The Rudolf Steiner Archive

My dear Friends,

The way of speaking about sound and tone which you will find in the customary description of modern Physics may be said to date back to the 15th century at the earliest. By such examples you will most readily confirm what I so often speak of more generally in Spiritual Science. Namely, before that turning-point in time, man's whole way of thinking was very different from what it then became.

The way we speak of the phenomena of sound and tone in the scholastic system of modern Physics came about only gradually. What first caught their attention was the velocity with which sound is propagated. To a first approximation it is not difficult to find what may be interpreted as the speed of propagation of sound. If a gun is fired at some distance from you, you see the flash of light in the distance and hear the report some time later, just as you hear the thunder after you see the lightning. If you neglect that there is such a thing as a velocity of light, you may then call the time that elapses between your perception of the impression of light and your perception of the sound, the time the sound has taken to go the corresponding distance. So you can calculate how quickly the sound advances in air—how far it goes, say, in a second—and you get something like a “velocity of propagation of sound”.

This was one of the earliest things to which men became attentive in this domain. They also became attentive to the so-called phenomena of resonance—sympathetic vibration. Leonardo da Vinci was among the first. If for example you twang a violin-string or the like, and another string attuned to it—or even quite a different object that happens to be so attuned—is there in the same room, the other will begin vibrating too. The Jesuits especially took up the study of these things. In the 17th century much was done for the science of sound or tone by the Jesuit Mersenne, who made important researches on what is called the ‘pitch’ of a musical note. A note contains three elements. It has first a certain intensity; secondly a certain pitch; thirdly a certain quality or colouring of sound. The problem is to ascertain what corresponds to the pitch,—to ascertain this from the point of view which, as I said, has gradually been adopted in modern time,—adopted most of all, perhaps, in this branch of Science. I have already drawn your attention to the fact which can indeed easily be ascertained. Whenever we perceive a sound or a musical note, there is always some oscillatory phenomenon that underlies it—or, shall we rather say, accompanies, runs parallel to it. The usual experiments can easily be reproduced, to demonstrate this oscillatory character of air or other bodies. Here is a tuning-fork with a point attached, which as it moves can make a mark in the layer of soot, deposited on this glass plate. We need not actually do all these experiments, but if we did strike the tuning-fork to begin with, the picture on the glass plate would reveal that this tuning-fork is executing regular movements. These forms of movement are naturally conveyed to the air and we may therefore say that when we hear any sounding body the air between it and us is in movement. Indeed we bring the air itself directly into movement in the instruments called pipes.

Now scientists have gradually discovered what kind of movement it is. It takes place in ‘longitudinal’ waves, as they are called. This too can be directly demonstrated. We kindle a note in this metallic tube, which we connect with another tube full of air, so that the movements of the metallic tube are communicated to this air. If we then put a very light and mobile dust into the tube that is filled with air, the mobility of the tiny spheres of dust enables us to recognize that the sound is propagated just in this way; first there arises a condensation, a densifying of the air; this will beat back again however as soon as the body oscillates the other way. So there arises a thinning-out, a dilution of the air. Then at the next forward beat of the metal the original condensation goes forward; so then dilutions and condensations alternate. We can thus prove by direct experiment that we are dealing with dilutions and condensations of the air. We really need not do all these experiments; they are at hand, if I may say so. What you can get from the text-books is not what I am here to shew.

It is significant indeed, how much was done for these branches of Physics, especially at the beginning of modern time, either by the Jesuits themselves, or else was set on foot by them through all their social connections. Now from this side there was always the strong tendency, above all things, not to enter spiritually into the processes of Nature,—not to penetrate to the spiritual in Nature. The spiritual should be reserved for the religious life. Among the Jesuits it was always looked upon as dangerous to apply to the phenomena of Nature spiritual forms of thought such as we have grown accustomed to through Goethe. They wanted to study Nature in purely materialistic ways,—not to approach Nature with the Spirit. In some respects therefore, the Jesuits were among the first to cultivate the materialistic ideas which are so prevalent today. Historically it is of course well-known, but people fail to reflect that this whole way of thinking, applied to Physics nowadays, is fundamentally a product of the said tendency, characteristically Roman-Catholic as indeed it is.

One of the main things we now have to discover is what happens when we perceive notes of different pitch. How do the external phenomena of vibration, which accompany the note, differ with respect to notes of different pitch? The answer can be shewn by such experiments as we are now about to demonstrate. You see this disc with its rows of holes. We can rotate it rapidly. Herr Stockmeyer will be so kind as to direct a stream of air on to the moving disc. (He did.) You can at once distinguish the different pitch of the two notes. How then did it arise? Nearer the centre of the disc are fewer holes,—40 in fact. When Herr Stockmeyer blew the stream of air on to here, every time it came upon a hole it went through, then in the intervening space it could not get through, then again it could, and so on. Again and again, by the quick motion of the disc, the next hole came where the last had been, and there arose as many beats as there were holes arriving at the place where the stream of air was going. Thus on the inner circle we got 40 beats, but on the outer we got 80 in the same period of time. The beats bring about the wave, the oscillations or vibrations. Thus in the same period of time we have 80 beats, 80 air-waves in the one case and 40 in the other. The note that arises when we have 80 oscillations is twice as high as the note that arises when we have 40. Sundry experiments of this kind shew how the pitch of the note is connected with the number of vibrations arising in the medium in which the sound is propagated.

Please take together what I have just been saying and what was said once before; it will then lead you to the following reflection. A single oscillation of condensation and attenuation gives, as regards the distance it has gone through, what we call the wave-length. If n such waves arise in a second and the length of each wave is s, the whole wave-movement must be advancing n times s in a second. The path, the distance therefore, through which the whole wave-movement advances in a second, is n times s. Now please recall what I said in an earlier lecture. I said that we must carefully distinguish all that is “phoronomical” on the one hand, and on the other hand all that which we do not merely think out in our own inner life of thought but which consists of outer realities. In effect, I said, outward realities can never be merely spatial, or arithmetical (able to be numbered and calculated), nor can they be mere displacements. Velocities on the other hand are outward realities,—they always are. And of course this remains so when we come to sound or tone. Neither the s nor the n can be experienced as an external reality, for the s is merely spatial while the n is a mere number. What is real is inherent in the velocity. The velocity contains the real being, the real entity which we are here describing as ‘sound’ or ‘tone’. If I now divide the velocity into two abstractions, in these abstractions I have no realities; I only have what is abstracted, separated out and divided from it. Such are the wave-lengths—the spatial magnitudes—and also the number n. If on the other hand I want to look at the reality of the sound—at what is real in the world outside myself,—then I must concentrate upon the inner faculty of the sound to have velocity. This then will lead me to a qualitative study of the sound, whereas the way of studying it which we have grown accustomed to in modern Physics is merely quantitative. In the theory of sound, in acoustics especially, we see how modern Physics is always prone to insert what can be stated and recorded in these extraneous, quantitative, spatial and temporal, kinematical and arithmetical forms, in place of the qualitative reality which finds expression simply and solely in a certain faculty of speed, or of velocity.

Today however, people no longer even notice how they sail off into materialistic channels even in the theory of sound. It is so evident, they may well argue, that the sound as such is not there outside us; outside us are only the oscillations. Could anything be clearer?—so they may well contend. There are the waves of condensation and attenuation. Then, when my ear is in the act of “hearing”, what is really there outside me are these condensations and attenuations; that unknown something within me (which the physicist of course need not go into,—it is not his department) therefore transforms the waves into subjective experiences,—transforms the vibrations of the vibrating bodies into the quality that is the ‘sound’ or ‘tone’. In all manner of variations you will find ever the same proposition. Outside us are the vibrations; in us are the effects of the vibrations—effects that are merely subjective. In course of time it has become part of their very flesh and bone, till such results emerge as you find quoted from Robert Hamerling for instance in my Riddles of Philosophy. Having absorbed and accepted the teachings of Physics, Hamerling says at the very outset: What we experience as the report of a gun, is, in the world outside us, no more nor less than a certain violent disturbance of the air. And from this premise Hamerling continues: Whoever does not believe that the sensory impression he experiences is only there in himself while in the world outside him is simply vibrating air or vibrating ether,—let him put down the book which Hamerling is writing; such books are not for him. Robert Hamerling even goes on to say: Whoever thinks that the picture which he obtains of a horse corresponds to an outward reality, understands nothing at all and had better close the book.

Such things, dear Friends, for once deserve to be followed to their logical conclusion. What would become of it if I treated you, who are now sitting here, according to this way of thinking (I do not say method, but way-of-thinking) which physicists have grown accustomed to apply to the phenomena of sound and light? This surely would be the outcome: You, all of you, now sitting here before me,—I only have you here before me through my own impressions, which (if this way of thought be true) are altogether subjective, since my sensations of light and sound are so. None of you are there outside me in the way I see you. Only the oscillations in the air, between you and me, lead me to the oscillations that are there in you, and I am led to the conclusion that all your inner being and life of soul—which, within you and for yourselves, is surely not to be denied—is not there at all. For me, this inner soul of everyone of you who are here seated is only the effect on my own psyche, while for the rest, all that is really there, seated on these benches, are so many heaps of vibrations. If you deny to light and sound the inner life and being which you experience in a seemingly subjective way, it is precisely as it would be if, having you here before me, I looked on all that is before me as merely part of my subjective life, and thus denied to you the experience of inner life and being.

What I have now been saying is indeed so obvious, so trite, that physicists and physiologists will naturally not presume that they could ever fall into such obvious mistakes. And yet they do. The whole distinction that is usually made of the subjective impression (or whatsoever is alleged to be subjective) from the objective process, amounts to this and nothing else. It is of course open to the physicist to be quite candid and to say: I, as physicist, am not proposing to investigate the sound or tone at all; I do not enter into what is qualitative. All I am out to investigate are the external, spatial processes (he will not have to call them “objective processes” for that again would beg the question). All I am out to investigate are the outwardly spatial processes, which of course also go on into my own body. These are the subject-matter of my researches. These I abstract from the totality; what is qualitative is no concern of mine. A man who speaks like this is at least candid and straightforward, only he must not then go on to say that the one is “objective” and the other “subjective”, or that the one is the “effect” of the other. What you experience in your soul,—when I experience it with you it is not the effect upon me of the vibrations of your brain. To see through a thing like that is of untold significance; nothing could be of greater importance for the requirements of the new age, not only in science but in the life of humanity at large.

We ought not to be too reluctant to go into deeper questions when dealing with these matters. How easily it can be argued that the uniquely oscillatory character of sound or tone is evident if only from the fact that if I twang a violin-string a second string in the same room, attuned to the same note, will resound too, this being due to the fact that the intervening medium propagates the accompanying oscillations. Yet we do not understand what is happening in such a case unless we bring it into connection with a more widespread phenomenon. I mean the following for instance,—it has in fact been observed.

You have a pendulum clock; you wind it up and start it. In the same room there is another pendulum clock; it must, admittedly, be of a certain type. This you do not wind up. In favourable circumstances you may observe that the second clock starts of its own accord. We will call this the “mutual sympathy” of phenomena; it can be investigated in a very wide domain. The last phenomenon of this type, still connected to some extent with the outer world, could be examined far more than it generally is, for it is very frequent. Times without number you may have this experience. You are at table with another person and he says something you yourself have just been thinking. You were thinking it but did not say it; he now utters it. It is the sympathetic going-together of events (or complexes of events) in some way attuned to one-another, which is here making itself felt in a highly spiritual realm. We need to recognize the whole range of continuity from the simple resonance of a violin-string which one may still interpret crudely and unspiritually within the sequence of outer material events, to these parallel phenomena which appear so much more spiritual—as when we experience one-another's thoughts.

Now we shall never gain insight into these things unless we have the will to see and understand how man himself is placed into the midst even of so-called physical Nature. A few days ago we were demonstrating and to some extent analyzing the human eye. Today we will do the same with the human ear. As we go inward in the eye, you will remember we come to the vitreous body, which, as we said, still has considerable vitality. Then there is the fluid between the lens and the cornea. As we go inward, we were saying, the eye gets ever more alive and vital, whereas the outer part is increasingly like a piece of physical apparatus. Now we can of course equally well describe the human ear, and in a purely external sense we may aver: Just as the light affects the eye and the optic nerve receives the stimulus, so do the oscillations of sound affect the ear. They go on into the external auditory canal and beat upon the drum which forms the inner end of this canal. Behind the drum are the minute bones or ossicles, called hammer, anvil and stirrup from their appearance. That which arises (speaking in terms of Physics) in the outer world and finds expression in waves of alternate compression and expansion in the air, is transmitted through this peculiar system of ossicles to the inner ear. There is the so-called cochlea, filled with a kind of fluid, and here the auditory nerve has its ending. Before the cochlea we come to the three semicircular canals,—their planes at right angles to each other according to the three dimensions of space. Thus we can imagine the sound penetrating here in the form of air-waves and transmitted by the ossicles until it comes into this fluid. There then it reaches the nerve and so affects the sentient brain. So we should have the eye as one sense-organ, the ear—another. We put them neatly side by side, and—for a further abstraction—we may even elaborate a general physiology of the senses and of sensation.

But it will not seem so simple if you recall what I said recently of the whole rhythm of the ascending and descending cerebrospinal fluid and how it interacts with what is taking place more externally in the outer air. Remember too what I was saying: a thing may look complete and self-contained when outwardly regarded, but we must not therefore take it to be a finished reality, for it need not be so at all. The rose I cut off from the shrub is no reality. It cannot be by itself. It can only come to existence by virtue of its connection with the whole rose-bush. If I think of it as a mere rose by itself, it is in truth an abstraction. I must go on to the totality—to the whole rose-bush at the very least. So too for hearing: the ear alone is no reality, though it is nearly always represented as such in this connection. What is transmitted inward through the ear must first interact in a certain way with the inner rhythm, manifested in the rise and fall of the cerebrospinal fluid. But we have still not reached the end. All this that takes its course in rhythm—and, as it were, includes the brain within its span—is also fundamental, in the real human being, to what appears in quite another part of our body, namely in the larynx and adjoining organs when we are speaking. There is the act of speaking,—its instruments quite obviously inserted into the breathing process, to which the rhythmic rise and fall of the cerebrospinal fluid is also due. In the whole rhythm which arises in you when you breathe, you can therefore insert on the one hand your active speaking and on the other hand your hearing. Then you will have a totality; it only comes to manifestation in a more intelligent or perceptive way in your hearing and in a more volitional way in your speaking. Once more, you only have a totality when you take together the more volitional element pulsating through the larynx and the more sensitive or intelligent that goes through the ear. To separate the ear on the one hand, the larynx on the other, is an abstraction; you have no real totality so long as you separate these two. The two belong together; this is a matter of fact and you need to see it. The physiological physicist or physical physiologist who studies the larynx and the ear apart from one-another proceeds as you would do if you cut up a human being so as to bring him to life instead of seeing things in living interaction.

If we have recognized the facts, this is what we shall see:—Consider what is left of the eye if I first take away the vitreous body and also the whole or at least part of what is here spread out—the retina (Figure IIIf). If I were able to remove all this, what would be left would be the ciliary muscle, the lens and the external liquid—the aqueous humour. What kind of organ would that represent? It would be an organ, my dear Friends, which I could never compare with the ear if I were thinking realistically, but only with the larynx. It is not a metamorphosis of the ear; it is a metamorphosis of the larynx. Only to touch upon the coarsest aspect: just as the muscles of the larynx take hold of the vocal chords, widening or narrowing the aperture between them, so do the ciliary muscles with the lens. The lens is inherently mobile and they take hold of it.

So far I should have separated-out what is larynx-like, so to speak, for the ethereal, even as the larynx is for the air. And if I now reinsert first the retina, then the vitreous body, and then for certain animals the pecten, which man only has etherically, or the falciform process, (blood-bearing organs, continued into the eye in certain lower animals),—this part alone I shall be able truly to relate to the ear. Such things as the expanding portions of the pecten, these I may rightly compare to what expands in the ear,—in the labyrinth and so on. Thus, at one level in the human body I have the eye. In its more inward parts it is a metamorphosed ear, enveloped from without by a metamorphosed larynx. If we take larynx and ear together as a single whole, we have a metamorphosed eye upon another level.

What I have now been pointing out will lead us presently along a most important path. We can have no real knowledge of these things if we relate them falsely to begin with by simply placing eye and ear side by side, whereas in truth the ear can only be compared to the part of the eye behind the lens—the inner and more vital part—while that which reaches farther forward and is more muscular in character must be related to the larynx. This of course makes the theory of metamorphosis more difficult. It is no use looking for metamorphoses in crude, external ways. You must be able to see into the inner dynamic qualities, for these are real.

If it be so however, my dear Friends, we shall no longer be able to conceive as parallel, without more ado, all that goes on in the phenomena of tone and sound on the one hand and on the other hand the phenomena of light. Having begun with the mistaken premise that eye and ear are equally sense-organs, we shall be no less mistaken in our approach to the related phenomena. My seeing in effect is fundamentally different from my hearing. When I am seeing, the same thing happens in my eye as when I hear and speak at the same time. Here, in a higher realm, an activity which can only be compared to the activity of speech accompanies the receptive activity as such—the perceiving, receiving activity of the eye. You will get nowhere in these realms unless you apprehend what is real. For if you once become aware that in the eye two things are welded together which are assigned to seemingly distinct organs of the body in sound or hearing, then you will realize that in seeing, in the eye, we have a kind of monologue,—as when you converse and come to an understanding with yourself. The eye always proceeds as you would do if you were listening intently and every time, to understand what you were hearing, you first repeated it aloud. Such is the eye's activity,—it is as though you were listening to someone and at the same time repeating what you heard, word for word. The other person says, “he writes”, but this does not suffice you. You first repeat aloud, “he writes”,—then and then only is the thing complete. So it is with the eye and the phenomena of light. What comes into our consciousness as an outcome of this whole complex—namely through the fact that we have the more vital, inner part of the eye to begin with—only becomes the full experience of sight, in that we reproduce it in the portion of the eye that corresponds to the larynx and that lies farther forward. Etherically we are talking to ourselves when we are seeing. The eye is engaged in a monologue, and it is wrong to compare the outcome of this monologue—in which the human being's own activity is already contained—with hearing alone, for this is but a single factor of the dual process.

I do believe, dear Friends, that if you work it through for yourselves this will give you much indeed. For it will shew you among other things how far astray materialistic Physics goes and how unreal it becomes in its study of the World, in that it starts by comparing what is not directly comparable—the eye and ear in this instance. It is this purely outward way of study—failing to look and see what are totalities and what are not—which leads away from any spiritual view of Nature. Think for example of what Goethe does at the conclusion of his Theory of Colour, where in the chapter on the “Ethical-Aesthetical Effects of Colour” he evolves the spiritual logically from what is physical. You will never do this if you take your start from the colour-theory of modern Physics.

Now I admit that sound or tone may cause misgivings. Is it not evident that in the outer world mere oscillations are going on when you hear sound? (In some such words it will be stated.) However, ask yourselves another question and then decide whether the very putting of it does not give the answer. Might it not be as follows? Suppose you had a globe or bell-jar, full of air, provided with an aperture and stopcock. Open the stopcock,—nothing will happen if the air inside has the same density as outside. But if there is a vacuum inside, plenty will happen. Air from outside will whistle in and fill the empty space. Will you then say that the air which the globe now contains came into being simply by virtue of what was going on inside the globe? No. You will say: This air has come in from outside, but the empty space—purely to describe the phenomenon as you see it—has somehow sucked it in.

So also when we turn this disc and blow against the holes, we create the conditions for a kind of suction to arise,—this is a true way to describe it. The tone, the sound that will appear when as I work the siren I cause the air to oscillate,—this tone is already in existence, only it is outside of space. It is not yet in space. The conditions for it to enter space are not given until I make them, even as the conditions for the outer air to get into the globe are not given until I make them. The outer air-waves can only be compared to the vacuum inside the globe, and what then grows audible can only be compared to what penetrates from the surrounding space into the vacuum inside when the conditions have been created for this to happen. In essence the air-waves have no more to do with the sound than that, where these waves are, a process of suction is produced to draw the sound from its non-spatial realm into the spatial. Of course the kind of sound, the particular tone that is drawn in, is modified by the kind of air-waves, but so too would it modify what happens in the evacuated globe if I made special-shaped channels in the aperture by which the air is to be drawn in. The air would then expand into the inner space along certain lines, of which an image was there. So have the processes of sound or tone their external image in the observed processes of oscillation.

You see from this, dear Friends, the fundamentals of a true Physical Science, which we aspire to, are not so easy to conceive. It is by no means enough to entertain a few mathematical notions about wave-movements or oscillations. We must make greater demands on the qualitative element in human thinking. If such demands are unfulfilled, we only get once more the picture of the World which is so worshipped in the Physics of today, and which is to reality as is a tissue-paper effigy to a living man.

My dear Friends,

I am sorry these explanations have had to be so improvised and brief, so that they scarcely go beyond mere aphorisms. It is inevitable. All I can do during these days is to give you a few points of view, with the intention of continuing when I am here again, so that in time these explanations may be rounded off, to give you something more complete. Tomorrow I will give a few concluding aspects, also enabling us to throw some light on the educational use of scientific knowledge. Now to prepare for tomorrow, I must today draw your attention to the development of electrical discoveries, beginning no doubt with things that are well-known to you from your school days. This will enable us, in tomorrow's lecture, to gain a more comprehensive view of Physics as a whole.

You know the elementary phenomena of electricity. A rod of glass, or it may be of resin, is made to develop a certain force by rubbing it with some material. The rod becomes, as we say, electrified; it will attract small bodies such as bits of paper. You know too what emerged from a more detailed observation of these phenomena. The forces proceeding from the glass rod, and from the rod of resin or sealing-wax, prove to be diverse. We can rub either rod, so that it gets electrified and will attract bits of paper. If the electrical permeation, brought about with the use of the glass rod, is of one kind, with the resinous rod it proves to be opposite in kind. Using the qualitative descriptions which these phenomena suggest, one speaks of vitreous and resinous electricities respectively; speaking more generally one calls them “positive” and “negative”. The vitreous is then the positive, the resinous the negative.

Now the peculiar thing is that positive electricity always induces and brings negative toward itself in some way. You know the phenomenon from the so-called Leyden Jar. This is a vessel with an electrifiable coating on the outside. Then comes an insulating layer (the substance of the vessel). Inside, there is another coating, connected with a metal rod, ending perhaps in a metallic knob (Figure IXa). If you electrify a metal rod and impart the electricity to the one coating, so that this coating will then evince the characteristic phenomena, say, of positive electricity, the other coating thereby becomes electrified negatively. Then, as you know, you can connect the one coating, imbued with positive, and the other, imbued with negative electricity, so as to bring about a connection of the electrical forces, positive and negative, with one another. You have to make connection so that the one electricity can be conducted out here, where it confronts the other. They confront each other with a certain tension, which they seek to balance out. A spark leaps across from the one to the other. We see how the electrical forces, when thus confronting one another, are in a certain tension, striving to resolve it. No doubt you have often witnessed the experiment.

Here is the Leyden Jar,—but we shall also need a two-pronged conductor to discharge it with. I will now charge it. The charge is not yet strong enough. You see the leaves repelling one another just a little. If we charged this sufficiently, the positive electricity would so induce the negative that if we brought them near enough together with a metallic discharger we should cause a spark to fly across the gap. Now you are also aware that this kind of electrification is called frictional electricity, since the force, whatever it may be, is brought about by friction. And—here again, I am presumably still recalling what you already know—it was only at the turn of the 18th and 19th centuries that they discovered, in addition to this “frictional electricity”, what is called “contact electricity”, thus opening up to modern Physics a domain which has become notably fruitful in the materialistic evolution of this science.

I need only remind you of the main principles. Galvani observed the leg of a frog which was in touch with metal plates and began twitching. He had discovered something of very great significance. He had found two things at once, truth to tell,—two things that should really be distinguished from one-another and are not yet quite properly distinguished, unhappily for Science, to this day. Galvani had discovered what Volta, a little later, was able to describe simply as “contact electricity”, namely the fact that when diverse metals are in contact, and their contact is also mediated by the proper liquids, an interaction arises—an interaction which can find expression in the form of an electric current from the one metal to the other. We have then the electric current, taking place to all appearances purely within the inorganic realm. But we have something else as well, if once again we turn attention to the discovery made by Galvani. We have what may in some sense be described as “physiological electricity”. It is a force of tension which is really always there between muscle and nerve and which can be awakened when electric currents are passed through them. So that in fact, that which Galvani had observed contained two things. One of them can be reproduced by purely inorganic methods, making electric currents by means of different metals with the help of liquids. The other thing which he observed is there in every organism and appears prominently in the electric fishes and certain other creatures. It is a state of tension between muscle and nerve, which, when it finds release, becomes to all appearances very like flowing electricity and its effects. It was then these discoveries which led upon the one hand to the great triumphs in materialistic science, and on the other hand provided the foundations for the immense and epoch-making technical developments which followed.

Now the fact is, the 19th century was chiefly filled with the idea that we must somehow find a single, abstract, unitary principle at the foundation of all the so-called “forces of Nature”. It was in this direction, as I said before that they interpreted what Julius Robert Mayer, the brilliant Heilbronn doctor had discovered. You will remember how we demonstrated it the other day. By mechanical force we turned a flywheel; this was attached to an apparatus whereby a mass of water was brought into inner mechanical activity. The water thereby became warmer, as we were able to shew. The effect produced—the development of warmth—may truly be attributed to the mechanical work that was done. All this was so developed and interpreted in course of time that they applied it to the most manifold phenomena of Nature,—nor was it difficult to do so within certain limits. One could release chemical forces and see how warmth arose in the process. Again, reversing the experiment which we have just described, warmth could be used in such a way as to evoke mechanical work,—as in the steam-engine and in a multitude of variations.

It was especially this so-called transformation of Nature's forces on which they riveted attention. They were encouraged to do this by what began in Julius Robert Mayer's work and then developed ever further. For it proves possible to calculate, down to the actual figures, how much warmth is needed to produce a given, measurable amount of work; and vice-versa, how much mechanical work is needed to produce a given, measurable amount of warmth or heat. So doing, they imagined—though to begin with surely there is no cause to think of it in this way—that the mechanical work, which we expended for example in making these vanes rotate in the water, has actually been transformed into the warmth. Again, they assumed that when warmth is applied in the steam-engine, this warmth is actually transformed into the mechanical work that emerges. The meditations of physicists during the 19th century kept taking this direction: they were always looking for the kinship between the diverse forces of Nature so-called,—trying to discover kinships which were to prove at last that some abstract, everywhere equal principle is at the bottom of them all, diverse and manifold as they appear. These tendencies were crowned to some extent when near the end of the century Heinrich Hertz, a physicist of some genius, discovered the so-called electric waves—here once again it was waves! It certainly seemed to justify the idea that the electricity that spreads through space is in some way akin to the light that spreads through space,—the latter too being already conceived at that time as a wave-movement in the ether.

That “electricity”—notably in the form of current electricity—cannot be grasped so simply with the help of primitive mechanical ideas, but makes it necessary to give our Physics a somewhat wider and more qualitative aspect,—this might already have been gathered from the existence of induction currents as they are called. Only to indicate it roughly: the flow of an electric current along a wire will cause a current to arise in a neighbouring wire, by the mere proximity of the one wire to the other. Electricity is thus able to take effect across space,—so we may somehow express it. Now Hertz made this very interesting discovery:—he found that the electrical influences or agencies do in fact spread out in space in a way quite akin to the spreading of waves, or to what could be imagined as such. He found for instance that if you generate an electric spark, much in the way we should be doing here, developing the necessary tension, you can produce the following result. Suppose we had a spark jumping across this gap. Then at some other point in space we could put two such “inductors”, as we may call them, opposite and at a suitable distance from one-another, and a spark would jump across here too.

This, after all, is a phenomenon not unlike what you would have if here for instance—Figure IXb—were a source of light and here a mirror. A cylinder of light is reflected, this is then gathered up again by a second mirror, and an image arises here. We may then say, the light spreads out in space and takes effect at a distance. In like manner. Hertz could now say that electricity spreads out and the effect of it is perceptible at a distance. Thus in his own conception and that of other scientists he had achieved pretty fair proof that with electricity something like a wave-movement is spreading out through space,—analogous to the way one generally imagines wave-movements to spread out. Even as light spreads out through space and takes effect at a distance, unfolding as it were, becoming manifest where it encounters other bodies, so too can the electric waves spread out, becoming manifest—taking effect once more—at a distance. You know how wireless telegraphy is based on this.

The favourite idea of 19th century physicists was once again fulfilled to some extent. For sound and light, they were imagining wave-trains, sequences of waves. Also for warmth as it spreads outward into space, they had begun to imagine wave-movements, since the phenomena of warmth are in fact similar in some respects. Now they could think the same of electricity; the waves had only to be imagined long by comparison. It seemed like incontrovertible proof that the way of thinking of 19th century Physics had been right.

Nevertheless, Hertz's experiments proved to be more like a closing chapter of the old. What happens in any sphere of life, can only properly be judged within that sphere. We have been undergoing social revolutions. They seem like great and shattering events in social life since we are looking rather intently in their direction. Look then at what has happened in Physics during the 1890's and the first fifteen years, say, of our century; you must admit that a revolution has here been going on, far greater in its domain than the external revolution in the social realm. It is no more nor less than that in Physics the old concepts are undergoing complete dissolution; only the physicists are still reluctant to admit it. Hertz's discoveries were still the twilight of the old, tending as they did to establish the old wave-theories even more firmly. What afterwards ensued, and was to some extent already on the way in his time, was to be revolutionary.

I refer now to those experiments where an electric current, which you can generate of course and lead to where you want it, is conducted through a glass tube from which the air has to a certain extent been pumped out, evacuated. The electric current, therefore, is made to pass through air of very high dilution. High tension is engendered in the tubes which you here see. In effect, the terminals from which the electricity will discharge into the tube are put far apart—as far as the length of the tube will allow. There is a pointed terminal at either end, one where the positive electricity will discharge (i.e. the positive pole) at the one end, so too the negative at the other. Between these points the electricity discharges; the coloured line which you are seeing is the path taken by the electricity. Thus we may say: What otherwise goes through the wires, appears in the form in which you see it here when it goes through the highly attenuated air. It becomes even more intense when the vacuum is higher. Look how a kind of movement is taking place from the one side and the other,—how the phenomenon gets modified. The electricity which otherwise flows through the wire: along a portion of its path we have been able, as it were, so to treat it that in its interplay with other factors it does at last reveal, to some extent, its inner essence. It shews itself, such as it is; it can no longer hide in the wire! Observe the green light on the glass; that is fluorescent light. I am sorry I cannot go into these phenomena in greater detail, but I should not get where I want to in this course if I did not go through them thus quickly. You see what is there going through the tube,—you see it in a highly dispersed condition in the highly attenuated air inside the tube.

Now the phenomena which thus appeared in tubes containing highly attenuated air or gas, called for more detailed study, in which many scientists engaged,—and among these was Crookes. Further experiments had to be made on the phenomena in these evacuated tubes, to get to know their conditions and reactions. Certain experiments, due among others to Crookes, bore witness to a very interesting fact. Now that they had at last exposed it—if I may so express myself—the inner character of electricity, which here revealed itself, proved to be very different from what they thought of light for instance being propagated in the form of wave-movements through the ether. What here revealed itself was clearly not propagated in that way. Whatever it is that is shooting through these tubes is in fact endowed with remarkable properties, strangely reminiscent of the properties of downright matter. Suppose you have a magnet or electromagnet. (I must again presume your knowledge of these things; I cannot go into them all from the beginning.) You can attract material objects with the magnet. Now the body of light that is going through this tube—this modified form, therefore, of electricity—has the same property. It too can be attracted by the electromagnet. Thus it behaves, in relation to a magnet, just as matter would behave. The magnetic field will modify what is here shooting through the tube.

Experiments of this kind led Crookes and others to the idea that what is there in the tube is not to be described as a wave-movement, propagated after the manner of the old wave-theories. Instead, they now imagined material particles to be shooting through the space inside the tube; these, as material particles, are then attracted by the magnetic force. Crookes therefore called that which is shot across there from pole to pole, (or howsoever we may describe it; something is there, demanding our consideration),—Crookes called it “radiant matter”. As a result of the extreme attenuation, he imagined, the matter that is left inside the tube has reached a state no longer merely gaseous but beyond the gaseous condition. He thinks of it as radiant matter—matter, the several particles of which are raying through space like the minutest specks of dust or spray, the single particles of which, when charged electrically, will shoot through space in this way. These particles themselves are then attracted by the electromagnetic force. Such was his line of thought: the very fact that they can thus be attracted shews that we have before us a last attenuated remnant of real matter, not a mere movement like the old-fashioned ether-movements.

It was the radiations (or what appeared as such) from the negative electric pole, known as the cathode, which lent themselves especially to these experiments. They called them “cathode rays”. Herewith the first breach had, so to speak, been made in the old physical conceptions. The process in these Hittorf tubes (Hittorf had been the first to make them, then came Geissler) was evidently due to something of a material kind—though in a very finely-divided condition—shooting through space. Not that they thereby knew what it was; in any case they did not pretend to know what so-called “matter” is. But the phenomena indicated that this was something somehow identifiable with matter,—of a material nature.

Crookes therefore was convinced that this was a kind of material spray, showering through space. The old wave-theory was shaken. However, fresh experiments now came to light, which in their turn seemed inconsistent with Crookes's theory. Lenard in 1893 succeeded in diverting the so-called rays that issue from this pole and carrying them outward. He inserted a thin wall of aluminium and led the rays out through this. The question arose: can material particles go through a material wall without more ado? So then the question had to be raised all over again:

Is it really material particles showering through space,—or is it something quite different after all? In course of time the physicists began to realize that it was neither the one nor the other: neither of the old conceptions—that of ether-waves, or that of matter—would suffice us here. The Hittorf tubes were enabling them, as it were, to pursue the electricity itself along its hidden paths. They had naturally hoped to find waves, but they found none. So they consoled themselves with the idea that it was matter shooting through space. This too now proved untenable.

At last they came to the conclusion which was in fact emerging from many and varied experiments, only a few characteristic examples of which I have been able to pick out. In effect, they said: It isn't waves, nor is it simply a fine spray of matter. It is flowing electricity itself; electricity as such is on the move. Electricity itself is flowing along here, but in its movement and in relation to other things—say, to a magnet—it shews some properties like those of matter. Shoot a material cannonball through the air and let it pass a magnet,—it will naturally be diverted So too is electricity. This is in favour of its being of a material nature. On the other hand, in going through a plate of aluminium without more ado, it shews that it isn't just matter. Matter would surely make a hole in going through other matter. So then they said: This is a stream of electricity as such. And now this flowing electricity shewed very strange phenomena. A clear direction was indeed laid out for further study, but in pursuing this direction they had the strangest experiences. Presently they found that streams were also going out from the other pole,—coming to meet the cathode rays. The other pole is called the anode; from it they now obtained the rays known as “canal rays”. In such a tube, they now imagined there to be two different kinds of ray, going in opposite directions.

One of the most interesting things was discovered in the 1890's by Roentgen ... From the cathode rays he produced a modified form of rays, now known as Roentgen rays or X-rays. They have the effect of electrifying certain bodies, and also shew characteristic reactions with magnetic and electric forces. Other discoveries followed. You know the Roentgen rays have the property of going through bodies without producing a perceptible disturbance; they go through flesh and bone in different ways and have thus proved of great importance to Anatomy and Physiology.

Now a phenomenon arose, making it necessary to think still further. The cathode rays or their modifications, when they impinge on glass or other bodies, call forth a kind of fluorescence; the materials become luminous under their influence. Evidently, said the scientists, the rays must here be undergoing further modification. So they were dealing already with many different kinds of rays. Those that first issued directly from the negative pole, proved to be modifiable by a number of other factors. They now looked round for bodies that should call forth such modifications in a very high degree—bodies that should especially transform the rays into some other form, e.g. into fluorescent rays. In pursuit of these researches it was presently discovered that there are bodies—uranium salts for example—which do not have to be irradiated at all, but under certain conditions will emit rays in their turn, quite of their own accord. It is their own inherent property to emit such rays. Prominent among these bodies were the kind that contain radium, as it is called.

Very strange properties these bodies have. They ray-out certain lines of force—so to describe it—which can be dealt with in a remarkable way. Say that we have a radium-containing body here, in a little vessel made of lead; we can examine the radiation with a magnet. We then find one part of the radiation separating off, being deflected pretty strongly in this direction by the magnet, so that it takes this form (Figure IXc). Another part stays unmoved, going straight on in this direction, while yet another is deflected in the opposite direction. The radiation, then, contains three elements. They no longer had names enough for all the different kinds! They therefore called the rays that will here be deflected towards the right, ß-rays; those that go straight on, γ-rays; and those are deflected in the opposite direction, α-rays.

Bringing a magnet near to the radiating body, studying these deflections and making certain computations, from the deflection one may now deduce the velocity of the radiation. The interesting fact emerges that the ß-rays have a velocity, say about nine-tenths the velocity of light, while the velocity of the α-rays is about one-tenth the velocity of light. We have therefore these explosions of force, if we may so describe them, which can be separated-out and analyzed and then reveal very striking differences of velocity.

Now I remind you how at the outset of these lectures we endeavoured in a purely spiritual way to understand the formula, v = s/t. We said that the real thing in space is the velocity; it is velocity which justifies us in saying that a thing is real. Here now you see what is exploding as it were, forth from the radiating body, characterized above all by the varying intensity and interplay of the velocities which it contains. Think what it signifies: in the same cylinder of force which is here raying forth, there is one element that wants to move nine times as fast as the other. One shooting force, tending to remain behind, makes itself felt as against the other that tends to go nine times as quickly. Now please pay heed a little to what the anthroposophists alone, we must suppose, have hitherto the right not to regard as sheer madness! Often and often, when speaking of the greatest activities in the Universe which we can comprehend, we had to speak of differences in velocity as the most essential thing. What is it brings about the most important things that play into the life of present time? It is the different velocities with which the normal, the Luciferic and the Ahrimanic spiritual activities work into one-another. It is that differences of velocity are there in the great spiritual streams to which the web and woof of the world is subjected. The scientific pathway which has opened out in the most recent times is compelling even Physics—though, to begin with, unconsciously—to go into differences of velocity in a way very similar to the way Spiritual Science had to do for the great all-embracing agencies of Cosmic Evolution.

Now we have not yet exhausted all that rays forth from this radium-body. The effects shew that there is also a raying-forth of the material itself. But the material thus emanated proves to be radium no longer. It presently reveals itself to be helium for instance—an altogether different substance. Thus we no longer have the conservation,—we have the metamorphosis of matter.

The phenomena to which I have been introducing you, all of them take their course in what may be described as the electrical domain. Moreover, all of them have one property in common. Their relation to ourselves is fundamentally different from that of the phenomena of sound or light for example, or even the phenomena of warmth. In light and sound and warmth we ourselves are swimming, so to speak, as was described in former lectures. The same cannot be said so simply of our relation to the electrical phenomena. We do not perceive electricity as a specific quality in the way we perceive light, for instance. Even when electricity is at last obliged to reveal itself, we perceive it by means of a phenomenon of light. This led to people's saying, what they have kept repeating: “There is no sense-organ for electricity in man.” The light has built for itself in man the eye—a sense-organ with which to see it. So has the sound, the ear. For warmth too, a kind of warmth-organ is built into man. For electricity, they say, there is nothing analogous. We perceive electricity indirectly.

We do, no doubt; but that is all that can be said of it till you go forward to the more penetrating form of Science which we are here at least inaugurating. In effect, when we expose ourselves to light, we swim in the element of light in such a way that we ourselves partake in it with our conscious life, or at least partially so. So do we in the case of warmth and in that of sound or tone. The same cannot be said of electricity. But now I ask you to remember what I have very often explained: as human beings we are in fact dual beings. That is however to put it crudely, for we are really threefold beings: beings of Thought, of Feeling and of Will. Moreover, as I have shewn again and again, it is only in our Thinking that we are really awake, whilst in our feelings we are dreaming and in our processes of will we are asleep—asleep even in the midst of waking life. We do not experience our processes of will directly. Where the essential Will is living, we are fast asleep. And now remember too, what has been pointed out during these lectures. Wherever in the formulae of Physics we write m for mass, we are in fact going beyond mere arithmetic—mere movement, space and time. We are including what is no longer purely geometrical or kinematical, and as I pointed out, this also corresponds to the transition of our consciousness into the state of sleep. We must be fully clear that this is so. Consider then this memberment of the human being; consider it with fully open mind, and you will then admit: Our experience of light, sound and warmth belongs—to a high degree at least, if not entirely—to the field which we comprise and comprehend with our sensory and thinking life. Above all is this true of the phenomena of light. An open-minded study of the human being shews that all these things are akin to our conscious faculties of soul. On the other hand, the moment we go on to the essential qualities of mass and matter, we are approaching what is akin to those forces which develop in us when we are sleeping. And we are going in precisely the same direction when we descend from the realm of light and sound and warmth into the realm of the electrical phenomena.

We have no direct experience of the phenomena of our own Will; all we are able to experience in consciousness is our thoughts about them. Likewise we have no direct experience of the electrical phenomena of Nature. We only experience what they deliver, what they send upward, to speak, into the realms of light and sound and warmth etc. For we are here crossing the same boundary as to the outer world, which we are crossing in ourselves when we descend from our thinking and idea-forming, conscious life into our life of Will. All that is light, and sound, and warmth, is then akin to our conscious life, while all that goes on in the realms of electricity and magnetism is akin—intimately akin—to our unconscious life of Will. Moreover the occurrence of physiological electricity in certain lower animals is but the symptom—becoming manifest somewhere in Nature—of a quite universal phenomenon which remains elsewhere unnoticed. Namely, wherever Will is working through the metabolism, there is working something very similar to the external phenomena of electricity and magnetism.

When in the many complicated ways—which we have only gone through in the barest outline in today's lecture—when in these complicated ways we go down into the realm of electrical phenomena, we are in fact descending into the very same realm into which we must descend whenever we come up against the simple element of mass. What are we doing then when we study electricity and magnetism? We are then studying matter, in all reality. It is into matter itself that you are descending when you study electricity and magnetism. And what an English philosopher has recently been saying is quite true—very true indeed. Formerly, he says, we tried to imagine in all kinds of ways, how electricity is based on matter. Now on the contrary we must assume, what we believe to be matter, to be in fact no more than flowing electricity. We used to think of matter as composed of atoms; now we must think of the electrons, moving through space and having properties like those we formerly attributed to matter.

In fact our scientists have taken the first step—they only do not yet admit it—towards the overcoming of matter. Moreover they have taken the first step towards the recognition of the fact that when in Nature we pass on from the phenomena of light, sound and warmth of those of electricity, we are descending—in the realm of Nature—into phenomena which are related to the former ones as is the Will in us to the life of Thought. This is the gist and conclusion of our studies for today, which I would fain impress upon your minds. After all, my main purpose in these lectures is to tell you what you will not find in the text-books. The text-book knowledge I may none the less bring forward, is only given as a foundation for the other.

My dear Friends,

I will now bring these few improvised hours of scientific study to a provisional conclusion. I want to give you a few guiding lines which may help you in developing such thoughts about Nature for yourselves, taking your start from characteristic facts which you can always make visible by experiment. In Science today—and this applies above all to the teacher—it is most important to develop a right way of thinking upon the facts and phenomena presented to us by Nature. You will remember what I was trying to shew yesterday in this connection. I shewed how since the 1890's physical science has so developed that materialism is being lifted right out of its bearings, so to speak, even by Physics itself. This is the point to remember above all in this connection.

The period when Science thought that it had golden proofs of the universality of waves and undulations was followed, as we say, by a new time. It was no longer possible to hold fast to the old wave-theories. The last three decades have in fact been revolutionary. One can imagine nothing more revolutionary in any realm than this most recent period has been in Physics. Impelled by the very facts that have not emerged, Physics has suffered no less a loss than the concept of matter itself in its old form. Out of the old ways of thinking, as we have seen, the phenomena of light had been brought into a very near relation to those of electricity and magnetism. Now the phenomena produced by the passage of electricity through tubes in which the air or gas was highly rarefied, led scientists to see in the raying light itself something like radiating electricity. I do not say that they were right, but this idea arose. It came about in this way:—The electric current until then had always been hidden as it were in wires, and one had little more to go on than Ohm's Law. Now one was able, so to speak, to get a glimpse of the electricity itself, for here it leaves the wire, jumps to the distant pole, and is no longer able as it were to conceal its content in the matter through which it passes.

The phenomena proved complicated. As we say yesterday, manifold types of radiation emerged. The first to be discovered were the so-called cathode rays, issuing from the negative pole of the Hittorf tube and making their way through the partial vacuum. In that they can be deflected by magnetic forces, they prove akin to what we should ordinarily feel to be material. Yet they are also evidently akin to what we see where radiations are at work. This kinship comes out most vividly when we catch the rays (or whatsoever it is that is issuing from the negative electric pole) upon a screen or other object, as we should do with light. Light throws a shadow. So do these radiations. Yet in this very experiment we are again establishing the near relation of these rays to the ordinary element of matter. For you can imagine that a bombardment is taking place from here (as we say yesterday, this is how Crookes thinks of the cathode rays). The “bombs” do not get through the screen which you put in the way; the space behind the screen is protected. This can be shewn by Crookes's experiment, interposing a screen in the way of the cathode rays.

We will here generate the electric current; we pass it through this tube in which the air is rarefied. It has its cathode or negative pole here, its anode or positive pole here. Sending the electricity through the tube, we are now getting the so-called cathode rays. We catch them on a screen shaped like a St. Andrew's cross. We let the cathode rays impinge on it, and on the other side you will see something like a shadow of the St. Andrew's cross, from which you may gather that the cross stops the rays. Observe it clearly, please. Inside the tube is the St. Andrew's cross. The cathode rays go along here; here they are stopped by the cross; the shadow of the cross becomes visible upon the wall of the vessel behind it. I will now bring the shadow which is thus made visible into the field of a magnet. I beg you to observe it now. You will find the shadow influenced by the magnetic field. You see then, just as I might attract a simple bit of iron with a magnet, so too, what here emerges like a kind of shadow behaves like external matter. It behaves materially.

Here then we have a type of rays which Crookes regards as “radiant matter”—as a form of matter neither solid, liquid or gaseous but even more attenuated,—revealing also that electricity itself, the current of electricity, behaves like simple matter. We have, as it were, been trying to look at the current of flowing electricity as such, and what we see seems very like the kind of effects we are accustomed to see in matter.

I will now shew you, what was not possible yesterday, the rays that issue from the other pole and that are called “canal rays”. You can distinguish the rays from the cathode, going in this direction, shimmering in a violet shade of colour, and the canal rays coming to meet them, giving a greenish light. The velocity of the canal rays is much smaller.

Finally I will shew you the kind of rays produced by this apparatus: they are revealed in that the glass becomes fluorescent when we send the current through. This is the kind of rays usually made visible by letting them fall upon a screen of barium platinocyanide. They have the property of making the glass intensely fluorescent. Please observe the glass. You see it shining with a very strong, greenish-yellow, fluorescent light. The rays that shew themselves in this way are the Roentgen rays or X-rays, mentioned yesterday. We observe this kind too, therefore.

Now I was telling you how in the further study of these things it appeared that certain entities, regarded as material substances, emit sheaves of rays—rays of three kinds, to begin with. We distinguished them as $\alpha$-, $\beta$-, and $\gamma$-rays (cf. the Figure IXc). They shew distinct properties. Moreover, yet another thing emerges from these materials, known as radium etc. It is the chemical element itself which as it were gives itself up completely. In sending out its radiation, it is transmuted. It changes into helium, for example; so it becomes something quite different from what it was before. We have to do no longer with stable and enduring matter but with a complete metamorphosis of phenomena.

Taking my start from these facts, I now want to unfold a point of view which may become for you an essential way, not only into these phenomena but into those of Nature generally. The Physics of the 19th century chiefly suffered from the fact that the inner activity, with which man sought to follow up the phenomena of Nature, was not sufficiently mobile in the human being himself. Above all, it was not able really to enter the facts of the outer world. In the realm of light, colours could be seen arising, but man had not enough inner activity to receive the world of colour into his forming of ideas, into his very thinking. Unable any longer to think the colours, scientists replaced the colours, which they could not think, by what they could,—namely by what was purely geometrical and kinematical—calculable waves in an unknown ether. This “ether” however, as you must see, proved a tricky fellow. Whenever you are on the point of catching it, it evades you. It will not answer the roll-call. In these experiments for instance, revealing all these different kinds of rays, the flowing electricity has become manifest to some extent, as a form of phenomenon in the outer world,—but the “ether” refuses to turn up. In fact it was not given to the 19th-century thinking to penetrate into the phenomena. But this is just what Physics will require from now on. We have to enter the phenomena themselves with human thinking. Now to this end certain ways will have to be opened up—most of all for the realm of Physics.

You see, the objective powers of the World, if I may put it so,—those that come to the human being rather than from him—have been obliging human thought to become rather more mobile (albeit, in a certain sense, from the wrong angle). What men regarded as most certain and secure, that they could most rely on, was that they could explain the phenomena so beautifully by means of arithmetic and geometry—by the arrangement of lines, surfaces and bodily forms in space. But the phenomena in these Hittorf tubes are compelling us to go more into the facts. Mere calculations begin to fail us here, if we still try to apply them in the same abstract way as in the old wave-theory.

Let me say something of the direction from which it first began, that we were somehow compelled to bring more movement into our geometrical and arithmetical thinking. Geometry, you know, was a very ancient science. The regularities and laws in line and triangle and quadrilateral etc.,—the way of thinking all these forms in pure Geometry—was a thing handed down from ancient time. This way of thinking was now applied to the external phenomena presented by Nature. Meanwhile however, for the thinkers of the 19th century, the Geometry itself began to grow uncertain. It happened in this way. Put yourselves back into your school days: you will remember how you were taught (and our good friends, the Waldorf teachers, will teach it too, needless to say; they cannot but do so),—you were undoubtedly taught that the three angles of a triangle (Figure Xa) together make a straight angle—an angle of 180°. Of course you know this. Now then we have to give our pupils some kind of proof, some demonstration of the fact. We do it by drawing a parallel to the base of the triangle through the vertex. We then say: the angle $\alpha$, which we have here, shews itself here again as ${\alpha }^{\prime }$. $\alpha$ and ${\alpha }^{\prime }$ are alternate angles and therefore equal. I can transfer this angle over here, then. Likewise this angle $\beta$, over here; again it remains the same.

The angle $\gamma$ stays where it is. If then I have $\gamma ={\gamma }^{\prime }$, $\alpha ={\alpha }^{\prime }$ and $\beta ={\beta }^{\prime }$, while ${\alpha }^{\prime }+\beta {;}^{\prime }+{\gamma }^{\prime }$ taken together give an angle of 180° as they obviously do, $\alpha +\beta +\gamma$ will do the same. Thus I can prove it so that you actually see it. A clearer or more graphic proof can scarcely be imagined.

However, what we are taking for granted is that this upper line A'B' is truly parallel to the lower line $AB$,—for this alone enables me to carry out the proof. Now in the whole of Euclid's Geometry there is no way of proving that two lines are really parallel, i.e. that they only meet at an infinite distance, or do not meet at all. They only look parallel so long as I hold fast to a space that is merely conceived in thought. I have no guarantee that it is so in any real space. I need only assume that the two lines meet, in reality, short of an infinite distance; then my whole proof, that the three angles together make 180°, breaks down. For I should then discover: whilst in the space which I myself construct in thought—the space of ordinary Geometry—the three angles of a triangle add up to 180° exactly, it is no longer so when I envisage another and perhaps more real space. The sum of the angles will no longer be 180°, but may be larger. That is to say, besides the ordinary geometry handed down to us from Euclid other geometries are possible, for which the sum of the three angles of a triangle is by no means 180°.

Nineteenth century thinking went a long way in this direction, especially since Lobachevsky, and from this starting-point the question could not but arise: Are then the processes of the real world—the world we see and examine with our senses—ever to be taken hold of in a fully valid way with geometrical ideas derived from a space of our own conceiving? We must admit: the space which we conceive in thought is only thought. Nice as it is to cherish the idea that what takes place outside us partly accords with what we figure-out about it, there is no guarantee that it really is so. There is no guarantee that what is going on in the outer world does really work in such a way that we can fully grasp it with the Euclidean Geometry which we ourselves think out. Might it not be—the facts alone can tell—might it not be that the processes outside are governed by quite another geometry, and it is only we who by our own way of thinking first translate this into Euclidean geometry and all the formulae thereof?

In a word, if we only go by the resources of Natural Science as it is today, we have at first no means whatever of deciding, how our own geometrical or kinematical ideas are related to what appears to us in outer Nature. We calculate Nature's phenomena in the realm of Physics—we calculate and draw them in geometrical figures. Yet, are we only drawing on the surface after all, or are we penetrating to what is real in Nature when we do so? What is there to tell? If people once begin to reflect deeply enough in modern Science—above all in Physics—they will then see that they are getting no further. They will only emerge from the blind alley if they first take the trouble to find out what is the origin of all our phoronomical—arithmetical, geometrical and kinematical—ideas. What is the origin of these, up to and including our ideas of movement purely as movement, but not including the forces? Whence do we get these ideas? We may commonly believe that we get them on the same basis as the ideas we gain when we go into the outer facts of Nature and work upon them with our reason. We see with our eyes and hear with our ears. All that our senses thus perceive,—we work upon it with our intellect in a more primitive way to begin with, without calculating, or drawing it geometrically, or analyzing the forms of movement. We have quite other categories of thought to go on when our intellect is thus at work on the phenomena seen by the senses. But if we now go further and begin applying to what goes on in the outer world the ideas of “scientific” arithmetic and algebra, geometry and kinematics, then we are doing far more—and something radically different. For we have certainly not gained these ideas from the outer world. We are applying ideas which we have spun out of our own inner life. Where then do these ideas come from? That is the cardinal question. Where do they come from? The truth is, these ideas come not from our intelligence—not from the intelligence which we apply when working up the ideas derived from sense-perception. They come in fact from the intelligent part of our Will. We make them with our Will-system—with the volitional part of our soul.

The difference is indeed immense between all the other ideas in which we live as intelligent beings and on the other hand the geometrical, arithmetical and kinematical ideas. The former we derive from our experience with the outer world; these on the other hand—the geometrical, the arithmetical ideas—rise up from the unconscious part of us, from the Will-part which has its outer organ in the metabolism. Our geometrical ideas above all spring from this realm; they come from the unconscious in the human being. And if you now apply these geometrical ideas (I will say “geometrical” henceforth to represent the arithmetical and algebraic too) to the phenomena of light or sound, then in your process of knowledge you are connecting, what arises from within you, with what you are perceiving from without. In doing so you remain utterly unconscious of the origin of the geometry you use. You unite it with the external phenomena, but you are quite unconscious of its source. So doing, you develop theories such as the wave-theory of light, or Newton's corpuscular theory,—it matters not which one it is. You develop theories by uniting what springs from the unconscious part of your being with what presents itself to you in conscious day-waking life. Yet the two things do not directly belong to one-another. They belong as little, my dear Friends, as the idea-forming faculty which you unfold when half-asleep belongs directly to the outer things which in your dreaming, half-asleep condition you perceive. In anthroposophical lectures I have often given instances of how the dream is wont to symbolize. An undergraduate dreams that at the door of the lecture-theatre he gets involved in a quarrel. The quarrel grows in violence; at last they challenge one-another to a duel. He goes on dreaming: the duel is arranged, they go out into the forest, he sees himself firing the shot,—and at the moment he wakes up. A chair has fallen over. This was the impact which projected itself forward into the dream. The idea-forming faculty has indeed somehow linked up with the outer phenomenon, but in a merely symbolizing way,—in no way consistent with the real object. So too, what in your geometrical and phoronomical thinking you fetch up from the subconscious part of your being, when you connect it with the phenomena of light. What you then do has no other value for reality than what finds expression in the dream when symbolizing an objective fact such as the fall and impact of the chair. All this elaboration of the outer world—optical, acoustic and even thermal to some extent (the phenomena of warmth)—by means of geometrical, arithmetical and kinematical thought-forms, is in point of fact a dreaming about Nature. Cool and sober as it may seem, it is a dream—a dreaming while awake. Moreover, until we recognize it for what it is, we shall not know where we are in our Natural Science, so that our Science gives us reality. What people fondly believe to be the most exact of Sciences, is modern mankind's dream of Nature.

But it is different when we go down from the phenomena of light and sound, via the phenomena of warmth, into the realm we are coming into with these rays and radiations, belonging as they do to the science of electricity. For we then come into connection with what in outer Nature is truly equivalent to the Will in Man. The realm of Will in Man is equivalent to this whole realm of action of the cathode rays, canal rays, Roentgen rays. $\alpha$-, $\beta$- and $\gamma$-rays and so on. It is from this very realm—which, once again, is in the human being the realm of Will,—it is from this that there arises what we possess in our mathematics, in our geometry, in our ideas of movement. These therefore are the realms, in Nature and in Man, which we may truly think of as akin to one-another. However, human thinking has in our time not yet gone far enough, really to think its way into these realms. Man of today can dream quite nicely, thinking out wave-theories and the like, but he is not yet able to enter with real mathematical perception into that realm of phenomena which is akin to the realm of human Will, in which geometry and arithmetic originate. For this, our arithmetical, algebraical and geometrical thinking must in themselves become more saturated with reality. It is along these lines that physical science should now seek to go.

Nowadays, if you converse with physicists who were brought up in the golden age of the old wave-theory, you will find many of them feeling a little uncanny about these new phenomena, in regard to which ordinary methods of calculation seem to break down in so many places. In recent times the physicists have had recourse to a new device. Plain-sailing arithmetical and geometrical methods proving inadequate, they now introduce a kind of statistical method. Taking their start more from the outer empirical data, they have developed numerical relations also empirical in kind. They then use the calculus of probabilities. Along these lines it is permissible to say: By all means let us calculate some law of Nature; it will hold good throughout a certain series, but then there comes a point where it no longer works.

There are indeed many things like this in modern Physics,—very significant moments where they lose hold of the thought, yet in the very act of losing it get more into reality. Conceivably for instance, starting from certain rigid ideas about the nature of a gas or air under the influence of warmth and in relation to its surroundings, a scientist of the past might have proved with mathematical certainty that air could not be liquefied. Yet air was liquefied, for at a certain point it emerged that the ideas which did indeed embrace the prevailing laws of a whole series of facts, ceased to hold good at the end of this series. Many examples might be cited. Reality today—especially in Physics—often compels the human being to admit this to himself: “You with your thinking, with your forming of ideas, no longer fully penetrate into reality; you must begin again from another angle.”

We must indeed; and to do this, my dear Friends, we must become aware of the kinship between all that comes from the human Will—whence come geometry and kinematics—and on the other hand what meets us outwardly in this domain that is somehow separated from us and only makes its presence known to us in the phenomena of the other pole. For in effect, all that goes on in these vacuum tubes makes itself known to us in phenomena of light, etc. Whatever is the electricity itself, flowing through there, is imperceptible in the last resort. Hence people say: If only we had a sixth sense—a sense for electricity—we should perceive it too, directly. That is of course wide of the mark. For it is only when you rise to Intuition, which has its ground in the Will, it is only then that you come into that region—even of the outer world—where electricity lives and moves. Moreover when you do so you perceive that in these latter phenomena you are in a way confronted by the very opposite than in the phenomena of sound or tone for instance. In sound or in musical tone, the very way man is placed into this world of sound and tone—as I explained in a former lecture—means that he enters into the sound or tone with his soul and only with his soul. What he then enters into with his body, is no more than what sucks-in the real essence of the sound or tone. I explained this some days ago; you will recall the analogy of the bell-jar from which the air has been pumped out. In sound or tone I am within what is most spiritual, while what the physicist observes (who of course cannot observe the spiritual nor the soul) is but the outer, so-called material concomitant, the movement of the wave. Not so in the phenomena of the realm we are now considering, my dear Friends. For as I enter into these, I have outside me not only the objective, so-called material element, but also what in the case of sound and tone is living in me—in the soul and spirit. The essence of the sound or tone is of course there in the outer world as well, but so am I. With these phenomena on the other hand, what in the case of sound could only be perceived in soul, is there in the same sphere in which—for sound—I should have no more than the material waves. I must now perceive physically, what in the case of sound or tone I can only perceive in the soul.

Thus in respect of the relation of man to the external world the perceptions of sound, and the perceptions of electrical phenomena for instance, are at the very opposite poles. When you perceive a sound you are dividing yourself as it were into a human duality. You swim in the elements of wave and undulation, the real existence of which can of course be demonstrated by quite external methods. Yet as you do so you become aware; herein is something far more than the mere material element. You are obliged to kindle your own inner life—your life of soul—to apprehend the tone itself. With your ordinary body—I draw it diagrammatically (the oval in Figure Xb)—you become aware of the undulations. You draw your ether—and astral body together, so that they occupy only a portion of your space. You then enjoy, what you are to experience of the sound or tone as such, in the thus inwarded and concentrated etheric-astral part of your being. It is quite different when you as human being meet the phenomena of this other domain, my dear Friends. In the first place there is no wave or undulation or anything like that for you to dive into; but you now feel impelled to expand what in the other case you concentrated (Figure Xc). In all directions, you drive your ether—and astral body out beyond your normal surface; you make them bigger, and in so doing you perceive these electrical phenomena.

Without including the soul and spirit of the human being, it will be quite impossible to gain a true or realistic conception of the phenomena of Physics. Ever-increasingly we shall be obliged to think in this way. The phenomena of sound and tone and light are akin to the conscious element of Thought and Ideation in ourselves, while those of electricity and magnetism are akin to the sub-conscious element of Will. Warmth is between the two. Even as Feeling is intermediate between Thought and Will, so is the outer warmth in Nature intermediate between light and sound on the one hand, electricity and magnetism on the other. Increasingly therefore, this must become the inner structure of our understanding of the phenomena of Nature. It can indeed become so if we follow up all that is latent in Goethe's Theory of Colour. We shall be studying the element of light and tone on the one hand, and of the very opposite of these—electricity and magnetism—on the other. As in the spiritual realm we differentiate between the Luciferic, that is akin to the quality of light, and the Ahrimanic, akin to electricity and magnetism, so also must we understand the structure of the phenomena of Nature. Between the two lies what we meet with in the phenomena of Warmth.

I have thus indicated a kind of pathway for this scientific realm,—a guiding line with which I wished provisionally to sum up the little that could be given in these few improvised hours. It had to be arranged so quickly that we have scarcely got beyond the good intentions we set before us. All I could give were a few hints and indications; I hope we shall soon be able to pursue them further. Yet, little as it is, I think what has been given may be of help to you—and notably to the Waldorf School teachers among you when imparting scientific notions to the children. You will of course not go about it in a fanatical way, for in such matters it is most essential to give the realities a chance to unfold. We must not get our children into difficulties. But this at least we can do: we can refrain from bringing into our teaching too many untenable ideas—ideas derived from the belief that the dream-picture which has been made of Nature represents actual reality. If you yourselves are imbued with the kind of scientific spirit with which these lectures—if we may take them as a fair example—have been pervaded, it will assuredly be of service to you in the whole way you speak with the children about natural phenomena.

Methodically too, you may derive some benefit. I am sorry it was necessary to go through the phenomena at such breakneck speed. Yet even so, you will have seen that there is a way of uniting what we see outwardly in our experiments with a true method of evoking thoughts and ideas, so that the human being does not merely stare at the phenomena but really thinks about them. If you arrange your lessons so as to get the children to think in connection with the experiments—discussing the experiments with them intelligently—you will develop a method, notably in the Science lessons, whereby these lessons will be very fruitful for the children who are entrusted to you.

Thus by the practical example of this course, I think I may have contributed to what was said in the educational lectures at the inception of the Waldorf School. I believe therefore that in arranging these scientific courses we shall also have done something for the good progress of our Waldorf School, which ought really to prosper after the good and very praiseworthy start which it has made. The School was meant as a beginning in a real work for the evolution of our humanity—a work that has its fount in new resources of the Spirit. This is the feeling we must have. So much is crumbling, of all that has developed hitherto in human evolution. Other and new developments must come in place of what is breaking down. This realization in our hearts and minds will give the consciousness we need for the Waldorf School. In Physics especially it becomes evident, how many of the prevailing ideas are in decay. More than one thinks, this is connected with the whole misery of our time. When people think sociologically, you quickly see where their thinking goes astray. Admittedly, here too most people fail to see it, but you can at least take notice of it; you know that sociological ways of thought will find their way into the social order of mankind. On the other hand, people fail to realize how deeply the ideas of Physics penetrate into the life of mankind. They do not know what havoc has in fact been wrought by the conceptions of modern Physics, terrible as these conceptions often are. In public lectures I have often quoted Hermann Grimm. Admittedly, he saw the scientific ideas of his time rather as one who looked upon them from outside. Yet he spoke not untruly when he said, future generations would find it difficult to understand that there was once a world so crazy as to explain the evolution of the Earth and Solar System by the theory of Kant and Laplace. To understand such scientific madness would not be easy for a future age, thought Hermann Grimm. Yet in our modern conceptions of inorganic Nature there are many features like the theory of Kant and Laplace. And you must realize how much is yet to do for the human beings of our time to get free of the ways of Kant and Konigsberg and all their kindred. How much will be to do in this respect, before they can advance to healthy, penetrating ways of thought!

Strange things one witnesses indeed from time to time, shewing how what is wrong on one side joins up with what is wrong on another. What of a thing like this? Some days ago—as one would say, by chance—I was presented with a reprint of a lecture by a German University professor. (He prides himself in this very lecture that there is in him something of Kant and Konigsberg!) It was a lecture in a Baltic University, on the relation of Physics and Technics, held on the 1st of May 1918,—please mark the date! This learned physicist of our time in peroration voices his ideal, saying in effect: The War has clearly shewn that we have not yet made the bond between Militarism and the scientific laboratory work of our Universities nearly close enough. For human progress to go on in the proper way, a far closer link must in future be forged between the military authorities and what is being done at our Universities. Questions of mobilization in future must include all that Science can contribute, to make the mobilization still more effective. At the beginning of the War we suffered greatly because the link was not yet close enough—the link which we must have in future, leading directly from the scientific places of research into the General Staffs of our armies.

Mankind, my dear Friends, must learn anew, and that in many fields. Once human beings make up their minds to learn anew in such a realm as Physics, they will be better prepared to learn anew in other fields as well. Those physicists who go on thinking in the old way, will never be so very far removed from the delightful coalition between the scientific laboratories and the General Staffs. How many things will have to alter! So may the Waldorf School be and remain a place where the new things which mankind needs can spring to life. In the expression of this hope, I will conclude our studies for the moment.

Rudolf Steiner, in all that he created and gave to the world, took his start from real needs,—never from theoretical programmes. Time and again, what he gave took its inception from the spiritual questions and interests of individuals or groups among his friends and pupils. Yet as the faculty to apprehend the spiritual aspect of the World first had to be rekindled and awakened in our time—a slow and gradual process—it must have signified a very great sacrifice and a severe hindrance for this universal spirit to bring the spiritual truths from infinite horizons into the narrower range of outlook of his contemporaries. This sacrifice he did not shun. Even into the anxiously constraining walls of earth 20th-century scientific thinking he brought the light of spiritual knowledge, and we who have received this cannot find adequate words in which to thank him. Our truest thanks must be the will to widen out our own horizon, thus making easier the teacher's task.

The Anthroposophical Movement within this 20th century is seeking to bring about a return from materialism to a spiritual understanding of the World. It is a good thing for mankind that in this Movement some individualities have also chosen the very hardest task, namely to lead again to spiritual sources that realm of human knowledge which has plunged most deeply into agnostic materialism—Natural Science. Future generations will surely be very grateful to the scientists—teachers of the Waldorf School at Stuttgart above all—who had the inner courage to put their questions to the great spiritual teacher.

We take this opportunity to thank those who have hitherto administered this spiritual treasure—who first revised and duplicated the notes of the lectures, thereby preserving them for posterity. We refer especially to the Waldorf School teachers E. A. K. Stockmeyer, Alexander Strakosch, and above all Dr. Eugen Kolisko and Dr. Walter Johannes Stein. My thanks are also due to Ehrenfried Pfeiffer of Dornach for his assistance in preparing the present edition.¹

It will be well for us to refer at this point to the following passages from Rudolf Steiner's Autobiography:—

“The anthroposophical period of my life-work began at a time when many people were feeling dissatisfied with the ways of knowledge of the immediate past. They looked for ways to get beyond that realm of existence to which the scientific era was restricted inasmuch as nothing was held valid as “secure knowledge” unless it could be grasped in mechanistic forms of thought. The strivings of many of our contemporaries towards some form of spiritual knowledge touched me deeply. There were biologists for instance such as Oskar Hertwig. Having begun his career as a disciple of Haeckel, he afterwards took leave of Darwinism, for he now felt that the driving forces recognized by the Darwinian school were inadequate to explain the facts of organic evolution. The longing of our time for knowledge seemed to me to find expression in such men as these. And yet it seemed to me this longing was oppressed by a heavy load—a burden due to the belief that only those things which we can investigate by means of the outer senses and then express in terms of measure, number and weight, constitute genuine knowledge. Men did not venture to unfold that inner activity of thought by means of which reality is experienced more intimately than by the physical senses. The most they did was to declare that with the kind of scientific explanation hitherto applied also to higher forms of reality—those of organic life for example—no fundamental progress was possible. If a more positive contribution was looked for,—if they were now to say what it is that works in the realm of life—they could bring forward only the vaguest notions.

“Those who were striving to transcend the mechanical explanation of the World generally lacked the courage to admit that if we want to overcome the mechanistic system we must also overcome the habits of thought which have led to it. The time was calling, yet called in vain, for a clear recognition of this kind. The orientation of our faculties of knowledge towards the outer senses enables us to penetrate what is mechanical in Nature. This mental tendency has become habitual throughout the second half of the 19th century. If the mechanical aspect of the world no longer satisfied us now, we ought not to expect to reach into higher regions in the identical frame of mind. The outer senses develop and awaken in the human being, so to speak, of their own accord; but on this basis we can only gain insight into the mechanical domain. If we desire to know more than this, we must by dint of our own efforts give to our deeper, latent faculties of knowledge the same development which Nature gives the powers of the senses. The faculties with which we apprehend what is mechanical are awake of their own accord; those that apply to higher realms of reality first need to be awakened.

“The time required, so it seemed to me, that in our striving after knowledge we should arrive at this clear recognition of our state, and I was happy when I saw any beginnings or indications that seemed to tend in this direction. . . .”

“There now exists a twofold outcome of the anthroposophical period of my life-work. There are my published books upon the one hand, while on the other hand there are a larger number of lecture-courses, printed at first for private circulation and available, to begin with, only to members of the Anthroposophical Society. These printed versions of my lectures are reports, more or less accurately made, which I was quite unable to correct for want of time. It would have pleased me best to let the spoken word remain as spoken word; but members wanted the lectures made available in printed form; so it was done. . . .

“To gain a picture of my own inner work, my unceasing effort to present the spiritual science of Anthroposophia to the prevailing consciousness of our time, one must have recourse to my published writings. In these I tried to come to terms with the modern striving after knowledge in its many aspects. Here I set forth, what in the realm of spiritual perception grew for me ever more fully and clearly into the edifice of “Anthroposophia”, admittedly imperfect as it still is in many ways. Herein I saw my essential task, in the fulfilment of which I only had to bear in mind what is required when communications from the spiritual world are imparted to the prevailing culture of our time. Yet side by side with this requirement I had to do full justice to another one, namely to meet the inner needs and spiritual longings that became manifest among the members of the Society.

“To this end the many lecture-courses were given in the Society; and this involved another circumstance. The lectures were attended by members only. Acquainted as they were already with the initial teachings of Anthroposophia, one could speak to them as to more advanced students. Thus the whole tenor of these member's lectures came to be different from what was possible in written books intended for the world at large. In these more intimate circles I might speak of many things in a form which I should certainly have had to change had I intended it for publication from the outset.

“Competent judgment on the content of these privately printed lectures will of course only be possible for those who are acquainted with the premisses of thought, taken for granted in those who heard them. For the great majority of these reprints, this implies at the very least some knowledge of the anthroposophical science of Man and of the essence of the great Universe as described in Anthroposophia; also a knowledge of ‘anthroposophical History’, for this too is an essential part of the communications from the spiritual world.”

Whoever reads the lectures here reproduced should bear the foregoing words in mind. If those who work with this lecture-course approach it with the will “to awaken in themselves the faculties of knowledge for higher forms of reality”, the time will surely come when the dead mechanistic picture of the world which the last century produced will be transcended—transcended above all by the most up-to-day, the most gifted and conscientious of our scientists, who will then see through the inherent impossibility and untruth of this world-picture. Then will the far more living and spiritual form of Science which Rudolf Steiner had in mind reveal its truth and beauty, also its ethical inspiring power. The Section calls to all its fellow-workers: Help the Goetheanum bring about the beginning of this new epoch even within the present century. For generations due to come at the end of the 20th century, let there be in existence a Science of Nature permeated with the living Spirit, permeated with the Christ-Impulse!

For the Natural Science Section at the Goetheanum
Guenther Wachsmuth.
Dornach, January 1925.

• 1. TRANSLATOR: The English version (1948) has been based on this edition, and on a number of corrections and improvements, issued a few years ago by Paul Eugen Schiller of Dornach.

At the beginning of this scientific lecture-course, Dr. Walter Johannes Stein read out the following quotations:—

From Goethe's Scientific Works, Kuerschner's Edition, Vol. 3 (1890), page XVII of the Introduction by Rudolf Steiner:—

“Needless to say I am not wanting to defend Goethe's Theory of Colour in every detail. It is the underlying principle which I would like to see maintained. Nor could it here be my task to derive from this principle the phenomena of colour which were not yet known in Goethe's time. I could only do justice to such a task if I were fortunate enough one day to have the leisure and the means to write a scientifically up-to-date Theory of Colour in Goethe's spirit.”

Fr. Vol. 1 of the same Edition (1883), page LXXXIV of the Introduction by Rudolf Steiner:

“May scientists and thinkers young in mind and in ideal—those above all who seek not only to extend the range of information but who look deep into the central issues of our life of knowledge—pay heed to what I have set forth and follow in large numbers, so to work out more fully and more perfectly what I have here attempted.”

From The Spiritual Guidance of Man and of Mankind by Rudolf Steiner (1911):

“In time to come there will be physicists and chemists whose teaching will not be such as now prevails under the influence of the Egypto-Chaldean Spirits that have remained behind, but who will teach that Matter is built up in the way in which the Christ has gradually ordained it. Even into the laws of Chemistry and Physics the Christ will be found. Thus will a spiritual form of Chemistry and Physics come to pass in future”.

My dear friends,

The present course of lectures will constitute a kind of continuation of the one given when I was last here. I will begin with those chapters of physics which are of especial importance for laying a satisfactory foundation for a scientific world view, namely the observations of heat relations in the world. Today I will try to lay out for you a kind of introduction to show the extent to which we can create a body of meaningful views of a physical sort within a general world view. This will show further how a foundation may be secured for a pedagogical impulse applicable to the teaching of science. Today we will therefore go as far as we can towards outlining a general introduction.

The theory of heat, so-called, has taken a form during the 19th century which has given a great deal of support to a materialistic view of the world. It has done so because in heat relationships it is very easy to turn one's glance away from the real nature of heat, from its being, and to direct it to the mechanical phenomena arising from heat.

Heat is first known through sensations of cold, warmth, lukewarm, etc. But man soon learns that there appears to be something vague about these sensations, something subjective. A simple experiment which can be made by anyone shows this fact.

Imagine you have a vessel filled with water of a definite temperature, $t$; on the right of it you have another vessel filled with water of a temperature $t-t_1$, that is of a temperature distinctly lower than the temperature in the first vessel. In addition, you have a vessel filled with water at a temperature $t+t_1$. When now, you hold your fingers in the two outer vessels you will note by your sensations the heat conditions in these vessels. You can then plunge your fingers which have been in the outer vessels into the central vessel and you will see that to the finger which has been in the cold water the water in the central vessel will feel warm, while to the finger which has been in the warm water, the water in the central vessel will feel cold. The same temperature therefore is experienced differently according to the temperature to which one has previously been exposed. Everyone knows that when he goes into a cellar, it may feel different in winter from the way it feels in summer. Even though the thermometer stands at the same point circumstances may be such that the cellar feels warm in the winter and cool in the summer. Indeed, the subjective experience of heat is not uniform and it is necessary to set an objective standard by which to measure the heat condition of any object or location. Now, I need not here go into the elementary phenomena or take up the elementary instruments for measuring heat. It must be assumed that you are acquainted with them. I will simply say that when the temperature condition is measured with a thermometer, there is a feeling that since we measure the degree above or below zero, we are getting an objective temperature measurement. In our thinking we consider that there is a fundamental difference between this objective determination in which we have no part and the subjective determination, where our own organization enters into the experience.

For all that the 19th century has striven to attain it may be said that this view on the matter was, from a certain point of view, fruitful and justified by its results. Now, however, we are in a time when people must pay attention to certain other things if they are to advance their way of thinking and their way of life. From science itself must come certain questions simply overlooked in such conclusions as those I have given. One question is this: Is there a difference, a real objective difference, between the determination of temperature by my organism and by a thermometer, or do I deceive myself for the sake of getting useful practical results when I bring such a difference into my ideas and concepts? This whole course will be designed to show why today such questions must be asked. From the principal questions it will be my object to proceed to those important considerations which have been overlooked owing to exclusive attention to the practical life. How they have been lost for us on account of the attention to technology you will see. I would like to impress you with the fact that we have completely lost our feeling for the real being of heat under the influence of certain ideas to be described presently. And, along with this loss, has gone the possibility of bringing this being of heat into relation with the human organism itself, a relation which must be all means be established in certain aspects of our life. To indicate to you in a merely preliminary way the bearing of these things on the human organism, I may call your attention to the fact that in many cases we are obliged today to measure the temperature of this organism, as for instance, when it is in a feverish condition. This will show you that the relation of the unknown being of heat to the human organism has considerable importance. Those extreme conditions as met with in chemical and technical processes will be dealt with subsequently. A proper attitude toward the relation of the unknown being of heat to the human organism has considerable importance. Those extreme conditions as met with in chemical and technical processes will be dealt with subsequently. A proper attitude toward the relation of the heat-being to the human organism cannot, however, be attained on the basis of a mechanical view of heat. The reason is, that in so doing, one neglects the fact that the various organs are quite different in their sensitiveness to this heat-being, that the heart, the liver, the lungs differ greatly in their capacity to react to the being of heat. Through the purely physical view of heat no foundation is laid for the real study of certain symptoms of disease, since the varying capacity to react to heat of the several organs of the body escapes attention. Today we are in no position to apply to the organic world the physical views built up in the course of the 19th century on the nature of heat. This is obvious to anyone who has an eye to see the harm done by modern physical research, so-called, in dealing with what might be designated the higher branches of knowledge of the living being. Certain questions must be asked, questions that call above everything for clear, lucid ideas. In the so-called “exact science,” nothing has done more harm than the introduction of confused ideas.

What then does it really mean when I say, if I put my fingers in the right and left hand vessels and then into a vessel with a liquid of an intermediate temperature, I get different sensations? Is there really something in the conceptual realm that is different from the so-called objective determination with the thermometer? Consider now, suppose you put thermometers in these two vessels in place of your fingers. You will then get different readings depending on whether you observe the thermometer in the one vessel or the other. If then you place the two thermometers instead of your fingers into the middle vessel, the mercury will act differently on the two. In the one it will rise; in the other it will fall. You see the thermometer does not behave differently from your sensations. For the setting up of a view of the phenomenon, there is no distinction between the two thermometers and the sensation from your finger. In both cases exactly the same thing occurs, namely a difference is shown from the immediately preceding conditions. And the thing our sensation depends on is that we do not within ourselves have any zero or reference point. If we had such a reference point then we would establish not merely the immediate sensation but would have apparatus to relate the temperature subjectively perceived, to such a reference point. We would then attach to the phenomenon just as we do with the thermometers something which really is not inherent in it, namely the variation from the reference point. You see, for the construction of our concept of the process there is no difference.

It is such questions as these that must be raised today if we are to clarify our ideas, or all the present ideas on these things are really confused. Do not imagine for a moment that this is of no consequence. Our whole life process is bound up with this fact that we have in us no temperature reference point. If we could establish such a reference point within ourselves, it would necessitate an entirely different state of consciousness, a different soul life. It is precisely because the reference point is hidden for us that we lead the kind of life we do.

You see, many things in life, in human life and in the animal organism, too, depend on the fact that we do not perceive certain processes. Think what you would have to do if you were obliged to experience subjectively everything that goes on in your organism. Suppose you had to be aware of all the details of the digestive process. A great deal pertaining to our condition of life rests on this fact that we do not bring into our consciousness certain things that take place in our organism. Among these things is that we do not carry within us a temperature reference point—we are not thermometers. A subjective-objective distinction such as is usually made is not therefore adequate for a comprehensive grasp of the physical.

It is this which has been the uncertain point in human thinking since the time of ancient Greeks. It had to be so, but it cannot remain so in the future. For the old Grecian philosophers, Zeno in particular, had already orientated human thinking about certain processes in a manner strikingly opposed to outer reality. I must call your attention to these things even at the risk of seeming pedantic. Let me recall to you the problem of Achilles and the tortoise, a problem I have often spoken about.

Let us assume we have the distance traveled by Achilles in a certain time $a$. This represents the rate at which he can travel. And here we have the tortoise $s$, who has a start on Achilles. Let us take the moment when Achilles gets to the point marked $1$. The tortoise is ahead of him. Since the problem stated that Achilles has to cover every point covered by the tortoise, the tortoise will always be a little ahead and Achilles can never catch up. But, the way people would consider it is this. You would say, yes, I understand the problem all right, but Achilles would soon catch the tortoise. The whole thing is absurd. But if we reason that Achilles must cover the same path as the tortoise and the tortoise is ahead, he will never catch the tortoise. Although people would say this is absurd, nevertheless the conclusion is absolutely necessary and nothing can be urged against it. It is not foolish to come to this conclusion but on the other hand, it is remarkably clever considering only the logic of the matter. It is a necessary conclusion and cannot be avoided. Now what does all this depend on? It depends on this: that as long as you think, you cannot think otherwise than the premise requires. As a matter of fact, you do not depend on thinking strictly, but instead you look at the reality and you realize that it is obvious that Achilles will soon catch the tortoise. And in doing this you uproot thinking by means of reality and abandon the pure thought process. There is no point in admitting the premises and then saying, “Anyone who thinks this way is stupid.” Through thinking alone we can get nothing out of the proposition but that Achilles will never catch the tortoise. And why not? Because when we apply our thinking absolutely to reality, then our conclusions are not in accord with the facts. They cannot be. When we turn our rationalistic thought on reality it does not help us at all that we establish so-called truths which turn out not to be true. For we must conclude if Achilles follows the tortoise that he passes through each point that the tortoise passes through. Ideally this is so; in reality he does nothing of the kind. His stride is greater than that of the tortoise. He does not pass through each point of the path of the tortoise. We must, therefore, consider what Achilles really does, and not simply limit ourselves to mere thinking. Then we come to a different result. People do not bother their heads about these things but in reality they are extraordinarily important. Today especially, in our present scientific development, they are extremely important. For only when we understand that much of our thinking misses the phenomena of nature if we go from observation to so-called explanation, only in this case will we get the proper attitude toward these things.

The observable, however, is something which only needs to be described. That I can do the following for instance, calls simply for a description: here I have a ball which will pass through this opening. We will now warm the ball slightly. Now you see it does not go through. It will only go through when it has cooled sufficiently. As soon as I cool it by pouring this cold water on it, the ball goes through again. This is the observation, and it is this observation that I need only describe. Let us suppose, however, that I begin to theorize. I will do so in a sketchy way with the object merely of introducing the matter. Here is the ball; it consists of a certain number of small parts—molecules, atoms, if you like. This is not observation, but something added to observation in theory. At this moment, I have left the observed and in doing so I assume an extremely tragic role. Only those who are in a position to have insight into these things can realize this tragedy. For you see, if you investigate whether Achilles can catch the tortoise, you may indeed begin by thinking “Achilles must pass over every point covered by the tortoise and can never catch it.” This may be strictly demonstrated. Then you can make an experiment. You place the tortoise ahead and Achilles or some other who does not run even so fast as Achilles, in the rear. And at any time you can show that observation furnishes the opposite of what you conclude from reasoning. The tortoise is soon caught.

When, however, you theorize about the sphere, as to how its atoms and molecules are arranged, and when you abandon the possibility of observation, you cannot in such a case look into the matter and investigate it—you can only theorize. And in this realm you will do no better than you did when you applied your thinking to the course of Achilles. That is to say, you carry the whole incompleteness of your logic into your thinking about something which cannot be made the object of observation. This is the tragedy. We build explanation upon explanation while at the same time we abandon observation, and think we have explained things simply because we have erected hypotheses and theories. And the consequence of this course of forced reliance on our mere thinking is that this same thinking fails us the moment we are able to observe. It no longer agrees with the observation.

You will remember I already pointed out this distinction in the previous course when I indicated the boundary between kinematics and mechanics. Kinematics describes mere motion phenomena or phenomena as expressed by equations, but it is restricted to verifying the data of observation.

The moment we pass over from kinematics to mechanics where force and mass concepts are brought in, at this moment, we cannot rely on thinking alone, but we begin simply to read off what is given from observation of the phenomena. With unaided thought we are not able to deal adequately even with the simplest physical process where mass plays a role. All the 19th century theories, abandoned now to a greater or lesser extent, are of such a nature that in order to verify them it would be necessary to make experiments with atoms and molecules. The fact that they have been shown to have a practical application in limited fields makes no difference. The principle applies to the small as well as to the large. You remember how I have often in my lectures called attention to something which enters into our considerations now wearing a scientific aspect. I have often said: From what the physicists have theorized about heat relations and from related things they get certain notions about the sun. They describe what they call the “physical conditions” on the sun and make certain claims that the facts support the description. Now I have often told you, the physicists would be tremendously surprised if they could really take a trip to the sun and could see that none of their theorizing based on terrestrial conditions agreed with the realities as found on the sun. These things have a very practical value at the present, a value for the development of science in our time. Just recently news has gone forth to the world that after infinite pains the findings of certain English investigators in regard to the bending of starlight in cosmic space have been confirmed and could now be presented before a learned society in Berlin. It was rightly stated there “the investigations of Einstein and others on the theory of relativity have received a certain amount of confirmation. But final confirmation could be secured only when sufficient progress had been made to make spectrum analysis showing the behavior of the light at the time of an eclipse of the sun. Then it would be possible to see what the instruments available at present failed to determine.” This was the information given at the last meeting of the Berlin Physical Society. It is remarkably interesting. Naturally the next step is to seek a way really to investigate the light of the sun by spectrum analysis. The method is to be by means of instruments not available today. Then certain things already deduced from modern scientific ideas may simply be confirmed. As you know it is thus with many things which have come along from time to time and been later clarified by physical experiments. But, people will learn to recognize the fact that it is simply impossible for men to carry over to conditions on the sun or to the cosmic spaces what may be calculated from those heat phenomena available to observation in the terrestrial sphere. It will be understood that the sun's corona and similar phenomena have antecedents not included in the observations made under terrestrial conditions. Just as our speculations lead us astray when we abandon observation and theorize our way through a world of atoms and molecules, so we fall into error when we go out into the macrocosm and carry over to the sun what we have determined from observations under earth conditions. Such a method has led to the belief that the sun is a kind of glowing gas ball, but the sun is not a glowing ball of gas by any means. Consider a moment, you have matter here on the earth. All matter on the earth has a certain degree of intensity in its action. This may be measured in one way or another, be density or the like, in any way you wish, it has a definite intensity of action. This may become zero. In other words, we may have empty space. But the end is not yet. That empty space is not the ultimate condition I may illustrate to you by the following: Assume to yourselves that you had a boy and that you said, “He is a rattle-brained fellow. I have made over a small property to him but he has begun to squander it. He cannot have less than zero. He may finally have nothing, but I comfort myself with the thought that he cannot go any further once he gets to zero!” But you may now have a disillusionment. The fellow begins to get into debt. Then he does not stop at zero; the thing gets worse than zero. It has a very real meaning. As his father, you really have less if he gets into debt than if he stopped when he had nothing.

The same sort of thing, now, applies to the condition on the sun. It is not usually considered as empty space but the greatest possible rarefaction is thought of and a rarefied glowing gas is postulated. But what we must do is to go to a condition of emptiness and then go beyond this. It is in a condition of negative material intensity. In the spot where the sun is will be found a hole in space. There is less there than empty space. Therefore all the effects to be observed in the sun must be considered as attractive forces not as pressures of the like. The sun's corona, for instance, must not be thought of as it is considered by the modern physicist. It must be considered in such a way that we have the consciousness not of forces radiating outward as appearances would indicate, but of attractive force from the hole in space, from the negation of matter. Here our logic fails us. Our thinking is not valid here, for the receptive organ or the sense organ through which we perceive it is our entire body. Our whole body corresponds in this sensation to the eye in the case of light. There is no isolated organ, we respond with our whole body to the heat conditions. The fact that we may use our finger to perceive a heat condition, for instance, does not militate against this fact. The finger corresponds to a portion of the eye. While the eye therefore is an isolated organ and functions as such to objectify the world of light as color, this is not the case for heat. We are heat organs in our entirety. On this account, however, the external condition that gives rise to heat does not come to us in so isolated a form as does the condition which gives rise to light. Our eye is objectified within our organism. We cannot perceive heat in an analogous manner to light because we are one with the heat. Imagine that you could not see colors with your eye but only different degrees of brightness, and that the colors as such remained entirely subjective, were only feelings. You would never see colors; you would speak of light and dark, but the colors would evoke in you no response and it is thus with the perception of heat. Those differences which you perceive in the case of light on account of the fact that your eye is an isolated organ, such differences you do not perceive at all in the case of heat. They live in you. Thus when you speak of blue and red, these colors are considered as objective. When the analogous phenomenon is met in the case of heat, that which corresponds to the blue and the red is within you. It is you yourself. Therefore you do not define it. This requires us to adopt an entirely different method for the observation of the objective being of heat from the method we use of the objective being of light. Nothing had so great a misleading effect on the observers of the 19th century as this general tendency to unify things schematically. You find everywhere in physiologies a “sense physiology.” Just as though there were such a thing! As though there were something of which it could be said, in general, “it holds for the ear as for the eye, or even for the sense of feeling or for the sense of heat. It is an absurdity to speak of a sense physiology and to say that a sense perception is this or that. It is possible only to speak of the perception of the eye by itself, or the perception of the ear by itself and likewise of our entire organism as heat sense organ, etc. They are very different things. Only meaningless abstractions result from a general consideration of the senses. But you find everywhere the tendency towards such a generalizing of these things. Conclusions result that would be humorous were they not so harmful to our whole life. If someone says—Here is a boy, another boy has given him a thrashing. Also then it is asserted—Yesterday he was whipped by his teacher; his teacher gave him a thrashing. In both cases there is a thrashing given; there is no difference. Am I to conclude from this that the bad boy who dealt out today's whipping and the teacher who administered yesterday's are moved by the same inner motives? That would be an absurdity; it would be impossible. But now, the following experiment is carried out: it is known that when light rays are allowed to fall on a concave mirror, under proper conditions they become parallel. When these are picked up by another concave mirror distant form the first they are concentrated and focused so that an intensified light appears at the focus. The same experiment is made with so-called heat rays. Again it may be demonstrated that these too can be focused—a thermometer will show it—and there is a point of high heat intensity produced. Here we have the same process as in the case of the light; therefore heat and light are fundamentally the same sort of thing. The thrashing of yesterday and the one of today are the same sort of thing. If a person came to such a conclusion in practical life, he would be considered a fool. In science, however, as it is pursued today, he is no fool, but a highly respected individual.

It is on account of things like this that we should strive for clear and lucid concepts, and without these we will not progress. Without them physics cannot contribute to a general world view. In the realm of physics especially it is necessary to attain to these obvious ideas.

You know quite well from what was made clear to you, at least to a certain extent, in my last course, that in the case of the phenomena of light, Goethe brought some degree of order into the physics of that particular class of facts, but no recognition has been given to him.

In the field of heat the difficulties that confront us are especially great. This is because in the time since Goethe the whole physical consideration of heat has been plunged into a chaos of theoretical considerations. In the 19th century the mechanical theory of heat as it is called has resulted in error upon error. It has applied concepts verifiable only by observation to a realm not accessible to observation. Everyone who believes himself able to think, but who in reality may not be able to do so, can propose theories. Such a one is the following: a gas enclosed in a vessel consists of particles. These particles are not at rest but in a state of continuous motion. Since these particles are in continuous motion and are small and conceived of as separated by relatively great distance, they do not collide with each other often but only occasionally. When they do so they rebound. Their motion is changed by this mutual bombardment. Now when one sums up all the various slight impacts there comes about a pressure on the wall of the vessel and through this pressure one can measure how great the temperature is. It is then asserted, “the gas particles in the vessel are in a certain state of motion, bombarding each other. The whole mass is in rapid motion, the particles bombarding each other and striking the wall. This gives rise to heat.” They may move faster and faster, strike the wall harder. Then it may be asked, what is heat? It is motion of these small particles. It is quite certain that under the influence of the facts such ideas have been fruitful, but only superficially. The entire method of thinking rests on one foundation. A great deal of pride is taken in this so-called “mechanical theory of heat,” for it seems to explain many things. For instance, it explains how when I rub my finger over a surface the effort I put forth, the pressure or work, is transformed into heat. I can turn heat back into work, in the steam engine for instance, where I secure motion by means of heat. A very convenient working concept has been built up along these lines. It is said that when we observe these things objectively going on in space, they are mechanical processes. The locomotive and the cars all move forward etc. When now, through some sort of work, I produce heat, what has really happened is that the outer observable motion has been transformed into motion of the ultimate particles. This is a convenient theory. It can be said that everything in the world is dependent on motion and we have merely transformation of observable motion into motion not observable. This latter we perceive as heat. But heat is in reality nothing but the impact and collision of the little gas particles striking each other and the walls of the vessel. The change into heat is as though the people in this whole audience suddenly began to move and collided with each other and with the walls etc. This is the Clausius theory of what goes on in a gas-filled space. This is the theory that has resulted from applying the method of the Achilles proposition to something not accessible to observation. It is not noticed that the same impossible grounds are taken as in the reasoning about Achilles and the tortoise. It is simply not as it is thought to be. Within a gas-filled space things are quite otherwise than we imagine them to be when we carry over the observable into the realm of the unobservable. My purpose today is to present this idea to you in an introductory way. From this consideration you can see that the fundamental method of thinking originated during the 19th century, begins to fail. For a large part of the method rests on the principle of calculating from observed facts by means of the differential concept. When the observed conditions in a gas-filled space are set down as differentials in accordance with the idea that we are dealing with the movements of ultimate particles, then the belief follows that by integrating something real is evolved. What must be understood is this: when we go from ordinary reckoning methods to differential equations, it is not possible to integrate forthwith without losing all contact with reality. This false notion of the relation of the integral to the differential has led the physics of the 19th century into wrong ideas of reality. It must be made clear that in certain instances one can set up differentials but what is obtained as a differential cannot be thought of as integrable without leading us into the realm of the ideal as opposed to the real. The understanding of this is of great importance in our relation to nature.

For you see, when I carry out a certain transformation period, I say that work is performed, heat produced and from this heat, work can again be secured by reversal of this process. But the processes of the organic cannot be reversed immediately. I will subsequently show the extent to which this reversal applies to the inorganic in the realm of heat in particular. There are also great inorganic processes that are not reversible, such as the plant processes. We cannot imagine a reversal of the process that goes on in the plant from the formation of roots, through the flower and fruit formation. The process takes its course from the seed to the setting of the fruit. It cannot be turned backwards like an inorganic process. This fact does not enter into our calculations. Even when we remain in the inorganic, there are certain macrocosmic processes for which our reckoning is not valid. Suppose you were able to set down a formula for the growth of a plant. It would be very complicated, but assume that you have such a formula. Certain terms in it could never be made negative because to do so would be to disagree with reality. In the face of the great phenomena of the world I cannot reverse reality. This does not apply, however, to reckoning. If I have today an eclipse of the moon I can simply calculate how in time past in the period of Thales, for instance, there was an eclipse of the moon. That is, in calculation only I can reverse the process, but in reality the process is not reversible. We cannot pass from the present state of the earth to former states—to an eclipse of the moon at the time of Thales, for instance, simply by reversing the process in calculation. A calculation may be made forward or backward, but usually reality does not agree with the calculation. The latter passes over reality. It must be defined to what extent our concepts and calculations are only conceptual in their content. In spite of the fact that they are reversible, there are no reversible processes in reality. This is important since we will see that the whole theory of heat is built on questions of the following sort: to what extent within nature are heat processes reversible and to what extent are they irreversible?

My dear friends,

Yesterday I touched upon the fact that bodies under the influence of heat expand. Today we will first consider how bodies, the solid bodies as we call them, expand when acted upon by the being of warmth. In order to impress these things upon our minds so that we can use them properly in pedagogy—and at this stage the matter is quite simple and elementary—we have set up this apparatus with an iron bar. We will heat the iron bar and make its expansion visible by noting the movements of this lever-arm over a scale. When I press here with my finger, the pointer moves upwards. (see drawing.)

You can see when we heat the rod, the pointer does move upwards which indicates for you the act that the rod expands. The pointer moves upwards at once. Also you notice that with continued heating the pointer moves more and more, showing that the expansion increases with the temperature. If instead of this rod I had another consisting of a different metal, and if we measured precisely the amount of the expansion, it would be found other than it is here. We would find that different substances expanded various amounts. Thus we would be able to establish at once that the expansion, the degree of elongation, depended on the substance. At this point we will leave out of account the fact that we are dealing with a cylinder and assume that we have a body of a certain length without breadth or thickness and turn our attention to the expansion in one direction only. To make the matter clear we may consider it as follows: here is a rod, considered simply $L_o$ the length of the rod at the original temperature, the starting temperature. The length attained by the rod when it is heated to a temperature $y$, we will indicate by $L$. Now I said that the rod expanded to various degrees depending upon the substance of which it is composed. We can express the amount of expansion to the original length of the rod. Let us denote this relative expansion by $\alpha$. Then we know the length of the rod after expansion. For the length $L$ after expansion may be considered as made up of the original length $L_o$ and the small addition to this length contributed by the expansion. This must be added on. Since I have denoted by $\alpha$ the fraction giving the ratio of the expansion and the original length, I get the expansion for a given substance by multiplying $L_o$ by $\alpha$. Also since the expansion is greater the higher the temperature, I have to multiply by the temperature $t$. Thus I can say the length of the rod after expansion is $L_o+L_o\alpha t$, which may be written $L_o(1+\alpha t)$. Stated in words: if I wish to determine the length of a rod expanded by heat, I must multiply the original length by a factor consisting of $1$ plus the temperature times the relative expansion of the substance under consideration. Physicists have called $\alpha$ the expansion coefficient of the substance considered. Now I have considered here a rod. Rods without breadth and thickness do not exist in reality. In reality bodies have three dimensions. If we proceed from the longitudinal expansion to the expansion of an assumed surface, the formula may be changed as follows: let us assume now that we are to observe the expansion of a surface instead of simply an expansion in one dimension. There is a surface. This surface extends in two directions, and after warming both will have increased in extent. We have therefore not only the longitudinal expansion to $L$ but also an increase in the breadth to $b$ to consider. Taking first the original length, $L_o$, we have as before the expansion in this direction to $L$ or

Considering now the breadth $b_o$ which expands to $b$, I must write down: $$b=b_o(1+\alpha t)$$

(It is obvious that the same rule will hold here as in the case of the length.) Now you know that the area of the surface is obtained by multiplying the length by the breadth. The original area I get by multiplying $b_o$ and $L_o$, and after expansion by multiplying $L_o(1+\alpha t)$ and $b_o(1+\alpha t)$

or

or

This gives the formula for the expansion of the surface. If now, you imagine thickness added to the surface, this thickness must be treated in the same manner and I can then write:

When you look at this formula I will ask you please to note the following: in the first two terms of you see $t$ raised no higher than the first power; in the third term you see the second, and in the fourth term it is raised to the third power. Note especially these last two terms of the formula for expansion. Observe that when we deal with the expansion of a three-dimensional body we obtain a formula containing the third power of the temperature. It is extremely important to keep in mind this fact that we come here upon the third power of the temperature.

Now I must always remember that we are here in the Waldorf School and everything must be presented in its relation to pedagogy. Therefore I will call your attention to the fact that the same introduction I have made here is presented very differently if you study it in the ordinary textbooks of physics. I will not well you how it is presented in the average textbook of physics. It would be said: $\alpha$ is a ratio. It is a fraction. The expansion is relatively very small as compared to the original length of the rod. When I have a fraction whose denominator is greater than its numerator, then when I square or cube it, I get a much smaller fraction. For if I square a third, I get a ninth and when I cube a third I get a twenty-seventh. That is, the third power is a very, very small fraction. $\alpha$ is a fraction whose denominator is usually very large. Therefore say most physics books: if I square $\alpha$ to get ${\alpha }^2$ or cube it to get ${\alpha }^3$ with which I multiply $t^3$ these are very small fractions and can simply be dropped out. The average physics text says: we simply drop these last terms of the expansion formula and write $l\cdot b\cdot d$—this is the volume and I will write is as $V$—the volume of an expanded body heated to a certain temperature is:

In this fashion is expressed the formula for the expansion of a solid body. It is simply considered that since the fraction $\alpha$ squared and cubed give such small quantities, these can be dropped out. You recognize this as the treatment in the physics texts. Now my friends, in doing this, the most important thing for a really informative theory of heat is stricken out. This will appear as we progress further. Expansion under the influence of heat is shown not only by solids but by fluids as well. Here we have a fluid colored so that you can see it. We will warm this colored fluid (See Figure 1b). Now you notice that after a short time the colored fluid rises and from that we can conclude that fluids expand just like solids. Since the colored fluid rises, therefore fluids expand when warmed.

Now we can in the same way investigate the expansion of a gaseous body. For this purpose we have here a vessel filled simply with air. (See Figure 2). We shut off the air in the vessel and warm it. Notice that here is a tube communicating with the vessel and containing a liquid whose level is the same in both arms of the tube. When we simply warm the air in the vessel, which air constitutes a gaseous body, you will see what happens. We will warm it by immersing the vessel in water heated to a temperature of 40°. (Note: temperatures in the lectures are given in degrees Celsius.) You will see, the mercury at once rises. Why does it rise? Because the gaseous body in the vessel expands. The air streams into the tube, presses on the mercury and the pressure forces the mercury column up into the tube. From this you see that the gaseous body has expanded. We may conclude that solid, liquid and gaseous bodies all expand under the influence of the being of heat, as yet unknown to us.

Now, however, a very important matter approaches us when we proceed from the study of the expansion of solids through the expansion of liquids to the expansion of a gas. I have already stated that $\alpha$, the relation of the expansion to the original length of the rod, differed for different substances. If by means of further experiments that cannot be performed here, we investigate $\alpha$ for various fluids, again we will find different values for various fluid substances. When however, we investigate $\alpha$ for gaseous bodies then a peculiar thing shows itself, namely that $\alpha$ is not different for various gases but that this expansion coefficient as it is called, is the same and has a constant value of about $1/273$. This fact is of tremendous importance. From it we see that as we advance from solid bodies to gases, genuinely new relations with heat appear. It appears that different gases are related to heat simply according to their property of being gases and not according to variations in the nature of the matter composing them. The condition of being a gas is, so to speak, a property which may be shared in common by all bodies. We see indeed, that for all gases known to us on earth, the property of being a gas gathers together into a unity this property of expanding. Keep in mind now that the facts of expansion under the influence of heat oblige us to say that as we proceed from solid bodies to gases, the different expansion values found in the case of solids are transformed into a kind of unity, or single power of expansion for gases. Thus if I may express myself cautiously, the solid condition may be said to be associated with an individualization of material condition. Modern physics pays scant attention to this circumstance. No attention is paid to it because the most important things are obscured by the fact of striking out certain values which cannot be adequately handled.

The history of the development of physics must be called in to a certain extent in order to gain insight into the things involved in a deeper insight into these matters. All the ideas current in the modern physics texts and ruling the methods by which the facts of physics are handled are really not old. They began for the most part in the 17th century and took their fundamental character from the new impulse given by a certain scientific spirit in Europe through Academia del Cimento in Florence. This was founded in 1667 and many experiments in quite different fields were carried out there, especially however, experiments dealing with heat, acoustics and tone. How recent our ordinary ideas are may be realized when we look up some of the special apparatus of the Academia del Cimento. It was there for instance, that the ground work for our modern thermometry was laid. It was at this academy that there was observed for the first time how the mercury behaves in a glass tube ending at the bottom in a closed cylinder, when the mercury filling the tube is warmed. Here, in the Academia del Cimento, it was first noticed that there is an apparent contradiction between the experiments where the expansion of liquids may be observed and another experiment. The generalization had been attained that liquids expand. But when the experiment was carried out with quicksilver it was noticed that it first fell when the tube was heated and after that began to rise. This was first explained in the 17th century, and quite simply, by saying: When heat is applied, the outer glass is heated at the start and expands. The space occupied by the quicksilver becomes greater. It sinks at first, and begins to rise only when the heat has penetrated into the mercury itself. Ideas of this sort have been current since the 17th century. At the same time, however, people were backward in a grasp of the real ideas necessary to understand physics, since this period, the Renaissance, found Europe little inclined to trouble itself with scientific concepts. It was the time set aside for the spread of Christianity. This in a certain sense, hindered the process of definite physical phenomena. For during the Renaissance, which carried with it an acquaintance with the ideas of ancient Greece, men were in somewhat the following situation. On the one hand encouraged by all and every kind of support, there arose institutions like the Academia del Cimento, where it was possible to experiment. The course of natural phenomena could be observed directly. On the other hand, people had become unaccustomed to construct concepts about things. They had lost the habit of really following things in thought. The old Grecian ideas were now taken up again, but they were no longer understood. Thus the concepts of fire or heat or as much of them as could be understood were assumed to be the same as were held by the ancient Greeks. And at this time was formed that great chasm between thought and what can be derived from the observation of experiments. This chasm has widened more and more since the 17th century. The art of experiment reached its full flower in the 19th century, but a development of clear, definite ideas did not parallel this flowering of the experimental art. And today, lacking the clear, definite ideas, we often stand perplexed before phenomena revealed in the course of time by unthinking experimentation. When the way has been found not only to experiment and to observe the outer results of the experiments but really to enter into the inner nature of the phenomena, then only can these results be made fruitful for human spiritual development.

Note now, when we penetrate into the inner being of natural phenomena then it becomes a matter of great importance that entirely different expansion relations enter in when we proceed from solids to gases. But until the whole body of our physical concepts is extended we will not really be able to evaluate such things as we have today drawn plainly from the facts themselves. To the facts, already brought out, another one of extraordinary importance must be added.

You know that a general rule can be stated as we have already stated it, namely if bodies are warmed they expand. If they are cooled again they contract. So that in general the law may be stated: “Through heating, bodies expand; through cooling they contract.” But you will recollect from your elementary physics that there are exceptions to this rule, and one exception that is of cardinal importance is the one in regard to water. When water is made to expand and contract, then a remarkable fact is come upon. If we have water at 80° say, and we cool it, it first contracts. That goes without saying, as it were. But when the water is cooled further it does not contract but expands again. Thus the ice that is formed from water—and we will speak further of this—since it is more expanded and therefore less dense than water, floats on the surface of the water. This is a striking phenomenon, that ice can float on the surface of the water! It comes about through the fact that water behaves irregularly and does not follow the general law of expansion and contraction. If this were not so, if we did not have this exception, the whole arrangement of nature would be peculiarly affected. If you observe a basin filled with water or a pond, you will see that even in the very cold winter weather, there is a coating of ice on the surface only and that this protects the underlying water from further cooling. Always there is an ice coating and underneath there is protected water. The irregularity that appears here is, to use a homely expression, of tremendous importance in the household of nature. Now the manner of forming a physical concept that we can depend on in this case must be strictly according to the principles laid down in the last course. We must avoid the path that leads to an Achilles-and-the-turtle conclusion. We must not forget the manifested facts and must experiment with the facts in mind, that is, we must remain in the field where the accessible facts are such as to enable us to determine something. Therefore, let us hold strictly to what is given and from this seek an explanation for the phenomena. We will especially hold fast to such things, given to observation, as expansion and irregularity in expansion like that of water (noting that it is associated with a fluid.) Such factual matters should be kept in mind and we must remain in the world of actualities. This is real Goetheanism.

Let us now consider this thing, which is not a theory but a demonstrable fact of the outer world. When matter passes into the gaseous condition there enters in a unification of properties for all the substances on the earth and with the passage to the solid condition there takes place an individualizing, a differentiation.

Now if we ask ourselves how it can come about that with the passage from the solid to the gaseous through the liquid state a unification takes place, we have a great deal of difficulty in answering on the basis of our available concepts. We must first, if we are to be able to remain in the realm of the demonstrable, put certain fundamental questions. We must first ask: Whence comes the possibility for expansion in bodies, followed finally by change into the gaseous state with its accompanying unification of properties?

You have only to look in a general way at all that is to be known about the physical processes on the earth in order to come to the following conclusion: Unless the action of the sun were present, we could not have all these phenomena taking place through heat. You must give attention to the enormous meaning that the being of the sun has for the phenomena of earth. And when you consider this which is simply a matter of fact, you are obliged to say: this unification of properties that takes place in the passage from the solid through the fluid and into the gaseous state, could not happen if the earth were left to itself. Only when we go beyond the merely earthly relations can we find a firm standpoint for our consideration of these things. When we admit this, however, we have made a very far reaching admission. For by putting the way of thinking of the Academia del Cimento and all that went with it in place of the above mentioned point of view, the old concepts still possible in Greece were robbed of all their super-earthly characteristics. And you will soon see, that purely from the facts, without any historical help, we are going to come back to these concepts. It will perhaps be easier to win way into your understanding if I make a short historical sketch at this time.

I have already said that the real meaning of those ideas and concepts of physical phenomena that were still prevalent in ancient Greece have been lost. Experimentation was started and without the inner thought process still gone through in ancient Greece, ideas and concepts were taken up parrot-fashion, as it were. Then all that the Greeks included in these physical concepts was forgotten. The Greeks had not simply said, “Solid, liquid, gaseous,” but what they expressed may be translated into our language as follows:

Whatever was solid was called in ancient Greek earth;
Whatever was fluid was called in ancient Greece water;
Whatever was gaseous was called in ancient Greece air.

It is quite erroneous to think that we carry our own meaning of the words earth, air and water over into old writings where Grecian influence was dominant, and assume that the corresponding words have the same meaning there. When in old writings, we come across the word water we must translate it by our word fluid; the word earth by our words solid bodies. Only in this way can we correctly translate old writings. But a profound meaning lies in this. The use of the word earth to indicate solid bodies implied especially that this solid condition falls under the laws ruling on the planet earth. (As stated above, we will come upon these things in following lectures from the fact themselves; they are presented today in this historical sketch simply to further your understanding of the matter.)

Solids were designated as earth because it was desired to convey this idea: When a body is solid it is under the influence of the earthly laws in every respect. On the other hand, when a body was spoken of as water, then it was not merely under the earthly laws but influenced by the entire planetary system. The forces active in fluid bodies, in water, spring not merely from the earth, but from the planetary system. The forces of Mercury, Mars, etc. are active in all that is fluid. But they act in such a way that they are oriented according to the relation of the planets and show a kind of resultant in the fluid.

The feeling was, thus, that only solid bodies, designated as earth, were under the earthly system of laws; and that when a body melted it was influenced from outside the earth. And when a gaseous body was called air, the feeling was that such a body was under the unifying influence of the sun, (these things are simply presented historically at this point,) this body was lifted out of the earthly and the planetary and stood under the unifying influence of the sun. Earthly air being were looked upon in this way, that their configuration, their inner arrangement and substance were principally the field for unifying forces of the sun.

You see, ancient physics had a cosmic character. It was willing to take account of the forces actually present in fact. For the Moon, Mercury, Mars, etc. are facts. But people lost the sources of this view of things and were at first not able to develop a need for new sources. Thus they could only conceive that since solid bodies in their expansion and in their whole configuration fell under the laws of the earth, that liquid and gaseous bodies must do likewise. You might say that it would never occur to a physicist to deny that the sun warmed the air, etc. He does not, indeed do this, but since he proceeds from concepts such as I characterized yesterday, which delineate the action of the sun according to ideas springing from observations on the earth, he therefore explains the sun in terrestrial terms instead of explaining the terrestrial in solar terms.

The essential thing is that the consciousness of certain things was completely lost in the period extending from the 15th to the 17th centuries. The consciousness that our earth is a member of the whole solar system and that consequently every single thing on the earth had to do with the whole solar system was lost. Also there was lost the feeling that the solidity of bodies arose, as it were, because the earthly emancipated itself from the cosmic, that it tore itself free to attain independent action while the gaseous, for example, the air, remained in its behavior under the unifying influence of the sun as it affected the earth as a whole. It is this which has led to the necessity of explaining things terrestrially which formerly received a cosmic explanation. Since man no longer sought for planetary forces acting when a solid body changes to a fluid, as when ice becomes fluid—changes to water—since the forces were no longer sought in the planetary system, they had to be placed within the body itself. It was necessary to rationalize and to theorize over the way in which the atoms and molecules were arranged in such a body. And to these unfortunate molecules and atoms had to be ascribed the ability from within to bring about the change from solid to liquid, from liquid to gas. Formerly such a change was considered as acting through the spatially given phenomena from the cosmic regions beyond the earth. It is in this way we must understand the transition of the concepts of physics as shown especially in the crass materialism of the Academia del Cimento which flowered in the ten year period between 1657 and 1667. You must picture to yourselves that this crass materialism arose through the gradual loss of ideas embodying the connection between the earthly and the cosmos beyond the earth. Today the necessity faces us again to realize this connection. It will not be possible, my friends, to escape from materialism unless we cease being Philistines just in this field of physics. The narrow-mindedness comes about just because we go from the concrete to the abstract, for no one loves abstractions more than the Philistine. He wishes to explain everything by a few formulae, a few abstract ideas. But physics cannot hope to advance if she continues to spin theories as has been the fashion ever since the materialism of the Academia del Cimento. We will only progress in such a field as that of the understanding of heat if we seek again to establish the connection between the terrestrial and the cosmic through wider and more comprehensive ideas than modern materialistic physics can furnish us.

My dear friends,

Today in order to press toward the goal of the first of these lectures, we will consider some of the relations between the being of heat and the so-called state of aggregation. By this state of aggregation I mean what I referred to yesterday as called in the ancient view of the physical world, earth, water, air. You are acquainted with the fact that earth, water, and air, or as they are called today, solid, fluid, and gaseous bodies may be transformed one into another. In this process however, a peculiar phenomenon shows itself so far as heat relations are concerned. I will first describe the phenomenon and then we will demonstrate it in a simple fashion. If we select any solid body and heat it, it will become warmer and warmer and finally come to a point where it will go over from the solid to the fluid condition. By means of a thermometer we can determine that as the body absorbs heat, its temperature rises. At the moment when the body begins to melt, to become fluid, the thermometer ceases rising. It remains stationary until the entire body has become fluid, and only begins to rise again when all of the solid is melted. Thus we can say: during the process of melting, the thermometer shows no increase in temperature. It must not be concluded from this however, that no heat is being absorbed. For if we discontinue heating, the process of melting will stop. (I will speak more of this subsequently.) Heat must be added in order to bring about melting, but the heat does not show itself in the form of an increase in temperature on the thermometer. The instrument begins to show an increase in temperature only when the melting has entirely finished, and the liquid formed from the solid begins to take up the heat. Let us consider this phenomenon carefully. For you see, this phenomenon shows discontinuity to exist in the process of temperature rise. We will collect a number of such facts and these can lead us to a comprehensive view of heat unless we go over to some reasoned-out theory. We have prepared here this solid body, sodium thiosulphate, which solid we will melt. You see here a temperature of about 25° C. Now we will proceed to heat this body and I will request someone to come up and watch the temperature to verify the fact that while the body is melting the temperature does not rise.(Note: The thermometer went to 48° C. which is the melting point of sodium thiosulphate, and remained there until the substance had melted.) Now the thermometer rises rapidly, since the melting is complete, although it remained stationary during the entire process of melting.

Suppose we illustrate this occurrence in a simple way, as follows: The temperature rise we will consider as a line sloping upward in this fashion (Fig. 1). Assume we have raised the temperature to the melting point as it is called. So far as the thermometer shows, the temperature again rises. It can be shown that through this further temperature rise, with its corresponding addition of heat, the liquid in question expands. Now if we heat such a melted body further, the temperature rises again from the point at which melting took place (dotted line.) It rises as long as the body remains fluid. We can then come upon another point at which the liquid begins to boil. Again we have the same phenomenon as before. The thermometer shows no further temperature rise until the entire liquid is vaporized. At the moment when the fluid has vaporized, we would find by holding the thermometer in the vapor that it again shows a temperature rise (dot-dash line.) You can see here that during vaporizing the instrument does not rise. There I find a second place where the thermometer remains stationary. (Note: the thermometer remained at 100° C. in a vessel of boiling water.)

Now I will ask you to add to the fact I have brought before you, another which you will know well from ordinary experience. If you consider solids, which form our starting point, you know that they hold their shape of themselves, whatever form is given them they maintain. If I place a solid here before you it remains as it is. If you select a fluid, that is, a body that has by the application of heat been made to go through the melting point, you know that I cannot handle it piece by piece, but it is necessary to place it in a vessel, and it takes the form of the vessel, forming a horizontal upper surface. (Fig. 3) If I select a gas—a body that has been vaporized by passing through the boiling point, I cannot keep it in an open vessel such as I use for the liquid, it will be lost. Such a gas or vapor I can hold only in a vessel closed in on all sides, otherwise the gas spreads out in all directions. (Fig. 4) This holds, at least for superficial observation, and we will consider the matter first in this way. And now I would ask you to make the following consideration of these things with me. We make this consideration in order to bring facts together so that we can reach a general conception of the nature of heat. Now have we determined the rise in temperature? We have determined it by means of the expansion of quicksilver. The expansion has taken place in space. And since at our ordinary temperature quicksilver is a liquid, we must keep clear in our minds that it is confined in a vessel, and the three dimensional expansion is summed up so that we get an expansion in that direction. By reducing the expansion of quicksilver in three dimensions to a single dimension, we have made this expansion measure the temperature rise.

Let us proceed from this observation which we have laid out as a fundamental and consider the following: Assume a line (Fig. 5) Naturally, a line can only exist in thought. And suppose on this line there lie a number of points $a$, $b$, $c$, $d$, etc. If you wish to reach these points you can remain in the line. If, for instance, you are at this point $a$ you can reach c by passing along the line. You can pass back again and again reach the point a. In brief, if I desire to reach the points $a$, $b$, $c$, $d$, I can do so and remain entirely in the line. The matter is otherwise when we consider the point $e$ or the point $f$. You cannot remain in the line if you wish to reach point $e$ or $f$. You must go outside to reach these points. You have to move along the line and then out of it to get to these points.

Now assume you have a surface, let us say the surface of the blackboard, and again I locate on the surface of this board a number of points; $a$, $b$, $c$, $d$. (Fig. 6) In order to reach these points you may remain always in the surface of the blackboard. If you are at this point $x$ you may trace your way to each of these points over a path that does not leave the blackboard. You cannot, however, if you wish to remain in the surface of the board, reach this point which is at a distance in front of the board. In this case you must leave the surface. This consideration leads to a view of the dimensionality of space from which one can say: To reach points in one dimension, movement in this single direction suffices, for those in two dimensions movement in two dimensions gives access to them. It is however, not possible to reach points outside a single dimension without leaving this dimension and likewise one cannot pass through points in three dimensions by moving about in a single plane. What is involved when I consider the points e and f in relation to the single dimension represented by points $a$, $b$, $c$ and $d$ ? Imagine a being who was able to observe only one dimension and who had no idea of a second or third dimension. Such a being would move in his one dimension just as you do in three dimensional space. If such a being carried the point $a$ to the position $b$ and the point then slipped off to e, at that moment the content of the point would simply vanish from the single dimension of the being. It would no longer exist for this being from the moment it left the single dimension of which he is aware. Likewise the points outside a surface would not exist for a being aware only of two dimensions. When a point dropped out of the plane, such a being would have no way of following it; the point would disappear form his space realm. What kind of a geometry would a unidimensional being have? He would have a one-dimensional geometry. He would be able to speak only of distance and the like, of the laws relating to such things as they applied in a single dimension. A two-dimensional being would be able to speak of the laws of plane figures and would have a two-dimensional geometry. We men have at the outset a three-dimensional geometry. A being with a unidimensional geometry would have no possibility of understanding what a point does when it leaves the single dimension. A being with a two-dimensional geometry would be unable to follow the motion of a point that left a surface and moved out in front of it as we supposed was the case when the point left a surface and moved out in front of it as we supposed was the case when the point left the surface of the blackboard. We men—I state again—have a three-dimensional geometry. Now I may just as well do what I am obliged to do on account of the reducing of the three-dimensional expansion of the quicksilver to a single dimension. I may draw two lines in two directions so as to form a system of axes, thus giving as in Fig. 7 an axis of abscissae and an axis of ordinates. At right angles to the plane of these two, suppose we have a third line which we will call a space line.¹ Just as soon as I come either to the melting point or the boiling point, at that moment I am not in a position to proceed with the line (Fig. 8). Theoretically or hypothetically there is no possibility of continuing the line. Let us assume that we can say, the rise of temperature is represented by this line. We can proceed along it and still have a point of connection with our ordinary world. But we do not as a matter of fact have such a point of connection. For when I draw this temperature curve and come to the melting or boiling point, I can only continue the curve from the same point ($x$, $x$ in Fig. 8). I had reached when the body had begun to melt or vaporize. You can see from this, that in regard to the melting or boiling point, I am in a position not different from that of the one-dimensional being when a point moves out of his first dimension into the second dimension, or of the two-dimensional being when a point disappears for him into the third dimension. When the point comes back again and starts from the same place, or as in Fig. 5 when the point moves out to one side and returns, then it is necessary to continue the line on in its one dimension. Considered simply as an observed phenomenon, when the temperature rise disappears at the melting and boiling point, it is as though my temperatures curve were broken, and I had to proceed after a time from the same point. But what is happening to the heat during this interruption falls outside the realm in which I draw my curve. Formally speaking, I may say that I can draw this on the space line. There is, at first considered—note I say at first—an analogy present between the disappearance of the point a from the first and into the second dimension and what happens to the temperature as shown by the thermometer when the instrument stands still at the melting point and the boiling point.

Now we have to bring another phenomenon in connection with this. Please note that in this linking together of phenomena we make progress, not in elaborating some kind of theory, but in bringing together phenomena so that they naturally illuminate each other. This is the distinction between the physics of Goethe that simply places phenomena side by side so that they throw light on each other, and modern physics which tends to go over into theories, and to add thought-out elaborations to the facts. For atoms and molecules are nothing else but fancies added to the facts.

Let us now consider another phenomenon along with this disappearance of the temperature recorded by the thermometer during the process of melting. This other phenomenon meets us when we look at yesterday's formula. This formula was written:

$$V=V_o(1+3\alpha t+3{\alpha }^2{t}^2+{\alpha }^3{t}^3)$$

You remember that I said yesterday you should pay especial attention to the last two terms. It is especially important for us at this time to consider $t^3$, the third power of the temperature. Imagine for a moment ordinary space. In this ordinary space you speak in mathematical terms of length, breadth, and thickness. These are actually the three dimensions of space. Now when we warm a rod, as we did yesterday, we can observe the expansion of this rod. We can also note the temperature of this rod. There is one thing we cannot bring about. We cannot bring it about that the rod while it is expanding, does not give off heat to its surroundings, that it does not stream out or radiate heat. This we cannot prevent. It is impossible for us to think—note the word—of a propagation of heat in one dimension. We can indeed think of a space extension in one dimension as one does in geometry in the case of a line. But we cannot under any circumstances imagine heat propagated along a line. When we consider this matter we cannot say that the propagation of heat is to be thought of as represented in space in reality by the line that I have drawn here. (Fig. 1) This curve does not express for me the whole process involved in the heat. Something else is active besides what I can deduce from the curve. And the activity of this something changes the entire nature and being of what is shown by this curve, which I am using as a symbol which may be considered equally well as a purely arithmetical or geometrical fact.

We have, thus, a peculiar situation. When we try to grasp the heat condition, in so far as the temperature shows this condition, by means of an ordinary geometrical line, we find it cannot be done. Now this has another bearing. Imagine for a moment that I have a line. This line has a certain length: $l$ (Fig. 9) I square this line, and then I can represent this $l^2$ by a square surface. Assume that I obtain $l^3$ then I can represent the third power by a cube, a solid body. But suppose I obtain the fourth power, $l^4$. How can I represent that? I can pass over from the line to the surface, from the surface to the solid, but what can I do by following this same method if I wish to represent the fourth power? I cannot do anything if I remain in our three-dimensional space. The mathematical consideration shows this. But we have seen that the heat condition in so far as it is revealed by temperature is not expressible in space terms. There is something else in it. If there were not, we could conceive of the heat condition passing along a rod as confined entirely to the rod. This, however, is impossible. The consequence of this is that when I really wish to work in this realm, I ought not to look upon the powers of $t$ in the same manner as the powers of a quantity measured in space. I cannot think about the powers of $t$ in the same way as those of $l$ or of any other mere space quantity. When, for instance, and I will consider this tomorrow hypothetically, when I have the first power and find it not expressible as a line, then the second power $t^2$ cannot be expressed as a surface and certainly the third power $t^3$ cannot be expressed as a solid. In purely mathematical space, it is only after I have obtained the third power that I get outside of ordinary space, but in this other case I am quite outside of ordinary space in the case of the second power and the third as well.

Therefore, you must realize that you have to conceive of $t$ as different entirely in its nature from space quantities. You must consider $t$ as something already squared, as a second power and the squared $t$ you must think of as of the third power, the cubed $t$ as of the fourth power. This takes us out of ordinary space. Consider now how this gives our formula a very special aspect. For the last member, which is in this super-space, forces me to go out of ordinary space. In such a case when I confine myself to reckoning I must go beyond three dimensional space for the last member of the formula. There is such a possibility in purely mathematical formulae.

When you observe a triangle and determine that it has three angles, you are dealing, at the start, with a conceived triangle. Since merely thinking about it is not enough to satisfy your senses, you draw it, but the drawing adds nothing to your idea. You have given, the sum of the angles is 180, or a right-angled triangle—the square of the hypotenuse equals the sum of the squares of the other two sides. These things are handled as I now handle the power of $t$.

Let us now go back and see what we have established as fact. This is the way it is done in geometry. It is always true that when I observe an actual triangle in bridge construction or elsewhere, the abstract idea verifies itself. What I have thought of in the abstract $t$ has at first a similarity with melting and vaporizing. (We will gradually get nearer to the essence of the reality.) Melting and vaporizing I could not express in terms of the three dimensions of space. The only way I could force them into the curve was to stop and then continue again. In order to prove the hypothesis that I made for you, it was necessary, in the case of the third power, the cube of the temperature, to go outside of three-dimensional space.

You see, I am showing you how we must, as it were, break a path if we wish to place together those phenomena which simply by being put side by side illustrate the being of heat and enable us to attain to an understanding similar to that reached in the preceding course of lectures on light.

The physicist Crookes approached this subject from entirely different hypotheses. It is significant that his considerations led him to a result similar to the one we have arrived at tentatively and whose validity we will establish in the next lectures. He also concluded the temperature changes had essentially to do with a kind of fourth dimension in space. It is important at this time to give attention to these things because the relativists, with Einstein at their head, feel obliged when they go outside of three-dimensional space, to consider time as the fourth dimension. Thus, in the Einstein formulae, everywhere one finds time as the fourth dimension. Crookes, on the other hand, considered the gain or loss of heat as the fourth dimension. So much for this side-light on historical development.

To these phenomena I would ask you now to add what I have formerly emphasized. I have said: An ordinary solid may be handled and it will keep its form, (Fig. 2). That is, it has a determinate boundary. A fluid must be poured into a vessel, (Fig. 3). It always forms a flat upper surface and for the rest takes the shape of the vessel. This is not so for a gas or vaporous body which extends itself in every direction. In order to hold it, I must put it into a vessel closed on all sides, (Fig. 4). This completely closed vessel gives it its form. Thus, in the case of a gas, I have a form only when I shut it in a vessel closed on all sides. The solid body possesses a form simply by virtue of the fact that it is a solid body. It has a form of itself, as it were. Considering the fluid as an intermediate condition, we will note that the solid and gaseous bodies may be described as opposites. The solid body provides for itself that which I must add to the gaseous body, namely the completely surrounding boundary.

Now, however, a peculiar thing occurs in the case of a gas. When you put a gas into a smaller volume (Fig. 10), using the same amount of gas but contracting the walls all around, you must use pressure. You have to exert pressure. This means nothing else but that you have to overcome the pressure of the gas. You do it by exerting pressure on the walls which give form to the gas. We may state the matter thus: that a gas which has the tendency to spread out in all directions is held together by the resistance of the bounding walls. This resistance is there of itself in the case of the solid body. So that, without any theorizing, but simply keeping in mind the quite obvious facts, I can define a polaric contrast between a gas and a solid body in the following way: That which I must add to the gas from the outside is present of itself in the solid. But now, if you cool the gas, you can pass back again to the boiling point and get a liquid from the vapor, and if you cool further to the melting point, you can get the solid from the liquid. That is to say, you are able by processes connected with the heat state to bring about a condition such that you no longer have to build the form from the outside, but the creation of form takes place of itself from within. Since I have done nothing but bring about a change in the heat condition, it is self-evident that form is related in some way to changes in the heat state. In a solid, something is present which is not present in a gas. If we hold a wall up against a solid, the solid does not of itself exert pressure against the wall unless we ourselves bring this about. When, however, we enclose a gas in a vessel, the gas presses against the solid wall. You see, we come upon the concept of pressure and have to bring this creation of pressure into relation with the heat condition. We have to say to ourselves: it is necessary to find the exact relation between the form of solid bodies, the diffusing tendency of gases and the opposition of the boundary walls that oppose this diffusion. When we know this relation we can hope really to press forward into the relation between heat and corporeality.

• 1. Referring again to the temperature rise diagram – tr.

My dear friends,

You will perhaps have noticed that in our considerations here, we are striving for a certain particular goal. We are trying to place together a series of phenomena taken from the realm of heat in such a manner that the real nature of warmth may be obvious to us from these phenomena. We have become acquainted in a general way with certain relations that meet us from within the realm of heat, and we have in particular observed the relation of this realm of the expansionability of bodies. We have followed this with an attempt to picture to ourselves mentally the nature of form in solid bodies, fluids and gaseous bodies. I have also spoken of the relation of heat to the changes produced in bodies in going from the solid to the fluid and from the fluid to the gaseous or vaporous condition. Now I wish to bring before you certain relations which come up when we have to do with gases or vapors. We already know that these are so connected with heat that by means of this we bring about the gaseous condition, and again, by appropriate change of temperature that we can obtain a liquid from a gas. Now you know that when we have a solid body, we cannot by any means interpenetrate this solid with another. The observation of such simple elementary relations is of enormous importance if we really wish to force our way through to the nature of heat. The experiment I will carry out here will show that water vapor produced here in this vessel passes through into this second vessel. And now having filled the second vessel with water vapor, we will produce in the first vessel another vapor whose formation you can follow by reason of the fact that it is colored. (The experiment was carried out.) You see that in spite of our having filled the vessel with water vapor, the other vapor goes into the space filled with the water vapor. That is, a gas does not prevent another gas from penetrating the space it occupies. We may make this clear to ourselves by saying that gaseous or vaporous bodies may to a certain extent interpenetrate each other.

I will now show you another phenomenon which will illustrate one more relation of heat to certain facts. We have here in the left hand tube, air which is in equilibrium with the outer air with which we are always surrounded. I must remind you that this outer air surrounding us is always under a certain pressure, the usual atmospheric pressure, and it exerts this pressure on us. Thus, we can say that air inside the left hand tube is under the same pressure as the outer air itself, which fact is shown by the similar level of mercury in the right and left hand tubes. You can see that on both right and left hand sides the mercury column is at the same height, and that since here on the right the tube is open to the atmosphere the air in the closed tube is at atmospheric pressure. We will now alter the conditions by bringing pressure on the air in the left hand tube, ($2$ × $p$). By doing this we have added to the usual atmospheric pressure, the pressure due to the higher mercury column. That is, we have simply added the weight of the mercury from here to here. (Fig. 1b from $a$ to $b$). By thus increasing the pressure exerted on this air by the pressure corresponding to the weight of the mercury column, the volume of the air in the left hand tube is, as you can see, made smaller. We can therefore say when we increase the pressure on the gas its volume decreases. We must extend this and consider it a general phenomenon that the space occupied by a gas and the pressure exerted on it have an inverse ratio to each other. The greater the pressure the smaller the volume, and the greater the volume the smaller must be the pressure acting on the gas. We can express this in the form of an equation where the volume $V_1$ divided by the volume $V_2$ equals the pressure $P_2$ divided by the pressure $P_1$.

From which it follows:

This expresses a relatively general law (we have to say relative and will see why later.) This may be stated as follows: volume and pressure of gases are so related that the volume-pressure product is a constant at constant temperature. As we have said, such phenomena as these must be placed side by side if we are to approach the nature of heat. And now, since our considerations are to be thought of as a basis for pedagogy we must consider the matter from two aspects. On the one hand, we must build up a knowledge of the method of thinking of modern physics and on the other, we must become acquainted with what must happen if we are to throw aside certain obstacles that modern physics places in the path to a real understanding of the nature of heat.

Please picture vividly to ourselves that when we consider the nature of heat we are necessarily dealing at the same time with volume increases, that is with changes in space and with alterations of pressure. In other words, mechanical facts meet us in our consideration of heat. I have to speak repeatedly in detail of these things although it is not customary to do this. Space changes, pressure changes. Mechanical facts meet us.

Now for physics, these facts that meet us when we consider heat are purely and simply mechanical facts. These mechanical occurrences are, as it were, the milieu in which heat is observed. The being of heat is left, so to speak, in the realm of the unknown and attention is focused on the mechanical phenomena which play themselves out under its influence. Since the perception of heat is alleged to be purely a subjective thing, the expansion of mercury, say, accompanying change of heat condition and of sensation of heat, is considered as something belonging in the realm of the mechanical. The dependence of gas pressure, for instance, on the temperature, which we will consider further, is thought of as essentially mechanical and the being of heat is left out of consideration. We saw yesterday that there is a good reason for this. For we saw that when we attempt to calculate heat, difficulties arise in the usual calculations and that we cannot, for example, handle the third power of the temperature in the same way as the third power of an ordinary quantity in space. And since modern physics has not appreciated the importance of the higher powers of the temperature, it has simply stricken them out of the expansion formulae I mentioned to you in former lectures.

Now you need only consider the following. You need consider only that in the sphere of outer nature heat always appears in external mechanical phenomena, primarily in space phenomena. Space phenomena are there to begin with and in them the heat appears. This it is, my dear friends, that constrains us to think of heat as we do of lines in space and that leads us to proceed from the first power of extension in space to the second power of the extension.

When we observe the first power of the extension, the line, and we wish to go over to the second power, we have to go out of the line. That is, we must add a second dimension to the first. The standard of measurement of the second power has to be thought of as entirely different from that of the first power. We have to proceed in an entirely similar fashion when we consider a temperature condition. The first power is, so to speak, present in the expansion. Change of temperature and expansion are so related that they may be expressed by rectilinear coordination (Fig. 2). I am obliged, when I wish to make the graph representing change in expansion with change in temperature, to add the axis of abscissae to the axis of ordinates. But this makes it necessary to consider what is appearing as temperature not as a first power but as a second power, and the second power as a third. When we deal with the third power of the temperature, we can no longer stay in our ordinary space. A simple consideration, dealing it is true with rather subtle distinctions, will show you that in dealing with the heat manifesting itself as the third power, we cannot limit ourselves to the three directions of space. It will show you how, the moment we deal with the third power, we are obliged, so far as heat effects are concerned, to go out of space.

In order to explain the phenomena, modern physics sets itself the problem of doing so and remaining within the three dimensional space.

You see, here we have an important point where physical science has to cross a kind of Rubicon to a higher view of the world. And one is obliged to emphasize the fact that since so little attempt is made to attain clarity at this point, a corresponding lack enters into the comprehensive world view.

Imagine to yourselves that physicists would so present these matters to their students as to show that one must leave ordinary space in which mechanical phenomena play when heat phenomena are to be observed. In such a case, these teachers of physics would call forth in their students, who are intelligent people since they find themselves able to study the subject, the idea that a person cannot really know it without leaving the three dimensional space. Then it would be much easier to place a higher world-view before people. For people in general, even if they were not students of physics, would say, “We cannot form a judgment on the matter, but those who have studied know that the human being must rise through the physics of space to other relations than the purely spatial relations.” Therefore so much depends on our getting into this science such ideas as those put forth in our considerations here. Then what is investigated would have an effect on a spiritually founded world view among people in general quite different from what it has now. The physicist announces that he explains all phenomena by means of purely mechanical facts. This causes people to say, “Well, there are only mechanical facts in space. Life must be a mechanical thing, soul phenomena must be mechanical and spiritual things must be mechanical.” “Exact sciences” will not admit the possibility of a spiritual foundation for the world. And “exact science” works as an especially powerful authority because they are not familiar with it. What people know, they pass their own judgment on and do not permit it to exercise such an authority. What they do not know they accept on authority. If more were done to popularize the so-called “rigidly exact science,” the authority of some of those who sit entrenched in possession of this exact science would practically disappear.

During the course of the 19th century there was added to the facts that we have already observed, another one of which I have spoken briefly. This is that mechanical phenomena not only appear in connection with the phenomena of heat, but that heat can be transformed into mechanical phenomena. This process you see in the ordinary steam locomotive where heat is applied and forward motion results. Also mechanical processes, friction and the like, can be transformed back again into heat since the mechanical processes, as it is said, bring about the appearance of heat. Thus mechanical processes and heat processes may be mutually transformed into each other.

We will sketch the matter today in a preliminary fashion and go into the details pertaining to this realm in subsequent lectures.

Further, it has been found that not only heat but electrical and chemical processes may be changed into mechanical processes And from this has been developed what has been called during the 19th century the “mechanical theory of heat.”

This mechanical theory of heat has as its principal postulate that heat and mechanical effects are mutually convertible one into the other. Now suppose we consider this idea somewhat closely. I am unable to avoid for you the consideration of these elementary things of the realm of physics. If we pass by the elementary things in our basic consideration, we will have to give up attaining any clarity in this realm of heat. We must therefore ask the questions: what does it really mean then when I say: Heat as it is applied in the steam engine shows itself as motion, as mechanical work? What does it mean when I draw from this idea: through heat, mechanical work is produced in the external world? Let us distinguish clearly between what we can establish as fact and the ideas which we add to these facts. We can establish the fact that a process subsequently is revealed as mechanical work, or shows itself as a mechanical process. Then the conclusion is drawn that the heat process, the heat as such, has been changed into a mechanical thing, into work.

Well now, my dear friends, if I come into this room and find the temperature such that I am comfortable, I may think to myself, perhaps unconsciously without saying it in words: In this room it is comfortable. I sit down at the desk and write something. Then following the same course of reasoning as has given rise to the mechanical theory of heat, I would say: I came into the room, the heat condition worked on me and what I wrote down is a consequence of this heat condition. Speaking in a certain sense I might say that if I had found the place cold like a cellar, I would have hurried out and would not have done this work of writing. If now I add to the above the conclusion that the heat conducted to me has been changed into the work I did, then obviously something has been left out of my thinking. I have left out all that which can only take place through myself. If I am to comprehend the whole reality I must insert into my judgment of it this which I have left out. The question now arises: When the corresponding conclusion is drawn in the realm of heat, by assuming that the motion of the locomotive is simply the transformed heat from the boiler, have I not fallen into the error noted above? That is, have I not committed the same fallacy as when I speak of a transformation of heat into an effect which can only take place because I myself am part of the picture? It may appear to be trivial to direct attention to such a thing as this, but it is just these trivialities that have been completely forgotten in the entire mechanical theory of heat. What is more, enormously important things depend on this. Two things are bound together here. First, when we pass over from the mechanical realm into the realm where heat is active we really have to leave three dimensional space, and then we have to consider that when external nature is observed, we simply do not have that which is interpolated in the case, where heat is changed over into my writing. When heat is changed into my writing, I can note from observation of my external bodily nature that something has been interpolated in the process. Suppose however, that I simply consider the fact that I must leave three dimensional space in order to relate the transformation of heat into mechanical effects. Then I can say, perhaps the most important factor involved in this change plays its part outside of three dimensional space. In the example that concerned myself which I gave you, the manner in which I entered into the process took place outside of three dimensions. And when I speak of simple transformation of heat into work I am guilty of the same superficiality as when I consider transformation of heat into a piece of written work and leave myself out.

This, however, leads to a very weighty consequence. For it requires me to consider in external nature even lifeless inorganic nature, a being not manifested in three dimensional space. This being, as it were, rules behind the three dimensions. Now this is very fundamental in relation to our studies of heat itself.

Since we have outlined the fundamentals of our conception of the realm of heat, we may look back again on something we have already indicated, namely on man's own relation to heat. We may compare the perception of heat to perception in other realms. I have already called attention to the fact that, for instance, when we perceive light, we note this perception of light to be bound up with a special organ. This organ is simply inserted into our body and we cannot, therefore, speak of being related to color and light with our whole organism, but our relation to it concerns a part of us only. Likewise with acoustical or sound phenomena, we are related to them with a portion of our organism, namely the organ of hearing. To the being of heat we are related through our entire organism. This fact, however, conditions our relation to the being of heat. We are related to it with our entire organism. And when we look more closely, when we try, as it were, to express these facts in terms of human consciousness, we are obliged to say, “We are really ourselves this heat being. In so far as we are men moving around in space, we are ourselves this heat being.” Imagine the temperature were to be raised a couple of hundred degrees; at that moment we could no longer be identical with it, and the same thing applies if you imagine it lowered several hundred degrees. Thus the heat condition belongs to that in which we continually live, but do not take up into our consciousness. We experience it as independent beings, but we do not experience it consciously. Only when some variation from the normal condition occurs, does it take conscious form.

Now with this fact a more inclusive one may be connected. It is this. You may say to yourselves when you contact a warm object and perceive the heat condition by means of your organism, that you can do it with the tip of your tongue, with the tip of your finger, you can do it with other parts of your organism: with the lobes of your ears, let us say. In fact, you can perceive the heat condition with your entire organism. But there is something else you can perceive with your entire organism. You can perceive anything exerting pressure. And here again, you are not limited strictly as you are in the case of the eye and color perception to a certain member of your entire organism. If would be very convenient if our heads, at least, were an exception to this rule of pressure perception; we would not then be made so uncomfortable from a rap on the head.

We can say there is an inner kinship between the nature of our relationship to the outer world perceived as heat and perceived as pressure. We have today spoken of pressure volume relations. We come back now to our own organism and find an inner kinship between our relation to heat and to pressure. Such a fact must be considered as a groundwork for what will follow.

But there is something else that must be taken into account as a preliminary to further observations. You know that in the most popular text books of physiology, a good deal of emphasis is laid on the fact that we have certain organs within our bodies by means of which we perceive the usual sense qualities. We have the eye for color, the ear for sound, the organ of taste for certain chemical processes, etc. We have spread over our entire organism, as it were, the undifferentiated heat organ, and the undifferentiated pressure organ.

Now, usually, attention is drawn to the fact that there are certain other things of which we are aware but for which we have no organs. Magnetism and electricity are known to us only through their effects and stand, as it were, outside of us, not immediately perceived. It is said sometimes that if we imagine our eyes were electrically sensitive instead of light sensitive, then when we turned them towards a telegraph wire we would perceive the streaming electricity in it. Electricity would be known not merely by its effects, but like light and color, would be immediately perceived. We cannot do this. We must therefore say: electricity is an example of something for whose immediate perception we have no organ. There are aspects of nature, thus, for which we have organs and aspects of nature for which we do not have organs. So it is said.

The question is whether perhaps a more unbiased observer would not come to a different conclusion from those whose view is expressed above. You all know, my dear friends, that what we call our ordinary passive concepts through which we apprehend the world, are closely bound up with the impressions received through the eye, the ear and somewhat less so with taste and smell impressions. If you will simply consider language, you may draw from it the summation of your conceptual life, and you will become aware that the words themselves used to represent our ideas are residues of our sense impressions. Even when we speak the very abstract word Sein (being), the derivation is from Ich habe gesehen, (I have seen.) What I have seen I can speak of as possessing “being.” In “being” there is included “what has been seen.” Now without becoming completely materialistic (and we will see later why it is not necessary to become so), it may be said that our conceptual world is really a kind of residue of seeing and hearing and to a lesser extent of smelling and tasting. (Those last two enter less into our higher sense impressions.) Through the intimate connection between our consciousness and our sense impressions, this consciousness is enabled to take up the passive concept world.

But within the soul nature, from another side, comes the will, and you remember how I have often told you in these anthroposophical lectures that man is really asleep so far as his will is concerned. He is, properly considered, awake only in the passive conceptual realm. What you will, you apprehend, only through these ideas or concepts. You have the idea. I will raise this glass. Now, in so far as your mental act contains ideas, it is a residue of sense impressions. You place before yourself in thought something which belongs entirely in the realm of the seen, and when you think of it, you have an image of something seen. Such an immediately derived image you cannot create from a will process proper, from what happens when you stretch out your arm and actually grasp the glass with your hand and raise it. That act is entirely outside of your consciousness. You are not aware of what happens between your consciousness and the delicate processes in your arm. Our unconsciousness of it is as complete as our unconsciousness between falling asleep and waking up. But something really is there and takes place, and can its existence be denied simply because it does not enter our consciousness? Those processes must be intimately bound up with us as human beings, because after all, it is we who raise the glass. Thus we are led in considering our human nature from that which is immediately alive in consciousness to will processes taking place, as it were, outside of consciousness. (Fig. 3) Imagine to yourselves that everything above this line is in the realm of consciousness. What is underneath is in the realm of will and is outside of consciousness. Starting from this point we proceed to the outer phenomena of nature and find our eye intimately connected with color phenomena, something which we can consciously apprehend; we find our ear intimately connected with sound, as something we can consciously apprehend. Tasting and smelling are, however, apprehended in a more dreamlike way. We have here something which is in the realm of consciousness and yet is intimately bound up with the outer world.

If now, we go to magnetic and electrical phenomena, the entity which is active in these is withdrawn from us in contrast with those phenomena of nature which have immediate connection with us through certain organs. This entity escapes us. Therefore, say the physicists and physiologists: we have no organ for it; it is cut off from us. It lies outside us. (Fig. 3 above) We have realms that we approach when we draw near the outer world—the realms of light and heat. How do electrical phenomena escape us? We can trace no connection between them and any of our organs. Within us we have the results of our working over of light and sound phenomena as residues in the form of ideas. When, however, we plunge down (Fig. 3 below), our own being disappears from us into will.

I will now tell you something a bit paradoxical, but think it over until tomorrow. Imagine we were not living men, but living rainbows, and that our consciousness dwelt in the green portion of the spectrum. On the one side we would trail off into unconsciousness in the yellow and red and this would escape us inwardly like our will. If we were rainbows, we would not perceive green, because that we are in our beings, we do not perceive immediately; we live it. We would touch the border of the real inner when we tried, as it were, to pass from the green to the yellow. We would say: I, as a rainbow, approach my red portion, but cannot take it up as a real inner experience; I approach my blue-violet, but it escapes me. If we were thinking rainbows, we would thus live in the green and have on the one side a blue-violet pole and on the other side a yellow-red pole. Similarly, we now as men are placed with our consciousness between what escapes us as external natural phenomena in the form of electricity and as inner phenomena in the form of will.

My dear friends,

I would have liked to carry out for you today some experiments to round out the series of facts that lead us to our goal. It is not possible to do so, however, and I must accordingly arrange my lecture somewhat differently from the way I intended. The reason for this is partly that the apparatus is not in working order and partly because we lack alcohol today, just as we lacked ice yesterday.

We will therefore take up in more detail the things that were begun yesterday. I will ask you to consider all these facts that were placed before you for the purpose of obtaining a survey of the relationships of various bodies to the being of heat. You will realize that certain typical phenomena meet us. We can say: These phenomena carry the impress of certain relations involving the being of heat, at first unknown to us. Heat and pressure exerted on a body or the state of aggregation that a body assumes according to its temperature, also the extent of space occupied, the volume, are examples. We are able on the one side, to see how a solid body melts, and can establish the fact that during the melting of the solid, no rise in temperature is measurable by the thermometer or any other temperature-measuring instrument. The temperature increase stands still, as it were, during the melting. On the other hand, we can see the change from a liquid to a gas, and there again we find the disappearance of the temperature increase and its reappearance when the whole body has passed into the gaseous condition. These facts make up a series that you can demonstrate for yourselves, and that you can follow with your eyes, your senses and with instruments. Yesterday, also, we called attention to certain inner experiences of the human being himself which he has under the influence of warmth and also under the influence of other sense qualities such as light and tone. But we saw that magnetism and electricity were not really sense impressions, at least not immediate sense impressions, because as ordinary physics says, there is no sense organ for these entities. We say, indeed, that so far as electrical and magnetic properties are concerned we come to know them through determining their effects, the attraction of bodies for instance, and the many other effects of electrical processes. But we have no immediate sense perception of electricity and magnetism as we have for tone and light.

We then noted particularly, and this must be emphasized, that our own passive concepts, by which we represent the world, are really a kind of distillation of the higher sense impressions. Wherever you make an examination you will find these higher concepts and will be able to convince yourselves that they are the distilled essence of the sense impressions. I illustrated this yesterday in the case of the concept of being. You can get echoes of tone in the picture of the conceptual realm, and you can everywhere see showing through how these concepts have borrowed from light . But there is one kind of concept where you cannot do this, as you will soon see. You cannot do it in the realm of the mathematical concepts. In so far as they are purely mathematical, there is no trace of the tonal or the visible. Now we must deceive ourselves here. Man is thinking of tone when he speaks of the wave number of sound vibrations. Naturally I do not refer to this sort of thing. I mean all that is obtained from pure mathematics. Such things, for instance, as the content of the proposition of Pythagoras, that the sum of the angles of a triangle is 180°, or that the whole is greater than the part, etc. The basis of our mathematical concepts does not relate itself to the seen or the heard, but it relates itself in the last analysis to our will impulse. Strange as it may seem to you at first, you will always find this fact when you look at these things from the psychological point of view, as it were. The human being who draws a triangle (the drawn triangle is only an externalization) is attaining in concept to an unfolding of the will around the three angles. There is an unfolding of action around three angles as shown by the motion of the hand or by walking, by turning of the body. The thing that you have within you as a will-concept, that in reality you carry into the pure mathematical concept. That is the essential distinction between mathematical concepts and other concepts. This is the distinction about which Kant and other philosophers waged such controversy. You can distinguish the inner determination of mathematical concepts. This distinction arises from the fact that mathematical concepts are so rigidly bound up with our own selves, that we carry our will nature into them. Only what subsists in the sphere of the will is brought into mathematical operations. This is what makes them seem so certain to us. What is not felt to be so intimately bound up with us, but is simply felt through an organ placed in a certain part of our make-up, that appears uncertain and empirical. This is the real distinction. Now, I wish to call your attention to a certain fact. When we dip down into the sphere of will, whence came, in a vague and glimmering way, the abstractions which make up the sum of our pure arithmetical and geometrical concepts, we enter the unknown region where the will rules, a region as completely unknown to us in the inner sense, as electricity and magnetism are in the outer sense. Yesterday I endeavored to illustrate this by asking you to imagine yourselves living, thinking rainbows with your consciousness in the green, in consequence of which you did not perceive the green but perceived the colors on each side of it, fading into the unknown. I compared the red to the dipping down inwardly into the unknown sphere of the will and the blue-violet to the outward extension into the spheres of electricity and magnetism and the like.

Now I am inserting at this point in our course this psychological-physiological point of view, as it might be called, because it is very essential for the future that people should be led back again to the relation of the human being to physical observations. Unless this relationship is established, the confusion that reigns at present cannot be eliminated. We will see this as we follow further the phenomena of heat. But it is not so easy to establish this relationship in the thinking of today. The reason is just this, that modern man cannot easily bridge the gap between what he perceives as outer space phenomena in the world, or better, as outer sense phenomena and what he experiences within. In these modern times there is such a pronounced dualism between all which we experience as knowledge of the outer world and what we experience inwardly, that it is extraordinarily difficult to bridge this gap, But the gap must be bridged if physics is to advance. To this end we must use the intuitive faculties rather than the rational when we relate something external to what goes on within man himself. Thus we can begin to grasp how we must orient ourselves, in observing phenomena so difficult as those arising from heat. Let me call your attention to the following:

Suppose you learn a poem by heart. You will, as you learn it, first find it necessary to become acquainted with the ideas that underlie the poem. At first you will always have the tendency, when you recite the poem, to let those ideas unroll in your mind. But you know that the more frequently you recite the poem, especially when there is a lapse of time between the recitations, the less intensely you are obliged to think of the ideas. There may come a time when it is not necessary to think at all, but simply to reel off the recitation mechanically. We never actually reach this point; do not wish to, in fact, but we approach the condition asymptotically as it were. Our feelings as human beings prevent us from reaching this stage of purely mechanical repetition, but it is thinkable that we would get to the point where we needed to think not at all, but when we spoke the first line the rest of the poem would follow without any thinking about it. You recognize the similarity between such a condition and the approach of the hyperbola to its asymptotes. But this leads us to the conception that when we speak a poem we are dealing with two different activities working simultaneously in our organism. We are dealing with a mechanical reeling-off of certain processes, and along with this go the processes included in our soul concepts. On the one hand, we have what we can properly speak of as playing itself out mechanically in space, and on the other hand, we have a soul process which is entirely non-spatial in nature.

When now, you fasten your attention simply on that which reels itself off mechanically, and you do this in thought, for instance, if you imagine you recited a poem in an unknown language, then you have simply the mechanical process. The instant you accompany this mechanical process with thinking, then you have an inner soul activity that cannot be brought out into space. You cannot express in space the thinking with which a man accompanies the recitation, as you can the mechanical processes of actual speaking, of the pronouncing of words.

Let me give you an analogy. When we follow the heating of a solid body up to the time it arrives at its melting point, the temperature becomes higher. We can see this on the thermometer. When the body begins to melt, the thermometer stands still until the melting is complete. There is an analogy between what we can follow with the thermometer, the outer physical process, and what we can follow physically in the spoken word. And there is an analogy also between what escapes us, and lies in the concepts of the reciter and what happens to the heat while the melting goes on. Here you see, we have an example where we can, by analogy, at least bridge the gap between an outer observation and something in the human being. In other realms than that of speech we do not have such ready examples to bridge the gap. This is because in speech there is, on the one hand, the possibility imaginable, at least, that a person could mechanically speak out something learned by heart. Or on the other hand, that the person would not speak at all but simply think about it and thus remove it entirely from the realm of space. In other spheres we do not have the opportunity to make this cleavage and see precisely how one activity passes over into another. Especially is this difficult when we wish to follow the nature of heat. In this case we have to set out to investigate physiologically and psychologically how heat behaves when we have taken it up into ourselves.

Yesterday, by way of illustration, I said to you: “I go into a room that is comfortably warmed, I sit down and write.” I cannot so directly find the inter-relationships between what I experience or feel when I go into the warm room. What goes on within me parallels the outer warmth, when I write my thoughts down. But I cannot determine the relationship so readily as I can between speaking something and thinking about it. Thus it is difficult to find the something within that corresponds to the outer sensation of warmth. It is a question of gradually approaching the concepts that will lead us further in this direction and in this connection I want to call your attention to something you know from your anthroposophy.

You know, when we make the attempt to extend our thinking by meditation, to increase its inner intensity, and so to work with our thoughts that we come again and again into the condition where we know we are using soul-forces without the help of the body, we notice a certain thing. We notice that in order to do this, our entire inner soul life has to change. With ordinary abstract thoughts man cannot enter the higher region of human soul life. There thoughts become picture-like and they have to be translated out of the imaginative element in order to get them into abstract form, if they are to be brought into the outer world which is not grasped by the imaginative element. But you need to understand a method of looking at these things, such as is presented, for instance, in my Occult Science. In this book the endeavor is to be as true to the facts as possible, and it is this which has so disturbed the people who are only able to think abstractly. For the attempt must be made to get things over into picture form, as I have done to some extent in the description of the Saturn and Sun states. There you will find purely picture concepts mixed in with the others. It is very hard for people to go over into the pictures, because these things cannot be put into the abstract form. The reason for this is that when we think abstractly, when we move within the narrow confines of concepts, in which people today are so much at home, and especially so in the realm of natural science, when we do this we are using ideas completely dependent on our bodies. We cannot, for instance, do without our bodies when we set out to think through the things set forth as laws in the physics books. There we must think in such a way that we use our bodies as instruments. When we rise to the sphere of the imagination, then the abstract ideas must be completely altered, because our inner soul life no longer uses the physical body.

Now you can take what I might call a comprehensive view of the realm of imaginative thought. This realm of imaginative thought has in us nothing to do with what is tied up in our outer corporeality. We rise to a region where we live as beings of soul and spirit without dependence on our corporeality. In other words, the instant we enter the realm of the imaginative, we leave space. We are then no longer in space.

Note now, this has an extremely important bearing. I have in the previous course, made a very definite differentiation between mere kinematics and what enters into our consideration as mechanical, such as mass, for instance. As long as I consider only kinematics, I need only think of things. I can write them down on a blackboard or a sheet of paper and complete the survey of motion and space so far as my thinking takes me. But in that case I must remain within what can be surveyed in terms of time and space. Why is this? This is so for a very definite reason. You must make the following clear to yourselves: All human beings, as they exist on earth, are as you yourselves, within time and space. They are bounded by a definite space and are related as space objects to other space objects. Therefore, when you speak of space, you are not able, considering the matter in an unprejudiced way, to take seriously the Kantian ideas. For if space were inside of us, then we could not ourselves be within space. We only think space is inside of us. We can free ourselves of this fancy, of this notion, if we consider the fact that this being-within space has a very real meaning for us. If space were inside of us, it would have no meaning for a person whether he were born in Moscow or Vienna. But where we are born has a very real significance. As a terrestrial-empirical person, I am quite completely a product of space facts. That is, as a human being, I belong to relations that form themselves in space. Likewise, with time, you would all be different persons if you had been born 20 years earlier. That is to say, your life does not have time inside of it, but time has your life within it. Thus as experiencing persons, you stand within time and space. And when we talk of time and space, or when we make a picture of will impulses, as I have explained we do in geometry, this is because we ourselves live inside of spatial and temporal relations, and are therefore quite definitely conditioned by them, and so are able, a priori, to speak of them as we do in mathematics. When you go over to the concept of mass, this is not so. The matter must then be put otherwise. In respect to mass, you are dealing with something quite special. You cannot say that you cut out a portion of time or space, but rather that you live in the general space mass and make it into your own mass. This mass then, is within you. It cannot be gainsaid that this mass with all its activities, all of its potentialities, is active inside of you; at this moment it falls into a different category from time and space so far as its relations to you are concerned. It is precisely because you yourself take part, as it were, with your inner being in the properties of the mass, because you take it up into your being, that it does not allow itself to be brought into consciousness like time and space. In the realm where the world gives us our own substance, we thus enter an unknown region. This is related to the fact that our will is, for instance, closely connected with the phenomena of mass inside us. But we are unconscious of these phenomena; we are asleep to them. And we are related to the will activity and accompany mass phenomena within us in no other way than we are to the world in general between going to sleep and waking up. We are not conscious of either one. Both these things are hidden from human consciousness, and in this respect, there is no immediate distinction between them.

Thus we gradually bring these things nearer to the human being. It is this that the physicists shy away from, the bringing of such things near to man. But in no other way can we obtain real concepts except by developing relationship between the human being and the world, a relationship that does not exist at the start, as in the case of time and space. We speak of time and space, let us say, out of our rational faculties, whence comes the remoteness of the mathematical and kinematical sciences. Of the things experienced merely through the senses, in an external fashion, things related to mass, we can at first speak only in an empirical fashion. But we can analyze the relation between the activity of a portion of mass within us and outer mass activity. As soon as we do this we can begin to deal with mass in the same way that we deal with the obvious relation between ourselves and time or ourselves and space. That is, we must grow inwardly into such relation with the world in our physical concepts, as we have for the mathematical or kinematical concepts.

It is a peculiar thing that, as we loosen ourselves from our own bodies in which all those things take place to which we are asleep, as we raise ourselves to imaginative concepts, we really take a step nearer the world. We approach always nearer to that which otherwise reigns in us unconsciously. There is no other way to enter into the objectivity of the facts than to push forward with our own developed inner soul forces. At the same time that we detach ourselves from our own materiality, we approach more and more closely to what is going on in the outside world.

However, it is not so easy to obtain even the most elementary experiences in this region, since a person must so transform himself that he pays attention to things that are not noticed at all under ordinary circumstances. But now, I will tell you something that will probably greatly astonish you. Let us suppose you have advanced further on the path of imaginative thinking. Suppose you have really begun to think imaginatively. You will then experience something that will astonish you. It will be much easier than it formerly was for you to recite in a merely mechanical way a poem that you have learned by heart. It will not be more difficult for you, but less so. If you examine your soul organism without prejudice and with care, you will at once find that you are more prone to recite a poem mechanically without thinking about it, if you have undergone an occult training than if you have not undergone such a training. You do not dislike this going over into the mechanical so strongly as you did before the occult development. It is such things as this that are not usually stated but are meant when it is said over and over again: The experiences you have in occult training are really opposed to the concepts that are ordinarily had before you enter occult training and thus it is, when the more advanced stage is reached, that one comes to look more lightly on the ideas of ordinary life. And therefore, anyone who advances in occultism is exposed to the danger of afterwards becoming a greater mechanist than before. An orderly occult training guards against this, but the tendency to become materialistic is quite marked in the very people who have undergone occult development. I will, by example, tell you why.

You see, in ordinary life, it is really, as the theorists say it is, the brain thinks. But ordinarily, a man does not actually experience this fact. It is quite possible in this ordinary life to carry out such a dialogue as I did in my childhood with a youthful friend who as a crass materialist and became more and more so. He would say, “When I think my brain does the thinking.” I would say to that: “ Yes, but when you are with me you always say, I will do this, I think. Why do you not say, my brain will do this, my brain thinks? You are always speaking an untruth.” The reason is that for the theoretical materialist, quite naturally, there does not exist the possibility of observing the processes in the brain. He cannot observe these physical processes. Therefore, materialism remains for him merely a theory.

The moment a person advances somewhat from imaginative to inspirational ideas, he becomes able really to observe the parallel processes in the brain. Then what goes on in the material part of the brain becomes really visible. Aside from the fact that it is extremely seductive, the things a person can observe in his own activity appear to him more and more wonderful to a high degree. For this activity of the brain is observable as something more wonderful than all that the theoretical materialists can describe about it. Therefore, the temperature comes to grow materialistic for the very reason that the activity of the human brain has become observable. Only one is, as has been said, protected from this.

But as I have explained to you these steps in occult development, I have at the same time showed you how this development creates the possibility of a deeper penetration into material processes. This is the extraordinary thing. He who functions in the spirit simply as an abstract thing, will be relatively powerless in the face of nature. He grows into contact with other natural phenomena as he has already grown into contact with time and space.

We must now set up on the one side, all the things we have just tried to place before our minds, and on the other side, those things that have met us from the realm of heat.

What has come to us from the realm of heat? Well, we followed the rise of temperature as we warmed a solid body to melting point. We showed how the temperature rise disappeared for a time, and then re-appeared until the body began to boil, to evaporate. When we extended our observations, another thing appeared. We could see that the gas produced passed over in all directions on its surroundings. (Fig. 1a), seeking to distribute itself in all directions, and could only be made to take on form if its own pressure were opposed by an equal and opposite pressure brought to bear from the outside. These things have been brought out by experiment and will be further cleared up by other experiments. The moment the temperature is lowered to the point where the body can solidify, it can give itself a form (Fig. 1b). When we experience temperature rise and fall, we experience what corresponds externally to form. We are experiencing the dissolution of form and the re-establishment of it. The gas shows us the dissolution, the solid pictures for us the establishment of form. We experience the transition between these two, also, and we experience it in an extremely interesting fashion. For, imagine to yourselves the solid and the gas and the liquid, the fluid body standing between. This liquid need not be enclosed by a vessel surrounding it completely, but only on the bottom and sides. On the upper side, the liquid forms its own surface perpendicular to the line between itself and the center of the earth. Thus we can say that we have here a transition form between the gas and the solid (Fig. 1c). In a gas we never have such a surface. In a liquid such as water, we have one surface formed. In the case of a solid, we have that all around the body which occurs in the liquid only on the upper surfaces.

Now this is an extremely interesting and significant relation. For it directs our attention to the fact that a solid body has over its entire surface something corresponding to the upper surface of a liquid, but that it determines the establishment of the surface on a body of water. It is at right angles to the line joining it to the center of the earth. The whole earth conditions the establishment of the surface. We can therefore say: In the case of water, each point within it has the same relation to the entire earth that the points in a solid have to something within the solid. The solid therefore includes something which in the case of water resides in the relation of the latter to the earth. The gas diffuses. The relation to the earth does not take part at all. It is out of the picture. Gases have no surface at all.

You will see from this that we are obliged to go back to an old conception. I called your attention in a previous lecture to the fact that the old Greek physicists called solid bodies Earth. They did this, not account of some superficial reason such as has been ascribed to them by people today, but they did it because they were conscious of the fact that the solid, of itself, takes care of that which is the case of water is taken care of by the earth as a whole. The solid takes into itself the role of the earthly. It is entirely justified to put the matter in this way: The earthly resides within a solid. In water it does not reside within, but the whole earth takes up the role of forming a surface on the liquid.

Thus you see, when we proceed from solid bodies to water, we are obliged to extend our considerations not only to what actually lies before us but in order to get an intelligent idea of the nature of water, we must extend them to include the water of the whole earth and to think of this as a unity in relation with the central point of the earth. To observe a “fragment” of water as a physical entity is absurd, just as much so as to consider a cut-off garment of my little finger as an organism. It would die at once. It only has meaning as an organism if it is considered in its relation to the whole organism. The meaning that the solid has in itself, can only be attached to water if we consider it in relation to the whole earth. And so it is with all liquids on earth.

And again, when we pass on from the fluid to the gaseous, we come to understand that the gaseous removes itself from the influence of the earth. It does not form surfaces. It partakes of everything which is not terrestrial. In other words, we must not merely look on the earth for the activities of a gas, we must bring in the environment of the earth to help us out, we must go out into space and seek there the forces involved. When we wish to learn the laws of the gaseous state, we become involved in nothing less than astronomical considerations.

Thus you see how these things are related to the whole terrestrial scheme when we examine the phenomena that we have up to this time simply gathered together. And when we come to such a point as the melting or boiling point, then there enter in things that must now appear to us as very significant. For, if we consider the melting point we pass from the terrestrial condition of the solid body where it determines its own form and relations, to something which includes the whole earth. The earth takes the sold captive when the latter goes over into the fluid state. From its own kingdom, the solid body enters the terrestrial kingdom as a whole when we reach the melting point. It ceases to have individuality. And when we carry the fluid body over into the gaseous condition, then we come to the point where the connection with the earth as shown by the formation of a liquid surface is loosened. The instant we go from a liquid to a gas, the body loosens itself from the earth, as it were, and enters the realm of the extra-terrestrial. When we consider a gas, the forces active in it are to be thought of as having escaped from the earth. Therefore, when we study these phenomena we cannot avoid passing from the ordinary physical-terrestrial into the cosmic. For we no longer are in contact with reality if our attention is not turned to what is actually working in the things themselves.

But now another phenomena meets us. Consider such a thing as the one you know very well and to which I have called your attention, namely that water behaves so remarkably, in that ice floats on water, or, stated otherwise, is less dense than water. When it goes over into the fluid condition its temperature rises, and it contracts and becomes denser. Only by virtue of this fact can ice float on the surface of the water. Here we have between zero and four degrees, water showing an exception to the general rule that we find when temperature increases, namely that bodies become less and less dense as they are warmed up. This range of four degrees, where water expands as the temperature is lowered, is very instructive. What do we learn from this range? We learn that the water sets up an opposition. As ice it is a solid body with a kind of individuality, but opposes the transition to an entirely different sphere. It is very necessary to consider such things. For then we begin to get an understanding as to why, under certain conditions, the temperature as determined by a thermometer disappears, say at the melting or boiling points. It disappears just as our bodily reality disappears when we rise to the realm of imagination. We will go into the matter a little more deeply, and it will not appear so paradoxical when we try to clear up further the following: What happens then, when a heat condition obliges us to raise the temperature to the third power, or in this case to go into the fourth dimension, thus passing out of space altogether? Let us at this time, put this proposition before our souls and tomorrow we ill speak further about it. Just as it is possible for our bodily activity to pass over into the spiritual when we enter the imaginative realm, so we can find a path leading from the external and visible in the realm of heat tot he phenomena that are pointed to by our thermometer when the temperature rise we are measuring with it disappears before our eyes. What process goes on behind this disappearance? That is the question which we are asking ourselves today. Tomorrow we will speak of it further.