Chapter VII: The Sources of the Principle of Energy (1)

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.

Experience teaches that the sense-elements [alpha beta gamma delta ...] into which the world may be decomposed, are subject to change. It tells us further, that certain of these elements are _connected_ 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: _The sensuous elements of the world ([alpha beta gamma delta ...]) show themselves to be interdependent._ 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.

As an example, we may take a mass of gas enclosed in a cylinder and possessed of a definite volume ([alpha]), which we change by a pressure ([beta]) on the piston, at the same time feeling the cylinder with our hand and receiving a sensation of heat ([gamma]). Increase of pressure diminishes the volume and increases the sensation of heat.

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 _agreement_ 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 _differences_ of related facts. Facts may be so nearly related as to contain the same kind of a [alpha beta gamma ...], but the relation be such that the [alpha beta gamma ...] of the one differ from the [alpha beta gamma ...] 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 _from one another_ the numbers which are the measures of these [alpha beta gamma ...], then we possess in such rules the _most general_ expression of a group of facts, as also that expression which corresponds to all its differences. This is the goal of quantitative investigation.

If this goal be reached what we have found is that between the [alpha beta gamma ...] 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 [alpha beta gamma ...]. If the former be smaller by one than the latter, then one portion of the [alpha beta gamma ...] is _uniquely_ determined by the other portion.

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 [alpha beta gamma ...] relations exist and of what kind they are. Further, only experience can tell that the relations that exist between the [alpha beta gamma ...] 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.

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 [alpha beta gamma ...] determines _uniquely_ another group [lambda mu nu ...]. Experience further teaches that changes of [alpha beta gamma ...] can be _reversed_. It is then a logical consequence of this observation, that every time that [alpha beta gamma ...] assume the same values this is also the case with [lambda mu nu ...]. Or, that purely _periodical_ changes of [alpha beta gamma ...] can produce no _permanent_ changes of [lambda mu nu ...]. If the group [lambda mu nu ...] is a mechanical group, then a perpetual motion is excluded.

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.

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.

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 whether the mechanical laws are broken directly or circuitously.

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 _formal conception_. As may be seen from the preceding exposition, the perpetual motion can be excluded without our employing or possessing the notion of _work_. The modern principle of energy results primarily from a _substantial_ 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 _absurd_ to suppose that the powers with which God has endowed matter can be destroyed."

Some writers have observed in such statements the attempt at a _metaphysical_ 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 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 _vividly felt the need of it_."

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, _mv_, as the equivalent of work, and did not light, until later, on the notion of living force (_mv²/2_). 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 _first_ present, and our conception of nature is subsequently gradually _adapted_ to it.

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 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.

FOOTNOTES:

[Footnote 39: Published in Vol. 5, No. I, of _The Monist_, October,
1894, being in part a re-elaboration of the treatise _Ueber die
Erhaltung der Arbeit_, Prague, 1872.]

[Footnote 40: _On Matter, Living Force, and Heat_, Joule:
_Scientific Papers_, London, 1884, I, p. 265.]

[Footnote 41: "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 _continuum et
aeternum motum efficient, quod est falsum_."]

[Footnote 42: "A igitur, (si ullo modo per naturam fieri possit)
locum sibi tributum non servato, ac delabatur in _D_; quibus positis
aqua quae ipsi _A_ succedit eandem ob causam deffluet in _D_, eademque
ab alia istinc expelletur, atque adeo aqua haec (cum ubique eadem
ratio sit) _motum instituet perpetuum, quod absurdum fuerit_."]

[Footnote 43: "Accipio, gradus velocitatis ejusdem mobilis super
diversas planorum inclinationes acquisitos tunc esse aequales, cum
eorundum planorum elevationes aequales sint."]

[Footnote 44: "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 _AB_ lungo due, o
tre braccia perpendicolare all'orizzonte, e nella parete segnate una
linea orizontale _DC_ segante a squadra il perpendicolo _AB_, il
quale sia lontano dalla parete due dita in circa, trasferendo poi il
filo _AB_ colla palla in _AC_, lasciata essa palla in libertà, la
quale primieramente vedrete scendere descrivendo l'arco _CBD_, e di
tanto trapassare il termine _B_, che scorrendo per l'arco _BD_
sormonterà fino quasi alla segnata parallela _CD_, 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 _B_ dalla
palla nello scendere per l'arco _CB_, fu tanto, che bastò a
risospingersi per un simile arco _BD_ alla medesima altezza; fatta,
e più volte reiterata cotale esperienza, voglio, che fiechiamo nella
parete rasente al perpendicolo _AB_ un chiodo come in _E_, ovvero in
_F_, che sporga in fuori cinque, o sei dita, e questo acciocchè il
filo _AC_ tornando come prima a riportar la palla _C_ per l'arco
_CB_, giunta che ella sia in _B_, inoppando il filo nel chiodo _E_,
sia costretta a camminare per la circonferenza _BG_ descritta in
torno al centro _E_, dal che vedremo quello, che potrà far quel
medesimo impeto, che dianzi concepizo nel medesimo termine _B_,
sospinse l'istesso mobile per l'arco _ED_ all'altezza
dell'orizzonale _CD_. Ora, Signori, voi vedrete con gusto condursi
la palla all'orizzontale nel punto _G_, e l'istesso accadere,
l'intoppo si metesse più basso, come in _F_, dove la palla
descriverebbe l'arco _BJ_, terminando sempre la sua salita
precisamente nella linea _CD_, e quando l'intoppe del chiodo fusse
tanto basso, che l'avanzo del filo sotto di lui non arivasse
all'altezza di _CD_ (il che accaderebbe, quando fusse più vicino al
punto _B_, che al segamento dell' _AB_ coll'orizzontale _CD_),
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 _CB_, _DB_ equali e
similmento posti, l'acquisto di momento fatto per la scesa nell'arco
_CB_, è il medesimo, che il fatto per la scesa dell'arco _DB_; ma il
momento acquistato in _B_ per l'arco _CB_ è potente a risospingere
in su il medesimo mobile per l'arco _BD_; adunque anco il momento
acquistato nella scesa _DB_ è eguale a quello, che sospigne
l'istesso mobile pel medesimo arco da _B_ in _D_, 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
_BD_, _BG_, _BJ_ sono eguali, poichè son fatti dal istesso medesimo
momento acquistato per la scesa _CB_, come mostra l'esperienza:
adunque tutti i momenti, che si acquistano per le scese negli archi
_DB_, _GB_, _JB_ sono eguali."]

[Footnote 45: "Constat jam, quod mobile ex quiete in _A_ descendens
per _AB_, gradus acquirit velocitatis juxta temporis ipsius
incrementum: gradum vero in _B_ 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 _BC_ fit reflexio: in horizontali
autem _GH_ aequabilis motus juxta gradum velocitatis ex _A_ in _B_
acquisitae in infinitum extenderetur."]

[Footnote 46: "Si gravitas non esset, neque aër motui corporum
officeret, unumquodque eorum, acceptum semel motum continuaturum
velocitate aequabili, secundum lineam rectam."]

[Footnote 47: "Si pondera quotlibet, vi gravitatis suae, moveri
incipiant; non posse centrum gravitatis ex ipsis compositae altius,
quam ubi incipiente motu reperiebatur, ascendere.

"Ipsa vero hypothesis nostra quominus scrupulum moveat, nihil aliud
sibi velle ostendemus, quam, quod nemo unquam negavit, gravia nempe
sursum non ferri.--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."]

[Footnote 48: "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."]

[Footnote 49: "Notato autem hic illud staticum axioma etiam locum
habere:

"Ut spatium agentis ad spatium patientis
Sic potentia patientis ad potentiam agentis."]

[Footnote 50: "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.

"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."]

[Footnote 51: _Traité de la lumière_, Leyden, 1690, p. 2.]

[Footnote 52: "L'on ne sçaurait douter que la lumière ne consiste
dans le _mouvement_ 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 _mouvement_, au moins dans la _vraye Philosophie_,
dans laquelle on conçoit la cause de tous les effets naturels par
des raisons de _mechanique_. Ce qu'il faut faire à mon avis, ou bien
renoncer à tout espérance de jamais rien comprendre dans la
Physique."]

[Footnote 53: _Sur la puissance motrice du feu_. (Paris, 1824.)]

[Footnote 54: "On objectra peut-être ici que le mouvement perpétuel,
démontré impossible par les _seules actions mécaniques_, ne l'est
peut-être pas lorsqu'on emploie l'influence soit de la _chaleur_,
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
_mouvements quelconques des corps_ et comme tels ne doivent-ils pas
être soumis aux lois générales de la mécanique?"]

[Footnote 55: By this is meant the temperature of a Celsius scale,
the zero of which is 273° below the melting-point of ice.]

[Footnote 56: I first drew attention to this fact in my treatise
_Ueber die Erhaltung der Arbeit_, 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 _Sitzungsberichte der Wiener_ _Akademie_,
December, 1892, entitled _Geschichte und Kritik des Carnot'schen
Wärmegesetzes_. Compare also the works of Popper (1884), Helm
(1887), Wronsky (1888), and Ostwald (1892).]

[Footnote 57: Sir William Thomson first consciously and
intentionally introduced (1848, 1851) a _mechanical_ measure of
temperature similar to the electric measure of potential.]

[Footnote 58: Compare my _Analysis of the Sensations_, Jena, 1886:
English translation, Chicago, 1897.]

[Footnote 59: 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 _actually_ performed in the passage of a
thermal condition _A_ to a condition _B_, alone deserves the name of
the _energy-value_ of this change of condition. In this way the
_arbitrary_ substantial conception of the processes would be
preserved and misapprehensions forestalled.]

THE ECONOMICAL NATURE OF PHYSICAL INQUIRY.[60]

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.

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[61] 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,[62] those very ideas still covertly lurk in the practices of modern spiritualism.

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 _physico-mechanical_ 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 really worthy of man, competent to overcome the old barbarism in the practical fields of life; it created the _Critique of Pure Reason_, 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.

The oft-quoted words of the great Laplace,[63] 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.

Gladly do we accord to the creator of the _Mécanique céleste_ 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 _new_ tasks, it 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.

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.

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, 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?

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 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.

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.

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. 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.

The principles which a prominent political economist, E. Hermann,[64] 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.

The communication of scientific knowledge always 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.

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 higher order, partly movable, partly stereotyped, which last on new editions of experience may become downright impediments.

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 _clear_. 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 _explained_ for us.

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 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.

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.

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, _The Economy of Manufactures and Machinery_.

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,--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.

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?

Our first knowledge, thus, is a product of the economy of self-preservation. By communication, the experience of _many_ persons, individually acquired at first, is collected in _one_. 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 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 _its_ amassment of wealth no one suffers the least loss. This, too, is its blessing, its freeing and saving power.

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

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 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 _ready_ experience fixed in the mosaic of the mind with which we meet new events, Kant calls an innate concept of the understanding (_Verstandesbegriff_).

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.

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 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.

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.

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 say that bodies or things are compendious mental symbols for groups of sensations--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.

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--a figure which is not a cube--we do so from a natural instinct of economy, which prefers to add to an old familiar conception a correction instead of forming an entirely new one. This is the process of all judgment.

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 sense-impressions. And the same is true of the molecules and atoms into which the chemical element is still further analysed.

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.

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.

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,--for example, that of Time, the producer and devourer of all things,--do not concern us here. We need only remind the reader that even Newton speaks of an _absolute_ time independent of all phenomena, and of an absolute space--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--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.[65] 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.[66]

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 _conceivable_ relation between its co-ordinates, of which only the real values have _geometrical_ significance. Similarly, the equations between the elements of _phenomena_ express a universal, mathematically conceivable relation. Here, however, for many values only certain directions of change are _physically_ admissible. As in the ellipse only certain _values_ satisfying the equation are realised, so in the physical world only certain _changes_ 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 _actual_ 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.

Let us endeavor now to summarise the results of 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.

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.[67] The condition of science, both in its origin and in its application, is a _great relative stability_ 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.

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, 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 _simplest_ and _most economical_ abstract expression of facts.

* * * * *

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.

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.

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 _elements_. 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 _foreign_ 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 _all_ elements, in so far as we regard them as dependent on this special part (our body), _sensations_. 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.[68]