Popular scientific lectures

APPENDIX.

A CONTRIBUTION TO THE HISTORY OF ACOUSTICS.[134]

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.

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.

The Histoire de l'Académie for 1700, p. 131, tells us that Sauveur had succeeded in making music an 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.

Sauveur regards the simplicity of the ratios obtaining between the rates of vibration of consonances as something universally known.[135] 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 [136] 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.

Sauveur's ideas took their development from his 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 (Harmonie Universelle, 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 (battemens), 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.[137] 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 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.

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.

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.

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 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 [138] probably, variously colored paper riders were placed at the nodes (noeuds) and ventral segments (ventres), and the division of the string due to the excitation of the overtones (sons harmoniques) belonging to its fundamental note (son fondamental) 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.[139] He himself mentions the simultaneous sounding of the overtones and the fundamental tone.[140] That he did not give the proper consideration to this circumstance was, as will afterwards be seen, fatal to his theory.

While studying beats Sauveur makes the remark that they are displeasing 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.[141]

"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.[142]

"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."[143]

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.

R. Smith [144] 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 to the modern theory, as appears from the following passage.[145]

"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 [146] is periodically confused and interrupted, with perfect ones which cannot beat, because the succession of their short cycles is never confused nor interrupted.

"The fluttering roughness 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 different kind from the smoother beats and undulations of tempered consonances; 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.

"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."

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 overtones, and the theory thus have attained the point of view of Helmholtz.

Reviewing the differences between Sauveur's and Helmholtz's theories, we find the following:

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 mathematical characteristic of consonance as well as a physical 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 physiological or psychological 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.

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 (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 strength 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, a sin(rt) and a sin[(r + ρ)(t + τ)], 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 r responds more weakly to the period r + ρ, the two amplitudes bearing to each other the proportion a: φa. Here φ decreases when ρ increases, and when ρ = 0 it becomes equal to 1, so that only the portion of the stimulus φa is subject to beats, and the portion (1-φ)a continues smoothly onward without disturbance.

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.

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 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 positive physiologico-psychological 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).

The positive physiological 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.

II.

REMARKS ON THE THEORY OF SPATIAL VISION.[147]

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.

Let us see how the theory stands with respect to the special facts involved.

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?

2. Why do we measure differently colored objects 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?

3. Why is it that differently colored figures of the same form reproduce one another and are recognised as the same?

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.

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.

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 undertake. The new principle could not fail to throw all psychology into the most dreadful confusion.

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 the same. 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.[148]

A few considerations will show the likelihood of the rôle of the muscle-sensations. The muscular apparatus of one eye is unsymmetrical. The two eyes together form a system which is vertical in symmetry. This already explains much.

1. The position 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 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.[149]

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 during 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.

Fig. 58.

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 a of the accompanying vertically-symmetrical figure induces in the left eye the same muscular feelings as the right-hand side b 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 quem dii oderunt 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.

Fig. 59.

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 form.

1. Of spatial figures we have already spoken.

2. We call two melodies like melodies when they 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.

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).

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.

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 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.

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.[150]

INDEX.

Absolute, temperature, 162;
time, 204;
forecasts, have no signification in science, 206.
Abstract, meaning of the term, 240.
Abstraction, 180, 200, 208, 231.
Acceleration, organ for forward, 299 et seq.
Accelerations, 204, 216, footnote, 225-226, 253.
Accident, logical and historical, in science, 160, 168, 170, 213;
in inventions and discoveries, 262 et seq.
Accord, the pure triple, 46.
Accumulators, electrical, 125 et seq.;
132, footnote.
Acoustic color, 36.
Acoustics, Sauveur on, 375 et seq.
Action and reaction, importance of the principle of, 191.
Adaptation, in organic and inorganic matter, 216, 229;
in scientific thought, 214-235.
Æsthetics, computation as a principle of, 34;
researches in, 89, footnote;
repetition, a principle of, 91.
Africa, 186, 234, 237.
Agreeable effects, due to repetition of sensations, 92, 97 et seq.
Agriculture, transition to, 265.
Air-gun, 135.
Alcohol and water, mixture of oil and, in Plateau's experiments, 4.
Algebra, economy of, 196.
Alien thoughts in science, 196.
All, the, 88.
Amontons, 174, 346.
Ampère, the word, 314.
Ampère's swimmer, 207.
Analogies, mechanical, 157, 160;
generally, 236-258.
Analogy, defined, 250.
Analysis, 188.
Analytical geometry, not necessary to physicians, 370, footnote.
Anatomic structures, transparent stereoscopic views of, 74.
Anatomy, character of research in, 255.
Andrieu, Jules, 49, footnote.
Animals, the psychical activity of, 190, 231;
the language of, 238;
their capacity for experience, 266 et seq.
Animism, 186, 187, 243, 254.
Anisotropic optical fields, 227.
Apparatus for producing movements of rotation, 287 et seq.
Arabesque, an inverted, 95.
Arabian Nights, 219.
Arago, 270.
Aral, the Sea of, 239.
Archæopteryx, 257.
Archimedes, 4, 237.
Arcimboldo, Giuseppe, 36.
Area, principle of least superficial, 10 et seq.
Ares, the bellowing of the wounded, 272.
Aristotelians, 283.
Aristotle, 348, 296.
Art, development of, 28 et seq.
Artillery, practical, 334-335.
Artistic value of scientific descriptions, 254.
Arts, practical, 108.
Ascent, heights of, 143-151.
Asia, 234.
Assyrians, the art of, 79.
Astronomer, measures celestial by terrestrial distances, 136.
Astronomy, antecedent to psychology, 90;
rigidity of its truths, 221.
Atomic theories, 104.
Atoms, 207.
Attention, the rôle of, in sensuous perception, 35 et seq.
Attraction, generally, 226;
of liquid particles, 13-14;
in electricity, 109 et seq.
Aubert, 298.
Audition. See Ear.
Austrian gymnasiums, 370, footnote.
Axioms, instinctive knowledge, 190.

Babbage, on the economy of machinery, 196.
Bach, 20.
Backwards, prophesying, 253.
Bacon, Lord, 48, 280.
Baer, C. E. von, 235.
Balance, electrical, 127, footnote;
torsion, 109, 168.
Balloon, a hydrogen, 199.
Barbarism and civilisation, 335 et seq.
Bass-clef, 101.
Bass, fundamental, 44.
Beats, 40-45, 377 et seq.
Beautiful, our notions of, variable, 99.
Beauty, objects of, in nature, 91.
Becker, J. K., 364, 369.
Beethoven, 39, 44.
Beginnings of science, 189, 191.
Belvedere Gallery at Vienna, 36.
Bernoulli, Daniel, on the conservation of living force, 149;
on the vibrations of strings, 249.
Bernoulli, James, on the centre of oscillation, 149.
Bernoulli, John, on the conservation of living force, 149;
on the principle of virtual velocities, 151.
Bible, parallel passages from, for language study, 356.
Binocular vision, 66 et seq.
Black, his theory of caloric, 138, 162;
on quantity of heat, 166, 174;
on latent heat, 167, 178;
researches in heat generally, 244.
Blind cat, 303.
Bodies, heavy, seek their places, 224 et seq.;
rotating, 285.
Body, a mental symbol for groups of sensations, 200-203;
the human, our knowledge of, 90.
Boltzmann, 236.
Booth, Mr., 77.
Borelli, 217.
Boulder, a granite, 233.
Bow-wave of ships and moving projectiles, 323 et seq.
Boys, 317.
Bradley, 273.
Brahman, the, 63.
Brain, localisation of functions in, 210.
Breuer, 272, 282 et seq., 293, 298, 300, 301, 303, 306.
Brewster, his stereoscope, 73.
Bridge, invention of the, 264, 268.
British Association, 108.
Brooklyn Bridge, 75, footnote.
Brown, Crum, 293, 301.
Building, our concepts directions for, 253;
facts the result of, 253;
science compared to, 257.
Building-stones, metrical units are, 253.
Busch, 328.
Business of a merchant, science compared to the, 16.
Butterfly, a, 22.

Calculating machines, their economical character, 196.
Caloric, theory of, stood in the way of scientific advancement, 138, 167.
Calypso, the island of, 351.
Canterbury, Archbishop of, 39.
Cantor, M., 361, footnote.
Capacity, electrical, 116 et seq., 123;
thermal, 123;
specific inductive, 117.
Capulets and Montagues, 87.
Cards, difficult games of, 357.
Carnot, S., excludes perpetual motion in heat, 156, 162;
his mechanical view of physics, 156;
on thermodynamics, 160 et seq.;
his principle, 162;
also, 191.
Carus, Dr. Paul, 265, footnote.
Casselli's telegraph, 26.
Cassini, 51.
Cauchy, character of the intellectual activity of a, 195.
Causal insight, awakened by science, 357.
Causality, 157-159, 190, 198 et seq., 221 et seq., 237, 253, 254.
Cause and effect, 198 et seq. See also Causality.
Centimetre-gramme-second system, 111.
Centre of gravity, must lie as low as possible for equilibrium to subsist, 15;
Torricelli's principle of, 150 et seq.
Centre of oscillation, 149.
Change, method of, in science, 230.
Changeable character of bodies, 202.
Changes, physical, how they occur, 205.
Character, a Universal Real, 192.
Character, like the forms of liquids, 3;
persons of, 24.
Charles the Fifth, 369.
Chemical, elements, 202;
symbols, 192;
current, 118.
Chemistry, character of research in, 255;
the method of thermodynamics in, 257.
Child, a, modes of thought of, 223;
looking into a moat, 208.
Child of the forest, his interpretation of new events, 218-219.
Childish questions, 199-200.
Children, the drawings of, 201-202.
Chinese language, economy of, 192;
study of, 354.
Chinese philosopher, an old, 186.
Chinese, speak with unwillingness of politics, 374;
the art of, 79-80.
Chosen, many are called but few are, 65.
Christ, saying of, 65.
Christianity, Latin introduced with, 311.
Christians and Jews, monotheism of the, 187.
Church and State, 88.
Cicero, 318.
Circe, 372.
Circle, the figure of least area with given periphery, 12.
Circular polarisation, 242.
Civilisation and barbarism, 335 et seq.
Civilisation, some phenomena of, explained by binocular vision, 74.
Civilised man, his modes of conception and interpretation, 219.
Clapeyron, 162.
Class-characters of animals, 255.
Classical, culture, the good and bad effects of, 347;
scholars, not the only educated people, 345.
Classics, on instruction in, 338-374;
the scientific, 368.
Classification in science, 255.
Clausius, on thermodynamics, 165;
on reversible cycles, 176.
Claviatur, Mach's, 42-43.
Club-law, 335.
Cochlea, the, a species of piano-forte, 19.
Cockchafer, 86.
Coefficient of self-induction, 250, 252.
Colophonium, solution of, 7.
Color, acoustic, 36.
Color-sensation, 210.
Color-signs, their economy, 192.
Colors, origin of the names of, 239.
Column, body moving behind a, 202.
Communication, its functions, import and fruits, 197, 238 et seq.;
by language, 237;
high importance of, 191 et seq.
Comparative physics, 239.
Comparison in science, 231, 238 et seq.
Computation, a principle of æsthetics, 34.
Concepts, abstract, defined, 250-252;
metrical, in electricity, 107 et seq.
Conceptual, meaning of the term, 240.
Conceptual thought, 192.
Concha, 18.
Condensers, electrical, 125 et seq. 132, footnote.
Conductors and non-conductors. See Electrical, etc.
Conformity in the deportment of the energies, 171-175.
Confusion of objects, cause of, 95.
Conic sections, 257.
Conical refraction, 29, 242.
Conservation of energy, 137 et seq. See Energy.
Conservation of weight or mass, 203.
Consonance, connexion of the simple natural numbers with, 33;
Euclid's definition of, 33;
explanation of, 42;
scientific definition of, 44;
and dissonance reduced to beats, 376, 370, 383.
Consonant intervals, 43.
Constancy of matter, 203.
Constant, the dielectric, 117.
Constants, the natural, 193.
Continuum of facts, 256 et seq.
Cornelius, 388, footnote.
Corti, the Marchese, his discovery of minute rods in the labyrinth of the ear, 19.
Coulomb, his electrical researches, 108, 109, 113;
his notion of quantity of electricity, 173;
his torsion-balance, 168.
Crew, Prof. Henry, 317, footnote.
Criticism, Socrates the father of scientific, 1, 16.
Critique of Pure Reason, Kant's, 188.
Crucible, derivation of the word, 49, footnote.
Crustacea, auditory filaments of, 29, 272, 302.
Cube of oil, 5.
Culture, ancient and modern, 344.
Currents, chemical, 118;
electrical, 118;
galvanic, 132;
measurement of electrical, 135-136;
of heat, 244, 249-250;
strength of, 250.
Curtius, 356.
Curved lines, their asymmetry, 98.
Curves, how their laws are investigated, 206.
Cycles, reversible, Clausius on, 176.
Cyclical processes, closed, 175.
Cyclops, 67.
Cyclostat, 298.
Cylinder, of oil, 6;
mass of gas enclosed in a, 179.

D'Alembert, on the causes of harmony, 34;
his principle, 142, 149, 154;
also 234, 279.
Danish schools, 338, footnote.
Darwin, his study of organic nature, 215 et seq.;
his methods of research, 216.
Deaf and dumb, not subject to giddiness, 299.
Deaf person, with a piano, analyses sounds, 27.
Death and life, 186.
Definition, compendious, 197.
Deiters, 19.
Delage, 298, 301, 302.
Democritus, his mechanical conception of the world, 155, 187.
Demonstration, character of, 362.
Deportment of the energies, conformity in the, 171-175.
Derivation, laws only methods of, 256.
Descent, Galileo's laws of, 193;
generally, 143 et seq., 204, 215.
Description, 108, 191, 236, 237;
a condition of scientific knowledge, 193;
direct and indirect, 240;
in physics, 197, 199.
Descriptive sciences, their resemblance to the abstract, 248.
Determinants, 195.
Diderot, 234.
Dielectric constant, the, 117.
Difference-engine, the, 196.
Differential coefficients, their relation to symmetry, 98.
Differential laws, 204.
Differential method, for detecting optical imperfections, 317.
Diffraction, 91, 194.
Diffusion, Fick's theory of, 249.
Discharge of Leyden jars, 114 et seq.
Discoveries, the gist of, 270, 375.
Discovery and invention, distinction between, 269.
Dissonance, explanation of, 42;
definition of, 33, 44. See Consonance.
Distances, estimation of, by the eye, 68 et seq.
Dogs, like tuning-forks, 23;
their mentality, 190.
Domenech, Abbé, 92.
Dramatic element in science, 243.
Drop of water, on a greased plate, 8;
on the end of a stick, 8;
in free descent, 8.
Dubois, 218.
Dubois-Reymond, 370, footnote.
Dufay, 271.
Dynamics, foundations of, 153 et seq.

Ear, researches in the theory of, 17 et seq.;
diagram of, 18;
its analysis of sounds, 20 et seq.;
a puzzle-lock, 28;
reflected in a mirror, 93;
no symmetry in its sensation, 103.
Earth, its oblateness not due to its original fluid condition, 2;
rotation of, 204;
internal disturbances of, 285.
Economical, nature of physical inquiry, 186;
procedure of the human mind, 186;
order of physics, 197;
schematism of science, 206;
tools of science, 207;
coefficient of dynamos, 133.
Economy, of the actions of nature, 15;
the purpose of science, 16;
of language, 191 et seq.;
of the industrial arts, 192;
of mathematics, 195-196;
of machinery, 196;
of self-preservation, our first knowledge derived from, 197;
generally, 186 et seq., 269.
Education, higher, 86;
liberal, 341 et seq., 371.
Efflux, liquid, 150.
Ego, its nature, 234-235.
Egypt, 234.
Egyptians, art of, 78 et seq., 201.
Eighteenth century, the scientific achievements of, 187, 188.
Eleatics, on motion, 158.
Electrical, attraction and repulsion, 109 et seq., 168;
capacity, 116 et seq.;
force, 110, 119, 168;
spark, 117, 127, 132, 133, 190;
energy, measurement of, 128 et seq., 169;
currents, conceptions of, 118, 132, 135-136, 226-227, 249, 250;
fluids, 112 et seq., 228;
pendulums, 110;
levels, 173;
potential, 121 et seq.;
quantity, 111, 118, 119.
Electricity, as a substance and as a motion, 170;
difference between the conceptions of heat and, 168 et seq.,
rôle of work in, 120 et seq.;
galvanic, 134.
See Electrical.
Electrometer, W. Thomson's absolute, 127, footnote.
Electrometers, 122, 127.
Electrostatic unit, 111.
Electrostatics, concepts of, 107 et seq.
Elements, interdependence of the sensuous, 179;
of bodies, 202;
of phenomena, equations between, 205;
of sensations, 200;
used instead of sensations, 208-209.
Ellipse, equation of, 205;
the word, 342.
Embryology, possible future state of, 257.
Energies, conformity in the deportment of, 171-175;
differences of, 175.
Energy, a metrical notion, 178;
conservation of, 137 et seq.;
defined, 139;
metaphysical establishment of the doctrine of, 183;
kinetic, 177;
potential, 128 et seq.;
substantial conception of, 164, 185, 244 et seq.;
conservation of, in electrical phenomena, 131 et seq.;
limits of principle of, 175;
principle of, in physics, 160-166;
sources of principle of, 179, 181;
thermal, 177;
Thomas Young on, 173.
Energy-value of heat, 178, footnote.
Enlightenment, the, 188.
Entropy, a metrical notion, 178.
Environment, stability of our, 206.
Equations for obtaining facts, 180;
between the elements of phenomena, 205.
Equilibrium, conditions of, in simple machines, 151;
figures of liquid, 4 et seq.;
general condition of, 15;
in the State, 15.
Etymology, the word, misused for entomology, 316.
Euclid, on consonance and dissonance, 33;
his geometry, 364.
Euler, on the causes of harmony, 34;
impression of the mathematical processes on, 196;
on the vibrations of strings, 249, 285, 376.
Euler and Hermann's principle, 149.
Euthyphron, questioned by Socrates, 1.
Evolute, the word, 342.
Evolution, theory of, as applied to ideas, 216 et seq.
Ewald, 298, 304.
Excluded perpetual motion, logical root of the principle of, 182.
Exner, S., 302, 305.
Experience, communication of, 191;
our ready, 199;
the principle of energy derived from, 179;
the wellspring of all knowledge of nature, 181;
incongruence between thought and, 206.
Experimental research, function of, 181.
Explanation, nature of, 194, 237, 362.
Eye, cannot analyse colors, 20;
researches in the theory of the, 18 et seq.;
loss of, as affecting vision, 98.
Eyes, purpose of, 66 et seq.;
their structure symmetrical not identical, 96.

Face, human, inverted, 95.
Facts and ideas, necessary to science, 231.
Facts, description of, 108;
agreement of, 180;
relations of, 180;
how represented, 206;
reflected in imagination, 220 et seq.;
the result of constructions, 253;
a continuum of, 256 et seq.;
equations for obtaining, 180.
Falling bodies, 204, 215;
Galileo on the law of, 143 et seq., 284.
Falling, cats, 303, footnote.
Falstaff, 309.
Familiar intermediate links of thought, 198.
Faraday, 191, 217, 237;
his conception of electricity, 114, 271.
Fechner, theory of Corti's fibres, 19 et seq.
Feeling, cannot be explained by motions of atoms, 208 et seq.
Fetishism, 186, 243, 254;
in our physical concepts, 187.
Fibres of Corti, 17 et seq.
Fick, his theory of diffusion, 249.
Figures, symmetry of, 92 et seq.
Figures of liquid equilibrium, 4 et seq.
Fire, use of, 264.
Fishes, 306.
Fixed note, determining of a, 377.
Fizeau, his determination of the velocity of light, 55 et seq.
Flats, reversed into sharps, 101.
Flouren's experiments, 272, 290.
Flower-girl, the baskets of a, 95.
Fluids, electrical, 112 et seq.
Force, electric, 110, 119, 168;
unit of 111;
living, 137, 149, 184;
generally 253.
See the related headings.
Forces, will compared to, 254.
Foreseeing events, 220 et seq.
Formal conceptions, rôle of, 183.
Formal need of a clear view of facts, 183, 246;
how far it corresponds to nature, 184.
Formative forces of liquids, 4.
Forms of liquids, 3 et seq.
Forward movement, sensation of, 300.
Forwards, prophesying, 253.
Foucault, 57, 70, 296.
Foucault and Toepler, method of, for detecting optical faults, 313 et seq., 320.
Foundation of scientific thought, primitive acts of knowledge, the, 190.
Fourier, on processes of heat, 249, 278.
Fox, a, 234.
Franklin's pane, 116.
Frary, 338, footnote.
Fraunhofer, 271.
Freezing-point, lowered by pressure, 162.
Fresnel, 271.
Fritsch, 321.
Frogs, larvæ of, not subject to vertigo, 298.
Froude, 333.
Frustra, misuse of the word, 345.
Future, science of the, 213.

Galileo, on the motion of pendulums, 21;
his attempted measurement of the velocity of light, 50 et seq.;
his exclusion of a perpetual motion, 143;
on velocities acquired in free descent, 143-147;
on the law of inertia, 146-147;
on virtual velocities, 150;
on work, 172;
his laws of descent, 193;
on falling bodies, 225;
great results of his study of nature, 214 et seq.;
his rude scientific implements, 215;
selections from his works for use in instruction, 368;
also 105, 182, 187, 237, 272, 274, 283.
Galle, observes the planet Neptune, 29.
Galvanic, electricity, 134;
current, 132;
dizziness, 291;
vertigo, 298.
Galvanoscope, 135.
Galvanotropism, 291.
Garda, Lake, 239.
Gas, the word, 264;
mass of, enclosed in a cylinder, 179.
Gases, tensions of, for scales of temperature, 174.
Gauss, on the foundations of dynamics, 154;
his principle, 154;
also, 108, 274.
Genius, 279, 280.
Geography, comparison in, 239.
Geometers, in our eyes, 72.
Geotropism, 289.
German schools and gymnasiums, 372, 373, 338, footnote.
Ghosts, photographic, 73.
Glass, invisible in a mixture of the same refrangibility, 312;
powdered, visible in a mixture of the same refrangibility, 312.
Glove, in a mirror, 93.
Goethe, quotations from, 9, 31, 49, 88;
on the cause of harmony, 35.
Goltz, 282, 291.
Gossot, 332.
Gothic cathedral, 94.
Gravitation, discovery of, 225 et seq.
Gravity, how to get rid of the effects of, in liquids, 4;
also 228.
Gray, Elisha, his telautograph, 26.
Greased plate, drop of water on a, 8.
Great minds, idiosyncrasies of, 247.
Greek language, scientific terms derivedfrom, 342-343;
common words derived from, 343, footnote;
still necessary for some professions, 346;
its literary wealth, 347-348;
narrowness and one-sidedness of its literature, 348-349;
its excessive study useless, 349-350;
its study sharpens the judgment, 357-358;
a knowledge of it not necessary to a liberal education, 371.
Greeks, their provinciality and narrow-mindedness, 349;
now only objects of historical research, 350.
Griesinger, 184.
Grimaldi, 270.
Grimm, 344, footnote.
Grunting fishes, 306.

Habitudes of thought, 199, 224, 227, 232.
Haeckel, 222, 235.
Hamilton, deduction of the conical refraction of light, 29.
Hankel, 364.
Harmonics, 38, 40.
Harmony, on the causes of, 32 et seq.;
laws of the theory of, explained, 30;
the investigation of the ancients concerning, 32;
generally, 103.
See Consonance.
Harris, electrical balance of, 127, footnote.
Hartwich, Judge, 343, 353, footnote.
Hat, a high silk, 24.
Hats, ladies', development of, 64.
Head-wave of a projectile, 323 et seq.
Hearing and orientation, relation between, 304 et seq.
Heat, a material substance, 177;
difference between the conceptions of electricity and, 168 et seq.;
substantial conception of, 243 et seq.;
Carnot on, 156, 160 et seq.;
Fourier on the conduction of, 249;
not necessarily a motion, 167, 170, 171;
mechanical equivalent of, 164, 167;
of liquefaction, 178;
quantity of, 166;
latent, 167, 178, 244;
specific, 166, 244;
the conceptions of, 160-171;
machine, 160;
a measure of electrical energy, 133 et seq.;
mechanical theory of, 133;
where does it come from? 200.
Heavy bodies, sinking of, 222.
Heights of ascent, 143-151.
Helm, 172.
Helmholtz, applies the principle of energy to electricity, 184;
his telestereoscope, 84;
his theory of Corti's fibres, 19 et seq.;
on harmony, 35, 99;
on the conservation of energy, 165, 247;
his method of thought, 247;
also 138, 305, 307, 375, 383.
Hensen, V., on the auditory function of the filaments of Crustacea, 29, 302.
Herbart, 386 et seq.
Herbartians, on motion, 158.
Herculaneum, art in, 80.
Heredity, in organic and inorganic matter, 216, footnote.
Hering, on development, 222;
on vision, 210.
Hermann, E., on the economy of the industrial arts, 192.
Hermann, L., 291.
Herodotus, 26, 234, 347, 350.
Hertz, his waves, 242;
his use of the phrase "prophesy," 253.
Herzen, 361, footnote.
Hindu mathematicians, their beautiful problems, 30.
Holtz's electric machine, 132.
Horse, 63.
Household, physics compared to a well-kept, 197.
Housekeeping in science and civil life, 198.
Hudson, the, 94.
Human beings, puzzle-locks, 27.
Human body, our knowledge of, 90.
Human mind, must proceed economically, 186.
Humanity, likened to a polyp-plant, 235.
Huygens, his mechanical view of physics, 155;
on the nature of light and heat, 155-156;
his principle of the heights of ascent, 149;
on the law of inertia and the motion of a compound pendulum, 147-149;
on the impossible perpetual motion, 147-148;
on work, 173;
selections from his works for use in instruction, 368;
his view of light, 227-228, 262.
Huygens, optical method for detecting imperfections in optical glasses 313.
Hydrogen balloon, 199.
Hydrostatics, Stevinus's principle of, 141.
Hypotheses, their rôle in explanation, 228 et seq.

Ichthyornis, 257.
Ichthyosaurus, 63.
Idea? what is a theoretical, 241.
Idealism, 209.
Ideas, a product of organic nature, 217 et seq.;
and facts, necessary to science, 231;
not all of life, 233;
their growth and importance, 233;
a product of universal evolution, 235;
the history of, 227 et seq.;
in great minds, 228;
the rich contents of, 197;
their unsettled character in common life, their clarification in science, 1-2.
Ideography, the Chinese, 192.
Imagery, mental, 253.
Imagination, facts reflected in, 220 et seq.
Inclined plane, law of, 140-141.
Incomprehensible, the, 186.
Indian, his modes of conception and interpretation, 218 et seq.
Individual, a thread on which pearls are strung, 234-235.
Industrial arts, economy of the, E. Hermann on, 192.
Inertia, law of, 143 et seq., 146 et seq., 216, footnote, 283 et seq.
Innate concepts of the understanding, Kant on, 199.
Innervation, visual, 99.
Inquirer, his division of labor, 105;
compared to a shoemaker, 105-106;
what constitutes the great, 191;
the true, seeks the truth everywhere, 63 et seq.;
the, compared to a wooer, 45.
Instinctive knowledge, 189, 190.
Instruction, aim of, the saving of experience, 191;
in the classics, mathematics, and sciences, 338-374;
limitation of matter of, 365 et seq.
Insulators, 130.
Integrals, 195.
Intellectual development, conditions of, 286 et seq.
Intentions, acts of nature compared to, 14-15.
Interconnexion of nature, 182.
Interdependence, of properties, 361;
of the sensuous elements of the world, 179.
Interference experiments with the head-wave of moving projectiles, 327-328.
International intercourse, established by Latin, 341.
International measures, 108.
Invention, discovery and, distinction between, 269.
Inventions, requisites for the development of, 266, 268 et seq.
Iron-filings, 220, 243.
Italian art, 234.

Jacobi, C. G. J., on mathematics, 280.
James, W., 275, 299.
Java, 163.
Jews and Christians, monotheism of the, 187.
Jolly, Professor von, 112, 274.
Joule, J. P., on the conservation of energy, 163-165, 167, 183;
his conception of energy, 245;
his metaphysics, 183, 246;
his method of thought, 247;
also 137, 138.
Journée, 317.
Judge, criminal, the natural philosopher compared to a, 48.
Judgment, essentially economy of thought, 201-202;
sharpened by languages and sciences, 357-358;
also 232-233, 238.
Juliet, Romeo and, 87.
Jupiter, its satellites employed in the determination of the velocity of light, 51 et seq.
Jurisprudence, Latin and Greek unnecessary for the study of, 346, footnote.

Kant, his hypothesis of the origin of the planetary system, 5;
his Critique of Pure Reason, 188;
on innate concepts of the understanding, 199;
on time, 204;
also footnote, 93.
Kepler, 187, 270.
Kinetic energy, 177.
Kirchhoff, his epistemological ideas, 257-258;
his definition of mechanics, 236, 258, 271, 273.
Knight, 289.
Knowledge, a product of organic nature, 217 et seq., 235;
instinctive, 190;
made possible by economy of thought, 198;
our first, derived from the economy of self-preservation, 197;
the theory of, 203;
our primitive acts of the foundation of science, 190.
Kocher, 328.
Koenig, measurement of the velocity of sound, 57 et seq.
Kölliker, 19.
Kopisch, 61.
Kreidl, 299, 302, 306;
his experiments, 272.
Krupp, 319.

Labels, the value of, 201.
Labor, the accumulation of, the foundation of wealth and power, 198;
inquirer's division of, 105, 258.
Labyrinth, of the ear, 18, 291, 305.
Lactantius, on the study of moral and physical science, 89.
Ladder of our abstraction, the, 208.
Ladies, their eyes, 71;
like tuning-forks, 23-24.
Lagrange, on Huygens's principle, 149;
on the principle of virtual velocities, 150-155;
character of the intellectual activity of a, 195, 278.
Lake-dwellers, 46, 271.
Lamp-shade, 70.
Lane's unit jar, 115.
Language, knowledge of the nature of, demanded by a liberal education, 356;
relationship between, and thought, 358;
communication by 237;
economy of, 191 et seq.;
human its character, 238;
of animals, 238;
instruction in, 338 et seq.;
its methods, 192.
Laplace, on the atoms of the brain, 188;
on the scientific achievements of the eighteenth century, 188;
his hypothesis of the origin of the planetary system, 5.
Latent heat, 167, 178, 244.
Latin city of Maupertuis, 339.
Latin, instruction in, 311 et seq.;
introduced with the Christian Church, 340;
the language of scholars, the medium of international intercourse, its power, utility, and final abandonment, 341-347;
the wealth of its literature, 348;
the excessive study of, 346, 349, 354, 355;
its power to sharpen the judgment, 357-358.
Lavish extravagance of science, 189.
Law, a, defined, 256;
a natural, not contained in the conformity of the energies, 175.
Law-maker, motives of not always discernible, 9.
Layard, 79.
Learning, its nature, 366 et seq.
Least superficial area, principle of, accounted for by the mutual attractions of liquid particles, 13-14;
illustrated by a pulley arrangement, 12-13;
also 9 et seq.
Leibnitz, on harmony, 33;
on international intercourse, 342, footnote.
Lessing, quotation from, 47.
Letters of the alphabet, their symmetry, 94, 97.
Level heights of work, 172-174.
Lever, a, in action, 222.
Leverrier, prediction of the planet Neptune, 29.
Leyden jar, 114.
Liberal education, a, 341 et seq., 359, 371.
Libraries, thoughts stored up in, 237.
Lichtenberg, on instruction, 276, 370.
Licius, a Chinese philosopher, 213.
Liebig, 163, 278.
Life and death, 186.
Light, history of as elucidating how theories obstruct research, 242;
Huygens's and Newton's views of, 227-228;
its different conceptions, 226;
rectilinear propagation of, 194;
rôle of, in vision, 81;
spatial and temporal periodicity of, explains optical phenomena, 194;
numerical velocity of, 58;
where does it go to? 199;
generally, 48 et seq.
Like effects in like circumstances, 199.
Likeness, 388, 391.
Lilliput, 84.
Lines, straight, their symmetry, 98;
curved, their asymmetry, 98;
of force, 249.
Links of thought, intermediate, 198.
Liquefaction, latent heat of, 178.
Liquid, efflux, law of, 150;
equilibrium, figures of, 4 et seq.;
the latter produced in open air, 7-8;
their beauty and multiplicity of form, 7, 8;
made permanent by melted colophonium, 7.
Liquids, forms of, 1-16;
difference between, and solids, 2;
their mobility and adaptiveness of form, 3;
the courtiers par excellence of the natural bodies, 3;
possess under certain circumstances forms of their own, 3.
Living force, 137, 184;
law of the conservation of, 149.
Lloyd, observation of the conical refraction of light, 29.
Lobster, of Lake Mohrin, the, 61.
Localisation, cerebral, 210.
Locke, on language and thought, 358.
Locomotive, steam in the boiler of, 219.
Loeb, J., 289, 291, 302.
Logarithms, 195, 219;
in music, 103-104.
Logical root, of the principle of energy, 181;
of the principle of excluded perpetual motion, 182.
Lombroso, 280.
Lucian, 347.

Macula acustica, 272.
Magic lantern, 96.
Magic powers of nature, 189.
Magical power of science, belief in the, 189.
Magnet, a, 220;
will compared to the pressure of a, 14;
coercive force of a, 216.
Magnetic needle, near a current, 207.
Magnetised bar of steel, 242-243.
Major and minor keys in music, 100 et seq.
Malus, 242.
Man, a fragment of nature's life, 49;
his life embraces others, 234.
Mann, 364.
Manuscript in a mirror, 93.
Maple syrup, statues of, on Moon, 4.
Marx, 35.
Material, the relations of work with heat and the consumption of, 245 et seq.
Mathematical methods, their character, 197-198.
Mathematics, economy of, 195;
on instruction in, 338-374;
C. G. J. Jacobi on, 280.
Matter, constancy of, 203;
its nature, 203;
the notion of, 213.
Maupertuis, his Latin city, 338.
Maximal and minimal problems, their rôle in physics, 14, footnote.
Mayer, J. R., his conception of energy, 245, 246;
his methods of thought, 247;
on the conservation of energy, 163, 164, 165, 167, 183, 184;
his metaphysical utterances, 183, 246;
also 138, 184, 191, 217, 271, 274.
Measurement, definition of, 206.
Measures, international, 108.
Mécanique céleste, 90, 188;
sociale, and morale, the, 90.
Mechanical, conception of the world, 105, 155 et seq., 188, 207;
energy, W. Thomson on waste of, 175;
analogies between —— and thermal energy, 17 et seq.;
equivalent of heat, electricity, etc., 164, 167 et seq.;
mythology, 207;
phenomena, physical events as, 182;
philosophy, 188;
physics, 155-160, 212;
substitution-value of heat, 178, footnote.
Mechanics, Kirchhoff's definition of, 236.
Medicine, students of, 326.
Melody, 101.
Melsens, 310, 327.
Memory, a treasure-house for comparison, 230;
common elements impressed upon the, 180;
its importance, 238;
science disburdens the, 193.
Mendelejeff, his periodical series, 256.
Mental, adaptation, 214-235;
completion of phenomena, 220;
imagery, 253;
imitation, our schematic, 199;
processes, economical, 195;
reproduction, 198;
visualisation, 250.
Mephistopheles, 88.
Mercantile principle, a miserly, at the basis of science, 15.
Mersenne, 377.
Mesmerism, the mental state of ordinary minds, 228.
Metaphysical establishment of doctrine of energy, 183.
Metaphysical spooks, 222.
Metrical, concepts of electricity, 107 et seq.;
notions, energy and entropy are, 178;
units, the building-stones of the physicist, 253.
Metronomes, 41.
Meyer, Lothar, his periodical series, 256.
Middle Ages, 243, 349.
Midsummer Night's Dream, 309.
Mill, John Stuart, 230.
Millers, school for, 326.
Mill-wheel, doing work, 161.
Mimicking facts in thought, 189, 193.
Minor and major keys in music, 100 et seq.
Mirror, symmetrical reversion of objects in, 92 et seq.
Miserly mercantile principle at the basis of science, 15.
Moat, child looking into, 208.
Modern scientists, adherents of the mechanical philosophy, 188.
Molecular theories, 104.
Molecules, 203, 207.
Molière, 234.
Momentum, 184.
Monocular vision, 98.
Monotheism of the Christians and Jews, 187.
Montagues and Capulets, 87.
Moon, eclipse of, 219;
lightness of bodies on, 4;
the study of the, 90, 284.
Moreau, 307.
Mosaic of thought, 192.
Motion, a perpetual, 181;
quantity of, 184;
the Eleatics on, 158;
Wundt on, 158;
the Herbartians on, 158.
Motions, natural and violent, 226;
their familiar character, 157.
Mountains of the earth, would crumble if very large, 3;
weight of bodies on, 112.
Mozart, 44, 279.
Müller, Johann, 291.
Multiplication-table, 195.
Multiplier, 132.
Music, band of, its tempo accelerated and retarded, 53;
the principle of repetition in, 99 et seq.;
its notation, mathematically illustrated, 103-104.
Musical notes, reversion of, 101 et seq.;
their economy, 192.
Musical scale, a species of one-dimensional space, 105.
Mystery, in physics, 222;
science can dispense with, 189.
Mysticism, numerical, 33;
in the principle of energy, 184.
Mythology, the mechanical, of philosophy, 207.

Nagel, von, 364.
Nansen, 296.
Napoleon, picture representing the tomb of, 36.
Nations, intercourse and ideas of, 336-337.
Natural constants, 193.
Natural law, a, not contained in the conformity of the energies, 175.
Natural laws, abridged descriptions, 193;
likened to type, 193.
Natural motions, 225.
Natural selection in scientific theories, 63, 218.
Nature, experience the well-spring of all knowledge of, 181;
fashions of, 64;
first knowledge of, instinctive, 189;
general interconnexion of, 182;
has many sides, 217;
her forces compared to purposes, 14-15;
likened to a good man of business, 15;
the economy of her actions, 15;
how she appears to other animals, 83 et seq.;
inquiry of, viewed as a torture, 48-49;
view of, as something designedly concealed from man, 49;
like a covetous tailor, 9-10;
magic powers of, 189;
our view of, modified by binocular vision, 82;
the experimental method a questioning of, 48.
Negro hamlet, the science of a, 237.
Neptune, prediction and discovery of the planet, 29.
New views, 296 et seq.
Newton, describes polarisation, 242;
expresses his wealth of thought in Latin, 341;
his discovery of gravitation, 225 et seq.;
his solution of dispersion, 362;
his principle of the equality of pressure and counterpressure, 191;
his view of light, 227-228;
on absolute time, 204;
selections from his works for use in instruction, 368;
also 270, 274, 279, 285, 289.
Nobility, they displace Latin, 342.
Notation, musical, mathematically illustrated, 103-104.
Numbers, economy of, 195;
their connexion with consonance, 32.
Numerical mysticism, 33.
Nursery, the questions of the, 199.

Observation, 310.
Observation, in science, 261.
Ocean-stream, 272.
Oettingen, Von, 103.
Ohm, on electric currents, 249.
Ohm, the word, 343.
Oil, alcohol, water, and, employed in Plateau's experiments, 4;
free mass of, assumes the shape of a sphere, 12;
geometrical figures of, 5 et seq.
One-eyed people, vision of, 98.
Ophthalmoscope, 18.
Optic nerves, 96.
Optimism and pessimism, 234.
Order of physics, 197.
Organ, bellows of an, 135.
Organic nature, results of Darwin's studies of, 215 et seq.
See Adaptation and Heredity.
Oriental world of fables, 273.
Orientation, sensations of, 282 et seq.
Oscillation, centre of, 147 et seq.
Ostwald, 172.
Otoliths, 301 et seq.
Overtones, 28, 40, 349.
Ozone, Schöbein's discovery of, 271.

Painted things, the difference between real and, 68.
Palestrina, 44.
Parameter, 257.
Partial tones, 390.
Particles, smallest, 104.
Pascheles, Dr. W., 285.
Paulsen, 338, 340, 373.
Pearls of life, strung on the individual as on a thread, 234-235.
Pencil surpasses the mathematician in intelligence, 196.
Pendulum, motion of a, 144 et seq.,
increased motion of, due to slight impulses, 21;
electrical, 110.
Percepts, of like form, 390.
Periodical, changes, 181;
series, 256.
Permanent, changes, 181, 199;
elements of the world, 194.
Perpetual motion, a, 181;
defined, 139;
impossibility of, 139 et seq.;
the principle of the, excluded, 140 et seq.;
excluded from general physics, 162.
Personality, its nature, 234-235.
Perspective, 76 et seq.;
contraction of, 74 et seq.;
distortion of, 77.
Pessimism and optimism, 234.
Pharaohs, 85.
Phenomenology, a universal physical, 250.
Philistine, modes of thought of, 223.
Philology, comparison in, 239.
Philosopher, an ancient, on the moral and physical sciences, 89.
Philosophy, its character at all times, 186;
mechanical, 155 et seq., 188, 207, 259 et seq.
Phonetic alphabets, their economy, 192.
Photography, by the electric spark, 318 et seq.
Photography of projectiles, 309-337.
Photography, stupendous advances of, 74.
Physical, concepts, fetishism in our, 187;
ideas and principles, their nature, 204;
inquiry, the economical nature of, 186;
research, object of 207, 209.
Physical phenomena, as mechanical phenomena, 182;
relations between, 205.
Physico-mechanical view of the world, 155, 187, 188, 207 et seq.
Physics, compared to a well-kept household, 197;
economical experience, 197;
the principles of, descriptive, 199;
the methods of, 209;
its method characterised, 211;
comparison in, 239;
the facts of, qualitatively homogeneous, 255;
how it began, 37;
helped by psychology, 104;
study of its own character, 189;
the goal of, 207, 209.
Physiological psychology, its methods, 211 et seq.
Physiology, its scope, 212.
Piano, its mirrored counterpart, 100 et seq.;
used to illustrate the facts of sympathetic vibration, 25 et seq.
Piano-player, a speaker compared to, 192.
Picture, physical, a, 110.
Pike, learns by experience, 267.
Pillars of Corti, 19.
Places, heavy bodies seek their, 224 et seq.
Planetary system, origin of, illustrated, 5.
Plasticity of organic nature, 216.
Plateau, his law of free liquid equilibrium, 9;
his method of getting rid of the effects of gravity, 4.
Plates of oil, thin, 6.
Plato, 347, 371.
Plautus, 347.
Playfair, 138.
Pleasant effects, cause of, 94 et seq.
Pliny, 349.
Poetry and science, 30, 31, 351.
Poinsot, on the foundations of mechanics, 152 et seq.
Polarisation, 91;
abstractly described by Newton, 242.
Politics, Chinese speak with unwillingness of, 374.
Pollak, 299.
Polyp plant, humanity likened to a, 235.
Pompeii, 234;
art in, 80.
Popper J., 172, 216.
Potential, social, 15;
electrical, 121 et seq.;
measurement of, 126;
fall of, 177;
swarm of notions in the idea of, 197;
its wide scope, 250.
Pottery, invention of, 263.
Prediction, 221 et seq.
Prejudice, the function, power, and dangers of, 232-233.
Preparatory schools, the defects of the German, 346-347;
what they should teach, 364 et seq.
Pressure of a stone or of a magnet, will compared to, 14;
also 157.
Primitive acts of knowledge the foundation of scientific thought, 190.
Problem, nature of a, 223.
Problems which are wrongly formulated, 308.
Process, Carnot's, 161 et seq.
Projectiles, the effects of the impact of, 310, 327-328;
seen with the naked eye, 311, 317;
measuring the velocity of, 332;
photography of, 309-337.
Prony's brake, 132.
Proof, nature of, 284.
Prophesying events, 220 et seq.
Psalms, quotation from the, 89.
Pseudoscope, Wheatstone's, 96.
Psychology, preceded by astronomy, 90;
how reached, 91 et seq.;
helps physical science, 104;
its method the same as that of physics, 207 et seq.
Pully arrangement, illustrating principle of least superficial area, 12-13.
Purkinje, 284, 285, 291, 299.
Purposes, the acts of nature compared to, 14-15;
nature pursues no, 66.
Puzzle-lock, a, 26.
Puzzles, 277.
Pyramid of oil, 6.
Pythagoras, his discovery of the laws of harmony, 32, 259.

Quality of tones, 36.
Quantitative investigation, the goal of, 180.
Quantity of electricity, 111, 118, 119, 167-170, 173;
of heat, 166, 167-171, 174, 177, 244;
of motion, 184.
Quests made of the inquirer, not by him, 30.
Quételet, 15, footnote.

Rabelais, 283.
Raindrop, form of, 3.
Rameau, 34.
Reaction and action, principle of, 191.
Reactions, disclosure of the connexion of, 270 et seq.
Realgymnasien, 365.
Realschulen, 365, 373.
Reason, stands above the senses, 105.
Reflex action, 210.
Reflexion, produces symmetrical reversion of objects, 93 et seq.
Refraction, 29, 193, 194, 208, 230, 231.
Reger, 328.
Reliefs, photographs of, 68.
Repetition, its rôle in æsthetics, 89, footnote, 91 et seq., 97, 98 et seq., 390.
Reproduction of facts in thought, 189, 193, 198, 253.
Repulsion, electric, 109 et seq., 168.
Research, function of experimental 181;
the aim of, 205.
Resemblances between facts, 255.
Resin, solution of, 7.
Resistance, laws of, for bodies travelling in air and fluids, 333 et seq.
Resonance, corporeal, 392.
Response of sonorous bodies, 25.
Retina, the corresponding spots of 98;
nerves of compared to fingers of a hand, 96 et seq.
Reversible processes, 161 et seq., 175, 176, 181, 182.
Rhine, the, 94.
Richard the Third, 77.
Riddles, 277.
Riders, 379.
Riegler, 319.
Riess, experiment with the thermo-electrometer, 133 et seq., 169.
Rigid connexions, 142.
Rind of a fruit, 190.
Rings of oil, illustrating formation of rings of Saturn, 5.
Ritter, 291, 299.
Rods of Corti, 19.
Rolph, W. H., 216.
Roman Church, Latin introduced with the, 340 et seq.
Romans, their provinciality and narrow-mindedness, 270.
Romeo and Juliet, 87.
Römer, Olaf, 51 et seq.
Roots, the nature of, in language, 252.
Rosetti, his experiment on the work required to develop electricity, 131.
Rotating bodies, 285.
Rotation, apparatus of, in physics, 59 et seq.;
sensations of, 288 et seq.
Rousseau, 336.
Rubber pyramid, illustrating the principle of least superficial area, 10-11.
Ruysdael, 279.

Sachs, Hans, 106.
Salcher, Prof. 319.
Salviati, 144.
Saturn, rings of, their formation illustrated, 5.
Saurians, 257.
Sauveur, on acoustics, 34, 375 et seq.
Savage, modes of conception and interpretation of a, 218 et seq.
Schäfer, K., 298.
Schlierenmethode, 317.
Schönbein's discovery of ozone, 271.
School-boy, copy-book of, 92.
Schoolmen, 214.
Schools, State-control of, 372 et seq.
Schopenhauer, 190.
Schultze, Max, 19.
Science, a miserly mercantile principle at its basis, 15;
compared to a business, 16;
viewed as a maximum or minimum problem, 16, footnote;
its process not greatly different from the intellectual activity of ordinary life, 16, footnote;
economy of its task, 16;
relation of, to poetry, 30, 31, 351;
the church of, 67;
beginnings of, 189, 191;
belief in the magical power of, 189;
can dispense with mystery, 189;
lavish extravagance of, 189;
economy of the terminology of, 192;
partly made up of the intelligence of others, 196;
stripped of mystery, 197;
its true power, 197;
the economical schematism of, 206;
the object of, 206;
the tools of, 207;
does not create facts, 211;
of the future, 213;
revolution in, dating from Galileo, 214 et seq.;
the natural foe of the marvellous, 224;
characterised, 227;
growth of, 237;
dramatic element in, 243;
described, 251;
its function, 253;
classification in, 255, 259 et seq.;
the way of discovery in, 316.
See also Physics.
Sciences, partition of the, 86;
the barriers and relations between the 257-258;
on instruction in the, 338-374.
Scientific, criticism, Socrates the father of, 1, 16;
discoveries, their fate, 138;
knowledge, involves description, 193;
thought, transformation and adaptation in, 214-235;
thought, advanced by new experiences, 223 et seq.;
thought, the difficulty of, 366;
terms, 342-343;
founded on primitive acts of knowledge, 190.
Scientists, stories about their ignorance, 342.
Screw, the, 62.
Sea-sickness, 284.
Secret computation, Leibnitz's, 33.
Seek their places, bodies, 226.
Self-induction, coefficient of, 250, 252.
Self-observation, 211.
Self-preservation, our first knowledge derived from the economy of, 197;
struggle for, among ideas, 228.
Semi-circular canals, 290 et seq.
Sensation of rounding a railway curve, 286.
Sensations, analysed, 251;
when similar, produce agreeable effects, 96;
their character, 200;
defined, 209;
of orientation, 282 et seq.
Sense-elements, 179.
Senses, theory of, 104;
the source of our knowledge of facts, 237.
Seventh, the troublesome, 46.
Shadow method, 313 et seq., 317 footnote.
Shadows, rôle of, in vision, 81.
Shakespeare, 278.
Sharps, reversed into flats, 101.
Shell, spherical, law of attraction for a, 124, footnote.
Shoemaker, inquirer compared to, 105-106.
Shooting, 309.
Shots, double report of, 229 et seq.
Similarity, 249.
Simony, 280.
Simplicity, a varying element in description, 254.
Sines, law of the, 193.
Sinking of heavy bodies, 222.
Sixth sense, 297.
Smith, R., on acoustics, 34, 381, 383.
Soap-films, Van der Mensbrugghe's experiment with, 11-12.
Soapsuds, films and figures of, 7.
Social potential, 15.
Socrates, the father of scientific criticism, 1, 16.
Sodium, 202.
Sodium-light, vibrations of, as a measure of time, 205.
Solidity, conception of, by the eye, 71 et seq.;
spatial, photographs of, 73.
Solids, and liquids, their difference merely one of degree, 2.
Sonorous bodies, 24 et seq.
Soret, J. P., 89.
Sounds, symmetry of, 99 et seq.;
generally, 22-47, 212.
Sound-waves rendered visible, 315 et seq.
Sources of the principle of energy, 179 et seq.
Space, 205;
sensation of, 210.
Spark, electric, 117, 127, 132, 133, 190.
Spatial vision, 386.
Species, stability of, a theory, 216.
Specific energies, 291.
Specific heat, 166, 244.
Specific inductive capacity, 117.
Spectral analysis of sound, 27.
Spectrum, mental associations of the, 190.
Speech, the instinct of, cultivated by languages, 354.
Spencer, 218, 222.
Sphere, a soft rotating, 2;
the figure of least surface, 12;
electrical capacity of, 123 et seq.
Spherical shell, law of attraction for 124, footnote.
Spiders, the eyes of, 67.
Spirits, as explanation of the world 186, 243.
Spiritualism, modern, 187.
Spooks, metaphysical, 222.
Squinting, 72.
Stability of our environment, 206.
Stallo, 336.
Stars, the fixed, 90.
State, benefits and evils of its control of the schools, 372 et seq.;
the Church and, 88.
Statical electricity, 134.
Stationary currents, 249.
Statoliths, 303.
Steam-engine, 160, 265.
Steeple-jacks, 75.
Stereoscope, Wheatstone and Brewster's, 73.
Stevinus, on the inclined plane, 140;
on hydrostatics, 141;
on the equilibrium of systems, 142;
discovers the principle of virtual velocities, 150;
characterisation of his thought, 142;
also 182, 187, 191.
Stone Age, 46, 321.
Störensen, 306.
Stove, primitive, 263.
Straight line, a, its symmetry, 98.
Straight, meaning of the word, 240.
Street, vista into a, 75.
Striae, in glass, 313.
Striate method, for detecting optical imperfections, 317.
Striking distance, 115, 127.
Strings, vibrations of, 249.
Struggle for existence among ideas, 217.
Substance, heat conceived as a, 177, 243 et seq.;
electricity as a, 170;
the source of our notion of, 199;
rôle of the notion of, 203, 244 et seq.;
energy conceived as a, 164, 185, 244 et seq.
Substitution-value of heat, 178, footnote.
Suetonius, 348.
Sulphur, specific inductive capacity of, 117.
Sun, human beings could not exist on, 3.
Swift, 84, 280.
Swimmer, Ampère's, 207.
Symmetry, definition of, 92;
figures of, 92 et seq.;
plane of, 94;
vertical and horizontal, 94;
in music, 99 et seq.
Sympathetic vibration, 22 et seq., 379.

Tailor, nature like a covetous, 9-10.
Tangent, the word, 263.
Taste, doubtful cultivation of, by the classics, 352-353;
of the ancients, 353.
Taylor, on the vibration of strings, 249.
Teaching, its nature, 366 et seq.
Telegraph, the word, 263.
Telescope, 262.
Telestereoscope, the, 84.
Temperament, even, in tuning, 47.
Temperature, absolute, 162;
differences of, 205;
differences of, viewed as level surfaces, 161;
heights of, 174;
scale of, derived from tensions of gases, 174.
Terence, 347.
Terms, scientific, 342-343.
Thales, 259.
Theories, their scope, function, and power, 241-242;
must be replaced by direct description, 248.
Thermal, energy, 174, 177;
capacity, 123, footnote.
Thermodynamics, 160 et seq.
Thermoelectrometer, Riess's, 133, 169.
Thing-in-itself, the, 200.
Things, mental symbols for groups of sensations, 200-201.
Thomson, James, on the lowering of the freezing-point of water by pressure, 162.
Thomson, W., his absolute electrometer, 127, footnote;
on thermodynamics, 162;
on the conservation of energy, 165;
on the mechanical measures of temperature, 174, footnote;
on waste of mechanical energy, 175;
also 108, 173, footnote.
Thought, habitudes of, 199, 224, 227, 232;
relationship between language and, 329;
incongruence between experience and, 206;
luxuriance of a fully developed, 58;
transformation in scientific, 214-235.
Thoughts, their development and the struggle for existence among them, 63;
importance of erroneous, 65;
as reproductions of facts, 107.
Thread, the individual a, on which pearls are strung, 234-235.
Tides, 283.
Timbre, 37, 38, 39.
Time, 178, 204, 205, footnote.
Toepler and Foucault, method of, for detecting optical faults, 313 et seq., 320.
Tone-figures, 91.
Tones, 22-47, 99 et seq., 212.
Torsion, moment of, 132.
Torsion-balance, Coulomb's, 109, 168.
Torricelli, on virtual velocities, 150;
his law of liquid efflux, 150;
on the atmosphere, 273.
Tourist, journey of, work of the inquirer compared to, 17, 29, 30.
Transatlantic cable, 108.
Transformation and adaptation in scientific thought, 214-235.
Transformation of ideas, 63.
Transformative law of the energies, 172.
Translation, difficulties of, 354.
Tree, conceptual life compared to a, 231.
Triangle, mutual dependence of the sides and angles of a, 179.
Triple accord, 46.
Truth, wooed by the inquirer, 45;
difficulty of its acquisition, 46.
Tumblers, resounding, 23.
Tuning-forks, explanation of their motion, 22 et seq.
Tylor, 186.
Tympanum, 18.
Type, natural laws likened to, 193;
words compared to, 191.

Ulysses, 347.
Understanding, what it means, 211.
Uniforms, do not fit heads, 369.
Unique determination, 181-182.
Unison, 43.
Unit, electrostatic, 111.
See Force and Work.
United States, 336.
Universal Real Character, a, 192.
Utility of physical science, 351.

Variation, the method of, in science, 230;
in biology, 216.
Velocity, of light, 48 et seq.;
of the descent of bodies, 143 et seq.;
meaning of, 204;
virtual, 149-155.
Verstandesbegriffe, 199.
Vertical, perception of the, 272, 286 et seq.;
symmetry, 389.
Vertigo, 285, 290.
Vestibule of the ear, 300.
Vibration, 22 et seq.
Vibration-figures, 91.
Vinci, Leonardo da, 278, 283.
Violent motions, 225.
Virtual velocities, 149-155.
Visibility, general conditions of, 312.
Vision, symmetry of our apparatus of, 96.
See Eye.
Visual nerves, 96.
Visualisation, mental, 250.
Volt, the word, 343.
Volta, 127, footnote, 134.
Voltaire, 260.
Voltaire's ingènu, 219.
Vowels, composed of simple musical notes, 26.

Wagner, Richard, 279.
Wald, F., 178, footnote.
Wallace, 216.
War, and peace, reflexions upon, 309, 335 et seq.
Waste of mechanical energy, W. Thomson on, 175.
Watches, experiment with, 41;
in a mirror, 93.
Water, jet of, resolved into drops, 60;
free, solid figures of, 8;
objects reflected in, 94, 191;
possible modes of measurement of, 170.
Watt, 266.
Wealth, the foundation of, 198.
Weapons, modern, 335.
Weber, 108, 306.
Weight of bodies, varies with their distance from the centre of the earth, 112.
Weismann, 216.
Wheatstone, his stereoscope, 73;
his pseudoscope, 96;
also 59.
Wheel, history and importance of, 61 et seq.
Whewell, on the formation of science, 231.
Whole, the, 204, footnote.
Why, the question, 199, 223.
Will, Schopenhauer on the, 190;
man's most familiar source of power, 243;
used to explain the world, 186;
forces compared to, 254;
compared to pressure, 14.
Windmill, a rotating, 53.
Wire frames and nets, for constructing liquid figures of equilibrium, 4 et seq.
Witchcraft, 187.
Wollaston, 284, 285.
Wonderful, science the natural foe of the, 224.
Woods, the relative distance of trees in, 68.
Wooer, inquirer compared to a, 45.
Words and sounds, 343.
Words, compared to type, 191.
Work, of liquid forces of attraction, 14;
in electricity, 173;
measure of, 119 et seq., 130, 223;
relation of, with heat, 162, 245 et seq.;
amount required to develop electricity, 131 et seq.;
produces various physical changes, 139;
substantial conception of, 183-184.
See Energy.
World, the, what it consists of, 208.
World-particles, 203.
Wronsky, 172.
Wundt, on causality and the axioms of physics, 157-159; 359 footnote.

Xenophon, 49, footnote.

Young, Thomas, on energy, 173.

Zelter, 35.
Zeuner, 171.
Zoölogy, comparison in, 239.