of energy will perhaps
contribute much to the removal of the mysticism
which still clings to this principle. With respect to
our formal need of a very simple, palpable, substantial
conception of the processes in our environment, it
remains an open question how far nature corresponds
to that need, or how far we can satisfy it. In one<span class="pagenum"><a name="Page_185" id="Page_185">[Pg 185]</a></span>
phase of the preceding discussions it would seem as
if the substantial notion of the principle of energy, like
Black's material conception of heat, has its natural
limits in facts, beyond which it can only be artificially
adhered to.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_186" id="Page_186">[Pg 186]</a></span></p>




<h2><a name="THE_ECONOMICAL_NATURE_OF" id="THE_ECONOMICAL_NATURE_OF">THE ECONOMICAL NATURE OF
PHYSICAL INQUIRY.</a><a name="FNanchor_60_60" id="FNanchor_60_60"></a><a href="#Footnote_60_60" class="fnanchor">[60]</a></h2>


<p>When the human mind, with its limited powers,
attempts to mirror in itself the rich life of the
world, of which it is itself only a small part, and which
it can never hope to exhaust, it has every reason for
proceeding economically. Hence that tendency, expressed
in the philosophy of all times, to compass by
a few organic thoughts the fundamental features of
reality. "Life understands not death, nor death life."
So spake an old Chinese philosopher. Yet in his unceasing
desire to diminish the boundaries of the incomprehensible,
man has always been engaged in attempts
to understand death by life and life by death.</p>

<p>Among the ancient civilised peoples, nature was
filled with demons and spirits having the feelings and
desires of men. In all essential features, this animistic
view of nature, as Tylor<a name="FNanchor_61_61" id="FNanchor_61_61"></a><a href="#Footnote_61_61" class="fnanchor">[61]</a> has aptly termed it, is shared
in common by the fetish-worshipper of modern Africa
and the most advanced nations of antiquity. As a
theory of the world it has never completely disappeared.
The monotheism of the Christians never fully
overcame it, no more than did that of the Jews. In
the belief in witchcraft and in the superstitions of the
sixteenth and seventeenth centuries, the centuries of
the rise of natural science, it assumed frightful pathological
dimensions. Whilst Stevinus, Kepler, and
Galileo were slowly rearing the fabric of modern physical
science, a cruel and relentless war was waged
with firebrand and rack against the devils that glowered
from every corner. To-day even, apart from all survivals
of that period, apart from the traces of fetishism
which still inhere in our physical concepts,<a name="FNanchor_62_62" id="FNanchor_62_62"></a><a href="#Footnote_62_62" class="fnanchor">[62]</a> those
very ideas still covertly lurk in the practices of modern
spiritualism.</p>
<p><span class="pagenum"><a name="Page_187" id="Page_187">[Pg 187]</a></span></p>

<p>By the side of this animistic conception of the
world, we meet from time to time, in different forms,
from Democritus to the present day, another view,
which likewise claims exclusive competency to comprehend
the universe. This view may be characterised
as the <i>physico-mechanical</i> view of the world. To-day,
that view holds, indisputably, the first place in the
thoughts of men, and determines the ideals and the
character of our times. The coming of the mind of
man into the full consciousness of its powers, in the
eighteenth century, was a period of genuine disillusionment.
It produced the splendid precedent of a life<span class="pagenum"><a name="Page_188" id="Page_188">[Pg 188]</a></span>
really worthy of man, competent to overcome the old
barbarism in the practical fields of life; it created the
<i>Critique of Pure Reason</i>, which banished into the realm
of shadows the sham-ideas of the old metaphysics; it
pressed into the hands of the mechanical philosophy
the reins which it now holds.</p>

<p>The oft-quoted words of the great Laplace,<a name="FNanchor_63_63" id="FNanchor_63_63"></a><a href="#Footnote_63_63" class="fnanchor">[63]</a> which
I will now give, have the ring of a jubilant toast to
the scientific achievements of the eighteenth century:
"A mind to which were given for a single instant all
the forces of nature and the mutual positions of all its
masses, if it were otherwise powerful enough to subject
these problems to analysis, could grasp, with a
single formula, the motions of the largest masses as
well as of the smallest atoms; nothing would be uncertain
for it; the future and the past would lie revealed
before its eyes." In writing these words, Laplace,
as we know, had also in mind the atoms of the
brain. That idea has been expressed more forcibly
still by some of his followers, and it is not too much
to say that Laplace's ideal is substantially that of the
great majority of modern scientists.</p>

<p>Gladly do we accord to the creator of the <i>Mécanique
céleste</i> the sense of lofty pleasure awakened in
him by the great success of the Enlightenment, to
which we too owe our intellectual freedom. But to-day,
with minds undisturbed and before <i>new</i> tasks, it<span class="pagenum"><a name="Page_189" id="Page_189">[Pg 189]</a></span>
becomes physical science to secure itself against self-deception
by a careful study of its character, so that
it can pursue with greater sureness its true objects.
If I step, therefore, beyond the narrow precincts of my
specialty in this discussion, to trespass on friendly
neighboring domains, I may plead in my excuse that
the subject-matter of knowledge is common to all domains
of research, and that fixed, sharp lines of demarcation
cannot be drawn.</p>

<p>The belief in occult magic powers of nature has
gradually died away, but in its place a new belief has
arisen, the belief in the magical power of science.
Science throws her treasures, not like a capricious
fairy into the laps of a favored few, but into the laps
of all humanity, with a lavish extravagance that no
legend ever dreamt of! Not without apparent justice,
therefore, do her distant admirers impute to her the
power of opening up unfathomable abysses of nature,
to which the senses cannot penetrate. Yet she who
came to bring light into the world, can well dispense
with the darkness of mystery, and with pompous show,
which she needs neither for the justification of her
aims nor for the adornment of her plain achievements.</p>

<p>The homely beginnings of science will best reveal
to us its simple, unchangeable character. Man acquires
his first knowledge of nature half-consciously
and automatically, from an instinctive habit of mimicking
and forecasting facts in thought, of supplementing
sluggish experience with the swift wings of thought,<span class="pagenum"><a name="Page_190" id="Page_190">[Pg 190]</a></span>
at first only for his material welfare. When he hears
a noise in the underbrush he constructs there, just as
the animal does, the enemy which he fears; when he
sees a certain rind he forms mentally the image of the
fruit which he is in search of; just as we mentally associate
a certain kind of matter with a certain line in
the spectrum or an electric spark with the friction of a
piece of glass. A knowledge of causality in this form
certainly reaches far below the level of Schopenhauer's
pet dog, to whom it was ascribed. It probably exists
in the whole animal world, and confirms that great
thinker's statement regarding the will which created
the intellect for its purposes. These primitive psychical
functions are rooted in the economy of our organism
not less firmly than are motion and digestion.
Who would deny that we feel in them, too, the elemental
power of a long practised logical and physiological
activity, bequeathed to us as an heirloom from
our forefathers?</p>

<p>Such primitive acts of knowledge constitute to-day
the solidest foundation of scientific thought. Our instinctive
knowledge, as we shall briefly call it, by virtue
of the conviction that we have consciously and
intentionally contributed nothing to its formation, confronts
us with an authority and logical power which
consciously acquired knowledge even from familiar
sources and of easily tested fallibility can never possess.
All so-called axioms are such instinctive knowledge.
Not consciously gained knowledge alone, but powerful<span class="pagenum"><a name="Page_191" id="Page_191">[Pg 191]</a></span>
intellectual instinct, joined with vast conceptive powers,
constitute the great inquirer. The greatest advances
of science have always consisted in some successful
formulation, in clear, abstract, and communicable terms,
of what was instinctively known long before, and of
thus making it the permanent property of humanity.
By Newton's principle of the equality of pressure and
counterpressure, whose truth all before him had felt, but
which no predecessor had abstractly formulated, mechanics
was placed by a single stroke on a higher level.
Our statement might also be historically justified by
examples from the scientific labors of Stevinus, S.
Carnot, Faraday, J. R. Mayer, and others.</p>

<p>All this, however, is merely the soil from which
science starts. The first real beginnings of science
appear in society, particularly in the manual arts,
where the necessity for the communication of experience
arises. Here, where some new discovery is to
be described and related, the compulsion is first felt of
clearly defining in consciousness the important and
essential features of that discovery, as many writers
can testify. The aim of instruction is simply the saving
of experience; the labor of one man is made to
take the place of that of another.</p>

<p>The most wonderful economy of communication is
found in language. Words are comparable to type,
which spare the repetition of written signs and thus
serve a multitude of purposes; or to the few sounds
of which our numberless different words are composed.<span class="pagenum"><a name="Page_192" id="Page_192">[Pg 192]</a></span>
Language, with its helpmate, conceptual thought, by
fixing the essential and rejecting the unessential, constructs
its rigid pictures of the fluid world on the plan
of a mosaic, at a sacrifice of exactness and fidelity but
with a saving of tools and labor. Like a piano-player
with previously prepared sounds, a speaker excites in
his listener thoughts previously prepared, but fitting
many cases, which respond to the speaker's summons
with alacrity and little effort.</p>

<p>The principles which a prominent political economist,
E. Hermann,<a name="FNanchor_64_64" id="FNanchor_64_64"></a><a href="#Footnote_64_64" class="fnanchor">[64]</a> has formulated for the economy of
the industrial arts, are also applicable to the ideas of
common life and of science. The economy of language
is augmented, of course, in the terminology of science.
With respect to the economy of written intercourse
there is scarcely a doubt that science itself will realise
that grand old dream of the philosophers of a Universal
Real Character. That time is not far distant.
Our numeral characters, the symbols of mathematical
analysis, chemical symbols, and musical notes, which
might easily be supplemented by a system of color-signs,
together with some phonetic alphabets now in
use, are all beginnings in this direction. The logical
extension of what we have, joined with a use of the
ideas which the Chinese ideography furnishes us, will
render the special invention and promulgation of a
Universal Character wholly superfluous.</p>

<p>The communication of scientific knowledge always<span class="pagenum"><a name="Page_193" id="Page_193">[Pg 193]</a></span>
involves description, that is, a mimetic reproduction
of facts in thought, the object of which is to replace
and save the trouble of new experience. Again, to
save the labor of instruction and of acquisition, concise,
abridged description is sought. This is really all
that natural laws are. Knowing the value of the acceleration
of gravity, and Galileo's laws of descent, we
possess simple and compendious directions for reproducing
in thought all possible motions of falling bodies.
A formula of this kind is a complete substitute
for a full table of motions of descent, because by means
of the formula the data of such a table can be easily
constructed at a moment's notice without the least
burdening of the memory.</p>

<p>No human mind could comprehend all the individual
cases of refraction. But knowing the index of refraction
for the two media presented, and the familiar
law of the sines, we can easily reproduce or fill out in
thought every conceivable case of refraction. The advantage
here consists in the disburdening of the memory;
an end immensely furthered by the written preservation
of the natural constants. More than this comprehensive
and condensed report about facts is not
contained in a natural law of this sort. In reality, the
law always contains less than the fact itself, because it
does not reproduce the fact as a whole but only in
that aspect of it which is important for us, the rest being
either intentionally or from necessity omitted.
Natural laws may be likened to intellectual type of a<span class="pagenum"><a name="Page_194" id="Page_194">[Pg 194]</a></span>
higher order, partly movable, partly stereotyped, which
last on new editions of experience may become downright
impediments.</p>

<p>When we look over a province of facts for the first
time, it appears to us diversified, irregular, confused,
full of contradictions. We first succeed in grasping
only single facts, unrelated with the others. The
province, as we are wont to say, is not <i>clear</i>. By and
by we discover the simple, permanent elements of the
mosaic, out of which we can mentally construct the
whole province. When we have reached a point where
we can discover everywhere the same facts, we no
longer feel lost in this province; we comprehend it
without effort; it is <i>explained</i> for us.</p>

<p>Let me illustrate this by an example. As soon as
we have grasped the fact of the rectilinear propagation
of light, the regular course of our thoughts stumbles
at the phenomena of refraction and diffraction. As soon
as we have cleared matters up by our index of refraction
we discover that a special index is necessary for
each color. Soon after we have accustomed ourselves
to the fact that light added to light increases its intensity,
we suddenly come across a case of total darkness
produced by this cause. Ultimately, however,
we see everywhere in the overwhelming multifariousness
of optical phenomena the fact of the spatial and
temporal periodicity of light, with its velocity of propagation
dependent on the medium and the period. This
tendency of obtaining a survey of a given province<span class="pagenum"><a name="Page_195" id="Page_195">[Pg 195]</a></span>
with the least expenditure of thought, and of representing
all its facts by some one single mental process,
may be justly termed an economical one.</p>

<p>The greatest perfection of mental economy is attained
in that science which has reached the highest
formal development, and which is widely employed in
physical inquiry, namely, in mathematics. Strange as
it may sound, the power of mathematics rests upon
its evasion of all unnecessary thought and on its wonderful
saving of mental operations. Even those arrangement-signs
which we call numbers are a system
of marvellous simplicity and economy. When we employ
the multiplication-table in multiplying numbers
of several places, and so use the results of old operations
of counting instead of performing the whole of
each operation anew; when we consult our table of
logarithms, replacing and saving thus new calculations
by old ones already performed; when we employ
determinants instead of always beginning afresh the
solution of a system of equations; when we resolve
new integral expressions into familiar old integrals;
we see in this simply a feeble reflexion of the intellectual
activity of a Lagrange or a Cauchy, who, with
the keen discernment of a great military commander,
substituted for new operations whole hosts of old ones.
No one will dispute me when I say that the most elementary
as well as the highest mathematics are economically-ordered
experiences of counting, put in forms
ready for use.</p><p><span class="pagenum"><a name="Page_196" id="Page_196">[Pg 196]</a></span></p>

<p>In algebra we perform, as far as possible, all numerical
operations which are identical in form once
for all, so that only a remnant of work is left for the
individual case. The use of the signs of algebra and
analysis, which are merely symbols of operations to
be performed, is due to the observation that we can
materially disburden the mind in this way and spare
its powers for more important and more difficult duties,
by imposing all mechanical operations upon the
hand. One result of this method, which attests its
economical character, is the construction of calculating
machines. The mathematician Babbage, the inventor
of the difference-engine, was probably the first who
clearly perceived this fact, and he touched upon it,
although only cursorily, in his work, <i>The Economy of
Manufactures and Machinery</i>.</p>

<p>The student of mathematics often finds it hard to
throw off the uncomfortable feeling that his science, in
the person of his pencil, surpasses him in intelligence,&mdash;an
impression which the great Euler confessed he
often could not get rid of. This feeling finds a sort of
justification when we reflect that the majority of the
ideas we deal with were conceived by others, often
centuries ago. In great measure it is really the intelligence
of other people that confronts us in science.
The moment we look at matters in this light, the uncanniness
and magical character of our impressions
cease, especially when we remember that we can think
over again at will any one of those alien thoughts.</p><p><span class="pagenum"><a name="Page_197" id="Page_197">[Pg 197]</a></span></p>

<p>Physics is experience, arranged in economical order.
By this order not only is a broad and comprehensive
view of what we have rendered possible, but also
the defects and the needful alterations are made manifest,
exactly as in a well-kept household. Physics
shares with mathematics the advantages of succinct
description and of brief, compendious definition, which
precludes confusion, even in ideas where, with no apparent
burdening of the brain, hosts of others are contained.
Of these ideas the rich contents can be produced
at any moment and displayed in their full perceptual
light. Think of the swarm of well-ordered notions
pent up in the idea of the potential. Is it wonderful
that ideas containing so much finished labor should
be easy to work with?</p>

<p>Our first knowledge, thus, is a product of the
economy of self-preservation. By communication, the
experience of <i>many</i> persons, individually acquired at
first, is collected in <i>one</i>. The communication of
knowledge and the necessity which every one feels of
managing his stock of experience with the least expenditure
of thought, compel us to put our knowledge in
economical forms. But here we have a clue which
strips science of all its mystery, and shows us what its
power really is. With respect to specific results it
yields us nothing that we could not reach in a sufficiently
long time without methods. There is no problem
in all mathematics that cannot be solved by direct
counting. But with the present implements of mathematics<span class="pagenum"><a name="Page_198" id="Page_198">[Pg 198]</a></span>
many operations of counting can be performed
in a few minutes which without mathematical methods
would take a lifetime. Just as a single human being,
restricted wholly to the fruits of his own labor, could
never amass a fortune, but on the contrary the accumulation
of the labor of many men in the hands of one is
the foundation of wealth and power, so, also, no knowledge
worthy of the name can be gathered up in a
single human mind limited to the span of a human life
and gifted only with finite powers, except by the most
exquisite economy of thought and by the careful
amassment of the economically ordered experience of
thousands of co-workers. What strikes us here as the
fruits of sorcery are simply the rewards of excellent
housekeeping, as are the like results in civil life. But
the business of science has this advantage over every
other enterprise, that from <i>its</i> amassment of wealth no
one suffers the least loss. This, too, is its blessing,
its freeing and saving power.</p>

<p>The recognition of the economical character of
science will now help us, perhaps, to understand better
certain physical notions.</p>

<p>Those elements of an event which we call "cause
and effect" are certain salient features of it, which are
important for its mental reproduction. Their importance
wanes and the attention is transferred to fresh
characters the moment the event or experience in
question becomes familiar. If the connexion of such
features strikes us as a necessary one, it is simply because<span class="pagenum"><a name="Page_199" id="Page_199">[Pg 199]</a></span>
the interpolation of certain intermediate links
with which we are very familiar, and which possess,
therefore, higher authority for us, is often attended
with success in our explanations. That <i>ready</i> experience
fixed in the mosaic of the mind with which we meet
new events, Kant calls an innate concept of the understanding
(<i>Verstandesbegriff</i>).</p>

<p>The grandest principles of physics, resolved into
their elements, differ in no wise from the descriptive
principles of the natural historian. The question,
"Why?" which is always appropriate where the explanation
of a contradiction is concerned, like all proper
habitudes of thought, can overreach itself and be asked
where nothing remains to be understood. Suppose we
were to attribute to nature the property of producing
like effects in like circumstances; just these like circumstances
we should not know how to find. Nature
exists once only. Our schematic mental imitation alone
produces like events. Only in the mind, therefore, does
the mutual dependence of certain features exist.</p>

<p>All our efforts to mirror the world in thought would
be futile if we found nothing permanent in the varied
changes of things. It is this that impels us to form the
notion of substance, the source of which is not different
from that of the modern ideas relative to the conservation
of energy. The history of physics furnishes
numerous examples of this impulse in almost all fields,
and pretty examples of it may be traced back to the
nursery. "Where does the light go to when it is put<span class="pagenum"><a name="Page_200" id="Page_200">[Pg 200]</a></span>
out?" asks the child. The sudden shrivelling up of a
hydrogen balloon is inexplicable to a child; it looks
everywhere for the large body which was just there
but is now gone.</p>

<p>Where does heat come from? Where does heat
go to? Such childish questions in the mouths of mature
men shape the character of a century.</p>

<p>In mentally separating a body from the changeable
environment in which it moves, what we really do
is to extricate a group of sensations on which our
thoughts are fastened and which is of relatively greater
stability than the others, from the stream of all our
sensations. Absolutely unalterable this group is not.
Now this, now that member of it appears and disappears,
or is altered. In its full identity it never recurs.
Yet the sum of its constant elements as compared
with the sum of its changeable ones, especially if we
consider the continuous character of the transition, is
always so great that for the purpose in hand the former
usually appear sufficient to determine the body's identity.
But because we can separate from the group
every single member without the body's ceasing to be
for us the same, we are easily led to believe that after
abstracting all the members something additional
would remain. It thus comes to pass that we form
the notion of a substance distinct from its attributes,
of a thing-in-itself, whilst our sensations are regarded
merely as symbols or indications of the properties of
this thing-in-itself. But it would be much better to<span class="pagenum"><a name="Page_201" id="Page_201">[Pg 201]</a></span>
say that bodies or things are compendious mental symbols
for groups of sensations&mdash;symbols that do not exist
outside of thought. Thus, the merchant regards
the labels of his boxes merely as indexes of their contents,
and not the contrary. He invests their contents,
not their labels, with real value. The same
economy which induces us to analyse a group and to
establish special signs for its component parts, parts
which also go to make up other groups, may likewise
induce us to mark out by some single symbol a whole
group.</p>

<p>On the old Egyptian monuments we see objects
represented which do not reproduce a single visual
impression, but are composed of various impressions.
The heads and the legs of the figures appear in profile,
the head-dress and the breast are seen from the
front, and so on. We have here, so to speak, a mean
view of the objects, in forming which the sculptor has
retained what he deemed essential, and neglected what
he thought indifferent. We have living exemplifications
of the processes put into stone on the walls of
these old temples, in the drawings of our children, and
we also observe a faithful analogue of them in the formation
of ideas in our own minds. Only in virtue of
some such facility of view as that indicated, are we
allowed to speak of a body. When we speak of a cube
with trimmed corners&mdash;a figure which is not a cube&mdash;we
do so from a natural instinct of economy, which
prefers to add to an old familiar conception a correction<span class="pagenum"><a name="Page_202" id="Page_202">[Pg 202]</a></span>
instead of forming an entirely new one. This is
the process of all judgment.</p>

<p>The crude notion of "body" can no more stand
the test of analysis than can the art of the Egyptians
or that of our little children. The physicist who sees
a body flexed, stretched, melted, and vaporised, cuts
up this body into smaller permanent parts; the chemist
splits it up into elements. Yet even an element is
not unalterable. Take sodium. When warmed, the
white, silvery mass becomes a liquid, which, when the
heat is increased and the air shut out, is transformed
into a violet vapor, and on the heat being still more
increased glows with a yellow light. If the name sodium
is still retained, it is because of the continuous
character of the transitions and from a necessary instinct
of economy. By condensing the vapor, the
white metal may be made to reappear. Indeed, even
after the metal is thrown into water and has passed
into sodium hydroxide, the vanished properties may
by skilful treatment still be made to appear; just as a
moving body which has passed behind a column and
is lost to view for a moment may make its appearance
after a time. It is unquestionably very convenient
always to have ready the name and thought for a
group of properties wherever that group by any possibility
can appear. But more than a compendious economical
symbol for these phenomena, that name and
thought is not. It would be a mere empty word for
one in whom it did not awaken a large group of well-ordered<span class="pagenum"><a name="Page_203" id="Page_203">[Pg 203]</a></span>
sense-impressions. And the same is true of
the molecules and atoms into which the chemical element
is still further analysed.</p>

<p>True, it is customary to regard the conservation of
weight, or, more precisely, the conservation of mass,
as a direct proof of the constancy of matter. But this
proof is dissolved, when we go to the bottom of it,
into such a multitude of instrumental and intellectual
operations, that in a sense it will be found to constitute
simply an equation which our ideas in imitating
facts have to satisfy. That obscure, mysterious lump
which we involuntarily add in thought, we seek for in
vain outside the mind.</p>

<p>It is always, thus, the crude notion of substance
that is slipping unnoticed into science, proving itself
constantly insufficient, and ever under the necessity of
being reduced to smaller and smaller world-particles.
Here, as elsewhere, the lower stage is not rendered
indispensable by the higher which is built upon it, no
more than the simplest mode of locomotion, walking,
is rendered superfluous by the most elaborate means of
transportation. Body, as a compound of light and
touch sensations, knit together by sensations of space,
must be as familiar to the physicist who seeks it, as to
the animal who hunts its prey. But the student of the
theory of knowledge, like the geologist and the astronomer,
must be permitted to reason back from the forms
which are created before his eyes to others which he
finds ready made for him.</p><p><span class="pagenum"><a name="Page_204" id="Page_204">[Pg 204]</a></span></p>

<p>All physical ideas and principles are succinct directions,
frequently involving subordinate directions,
for the employment of economically classified experiences,
ready for use. Their conciseness, as also the
fact that their contents are rarely exhibited in full,
often invests them with the semblance of independent
existence. Poetical myths regarding such ideas,&mdash;for
example, that of Time, the producer and devourer of
all things,&mdash;do not concern us here. We need only
remind the reader that even Newton speaks of an <i>absolute</i>
time independent of all phenomena, and of an
absolute space&mdash;views which even Kant did not shake
off, and which are often seriously entertained to-day.
For the natural inquirer, determinations of time are
merely abbreviated statements of the dependence of
one event upon another, and nothing more. When
we say the acceleration of a freely falling body is 9·810
metres per second, we mean the velocity of the body
with respect to the centre of the earth is 9·810 metres
greater when the earth has performed an additional
86400th part of its rotation&mdash;a fact which itself can be
determined only by the earth's relation to other heavenly
bodies. Again, in velocity is contained simply a
relation of the position of a body to the position of
the earth.<a name="FNanchor_65_65" id="FNanchor_65_65"></a><a href="#Footnote_65_65" class="fnanchor">[65]</a> Instead of referring events to the earth
we may refer them to a clock, or even to our internal
sensation of time. Now, because all are connected,
and each may be made the measure of the rest, the illusion
easily arises that time has significance independently
of all.<a name="FNanchor_66_66" id="FNanchor_66_66"></a><a href="#Footnote_66_66" class="fnanchor">[66]</a></p>

<p><span class="pagenum"><a name="Page_205" id="Page_205">[Pg 205]</a></span></p><p>The aim of research is the discovery of the equations
which subsist between the elements of phenomena.
The equation of an ellipse expresses the universal
<i>conceivable</i> relation between its co-ordinates, of which
only the real values have <i>geometrical</i> significance.
Similarly, the equations between the elements of <i>phenomena</i>
express a universal, mathematically conceivable
relation. Here, however, for many values only
certain directions of change are <i>physically</i> admissible.
As in the ellipse only certain <i>values</i> satisfying the
equation are realised, so in the physical world only
certain <i>changes</i> of value occur. Bodies are always accelerated
towards the earth. Differences of temperature,
left to themselves, always grow less; and so on.
Similarly, with respect to space, mathematical and
physiological researches have shown that the space of
experience is simply an <i>actual</i> case of many conceivable
cases, about whose peculiar properties experience
alone can instruct us. The elucidation which this idea
diffuses cannot be questioned, despite the absurd uses
to which it has been put.</p>

<p>Let us endeavor now to summarise the results of<span class="pagenum"><a name="Page_206" id="Page_206">[Pg 206]</a></span>
our survey. In the economical schematism of science
lie both its strength and its weakness. Facts are always
represented at a sacrifice of completeness and
never with greater precision than fits the needs of the
moment. The incongruence between thought and experience,
therefore, will continue to subsist as long as
the two pursue their course side by side; but it will
be continually diminished.</p>

<p>In reality, the point involved is always the completion
of some partial experience; the derivation of
one portion of a phenomenon from some other. In
this act our ideas must be based directly upon sensations.
We call this measuring.<a name="FNanchor_67_67" id="FNanchor_67_67"></a><a href="#Footnote_67_67" class="fnanchor">[67]</a> The condition of
science, both in its origin and in its application, is a
<i>great relative stability</i> of our environment. What it
teaches us is interdependence. Absolute forecasts,
consequently, have no significance in science. With
great changes in celestial space we should lose our
co-ordinate systems of space and time.</p>

<p>When a geometer wishes to understand the form of
a curve, he first resolves it into small rectilinear elements.
In doing this, however, he is fully aware that
these elements are only provisional and arbitrary devices
for comprehending in parts what he cannot comprehend
as a whole. When the law of the curve is
found he no longer thinks of the elements. Similarly,
it would not become physical science to see in its self-created,<span class="pagenum"><a name="Page_207" id="Page_207">[Pg 207]</a></span>
changeable, economical tools, molecules and
atoms, realities behind phenomena, forgetful of the
lately acquired sapience of her older sister, philosophy,
in substituting a mechanical mythology for the old
animistic or metaphysical scheme, and thus creating
no end of suppositious problems. The atom must remain
a tool for representing phenomena, like the
functions of mathematics. Gradually, however, as
the intellect, by contact with its subject-matter, grows
in discipline, physical science will give up its mosaic
play with stones and will seek out the boundaries and
forms of the bed in which the living stream of phenomena
flows. The goal which it has set itself is the
<i>simplest</i> and <i>most economical</i> abstract expression of facts.</p>
<p><span class="pagenum"><a name="Page_208" id="Page_208">[Pg 208]</a></span></p><p><span class="pagenum"><a name="Page_209" id="Page_209">[Pg 209]</a></span></p><p><span class="pagenum"><a name="Page_210" id="Page_210">[Pg 210]</a></span></p><p><span class="pagenum"><a name="Page_211" id="Page_211">[Pg 211]</a></span></p><p><span class="pagenum"><a name="Page_212" id="Page_212">[Pg 212]</a></span></p><p><span class="pagenum"><a name="Page_213" id="Page_213">[Pg 213]</a></span></p><p><span class="pagenum"><a name="Page_214" id="Page_214">[Pg 214]</a></span></p>
<hr class="tb" />

<p>The question now remains, whether the same
method of research which till now we have tacitly restricted
to physics, is also applicable in the psychical
domain. This question will appear superfluous to the
physical inquirer. Our physical and psychical views
spring in exactly the same manner from instinctive
knowledge. We read the thoughts of men in their
acts and facial expressions without knowing how.
Just as we predict the behavior of a magnetic needle
placed near a current by imagining Ampère's swimmer
in the current, similarly we predict in thought the
acts and behavior of men by assuming sensations, feelings,
and wills similar to our own connected with their
bodies. What we here instinctively perform would
appear to us as one of the subtlest achievements of
science, far outstripping in significance and ingenuity
Ampère's rule of the swimmer, were it not that every
child unconsciously accomplished it. The question
simply is, therefore, to grasp scientifically, that is, by
conceptional thought, what we are already familiar
with from other sources. And here much is to be
accomplished. A long sequence of facts is to be disclosed
between the physics of expression and movement
and feeling and thought.</p>

<p>We hear the question, "But how is it possible to
explain feeling by the motions of the atoms of the
brain?" Certainly this will never be done, no more
than light or heat will ever be deduced from the law
of refraction. We need not deplore, therefore, the
lack of ingenious solutions of this question. The problem
is not a problem. A child looking over the walls
of a city or of a fort into the moat below sees with
astonishment living people in it, and not knowing of
the portal which connects the wall with the moat, cannot
understand how they could have got down from
the high ramparts. So it is with the notions of physics.
We cannot climb up into the province of psychology
by the ladder of our abstractions, but we can climb
down into it.</p>

<p>Let us look at the matter without bias. The world
consists of colors, sounds, temperatures, pressures,
spaces, times, and so forth, which now we shall not
call sensations, nor phenomena, because in either term
an arbitrary, one-sided theory is embodied, but simply
<i>elements</i>. The fixing of the flux of these elements,
whether mediately or immediately, is the real object of
physical research. As long as, neglecting our own
body, we employ ourselves with the interdependence
of those groups of elements which, including men and
animals, make up <i>foreign</i> bodies, we are physicists.
For example, we investigate the change of the red
color of a body as produced by a change of illumination.
But the moment we consider the special influence
on the red of the elements constituting our
body, outlined by the well-known perspective with
head invisible, we are at work in the domain of physiological
psychology. We close our eyes, and the red
together with the whole visible world disappears.
There exists, thus, in the perspective field of every sense
a portion which exercises on all the rest a different
and more powerful influence than the rest upon one
another. With this, however, all is said. In the light
of this remark, we call <i>all</i> elements, in so far as we regard
them as dependent on this special part (our body),
<i>sensations</i>. That the world is our sensation, in this
sense, cannot be questioned. But to make a system
of conduct out of this provisional conception, and to
abide its slaves, is as unnecessary for us as would be
a similar course for a mathematician who, in varying a
series of variables of a function which were previously
assumed to be constant, or in interchanging the independent
variables, finds his method to be the source
of some very surprising ideas for him.<a name="FNanchor_68_68" id="FNanchor_68_68"></a><a href="#Footnote_68_68" class="fnanchor">[68]</a></p>

<p>If we look at the matter in this unbiassed light it
will appear indubitable that the method of physiological
psychology is none other than that of physics;
what is more, that this science is a part of physics.
Its subject-matter is not different from that of physics.
It will unquestionably determine the relations
the sensations bear to the physics of our body. We
have already learned from a member of this academy
(Hering) that in all probability a sixfold manifoldness
of the chemical processes of the visual substance corresponds
to the sixfold manifoldness of color-sensation,
and a threefold manifoldness of the physiological processes
to the threefold manifoldness of space-sensations.
The paths of reflex actions and of the will are
followed up and disclosed; it is ascertained what region
of the brain subserves the function of speech,
what region the function of locomotion, etc. That
which still clings to our body, namely, our thoughts,
will, when those investigations are finished, present no
difficulties new in principle. When experience has
once clearly exhibited these facts and science has
marshalled them in economic and perspicuous order,
there is no doubt that we shall <i>understand</i> them. For
other "understanding" than a mental mastery of facts
never existed. Science does not create facts from facts,
but simply <i>orders</i> known facts.</p>

<p>Let us look, now, a little more closely into the modes
of research of physiological psychology. We have a
very clear idea of how a body moves in the space encompassing
it. With our optical field of sight we are
very familiar. But we are unable to state, as a rule,
how we have come by an idea, from what corner of
our intellectual field of sight it has entered, or by what
region the impulse to a motion is sent forth. Moreover,
we shall never get acquainted with this mental
field of view from self-observation alone. Self-observation,
in conjunction with physiological research,
which seeks out physical connexions, can put this field
of vision in a clear light before us, and will thus first
really reveal to us our inner man.</p>

<p>Primarily, natural science, or physics, in its widest
sense, makes us acquainted with only the firmest connexions
of groups of elements. Provisorily, we may
not bestow too much attention on the single constituents
of those groups, if we are desirous of retaining a
comprehensible whole. Instead of equations between
the primitive variables, physics gives us, as much the
easiest course, equations between <i>functions</i> of those
variables. Physiological psychology teaches us how
to separate the visible, the tangible, and the audible
from bodies&mdash;a labor which is subsequently richly requited,
as the division of the subjects of physics well
shows. Physiology further analyses the visible into
light and space sensations; the first into colors, the
last also into their component parts; it resolves noises
into sounds, these into tones, and so on. Unquestionably
this analysis can be carried much further than it
has been. It will be possible in the end to exhibit the
common elements at the basis of very abstract but
definite logical acts of like form,&mdash;elements which the
acute jurist and mathematician, as it were, <i>feels</i> out,
with absolute certainty, where the uninitiated hears
only empty words. Physiology, in a word, will reveal
to us the true real elements of the world. Physiological
psychology bears to physics in its widest sense a relation
similar to that which chemistry bears to physics
in its narrowest sense. But far greater than the mutual
support of physics and chemistry will be that
which natural science and psychology will render each
other. And the results that shall spring from this
union will, in all likelihood, far outstrip those of the
modern mechanical physics.</p>

<p>What those ideas are with which we shall comprehend
the world when the closed circuit of physical and
psychological facts shall lie complete before us, (that
circuit of which we now see only two disjoined parts,)
cannot be foreseen at the outset of the work. The
men will be found who will see what is right and
will have the courage, instead of wandering in the
intricate paths of logical and historical accident, to
enter on the straight ways to the heights from which
the mighty stream of facts can be surveyed. Whether
the notion which we now call matter will continue to
have a scientific significance beyond the crude purposes
of common life, we do not know. But we certainly
shall wonder how colors and tones which were
such innermost parts of us could suddenly get lost in
our physical world of atoms; how we could be suddenly
surprised that something which outside us simply
clicked and beat, in our heads should make light
and music; and how we could ask whether matter can
feel, that is to say, whether a mental symbol for a
group of sensations can feel?</p>

<p>We cannot mark out in hard and fast lines the
science of the future, but we can foresee that the rigid
walls which now divide man from the world will gradually
disappear; that human beings will not only confront
each other, but also the entire organic and so-called
lifeless world, with less selfishness and with livelier
sympathy. Just such a presentiment as this perhaps
possessed the great Chinese philosopher Licius
some two thousand years ago when, pointing to a heap
of mouldering human bones, he said to his scholars in
the rigid, lapidary style of his tongue: "These and I
alone have the knowledge that we neither live nor are
dead."</p>




<h2><a name="ON_TRANSFORMATION_AND_ADAPTATION" id="ON_TRANSFORMATION_AND_ADAPTATION">ON TRANSFORMATION AND ADAPTATION
IN SCIENTIFIC THOUGHT.</a><a name="FNanchor_69_69" id="FNanchor_69_69"></a><a href="#Footnote_69_69" class="fnanchor">[69]</a></h2>


<p>It was towards the close of the sixteenth century
that Galileo with a superb indifference to the dialectic
arts and sophistic subtleties of the Schoolmen of
his time, turned the attention of his brilliant mind
to nature. By nature his ideas were transformed and
released from the fetters of inherited prejudice. At
once the mighty revolution was felt, that was therewith
effected in the realm of human thought&mdash;felt indeed in
circles far remote and wholly unrelated to the sphere
of science, felt in strata of society that hitherto had
only indirectly recognised the influence of scientific
thought.</p>
<p><span class="pagenum"><a name="Page_215" id="Page_215">[Pg 215]</a></span></p>
<p>And how great and how far-reaching that revolution
was! From the beginning of the seventeenth century
till its close we see arising, at least in embryo,
almost all that plays a part in the natural and technical
science of to-day, almost all that in the two centuries
following so wonderfully transformed the facial
appearance of the earth, and all that is moving onward
in process of such mighty evolution to-day. And all
this, the direct result of Galilean ideas, the direct outcome
of that freshly awakened sense for the investigation
of natural phenomena which taught the Tuscan
philosopher to form the concept and the law of falling
bodies from the <i>observation</i> of a falling stone! Galileo
began his investigations without an implement worthy
of the name; he measured time in the most primitive
way, by the efflux of water. Yet soon afterwards the
telescope, the microscope, the barometer, the thermometer,
the air-pump, the steam engine, the pendulum,
and the electrical machine were invented in rapid
succession. The fundamental theorems of dynamical
science, of optics, of heat, and of electricity were all
disclosed in the century that followed Galileo.</p>

<p>Of scarcely less importance, it seems, was that
movement which was prepared for by the illustrious
biologists of the hundred years just past, and formally
begun by the late Mr. Darwin. Galileo quickened the
sense for the simpler phenomena of <i>inorganic</i> nature.
And with the same simplicity and frankness that
marked the efforts of Galileo, and without the aid of<span class="pagenum"><a name="Page_216" id="Page_216">[Pg 216]</a></span>
technical or scientific instruments, without physical or
chemical experiment, but solely by the power of
thought and observation, Darwin grasps a new property
of <i>organic</i> nature&mdash;which we may briefly call its
<i>plasticity</i>.<a name="FNanchor_70_70" id="FNanchor_70_70"></a><a href="#Footnote_70_70" class="fnanchor">[70]</a> With the same directness of purpose, Darwin,
too, pursues his way. With the same candor
and love of truth, he points out the strength and the
weakness of his demonstrations. With masterly equanimity
he holds aloof from the discussion of irrelevant
subjects and wins alike the admiration of his adherents
and of his adversaries.</p>

<p>Scarcely thirty years have elapsed<a name="FNanchor_71_71" id="FNanchor_71_71"></a><a href="#Footnote_71_71" class="fnanchor">[71]</a> since Darwin first
propounded the principles of his theory of evolution.<span class="pagenum"><a name="Page_217" id="Page_217">[Pg 217]</a></span>
Yet, already we see his ideas firmly rooted in every
branch of human thought, however remote. Everywhere,
in history, in philosophy, even in the physical
sciences, we hear the watchwords: heredity, adaptation,
selection. We speak of the struggle for existence
among the heavenly bodies and of the struggle for existence
in the world of molecules.<a name="FNanchor_72_72" id="FNanchor_72_72"></a><a href="#Footnote_72_72" class="fnanchor">[72]</a></p>

<p>The impetus given by Galileo to scientific thought
was marked in every direction; thus, his pupil, Borelli,
founded the school of exact medicine, from
whence proceeded even distinguished mathematicians.
And now Darwinian ideas, in the same way, are animating
all provinces of research. It is true, nature is
not made up of two distinct parts, the inorganic and
the organic; nor must these two divisions be treated
perforce by totally distinct methods. Many <i>sides</i>, however,
nature has. Nature is like a thread in an intricate
tangle, which must be followed and traced, now from
this point, now from that. But we must never imagine,&mdash;and
this physicists have learned from Faraday and
J. R. Mayer,&mdash;that progress along paths once entered
upon is the <i>only</i> means of reaching the truth.</p>

<p>It will devolve upon the specialists of the future to
determine the relative tenability and fruitfulness of the
Darwinian ideas in the different provinces. Here I
wish simply to consider the growth of natural <i>knowledge</i>
in the light of the theory of evolution. For knowledge,
too, is a product of organic nature. And although<span class="pagenum"><a name="Page_218" id="Page_218">[Pg 218]</a></span>
ideas, as such, do not comport themselves in all respects
like independent organic individuals, and although
violent comparisons should be avoided, still, if Darwin
reasoned rightly, the general imprint of evolution and
transformation must be noticeable in ideas also.</p>

<p>I shall waive here the consideration of the fruitful
topic of the transmission of ideas or rather of the
transmission of the aptitude for certain ideas.<a name="FNanchor_73_73" id="FNanchor_73_73"></a><a href="#Footnote_73_73" class="fnanchor">[73]</a> Nor
would it come within my province to discuss psychical
evolution in any form, as Spencer<a name="FNanchor_74_74" id="FNanchor_74_74"></a><a href="#Footnote_74_74" class="fnanchor">[74]</a> and many other
modern psychologists have done, with varying success.
Neither shall I enter upon a discussion of the
struggle for existence and of natural selection among
scientific theories.<a name="FNanchor_75_75" id="FNanchor_75_75"></a><a href="#Footnote_75_75" class="fnanchor">[75]</a> We shall consider here only such
processes of transformation as every student can easily
observe in his own mind.</p>
<p><span class="pagenum"><a name="Page_219" id="Page_219">[Pg 219]</a></span></p><p><span class="pagenum"><a name="Page_220" id="Page_220">[Pg 220]</a></span></p><p><span class="pagenum"><a name="Page_221" id="Page_221">[Pg 221]</a></span></p><p><span class="pagenum"><a name="Page_222" id="Page_222">[Pg 222]</a></span></p><p><span class="pagenum"><a name="Page_223" id="Page_223">[Pg 223]</a></span></p><p><span class="pagenum"><a name="Page_224" id="Page_224">[Pg 224]</a></span></p><p><span class="pagenum"><a name="Page_225" id="Page_225">[Pg 225]</a></span></p><p><span class="pagenum"><a name="Page_226" id="Page_226">[Pg 226]</a></span></p><p><span class="pagenum"><a name="Page_227" id="Page_227">[Pg 227]</a></span></p><p><span class="pagenum"><a name="Page_228" id="Page_228">[Pg 228]</a></span></p><p><span class="pagenum"><a name="Page_229" id="Page_229">[Pg 229]</a></span></p><p><span class="pagenum"><a name="Page_230" id="Page_230">[Pg 230]</a></span></p><p><span class="pagenum"><a name="Page_231" id="Page_231">[Pg 231]</a></span></p><p><span class="pagenum"><a name="Page_232" id="Page_232">[Pg 232]</a></span></p><p><span class="pagenum"><a name="Page_233" id="Page_233">[Pg 233]</a></span></p>
<hr class="tb" />

<p>The child of the forest picks out and pursues with
marvellous acuteness the trails of animals. He outwits
and overreaches his foes with surpassing cunning.
He is perfectly at home in the sphere of his peculiar
experience. But confront him with an unwonted phenomenon;
place him face to face with a technical product
of modern civilisation, and he will lapse into impotency
and helplessness. Here are facts which he
does not comprehend. If he endeavors to grasp their
meaning, he misinterprets them. He fancies the moon,
when eclipsed, to be tormented by an evil spirit. To
his mind a puffing locomotive is a living monster. The
letter accompanying a commission with which he is
entrusted, having once revealed his thievishness, is in
his imagination a conscious being, which he must hide
beneath a stone, before venturing to commit a fresh
trespass. Arithmetic to him is like the art of the
geomancers in the Arabian Nights,&mdash;an art which is
able to accomplish every imaginable impossibility.
And, like Voltaire's <i>ingénu</i>, when placed in our social
world, he plays, as we think, the maddest pranks.</p>

<p>With the man who has made the achievements of
modern science and civilisation his own, the case is
quite different. He sees the moon pass temporarily
into the shadow of the earth. He feels in his thoughts
the water growing hot in the boiler of the locomotive;
he feels also the increase of the tension which pushes
the piston forward. Where he is not able to trace the
direct relation of things he has recourse to his yard-stick
and table of logarithms, which aid and facilitate
his thought without predominating over it. Such opinions
as he cannot concur in, are at least known to him,
and he knows how to meet them in argument.</p>

<p>Now, wherein does the difference between these
two men consist? The train of thought habitually
employed by the first one does not correspond to the
facts that he sees. He is surprised and nonplussed
at every step. But the thoughts of the second man
follow and anticipate events, his thoughts have become
adapted or accommodated to the larger field of
observation and activity in which he is located; he conceives
things as they are. The Indian's sphere of experience,
however, is quite different; his bodily organs
of sense are in constant activity; he is ever intensely
alert and on the watch for his foes; or, his entire attention
and energy are engaged in procuring sustenance.
Now, how can such a creature project his mind
into futurity, foresee or prophesy? This is not possible
until our fellow-beings have, in a measure, relieved
us of our concern for existence. It is then that we
acquire freedom for observation, and not infrequently
too that narrowness of thought which society helps and
teaches us to disregard.</p>

<p>If we move for a time within a fixed circle of phenomena
which recur with unvarying uniformity, our
thoughts gradually adapt themselves to our environment;
our ideas reflect unconsciously our surroundings.
The stone we hold in our hand, when dropped,
not only falls to the ground in reality; it also falls in
our thoughts. Iron-filings dart towards a magnet in
imagination as well as in fact, and, when thrown into
a fire, they grew hot in conception as well.</p>

<p>The impulse to complete mentally a phenomenon
that has been only partially observed, has not its origin
in the phenomenon itself; of this fact, we are fully
sensible. And we well know that it does not lie within
the sphere of our volition. It seems to confront us
rather as a power and a law imposed from without
and controlling both thought and facts.</p>

<p>The fact that we are able by the help of this law to
prophesy and forecast, merely proves a sameness or
uniformity of environment sufficient to effect a mental
adaptation of this kind. A necessity of fulfilment,
however, is not contained in this compulsory principle
which controls our thoughts; nor is it in any way determined
by the possibility of prediction. We are always
obliged, in fact, to await the completion of what
has been predicted. Errors and departures are constantly
discernible, and are slight only in provinces of
great rigid constancy, as in astronomy.</p>

<p>In cases where our thoughts follow the connexion
of events with ease, and in instances where we positively
forefeel the course of a phenomenon, it is natural
to fancy that the latter is determined by and must
conform to our thoughts. But the belief in that mysterious
agency called <i>causality</i>, which holds thought and
event in unison, is violently shaken when a person first
enters a province of inquiry in which he has previously
had no experience. Take for instance the strange
interaction of electric currents and magnets, or the
reciprocal action of currents, which seem to defy all
the resources of mechanical science. Let him be confronted
with such phenomena and he will immediately
feel himself forsaken by his power of prediction; he
will bring nothing with him into this strange field of
events but the hope of soon being able to adapt his
ideas to the new conditions there presented.</p>

<p>A person constructs from a bone the remaining
anatomy of an animal; or from the visible part of a
half-concealed wing of a butterfly he infers and reconstructs
the part concealed. He does so with a feeling
of highest confidence in the accuracy of his results;
and in these processes we find nothing preternatural
or transcendent. But when physicists adapt their
thoughts to conform to the dynamical course of events
in time, we invariably surround their investigations
with a metaphysical halo; yet these latter adaptations
bear quite the same character as the former, and our
only reason for investing them with a metaphysical
garb, perhaps, is their high practical value.<a name="FNanchor_76_76" id="FNanchor_76_76"></a><a href="#Footnote_76_76" class="fnanchor">[76]</a></p>

<p>Let us consider for a moment what takes place
when the field of observation to which our ideas have
been adapted and now conform, becomes enlarged.
We had, let us say, always seen heavy bodies sink
when their support was taken away; we had also seen,
perhaps, that the sinking of heavier bodies forced
lighter bodies upwards. But now we see a lever in
action, and we are suddenly struck with the fact that
a lighter body is lifting another of much greater weight.
Our customary train of thought demands its rights;
the new and unwonted event likewise demands its
rights. From this conflict between thought and fact
the <i>problem</i> arises; out of this partial contrariety springs
the question, "Why?" With the new adaptation to the
enlarged field of observation, the problem disappears,
or, in other words, is solved. In the instance cited,
we must adopt the habit of always considering the
mechanical work performed.</p>

<p>The child just awakening into consciousness of the
world, knows no problem. The bright flower, the
ringing bell, are all new to it; yet it is surprised at
nothing. The out and out Philistine, whose only
thoughts lie in the beaten path of his every-day pursuits,
likewise has no problems. Everything goes its
wonted course, and if perchance a thing go wrong at
times, it is at most a mere object of curiosity and
not worth serious consideration. In fact, the question
"Why?" loses all warrant in relations where we are
familiar with every aspect of events. But the capable
and talented young man has his head full of problems;
he has acquired, to a greater or less degree, certain
habitudes of thought, and at the same time he is constantly
observing what is new and unwonted, and in
his case there is no end to the questions, "Why?"</p>

<p>Thus, the factor which most promotes scientific
thought is the gradual widening of the field of experience.
We scarcely notice events we are accustomed
to; the latter do not really develop their intellectual
significance until placed in contrast with something to
which we are unaccustomed. Things that at home
are passed by unnoticed, delight us when abroad,
though they may appear in only slightly different forms.
The sun shines with heightened radiance, the flowers
bloom in brighter colors, our fellow-men accost us
with lighter and happier looks. And, returning home,
we find even the old familiar scenes more inspiring
and suggestive than before.</p>

<p>Every motive that prompts and stimulates us to
modify and transform our thoughts, proceeds from
what is new, uncommon, and not understood. Novelty
excites wonder in persons whose fixed habits of thought
are shaken and disarranged by what they see. But the
element of wonder never lies in the phenomenon or
event observed; its place is in the person observing.
People of more vigorous mental type aim at once at an
<i>adaptation of thought</i> that will conform to what they
have observed. Thus does science eventually become
the natural foe of the wonderful. The sources of the
marvellous are unveiled, and surprise gives way to
calm interpretation.</p>

<p>Let us consider such a mental transformative process
in detail. The circumstance that heavy bodies
fall to the earth appears perfectly natural and regular.
But when a person observes that wood floats upon
water, and that flames and smoke rise in the air, then
the contrary of the first phenomenon is presented.
An olden theory endeavors to explain these facts by imputing
to substances the power of volition, as that attribute
which is most familiar to man. It asserted
that every substance seeks its proper place, heavy
bodies tending downwards and light ones upwards.
It soon turned out, however, that even smoke had
weight, that it, too, sought its place below, and that
it was forced upwards only because of the downward
tendency of the air, as wood is forced to the surface of
water because the water exerts the greater downward
pressure.</p>

<p>Again, we see a body thrown into the air. It ascends.
How is it that it does not seek its proper place? Why
does the velocity of its "violent" motion decrease as
it rises, while that of its "natural" fall increases as it
descends. If we mark closely the relation between
these two facts, the problem will solve itself. We shall
see, as Galileo did, that the decrease of velocity in
rising and the increase of velocity in falling are one
and the same phenomenon, viz., an increase of velocity
towards the earth. Accordingly, it is not a place
that is assigned to the body, but an increase of velocity
towards the earth.</p>

<p>By this idea the movements of heavy bodies are
rendered perfectly familiar. Newton, now, firmly
grasping this new way of thinking, sees the moon and
the planets moving in their paths upon principles similar
to those which determine the motion of a projectile
thrown into the air. Yet the movements of the
planets were marked by peculiarities which compelled
him once more to modify slightly his customary mode
of thought. The heavenly bodies, or rather the parts
composing them, do not move with constant accelerations
towards each other, but "attract each other,"
directly as the mass and inversely as the square of the
distance.</p>

<p>This latter notion, which includes the one applying
to terrestrial bodies as a special case, is, as we see,
quite different from the conception from which we
started. How limited in scope was the original idea
and to what a multitude of phenomena is not the present
one applicable! Yet there is a trace, after all,
of the "search for place" in the expression "attraction."
And it would be folly, indeed, for us to avoid,
with punctilious dread, this conception of "attraction"
as bearing marks of its pedigree. It is the historical
base of the Newtonian conception and it still continues
to direct our thoughts in the paths so long familiar to
us. Thus, the happiest ideas do not fall from heaven,
but spring from notions already existing.</p>

<p>Similarly, a ray of light was first regarded as a continuous
and homogeneous straight line. It then became
the path of projection for minute missiles; then
an aggregate of the paths of countless different kinds
of missiles. It became periodic; it acquired various
sides; and ultimately it even lost its motion in a
straight line.</p>

<p>The electric current was conceived originally as
the flow of a hypothetical fluid. To this conception
was soon added the notion of a chemical current, the
notion of an electric, magnetic, and anisotropic optical
field, intimately connected with the path of the current.
And the richer a conception becomes in following
and keeping pace with facts, the better adapted it
is to anticipate them.</p>

<p>Adaptive processes of this kind have no assignable
beginning, inasmuch as every problem that incites
to new adaptation, presupposes a fixed habitude of
thought. Moreover, they have no visible end; in so
far as experience never ceases. Science, accordingly,
stands midway in the evolutionary process; and science
may advantageously direct and promote this process,
but it can never take its place. That science is inconceivable
the principles of which would enable a person
with no experience to construct the world of experience,
without a knowledge of it. One might just as
well expect to become a great musician, solely by the
aid of theory, and without musical experience; or to
become a painter by following the directions of a text-book.</p>

<p>In glancing over the history of an idea with which
we have become perfectly familiar, we are no longer
able to appreciate the full significance of its growth.
The deep and vital changes that have been effected in
the course of its evolution, are recognisable only from
the astounding narrowness of view with which great
contemporary scientists have occasionally opposed
each other. Huygens's wave-theory of light was incomprehensible
to Newton, and Newton's idea of universal
gravity was unintelligible to Huygens. But a
century afterwards both notions were reconcilable,
even in ordinary minds.</p>

<p>On the other hand, the original creations of pioneer
intellects, unconsciously formed, do not assume
a foreign garb; their form is their own. In them,
childlike simplicity is joined to the maturity of manhood,
and they are not to be compared with processes
of thought in the average mind. The latter are carried
on as are the acts of persons in the state of mesmerism,
where actions involuntarily follow the images which
the words of other persons suggest to their minds.</p>

<p>The ideas that have become most familiar through
long experience, are the very ones that intrude themselves
into the conception of every new fact observed.
In every instance, thus, they become involved in a
struggle for self-preservation, and it is just they that
are seized by the inevitable process of transformation.</p>

<p>Upon this process rests substantially the method
of explaining by hypothesis new and uncomprehended
phenomena. Thus, instead of forming entirely new
notions to explain the movements of the heavenly
bodies and the phenomena of the tides, we imagine the
material particles composing the bodies of the universe
to possess weight or gravity with respect to one another.
Similarly, we imagine electrified bodies to be
freighted with fluids that attract and repel, or we conceive
the space between them to be in a state of elastic
tension. In so doing, we substitute for new ideas
distinct and more familiar notions of old experience&mdash;notions
which to a great extent run unimpeded in their
courses, although they too must suffer partial transformation.</p>

<p>The animal cannot construct new members to perform
every new function that circumstances and fate
demand of it. On the contrary it is obliged to make
use of those it already possesses. When a vertebrate
animal chances into an environment where it must
learn to fly or swim, an additional pair of extremities is
not grown for the purpose. On the contrary, the animal
must adapt and transform a pair that it already
has.</p>

<p>The construction of hypotheses, therefore, is not
the product of artificial scientific methods. This process
is unconsciously carried on in the very infancy of
science. Even later, hypotheses do not become detrimental
and dangerous to progress except when more
reliance is placed on them than on the facts themselves;
when the contents of the former are more
highly valued than the latter, and when, rigidly adhering
to hypothetical notions, we overestimate the
ideas we possess as compared with those we have to
acquire.</p>

<p>The extension of our sphere of experience always
involves a transformation of our ideas. It matters not
whether the face of nature becomes actually altered,
presenting new and strange phenomena, or whether
these phenomena are brought to light by an intentional
or accidental turn of observation. In fact, all the varied
methods of scientific inquiry and of purposive
mental adaptation enumerated by John Stuart Mill,
those of observation as well as those of experiment,
are ultimately recognisable as forms of one fundamental
method, the method of change, or variation. It is
through change of circumstances that the natural philosopher
learns. This process, however, is by no means
confined to the investigator of nature. The historian,
the philosopher, the jurist, the mathematician, the
artist, the æsthetician,<a name="FNanchor_77_77" id="FNanchor_77_77"></a><a href="#Footnote_77_77" class="fnanchor">[77]</a> all illuminate and unfold their
ideas by producing from the rich treasures of memory
similar, but different, cases; thus, they observe and
experiment in their thoughts. Even if all sense-experience
should suddenly cease, the events of the days
past would meet in different attitudes in the mind
and the process of adaptation would still continue&mdash;a
process which, in contradistinction to the adaptation
of thoughts to facts in practical spheres, would be
strictly theoretical, being an adaptation of thoughts to
thoughts.</p>

<p>The method of change or variation brings before us
like cases of phenomena, having partly the same and
partly different elements. It is only by comparing
different cases of refracted light at changing angles of
incidence that the common factor, the constancy of
the refractive index, is disclosed. And only by comparing
the refractions of light of different colors, does
the difference, the inequality of the indices of refraction,
arrest the attention. Comparison based upon
change leads the mind simultaneously to the highest
abstractions and to the finest distinctions.</p>

<p>Undoubtedly, the animal also is able to distinguish
between the similar and dissimilar of two cases. Its
consciousness is aroused by a noise or a rustling, and
its motor centre is put in readiness. The sight of the
creature causing the disturbance, will, according to its
size, provoke flight or prompt pursuit; and in the latter
case, the more exact distinctions will determine the
mode of attack. But man alone attains to the faculty
of voluntary and conscious comparison. Man alone
can, by his power of abstraction, rise, in one moment,
to the comprehension of principles like the conservation
of mass or the conservation of energy, and in the
next observe and mark the arrangement of the iron
lines in the spectrum. In thus dealing with the objects
of his conceptual life, his ideas unfold and expand,
like his nervous system, into a widely ramified
and organically articulated tree, on which he may follow
every limb to its farthermost branches, and, when
occasion demands, return to the trunk from which he
started.</p>

<p>The English philosopher Whewell has remarked
that two things are requisite to the formation of science:
facts and ideas. Ideas alone lead to empty
speculation; mere facts can yield no organic knowledge.
We see that all depends upon the capacity of
adapting existing notions to fresh facts.</p>

<p>Over-readiness to yield to every new fact prevents
fixed habits of thought from arising. Excessively rigid
habits of thought impede freedom of observation. In
the struggle, in the compromise between judgment
and prejudgment (prejudice), if we may use the term,
our understanding of things broadens.</p>

<p>Habitual judgment, applied to a new case without
antecedent tests, we call prejudgment or prejudice.
Who does not know its terrible power! But we think
less often of the importance and utility of prejudice.
Physically, no one could exist, if he had to guide and
regulate the circulation, respiration, and digestion of
his body by conscious and purposive acts. So, too,
no one could exist intellectually if he had to form judgments
on every passing experience, instead of allowing
himself to be controlled by the judgments he has
already formed. Prejudice is a sort of reflex motion
in the province of intelligence.</p>

<p>On prejudices, that is, on habitual judgments not
tested in every case to which they are applied, reposes
a goodly portion of the thought and work of the natural
scientist. On prejudices reposes most of the conduct
of society. With the sudden disappearance of
prejudice society would hopelessly dissolve. That
prince displayed a deep insight into the power of intellectual
habit, who quelled the loud menaces and
demands of his body-guard for arrears of pay and compelled
them to turn about and march, by simply pronouncing
the regular word of command; he well knew
that they would be unable to resist that.</p>

<p>Not until the discrepancy between habitual judgments
and facts becomes great is the investigator implicated
in appreciable illusion. Then tragic complications
and catastrophes occur in the practical life of
individuals and nations&mdash;crises where man, placing
custom above life, instead of pressing it into the service
of life, becomes the victim of his error. The very
power which in intellectual life advances, fosters, and
sustains us, may in other circumstances delude and
destroy us.</p>
<p><span class="pagenum"><a name="Page_234" id="Page_234">[Pg 234]</a></span></p><p><span class="pagenum"><a name="Page_235" id="Page_235">[Pg 235]</a></span></p><p><span class="pagenum"><a name="Page_236" id="Page_236">[Pg 236]</a></span></p>
<hr class="tb" />

<p>Ideas are not all of life. They are only momentary
efflorescences of light, designed to illuminate the paths
of the will. But as delicate reagents on our organic
evolution our ideas are of paramount importance. No
theory can gainsay the vital transformation which we
feel taking place within us through their agency. Nor
is it necessary that we should have a proof of this process.
We are immediately assured of it.</p>

<p>The transformation of ideas thus appears as a part
of the general evolution of life, as a part of its adaptation
to a constantly widening sphere of action. A
granite boulder on a mountain-side tends towards the
earth below. It must abide in its resting-place for
thousands of years before its support gives way. The
shrub that grows at its base is farther advanced; it
accommodates itself to summer and winter. The fox
which, overcoming the force of gravity, creeps to the
summit where he has scented his prey, is freer in his
movements than either. The arm of man reaches
further still; and scarcely anything of note happens
in Africa or Asia that does not leave an imprint upon
his life. What an immense portion of the life of
other men is reflected in ourselves; their joys, their
affections, their happiness and misery! And this too,
when we survey only our immediate surroundings,
and confine our attention to modern literature. How
much more do we experience when we travel through
ancient Egypt with Herodotus, when we stroll through
the streets of Pompeii, when we carry ourselves back
to the gloomy period of the crusades or to the golden
age of Italian art, now making the acquaintance of a
physician of Molière, and now that of a Diderot or of
a D'Alembert. What a great part of the life of others,
of their character and their purpose, do we not absorb
through poetry and music! And although they only
gently touch the chords of our emotions, like the memory
of youth softly breathing upon the spirit of an
aged man, we have nevertheless lived them over again
in part. How great and comprehensive does self become
in this conception; and how insignificant the
person! Egoistical systems both of optimism and pessimism
perish with their narrow standard of the import
of intellectual life. We feel that the real pearls
of life lie in the ever changing contents of consciousness,
and that the person is merely an indifferent symbolical
thread on which they are strung.<a name="FNanchor_78_78" id="FNanchor_78_78"></a><a href="#Footnote_78_78" class="fnanchor">[78]</a></p>

<p>We are prepared, thus, to regard ourselves and
every one of our ideas as a product and a subject of
universal evolution; and in this way we shall advance
sturdily and unimpeded along the paths which the
future will throw open to us.<a name="FNanchor_79_79" id="FNanchor_79_79"></a><a href="#Footnote_79_79" class="fnanchor">[79]</a></p>




<h2><a name="ON_THE_PRINCIPLE_OF_COMPARISON" id="ON_THE_PRINCIPLE_OF_COMPARISON">ON THE PRINCIPLE OF COMPARISON
IN PHYSICS.</a><a name="FNanchor_80_80" id="FNanchor_80_80"></a><a href="#Footnote_80_80" class="fnanchor">[80]</a></h2>


<p>Twenty years ago when Kirchhoff defined the object
of mechanics as the "description, in complete
and very simple terms, of the motions occurring in nature,"
he produced by the statement a peculiar impression.
Fourteen years subsequently, Boltzmann, in the
life-like picture which he drew of the great inquirer,
could still speak of the universal astonishment at this
novel method of treating mechanics, and we meet with
epistemological treatises to-day, which plainly show
how difficult is the acceptance of this point of view. A
modest and small band of inquirers there were, however,
to whom Kirchhoff's few words were tidings of a
welcome and powerful ally in the epistemological field.</p>

<p>Now, how does it happen that we yield our assent
so reluctantly to the philosophical opinion of an inquirer
for whose scientific achievements we have only
words of praise? One reason probably is that few inquirers
can find time and leisure, amid the exacting<span class="pagenum"><a name="Page_237" id="Page_237">[Pg 237]</a></span>
employments demanded for the acquisition of new
knowledge, to inquire closely into that tremendous
psychical process by which science is formed. Further,
it is inevitable that much should be put into Kirchhoff's
rigid words that they were not originally intended to
convey, and that much should be found wanting in
them that had always been regarded as an essential
element of scientific knowledge. What can mere description
accomplish? What has become of explanation,
of our insight into the causal connexion of things?</p>
<p><span class="pagenum"><a name="Page_238" id="Page_238">[Pg 238]</a></span></p><p><span class="pagenum"><a name="Page_239" id="Page_239">[Pg 239]</a></span></p>
<hr class="tb" />

<p>Permit me, for a moment, to contemplate not the
results of science, but the mode of its <i>growth</i>, in a
frank and unbiassed manner. We know of only <i>one</i>
source of <i>immediate revelation</i> of scientific facts&mdash;<i>our
senses</i>. Restricted to this source alone, thrown wholly
upon his own resources, obliged to start always anew,
what could the isolated individual accomplish? Of a
stock of knowledge so acquired the science of a distant
negro hamlet in darkest Africa could hardly give
us a sufficiently humiliating conception. For there
that veritable miracle of thought-transference has already
begun its work, compared with which the miracles
of the spiritualists are rank monstrosities&mdash;<i>communication
by language</i>. Reflect, too, that by means
of the magical characters which our libraries contain
we can raise the spirits of the "the sovereign dead of
old" from Faraday to Galileo and Archimedes, through
ages of time&mdash;spirits who do not dismiss us with ambiguous
and derisive oracles, but tell us the best they
know; then shall we feel what a stupendous and indispensable
factor in the formation of science <i>communication</i>
is. Not the dim, half-conscious <i>surmises</i>
of the acute observer of nature or critic of humanity
belong to science, but only that which they possess
clearly enough to <i>communicate</i> to others.</p>

<p>But how, now, do we go about this communication
of a newly acquired experience, of a newly observed
fact? As the different calls and battle-cries of gregarious
animals are unconsciously formed signs for
a common observation or action, irrespective of the
causes which produce such action&mdash;a fact that already
involves the germ of the concept; so also the words
of human language, which is only more highly specialised,
are names or signs for universally known
facts, which all can observe or have observed. If the
mental representation, accordingly, follows the new
fact at once and <i>passively</i>, then that new fact must, of
itself, immediately be constituted and represented in
thought by facts already universally known and commonly
observed. Memory is always ready to put forward
for <i>comparison</i> known facts which resemble the
new event, or agree with it in certain features, and
so renders possible that elementary internal judgment
which the mature and definitively formulated judgment
soon follows.</p>

<p>Comparison, as the fundamental condition of communication,
is the most powerful inner vital element
of science. The zoölogist sees in the bones of the
wing-membranes of bats, fingers; he compares the
bones of the cranium with the vertebræ, the embryos
of different organisms with one another, and the different
stages of development of the same organism
with one another. The geographer sees in Lake Garda
a fjord, in the Sea of Aral a lake in process of drying
up. The philologist compares different languages with
one another, and the formations of the same language
as well. If it is not customary to speak of comparative
physics in the same sense that we speak of comparative
anatomy, the reason is that in a science of
such great experimental activity the attention is turned
away too much from the <i>contemplative</i> element. But
like all other sciences, physics lives and grows by
comparison.</p>
<p><span class="pagenum"><a name="Page_240" id="Page_240">[Pg 240]</a></span></p><p><span class="pagenum"><a name="Page_241" id="Page_241">[Pg 241]</a></span></p>
<hr class="tb" />

<p>The manner in which the result of the comparison
finds expression in the communication, varies of course
very much. When we say that the colors of the spectrum
are red, yellow, green, blue, and violet, the designations
employed may possibly have been derived
from the technology of tattooing, or they may subsequently
have acquired the significance of standing for
the colors of the rose, the lemon, the leaf, the corn-flower,
and the violet. From the frequent repetition
of such comparisons, however, made under the most
manifold circumstances, the inconstant features, as
compared with the permanent congruent features, get
so obliterated that the latter acquire a fixed significance
independent of every object and connexion, or take on
as we say an <i>abstract</i> or <i>conceptual</i> import. No one
thinks at the word "red" of any other agreement with
the rose than that of color, or at the word "straight"
of any other property of a stretched cord than the
sameness of direction. Just so, too, numbers, originally
the names of the fingers of the hands and feet,
from being used as arrangement-signs for all kinds of
objects, were lifted to the plane of abstract concepts.
A verbal report (communication) of a fact that uses
only these purely abstract implements, we call a <i>direct
description</i>.</p>

<p>The direct description of a fact of any great extent
is an irksome task, even where the requisite notions
are already completely developed. What a simplification
it involves if we can say, the fact <i>A</i> now
considered comports itself, not in <i>one</i>, but in <i>many</i> or
in <i>all</i> its features, like an old and well-known fact <i>B</i>.
The moon comports itself as a heavy body does with
respect to the earth; light like a wave-motion or an
electric vibration; a magnet, as if it were laden with
gravitating fluids, and so on. We call such a description,
in which we appeal, as it were, to a description
already and elsewhere formulated, or perhaps still to
be precisely formulated, an <i>indirect description</i>. We
are at liberty to supplement this description, gradually,
by direct description, to correct it, or to replace it altogether.
We see, thus, without difficulty, that what is
called a <i>theory</i> or a <i>theoretical idea</i>, falls under the
category of what is here termed indirect description.</p>
<p><span class="pagenum"><a name="Page_242" id="Page_242">[Pg 242]</a></span></p><p><span class="pagenum"><a name="Page_243" id="Page_243">[Pg 243]</a></span></p><p><span class="pagenum"><a name="Page_244" id="Page_244">[Pg 244]</a></span></p><p><span class="pagenum"><a name="Page_245" id="Page_245">[Pg 245]</a></span></p><p><span class="pagenum"><a name="Page_246" id="Page_246">[Pg 246]</a></span></p><p><span class="pagenum"><a name="Page_247" id="Page_247">[Pg 247]</a></span></p><p><span class="pagenum"><a name="Page_248" id="Page_248">[Pg 248]</a></span></p><p><span class="pagenum"><a name="Page_249" id="Page_249">[Pg 249]</a></span></p><p><span class="pagenum"><a name="Page_250" id="Page_250">[Pg 250]</a></span></p><p><span class="pagenum"><a name="Page_251" id="Page_251">[Pg 251]</a></span></p><p><span class="pagenum"><a name="Page_252" id="Page_252">[Pg 252]</a></span></p>
<hr class="tb" />

<p>What, now, is a theoretical idea? Whence do we
get it? What does it accomplish for us? Why does it
occupy a higher place in our judgment than the mere
holding fast to a fact or an observation? Here, too,
memory and comparison alone are in play. But instead
of <i>a single</i> feature of resemblance culled from
memory, in this case <i>a great system</i> of resemblances
confronts us, a well-known physiognomy, by means of
which the new fact is immediately transformed into an
old acquaintance. Besides, it is in the power of the
idea to offer us more than we actually see in the new
fact, at the first moment; it can extend the fact, and
enrich it with features which we are first induced to
<i>seek</i> from such suggestions, and which are often actually
found. It is this <i>rapidity</i> in extending knowledge
that gives to theory a preference over simple observation.
But that preference is wholly <i>quantitative</i>.
Qualitatively, and in real essential points, theory differs
from observation neither in the mode of its origin
nor in its last results.</p>

<p>The adoption of a theory, however, always involves
a danger. For a theory puts in the place of a fact <i>A</i>
in thought, always a <i>different</i>, but simpler and more
familiar fact <i>B</i>, which in <i>some</i> relations can mentally
represent <i>A</i>, but for the very reason that it is different,
in other relations cannot represent it. If now, as
may readily happen, sufficient care is not exercised,
the most fruitful theory may, in special circumstances,
become a downright obstacle to inquiry. Thus, the
emission-theory of light, in accustoming the physicist
to think of the projectile path of the "light-particles"
as an undifferentiated straight-line, demonstrably impeded
the discovery of the periodicity of light. By
putting in the place of light the more familiar phenomena
of sound, Huygens renders light in many of
its features a familiar event, but with respect to polarisation,
which lacks the longitudinal waves with which
alone he was acquainted, it had for him a doubly
strange aspect. He is unable thus to grasp in abstract
thought the fact of polarisation, which is before his
eyes, whilst Newton, merely by adapting to the observation
his thoughts, and putting this question, "<i>Annon
radiorum luminis diversa sunt latera?</i>" abstractly
grasped polarisation, that is, directly described it, a
century before Malus. On the other hand, if the
agreement of the fact with the idea theoretically representing
it, extends further than its inventor originally
anticipated, then we may be led by it to unexpected
discoveries, of which conical refraction, circular polarisation
by total reflexion, Hertz's waves offer ready
examples, in contrast to the illustrations given above.</p>

<p>Our insight into the conditions indicated will be
improved, perhaps, by contemplating the development
of some theory or other more in detail. Let us consider
a magnetised bar of steel by the side of a second
unmagnetised bar, in all other respects the same. The
second bar gives no indication of the presence of iron-filings;
the first attracts them. Also, when the iron-filings
are absent, we must think of the magnetised
bar as in a different condition from that of the unmagnetised.
For, that the mere presence of the iron-filings
does not induce the phenomenon of attraction is proved
by the second unmagnetised bar. The ingenuous man,
who finds in his will, as his most familiar source of
power, the best facilities for comparison, conceives a
species of <i>spirit</i> in the magnet. The behavior of a
warm body or of an <i>electrified</i> body suggests similar
ideas. This is the point of view of the oldest theory,
<i>fetishism</i>, which the inquirers of the early Middle
Ages had not yet overcome, and which in its last vestiges,
in the conception of forces, still flourishes in
modern physics. We see, thus, the <i>dramatic</i> element
need no more be absent in a scientific description, than
in a thrilling novel.</p>

<p>If, on subsequent examination, it be observed that
a cold body, in contact with a hot body, warms itself,
so to speak, <i>at the expense</i> of the hot body; further,
that when the substances are the same, the cold body,
which, let us say, has twice the mass of the other,
gains only half the number of degrees of temperature
that the other loses, a wholly new impression arises.
The demoniac character of the event vanishes, for the
supposed spirit acts not by caprice, but according to
fixed laws. In its place, however, <i>instinctively</i> the
notion of a <i>substance</i> is substituted, part of which flows
over from the one body to the other, but the total
amount of which, representable by the sum of the products
of the masses into the respective changes of
temperature, remains constant. Black was the first to
be <i>powerfully</i> struck with this resemblance of thermal
processes to the motion of a substance, and under its
guidance discovered the specific heat, the heat of fusion,
and the heat of vaporisation of bodies. Gaining
strength and fixity, however, from these successes,
this notion of substance subsequently stood in the way
of scientific advancement. It blinded the eyes of the
successors of Black, and prevented them from seeing
the manifest fact, which every savage knows, that heat
is <i>produced</i> by friction. Fruitful as that notion was
for Black, helpful as it still is to the learner to-day in
Black's special field, permanent and universal validity
as a <i>theory</i> it could never maintain. But what is essential,
conceptually, in it, viz., the constancy of the product-sum
above mentioned, retains its value and may
be regarded as a <i>direct description</i> of Black's facts.</p>

<p>It stands to reason that those theories which push
themselves forward unsought, instinctively, and wholly
of their own accord, should have the greatest power,
should carry our thoughts most with them, and exhibit
the staunchest powers of self-preservation. On the
other hand, it may also be observed that when critically
scrutinised such theories are extremely apt to
lose their cogency. We are constantly busied with
"substance," its modes of action have stamped themselves
indelibly upon our thoughts, our vividest and
clearest reminiscences are associated with it. It should
cause us no surprise, therefore, that Robert Mayer and
Joule, who gave the final blow to Black's substantial
conception of heat, should have re-introduced the
same notion of substance in a more abstract and modified
form, only applying to a much more extensive
field.</p>

<p>Here, too, the psychological circumstances which
impart to the new conception its power, lie clearly before
us. By the unusual redness of the venous blood
in tropical climates Mayer's attention is directed to
the lessened expenditure of internal heat and to the
proportionately lessened <i>consumption of material</i> by the
human body in those climates. But as every effort of
the human organism, including its mechanical work,
is connected with the consumption of material, and as
work by friction can engender heat, therefore heat and
work appear in kind equivalent, and between them a
proportional relation must subsist. Not <i>every</i> quantity,
but the appropriately calculated <i>sum</i> of the two, as
connected with a proportionate consumption of material,
appears <i>substantial</i>.</p>

<p>By exactly similar considerations, relative to the
economy of the galvanic element, Joule arrived at his
view; he found experimentally that the sum of the
heat evolved in the circuit, of the heat consumed in the
combustion of the gas developed, of the electro-magnetic
work of the current, properly calculated,&mdash;in
short, the sum of all the effects of the battery,&mdash;is connected
with a proportionate consumption of zinc. Accordingly,
this sum itself has a substantial character.</p>

<p>Mayer was so absorbed with the view attained,
that the indestructibility of <i>force</i>, in our phraseology
<i>work</i>, appeared to him <i>a priori</i> evident. "The creation
or annihilation of a force," he says, "lies without
the province of human thought and power." Joule
expressed himself to a similar effect: "It is manifestly
absurd to suppose that the powers with which God
has endowed matter can be destroyed." Strange to
say, on the basis of such utterances, not Joule, but
Mayer, was stamped as a metaphysician. We may
be sure, however, that both men were merely giving
expression, and that half-unconsciously, to a powerful
<i>formal</i> need of the new simple view, and that both
would have been extremely surprised if it had been
proposed to them that their principle should be submitted
to a philosophical congress or ecclesiastical
synod for a decision upon its validity. But with all
agreements, the attitude of these two men, in other
respects, was totally different. Whilst Mayer represented
this <i>formal</i> need with all the stupendous instinctive
force of genius, we might say almost with the
ardor of fanaticism, yet was withal not wanting in the
conceptive ability to compute, prior to all other inquirers,
the mechanical equivalent of heat from old
physical constants long known and at the disposal of
all, and so to set up for the new doctrine a programme
embracing all physics and physiology; Joule, on the
other hand, applied himself to the exact verification of
the doctrine by beautifully conceived and masterfully
executed experiments, extending over all departments
of physics. Soon Helmholtz too attacked the problem,
in a totally independent and characteristic manner.
After the professional virtuosity with which this physicist
grasped and disposed of all the points unsettled
by Mayer's programme and more besides, what especially
strikes us is the consummate critical lucidity of
this young man of twenty-six years. In his exposition
is wanting that vehemence and impetuosity which
marked Mayer's. The principle of the conservation
of energy is no self-evident or <i>a priori</i> proposition for
him. What follows, on the assumption that that proposition
obtains? In this hypothetical form, he subjugates
his matter.</p>

<p>I must confess, I have always marvelled at the
æsthetic and ethical taste of many of our contemporaries
who have managed to fabricate out of this relation
of things, odious national and personal questions,
instead of praising the good fortune that made <i>several</i>
such men work together and of rejoicing at the instructive
diversity and idiosyncrasies of great minds
fraught with such rich consequences for us.</p>

<p>We know that still another theoretical conception
played a part in the development of the principle of
energy, which Mayer held aloof from, namely, the conception
that heat, as also the other physical processes,
are due to motion. But once the principle of energy
has been reached, these auxiliary and transitional theories
discharge no essential function, and we may regard
the principle, like that which Black gave, as a
contribution to the <i>direct description</i> of a widely extended
domain of facts.</p>

<p>It would appear from such considerations not only
advisable, but even necessary, with all due recognition
of the helpfulness of theoretic ideas in research,
yet gradually, as the new facts grow familiar, to substitute
for indirect description <i>direct</i> description, which
contains nothing that is unessential and restricts itself
absolutely to the abstract apprehension of facts. We
might almost say, that the descriptive sciences, so
called with a tincture of condescension, have, in respect
of scientific character, outstripped the physical
expositions lately in vogue. Of course, a virtue has
been made of necessity here.</p>

<p>We must admit, that it is not in our power to describe
directly every fact, on the moment. Indeed,
we should succumb in utter despair if the whole wealth
of facts which we come step by step to know, were
presented to us all at once. Happily, only detached
and unusual features first strike us, and such we bring
nearer to ourselves by <i>comparison</i> with every-day
events. Here the notions of the common speech are
first developed. The comparisons then grow more
manifold and numerous, the fields of facts compared
more extensive, the concepts that make direct description
possible, proportionately more general and more
abstract.</p>

<p>First we become familiar with the motion of freely
falling bodies. The concepts of force, mass, and work
are then carried over, with appropriate modifications,
to the phenomena of electricity and magnetism. A
stream of water is said to have suggested to Fourier
the first distinct picture of currents of heat. A special
case of vibrations of strings investigated by Taylor,
cleared up for him a special case of the conduction of
heat. Much in the same way that Daniel Bernoulli
and Euler constructed the most diverse forms of vibrations
of strings from Taylor's cases, so Fourier constructs
out of simple cases of conduction the most
multifarious motions of heat; and that method has
extended itself over the whole of physics. Ohm forms
his conception of the electric current in imitation of
Fourier's. The latter, also, adopts Fick's theory of
diffusion. In an analogous manner a conception of
the magnetic current is developed. All sorts of stationary
currents are thus made to exhibit common
features, and even the condition of complete equilibrium
in an extended medium shares these features
with the dynamical condition of equilibrium of a stationary
current. Things as remote as the magnetic
lines of force of an electric current and the stream-lines
of a frictionless liquid vortex enter in this way
into a peculiar relationship of similarity. The concept
of potential, originally enunciated for a restricted
province, acquires a wide-reaching applicability.
Things as dissimilar as pressure, temperature,
and electromotive force, now show points of agreement
in relation to ideas derived by definite methods
from that concept: viz., fall of pressure, fall of temperature,
fall of potential, as also with the further notions
of liquid, thermal, and electric strength of current.
That relationship between systems of ideas in
which the dissimilarity of every two homologous concepts
as well as the agreement in logical relations
of every two homologous pairs of concepts, is clearly
brought to light, is called an <i>analogy</i>. It is an effective
means of mastering heterogeneous fields of facts in
unitary comprehension. The path is plainly shown in
which <i>a universal physical phenomenology</i> embracing all
domains, will be developed.</p>

<p>In the process described we attain for the first time
to what is indispensable in the direct description of
broad fields of fact&mdash;the wide-reaching <i>abstract concept</i>.
And now I must put a question smacking of the school-master,
but unavoidable: What is a concept? Is it a
hazy representation, admitting withal of mental visualisation?
No. Mental visualisation accompanies it
only in the simplest cases, and then merely as an adjunct.
Think, for example, of the "coefficient of self-induction,"
and seek for its visualised mental image.
Or is, perhaps, the concept a mere word? The adoption
of this forlorn idea, which has been actually proposed
of late by a reputed mathematician would only
throw us back a thousand years into the deepest scholasticism.
We must, therefore, reject it.</p>

<p>The solution is not far to seek. We must not think
that sensation, or representation, is a purely passive
process. The lowest organisms respond to it with a
simple reflex motion, by engulfing the prey which approaches
them. In higher organisms the centripetal
stimulus encounters in the nervous system obstacles
and aids which modify the centrifugal process. In still
higher organisms, where prey is pursued and examined,
the process in question may go through extensive
paths of circular motions before it comes to relative
rest. Our own life, too, is enacted in such
processes; all that we call science may be regarded
as parts, or middle terms, of such activities.</p>

<p>It will not surprise us now if I say: the definition
of a concept, and, when it is very familiar, even its
name, is an <i>impulse</i> to some accurately determined,
often complicated, critical, comparative, or constructive
<i>activity</i>, the usually sense-perceptive result of
which is a term or member of the concept's scope. It
matters not whether the concept draws the attention
only to one certain sense (as sight) or to a phase of a
sense (as color, form), or is the starting-point of a
complicated action; nor whether the activity in question
(chemical, anatomical, and mathematical operations)
is muscular or technical, or performed wholly
in the imagination, or only intimated. The concept is
to the physicist what a musical note is to a piano-player.
A trained physicist or mathematician reads a
memoir like a musician reads a score. But just as the
piano-player must first learn to move his fingers singly
and collectively, before he can follow his notes without
effort, so the physicist or mathematician must go
through a long apprenticeship before he gains control,
so to speak, of the manifold delicate innervations
of his muscles and imagination. Think of how frequently
the beginner in physics or mathematics performs
more, or less, than is required, or of how frequently
he conceives things differently from what they
are! But if, after having had sufficient discipline, he
lights upon the phrase "coefficient of self-induction,"
he knows immediately what that term requires of him.
Long and thoroughly practised <i>actions</i>, which have
their origin in the necessity of comparing and representing
facts by other facts, are thus the very kernel
of concepts. In fact, positive and philosophical philology
both claim to have established that all roots
represent concepts and stood originally for muscular
activities alone. The slow assent of physicists to
Kirchhoff's dictum now becomes intelligible. They
best could feel the vast amount of individual labor,
theory, and skill required before the ideal of direct
description could be realised.</p>
<p><span class="pagenum"><a name="Page_253" id="Page_253">[Pg 253]</a></span></p><p><span class="pagenum"><a name="Page_254" id="Page_254">[Pg 254]</a></span></p><p><span class="pagenum"><a name="Page_255" id="Page_255">[Pg 255]</a></span></p><p><span class="pagenum"><a name="Page_256" id="Page_256">[Pg 256]</a></span></p><p><span class="pagenum"><a name="Page_257" id="Page_257">[Pg 257]</a></span></p><p><span class="pagenum"><a name="Page_258" id="Page_258">[Pg 258]</a></span></p><p><span class="pagenum"><a name="Page_259" id="Page_259">[Pg 259]</a></span></p>
<hr class="tb" />

<p>Suppose, now, the ideal of a given province of
facts is reached. Does description accomplish all that
the inquirer can ask? In my opinion, it does. Description
is a building up of facts in thought, and this
building up is, in the experimental sciences, often the
condition of actual execution. For the physicist, to
take a special case, the metrical units are the building-stones,
the concepts the directions for building, and
the facts the result of the building. Our mental
imagery is almost a complete substitute for the fact,
and by means of it we can ascertain all the fact's properties.
We do not know that worst which we ourselves
have made.</p>

<p>People require of science that it should <i>prophesy</i>,
and Hertz uses that expression in his posthumous
<i>Mechanics</i>. But, natural as it is, the expression is too
narrow. The geologist and the palæontologist, at times
the astronomer, and always the historian and the philologist,
prophesy, so to speak, <i>backwards</i>. The descriptive
sciences, like geometry and mathematics, prophesy
neither forward or backwards, but seek from given
conditions the conditioned. Let us say rather: <i>Science
completes in thought facts that are only partly given</i>.
This is rendered possible by description, for description
presupposes the interdependence of the descriptive
elements: otherwise nothing would be described.</p>

<p>It is said, description leaves the sense of causality
unsatisfied. In fact, many imagine they understand
motions better when they picture to themselves the
pulling forces; and yet the <i>accelerations</i>, the facts,
accomplish more, without superfluous additions. I
hope that the science of the future will discard the
idea of cause and effect, as being formally obscure;
and in my feeling that these ideas contain a strong
tincture of fetishism, I am certainly not alone. The
more proper course is, <i>to regard the abstract determinative
elements of a fact as interdependent</i>, in a purely logical
way, as the mathematician or geometer does.
True, by comparison with the will, forces are brought
nearer to our feeling; but it may be that ultimately the
will itself will be made clearer by comparison with the
accelerations of masses.</p>

<p>If we are asked, candidly, when is a fact <i>clear</i> to
us, we must say "when we can reproduce it by very
<i>simple</i> and very familiar intellectual operations, such
as the construction of accelerations, or the geometrical
summation of accelerations, and so forth." The
requirement of <i>simplicity</i> is of course to the expert
a different matter from what it is to the novice. For
the first, description by a system of differential equations
is sufficient; for the second, a gradual construction
out of elementary laws is required. The first
discerns at once the connexion of the two expositions.
Of course, it is not disputed that the <i>artistic</i> value of
materially equivalent descriptions may not be different.</p>

<p>Most difficult is it to persuade strangers that the
grand universal laws of physics, such as apply indiscriminately
to material, electrical, magnetic, and other
systems, are not essentially different from descriptions.
As compared with many sciences, physics occupies in
this respect a position of vantage that is easily explained.
Take, for example, anatomy. As the anatomist
in his quest for agreements and differences in
animals ascends to ever higher and higher <i>classifications</i>,
the individual facts that represent the ultimate
terms of the system, are still so different that they
must be <i>singly</i> noted. Think, for example, of the common
marks of the Vertebrates, of the class-characters
of Mammals and Birds on the one hand and of Fishes
on the other, of the double circulation of the blood on
the one hand and of the single on the other. In the
end, always <i>isolated</i> facts remain, which show only a
<i>slight</i> likeness to one another.</p>

<p>A science still more closely allied to physics, chemistry,
is often in the same strait. The abrupt change
of the qualitative properties, in all likelihood conditioned
by the slight stability of the intermediate states,
the remote resemblance of the co-ordinated facts of
chemistry render the treatment of its data difficult.
Pairs of bodies of different qualitative properties unite
in different mass-ratios; but no connexion between
the first and the last is to be noted, at first.</p>

<p>Physics, on the other hand, reveals to us wide domains
of <i>qualitatively homogeneous</i> facts, differing from
one another only in the number of equal parts into
which their characteristic marks are divisible, that is,
differing only <i>quantitatively</i>. Even where we have to
deal with qualities (colors and sounds), quantitative
characters of those qualities are at our disposal. Here
the classification is so simple a task that it rarely impresses
us as such, whilst in infinitely fine gradations,
in a <i>continuum of facts</i>, our number-system is ready beforehand
to follow as far as we wish. The co-ordinated
facts are here extremely similar and very closely affined,
as are also their descriptions which consist in
the determination of the numerical measures of one
given set of characters from those of a different set by
means of familiar mathematical operations&mdash;methods
of derivation. Thus, the common characteristics of
all descriptions can be found here; and with them a
succinct, comprehensive description, or a rule for the
construction of all single descriptions, is assigned,&mdash;and
this we call <i>law</i>. Well-known examples are the
formulæ for freely falling bodies, for projectiles, for
central motion, and so forth. If physics apparently
accomplishes more by its methods than other sciences,
we must remember that in a sense it has presented to
it much simpler problems.</p>

<p>The remaining sciences, whose facts also present a
physical side, need not be envious of physics for this
superiority; for all its acquisitions ultimately redound
to their benefit as well. But also in other ways this
mutual help shall and must change. Chemistry has advanced
very far in making the methods of physics her
own. Apart from older attempts, the periodical series
of Lothar Meyer and Mendelejeff are a brilliant and
adequate means of producing an easily surveyed system
of facts, which by gradually becoming complete,
will take the place almost of a continuum of facts.
Further, by the study of solutions, of dissociation, in
fact generally of phenomena which present a continuum
of cases, the methods of thermodynamics have
found entrance into chemistry. Similarly we may hope
that, at some future day, a mathematician, letting the
fact-continuum of embryology play before his mind,
which the palæontologists of the future will supposedly
have enriched with more intermediate and derivative
forms between Saurian and Bird than the isolated
Pterodactyl, Archæopteryx, Ichthyornis, and so forth,
which we now have&mdash;that such a mathematician shall
transform, by the variation of a few parameters, as in
a dissolving view, one form into another, just as we
transform one conic section into another.</p>

<p>Reverting now to Kirchhoff's words, we can come
to some agreement regarding their import. Nothing
can be built without building-stones, mortar, scaffolding,
and a builder's skill. Yet assuredly the wish is
well founded, that will show to posterity the complete
structure in its finished form, bereft of unsightly scaffolding.
It is the pure logical and æsthetic sense of the
mathematician that speaks out of Kirchhoff's words.
Modern expositions of physics aspire after his ideal;
that, too, is intelligible. But it would be a poor didactic
shift, for one whose business it was to train
architects, to say: "Here is a splendid edifice; if thou
wouldst really build, go thou and do likewise".</p>

<p>The barriers between the special sciences, which
make division of work and concentration possible, but
which appear to us after all as cold and conventional
restrictions, will gradually disappear. Bridge upon
bridge is thrown over the gaps. Contents and methods,
even of the remotest branches, are compared.
When the Congress of Natural Scientists shall meet a
hundred years hence, we may expect that they will
represent a unity in a higher sense than is possible to-day,
not in sentiment and aim alone, but in method
also. In the meantime, this great change will be
helped by our keeping constantly before our minds the
fact of the intrinsic relationship of all research, which
Kirchhoff characterised with such classical simplicity.</p>




<h2><a name="THE_PART_PLAYED_BY_ACCIDENT_IN" id="THE_PART_PLAYED_BY_ACCIDENT_IN">THE PART PLAYED BY ACCIDENT IN
INVENTION AND DISCOVERY.</a><a name="FNanchor_81_81" id="FNanchor_81_81"></a><a href="#Footnote_81_81" class="fnanchor">[81]</a></h2>


<p>It is characteristic of the naïve and sanguine
beginnings of thought in youthful men and
nations, that all problems are held to be soluble and
fundamentally intelligible on the first appearance of
success. The sage of Miletus, on seeing plants take
their rise from moisture, believed he had comprehended
the whole of nature, and he of Samos, on discovering
that definite numbers corresponded to the
lengths of harmonic strings, imagined he could exhaust
the nature of the world by means of numbers.
Philosophy and science in such periods are blended.
Wider experience, however, speedily discloses the
error of such a course, gives rise to criticism, and
leads to the division and ramification of the sciences.</p>

<p>At the same time, the necessity of a broad and
general view of the world remains; and to meet this
need philosophy parts company with special inquiry.<span class="pagenum"><a name="Page_260" id="Page_260">[Pg 260]</a></span>
It is true, the two are often found united in gigantic
personalities. But as a rule their ways diverge more
and more widely from each other. And if the estrangement
of philosophy from science can reach a point
where data unworthy of the nursery are not deemed too
scanty as foundations of the world, on the other hand
the thorough-paced specialist may go to the extreme
of rejecting point-blank the possibility of a broader
view, or at least of deeming it superfluous, forgetful
of Voltaire's apophthegm, nowhere more applicable
than here, <i>Le superflu&mdash;chose très nécessaire</i>.</p>

<p>It is true, the history of philosophy, owing to the
insufficiency of its constructive data, is and must be
largely a history of error. But it would be the height
of ingratitude on our part to forget that the seeds of
thoughts which still fructify the soil of special research,
such as the theory of irrationals, the conceptions
of conservation, the doctrine of evolution, the
idea of specific energies, and so forth, may be traced
back in distant ages to philosophical sources. Furthermore,
to have deferred or abandoned the attempt
at a broad philosophical view of the world from a full
knowledge of the insufficiency of our materials, is
quite a different thing from never having undertaken
it at all. The revenge of its neglect, moreover, is
constantly visited upon the specialist by his committal
of the very errors which philosophy long ago exposed.
As a fact, in physics and physiology, particularly
during the first half of this century, are to be<span class="pagenum"><a name="Page_261" id="Page_261">[Pg 261]</a></span>
met intellectual productions which for naïve simplicity
are not a jot inferior to those of the Ionian school, or
to the Platonic ideas, or to that much reviled ontological
proof.</p>

<p>Latterly, there has been evidence of a gradual
change in the situation. Recent philosophy has set
itself more modest and more attainable ends; it is
no longer inimical to special inquiry; in fact, it is
zealously taking part in that inquiry. On the other
hand, the special sciences, mathematics and physics,
no less than philology, have become eminently philosophical.
The material presented is no longer accepted
uncritically. The glance of the inquirer is
bent upon neighboring fields, whence that material
has been derived. The different special departments
are striving for closer union, and gradually the conviction
is gaining ground that philosophy can consist
only of mutual, complemental criticism, interpenetration,
and union of the special sciences into a consolidated
whole. As the blood in nourishing the body
separates into countless capillaries, only to be collected
again and to meet in the heart, so in the science
of the future all the rills of knowledge will gather
more and more into a common and undivided stream.</p>

<p>It is this view&mdash;not an unfamiliar one to the present
generation&mdash;that I purpose to advocate. Entertain
no hope, or rather fear, that I shall construct
systems for you. I shall remain a natural inquirer.
Nor expect that it is my intention to skirt all the<span class="pagenum"><a name="Page_262" id="Page_262">[Pg 262]</a></span>
fields of natural inquiry. I can attempt to be your
guide only in that branch which is familiar to me, and
even there I can assist in the furtherment of only a
small portion of the allotted task. If I shall succeed
in rendering plain to you the relations of physics,
psychology, and the theory of knowledge, so that you
may draw from each profit and light, redounding to
the advantage of each, I shall regard my work as not
having been in vain. Therefore, to illustrate by an
example how, consonantly with my powers and views,
I conceive such inquiries should be conducted, I shall
treat to-day, in the form of a brief sketch, of the following
special and limited subject&mdash;of <i>the part which
accidental circumstances play in the development of inventions
and discoveries</i>.</p>
<p><span class="pagenum"><a name="Page_263" id="Page_263">[Pg 263]</a></span></p><p><span class="pagenum"><a name="Page_264" id="Page_264">[Pg 264]</a></span></p><p><span class="pagenum"><a name="Page_265" id="Page_265">[Pg 265]</a></span></p><p><span class="pagenum"><a name="Page_266" id="Page_266">[Pg 266]</a></span></p><p><span class="pagenum"><a name="Page_267" id="Page_267">[Pg 267]</a></span></p><p><span class="pagenum"><a name="Page_268" id="Page_268">[Pg 268]</a></span></p><p><span class="pagenum"><a name="Page_269" id="Page_269">[Pg 269]</a></span></p><p><span class="pagenum"><a name="Page_270" id="Page_270">[Pg 270]</a></span></p><p><span class="pagenum"><a name="Page_271" id="Page_271">[Pg 271]</a></span></p><p><span class="pagenum"><a name="Page_272" id="Page_272">[Pg 272]</a></span></p><p><span class="pagenum"><a name="Page_273" id="Page_273">[Pg 273]</a></span></p><p><span class="pagenum"><a name="Page_274" id="Page_274">[Pg 274]</a></span></p><p><span class="pagenum"><a name="Page_275" id="Page_275">[Pg 275]</a></span></p><p><span class="pagenum"><a name="Page_276" id="Page_276">[Pg 276]</a></span></p><p><span class="pagenum"><a name="Page_277" id="Page_277">[Pg 277]</a></span></p><p><span class="pagenum"><a name="Page_278" id="Page_278">[Pg 278]</a></span></p><p><span class="pagenum"><a name="Page_279" id="Page_279">[Pg 279]</a></span></p><p><span class="pagenum"><a name="Page_280" id="Page_280">[Pg 280]</a></span></p><p><span class="pagenum"><a name="Page_281" id="Page_281">[Pg 281]</a></span></p><p><span class="pagenum"><a name="Page_282" id="Page_282">[Pg 282]</a></span></p>
<hr class="tb" />

<p>When we Germans say of a man that he was not
the inventor of gunpowder,<a name="FNanchor_82_82" id="FNanchor_82_82"></a><a href="#Footnote_82_82" class="fnanchor">[82]</a> we impliedly cast a grave
suspicion on his abilities. But the expression is not
a felicitous one, as there is probably no invention in
which deliberate thought had a smaller, and pure luck
a larger, share than in this. It is well to ask, Are we
justified in placing a low estimate on the achievement
of an inventor because accident has assisted him in
his work? Huygens, whose discoveries and inventions
are justly sufficient to entitle him to an opinion
in such matters, lays great emphasis on this factor.
He asserts that a man capable of inventing the telescope
without the concurrence of accident must have
been gifted with superhuman genius.<a name="FNanchor_83_83" id="FNanchor_83_83"></a><a href="#Footnote_83_83" class="fnanchor">[83]</a></p>

<p>A man living in the midst of civilisation finds himself
surrounded by a host of marvellous inventions,
considering none other than the means of satisfying
the needs of daily life. Picture such a man transported
to the epoch preceding the invention of these
ingenious appliances, and imagine him undertaking
in a serious manner to comprehend their origin. At
first the intellectual power of the men capable of producing
such marvels will strike him as incredible, or,
if we adopt the ancient view, as divine. But his astonishment
is considerably allayed by the disenchanting
yet elucidative revelations of the history of primitive
culture, which to a large extent prove that these
inventions took their rise very slowly and by imperceptible
degrees.</p>

<p>A small hole in the ground with fire kindled in it
constituted the primitive stove. The flesh of the
quarry, wrapped with water in its skin, was boiled by
contact with heated stones. Cooking by stones was
also done in wooden vessels. Hollow gourds were
protected from the fire by coats of clay. Thus, from
the burned clay accidentally originated the enveloping
pot, which rendered the gourd superfluous, although
for a long time thereafter the clay was still spread
over the gourd, or pressed into woven wicker-work
before the potter's art assumed its final independence.
Even then the wicker-work ornament was retained, as
a sort of attest of its origin.</p>

<p>We see, thus, it is by accidental circumstances, or
by such as lie without our purpose, foresight, and
power, that man is gradually led to the acquaintance
of improved means of satisfying his wants. Let the
reader picture to himself the genius of a man who
could have foreseen without the help of accident that
clay handled in the ordinary manner would produce a
useful cooking utensil! The majority of the inventions
made in the early stages of civilisation, including
language, writing, money, and the rest, could not
have been the product of deliberate methodical reflexion
for the simple reason that no idea of their
value and significance could have been had except
from practical use. The invention of the bridge may
have been suggested by the trunk of a tree which had
fallen athwart a mountain-torrent; that of the tool by
the use of a stone accidentally taken into the hand to
crack nuts. The use of fire probably started in and
was disseminated from regions where volcanic eruptions,
hot springs, and burning jets of natural gas
afforded opportunity for quietly observing and turning
to practical account the properties of fire. Only
after that had been done could the significance of the
fire-drill be appreciated, an instrument which was
probably discovered from boring a hole through a
piece of wood. The suggestion of a distinguished inquirer
that the invention of the fire-drill originated on
the occasion of a religious ceremony is both fantastic
and incredible. And as to the use of fire, we should
no more attempt to derive that from the invention of
the fire-drill than we should from the invention of sulphur
matches. Unquestionably the opposite course
was the real one.<a name="FNanchor_84_84" id="FNanchor_84_84"></a><a href="#Footnote_84_84" class="fnanchor">[84]</a></p>

<p>Similar phenomena, though still largely veiled in
obscurity, mark the initial transition of nations from
a hunting to a nomadic life and to agriculture.<a name="FNanchor_85_85" id="FNanchor_85_85"></a><a href="#Footnote_85_85" class="fnanchor">[85]</a> We
shall not multiply examples, but content ourselves
with the remark that the same phenomena recur in
historical times, in the ages of great technical inventions,
and, further, that regarding them the most
whimsical notions have been circulated&mdash;notions which
ascribe to accident an unduly exaggerated part, and
one which in a psychological respect is absolutely impossible.
The observation of steam escaping from a
tea-kettle and of the clattering of the lid is supposed
to have led to the invention of the steam-engine. Just
think of the gap between this spectacle and the conception
of the performance of great mechanical work
by steam, for a man totally ignorant of the steam-engine!
Let us suppose, however, that an engineer,
versed in the practical construction of pumps, should
accidentally dip into water an inverted bottle that had
been filled with steam for drying and still retained its
steam. He would see the water rush violently into
the bottle, and the idea would very naturally suggest
itself of founding on this experience a convenient and
useful atmospheric steam-pump, which by imperceptible
degrees, both psychologically possible and immediate,
would then undergo a natural and gradual transformation
into Watt's steam-engine.</p>

<p>But granting that the most important inventions
are brought to man's notice accidentally and in ways
that are beyond his foresight, yet it does not follow
that accident alone is sufficient to produce an invention.
The part which man plays is by no means a
passive one. Even the first potter in the primeval
forest must have felt some stirrings of genius within
him. In all such cases, the inventor is obliged <i>to take
note</i> of the new fact, he must discover and grasp its
advantageous feature, and must have the power to
turn that feature to account in the realisation of his
purpose. He must <i>isolate</i> the new feature, impress it
upon his memory, unite and interweave it with the
rest of his thought; in short, he must possess the capacity
<i>to profit by experience</i>.</p>

<p>The capacity to profit by experience might well be
set up as a test of intelligence. This power varies
considerably in men of the same race, and increases
enormously as we advance from the lower animals to
man. The former are limited in this regard almost
entirely to the reflex actions which they have inherited
with their organism, they are almost totally incapable
of individual experience, and considering their simple
wants are scarcely in need of it. The ivory-snail
(<i>Eburna spirata</i>) never learns to avoid the carnivorous
Actinia, no matter how often it may wince under the
latter's shower of needles, apparently having no memory
for pain whatever.<a name="FNanchor_86_86" id="FNanchor_86_86"></a><a href="#Footnote_86_86" class="fnanchor">[86]</a> A spider can be lured forth
repeatedly from its hole by touching its web with a
tuning-fork. The moth plunges again and again into
the flame which has burnt it. The humming-bird
hawk-moth<a name="FNanchor_87_87" id="FNanchor_87_87"></a><a href="#Footnote_87_87" class="fnanchor">[87]</a> dashes repeatedly against the painted
roses of the wall-paper, like the unhappy and desperate
thinker who never wearies of attacking the same
insoluble chimerical problem. As aimlessly almost as
Maxwell's gaseous molecules and in the same unreasoning
manner common flies in their search for light
and air stream against the glass pane of a half-opened
window and remain there from sheer inability to find
their way around the narrow frame. But a pike separated
from the minnows of his aquarium by a glass
partition, learns after the lapse of a few months,
though only after having butted himself half to death,
that he cannot attack these fishes with impunity.
What is more, he leaves them in peace even after the
removal of the partition, though he will bolt a strange
fish at once. Considerable memory must be attributed
to birds of passage, a memory which, probably
owing to the absence of disturbing thoughts, acts with
the precision of that of some idiots. Finally, the
susceptibility to training evinced by the higher vertebrates
is indisputable proof of the ability of these animals
to profit by experience.</p>

<p>A powerfully developed <i>mechanical</i> memory, which
recalls vividly and faithfully old situations, is sufficient
for avoiding definite particular dangers, or for taking
advantage of definite particular opportunities. But
more is required for the development of <i>inventions</i>.
More extensive chains of images are necessary here,
the excitation by mutual contact of widely different
trains of ideas, a more powerful, more manifold, and
richer connexion of the contents of memory, a more
powerful and impressionable psychical life, heightened
by use. A man stands on the bank of a mountain-torrent,
which is a serious obstacle to him. He remembers
that he has crossed just such a torrent before
on the trunk of a fallen tree. Hard by trees are
growing. He has often moved the trunks of fallen
trees. He has also felled trees before, and then moved
them. To fell trees he has used sharp stones. He
goes in search of such a stone, and as the old situations
that crowd into his memory and are held there in living
reality by the definite powerful interest which he
has in crossing just this torrent,&mdash;as these impressions
are made to pass before his mind in the <i>inverse order</i> in
which they were here evoked, he invents the bridge.</p>

<p>There can be no doubt but the higher vertebrates
adapt their actions in some moderate degree to circumstances.
The fact that they give no appreciable
evidence of advance by the accumulation of inventions,
is satisfactorily explained by a difference of degree
or intensity of intelligence as compared with
man; the assumption of a difference of kind is not
necessary. A person who saves a little every day, be
it ever so little, has an incalculable advantage over
him who daily squanders that amount, or is unable to
keep what he has accumulated. A slight quantitative
difference in such things explains enormous differences
of advancement.</p>

<p>The rules which hold good in prehistoric times
also hold good in historical times, and the remarks
made on invention may be applied almost without
modification to discovery; for the two are distinguished
solely by the use to which the new knowledge
is put. In both cases the investigator is concerned
with some <i>newly observed</i> relation of new or old properties,
abstract or concrete. It is observed, for example,
that a substance which gives a chemical reaction
<i>A</i> is also the cause of a chemical reaction <i>B</i>. If this
observation fulfils no purpose but that of furthering
the scientist's insight, or of removing a source of intellectual
discomfort, we have a discovery; but an invention,
if in using the substance giving the reaction
<i>A</i> to produce the desired reaction <i>B</i>, we have a practical
end in view, and seek to remove a source of material
discomfort. The phrase, <i>disclosure of the connexion
of reactions</i>, is broad enough to cover discoveries
and inventions in all departments. It embraces the
Pythagorean proposition, which is a combination of a
geometrical and an arithmetical reaction, Newton's
discovery of the connexion of Kepler's motions with
the law of the inverse squares, as perfectly as it does
the detection of some minute but appropriate alteration
in the construction of a tool, or of some appropriate
change in the methods of a dyeing establishment.</p>

<p>The disclosure of new provinces of facts before
unknown can only be brought about by accidental circumstances,
under which are <i>remarked</i> facts that commonly
go unnoticed. The achievement of the discoverer
here consists in his <i>sharpened attention</i>, which
detects the uncommon features of an occurrence and
their determining conditions from their most evanescent
marks,<a name="FNanchor_88_88" id="FNanchor_88_88"></a><a href="#Footnote_88_88" class="fnanchor">[88]</a> and discovers means of submitting them
to exact and full observation. Under this head belong
the first disclosures of electrical and magnetic
phenomena, Grimaldi's observation of interference,
Arago's discovery of the increased check suffered by a
magnetic needle vibrating in a copper envelope as
compared with that observed in a bandbox, Foucault's
observation of the stability of the plane of vibration
of a rod accidentally struck while rotating in a turning-lathe,
Mayer's observation of the increased redness
of venous blood in the tropics, Kirchhoff's observation
of the augmentation of the <i>D</i>-line in the solar
spectrum by the interposition of a sodium lamp,
Schönbein's discovery of ozone from the phosphoric
smell emitted on the disruption of air by electric
sparks, and a host of others. All these facts, of which
unquestionably many were <i>seen</i> numbers of times before
they were <i>noticed</i>, are examples of the inauguration
of momentous discoveries by accidental circumstances,
and place the importance of strained attention
in a brilliant light.</p>

<p>But not only is a significant part played in the beginning
of an inquiry by co-operative circumstances
beyond the foresight of the investigator; their influence
is also active in its prosecution. Dufay, thus, whilst
following up the behavior of <i>one</i> electrical state which
he had assumed, discovers the existence of <i>two</i>. Fresnel
learns by accident that the interference-bands received
on ground glass are seen to better advantage
in the open air. The diffraction-phenomenon of two
slits proved to be considerably different from what
Fraunhofer had anticipated, and in following up this
circumstance he was led to the important discovery of
grating-spectra. Faraday's induction-phenomenon departed
widely from the initial conception which occasioned
his experiments, and it is precisely this deviation
that constitutes his real discovery.</p>

<p>Every man has pondered on some subject. Every
one of us can multiply the examples cited, by less illustrious
ones from his own experience. I shall cite
but one. On rounding a railway curve once, I accidentally
remarked a striking apparent inclination of
the houses and trees. I inferred that the direction of
the total resultant <i>physical</i> acceleration of the body
reacts <i>physiologically</i> as the vertical. Afterwards, in
attempting to inquire more carefully into this phenomenon,
and this only, in a large whirling machine,
the collateral phenomena conducted me to the sensation
of angular acceleration, vertigo, Flouren's experiments
on the section of the semi-circular canals
etc., from which gradually resulted views relating to
sensations of direction which are also held by Breuer
and Brown, which were at first contested on all hands,
but are now regarded on many sides as correct, and
which have been recently enriched by the interesting
inquiries of Breuer concerning the <i>macula acustica</i>, and
Kreidel's experiments with magnetically orientable
crustacea.<a name="FNanchor_89_89" id="FNanchor_89_89"></a><a href="#Footnote_89_89" class="fnanchor">[89]</a> Not disregard of accident but a direct and
purposeful employment of it advances research.</p>

<p>The more powerful the psychical connexion of the
memory pictures is,&mdash;and it varies with the individual
and the mood,&mdash;the more apt is the same accidental
observation to be productive of results. Galileo knows
that the air has weight; he also knows of the "resistance
to a vacuum," expressed both in weight and
in the height of a column of water. But the two ideas
dwelt asunder in his mind. It remained for Torricelli
to vary the specific gravity of the liquid measuring the
pressure, and not till then was the air included in the
list of pressure-exerting fluids. The reversal of the
lines of the spectrum was seen repeatedly before
Kirchhoff, and had been mechanically explained. But
it was left for his penetrating vision to discern the
evidence of the connexion of this phenomenon with
questions of heat, and to him alone through persistent
labor was revealed the sweeping significance of the
fact for the mobile equilibrium of heat. Supposing,
then, that such a rich organic connexion of the elements
of memory exists, and is the prime distinguishing
mark of the inquirer, next in importance certainly
is that <i>intense interest</i> in a definite object, in a definite
idea, which fashions advantageous combinations of
thought from elements before disconnected, and obtrudes
that idea into every observation made, and into
every thought formed, making it enter into relationship
with all things. Thus Bradley, deeply engrossed
with the subject of aberration, is led to its solution
by an exceedingly unobtrusive experience in crossing
the Thames. It is permissible, therefore, to ask
whether accident leads the discoverer, or the discoverer
accident, to a successful outcome in scientific
quests.</p>

<p>No man should dream of solving a great problem
unless he is so thoroughly saturated with his subject
that everything else sinks into comparative insignificance.
During a hurried meeting with Mayer in Heidelberg
once, Jolly remarked, with a rather dubious
implication, that if Mayer's theory were correct water
could be warmed by shaking. Mayer went away without
a word of reply. Several weeks later, and now
unrecognised by Jolly, he rushed into the latter's presence
exclaiming: "Es ischt aso!" (It is so, it is
so!) It was only after considerable explanation that
Jolly found out what Mayer wanted to say. The incident
needs no comment.<a name="FNanchor_90_90" id="FNanchor_90_90"></a><a href="#Footnote_90_90" class="fnanchor">[90]</a></p>

<p>A person deadened to sensory impressions and
given up solely to the pursuit of his own thoughts,
may also light on an idea that will divert his mental
activity into totally new channels. In such cases it is
a psychical accident, an intellectual experience, as
distinguished from a physical accident, to which the
person owes his discovery&mdash;a discovery which is here
made "deductively" by means of mental copies of the
world, instead of experimentally. <i>Purely</i> experimental
inquiry, moreover, does not exist, for, as Gauss says,
virtually we always experiment with our thoughts.
And it is precisely that constant, corrective interchange
or intimate union of experiment and deduction,
as it was cultivated by Galileo in his <i>Dialogues</i>
and by Newton in his <i>Optics</i>, that is the foundation of
the benign fruitfulness of modern scientific inquiry as
contrasted with that of antiquity, where observation
and reflexion ofttimes pursued their respective courses
like two strangers.</p>

<p>We have to wait for the appearance of a favorable
physical accident. The movement of our thoughts
obeys the law of association. In the case of meagre
experience the result of this law is simply the mechanical
reproduction of definite sensory experiences. On
the other hand, if the psychical life is subjected to the
incessant influences of a powerful and rich experience,
then every representative element in the mind is connected
with so many others that the actual and natural
course of the thoughts is easily influenced and determined
by insignificant circumstances, which accidentally
are decisive. Hereupon, the process termed imagination
produces its protean and infinitely diversified
forms. Now what can we do to guide this process,
seeing that the combinatory law of the images is without
our reach? Rather let us ask, what influence can
a powerful and constantly recurring idea exert on the
movement of our thoughts? According to what has
preceded, the answer is involved in the question itself.
The <i>idea</i> dominates the thought of the inquirer, not
the latter the former.</p>

<p>Let us see, now, if we can acquire a profounder
insight into the process of discovery. The condition
of the discoverer is, as James has aptly remarked, not
unlike the situation of a person who is trying to remember
something that he has forgotten. Both are
sensible of a gap, and have only a remote presentiment
of what is missing. Suppose I meet in a company
a well-known and affable gentleman whose name
I have forgotten, and who to my horror asks to be introduced
to some one. I set to work according to
Lichtenberg's rule, and run down the alphabet in
search of the initial letter of his name. A vague sympathy
holds me at the letter <i>G</i>. Tentatively I add the
second letter and am arrested at <i>e</i>, and long before I
have tried the third letter <i>r</i>, the name "Gerson" sounds
sonorously upon my ear, and my anguish is gone.
While taking a walk I meet a gentleman from whom
I receive a communication. On returning home, and
in attending to weightier affairs, the matter slips my
mind. Moodily, but in vain, I ransack my memory.
Finally I observe that I am going over my walk again
in thought. On the street corner in question the self-same
gentleman stands before me and repeats his
communication. In this process are successively recalled
to consciousness all the percepts which were
connected with the percept that was lost, and with
them, finally, that, too, is brought to light. In the
first case&mdash;where the experience had already been
made and is permanently impressed on our thought&mdash;a
<i>systematic</i> procedure is both possible and easy, for
we know that a name must be composed of a limited
number of sounds. But at the same time it should be
observed that the labor involved in such a combinatorial
task would be enormous if the name were long
and the responsiveness of the mind weaker.</p>

<p>It is often said, and not wholly without justification,
that the scientist has solved a <i>riddle</i>. Every problem
in geometry may be clothed in the garb of a <i>riddle</i>.
Thus: "What thing is that <i>M</i> which has the properties
<i>A</i>, <i>B</i>, <i>C</i>?" "What circle is that which touches
the straight lines <i>A</i>, <i>B</i>, but touches <i>B</i> in the point <i>C</i>?"
The first two conditions marshal before the imagination
the group of circles whose centres lie in the line
of symmetry of <i>A</i>, <i>B</i>. The third condition reminds
us of all the circles having centres in the straight line
that stands at right angles to <i>B</i> in <i>C</i>. The <i>common</i>
term, or common terms, of the two groups of images
solves the riddle&mdash;satisfies the problem. Puzzles dealing
with things or words induce similar processes, but
the memory in such cases is exerted in many directions
and more varied and less clearly ordered provinces
of ideas are surveyed. The difference between
the situation of a geometer who has a construction to
make, and that of an engineer, or a scientist, confronted
with a problem, is simply this, that the first
moves in a field with which he is thoroughly acquainted,
whereas the two latter are obliged to familiarise
themselves with this field subsequently, and in
a measure far transcending what is commonly required.
In this process the mechanical engineer has
at least always a definite goal before him and definite
means to accomplish his aim, whilst in the case of the
scientist that aim is in many instances presented only
in vague and general outlines. Often the very formulation
of the riddle devolves on him. Frequently it
is not until the aim has been reached that the broader
outlook requisite for systematic procedure is obtained.
By far the larger portion of his success, therefore, is
contingent on luck and instinct. It is immaterial, so
far as its character is concerned, whether the process
in question is brought rapidly to a conclusion in the
brain of one man, or whether it is spun out for centuries
in the minds of a long succession of thinkers.
The same relation that a word solving a riddle bears
to that riddle is borne by the modern conception of
light to the facts discovered by Grimaldi, Römer,
Huygens, Newton, Young, Malus, and Fresnel, and
only by the help of this slowly developed conception
is our mental vision enabled to embrace the broad
domain of facts in question.</p>

<p>A welcome complement to the discoveries which
the history of civilisation and comparative psychology
have furnished, is to be found in the confessions of
great scientists and artists. Scientists <i>and</i> artists, we
might say, for Liebig boldly declared there was no
essential difference between the two. Are we to regard
Leonardo da Vinci as a scientist or as an artist?
If the artist builds up his work from a few motives,
the scientist discovers the motives which permeate
reality. If scientists like Lagrange or Fourier are in
a certain measure artists in the presentation of their
results, on the other hand, artists like Shakespeare or
Ruysdael are scientists in the insight which must
have preceded their creations.</p>

<p>Newton, when questioned about his methods of
work, could give no other answer but that he was
wont to ponder again and again on a subject; and
similar utterances are accredited to D'Alembert and
Helmholtz. Scientists and artists both recommend
persistent labor. After the repeated survey of a field
has afforded opportunity for the interposition of advantageous
accidents, has rendered all the traits that
suit with the mood or the dominant thought more
vivid, and has gradually relegated to the background
all things that are inappropriate, making their future
appearance impossible; then from the teeming, swelling
host of fancies which a free and high-flown imagination
calls forth, suddenly that particular form
arises to the light which harmonises perfectly with
the ruling idea, mood, or design. Then it is that that
which has resulted slowly as the result of a gradual
selection, appears as if it were the outcome of a deliberate
act of creation. Thus are to be explained the
statements of Newton, Mozart, Richard Wagner, and
others, when they say that thoughts, melodies, and
harmonies had poured in upon them, and that they
had simply retained the right ones. Undoubtedly,
the man of genius, too, consciously or instinctively,
pursues systematic methods wherever it is possible;
but in his delicate presentiment he will omit many a
task or abandon it after a hasty trial on which a less
endowed man would squander his energies in vain.
Thus, the genius accomplishes<a name="FNanchor_91_91" id="FNanchor_91_91"></a><a href="#Footnote_91_91" class="fnanchor">[91]</a> in a brief space of
time undertakings for which the life of an ordinary
man would far from suffice. We shall hardly go astray
if we regard genius as only a slight deviation from
the average mental endowment&mdash;as possessing simply
a greater sensitiveness of cerebral reaction and a
greater swiftness of reaction. The men who, obeying
their inner impulses, make sacrifices for an idea instead
of advancing their material welfare, may appear
to the full-blooded Philistine as fools; yet we shall
scarcely adopt Lombroso's view, that genius is to be
regarded as a disease, although it is unfortunately
true that the sensitive brains and fragile constitutions
succumb most readily to sickness.</p>

<p>The remark of C. G. J. Jacobi that mathematics
is slow of growth and only reaches the truth by long
and devious paths, that the way to its discovery must
be prepared for long beforehand, and that then the
truth will make its long-deferred appearance as if impelled
by some divine necessity<a name="FNanchor_92_92" id="FNanchor_92_92"></a><a href="#Footnote_92_92" class="fnanchor">[92]</a>&mdash;all this holds true
of every science. We are astounded often to note
that it required the combined labors of many eminent
thinkers for a full century to reach a truth which it
takes us only a few hours to master and which once
acquired seems extremely easy to reach under the
right sort of circumstances. To our humiliation we
learn that even the greatest men are born more for
life than for science. The extent to which even they
are indebted to accident&mdash;to that singular conflux of
the physical and the psychical life in which the continuous
but yet imperfect and never-ending adaptation
of the latter to the former finds its distinct expression&mdash;that
has been the subject of our remarks to-day.
Jacobi's poetical thought of a divine necessity acting
in science will lose none of its loftiness for us if we
discover in this necessity the same power that destroys
the unfit and fosters the fit. For loftier, nobler,
and more romantic than poetry is the truth and the
reality.</p>




<h2><a name="ON_SENSATIONS_OF_ORIENTATION93" id="ON_SENSATIONS_OF_ORIENTATION93">ON SENSATIONS OF ORIENTATION.</a><a name="FNanchor_93_93" id="FNanchor_93_93"></a><a href="#Footnote_93_93" class="fnanchor">[93]</a></h2>


<p>Through the co-operation of a succession of inquirers,
among whom are particularly to be mentioned
Goltz of Strassburg and Breuer of Vienna,
considerable advances have been made during the
last twenty-five years in our knowledge of the means
by which we ascertain our position in space and the
direction of our motion, or orient ourselves, as the
phrase goes. I presume that you are already acquainted
with the physiological part of the processes
with which our sensations of movement, or, more generally
speaking, our sensations of orientation, are connected.
Here I shall consider more particularly the
physical side of the matter. In fact, I was originally
led to the consideration of these questions by the
observation of extremely simple and perfectly well-known
physical facts, before I had any great acquaintance
with physiology and while pursuing unbiasedly
my natural thoughts; and I am of the conviction that<span class="pagenum"><a name="Page_283" id="Page_283">[Pg 283]</a></span>
the way which I have pursued, and which is entirely
free from hypotheses, will, if you will follow my exposition,
be that of easiest acquisition for the most of
you.</p>

<p>No man of sound common sense could ever have
doubted that a pressure or force is requisite to set a
body in motion in a given direction and that a contrary
pressure is required to stop suddenly a body in
motion. Though the law of inertia was first formulated
with anything like exactness by Galileo, the
facts at the basis of it were known long previously to
men of the stamp of Leonardo da Vinci, Rabelais,
and others, and were illustrated by them with appropriate
experiments. Leonardo knew that by a swift
stroke with a ruler one can knock out from a vertical
column of checkers a single checker without over-throwing
the column. The experiment with a coin
resting on a piece of pasteboard covering a goblet,
which falls into the goblet when the pasteboard is
jerked away, like all experiments of the kind, is certainly
very old.</p>

<p>With Galileo the experience in question assumes
greater clearness and force. In the famous dialogue
on the Copernican system which cost him his freedom,
he explains the tides in an unfelicitous, though
in principle correct manner, by the analogue of a
platter of water swung to and fro. In opposition to
the Aristotelians of his time, who believed the descent<span class="pagenum"><a name="Page_284" id="Page_284">[Pg 284]</a></span>
of a heavy body could be accelerated by the
superposition of another heavy body, he asserted that
a body could never be accelerated by one lying upon
it unless the first in some way impeded the superposed
body in its descent. To seek to press a falling
body by means of another placed upon it, is as senseless
as trying to prod a man with a lance when the man
is speeding away from one with the same velocity as
the lance. Even this little excursion into physics can
explain much to us. You know the peculiar sensation
which one has in falling, as when one jumps from a
high springboard into the water, and which is also
experienced in some measure at the beginning of the
descent of elevators and swings. The reciprocal gravitational
pressure of the different parts of our body,
which is certainly felt in some manner, vanishes in
free descent, or, in the case of the elevator, is diminished
on the beginning of the descent. A similar sensation
would be experienced if we were suddenly
transported to the moon where the acceleration of
gravity is much less than upon the earth. I was led
to these considerations in 1866 by a suggestion in
physics, and having also taken into account the alterations
of the blood-pressure in the cases in question,
I found I coincided without knowing it with Wollaston
and Purkinje. The first as early as 1810 in his Croonian
lecture had touched on the subject of sea-sickness
and explained it by alterations of the blood-pressure,<span class="pagenum"><a name="Page_285" id="Page_285">[Pg 285]</a></span>
and later had laid similar considerations at the
basis of his explanation of vertigo (1820-1826).<a name="FNanchor_94_94" id="FNanchor_94_94"></a><a href="#Footnote_94_94" class="fnanchor">[94]</a></p>

<p>Newton was the first to enunciate with perfect
generality that a body can change the velocity and
direction of its motion only by the action of a force,
or the action of a second body. A corollary of this
law which was first expressly deduced by Euler is
that a body can never be set <i>rotating</i> or made to cease
rotating of itself but only by forces and other bodies.
For example, turn an open watch which has run down
freely backwards and forwards in your hand. The
balance-wheel will not fully catch the rapid rotations,
it does not even respond fully to the elastic force of
the spring which proves too weak to carry the wheel
entirely with it.</p>

<p>Let us consider now that whether we move ourselves
by means of our legs, or whether we are
moved by a vehicle or a boat, at first only a part
of our body is directly moved and the rest of it is
afterwards set in motion by the first part. We see
that pressures, pulls, and tensions are always produced
between the parts of the body in this action,
which pressures, pulls, and tensions give rise to sensations
by which the forward or rotary movements in
which we are engaged are made perceptible.<a name="FNanchor_95_95" id="FNanchor_95_95"></a><a href="#Footnote_95_95" class="fnanchor">[95]</a> But it
is quite natural that sensations so familiar should be
little noticed and that attention should be drawn to
them only under special circumstances when they occur
unexpectedly or with unusual strength.</p>

<p><span class="pagenum"><a name="Page_286" id="Page_286">[Pg 286]</a></span></p>
<div class="figcenter" style="width: 450px;">
<img src="images/i_296.jpg" width="450" height="330" alt="" />
<span class="caption">Fig. 45.</span>
</div>

<p>Thus my attention was drawn to this point by the
sensation of falling and subsequently by another singular
occurrence. I was rounding a sharp railway
curve once when I suddenly saw all the trees, houses,
and factory chimneys along the track swerve from the
vertical and assume a strikingly inclined position.
What had hitherto appeared to me perfectly natural,
namely, the fact that we distinguish the vertical so
perfectly and sharply from every other direction, now<span class="pagenum"><a name="Page_287" id="Page_287">[Pg 287]</a></span>
struck me as enigmatical. Why is it that the same
direction can now appear vertical to me and now cannot?
By what is the vertical distinguished for us?
(Compare Figure 45.)</p>

<p>The rails are raised on the convex or outward side
of the track in order to insure the stability of the carriage
as against the action of the centrifugal force, the
whole being so arranged that the combination of the
force of gravity with the centrifugal force of the train
shall give rise to a force perpendicular to the plane
of the rails.</p>

<p>Let us assume, now, that under all circumstances
we somehow sense the direction of the total resultant
mass-acceleration whencesoever it may arise as the
vertical. Then both the ordinary and the extraordinary
phenomena will be alike rendered intelligible.<a name="FNanchor_96_96" id="FNanchor_96_96"></a><a href="#Footnote_96_96" class="fnanchor">[96]</a></p>

<p>I was now desirous of putting the view I had
reached to a more convenient and exact test than was
possible on a railway journey where one has no control
over the determining circumstances and cannot
alter them at will. I accordingly had the simple apparatus
constructed which is represented in Figure 46.</p>

<p>In a large frame <i>BB</i>, which is fastened to the walls,
rotates about a vertical axis <i>AA</i> a second frame <i>RR</i>,
and within the latter a third one <i>rr</i>, which can be set<span class="pagenum"><a name="Page_288" id="Page_288">[Pg 288]</a></span>
at any distance and position from the axis, made stationary
or movable, and is provided with a chair for
the observer.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_298.jpg" width="600" height="385" alt="" />
<span class="caption">Fig. 46.</span>
</div>

<p>From Mach's <i>Bewegungsempfindungen</i>, Leipsic, Engelmann, 1875.]</p>

<p>The observer takes his seat in the chair and to
prevent disturbances of judgment is enclosed in a paper
box. If the observer together with the frame <i>rr</i>
be then set in uniform rotation, he will feel and see
the beginning of the rotation both as to direction and
amount very distinctly although every outward visible
or tangible point of reference is wanting. If the motion
be uniformly continued the sensation of rotation
will gradually cease entirely and the observer will imagine
himself at rest. But if <i>rr</i> be placed outside the
axis of rotation, at once on the rotation beginning, a
strikingly apparent, palpable, actually visible inclination
of the entire paper box is produced, slight when<span class="pagenum"><a name="Page_289" id="Page_289">[Pg 289]</a></span>
the rotation is slow, strong when the rotation is rapid,
and continuing as long as the rotation lasts. It is absolutely
impossible for the observer to escape perceiving
the inclination, although here also all outward
points of reference are wanting. If the observer, for
example, is seated so as to look towards the axis, he
will feel the box strongly tipped backwards, as it necessarily
must be if the direction of the total resultant
force is perceived as the vertical. For other positions
of the observer the situation is similar.<a name="FNanchor_97_97" id="FNanchor_97_97"></a><a href="#Footnote_97_97" class="fnanchor">[97]</a></p>

<p>Once, while performing one of these experiments,
and after rotating so long that I was no longer conscious
of the movement, I suddenly caused the apparatus
to be stopped, whereupon I immediately felt
and saw myself with the whole box rapidly flung round
in rotation in the opposite direction, although I knew
that the whole apparatus was at rest and every outward
point of reference for the perception of motion
was wanting. Every one who disbelieves in sensations
of movement should be made acquainted with
these phenomena. Had Newton known them and had
he ever observed how we may actually imagine ourselves
turned and displaced in space without the assistance
of stationary bodies as points of reference, he
would certainly have been confirmed more than ever<span class="pagenum"><a name="Page_290" id="Page_290">[Pg 290]</a></span>
in his unfortunate speculations regarding absolute
space.</p>

<p>The sensation of rotation in the opposite direction
after the apparatus has been stopped, slowly and gradually
ceases. But on accidentally inclining my head
once during this occurrence, the axis of apparent rotation
was also observed to incline in exactly the same
manner both as to direction and as to amount. It is
accordingly clear that the acceleration or retardation
of rotation is felt. The acceleration operates as a
stimulus. The sensation, however, like almost all
sensations, though it gradually decreases, lasts perceptibly
longer than the stimulus. Hence the long
continued apparent rotation after the stopping of the
apparatus. The organ, however, which causes the
persistence of this sensation must have its seat in the
<i>head</i>, since otherwise the axis of apparent rotation
could not assume the same motion as the head.</p>

<p>If I were to say, now, that a light had flashed
upon me in making these last observations, the expression
would be a feeble one. I ought to say I experienced
a perfect illumination. My juvenile experiences
of vertigo occurred to me. I remembered
Flourens's experiments relative to the section of the
semi-circular canals of the labyrinths of doves and
rabbits, where this inquirer had observed phenomena
similar to vertigo, but which he preferred to interpret,
from his bias to the acoustic theory of the labyrinth,
as the expression of painful auditive disturbances. I<span class="pagenum"><a name="Page_291" id="Page_291">[Pg 291]</a></span>
saw that Goltz had nearly but not quite hit the bull's
eye with his theory of the semi-circular canals. This
inquirer, who, from his happy habit of following his
own natural thoughts without regard for tradition,
has cleared up so much in science, spoke, as early as
1870, on the ground of experiments, as follows: "It
is uncertain whether the semi-circular canals are auditive
organs or not. In any event they form an apparatus
which serves for the preservation of equilibrium.
They are, so to speak, the sense-organs of equilibrium
of the head and indirectly of the whole body." I
remembered the galvanic dizziness which had been
observed by Ritter and Purkinje on the passage of a
current through the head, when the persons experimented
upon imagined they were falling towards the
cathode. The experiment was immediately repeated,
and sometime later (1874) I was enabled to demonstrate
the same objectively with fishes, all of which
placed themselves sidewise and in the same direction
in the field of the current as if at command.<a name="FNanchor_98_98" id="FNanchor_98_98"></a><a href="#Footnote_98_98" class="fnanchor">[98]</a> Müller's
doctrine of specific energies now appeared to me
to bring all these new and old observations into a simple,
connected unity.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_302.jpg" width="600" height="406" alt="" />
<span class="caption">Fig. 47.</span>
</div>

<p>The labyrinth of a dove (stereoscopically reproduced), from R. Ewald,
<i>Nervus Octavus</i>, Wiesbaden, Bergmann, 1892.]</p>

<p>Let us picture to ourselves the labyrinth of the
ear with its three semi-circular canals lying in three
mutually perpendicular planes (Comp. Fig. 47), the<span class="pagenum"><a name="Page_292" id="Page_292">[Pg 292]</a></span>
mysterious position of which inquirers have endeavored
to explain in every possible and impossible way.
Let us conceive the nerves of the ampullæ, or the dilated
extensions of the semi-circular canals, equipped
with a capacity for responding to every imaginable
stimulus with a sensation of rotation just as the nerves
of the retina of the eye when excited by pressures,
by electrical or chemical stimuli always respond with
the sensation of light; let us picture to ourselves,
further, that the usual excitation of the ampullæ
nerves is produced by the inertia of the contents of
the semi-circular canals, which contents on suitable
rotations in the plane of the semi-circular canal are
left behind in the motion, or at least have a tendency<span class="pagenum"><a name="Page_293" id="Page_293">[Pg 293]</a></span>
to remain behind and consequently exert a pressure.
It will be seen that on this supposition all the single
facts which without the theory appear as so many
different individual phenomena, become from this single
point of view clear and intelligible.</p>

<p>I had the satisfaction, immediately after the communication
in which I set forth this idea,<a name="FNanchor_99_99" id="FNanchor_99_99"></a><a href="#Footnote_99_99" class="fnanchor">[99]</a> of seeing a
paper by Breuer appear<a name="FNanchor_100_100" id="FNanchor_100_100"></a><a href="#Footnote_100_100" class="fnanchor">[100]</a> in which this author had
arrived by entirely different methods at results that
agreed in all essential points with my own. A few
weeks later appeared the researches of Crum Brown
of Edinburgh, whose methods were even still nearer
mine. Breuer's paper was far richer in physiological
respects than mine, and he had particularly gone
into greater detail in his investigation of the collateral
effects of the reflex motions and orientation of
the eyes in the phenomena under consideration.<a name="FNanchor_101_101" id="FNanchor_101_101"></a><a href="#Footnote_101_101" class="fnanchor">[101]</a> In
addition certain experiments which I had suggested in
my paper as a test of the correctness of the view
in question had already been performed by Breuer.
Breuer has also rendered services of the highest order
in the further elaboration of this field. But in a
physical regard, my paper was, of course, more complete.</p>

<p>In order to portray to the eye the behavior of the
semi-circular canals, I have constructed here a little<span class="pagenum"><a name="Page_294" id="Page_294">[Pg 294]</a></span>
apparatus. (See Fig. 48.) The large rotatable disc
represents the osseous semi-circular canal, which is
continuous with the bones of the head; the small disc,
which is free to rotate on the axis of the first, represents
the mobile and partly liquid contents of the semi-circular
canal. On rotating the large disc, the small
disc as you see remains
behind. I
have to turn some
time before the
small disc is carried
along with the large
one by friction. But
if I now stop the
large disc the small
disc as you see continues
to rotate.</p>

<div class="figcenter" style="width: 350px;">
<img src="images/i_304.jpg" width="350" height="497" alt="" />
<span class="caption">Fig. 48.</span>
</div>

<p>Model representing the action of the semi-circular
canals.]</p>

<p>Simply assume
now that the rotation
of the small
disc, say in the direction
of the hands
of a watch, would
give rise to a sensation of rotation in the opposite
direction, and conversely, and you already understand
a good portion of the facts above set forth.
The explanation still holds, even if the small disc
does not perform appreciable rotations but is checked
by a contrivance similar to an elastic spring, the tension<span class="pagenum"><a name="Page_295" id="Page_295">[Pg 295]</a></span>
of which disengages a sensation. Conceive, now,
three such contrivances with their mutually perpendicular
planes of rotation joined together so as to
form a single apparatus; then to this apparatus as a
whole, no rotation can be imparted without its being
indicated by the small mobile discs or by the springs
which are attached to them. Conceive both the right
and the left ear equipped with such an apparatus, and
you will find that it answers all the purposes of the
semi-circular canals, which you see represented stereoscopically
in Fig. 47 for the ear of a dove.</p>

<p>Of the many experiments which I have made on
my own person, and the results of which could be
predicted by the new view according to the behavior
of the model and consequently according to the rules
of mechanics, I shall cite but one. I fasten a horizontal
board in the frame <i>RR</i> of my rotatory apparatus,
lie down upon the same with my right ear upon the
board, and cause the apparatus to be uniformly rotated.
As soon as I no longer perceive the rotation,
I turn around upon my left ear and immediately the
sensation of rotation again starts up with marked vividness.
The experiment can be repeated as often as
one wishes. A slight turn of the head even is sufficient
for reviving the sensation of rotation which in
the perfectly quiescent state at once disappears altogether.</p>

<p>We will imitate the experiment on the model. I
turn the large disc until finally the small disc is carried<span class="pagenum"><a name="Page_296" id="Page_296">[Pg 296]</a></span>
along with it. If, now, while the rotation continues
uniform, I burn off a little thread which you
see here, the small disc will be flipped round by a
spring into its own plane 180°, so as now to present
its opposite side to you, when the rotation at once begins
in the opposite direction.</p>

<p>We have consequently a very simple means for determining
whether one is actually the subject or not
of uniform and imperceptible rotations. If the earth
rotated much more rapidly than it really does, or if
our semi-circular canals were much more sensitive, a
Nansen sleeping at the North Pole would be waked
by a sensation of rotation every time he turned over.
Foucault's pendulum experiment as a demonstration
of the earth's rotation would be superfluous under
such circumstances. The only reason we cannot prove
the rotation of the earth with the help of our model,
lies in the small angular velocity of the earth and in
the consequent liability to great experimental errors.<a name="FNanchor_102_102" id="FNanchor_102_102"></a><a href="#Footnote_102_102" class="fnanchor">[102]</a></p>

<p>Aristotle has said that "The sweetest of all
things is knowledge." And he is right. But if you
were to suppose that the <i>publication</i> of a new view
were productive of unbounded sweetness, you would
be mightily mistaken. No one disturbs his fellow-men
with a new view unpunished. Nor should the fact be
made a subject of reproach to these fellow-men. To<span class="pagenum"><a name="Page_297" id="Page_297">[Pg 297]</a></span>
presume to revolutionise the current way of thinking
with regard to any question, is no pleasant task, and
above all not an easy one. They who have advanced
new views know best what serious difficulties stand in
their way. With honest and praiseworthy zeal, men
set to work in search of everything that does not
suit with them. They seek to discover whether they
cannot explain the facts better or as well, or approximately
as well, by the traditional views. And that,
too, is justified. But at times some extremely artless
animadversions are heard that almost nonplus us.
"If a sixth sense existed it could not fail to have
been discovered thousands of years ago." Indeed;
there was a time, then, when only seven planets could
have existed! But I do not believe that any one will
lay any weight on the philological question whether
the set of phenomena which we have been considering
should be called a sense. The phenomena will not
disappear when the name disappears. It was further
said to me that animals exist which have no labyrinth,
but which can yet orientate themselves, and that consequently
the labyrinth has nothing to do with orientation.
We do not walk forsooth with our legs, because
snakes propel themselves without them!</p>

<p>But if the promulgator of a new idea cannot hope
for any great pleasure from its publication, yet the
critical process which his views undergo is extremely
helpful to the subject-matter of them. All the defects
which necessarily adhere to the new view are gradually<span class="pagenum"><a name="Page_298" id="Page_298">[Pg 298]</a></span>
discovered and eliminated. Over-rating and exaggeration
give way to more sober estimates. And
so it came about that it was found unpermissible to
attribute all functions of orientation exclusively to the
labyrinth. In these critical labors Delage, Aubert,
Breuer, Ewald, and others have rendered distinguished
services. It can also not fail to happen that
fresh facts become known in this process which could
have been predicted by the new view, which actually
were predicted in part, and which consequently furnish
a support for the new view. Breuer and Ewald
succeeded in electrically and mechanically exciting
the labyrinth, and even single parts of the labyrinth,
and thus in producing the movements that belong to
such stimuli. It was shown that when the semi-circular
canals were absent vertigo could not be produced,
when the entire labyrinth was removed the orientation
of the head was no longer possible, that without
the labyrinth galvanic vertigo could not be induced. I
myself constructed as early as 1875 an apparatus for
observing animals in rotation, which was subsequently
reinvented in various forms and has since received the
name of "cyclostat."<a name="FNanchor_103_103" id="FNanchor_103_103"></a><a href="#Footnote_103_103" class="fnanchor">[103]</a> In experiments with the most
varied kinds of animals it was shown that, for example,
the larvæ of frogs are not subject to vertigo until
their semi-circular canals which at the start are wanting
are developed (K. Schäfer). A large percentage
of the deaf and dumb are afflicted with grave affections<span class="pagenum"><a name="Page_299" id="Page_299">[Pg 299]</a></span>
of the labyrinth. The American psychologist,
William James, has made whirling experiments with
many deaf and dumb subjects, and in a large number
of them found that susceptibility to giddiness is wanting.
He also found that many deaf and dumb people
on being ducked under water, whereby they lose their
weight and consequently have no longer the full assistance
of their muscular sense, utterly lose their
sense of position in space, do not know which is up
and which is down, and are thrown into the greatest
consternation,&mdash;results which do not occur in normal
men. Such facts are convincing proof that we do not
orientate ourselves entirely by means of the labyrinth,
important as it is for us. Dr. Kreidl has made experiments
similar to those of James and found that
not only is vertigo absent in deaf and dumb people
when whirled about, but that also the reflex movements
of the eyes which are normally induced by the
labyrinth are wanting. Finally, Dr. Pollak has found
that galvanic vertigo does not exist in a large percentage
of the deaf and dumb. Neither the jerking
movements nor the uniform movements of the eyes
were observed which normal human beings exhibit in
the Ritter and Purkinje experiment.</p>

<p>After the physicist has arrived at the idea that the
semi-circular canals are the organ of sensation of rotation
or of angular acceleration, he is next constrained
to ask for the organs that mediate the sensation
of acceleration noticed in forward movements.<span class="pagenum"><a name="Page_300" id="Page_300">[Pg 300]</a></span>
In searching for an organ for this function, he of
course is not apt to select one that stands in no anatomical
and spatial relation with the semi-circular
canals. And in addition there are physiological considerations
to be weighed. The preconceived opinion
once having been abandoned that the <i>entire</i> labyrinth
is auditory in its function, there remains after the
cochlea is reserved for sensations of tone and the
semi-circular canals for the sensation of angular acceleration,
the vestibule for the discharge of additional
functions. The vestibule, particularly the part of it
known as the sacculus, appeared to me, by reason of
the so-called otoliths which it contains, eminently
adapted for being the organ of sensation of forward
acceleration or of the position of the head. In this
conjecture I again closely coincided with Breuer.</p>

<p>That a sensation of position, of direction and
amount of mass-acceleration exists, our experience in
elevators as well as of movement in curved paths is
sufficient proof. I have also attempted to produce and
destroy suddenly great velocities of forward movement
by means of various contrivances of which I
shall mention only one here. If, while enclosed in
the paper box of my large whirling apparatus at some
distance from the axis, my body is in uniform rotation
which I no longer feel, and I then loosen the connexions
of the frame <i>rr</i> with <i>R</i> thus making the former
moveable and I then suddenly stop the larger frame,
my forward motion is abruptly impeded while the<span class="pagenum"><a name="Page_301" id="Page_301">[Pg 301]</a></span>
frame <i>rr</i> continues to rotate. I imagine now that I
am speeding on in a straight line in a direction opposite
to that of the checked motion. Unfortunately, for
many reasons it cannot be proved convincingly that
the organ in question has its seat in the head. According
to the opinion of Delage, the labyrinth has
nothing to do with this particular sensation of movement.
Breuer, on the other hand, is of the opinion
that the organ of forward movement in man is stunted
and the persistence of the sensation in question is too
brief to permit our instituting experiments as obvious
as in the case of rotation. In fact, Crum Brown once
observed while in an irritated condition peculiar vertical
phenomena in his own person, which were all
satisfactorily explained by an abnormally long persistence
of the sensation of rotation, and I myself in an
analogous case on the stopping of a railway train felt
the apparent backward motion in striking intensity
and for an unusual length of time.</p>

<p>There is no doubt whatever that we feel changes
of vertical acceleration, and it will appear from the
following extremely probable that the otoliths of the
vestibule are the sense-organ for the <i>direction</i> of the
mass-acceleration. It will then be incompatible with
a really logical view to regard the latter as incapable
of sensing horizontal accelerations.</p>

<p>In the lower animals the analogue of the labyrinth
is shrunk to a little vesicle filled with a liquid and
containing tiny crystals, auditive stones, or otoliths, of<span class="pagenum"><a name="Page_302" id="Page_302">[Pg 302]</a></span>
greater specific gravity, suspended on minute hairs.
These crystals appear physically well adapted for indicating
both the direction of gravity and the direction
of incipient movements. That they discharge the former
function, Delage was the first to convince himself
by experiments with lower animals which on the removal
of the otoliths utterly lost their bearings and
could no longer regain their normal position. Loeb
also found that fishes without labyrinths swim now on
their bellies and now on their backs. But the most
remarkable, most beautiful, and most convincing experiment
is that which Dr. Kreidl instituted with
crustaceans. According to Hensen, certain Crustacea
on sloughing spontaneously introduce fine grains of
sand as auditive stones into their otolith vesicle. At
the ingenious suggestion of S. Exner, Dr. Kreidl constrained
some of these animals to put up with iron
filings (<i>ferrum limatum</i>). If the pole of an electro-magnet
be brought near the animal, it will at once
turn its back away from the pole accompanying the
movement with appropriate reflex motions of the eye
the moment the current is closed, exactly as if gravity
had been brought to bear upon the animal in the
same direction as the magnetic force.<a name="FNanchor_104_104" id="FNanchor_104_104"></a><a href="#Footnote_104_104" class="fnanchor">[104]</a> This, in fact,
is what should be expected from the function ascribed
to the otoliths. If the eyes be covered with asphalt<span class="pagenum"><a name="Page_303" id="Page_303">[Pg 303]</a></span>
varnish, and the auditive sacs removed, the crustaceans
lose their sense of direction utterly, tumble
head over heels, lie on their side or their back indifferently.
This does not happen when the eyes only
are covered. For vertebrates, Breuer has demonstrated
by searching investigations that the otoliths,
or better, statoliths, slide in three planes parallel to
the planes of the semi-circular canals, and are consequently
perfectly adapted for indicating changes
both in the amount and the direction of the mass-acceleration.<a name="FNanchor_105_105" id="FNanchor_105_105"></a><a href="#Footnote_105_105" class="fnanchor">[105]</a></p>

<p>I have already remarked that not every function
of orientation can be ascribed exclusively to the labyrinth.
The deaf and dumb who have to be immersed
in water, and the crustaceans who must have their
eyes closed if they are to be perfectly disorientated, are
proof of this fact. I saw a blind cat at Hering's laboratory
which to one who was not a very attentive observer
behaved exactly like a seeing cat. It played
nimbly with objects rolling on the floor, stuck its head
inquisitively into open drawers, sprang dexterously
upon chairs, ran with perfect accuracy through open<span class="pagenum"><a name="Page_304" id="Page_304">[Pg 304]</a></span>
doors, and never bumped against closed ones. The
visual sense had here been rapidly replaced by the
tactual and auditive senses. And it appears from
Ewald's investigations that even after the labyrinths
have been removed, animals gradually learn to move
about again quite in the normal fashion, presumably
because the eliminated function of the labyrinth is
now performed by some part of the brain. A certain
peculiar weakness of the muscles alone is perceptible
which Ewald ascribes to the absence of the stimulus
which is otherwise constantly emitted by the labyrinth
(the labyrinth-tonus). But if the part of the
brain which discharges the deputed function be removed,
the animals are again completely disorientated
and absolutely helpless.</p>

<p>It may be said that the views enunciated by Breuer,
Crum Brown and myself in 1873 and 1874, and which
are substantially a fuller and richer development of
Goltz's idea, have upon the whole been substantiated.
At least they have exercised a helpful and stimulative
influence. New problems have of course arisen in the
course of the investigation which still await solution,
and much work remains to be done. At the same
time we see how fruitful the renewed co-operation of
the various special departments of science may become
after a period of isolation and invigorating labor
apart.</p>

<p>I may be permitted, therefore, to consider the relation
between hearing and orientation from another<span class="pagenum"><a name="Page_305" id="Page_305">[Pg 305]</a></span>
and more general point of view. What we call the
auditive organ is in the lower animals simply a sac
containing auditive stones. As we ascend the scale,
1, 2, 3 semi-circular canals gradually develop from
them, whilst the structure of the otolith organ itself
becomes more complicated. Finally, in the higher
vertebrates, and particularly in the mammals, a part
of the latter organ (the lagena) becomes the cochlea,
which Helmholtz explained as the organ for sensations
of tone. In the belief that the entire labyrinth
was an auditive organ, Helmholtz, contrary to the results
of his own masterly analysis, originally sought
to interpret another part of the labyrinth as the organ
of noises. I showed a long time ago (1873) that every
tonal stimulus by shortening the duration of the excitation
to a few vibrations, gradually loses its character
of pitch and takes on that of a sharp, dry report or
noise.<a name="FNanchor_106_106" id="FNanchor_106_106"></a><a href="#Footnote_106_106" class="fnanchor">[106]</a> All the intervening stages between tones and
noises can be exhibited. Such being the case, it will
hardly be assumed that one organ is suddenly and at
some given point replaced in function by another. On
the basis of different experiments and reasonings S.
Exner also regards the assumption of a special organ
for the sensing of noises as unnecessary.</p>

<p>If we will but reflect how small a portion of the
labyrinth of higher animals is apparently in the service
of the sense of hearing, and how large, on the other<span class="pagenum"><a name="Page_306" id="Page_306">[Pg 306]</a></span>
hand, the portion is which very likely serves the purposes
of orientation, how much the first anatomical
beginnings of the auditive sac of lower animals resemble
that part of the fully developed labyrinth which
does not hear, the view is irresistibly suggested which
Breuer and I (1874, 1875) expressed, that the auditive
organ took its development from an organ for sensing
movements by adaptation to weak periodic motional
stimuli, and that many apparatuses in the lower animals
which are held to be organs of hearing are not
auditive organs at all.<a name="FNanchor_107_107" id="FNanchor_107_107"></a><a href="#Footnote_107_107" class="fnanchor">[107]</a></p>

<p>This view appears to be perceptibly gaining
ground. Dr. Kreidl by skilfully-planned experiments
has arrived at the conclusion that even fishes do not
hear, whereas E. H. Weber, in his day, regarded the
ossicles which unite the air-bladder of fishes with the
labyrinth as organs expressly designed for conducting
sound from the former to the latter.<a name="FNanchor_108_108" id="FNanchor_108_108"></a><a href="#Footnote_108_108" class="fnanchor">[108]</a> Störensen has
investigated the excitation of sounds by the air-bladder
of fishes, as also the conduction of shocks through
Weber's ossicles. He regards the air-bladder as particularly
adapted for receiving the noises made by
other fishes and conducting them to the labyrinth.
He has heard the loud grunting tones of the fishes
in South American rivers, and is of the opinion that
they allure and find each other in this manner. According
to these views certain fishes are neither deaf
nor dumb.<a name="FNanchor_109_109" id="FNanchor_109_109"></a><a href="#Footnote_109_109" class="fnanchor">[109]</a> The question here involved might be
solved perhaps by sharply distinguishing between the
sensation of hearing proper, and the perception of
shocks. The first-mentioned sensation may, even in
the case of many vertebrates, be extremely restricted,
or perhaps even absolutely wanting. But besides the
auditive function, Weber's ossicles may perfectly well
discharge some other function. Although, as Moreau
has shown, the air-bladder itself is not an organ of
equilibrium in the simple physical sense of Borelli,
yet doubtless some function of this character is still
reserved for it. The union with the labyrinth favors
this conception, and so a host of new problems rises
here before us.</p>
<p><span class="pagenum"><a name="Page_307" id="Page_307">[Pg 307]</a></span></p>
<p>I should like to close with a reminiscence from the
year 1863. Helmholtz's <i>Sensations of Tone</i> had just
been published and the function of the cochlea now
appeared clear to the whole world. In a private conversation
which I had with a physician, the latter declared
it to be an almost hopeless undertaking to seek
to fathom the function of the other parts of the labyrinth,
whereas I in youthful boldness maintained that
the question could hardly fail to be solved, and that
very soon, although of course I had then no glimmering
of how it was to be done. Ten years later the
question was substantially solved.</p>

<p>To-day, after having tried my powers frequently
and in vain on many questions, I no longer believe<span class="pagenum"><a name="Page_308" id="Page_308">[Pg 308]</a></span>
that we can make short work of the problems of science.
Nevertheless, I should not consider an "ignorabimus"
as an expression of modesty, but rather as
the opposite. That expression is a suitable one only
with regard to problems which are wrongly formulated
and which are therefore not problems at all.
Every real problem can and will be solved in due
course of time without supernatural divination, entirely
by accurate observation and close, searching
thought.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_309" id="Page_309">[Pg 309]</a></span></p>




<h2><a name="ON_SOME_PHENOMENA_ATTENDING" id="ON_SOME_PHENOMENA_ATTENDING">ON SOME PHENOMENA ATTENDING
THE FLIGHT OF PROJECTILES.</a><a name="FNanchor_110_110" id="FNanchor_110_110"></a><a href="#Footnote_110_110" class="fnanchor">[110]</a></h2>


<blockquote><p>"I have led my ragamuffins where they were
peppered."&mdash;<i>Falstaff.</i></p>

<p>"He goes but to see a noise that he heard."&mdash;<i>Midsummer
Night's Dream.</i></p></blockquote>

<p>To shoot, in the shortest time possible, as many
holes as possible in one another's bodies, and
not always for exactly pardonable objects and ideals,
seems to have risen to the dignity of a duty with modern
men, who, by a singular inconsistency, and in
subservience to a diametrically contrary ideal, are
bound by the equally holy obligation of making these
holes as small as possible, and, when made, of stopping
them up and of healing them as speedily as
possible. Since, then, shooting and all that appertains
thereto, is a very important, if not the most important,
affair of modern life, you will doubtless not be averse
to giving your attention for an hour to some experiments
which have been undertaken, not for advancing
the ends of war, but for promoting the ends of science,<span class="pagenum"><a name="Page_310" id="Page_310">[Pg 310]</a></span>
and which throw some light on the phenomena
attending the flight of projectiles.</p>

<p>Modern science strives to construct its picture of
the world not from speculations but so far as possible
from facts. It verifies its constructs by recourse to
observation. Every newly observed fact completes
its world-picture, and every divergence of a construct
from observation points to some imperfection, to some
lacuna in it. What is seen is put to the test of, and
supplemented by, what is thought, which is again
naught but the result of things previously seen. It
is always peculiarly fascinating, therefore, to subject
to direct verification by observation, that is, to render
palpable to the senses, something which we have only
theoretically excogitated or theoretically surmised.</p>

<p>In 1881, on hearing in Paris the lecture of the Belgian
artillerist Melsens, who hazarded the conjecture
that projectiles travelling at a high rate of speed carry
masses of compressed air before them which are instrumental
in producing in bodies struck by the projectiles
certain well-known facts of the nature of explosions,
the desire arose in me of experimentally testing
his conjecture and of rendering the phenomenon,
if it really existed, perceptible. The desire was the
stronger as I could say that all the means for realising
it existed, and that I had in part already used and
tested them for other purposes.</p>

<p>And first let us get clear regarding the difficulties
which have to be surmounted. Our task is that of<span class="pagenum"><a name="Page_311" id="Page_311">[Pg 311]</a></span>
observing a bullet or other projectile which is rushing
through space at a velocity of many hundred yards a
second, together with the disturbances which the bullet
causes in the surrounding atmosphere. Even the
opaque solid body itself, the projectile, is only exceptionally
visible under such circumstances&mdash;only when
it is of considerable size and when we see its line of
flight in strong perspective abridgement so that the
velocity is apparently diminished. We see a large
projectile quite clearly when we stand behind the cannon
and look steadily along its line of flight or in the
less pleasant case when the projectile is speeding towards
us. There is, however, a very simple and effective
method of observing swiftly moving bodies with as
little trouble as if they were held at rest at some point
in their path. The method is that of illumination by
a brilliant electric spark of extremely short duration
in a dark room. But since, for the full intellectual
comprehension of a picture presented to the eye, a
certain, not inconsiderable interval of time is necessary,
the method of instantaneous photography will
naturally also be employed. The pictures, which are
of extremely minute duration, are thus permanently
recorded and can be examined and analysed at one's
convenience and leisure.</p>

<p>With the difficulty just mentioned is associated
still another and greater difficulty which is due to the
air. The atmosphere in its usual condition is generally
not visible even when at rest. But the task presented<span class="pagenum"><a name="Page_312" id="Page_312">[Pg 312]</a></span>
to us is to render visible masses of air which
in addition are moving with a high velocity.</p>

<p>To be visible, a body must either emit light itself,
must shine, or must affect in some way the light which
falls upon it, must take up that light entirely or partly,
absorb it, or must have a deflective effect upon it, that
is, reflect or refract it. We cannot see the air as we
can a flame, for it shines only exceptionally, as in a
Geissler's tube. The atmosphere is extremely transparent
and colorless; it cannot be seen, therefore, as
a dark or colored body can, or as chlorine gas can,
or vapor of bromine or iodine. Air, finally, has so
small an index of refraction and so small a deflective
influence upon light, that the refractive effect is commonly
imperceptible altogether.</p>

<p>A glass rod is visible in air or in water, but it is
almost invisible in a mixture of benzol and bisulphuret
of carbon, which has the same mean index of refraction
as the glass. Powdered glass in the same mixture
has a vivid coloring, because owing to the decomposition
of the colors the indices are the same
for only one color which traverses the mixture unimpeded,
whilst the other colors undergo repeated reflexions.<a name="FNanchor_111_111" id="FNanchor_111_111"></a><a href="#Footnote_111_111" class="fnanchor">[111]</a></p>

<p>Water is invisible in water, alcohol in alcohol. But
if alcohol be mixed with water the flocculent streaks
of the alcohol in the water will be seen at once and<span class="pagenum"><a name="Page_313" id="Page_313">[Pg 313]</a></span>
<i>vice versa</i>. And in like manner the air, too, under
favorable circumstances, may be seen. Over a roof
heated by the burning sun, a tremulous wavering of
objects is noticeable, as there is also over red-hot
stoves, radiators, and registers. In all these cases
tiny flocculent masses of hot and cold air, of slightly
differing refrangibility, are mingled together.</p>

<p>In like manner the more highly refracting parts of
non-homogeneous masses of glass, the so-called striæ
or imperfections of the glass, are readily detectible
among the less refracting parts which constitute the
bulk of the same. Such glasses are unserviceable for
optical purposes, and special attention has been devoted
to the investigation of the methods for eliminating
or avoiding these defects. The result has been
the development of an extremely delicate method for
detecting optical faults&mdash;the so-called method of Foucault
and Toepler&mdash;which is suitable also for our
present purpose.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_324.jpg" width="600" height="169" alt="" />
<span class="caption">Fig. 49.</span>
</div>

<p>Even Huygens when trying to detect the presence
of striæ in polished glasses viewed them under oblique
illumination, usually at a considerable distance, so as
to give full scope to the aberrations, and had recourse
for greater exactitude to a telescope. But the method
was carried to its highest pitch of perfection in 1867
by Toepler who employed the following procedure:
A small luminous source <i>a</i> (Fig. 49) illuminates a lens
<i>L</i> which throws an image <i>b</i> of the luminous source.
If the eye be so placed that the image falls on the<span class="pagenum"><a name="Page_314" id="Page_314">[Pg 314]</a></span>
pupil, the entire lens, if perfect, will appear equally
illuminated, for the reason that all points of it send
out rays to the eye. Coarse imperfections of form or
of homogeneity are rendered visible only in case the
aberrations are so large that the light from many spots
passes by the pupil of the eye. But if the image <i>b</i> be
partly intercepted by the edge of a small slide, then
those spots in the lens as thus partly darkened will
appear brighter whose light by its greater aberrations
still reaches the eye in spite of the intercepting slide,
while those spots will appear darker which in consequence
of aberration in the other direction throw their
light entirely upon the slide. This artifice of the intercepting
slide which had previously been employed
by Foucault for the investigation of the optical imperfections
of mirrors enhances enormously the delicacy
of the method, which is still further augmented by
Toepler's employment of a telescope behind the slide.
Toepler's method, accordingly, enjoys all the advantages
of the Huygens and the Foucault procedure
combined. It is so delicate that the minutest irregularities
in the air surrounding the lens can be rendered
distinctly visible, as I shall show by an example. I<span class="pagenum"><a name="Page_315" id="Page_315">[Pg 315]</a></span>
place a candle before the lens <i>L</i> (Fig. 50) and so arrange
a second lens <i>M</i> that the flame of the candle is
imaged upon the screen <i>S</i>. As soon as the intercepting
slide is pushed into the focus, <i>b</i>, of the light issuing
from <i>a</i>, you see the images of the changes of
density and the images of the movements induced in
the air by the flame quite distinctly upon the screen.
The distinctness of the phenomenon as a whole depends
upon the position of the intercepting slide <i>b</i>.
The removal of <i>b</i> increases the illumination but decreases
the distinctness. If the luminous source <i>a</i> be
removed, we see the image of the candle flame only
upon the screen <i>S</i>. If we remove the flame and allow
<i>a</i> to continue shining, the screen <i>S</i> will appear uniformly
illuminated.</p>

<div class="figcenter" style="width: 700px;">
<img src="images/i_325.jpg" width="700" height="265" alt="" />
<span class="caption">Fig. 50.</span>
</div>

<p>After Toepler had sought long and in vain to render
the irregularities produced in air by sound-waves
visible by this principle, he was at last conducted to
his goal by the favorable circumstances attending the
production of electric sparks. The waves generated
in the air by electric sparks and accompanying the
explosive snapping of the same, are of sufficiently<span class="pagenum"><a name="Page_316" id="Page_316">[Pg 316]</a></span>
short period and sufficiently powerful to be rendered
visible by these methods. Thus we see how by a
careful regard for the merest and most shadowy indications
of a phenomenon and by slight progressive
and appropriate alterations of the circumstances and
the methods, ultimately the most astounding results
can be attained. Consider, for example, two such
phenomena as the rubbing of amber and the electric
lighting of modern streets. A person ignorant of the
myriad minute links that join these two things together,
will be absolutely nonplussed at their connexion,
and will comprehend it no more than the ordinary
observer who is unacquainted with embryology, anatomy,
and paleontology will understand the connexion
between a saurian and a bird. The high value and
significance of the co-operation of inquirers through
centuries, where each has but to take up the thread of
work of his predecessors and spin it onwards, is rendered
forcibly evident by such examples. And such
knowledge destroys, too, in the clearest manner imaginable
that impression of the marvellous which the
spectator may receive from science, and at the same
time is a most salutary admonishment to the worker
in science against superciliousness. I have also to
add the sobering remark that all our art would be in
vain did not nature herself afford at least some slight
guiding threads leading from a hidden phenomenon
into the domain of the observable. And so it need
not surprise us that once under particularly favorable<span class="pagenum"><a name="Page_317" id="Page_317">[Pg 317]</a></span>
circumstances an extremely powerful sound-wave
which had been caused by the explosion of several
hundred pounds of dynamite threw a directly visible
shadow in the sunlight, as Boys has recently told us.
If the sound-waves were absolutely without influence
upon the light, this could not have occurred, and all
our artifices would then, too, be in vain. And so,
similarly, the phenomenon accompanying projectiles
which I am about to show you was once in a very imperfect
manner incidentally seen by a French artillerist,
Journée, while that observer was simply following
the line of flight of a projectile with a telescope, just
as also the undulations produced by candle flames are
in a weak degree directly visible and in the bright sunlight
are imaged in shadowy waves upon a uniform
white background.</p>

<p><i>Instantaneous illumination</i> by the electric spark,
the method of rendering visible small optical differences
or striæ, which may hence be called the <i>striate</i>,
or <i>differential</i>, method,<a name="FNanchor_112_112" id="FNanchor_112_112"></a><a href="#Footnote_112_112" class="fnanchor">[112]</a> invented by Foucault and
Toepler, and finally the <i>recording</i> of the image by a <i>photographic</i>
plate,&mdash;these therefore are the chief means
which are to lead us to our goal.</p>
<p><span class="pagenum"><a name="Page_318" id="Page_318">[Pg 318]</a></span></p>
<p>I instituted my first experiments in the summer of
1884 with a target-pistol, shooting the bullet through
a striate field as described above, and taking care that
the projectile whilst in the field should disengage an
illuminating electric spark from a Leyden jar or Franklin's
pane, which spark produced a photographic impression
of the projectile upon a plate, especially arranged
for the purpose. I obtained the image of the
projectile at once and without difficulty. I also readily
obtained, with the still rather defective dry plate which
I was using, exceedingly delicate images of the sound-waves
(spark-waves). But no atmospheric condensation
produced by the projectile was visible. I now
determined the velocity of my projectile and found it
to be only 240 metres per second, or considerably less
than the velocity of sound (which is 340 metres per
second). I saw immediately that under such circumstances
no noticeable compression of the air could be
produced, for any atmospheric compression must of
necessity travel forward at the same speed with sound
(340 metres per second) and consequently would be
always ahead of and speeding away from the projectile.</p>

<p>I was so thoroughly convinced, however, of the
existence of the supposed phenomenon at a velocity
exceeding 340 metres per second, that I requested<span class="pagenum"><a name="Page_319" id="Page_319">[Pg 319]</a></span>
Professor Salcher, of Fiume, an Austrian port on the
Gulf of Quarnero, to undertake the experiment with
projectiles travelling at a high rate of speed. In the
summer of 1886 Salcher in conjunction with Professor
Riegler conducted in a spacious and suitable apartment
placed at their disposal by the Directors of the
Royal Imperial Naval Academy, experiments of the
kind indicated and conforming in method exactly to
those which I had instituted, with the precise results
expected. The phenomenon, in fact, accorded perfectly
with the <i>a priori</i> sketch of it which I had drafted
previously to the experiment. As the experimenting
was continued, new and unforeseen features made their
appearance.</p>

<p>It would be unfair, of course, to expect from the
very first experiments faultless and highly distinct photographs.
It was sufficient that success was secured
and that I had convinced myself that further labor
and expenditure would not be vain. And on this
score I am greatly indebted to the two gentlemen
above mentioned.</p>

<p>The Austrian Naval Department subsequently
placed a cannon at Salcher's disposal in Pola, an
Adriatic seaport, and I myself, together with my son,
then a student of medicine, having received and accepted
a courteous invitation from Krupp, repaired to
Meppen, a town in Hanover, where we conducted
with only the necessary apparatus several experiments
on the open artillery range. All these experiments<span class="pagenum"><a name="Page_320" id="Page_320">[Pg 320]</a></span>
furnished tolerably good and complete pictures. Some
little progress, too, was made. The outcome of our
experience on both artillery ranges, however, was the
settled conviction that really good results could be
obtained only by the most careful conduct of the experiments
in a laboratory especially adapted to the
purpose. The expensiveness of the experiments on
a large scale was not the determining consideration
here, for the size of the projectile is indifferent. Given
the same velocity and the results are quite similar,
whether the projectiles are large or small. On the
other hand, in a laboratory the experimenter has perfect
control over the initial velocity, which, provided
the proper equipment is at hand, can be altered at
will simply by altering the charge and the weight of
the projectile. The requisite experiments were accordingly
conducted by me in my laboratory at Prague,
partly in conjunction with my son and partly afterwards
by him alone. The latter are the most perfect
and I shall accordingly speak in detail here of
these only.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_331.jpg" width="600" height="341" alt="" />
<span class="caption">Fig. 51.</span>
</div>

<p>Picture to yourself an apparatus for detecting optical
striæ set up in a dark room. In order not to
make the description too complicated, I shall give the
essential features only of the apparatus, leaving out
of account altogether the minuter details which are
rather of consequence for the technical performance
of the experiment than for its understanding. We
suppose the projectile speeding on its path, accordingly,<span class="pagenum"><a name="Page_321" id="Page_321">[Pg 321]</a></span>
through the field of our differential optical apparatus.
On reaching the centre of the field (Fig. 51)
the projectile disengages an illuminating electric spark
<i>a</i>, and the image of the projectile, so produced, is photographically
impressed upon the plate of the camera
behind the intercepting slide <i>b</i>. In the last and
best experiments the lens <i>L</i> was replaced by a spherical
silvered-glass mirror made by K. Fritsch (formerly
Prokesch) of Vienna, whereby the apparatus was
naturally more complicated than it appears in our diagram.
The projectile having been carefully aimed
passes in crossing the differential field between two
vertical isolated wires which are connected with the
two coatings of a Leyden jar, and completely filling
the space between the wires discharges the jar. In
the axis of the differential apparatus the circuit has a
second gap <i>a</i> which furnishes the illuminating spark,
the image of which falls on the intercepting slide <i>b</i>.
The wires in the differential field having occasioned<span class="pagenum"><a name="Page_322" id="Page_322">[Pg 322]</a></span>
manifold disturbances were subsequently done away
with. In the new arrangement the projectile passes
through a ring (see dotted line, Fig. 51), to the air in
which it imparts a sharp impulse which travels forward
in the tube <i>r</i> as a sound-wave having the approximate
velocity of 340 metres per second, topples
over through the aperture of an electric screen the
flame of a candle situated at the other opening of the
tube, and so discharges the jar. The length of the
tube <i>r</i> is so adjusted that the discharge occurs the
moment the projectile enters the centre of the now
fully clear and free field of vision. We will also leave
out of account the fact that to secure fully the success
of the experiment, a large jar is first discharged
by the flame, and that by the agency of this first discharge
the discharge of a second small jar having a
spark of very short period which furnishes the spark
really illuminating the projectile is effected. Sparks
from large jars have an appreciable duration, and
owing to the great velocity of the projectiles furnish
blurred photographs only. By carefully husbanding
the light of the differential apparatus, and owing to
the fact that much more light reaches the photographic
plate in this way than would otherwise reach
it, we can obtain beautiful, strong, and sharp photographs
with incredibly small sparks. The contours of
the pictures appear as very delicate and very sharp,
closely adjacent double lines. From their distance
from one another, and from the velocity of the projectile,<span class="pagenum"><a name="Page_323" id="Page_323">[Pg 323]</a></span>
the duration of the illumination, or of the spark,
is found to be 1/800000 of a second. It is evident,
therefore, that experiments with mechanical snap
slides can furnish no results worthy of the name.</p>

<div class="figcenter" style="width: 400px;">
<img src="images/i_333.jpg" width="400" height="517" alt="" />
<span class="caption">Fig. 52.</span>
</div>

<p>Let us consider now first the picture of a projectile
in the rough, as represented in Figure 52, and
then let us examine it in its photographic form as seen
in Figure 53. The latter picture is of a shot from an
Austrian Mannlicher rifle. If I were not to tell you
what the picture represented you would very likely
imagine it to be a bird's eye view of a boat <i>b</i> moving
swiftly through the water. In front you see the bow-wave
and behind the body a phenomenon <i>k</i> which
closely resembles the eddies formed in the wake of a<span class="pagenum"><a name="Page_324" id="Page_324">[Pg 324]</a></span>
ship. And as a matter of fact the dark hyperboloid
arc which streams from the tip of the projectile really
is a compressed wave of air exactly analogous to the
bow-wave produced by a ship moving through the
water, with the exception that the wave of air is not
a surface-wave. The air-wave is produced in atmospheric
space and encompasses the projectile in the
form of a shell on all sides. The wave is visible for
the same reason that the heated shell of air surrounding
the candle flame of our former experiments is visible.
And the cylinder of friction-heated air which the
projectile throws off in the form of vortex rings really
does answer to the water in the wake of a vessel.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_334.jpg" width="300" height="363" alt="" />
<span class="caption">Fig. 53. Photograph of a blunted projectile.]</span>
</div>

<p><span class="pagenum"><a name="Page_325" id="Page_325">[Pg 325]</a></span></p>

<p>Now just as a slowly moving boat produces no
bow-wave, but the bow-wave is seen only when the
boat moves with a speed which is greater than the
velocity of propagation of surface-waves in water, so,
in like manner, no wave of compression is visible in
front of a projectile so long as the speed of the projectile
is less than the velocity of sound. But if the
speed of the projectile reaches and exceeds the velocity
of sound, then the head-wave, as we shall call it,
augments noticeably in power, and is more and more
extended, that is, the angle made by the contours of
the wave with the direction of flight is more and more
diminished, just as when the speed of a boat is increased
a similar phenomenon is noticed in connexion
with the bow-wave. In fact, we can from an instantaneous
photograph so taken approximately estimate
the speed with which the projectile is travelling.</p>

<p>The explanation of the bow-wave of a ship and
that of the head-wave of a body travelling in atmospheric
space both repose upon the same principle,
long ago employed by Huygens. Conceive a number
of pebbles to be cast into a pond of water at regular
intervals in such wise that all the spots struck are situate
in the same straight line, and that every spot
subsequently struck lies a short space farther to the
right. The spots first struck will furnish then the
wave-circles which are widest, and all of them together
will, at the points where they are thickest,
form a sort of cornucopia closely resembling the bow-wave.<span class="pagenum"><a name="Page_326" id="Page_326">[Pg 326]</a></span>
(Fig. 54.) The resemblance is greater the
smaller the pebbles are, and the more quickly they
succeed each other. If a rod be dipped into the water
and quickly carried along its surface, the falling of
the pebbles will then take place, so to speak, uninterruptedly,
and we shall have a real bow-wave. If we
put the compressed air-wave in the place of the surface-waves
of the water, we shall have the head-wave
of the projectile.</p>

<div class="figcenter" style="width: 600px;">
<img src="images/i_336.jpg" width="600" height="265" alt="" />
<span class="caption">Fig. 54.</span>
</div>

<p>You may be disposed to say now, it is all very
pretty and interesting to observe a projectile in its
flight, but of what practical use is it?</p>

<p>It is true, I reply, one cannot <i>wage war</i> with photographed
projectiles. And I have likewise often had
to say to medical students attending my lectures on
physics, when they inquired for the practical value of
some physical observation, "You cannot, gentlemen,
cure diseases with it." I had also once to give my
opinion regarding how much physics should be taught
at a school for millers, supposing the instruction
there to be confined <i>exactly</i> to what was necessary for<span class="pagenum"><a name="Page_327" id="Page_327">[Pg 327]</a></span>
a miller. I was obliged to reply: "A miller always
<i>needs</i> exactly as much physics as he <i>knows</i>." Knowledge
which one does not possess one cannot use.</p>

<p>Let us forego entirely the consideration that as a
general thing every scientific advance, every new
problem elucidated, every extension or enrichment of
our knowledge of facts, affords a better foundation for
practical pursuits. Let us rather put the special
question, Is it not possible to derive some really practical
knowledge from our theoretical acquaintance
with the phenomena which take place in the space
surrounding a projectile?</p>

<p>No physicist who has ever studied waves of sound
or photographed them will have the least doubt regarding
the sound-wave character of the atmospheric
condensation encompassing the head of a flying projectile.
We have therefore, without ado, called this
condensation the head-wave.</p>

<p>Knowing this, it follows that the view of Melsens
according to which the projectile carries along with
it masses of air which it forces into the bodies struck,
is untenable. A forward-moving sound-wave is not a
forward-moving mass of matter but a forward-moving
form of motion, just as a water-wave or the waves of
a field of wheat are only forward-moving forms of motion
and not movements of masses of water or masses
of wheat.</p>

<p>By interference-experiments, on which I cannot
touch here but which will be found roughly represented<span class="pagenum"><a name="Page_328" id="Page_328">[Pg 328]</a></span>
in Figure 55, it was found that the bell-shaped
head-wave in question is an extremely thin shell and
that the condensations of the same are quite moderate,
scarcely exceeding two-tenths of an atmosphere.
There can be no question, therefore, of explosive effects
in the body struck by the projectile through so
slight a degree of atmospheric compression. The
phenomena attending wounds from rifle balls, for example,
are not to be explained as Melsens and Busch
explain them, but are due, as Kocher and Reger maintain,
to the effects of the impact of the projectile itself.</p>

<div class="figright" style="width: 400px;">
<img src="images/i_338.jpg" width="400" height="394" alt="" />
<span class="caption">Fig. 55.</span>
</div>

<p>A simple experiment will show how insignificant is
the part played by the friction of the air, or the supposed
conveyance of the air along with the moving
projectile. If the photograph of the projectile be<span class="pagenum"><a name="Page_329" id="Page_329">[Pg 329]</a></span>
taken while passing through a flame, i. e., a visible
gas, the flame will be seen to be, not torn and deformed,
but smoothly and cleanly perforated, like any
solid body. Within and around the flame the contours
of the head-wave will be seen. The flickering,
the extinction of the flame, etc., take place only after
the projectile has travelled on a considerable distance
in its path, and is then affected by the powder gases
which hurry after the bullet or by the air preceding
the powder-gases.</p>

<p>The physicist who examines the head-wave and
recognises its sound-wave character also sees that the
wave in question is of the same kind with the short
sharp waves produced by electric sparks, that it is a
<i>noise</i>-wave. Hence, whenever any portion of the head-wave
strikes the ear it will be heard as a report. Appearances
point to the conclusion that the projectile
carries this report along with it. In addition to this
report, which advances with the velocity of the projectile
and so usually travels at a speed greater than the
velocity of sound, there is also to be heard the report
of the exploding powder which travels forward with
the ordinary velocity of sound. Hence two explosions
will be heard, each distinct in time. The circumstance
that this fact was long misconstrued by
practical observers but when actually noticed frequently
received grotesque explanations and that ultimately
my view was accepted as the correct one, appears
to me in itself a sufficient justification that<span class="pagenum"><a name="Page_330" id="Page_330">[Pg 330]</a></span>
researches such as we are here speaking of are not utterly
superfluous even in practical directions. That
the flashes and sounds of discharging artillery are
used for estimating the distances of batteries is well
known, and it stands to reason that any unclear theoretical
conception of the facts here involved will seriously
affect the correctness of practical calculations.</p>

<p>It may appear astonishing to a person hearing it
for the first time, that a single shot has a double report
due to two different velocities of propagation.
But the reflexion that projectiles whose velocity is less
than the velocity of sound produce no head-waves (because
every impulse imparted to the air travels forward,
that is, ahead, with exactly the velocity of
sound), throws full light when logically developed
upon the peculiar circumstance above mentioned. If
the projectile moves faster than sound, the air ahead
of it cannot recede from it quickly enough. The air
is condensed and warmed, and thereupon, as all know,
the velocity of sound is augmented until the head-wave
travels forward as rapidly as the projectile itself, so
that there is no need whatever of any additional augmentation
of the velocity of propagation. If such a
wave were left entirely to itself, it would increase in
length and soon pass into an ordinary sound-wave,
travelling with less velocity. But the projectile is always
behind it and so maintains it at its proper density
and velocity. Even if the projectile penetrates a
piece of cardboard or a board of wood, which catches<span class="pagenum"><a name="Page_331" id="Page_331">[Pg 331]</a></span>
and obstructs the head-wave, there will, as Figure 56
shows, immediately appear at the emerging apex a
newly formed, not to say newly born, head-wave. We
may observe on the cardboard the reflexion and diffraction
of the head-wave, and by means of a flame
its refraction, so that no doubt as to its nature can remain.</p>

<div class="figright" style="width: 350px;">
<img src="images/i_341.jpg" width="350" height="411" alt="" />
<span class="caption">Fig. 56.</span>
</div>

<p>Permit me, now, to illustrate the most essential of
the points that I have just adduced, by means of a few
rough drawings taken from older and less perfect photographs.</p>

<p>In the sketch of Figure 57 you see the projectile,
which has just left the barrel of the rifle, touch a wire
and disengage the illuminating spark. At the apex of<span class="pagenum"><a name="Page_332" id="Page_332">[Pg 332]</a></span>
the projectile you already see the beginnings of a
powerful head-wave, and in front of the wave a transparent
fungiform cluster. This latter is the air which
has been forced out of the barrel by the projectile.
Circular sound-waves, noise-waves, which are soon
overtaken by the projectile, also issue from the barrel.
But behind the projectile opaque puffs of powder-gas
rush forth. It is scarcely necessary to add that many
other questions in ballistics may be studied by this
method, as, for example, the movement of the gun-carriage.</p>

<div class="figleft" style="width: 450px;">
<img src="images/i_342.jpg" width="450" height="354" alt="" />
<span class="caption">Fig. 57.</span>
</div>

<p>A distinguished French artillerist, M. Gossot, has
applied the views of the head-wave here given in quite
a different manner. The practice in measuring the
velocity of projectiles is to cause the projectile to pass
through wire screens placed at different points in its
path, and by the tearing of these screens to give rise<span class="pagenum"><a name="Page_333" id="Page_333">[Pg 333]</a></span>
to electro-magnetic time-signals on falling slabs or
rotating drums. Gossot caused these signals to be
made directly by the impact of the head-wave, did
away thus with the wire screens, and carried the
method so far as to be able to measure the velocities
of projectiles travelling in high altitudes, where the
use of wire screens was quite out of the question.</p>

<p>The laws of the resistance of fluids and of air to
bodies travelling in them form an extremely complicated
problem, which can be reasoned out very simply
and prettily as a matter of pure philosophy but
practice offers not a few difficulties. The same
body having the velocity 2, 3, 4 ... displaces in the
same interval 2, 3, 4 ... times the same mass of air,
or the same mass of fluid, and imparts to it <i>in addition</i>
2, 3, 4 ... times the same velocity. But for this,
plainly, 4, 9, 16 ... times the original force is required.
Hence, the resistance, it is said, increases
with the square of the velocity. This is all very pretty
and simple and obvious. But practice and theory are
at daggers' points here. Practice tells us that when
we increase the velocity, the law of the resistance
changes. For every portion of the velocity the law is
different.</p>

<p>The studies of the talented English naval architect,
Froude, have thrown light upon this question.
Froude has shown that the resistance is conditioned
by a combination of the most multifarious phenomena.
A ship in motion is subjected to the friction of<span class="pagenum"><a name="Page_334" id="Page_334">[Pg 334]</a></span>
the water. It causes eddies and it generates in addition
waves which radiate outward from it. Every one
of these phenomena are dependent upon the velocity
in some different manner, and it is consequently not
astonishing that the law of the resistance should be a
complicated one.</p>

<p>The preceding observations suggest quite analogous
reflexions for projectiles. Here also we have friction,
the formation of eddies, and the generation of
waves. Here, also, therefore, we should not be surprised
at finding the law of the resistance of the air a
complicated one, nor puzzled at learning that in actuality
the law of resistance changes as soon as the
speed of the projectile exceeds the velocity of sound,
for this is the precise point at which one important
element of the resistance, namely, the formation of
waves, first comes into play.</p>

<p>No one doubts that a pointed bullet pierces the
air with less resistance than a blunt bullet. The
photographs themselves show that the head-wave is
weaker for a pointed projectile. It is not impossible,
similarly, that forms of bullets will be invented which
generate fewer eddies, etc., and that we shall study
these phenomena also by photography. I am of opinion
from the few experiments which I have made in
this direction that not much more can be done by
changing the form of the projectile when the velocity
is very great, but I have not gone into the question
thoroughly. Researches of the kind we are considering<span class="pagenum"><a name="Page_335" id="Page_335">[Pg 335]</a></span>
can certainly not be detrimental to practical artillery,
and it is no less certain that experiments by artillerists
on a large scale will be of undoubted benefit
to physics.</p>

<p>No one who has had the opportunity of studying
modern guns and projectiles in their marvellous perfection,
their power and precision, can help confessing
that a high technical and scientific achievement has
found its incarnation in these objects. We may surrender
ourselves so completely to this impression as
to forget for a moment the terrible purposes they
serve.</p>

<p>Permit me, therefore, before we separate, to say a
few words on this glaring contrast. The greatest man
of war and of silence which the present age has produced
once asserted that perpetual peace is a dream,
and not a beautiful dream at that. We may accord
to this profound student of mankind a judgment in
these matters and can also appreciate the soldier's
horror of stagnation from all too lengthy peace. But
it requires a strong belief in the insuperableness of
mediæval barbarism to hope for and to expect no
great improvement in international relations. Think
of our forefathers and of the times when club law
ruled supreme, when within the same country and the
same state brutal assaults and equally brutal self-defence
were universal and self-evident. This state
of affairs grew so oppressive that finally a thousand
and one circumstances compelled people to put an<span class="pagenum"><a name="Page_336" id="Page_336">[Pg 336]</a></span>
end to it, and the cannon had most to say in accomplishing
the work. Yet the rule of club law was not
abolished so quickly after all. It had simply passed
to other clubs. We must not abandon ourselves to
dreams of the Rousseau type. Questions of law will
in a sense forever remain questions of might. Even
in the United States where every one is as a matter
of principle entitled to the same privileges, the ballot
according to Stallo's pertinent remark is but a milder
substitute for the club. Nor need I tell you that
many of our own fellow-citizens are still enamored of
the old original methods. Very, very gradually, however,
as civilisation progresses, the intercourse of men
takes on gentler forms, and no one who really knows
the good old times will ever honestly wish them back
again, however beautifully they may be painted and
rhymed about.</p>

<p>In the intercourse of the nations, however, the old
club law still reigns supreme. But since its rule is
taxing the intellectual, the moral, and the material resources
of the nations to the utmost and constitutes
scarcely less a burden in peace than in war, scarcely
less a yoke for the victor than for the vanquished, it
must necessarily grow more and more unendurable.
Reason, fortunately, is no longer the exclusive possession
of those who modestly call themselves the
upper ten thousand. Here, as everywhere, the evil
itself will awaken the intellectual and ethical forces
which are destined to mitigate it. Let the hate of<span class="pagenum"><a name="Page_337" id="Page_337">[Pg 337]</a></span>
races and of nationalities run riot as it may, the intercourse
of nations will still increase and grow more intimate.
By the side of the problems which separate
nations, the great and common ideals which claim the
exclusive powers of the men of the future appear one
after another in greater distinctness and in greater
might.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_338" id="Page_338">[Pg 338]</a></span></p>




<h2><a name="ON_INSTRUCTION_IN_THE_CLASSICS" id="ON_INSTRUCTION_IN_THE_CLASSICS">ON INSTRUCTION IN THE CLASSICS
AND THE SCIENCES.</a><a name="FNanchor_113_113" id="FNanchor_113_113"></a><a href="#Footnote_113_113" class="fnanchor">[113]</a></h2>


<p>Perhaps the most fantastic proposition that Maupertuis,<a name="FNanchor_114_114" id="FNanchor_114_114"></a><a href="#Footnote_114_114" class="fnanchor">[114]</a>
the renowned president of the Berlin
Academy, ever put forward for the approval of his
contemporaries was that of founding a city in which,
to instruct and discipline young students, only Latin
should be spoken. Maupertuis's Latin city remained
an idle wish. But for centuries Latin and Greek <i>institutions</i>
exist in which our children spend a goodly
portion of their days, and whose atmosphere constantly
surrounds them, even when without their walls.</p>
<p><span class="pagenum"><a name="Page_339" id="Page_339">[Pg 339]</a></span></p>
<p>For centuries instruction in the ancient languages
has been zealously cultivated. For centuries its necessity
has been alternately championed and contested.
More strongly than ever are authoritative voices now
raised against the preponderance of instruction in the
classics and in favor of an education more suited to
the needs of the time, especially for a more generous
treatment of mathematics and the natural sciences.</p>

<p>In accepting your invitation to speak here on the
relative educational value of the classical and the
mathematico-physical sciences in colleges and high
schools, I find my justification in the duty and the
necessity laid upon every teacher of forming from his
own experiences an opinion upon this important question,
as partly also in the special circumstance that in
my youth I was personally under the influence of
school-life for only a short time, just previous to my
entering the university, and had, therefore, ample opportunity
to observe the effects of widely different
methods upon my own person.</p>

<p>Passing now, to a review of the arguments which
the advocates of instruction in the classics advance,<span class="pagenum"><a name="Page_340" id="Page_340">[Pg 340]</a></span>
and of what the adherents of instruction in the physical
sciences in their turn adduce, we find ourselves in
rather a perplexing position with respect to the arguments
of the first named. For these have been different
at different times, and they are even now of a very
multifarious character, as must be where men advance,
in favor of an institution that exists and which they are
determined to retain at any cost, everything they can
possibly think of. We shall find here much that has
evidently been brought forward only to impress the
minds of the ignorant; much, too, that was advanced
in good faith and which is not wholly without foundation.
We shall get a fair idea of the reasoning employed
by considering, first, the arguments that have grown
out of the historical circumstances connected with the
original introduction of the classics, and, lastly, those
which were subsequently adduced as accidental afterthoughts.</p>
<p><span class="pagenum"><a name="Page_341" id="Page_341">[Pg 341]</a></span></p><p><span class="pagenum"><a name="Page_342" id="Page_342">[Pg 342]</a></span></p><p><span class="pagenum"><a name="Page_343" id="Page_343">[Pg 343]</a></span></p><p><span class="pagenum"><a name="Page_344" id="Page_344">[Pg 344]</a></span></p><p><span class="pagenum"><a name="Page_345" id="Page_345">[Pg 345]</a></span></p><p><span class="pagenum"><a name="Page_346" id="Page_346">[Pg 346]</a></span></p><p><span class="pagenum"><a name="Page_347" id="Page_347">[Pg 347]</a></span></p>
<hr class="tb" />

<p>Instruction in Latin, as Paulsen<a name="FNanchor_115_115" id="FNanchor_115_115"></a><a href="#Footnote_115_115" class="fnanchor">[115]</a> has minutely
shown, was introduced by the Roman Church along
with Christianity. With the Latin language were also
transmitted the scant and meagre remnants of ancient
science. Whoever wished to acquire this ancient education,
then the only one worthy of the name, for him
the Latin language was the only and indispensable
means; such a person had to learn Latin to rank
among educated people.</p>

<p>The wide-spread influence of the Roman Church
wrought many and various results. Among those for
which all are glad, we may safely count the establishment
of a sort of <i>uniformity</i> among the nations and of a
regular international intercourse by means of the Latin
language, which did much to unite the nations in the
common work of civilisation, carried on from the fifteenth
to the eighteenth century. The Latin language
was thus long the language of scholars, and instruction
in Latin the road to a liberal education&mdash;a shibboleth
still employed, though long inappropriate.</p>

<p>For scholars as a class, it is to be regretted, perhaps,
that Latin has ceased to be the medium of international
communication. But the attributing of the
loss of this function by the Latin language to its incapacity
to accommodate itself to the numerous new
ideas and conceptions which have arisen in the course
of the development of science is, in my opinion, wholly
erroneous. It would be difficult to find a modern
scientist who had enriched science with as many new
ideas as Newton has, yet Newton knew how to express
those ideas very correctly and precisely in the
Latin language. If this view were correct, it would
also hold true of every living language. Originally
every language has to adapt itself to new ideas.</p>

<p>It is far more likely that Latin was displaced as
the literary vehicle of science by the influence of the
nobility. By their desire to enjoy the fruits of literature
and science, through a less irksome medium than
Latin, the nobility performed for the people at large
an undeniable service. For the days were now past
when acquaintance with the language and literature of
science was restricted to a caste, and in this step, perhaps,
was made the most important advance of modern
times. To-day, when international intercourse is firmly
established in spite of the many languages employed,
no one would think of reintroducing Latin.<a name="FNanchor_116_116" id="FNanchor_116_116"></a><a href="#Footnote_116_116" class="fnanchor">[116]</a></p>

<p>The facility with which the ancient languages lend
themselves to the expression of new ideas is evidenced
by the fact that the great majority of our scientific
ideas, as survivals of this period of Latin intercourse,
bear Latin and Greek designations, while in great
measure scientific ideas are even now invested with
names from these sources. But to deduce from the
existence and use of such terms the necessity of still
learning Latin and Greek on the part of all who employ
them is carrying the conclusion too far. All terms,
appropriate and inappropriate,&mdash;and there are a large
number of inappropriate and monstrous combinations
in science,&mdash;rest on convention. The essential thing
is, that people should associate with the sign the precise
idea that is designated by it. It matters little
whether a person can correctly derive the words <i>telegraph</i>,
<i>tangent</i>, <i>ellipse</i>, <i>evolute</i>, etc., if the correct idea
is present in his mind when he uses them. On the
other hand, no matter how well he may know their etymology,
his knowledge will be of little use to him if
the correct idea is absent. Ask the average and fairly
educated classical scholar to translate a few lines for
you from Newton's <i>Principia</i>, or from Huygens's <i>Horologium</i>,
and you will discover at once what an extremely
subordinate rôle the mere knowledge of language
plays in such things. Without its associated
thought a word remains a mere sound. The fashion of
employing Greek and Latin designations&mdash;for it can
be termed nothing else&mdash;has a natural root in history;
it is impossible for the practice to disappear suddenly,
but it has fallen of late considerably into disuse. The
terms <i>gas</i>, <i>ohm</i>, <i>Ampère</i>, <i>volt</i>, etc., are in international
use, but they are not Latin nor Greek. Only the person
who rates the unessential and accidental husk
higher than its contents, can speak of the necessity of
learning Latin or Greek for such reasons, to say nothing
of spending eight or ten years on the task. Will
not a dictionary supply in a few seconds all the information
we wish on such subjects?<a name="FNanchor_117_117" id="FNanchor_117_117"></a><a href="#Footnote_117_117" class="fnanchor">[117]</a></p>

<p>It is indisputable that our modern civilisation took
up the threads of the ancient civilisation, that at
many points it begins where the latter left off, and
that centuries ago the remains of the ancient culture
were the only culture existing in Europe. Then, of
course, a classical education really was the liberal education,
the higher education, the ideal education, for
it was the <i>sole</i> education. But when the same claim
is now raised in behalf of a classical education, it must
be uncompromisingly contested as bereft of all foundation.
For our civilisation has gradually attained
its independence; it has lifted itself far above the ancient
civilisation, and has entered generally new directions
of progress. Its note, its characteristic feature,
is the enlightenment that has come from the great
mathematical and physical researches of the last centuries,
and which has permeated not only the practical
arts and industries but is also gradually finding
its way into all fields of thought, including philosophy
and history, sociology and linguistics. Those traces
of ancient views that are still discoverable in philosophy,
law, art, and science, operate more as hindrances
than helps, and will not long stand before the development
of independent and more natural views.</p>

<p>It ill becomes classical scholars, therefore, to regard
themselves, at this day, as the educated class
<i>par excellence</i>, to condemn as uneducated all persons
who do not understand Latin and Greek, to complain
that with such people profitable conversations are not
to be carried on, etc. The most delectable stories
have got into circulation, illustrative of the defective
education of scientists and engineers. A renowned
inquirer, for example, is said to have once announced
his intention of holding a free course of university lectures,
with the word "frustra"; an engineer who spent
his leisure hours in collecting insects is said to have
declared that he was studying "etymology." It is
true, incidents of this character make us shudder or
smile, according to our mood or temperament. But
we must admit, the next moment, that in giving way
to such feelings we have merely succumbed to a childish
prejudice. A lack of tact but certainly no lack of
education is displayed in the use of such half-understood
expressions. Every candid person will confess
that there are many branches of knowledge about which
he had better be silent. We shall not be so uncharitable
as to turn the tables and discuss the impression
that classical scholars might make on a scientist or
engineer, in speaking of science. Possibly many ludicrous
stories might be told of them, and of far more
serious import, which should fully compensate for the
blunders of the other party.</p>

<p>The mutual severity of judgment which we have
here come upon, may also forcibly bring home to us
how really scarce a true liberal culture is. We may
detect in this mutual attitude, too, something of that
narrow, mediæval arrogance of caste, where a man
began, according to the special point of view of the
speaker, with the scholar, the soldier, or the nobleman.
Little sense or appreciation is to be found in it for the
<i>common</i> task of humanity, little feeling for the need of
mutual assistance in the great work of civilisation,
little breadth of mind, little truly liberal culture.</p>

<p>A knowledge of Latin, and partly, also, a knowledge
of Greek, is still a necessity for the members of
a few professions by nature more or less directly concerned
with the civilisations of antiquity, as for lawyers,
theologians, philologists, historians, and generally
for a small number of persons, among whom
from time to time I count myself, who are compelled
to seek for information in the Latin literature of the
centuries just past.<a name="FNanchor_118_118" id="FNanchor_118_118"></a><a href="#Footnote_118_118" class="fnanchor">[118]</a> But that all young persons in
search of a higher education should pursue for this
reason Latin and Greek to such excess; that persons
intending to become physicians and scientists should
come to the universities defectively educated, or even
miseducated; and that they should be compelled to
come only from schools that do <i>not</i> supply them with
the proper preparatory knowledge is going a little bit
too far.</p>
<p><span class="pagenum"><a name="Page_348" id="Page_348">[Pg 348]</a></span></p><p><span class="pagenum"><a name="Page_349" id="Page_349">[Pg 349]</a></span></p><p><span class="pagenum"><a name="Page_350" id="Page_350">[Pg 350]</a></span></p><p><span class="pagenum"><a name="Page_351" id="Page_351">[Pg 351]</a></span></p><p><span class="pagenum"><a name="Page_352" id="Page_352">[Pg 352]</a></span></p>
<hr class="tb" />

<p>After the conditions which had given to the study
of Latin and Greek their high import had ceased to
exist, the traditional curriculum, naturally, was retained.
Then, the different effects of this method of
education, good and bad, which no one had thought of
at its introduction, were realised and noted. As natural,
too, was it that those who had strong interests
in the preservation of these studies, from knowing no
others or from living by them, or for still other reasons,
should emphasise the <i>good</i> results of such instruction.
They pointed to the good effects as if they
had been consciously aimed at by the method and could
be attained only through its agency.</p>

<p>One real benefit that students might derive from
a rightly conducted course in the classics would be
the opening up of the rich literary treasures of antiquity,
and intimacy with the conceptions and views
of the world held by two advanced nations. A person
who has read and understood the Greek and Roman
authors has felt and experienced more than one who is
restricted to the impressions of the present. He sees
how men placed in different circumstances judge quite
differently of the same things from what we do to-day.
His own judgments will be rendered thus more independent.
Again, the Greek and Latin authors are indisputably
a rich fountain of recreation, of enlightenment,
and of intellectual pleasure after the day's toil, and
the individual, not less than civilised humanity generally,
will remain grateful to them for all time. Who
does not recall with pleasure the wanderings of Ulysses,
who does not listen joyfully to the simple narratives
of Herodotus, who would ever repent of having
made the acquaintance of Plato's Dialogues, or of
having tasted Lucian's divine humor? Who would
give up the glances he has obtained into the private
life of antiquity from Cicero's letters, from Plautus or
Terence? To whom are not the portraits of Suetonius
undying reminiscences? Who, in fact, would throw
away <i>any</i> knowledge he had once gained?</p>

<p>Yet people who draw from these sources only, who
know only this culture, have surely no right to dogmatise
about the value of some other culture. As objects
of research for individuals, this literature is extremely
valuable, but it is a different question whether
it is equally valuable as the almost exclusive means of
education of our youth.</p>

<p>Do not other nations and other literatures exist
from which we ought to learn? Is not nature herself
our first school-mistress? Are our highest models always
to be the Greeks, with their narrow provinciality
of mind, that divided the world into "Greeks and barbarians,"
with their superstitions, with their eternal
questioning of oracles? Aristotle with his incapacity
to learn from facts, with his word-science; Plato with
his heavy, interminable dialogues, with his barren, at
times childish, dialectics&mdash;are they unsurpassable?<a name="FNanchor_119_119" id="FNanchor_119_119"></a><a href="#Footnote_119_119" class="fnanchor">[119]</a>
The Romans with their apathy, their pompous externality,
set off by fulsome and bombastic phrases, with
their narrow-minded, philistine philosophy, with their
frenzied sensuality, with their cruel and bestial indulgence
in animal and man baiting, with their outrageous
maltreatment and plundering of their subjects&mdash;are
they patterns worthy of imitation? Or shall, perhaps,
our science edify itself with the works of Pliny who
cites midwives as authorities and himself stands on
their point of view?</p>

<p>Besides, if an acquaintance with the ancient world
really were attained, we might come to some settlement
with the advocates of classical education. But it
is words and forms, and forms and words only, that
are supplied to our youth; and even collateral subjects
are forced into the strait-jacket of the same
rigid method and made a science of words, sheer feats
of mechanical memory. Really, we feel ourselves set
back a thousand years into the dull cloister-cells of the
Middle Ages.</p>

<p>This must be changed. It is possible to get acquainted
with the views of the Greeks and Romans by
a shorter road than the intellect deadening process
of eight or ten years of declining, conjugating, analysing,
and extemporisation. There are to-day plenty of
educated persons who have acquired through good
translations vivider, clearer, and more just views of
classical antiquity than the graduates of our gymnasiums
and colleges.<a name="FNanchor_120_120" id="FNanchor_120_120"></a><a href="#Footnote_120_120" class="fnanchor">[120]</a></p>

<p>For us moderns, the Greeks and the Romans are
simply two objects of archæological and historical research
like all others. If we put them before our
youth in fresh and living pictures, and not merely in
words and syllables, the effect will be assured. We
derive a totally different enjoyment from the Greeks
when we approach them after a study of the results
of modern research in the history of civilisation. We
read many a chapter of Herodotus differently when we
attack his works equipped with a knowledge of natural
science, and with information about the stone age and
the lake-dwellers. What our classical institutions <i>pretend</i>
to give can and actually will be given to our youth
with much more fruitful results by competent <i>historical</i>
instruction, which must supply, not names and numbers
alone, nor the mere history of dynasties and wars,
but be in every sense of the word a true history of
civilisation.</p>

<p>The view still widely prevails that although all
"higher, ideal culture," all extension of our view of
the world, is acquired by philological and in a lesser
degree by historical studies, still the mathematics and
natural sciences should not be neglected on account
of their usefulness. This is an opinion to which I must
refuse my assent. It were strange if man could learn
more, could draw more intellectual nourishment, from
the shards of a few old broken jugs, from inscribed
stones, or yellow parchments, than from all the rest
of nature. True, man is man's first concern, but he
is not his sole concern.</p>

<p>In ceasing to regard man as the centre of the world;
in discovering that the earth is a top whirled about
the sun, which speeds off with it into infinite space;
in finding that in the fixed stars the same elements
exist as on earth; in meeting everywhere the same
processes of which the life of man is merely a vanishingly
small part&mdash;in such things, too, is a widening of
our view of the world, and edification, and poetry.
There are here perhaps grander and more significant
facts than the bellowing of the wounded Ares, or the
charming island of Calypso, or the ocean-stream engirdling
the earth. He only should speak of the relative
value of these two domains of thought, of their
poetry, who knows both.</p>

<p>The "utility" of physical science is, in a measure,
only a <i>collateral</i> product of that flight of the intellect
which produced science. No one, however, should
underrate the utility of science who has shared in the
realisation by modern industrial art of the Oriental
world of fables, much less one upon whom those treasures
have been poured, as it were, from the fourth dimension,
without his aid or understanding.</p>

<p>Nor may we believe that science is useful only to
the practical man. Its influence permeates all our affairs,
our whole life; everywhere its ideas are decisive.
How differently does the jurist, the legislator, or the
political economist think, who knows, for example,
that a square mile of the most fertile soil can support
with the solar heat annually consumed only a definite
number of human beings, which no art or science can
increase. Many economical theories, which open new
air-paths of progress, air-paths in the literal sense of
the word, would be made impossible by such knowledge.</p>
<p><span class="pagenum"><a name="Page_353" id="Page_353">[Pg 353]</a></span></p><p><span class="pagenum"><a name="Page_354" id="Page_354">[Pg 354]</a></span></p>
<hr class="tb" />

<p>The eulogists of classical education love to emphasise
the cultivation of taste which comes from employment
with the ancient models. I candidly confess
that there is something absolutely revolting in this to
me. To form the taste, then, our youths must sacrifice
ten years of their life! Luxury takes precedence over
necessity. Have the future generations, in the face
of the difficult problems, the great social questions,
which they must meet, and that with strengthened
mind and heart, no more important duties to fulfil than
these?</p>

<p>But let us assume that this end were desirable.
Can taste be formed by rules and precepts? Do not
ideals of beauty change? Is it not a stupendous absurdity
to force one's self artificially to admire things
which, with all their historical interest, with all their
beauty in individual points, are for the most part
foreign to the rest of our thoughts and feelings, provided
we have such of <i>our own</i>. A nation that is
truly such, has its own taste and will not go to others
for it. And every individual perfect man has his own
taste.<a name="FNanchor_121_121" id="FNanchor_121_121"></a><a href="#Footnote_121_121" class="fnanchor">[121]</a></p>

<p>And what, after all, does this cultivation of taste
consist in? In the acquisition of the personal literary
style of a few select authors! What should we think
of a people that would force its youth a thousand
years from now, by years of practice, to master the
tortuous or bombastic style of some successful lawyer
or politician of to-day? Should we not justly accuse
them of a woful lack of taste?</p>

<p>The evil effects of this imagined cultivation of the
taste find expression often enough. The young <i>savant</i>
who regards the composition of a scientific essay as a
rhetorical exercise instead of a simple and unadorned
presentation of the facts and the truth, still sits unconsciously
on the school-bench, and still unwittingly represents
the point of view of the Romans, by whom the
elaboration of speeches was regarded as a serious scientific (!)
employment.</p>
<p><span class="pagenum"><a name="Page_355" id="Page_355">[Pg 355]</a></span></p><p><span class="pagenum"><a name="Page_356" id="Page_356">[Pg 356]</a></span></p><p><span class="pagenum"><a name="Page_357" id="Page_357">[Pg 357]</a></span></p>
<hr class="tb" />

<p>Far be it from me to underrate the value of the development
of the instinct of speech and of the increased
comprehension of our own language which comes from
philological studies. By the study of a foreign language,
especially of one which differs widely from ours,
the signs and forms of words are first clearly distinguished
from the thoughts which they express. Words
of the closest possible correspondence in different languages
never coincide absolutely with the ideas they
stand for, but place in relief slightly different aspects
of the same thing, and by the study of language the
attention is directed to these shades of difference. But
it would be far from admissible to contend that the
study of Latin and Greek is the most fruitful and natural,
let alone the <i>only</i>, means of attaining this end.
Any one who will give himself the pleasure of a few
hours' companionship with a Chinese grammar; who
will seek to make clear to himself the mode of speech
and thought of a people who never advanced as far as
the analysis of articulate sounds, but stopped at the
analysis of syllables, to whom our alphabetical characters,
therefore, are an inexplicable puzzle, and who
express all their rich and profound thoughts by means
of a few syllables with variable emphasis and position,&mdash;such
a person, perhaps, will acquire new, and extremely
elucidative ideas upon the relation of language
and thought. But should our children, therefore,
study Chinese? Certainly not. No more, then,
should they be burdened with Latin, at least in the
measure they are.</p>

<p>It is a beautiful achievement to reproduce a Latin
thought in a modern language with the maximum fidelity
of meaning and expression&mdash;for the <i>translator</i>.
Moreover, we shall be very grateful to the translator
for his performance. But to demand this feat of every
educated man, without consideration of the sacrifice of
time and labor which it entails, is unreasonable. And
for this very reason, as classical teachers admit, that
ideal is never perfectly attained, except in rare cases
with scholars possessed of special talents and great
industry. Without slurring, therefore, the high importance
of the study of the ancient languages as a
profession, we may yet feel sure that the instinct for
speech which is part of every liberal education can,
and must, be acquired in a different way. Should we,
indeed, be forever lost if the Greeks had not lived before
us?</p>

<p>The fact is, we must carry our demands further
than the representatives of classical philology. We
must ask of every educated man a fair scientific conception
of the nature and value of language, of the
formation of language, of the alteration of the meaning
of roots, of the degeneration of fixed forms of
speech to grammatical forms, in brief, of all the main
results of modern comparative philology. We should
judge that this were attainable by a careful study of
our mother tongue and of the languages next allied to
it, and subsequently of the more ancient tongues from
which the former are derived. If any one object that
this is too difficult and entails too much labor, I should
advise such a person to place side by side an English,
a Dutch, a Danish, a Swedish, and a German Bible, and
to compare a few lines of them; he will be amazed at
the multitude of suggestions that offer themselves.<a name="FNanchor_122_122" id="FNanchor_122_122"></a><a href="#Footnote_122_122" class="fnanchor">[122]</a>
In fact, I believe that a really progressive, fruitful, rational,
and instructive study of languages can be conducted
only on this plan. Many of my audience will
remember, perhaps, the bright and encouraging effect,
like that of a ray of sunlight on a gloomy day, which
the meagre and furtive remarks on comparative philology
in Curtius's Greek grammar wrought in that
barren and lifeless desert of verbal quibbles.</p>
<p><span class="pagenum"><a name="Page_358" id="Page_358">[Pg 358]</a></span></p><p><span class="pagenum"><a name="Page_359" id="Page_359">[Pg 359]</a></span></p>
<hr class="tb" />

<p>The principal result obtained by the present method
of studying the ancient languages is that which comes
from the student's employment with their complicated
grammars. It consists in the sharpening of the attention
and in the exercise of the judgment by the practice
of subsuming special cases under general rules,
and of distinguishing between different cases. Obviously,
the same result can be reached by many other
methods; for example, by difficult games of cards.
Every science, the mathematics and the physical sciences
included, accomplish as much, if not more, in
this disciplining of the judgment. In addition, the
matter treated by those sciences has a much higher intrinsic
interest for young people, and so engages spontaneously
their attention; while on the other hand they
are elucidative and useful in other directions in which
grammar can accomplish nothing.</p>

<p>Who cares, so far as the matter of it is concerned,
whether we say <i>hominum</i> or <i>hominorum</i> in the genitive
plural, interesting as the fact may be for the philologist?
And who would dispute that the intellectual
need of causal insight is awakened not by grammar
but by the natural sciences?</p>

<p>It is not our intention, therefore, to gainsay in the
least the good influence which the study of Latin and
Greek grammar <i>also</i> exercises on the sharpening of the
judgment. In so far as the study of words as such
must greatly promote lucidity and accuracy of expression,
in so far as Latin and Greek are not yet
wholly indispensable to many branches of knowledge,
we willingly concede to them a place in our schools,
but would demand that the disproportionate amount of
time allotted to them, wrongly withdrawn from other
useful studies, should be considerably curtailed. That
in the end Latin and Greek will not be employed as
the universal means of education, we are fully convinced.
They will be relegated to the closet of the
scholar or professional philologist, and gradually make
way for the modern languages and the modern science
of language.</p>

<p>Long ago Locke reduced to their proper limits the
exaggerated notions which obtained of the close connexion
of thought and speech, of logic and grammar,
and recent investigators have established on still surer
foundations his views. How little a complicated grammar
is necessary for expressing delicate shades of
thought is demonstrated by the Italians and French,
who, although they have almost totally discarded the
grammatical redundancies of the Romans, are yet not
surpassed by the latter in accuracy of thought, and
whose poetical, but especially whose scientific literature,
as no one will dispute, can bear favorable comparison
with the Roman.</p>

<p>Reviewing again the arguments advanced in favor
of the study of the ancient languages, we are obliged
to say that in the main and as applied to the present,
they are wholly devoid of force. In so far as the
aims which this study theoretically pursues are still
worthy of attainment, they appear to us as altogether
too narrow, and are surpassed in this only by the
means employed. As almost the sole, indisputable result
of this study we must count the increase of the
student's skill and precision in expression. One inclined
to be uncharitable might say that our gymnasiums
and classical academies turn out men who can
speak and write, but, unfortunately, have little to write
or speak about. Of that broad, liberal view, of that
famed universal culture, which the classical curriculum
is supposed to yield, serious words need not be lost.
This culture might, perhaps, more properly be termed
the contracted or lopsided culture.</p>
<p><span class="pagenum"><a name="Page_360" id="Page_360">[Pg 360]</a></span></p><p><span class="pagenum"><a name="Page_361" id="Page_361">[Pg 361]</a></span></p><p><span class="pagenum"><a name="Page_362" id="Page_362">[Pg 362]</a></span><span class="pagenum"><a name="Page_363" id="Page_363">[Pg 363]</a></span></p><p><span class="pagenum"><a name="Page_364" id="Page_364">[Pg 364]</a></span></p><p><span class="pagenum"><a name="Page_365" id="Page_365">[Pg 365]</a></span></p><p><span class="pagenum"><a name="Page_366" id="Page_366">[Pg 366]</a></span></p><p><span class="pagenum"><a name="Page_367" id="Page_367">[Pg 367]</a></span></p><p><span class="pagenum"><a name="Page_368" id="Page_368">[Pg 368]</a></span></p><p><span class="pagenum"><a name="Page_369" id="Page_369">[Pg 369]</a></span></p><p><span class="pagenum"><a name="Page_370" id="Page_370">[Pg 370]</a></span><span class="pagenum"><a name="Page_371" id="Page_371">[Pg 371]</a></span></p>
<hr class="tb" />

<p>While considering the study of languages we threw
a few side glances at mathematics and the natural sciences.
Let us now inquire whether these, as branches
of study, cannot accomplish much that is to be attained
in no other way. I shall meet with no contradiction
when I say that without at least an elementary mathematical
and scientific education a man remains a total
stranger in the world in which he lives, a stranger in
the civilisation of the time that bears him. Whatever
he meets in nature, or in the industrial world, either
does not appeal to him at all, from his having neither
eye nor ear for it, or it speaks to him in a totally unintelligible
language.</p>

<p>A real understanding of the world and its civilisation,
however, is not the only result of the study of
mathematics and the physical sciences. Much more
essential for the preparatory school is the <i>formal</i> cultivation
which comes from these studies, the strengthening
of the reason and the judgment, the exercise
of the imagination. Mathematics, physics, chemistry,
and the so-called descriptive sciences are so much
alike in this respect, that, apart from a few points, we
need not separate them in our discussion.</p>

<p>Logical sequence and continuity of ideas, so necessary
for fruitful thought, are <i>par excellence</i> the results of
mathematics; the ability to follow facts with thoughts,
that is, to observe or collect experiences, is chiefly developed
by the natural sciences. Whether we notice
that the sides and the angles of a triangle are connected
in a definite way, that an equilateral triangle possesses
certain definite properties of symmetry, or whether we
notice the deflexion of a magnetic needle by an electric
current, the dissolution of zinc in diluted sulphuric
acid, whether we remark that the wings of a butterfly
are slightly colored on the under, and the fore-wings
of the moth on the upper, surface: indiscriminately
here we proceed from <i>observations</i>, from individual
acts of immediate intuitive knowledge. The field of
observation is more restricted and lies closer at hand
in mathematics; it is more varied and broader but
more difficult to compass in the natural sciences. The
essential thing, however, is for the student to learn to
make observations in all these fields. The philosophical
question whether our acts of knowledge in mathematics
are of a special kind is here of no importance
for us. It is true, of course, that the observation can
be practised by languages also. But no one, surely,
will deny, that the concrete, living pictures presented
in the fields just mentioned possess different
and more powerful attractions for the mind of the
youth than the abstract and hazy figures which language
offers, and on which the attention is certainly not
so spontaneously bestowed, nor with such good results.<a name="FNanchor_123_123" id="FNanchor_123_123"></a><a href="#Footnote_123_123" class="fnanchor">[123]</a></p>

<p>Observation having revealed the different properties
of a given geometrical or physical object, it is discovered
that in many cases these properties <i>depend</i> in
some way upon one another. This interdependence
of properties (say that of equal sides and equal angles
at the base of a triangle, the relation of pressure to
motion,) is nowhere so distinctly marked, nowhere is
the necessity and permanency of the interdependence
so plainly noticeable, as in the fields mentioned.
Hence the continuity and logical consequence of the
ideas which we acquire in those fields. The relative
simplicity and perspicuity of geometrical and physical
relations supply here the conditions of natural and
easy progress. Relations of equal simplicity are not
met with in the fields which the study of language
opens up. Many of you, doubtless, have often wondered
at the little respect for the notions of cause and
effect and their connexion that is sometimes found
among professed representatives of the classical studies.
The explanation is probably to be sought in the
fact that the analogous relation of motive and action
familiar to them from their studies, presents nothing
like the clear simplicity and determinateness that the
relation of cause and effect does.</p>

<p>That perfect mental grasp of all possible cases,
that economical order and organic union of the thoughts
which comes from it, which has grown for every one
who has ever tasted it a permanent need which he
seeks to satisfy in every new province, can be developed
only by employment with the relative simplicity of
mathematical and scientific investigations.</p>

<p>When a set of facts comes into apparent conflict
with another set of facts, and a problem is presented,
its solution consists ordinarily in a more refined distinction
or in a more extended view of the facts, as
may be aptly illustrated by Newton's solution of the
problem of dispersion. When a new mathematical or
scientific fact is <i>demonstrated</i>, or <i>explained</i>, such demonstration
also rests simply upon showing the connexion
of the new fact with the facts already known; for
example, that the radius of a circle can be laid off as
chord exactly six times in the circle is explained or
proved by dividing the regular hexagon inscribed in
the circle into equilateral triangles. That the quantity
of heat developed in a second in a wire conveying an
electric current is quadrupled on the doubling of the
strength of the current, we explain from the doubling of
the fall of the potential due to the doubling of the
current's intensity, as also from the doubling of the
quantity flowing through, in a word, from the quadrupling
of the work done. In point of principle, explanation
and direct proof do not differ much.</p>

<p>He who solves scientifically a geometrical, physical,
or technical problem, easily remarks that his
procedure is a <i>methodical</i> mental quest, rendered possible
by the economical order of the province&mdash;a simplified
purposeful quest as contrasted with unmethodical,
unscientific guess-work. The geometer, for example,
who has to construct a circle touching two given
straight lines, casts his eye over the relations of symmetry
of the desired construction, and seeks the centre
of his circle solely in the line of symmetry of the two
straight lines. The person who wants a triangle of
which two angles and the sum of the sides are given,
grasps in his mind the determinateness of the form of
this triangle and restricts his search for it to a certain
group of triangles of the <i>same form</i>. Under very different
circumstances, therefore, the simplicity, the intellectual
perviousness, of the subject-matter of mathematics
and natural science is felt, and promotes both
the discipline and the self-confidence of the reason.</p>

<p>Unquestionably, much more will be attained by instruction
in the mathematics and the natural sciences
than now is, when more natural methods are adopted.
One point of importance here is that young students
should not be spoiled by premature abstraction, but
should be made acquainted with their material from
living pictures of it before they are made to work with
it by purely ratiocinative methods. A good stock of
geometrical experience could be obtained, for example,
from geometrical drawing and from the practical
construction of models. In the place of the unfruitful
method of Euclid, which is only fit for special, restricted
uses, a broader and more conscious method
must be adopted, as Hankel has pointed out.<a name="FNanchor_124_124" id="FNanchor_124_124"></a><a href="#Footnote_124_124" class="fnanchor">[124]</a> Then,
if, on reviewing geometry, and after it presents no
substantial difficulties, the more general points of view,
the principles of scientific method are placed in relief
and brought to consciousness, as Von Nagel,<a name="FNanchor_125_125" id="FNanchor_125_125"></a><a href="#Footnote_125_125" class="fnanchor">[125]</a> J. K.
Becker,<a name="FNanchor_126_126" id="FNanchor_126_126"></a><a href="#Footnote_126_126" class="fnanchor">[126]</a> Mann,<a name="FNanchor_127_127" id="FNanchor_127_127"></a><a href="#Footnote_127_127" class="fnanchor">[127]</a> and others have well done, fruitful
results will be surely attained. In the same way,
the subject-matter of the natural sciences should be
made familiar by pictures and experiment before a
profounder and reasoned grasp of these subjects is
attempted. Here the emphasis of the more general
points of view is to be postponed.</p>

<p>Before my present audience it would be superfluous
for me to contend further that mathematics and natural
science are justified constituents of a sound education,&mdash;a
claim that even philologists, after some
resistance, have conceded. Here I may count upon
assent when I say that mathematics and the natural
sciences pursued alone as means of instruction yield a
richer education in matter and form, a more general
education, an education better adapted to the needs
and spirit of the time,&mdash;than the philological branches
pursued alone would yield.</p>

<p>But how shall this idea be realised in the curricula
of our intermediate educational institutions? It is unquestionable
in my mind that the German <i>Realschulen</i>
and <i>Realgymnasien</i>, where the exclusive classical course
is for the most part replaced by mathematics, science,
and modern languages, give the <i>average</i> man a more
timely education than the gymnasium proper, although
they are not yet regarded as fit preparatory schools for
future theologians and professional philologists. The
German gymnasiums are too one-sided. With these
the first changes are to be made; of these alone we
shall speak here. Possibly a <i>single</i> preparatory school,
suitably planned, might serve all purposes.</p>

<p>Shall we, then, in our gymnasiums fill out the hours
of study which stand at our disposal, or are still to be
wrested from the classicists, with as great and as varied
a quantity of mathematical and scientific matter
as possible? Expect no such proposition from me.
No one will suggest such a course who has himself
been actively engaged in scientific thought. Thoughts
can be awakened and fructified as a field is fructified
by sunshine and rain. But thoughts cannot be juggled
out and worried out by heaping up materials and
the hours of instruction, nor by any sort of precepts:
they must grow naturally of their own free accord.
Furthermore, thoughts cannot be accumulated beyond
a certain limit in a single head, any more than the produce
of a field can be increased beyond certain limits.</p>

<p>I believe that the amount of matter necessary for a
useful education, such as should be offered to <i>all</i> the
pupils of a preparatory school, is very small. If I had
the requisite influence, I should, in all composure, and
fully convinced that I was doing what was best, first
greatly curtail in the lower classes the amount of matter
in both the classical and the scientific courses; I
should cut down considerably the number of the school
hours and the work done outside the school. I am
not with many teachers of opinion that ten hours work
a day for a child is not too much. I am convinced
that the mature men who offer this advice so lightly
are themselves unable to give their attention successfully
for as long a time to any subject that is new to
them, (for example, to elementary mathematics or
physics,) and I would ask every one who thinks the
contrary to make the experiment upon himself. Learning
and teaching are not routine office-work that can
be kept up mechanically for long periods. But even
such work tires in the end. If our young men are
not to enter the universities with blunted and impoverished
minds, if they are not to leave in the preparatory
schools their vital energy, which they should
there gather, great changes must be made. Waiving
the injurious effects of overwork upon the body, the
consequences of it for the mind seem to me positively
dreadful.</p>

<p>I know of nothing more terrible than the poor creatures
who have learned too much. Instead of that
sound powerful judgment which would probably have
grown up if they had learned nothing, their thoughts
creep timidly and hypnotically after words, principles,
and formulæ, constantly by the same paths. What
they have acquired is a spider's web of thoughts too
weak to furnish sure supports, but complicated enough
to produce confusion.</p>

<p>But how shall better methods of mathematical and
scientific education be combined with the decrease of
the subject-matter of instruction? I think, by abandoning
systematic instruction altogether, at least in so
far as that is required of <i>all</i> young pupils. I see no
necessity whatever that the graduates of our high
schools and preparatory schools should be little philologists,
and at the same time little mathematicians,
physicists, and botanists; in fact, I do not see the possibility
of such a result. I see in the endeavor to attain
this result, in which every instructor seeks for his
own branch a place apart from the others, the main
mistake of our whole system. I should be satisfied if
every young student could come into living contact
with and pursue to their ultimate logical consequences
merely a <i>few</i> mathematical or scientific discoveries.
Such instruction would be mainly and naturally associated
with selections from the great scientific classics.
A few powerful and lucid ideas could thus be made
to take root in the mind and receive thorough elaboration.
This accomplished, our youth would make a
different showing from what they do to-day.<a name="FNanchor_128_128" id="FNanchor_128_128"></a><a href="#Footnote_128_128" class="fnanchor">[128]</a></p>

<p>What need is there, for example, of burdening the
head of a young student with all the details of botany?
The student who has botanised under the guidance of
a teacher finds on all hands, not indifferent things, but
known or unknown things, by which he is stimulated,
and his gain made permanent. I express here, not my
own, but the opinion of a friend, a practical teacher.
Again, it is not at all necessary that all the matter that
is offered in the schools should be learned. The best
that we have learned, that which has remained with
us for life, outlived the test of examination. How can
the mind thrive when matter is heaped on matter, and
new materials piled constantly on old, undigested materials?
The question here is not so much that of the
accumulation of positive knowledge as of intellectual
discipline. It seems also unnecessary that <i>all</i> branches
should be treated at school, and that exactly the same
studies should be pursued in all schools. A single
philological, a single historical, a single mathematical,
a single scientific branch, pursued as common subjects
of instruction for all pupils, are sufficient to accomplish
all that is necessary for the intellectual development.
On the other hand, a wholesome mutual stimulus
would be produced by this greater variety in the
positive culture of men. Uniforms are excellent for
soldiers, but they will not fit heads. Charles V. learned
this, and it should never be forgotten. On the contrary,
teachers and pupils both need considerable latitude, if
they are to yield good results.</p>

<p>With John Karl Becker I am of the opinion that
the utility and amount for individuals of every study
should be precisely determined. All that exceeds this
amount should be unconditionally banished from the
lower classes. With respect to mathematics, Becker,<a name="FNanchor_129_129" id="FNanchor_129_129"></a><a href="#Footnote_129_129" class="fnanchor">[129]</a>
in my judgment, has admirably solved this question.</p>

<p>With respect to the upper classes the demand assumes
a different form. Here also the amount of matter
obligatory on all pupils ought not to exceed a certain
limit. But in the great mass of knowledge that a
young man must acquire to-day for his profession it is
no longer just that ten years of his youth should be
wasted with mere preludes. The upper classes should
supply a truly useful preparation for the professions,
and should not be modelled upon the wants merely of
future lawyers, ministers, and philologists. Again, it
would be both foolish and impossible to attempt to
prepare the same person properly for all the different
professions. In such case the function of the schools
would be, as Lichtenberg feared, simply to select the
persons best fitted for being drilled, whilst precisely the
finest special talents, which do not submit to indiscriminate
discipline, would be excluded from the contest.
Hence, a certain amount of liberty in the choice
of studies must be introduced in the upper classes, by
means of which it will be free for every one who is clear
about the choice of his profession to devote his chief
attention either to the study of the philologico-historical
or to that of the mathematico-scientific branches.
Then the matter now treated could be retained, and in
some branches, perhaps, judiciously extended,<a name="FNanchor_130_130" id="FNanchor_130_130"></a><a href="#Footnote_130_130" class="fnanchor">[130]</a> without
burdening the scholar with many branches or increasing
the number of the hours of study. With more
homogeneous work the student's capacity for work increases,
one part of his labor supporting the other
instead of obstructing it. If, however, a young man
should subsequently choose a different profession, then
it is <i>his</i> business to make up what he has lost. No
harm certainly will come to society from this change,
nor could it be regarded as a misfortune if philologists
and lawyers with mathematical educations or physical
scientists with classical educations should now and
then appear.</p>
<p><span class="pagenum"><a name="Page_372" id="Page_372">[Pg 372]</a></span></p><p><span class="pagenum"><a name="Page_373" id="Page_373">[Pg 373]</a></span><span class="pagenum"><a name="Page_374" id="Page_374">[Pg 374]</a></span></p><p><span class="pagenum"><a name="Page_375" id="Page_375">[Pg 375]</a></span></p>
<hr class="tb" />

<p>The view is now wide-spread that a Latin and
Greek education no longer meets the general wants of
the times, that a more opportune, a more "liberal"
education exists. The phrase, "a liberal education,"
has been greatly misused. A truly liberal education is
unquestionably very rare. The <i>schools</i> can hardly offer
such; at best they can only bring home to the student
the necessity of it. It is, then, his business to acquire,
as best he can, a more or less liberal education. It
would be very difficult, too, at any one time to give a
definition of a "liberal" education which would satisfy
every one, still more difficult to give one which would
hold good for a hundred years. The educational
ideal, in fact, varies much. To one, a knowledge of
classical antiquity appears not too dearly bought "with
early death." We have no objection to this person,
or to those who think like him, pursuing their ideal
after their own fashion. But we may certainly protest
strongly against the realisation of such ideals on our
own children. Another,&mdash;Plato, for example,&mdash;puts
men ignorant of geometry on a level with animals.<a name="FNanchor_131_131" id="FNanchor_131_131"></a><a href="#Footnote_131_131" class="fnanchor">[131]</a>
If such narrow views had the magical powers of the
sorceress Circe, many a man who perhaps justly
thought himself well educated would become conscious
of a not very flattering transformation of himself.
Let us seek, therefore, in our educational system
to meet the wants of the present, and not establish
prejudices for the future.</p>

<p>But how does it come, we must ask, that institutions
so antiquated as the German gymnasiums could
subsist so long in opposition to public opinion? The
answer is simple. The schools were first organised by
the Church; since the Reformation they have been in
the hands of the State. On so large a scale, the plan
presents many advantages. Means can be placed at
the disposal of education such as no private source, at
least in Europe, could furnish. Work can be conducted
upon the same plan in many schools, and so
experiments made of extensive scope which would be
otherwise impossible. A single man with influence
and ideas can under such circumstances do great
things for the promotion of education.</p>

<p>But the matter has also its reverse aspect. The
party in power works for its own interests, uses the
schools for its special purposes. Educational competition
is excluded, for all successful attempts at improvement
are impossible unless undertaken or permitted
by the State. By the uniformity of the people's
education, a prejudice once in vogue is permanently
established. The highest intelligences, the strongest
wills cannot overthrow it suddenly. In fact, as everything
is adapted to the view in question, a sudden
change would be physically impossible. The two
classes which virtually hold the reins of power in the
State, the jurists and theologians, know only the one-sided,
predominantly classical culture which they have
acquired in the State schools, and would have this culture
alone valued. Others accept this opinion from
credulity; others, underestimating their true worth for
society, bow before the power of the prevalent opinion;
others, again, affect the opinion of the ruling
classes even against their better judgment, so as to
abide on the same plane of respect with the latter. I
will make no charges, but I must confess that the deportment
of medical men with respect to the question
of the qualification of graduates of your <i>Realschulen</i>
has frequently made that impression upon me. Let
us remember, finally, that an influential statesman,
even within the boundaries which the law and public
opinion set him, can do serious harm to the cause
of education by considering his own one-sided views
infallible, and in enforcing them recklessly and inconsiderately&mdash;which
not only <i>can</i> happen, but has, repeatedly,
happened.<a name="FNanchor_132_132" id="FNanchor_132_132"></a><a href="#Footnote_132_132" class="fnanchor">[132]</a> The monopoly of education by
the State<a name="FNanchor_133_133" id="FNanchor_133_133"></a><a href="#Footnote_133_133" class="fnanchor">[133]</a> thus assumes in our eyes a somewhat different
aspect. And to revert to the question above asked,
there is not the slightest doubt that the German gymnasiums
in their present form would have ceased to
exist long ago if the State had not supported them.</p>

<p>All this must be changed. But the change will
not be made of itself, nor without our energetic interference,
and it will be made slowly. But the path is
marked out for us, the will of the people must acquire
and exert upon our school legislation a greater and
more powerful influence. Furthermore, the questions
at issue must be publicly and candidly discussed that
the views of the people may be clarified. All who feel
the insufficiency of the existing <i>régime</i> must combine
into a powerful organisation that their views may
acquire impressiveness and the opinions of the individual
not die away unheard.</p>

<p>I recently read, gentlemen, in an excellent book of
travels, that the Chinese speak with unwillingness of
politics. Conversations of this sort are usually cut
short with the remark that they may bother about such
things whose business it is and who are paid for it.
Now it seems to me that it is not only the business of
the State, but a very serious concern of all of us, how
our children shall be educated in the public schools
at <i>our</i> cost.</p>




<h2><a name="APPENDIX" id="APPENDIX">APPENDIX.</a></h2>

<p class="center">I.</p>

<h3><a id="A_CONTRIBUTION_TO_THE_HISTORY_OF_ACOUSTICS"></a>A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.<a name="FNanchor_134_134" id="FNanchor_134_134"></a><a href="#Footnote_134_134" class="fnanchor">[134]</a></h3>


<p>While searching for papers by Amontons, several
volumes of the Memoirs of the Paris Academy
for the first years of the eighteenth century, fell
into my hands. It is difficult to portray the delight
which one experiences in running over the leaves of
these volumes. One sees as an actual spectator almost
the rise of the most important discoveries and witnesses
the progress of many fields of knowledge from
almost total ignorance to relatively perfect clearness.</p>

<p>I propose to discuss here the fundamental researches
of Sauveur in Acoustics. It is astonishing
how extraordinarily near Sauveur was to the view
which Helmholtz was the first to adopt in its full extent
a hundred and fifty years later.</p>

<p>The <i>Histoire de l'Académie</i> for 1700, p. 131, tells
us that Sauveur had succeeded in making music an<span class="pagenum"><a name="Page_376" id="Page_376">[Pg 376]</a></span>
object of scientific research, and that he had invested
the new science with the name of "acoustics." On
five successive pages a number of discoveries are recorded
which are more fully discussed in the volume
for the year following.</p>

<p>Sauveur regards the <i>simplicity</i> of the ratios obtaining
between the rates of vibration of consonances as
something universally known.<a name="FNanchor_135_135" id="FNanchor_135_135"></a><a href="#Footnote_135_135" class="fnanchor">[135]</a> He is in hope, by
further research, of determining the chief rules of musical
composition and of fathoming the "metaphysics
of the agreeable," the main law of which he asserts
to be the union of "simplicity with multiplicity."
Precisely as Euler<a name="FNanchor_136_136" id="FNanchor_136_136"></a><a href="#Footnote_136_136" class="fnanchor">[136]</a> did a number of years later, he
regards a consonance as more perfect according as
the ratio of its vibrational rates is expressed in smaller
whole numbers, because the smaller these whole numbers
are the oftener the vibrations of the two tones
coincide, and hence the more readily they are apprehended.
As the limit of consonance, he takes the
ratio 5:6, although he does not conceal the fact that
practice, sharpened attention, habit, taste, and even
prejudice play collateral rôles in the matter, and that
consequently the question is not a purely scientific
one.</p>

<p>Sauveur's ideas took their development from his<span class="pagenum"><a name="Page_377" id="Page_377">[Pg 377]</a></span>
having instituted at all points more exact quantitative
investigations than his predecessors. He is first desirous
of determining as the foundation of musical
tuning a fixed note of one hundred vibrations which
can be reproduced at any time; the fixing of the notes
of musical instruments by the common tuning pipes
then in use with rates of vibration unknown, appearing
to him inadequate. According to Mersenne (<i>Harmonie
Universelle</i>, 1636), a given cord seventeen feet
long and weighted with eight pounds executes eight
visible vibrations in a second. By diminishing its
length then in a given proportion we obtain a proportionately
augmented rate of vibration. But this procedure
appears too uncertain to Sauveur, and he employs
for his purpose the beats (<i>battemens</i>), which were
known to the organ-makers of his day, and which he
correctly explains as due to the alternate coincidence
and non-coincidence of the same vibrational phases of
differently pitched notes.<a name="FNanchor_137_137" id="FNanchor_137_137"></a><a href="#Footnote_137_137" class="fnanchor">[137]</a> At every coincidence there
is a swelling of the sound, and hence the number of
beats per second will be equal to the difference of the
rates of vibration. If we tune two of three organ-pipes
to the remaining one in the ratio of the minor and major
third, the mutual ratio of the rates of vibration of
the first two will be as 24: 25, that is to say, for every
24 vibrations to the lower note there will be 25 to the
higher, and one beat. If the two pipes give together<span class="pagenum"><a name="Page_378" id="Page_378">[Pg 378]</a></span>
four beats in a second, then the higher has the fixed
tone of 100 vibrations. The open pipe in question
will consequently be five feet in length. We also determine
by this procedure the absolute rates of vibration
of all the other notes.</p>

<p>It follows at once that a pipe eight times as long
or 40 feet in length will yield a vibrational rate of
12-1/2, which Sauveur ascribes to the lowest audible
tone, and further also that a pipe 64 times as small
will execute 6,400 vibrations, which Sauveur took for
the highest audible limit. The author's delight at his
successful enumeration of the "imperceptible vibrations"
is unmistakably asserted here, and it is justified
when we reflect that to-day even Sauveur's principle,
slightly modified, constitutes the simplest and most
delicate means we have for exactly determining rates
of vibration. Far more important still, however, is a
second observation which Sauveur made while studying
beats, and to which we shall revert later.</p>

<p>Strings whose lengths can be altered by movable
bridges are much easier to handle than pipes in such
investigations, and it was natural that Sauveur should
soon resort to their use.</p>

<p>One of his bridges accidentally not having been
brought into full and hard contact with the string,
and consequently only imperfectly impeding the vibrations,
Sauveur discovered the harmonic overtones of
the string, at first by the unaided ear, and concluded
from this fact that the string was divided into aliquot<span class="pagenum"><a name="Page_379" id="Page_379">[Pg 379]</a></span>
parts. The string when plucked, and when the bridge
stood at the third division for example, yielded the
twelfth of its fundamental note. At the suggestion
of some academician<a name="FNanchor_138_138" id="FNanchor_138_138"></a><a href="#Footnote_138_138" class="fnanchor">[138]</a> probably, variously colored
paper riders were placed at the nodes (<i>noeuds</i>) and
ventral segments (<i>ventres</i>), and the division of the
string due to the excitation of the overtones (<i>sons
harmoniques</i>) belonging to its fundamental note (<i>son
fondamental</i>) thus rendered visible. For the clumsy
bridge the more convenient feather or brush was soon
substituted.
.
While engaged in these investigations Sauveur also
observed the sympathetic vibration of a string induced
by the excitation of a second one in unison with it.
He also discovered that the overtone of a string can
respond to another string tuned to its note. He even
went further and discovered that on exciting one string
the overtone which it has in common with another,
differently pitched string can be produced on that
other; for example, on strings having for their vibrational
ratio 3:4, the fourth of the lower and the third
of the higher may be made to respond. It follows indisputably
from this that the excited string yields
overtones simultaneously with its fundamental tone.
Previously to this Sauveur's attention had been drawn
by other observers to the fact that the overtones of
musical instruments can be picked out by attentive
listening, particularly in the night.<a name="FNanchor_139_139" id="FNanchor_139_139"></a><a href="#Footnote_139_139" class="fnanchor">[139]</a> He himself mentions<span class="pagenum"><a name="Page_380" id="Page_380">[Pg 380]</a></span>
the simultaneous sounding of the overtones and
the fundamental tone.<a name="FNanchor_140_140" id="FNanchor_140_140"></a><a href="#Footnote_140_140" class="fnanchor">[140]</a> That he did not give the
proper consideration to this circumstance was, as will
afterwards be seen, fatal to his theory.</p>

<p>While studying beats Sauveur makes the remark
that they are <i>displeasing</i> to the ear. He held the beats
were distinctly audible only when less than six occurred
in a second. Larger numbers were not distinctly
perceptible and gave rise accordingly to no
disturbance. He then attempts to reduce the difference
between consonance and dissonance to a question
of beats. Let us hear his own words.<a name="FNanchor_141_141" id="FNanchor_141_141"></a><a href="#Footnote_141_141" class="fnanchor">[141]</a></p>

<blockquote><p>"Beats are unpleasing to the ear because of the unevenness
of the sound, and it may be held with much plausibility that the
reason why octaves are so pleasing is that we never hear their
beats.<a name="FNanchor_142_142" id="FNanchor_142_142"></a><a href="#Footnote_142_142" class="fnanchor">[142]</a></p>

<p>"In following out this idea, we find that the chords whose
beats we cannot hear are precisely those which the musicians call
consonances and that those whose beats are heard are the dissonances,
and that when a chord is a dissonance in one octave and a
consonance in another, it beats in the one and does not beat in the
other. Consequently it is called an imperfect consonance. It is
very easy by the principles of M. Sauveur, here established, to ascertain
what chords beat and in what octaves, above or below the
fixed note. If this hypothesis be correct, it will disclose the true
source of the rules of composition, hitherto unknown to science,
and given over almost entirely to judgment by the ear. These
sorts of natural judgment, marvellous though they may sometimes
appear, are not so but have very real causes, the knowledge of
which belongs to science, provided it can gain possession thereof."<a name="FNanchor_143_143" id="FNanchor_143_143"></a><a href="#Footnote_143_143" class="fnanchor">[143]</a></p></blockquote>

<p><span class="pagenum"><a name="Page_381" id="Page_381">[Pg 381]</a></span></p><p>Sauveur thus correctly discerns in beats the cause
of the disturbance of consonance, to which all disharmony
is "probably" to be referred. It will be seen,
however, that according to his view all distant intervals
must necessarily be consonances and all near intervals
dissonances. He also overlooks the absolute
difference in point of principle between his old view,
mentioned at the outset, and his new view, rather attempting
to obliterate it.</p>

<p>R. Smith<a name="FNanchor_144_144" id="FNanchor_144_144"></a><a href="#Footnote_144_144" class="fnanchor">[144]</a> takes note of the theory of Sauveur and
calls attention to the first of the above-mentioned defects.
Being himself essentially involved in the old
view of Sauveur, which is usually attributed to Euler,
he yet approaches in his criticism a brief step nearer<span class="pagenum"><a name="Page_382" id="Page_382">[Pg 382]</a></span>
to the modern theory, as appears from the following
passage.<a name="FNanchor_145_145" id="FNanchor_145_145"></a><a href="#Footnote_145_145" class="fnanchor">[145]</a></p>

<blockquote><p>"The truth is, this gentleman confounds the distinction between
perfect and imperfect consonances, by comparing imperfect
consonances which beat because the succession of their short cycles<a name="FNanchor_146_146" id="FNanchor_146_146"></a><a href="#Footnote_146_146" class="fnanchor">[146]</a>
is periodically confused and interrupted, with perfect ones
which cannot beat, because the succession of their short cycles is
never confused nor interrupted.</p>

<p>"The <i>fluttering roughness</i> above mentioned is perceivable
in all other perfect consonances, in a smaller degree in proportion
as their cycles are shorter and simpler, and their pitch is higher;
and is of a <i>different kind</i> from the <i>smoother beats</i> and undulations
of <i>tempered consonances</i>; because we can alter the rate of
the latter by altering the temperament, but not of the former, the
consonance being perfect at a given pitch: And because a judicious
ear can often hear, at the same time, both the flutterings and the
beats of a tempered consonance; sufficiently distinct from each
other.</p>

<p>"For nothing gives greater offence to the hearer, though ignorant
of the cause of it, than those rapid, piercing beats of high
and loud sounds, which make imperfect consonances with one another.
And yet a few slow beats, like the slow undulations of a
close shake now and then introduced, are far from being disagreeable."</p></blockquote>

<p>Smith is accordingly clear that other "roughnesses"
exist besides the beats which Sauveur considered,
and if the investigations had been continued
on the basis of Sauveur's idea, these additional roughnesses
would have turned out to be the beats of the<span class="pagenum"><a name="Page_383" id="Page_383">[Pg 383]</a></span>
overtones, and the theory thus have attained the
point of view of Helmholtz.</p>

<p>Reviewing the differences between Sauveur's and
Helmholtz's theories, we find the following:</p>

<p>1. The theory according to which consonance depends
on the frequent and regular coincidence of vibrations
and their ease of enumeration, appears from
the new point of view inadmissible. The simplicity
of the ratios obtaining between the rates of vibration
is indeed a <i>mathematical</i> characteristic of consonance
as well as a <i>physical</i> condition thereof, for the reason
that the coincidence of the overtones as also their
further physical and physiological consequences is
connected with this fact. But no <i>physiological</i> or <i>psychological</i>
explanation of consonance is given by this
fact, for the simple reason that in the acoustic nerve-process
nothing corresponding to the periodicity of
the sonant stimulus is discoverable.</p>

<p>2. In the recognition of beats as a disturbance of
consonance, both theories agree. Sauveur's theory,
however, does not take into account the fact that
clangs, or musical sounds generally, are composite
and that the disturbance in the consonances of distant
intervals principally arise from the beats of the overtones.
Furthermore, Sauveur was wrong in asserting
that the number of beats must be less than six in a
second in order to produce disturbances. Even Smith
knows that very slow beats are not a cause of disturbance,
and Helmholtz found a much higher number<span class="pagenum"><a name="Page_384" id="Page_384">[Pg 384]</a></span>
(33) for the maximum of disturbance. Finally, Sauveur
did not consider that although the number of
beats increases with the recession from unison, yet
their <i>strength</i> is diminished. On the basis of the
principle of specific energies and of the laws of sympathetic
vibration the new theory finds that two atmospheric
motions of like amplitude but different periods,
<i>a</i> sin(<i>rt</i>) and <i>a</i> sin[(<i>r</i> + &#961;)(<i>t</i> + &#964;)], cannot be
communicated with the same amplitude to the same
nervous end-organ. On the contrary, an end-organ
that reacts best to the period <i>r</i> responds more weakly
to the period <i>r</i> + &#961;, the two amplitudes bearing to each
other the proportion <i>a</i>: &#966;<i>a</i>. Here &#966; decreases when
&#961; increases, and when &#961; = 0 it becomes equal to 1, so
that only the portion of the stimulus &#966;<i>a</i> is subject to
beats, and the portion (1-&#966;)<i>a</i> continues smoothly
onward without disturbance.</p>

<p>If there is any moral to be drawn from the history
of this theory, it is that considering how near Sauveur's
errors were to the truth, it behooves us to exercise
some caution also with regard to the new theory.
And in reality there seems to be reason for
doing so.</p>

<p>The fact that a musician will never confound a
more perfectly consonant chord on a poorly tuned
piano with a less perfectly consonant chord on a well
tuned piano, although the roughness in the two cases
may be the same, is sufficient indication that the degree
of roughness is not the only characteristic of a<span class="pagenum"><a name="Page_385" id="Page_385">[Pg 385]</a></span>
harmony. As the musician knows, even the harmonic
beauties of a Beethoven sonata are not easily effaced
on a poorly tuned piano; they scarcely suffer more
than a Raphael drawing executed in rough unfinished
strokes. The <i>positive physiologico-psychological</i> characteristic
which distinguishes one harmony from another
is not given by the beats. Nor is this characteristic
to be found in the fact that, for example, in sounding
a major third the fifth partial tone of the lower note
coincides with the fourth of the higher note. This
characteristic comes into consideration only for the
investigating and abstracting reason. If we should
regard it also as characteristic of the sensation, we
should lapse into a fundamental error which would
be quite analogous to that cited in (1).</p>

<p>The <i>positive physiological</i> characteristics of the intervals
would doubtless be speedily revealed if it were
possible to conduct aperiodic, for example galvanic,
stimuli to the single sound-sensing organs, in which
case the beats would be totally eliminated. Unfortunately
such an experiment can hardly be regarded as
practicable. The employment of acoustic stimuli of
short duration and consequently also free from beats,
involves the additional difficulty of a pitch not precisely
determinable.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_386" id="Page_386">[Pg 386]</a></span></p>




<p class="center">II.</p>

<h3><a id="REMARKS_ON_THE_THEORY_OF_SPATIAL_VISION"></a>REMARKS ON THE THEORY OF SPATIAL VISION.<a name="FNanchor_147_147" id="FNanchor_147_147"></a><a href="#Footnote_147_147" class="fnanchor">[147]</a></h3>


<p>According to Herbart, spatial vision rests on reproduction-series.
In such an event, of course, and
if the supposition is correct, the magnitudes of the
residua with which the percepts or representations
are coalesced (the helps to coalescence) are of cardinal
influence. Furthermore, since the coalescences
must first be fully perfected before they make their
appearance, and since upon their appearance the inhibitory
ratios are brought into play, ultimately, then,
if we leave out of account the accidental order of time
in which the percepts are given, everything in spatial
vision depends on the oppositions and affinities, or,
in brief, on the qualities of the percepts, which enter
into series.</p>

<p>Let us see how the theory stands with respect to
the special facts involved.</p>

<p>1. If intersecting series only, running anteriorly
and posteriorly, are requisite for the production of
spatial sensation, why are not analogues of them found
in all the senses?</p>

<p>2. Why do we measure differently colored objects<span class="pagenum"><a name="Page_387" id="Page_387">[Pg 387]</a></span>
and variegated objects with one and the same spatial
measure? How do we recognise differently colored
objects as the same in size? Where do we get our
measure of space from and what is it?</p>

<p>3. Why is it that differently colored figures of the
same form reproduce one another and are recognised
as the same?</p>

<p>Here are difficulties enough. Herbart is unable to
solve them by his theory. The unprejudiced student
sees at once that his "inhibition by reason of form"
and "preference by reason of form" are absolutely
impossible. Think of Herbart's example of the red
and black letters.</p>

<p>The "help to coalescence" is a passport, so to
speak, made out to the name and person of the percept.
A percept which is coalesced with another cannot
reproduce all others qualitatively different from it
for the simple reason that the latter are in like manner
coalesced with one another. Two qualitatively different
series certainly do not reproduce themselves because
they present the same order of degree of coalescence.</p>

<p>If it is certain that only things simultaneous and
things which are alike are reproduced, a basic principle
of Herbart's psychology which even the most
absolute empiricists will not deny, nothing remains
but to modify the theory of spatial perception or to
invent in its place a new principle in the manner indicated,
a step which hardly any one would seriously<span class="pagenum"><a name="Page_388" id="Page_388">[Pg 388]</a></span>
undertake. The new principle could not fail to throw
all psychology into the most dreadful confusion.</p>

<p>As to the modification which is needed there can
be hardly any doubt as to how in the face of the facts
and conformably to Herbart's own principles it is to
be carried out. If two differently colored figures of
equal size reproduce each other and are recognised as
equal, the result can be due to nothing but to the existence
in both series of presentations of a presentation
or percept which is qualitatively <i>the same</i>. The
colors are different. Consequently, like or equal percepts
must be connected with the colors which are
yet independent of the colors. We have not to look
long for them, for they are the like effects of the muscular
feelings of the eye when confronted by the two
figures. We might say we reach the vision of space
by the registering of light-sensations in a schedule of
graduated muscle-sensations.<a name="FNanchor_148_148" id="FNanchor_148_148"></a><a href="#Footnote_148_148" class="fnanchor">[148]</a></p>

<p>A few considerations will show the likelihood of
the rôle of the muscle-sensations. The muscular apparatus
of <i>one</i> eye is unsymmetrical. The two eyes
together form a system which is vertical in symmetry.
This already explains much.</p>

<p>1. The <i>position</i> of a figure influences its view. According
to the position in which objects are viewed
different muscle-sensations come into play and the
impression is altered. To recognise inverted letters<span class="pagenum"><a name="Page_389" id="Page_389">[Pg 389]</a></span>
as such long experience is required. The best proof
of this are the letters d, b, p, q, which are represented
by the same figure in different positions and yet are
always distinguished as different.<a name="FNanchor_149_149" id="FNanchor_149_149"></a><a href="#Footnote_149_149" class="fnanchor">[149]</a></p>

<p>2. It will not escape the attentive observer that for
the same reasons and even with the same figure and
in the same position the fixation point is also decisive.
The figure seems to change <i>during</i> the act of vision.
For example, an eight-pointed star constructed by
successively joining in a regular octagon the first corner
with the fourth, the fourth with the seventh, etc.,
skipping in every case two corners, assumes alternately,
according to where we suffer the centre of vision
to rest, a predominantly architectonic or a freer
and more open character. Vertical and horizontal
lines are always differently apprehended from what
oblique lines are.</p>

<div class="figleft" style="width: 300px;">
<img src="images/i_399.jpg" width="300" height="130" alt="" />
<span class="caption">Fig. 58.</span>
</div>

<p>3. The reason why we prefer vertical symmetry
and regard it as something special in its kind, whereas
we do not recognise
horizontal symmetry
at all immediately, is
due to the vertical
symmetry of the muscular
apparatus of the eye. The left-hand side <i>a</i> of
the accompanying vertically-symmetrical figure induces
in the left eye the same muscular feelings as the<span class="pagenum"><a name="Page_390" id="Page_390">[Pg 390]</a></span>
right-hand side <i>b</i> does in the right eye. The pleasing
effect of symmetry has its cause primarily in the repetition
of muscular feelings. That a repetition actually
occurs here, sometimes sufficiently marked in character
as to lead to the confounding of objects, is
proved apart from the theory by the fact which is
familiar to every one <i>quem dii oderunt</i> that children
frequently reverse figures from the right to the left,
but never from above downwards; for example, write
ε instead of 3 until they finally come to notice the
slight difference. Figure 50 shows how pleasing the
repetition of muscular
feelings may be. As
will be readily understood,
vertical and horizontal lines exhibit relations
similar to symmetrical figures which are immediately
disturbed when oblique positions are chosen for the
lines. Compare what Helmholtz says regarding the
repetition and coincidence of partial tones.</p>

<div class="figright" style="width: 400px;">
<img src="images/i_400.jpg" width="400" height="88" alt="" />
<span class="caption">Fig. 59.</span>
</div>

<p>I may be permitted to add a general remark. It
is a quite universal phenomenon in psychology that
certain qualitatively quite different series of percepts
mutually awaken and reproduce one another and in a
certain aspect produce the appearance of sameness or
similarity. We say of such series that they are of
like or of similar form, naming their abstracted likeness
<i>form</i>.</p>

<blockquote><p>1. Of spatial figures we have already spoken.</p>

<p>2. We call two melodies like melodies when they<span class="pagenum"><a name="Page_391" id="Page_391">[Pg 391]</a></span>
present the same succession of pitch-ratios;
the absolute pitch (or key) may be as different
as can be. We can so select the melodies that
not even two partial tones of the notes in each
are common. Yet we recognise the melodies
as alike. And, what is more, we notice the
form of the melody more readily and recognise
it again more easily than the key (the absolute
pitch) in which it was played.</p>

<p>3. We recognise in two different melodies the
same rhythm no matter how different the melodies
may be otherwise. We know and recognise
the rhythm more easily even than the absolute
duration (the tempo).</p></blockquote>

<p>These examples will suffice. In all these and in
all similar cases the recognition and likeness cannot
depend upon the qualities of the percepts, for these
are different. On the other hand recognition, conformably
to the principles of psychology, is possible
only with percepts which are the same in quality.
Consequently there is no other escape than to imagine
the qualitatively unlike percepts of the two series as
necessarily connected with other percepts which are
qualitatively alike.</p>

<p>Since in differently colored figures of like form, like
muscular feelings are necessarily induced if the figures
are recognised as alike, so there must necessarily lie
at the basis of all forms also, and we might even say
at the basis of all abstractions, percepts of a peculiar<span class="pagenum"><a name="Page_392" id="Page_392">[Pg 392]</a></span>
quality. And this holds true for space and form as
well as for time, rhythm, pitch, the form of melodies,
intensity, etc. But whence is psychology to derive all
these qualities? Have no fear, they will all be found,
as were the sensations of muscles for the theory of
space. The organism is at present still rich enough
to meet all the requirements of psychology in this direction,
and it is even time to give serious ear to the
question of "corporeal resonance" which psychology
so loves to dwell on.</p>

<p>Different psychical qualities appear to bear a very
intimate mutual relation to one another. Special research
on the subject, as well also as the demonstration
that this remark may be generally employed in
physics, will follow later.<a name="FNanchor_150_150" id="FNanchor_150_150"></a><a href="#Footnote_150_150" class="fnanchor">[150]</a></p>
<p><span class="pagenum"><a name="Page_393" id="Page_393">[Pg 393]</a></span></p>



<h2><a name="INDEX" id="INDEX">INDEX.</a></h2>


<ul class="IX"><li>Absolute, temperature, <a href="#Page_162">162</a>;</li>
<li><span style="margin-left: 1em;">time, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">forecasts, have no signification in science, <a href="#Page_206">206</a>.</span></li>

<li>Abstract, meaning of the term, <a href="#Page_240">240</a>.</li>

<li>Abstraction, <a href="#Page_180">180</a>, <a href="#Page_200">200</a>, <a href="#Page_208">208</a>, <a href="#Page_231">231</a>.</li>

<li>Acceleration, organ for forward, <a href="#Page_299">299</a> et seq.</li>

<li>Accelerations, <a href="#Page_204">204</a>, <a href="#Page_216">216</a>, footnote, <a href="#Page_225">225</a>-<a href="#Page_226">226</a>, <a href="#Page_253">253</a>.</li>

<li>Accident, logical and historical, in science, <a href="#Page_160">160</a>, <a href="#Page_168">168</a>, <a href="#Page_170">170</a>, <a href="#Page_213">213</a>;</li>
<li><span style="margin-left: 1em;">in inventions and discoveries, <a href="#Page_262">262</a> et seq.</span></li>

<li>Accord, the pure triple, <a href="#Page_46">46</a>.</li>

<li>Accumulators, electrical, <a href="#Page_125">125</a> et seq.;</li>
<li><span style="margin-left: 1em;"><a href="#Page_132">132</a>, footnote.</span></li>

<li>Acoustic color, <a href="#Page_36">36</a>.</li>

<li>Acoustics, Sauveur on, <a href="#Page_375">375</a> et seq.</li>

<li>Action and reaction, importance of the principle of, <a href="#Page_191">191</a>.</li>

<li>Adaptation, in organic and inorganic matter, <a href="#Page_216">216</a>, <a href="#Page_229">229</a>;</li>
<li><span style="margin-left: 1em;">in scientific thought, <a href="#Page_214">214</a>-<a href="#Page_235">235</a>.</span></li>

<li>Æsthetics, computation as a principle of, <a href="#Page_34">34</a>;</li>
<li><span style="margin-left: 1em;">researches in, <a href="#Page_89">89</a>, footnote;</span></li>
<li><span style="margin-left: 1em;">repetition, a principle of, <a href="#Page_91">91</a>.</span></li>

<li>Africa, <a href="#Page_186">186</a>, <a href="#Page_234">234</a>, <a href="#Page_237">237</a>.</li>

<li>Agreeable effects, due to repetition of sensations, <a href="#Page_92">92</a>, <a href="#Page_97">97</a> et seq.</li>

<li>Agriculture, transition to, <a href="#Page_265">265</a>.</li>

<li>Air-gun, <a href="#Page_135">135</a>.</li>

<li>Alcohol and water, mixture of oil and, in Plateau's experiments, <a href="#Page_4">4</a>.</li>

<li>Algebra, economy of, <a href="#Page_196">196</a>.</li>

<li>Alien thoughts in science, <a href="#Page_196">196</a>.</li>

<li>All, the, <a href="#Page_88">88</a>.</li>

<li>Amontons, <a href="#Page_174">174</a>, <a href="#Page_346">346</a>.</li>

<li>Ampère, the word, <a href="#Page_314">314</a>.</li>

<li>Ampère's swimmer, <a href="#Page_207">207</a>.</li>

<li>Analogies, mechanical, <a href="#Page_157">157</a>, <a href="#Page_160">160</a>;</li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_236">236</a>-<a href="#Page_258">258</a>.</span></li>

<li>Analogy, defined, <a href="#Page_250">250</a>.</li>

<li>Analysis, <a href="#Page_188">188</a>.</li>

<li>Analytical geometry, not necessary to physicians, <a href="#Page_370">370</a>, footnote.</li>

<li>Anatomic structures, transparent stereoscopic views of, <a href="#Page_74">74</a>.</li>

<li>Anatomy, character of research in, <a href="#Page_255">255</a>.</li>

<li>Andrieu, Jules, <a href="#Page_49">49</a>, footnote.</li>

<li>Animals, the psychical activity of, <a href="#Page_190">190</a>, <a href="#Page_231">231</a>;</li>
<li><span style="margin-left: 1em;">the language of, <a href="#Page_238">238</a>;</span></li>
<li><span style="margin-left: 1em;">their capacity for experience, <a href="#Page_266">266</a> et seq.</span></li>

<li>Animism, <a href="#Page_186">186</a>, <a href="#Page_187">187</a>, <a href="#Page_243">243</a>, <a href="#Page_254">254</a>.</li>

<li>Anisotropic optical fields, <a href="#Page_227">227</a>.</li>

<li>Apparatus for producing movements of rotation, <a href="#Page_287">287</a> et seq.</li>

<li>Arabesque, an inverted, <a href="#Page_95">95</a>.</li>

<li>Arabian Nights, <a href="#Page_219">219</a>.</li>

<li>Arago, <a href="#Page_270">270</a>.</li>

<li>Aral, the Sea of, <a href="#Page_239">239</a>.</li>

<li>Archæopteryx, <a href="#Page_257">257</a>.</li>

<li>Archimedes, <a href="#Page_4">4</a>, <a href="#Page_237">237</a>.</li>

<li>Arcimboldo, Giuseppe, <a href="#Page_36">36</a>.</li>

<li>Area, principle of least superficial, <a href="#Page_10">10</a> et seq.</li>

<li>Ares, the bellowing of the wounded, <a href="#Page_272">272</a>.</li>

<li>Aristotelians, <a href="#Page_283">283</a>.</li>

<li>Aristotle, <a href="#Page_348">348</a>, <a href="#Page_296">296</a>.</li>

<li>Art, development of, <a href="#Page_28">28</a> et seq.</li>

<li>Artillery, practical, <a href="#Page_334">334</a>-<a href="#Page_335">335</a>.</li>

<li>Artistic value of scientific descriptions, <a href="#Page_254">254</a>.</li>

<li>Arts, practical, <a href="#Page_108">108</a>.</li>

<li>Ascent, heights of, <a href="#Page_143">143</a>-<a href="#Page_151">151</a>.</li>

<li><span class="pagenum"><a name="Page_394" id="Page_394">[Pg 394]</a></span>Asia, 234.</li>

<li>Assyrians, the art of, <a href="#Page_79">79</a>.</li>

<li>Astronomer, measures celestial by terrestrial distances, <a href="#Page_136">136</a>.</li>

<li>Astronomy, antecedent to psychology, <a href="#Page_90">90</a>;</li>
<li><span style="margin-left: 1em;">rigidity of its truths, <a href="#Page_221">221</a>.</span></li>

<li>Atomic theories, <a href="#Page_104">104</a>.</li>

<li>Atoms, <a href="#Page_207">207</a>.</li>

<li>Attention, the rôle of, in sensuous perception, <a href="#Page_35">35</a> et seq.</li>

<li>Attraction, generally, <a href="#Page_226">226</a>;</li>
<li><span style="margin-left: 1em;">of liquid particles, <a href="#Page_13">13</a>-<a href="#Page_14">14</a>;</span></li>
<li><span style="margin-left: 1em;">in electricity, <a href="#Page_109">109</a> et seq.</span></li>

<li>Aubert, <a href="#Page_298">298</a>.</li>

<li>Audition. See <i>Ear</i>.</li>

<li>Austrian gymnasiums, <a href="#Page_370">370</a>, footnote.</li>

<li>Axioms, instinctive knowledge, <a href="#Page_190">190</a>.</li>

</ul>

<ul class="IX"><li>Babbage, on the economy of machinery, <a href="#Page_196">196</a>.</li>

<li>Bach, <a href="#Page_20">20</a>.</li>

<li>Backwards, prophesying, <a href="#Page_253">253</a>.</li>

<li>Bacon, Lord, <a href="#Page_48">48</a>, <a href="#Page_280">280</a>.</li>

<li>Baer, C. E. von, <a href="#Page_235">235</a>.</li>

<li>Balance, electrical, <a href="#Page_127">127</a>, footnote;</li>
<li><span style="margin-left: 1em;">torsion, <a href="#Page_109">109</a>, <a href="#Page_168">168</a>.</span></li>

<li>Balloon, a hydrogen, <a href="#Page_199">199</a>.</li>

<li>Barbarism and civilisation, <a href="#Page_335">335</a> et seq.</li>

<li>Bass-clef, <a href="#Page_101">101</a>.</li>

<li>Bass, fundamental, <a href="#Page_44">44</a>.</li>

<li>Beats, <a href="#Page_40">40</a>-<a href="#Page_45">45</a>, <a href="#Page_377">377</a> et seq.</li>

<li>Beautiful, our notions of, variable, <a href="#Page_99">99</a>.</li>

<li>Beauty, objects of, in nature, <a href="#Page_91">91</a>.</li>

<li>Becker, J. K., <a href="#Page_364">364</a>, <a href="#Page_369">369</a>.</li>

<li>Beethoven, <a href="#Page_39">39</a>, <a href="#Page_44">44</a>.</li>

<li>Beginnings of science, <a href="#Page_189">189</a>, <a href="#Page_191">191</a>.</li>

<li>Belvedere Gallery at Vienna, <a href="#Page_36">36</a>.</li>

<li>Bernoulli, Daniel, on the conservation of living force, <a href="#Page_149">149</a>;</li>
<li><span style="margin-left: 1em;">on the vibrations of strings, <a href="#Page_249">249</a>.</span></li>

<li>Bernoulli, James, on the centre of oscillation, <a href="#Page_149">149</a>.</li>

<li>Bernoulli, John, on the conservation of living force, <a href="#Page_149">149</a>;</li>
<li><span style="margin-left: 1em;">on the principle of virtual velocities, <a href="#Page_151">151</a>.</span></li>

<li>Bible, parallel passages from, for language study, <a href="#Page_356">356</a>.</li>

<li>Binocular vision, <a href="#Page_66">66</a> et seq.</li>

<li>Black, his theory of caloric, <a href="#Page_138">138</a>, <a href="#Page_162">162</a>;</li>
<li><span style="margin-left: 1em;">on quantity of heat, <a href="#Page_166">166</a>, <a href="#Page_174">174</a>;</span></li>
<li><span style="margin-left: 1em;">on latent heat, <a href="#Page_167">167</a>, <a href="#Page_178">178</a>;</span></li>
<li><span style="margin-left: 1em;">researches in heat generally, <a href="#Page_244">244</a>.</span></li>

<li>Blind cat, <a href="#Page_303">303</a>.</li>

<li>Bodies, heavy, seek their places, <a href="#Page_224">224</a> et seq.;</li>
<li><span style="margin-left: 1em;">rotating, <a href="#Page_285">285</a>.</span></li>

<li>Body, a mental symbol for groups of sensations, <a href="#Page_200">200</a>-<a href="#Page_203">203</a>;</li>
<li><span style="margin-left: 1em;">the human, our knowledge of, <a href="#Page_90">90</a>.</span></li>

<li>Boltzmann, <a href="#Page_236">236</a>.</li>

<li>Booth, Mr., <a href="#Page_77">77</a>.</li>

<li>Borelli, <a href="#Page_217">217</a>.</li>

<li>Boulder, a granite, <a href="#Page_233">233</a>.</li>

<li>Bow-wave of ships and moving projectiles, <a href="#Page_323">323</a> et seq.</li>

<li>Boys, <a href="#Page_317">317</a>.</li>

<li>Bradley, <a href="#Page_273">273</a>.</li>

<li>Brahman, the, <a href="#Page_63">63</a>.</li>

<li>Brain, localisation of functions in, <a href="#Page_210">210</a>.</li>

<li>Breuer, <a href="#Page_272">272</a>, <a href="#Page_282">282</a> et seq., <a href="#Page_293">293</a>, <a href="#Page_298">298</a>, <a href="#Page_300">300</a>, <a href="#Page_301">301</a>, <a href="#Page_303">303</a>, <a href="#Page_306">306</a>.</li>

<li>Brewster, his stereoscope, <a href="#Page_73">73</a>.</li>

<li>Bridge, invention of the, <a href="#Page_264">264</a>, <a href="#Page_268">268</a>.</li>

<li>British Association, <a href="#Page_108">108</a>.</li>

<li>Brooklyn Bridge, <a href="#Page_75">75</a>, footnote.</li>

<li>Brown, Crum, <a href="#Page_293">293</a>, <a href="#Page_301">301</a>.</li>

<li>Building, our concepts directions for, <a href="#Page_253">253</a>;</li>
<li><span style="margin-left: 1em;">facts the result of, <a href="#Page_253">253</a>;</span></li>
<li><span style="margin-left: 1em;">science compared to, <a href="#Page_257">257</a>.</span></li>

<li>Building-stones, metrical units are, <a href="#Page_253">253</a>.</li>

<li>Busch, <a href="#Page_328">328</a>.</li>

<li>Business of a merchant, science compared to the, <a href="#Page_16">16</a>.</li>

<li>Butterfly, a, <a href="#Page_22">22</a>.</li>
</ul>

<ul class="IX"><li>Calculating machines, their economical character, <a href="#Page_196">196</a>.</li>

<li>Caloric, theory of, stood in the way of scientific advancement, <a href="#Page_138">138</a>, <a href="#Page_167">167</a>.</li>

<li>Calypso, the island of, <a href="#Page_351">351</a>.</li>

<li>Canterbury, Archbishop of, <a href="#Page_39">39</a>.</li>

<li>Cantor, M., <a href="#Page_361">361</a>, footnote.</li>

<li>Capacity, electrical, <a href="#Page_116">116</a> et seq., <a href="#Page_123">123</a>;</li>
<li><span style="margin-left: 1em;">thermal, <a href="#Page_123">123</a>;</span></li>
<li><span style="margin-left: 1em;">specific inductive, <a href="#Page_117">117</a>.</span></li>

<li>Capulets and Montagues, <a href="#Page_87">87</a>.</li>

<li>Cards, difficult games of, <a href="#Page_357">357</a>.</li>

<li>Carnot, S., excludes perpetual motion in heat, <a href="#Page_156">156</a>, <a href="#Page_162">162</a>;</li>
<li><span style="margin-left: 1em;">his mechanical view of physics, <a href="#Page_156">156</a>;</span></li>
<li><span class="pagenum"><a name="Page_395" id="Page_395">[Pg 395]</a></span><span style="margin-left: 1em;">on thermodynamics, <a href="#Page_160">160</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">his principle, <a href="#Page_162">162</a>;</span></li>
<li><span style="margin-left: 1em;">also, <a href="#Page_191">191</a>.</span></li>

<li>Carus, Dr. Paul, <a href="#Page_265">265</a>, footnote.</li>

<li>Casselli's telegraph, <a href="#Page_26">26</a>.</li>

<li>Cassini, <a href="#Page_51">51</a>.</li>

<li>Cauchy, character of the intellectual activity of a, <a href="#Page_195">195</a>.</li>

<li>Causal insight, awakened by science, <a href="#Page_357">357</a>.</li>

<li>Causality, <a href="#Page_157">157</a>-<a href="#Page_159">159</a>, <a href="#Page_190">190</a>, <a href="#Page_198">198</a> et seq., <a href="#Page_221">221</a> et seq., <a href="#Page_237">237</a>, <a href="#Page_253">253</a>, <a href="#Page_254">254</a>.</li>

<li>Cause and effect, <a href="#Page_198">198</a> et seq. See also <i>Causality</i>.</li>

<li>Centimetre-gramme-second system, <a href="#Page_111">111</a>.</li>

<li>Centre of gravity, must lie as low as possible for equilibrium to subsist, <a href="#Page_15">15</a>;</li>
<li><span style="margin-left: 1em;">Torricelli's principle of, <a href="#Page_150">150</a> et seq.</span></li>

<li>Centre of oscillation, <a href="#Page_149">149</a>.</li>

<li>Change, method of, in science, <a href="#Page_230">230</a>.</li>

<li>Changeable character of bodies, <a href="#Page_202">202</a>.</li>

<li>Changes, physical, how they occur, <a href="#Page_205">205</a>.</li>

<li>Character, a Universal Real, <a href="#Page_192">192</a>.</li>

<li>Character, like the forms of liquids, <a href="#Page_3">3</a>;</li>
<li><span style="margin-left: 1em;">persons of, <a href="#Page_24">24</a>.</span></li>

<li>Charles the Fifth, <a href="#Page_369">369</a>.</li>

<li>Chemical, elements, <a href="#Page_202">202</a>;</li>
<li><span style="margin-left: 1em;">symbols, <a href="#Page_192">192</a>;</span></li>
<li><span style="margin-left: 1em;">current, <a href="#Page_118">118</a>.</span></li>

<li>Chemistry, character of research in, <a href="#Page_255">255</a>;</li>
<li><span style="margin-left: 1em;">the method of thermodynamics in, <a href="#Page_257">257</a>.</span></li>

<li>Child, a, modes of thought of, <a href="#Page_223">223</a>;</li>
<li><span style="margin-left: 1em;">looking into a moat, <a href="#Page_208">208</a>.</span></li>

<li>Child of the forest, his interpretation of new events, <a href="#Page_218">218</a>-<a href="#Page_219">219</a>.</li>

<li>Childish questions, <a href="#Page_199">199</a>-<a href="#Page_200">200</a>.</li>

<li>Children, the drawings of, <a href="#Page_201">201</a>-<a href="#Page_202">202</a>.</li>

<li>Chinese language, economy of, <a href="#Page_192">192</a>;</li>
<li><span style="margin-left: 1em;">study of, <a href="#Page_354">354</a>.</span></li>

<li>Chinese philosopher, an old, <a href="#Page_186">186</a>.</li>

<li>Chinese, speak with unwillingness of politics, <a href="#Page_374">374</a>;</li>
<li><span style="margin-left: 1em;">the art of, <a href="#Page_79">79</a>-<a href="#Page_80">80</a>.</span></li>

<li>Chosen, many are called but few are, <a href="#Page_65">65</a>.</li>

<li>Christ, saying of, <a href="#Page_65">65</a>.</li>

<li>Christianity, Latin introduced with, <a href="#Page_311">311</a>.</li>

<li>Christians and Jews, monotheism of the, <a href="#Page_187">187</a>.</li>

<li>Church and State, <a href="#Page_88">88</a>.</li>

<li>Cicero, <a href="#Page_318">318</a>.</li>

<li>Circe, <a href="#Page_372">372</a>.</li>

<li>Circle, the figure of least area with given periphery, <a href="#Page_12">12</a>.</li>

<li>Circular polarisation, <a href="#Page_242">242</a>.</li>

<li>Civilisation and barbarism, <a href="#Page_335">335</a> et seq.</li>

<li>Civilisation, some phenomena of, explained by binocular vision, <a href="#Page_74">74</a>.</li>

<li>Civilised man, his modes of conception and interpretation, <a href="#Page_219">219</a>.</li>

<li>Clapeyron, <a href="#Page_162">162</a>.</li>

<li>Class-characters of animals, <a href="#Page_255">255</a>.</li>

<li>Classical, culture, the good and bad effects of, <a href="#Page_347">347</a>;</li>
<li><span style="margin-left: 1em;">scholars, not the only educated people, <a href="#Page_345">345</a>.</span></li>

<li>Classics, on instruction in, <a href="#Page_338">338</a>-<a href="#Page_374">374</a>;</li>
<li><span style="margin-left: 1em;">the scientific, <a href="#Page_368">368</a>.</span></li>

<li>Classification in science, <a href="#Page_255">255</a>.</li>

<li>Clausius, on thermodynamics, <a href="#Page_165">165</a>;</li>
<li><span style="margin-left: 1em;">on reversible cycles, <a href="#Page_176">176</a>.</span></li>

<li>Claviatur, Mach's, <a href="#Page_42">42</a>-<a href="#Page_43">43</a>.</li>

<li>Club-law, <a href="#Page_335">335</a>.</li>

<li>Cochlea, the, a species of piano-forte, <a href="#Page_19">19</a>.</li>

<li>Cockchafer, <a href="#Page_86">86</a>.</li>

<li>Coefficient of self-induction, <a href="#Page_250">250</a>, <a href="#Page_252">252</a>.</li>

<li>Colophonium, solution of, <a href="#Page_7">7</a>.</li>

<li>Color, acoustic, <a href="#Page_36">36</a>.</li>

<li>Color-sensation, <a href="#Page_210">210</a>.</li>

<li>Color-signs, their economy, <a href="#Page_192">192</a>.</li>

<li>Colors, origin of the names of, <a href="#Page_239">239</a>.</li>

<li>Column, body moving behind a, <a href="#Page_202">202</a>.</li>

<li>Communication, its functions, import and fruits, <a href="#Page_197">197</a>, <a href="#Page_238">238</a> et seq.;</li>
<li><span style="margin-left: 1em;">by language, <a href="#Page_237">237</a>;</span></li>
<li><span style="margin-left: 1em;">high importance of, <a href="#Page_191">191</a> et seq.</span></li>

<li>Comparative physics, <a href="#Page_239">239</a>.</li>

<li>Comparison in science, <a href="#Page_231">231</a>, <a href="#Page_238">238</a> et seq.</li>

<li>Computation, a principle of æsthetics, <a href="#Page_34">34</a>.</li>

<li>Concepts, abstract, defined, <a href="#Page_250">250</a>-<a href="#Page_252">252</a>;</li>
<li><span style="margin-left: 1em;">metrical, in electricity, <a href="#Page_107">107</a> et seq.</span></li>

<li>Conceptual, meaning of the term, <a href="#Page_240">240</a>.</li>

<li>Conceptual thought, <a href="#Page_192">192</a>.</li>

<li>Concha, <a href="#Page_18">18</a>.</li>

<li>Condensers, electrical, <a href="#Page_125">125</a> et seq. <a href="#Page_132">132</a>, footnote.</li>

<li><span class="pagenum"><a name="Page_396" id="Page_396">[Pg 396]</a></span>Conductors and non-conductors. See <i>Electrical</i>, etc.</li>

<li>Conformity in the deportment of the energies, <a href="#Page_171">171</a>-<a href="#Page_175">175</a>.</li>

<li>Confusion of objects, cause of, <a href="#Page_95">95</a>.</li>

<li>Conic sections, <a href="#Page_257">257</a>.</li>

<li>Conical refraction, <a href="#Page_29">29</a>, <a href="#Page_242">242</a>.</li>

<li>Conservation of energy, <a href="#Page_137">137</a> et seq. See <i>Energy</i>.</li>

<li>Conservation of weight or mass, <a href="#Page_203">203</a>.</li>

<li>Consonance, connexion of the simple natural numbers with, <a href="#Page_33">33</a>;</li>
<li><span style="margin-left: 1em;">Euclid's definition of, <a href="#Page_33">33</a>;</span></li>
<li><span style="margin-left: 1em;">explanation of, <a href="#Page_42">42</a>;</span></li>
<li><span style="margin-left: 1em;">scientific definition of, <a href="#Page_44">44</a>;</span></li>
<li><span style="margin-left: 1em;">and dissonance reduced to beats, <a href="#Page_376">376</a>, <a href="#Page_370">370</a>, <a href="#Page_383">383</a>.</span></li>

<li>Consonant intervals, <a href="#Page_43">43</a>.</li>

<li>Constancy of matter, <a href="#Page_203">203</a>.</li>

<li>Constant, the dielectric, <a href="#Page_117">117</a>.</li>

<li>Constants, the natural, <a href="#Page_193">193</a>.</li>

<li>Continuum of facts, <a href="#Page_256">256</a> et seq.</li>

<li>Cornelius, <a href="#Page_388">388</a>, footnote.</li>

<li>Corti, the Marchese, his discovery of minute rods in the labyrinth of the ear, <a href="#Page_19">19</a>.</li>

<li>Coulomb, his electrical researches, <a href="#Page_108">108</a>, <a href="#Page_109">109</a>, <a href="#Page_113">113</a>;</li>
<li><span style="margin-left: 1em;">his notion of quantity of electricity, <a href="#Page_173">173</a>;</span></li>
<li><span style="margin-left: 1em;">his torsion-balance, <a href="#Page_168">168</a>.</span></li>

<li>Crew, Prof. Henry, <a href="#Page_317">317</a>, footnote.</li>

<li>Criticism, Socrates the father of scientific, <a href="#Page_1">1</a>, <a href="#Page_16">16</a>.</li>

<li><i>Critique of Pure Reason</i>, Kant's, <a href="#Page_188">188</a>.</li>

<li>Crucible, derivation of the word, <a href="#Page_49">49</a>, footnote.</li>

<li>Crustacea, auditory filaments of, <a href="#Page_29">29</a>, <a href="#Page_272">272</a>, <a href="#Page_302">302</a>.</li>

<li>Cube of oil, <a href="#Page_5">5</a>.</li>

<li>Culture, ancient and modern, <a href="#Page_344">344</a>.</li>

<li>Currents, chemical, <a href="#Page_118">118</a>;</li>
<li><span style="margin-left: 1em;">electrical, <a href="#Page_118">118</a>;</span></li>
<li><span style="margin-left: 1em;">galvanic, <a href="#Page_132">132</a>;</span></li>
<li><span style="margin-left: 1em;">measurement of electrical, <a href="#Page_135">135</a>-<a href="#Page_136">136</a>;</span></li>
<li><span style="margin-left: 1em;">of heat, <a href="#Page_244">244</a>, <a href="#Page_249">249</a>-<a href="#Page_250">250</a>;</span></li>
<li><span style="margin-left: 1em;">strength of, <a href="#Page_250">250</a>.</span></li>

<li>Curtius, <a href="#Page_356">356</a>.</li>

<li>Curved lines, their asymmetry, <a href="#Page_98">98</a>.</li>

<li>Curves, how their laws are investigated, <a href="#Page_206">206</a>.</li>

<li>Cycles, reversible, Clausius on, <a href="#Page_176">176</a>.</li>

<li>Cyclical processes, closed, <a href="#Page_175">175</a>.</li>

<li>Cyclops, <a href="#Page_67">67</a>.</li>

<li>Cyclostat, <a href="#Page_298">298</a>.</li>

<li>Cylinder, of oil, <a href="#Page_6">6</a>;</li>
<li><span style="margin-left: 1em;">mass of gas enclosed in a, <a href="#Page_179">179</a>.</span></li>

</ul>

<ul class="IX"><li>D'Alembert, on the causes of harmony, <a href="#Page_34">34</a>;</li>
<li><span style="margin-left: 1em;">his principle, <a href="#Page_142">142</a>, <a href="#Page_149">149</a>, <a href="#Page_154">154</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_234">234</a>, <a href="#Page_279">279</a>.</span></li>

<li>Danish schools, <a href="#Page_338">338</a>, footnote.</li>

<li>Darwin, his study of organic nature, <a href="#Page_215">215</a> et seq.;</li>
<li><span style="margin-left: 1em;">his methods of research, <a href="#Page_216">216</a>.</span></li>

<li>Deaf and dumb, not subject to giddiness, <a href="#Page_299">299</a>.</li>

<li>Deaf person, with a piano, analyses sounds, <a href="#Page_27">27</a>.</li>

<li>Death and life, <a href="#Page_186">186</a>.</li>

<li>Definition, compendious, <a href="#Page_197">197</a>.</li>

<li>Deiters, <a href="#Page_19">19</a>.</li>

<li>Delage, <a href="#Page_298">298</a>, <a href="#Page_301">301</a>, <a href="#Page_302">302</a>.</li>

<li>Democritus, his mechanical conception of the world, <a href="#Page_155">155</a>, <a href="#Page_187">187</a>.</li>

<li>Demonstration, character of, <a href="#Page_362">362</a>.</li>

<li>Deportment of the energies, conformity in the, <a href="#Page_171">171</a>-<a href="#Page_175">175</a>.</li>

<li>Derivation, laws only methods of, <a href="#Page_256">256</a>.</li>

<li>Descent, Galileo's laws of, <a href="#Page_193">193</a>;</li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_143">143</a> et seq., <a href="#Page_204">204</a>, <a href="#Page_215">215</a>.</span></li>

<li>Description, <a href="#Page_108">108</a>, <a href="#Page_191">191</a>, <a href="#Page_236">236</a>, <a href="#Page_237">237</a>;</li>
<li><span style="margin-left: 1em;">a condition of scientific knowledge, <a href="#Page_193">193</a>;</span></li>
<li><span style="margin-left: 1em;">direct and indirect, <a href="#Page_240">240</a>;</span></li>
<li><span style="margin-left: 1em;">in physics, <a href="#Page_197">197</a>, <a href="#Page_199">199</a>.</span></li>

<li>Descriptive sciences, their resemblance to the abstract, <a href="#Page_248">248</a>.</li>

<li>Determinants, <a href="#Page_195">195</a>.</li>

<li>Diderot, <a href="#Page_234">234</a>.</li>

<li>Dielectric constant, the, <a href="#Page_117">117</a>.</li>

<li>Difference-engine, the, <a href="#Page_196">196</a>.</li>

<li>Differential coefficients, their relation to symmetry, <a href="#Page_98">98</a>.</li>

<li>Differential laws, <a href="#Page_204">204</a>.</li>

<li>Differential method, for detecting optical imperfections, <a href="#Page_317">317</a>.</li>

<li>Diffraction, <a href="#Page_91">91</a>, <a href="#Page_194">194</a>.</li>

<li>Diffusion, Fick's theory of, <a href="#Page_249">249</a>.</li>

<li>Discharge of Leyden jars, <a href="#Page_114">114</a> et seq.</li>

<li>Discoveries, the gist of, <a href="#Page_270">270</a>, <a href="#Page_375">375</a>.</li>

<li>Discovery and invention, distinction between, <a href="#Page_269">269</a>.</li>

<li>Dissonance, explanation of, <a href="#Page_42">42</a>;</li>
<li><span style="margin-left: 1em;">definition of, <a href="#Page_33">33</a>, <a href="#Page_44">44</a>. See <i>Consonance</i>.</span></li>

<li>Distances, estimation of, by the eye, <a href="#Page_68">68</a> et seq.</li>

<li>Dogs, like tuning-forks, <a href="#Page_23">23</a>;</li>
<li><span class="pagenum"><a name="Page_397" id="Page_397">[Pg 397]</a></span><span style="margin-left: 1em;">their mentality, <a href="#Page_190">190</a>.</span></li>

<li>Domenech, Abbé, <a href="#Page_92">92</a>.</li>

<li>Dramatic element in science, <a href="#Page_243">243</a>.</li>

<li>Drop of water, on a greased plate, <a href="#Page_8">8</a>;</li>
<li><span style="margin-left: 1em;">on the end of a stick, <a href="#Page_8">8</a>;</span></li>
<li><span style="margin-left: 1em;">in free descent, <a href="#Page_8">8</a>.</span></li>

<li>Dubois, <a href="#Page_218">218</a>.</li>

<li>Dubois-Reymond, <a href="#Page_370">370</a>, footnote.</li>

<li>Dufay, <a href="#Page_271">271</a>.</li>

<li>Dynamics, foundations of, <a href="#Page_153">153</a> et seq.</li>

</ul>

<ul class="IX"><li>Ear, researches in the theory of, <a href="#Page_17">17</a> et seq.;</li>
<li><span style="margin-left: 1em;">diagram of, <a href="#Page_18">18</a>;</span></li>
<li><span style="margin-left: 1em;">its analysis of sounds, <a href="#Page_20">20</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">a puzzle-lock, <a href="#Page_28">28</a>;</span></li>
<li><span style="margin-left: 1em;">reflected in a mirror, <a href="#Page_93">93</a>;</span></li>
<li><span style="margin-left: 1em;">no symmetry in its sensation, <a href="#Page_103">103</a>.</span></li>

<li>Earth, its oblateness not due to its original fluid condition, <a href="#Page_2">2</a>;</li>
<li><span style="margin-left: 1em;">rotation of, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">internal disturbances of, <a href="#Page_285">285</a>.</span></li>

<li>Economical, nature of physical inquiry, <a href="#Page_186">186</a>;</li>
<li><span style="margin-left: 1em;">procedure of the human mind, <a href="#Page_186">186</a>;</span></li>
<li><span style="margin-left: 1em;">order of physics, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">schematism of science, <a href="#Page_206">206</a>;</span></li>
<li><span style="margin-left: 1em;">tools of science, <a href="#Page_207">207</a>;</span></li>
<li><span style="margin-left: 1em;">coefficient of dynamos, <a href="#Page_133">133</a>.</span></li>

<li>Economy, of the actions of nature, <a href="#Page_15">15</a>;</li>
<li><span style="margin-left: 1em;">the purpose of science, <a href="#Page_16">16</a>;</span></li>
<li><span style="margin-left: 1em;">of language, <a href="#Page_191">191</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">of the industrial arts, <a href="#Page_192">192</a>;</span></li>
<li><span style="margin-left: 1em;">of mathematics, <a href="#Page_195">195</a>-<a href="#Page_196">196</a>;</span></li>
<li><span style="margin-left: 1em;">of machinery, <a href="#Page_196">196</a>;</span></li>
<li><span style="margin-left: 1em;">of self-preservation, our first knowledge derived from, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_186">186</a> et seq., <a href="#Page_269">269</a>.</span></li>

<li>Education, higher, <a href="#Page_86">86</a>;</li>
<li><span style="margin-left: 1em;">liberal, <a href="#Page_341">341</a> et seq., <a href="#Page_371">371</a>.</span></li>

<li>Efflux, liquid, <a href="#Page_150">150</a>.</li>

<li>Ego, its nature, <a href="#Page_234">234</a>-<a href="#Page_235">235</a>.</li>

<li>Egypt, <a href="#Page_234">234</a>.</li>

<li>Egyptians, art of, <a href="#Page_78">78</a> et seq., <a href="#Page_201">201</a>.</li>

<li>Eighteenth century, the scientific achievements of, <a href="#Page_187">187</a>, <a href="#Page_188">188</a>.</li>

<li>Eleatics, on motion, <a href="#Page_158">158</a>.</li>

<li>Electrical, attraction and repulsion, <a href="#Page_109">109</a> et seq., <a href="#Page_168">168</a>;</li>
<li><span style="margin-left: 1em;">capacity, <a href="#Page_116">116</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">force, <a href="#Page_110">110</a>, <a href="#Page_119">119</a>, <a href="#Page_168">168</a>;</span></li>
<li><span style="margin-left: 1em;">spark, <a href="#Page_117">117</a>, <a href="#Page_127">127</a>, <a href="#Page_132">132</a>, <a href="#Page_133">133</a>, <a href="#Page_190">190</a>;</span></li>
<li><span style="margin-left: 1em;">energy, measurement of, <a href="#Page_128">128</a> et seq., <a href="#Page_169">169</a>;</span></li>
<li><span style="margin-left: 1em;">currents, conceptions of, <a href="#Page_118">118</a>, <a href="#Page_132">132</a>, <a href="#Page_135">135</a>-<a href="#Page_136">136</a>, <a href="#Page_226">226</a>-<a href="#Page_227">227</a>, <a href="#Page_249">249</a>, <a href="#Page_250">250</a>;</span></li>
<li><span style="margin-left: 1em;">fluids, <a href="#Page_112">112</a> et seq., <a href="#Page_228">228</a>;</span></li>
<li><span style="margin-left: 1em;">pendulums, <a href="#Page_110">110</a>;</span></li>
<li><span style="margin-left: 1em;">levels, <a href="#Page_173">173</a>;</span></li>
<li><span style="margin-left: 1em;">potential, <a href="#Page_121">121</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">quantity, <a href="#Page_111">111</a>, <a href="#Page_118">118</a>, <a href="#Page_119">119</a>.</span></li>

<li>Electricity, as a substance and as a motion, <a href="#Page_170">170</a>;</li>
<li><span style="margin-left: 1em;">difference between the conceptions of heat and, <a href="#Page_168">168</a> et seq.,</span></li>
<li><span style="margin-left: 1em;">rôle of work in, <a href="#Page_120">120</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">galvanic, <a href="#Page_134">134</a>.</span></li>
<li><span style="margin-left: 1em;">See <i>Electrical</i>.</span></li>

<li>Electrometer, W. Thomson's absolute, <a href="#Page_127">127</a>, footnote.</li>

<li>Electrometers, <a href="#Page_122">122</a>, <a href="#Page_127">127</a>.</li>

<li>Electrostatic unit, <a href="#Page_111">111</a>.</li>

<li>Electrostatics, concepts of, <a href="#Page_107">107</a> et seq.</li>

<li>Elements, interdependence of the sensuous, <a href="#Page_179">179</a>;</li>
<li><span style="margin-left: 1em;">of bodies, <a href="#Page_202">202</a>;</span></li>
<li><span style="margin-left: 1em;">of phenomena, equations between, <a href="#Page_205">205</a>;</span></li>
<li><span style="margin-left: 1em;">of sensations, <a href="#Page_200">200</a>;</span></li>
<li><span style="margin-left: 1em;">used instead of sensations, <a href="#Page_208">208</a>-<a href="#Page_209">209</a>.</span></li>

<li>Ellipse, equation of, <a href="#Page_205">205</a>;</li>
<li><span style="margin-left: 1em;">the word, <a href="#Page_342">342</a>.</span></li>

<li>Embryology, possible future state of, <a href="#Page_257">257</a>.</li>

<li>Energies, conformity in the deportment of, <a href="#Page_171">171</a>-<a href="#Page_175">175</a>;</li>
<li><span style="margin-left: 1em;">differences of, <a href="#Page_175">175</a>.</span></li>

<li>Energy, a metrical notion, <a href="#Page_178">178</a>;</li>
<li><span style="margin-left: 1em;">conservation of, <a href="#Page_137">137</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">defined, <a href="#Page_139">139</a>;</span></li>
<li><span style="margin-left: 1em;">metaphysical establishment of the doctrine of, <a href="#Page_183">183</a>;</span></li>
<li><span style="margin-left: 1em;">kinetic, <a href="#Page_177">177</a>;</span></li>
<li><span style="margin-left: 1em;">potential, <a href="#Page_128">128</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">substantial conception of, <a href="#Page_164">164</a>, <a href="#Page_185">185</a>, <a href="#Page_244">244</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">conservation of, in electrical phenomena, <a href="#Page_131">131</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">limits of principle of, <a href="#Page_175">175</a>;</span></li>
<li><span style="margin-left: 1em;">principle of, in physics, <a href="#Page_160">160</a>-<a href="#Page_166">166</a>;</span></li>
<li><span style="margin-left: 1em;">sources of principle of, <a href="#Page_179">179</a>, <a href="#Page_181">181</a>;</span></li>
<li><span style="margin-left: 1em;">thermal, <a href="#Page_177">177</a>;</span></li>
<li><span style="margin-left: 1em;">Thomas Young on, <a href="#Page_173">173</a>.</span></li>

<li>Energy-value of heat, <a href="#Page_178">178</a>, footnote.</li>

<li>Enlightenment, the, <a href="#Page_188">188</a>.</li>

<li>Entropy, a metrical notion, <a href="#Page_178">178</a>.</li>

<li>Environment, stability of our, <a href="#Page_206">206</a>.</li>

<li>Equations for obtaining facts, <a href="#Page_180">180</a>;</li>
<li><span style="margin-left: 1em;">between the elements of phenomena, <a href="#Page_205">205</a>.</span></li>

<li>Equilibrium, conditions of, in simple machines, <a href="#Page_151">151</a>;</li>
<li><span style="margin-left: 1em;">figures of liquid, <a href="#Page_4">4</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">general condition of, <a href="#Page_15">15</a>;</span></li>
<li><span style="margin-left: 1em;">in the State, <a href="#Page_15">15</a>.</span></li>

<li>Etymology, the word, misused for entomology, <a href="#Page_316">316</a>.</li>

<li>Euclid, on consonance and dissonance, <a href="#Page_33">33</a>;</li>
<li><span style="margin-left: 1em;">his geometry, <a href="#Page_364">364</a>.</span></li>

<li><span class="pagenum"><a name="Page_398" id="Page_398">[Pg 398]</a></span>Euler, on the causes of harmony, <a href="#Page_34">34</a>;</li>
<li><span style="margin-left: 1em;">impression of the mathematical processes on, <a href="#Page_196">196</a>;</span></li>
<li><span style="margin-left: 1em;">on the vibrations of strings, <a href="#Page_249">249</a>, <a href="#Page_285">285</a>, <a href="#Page_376">376</a>.</span></li>

<li>Euler and Hermann's principle, <a href="#Page_149">149</a>.</li>

<li>Euthyphron, questioned by Socrates, <a href="#Page_1">1</a>.</li>

<li>Evolute, the word, <a href="#Page_342">342</a>.</li>

<li>Evolution, theory of, as applied to ideas, <a href="#Page_216">216</a> et seq.</li>

<li>Ewald, <a href="#Page_298">298</a>, <a href="#Page_304">304</a>.</li>

<li>Excluded perpetual motion, logical root of the principle of, <a href="#Page_182">182</a>.</li>

<li>Exner, S., <a href="#Page_302">302</a>, <a href="#Page_305">305</a>.</li>

<li>Experience, communication of, <a href="#Page_191">191</a>;</li>
<li><span style="margin-left: 1em;">our ready, <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">the principle of energy derived from, <a href="#Page_179">179</a>;</span></li>
<li><span style="margin-left: 1em;">the wellspring of all knowledge of nature, <a href="#Page_181">181</a>;</span></li>
<li><span style="margin-left: 1em;">incongruence between thought and, <a href="#Page_206">206</a>.</span></li>

<li>Experimental research, function of, <a href="#Page_181">181</a>.</li>

<li>Explanation, nature of, <a href="#Page_194">194</a>, <a href="#Page_237">237</a>, <a href="#Page_362">362</a>.</li>

<li>Eye, cannot analyse colors, <a href="#Page_20">20</a>;</li>
<li><span style="margin-left: 1em;">researches in the theory of the, <a href="#Page_18">18</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">loss of, as affecting vision, <a href="#Page_98">98</a>.</span></li>

<li>Eyes, purpose of, <a href="#Page_66">66</a> et seq.;</li>
<li><span style="margin-left: 1em;">their structure symmetrical not identical, <a href="#Page_96">96</a>.</span></li>

</ul>

<ul class="IX"><li>Face, human, inverted, <a href="#Page_95">95</a>.</li>

<li>Facts and ideas, necessary to science, <a href="#Page_231">231</a>.</li>

<li>Facts, description of, <a href="#Page_108">108</a>;</li>
<li><span style="margin-left: 1em;">agreement of, <a href="#Page_180">180</a>;</span></li>
<li><span style="margin-left: 1em;">relations of, <a href="#Page_180">180</a>;</span></li>
<li><span style="margin-left: 1em;">how represented, <a href="#Page_206">206</a>;</span></li>
<li><span style="margin-left: 1em;">reflected in imagination, <a href="#Page_220">220</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the result of constructions, <a href="#Page_253">253</a>;</span></li>
<li><span style="margin-left: 1em;">a continuum of, <a href="#Page_256">256</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">equations for obtaining, <a href="#Page_180">180</a>.</span></li>

<li>Falling bodies, <a href="#Page_204">204</a>, <a href="#Page_215">215</a>;</li>
<li><span style="margin-left: 1em;">Galileo on the law of, <a href="#Page_143">143</a> et seq., <a href="#Page_284">284</a>.</span></li>

<li>Falling, cats, <a href="#Page_303">303</a>, footnote.</li>

<li>Falstaff, <a href="#Page_309">309</a>.</li>

<li>Familiar intermediate links of thought, <a href="#Page_198">198</a>.</li>

<li>Faraday, <a href="#Page_191">191</a>, <a href="#Page_217">217</a>, <a href="#Page_237">237</a>;</li>
<li><span style="margin-left: 1em;">his conception of electricity, <a href="#Page_114">114</a>, <a href="#Page_271">271</a>.</span></li>

<li>Fechner, theory of Corti's fibres, <a href="#Page_19">19</a> et seq.</li>

<li>Feeling, cannot be explained by motions of atoms, <a href="#Page_208">208</a> et seq.</li>

<li>Fetishism, <a href="#Page_186">186</a>, <a href="#Page_243">243</a>, <a href="#Page_254">254</a>;</li>
<li><span style="margin-left: 1em;">in our physical concepts, <a href="#Page_187">187</a>.</span></li>

<li>Fibres of Corti, <a href="#Page_17">17</a> et seq.</li>

<li>Fick, his theory of diffusion, <a href="#Page_249">249</a>.</li>

<li>Figures, symmetry of, <a href="#Page_92">92</a> et seq.</li>

<li>Figures of liquid equilibrium, <a href="#Page_4">4</a> et seq.</li>

<li>Fire, use of, <a href="#Page_264">264</a>.</li>

<li>Fishes, <a href="#Page_306">306</a>.</li>

<li>Fixed note, determining of a, <a href="#Page_377">377</a>.</li>

<li>Fizeau, his determination of the velocity of light, <a href="#Page_55">55</a> et seq.</li>

<li>Flats, reversed into sharps, <a href="#Page_101">101</a>.</li>

<li>Flouren's experiments, <a href="#Page_272">272</a>, <a href="#Page_290">290</a>.</li>

<li>Flower-girl, the baskets of a, <a href="#Page_95">95</a>.</li>

<li>Fluids, electrical, <a href="#Page_112">112</a> et seq.</li>

<li>Force, electric, <a href="#Page_110">110</a>, <a href="#Page_119">119</a>, <a href="#Page_168">168</a>;</li>
<li><span style="margin-left: 1em;">unit of <a href="#Page_111">111</a>;</span></li>
<li><span style="margin-left: 1em;">living, <a href="#Page_137">137</a>, <a href="#Page_149">149</a>, <a href="#Page_184">184</a>;</span></li>
<li><span style="margin-left: 1em;">generally <a href="#Page_253">253</a>.</span></li>
<li><span style="margin-left: 1em;">See the related headings.</span></li>

<li>Forces, will compared to, <a href="#Page_254">254</a>.</li>

<li>Foreseeing events, <a href="#Page_220">220</a> et seq.</li>

<li>Formal conceptions, rôle of, <a href="#Page_183">183</a>.</li>

<li>Formal need of a clear view of facts, <a href="#Page_183">183</a>, <a href="#Page_246">246</a>;</li>
<li><span style="margin-left: 1em;">how far it corresponds to nature, <a href="#Page_184">184</a>.</span></li>

<li>Formative forces of liquids, <a href="#Page_4">4</a>.</li>

<li>Forms of liquids, <a href="#Page_3">3</a> et seq.</li>

<li>Forward movement, sensation of, <a href="#Page_300">300</a>.</li>

<li>Forwards, prophesying, <a href="#Page_253">253</a>.</li>

<li>Foucault, <a href="#Page_57">57</a>, <a href="#Page_70">70</a>, <a href="#Page_296">296</a>.</li>

<li>Foucault and Toepler, method of, for detecting optical faults, <a href="#Page_313">313</a> et seq., <a href="#Page_320">320</a>.</li>

<li>Foundation of scientific thought, primitive acts of knowledge, the, <a href="#Page_190">190</a>.</li>

<li>Fourier, on processes of heat, <a href="#Page_249">249</a>, <a href="#Page_278">278</a>.</li>

<li>Fox, a, <a href="#Page_234">234</a>.</li>

<li>Franklin's pane, <a href="#Page_116">116</a>.</li>

<li>Frary, <a href="#Page_338">338</a>, footnote.</li>

<li>Fraunhofer, <a href="#Page_271">271</a>.</li>

<li>Freezing-point, lowered by pressure, <a href="#Page_162">162</a>.</li>

<li>Fresnel, <a href="#Page_271">271</a>.</li>

<li>Fritsch, <a href="#Page_321">321</a>.</li>

<li>Frogs, larvæ of, not subject to vertigo, <a href="#Page_298">298</a>.</li>

<li>Froude, <a href="#Page_333">333</a>.</li>

<li>Frustra, misuse of the word, <a href="#Page_345">345</a>.</li>

<li>Future, science of the, <a href="#Page_213">213</a>.</li>

</ul>

<ul class="IX"><li>Galileo, on the motion of pendulums, <a href="#Page_21">21</a>;</li>
<li><span style="margin-left: 1em;">his attempted measurement of the velocity of light, <a href="#Page_50">50</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">his exclusion of a perpetual motion, <a href="#Page_143">143</a>;</span></li>
<li><span style="margin-left: 1em;">on velocities acquired in free descent, <a href="#Page_143">143</a>-<a href="#Page_147">147</a>;</span></li>
<li><span style="margin-left: 1em;">on the law of inertia, <a href="#Page_146">146</a>-<a href="#Page_147">147</a>;</span></li>
<li><span style="margin-left: 1em;">on virtual velocities, <a href="#Page_150">150</a>;</span></li>
<li><span style="margin-left: 1em;">on work, <a href="#Page_172">172</a>;</span></li>
<li><span style="margin-left: 1em;">his laws of descent, <a href="#Page_193">193</a>;</span></li>
<li><span style="margin-left: 1em;">on falling bodies, <a href="#Page_225">225</a>;</span></li>
<li><span style="margin-left: 1em;">great results of his study of nature, <a href="#Page_214">214</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">his rude scientific implements, <a href="#Page_215">215</a>;</span></li>
<li><span style="margin-left: 1em;">selections from his works for use in instruction, <a href="#Page_368">368</a>;</span></li>
<li><span class="pagenum"><a name="Page_399" id="Page_399">[Pg 399]</a></span><span style="margin-left: 1em;">also <a href="#Page_105">105</a>, <a href="#Page_182">182</a>, <a href="#Page_187">187</a>, <a href="#Page_237">237</a>, <a href="#Page_272">272</a>, <a href="#Page_274">274</a>, <a href="#Page_283">283</a>.</span></li>

<li>Galle, observes the planet Neptune, <a href="#Page_29">29</a>.</li>

<li>Galvanic, electricity, <a href="#Page_134">134</a>;</li>
<li><span style="margin-left: 1em;">current, <a href="#Page_132">132</a>;</span></li>
<li><span style="margin-left: 1em;">dizziness, <a href="#Page_291">291</a>;</span></li>
<li><span style="margin-left: 1em;">vertigo, <a href="#Page_298">298</a>.</span></li>

<li>Galvanoscope, <a href="#Page_135">135</a>.</li>

<li>Galvanotropism, <a href="#Page_291">291</a>.</li>

<li>Garda, Lake, <a href="#Page_239">239</a>.</li>

<li>Gas, the word, <a href="#Page_264">264</a>;</li>
<li><span style="margin-left: 1em;">mass of, enclosed in a cylinder, <a href="#Page_179">179</a>.</span></li>

<li>Gases, tensions of, for scales of temperature, <a href="#Page_174">174</a>.</li>

<li>Gauss, on the foundations of dynamics, <a href="#Page_154">154</a>;</li>
<li><span style="margin-left: 1em;">his principle, <a href="#Page_154">154</a>;</span></li>
<li><span style="margin-left: 1em;">also, <a href="#Page_108">108</a>, <a href="#Page_274">274</a>.</span></li>

<li>Genius, <a href="#Page_279">279</a>, <a href="#Page_280">280</a>.</li>

<li>Geography, comparison in, <a href="#Page_239">239</a>.</li>

<li>Geometers, in our eyes, <a href="#Page_72">72</a>.</li>

<li>Geotropism, <a href="#Page_289">289</a>.</li>

<li>German schools and gymnasiums, <a href="#Page_372">372</a>, <a href="#Page_373">373</a>, <a href="#Page_338">338</a>, footnote.</li>

<li>Ghosts, photographic, <a href="#Page_73">73</a>.</li>

<li>Glass, invisible in a mixture of the same refrangibility, <a href="#Page_312">312</a>;</li>
<li><span style="margin-left: 1em;">powdered, visible in a mixture of the same refrangibility, <a href="#Page_312">312</a>.</span></li>

<li>Glove, in a mirror, <a href="#Page_93">93</a>.</li>

<li>Goethe, quotations from, <a href="#Page_9">9</a>, <a href="#Page_31">31</a>, <a href="#Page_49">49</a>, <a href="#Page_88">88</a>;</li>
<li><span style="margin-left: 1em;">on the cause of harmony, <a href="#Page_35">35</a>.</span></li>

<li>Goltz, <a href="#Page_282">282</a>, <a href="#Page_291">291</a>.</li>

<li>Gossot, <a href="#Page_332">332</a>.</li>

<li>Gothic cathedral, <a href="#Page_94">94</a>.</li>

<li>Gravitation, discovery of, <a href="#Page_225">225</a> et seq.</li>

<li>Gravity, how to get rid of the effects of, in liquids, <a href="#Page_4">4</a>;</li>
<li><span style="margin-left: 1em;">also <a href="#Page_228">228</a>.</span></li>

<li>Gray, Elisha, his telautograph, <a href="#Page_26">26</a>.</li>

<li>Greased plate, drop of water on a, <a href="#Page_8">8</a>.</li>

<li>Great minds, idiosyncrasies of, <a href="#Page_247">247</a>.</li>

<li>Greek language, scientific terms derivedfrom, <a href="#Page_342">342</a>-<a href="#Page_343">343</a>;</li>
<li><span style="margin-left: 1em;">common words derived from, <a href="#Page_343">343</a>, footnote;</span></li>
<li><span style="margin-left: 1em;">still necessary for some professions, <a href="#Page_346">346</a>;</span></li>
<li><span style="margin-left: 1em;">its literary wealth, <a href="#Page_347">347</a>-<a href="#Page_348">348</a>;</span></li>
<li><span style="margin-left: 1em;">narrowness and one-sidedness of its literature, <a href="#Page_348">348</a>-<a href="#Page_349">349</a>;</span></li>
<li><span style="margin-left: 1em;">its excessive study useless, <a href="#Page_349">349</a>-<a href="#Page_350">350</a>;</span></li>
<li><span style="margin-left: 1em;">its study sharpens the judgment, <a href="#Page_357">357</a>-<a href="#Page_358">358</a>;</span></li>
<li><span style="margin-left: 1em;">a knowledge of it not necessary to a liberal education, <a href="#Page_371">371</a>.</span></li>

<li>Greeks, their provinciality and narrow-mindedness, <a href="#Page_349">349</a>;</li>
<li><span style="margin-left: 1em;">now only objects of historical research, <a href="#Page_350">350</a>.</span></li>

<li>Griesinger, <a href="#Page_184">184</a>.</li>

<li>Grimaldi, <a href="#Page_270">270</a>.</li>

<li>Grimm, <a href="#Page_344">344</a>, footnote.</li>

<li>Grunting fishes, <a href="#Page_306">306</a>.</li>

</ul>

<ul class="IX"><li>Habitudes of thought, <a href="#Page_199">199</a>, <a href="#Page_224">224</a>, <a href="#Page_227">227</a>, <a href="#Page_232">232</a>.</li>

<li>Haeckel, <a href="#Page_222">222</a>, <a href="#Page_235">235</a>.</li>

<li>Hamilton, deduction of the conical refraction of light, <a href="#Page_29">29</a>.</li>

<li>Hankel, <a href="#Page_364">364</a>.</li>

<li>Harmonics, <a href="#Page_38">38</a>, <a href="#Page_40">40</a>.</li>

<li>Harmony, on the causes of, <a href="#Page_32">32</a> et seq.;</li>
<li><span style="margin-left: 1em;">laws of the theory of, explained, <a href="#Page_30">30</a>;</span></li>
<li><span style="margin-left: 1em;">the investigation of the ancients concerning, <a href="#Page_32">32</a>;</span></li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_103">103</a>.</span></li>
<li><span style="margin-left: 1em;">See <i>Consonance</i>.</span></li>

<li>Harris, electrical balance of, <a href="#Page_127">127</a>, footnote.</li>

<li>Hartwich, Judge, <a href="#Page_343">343</a>, <a href="#Page_353">353</a>, footnote.</li>

<li>Hat, a high silk, <a href="#Page_24">24</a>.</li>

<li>Hats, ladies', development of, <a href="#Page_64">64</a>.</li>

<li>Head-wave of a projectile, <a href="#Page_323">323</a> et seq.</li>

<li>Hearing and orientation, relation between, <a href="#Page_304">304</a> et seq.</li>

<li>Heat, a material substance, <a href="#Page_177">177</a>;</li>
<li><span style="margin-left: 1em;">difference between the conceptions of electricity and, <a href="#Page_168">168</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">substantial conception of, <a href="#Page_243">243</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">Carnot on, <a href="#Page_156">156</a>, <a href="#Page_160">160</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">Fourier on the conduction of, <a href="#Page_249">249</a>;</span></li>
<li><span style="margin-left: 1em;">not necessarily a motion, <a href="#Page_167">167</a>, <a href="#Page_170">170</a>, <a href="#Page_171">171</a>;</span></li>
<li><span style="margin-left: 1em;">mechanical equivalent of, <a href="#Page_164">164</a>, <a href="#Page_167">167</a>;</span></li>
<li><span style="margin-left: 1em;">of liquefaction, <a href="#Page_178">178</a>;</span></li>
<li><span style="margin-left: 1em;">quantity of, <a href="#Page_166">166</a>;</span></li>
<li><span style="margin-left: 1em;">latent, <a href="#Page_167">167</a>, <a href="#Page_178">178</a>, <a href="#Page_244">244</a>;</span></li>
<li><span style="margin-left: 1em;">specific, <a href="#Page_166">166</a>, <a href="#Page_244">244</a>;</span></li>
<li><span style="margin-left: 1em;">the conceptions of, <a href="#Page_160">160</a>-<a href="#Page_171">171</a>;</span></li>
<li><span style="margin-left: 1em;">machine, <a href="#Page_160">160</a>;</span></li>
<li><span style="margin-left: 1em;">a measure of electrical energy, <a href="#Page_133">133</a> et seq.;</span></li>
<li><span class="pagenum"><a name="Page_400" id="Page_400">[Pg 400]</a></span><span style="margin-left: 1em;">mechanical theory of, <a href="#Page_133">133</a>;</span></li>
<li><span style="margin-left: 1em;">where does it come from? <a href="#Page_200">200</a>.</span></li>

<li>Heavy bodies, sinking of, <a href="#Page_222">222</a>.</li>

<li>Heights of ascent, <a href="#Page_143">143</a>-<a href="#Page_151">151</a>.</li>

<li>Helm, <a href="#Page_172">172</a>.</li>

<li>Helmholtz, applies the principle of energy to electricity, <a href="#Page_184">184</a>;</li>
<li><span style="margin-left: 1em;">his telestereoscope, <a href="#Page_84">84</a>;</span></li>
<li><span style="margin-left: 1em;">his theory of Corti's fibres, <a href="#Page_19">19</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">on harmony, <a href="#Page_35">35</a>, <a href="#Page_99">99</a>;</span></li>
<li><span style="margin-left: 1em;">on the conservation of energy, <a href="#Page_165">165</a>, <a href="#Page_247">247</a>;</span></li>
<li><span style="margin-left: 1em;">his method of thought, <a href="#Page_247">247</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_138">138</a>, <a href="#Page_305">305</a>, <a href="#Page_307">307</a>, <a href="#Page_375">375</a>, <a href="#Page_383">383</a>.</span></li>

<li>Hensen, V., on the auditory function of the filaments of Crustacea, <a href="#Page_29">29</a>, <a href="#Page_302">302</a>.</li>

<li>Herbart, <a href="#Page_386">386</a> et seq.</li>

<li>Herbartians, on motion, <a href="#Page_158">158</a>.</li>

<li>Herculaneum, art in, <a href="#Page_80">80</a>.</li>

<li>Heredity, in organic and inorganic matter, <a href="#Page_216">216</a>, footnote.</li>

<li>Hering, on development, <a href="#Page_222">222</a>;</li>
<li><span style="margin-left: 1em;">on vision, <a href="#Page_210">210</a>.</span></li>

<li>Hermann, E., on the economy of the industrial arts, <a href="#Page_192">192</a>.</li>

<li>Hermann, L., <a href="#Page_291">291</a>.</li>

<li>Herodotus, <a href="#Page_26">26</a>, <a href="#Page_234">234</a>, <a href="#Page_347">347</a>, <a href="#Page_350">350</a>.</li>

<li>Hertz, his waves, <a href="#Page_242">242</a>;</li>
<li><span style="margin-left: 1em;">his use of the phrase "prophesy," <a href="#Page_253">253</a>.</span></li>

<li>Herzen, <a href="#Page_361">361</a>, footnote.</li>

<li>Hindu mathematicians, their beautiful problems, <a href="#Page_30">30</a>.</li>

<li>Holtz's electric machine, <a href="#Page_132">132</a>.</li>

<li>Horse, <a href="#Page_63">63</a>.</li>

<li>Household, physics compared to a well-kept, <a href="#Page_197">197</a>.</li>

<li>Housekeeping in science and civil life, <a href="#Page_198">198</a>.</li>

<li>Hudson, the, <a href="#Page_94">94</a>.</li>

<li>Human beings, puzzle-locks, <a href="#Page_27">27</a>.</li>

<li>Human body, our knowledge of, <a href="#Page_90">90</a>.</li>

<li>Human mind, must proceed economically, <a href="#Page_186">186</a>.</li>

<li>Humanity, likened to a polyp-plant, <a href="#Page_235">235</a>.</li>

<li>Huygens, his mechanical view of physics, <a href="#Page_155">155</a>;</li>
<li><span style="margin-left: 1em;">on the nature of light and heat, <a href="#Page_155">155</a>-<a href="#Page_156">156</a>;</span></li>
<li><span style="margin-left: 1em;">his principle of the heights of ascent, <a href="#Page_149">149</a>;</span></li>
<li><span style="margin-left: 1em;">on the law of inertia and the motion of a compound pendulum, <a href="#Page_147">147</a>-<a href="#Page_149">149</a>;</span></li>
<li><span style="margin-left: 1em;">on the impossible perpetual motion, <a href="#Page_147">147</a>-<a href="#Page_148">148</a>;</span></li>
<li><span style="margin-left: 1em;">on work, <a href="#Page_173">173</a>;</span></li>
<li><span style="margin-left: 1em;">selections from his works for use in instruction, <a href="#Page_368">368</a>;</span></li>
<li><span style="margin-left: 1em;">his view of light, <a href="#Page_227">227</a>-<a href="#Page_228">228</a>, <a href="#Page_262">262</a>.</span></li>

<li>Huygens, optical method for detecting imperfections in optical glasses <a href="#Page_313">313</a>.</li>

<li>Hydrogen balloon, <a href="#Page_199">199</a>.</li>

<li>Hydrostatics, Stevinus's principle of, <a href="#Page_141">141</a>.</li>

<li>Hypotheses, their rôle in explanation, <a href="#Page_228">228</a> et seq.</li>

</ul>

<ul class="IX"><li>Ichthyornis, <a href="#Page_257">257</a>.</li>

<li>Ichthyosaurus, <a href="#Page_63">63</a>.</li>

<li>Idea? what is a theoretical, <a href="#Page_241">241</a>.</li>

<li>Idealism, <a href="#Page_209">209</a>.</li>

<li>Ideas, a product of organic nature, <a href="#Page_217">217</a> et seq.;</li>
<li><span style="margin-left: 1em;">and facts, necessary to science, <a href="#Page_231">231</a>;</span></li>
<li><span style="margin-left: 1em;">not all of life, <a href="#Page_233">233</a>;</span></li>
<li><span style="margin-left: 1em;">their growth and importance, <a href="#Page_233">233</a>;</span></li>
<li><span style="margin-left: 1em;">a product of universal evolution, <a href="#Page_235">235</a>;</span></li>
<li><span style="margin-left: 1em;">the history of, <a href="#Page_227">227</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">in great minds, <a href="#Page_228">228</a>;</span></li>
<li><span style="margin-left: 1em;">the rich contents of, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">their unsettled character in common life, their clarification in science, <a href="#Page_1">1</a>-<a href="#Page_2">2</a>.</span></li>

<li>Ideography, the Chinese, <a href="#Page_192">192</a>.</li>

<li>Imagery, mental, <a href="#Page_253">253</a>.</li>

<li>Imagination, facts reflected in, <a href="#Page_220">220</a> et seq.</li>

<li>Inclined plane, law of, <a href="#Page_140">140</a>-<a href="#Page_141">141</a>.</li>

<li>Incomprehensible, the, <a href="#Page_186">186</a>.</li>

<li>Indian, his modes of conception and interpretation, <a href="#Page_218">218</a> et seq.</li>

<li>Individual, a thread on which pearls are strung, <a href="#Page_234">234</a>-<a href="#Page_235">235</a>.</li>

<li>Industrial arts, economy of the, E. Hermann on, <a href="#Page_192">192</a>.</li>

<li>Inertia, law of, <a href="#Page_143">143</a> et seq., <a href="#Page_146">146</a> et seq., <a href="#Page_216">216</a>, footnote, <a href="#Page_283">283</a> et seq.</li>

<li>Innate concepts of the understanding, Kant on, <a href="#Page_199">199</a>.</li>

<li>Innervation, visual, <a href="#Page_99">99</a>.</li>

<li>Inquirer, his division of labor, <a href="#Page_105">105</a>;</li>
<li><span style="margin-left: 1em;">compared to a shoemaker, <a href="#Page_105">105</a>-<a href="#Page_106">106</a>;</span></li>
<li><span style="margin-left: 1em;">what constitutes the great, <a href="#Page_191">191</a>;</span></li>
<li><span style="margin-left: 1em;">the true, seeks the truth everywhere, <a href="#Page_63">63</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the, compared to a wooer, <a href="#Page_45">45</a>.</span></li>

<li>Instinctive knowledge, <a href="#Page_189">189</a>, <a href="#Page_190">190</a>.</li>

<li><span class="pagenum"><a name="Page_401" id="Page_401">[Pg 401]</a></span>Instruction, aim of, the saving of experience, <a href="#Page_191">191</a>;</li>
<li><span style="margin-left: 1em;">in the classics, mathematics, and sciences, <a href="#Page_338">338</a>-<a href="#Page_374">374</a>;</span></li>
<li><span style="margin-left: 1em;">limitation of matter of, <a href="#Page_365">365</a> et seq.</span></li>

<li>Insulators, <a href="#Page_130">130</a>.</li>

<li>Integrals, <a href="#Page_195">195</a>.</li>

<li>Intellectual development, conditions of, <a href="#Page_286">286</a> et seq.</li>

<li>Intentions, acts of nature compared to, <a href="#Page_14">14</a>-<a href="#Page_15">15</a>.</li>

<li>Interconnexion of nature, <a href="#Page_182">182</a>.</li>

<li>Interdependence, of properties, <a href="#Page_361">361</a>;</li>
<li><span style="margin-left: 1em;">of the sensuous elements of the world, <a href="#Page_179">179</a>.</span></li>

<li>Interference experiments with the head-wave of moving projectiles, <a href="#Page_327">327</a>-<a href="#Page_328">328</a>.</li>

<li>International intercourse, established by Latin, <a href="#Page_341">341</a>.</li>

<li>International measures, <a href="#Page_108">108</a>.</li>

<li>Invention, discovery and, distinction between, <a href="#Page_269">269</a>.</li>

<li>Inventions, requisites for the development of, <a href="#Page_266">266</a>, <a href="#Page_268">268</a> et seq.</li>

<li>Iron-filings, <a href="#Page_220">220</a>, <a href="#Page_243">243</a>.</li>

<li>Italian art, <a href="#Page_234">234</a>.</li>

</ul>

<ul class="IX"><li>Jacobi, C. G. J., on mathematics, <a href="#Page_280">280</a>.</li>

<li>James, W., <a href="#Page_275">275</a>, <a href="#Page_299">299</a>.</li>

<li>Java, <a href="#Page_163">163</a>.</li>

<li>Jews and Christians, monotheism of the, <a href="#Page_187">187</a>.</li>

<li>Jolly, Professor von, <a href="#Page_112">112</a>, <a href="#Page_274">274</a>.</li>

<li>Joule, J. P., on the conservation of energy, <a href="#Page_163">163</a>-<a href="#Page_165">165</a>, <a href="#Page_167">167</a>, <a href="#Page_183">183</a>;</li>
<li><span style="margin-left: 1em;">his conception of energy, <a href="#Page_245">245</a>;</span></li>
<li><span style="margin-left: 1em;">his metaphysics, <a href="#Page_183">183</a>, <a href="#Page_246">246</a>;</span></li>
<li><span style="margin-left: 1em;">his method of thought, <a href="#Page_247">247</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_137">137</a>, <a href="#Page_138">138</a>.</span></li>

<li>Journée, <a href="#Page_317">317</a>.</li>

<li>Judge, criminal, the natural philosopher compared to a, <a href="#Page_48">48</a>.</li>

<li>Judgment, essentially economy of thought, <a href="#Page_201">201</a>-<a href="#Page_202">202</a>;</li>
<li><span style="margin-left: 1em;">sharpened by languages and sciences, <a href="#Page_357">357</a>-<a href="#Page_358">358</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_232">232</a>-<a href="#Page_233">233</a>, <a href="#Page_238">238</a>.</span></li>

<li>Juliet, Romeo and, <a href="#Page_87">87</a>.</li>

<li>Jupiter, its satellites employed in the determination of the velocity of light, <a href="#Page_51">51</a> et seq.</li>

<li>Jurisprudence, Latin and Greek unnecessary for the study of, <a href="#Page_346">346</a>, footnote.</li>

</ul>

<ul class="IX"><li>Kant, his hypothesis of the origin of the planetary system, <a href="#Page_5">5</a>;</li>
<li><span style="margin-left: 1em;">his <i>Critique of Pure Reason</i>, <a href="#Page_188">188</a>;</span></li>
<li><span style="margin-left: 1em;">on innate concepts of the understanding, <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">on time, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">also footnote, <a href="#Page_93">93</a>.</span></li>

<li>Kepler, <a href="#Page_187">187</a>, <a href="#Page_270">270</a>.</li>

<li>Kinetic energy, <a href="#Page_177">177</a>.</li>

<li>Kirchhoff, his epistemological ideas, <a href="#Page_257">257</a>-<a href="#Page_258">258</a>;</li>
<li><span style="margin-left: 1em;">his definition of mechanics, <a href="#Page_236">236</a>, <a href="#Page_258">258</a>, <a href="#Page_271">271</a>, <a href="#Page_273">273</a>.</span></li>

<li>Knight, <a href="#Page_289">289</a>.</li>

<li>Knowledge, a product of organic nature, <a href="#Page_217">217</a> et seq., <a href="#Page_235">235</a>;</li>
<li><span style="margin-left: 1em;">instinctive, <a href="#Page_190">190</a>;</span></li>
<li><span style="margin-left: 1em;">made possible by economy of thought, <a href="#Page_198">198</a>;</span></li>
<li><span style="margin-left: 1em;">our first, derived from the economy of self-preservation, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">the theory of, <a href="#Page_203">203</a>;</span></li>
<li><span style="margin-left: 1em;">our primitive acts of the foundation of science, <a href="#Page_190">190</a>.</span></li>

<li>Kocher, <a href="#Page_328">328</a>.</li>

<li>Koenig, measurement of the velocity of sound, <a href="#Page_57">57</a> et seq.</li>

<li>Kölliker, <a href="#Page_19">19</a>.</li>

<li>Kopisch, <a href="#Page_61">61</a>.</li>

<li>Kreidl, <a href="#Page_299">299</a>, <a href="#Page_302">302</a>, <a href="#Page_306">306</a>;</li>
<li><span style="margin-left: 1em;">his experiments, <a href="#Page_272">272</a>.</span></li>

<li>Krupp, <a href="#Page_319">319</a>.</li>

</ul>

<ul class="IX"><li>Labels, the value of, <a href="#Page_201">201</a>.</li>

<li>Labor, the accumulation of, the foundation of wealth and power, <a href="#Page_198">198</a>;</li>
<li><span style="margin-left: 1em;">inquirer's division of, <a href="#Page_105">105</a>, <a href="#Page_258">258</a>.</span></li>

<li>Labyrinth, of the ear, <a href="#Page_18">18</a>, <a href="#Page_291">291</a>, <a href="#Page_305">305</a>.</li>

<li>Lactantius, on the study of moral and physical science, <a href="#Page_89">89</a>.</li>

<li>Ladder of our abstraction, the, <a href="#Page_208">208</a>.</li>

<li>Ladies, their eyes, <a href="#Page_71">71</a>;</li>
<li><span style="margin-left: 1em;">like tuning-forks, <a href="#Page_23">23</a>-<a href="#Page_24">24</a>.</span></li>

<li>Lagrange, on Huygens's principle, <a href="#Page_149">149</a>;</li>
<li><span style="margin-left: 1em;">on the principle of virtual velocities, <a href="#Page_150">150</a>-<a href="#Page_155">155</a>;</span></li>
<li><span style="margin-left: 1em;">character of the intellectual activity of a, <a href="#Page_195">195</a>, <a href="#Page_278">278</a>.</span></li>

<li>Lake-dwellers, <a href="#Page_46">46</a>, <a href="#Page_271">271</a>.</li>

<li>Lamp-shade, <a href="#Page_70">70</a>.</li>

<li>Lane's unit jar, <a href="#Page_115">115</a>.</li>

<li>Language, knowledge of the nature of, demanded by a liberal education, <a href="#Page_356">356</a>;</li>
<li><span style="margin-left: 1em;">relationship between, and thought, <a href="#Page_358">358</a>;</span></li>
<li><span style="margin-left: 1em;">communication by <a href="#Page_237">237</a>;</span></li>
<li><span style="margin-left: 1em;">economy of, <a href="#Page_191">191</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">human its character, <a href="#Page_238">238</a>;</span></li>
<li><span class="pagenum"><a name="Page_402" id="Page_402">[Pg 402]</a></span><span style="margin-left: 1em;">of animals, <a href="#Page_238">238</a>;</span></li>
<li><span style="margin-left: 1em;">instruction in, <a href="#Page_338">338</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">its methods, <a href="#Page_192">192</a>.</span></li>

<li>Laplace, on the atoms of the brain, <a href="#Page_188">188</a>;</li>
<li><span style="margin-left: 1em;">on the scientific achievements of the eighteenth century, <a href="#Page_188">188</a>;</span></li>
<li><span style="margin-left: 1em;">his hypothesis of the origin of the planetary system, <a href="#Page_5">5</a>.</span></li>

<li>Latent heat, <a href="#Page_167">167</a>, <a href="#Page_178">178</a>, <a href="#Page_244">244</a>.</li>

<li>Latin city of Maupertuis, <a href="#Page_339">339</a>.</li>

<li>Latin, instruction in, <a href="#Page_311">311</a> et seq.;</li>
<li><span style="margin-left: 1em;">introduced with the Christian Church, <a href="#Page_340">340</a>;</span></li>
<li><span style="margin-left: 1em;">the language of scholars, the medium of international intercourse, its power, utility, and final abandonment, <a href="#Page_341">341</a>-<a href="#Page_347">347</a>;</span></li>
<li><span style="margin-left: 1em;">the wealth of its literature, <a href="#Page_348">348</a>;</span></li>
<li><span style="margin-left: 1em;">the excessive study of, <a href="#Page_346">346</a>, <a href="#Page_349">349</a>, <a href="#Page_354">354</a>, <a href="#Page_355">355</a>;</span></li>
<li><span style="margin-left: 1em;">its power to sharpen the judgment, <a href="#Page_357">357</a>-<a href="#Page_358">358</a>.</span></li>

<li>Lavish extravagance of science, <a href="#Page_189">189</a>.</li>

<li>Law, a, defined, <a href="#Page_256">256</a>;</li>
<li><span style="margin-left: 1em;">a natural, not contained in the conformity of the energies, <a href="#Page_175">175</a>.</span></li>

<li>Law-maker, motives of not always discernible, <a href="#Page_9">9</a>.</li>

<li>Layard, <a href="#Page_79">79</a>.</li>

<li>Learning, its nature, <a href="#Page_366">366</a> et seq.</li>

<li>Least superficial area, principle of, accounted for by the mutual attractions of liquid particles, <a href="#Page_13">13</a>-<a href="#Page_14">14</a>;</li>
<li><span style="margin-left: 1em;">illustrated by a pulley arrangement, <a href="#Page_12">12</a>-<a href="#Page_13">13</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_9">9</a> et seq.</span></li>

<li>Leibnitz, on harmony, <a href="#Page_33">33</a>;</li>
<li><span style="margin-left: 1em;">on international intercourse, <a href="#Page_342">342</a>, footnote.</span></li>

<li>Lessing, quotation from, <a href="#Page_47">47</a>.</li>

<li>Letters of the alphabet, their symmetry, <a href="#Page_94">94</a>, <a href="#Page_97">97</a>.</li>

<li>Level heights of work, <a href="#Page_172">172</a>-<a href="#Page_174">174</a>.</li>

<li>Lever, a, in action, <a href="#Page_222">222</a>.</li>

<li>Leverrier, prediction of the planet Neptune, <a href="#Page_29">29</a>.</li>

<li>Leyden jar, <a href="#Page_114">114</a>.</li>

<li>Liberal education, a, <a href="#Page_341">341</a> et seq., <a href="#Page_359">359</a>, <a href="#Page_371">371</a>.</li>

<li>Libraries, thoughts stored up in, <a href="#Page_237">237</a>.</li>

<li>Lichtenberg, on instruction, <a href="#Page_276">276</a>, <a href="#Page_370">370</a>.</li>

<li>Licius, a Chinese philosopher, <a href="#Page_213">213</a>.</li>

<li>Liebig, <a href="#Page_163">163</a>, <a href="#Page_278">278</a>.</li>

<li>Life and death, <a href="#Page_186">186</a>.</li>

<li>Light, history of as elucidating how theories obstruct research, <a href="#Page_242">242</a>;</li>
<li><span style="margin-left: 1em;">Huygens's and Newton's views of, <a href="#Page_227">227</a>-<a href="#Page_228">228</a>;</span></li>
<li><span style="margin-left: 1em;">its different conceptions, <a href="#Page_226">226</a>;</span></li>
<li><span style="margin-left: 1em;">rectilinear propagation of, <a href="#Page_194">194</a>;</span></li>
<li><span style="margin-left: 1em;">rôle of, in vision, <a href="#Page_81">81</a>;</span></li>
<li><span style="margin-left: 1em;">spatial and temporal periodicity of, explains optical phenomena, <a href="#Page_194">194</a>;</span></li>
<li><span style="margin-left: 1em;">numerical velocity of, <a href="#Page_58">58</a>;</span></li>
<li><span style="margin-left: 1em;">where does it go to? <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_48">48</a> et seq.</span></li>

<li>Like effects in like circumstances, <a href="#Page_199">199</a>.</li>

<li>Likeness, <a href="#Page_388">388</a>, <a href="#Page_391">391</a>.</li>

<li>Lilliput, <a href="#Page_84">84</a>.</li>

<li>Lines, straight, their symmetry, <a href="#Page_98">98</a>;</li>
<li><span style="margin-left: 1em;">curved, their asymmetry, <a href="#Page_98">98</a>;</span></li>
<li><span style="margin-left: 1em;">of force, <a href="#Page_249">249</a>.</span></li>

<li>Links of thought, intermediate, <a href="#Page_198">198</a>.</li>

<li>Liquefaction, latent heat of, <a href="#Page_178">178</a>.</li>

<li>Liquid, efflux, law of, <a href="#Page_150">150</a>;</li>
<li><span style="margin-left: 1em;">equilibrium, figures of, <a href="#Page_4">4</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the latter produced in open air, <a href="#Page_7">7</a>-<a href="#Page_8">8</a>;</span></li>
<li><span style="margin-left: 1em;">their beauty and multiplicity of form, <a href="#Page_7">7</a>, <a href="#Page_8">8</a>;</span></li>
<li><span style="margin-left: 1em;">made permanent by melted colophonium, <a href="#Page_7">7</a>.</span></li>

<li>Liquids, forms of, <a href="#Page_1">1</a>-<a href="#Page_16">16</a>;</li>
<li><span style="margin-left: 1em;">difference between, and solids, <a href="#Page_2">2</a>;</span></li>
<li><span style="margin-left: 1em;">their mobility and adaptiveness of form, <a href="#Page_3">3</a>;</span></li>
<li><span style="margin-left: 1em;">the courtiers <i>par excellence</i> of the natural bodies, <a href="#Page_3">3</a>;</span></li>
<li><span style="margin-left: 1em;">possess under certain circumstances forms of their own, <a href="#Page_3">3</a>.</span></li>

<li>Living force, <a href="#Page_137">137</a>, <a href="#Page_184">184</a>;</li>
<li><span style="margin-left: 1em;">law of the conservation of, <a href="#Page_149">149</a>.</span></li>

<li>Lloyd, observation of the conical refraction of light, <a href="#Page_29">29</a>.</li>

<li>Lobster, of Lake Mohrin, the, <a href="#Page_61">61</a>.</li>

<li>Localisation, cerebral, <a href="#Page_210">210</a>.</li>

<li>Locke, on language and thought, <a href="#Page_358">358</a>.</li>

<li>Locomotive, steam in the boiler of, <a href="#Page_219">219</a>.</li>

<li>Loeb, J., <a href="#Page_289">289</a>, <a href="#Page_291">291</a>, <a href="#Page_302">302</a>.</li>

<li>Logarithms, <a href="#Page_195">195</a>, <a href="#Page_219">219</a>;</li>
<li><span style="margin-left: 1em;">in music, <a href="#Page_103">103</a>-<a href="#Page_104">104</a>.</span></li>

<li>Logical root, of the principle of energy, <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">of the principle of excluded perpetual motion, <a href="#Page_182">182</a>.</span></li>

<li>Lombroso, <a href="#Page_280">280</a>.</li>

<li>Lucian, <a href="#Page_347">347</a>.</li>

</ul>

<ul class="IX"><li><i>Macula acustica</i>, <a href="#Page_272">272</a>.</li>

<li>Magic lantern, <a href="#Page_96">96</a>.</li>

<li>Magic powers of nature, <a href="#Page_189">189</a>.</li>

<li><span class="pagenum"><a name="Page_403" id="Page_403">[Pg 403]</a></span>Magical power of science, belief in the, <a href="#Page_189">189</a>.</li>

<li>Magnet, a, <a href="#Page_220">220</a>;</li>
<li><span style="margin-left: 1em;">will compared to the pressure of a, <a href="#Page_14">14</a>;</span></li>
<li><span style="margin-left: 1em;">coercive force of a, <a href="#Page_216">216</a>.</span></li>

<li>Magnetic needle, near a current, <a href="#Page_207">207</a>.</li>

<li>Magnetised bar of steel, <a href="#Page_242">242</a>-<a href="#Page_243">243</a>.</li>

<li>Major and minor keys in music, <a href="#Page_100">100</a> et seq.</li>

<li>Malus, <a href="#Page_242">242</a>.</li>

<li>Man, a fragment of nature's life, <a href="#Page_49">49</a>;</li>
<li><span style="margin-left: 1em;">his life embraces others, <a href="#Page_234">234</a>.</span></li>

<li>Mann, <a href="#Page_364">364</a>.</li>

<li>Manuscript in a mirror, <a href="#Page_93">93</a>.</li>

<li>Maple syrup, statues of, on Moon, <a href="#Page_4">4</a>.</li>

<li>Marx, <a href="#Page_35">35</a>.</li>

<li>Material, the relations of work with heat and the consumption of, <a href="#Page_245">245</a> et seq.</li>

<li>Mathematical methods, their character, <a href="#Page_197">197</a>-<a href="#Page_198">198</a>.</li>

<li>Mathematics, economy of, <a href="#Page_195">195</a>;</li>
<li><span style="margin-left: 1em;">on instruction in, <a href="#Page_338">338</a>-<a href="#Page_374">374</a>;</span></li>
<li><span style="margin-left: 1em;">C. G. J. Jacobi on, <a href="#Page_280">280</a>.</span></li>

<li>Matter, constancy of, <a href="#Page_203">203</a>;</li>
<li><span style="margin-left: 1em;">its nature, <a href="#Page_203">203</a>;</span></li>
<li><span style="margin-left: 1em;">the notion of, <a href="#Page_213">213</a>.</span></li>

<li>Maupertuis, his Latin city, <a href="#Page_338">338</a>.</li>

<li>Maximal and minimal problems, their rôle in physics, <a href="#Page_14">14</a>, footnote.</li>

<li>Mayer, J. R., his conception of energy, <a href="#Page_245">245</a>, <a href="#Page_246">246</a>;</li>
<li><span style="margin-left: 1em;">his methods of thought, <a href="#Page_247">247</a>;</span></li>
<li><span style="margin-left: 1em;">on the conservation of energy, <a href="#Page_163">163</a>, <a href="#Page_164">164</a>, <a href="#Page_165">165</a>, <a href="#Page_167">167</a>, <a href="#Page_183">183</a>, <a href="#Page_184">184</a>;</span></li>
<li><span style="margin-left: 1em;">his metaphysical utterances, <a href="#Page_183">183</a>, <a href="#Page_246">246</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_138">138</a>, <a href="#Page_184">184</a>, <a href="#Page_191">191</a>, <a href="#Page_217">217</a>, <a href="#Page_271">271</a>, <a href="#Page_274">274</a>.</span></li>

<li>Measurement, definition of, <a href="#Page_206">206</a>.</li>

<li>Measures, international, <a href="#Page_108">108</a>.</li>

<li>Mécanique céleste, <a href="#Page_90">90</a>, <a href="#Page_188">188</a>;</li>
<li><span style="margin-left: 1em;">sociale, and morale, the, <a href="#Page_90">90</a>.</span></li>

<li>Mechanical, conception of the world, <a href="#Page_105">105</a>, <a href="#Page_155">155</a> et seq., <a href="#Page_188">188</a>, <a href="#Page_207">207</a>;</li>
<li><span style="margin-left: 1em;">energy, W. Thomson on waste of, <a href="#Page_175">175</a>;</span></li>
<li><span style="margin-left: 1em;">analogies between &mdash;&mdash; and thermal energy, <a href="#Page_17">17</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">equivalent of heat, electricity, etc., <a href="#Page_164">164</a>, <a href="#Page_167">167</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">mythology, <a href="#Page_207">207</a>;</span></li>
<li><span style="margin-left: 1em;">phenomena, physical events as, <a href="#Page_182">182</a>;</span></li>
<li><span style="margin-left: 1em;">philosophy, <a href="#Page_188">188</a>;</span></li>
<li><span style="margin-left: 1em;">physics, <a href="#Page_155">155</a>-<a href="#Page_160">160</a>, <a href="#Page_212">212</a>;</span></li>
<li><span style="margin-left: 1em;">substitution-value of heat, <a href="#Page_178">178</a>, footnote.</span></li>

<li>Mechanics, Kirchhoff's definition of, <a href="#Page_236">236</a>.</li>

<li>Medicine, students of, <a href="#Page_326">326</a>.</li>

<li>Melody, <a href="#Page_101">101</a>.</li>

<li>Melsens, <a href="#Page_310">310</a>, <a href="#Page_327">327</a>.</li>

<li>Memory, a treasure-house for comparison, <a href="#Page_230">230</a>;</li>
<li><span style="margin-left: 1em;">common elements impressed upon the, <a href="#Page_180">180</a>;</span></li>
<li><span style="margin-left: 1em;">its importance, <a href="#Page_238">238</a>;</span></li>
<li><span style="margin-left: 1em;">science disburdens the, <a href="#Page_193">193</a>.</span></li>

<li>Mendelejeff, his periodical series, <a href="#Page_256">256</a>.</li>

<li>Mental, adaptation, <a href="#Page_214">214</a>-<a href="#Page_235">235</a>;</li>
<li><span style="margin-left: 1em;">completion of phenomena, <a href="#Page_220">220</a>;</span></li>
<li><span style="margin-left: 1em;">imagery, <a href="#Page_253">253</a>;</span></li>
<li><span style="margin-left: 1em;">imitation, our schematic, <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">processes, economical, <a href="#Page_195">195</a>;</span></li>
<li><span style="margin-left: 1em;">reproduction, <a href="#Page_198">198</a>;</span></li>
<li><span style="margin-left: 1em;">visualisation, <a href="#Page_250">250</a>.</span></li>

<li>Mephistopheles, <a href="#Page_88">88</a>.</li>

<li>Mercantile principle, a miserly, at the basis of science, <a href="#Page_15">15</a>.</li>

<li>Mersenne, <a href="#Page_377">377</a>.</li>

<li>Mesmerism, the mental state of ordinary minds, <a href="#Page_228">228</a>.</li>

<li>Metaphysical establishment of doctrine of energy, <a href="#Page_183">183</a>.</li>

<li>Metaphysical spooks, <a href="#Page_222">222</a>.</li>

<li>Metrical, concepts of electricity, <a href="#Page_107">107</a> et seq.;</li>
<li><span style="margin-left: 1em;">notions, energy and entropy are, <a href="#Page_178">178</a>;</span></li>
<li><span style="margin-left: 1em;">units, the building-stones of the physicist, <a href="#Page_253">253</a>.</span></li>

<li>Metronomes, <a href="#Page_41">41</a>.</li>

<li>Meyer, Lothar, his periodical series, <a href="#Page_256">256</a>.</li>

<li>Middle Ages, <a href="#Page_243">243</a>, <a href="#Page_349">349</a>.</li>

<li>Midsummer Night's Dream, <a href="#Page_309">309</a>.</li>

<li>Mill, John Stuart, <a href="#Page_230">230</a>.</li>

<li>Millers, school for, <a href="#Page_326">326</a>.</li>

<li>Mill-wheel, doing work, <a href="#Page_161">161</a>.</li>

<li>Mimicking facts in thought, <a href="#Page_189">189</a>, <a href="#Page_193">193</a>.</li>

<li>Minor and major keys in music, <a href="#Page_100">100</a> et seq.</li>

<li>Mirror, symmetrical reversion of objects in, <a href="#Page_92">92</a> et seq.</li>

<li>Miserly mercantile principle at the basis of science, <a href="#Page_15">15</a>.</li>

<li>Moat, child looking into, <a href="#Page_208">208</a>.</li>

<li>Modern scientists, adherents of the mechanical philosophy, <a href="#Page_188">188</a>.</li>

<li>Molecular theories, <a href="#Page_104">104</a>.</li>

<li>Molecules, <a href="#Page_203">203</a>, <a href="#Page_207">207</a>.</li>

<li>Molière, <a href="#Page_234">234</a>.</li>

<li>Momentum, <a href="#Page_184">184</a>.</li>

<li>Monocular vision, <a href="#Page_98">98</a>.</li>

<li><span class="pagenum"><a name="Page_404" id="Page_404">[Pg 404]</a></span>Monotheism of the Christians and Jews, <a href="#Page_187">187</a>.</li>

<li>Montagues and Capulets, <a href="#Page_87">87</a>.</li>

<li>Moon, eclipse of, <a href="#Page_219">219</a>;</li>
<li><span style="margin-left: 1em;">lightness of bodies on, <a href="#Page_4">4</a>;</span></li>
<li><span style="margin-left: 1em;">the study of the, <a href="#Page_90">90</a>, <a href="#Page_284">284</a>.</span></li>

<li>Moreau, <a href="#Page_307">307</a>.</li>

<li>Mosaic of thought, <a href="#Page_192">192</a>.</li>

<li>Motion, a perpetual, <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">quantity of, <a href="#Page_184">184</a>;</span></li>
<li><span style="margin-left: 1em;">the Eleatics on, <a href="#Page_158">158</a>;</span></li>
<li><span style="margin-left: 1em;">Wundt on, <a href="#Page_158">158</a>;</span></li>
<li><span style="margin-left: 1em;">the Herbartians on, <a href="#Page_158">158</a>.</span></li>

<li>Motions, natural and violent, <a href="#Page_226">226</a>;</li>
<li><span style="margin-left: 1em;">their familiar character, <a href="#Page_157">157</a>.</span></li>

<li>Mountains of the earth, would crumble if very large, <a href="#Page_3">3</a>;</li>
<li><span style="margin-left: 1em;">weight of bodies on, <a href="#Page_112">112</a>.</span></li>

<li>Mozart, <a href="#Page_44">44</a>, <a href="#Page_279">279</a>.</li>

<li>Müller, Johann, <a href="#Page_291">291</a>.</li>

<li>Multiplication-table, <a href="#Page_195">195</a>.</li>

<li>Multiplier, <a href="#Page_132">132</a>.</li>

<li>Music, band of, its <i>tempo</i> accelerated and retarded, <a href="#Page_53">53</a>;</li>
<li><span style="margin-left: 1em;">the principle of repetition in, <a href="#Page_99">99</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">its notation, mathematically illustrated, <a href="#Page_103">103</a>-<a href="#Page_104">104</a>.</span></li>

<li>Musical notes, reversion of, <a href="#Page_101">101</a> et seq.;</li>
<li><span style="margin-left: 1em;">their economy, <a href="#Page_192">192</a>.</span></li>

<li>Musical scale, a species of one-dimensional space, <a href="#Page_105">105</a>.</li>

<li>Mystery, in physics, <a href="#Page_222">222</a>;</li>
<li><span style="margin-left: 1em;">science can dispense with, <a href="#Page_189">189</a>.</span></li>

<li>Mysticism, numerical, <a href="#Page_33">33</a>;</li>
<li><span style="margin-left: 1em;">in the principle of energy, <a href="#Page_184">184</a>.</span></li>

<li>Mythology, the mechanical, of philosophy, <a href="#Page_207">207</a>.</li>

</ul>

<ul class="IX"><li>Nagel, von, <a href="#Page_364">364</a>.</li>

<li>Nansen, <a href="#Page_296">296</a>.</li>

<li>Napoleon, picture representing the tomb of, <a href="#Page_36">36</a>.</li>

<li>Nations, intercourse and ideas of, <a href="#Page_336">336</a>-<a href="#Page_337">337</a>.</li>

<li>Natural constants, <a href="#Page_193">193</a>.</li>

<li>Natural law, a, not contained in the conformity of the energies, <a href="#Page_175">175</a>.</li>

<li>Natural laws, abridged descriptions, <a href="#Page_193">193</a>;</li>
<li><span style="margin-left: 1em;">likened to type, <a href="#Page_193">193</a>.</span></li>

<li>Natural motions, <a href="#Page_225">225</a>.</li>

<li>Natural selection in scientific theories, <a href="#Page_63">63</a>, <a href="#Page_218">218</a>.</li>

<li>Nature, experience the well-spring of all knowledge of, <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">fashions of, <a href="#Page_64">64</a>;</span></li>
<li><span style="margin-left: 1em;">first knowledge of, instinctive, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">general interconnexion of, <a href="#Page_182">182</a>;</span></li>
<li><span style="margin-left: 1em;">has many sides, <a href="#Page_217">217</a>;</span></li>
<li><span style="margin-left: 1em;">her forces compared to purposes, <a href="#Page_14">14</a>-<a href="#Page_15">15</a>;</span></li>
<li><span style="margin-left: 1em;">likened to a good man of business, <a href="#Page_15">15</a>;</span></li>
<li><span style="margin-left: 1em;">the economy of her actions, <a href="#Page_15">15</a>;</span></li>
<li><span style="margin-left: 1em;">how she appears to other animals, <a href="#Page_83">83</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">inquiry of, viewed as a torture, <a href="#Page_48">48</a>-<a href="#Page_49">49</a>;</span></li>
<li><span style="margin-left: 1em;">view of, as something designedly concealed from man, <a href="#Page_49">49</a>;</span></li>
<li><span style="margin-left: 1em;">like a covetous tailor, <a href="#Page_9">9</a>-<a href="#Page_10">10</a>;</span></li>
<li><span style="margin-left: 1em;">magic powers of, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">our view of, modified by binocular vision, <a href="#Page_82">82</a>;</span></li>
<li><span style="margin-left: 1em;">the experimental method a questioning of, <a href="#Page_48">48</a>.</span></li>

<li>Negro hamlet, the science of a, <a href="#Page_237">237</a>.</li>

<li>Neptune, prediction and discovery of the planet, <a href="#Page_29">29</a>.</li>

<li>New views, <a href="#Page_296">296</a> et seq.</li>

<li>Newton, describes polarisation, <a href="#Page_242">242</a>;</li>
<li><span style="margin-left: 1em;">expresses his wealth of thought in Latin, <a href="#Page_341">341</a>;</span></li>
<li><span style="margin-left: 1em;">his discovery of gravitation, <a href="#Page_225">225</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">his solution of dispersion, <a href="#Page_362">362</a>;</span></li>
<li><span style="margin-left: 1em;">his principle of the equality of pressure and counterpressure, <a href="#Page_191">191</a>;</span></li>
<li><span style="margin-left: 1em;">his view of light, <a href="#Page_227">227</a>-<a href="#Page_228">228</a>;</span></li>
<li><span style="margin-left: 1em;">on absolute time, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">selections from his works for use in instruction, <a href="#Page_368">368</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_270">270</a>, <a href="#Page_274">274</a>, <a href="#Page_279">279</a>, <a href="#Page_285">285</a>, <a href="#Page_289">289</a>.</span></li>

<li>Nobility, they displace Latin, <a href="#Page_342">342</a>.</li>

<li>Notation, musical, mathematically illustrated, <a href="#Page_103">103</a>-<a href="#Page_104">104</a>.</li>

<li>Numbers, economy of, <a href="#Page_195">195</a>;</li>
<li><span style="margin-left: 1em;">their connexion with consonance, <a href="#Page_32">32</a>.</span></li>

<li>Numerical mysticism, <a href="#Page_33">33</a>.</li>

<li>Nursery, the questions of the, <a href="#Page_199">199</a>.</li>

</ul>

<ul class="IX"><li>Observation, <a href="#Page_310">310</a>.</li>

<li>Observation, in science, <a href="#Page_261">261</a>.</li>

<li>Ocean-stream, <a href="#Page_272">272</a>.</li>

<li>Oettingen, Von, <a href="#Page_103">103</a>.</li>

<li>Ohm, on electric currents, <a href="#Page_249">249</a>.</li>

<li>Ohm, the word, <a href="#Page_343">343</a>.</li>

<li>Oil, alcohol, water, and, employed in Plateau's experiments, <a href="#Page_4">4</a>;</li>
<li><span style="margin-left: 1em;">free mass of, assumes the shape of a sphere, <a href="#Page_12">12</a>;</span></li>
<li><span style="margin-left: 1em;">geometrical figures of, <a href="#Page_5">5</a> et seq.</span></li>

<li>One-eyed people, vision of, <a href="#Page_98">98</a>.</li>

<li>Ophthalmoscope, <a href="#Page_18">18</a>.</li>

<li>Optic nerves, <a href="#Page_96">96</a>.</li>

<li><span class="pagenum"><a name="Page_405" id="Page_405">[Pg 405]</a></span>Optimism and pessimism, <a href="#Page_234">234</a>.</li>

<li>Order of physics, <a href="#Page_197">197</a>.</li>

<li>Organ, bellows of an, <a href="#Page_135">135</a>.</li>

<li>Organic nature, results of Darwin's studies of, <a href="#Page_215">215</a> et seq.</li>
<li><span style="margin-left: 1em;">See <i>Adaptation</i> and <i>Heredity</i>.</span></li>

<li>Oriental world of fables, <a href="#Page_273">273</a>.</li>

<li>Orientation, sensations of, <a href="#Page_282">282</a> et seq.</li>

<li>Oscillation, centre of, <a href="#Page_147">147</a> et seq.</li>

<li>Ostwald, <a href="#Page_172">172</a>.</li>

<li>Otoliths, <a href="#Page_301">301</a> et seq.</li>

<li>Overtones, <a href="#Page_28">28</a>, <a href="#Page_40">40</a>, <a href="#Page_349">349</a>.</li>

<li>Ozone, Schöbein's discovery of, <a href="#Page_271">271</a>.</li>

</ul>

<ul class="IX"><li>Painted things, the difference between real and, <a href="#Page_68">68</a>.</li>

<li>Palestrina, <a href="#Page_44">44</a>.</li>

<li>Parameter, <a href="#Page_257">257</a>.</li>

<li>Partial tones, <a href="#Page_390">390</a>.</li>

<li>Particles, smallest, <a href="#Page_104">104</a>.</li>

<li>Pascheles, Dr. W., <a href="#Page_285">285</a>.</li>

<li>Paulsen, <a href="#Page_338">338</a>, <a href="#Page_340">340</a>, <a href="#Page_373">373</a>.</li>

<li>Pearls of life, strung on the individual as on a thread, <a href="#Page_234">234</a>-<a href="#Page_235">235</a>.</li>

<li>Pencil surpasses the mathematician in intelligence, <a href="#Page_196">196</a>.</li>

<li>Pendulum, motion of a, <a href="#Page_144">144</a> et seq.,</li>
<li><span style="margin-left: 1em;">increased motion of, due to slight impulses, <a href="#Page_21">21</a>;</span></li>
<li><span style="margin-left: 1em;">electrical, <a href="#Page_110">110</a>.</span></li>

<li>Percepts, of like form, <a href="#Page_390">390</a>.</li>

<li>Periodical, changes, <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">series, <a href="#Page_256">256</a>.</span></li>

<li>Permanent, changes, <a href="#Page_181">181</a>, <a href="#Page_199">199</a>;</li>
<li><span style="margin-left: 1em;">elements of the world, <a href="#Page_194">194</a>.</span></li>

<li>Perpetual motion, a, <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">defined, <a href="#Page_139">139</a>;</span></li>
<li><span style="margin-left: 1em;">impossibility of, <a href="#Page_139">139</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the principle of the, excluded, <a href="#Page_140">140</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">excluded from general physics, <a href="#Page_162">162</a>.</span></li>

<li>Personality, its nature, <a href="#Page_234">234</a>-<a href="#Page_235">235</a>.</li>

<li>Perspective, <a href="#Page_76">76</a> et seq.;</li>
<li><span style="margin-left: 1em;">contraction of, <a href="#Page_74">74</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">distortion of, <a href="#Page_77">77</a>.</span></li>

<li>Pessimism and optimism, <a href="#Page_234">234</a>.</li>

<li>Pharaohs, <a href="#Page_85">85</a>.</li>

<li>Phenomenology, a universal physical, <a href="#Page_250">250</a>.</li>

<li>Philistine, modes of thought of, <a href="#Page_223">223</a>.</li>

<li>Philology, comparison in, <a href="#Page_239">239</a>.</li>

<li>Philosopher, an ancient, on the moral and physical sciences, <a href="#Page_89">89</a>.</li>

<li>Philosophy, its character at all times, <a href="#Page_186">186</a>;</li>
<li><span style="margin-left: 1em;">mechanical, <a href="#Page_155">155</a> et seq., <a href="#Page_188">188</a>, <a href="#Page_207">207</a>, <a href="#Page_259">259</a> et seq.</span></li>

<li>Phonetic alphabets, their economy, <a href="#Page_192">192</a>.</li>

<li>Photography, by the electric spark, <a href="#Page_318">318</a> et seq.</li>

<li>Photography of projectiles, <a href="#Page_309">309</a>-<a href="#Page_337">337</a>.</li>

<li>Photography, stupendous advances of, <a href="#Page_74">74</a>.</li>

<li>Physical, concepts, fetishism in our, <a href="#Page_187">187</a>;</li>
<li><span style="margin-left: 1em;">ideas and principles, their nature, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">inquiry, the economical nature of, <a href="#Page_186">186</a>;</span></li>
<li><span style="margin-left: 1em;">research, object of <a href="#Page_207">207</a>, <a href="#Page_209">209</a>.</span></li>

<li>Physical phenomena, as mechanical phenomena, <a href="#Page_182">182</a>;</li>
<li><span style="margin-left: 1em;">relations between, <a href="#Page_205">205</a>.</span></li>

<li>Physico-mechanical view of the world, <a href="#Page_155">155</a>, <a href="#Page_187">187</a>, <a href="#Page_188">188</a>, <a href="#Page_207">207</a> et seq.</li>

<li>Physics, compared to a well-kept household, <a href="#Page_197">197</a>;</li>
<li><span style="margin-left: 1em;">economical experience, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">the principles of, descriptive, <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">the methods of, <a href="#Page_209">209</a>;</span></li>
<li><span style="margin-left: 1em;">its method characterised, <a href="#Page_211">211</a>;</span></li>
<li><span style="margin-left: 1em;">comparison in, <a href="#Page_239">239</a>;</span></li>
<li><span style="margin-left: 1em;">the facts of, qualitatively homogeneous, <a href="#Page_255">255</a>;</span></li>
<li><span style="margin-left: 1em;">how it began, <a href="#Page_37">37</a>;</span></li>
<li><span style="margin-left: 1em;">helped by psychology, <a href="#Page_104">104</a>;</span></li>
<li><span style="margin-left: 1em;">study of its own character, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">the goal of, <a href="#Page_207">207</a>, <a href="#Page_209">209</a>.</span></li>

<li>Physiological psychology, its methods, <a href="#Page_211">211</a> et seq.</li>

<li>Physiology, its scope, <a href="#Page_212">212</a>.</li>

<li>Piano, its mirrored counterpart, <a href="#Page_100">100</a> et seq.;</li>
<li><span style="margin-left: 1em;">used to illustrate the facts of sympathetic vibration, <a href="#Page_25">25</a> et seq.</span></li>

<li>Piano-player, a speaker compared to, <a href="#Page_192">192</a>.</li>

<li>Picture, physical, a, <a href="#Page_110">110</a>.</li>

<li>Pike, learns by experience, <a href="#Page_267">267</a>.</li>

<li>Pillars of Corti, <a href="#Page_19">19</a>.</li>

<li>Places, heavy bodies seek their, <a href="#Page_224">224</a> et seq.</li>

<li>Planetary system, origin of, illustrated, <a href="#Page_5">5</a>.</li>

<li>Plasticity of organic nature, <a href="#Page_216">216</a>.</li>

<li>Plateau, his law of free liquid equilibrium, <a href="#Page_9">9</a>;</li>
<li><span style="margin-left: 1em;">his method of getting rid of the effects of gravity, <a href="#Page_4">4</a>.</span></li>

<li>Plates of oil, thin, <a href="#Page_6">6</a>.</li>

<li>Plato, <a href="#Page_347">347</a>, <a href="#Page_371">371</a>.</li>

<li>Plautus, <a href="#Page_347">347</a>.</li>

<li>Playfair, <a href="#Page_138">138</a>.</li>

<li><span class="pagenum"><a name="Page_406" id="Page_406">[Pg 406]</a></span>Pleasant effects, cause of, <a href="#Page_94">94</a> et seq.</li>

<li>Pliny, <a href="#Page_349">349</a>.</li>

<li>Poetry and science, <a href="#Page_30">30</a>, <a href="#Page_31">31</a>, <a href="#Page_351">351</a>.</li>

<li>Poinsot, on the foundations of mechanics, <a href="#Page_152">152</a> et seq.</li>

<li>Polarisation, <a href="#Page_91">91</a>;</li>
<li><span style="margin-left: 1em;">abstractly described by Newton, <a href="#Page_242">242</a>.</span></li>

<li>Politics, Chinese speak with unwillingness of, <a href="#Page_374">374</a>.</li>

<li>Pollak, <a href="#Page_299">299</a>.</li>

<li>Polyp plant, humanity likened to a, <a href="#Page_235">235</a>.</li>

<li>Pompeii, <a href="#Page_234">234</a>;</li>
<li><span style="margin-left: 1em;">art in, <a href="#Page_80">80</a>.</span></li>

<li>Popper J., <a href="#Page_172">172</a>, <a href="#Page_216">216</a>.</li>

<li>Potential, social, <a href="#Page_15">15</a>;</li>
<li><span style="margin-left: 1em;">electrical, <a href="#Page_121">121</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">measurement of, <a href="#Page_126">126</a>;</span></li>
<li><span style="margin-left: 1em;">fall of, <a href="#Page_177">177</a>;</span></li>
<li><span style="margin-left: 1em;">swarm of notions in the idea of, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">its wide scope, <a href="#Page_250">250</a>.</span></li>

<li>Pottery, invention of, <a href="#Page_263">263</a>.</li>

<li>Prediction, <a href="#Page_221">221</a> et seq.</li>

<li>Prejudice, the function, power, and dangers of, <a href="#Page_232">232</a>-<a href="#Page_233">233</a>.</li>

<li>Preparatory schools, the defects of the German, <a href="#Page_346">346</a>-<a href="#Page_347">347</a>;</li>
<li><span style="margin-left: 1em;">what they should teach, <a href="#Page_364">364</a> et seq.</span></li>

<li>Pressure of a stone or of a magnet, will compared to, <a href="#Page_14">14</a>;</li>
<li><span style="margin-left: 1em;">also <a href="#Page_157">157</a>.</span></li>

<li>Primitive acts of knowledge the foundation of scientific thought, <a href="#Page_190">190</a>.</li>

<li>Problem, nature of a, <a href="#Page_223">223</a>.</li>

<li>Problems which are wrongly formulated, <a href="#Page_308">308</a>.</li>

<li>Process, Carnot's, <a href="#Page_161">161</a> et seq.</li>

<li>Projectiles, the effects of the impact of, <a href="#Page_310">310</a>, <a href="#Page_327">327</a>-<a href="#Page_328">328</a>;</li>
<li><span style="margin-left: 1em;">seen with the naked eye, <a href="#Page_311">311</a>, <a href="#Page_317">317</a>;</span></li>
<li><span style="margin-left: 1em;">measuring the velocity of, <a href="#Page_332">332</a>;</span></li>
<li><span style="margin-left: 1em;">photography of, <a href="#Page_309">309</a>-<a href="#Page_337">337</a>.</span></li>

<li>Prony's brake, <a href="#Page_132">132</a>.</li>

<li>Proof, nature of, <a href="#Page_284">284</a>.</li>

<li>Prophesying events, <a href="#Page_220">220</a> et seq.</li>

<li>Psalms, quotation from the, <a href="#Page_89">89</a>.</li>

<li>Pseudoscope, Wheatstone's, <a href="#Page_96">96</a>.</li>

<li>Psychology, preceded by astronomy, <a href="#Page_90">90</a>;</li>
<li><span style="margin-left: 1em;">how reached, <a href="#Page_91">91</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">helps physical science, <a href="#Page_104">104</a>;</span></li>
<li><span style="margin-left: 1em;">its method the same as that of physics, <a href="#Page_207">207</a> et seq.</span></li>

<li>Pully arrangement, illustrating principle of least superficial area, <a href="#Page_12">12</a>-<a href="#Page_13">13</a>.</li>

<li>Purkinje, <a href="#Page_284">284</a>, <a href="#Page_285">285</a>, <a href="#Page_291">291</a>, <a href="#Page_299">299</a>.</li>

<li>Purposes, the acts of nature compared to, <a href="#Page_14">14</a>-<a href="#Page_15">15</a>;</li>
<li><span style="margin-left: 1em;">nature pursues no, <a href="#Page_66">66</a>.</span></li>

<li>Puzzle-lock, a, <a href="#Page_26">26</a>.</li>

<li>Puzzles, <a href="#Page_277">277</a>.</li>

<li>Pyramid of oil, <a href="#Page_6">6</a>.</li>

<li>Pythagoras, his discovery of the laws of harmony, <a href="#Page_32">32</a>, <a href="#Page_259">259</a>.</li>

</ul>

<ul class="IX"><li>Quality of tones, <a href="#Page_36">36</a>.</li>

<li>Quantitative investigation, the goal of, <a href="#Page_180">180</a>.</li>

<li>Quantity of electricity, <a href="#Page_111">111</a>, <a href="#Page_118">118</a>, <a href="#Page_119">119</a>, <a href="#Page_167">167</a>-<a href="#Page_170">170</a>, <a href="#Page_173">173</a>;</li>
<li><span style="margin-left: 1em;">of heat, <a href="#Page_166">166</a>, <a href="#Page_167">167</a>-<a href="#Page_171">171</a>, <a href="#Page_174">174</a>, <a href="#Page_177">177</a>, <a href="#Page_244">244</a>;</span></li>
<li><span style="margin-left: 1em;">of motion, <a href="#Page_184">184</a>.</span></li>

<li>Quests made of the inquirer, not by him, <a href="#Page_30">30</a>.</li>

<li>Quételet, <a href="#Page_15">15</a>, footnote.</li>

</ul>

<ul class="IX"><li>Rabelais, <a href="#Page_283">283</a>.</li>

<li>Raindrop, form of, <a href="#Page_3">3</a>.</li>

<li>Rameau, <a href="#Page_34">34</a>.</li>

<li>Reaction and action, principle of, <a href="#Page_191">191</a>.</li>

<li>Reactions, disclosure of the connexion of, <a href="#Page_270">270</a> et seq.</li>

<li>Realgymnasien, <a href="#Page_365">365</a>.</li>

<li>Realschulen, <a href="#Page_365">365</a>, <a href="#Page_373">373</a>.</li>

<li>Reason, stands above the senses, <a href="#Page_105">105</a>.</li>

<li>Reflex action, <a href="#Page_210">210</a>.</li>

<li>Reflexion, produces symmetrical reversion of objects, <a href="#Page_93">93</a> et seq.</li>

<li>Refraction, <a href="#Page_29">29</a>, <a href="#Page_193">193</a>, <a href="#Page_194">194</a>, <a href="#Page_208">208</a>, <a href="#Page_230">230</a>, <a href="#Page_231">231</a>.</li>

<li>Reger, <a href="#Page_328">328</a>.</li>

<li>Reliefs, photographs of, <a href="#Page_68">68</a>.</li>

<li>Repetition, its rôle in æsthetics, <a href="#Page_89">89</a>, footnote, <a href="#Page_91">91</a> et seq., <a href="#Page_97">97</a>, <a href="#Page_98">98</a> et seq., <a href="#Page_390">390</a>.</li>

<li>Reproduction of facts in thought, <a href="#Page_189">189</a>, <a href="#Page_193">193</a>, <a href="#Page_198">198</a>, <a href="#Page_253">253</a>.</li>

<li>Repulsion, electric, <a href="#Page_109">109</a> et seq., <a href="#Page_168">168</a>.</li>

<li>Research, function of experimental <a href="#Page_181">181</a>;</li>
<li><span style="margin-left: 1em;">the aim of, <a href="#Page_205">205</a>.</span></li>

<li>Resemblances between facts, <a href="#Page_255">255</a>.</li>

<li>Resin, solution of, <a href="#Page_7">7</a>.</li>

<li>Resistance, laws of, for bodies travelling in air and fluids, <a href="#Page_333">333</a> et seq.</li>

<li>Resonance, corporeal, <a href="#Page_392">392</a>.</li>

<li>Response of sonorous bodies, <a href="#Page_25">25</a>.</li>

<li>Retina, the corresponding spots of <a href="#Page_98">98</a>;</li>
<li><span class="pagenum"><a name="Page_407" id="Page_407">[Pg 407]</a></span><span style="margin-left: 1em;">nerves of compared to fingers of a hand, <a href="#Page_96">96</a> et seq.</span></li>

<li>Reversible processes, <a href="#Page_161">161</a> et seq., <a href="#Page_175">175</a>, <a href="#Page_176">176</a>, <a href="#Page_181">181</a>, <a href="#Page_182">182</a>.</li>

<li>Rhine, the, <a href="#Page_94">94</a>.</li>

<li>Richard the Third, <a href="#Page_77">77</a>.</li>

<li>Riddles, <a href="#Page_277">277</a>.</li>

<li>Riders, <a href="#Page_379">379</a>.</li>

<li>Riegler, <a href="#Page_319">319</a>.</li>

<li>Riess, experiment with the thermo-electrometer, <a href="#Page_133">133</a> et seq., <a href="#Page_169">169</a>.</li>

<li>Rigid connexions, <a href="#Page_142">142</a>.</li>

<li>Rind of a fruit, <a href="#Page_190">190</a>.</li>

<li>Rings of oil, illustrating formation of rings of Saturn, <a href="#Page_5">5</a>.</li>

<li>Ritter, <a href="#Page_291">291</a>, <a href="#Page_299">299</a>.</li>

<li>Rods of Corti, <a href="#Page_19">19</a>.</li>

<li>Rolph, W. H., <a href="#Page_216">216</a>.</li>

<li>Roman Church, Latin introduced with the, <a href="#Page_340">340</a> et seq.</li>

<li>Romans, their provinciality and narrow-mindedness, <a href="#Page_270">270</a>.</li>

<li>Romeo and Juliet, <a href="#Page_87">87</a>.</li>

<li>Römer, Olaf, <a href="#Page_51">51</a> et seq.</li>

<li>Roots, the nature of, in language, <a href="#Page_252">252</a>.</li>

<li>Rosetti, his experiment on the work required to develop electricity, <a href="#Page_131">131</a>.</li>

<li>Rotating bodies, <a href="#Page_285">285</a>.</li>

<li>Rotation, apparatus of, in physics, <a href="#Page_59">59</a> et seq.;</li>
<li><span style="margin-left: 1em;">sensations of, <a href="#Page_288">288</a> et seq.</span></li>

<li>Rousseau, <a href="#Page_336">336</a>.</li>

<li>Rubber pyramid, illustrating the principle of least superficial area, <a href="#Page_10">10</a>-<a href="#Page_11">11</a>.</li>

<li>Ruysdael, <a href="#Page_279">279</a>.</li>
</ul>

<ul class="IX">
<li>Sachs, Hans, <a href="#Page_106">106</a>.</li>

<li>Salcher, Prof. <a href="#Page_319">319</a>.</li>

<li>Salviati, <a href="#Page_144">144</a>.</li>

<li>Saturn, rings of, their formation illustrated, <a href="#Page_5">5</a>.</li>

<li>Saurians, <a href="#Page_257">257</a>.</li>

<li>Sauveur, on acoustics, <a href="#Page_34">34</a>, <a href="#Page_375">375</a> et seq.</li>

<li>Savage, modes of conception and interpretation of a, <a href="#Page_218">218</a> et seq.</li>

<li>Schäfer, K., <a href="#Page_298">298</a>.</li>

<li><i>Schlierenmethode</i>, <a href="#Page_317">317</a>.</li>

<li>Schönbein's discovery of ozone, <a href="#Page_271">271</a>.</li>

<li>School-boy, copy-book of, <a href="#Page_92">92</a>.</li>

<li>Schoolmen, <a href="#Page_214">214</a>.</li>

<li>Schools, State-control of, <a href="#Page_372">372</a> et seq.</li>

<li>Schopenhauer, <a href="#Page_190">190</a>.</li>

<li>Schultze, Max, <a href="#Page_19">19</a>.</li>

<li>Science, a miserly mercantile principle at its basis, <a href="#Page_15">15</a>;</li>
<li><span style="margin-left: 1em;">compared to a business, <a href="#Page_16">16</a>;</span></li>
<li><span style="margin-left: 1em;">viewed as a maximum or minimum problem, <a href="#Page_16">16</a>, footnote;</span></li>
<li><span style="margin-left: 1em;">its process not greatly different from the intellectual activity of ordinary life, <a href="#Page_16">16</a>, footnote;</span></li>
<li><span style="margin-left: 1em;">economy of its task, <a href="#Page_16">16</a>;</span></li>
<li><span style="margin-left: 1em;">relation of, to poetry, <a href="#Page_30">30</a>, <a href="#Page_31">31</a>, <a href="#Page_351">351</a>;</span></li>
<li><span style="margin-left: 1em;">the church of, <a href="#Page_67">67</a>;</span></li>
<li><span style="margin-left: 1em;">beginnings of, <a href="#Page_189">189</a>, <a href="#Page_191">191</a>;</span></li>
<li><span style="margin-left: 1em;">belief in the magical power of, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">can dispense with mystery, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">lavish extravagance of, <a href="#Page_189">189</a>;</span></li>
<li><span style="margin-left: 1em;">economy of the terminology of, <a href="#Page_192">192</a>;</span></li>
<li><span style="margin-left: 1em;">partly made up of the intelligence of others, <a href="#Page_196">196</a>;</span></li>
<li><span style="margin-left: 1em;">stripped of mystery, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">its true power, <a href="#Page_197">197</a>;</span></li>
<li><span style="margin-left: 1em;">the economical schematism of, <a href="#Page_206">206</a>;</span></li>
<li><span style="margin-left: 1em;">the object of, <a href="#Page_206">206</a>;</span></li>
<li><span style="margin-left: 1em;">the tools of, <a href="#Page_207">207</a>;</span></li>
<li><span style="margin-left: 1em;">does not create facts, <a href="#Page_211">211</a>;</span></li>
<li><span style="margin-left: 1em;">of the future, <a href="#Page_213">213</a>;</span></li>
<li><span style="margin-left: 1em;">revolution in, dating from Galileo, <a href="#Page_214">214</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the natural foe of the marvellous, <a href="#Page_224">224</a>;</span></li>
<li><span style="margin-left: 1em;">characterised, <a href="#Page_227">227</a>;</span></li>
<li><span style="margin-left: 1em;">growth of, <a href="#Page_237">237</a>;</span></li>
<li><span style="margin-left: 1em;">dramatic element in, <a href="#Page_243">243</a>;</span></li>
<li><span style="margin-left: 1em;">described, <a href="#Page_251">251</a>;</span></li>
<li><span style="margin-left: 1em;">its function, <a href="#Page_253">253</a>;</span></li>
<li><span style="margin-left: 1em;">classification in, <a href="#Page_255">255</a>, <a href="#Page_259">259</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">the way of discovery in, <a href="#Page_316">316</a>.</span></li>
<li><span style="margin-left: 1em;">See also <i>Physics</i>.</span></li>

<li>Sciences, partition of the, <a href="#Page_86">86</a>;</li>
<li><span style="margin-left: 1em;">the barriers and relations between the <a href="#Page_257">257</a>-<a href="#Page_258">258</a>;</span></li>
<li><span style="margin-left: 1em;">on instruction in the, <a href="#Page_338">338</a>-<a href="#Page_374">374</a>.</span></li>

<li>Scientific, criticism, Socrates the father of, <a href="#Page_1">1</a>, <a href="#Page_16">16</a>;</li>
<li><span style="margin-left: 1em;">discoveries, their fate, <a href="#Page_138">138</a>;</span></li>
<li><span style="margin-left: 1em;">knowledge, involves description, <a href="#Page_193">193</a>;</span></li>
<li><span style="margin-left: 1em;">thought, transformation and adaptation in, <a href="#Page_214">214</a>-<a href="#Page_235">235</a>;</span></li>
<li><span style="margin-left: 1em;">thought, advanced by new experiences, <a href="#Page_223">223</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">thought, the difficulty of, <a href="#Page_366">366</a>;</span></li>
<li><span style="margin-left: 1em;">terms, <a href="#Page_342">342</a>-<a href="#Page_343">343</a>;</span></li>
<li><span style="margin-left: 1em;">founded on primitive acts of knowledge, <a href="#Page_190">190</a>.</span></li>

<li>Scientists, stories about their ignorance, <a href="#Page_342">342</a>.</li>

<li>Screw, the, <a href="#Page_62">62</a>.</li>

<li>Sea-sickness, <a href="#Page_284">284</a>.</li>

<li>Secret computation, Leibnitz's, <a href="#Page_33">33</a>.</li>

<li>Seek their places, bodies, <a href="#Page_226">226</a>.</li>

<li>Self-induction, coefficient of, <a href="#Page_250">250</a>, <a href="#Page_252">252</a>.</li>

<li>Self-observation, <a href="#Page_211">211</a>.</li>

<li><span class="pagenum"><a name="Page_408" id="Page_408">[Pg 408]</a></span>Self-preservation, our first knowledge derived from the economy of, <a href="#Page_197">197</a>;</li>
<li><span style="margin-left: 1em;">struggle for, among ideas, <a href="#Page_228">228</a>.</span></li>

<li>Semi-circular canals, <a href="#Page_290">290</a> et seq.</li>

<li>Sensation of rounding a railway curve, <a href="#Page_286">286</a>.</li>

<li>Sensations, analysed, <a href="#Page_251">251</a>;</li>
<li><span style="margin-left: 1em;">when similar, produce agreeable effects, <a href="#Page_96">96</a>;</span></li>
<li><span style="margin-left: 1em;">their character, <a href="#Page_200">200</a>;</span></li>
<li><span style="margin-left: 1em;">defined, <a href="#Page_209">209</a>;</span></li>
<li><span style="margin-left: 1em;">of orientation, <a href="#Page_282">282</a> et seq.</span></li>

<li>Sense-elements, <a href="#Page_179">179</a>.</li>

<li>Senses, theory of, <a href="#Page_104">104</a>;</li>
<li><span style="margin-left: 1em;">the source of our knowledge of facts, <a href="#Page_237">237</a>.</span></li>

<li>Seventh, the troublesome, <a href="#Page_46">46</a>.</li>

<li>Shadow method, <a href="#Page_313">313</a> et seq., <a href="#Page_317">317</a> footnote.</li>

<li>Shadows, rôle of, in vision, <a href="#Page_81">81</a>.</li>

<li>Shakespeare, <a href="#Page_278">278</a>.</li>

<li>Sharps, reversed into flats, <a href="#Page_101">101</a>.</li>

<li>Shell, spherical, law of attraction for a, <a href="#Page_124">124</a>, footnote.</li>

<li>Shoemaker, inquirer compared to, <a href="#Page_105">105</a>-<a href="#Page_106">106</a>.</li>

<li>Shooting, <a href="#Page_309">309</a>.</li>

<li>Shots, double report of, <a href="#Page_229">229</a> et seq.</li>

<li>Similarity, <a href="#Page_249">249</a>.</li>

<li>Simony, <a href="#Page_280">280</a>.</li>

<li>Simplicity, a varying element in description, <a href="#Page_254">254</a>.</li>

<li>Sines, law of the, <a href="#Page_193">193</a>.</li>

<li>Sinking of heavy bodies, <a href="#Page_222">222</a>.</li>

<li>Sixth sense, <a href="#Page_297">297</a>.</li>

<li>Smith, R., on acoustics, <a href="#Page_34">34</a>, <a href="#Page_381">381</a>, <a href="#Page_383">383</a>.</li>

<li>Soap-films, Van der Mensbrugghe's experiment with, <a href="#Page_11">11</a>-<a href="#Page_12">12</a>.</li>

<li>Soapsuds, films and figures of, <a href="#Page_7">7</a>.</li>

<li>Social potential, <a href="#Page_15">15</a>.</li>

<li>Socrates, the father of scientific criticism, <a href="#Page_1">1</a>, <a href="#Page_16">16</a>.</li>

<li>Sodium, <a href="#Page_202">202</a>.</li>

<li>Sodium-light, vibrations of, as a measure of time, <a href="#Page_205">205</a>.</li>

<li>Solidity, conception of, by the eye, <a href="#Page_71">71</a> et seq.;</li>
<li><span style="margin-left: 1em;">spatial, photographs of, <a href="#Page_73">73</a>.</span></li>

<li>Solids, and liquids, their difference merely one of degree, <a href="#Page_2">2</a>.</li>

<li>Sonorous bodies, <a href="#Page_24">24</a> et seq.</li>

<li>Soret, J. P., <a href="#Page_89">89</a>.</li>

<li>Sounds, symmetry of, <a href="#Page_99">99</a> et seq.;</li>
<li><span style="margin-left: 1em;">generally, <a href="#Page_22">22</a>-<a href="#Page_47">47</a>, <a href="#Page_212">212</a>.</span></li>

<li>Sound-waves rendered visible, <a href="#Page_315">315</a> et seq.</li>

<li>Sources of the principle of energy, <a href="#Page_179">179</a> et seq.</li>

<li>Space, <a href="#Page_205">205</a>;</li>
<li><span style="margin-left: 1em;">sensation of, <a href="#Page_210">210</a>.</span></li>

<li>Spark, electric, <a href="#Page_117">117</a>, <a href="#Page_127">127</a>, <a href="#Page_132">132</a>, <a href="#Page_133">133</a>, <a href="#Page_190">190</a>.</li>

<li>Spatial vision, <a href="#Page_386">386</a>.</li>

<li>Species, stability of, a theory, <a href="#Page_216">216</a>.</li>

<li>Specific energies, <a href="#Page_291">291</a>.</li>

<li>Specific heat, <a href="#Page_166">166</a>, <a href="#Page_244">244</a>.</li>

<li>Specific inductive capacity, <a href="#Page_117">117</a>.</li>

<li>Spectral analysis of sound, <a href="#Page_27">27</a>.</li>

<li>Spectrum, mental associations of the, <a href="#Page_190">190</a>.</li>

<li>Speech, the instinct of, cultivated by languages, <a href="#Page_354">354</a>.</li>

<li>Spencer, <a href="#Page_218">218</a>, <a href="#Page_222">222</a>.</li>

<li>Sphere, a soft rotating, <a href="#Page_2">2</a>;</li>
<li><span style="margin-left: 1em;">the figure of least surface, <a href="#Page_12">12</a>;</span></li>
<li><span style="margin-left: 1em;">electrical capacity of, <a href="#Page_123">123</a> et seq.</span></li>

<li>Spherical shell, law of attraction for <a href="#Page_124">124</a>, footnote.</li>

<li>Spiders, the eyes of, <a href="#Page_67">67</a>.</li>

<li>Spirits, as explanation of the world <a href="#Page_186">186</a>, <a href="#Page_243">243</a>.</li>

<li>Spiritualism, modern, <a href="#Page_187">187</a>.</li>

<li>Spooks, metaphysical, <a href="#Page_222">222</a>.</li>

<li>Squinting, <a href="#Page_72">72</a>.</li>

<li>Stability of our environment, <a href="#Page_206">206</a>.</li>

<li>Stallo, <a href="#Page_336">336</a>.</li>

<li>Stars, the fixed, <a href="#Page_90">90</a>.</li>

<li>State, benefits and evils of its control of the schools, <a href="#Page_372">372</a> et seq.;</li>
<li><span style="margin-left: 1em;">the Church and, <a href="#Page_88">88</a>.</span></li>

<li>Statical electricity, <a href="#Page_134">134</a>.</li>

<li>Stationary currents, <a href="#Page_249">249</a>.</li>

<li>Statoliths, <a href="#Page_303">303</a>.</li>

<li>Steam-engine, <a href="#Page_160">160</a>, <a href="#Page_265">265</a>.</li>

<li>Steeple-jacks, <a href="#Page_75">75</a>.</li>

<li>Stereoscope, Wheatstone and Brewster's, <a href="#Page_73">73</a>.</li>

<li>Stevinus, on the inclined plane, <a href="#Page_140">140</a>;</li>
<li><span style="margin-left: 1em;">on hydrostatics, <a href="#Page_141">141</a>;</span></li>
<li><span style="margin-left: 1em;">on the equilibrium of systems, <a href="#Page_142">142</a>;</span></li>
<li><span style="margin-left: 1em;">discovers the principle of virtual velocities, <a href="#Page_150">150</a>;</span></li>
<li><span style="margin-left: 1em;">characterisation of his thought, <a href="#Page_142">142</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_182">182</a>, <a href="#Page_187">187</a>, <a href="#Page_191">191</a>.</span></li>

<li>Stone Age, <a href="#Page_46">46</a>, <a href="#Page_321">321</a>.</li>

<li>Störensen, <a href="#Page_306">306</a>.</li>

<li>Stove, primitive, <a href="#Page_263">263</a>.</li>

<li>Straight line, a, its symmetry, <a href="#Page_98">98</a>.</li>

<li>Straight, meaning of the word, <a href="#Page_240">240</a>.</li>

<li><span class="pagenum"><a name="Page_409" id="Page_409">[Pg 409]</a></span>Street, vista into a, <a href="#Page_75">75</a>.</li>

<li>Striae, in glass, <a href="#Page_313">313</a>.</li>

<li>Striate method, for detecting optical imperfections, <a href="#Page_317">317</a>.</li>

<li>Striking distance, <a href="#Page_115">115</a>, <a href="#Page_127">127</a>.</li>

<li>Strings, vibrations of, <a href="#Page_249">249</a>.</li>

<li>Struggle for existence among ideas, <a href="#Page_217">217</a>.</li>

<li>Substance, heat conceived as a, <a href="#Page_177">177</a>, <a href="#Page_243">243</a> et seq.;</li>
<li><span style="margin-left: 1em;">electricity as a, <a href="#Page_170">170</a>;</span></li>
<li><span style="margin-left: 1em;">the source of our notion of, <a href="#Page_199">199</a>;</span></li>
<li><span style="margin-left: 1em;">rôle of the notion of, <a href="#Page_203">203</a>, <a href="#Page_244">244</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">energy conceived as a, <a href="#Page_164">164</a>, <a href="#Page_185">185</a>, <a href="#Page_244">244</a> et seq.</span></li>

<li>Substitution-value of heat, <a href="#Page_178">178</a>, footnote.</li>

<li>Suetonius, <a href="#Page_348">348</a>.</li>

<li>Sulphur, specific inductive capacity of, <a href="#Page_117">117</a>.</li>

<li>Sun, human beings could not exist on, <a href="#Page_3">3</a>.</li>

<li>Swift, <a href="#Page_84">84</a>, <a href="#Page_280">280</a>.</li>

<li>Swimmer, Ampère's, <a href="#Page_207">207</a>.</li>

<li>Symmetry, definition of, <a href="#Page_92">92</a>;</li>
<li><span style="margin-left: 1em;">figures of, <a href="#Page_92">92</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">plane of, <a href="#Page_94">94</a>;</span></li>
<li><span style="margin-left: 1em;">vertical and horizontal, <a href="#Page_94">94</a>;</span></li>
<li><span style="margin-left: 1em;">in music, <a href="#Page_99">99</a> et seq.</span></li>

<li>Sympathetic vibration, <a href="#Page_22">22</a> et seq., <a href="#Page_379">379</a>.</li>

</ul>

<ul class="IX"><li>Tailor, nature like a covetous, <a href="#Page_9">9</a>-<a href="#Page_10">10</a>.</li>

<li>Tangent, the word, <a href="#Page_263">263</a>.</li>

<li>Taste, doubtful cultivation of, by the classics, <a href="#Page_352">352</a>-<a href="#Page_353">353</a>;</li>
<li><span style="margin-left: 1em;">of the ancients, <a href="#Page_353">353</a>.</span></li>

<li>Taylor, on the vibration of strings, <a href="#Page_249">249</a>.</li>

<li>Teaching, its nature, <a href="#Page_366">366</a> et seq.</li>

<li>Telegraph, the word, <a href="#Page_263">263</a>.</li>

<li>Telescope, <a href="#Page_262">262</a>.</li>

<li>Telestereoscope, the, <a href="#Page_84">84</a>.</li>

<li>Temperament, even, in tuning, <a href="#Page_47">47</a>.</li>

<li>Temperature, absolute, <a href="#Page_162">162</a>;</li>
<li><span style="margin-left: 1em;">differences of, <a href="#Page_205">205</a>;</span></li>
<li><span style="margin-left: 1em;">differences of, viewed as level surfaces, <a href="#Page_161">161</a>;</span></li>
<li><span style="margin-left: 1em;">heights of, <a href="#Page_174">174</a>;</span></li>
<li><span style="margin-left: 1em;">scale of, derived from tensions of gases, <a href="#Page_174">174</a>.</span></li>

<li>Terence, <a href="#Page_347">347</a>.</li>

<li>Terms, scientific, <a href="#Page_342">342</a>-<a href="#Page_343">343</a>.</li>

<li>Thales, <a href="#Page_259">259</a>.</li>

<li>Theories, their scope, function, and power, <a href="#Page_241">241</a>-<a href="#Page_242">242</a>;</li>
<li><span style="margin-left: 1em;">must be replaced by direct description, <a href="#Page_248">248</a>.</span></li>

<li>Thermal, energy, <a href="#Page_174">174</a>, <a href="#Page_177">177</a>;</li>
<li><span style="margin-left: 1em;">capacity, <a href="#Page_123">123</a>, footnote.</span></li>

<li>Thermodynamics, <a href="#Page_160">160</a> et seq.</li>

<li>Thermoelectrometer, Riess's, <a href="#Page_133">133</a>, <a href="#Page_169">169</a>.</li>

<li>Thing-in-itself, the, <a href="#Page_200">200</a>.</li>

<li>Things, mental symbols for groups of sensations, <a href="#Page_200">200</a>-<a href="#Page_201">201</a>.</li>

<li>Thomson, James, on the lowering of the freezing-point of water by pressure, <a href="#Page_162">162</a>.</li>

<li>Thomson, W., his absolute electrometer, <a href="#Page_127">127</a>, footnote;</li>
<li><span style="margin-left: 1em;">on thermodynamics, <a href="#Page_162">162</a>;</span></li>
<li><span style="margin-left: 1em;">on the conservation of energy, <a href="#Page_165">165</a>;</span></li>
<li><span style="margin-left: 1em;">on the mechanical measures of temperature, <a href="#Page_174">174</a>, footnote;</span></li>
<li><span style="margin-left: 1em;">on waste of mechanical energy, <a href="#Page_175">175</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_108">108</a>, <a href="#Page_173">173</a>, footnote.</span></li>

<li>Thought, habitudes of, <a href="#Page_199">199</a>, <a href="#Page_224">224</a>, <a href="#Page_227">227</a>, <a href="#Page_232">232</a>;</li>
<li><span style="margin-left: 1em;">relationship between language and, <a href="#Page_329">329</a>;</span></li>
<li><span style="margin-left: 1em;">incongruence between experience and, <a href="#Page_206">206</a>;</span></li>
<li><span style="margin-left: 1em;">luxuriance of a fully developed, <a href="#Page_58">58</a>;</span></li>
<li><span style="margin-left: 1em;">transformation in scientific, <a href="#Page_214">214</a>-<a href="#Page_235">235</a>.</span></li>

<li>Thoughts, their development and the struggle for existence among them, <a href="#Page_63">63</a>;</li>
<li><span style="margin-left: 1em;">importance of erroneous, <a href="#Page_65">65</a>;</span></li>
<li><span style="margin-left: 1em;">as reproductions of facts, <a href="#Page_107">107</a>.</span></li>

<li>Thread, the individual a, on which pearls are strung, <a href="#Page_234">234</a>-<a href="#Page_235">235</a>.</li>

<li>Tides, <a href="#Page_283">283</a>.</li>

<li>Timbre, <a href="#Page_37">37</a>, <a href="#Page_38">38</a>, <a href="#Page_39">39</a>.</li>

<li>Time, <a href="#Page_178">178</a>, <a href="#Page_204">204</a>, <a href="#Page_205">205</a>, footnote.</li>

<li>Toepler and Foucault, method of, for detecting optical faults, <a href="#Page_313">313</a> et seq., <a href="#Page_320">320</a>.</li>

<li>Tone-figures, <a href="#Page_91">91</a>.</li>

<li>Tones, <a href="#Page_22">22</a>-<a href="#Page_47">47</a>, <a href="#Page_99">99</a> et seq., <a href="#Page_212">212</a>.</li>

<li>Torsion, moment of, <a href="#Page_132">132</a>.</li>

<li>Torsion-balance, Coulomb's, <a href="#Page_109">109</a>, <a href="#Page_168">168</a>.</li>

<li>Torricelli, on virtual velocities, <a href="#Page_150">150</a>;</li>
<li><span style="margin-left: 1em;">his law of liquid efflux, <a href="#Page_150">150</a>;</span></li>
<li><span style="margin-left: 1em;">on the atmosphere, <a href="#Page_273">273</a>.</span></li>

<li>Tourist, journey of, work of the inquirer compared to, <a href="#Page_17">17</a>, <a href="#Page_29">29</a>, <a href="#Page_30">30</a>.</li>

<li>Transatlantic cable, <a href="#Page_108">108</a>.</li>

<li>Transformation and adaptation in scientific thought, <a href="#Page_214">214</a>-<a href="#Page_235">235</a>.</li>

<li>Transformation of ideas, <a href="#Page_63">63</a>.</li>

<li>Transformative law of the energies, <a href="#Page_172">172</a>.</li>

<li><span class="pagenum"><a name="Page_410" id="Page_410">[Pg 410]</a></span>Translation, difficulties of, <a href="#Page_354">354</a>.</li>

<li>Tree, conceptual life compared to a, <a href="#Page_231">231</a>.</li>

<li>Triangle, mutual dependence of the sides and angles of a, <a href="#Page_179">179</a>.</li>

<li>Triple accord, <a href="#Page_46">46</a>.</li>

<li>Truth, wooed by the inquirer, <a href="#Page_45">45</a>;</li>
<li><span style="margin-left: 1em;">difficulty of its acquisition, <a href="#Page_46">46</a>.</span></li>

<li>Tumblers, resounding, <a href="#Page_23">23</a>.</li>

<li>Tuning-forks, explanation of their motion, <a href="#Page_22">22</a> et seq.</li>

<li>Tylor, <a href="#Page_186">186</a>.</li>

<li>Tympanum, <a href="#Page_18">18</a>.</li>

<li>Type, natural laws likened to, <a href="#Page_193">193</a>;</li>
<li><span style="margin-left: 1em;">words compared to, <a href="#Page_191">191</a>.</span></li>

</ul>

<ul class="IX"><li>Ulysses, <a href="#Page_347">347</a>.</li>

<li>Understanding, what it means, <a href="#Page_211">211</a>.</li>

<li>Uniforms, do not fit heads, <a href="#Page_369">369</a>.</li>

<li>Unique determination, <a href="#Page_181">181</a>-<a href="#Page_182">182</a>.</li>

<li>Unison, <a href="#Page_43">43</a>.</li>

<li>Unit, electrostatic, <a href="#Page_111">111</a>.</li>
<li><span style="margin-left: 1em;">See <i>Force</i> and <i>Work</i>.</span></li>

<li>United States, <a href="#Page_336">336</a>.</li>

<li>Universal Real Character, a, <a href="#Page_192">192</a>.</li>

<li>Utility of physical science, <a href="#Page_351">351</a>.</li>

</ul>

<ul class="IX"><li>Variation, the method of, in science, <a href="#Page_230">230</a>;</li>
<li><span style="margin-left: 1em;">in biology, <a href="#Page_216">216</a>.</span></li>

<li>Velocity, of light, <a href="#Page_48">48</a> et seq.;</li>
<li><span style="margin-left: 1em;">of the descent of bodies, <a href="#Page_143">143</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">meaning of, <a href="#Page_204">204</a>;</span></li>
<li><span style="margin-left: 1em;">virtual, <a href="#Page_149">149</a>-<a href="#Page_155">155</a>.</span></li>

<li><i>Verstandesbegriffe</i>, <a href="#Page_199">199</a>.</li>

<li>Vertical, perception of the, <a href="#Page_272">272</a>, <a href="#Page_286">286</a> et seq.;</li>
<li><span style="margin-left: 1em;">symmetry, <a href="#Page_389">389</a>.</span></li>

<li>Vertigo, <a href="#Page_285">285</a>, <a href="#Page_290">290</a>.</li>

<li>Vestibule of the ear, <a href="#Page_300">300</a>.</li>

<li>Vibration, <a href="#Page_22">22</a> et seq.</li>

<li>Vibration-figures, <a href="#Page_91">91</a>.</li>

<li>Vinci, Leonardo da, <a href="#Page_278">278</a>, <a href="#Page_283">283</a>.</li>

<li>Violent motions, <a href="#Page_225">225</a>.</li>

<li>Virtual velocities, <a href="#Page_149">149</a>-<a href="#Page_155">155</a>.</li>

<li>Visibility, general conditions of, <a href="#Page_312">312</a>.</li>

<li>Vision, symmetry of our apparatus of, <a href="#Page_96">96</a>.</li>
<li><span style="margin-left: 1em;">See <i>Eye</i>.</span></li>

<li>Visual nerves, <a href="#Page_96">96</a>.</li>

<li>Visualisation, mental, <a href="#Page_250">250</a>.</li>

<li>Volt, the word, <a href="#Page_343">343</a>.</li>

<li>Volta, <a href="#Page_127">127</a>, footnote, <a href="#Page_134">134</a>.</li>

<li>Voltaire, <a href="#Page_260">260</a>.</li>

<li>Voltaire's ingènu, <a href="#Page_219">219</a>.</li>

<li>Vowels, composed of simple musical notes, <a href="#Page_26">26</a>.</li>

</ul>

<ul class="IX"><li>Wagner, Richard, <a href="#Page_279">279</a>.</li>

<li>Wald, F., <a href="#Page_178">178</a>, footnote.</li>

<li>Wallace, <a href="#Page_216">216</a>.</li>

<li>War, and peace, reflexions upon, <a href="#Page_309">309</a>, <a href="#Page_335">335</a> et seq.</li>

<li>Waste of mechanical energy, W. Thomson on, <a href="#Page_175">175</a>.</li>

<li>Watches, experiment with, <a href="#Page_41">41</a>;</li>
<li><span style="margin-left: 1em;">in a mirror, <a href="#Page_93">93</a>.</span></li>

<li>Water, jet of, resolved into drops, <a href="#Page_60">60</a>;</li>
<li><span style="margin-left: 1em;">free, solid figures of, <a href="#Page_8">8</a>;</span></li>
<li><span style="margin-left: 1em;">objects reflected in, <a href="#Page_94">94</a>, <a href="#Page_191">191</a>;</span></li>
<li><span style="margin-left: 1em;">possible modes of measurement of, <a href="#Page_170">170</a>.</span></li>

<li>Watt, <a href="#Page_266">266</a>.</li>

<li>Wealth, the foundation of, <a href="#Page_198">198</a>.</li>

<li>Weapons, modern, <a href="#Page_335">335</a>.</li>

<li>Weber, <a href="#Page_108">108</a>, <a href="#Page_306">306</a>.</li>

<li>Weight of bodies, varies with their distance from the centre of the earth, <a href="#Page_112">112</a>.</li>

<li>Weismann, <a href="#Page_216">216</a>.</li>

<li>Wheatstone, his stereoscope, <a href="#Page_73">73</a>;</li>
<li><span style="margin-left: 1em;">his pseudoscope, <a href="#Page_96">96</a>;</span></li>
<li><span style="margin-left: 1em;">also <a href="#Page_59">59</a>.</span></li>

<li>Wheel, history and importance of, <a href="#Page_61">61</a> et seq.</li>

<li>Whewell, on the formation of science, <a href="#Page_231">231</a>.</li>

<li>Whole, the, <a href="#Page_204">204</a>, footnote.</li>

<li>Why, the question, <a href="#Page_199">199</a>, <a href="#Page_223">223</a>.</li>

<li>Will, Schopenhauer on the, <a href="#Page_190">190</a>;</li>
<li><span style="margin-left: 1em;">man's most familiar source of power, <a href="#Page_243">243</a>;</span></li>
<li><span style="margin-left: 1em;">used to explain the world, <a href="#Page_186">186</a>;</span></li>
<li><span style="margin-left: 1em;">forces compared to, <a href="#Page_254">254</a>;</span></li>
<li><span style="margin-left: 1em;">compared to pressure, <a href="#Page_14">14</a>.</span></li>

<li>Windmill, a rotating, <a href="#Page_53">53</a>.</li>

<li>Wire frames and nets, for constructing liquid figures of equilibrium, <a href="#Page_4">4</a> et seq.</li>

<li>Witchcraft, <a href="#Page_187">187</a>.</li>

<li>Wollaston, <a href="#Page_284">284</a>, <a href="#Page_285">285</a>.</li>

<li>Wonderful, science the natural foe of the, <a href="#Page_224">224</a>.</li>

<li>Woods, the relative distance of trees in, <a href="#Page_68">68</a>.</li>

<li>Wooer, inquirer compared to a, <a href="#Page_45">45</a>.</li>

<li>Words and sounds, <a href="#Page_343">343</a>.</li>

<li>Words, compared to type, <a href="#Page_191">191</a>.</li>

<li><span class="pagenum"><a name="Page_411" id="Page_411">[Pg 411]</a></span>Work, of liquid forces of attraction, <a href="#Page_14">14</a>;</li>
<li><span style="margin-left: 1em;">in electricity, <a href="#Page_173">173</a>;</span></li>
<li><span style="margin-left: 1em;">measure of, <a href="#Page_119">119</a> et seq., <a href="#Page_130">130</a>, <a href="#Page_223">223</a>;</span></li>
<li><span style="margin-left: 1em;">relation of, with heat, <a href="#Page_162">162</a>, <a href="#Page_245">245</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">amount required to develop electricity, <a href="#Page_131">131</a> et seq.;</span></li>
<li><span style="margin-left: 1em;">produces various physical changes, <a href="#Page_139">139</a>;</span></li>
<li><span style="margin-left: 1em;">substantial conception of, <a href="#Page_183">183</a>-<a href="#Page_184">184</a>.</span></li>
<li><span style="margin-left: 1em;">See <i>Energy</i>.</span></li>

<li>World, the, what it consists of, <a href="#Page_208">208</a>.</li>

<li>World-particles, <a href="#Page_203">203</a>.</li>

<li>Wronsky, <a href="#Page_172">172</a>.</li>

<li>Wundt, on causality and the axioms of physics, <a href="#Page_157">157</a>-<a href="#Page_159">159</a>; <a href="#Page_359">359</a> footnote.</li>

</ul>

<ul class="IX"><li>Xenophon, <a href="#Page_49">49</a>, footnote.</li>

</ul>

<ul class="IX"><li>Young, Thomas, on energy, <a href="#Page_173">173</a>.</li>

</ul>

<ul class="IX"><li>Zelter, <a href="#Page_35">35</a>.</li>

<li>Zeuner, <a href="#Page_171">171</a>.</li>

<li>Zoölogy, comparison in, <a href="#Page_239">239</a>.</li></ul>

<p><span class="pagenum"><a name="Page_412" id="Page_412">[Pg 412]</a><br /><a name="Page_413" id="Page_413">[Pg 413]</a></span></p>

<hr class="chap" />


<h2>THE SCIENCE OF MECHANICS.</h2>

<p class="center">A CRITICAL AND HISTORICAL EXPOSITION OF ITS
PRINCIPLES.</p>

<p class="center bold">By DR. ERNST MACH.</p>

<p class="center small">PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.</p>

<p class="center">Translated from the Second German Edition<br />
<span class="bold">By THOMAS J. McCORMACK.</span></p>

<hr/>
<p class="center">250 Cuts. 534 Pages. Half Morocco, Gilt Top, Marginal Analyses.<br/>
Exhaustive Index. Price $2.50.</p>


<hr/>

<h3><a name="TABLE_OF_CONTENTS2" id="TABLE_OF_CONTENTS2">TABLE OF CONTENTS.</a></h3>


<p><span class="smcap">Statics.</span></p>

<ul class="IX"><li>The Lever.</li>

<li>The Inclined Plane.</li>

<li>The Composition of Forces.</li>

<li>Virtual Velocities.</li>

<li>Statics in Their Application to Fluids.</li>

<li>Statics in Their Application to Gases.</li></ul>



<p><span class="smcap">Dynamics.</span></p>

<ul class="IX"><li>Galileo's Achievements.</li>

<li>Achievements of Huygens.</li>

<li>Achievements of Newton.</li>

<li>Principle of Reaction.</li>

<li>Criticism of the Principle of Reaction
and of the Concept of Mass.</li>

<li>Newton's Views of Time, Space, and
Motion.</li>

<li>Critique of the Newtonian Enunciations.</li>

<li>Retrospect of the Development of
Dynamics.</li></ul>



<p><span class="smcap">The Extension of the Principles of Mechanics.</span></p>

<ul class="IX"><li>Scope of the Newtonian Principles.</li>

<li>Formulæ and Units of Mechanics.</li>

<li>Conservation of Momentum, Conservation
of the Centre of Gravity,
and Conservation of Areas.</li>

<li>Laws of Impact.</li>

<li>D'Alembert's Principle.</li>

<li>Principle of <i>Vis Viva</i>.</li>

<li>Principle of Least Constraint.</li>

<li>Principle of Least Action.</li>

<li>Hamilton's Principle.</li>

<li>Hydrostatic and Hydrodynamic
Questions.</li></ul>



<p><span class="smcap">Formal Development of Mechanics.</span></p>

<ul class="IX"><li>The Isoperimetrical Problems.</li>

<li>Theological, Animistic, and Mystical
Points of View in Mechanics.</li>

<li>Analytical Mechanics.</li>

<li>The Economy of Science.</li></ul>


<p><span class="smcap">The Relation of Mechanics to Other Departments of Knowledge.</span></p>

<ul class="IX">
<li>Relations of Mechanics to Physics.</li>

<li>Relations of Mechanics to Physiology.</li></ul>


<p><span class="pagenum"><a name="Page_414" id="Page_414">[Pg 414]</a></span></p>




<h2><a name="PRESS_NOTICES" id="PRESS_NOTICES">PRESS NOTICES.</a></h2>


<p>"The appearance of a translation into English of this remarkable book
should serve to revivify in this country [England] the somewhat stagnating
treatment of its subject, and should call up the thoughts which puzzle us when
we think of them, and that is not sufficiently often.... Professor Mach is a
striking instance of the combination of great mathematical knowledge with
experimental skill, as exemplified not only by the elegant illustrations of mechanical
principles which abound in this treatise, but also from his brilliant
experiments on the photography of bullets.... A careful study of Professor
Mach's work, and a treatment with more experimental illustration, on the
lines laid down in the interesting diagrams of his <i>Science of Mechanics</i>, will
do much to revivify theoretical mechanical science, as developed from the
elements by rigorous logical treatment."&mdash;Prof. A. G. Greenhill, in <i>Nature</i>,
London.</p>

<p>"Those who are curious to learn how the principles of mechanics have
been evolved, from what source they take their origin, and how far they can
be deemed of positive and permanent value, will find Dr. Mach's able treatise
entrancingly interesting.... The book is a remarkable one in many respects,
while the mixture of history with the latest scientific principles and
absolute mathematical deductions makes it exceedingly attractive."&mdash;<i>Mechanical
World</i>, Manchester and London, England.</p>

<p>"Mach's Mechanics is unique. It is not a text-book, but forms a useful
supplement to the ordinary text-book. The latter is usually a skeleton outline,
full of mathematical symbols and other abstractions. Mach's book has
'muscle and clothing,' and being written from the historical standpoint, introduces
the leading contributors in succession, tells what they did and how
they did it, and often what manner of men they were. Thus it is that the
pages glow, as it were, with a certain humanism, quite delightful in a scientific
book.... The book is handsomely printed, and deserves a warm reception
from all interested in the progress of science."&mdash;<i>The Physical Review</i>, New
York and London.</p>

<p>"Mr. T. J. McCormack, by his effective translation, where translation
was no light task, of this masterly treatise upon the earliest and most fundamental
of the sciences, has rendered no slight service to the English speaking
student. The German and English languages are generally accounted
second to none in their value as instruments for the expression of scientific
thought; but the conversion bodily of an abstruse work from one into the
other, so as to preserve all the meaning and spirit of the original and to set it
easily and naturally into its new form, is a task of the greatest difficulty, and<span class="pagenum"><a name="Page_415" id="Page_415">[Pg 415]</a></span>
when performed so well as in the present instance, merits great commendation.
Dr. Mach has created for his own works the severest possible standard
of judgment. To expect no more from the books of such a master than from
the elementary productions of an ordinary teacher in the science would be
undue moderation. Our author has lifted what, to many of us, was at one
time a course of seemingly unprofitable mental gymnastics, encompassed
only at vast expenditure of intellectual effort, into a study possessing a deep
philosophical value and instinct with life and interest. 'No profit grows
where is no pleasure ta'en,' and the emancipated collegian will turn with
pleasure from the narrow methods of the text-book to where the science is
made to illustrate, by a treatment at once broad and deep, the fundamental
connexion between all the physical sciences, taken together."&mdash;<i>The Mining
Journal</i>, London, England.</p>

<p>"As a history of mechanics, the work is admirable."&mdash;<i>The Nation</i>, New
York.</p>

<p>"An excellent book, admirably illustrated."&mdash;<i>The Literary World</i>, London,
England.</p>

<p>"Sets forth the elements of its subject with a lucidity, clearness, and
force unknown in the mathematical text-books ... is admirably fitted to
serve students as an introduction on historical lines to the principles of mechanical
science."&mdash;<i>Canadian Mining and Mechanical Review</i>, Ottawa, Can.</p>

<p>"A masterly book.... To any one who feels that he does not know as
much as he ought to about physics, we can commend it most heartily as a
scholarly and able treatise ... both interesting and profitable."&mdash;A. M.
Wellington, in <i>Engineering News</i>, New York.</p>

<p>"The book as a whole is unique, and is a valuable addition to any library
of science or philosophy.... Reproductions of quaint old portraits and
vignettes give piquancy to the pages. The numerous marginal titles form a
complete epitome of the work; and there is that invaluable adjunct, a good
index. Altogether the publishers are to be congratulated upon producing a
technical work that is thoroughly attractive in its make-up."&mdash;Prof. D. W.
Hering, in <i>Science</i>.</p>

<p>"There is one other point upon which this volume should be commended,
and that is the perfection of the translation. It is a common fault that books
of the greatest interest and value in the original are oftenest butchered or
made ridiculous by a clumsy translator. The present is a noteworthy exception."&mdash;<i>Railway
Age</i>.</p><p><span class="pagenum"><a name="Page_416" id="Page_416">[Pg 416]</a></span></p>

<p>"The book is admirably printed and bound.... The presswork is unexcelled
by any technical books that have come to our hands for some time,
and the engravings and figures are all clearly and well executed."&mdash;<i>Railroad
Gazette</i>.</p>




<h2><a name="TESTIMONIALS_OF_PROMINENT_EDUCATORS" id="TESTIMONIALS_OF_PROMINENT_EDUCATORS">TESTIMONIALS OF PROMINENT EDUCATORS.</a></h2>


<p>"I am delighted with Professor Mach's <i>Science of Mechanics</i>."&mdash;<i>M. E.
Cooley</i>, Professor of Mechanical Engineering, Ann Arbor, Mich.</p>

<p>"You have done a good service to science in publishing Mach's <i>Science
of Mechanics</i> in English. I shall take every opportunity to recommend it to
young students as a source of much interesting information and inspiration."&mdash;<i>M.
I. Pupin</i>, Professor of Mechanics, Columbia College, New York.</p>

<p>"Mach's <i>Science of Mechanics</i> is an admirable ... book."&mdash;<i>Prof. E. A.
Fuertes</i>, Director of the College of Civil Engineering of Cornell University,
Ithaca, N. Y.</p>

<p>"I congratulate you upon producing the work in such good style and in
so good a translation. I bought a copy of it a year ago, very shortly after you
issued it. The book itself is deserving of the highest admiration; and you
are entitled to the thanks of all English-speaking physicists for the publication
of this translation."&mdash;<i>D. W. Hering</i>, Professor of Physics, University of
the City of New York, New York.</p>

<p>"I have read Mach's <i>Science of Mechanics</i> with great pleasure. The book
is exceedingly interesting."&mdash;<i>W. F. Magie</i>, Professor of Physics, Princeton
University, Princeton, N. J.</p>

<p>"The <i>Science of Mechanics</i> by Mach, translated by T. J. McCormack, I
regard as a most valuable work, not only for acquainting the student with the
history of the development of Mechanics, but as serving to present to him
most favorably the fundamental ideas of Mechanics and their rational connexion
with the highest mathematical developments. It is a most profitable
book to read along with the study of a text-book of Mechanics, and I shall
take pleasure in recommending its perusal by my students."&mdash;<i>S. W. Robinson</i>,
Professor of Mechanical Engineering, Ohio State University, Columbus, Ohio.</p>

<p>"I am delighted with Mach's 'Mechanics.' I will call the attention to
it of students and instructors who have the Mechanics or Physics to study or
teach."&mdash;<i>J. E. Davies</i>, University of Wisconsin, Madison, Wis.</p>

<p>"There can be but one opinion as to the value of Mach's work in this
translation. No instructor in physics should be without a copy of it."&mdash;<i>Henry
Crew</i>, Professor of Physics in the Northwestern University, Evanston, Ill.</p><hr class="chap" /><p><span class="pagenum"><a name="Page_417" id="Page_417">[Pg 417]</a></span></p>




<h2><a name="POPULAR_SCIENTIFIC_LECTURES" id="POPULAR_SCIENTIFIC_LECTURES">POPULAR SCIENTIFIC LECTURES.</a></h2>

<p class="center">A PORTRAYAL OF THE SPIRIT AND METHODS
OF SCIENCE.</p>

<p class="center bold">By DR. ERNST MACH.</p>

<p class="small center">PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.</p>

<p class="center bold">Translated by THOMAS J. McCORMACK.</p>

<p class="center"><i>Third Edition, Revised Throughout and Greatly Enlarged.</i></p>

<hr/>

<p class="center">Cloth, Gilt Top. Exhaustively Indexed. Pages, 415. Cuts, 59. Price, $1.50.</p>

<hr/>


<h3><a name="TITLES_OF_THE_LECTURES" id="TITLES_OF_THE_LECTURES">TITLES OF THE LECTURES.</a></h3>


<ul class="IX"><li>The Forms of Liquids.</li>

<li>The Fibres of Corti.</li>

<li>On the Causes of Harmony.</li>

<li>On the Velocity of Light.</li>

<li>Why Has Man Two Eyes?</li>

<li>On Symmetry.</li>

<li>On the Fundamental Concepts of Static Electricity.</li>

<li>On the Principle of the Conservation of Energy.</li>

<li>On the Economical Nature of Physical Inquiry.</li>

<li>On the Principle of Comparison in Physics.</li>

<li>On the Part Played by Accident in Invention and Discovery.</li>

<li>On Sensations of Orientation.</li>

<li>On the Relative Educational Value of the Classics and the Mathematico-Physical Sciences.</li>

<li>A Contribution to the History of Acoustics.</li>

<li>Remarks on the Theory of Spatial Vision.</li>

<li>On Transformation and Adaptation in Scientific Thought.</li></ul>



<h2>PRESS NOTICES.</h2>

<p>"A most fascinating volume, treating of phenomena in which all are interested,
in a delightful style and with wonderful clearness. For lightness
of touch and yet solid value of information the chapter 'Why Has Man Two
Eyes?' has scarcely a rival in the whole realm of popular scientific writing."&mdash;<i>The
Boston Traveller</i>.</p>

<p>"Truly remarkable in the insight they give into the relationship of the
various fields cultivated under the name of Physics.... A vein of humor is
met here and there reminding the reader of Heaviside, never offending one's
taste. These features, together with the lightness of touch with which Mr.
McCormack has rendered them, make the volume one that may be fairly
called rare. The spirit of the author is preserved in such attractive, really
delightful, English that one is assured nothing has been lost by translation."&mdash;Prof.
Henry Crew, in <i>The Astrophysical Journal</i>.</p><p><span class="pagenum"><a name="Page_418" id="Page_418">[Pg 418]</a></span></p>

<p>"A very delightful and useful book.... The author treats some of the
most recondite problems of natural science, in so charmingly untechnical a
way, with such a wealth of bright illustration, as makes his meaning clear to
the person of ordinary intelligence and education.... This is a work that
should find a place in every library, and that people should be encouraged to
read."&mdash;<i>Daily Picayune</i>, New Orleans.</p>

<p>"In his translation Mr. McCormack has well preserved the frank, simple,
and pleasing style of this famous lecturer on scientific topics. Professor
Mach deals with the live facts, the salient points of science, and not with its
mysticism or dead traditions. He uses the simplest of illustrations and expresses
himself clearly, tersely, and with a delightful freshness that makes
entertaining reading of what in other hands would be dull and prosy."&mdash;<i>Engineering
News</i>, N. Y.</p>

<p>"The general reader is led by plain and easy steps along a delightful way
through what would be to him without such a help a complicated maze of
difficulties. Marvels are invented and science is revealed as the natural foe
to mysteries."&mdash;<i>The Chautauquan</i>.</p>

<p>"The beautiful quality of the work is not marred by abstruse discussions
which would require a scientist to fathom, but is so simple and so clear that
it brings us into direct contact with the matter treated."&mdash;<i>The Boston Post</i>.</p>

<p>"A masterly exposition of important scientific truths."&mdash;<i>Scotsman</i>, Edinburgh.</p>

<p>"These lectures by Dr. Mach are delightfully simple and frank; there is
no dryness or darkness of technicalities, and science and common life do not
seem separated by a gulf.... The style is admirable, and the whole volume
seems gloriously alive and human."&mdash;<i>Providence Journal</i>, R. I.</p>

<p>"The non-scientific reader who desires to learn something of modern
scientific theories, and the reasons for their existence, cannot do better than
carefully study these lectures. The English is excellent throughout, and reflects
great credit on the translator."&mdash;<i>Manufacturer and Builder</i>.</p>

<p>"We like the quiet, considerate intelligence of these lectures."&mdash;<i>Independent</i>,
New York.</p>

<p>"Professor Mach's lectures are so pleasantly written and illumined with
such charm of illustration that they have all the interest of lively fiction."&mdash;<i>New
York Com. Advertiser</i>.</p>

<p>"The literary and philosophical suggestiveness of the book is very rich."
<i>Hartford Seminary Record</i>.</p><p><span class="pagenum"><a name="Page_419" id="Page_419">[Pg 419]</a></span></p>

<p>"All are presented so skilfully that one can imagine that Professor Mach's
hearers departed from his lecture-room with the conviction that science was
a matter for abecedarians. Will please those who find the fairy tales of
science more absorbing than fiction."&mdash;<i>The Pilot</i>, Boston.</p>

<p>"Professor Mach ... is a master in physics.... His book is a good one
and will serve a good purpose, both for instruction and suggestion."&mdash;Prof.
A. E. Dolbear, in <i>The Dial</i>.</p>

<p>"The most beautiful ideas are unfolded in the exposition."&mdash;<i>Catholic
World</i>, New York.</p>




<h2><a name="THE_ANALYSIS_OF_THE_SENSATIONS" id="THE_ANALYSIS_OF_THE_SENSATIONS">THE ANALYSIS OF THE SENSATIONS</a></h2>

<p class="center bold">By DR. ERNST MACH.</p>

<p class="center small">PROFESSOR OF THE HISTORY AND THEORY OF INDUCTIVE SCIENCE IN THE
UNIVERSITY OF VIENNA.</p>

<hr/>

<p class="center">Pages, 208. Illustrations, 37. Indexed.</p>

<p class="center">(Price, Cloth, $1.25.)</p>

<hr/>


<h3><a name="CONTENTS" id="CONTENTS">CONTENTS.</a></h3>


<ul class="IX"><li>Introductory: Antimetaphysical.</li>

<li>The Chief Points of View for the Investigation
of the Senses.</li>

<li>The Space-Sensations of the Eye.</li>

<li>Space-Sensation, Continued.</li>

<li>The Relations of the Sight-Sensations
to One Another and to the
Other Psychical Elements.</li>

<li>The Sensation of Time.</li>

<li>The Sensation of Sound.</li>

<li>Influence of the Preceding Investigations
on the Mode of Conceiving
Physics.</li></ul>

<hr/>

<p>"A wonderfully original little book. Like everything he writes a work of
genius."&mdash;<i>Prof. W. James</i> of Harvard.</p>

<p>"I consider each work of Professor Mach a distinct acquisition to a
library of science."&mdash;<i>Prof. D. W. Hering</i>, New York University.</p>

<p>"There is no work known to the writer which, in its general scientific
bearings, is more likely to repay richly thorough study. We are all interested
in nature in one way or another, and our interests can only be heightened
and clarified by Mach's wonderfully original and wholesome book. It is not
saying too much to maintain that every intelligent person should have a copy
of it,&mdash;and should study that copy."&mdash;<i>Prof. J. E. Trevor</i>, Cornell.</p>

<p>"Students may here make the acquaintance of some of the open questions
of sensation and at the same time take a lesson in the charm of scientific
modesty that can hardly be excelled."&mdash;<i>Prof. E. C. Sanford</i>, Clark University.</p>

<p>"It exhibits keen observation and acute thought, with many new and interesting
experiments by way of illustration. Moreover, the style is light
and even lively&mdash;a rare merit in a German prose work, and still rarer in a
translation of one."&mdash;<i>The Literary World</i>, London.</p>

<hr/>

<p class="center">CHICAGO:<br />
<span class="bold">The Open Court Publishing Company</span><br />
324 DEARBORN STREET.</p>

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<p><span class="pagenum"><a name="Page_420" id="Page_420">[Pg 420]</a></span></p>




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<p>FOOTNOTES:</p>

<div class="footnote"><p><a name="Footnote_1_1" id="Footnote_1_1"></a><a href="#FNanchor_1_1"><span class="label">[1]</span></a> <i>Statique expérimentale et théorique des liquids</i>, 1873. See also <i>The Science
of Mechanics</i>, p. 384 et seqq., The Open Court Publishing Co., Chicago, 1893.</p></div>

<div class="footnote"><p><a name="Footnote_2_2" id="Footnote_2_2"></a><a href="#FNanchor_2_2"><span class="label">[2]</span></a> Compare Mach, <i>Ueber die Molecularwirkung der Flüssigkeiten</i>, Reports
of the Vienna Academy, 1862.</p></div>

<div class="footnote"><p><a name="Footnote_3_3" id="Footnote_3_3"></a><a href="#FNanchor_3_3"><span class="label">[3]</span></a> In almost all branches of physics that are well worked out such maximal
and minimal problems play an important part.</p></div>

<div class="footnote"><p><a name="Footnote_4_4" id="Footnote_4_4"></a><a href="#FNanchor_4_4"><span class="label">[4]</span></a> Compare Mach, <i>Vorträge über Psychophysik</i>, Vienna, 1863, page 41; <i>Compendium
der Physik für Mediciner</i>, Vienna, 1863, page 234; and also <i>The Science
of Mechanics</i>, Chicago, 1893, pp. 84 and 464.</p></div>

<div class="footnote"><p><a name="Footnote_5_5" id="Footnote_5_5"></a><a href="#FNanchor_5_5"><span class="label">[5]</span></a> Like reflexions are found in Quételet, <i>Du système sociale</i>.</p></div>

<div class="footnote"><p><a name="Footnote_6_6" id="Footnote_6_6"></a><a href="#FNanchor_6_6"><span class="label">[6]</span></a> For the full development of this idea see the essay "On the Economical
Nature of Physical Inquiry," p. 186, and the chapter on "The Economy of
Science," in my <i>Mechanics</i> (Chicago: The Open Court Publishing Company,
1893), p. 481.</p></div>

<div class="footnote"><p><a name="Footnote_7_7" id="Footnote_7_7"></a><a href="#FNanchor_7_7"><span class="label">[7]</span></a> Science may be regarded as a maximum or minimum problem, exactly
as the business of the merchant. In fact, the intellectual activity of natural
inquiry is not so greatly different from that exercised in ordinary life as is
usually supposed.</p></div>

<div class="footnote"><p><a name="Footnote_8_8" id="Footnote_8_8"></a><a href="#FNanchor_8_8"><span class="label">[8]</span></a> This experiment, with its associated reflexions, is due to Galileo.</p></div>

<div class="footnote"><p><a name="Footnote_9_9" id="Footnote_9_9"></a><a href="#FNanchor_9_9"><span class="label">[9]</span></a> A development of the theory of musical audition differing in many
points from the theory of Helmholtz here expounded, will be found in my
<i>Contributions to the Analysis of the Sensations</i> (English translation by C. M.
Williams), Chicago, The Open Court Publishing Company, 1897.</p></div>

<div class="footnote"><p><a name="Footnote_10_10" id="Footnote_10_10"></a><a href="#FNanchor_10_10"><span class="label">[10]</span></a> Sauveur also set out from Leibnitz's idea, but arrived by independent
researches at a different theory, which was very near to that of Helmholtz.
Compare on this point Sauveur, <i>Mémoires de l'Académie des Sciences</i>, Paris,
1700-1705, and R. Smith, <i>Harmonics</i>, Cambridge, 1749. (See <i>Appendix</i>, p. 346.)</p></div>

<div class="footnote"><p><a name="Footnote_11_11" id="Footnote_11_11"></a><a href="#FNanchor_11_11"><span class="label">[11]</span></a> According to Mr. Jules Andrieu, the idea that nature must be tortured
to reveal her secrets is preserved in the name <i>crucible</i>&mdash;from the Latin <i>crux</i>,
a cross. But, more probably, <i>crucible</i> is derived from some Old French or
Teutonic form, as <i>cruche</i>, <i>kroes</i>, <i>krus</i>, etc., a pot or jug (cf. Modern English
<i>crock</i>, <i>cruse</i>, and German <i>Krug</i>).&mdash;<i>Trans.</i></p></div>

<div class="footnote"><p><a name="Footnote_12_12" id="Footnote_12_12"></a><a href="#FNanchor_12_12"><span class="label">[12]</span></a> Xenophon, Memorabilia iv, 7, puts into the mouth of Socrates these
words: &#959;&#8020;&#964;&#949; &#947;&#8048;&#961; &#949;&#8017;&#961;&#949;&#964;&#8048; &#7936;&#957;&#952;&#961;&#8061;&#960;&#959;&#953;&#962; &#945;&#8016;&#964;&#8048; &#7952;&#957;&#8057;&#956;&#953;&#950;&#949;&#957; &#949;&#7990;&#957;&#945;&#953;, &#959;&#8020;&#964;&#949; &#967;&#945;&#961;&#8055;&#950;&#949;&#963;&#952;&#945;&#953;
&#952;&#949;&#959;&#8150;&#962; &#7938;&#957; &#7969;&#947;&#949;&#8150;&#964;&#959; &#964;&#8056;&#957; &#950;&#951;&#964;&#959;&#8166;&#957;&#964;&#945; &#7939; &#7952;&#954;&#949;&#8150;&#957;&#959;&#953; &#963;&#945;&#966;&#951;&#957;&#8055;&#963;&#945;&#953; &#959;&#8016;&#954; &#7952;&#946;&#959;&#965;&#955;&#8053;&#952;&#951;&#963;&#945;&#957;.</p></div>

<div class="footnote"><p><a name="Footnote_13_13" id="Footnote_13_13"></a><a href="#FNanchor_13_13"><span class="label">[13]</span></a> Galilei, <i>Discorsi e dimostrazione matematiche</i>. Leyden, 1638. <i>Dialogo
Primo.</i></p></div>

<div class="footnote"><p><a name="Footnote_14_14" id="Footnote_14_14"></a><a href="#FNanchor_14_14"><span class="label">[14]</span></a> In the same way, the pitch of a locomotive-whistle is higher as the
locomotive rapidly approaches an observer, and lower when rapidly leaving
him than if the locomotive were at rest.&mdash;<i>Trans.</i></p></div>

<div class="footnote"><p><a name="Footnote_15_15" id="Footnote_15_15"></a><a href="#FNanchor_15_15"><span class="label">[15]</span></a> A kilometre is 0.621 or nearly five-eighths of a statute mile.</p></div>

<div class="footnote"><p><a name="Footnote_16_16" id="Footnote_16_16"></a><a href="#FNanchor_16_16"><span class="label">[16]</span></a> Observe, also, the respect in which the wheel is held in India, Japan
and other Buddhistic countries, as the emblem of power, order, and law, and
of the superiority of mind over matter. The consciousness of the importance of
this invention seems to have lingered long in the minds of these nations.&mdash;<i>Tr.</i></p></div>

<div class="footnote"><p><a name="Footnote_17_17" id="Footnote_17_17"></a><a href="#FNanchor_17_17"><span class="label">[17]</span></a> This effect is particularly noticeable in the size of workmen on high
chimneys and church-steeples&mdash;"steeple Jacks." When the cables were slung
from the towers of the Brooklyn bridge (277 feet high), the men sent out in
baskets to paint them, appeared, against the broad background of heaven and
water, like flies.&mdash;<i>Trans.</i></p></div>

<div class="footnote"><p><a name="Footnote_18_18" id="Footnote_18_18"></a><a href="#FNanchor_18_18"><span class="label">[18]</span></a> See Joh. Müller, <i>Vergleichende Physiologie des Gesichtssinnes</i>, Leipsic,
1826.</p></div>

<div class="footnote"><p><a name="Footnote_19_19" id="Footnote_19_19"></a><a href="#FNanchor_19_19"><span class="label">[19]</span></a> Delivered before the German Casino of Prague, in the winter of 1871.
</p>
<p>
A fuller treatment of the problems of this lecture will be found in my <i>Contributions
to the Analysis of the Sensations</i> (Jena, 1886), English Translation,
Chicago, 1895. J. P. Soret, <i>Sur la perception du beau</i> (Geneva, 1892), also regards
repetition as a principle of æsthetics. His discussions of the <i>æsthetical</i>
side of the subject are much more detailed than mine. But with respect to
the psychological and physiological foundation of the principle, I am convinced
that the <i>Contributions to the Analysis of the Sensations</i> go deeper.&mdash;<span class="smcap">Mach</span>
(1894).</p></div>

<div class="footnote"><p><a name="Footnote_20_20" id="Footnote_20_20"></a><a href="#FNanchor_20_20"><span class="label">[20]</span></a> Kant, in his <i>Prolegomena zu jeder künftigen Metaphysik</i>, also refers to
this fact, but for a different purpose.</p></div>

<div class="footnote"><p><a name="Footnote_21_21" id="Footnote_21_21"></a><a href="#FNanchor_21_21"><span class="label">[21]</span></a> Compare Mach, <i>Fichte's Zeitschrift für Philosophie</i>, 1864, p. 1.</p></div>

<div class="footnote"><p><a name="Footnote_22_22" id="Footnote_22_22"></a><a href="#FNanchor_22_22"><span class="label">[22]</span></a> The fact that the first and second differential coefficients of a curve are
directly seen, but the higher coefficients not, is very simply explained. The
first gives the position of the tangent, the declination of the straight line from
the position of symmetry, the second the declination of the curve from the
straight line. It is, perhaps, not unprofitable to remark here that the ordinary
method of testing rulers and plane surfaces (by reversed applications)
ascertains the deviation of the object from symmetry to itself.</p></div>

<div class="footnote"><p><a name="Footnote_23_23" id="Footnote_23_23"></a><a href="#FNanchor_23_23"><span class="label">[23]</span></a> See the lecture <i>On the Causes of Harmony</i>.</p></div>

<div class="footnote"><p><a name="Footnote_24_24" id="Footnote_24_24"></a><a href="#FNanchor_24_24"><span class="label">[24]</span></a> A. von Oettingen, <i>Harmoniesystem in dualer Entwicklung</i>. Leipsic and
Dorpat, 1866.</p></div>

<div class="footnote"><p><a name="Footnote_25_25" id="Footnote_25_25"></a><a href="#FNanchor_25_25"><span class="label">[25]</span></a> Compare Mach's <i>Zur Theorie des Gehörorgans</i>, Vienna Academy, 1863.</p></div>

<div class="footnote"><p><a name="Footnote_26_26" id="Footnote_26_26"></a><a href="#FNanchor_26_26"><span class="label">[26]</span></a> A lecture delivered at the International Electrical Exhibition, in Vienna,
on September 4, 1883.</p></div>

<div class="footnote"><p><a name="Footnote_27_27" id="Footnote_27_27"></a><a href="#FNanchor_27_27"><span class="label">[27]</span></a> If the two bodies were oppositely electrified they would exert attractions
upon each other.</p></div>

<div class="footnote"><p><a name="Footnote_28_28" id="Footnote_28_28"></a><a href="#FNanchor_28_28"><span class="label">[28]</span></a> The quantity which flows off is in point of fact less than <i>q</i>. It would be
equal to the quantity <i>q</i> only if the inner coating of the jar were wholly encompassed
by the outer coating.</p></div>

<div class="footnote"><p><a name="Footnote_29_29" id="Footnote_29_29"></a><a href="#FNanchor_29_29"><span class="label">[29]</span></a> Rigorously, of course, this is not correct. First, it is to be noted that the
jar <i>L</i> is discharged simultaneously with the electrode of the machine. The
jar <i>F</i>, on the other hand, is always discharged simultaneously with the outer
coating of the jar <i>L</i>. Hence, if we call the capacity of the electrode of the
machine <i>E</i>, that of the unit jar <i>L</i>, that of the outer coating of <i>L</i>, <i>A</i>, and that of
the principal jar <i>F</i>, then this equation would exist for the example in the text:
(<i>F</i> + <i>A</i>)/(<i>L</i> + <i>E</i>) = 5. A cause of further departure from absolute exactness is
the residual charge.</p></div>

<div class="footnote"><p><a name="Footnote_30_30" id="Footnote_30_30"></a><a href="#FNanchor_30_30"><span class="label">[30]</span></a> Making allowance for the corrections indicated in the preceding footnote,
I have obtained for the dielectric constant of sulphur the number 3.2,
which agrees practically with the results obtained by more delicate methods.
For the highest attainable precision one should by rights immerse the two
plates of the condenser first wholly in air and then wholly in sulphur, if the
ratio of the capacities is to correspond to the dielectric constant. In point of
fact, however, the error which arises from inserting simply a plate of sulphur
that exactly fills the space between the two plates, is of no consequence.</p></div>

<div class="footnote"><p><a name="Footnote_31_31" id="Footnote_31_31"></a><a href="#FNanchor_31_31"><span class="label">[31]</span></a> As this definition in its simple form is apt to give rise to misunderstandings,
elucidations are usually added to it. It is clear that we cannot lift a
quantity of electricity to <i>K</i>, without changing the distribution on <i>K</i> and the
potential on <i>K</i>. Hence, the charges on <i>K</i> must be conceived as fixed, and so
small a quantity raised that no appreciable change is produced by it. Taking
the work thus expended as many times as the small quantity in question is
contained in the unit of quantity, we shall obtain the potential. The potential
of a body <i>K</i> may be briefly and precisely defined as follows: If we expend
the element of work <i>dW</i> to raise the element of positive quantity <i>dQ</i> from the
earth to the conductor, the potential of a conductor <i>K</i> will be given by <i>V</i> =
<i>dW</i>/<i>dQ</i>.</p></div>

<div class="footnote"><p><a name="Footnote_32_32" id="Footnote_32_32"></a><a href="#FNanchor_32_32"><span class="label">[32]</span></a> In this article the solidus or slant stroke is used for the usual fractional
sign of division. Where plus or minus signs occur in the numerator or denominator,
brackets or a vinculum is used.&mdash;<i>Tr.</i></p></div>

<div class="footnote"><p><a name="Footnote_33_33" id="Footnote_33_33"></a><a href="#FNanchor_33_33"><span class="label">[33]</span></a> A sort of agreement exists between the notions of thermal and electrical
capacity, but the difference between the two ideas also should be carefully
borne in mind. The thermal capacity of a body depends solely upon that body
itself. The electrical capacity of a body <i>K</i> is influenced by all bodies in its
vicinity, inasmuch as the charge of these bodies is able to alter the potential
of <i>K</i>. To give, therefore, an unequivocal significance to the notion of the capacity
(<i>C</i>) of a body <i>K</i>, <i>C</i> is defined as the relation <i>Q</i>/<i>V</i> for the body <i>K</i> in a
certain given position of all neighboring bodies, and during connexion of all
neighboring conductors with the earth. In practice the situation is much
simpler. The capacity, for example, of a jar, the inner coating of which is
almost enveloped by its outer coating, communicating with the ground, is not
sensibly affected by charged or uncharged adjacent conductors.</p></div>

<div class="footnote"><p><a name="Footnote_34_34" id="Footnote_34_34"></a><a href="#FNanchor_34_34"><span class="label">[34]</span></a> These formulæ easily follow from Newton's theorem that a homogeneous
spherical shell, whose elements obey the law of the inverse squares, exerts no
force whatever on points within it but acts on points without as if the whole
mass were concentrated at its centre. The formulæ next adduced also flow
from this proposition.</p></div>

<div class="footnote"><p><a name="Footnote_35_35" id="Footnote_35_35"></a><a href="#FNanchor_35_35"><span class="label">[35]</span></a> The energy of a sphere of radius <i>r</i> charged with the quantity <i>q</i> is
1/2(<i>q</i><sup>2</sup>/<i>r</i>). If the radius increase by the space <i>dr</i> a loss of energy occurs, and
the work done is 1/2(<i>q</i><sup>2</sup>/<i>r</i><sup>2</sup>)<i>dr</i>. Letting <i>p</i> denote the uniform electrical pressure
on unit of surface of the sphere, the work done is also 4<i>r</i><sup>2</sup>&#960;<i>pdr</i>. Hence
<i>p</i> = (1/8<i>r</i><sup>2</sup>&#960;)(<i>q</i><sup>2</sup>/<i>r</i><sup>2</sup>). Subjected to the same superficial pressure on all sides,
say in a fluid, our half sphere would be an equilibrium. Hence we must make
the pressure <i>p</i> act on the surface of the great circle to obtain the effect on the
balance, which is <i>r</i><sup>2</sup>&#960;<i>p</i> = 1/8(<i>q</i><sup>2</sup>/<i>r</i><sup>2</sup>) = 1/8<i>V</i><sup>2</sup>.</p></div>

<div class="footnote"><p><a name="Footnote_36_36" id="Footnote_36_36"></a><a href="#FNanchor_36_36"><span class="label">[36]</span></a> The arrangement described is for several reasons not fitted for the actual
measurement of potential. Thomson's absolute electrometer is based upon
an ingenious modification of the electrical balance of Harris and Volta. Of
two large plane parallel plates, one communicates with the earth, while the
other is brought to the potential to be measured. A small movable superficial
portion <i>f</i> of this last hangs from the balance for the determination of the
attraction <i>P</i>. The distance of the plates from each other being <i>D</i> we get <i>V</i> =
<i>D</i>&#8730;(8&#960;<i>P</i>/<i>f</i>).</p></div>

<div class="footnote"><p><a name="Footnote_37_37" id="Footnote_37_37"></a><a href="#FNanchor_37_37"><span class="label">[37]</span></a> This moment of torsion needs a supplementary correction, on account of
the vertical electric attraction of the excited disks. This is done by changing
the weight of the disk by means of additional weights and by making a second
reading of the angles of deflexion.</p></div>

<div class="footnote"><p><a name="Footnote_38_38" id="Footnote_38_38"></a><a href="#FNanchor_38_38"><span class="label">[38]</span></a> The jar in our experiment acts like an accumulator, being charged by a
dynamo machine. The relation which obtains between the expended and the
available work may be gathered from the following simple exposition. A
Holtz machine <i>H</i> (Fig. 40) is charging a unit jar <i>L</i>, which after <i>n</i> discharges
of quantity <i>q</i> and potential <i>v</i>, charges the jar <i>F</i> with the quantity <i>Q</i> at the potential
<i>V</i>. The energy of the unit-jar discharges is lost and that of the jar <i>F</i>
alone is left. Hence the ratio of the available work to the total work expended
is
</p>
<p>
<i>½QV/[½QV + (n/2)qv]</i> and as <i>Q</i> = <i>nq</i>, also <i>V/(V + v)</i>.
</p>

<p>
If, now, we interpose no unit jar, still the parts of the machine and the wires
of conduction are themselves virtually such unit jars and the formula still
subsists <i>V</i>/(<i>V</i> + &#931;<i>v</i>), in which &#931;<i>v</i> represents the sum of all the successively introduced
differences of potential in the circuit of connexion.</p></div>

<div class="footnote"><p><a name="Footnote_39_39" id="Footnote_39_39"></a><a href="#FNanchor_39_39"><span class="label">[39]</span></a> Published in Vol. 5, No. I, of <i>The Monist</i>, October, 1894, being in part
a re-elaboration of the treatise <i>Ueber die Erhaltung der Arbeit</i>, Prague, 1872.</p></div>

<div class="footnote"><p><a name="Footnote_40_40" id="Footnote_40_40"></a><a href="#FNanchor_40_40"><span class="label">[40]</span></a> <i>On Matter, Living Force, and Heat</i>, Joule: <i>Scientific Papers</i>, London,
1884, I, p. 265.</p></div>

<div class="footnote"><p><a name="Footnote_41_41" id="Footnote_41_41"></a><a href="#FNanchor_41_41"><span class="label">[41]</span></a> "Atqui hoc si sit, globorum series sive corona eundem situm cum priore
habebit, eademque de causa octo globi sinistri ponderosiores erunt sex dextris,
ideoque rursus octo illi descendent, sex illi ascendent, istique globi ex sese
<i>continuum et aeternum motum efficient, quod est falsum</i>."</p></div>

<div class="footnote"><p><a name="Footnote_42_42" id="Footnote_42_42"></a><a href="#FNanchor_42_42"><span class="label">[42]</span></a> "A igitur, (si ullo modo per naturam fieri possit) locum sibi tributum
non servato, ac delabatur in <i>D</i>; quibus positis aqua quae ipsi <i>A</i> succedit eandem
ob causam deffluet in <i>D</i>, eademque ab alia istinc expelletur, atque adeo
aqua haec (cum ubique eadem ratio sit) <i>motum instituet perpetuum, quod absurdum
fuerit</i>."</p></div>

<div class="footnote"><p><a name="Footnote_43_43" id="Footnote_43_43"></a><a href="#FNanchor_43_43"><span class="label">[43]</span></a> "Accipio, gradus velocitatis ejusdem mobilis super diversas planorum
inclinationes acquisitos tunc esse aequales, cum eorundum planorum elevationes
aequales sint."</p></div>

<div class="footnote"><p><a name="Footnote_44_44" id="Footnote_44_44"></a><a href="#FNanchor_44_44"><span class="label">[44]</span></a> "Voi molto probabilmente discorrete, ma oltre al veri simile voglio con
una esperienza crescer tanto la probabilità, che poco gli manchi all'agguagliarsi
ad una ben necessaria dimostrazione. Figuratevi questo foglio essere
una parete eretta all'orizzonte, e da un chiodo fitto in essa pendere una palla
di piombo d'un'oncia, o due, sospesa dal sottil filo <i>AB</i> lungo due, o tre braccia
perpendicolare all'orizzonte, e nella parete segnate una linea orizontale <i>DC</i>
segante a squadra il perpendicolo <i>AB</i>, il quale sia lontano dalla parete due
dita in circa, trasferendo poi il filo <i>AB</i> colla palla in <i>AC</i>, lasciata essa palla in
libertà, la quale primieramente vedrete scendere descrivendo l'arco <i>CBD</i>, e
di tanto trapassare il termine <i>B</i>, che scorrendo per l'arco <i>BD</i> sormonterà fino
quasi alla segnata parallela <i>CD</i>, restando di per vernirvi per piccolissimo intervallo,
toltogli il precisamente arrivarvi dall'impedimento dell'aria, e del
filo. Dal che possiamo veracemente concludere, che l'impeto acquistato nel
punto <i>B</i> dalla palla nello scendere per l'arco <i>CB</i>, fu tanto, che bastò a risospingersi
per un simile arco <i>BD</i> alla medesima altezza; fatta, e più volte reiterata
cotale esperienza, voglio, che fiechiamo nella parete rasente al perpendicolo
<i>AB</i> un chiodo come in <i>E</i>, ovvero in <i>F</i>, che sporga in fuori cinque, o
sei dita, e questo acciocchè il filo <i>AC</i> tornando come prima a riportar la palla
<i>C</i> per l'arco <i>CB</i>, giunta che ella sia in <i>B</i>, inoppando il filo nel chiodo <i>E</i>, sia
costretta a camminare per la circonferenza <i>BG</i> descritta in torno al centro <i>E</i>,
dal che vedremo quello, che potrà far quel medesimo impeto, che dianzi concepizo
nel medesimo termine <i>B</i>, sospinse l'istesso mobile per l'arco <i>ED</i> all'altezza
dell'orizzonale <i>CD</i>. Ora, Signori, voi vedrete con gusto condursi la
palla all'orizzontale nel punto <i>G</i>, e l'istesso accadere, l'intoppo si metesse
più basso, come in <i>F</i>, dove la palla descriverebbe l'arco <i>BJ</i>, terminando sempre
la sua salita precisamente nella linea <i>CD</i>, e quando l'intoppe del chiodo
fusse tanto basso, che l'avanzo del filo sotto di lui non arivasse all'altezza di
<i>CD</i> (il che accaderebbe, quando fusse più vicino al punto <i>B</i>, che al segamento
dell' <i>AB</i> coll'orizzontale <i>CD</i>), allora il filo cavalcherebbe il chiodo, e
segli avolgerebbe intorno. Questa esperienza non lascia luogo di dubitare
della verità del supposto: imperocchè essendo li due archi <i>CB</i>, <i>DB</i> equali e
similmento posti, l'acquisto di momento fatto per la scesa nell'arco <i>CB</i>, è il
medesimo, che il fatto per la scesa dell'arco <i>DB</i>; ma il momento acquistato
in <i>B</i> per l'arco <i>CB</i> è potente a risospingere in su il medesimo mobile per l'arco
<i>BD</i>; adunque anco il momento acquistato nella scesa <i>DB</i> è eguale a quello,
che sospigne l'istesso mobile pel medesimo arco da <i>B</i> in <i>D</i>, sicche universal-mente
ogni memento acquistato per la scesa d'un arco è eguale a quello, che
può far risalire l'istesso mobile pel medesimo arco: ma i momenti tutti che
fanno resalire per tutti gli archi <i>BD</i>, <i>BG</i>, <i>BJ</i> sono eguali, poichè son fatti
dal istesso medesimo momento acquistato per la scesa <i>CB</i>, come mostra
l'esperienza: adunque tutti i momenti, che si acquistano per le scese negli
archi <i>DB</i>, <i>GB</i>, <i>JB</i> sono eguali."</p></div>

<div class="footnote"><p><a name="Footnote_45_45" id="Footnote_45_45"></a><a href="#FNanchor_45_45"><span class="label">[45]</span></a> "Constat jam, quod mobile ex quiete in <i>A</i> descendens per <i>AB</i>, gradus
acquirit velocitatis juxta temporis ipsius incrementum: gradum vero in <i>B</i>
esse maximum acquisitorum, et suapte natura immutabiliter impressum, sublatis
scilicet causis accelerationis novae, aut retardationis: accelerationis inquam,
si adhuc super extenso plano ulterius progrederetur; retardationis
vero, dum super planum acclive <i>BC</i> fit reflexio: in horizontali autem <i>GH</i>
aequabilis motus juxta gradum velocitatis ex <i>A</i> in <i>B</i> acquisitae in infinitum
extenderetur."</p></div>

<div class="footnote"><p><a name="Footnote_46_46" id="Footnote_46_46"></a><a href="#FNanchor_46_46"><span class="label">[46]</span></a> "Si gravitas non esset, neque aër motui corporum officeret, unumquodque
eorum, acceptum semel motum continuaturum velocitate aequabili, secundum
lineam rectam."</p></div>

<div class="footnote"><p><a name="Footnote_47_47" id="Footnote_47_47"></a><a href="#FNanchor_47_47"><span class="label">[47]</span></a> "Si pondera quotlibet, vi gravitatis suae, moveri incipiant; non posse
centrum gravitatis ex ipsis compositae altius, quam ubi incipiente motu reperiebatur,
ascendere.
</p>
<p>
"Ipsa vero hypothesis nostra quominus scrupulum moveat, nihil aliud
sibi velle ostendemus, quam, quod nemo unquam negavit, gravia nempe sursum
non ferri.&mdash;Et sane, si hac eadem uti scirent novorum operum machinatores,
qui motum perpetuum irrito conatu moliuntur, facile suos ipsi errores
deprehenderent, intelligerentque rem eam mechanica ratione haud quaquam
possibilem esse."</p></div>

<div class="footnote"><p><a name="Footnote_48_48" id="Footnote_48_48"></a><a href="#FNanchor_48_48"><span class="label">[48]</span></a> "Si pendulum e pluribus ponderibus compositum, atque e quiete dimissum,
partem quamcunque oscillationis integrae confecerit, atque inde porro
intelligantur pondera ejus singula, relicto communi vinculo, celeritates acquisitas
sursum convertere, ac quousque possunt ascendere; hoc facto centrum
gravitatis ex omnibus compositae, ad eandem altitudinem reversum erit, quam
ante inceptam oscillationem obtinebat."</p></div>

<div class="footnote"><p><a name="Footnote_49_49" id="Footnote_49_49"></a><a href="#FNanchor_49_49"><span class="label">[49]</span></a> "Notato autem hic illud staticum axioma etiam locum habere:
</p>

<div class="poem"><div class="stanza">
<span class="i0">"Ut spatium agentis ad spatium patientis<br /></span>
<span class="i0">Sic potentia patientis ad potentiam agentis."<br /></span>
</div></div>
</div>

<div class="footnote"><p><a name="Footnote_50_50" id="Footnote_50_50"></a><a href="#FNanchor_50_50"><span class="label">[50]</span></a> "Cependant, comme dans cet ouvrage on ne fut d'abord attentif qu'à
considérer ce beau développement de la mécanique qui semblait sortir tout
entière d'une seule et même formule, on crut naturellement que la science etait
faite, et qu'il ne restait plus qu'à chercher la démonstration du principe des
vitesses virtuelles. Mais cette recherche ramena toutes les difficultés qu'on
avait franchies par le principe même. Cette loi si générale, où se mêlent des
idées vagues et étrangères de mouvements infinement petits et de perturbation
d'équilibre, ne fit en quelque sorte que s'obsurcir à l'examen; et le livre de
Lagrange n'offrant plus alors rien de clair que la marche des calculs, on vit
bien que les nuages n'avaient paru levé sur le cours de la mécanique que
parcequ'ils étaient, pour ainsi dire, rassemblés à l'origine même do cette
science.
</p>
<p>
"Une démonstration générale du principe des vitesses virtuelles devait
au fond revenir a établir le mécanique entière sur une autre base: car la demonstration
d'une loi qui embrasse toute une science ne peut être autre chose
qua la reduction de cette science à une autre loi aussi générale, mais évidente,
ou du moins plus simple que la première, et qui partant la rende inutile."</p></div>

<div class="footnote"><p><a name="Footnote_51_51" id="Footnote_51_51"></a><a href="#FNanchor_51_51"><span class="label">[51]</span></a> <i>Traité de la lumière</i>, Leyden, 1690, p. 2.</p></div>

<div class="footnote"><p><a name="Footnote_52_52" id="Footnote_52_52"></a><a href="#FNanchor_52_52"><span class="label">[52]</span></a> "L'on ne sçaurait douter que la lumière ne consiste dans le <i>mouvement</i> de
certaine matière. Car soit qu'on regarde sa production, on trouve qu'içy sur
la terre c'est principalement le feu et la flamme qui l'engendrent, lesquels
contient sans doute des corps qui sont dans un mouvement rapide, puis qu'ils
dissolvent et fondent plusieurs autres corps des plus solides: soit qu'on regarde
ses effets, on voit que quand la lumière est ramasseé, comme par des
miroires concaves, elle a la vertu de brûler comme le feu. c-est-à-dire qu'elle
desunit les parties des corps; ce qui marque assurément du <i>mouvement</i>, au
moins dans la <i>vraye Philosophie</i>, dans laquelle on conçoit la cause de tous les
effets naturels par des raisons de <i>mechanique</i>. Ce qu'il faut faire à mon avis,
ou bien renoncer à tout espérance de jamais rien comprendre dans la Physique."</p></div>

<div class="footnote"><p><a name="Footnote_53_53" id="Footnote_53_53"></a><a href="#FNanchor_53_53"><span class="label">[53]</span></a> <i>Sur la puissance motrice du feu</i>. (Paris, 1824.)</p></div>

<div class="footnote"><p><a name="Footnote_54_54" id="Footnote_54_54"></a><a href="#FNanchor_54_54"><span class="label">[54]</span></a> "On objectra peut-être ici que le mouvement perpétuel, démontré impossible
par les <i>seules actions mécaniques</i>, ne l'est peut-être pas lorsqu'on
emploie l'influence soit de la <i>chaleur</i>, soit de l'électricité; mais pent-on concevoir
les phénomènes de la chaleur et de l'électricité comme dus à autre
chose qu'à des <i>mouvements quelconques des corps</i> et comme tels ne doivent-ils
pas être soumis aux lois générales de la mécanique?"</p></div>

<div class="footnote"><p><a name="Footnote_55_55" id="Footnote_55_55"></a><a href="#FNanchor_55_55"><span class="label">[55]</span></a> By this is meant the temperature of a Celsius scale, the zero of which is
273° below the melting-point of ice.</p></div>

<div class="footnote"><p><a name="Footnote_56_56" id="Footnote_56_56"></a><a href="#FNanchor_56_56"><span class="label">[56]</span></a> I first drew attention to this fact in my treatise <i>Ueber die Erhaltung der
Arbeit</i>, Prague, 1872. Before this, Zeuner had pointed out the analogy between
mechanical and thermal energy. I have given a more extensive development
of this idea in a communication to the <i>Sitzungsberichte der Wiener</i>
<i>Akademie</i>, December, 1892, entitled <i>Geschichte und Kritik des Carnot'schen
Wärmegesetzes</i>. Compare also the works of Popper (1884), Helm (1887),
Wronsky (1888), and Ostwald (1892).</p></div>

<div class="footnote"><p><a name="Footnote_57_57" id="Footnote_57_57"></a><a href="#FNanchor_57_57"><span class="label">[57]</span></a> Sir William Thomson first consciously and intentionally introduced
(1848, 1851) a <i>mechanical</i> measure of temperature similar to the electric measure
of potential.</p></div>

<div class="footnote"><p><a name="Footnote_58_58" id="Footnote_58_58"></a><a href="#FNanchor_58_58"><span class="label">[58]</span></a> Compare my <i>Analysis of the Sensations</i>, Jena, 1886: English translation,
Chicago, 1897.</p></div>

<div class="footnote"><p><a name="Footnote_59_59" id="Footnote_59_59"></a><a href="#FNanchor_59_59"><span class="label">[59]</span></a> A better terminology appears highly desirable in the place of the usual
misleading one. Sir William Thomson (1852) appears to have felt this need,
and it has been clearly expressed by F. Wald (1889). We should call the work
which corresponds to a vanished quantity of heat its mechanical substitution-value;
while that work which can be <i>actually</i> performed in the passage of a
thermal condition <i>A</i> to a condition <i>B</i>, alone deserves the name of the <i>energy-value</i>
of this change of condition. In this way the <i>arbitrary</i> substantial conception
of the processes would be preserved and misapprehensions forestalled.</p></div>

<div class="footnote"><p><a name="Footnote_60_60" id="Footnote_60_60"></a><a href="#FNanchor_60_60"><span class="label">[60]</span></a> An address delivered before the anniversary meeting of the Imperial
Academy of Sciences, at Vienna, May 25, 1882.</p></div>

<div class="footnote"><p><a name="Footnote_61_61" id="Footnote_61_61"></a><a href="#FNanchor_61_61"><span class="label">[61]</span></a> <i>Primitive Culture.</i></p></div>

<div class="footnote"><p><a name="Footnote_62_62" id="Footnote_62_62"></a><a href="#FNanchor_62_62"><span class="label">[62]</span></a> Tylor, <i>loc. cit.</i></p></div>

<div class="footnote"><p><a name="Footnote_63_63" id="Footnote_63_63"></a><a href="#FNanchor_63_63"><span class="label">[63]</span></a> <i>Essai philosophique sur les probabilités</i>. 6th Ed. Paris, 1840, p. 4. The
necessary consideration of the initial velocities is lacking in this formulation.</p></div>

<div class="footnote"><p><a name="Footnote_64_64" id="Footnote_64_64"></a><a href="#FNanchor_64_64"><span class="label">[64]</span></a> <i>Principien der Wirthschaftslehre</i>, Vienna, 1873.</p></div>

<div class="footnote"><p><a name="Footnote_65_65" id="Footnote_65_65"></a><a href="#FNanchor_65_65"><span class="label">[65]</span></a> It is clear from this that all so-called elementary (differential) laws involve
a relation to the whole.</p></div>

<div class="footnote"><p><a name="Footnote_66_66" id="Footnote_66_66"></a><a href="#FNanchor_66_66"><span class="label">[66]</span></a> If it be objected, that in the case of perturbations of the velocity of rotation
of the earth, we could be sensible of such perturbations, and being obliged
to have some measure of time, we should resort to the period of vibration of
the waves of sodium light,&mdash;all that this would show is that for practical reasons
we should select that event which best served us as the <i>simplest</i> common
measure of the others.</p></div>

<div class="footnote"><p><a name="Footnote_67_67" id="Footnote_67_67"></a><a href="#FNanchor_67_67"><span class="label">[67]</span></a> Measurement, in fact, is the definition of one phenomenon by another
(standard) phenomenon.</p></div>

<div class="footnote"><p><a name="Footnote_68_68" id="Footnote_68_68"></a><a href="#FNanchor_68_68"><span class="label">[68]</span></a> I have represented the point of view here taken for more than thirty
years and developed it in various writings (<i>Erhaltung der Arbeit</i>, 1872, parts
of which are published in the article on <i>The Conservation of Energy</i> in this
collection; <i>The Forms of Liquids</i>, 1872, also published in this collection; and
the <i>Bewegungsempfindungen</i>, 1875). The idea, though known to philosophers,
is unfamiliar to the majority of physicists. It is a matter of deep regret to me,
therefore, that the title and author of a small tract which accorded with my
views in numerous details and which I remember having caught a glance of
in a very busy period (1879-1880), have so completely disappeared from my
memory that all efforts to obtain a clue to them have hitherto been fruitless.</p></div>

<div class="footnote"><p><a name="Footnote_69_69" id="Footnote_69_69"></a><a href="#FNanchor_69_69"><span class="label">[69]</span></a> Inaugural Address, delivered on assuming the Rectorate of the University
of Prague, October 18, 1883.
</p>
<p>
The idea presented in this essay is neither new nor remote. I have touched
upon it myself on several occasions (first in 1867), but have never made it the
subject of a formal disquisition. Doubtless, others, too, have treated it; it
lies, so to speak, in the air. However, as many of my illustrations were well
received, although known only in an imperfect form from the lecture itself
and the newspapers, I have, contrary to my original intention, decided to
publish it. It is not my intention to trespass here upon the domain of biology.
My statements are to be taken merely as the expression of the fact that no one
can escape the influence of a great and far-reaching idea.</p></div>

<div class="footnote"><p><a name="Footnote_70_70" id="Footnote_70_70"></a><a href="#FNanchor_70_70"><span class="label">[70]</span></a> At first sight an apparent contradiction arises from the admission of both
heredity and adaptation; and it is undoubtedly true that a strong disposition
to heredity precludes great capability of adaptation. But imagine the organism
to be a plastic mass which retains the form transmitted to it by former
influences until new influences modify it; the <i>one</i> property of <i>plasticity</i> will
then represent capability of adaptation as well as power of heredity. Analogous
to this is the case of a bar of magnetised steel of high coercive force:
the steel retains its magnetic properties until a new force displaces them.
Take also a body in motion: the body retains the velocity acquired in (<i>inherited</i>
from) the interval of time just preceding, except it be changed in the
next moment by an accelerating force. In the case of the body in motion the
<i>change</i> of velocity (<i>Abänderung</i>) was looked upon as a matter of course, while
the discovery of the principle of <i>inertia</i> (or persistence) created surprise; in
Darwin's case, on the contrary, <i>heredity</i> (or persistence) was taken for granted,
while the principle of <i>variation</i> (<i>Abänderung</i>) appeared novel.
</p>
<p>
Fully adequate views are, of course, to be reached only by a study of the
original facts emphasised by Darwin, and not by these analogies. The example
referring to motion, if I am not mistaken, I first heard, in conversation,
from my friend J. Popper, Esq., of Vienna.
</p>
<p>
Many inquirers look upon the stability of the species as something settled,
and oppose to it the Darwinian theory. But the stability of the species is itself
a "theory." The essential modifications which Darwin's views also are
undergoing will be seen from the works of Wallace [and Weismann], but more
especially from a book of W. H. Rolph, <i>Biologische Probleme</i>, Leipsic, 1882.
Unfortunately, this last talented investigator is no longer numbered among
the living.</p></div>

<div class="footnote"><p><a name="Footnote_71_71" id="Footnote_71_71"></a><a href="#FNanchor_71_71"><span class="label">[71]</span></a> Written in 1883.</p></div>

<div class="footnote"><p><a name="Footnote_72_72" id="Footnote_72_72"></a><a href="#FNanchor_72_72"><span class="label">[72]</span></a> See Pfaundler, <i>Pogg. Ann., Jubelband</i>, p. 182.</p></div>

<div class="footnote"><p><a name="Footnote_73_73" id="Footnote_73_73"></a><a href="#FNanchor_73_73"><span class="label">[73]</span></a> See the beautiful discussions of this point in Hering's <i>Memory as a General
Function of Organised Matter</i> (1870), Chicago, The Open Court Publishing
Co., 1887. Compare also Dubois, <i>Ueber die Uebung</i>, Berlin, 1881.</p></div>

<div class="footnote"><p><a name="Footnote_74_74" id="Footnote_74_74"></a><a href="#FNanchor_74_74"><span class="label">[74]</span></a> Spencer, <i>The Principles of Psychology</i>. London, 1872.</p></div>

<div class="footnote"><p><a name="Footnote_75_75" id="Footnote_75_75"></a><a href="#FNanchor_75_75"><span class="label">[75]</span></a> See the article <i>The Velocity of Light</i>, page 63.</p></div>

<div class="footnote"><p><a name="Footnote_76_76" id="Footnote_76_76"></a><a href="#FNanchor_76_76"><span class="label">[76]</span></a> I am well aware that the endeavor to confine oneself in natural research
to <i>facts</i> is often censured as an exaggerated fear of metaphysical spooks.
But I would observe, that, judged by the mischief which they have wrought,
the metaphysical, of all spooks, are the least fabulous. It is not to be denied
that many forms of thought were not originally acquired by the individual, but
were antecedently formed, or rather prepared for, in the development of the
species, in some such way as Spencer, Haeckel, Hering, and others have
supposed, and as I myself have hinted on various occasions.</p></div>

<div class="footnote"><p><a name="Footnote_77_77" id="Footnote_77_77"></a><a href="#FNanchor_77_77"><span class="label">[77]</span></a> Compare, for example, <i>Schiller, Zerstreute Betrachtungen über verschiedene
ästhetische Gegenstände</i>.</p></div>

<div class="footnote"><p><a name="Footnote_78_78" id="Footnote_78_78"></a><a href="#FNanchor_78_78"><span class="label">[78]</span></a> We must not be deceived in imagining that the happiness of other people
is not a very considerable and essential portion of our own. It is common
capital, which cannot be created by the individual, and which does not perish
with him. The formal and material limitation of the <i>ego</i> is necessary and sufficient
only for the crudest practical objects, and cannot subsist in a broad conception.
Humanity in its entirety may be likened to a polyp-plant. The
material and organic bonds of individual union have, indeed, been severed;
they would only have impeded freedom of movement and evolution. But the
ultimate aim, the psychical connexion of the whole, has been attained in a
much higher degree through the richer development thus made possible.</p></div>

<div class="footnote"><p><a name="Footnote_79_79" id="Footnote_79_79"></a><a href="#FNanchor_79_79"><span class="label">[79]</span></a> C. E. von Baer, the subsequent opponent of Darwin and Haeckel, has
discussed in two beautiful addresses (<i>Das allgemeinste Gesetz der Natur in
aller Entwickelung</i>, and <i>Welche Auffassung der lebenden Natur ist die richtige,
und wie ist diese Auffassung auf die Entomologie anzuwenden</i>?) the
narrowness of the view which regards an animal in its existing state as
finished and complete, instead of conceiving it as a phase in the series of evolutionary
forms and regarding the species itself as a phase of the development
of the animal world in general.</p></div>

<div class="footnote"><p><a name="Footnote_80_80" id="Footnote_80_80"></a><a href="#FNanchor_80_80"><span class="label">[80]</span></a> An address delivered before the General Session of the German Association
of Naturalists and Physicians, at Vienna, Sept. 24, 1894.</p></div>

<div class="footnote"><p><a name="Footnote_81_81" id="Footnote_81_81"></a><a href="#FNanchor_81_81"><span class="label">[81]</span></a> Inaugural lecture delivered on assuming the Professorship of the History
and Theory of Inductive Science in the University of Vienna, October
21, 1895.</p></div>

<div class="footnote"><p><a name="Footnote_82_82" id="Footnote_82_82"></a><a href="#FNanchor_82_82"><span class="label">[82]</span></a> The phrase is, <i>Er hat das Pulver nicht erfunden</i>.</p></div>

<div class="footnote"><p><a name="Footnote_83_83" id="Footnote_83_83"></a><a href="#FNanchor_83_83"><span class="label">[83]</span></a> "Quod si quis tanta industria exstitisset, ut ex naturae principiis at geometria
hanc rem eruere potuisset, eum ego supra mortalium sortem ingenio
valuisse dicendum crederem. Sed hoc tantum abest, ut fortuito reperti artificii
rationem non adhuc satis explicari potuerint viri doctissimi."&mdash;Hugenii
Dioptrica (de telescopiis).</p></div>

<div class="footnote"><p><a name="Footnote_84_84" id="Footnote_84_84"></a><a href="#FNanchor_84_84"><span class="label">[84]</span></a> I must not be understood as saying that the fire-drill has played no part
in the worship of fire or of the sun.</p></div>

<div class="footnote"><p><a name="Footnote_85_85" id="Footnote_85_85"></a><a href="#FNanchor_85_85"><span class="label">[85]</span></a> Compare on this point the extremely interesting remarks of Dr. Paul
Carus in his <i>Philosophy of the Tool</i>, Chicago, 1893.</p></div>

<div class="footnote"><p><a name="Footnote_86_86" id="Footnote_86_86"></a><a href="#FNanchor_86_86"><span class="label">[86]</span></a> Möbius, <i>Naturwissenschaftlicher Verein für Schleswig-Holstein</i>, Kiel,
1893, p. 113 et seq.</p></div>

<div class="footnote"><p><a name="Footnote_87_87" id="Footnote_87_87"></a><a href="#FNanchor_87_87"><span class="label">[87]</span></a> I am indebted for this observation to Professor Hatscheck.</p></div>

<div class="footnote"><p><a name="Footnote_88_88" id="Footnote_88_88"></a><a href="#FNanchor_88_88"><span class="label">[88]</span></a> Cf. Hoppe, <i>Entdecken und Finden</i>, 1870.</p></div>

<div class="footnote"><p><a name="Footnote_89_89" id="Footnote_89_89"></a><a href="#FNanchor_89_89"><span class="label">[89]</span></a> See the lecture "Sensations of Orientation," p. 282 et seq.</p></div>

<div class="footnote"><p><a name="Footnote_90_90" id="Footnote_90_90"></a><a href="#FNanchor_90_90"><span class="label">[90]</span></a> This story was related to me by Jolly, and subsequently repeated in a
letter from him.</p></div>

<div class="footnote"><p><a name="Footnote_91_91" id="Footnote_91_91"></a><a href="#FNanchor_91_91"><span class="label">[91]</span></a> I do not know whether Swift's academy of schemers in Lagado, in
which great discoveries and inventions were made by a sort of verbal game
of dice, was intended as a satire on Francis Bacon's method of making discoveries
by means of huge synoptic tables constructed by scribes. It certainly
would not have been ill-placed.</p></div>

<div class="footnote"><p><a name="Footnote_92_92" id="Footnote_92_92"></a><a href="#FNanchor_92_92"><span class="label">[92]</span></a> "Crescunt disciplinae lente tardeque; per varios errores sero pervenitur
ad veritatem. Omnia praeparata esse debent diuturno et assiduo labore
ad introitum veritatis novae. Jam illa certo temporis momento divina quadam
necessitate coacta emerget."
</p>
<p>
Quoted by Simony, <i>In ein ringförmiges Band einen Knoten zu machen</i>,
Vienna, 1881, p. 41.</p></div>

<div class="footnote"><p><a name="Footnote_93_93" id="Footnote_93_93"></a><a href="#FNanchor_93_93"><span class="label">[93]</span></a> A lecture delivered on February 24, 1897, before the <i>Verein zur Verbreitung
naturwissenschaftlicher Kenntnisse in Wien</i>.</p></div>

<div class="footnote"><p><a name="Footnote_94_94" id="Footnote_94_94"></a><a href="#FNanchor_94_94"><span class="label">[94]</span></a> Wollaston, <i>Philosophical Transactions, Royal Society</i>, 1810. In the same
place Wollaston also describes and explains the creaking of the muscles.
My attention was recently called to this work by Dr. W. Pascheles.&mdash;Cf. also
Purkinje, <i>Prager medicin</i>. <i>Jahrbücher</i>, Bd. 6, Wien, 1820.</p></div>

<div class="footnote"><p><a name="Footnote_95_95" id="Footnote_95_95"></a><a href="#FNanchor_95_95"><span class="label">[95]</span></a> Similarly many external forces do not act at once on all parts of the
earth, and the internal forces which produce deformations act at first immediately
only upon limited parts. If the earth were a feeling being, the tides
and other terrestrial events would provoke in it similar sensations to those
of our movements. Perhaps the slight alterations of the altitude of the
pole which are at present being studied are connected with the continual
slight deformations of the central ellipsoid occasioned by seismical happenings.</p></div>

<div class="footnote"><p><a name="Footnote_96_96" id="Footnote_96_96"></a><a href="#FNanchor_96_96"><span class="label">[96]</span></a> For the popular explanation by unconscious inference the matter is extremely
simple. We regard the railway carriage as vertical and unconsciously
infer the inclination of the trees. Of course the opposite conclusion that we
regard the trees as vertical and infer the inclination of the carriage, unfortunately,
is equally clear on this theory.</p></div>

<div class="footnote"><p><a name="Footnote_97_97" id="Footnote_97_97"></a><a href="#FNanchor_97_97"><span class="label">[97]</span></a> It will be observed that my way of thinking and experimenting here is
related to that which led Knight to the discovery and investigation of the
geotropism of plants. <i>Philosophical Transactions</i>, January 9, 1806. The relations
between vegetable and animal geotropism have been more recently investigated
by J. Loeb.</p></div>

<div class="footnote"><p><a name="Footnote_98_98" id="Footnote_98_98"></a><a href="#FNanchor_98_98"><span class="label">[98]</span></a> This experiment is doubtless related to the galvanotropic experiment
with the larvæ of frogs described ten years later by L. Hermann. Compare
on this point my remarks in the <i>Anzeiger der Wiener Akademie</i>, 1886, No. 21.
Recent experiments in galvanotropism are due to J. Loeb.</p></div>

<div class="footnote"><p><a name="Footnote_99_99" id="Footnote_99_99"></a><a href="#FNanchor_99_99"><span class="label">[99]</span></a> <i>Wiener Akad.</i>, 6 November, 1873.</p></div>

<div class="footnote"><p><a name="Footnote_100_100" id="Footnote_100_100"></a><a href="#FNanchor_100_100"><span class="label">[100]</span></a> <i>Wiener Gesellschaft der Aerzte</i>, 14 November, 1874.</p></div>

<div class="footnote"><p><a name="Footnote_101_101" id="Footnote_101_101"></a><a href="#FNanchor_101_101"><span class="label">[101]</span></a> I have made a contribution to this last question in my <i>Analysis of the
Sensations</i>, (1886), English translation, 1897.</p></div>

<div class="footnote"><p><a name="Footnote_102_102" id="Footnote_102_102"></a><a href="#FNanchor_102_102"><span class="label">[102]</span></a> In my <i>Grundlinien der Lehre von den Bewegungsempfindungen</i>, 1875,
the matter occupying lines 4 to 13 of page 20 from below, which rests on an
error, is, as I have also elsewhere remarked, to be stricken out. For another
experiment related to that of Foucault, compare my <i>Mechanics</i>, p. 303.</p></div>

<div class="footnote"><p><a name="Footnote_103_103" id="Footnote_103_103"></a><a href="#FNanchor_103_103"><span class="label">[103]</span></a> <i>Anzeiger der Wiener Akad.</i>, 30 December, 1875.</p></div>

<div class="footnote"><p><a name="Footnote_104_104" id="Footnote_104_104"></a><a href="#FNanchor_104_104"><span class="label">[104]</span></a> The experiment was specially interesting for me as I had already attempted
in 1874, although with very little confidence and without success, to
excite electromagnetically my own labyrinth through which I had caused a
current to pass.</p></div>

<div class="footnote"><p><a name="Footnote_105_105" id="Footnote_105_105"></a><a href="#FNanchor_105_105"><span class="label">[105]</span></a> Perhaps the discussion concerning the peculiarity of cats always falling
on their feet, which occupied the Parisian Academy, and, incidentally, Parisian
society a few years ago, will be remembered here. I believe that the
questions which arose are disposed of by the considerations advanced in my
<i>Bewegungsempfindungen</i> (1875). I also partly gave, as early as 1866, the apparatus
conceived by the Parisian scientists to illustrate the phenomena in
question. One difficulty was left untouched in the Parisian debate. The
otolith apparatus of the cat can render it no service in <i>free</i> descent. The
cat, however, while at rest, doubtless knows its position in space and is instinctively
conscious of the amount of movement which will put it on its feet.</p></div>

<div class="footnote"><p><a name="Footnote_106_106" id="Footnote_106_106"></a><a href="#FNanchor_106_106"><span class="label">[106]</span></a> See the Appendix to the English edition of my <i>Analysis of the Sensations</i>,
Chicago, 1897.</p></div>

<div class="footnote"><p><a name="Footnote_107_107" id="Footnote_107_107"></a><a href="#FNanchor_107_107"><span class="label">[107]</span></a> Compare my <i>Analysis of Sensations</i>, p. 123 ff.</p></div>

<div class="footnote"><p><a name="Footnote_108_108" id="Footnote_108_108"></a><a href="#FNanchor_108_108"><span class="label">[108]</span></a> E. H. Weber, <i>De aure et auditu hominis et animalium</i>, Lipsiae, 1820.</p></div>

<div class="footnote"><p><a name="Footnote_109_109" id="Footnote_109_109"></a><a href="#FNanchor_109_109"><span class="label">[109]</span></a> Störensen, <i>Journ. Anat. Phys.</i>, London, Vol. 29 (1895).</p></div>

<div class="footnote"><p><a name="Footnote_110_110" id="Footnote_110_110"></a><a href="#FNanchor_110_110"><span class="label">[110]</span></a> A lecture delivered on Nov. 10, 1897.</p></div>

<div class="footnote"><p><a name="Footnote_111_111" id="Footnote_111_111"></a><a href="#FNanchor_111_111"><span class="label">[111]</span></a> Christiansen, <i>Wiedemann's Annalen</i>, XXIII. S. 298, XXIV., p. 439 (1884-1885).</p></div>

<div class="footnote"><p><a name="Footnote_112_112" id="Footnote_112_112"></a><a href="#FNanchor_112_112"><span class="label">[112]</span></a> The German phrase is <i>Schlierenmethode</i>, by which term the method is
known even by American physicists. It is also called in English the "shadow-method."
But a term is necessary which will cover all the derivatives, and
so we have employed alternatively the words <i>striate</i> and <i>differential</i>. The
etymology of <i>schlieren</i>, it would seem, is uncertain. Its present use is derived
from its technological signification in glass-manufacturing, where by <i>die
Schlieren</i> are meant the wavy streaks and imperfections in glass. Hence its
application to the method for detecting small optical <i>differences</i> and faults
generally. Professor Crew of Evanston suggests to the translator that <i>schlieren</i>
may be related to our <i>slur</i> (L. G., <i>slüren</i>, to trail, to draggle), a conjecture
which is doubtless correct and agrees both with the meaning of <i>schlieren</i> as
given in the large German dictionaries and with the intransitive use of our
own verb <i>slur</i>, the faults in question being conceived as "trailings," "streakings,"
etc.&mdash;<i>Trans.</i></p></div>

<div class="footnote"><p><a name="Footnote_113_113" id="Footnote_113_113"></a><a href="#FNanchor_113_113"><span class="label">[113]</span></a> An address delivered before the Congress of Delegates of the German
Realschulmännerverein, at Dortmund, April 16, 1886. The full title of the
address reads: "On the Relative Educational Value of the Classics and the
Mathematico-Physical Sciences in Colleges and High Schools."
</p>
<p>
Although substantially contained in an address which I was to have made
at the meeting of Natural Scientists at Salzburg in 1881 (deferred on account
of the Paris Exposition), and in the Introduction to a course of lectures on
"Physical Instruction in Preparatory Schools," which I delivered in 1883, the
invitation of the German Realschulmännerverein afforded me the first opportunity
of putting my views upon this subject before a large circle of readers.
Owing to the place and circumstances of delivery, my remarks apply of course,
primarily, only to German schools, but, with slight modifications, made in
this translation, are not without force for the institutions of other countries.
In giving here expression to a strong personal conviction formed long ago, it
is a matter of deep satisfaction to me to find that they agree in many points
with the views recently advanced in independent form by Paulsen (<i>Geschichte
des gelehrten Unterrichts</i>, Leipsic, 1885) and Frary (<i>La question du latin</i>,
Paris, Cerf, 1885). It is not my desire nor effort here to say much that is new,
but merely to contribute my mite towards bringing about the inevitable revolution
now preparing in the world of elementary instruction. In the opinion
of experienced educationists the first result of that revolution will be to make
Greek and mathematics alternately optional subjects in the higher classes of
the German Gymnasium and in the corresponding institutions of other countries,
as has been done in the splendid system of instruction in Denmark. The
gap between the German classical Gymnasium and the German Realgymnasium,
or between classical and scientific schools generally, can thus be bridged
over, and the remaining inevitable transformations will then be accomplished
in relative peace and quiet. (Prague, May, 1886.)</p></div>

<div class="footnote"><p><a name="Footnote_114_114" id="Footnote_114_114"></a><a href="#FNanchor_114_114"><span class="label">[114]</span></a> Maupertuis, <i>&OElig;uvres</i>, Dresden, 1752, p. 339.</p></div>

<div class="footnote"><p><a name="Footnote_115_115" id="Footnote_115_115"></a><a href="#FNanchor_115_115"><span class="label">[115]</span></a> F. Paulsen, <i>Geschichte des gelehrten Unterrichts</i>, Leipsic, 1885.</p></div>

<div class="footnote"><p><a name="Footnote_116_116" id="Footnote_116_116"></a><a href="#FNanchor_116_116"><span class="label">[116]</span></a> There is a peculiar irony of fate in the fact that while Leibnitz was casting
about for a new vehicle of universal linguistic intercourse, the Latin language
which still subserved this purpose the best of all, was dropping more
and more out of use, and that Leibnitz himself contributed not the least to
this result.</p></div>

<div class="footnote"><p><a name="Footnote_117_117" id="Footnote_117_117"></a><a href="#FNanchor_117_117"><span class="label">[117]</span></a> As a rule, the human brain is too much, and wrongly, burdened with
things which might be more conveniently and accurately preserved in books
where they could be found at a moment's notice. In a recent letter to me
from Düsseldorf, Judge Hartwich writes:
</p>
<p>
"A host of words exist which are out and out Latin or Greek, yet are employed
with perfect correctness by people of good education who never had
the good luck to be taught the ancient languages. For example, words like
'dynasty.' ... The child learns such words as parts of the common stock of
speech, or even as parts of his mother-tongue, just as he does the words
'father,' 'mother,' 'bread,' 'milk.' Does the ordinary mortal know the etymology
of these Saxon words? Did it not require the almost incredible
industry of the Grimms and other Teutonic philologists to throw the merest
glimmerings of light upon the origin and growth of our own mother-tongue?
Besides, do not thousands of people of so-called classical education use
every moment hosts of words of foreign origin whose derivation they do not
know? Very few of them think it worth while to look up such words in the
dictionaries, although they love to maintain that people should study the
ancient languages for the sake of etymology alone."</p></div>

<div class="footnote"><p><a name="Footnote_118_118" id="Footnote_118_118"></a><a href="#FNanchor_118_118"><span class="label">[118]</span></a> Standing remote from the legal profession I should not have ventured to
declare that the study of Greek was not necessary for the jurists; yet this
view was taken in the debate that followed this lecture by professional jurists
of high standing. According to this opinion, the preparatory education obtained
in the German Realgymnasium would also be sufficient for the future
jurists and insufficient only for theologians and philologists. [In England and
America not only is Greek not necessary, but the law-Latin is so peculiar that
even persons of <i>good</i> classical education cannot understand it.&mdash;<i>Tr.</i>]</p></div>

<div class="footnote"><p><a name="Footnote_119_119" id="Footnote_119_119"></a><a href="#FNanchor_119_119"><span class="label">[119]</span></a> In emphasising here the weak sides of the writings of Plato and Aristotle,
forced on my attention while reading them in German translations, I, of
course, have no intention of underrating the great merits and the high historical
importance of these two men. Their importance must not be measured
by the fact that our speculative philosophy still moves to a great extent
in their paths of thought. The more probable conclusion is that this branch
has made very little progress in the last two thousand years. Natural science
also was implicated for centuries in the meshes of the Aristotelian thought,
and owes its rise mainly to having thrown off those fetters.</p></div>

<div class="footnote"><p><a name="Footnote_120_120" id="Footnote_120_120"></a><a href="#FNanchor_120_120"><span class="label">[120]</span></a> I would not for a moment contend that we derive exactly the same profit
from reading a Greek author in a translation as from reading him in the original;
but the difference, the excess of gain in the second case, appears to me,
and probably will to most men who are not professional philologists, to be
too dearly bought with the expenditure of eight years of valuable time.</p></div>

<div class="footnote"><p><a name="Footnote_121_121" id="Footnote_121_121"></a><a href="#FNanchor_121_121"><span class="label">[121]</span></a> "The temptation," Judge Hartwich writes, "to regard the 'taste' of the
ancients as so lofty and unsurpassable appears to me to have its chief origin
in the fact that the ancients were unexcelled in the representation of the
nude. First, by their unremitting care of the human body they produced
splendid models; and secondly, in their gymnasiums and in their athletic
games they had these models constantly before their eyes. No wonder, then,
that their statues still excite our admiration! For the form, the ideal of the
human body has not changed in the course of the centuries. But with intellectual
matters it is totally different; they change from century to century,
nay, from decennium to decennium. It is very natural now, that people
should unconsciously apply what is thus so easily seen, namely, the works of
sculpture, as a universal criterion of the highly developed taste of the ancients&mdash;a
fallacy against which people cannot, in my judgment, be too strongly
warned."</p></div>

<div class="footnote"><p><a name="Footnote_122_122" id="Footnote_122_122"></a><a href="#FNanchor_122_122"><span class="label">[122]</span></a> English: "In the beginning God created the heaven and the earth.
And the earth was without form and void; and darkness was upon the face
of the deep. And the spirit of God moved upon the face of the waters."&mdash;Dutch:
"In het begin schiep God den hemel en de aarde. De aarde nu was
woest en ledig, en duisternis was op den afgrond; en de Geest Gods zwefde
op de wateren."&mdash;Danish: "I Begyndelsen skabte Gud Himmelen og Jorden.
Og Jorden var ode og tom, og der var morkt ovenover Afgrunden, og
Guds Aand svoevede ovenover Vandene."&mdash;Swedish: "I begynnelsen skapade
Gud Himmel och Jord. Och Jorden war öde och tom, och mörker war
pä djupet, och Gods Ande swäfde öfwer wattnet."&mdash;German: "Am Anfang
schuf Gott Himmel und Erde. Und die Erde war wüst und leer, und es war
finster auf der Tiefe; und der Geist Gottes schwebte auf dem Wasser."</p></div>

<div class="footnote"><p><a name="Footnote_123_123" id="Footnote_123_123"></a><a href="#FNanchor_123_123"><span class="label">[123]</span></a> Compare Herzen's excellent remarks, <i>De l'enseignement secondaire dans
la Suisse romande</i>, Lausanne, 1886.</p></div>

<div class="footnote"><p><a name="Footnote_124_124" id="Footnote_124_124"></a><a href="#FNanchor_124_124"><span class="label">[124]</span></a> <i>Geschichte der Mathematik</i>, Leipsic, 1874.</p></div>

<div class="footnote"><p><a name="Footnote_125_125" id="Footnote_125_125"></a><a href="#FNanchor_125_125"><span class="label">[125]</span></a> <i>Geometrische Analyse</i>, Ulm, 1886.</p></div>

<div class="footnote"><p><a name="Footnote_126_126" id="Footnote_126_126"></a><a href="#FNanchor_126_126"><span class="label">[126]</span></a> In his text-books of elementary mathematics</p></div>

<div class="footnote"><p><a name="Footnote_127_127" id="Footnote_127_127"></a><a href="#FNanchor_127_127"><span class="label">[127]</span></a> <i>Abhandlungen aus dem Gebiete der Mathematik</i>, Würzburg, 1883.</p></div>

<div class="footnote"><p><a name="Footnote_128_128" id="Footnote_128_128"></a><a href="#FNanchor_128_128"><span class="label">[128]</span></a> My idea here is an appropriate selection of readings from Galileo, Huygens,
Newton, etc. The choice is so easily made that there can be no question
of difficulties. The contents would be discussed with the students, and
the original experiments performed with them. Those scholars alone should
receive this instruction in the upper classes who did not look forward to systematical
instruction in the physical sciences. I do not make this proposition
of reform here for the first time. I have no doubt, moreover, that such radical
changes will only be slowly introduced.</p></div>

<div class="footnote"><p><a name="Footnote_129_129" id="Footnote_129_129"></a><a href="#FNanchor_129_129"><span class="label">[129]</span></a> <i>Die Mathematik als Lehrgegenstand des Gymnasiums</i>, Berlin, 1883.</p></div>

<div class="footnote"><p><a name="Footnote_130_130" id="Footnote_130_130"></a><a href="#FNanchor_130_130"><span class="label">[130]</span></a> Wrong as it is to burden future physicians and scientists with Greek for
the sake of the theologians and philologists, it would be just as wrong to compel
theologians and philologists, on account of the physicians, to study such
subjects as analytical geometry. Moreover, I cannot believe that ignorance
of analytical geometry would be a serious hindrance to a physician that was
otherwise well versed in quantitative thought. No special advantage generally
is observable in the graduates of the Austrian gymnasiums, all of whom have
studied analytical geometry. [Refers to an assertion of Dubois-Reymond.]</p></div>

<div class="footnote"><p><a name="Footnote_131_131" id="Footnote_131_131"></a><a href="#FNanchor_131_131"><span class="label">[131]</span></a> Compare M. Cantor, <i>Geschichte der Mathematik</i>, Leipsic, 1880, Vol. I. p.
193.</p></div>

<div class="footnote"><p><a name="Footnote_132_132" id="Footnote_132_132"></a><a href="#FNanchor_132_132"><span class="label">[132]</span></a> Compare Paulsen, <i>l. c.</i>, pp. 607, 688.</p></div>

<div class="footnote"><p><a name="Footnote_133_133" id="Footnote_133_133"></a><a href="#FNanchor_133_133"><span class="label">[133]</span></a> It is to be hoped that the Americans will jealously guard their schools
and universities against the influence of the State.</p></div>

<div class="footnote"><p><a name="Footnote_134_134" id="Footnote_134_134"></a><a href="#FNanchor_134_134"><span class="label">[134]</span></a> This article, which appeared in the Proceedings of the German Mathematical
Society of Prague for the year 1892, is printed as a supplement to the
article on "The Causes of Harmony," at page 32.</p></div>

<div class="footnote"><p><a name="Footnote_135_135" id="Footnote_135_135"></a><a href="#FNanchor_135_135"><span class="label">[135]</span></a> The present exposition is taken from the volumes for 1700 (published in
1703) and for 1701 (published in 1704), and partly also from the <i>Histoire de
l'Académie</i> and partly from the <i>Mémoires</i>. Sauveur's later works enter less
into consideration here.</p></div>

<div class="footnote"><p><a name="Footnote_136_136" id="Footnote_136_136"></a><a href="#FNanchor_136_136"><span class="label">[136]</span></a> Euler, <i>Tentamen novae theoriae musicae</i>, Petropoli, 1739.</p></div>

<div class="footnote"><p><a name="Footnote_137_137" id="Footnote_137_137"></a><a href="#FNanchor_137_137"><span class="label">[137]</span></a> In attempting to perform his experiment of beats before the Academy,
Sauveur was not quite successful. <i>Histoire de l'Académie</i>, Année 1700, p. 136.</p></div>

<div class="footnote"><p><a name="Footnote_138_138" id="Footnote_138_138"></a><a href="#FNanchor_138_138"><span class="label">[138]</span></a> <i>Histoire de l'Académie</i>, Année 1701, p. 134.</p></div>

<div class="footnote"><p><a name="Footnote_139_139" id="Footnote_139_139"></a><a href="#FNanchor_139_139"><span class="label">[139]</span></a> <i>Ibid.</i>, p. 298.</p></div>

<div class="footnote"><p><a name="Footnote_140_140" id="Footnote_140_140"></a><a href="#FNanchor_140_140"><span class="label">[140]</span></a> <i>Histoire de l'Académie</i>, Année 1702, p. 91.</p></div>

<div class="footnote"><p><a name="Footnote_141_141" id="Footnote_141_141"></a><a href="#FNanchor_141_141"><span class="label">[141]</span></a> From the <i>Histoire de l'Académie</i>, Année 1700, p. 139.</p></div>

<div class="footnote"><p><a name="Footnote_142_142" id="Footnote_142_142"></a><a href="#FNanchor_142_142"><span class="label">[142]</span></a> Because all octaves in use in music offer too great differences of rates
of vibration.</p></div>

<div class="footnote"><p><a name="Footnote_143_143" id="Footnote_143_143"></a><a href="#FNanchor_143_143"><span class="label">[143]</span></a> "Les battemens ne plaisent pas à l'Oreille, à cause de l'inégalité du son,
et l'on peut croire avec beaucoup d'apparence que ce qui rend les Octaves si
agréables, c'est qu'on n'y entend jamais de battemens.
</p>
<p>
"En suivant cette idée, on trouve que les accords dont on ne peut entendre
les battemens, sont justement ceux que les Musiciens traitent de Consonances,
et que ceux dont les battemens se font sentir, sont les Dissonances, et que
quand un accord est Dissonance dans une certaine octave et Consonance
dans une autre, c'est qu'il bat dans l'une, et qu'il ne bat pas dans l'autre.
Aussi est il traité de Consonance imparfaite. Il est fort aisé par les principes
de Mr. Sauveur qu'on a établis ici, de voir quels accords battent, et dans
quelles Octaves au-dessus on au-dessous du son fixe. Si cette hypothèse est
vraye, elle découvrira la véritable source des Règles de la composition, inconnue
jusqu'à présent à la Philosophie, qui s'en remettait presque entièrement
au jugement de l'Oreille. Ces sortes de jugemens naturels, quelque
bisarres qu'ils paroissent quelquefois, ne le sont point, ils ont des causes
très réelles, dont la connaissance appartient à la Philosophie, pourvue qu'elle
s'en puisse mettre en possession."</p></div>

<div class="footnote"><p><a name="Footnote_144_144" id="Footnote_144_144"></a><a href="#FNanchor_144_144"><span class="label">[144]</span></a> <i>Harmonics or the Philosophy of Musical Sounds</i>, Cambridge, 1749. I saw
this book only hastily in 1864 and drew attention to it in a work published in
1866. I did not come into its actual possession until three years ago and then
only did I learn its exact contents.</p></div>

<div class="footnote"><p><a name="Footnote_145_145" id="Footnote_145_145"></a><a href="#FNanchor_145_145"><span class="label">[145]</span></a> <i>Harmonics</i>, pp. 118 and 243.</p></div>

<div class="footnote"><p><a name="Footnote_146_146" id="Footnote_146_146"></a><a href="#FNanchor_146_146"><span class="label">[146]</span></a> "Short cycle" is the period in which the same phases of the two co-operant
tones are repeated.</p></div>

<div class="footnote"><p><a name="Footnote_147_147" id="Footnote_147_147"></a><a href="#FNanchor_147_147"><span class="label">[147]</span></a> This article, designed to illustrate historically that on Symmetry, at
page 89, first appeared in Fichte's <i>Zeitschrift für Philosophie</i>, for 1865.</p></div>

<div class="footnote"><p><a name="Footnote_148_148" id="Footnote_148_148"></a><a href="#FNanchor_148_148"><span class="label">[148]</span></a> Comp. Cornelius, <i>Ueber das Sehen</i>; Wundt, <i>Theorie der Sinneswahrnehmung</i>.</p></div>

<div class="footnote"><p><a name="Footnote_149_149" id="Footnote_149_149"></a><a href="#FNanchor_149_149"><span class="label">[149]</span></a> Comp. Mach, <i>Ueber das Sehen von Lagen and Winkeln</i>. <i>Sitzungsb. der
Wiener Akademie</i>, 1861.</p></div>

<div class="footnote"><p><a name="Footnote_150_150" id="Footnote_150_150"></a><a href="#FNanchor_150_150"><span class="label">[150]</span></a> Comp. Mach, <i>Zur Theorie des Gehörorgans</i>. <i>Sitsungsber, der Wiener
Akad.</i>, 1863.&mdash;<i>Ueber einige Erscheinungen der physiolog. Akustik.</i> <i>Ibid.</i>, 1864.</p></div>









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