The Rudolf Steiner Archive

You will have seen how we are trying in these lectures to prepare the ground for an adequate World-picture. As I have pointed out again and again, the astronomical phenomena themselves impel us to advance from the merely quantitative to the qualitative aspect. Under the influence of Natural Science there is a tendency, in modern scholarship altogether, to neglect the qualitative side and to translate what is really qualitative into quantitative terms, or at least into rigid forms. For when we study things from a formal aspect we tend to pass quite involuntarily into rigid forms, even if we went to keep them mobile. But the question is, whether an adequate understanding of the phenomena of the Universe is possible at all in terms of rigid, formal concepts. We cannot build an astronomical World-picture until this question has been answered.

This proneness to the quantitative, abstracting from the qualitative aspect, has led to a downright mania for abstraction which is doing no little harm in scientific life, for it leads right away from reality. People will calculate for instance under what conditions, if two sound-waves are emitted one after the other, the sound omitted later will be heard before the other. All that is necessary is the trifling detail that we ourselves should be moving with a velocity greater than that of sound. But anyone who thinks in keeping with real life instead of letting his thoughts and concepts run away from the reality, will, when he finds them incompatible with the conditions of man's co-existence with his environment, stop forming concepts in this direction. He cannot but do so. There is no sense whatever in formulating concepts for situations in which one can never be.

To be a spiritual scientist one must educate oneself to look at things in this way. The spiritual scientist will always want his concepts to be united with reality. He does not want to form concepts remote from reality, going off at a tangent,—or at least not for long. He brings them back to reality again and again. The harm that is done by the wrong kinds of hypothesis in modern time is due above all to the deficient feeling for the reality in which one lives. A conception of the world free of hypotheses, for which we strive and ought to strive, would be achieved far more quickly if we could only permeate ourselves with this sense of reality. And we should then be prepared, really to see what the phenomenal world presents. In point of fact this is not done today. If the phenomena were looked at without prejudice, quite another world-picture would arise than the world-pictures of contemporary science, from which far-fetched conclusions are deduced to no real purpose, piling one unreality upon another in merely hypothetical thought-structures.

Starting from this and from what was given yesterday, I must again introduce certain concepts which may not seem at first to be connected with our subject, though in the further course you will see that they too are necessary for the building of a true World-picture. I shall again refer to what was said yesterday in connection with the Ice-ages and with the evolution of the Earth altogether. To begin with however, we will take our start from another direction.

Our life of knowledge is made up of the sense-impressions we receive and of what comes into being when we assimilate the sense-impressions in our inner mental life. Rightly and naturally, we distinguish in our cognitional life the sense-perceptions as such and the inner life of ‘ideas’—mental pictures. To approach the reality of this domain we must being by forming these two concepts: That of the sense-perception pure and simple, and of the sense-perception transformed and assimilated into a mental picture.

It is important to see without prejudice, what is the real difference between our cognitional life insofar as this is permeated with actual sense-perceptions and insofar as it consists of mere mental picture. We need to see these things not merely side by side in an indifferent way; we need to recognize the subtle differences of quality and intensity with which they come into our inner life.

If we compare the realm of our sense-perceptions—the way in which we experience them—with our dream-life, we shall of course observe an essential qualitative difference between the two. But it is not the same as regards our inner life of ideas and mental pictures. I am referring now, not to their content but to their inner quality. Concerning this, the content—permeated as it is with reminiscences of sense-perceptions—easily deludes us. Leaving aside the actual content and looking only at its inner quality and character—the whole way we experience it,—there is no qualitative difference between our inner life in ideas and mental pictures and our life of dreams. Think of our waking life by day, or all that is present in the field of our consciousness in that we open our senses to the outer world and are thereby active in our inner life, forming mental pictures and ideas. In all this forming of mental pictures we have precisely the same kind of inner activity as in our dream-life; the only thing that is added to it is the content determined by sense-perception.

This also helps us realize that man's life of ideation—his forming of mental pictures—is a more inward process than sense-perception. Even the structure of our sense-organs—the way they are built into the body—shows it. The processes in which we live by virtue of these organs are not a little detached from the rest of the bodily organic life. As a pure matter of fact, it is far truer to describe the life of our senses as a gulf-like penetration of the outer world into our body (Fig. 1) than as something primarily contained within the latter. Once more, it is truer to the facts to say that through the eye, for instance, we experience a gulf-like entry of the outer world. The relative detachment of the sense-organs enables us consciously to share in the domain of the outer world. Our most characteristic organs of sense are precisely the part of us which is least closely bound to the inner life and organization of the body. Our inner life of ideation on the other hand—our forming of mental pictures—is very closely bound to it. Ideation therefore is quite another element in our cognitional life than sense-perception as such. (Remember always that I am thinking of these processes such as they are at the present stage in human evolution.)

Now think again of what I spoke of yesterday—the evolution of the life of knowledge from one Ice-Age to another. Looking back in time, you will observe that the whole interplay of sense-perceptions with the inner life of ideation—the forming of mental pictures—has undergone a change since the last Ice-Age. If you perceive the very essence of that metamorphosis in the life of knowledge which I was describing yesterday, then you will realize that in the times immediately after the decline of the Ice-Age the human life of cognition took its start from quite another quality of experience than we have today. To describe it more definitely; whilst our cognitional life has become more permeated and determined by the senses and all that we receive from them, what we do not receive from the senses—what we received long, long ago through quite another way of living with the outer world—has faded out and vanished, ever more as time went on. This other quality—this other way of living with the world—belongs however to this day to our ideas and mental pictures. In quality they are like dreams. Fro in our dreams we have a feeling of being given up to, surrendered to the world around us. We have the same kind of experience in our mental pictures. While forming mental pictures we do not really differentiate between ourselves and the world that then surrounds us; we are quite given up to the latter. Only in the act of sense-perception do we separate ourselves from the surrounding world. Now this is just what happened to the whole character of man's cognitional life since the last Ice-Age. Self-consciousness was kindled. Again and again the feeling of the “I” lit up, and this became ever more so.

What do we come to therefore, as we go back in evolution beyond the last Ice-Age? (We are not making hypotheses; we are observing what really happened.) We come to a human life of soul, not only more dream-like than that of today, but akin to our present life of ideation rather than to our life in actual sense-perception. Now ideation—once again, the forming of mental pictures—is more closely bound to the bodily nature than is the life of the senses. Therefore what lives and works in this realm will find expression rather within the bodily nature than independently of the latter. Remembering what was said in the last few lectures, this will then lead you from the daily to the yearly influences of the surrounding world. The daily influences, as I showed, are those which tend to form our conscious picture of the world, whereas the yearly influences affect our bodily nature as such. Hence if we trace what has been going on in man's inner life, as we go back in time we are led from the conscious life of soul deeper and deeper into the bodily organic life.

In other works; before the last Ice-Age the course of the year and the seasons had a far greater influence on man than after. Man, once again, is the reagent whereby we can discern the cosmic influences which surround the Earth. Only when this is seen can we form true ideas of the relations—including even those of movement—between the Earth and the surrounding heavenly bodies. To penetrate the phenomena of movement in the Heavens, we have to take our start from man—man, the most sensitive of instruments, if I may call him so. And to this end we need to know man; we must be able to discern what belongs to the one realm, namely the influences of the day, and to the other, the influences of the year.

Those who have made a more intensive study of Anthroposophical Science may be reminded here of what I have often described from spiritual perception; the conditions of life in old Atlantis, that is before the last Ice-Age. For I was there describing from another aspect—namely from direct spiritual sight—the very same things which we are here approaching more by the light of reason, taking our start from the facts of the external world.

We are led back then to a kind of interplay between the Earth and its celestial environment which gave men an inner life of ideation—mental pictures—and which was afterwards transmuted in such a way as to give rise to the life of sense-perception in its present form. (The life of the senses as such is of course a much wider concept; we are here referring to the form it takes in present time.)

But we must make a yet more subtle distinction. It is true that self-consciousness or Ego-consciousness, such as we have it in our ordinary life today, is only kindled in us in the moment of awakening. Self-consciousness trikes in upon us the moment we awaken. It is our relation to the outer world—that relation to it, into which we enter by the use of our senses—to which we owe our self-consciousness. But if we really analyze what it is that thus strikes in upon us, we shall perceive the following. If our inner life in mental pictures retained its dream-like quality and only the life of the senses were added to it, something would still be lacking. Our concepts would remain like the concepts of fantasy or fancy (I do not say identical with these, but like them). We should not get the sharply outlined concepts which we need for outer life. Simultaneously therefore with the life of the senses, something flows into us from the outer world which gives sharp outlines and contours to the mental pictures of our every-day cognitional life. This too is given to us by the outer world. Were it not for this, the mere interplay of sensory effects with the forming of ideas and mental pictures would bring about in us a life of fantasy or fancy and nothing more; we should never achieve the sharp precision of every-day waking life.

Now let us look at the different phenomena quite simply in Goethe's way, or—as has since been said, rather more abstractly—in Kizchhoff's way. Before doing so I must however make another incidental remark, Scientists nowadays speak of a “physiology of the senses”, and even try to build on this foundation a “psychology of the senses”, of which there are different schools. But if you see things as they are, you will find little reality under these headings. In effect, our senses are so radically different from one-another that a “Physiology of the senses”, claiming to treat them all together, can at more be highly abstract. All that emerges, in the last resort, is a rather scanty and even then very questionable physiology and psychology of the sense of touch, which is transferred by analogy to the other senses. If you look for what is real, you will require a distinct physiology and a distinct psychology for every one of the senses.

Provided we remember this, we may proceed. With all the necessary qualifications, we can then say the following. Look at the human eye. (I cannot now repeat the elementary details which you can find in any scientific text-book.) Look at the human eye, one of the organs giving us impressions of the outer world,—sense-impressions and also what gives them form and contour. These impressions, received through the eye, are—once again—connected with all the mental pictures which we then make of them in our inner life.

Let us now make the clear distinction, so as to perceive what underlies the sharp outline and configuration which makes our mental images more than mere pictures of fancy, giving them clear and precise outline. We will distinguish this from the whole realm of imagery where this clarity and sharpness is not to be found,—where in effect we should be living in fantasies. Even through what we experience with the help of our sense-organs—and what our inner faculty of ideation makes of it—we should still be floating in a realm of fancies. It is through the outer world that all this imagery receives clear outline, finished contours. It is through something from the outer world, which in a certain way comes into a definite relation to our eye.

And now look around. Transfer, what we have thus recognized as regards the human eye, to the human being as a whole. Look for it, simply and empirically, in the human being as a whole. Where do we find—though in a metamorphosed form—what makes a similar impression? We find it in the process of fertilization. The relation of the human being as a whole—the female human body—to the environment is, in a metamorphosed form, the same as the relation of the eye to the environment. To one who is ready to enter into these things it will be fully clear. Only translated, one might say, into the material domain, the female life is the life of fantasy or fancy of the Universe, whereas the male is that which forms the contours and sharp outlines. It is the male which transforms the undetermined life of fancy into a life of determined form and outline. Seen in the way we have described in today's lecture, the process of sight is none other than a direct metamorphosis of that of fertilization; and vice-versa.

We cannot reach workable ideas about the Universe without entering into such things as these. I am only sorry that I can do no more than indicate them, but after all, these lectures are meant as a stimulus to further work. This I conceive to be the purpose of such lectures; as an outcome, every one of you should be able to go on working in one or other of the directions indicated. I only want to show the directions; they can be followed up in diverse ways. There are indeed countless possibilities in our time, to carry scientific methods of research into new directions. Only we need to lay more stress on the qualitative aspects, even in those domains where one has grown accustomed to a mere quantitative treatment.

What do we do, in quantitative treatment? Mathematics is the obvious example; ‘Phoronomy’ (Kinematics) is another. We ourselves first develop such a science, and we then look to find its truths in the external, empirical reality. But in approaching the empirical reality in its completeness we need more than this. We need a richer content to approach it with, than merely mathematical and phoronomical ideas. Approach the world with the premises of Phoronomy and Mathematics, and we shall naturally find starry worlds, or developmental mechanisms as the case may be, phoronomically and mathematically ordered. We shall find other contents in the world if once we take our start from other realms than the mathematical and phoronomical. Even in experimental research we shall do so.

The clear differentiation between the life of the senses and the organic life of the human being as a whole had not yet taken place in the time preceding the last Ice-Age. The human being still enjoyed a more synthetic, more ‘single’ organic life. Since the last Ice-Age man's organic life has undergone, as one might say, a very real ‘analysis’. This too is an indication that the relation of the Earth to the Sun was different before the last Ice-Age from what it afterwards became. This is the kind of premise from which we have to take our start, so as to reach genuine pictures and ideas about the Universe in its relation to the Earth and man.

Moreover our attention is here drawn to another question, my dear Friends. To what extent is ‘Euclidean space’—the name, of course, does not matter—I mean the space which is characterized by three rigid directions at right angles to each other. This, surely, is a rough and ready definition of Euclidean space. I might also call it ‘Kantian space’, for Kant's arguments are based on this assumption. Now as regards this Euclidean—or, if you will, Kantian—space we have to put the question: Does it correspond to a reality, or is it only a thought-picture, an abstraction? After all, it might well be that there is really no such thing as this rigid space. Now you will have to admit; when we do analytical geometry we start with the assumption that the X-, Y- and Z-axes may be taken in this immobile way. We assume that this inner rigidity of the X, Y and Z has something to do with the real world. What if there were nothing after all, in the realms of reality, to justify our setting up the three coordinate axes of analytical geometry in this rigid way? Then too the whole of our Euclidean Mathematics would be at most a kind of approximation to the reality—an approximation which we ourselves develop in our inner life,—convenient framework with which to approach it in the first place. It would not hold out any promise, when applied to the real world, to give us real information.

The question now is, are there any indications pointing in this direction,—suggesting, in effect, that this rigidity of space can not, after all, be maintained? I know, what I am here approaching will cause great difficulty to many people of today, for the simple reason that they do not keep step with reality in their thinking. They think you can rely upon an endless chain of concepts, deducing one thing logically from another, drawing logical and mathematical conclusions without limit. In contrast to this tendency in science nowadays, we have to learn to think with the reality,—not to permit ourselves merely to entertain a thought-picture without at least looking to see whether or not it is in accord with reality. So in this instance, we should investigate. Perhaps after all, by looking into the world of concrete things, there is some way of reaching a more qualitative determination of space.

I am aware, my dear Friends, that the ideas I shall now set forth will meet with great resistance. Yet it is necessary to draw attention to such things. The theory of evolution has entered ever more into the different fields of science. They even began applying it to Astronomy. (This phase, perhaps, is over now, but it was so a little while ago.) They began to speak of a kind of natural selection. Then as the radical Darwinians would do for living organisms, so they began to attribute the genesis of heavenly bodies to a kind of natural selection, as though the eventual form of our solar system had arisen by selection from among all the bodies that had first been ejected. Even this theory was once put forward. There is this p to the whole Universe the leading ideas that have once been gaining some particular domain of science.

So too it came about that man was simply placed at the latter end of the evolutionary series of the animal kingdom. Human morphology, physiology etc. were thus interpreted. But the question is whether this kind of investigation can do justice to man's organization in its totality. For, to begin with, it omits what is most striking and essential even from a purely empirical point of view. One saw the evolutionists of Haechel's school simply counting how many bones, muscles and so on man and the higher animals respectively possess. Counting in that way, one can hardly do otherwise than put man at the end of the animal kingdom. Yet it is quite another matter when you envisage what is evident for all eyes to see, namely that the spine of man is vertical while that of the animal is mainly horizontal. Approximate though this may be, it is definite and evident. The deviations in certain animals—looked into empirically—will prove to be of definite significance in each single case. Where the direction of the spine is turned towards the vertical, corresponding changes are called forth in the animal as a whole. But the essential thing is to observe this very characteristic difference between man and animal. The human spine follows the vertical direction of the radius of the Earth, whereas the animal spine is parallel to the Earth's surface. Here you have purely spatial phenomena with a quite evident inner differentiation, inasmuch as they apply to the whole figure and formation of the animal and man. Taking our start from the realities of the world, we cannot treat the horizontal in the same way as the vertical. Enter into the reality of space—see what is happening in space, such as it really is,—you cannot possibly regard the horizontal as though it were equivalent or interchangeable with the vertical dimension.

Now there is a further consequence of this. Look at the animal form and at the form of man. We will take our start from the animal, and please fill in for yourselves on some convenient occasion what I shall now be indicating. I mean, observe and contemplate for yourselves the skeleton of an mammal. The usual reflections in this realm are not nearly concrete enough; they do not enter thoroughly enough into the details.

Consider then the skeleton of an animal. I will go no farther than the skeleton, but what I say of this is true in an even higher degree of the other parts and systems in the human and animal body. Look at the obvious differentiation, comparing the skull with the opposite end of the animal. If you do this with morphological insight, you will perceive characteristic harmonies or agreements, and also characteristic diversities. Here is a line of research which should be followed in far greater detail. Here is something to be seen and recognized, which will lead far more deeply into realty than scientists today are wont to go.

It lies in the very nature of these lectures that I can only hint at such things, leaving out many an intervening link. I must appeal to your own intuition, trusting you to think it out and fill in what is missing between one lecture and the next. You will then see how all these things are connected. If I did otherwise in these few lectures, we should not reach the desired end.

Diagrammatically now (Fig. 2), let this be the animal form. If after going into an untold number of intervening links in the investigation, you put the question: ‘What is the characteristic difference of the front and the back, the head and the tail end due to?’, you will reach a very interesting conclusion. Namely you will connect the differentiation of the front end with the influences of the Sun. Here is the Earth (Fig. 3). You have an animal on the side of the Earth exposed to the Sun. Now take the side of the Earth that is turned away from the Sun. In one way or another it will come about that the animal is on this other side. Here too the Sun's rays will be influencing the animal, but the earth is now between. In the one case the rays of the Sun are working on the animal directly; in the other case indirectly, inasmuch as the Earth is between and the Sun's rays first have to pass through the Earth (Fig. 3).

Expose the animal form to the direct influence of the Sun and you get the head. Expose the animal to those rays of the Sun which have first gone through the Earth and you get the opposite pole to the head. Study the skull, so as to recognize in it the direct outcome of the influences of the Sun. Study the forms, the whole morphology of the opposite pole, so as to recognize the working of the Sun's rays before which the Earth is interposed—the indirect rays of the Sun. Thus the morphology of the animal itself draws our attention to a certain interrelation between Earth and Sun. For a true knowledge of the mutual relations of Earth and Sun we must create the requisite conditions, not by the mere visual appearance (even though the eye be armed with telescopes), but by perceiving also how the animal is formed—how the whole animal form comes into being.

Now think again of how the human spine is displaced through right angle in relation to the animal. All the effects which we have been describing will undergo further modification where man is concerned. The influences of the Sun will therefore be different in man than in the animal. The way it works in man will be like a resultant (Fig. 4). That is to say, if we symbolize the horizontal line—whether it represent the direct or the indirect influence of the Sun—by this length, we shall have to say; here is a vertical line; this also will be acting. And we shall only get what really works in man by forming the resultant of the two.

Suppose in other words that we are led to relate animal formation quite fundamentally to some form of cosmic movement—say, a rotation of the Sun about the Earth, or a rotation of the Earth about its own axis. If then this movement underlies animal formation, we shall be led inevitably to attribute to the Earth or to the Sun yet another movement, related to the forming of man himself,—a movement which, for its ultimate effect, unites to a resultant with the first. From what emerges in man and in the animal we must derive the basis for a true recognition of the mutual movements among the heavenly bodies.

The study of Astronomy will thus be lifted right out of its present limited domain, where one merely takes the outward visual appearance, even if calling in the aid of telescopes, mathematical calculations and mechanics. It will be lifted into what finds expression in this most sensitive of instruments, the living body. The forming forces working in the animal, and then again in man, are a clear indication of the real movements in celestial space.

This is indeed a kind of qualitative Mathematics. How, then, shall we metamorphose the idea when we pass on from the animal to the plant? We can no longer make use of either of the two directions we have hitherto been using. Admittedly, it might appear as though the vertical direction of the plant coincided with that of the human spine. From the aspect of Euclidean space it does, no doubt (Euclidean space, that is to say, not with respect to detailed configuration but simply with respect to its rigidity.) But it will not be the same in an inherently mobile space. I mean a space, the dimensions of which are so inherently mobile that in the relevant equations, for example, we cannot merely equate the \(x\)- and the \(y\)-dimensions: \(y = ƒ(x)\). (The equation might be written very differently from this. You will see what I intend more from the words I use than from the symbols; it is by no means easy to express in mathematical form.) In a co-ordinate system answering to what I now intend, it would no longer be permissible to measure the ordinates with the same inherent measures as the abscissae. We could not keep the measures rigid when passing from the one to the other. We should be led in this way from the rigid co-ordinate system of Euclidean space to a co-ordinate system that is inherently mobile.

And if we now once more ask the question: How are the vertical directions of plant growth and of human growth respectively related?—we shall be led to differentiate one vertical from another. The question is, then, how to find the way to a different idea of space from the rigid one of Euclid. For it may well be that the celestial phenomena can only be understood in terms of quite another kind of space—neither Euclidean, nor any abstractly conceived space of modern Mathematics, but a form of space derived from the reality itself. if this is so, then there is no alternative; it is in such a space and not in the rigid space of Euclid that we shall have to understand them.

Thus we are led into quite other realms, namely to the Ice-Age on the one hand and on the other to a much needed reform of the Euclidean idea of space. But this reform will be in a different spirit than in the work of Minkowski and others. Simply in contemplating the given facts and trying to build up a science free of hypotheses, we are confronted with the need for a thoroughgoing revision of the concept of space itself. Of these things we shall speak again tomorrow.

To lead our present studies to a fruitful conclusion we must still pursue the rather subtle course I have been adopting, bringing together a great variety of ideas from different fields. For this reason we shall have to continue with this course also while the other course¹ is going on—between the 11th and 15th January. We must arrange the times by agreement with the Waldorf School. There is so much to bring in that we shall need these days too. Now I am also well aware how many queries, doubts and problems may be arising in connection with this subject. Please prepare whatever questions you would like to put, if you need further elucidation. I will then try to incorporate the answers in one of next week's lectures, so as to make the picture more complete. Working in this way we shall be able to continue as heretofore, bringing in what I would call the subtler aspects of our theme.

Let us envisage once again the course we have been pursuing. Our aim is to gain a deeper understanding of Astronomy—the science of the Heavens—in connection with phenomena on Earth. To begin with, we pointed out that as a rule the Astronomy of our time only takes into account what is observed directly with the outer senses aided, no doubt, by optical instruments and the like. Such, in the main, were all the data hitherto adduced when seeking to explain and understand the phenomena of the Heavens. They took their start from the ‘apparent movements’, as they would now be called, or the celestial bodies. First they considered the apparent movement of the starry Heavens as a whole around the Earth and the apparent movement of the Sun. Then they observed the very strange paths described by the Planets. Such, in effect, is the immediate visual appearance; portions of the planetary paths look like loops (Fig. 1) the planet moves along here, reverses and goes back, and then forward again, here ... And now they reasoned; if the Earth itself is moving and we have no direct perception of this movement, the real movements of the heavenly bodies cannot but be different from the visual appearance. Interpreting along these lines—applying mathematical and geometrical laws—they arrived at an idea of what the ‘real’ movements might be like. So they arrived at the Copernican system and at its subsequent modifications. Such, in the main, were the methods of cognition used; first, what our senses when looking out into the Heavens, and then the intellectual assimilation, the reasoned interpretation of these sense-impression.

We then pointed out that this procedure can never lead to the adequate penetration of the celestial phenomena, if only for the reason that the mathematical method itself is insufficient. We begin our calculations along certain lines and are then brought to a stop. For as I was reminding you, the ratios between the periods of revolution of the several planets are incommensurable numbers,—incommensurable magnitudes. By calculation therefore, we do not reach the innermost structure of the celestial phenomena. Sooner or later we have to leave off.

It follows that we must adopt a different method. We have to take our start not only from what man observes when he looks out into the Universe with his senses; we must take man as a whole in his connection with the Universe, and perhaps not only man, but other creatures too,—the kingdoms of Nature upon Earth. All these things we pointed out, and I then showed how the whole organization of man can be seen in relation to certain phenomena in the evolution of the Earth, namely the Ice-Ages in their rhythmical recurrence. They also have to do with the inner evolution of man and of mankind. This too, I said, will give us indications of what the real movements in celestial space may be. These are the kind of things we must pursue.

Before continuing the rather more formal lines of thought with which we ended yesterday's lecture, let us consider once again this connection of man's evolution with the evolution of the Earth through the Ice-Ages. We saw that the special kind of knowledge or of cognitional life which the man of present time calls his own has only come into being since the last Ice-Age. Moreover all the civilization-epochs, of which I have so often told, have taken place since then—namely the Ancient Indian, the Persian, the Egypto—Chaldean, the Graeco-Latin and then the epoch in which we are now living. Before the last Ice-Age, we said, there must have been developing in human nature what in the man of today is more withdrawn, less at the surface of his nature, namely his power of ideation—the forming of mental pictures. The inner quality, we said, of this part of our inner life is truly to be understood only if we compare it with our dream-life. It is through sense-perception that our mental pictures receive clear and firm configuration and, as it were, a fully saturated content. The mental pictures are being formed in a more inward region of our bodily organic life—farther back, as it were behind the sense-perceptions,—and this activity is dim and hazy like our dream-life. Our forming of mental pictures would be as dim as it is in dreams, if the experiences of the senses did not strike in upon us every time we awaken. (We may allow the supposition, to help explain what is meant.)

More dim and hazy than our life in sense-perception, this inner life of ideation, mental imagery, is related to those earlier phases in the evolution of man's nature which preceded the last glacial epoch, or which—to speak in anthroposophical terms—belonged to old Atlantis.

What must it then have been like for man? In the first place he must have had a far more intimate inner connection with the surrounding world than he has today through sense-perception. We can control our sense-perception with our will. It is with our will at any rate that we direct the vision of our eyes, and by deliberate attention we can go even farther in governing our sense-perception by our own will. At all events, our will is very much at work in our sense—perceptions, making us to a large extent independent of the outer world. We orientate ourselves by our own arbitrary choice. Now this in only possible because as human beings we have in a way emancipated ourselves from the Universe. Before the last Ice-Age we cannot have been thus emancipated. (I say ‘cannot have been’ since I am wanting now to speak from the empirical aspect of external Science.) During that time, as we have seen, the power of ideation—the forming of mental images—was especially developed, and in his inner conditions man must have been far more dependent on all that was going on around him. Today we see the world around us shining in the sunlight, but the way we see it is considerably subject to the inner culture and control of our own life of will. In Atlantean time the way man was given up to the outer world must have been somehow dependent on the illumined Earth and its illumined objects, and then again—at night-time when the Sun was not shining—on the darkness, the gloaming. He must in other words have experienced periodic alternations in this respect. His inner life of mental imagery, which as we saw was then in process of development must alternately have been lighting up and ebbing down again. This inner periodicity, brought about by man's relation to the surrounding Universe, was indeed not unlike the peculiar periodicity of woman's organic functions of which we spoke before, which is related to the Lunar phases though only as regards length of time. This inner functioning of the woman's nature (I said, you will remember, it is there in man too but in a more inward way and therefore less easily perceived) was at one time actually linked with the corresponding events in the outer Universe. It then became emancipated—a property of human nature on its own,—so that what now goes on in the human being in this respect need not coincide with the outer events. yet the periodicity—the sequence of phases—remains the same as it was when the one coincided with the other.

Something quite similar is true of the rhythmic alternation in our inner life—in our ideation, our forming of mental images. The whole way we are organized in this respect, implanted in us in a far distant past, is to this day more or less independent of the life of the outer senses. Day by day we undergo an inner rhythm, our powers of mental imagery alternately lighting up and growing more dim; it is a daily ebb and flow. We only fail to notice it, since it is far less intense than that other periodicity which runs parallel to the Lunar phases. Nevertheless, in our head-organization to this day we have an alternation between a brighter and a dimmer kind of life. We carry in our head a rhythmic life. We are at one time more and at another less inclined to meet our sense-perceptions actively from within. It is a 24-hour rhythmic alteration. It would be interesting to observe—it might even be recorded in graphically—how human being vary as regards this inner period of the head, the forces of ideation and mental imagery alternating between brighter and more lively and then again dimmer and more sleepy times. The dim and sleepy times represent, so to speak, the inner night of the head, the brighter ones the inner day, but it does not coincide with the external alternation of day and night. It is an inner alternation of light and darkness, or relatively bright and dim conditions. And people vary in this respect. One human being has this inner alternation of light and dark in such a way that he tends rather to connect the lighter period of his mental image-forming power with his sense-perceptions. Another tends to it with the darker. Individuals are organized in one way or the other, and differ accordingly as to their power of observing the outer world. One human being will be inclined sharply to focus the phenomena of the outer world; another tends to do so less,—is more inclined to an inner brooding. All this is due to the alternating conditions I have been describing. Notably as educators, my dear Friends, we should cultivate the habit of observing things like this. They will be valuable signposts, indicating how we should treat the individual children both in our teaching and in education generally.

What interests us however here and now is the fact that man thus makes inward, as it were, what he once underwent in direct mutual relation with the outer world; so that it now works in him as an inner rhythm, the phases no longer coinciding with the outer yet still retaining the periodicity Before the Ice-Age, man's periods of brighter and more intimate participation in the surrounding Universe,. and then of dim withdrawal into himself, will have coincided regularly with the processes of the outer world. He still retains an echo of this rhythm, which in those long-ago times proceed from his living-together with the Universe around him, where at one moment his consciousness was lightened and filled with pictures while at another he withdrew into himself, brooding over the pictures. It is an echo of this latter state whenever we today are inclined to brood more or less melancholically in our own inner life. Once again therefore, what man experienced in and with the world in those older times has been driven farther back into his inner bodily nature, while at the outer periphery a new development has taken place in his faculties of sense-perception. He had these faculties, of course in earlier epochs too, but not developed in the way they now are.

While looking thus at what has taken place in man through his connection with the phenomena of the world around him, we are in fact looking into the Universe itself. Man then becomes the reagent for a true judgment of the phenomena of the Universe. But to complete this we need the other kingdoms of Nature too. Here I should like to draw your attention to something well-known and evident to everyone, the essential significance of which, however, remains unrecognized.

Consider the annual plant,—the characteristic cycle of its development. We see in it quite evidently what I was mentioning yesterday—the direct and indirect influences of the Sun. Where the Sun works directly, the flower comes into being; where the Sun works in such a way that the Earth comes in between, we get the root. The plant too makes manifest what we were speaking of yesterday as regards the animal and then applied in another way to man.

Yet we shall only see the full significance of this if we relate it to another fact. There are perennial plants too. What is the relation of the perennial plant to the annual, as regards the way in which plant-growth belongs to the Earth as a whole? The perennial retains its stem or trunk, and the truth is: Year by year a new world of plants springs, so to speak, from the trunk itself. Of course it is modified and metamorphosed, yet it is a vegetation growing on the trunk, which in its turn grows out of the Earth (Fig. 2). If you have morphological perception you will see it as clearly as can be,—it almost goes without saying. Here on the left I have the surface of the Earth, and the annual plant springing from it. Here on the right is the stem or trunk of the perennial, from which new vegetation, new plant-growth springs in each succeeding year. I must image something or other (to leave it vague, for the moment) continued from the Earth into the trunk. I must say to myself—what this plant here (Fig. 2 on the left) is growing on, must somehow be there in the trunk too (on the right). In other words there must be some element of the Earth—whatever it may be—entering into the trunk. I have no right to regard the trunk of the perennial as a thing apart, not belonging to the Earth; rather must I regard it as a modified portion of the Earth itself. Only then shall I be seeing it rightly; only then shall I discern the inner relationships, such as they really are. Something is there in the perennial plant, which otherwise is only in the Earth. It is through this that the plant becomes perennial. In effect, precisely by taking something of the Earth into itself it frees itself from dependence on the yearly course of the Sun. For we may truly say: The perennial wrests itself away from its dependence on the Sun's yearly course. it emancipates itself from the yearly course of the Sun, in that it forms the trunk, receiving into its own Nature—becoming able, as it were, to do for itself what otherwise could only come about through the working of the whole cosmic environment.

Do we not here see prefigured in the plant world, what I was just describing with regard to man in preglacial times? For in those times, as I was showing, the inner rhythm of the man's ideation—his life in mental pictures—developed by relation to the surrounding world. What then lived in the mutual relation between man and the surrounding world has since become a feature of his own inner life. There is an indication of the same kind of change in the plant kingdom, in that the annual is changed to a perennial. This is indeed a universal tendency in evolution; the living entities are on the way to emancipation from their original connections with the surrounding world.

Seeing the perennials arising, we have to say: It is as though the plant, when it becomes perennial, had learned something it you will allow the expression—learned from the time when it depended on cosmic environment, something which it can now do for itself. Now it is able of itself to bring forth fresh plant-shoots year by year. We do not reach an understanding of the phenomena of the world by merely staring at the things that happen to be side by side, or that are crowded into the field of view under the microscope. We have to see the larger whole and recognize the single phenomena in their connection with it.

Look at it all once more. The annual plant is given up to the cycle of the year, with all the changing relations to the Cosmos which this involves. This influence of the Cosmos beings to fade away in the perennial. In the perennial, what would otherwise vanish in the further course of the year is, as it were, preserved. In the trunk we see springing from the ground the working of the year, made permanent and lasting. This transition of what was first connected with the outer Universe into a more inward way of working we see it throughout the whole range of Nature's phenomena, in so far as they are cosmic. Hence too there are phenomena in which we can more quickly find the living connections between our Earth and the wider Cosmos, whilst there are others in which the cosmic influences are more concealed. We need to find out which of them are sensitive reagents, telling of the cosmic influences. The annual plant will tell us of the Earth's connection with Cosmos, the perennial will not be able to tell us much.

Again, the relation of the animal to man can give us an important clue. Look at the animal's development. (Though we might also include it, we will for the moment disregard the embryonic life.) The animal is born and grows up to a certain limit. It reaches puberty. Look at the animal's whole life, until puberty and beyond. Without any added hypotheses—taking the simple facts—you must admit that it is strange, what happens to the animal once puberty has been attained. For in a way the animal is finished then, so far as the earthly world is concerned. Any such statement is of course an approximation to the truth, needless to say; yet in the main we must admit that in the animal no further progression is to be seen, not after puberty. Puberty is the important goal of animal development. The immediate consequence of puberty—all that happens as an outcome of it—is there of course, but we cannot allege that anything takes place thence forward, deserving to be called a true progression.

With man it is different. Man remains capable of development far beyond puberty; but the development becomes more inward. Indeed it would be very sad for man if in his human nature he were to end his development at puberty in the way animals do. Man goes beyond this. He holds something in reserve by means of which he can go farther,—can undertake quite other journeys, unconnected with sexual maturity or puberty. This again is not unlike the “inwarding” of the cycle of the year in the perennial as against the annual plant. What is in evidence in the animal when puberty is reached, we see it transmuted into a more inward process in man, from puberty onward. Something therefore is at work in man, that is related to a cosmic process in his development from birth until puberty, and that then gets emancipated from the Cosmos—just as it does in the perennial plant—when puberty has been outgrown.

Here then you have a subtler way of estimating the phenomena among the kingdoms of Nature; so will you presently find signposts, indicating the connections between the creatures upon Earth and the Cosmos. We see how, when the cosmic influences cease as such, they are transplanted into the inner nature of the several creatures. We will take note of this and set it on one side for the moment; later we shall find the synthesis between this and quite another aspect.

Let us now take up again what I have frequently mentioned: The incommensurable ratios between the periods of revolution of the planets of the solar system. We may ask, what would the outcome be if they were commensurable? Cumulative disturbances would arise, whereby the planetary system would be brought to a standstill. This can be proved by calculation, though it would lead too far afield to do it now. Only the incommensurability between the periods of revolution enables the planetary system, so to speak, to stay alive. In other words, the solar system contains among other things a condition even tending to a standstill. It is precisely this condition which we are calculating. When in our calculations we get to the end of our tether, there is the incommensurable—and there, withal, is the very life of the planetary system! We are in a strange predicament when calculating the planetary system. If it were such that we could fully calculate it, it would die,—nay, as I said before, would have died long ago. It lives by virtue of the face that we can not calculate it fully. What is alive in the planetary system is precisely what we cannot calculate.

Now upon what do we base these calculations, from which once more, if we could pursue them to the end, we must deduce the inevitable death of the whole system? We base them on the force of gravitation—universal gravitation. Suppose we take our start from gravitation and nothing more, and think it out consistently. We get the picture of a planetary system subject to the force of gravitation. Then indeed we do arrive at commensurable ratios. But the planetary system would inevitably die. We calculate, in other words, to the extent that death prevails in the planetary system, basing our calculations on the force of gravity. In other words there must be something in the planetary system—different from gravitation—to which the incommensurability is due.

The planetary orbits can be brought into accord with the force of gravity very nicely, even as to their genesis, but their periods of revolution would then have to be commensurable. Now there is something which cannot be brought into accord with the force of gravitation, and which moreover does not so tidily fit into our planetary system. I mean what reveals itself in the cometary bodies. The comets play a very strange part in the system, and they have recently been leading scientists to some unusual ideas.

I leave aside the kind of explanations which often tend to arise, where anything most recently discovered is seized on to explain phenomena in other fields. In physiology for instance there was a time when they were fond of comparing the so-called sensory nerves to telegraph-wires leading in from the periphery. Through some central switch or commutator the impulse was supposed to be transmitted, leading to impulses and acts of will. From the centripetal nerves it was supposed to be switched over to the centrifugal; they compared it all to a telegraphic system. Maybe one day something quite different from telegraph-wires will be invented and by this way of thinking quite another picture will be applied to the same thing. So do the scientific fashions change. Whatever happens to have been discovered is quickly seized on as a handy way of explaining the phenomena in other fields. Much as they do in medicine! Scarcely has any new thing been found,—it is “discovered” to be a valuable remedy, though little thought is given to the inner reasons. Now that we have X-rays, X-rays are the remedy to use; we only use them because we happen to have found them. It is as though men let themselves be swept along chaotically, willy-nilly by whatever happens to turn up from time to time.

So for the comets: By spectroscopic investigation and by comparison with the corresponding results for the planets, the idea arose that the phenomena might be explained electromagnetically. Such ideas will at most lead to analogies, which may no doubt have some connection with the reality, but which will certainly not satisfy us if we are looking into it more deeply.

Yet as I said, leaving this aside, there was one thing which emerged quite inevitably when the phenomena of comets were studied in more detail. While for the rest of the planetary system they always speak of gravitational forces, the peculiar position of the comet's tail in relation to the Sun inevitably drove the scientists to speak of forces of repulsion from the Sun—forces, as it were of recoil. The terminology is not the main point; it will of course vary with the prevailing fashion. The point is that science was here obliged to look for something in addition to—and indeed opposite to gravity.

In effect, with the comets something different enters our planetary system,—something which in its nature is in a way opposite to the inner structure of the planetary system as such. Hence it is understandable that for long ages the riddle of the comets gave rise to manifold superstitions. Men had a feeling that in the courses of the planets laws of Nature, inherently belonging to our planetary system, find expression, while with the comets something contrary comes in. Here something disparate and diverse makes its way into our planetary system. Thus they inclined to see the planetary phenomena as an embodiment of normal laws of Nature, and to regard the cometary apparitions as something contrary to these normal laws. There were times—though not the most ancient times—when comets were associated, as it were, with moral forces flying through the Universe, scourges for sinful man.

Today we rightly look on that as superstition. Yet even Hegel could not quite escape associating the comets with something not quite explicable or only half explicable by ordinary means. The 19th century, of course, no longer believed the comets to appear like judges to chastise mankind. Yet in the early 19th century they had statistics purporting to connect them with good and bad vintage years. These too occur somewhat irregularly; their sequence does not seem to follow regular laws of Nature. And even Hegel did not quite escape this conclusion. He though it plausible that the appearance or non-appearance of comets should have to do with the good and bad vintage years.

The standpoint of the people of today—at least, of those who share the normal scientific outlook—is that our planetary system has nothing to fear from the comets. Yet the phenomena which they evoke within this planetary system somehow have little inner connection with it. Like cosmic vagrants they seem to come from very distant regions into the near neighborhood of our Sun. Here they call forth certain phenomena, indicating forces of repulsion from the Sun. The phenomena appear, was and wane, and vanish.

There was a man who still had a certain fund of wisdom where by he contemplated the Universe not only with his intellect but with the whole human being. He still had some intuitive perception of the phenomena of the Heavens. I refer to Kepler. He was the author of a strange saying about the comets—a saying which gives food for thought to anyone who is at all sensitive to Kepler; way of though and mood of soul. We spoke of his three Laws—a work of genius, when one considers the ideas and the data which were accessible in his time. Kepler arrived at his Laws out of a feeling for the inner harmony of the planetary system. For him it was no mere dry calculation; it was a feeling of harmony. He felt has three planetary Laws as a last quantitative expression of something qualitative—the harmony pervading the whole planetary system. And out of this same feeling he made a statement about the comets, the deep significance of which one feels if one is able to enter into such things at all. Kepler said: In the great Universe—even the Universe into which we look by night—there are as many comets as there are fishes in the ocean. We only see very, very few among them, while all the rest remain invisible, either because they are too small or for some other reason. Even external research has tended to confirm Kepler's saying. The comets seen were recorded even in olden time and it is possible to compare the number. Since the invention of the telescope ever so many more have been seen than before. Also when looking out into the starry Heavens under different conditions of illumination—that is to say, making provision for extreme darkness—a larger number of comets are recorded than otherwise. Even empirical research therefore comes near to what Kepler exclaimed, inspired as he was by a deep feeling for Nature.

Now if one speaks at all of a connection between the Cosmos and what happens on the Earth, it surely is not right to dwell one-sidedly on the relation to our Earth of the other planets of our system and to omit the heavenly bodies which come and go as the comets do. It is especially one-sided since we must now admit that the comets give rise to phenomena indicating the presence of quite other forces—forces opposite in kind to those to which we usually attribute the coherence of our planetary system. The comets do in fact bring something opposite into our system, and if we follow it up we must admit that this too is of great significance. Something in some way opposite in nature to the force which holds it together, comes with the comets into our planetary system.

In an earlier lecture-course about natural phenomena I drew attention to something of which I must here remind you. Those who were present—the course was mainly about Heat or Warmth²—will no doubt recall it. I said that when we look at the phenomena of warmth in their relation to other phenomena of the Universe we are obliged to form a far more concrete idea of the Ether, of which the physicists generally speak in rather hypothetical terms. I said that in the formulae of Physics, wherever the force of pressure occurs as regards ponderable matter, we have to replace it by a force of suction as regards the ether. In other words, if we insert a plus sign for the intensity of a force in the realm of ponderable matter, we must give a minus sign to the corresponding intensity in the ether. I suggested that the well-known formulae should be looked through with this end in view; for one would see how remarkably, when this is done, they harmonize with the phenomena of Nature.

Take for example that whole game of thought, if I may call it so, the Kinetic Theory of Gases, of of Heat itself,—the molecules impinging on each other and on the walls of the containing vessel. Take all this brutal play of mutual impact and recoil which is supposed to represent the thermal condition of gas. Instead of this phenomena will become clear and penetrable the moment we perceive that within warmth itself there are two conditions. akin to the conditions that prevail in ponderable matter; the other must be thought of as akin to the ether. Warmth is in this respect different from Air or Light. For light, if we are calculating truly we must use the negative sign throughout. Whatever in our formulae is to represent the effects of light, must bear a negative sign. For air or gas the sign must be positive. For warmth on the other hand, the positive and negative will have to alternate. What we are wont to distinguish as conducted heat, radiant heat and so on will only then become clear and transparent.

Within the realm of matter itself, these things reveal the need for a qualitative transition from the positive to the negative in characterizing the different kinds of force. And we now see, very significantly, how for the planetary system we also have to pass from the positive—that is, gravitation—to the corresponding negative, the repelling force.

One more thing I will say today, if only to formulate the problem. For the moment I will carry it no further, but only put the problem; we shall have time to go into these things in later lectures. Now that we have ascertained all this about the cometary bodies, let me compare the relation between our planetary system and the comets to what is there in the ovum, the female germ-cell, in its relation to the male element, the fertilizing sperm. Try to imagine, try to visualize the two processes, as you might actually see them. There is the planetary system; it receives something new into itself, namely the effects of a comet. There is the ovum; it receives into itself the fertilizing effect of the male cell, the spermatozoid.

Look at the two phenomena side by side without prejudice, as you might do in ordinary life when you see two things obviously comparable, side by side. Do you not find plenty of comparable features when you contemplate these two? I do not mean to set up any theory or hypothesis, I only want to indicate what you will see for yourselves if you once look at these things in their true connection.

Taking our start from this, tomorrow we may hope to enter into more concrete and more detailed aspects.

• 1. Examples of the relation of Spiritual Science to the different branches of Science. Four lectures to students, Stuttgart, 11th to 15th January, 1921. Published (in the original German) in the Swiss periodical “Gegenwart”, Vol. 14, nos. 2 to 8, Berne, 1952.
• 2. Stuttgart, 1st to 14th March 1920, generally known as the “Second Scientific Lecture-Course”. Issued (in the original German) by the Science Section of the Goetheanum, Dornach, Switzerland, 1925.

We have now reached a point in our studies from which we must proceed with extreme caution, in order to see where there is a danger of allowing our thought to depart from reality and to see also when we are avoiding this danger, by keeping within the bounds of what is real.

Last time, we suggested the comparison of two facts: The appearance within the planetary system of the cometary phenomena, and, alas within the planetary system, though perhaps not bearing quite the same relationship to it, all that we observe in the phenomena of fertilization. In order, however, to come to ideas about this which are at all justified, we must first see whether it is indeed possible to find connections between two so widely separated things, with which we are confronted in the external world of facts. In scientific method, we shall not make real progress, unless we can refer from one realm of facts to another, manifesting something of a similar nature and thus leading us on.

We have seen how on the one hand we have to use the element of figure and form, the mathematical, and then how we are again and again impelled to come to terms in one way or another with the qualitative aspect, in some way to find a qualitative approach. And so today we will bring in something which arises in regard to man if one really studies this man, who is, after all, in some way an image of the heavenly phenomena,—as the many statements in these lectures may enable us to deduce. Yet we still have to establish in what way he is this image. If this is what he is, we must first of all gain a clear understanding of man himself. We must understand the picture from which we intend to take our start,—understand its inner perspective. Just as in looking at a painting one must know what a foreshortening means, and so on, in order to pass from the picture to the real spatial relationships and to relate the picture to what it represents in reality, so, if we would approach reality in the universe, interpreting it through man, we must first be clear about man. Now it is, extraordinarily difficult, as a human being, to come near to the human being with palpable ideas. Therefore, I should like today to bring before your souls what I might call “palpably impalpable” thought-pictures arising from quite simple foundations, ideas with which most of you are probably already well acquainted, but which we must nevertheless bring before our minds in a certain connection. These ideas, which seem in part to be quite easy to grasp and yet again, beyond certain limits, to elude our comprehension, will afford us a means of orientation in the striving to take hold of the outer world through ideas.

It may appear somewhat forced to keep emphasizing the necessity of referring back to man's life of pictorial imagination in order to understand the phenomena of the heavens. But after all it is obvious that however carefully we may describe the heavenly phenomena, we have, to begin with, nothing more than a form of optical picture, permeated with mathematical thoughts. What Astronomy gives us has fundamentally the character of a picture. To be on the right path, we must therefore concern ourselves with the arising of the picture in man, otherwise we shall gain no true relationship to what Astronomy can say to us. And so I should like today to proceed from some quite simple mathematics and to show you how, in a different domain from that to which we were led through the ratios of the periods of revolution of the planets, there appears within Mathematics itself this element of the incomprehensible, the impalpable. We meet with it when in a certain connection we study quite familiar curves. (As I said, many of you already know what I am about to describe, I only want to elucidate the subject today from a particular aspect.)

Consider the Ellipse, with its two foci \(A\) and \(B\), and you know that it is a definition of the ellipse that for any point \(M\) of the curve, the sum of its distances \((a + b)\) from the two foci remains constant. It is characteristic of the ellipse, that the sum of the distances of any one of its points from two fixed points, the two foci, remains constant (Fig. 1).

Then we have a second curve, the Hyperbola (Fig. 2). You know that it has two branches. It is defined in that the difference of the distances of any point of the curve from the two foci, \((b - a)\) is a constant magnitude. In the ellipse, then, we have the curve of the constant sum, in the hyperbola, the curve of constant difference, and we must now ask: What is the curve of constant product?

I have often drawn attention to this: The curve of constant product is the so-called Curve of Cassini (Fig. 3). We find it when, having two points, \(A\) and \(B\), we consider a point M in regard to its distances from \(A\) and \(B\), and establish the condition that the two distances \(AM\) and \(BM\) multiplied together should equal a constant magnitude. For the sake of simplicity in the calculation, I will call the constant magnitude \(b^2\) and the distance \(AB\), \(2a\). If we take the mid-point between \(a\) and \(b\) as the center of the axes of a co-ordinate system and calculate the ordinates for each point that fulfills these conditions,—take \(C\) as the center of the co-ordinate system and let the point whose ordinate we will call y move round so that for each point of the curve \(AM•BM = b^2\), we get the following equation. (I will only give you the result, for the simply reason that everyone can easily work out the calculation for himself; it is to be found in any mathematical text-book relating to the subject.) We find for \(y\) the value:

Taking here into account that we cannot use the negative sign because we should then have an imaginary y, and considering therefore taking only the positive sign, we have:

If we then draw the corresponding curve, we have a curve, rather like but not identical with an ellipse, called the curve of Cassini (Fig. 4). It is symmetrical to the left and right of the ordinate axis and about and below the abscissa axis.

But now, this curve has various forms, and for us at any rate this is the important thing about it. The curve has different forms, according to whether b, as I have taken it here, is greater than a, equal to a, or less than a. The curve I have just drawn arises when b ˃ a, and furthermore when another condition is fulfilled, namely, that b is also greater than or equal to a √2. Moreover, when b ˃ a√2, there is a distinct curvature above and below, If b = a√2, then at this point above and below, the line of the curve becomes straightened,m it flattens so much that it almost becomes a straight line (Fig. 4). If, however, b ˂ a√2, then the whole course of the curve is changed and it takes on this form (Fig. 5). And if b = a, the curve passes over into a quite special form, it changes into this form (Fig. 6). It runs back into itself, cuts through itself and comes out on the other side, and we obtain the special form of the Lemniscate. The lemniscate, then, is a special form of Curve of Cassini—these curves are so named after their discoverer. The particular form assumed by the curve is determined by the ratio between the constant magnitudes which appear in the equation characterizing the curve. In the equation, we have only these two constant magnitudes, b and a, and the form of the curve depends on the ratio between them.

Then the third case is possible, that b ˂ a. If b ˂ a, we can still find values for the curve. We can always solve the equation and obtain values for the curve, ordinates and abscissae, even when b is smaller than a, only the curve then undergoes yet another metamorphosis. For when b ˂ a, we find two branches of the curve, which look something like this (Fig. 7). We have a discontinuous curve. And here we come to the point where the mathematics itself confronts us with what I called the “palpably impalpable”, something that is difficult to grasp in space. For in the sense of the mathematical equation, this is not two curves, but one; it is a single curve in exactly the same way as all these are single curves (Figs. 3 through 5). In this one (the lemniscate) there is already a transition. The point which describes the curve takes this path, goes round underneath, cuts its previous path here and continues on here (Fig. 7). Here, we must picture the following: If we let the point M move along this line, it does not simply cross over from one side to the other,—it does not do this. It runs along the path just as in the other curves, describes a curve here, but then manages to turn up again here (Fig. 7) You see, that which carries the point along the line disappears here in the middle. If you want to understand the curve you can only imagine that it disappears in the middle. If you try to form a continuous mental picture of this curve, what must you do?

It is quite easy, is it not, to imagine curves such as thes. (I only say this in parenthesis for the ordinary philistine!) You can go on imagining points along the curve and you do not find that the picture breaks off. Here (in the lemniscate) admittedly, you have to modify the comfortable way of simply going round and round, but still it goes on continuously. You can keep hold of the mental picture. But now, when you come to this curve (Fig. 7), which is not so commonplace, and you want to image it, then, in order to keep the continuity of the idea you will have to say: Space no longer gives me a point of support. In crossing over to the other branch in my imagination, unless I break the continuity and regard the one branch as independent of the other, I must go out of space; I cannot remain in space. So you see, Mathematics itself provides us with facts which oblige us to go out of space, if we would preserve the continuity of the idea. The reality itself demands of us that in our ideas we go out of space. Even in Mathematics therefore we are confronted with something which shows us that in some way we must leave space behind, if the pure idea is to follow its right path. Having ourselves and going the idea is beginning to think the process through, we must go on thinking in such a way that space is no longer of any help to us. If this were not so, we should not be able to calculate all possibilities in the equation.

In pursuing similar line of thought, we meet with other instances of this kind. I will only draw your attention to the next step, which ensures if one things as follows. The ellipse is the locus of the constant sum,—it is defined by the fact that is is the curve of constant sum. The hyperbola is the curve of constant difference. The curve of Cassini in its various forms is the curve of constant product. There must then be a curve of constant quotient also, if we have here A, here B, here a point M, and then a constant quotient to be formed through the division of BM by AM. We must be able to find different points, M 1, M 2, etc., for which

etc. are equal to one another and always equal to a constant number. This curve is, in fact, the Circle. If we look for the points M1, M2 etc. we find a circle which has this particular relationship to thee points A and B (Fig. 8). So that we can say: Besides the usual, simple definition of a circle,—namely, that it is the locus of a point whose distance from a fixed point remains constant,—there is another definition. The circle is that curve, very point of which fulfills the condition that its distances from two fixed points maintain a constant quotient.

Now, in considering the circle in this way there is something else to be observed. For you see, if we express this

(it could of course be expressed in some other way), we always obtain corresponding values in the equation, and we can find the circle. In doing this we find different forms of the circle (that is, different proportions between the radius of the circle and the length of the straight line AB), according to the proportion of m to n. These different forms of the circle behave in such a way that their curvature becomes less and less. When \(n\) is much greater than m, we find a circle with a very strong curvature; when n is not so much greater, the curvature is less. The circle becomes larger and larger the smaller the difference between n and m. And if we follow this proportion of m to n still further, the circle gradually passes over into a straight line. You can follow this in the equation. It passes over into the ordinate axis itself. The circle becomes the ordinate axis when \(m=n\), that is, when the quotient \(m/n=1\). In this way the circle gradually changes into the ordinate axis, into a straight line.

You need not be particularly astonished at this. It is quite possible to imagine. But something very different happens it we wish to follow the process still further. The circle has flattened more and more, and through becoming flatter from within, as it were, it changes into a straight line. It does this because the constant ratio in the equation undergoes a change. Through this the circle becomes a straight line. But this constant ratio can of course grow beyond \(1\), so that the arcs of the circles appear here (on the left of the \(y\)-axis). What must we do, however, if we try to follow it in our imagination? We have to do something quite peculiar. We have, in fact, to think of a circle which is not curved towards the inside, but is curved towards the outside. Of course, I cannot draw this circle, but it is possible to think of a circle which is curved towards the outside.¹ In an ordinary circle the curvature is towards the inside, it is not? If we follow the line round it returns into itself. But defining the circle in this other way, if we use the necessary constant, we obtain a straight line. The curvature is still on this side (right of the \(y\)-axis). But it now makes things not nearly so comfortable for us as before! Previously, the curvature always turned towards the center of the circle, while now (in the case of the straight line), we are shown that the center is somewhere in the infinite distance, as one says. Following on from this, there arises for us the idea of a circle which is curved towards the outside. Its curvature is then no longer as it is here (Fig. 9a)—that would be the ordinary, commonplace, philistine circle,—but its curvature is here (Fig. 9b). Therefore, the inside of this circle is not here; this is the outside; the inside of this circle (Fig. 9c) is to the right.

Now compare what I have just put before you. I have described the curve of Cassini, with its various forms, the lemniscate and the form in which there are two branches. And now we have pictured the circle in such a way that at one time it is curved in the familiar way, with the inside here and the outside here; while in a second form of circle (in drawing it we are only indicating what is meant) we find that the curvature is this way round, with an inside here and an outside here. Comparing it with the Cassini curve, the first form of the circle would correspond to the closed forms, as far as the lemniscate. After this we have another kind of circle, which must be thought of in the other direction, being curved this way, with the inside here and the outside here. You see, when we are concerned with the constant product we find forms of the curve of Cassini where, it is true, we are thrown out of space, yet we can still draw the other branch on the other side. The other branch is once more in space, although in order to pass from the one to the other we are thrown out of space. Here, in the case of the circle, however, the matter becomes still more difficult. In the transition from circle to straight line we are, indeed, thrown out of space, and moreover, we can no longer draw a self-contained form at all. This we are unable to do. In passing over from the curve of constant product to the curve of constant quotient, we are only just able to indicate the thought spatially.

It is extraordinarily important that we concern ourselves with the creating of ideas which, as it were, will still slip into such curve-forms. I am convinced that most people who concern themselves with mathematics take note of such discontinuities, but then make the thought more comfortable by simply holding to the formula and not passing on to what should accompany the mathematical formula in true continuity of thought. I have also never seen that in the treatment of Mathematics as subject matter for education any great value is laid upon the forming of such thoughts in imagination.—I do not know,—I ask the mathematicians present, Herr Blümel, Herr Baravalle, if this is so; whether in modern University education any importance is attached to this? (Dr. Unger here mentioned the use of the cinema.) Yes, but that is a pretense. It is only possible to represent such things within empirical space by means of the cinema or in similar ways, it some sort of deception is introduced. It cannot be pictured fully in real space without the effect being achieved through some form of deception. The point is, whether there is anywhere in the sphere of reality something which obliges us to think realistically in terms of such curves. This is the question I am now asking. Before passing on, however, to describe what might perhaps correspond to these things in the realm of reality, I should like to add something which may perhaps make it easier for you to pass transition from these abstract ideas to the reality. It is the following.

You can set another problem in the sphere of theoretical Astronomy, theoretical Physics. You can say: Let us suppose that here as \(A\), is a source of light, and this source of light in a illumines a point \(M\) (Fig. 10). The strength of the light shining from \(M\) is observed from \(B\). That is, with the necessary optical instruments, observation is made from \(B\) of the strength of the light shining from the point \(M\), which is illumined from \(A\). And of course, the strength of the light would vary, according to the distance between \(B\) and \(M\). But there is a path which could be described by the point \(M\), such that, being illumined from A, it always shines back to \(B\) with the same intensity. There is such a path; and we can therefore ask: What must be the locus of a point, illumined from a fixed point \(A\), such that, seen from another fixed point \(B\), its light is always of the same intensity? This curve—the curve in which such a point would have to move—is the curve of Cassini! From this you see that something which takes on a qualitative nature is set into spatial connection, fitting into a complicated curve. The quality that we must see in the beam of light—for the intensity of light is a quality—depends in this case on the element of form in the spatial relationships.

I only wished to bring this forward for you to see that there is at least some way of leading over from what can be grasped in geometrical form to what is qualitative. This way is a long one, and what we will now discuss is something to which I want to draw your attention, although it would take months to present in all detail. You must be fully aware that I only intend to give you guiding lines; it is left to you to develop them further and to go into all the details which would testify to the truth of what is said. For you see, the connection which must be formed between spiritual science and empirical sciences of today demands very far-reaching and extensive work. But when lines of direction are once given, this work can to some extent be undertaken and carried forward. It is at all events possible. One must only be able in a quite definite way to penetrate into the empirical phenomena.

If we now tackle the problem from quite another angle,—we have sought to some degree to understand it from the mathematical aspect, then, to anyone who is studying the human organism, there is something which cannot escape unnoticed, something which has often been brought forward in our circle, especially in the talks which accompanied the course of lectures on Medicine in Dornach in the spring of 1920. It is not to be overlooked that certain relationships exist between the organisation of the head and the rest of the human organisation, for example the metabolism. There is indeed a connection, indefinable to begin with, between what takes place in the third system of the human being—in all the organs of metabolism—and what takes place in the head. The relationship is there, but it is hard to formulate. Clearly as it emerges in various phenomena,—for example, it is obvious that certain illnesses are connected with skull or head deformities and the like, and these things can easily be traced by one who tries to follow them with biological reasoning,—it nevertheless difficult to grasp this relationship in imagination. People do not usually get beyond the point of saying that there must be some sort of connection between what takes place in the head, for instance, and in the rest of the human organism. It is a picture which is difficult to form, just because it is so very hard for people to make the transition from the quantitative aspect to the qualitative. If we are not educated through spiritual-scientific methods to find this transition, quite independently of what outer experience offers,—to extend to what is qualitative the kind of thought we use for what is quantitative, if we do not methodically train ourselves to do this, then, my dear friends, there will always be an apparent limit to our understanding of the external phenomena.

Let me indicate but one way in which you can train yourselves methodologically to think the qualitative in a similar way as you think the quantitative. You are all acquainted with the phenomenon of the solar spectrum, the usual continuous spectrum. You know that we have there the transition of colour from red to violet. You know, too, that Goethe wrestled with the problem of how this spectrum is in a sense the reverse of what must arise if darkness be allowed to pass through the prism in the same way as is usually done with light. The result is a kind of inverted spectrum, and as you know Goethe arranged this experiment also. In the ordinary spectrum, the green passes over on the one side towards the violet and on the other towards the red; whereas in the spectrum obtained by Goethe in applying a strip of darkness to the prism there is peach-blossom in the middle and then again red on the one side and violet on the other (Fig. 11). The two colour bands are obtained, the centres of which are opposite to one another, qualitatively opposite, and both bands seem to stretch away as it were into infinity. But now, one can imagine that this axis, the longitudinal axis of the ordinary spectrum, is not simply a straight line, but a circle, as indeed every straight line is a circle. If this straight line is a circle, it returns into itself, and we can consider the point where the peach-blossom appears to be the same point as the one in which the violet, stretching to the right, meets the red, which stretches to the left. They meet in the infinite distance to the right and left. If we were to succeed—maybe you know that one of the first experiments to be made in our newly established physical laboratory is to be in this direction—if we were to succeed in bending the spectrum in a certain way into itself, then even those who are not willing to grasp the matter to begin with in pure thought will be able to see that we are here concerned with something real and of a qualitative nature.

We come to certain limiting ideas in Mathematics, where—as in Synthetic Geometry—we are obliged to regard the straight line as a circle in a quite real though inner sense; where we are obliged to admit of the infinitely distant point of a straight line as being only one point; or to understand as bounding a plane, not some line above and then again below, but a single straight line; or to think of the boundary of infinite space, not in the nature of something spherical, but as a plane. Such ideas, however, also become, in a way, limiting ideas for sense-perceptible empirical reality, and we are made to realise it if we insist on restricting ourselves to sense-perceptible reality.

This brings us to something which would otherwise always remain perpetually in the dark. I have already mentioned it. It invites us really to think-through the thought-pictures to which we come when we allow the lemniscate-form of the Cassini curve to pass over into the double-branched form,—the form with the two branches for which we must go out of space,—and them compare this with what confronts us in the empirical reality.

You are indeed already doing this, my dear friends, when you apply Mathematics in one way or another to the empirical reality. You call a triangle a triangle, because you have first constructed it mathematically. You apply to the outer form what has been evolved in an inner constructive way within you. The process I have just described is only more complicated, but it is the same process when you think of the two branches of that particular form of the Cassini curve as one. Apply this thought to the correspondence between the human head and the rest of the human organism and you will have to realise that in the head there is a connection with the remaining organism of precisely such a character as is expressed by the equation which requires, not a continuous curve, but a discontinuous one. This cannot be followed anatomically; you must go out beyond what the body comprises physically, if you would find the connection of what comes to expression in the head with what comes to expression in the metabolic system. It is essential to approach the human organism with thoughts which are quite unattainable if for every element of the thought you insist on an entire correspondence within the sense-perceptible empirical realm. We must reach out to something else, beyond the sense-perceptible empirical realm, if we are to find what this relationship really is within the human being.

Such a study, if one really gives oneself up to it and carried it out methodically, is extraordinarily rich in its results. The human organisation is of such a nature that it cannot be embraced by the anatomical approach alone. Just as we are driven out of space in the Cassini curve, so in the study of man we are driven out of the body, by the method of study itself. You see, it is quite possible to understand in the first place in thought, that in a study of the whole man we are driven out of the realm of what can be grasped in a physical-empirical sense. To put forward such things is no offence against scientific principles. Such ideas are far removed from the purely hypothetical fantasies which are often entertained in connection with natural phenomena, for they refer to the whole way in which man is membered into the universe. You are not looking for something which is otherwise non-existent, but rather for something which is exactly the same as what is expressed in the relationship between a man thinking mathematically and the empirical reality.

It is not a question of looking for hypotheses which in the end are unjustifiable; it is a question, since the reality is obviously complicated, of looking for other cognitive relations to the inner reality, in addition to the simple relation of mathematical man to empirical reality. When once you have accepted such thoughts, you will also be led to ask whether what takes place outside the human being in other domains besides the astronomical,—for example, in those phenomena which we call the chemical and physical,—whether those same phenomena, which we regard as chemical phenomena outside of man, take the same course within man, when he is alive, as they do outside him, or whether here, too, a transition is necessary which leads in some way out of space.

Now consider the important question arising out of this. Suppose we have here some kind of chemical phenomena and here the boundary leading over to the inside of the human being (Fig. 13). Supposing that this chemical phenomenon were able to call forth another, so that the human being reacted here (inside); then, if we remain in the field of the empirical, space would of course be the mediator. If, however, the continuance of this phenomenon within the human being comes about by virtue of the fact, say, that the human being is nourished by food, and the processes already taking place outside him continue inside him, then the question arises: Does the force which is at work in the chemical process remain in the same space when it taking place within man as when it is taking its course outside him? Or must we perhaps go out of space?

And there you have what is analogous to the circle which changes over into a straight line. If you look for its other form, where what is usually turned outward is now turned inward, you are entirely outside of space.

The question is, whether we do not need such ideas as these, thought-pictures which, while remaining continuous, go right out of space,—when we follow the course of what happens outwardly, outside of man, into the interior of the human being. The only thing to be said against such things, my dear friends, is that they certainly impose greater demands on the human capacity of understanding than the ideas with which he phenomena are approached today. They might therefore be rather awkward in University education. They are, no doubt, thoroughly awkward, for they imply that before approaching the phenomena we must awaken in ourselves what will enable us to understand them. Nothing like this exists in our educational system today; but it must come, it must certainly come, otherwise simply in speaking of a phenomenon we get into the greatest disparities, without in any way seeing the reality. Just think what happens when someone observes the circle as it curves to this side (Fig. 9a), and then sees how it curves to this side (Fig. 9b), but then remains a philistine and simple does not conceive that the circle now curves towards the other side. He says: This is impossible, the circle cannot curve this way; I must put the curvature this way round, I must simply place myself on the other side. What he is speaking about seems to be one and the same thing; but he has changed his point of view.

In this way today we make matters simple, in describing what is within the human being in comparison with what takes place in Nature outside him. We say: What is within man does not exist at all; I must simply place myself within man and say that the curvature is facing this way (Fig. 9c). I will then consider what is inside, without taking into account that I have reversed the curvature. I will make the interior of the human being into an outer Nature. I simply imagine outer Nature to continue through the skin into the interior. I turn myself round, because I am not willing to admit the other form of curvature, and then I theorise. That is the trick which is performed today, only in order to adhere to more comfortable motions. There is no desire to accent what is real; in order not to have to do so, we simply turn ourselves round, and—this is now a comparison—instead of looking at the human from in front, we look at Nature from behind and thus arrive in this way at all the various theories concerning man.

We will continue, then, tomorrow.

• 1. If it were drawn it would look like an ordinary circle, only one would have to bear in mind that “outside” and “inside” had changed places. (Editor's note.)

Taking my start yesterday from certain considerations in the realm of form, I showed how the connections should be thought of between the processes of the human metabolic system and the processes of the head, the nervous system, or whatever you wish to call it in the sense of the indications given in my book Riddles of the Soul (Von Seelenrätseln).

It would be regarded as quite out of the question to study the movements of a magnet-needle on the Earth's surface in such a way as to try to explain these movements solely out of what can be observed within the space occupied by the needle. The movements of the magnet-needle are, as you know, brought into connection with the magnetism of the Earth. We connect the momentary direction of the needle with the direction of the Earth's magnetism, that is, with the line of direction which can be drawn between the north and south magnetic poles of the Earth. When it is a question of explaining the phenomena presented by the magnetic needle, we go out of the region of the needle itself and try to enter, with the facts that have been collected towards an explanation, into the totality which alone affords the opportunity to explain phenomena, the manifestations of which belong to this totality. This rule of method is certainly observed in regard to some phenomena,—to those, I should say, the significance of which is fairly obvious. But it is not observed when it is a question of explaining and understanding more complicated phenomena.

Just as it is impossible to explain the phenomena of the magnetic needle from the needle itself, it is equally and fundamentally impossible to explain the phenomena relating to the organism from out of the organism itself, or from connections which do not belong to a totality, to a whole. And just for this reason, because there is so little inclination to reach the realm of totalities in order to find explanations, we arrive at those results put forward by the modern scientific method in which the wider connections are almost entirely left out of the picture. This method encloses the phenomena, whatever they may be, within the field of vision of the microscope; while the celestial phenomena are restricted to what is observable externally, with the help of instruments. In seeking for explanations, no attempt is made to consider the necessity of reaching out to the surrounding totality within which a phenomenon is localised. Only when we become familiar with this quite indispensable principle of method, are we in a position to bring our judgment to bear upon such things as I was describing to you yesterday. Only in this way shall we grow able to estimate how such realms of phenomena as are met within the human organism will appear, when truly recognised in the totality to which they properly belong.

Remember what I described at the very beginning of this course of lectures. I drew your attention to the fact that the principle of metamorphosis as it appeared first in the work of Goethe and Oken must be modified if it is truly to be applied to man. The attempt was made—and it was made with genius on the part of Goethe—to derive the formation of the bones of the skull from that of the vertebrae. These investigations were continued by others in a way more akin to 19th-century method, and the progress of the method of investigation (I will not now decide whether it was a step forward or not) can be studied by comparing how this problem of the metamorphosis of one form of bone into another was conceived on the one hand by Goethe and Oken and on the other, for example, by the anatomist Gegenbauer.

These things are only to be set on a real basis, if one knows (as I said, I have already mentioned this in the course of these lectures, but we will now link on to it again) how two types of bone in the human organism (not the animal, but the human organism), most widely separated from the point of view of their morphology, are actually related to one another. Bones far removed from one another in the aspect of their form would be a tubular or long bone—femur or humerus, for example,—and a skull-bone. To make a superficial comparison, without really entering into the inner nature of the form and bringing a whole range of phenomena into connection with it, is not enough to reveal the morphological relationship between two polar opposite bones—polar opposite, once more, in regard to their form. We only begin to perceive it if we compare the inner surface of a tubular bone with the outer surface of a skull-bone. Only thus do we get the true correspondence (Fig. 1) which we must have in order to establish the morphological relation. The inner surface of the tubular bone corresponds morphologically to the outer surface of the skull-bone. The skull-bone can be derived from the tubular bone if we picture it as being reversed, to begin with, according to the principle of the turning-inside-out of a glove. In the glove, however, when I turn the outer surface to the inside and the inner to the outside, I get a form similar to the original one. But if in the moment of turning the inside of the tubular bone to the outside, certain forces of tension come into play and mutual relationships of the forces change in such a way that the form which was inside and has now been turned outward alters the shape and distribution of its surface, then we obtain, through inversion on the principle of the turning-inside-out of a glove, the outer surface of the skull bone as derived from the inner surface of the tubular bone. From this you can conclude as follows. The inner space of the tubular bone, this compressed inner space, corresponds in regard to the human skull to the entire outer world. You must consider as related in their influence upon the human being: The outer universe, forming the outside of his head, and what works within, tending from within toward the inner surface of the tubular bone. These you must see to belong together. You must regard the world in the inside of the tubular bone as a kind of inversion of the world surrounding us outside.

There, for the bones in the first place, you have the true principle of metamorphosis! The other bones are intermediary forms; morphologically, they mediate between the two opposite extremes, which represent a complete inversion, accompanied by a change in the forces determining the surface. The idea must however be extended to the entire human organism. In one way, it comes to expression most clearly in the bones; but in all the human organs we must distinguish between two opposing factors,—that which works outward from an unknown interior, as we will call it for the moment, and that which works inward from without. The latter corresponds to all that surrounds us human beings on the planet Earth.

The tubular bone and the skull-bone represent indeed a remarkable polarity. Take the tubular bone and think of this centre-line (Fig. 2). This line is in a way the place of origin of what works outward, in a direction perpendicular to the inner surface of the bone (Fig. 3). If you now think of what envelops the human skull, you have what corresponds to the central line of the tubular bone. But how must you draw the counterpart of this line? You must draw it somewhere as a circle, or more exactly, as a spherical surface, far way at some indeterminate distance (Fig. 4). All the lines which can be drawn from the centre-line of the tubular bone towards it inner surface (Fig. 3). correspond, in regard to the skull-bone, to all the lines which can be drawn from a spherical surface as though to meet in the centre of the Earth (Fig. 4). In this way you find a connection—approximate, needless to say—between a straight line, or a system of straight lines, passing through a tubular bone and bearing a certain relation to the vertical axis of the body, the direction of which coincides, in fact, with that of the Earth's radius and a sphere surrounding the Earth at an indeterminate distance. In other words, the connection is as follows. The radius of the Earth has the same cosmic value in regard to the vertical posture of the human organism, perpendicular to the surface of the Earth, as a spherical surface, a cosmic spherical surface has in regard to the skull organisation. This, however, is the same contrast which you experience within yourself if you make yourself aware of the feeling of being inside your own organism and experiencing of the outer world at the same time. This is the polarity you reach if you compare your feeling of self—that feeling of self which is really based on the fact that in normal life you can depend upon your bodily organisation, that you do not become giddy, but keeping a right relation to the force of gravity—with all that is present in your consciousness in connection with what you see around you through the senses, even as far away as the stars.

Putting all this together, you will be able to say: There is the same relation between this feeling of being in yourself and the feeling of consciousness you have in perceiving the outer world as there is between the structure of your body and of your skull. We are thus led to the relationship between what we might call: Earthly influence upon man, of such a character that it works in the direction of the Earth's radius, and what we might call: The influence which makes itself felt in the entire circumference of our life of consciousness, and which we must look for in the sphere, in what really is for us the inner wall, the inner surface, of a hollow sphere. This polarity prevails in our normal day-waking conscious life. It is this polarity which, roughly speaking—if we leave out of account what is in our consciousness as a result of observating our earthly environment—we may look upon as the contrast between the starry sphere and earthly consciousness, earthly feeling of ourselves,—Earth-impulse living in us. If we compare this impulse of Earth, this radial Earth-impulse, to our consciousness of the vast sphere,—if we observe how this polarity, prevails in normal waking consciousness, we shall perceive that it is always there, living in us, playing its part in our conscious life. We live far more in this polarity than we are wont to think. It is always present and we live within it. The connection between the forming of mental images and the life of will can be really studied in no other way than by considering the contrast between ‘sphere’ and ‘radius’. In psychology, too, we should come to truer results with regard to the connection of our world of ideas and mental pictures, manifold and extensive as it is, with the more unified world of our will, if a similar relationship were sought between them as is symbolised in the relation of the surface-area of a sphere with the corresponding radius.

Now, my dear friends, let us look at all this which is at work in our day-waking consciousness, forming the content of our soul-life, let us now consider how it takes its course when we are in quite a different situation. In effect, how does it work upon us during the time of the embryonic life? We can well imagine, indeed we must imagine that the same polarity will be at work here too, only in another way. During the embryonic period, we do not direct towards the outer world the same activity which afterwards dims down this polarity to a pictorial one; at this time, the polarity affects all that is formative in our organisation, in a much more real way than when, in picture form, it becomes active in our life of mind and soul. If therefore we project the activity of consciousness back in time to the embryonic period, then one might say that in the embryonic life we have what we otherwise have in the activity of consciousness, but we have it at a more intensive, more realistic stage. Just as we clearly see the relation of sphere and radius in our consciousness, so to reach any real result, we must look for this same polarity of heavenly sphere and earthly activity in what happens in the embryonic life. In other words, we must look for the genesis of human embryonic life by finding a resultant between what takes place out in the starry world—an activity in the ‘sphere’—and what takes place in man as a result of the radial Earth-activity.

What I have just described must be taken into account with the same inner necessity of method as the Earth's magnetism is in connection with the magnetic needle. There may be much that is hypothetical even in this, but I will not go into it now. I only wish to point out: We have no right to restrict our considerations to the embryo alone,—to explain the processes taking place within it simply out of the embryo itself. In just the same way as we have no right too explain the phenomenon of the magnet out of itself alone, so too, we have no right to explain the form and development of the embryo purely on the basis of the embryo itself. In attempting to explain the embryo we must take these two opposites into account. As we take the Earth's magnetism into account in connection with the magnet, so must we observe the polarity of sphere and radial activity, in order to understand what is developing in the embryo,—which, when the embryo is born, fades into the pictorial quality of the experience of consciousness. The point is, we must learn to see the relationship which exists in man between tubular or long bone and skull-bone in the other systems too—in muscle and nerve, and so on;—and when we do study this polarity, we are led out into the life of the Cosmos. Consider how closely related (as described in my book “Riddles of the Soul”) is the whole essence and content of the human metabolic system with what I have now characterised as being under the influence of the ‘radial’ element, and how closely related is the head system to what I have just described as being under the influence of the ‘sphere’. Then you will say: We must distinguish in the human being what conditions his sensory nature and what conditions his metabolic life; moreover, these two elements are related to one another as heavenly sphere to earthly activity.

We must therefore look for the product of the celestial activity in what we bear in our head organisation and for what unites to a resultant with this, the activity belonging to the Earth—tending, as it were, towards the centre of the Earth—in our metabolism. These two realms of activity and influence fall apart in man; it is as thought they represent two Ice Ages, and the middle realm, the rhythmic realm, mediates between them. In the rhythmic system we actually have something,—if I may so express myself,—which is a realm of mutual interplay between Earth and Heaven.

And now if we wish to go further, we must consider various other relationships which reveal themselves to us in the realm of reality. I will now draw your attention to something very intimately connected with what I have just been describing.

There is the familiar membering of the outer world which surrounds us and to which we as physical man belong; we divide it into mineral kingdom, plant kingdom, animal kingdom, and regard man as the culmination of this external world of Nature. Now, if we would obtain a clearer view of what we have described in connection with the working of the celestial phenomena, we must turn our attention to yet another thing.

It is not to be denied—it is indeed quite obvious to any prejudiced observer—that with our human organisation as it is now, in the present phase of the cosmic evolution of humanity, we are, in regard to our capacities of knowledge, entirely adapted to the mineral kingdom. Take the kind of laws we seek in Nature; and you will agree that we are certainly not adapted to all aspects of our environment. To put it curtly, all that we really understand is the mineral kingdom. Hence all the efforts to refer the other kingdoms of Nature back to the laws of the mineral domain. After all, it is because of this that such confusion has arisen with regard to mechanism and vitalism. To the ordinary view which is ours toady, life remains either a vague hypothesis, as it was in earlier times, or else its manifestations are explained in terms of the mechanical, the mineral. The ideal, to reach an understanding of life, is unaccompanied by any recognition of the fact that life must be understood as life; on the contrary, the fundamental aim is to refer life back to the laws of the mineral realm. Precisely this betrays a vague awareness of the fact that man's faculties of knowledge are only adapted to understand the mineral kingdom and not the plant nor animal.

Now when we study on the one hand the mineral kingdom itself and on the other hand its counterpart, namely, our own knowledge of the mineral kingdom, in that these two correspond to one another, we shall be compelled,—since as explained just now we must relate all our life of knowledge to the heavenly sphere, also to bring into connection with the heavenly sphere, in some way, that to which our knowledge is related, namely the mineral kingdom. We must admit: In regard to our head organisation, we are organised from the celestial sphere; therefore what underlies the forces of the mineral kingdom must also be organised from the celestial sphere in some way. Compare then what you have to your sphere of understanding—the whole compass of your knowledge of the mineral kingdom—with what is actually there in the mineral kingdom in the outer world, and you will be led to say: What is thus within you relates to what is in the mineral kingdom outside you, as picture to reality.

Now we must think of this relationship more concretely than in the form of picture and reality, and we are helped to do so by what I said before. Our attention is drawn to what underlies the human metabolic system and to the forces active there, forces which are connected with the pole of earthly activity, typified by the radius. In seeking for the polar opposite, within ourselves, to that part of our organisation which forms the basis for our life of knowledge, we are directed from the encompassing Sphere to the Earth. The radii converge to the middle point of the Earth. In the radial element we have something by which we feel ourselves, which gives us the feeling of being real. This is not what fills us with pictures in which we are merely conscious; this is what gives us the experience of ourselves as a reality. When we really experience this contrast, we come into the sphere of the mineral kingdom. We are led from what is organised only for the picture to what is organised for the reality. In other words: In connection with the cause and origin of our life of knowledge, we are led to the wide, encompassing sphere,—we concave it in the first place as a sphere,—whereas, in following the radii of the sphere towards the middle of the Earth, we are led to the middle point of the Earth as the other pole.

Thinking this out in more detail, we might say: Well, according to the Ptolemaic conception for example, out there is the blue sphere, on it a point (Fig. 5)—we should have to think of a polar point in the centre of the Earth. Every point of the sphere would have its reflected point in the Earth's centre. But, or course, it is not to be understood like that. (I shall speak more in detail later on; to what extent these things correspond exactly is not the question for the moment.) The stars, in effect, would be here (Fig. 6). So that in thinking of the sphere concentrated in the centre of the Earth, we should have to think of it in the following way: The pole of this star is here, of this one here, and so on (Fig. 6). We come, then, to a complete mirroring of what is outside in the interior of the Earth.

Picturing this in regard to each individual planet, we have, say, Jupiter and then a polar Jupiter’ within the Earth. We come to something which works outward from within the Earth in the way that Jupiter works in the Earth's environment. We arrive at a mirroring (in reality it is the opposite way round, but I will now describe it like this), a mirroring of what is outside the Earth into the interior of the Earth. And if we see the effect of this reflection in the forms of the minerals then we must also see the effect of what works in the cosmic sphere itself in forming our faculty of understanding the minerals. In other words: We can think of the whole celestial sphere as being mirrored in the Earth: We conceive the mineral kingdom of the Earth as an outcome of this reflection, and we conceive that what lives within us, enabling us to understand the mineral kingdom, comes from what surrounds us out in the celestial space. Meanwhile the realities we grasp by means of this faculty of understanding come from within the Earth.

You need only follow up this idea and then cast a glance at man, at the human countenance, and, if you really look at this human countenance, you will hardly be able to doubt that in it something is expressed of the celestial sphere, and that there also appears in it what is present as pictorial experience in the soul, namely the forces which rise up into the realm of soul activity from the realm of bodily activity, after having been at work more intensively in this bodily realm during embryonic life. Thus we find a connection between what is out side us in outer reality, and our own organisation for the understanding of this outer reality. We can say: The cosmos produces the outer reality, and our power to understand this outer reality is organised physically by virtue of the fact that the cosmic sphere is only active in us now for our faculty of knowledge. Therefore we must distinguish, in the genesis of the Earth as well, between two phases: One in which active forces work in such a way that the real Earth itself is created, and then a later phase of evolution, in which the forces work so as to create the human faculty for understanding the realities of the Earth.

Only in this way, my dear friends, do we really come near to an understanding of the Universe.

You may say: Well and good, but this method of understanding is less secure than the method used today with the aid of microscope and telescope. It may be that to some people it appears less secure. But if things are so constituted that we cannot reach the realities with the methods in favour today, then we are faced with the absolute necessity of comprehending the reality with other modes of understanding and we shall have to get used to developing those other methods. It is of no avail to say, you will have nothing to do with such lines of thought, since they appear too uncertain. What if this degree of certainty alone were possible! However, if you really follow up this line of thought, you will see that the degree of certainty is just as great as in your conception of a real triangle in the outer world when you take hold of it in thought with the inner idea of construction of a triangle. It is the same principle, the same manner of comprehending outer reality in the one case as in the other. This should be borne in mind.

Certainly, the question arises: Taking these thoughts, as I have here developed them, it is possible to become clear in a general way about such connections, but how can one reach a more definite comprehension of these things? For only in a much more definite form can they be of use in helping us to grasp the realm of reality. In order to go into this, I must draw your attention to something else.

Let us return to what I aid yesterday, for example, in regard to the Cassini curve. We know that this curve has three, or, if you like, four forms. You remember, the Cassini curve is determined as follows. Given two points \(A\) and \(B\), I will call the distance between them \(2a\); then any point of the curve will be such that \(AM-MB=b2\), that is, a constant. And I obtain the various forms of the Cassini Curve according to whether \(a\), that is, half the distance between the foci, is greater than, equal to, or less than \(b\). I obtain the lemniscate when \(a=b\), and the discontinuous curve when a is greater than \(b\).

Imagine now that I wanted not only to solve this geometrical problem, assuming two constant magnitudes a and b and then setting up equations to determine the distances of M from A and B. Suppose I wanted to do more than this, namely, to move in the plane from one form of line or curve to another by treating as variable magnitudes those magnitudes which remain constant for a particular curve. In the picture (Lecture IX, Fig. 3) after all, we only envisaged certain limiting positions with a greater or smaller than b. Between these there are an infinite number of possibilities. I can pass over quite continuously to the construction of one form of the Cassini curve after another. And I shall obtain these different forms if, let us say, to the variability of the first order, say between \(y\) and \(x\). I add a variability of the second order; that is, if I allow my construction of the curves as they pass over from one to the other continuously, to take its course in such a way that a remains a function of \(b\).

What am I doing when I do this? I am constructing curves in such a way that I create a continuous, moving system of Cassini curves passing over via the lemniscate into the discontinuous forms, not at random, but by basing it on a variability of the second order, in that I bring the constants of the curves themselves into relationship with one another so that a is a function of \(b\), \(a=φ(b)\). Mathematically, it is of course perfectly feasible. But what do we obtain by it? Just think, by means of it I obtain the condition for the character of a surface such that there is a qualitative difference even mathematically speaking, in all its points. At every point another quality is present. I cannot comprehend the surface obtained like this in the same way as I comprehend some abstract Euclidean plane. I must look upon it as a surface which is differentiated within itself. And if by rotation I create three-dimensional forms then I should obtain bodies differentiated within themselves.

If you think of what I said yesterday, namely, that the Cassini Curve is also the curve in which a point must move in space if, illuminated from a point \(B\), it reflects the light to a point \(A\) with constant intensity; and if you also bear in mind that the constancy underlying the curve here brings about a relation between the effects of light at different points; then, just as in this instance certain light-effects result from the relation of the constants, so one can also imagine that a system of light-effects would follow if a variability of the second order were added to the variability of the first. In this way you can create, even in mathematics itself, a process of transition from the quantitative to the qualitative aspect.

These attempts must indeed be made in order to find a way of transition from quantity to quality,—and this endeavour we must not abandon. For a start can be made from what it is that we are really doing when we form an inner connection between the function within the variability of the second order and the function within variability of the first order. (It has nothing to do with the expression “order”, as it is familiarly used; but you will understand me, as I have explained the whole thing from the beginning.) By turning our attention to this relationship between what I have called first and second order, we shall gradually come to see that our equations must be formed differently, according to whether we are taking into account, for example, what in an ordinary bodily surface lies between the surface and our eye, or what lies behind the surface of the body. For a relationship not unlike this between the variability's of the first order and of the second order, exists between what I must consider as being between myself and the surface of a quite ordinary body and what lies behind the surface of the body. For example, suppose we are trying to understand the so-called reflection of the rays of light,—what we observe when there is a reflecting surface. It is a process taking place, to begin with, between the observer and the surface of the body. Suppose that I conceive this as a confluence of equations taking their course between me and the surface of the body in a variability of the first order, and then, in this connection consider what is at work behind the surface so as to bring about the reflection as an equation in the variability of the second order. I shall arrive at quite other formulae than are now applied according to purely mechanical laws,—omitting phases of vibration and so on—when dealing with reflection and refraction.

In this way the possibility would be reached of creating a form of mathematics capable of dealing with realities; and it is essential for this to happen, if we would find explanations particularly in the realm of astronomical phenomena. In regard to the external world, we have before us what takes place between the surface of the Earth-body and ourselves. When, however, we contemplate the celestial phenomena—say, a loop of Venus—trivially speaking we also have before us something which takes place between us and some other thing; yet the reality confronting us in this case is in fact like the realm beyond the sphere in its relation to what is within the central point. However we look to the phenomena of the heavens, we must recognise that we cannot study them simply according to the laws of centric forces, but that we must regard them in the light of laws which are related to the laws of centric forces as is the sphere to the radius.

If, then, we would reach an interpretation at all of the celestial phenomena, we must not arrange the calculations in such a way that they are a picture of the kind of calculations used in mechanics in the development of the laws of centric forces; but we must formulate the calculations, and also the geometrical forms involved, so that they relate to mechanics as sphere relates to radius. It will then become apparent (and we will speak about this next time) that we need: In the first place, the manner of thinking of mechanics and phoronomy, which has essentially to do with centric forces, and secondly, in addition to this system, another, which has to do with rotating movements, with shearing movements and with deforming movements. Only then, when we apply the meta-mechanical, meta-phoronomical system for the rotating, shearing and deforming movements, just as we now apply the familiar system of mechanics and phoronomy to the centric forces and centric phenomena of movement, only then shall we arrive at an explanation of the celestial phenomena, taking our start from what lies empirically before us.

We have now gained the most essential premises for a study of some aspects at least of celestial and also of earthly-physical phenomena. In human nature, once again, we have the very significant contrast (to ascertain which, as you will readily understand, we must leave the animal out of account to begin with)—the contrast between the organisation of the head and that of the metabolic system including the limbs. As we have seen, if we wish to relate Man to the Cosmos, we must assign the metabolic system to what is earthly,—what comes to man in a radial direction. Whereas we must assign the forming of the head to all that derives from the great Sphere,—that sends its lines of influence, as it were from the celestial Sphere towards the centre of the Earth, even as the radius reaches outward with its lines of influence to its surroundings. We saw this in the construction of the typical long bones or tubular bones by contrast to the skull-bones, the latter being sphere-like, or like a sector of a sphere.

Envisaging this difference, we must relate it, to begin with, to what appears to us in the relation of the Earth to the Celestial Sphere. You are of course aware, how the scientific consciousness of our time departs from what the naive human being, untouched by any learning, would judge of the appearance of the celestial sphere, the movements of the stars upon it, and so on. We speak of the ‘Apparent aspect’ of the celestial vault. In contrast to it, as you know, we have a picture—a World-picture—gained in a fairly complicated way by interpreting the apparent movements, and so on. Upon this picture—the form of picture which has evolved through the great changes in cosmology since the Copernican era—we are wont to base all our considerations of celestial phenomena.

Today I take it to be generally realised that this World-picture does not represent absolute reality. We can no longer maintain: What is presented to us by this picture, say, as the planetary movements or as the Sun's relation to the Planets, is the true form of the underlying reality, while what the eye beholds is mere appearance. I hardly think any competent person would adopt this standpoint nowadays. Yet he will still have a feeling that he at least gets nearer to a true conception when he proceeds from the apparent picture of the celestial movements—fraught, he will say with illusionary factors (yet after all, we must admit, objectively observed)—to the interpretation of it by mathematical Astronomy.

The question now is, do we really gain a comprehensive view of the phenomena in question if we only base our picture of the World on this, the customary kind of interpretation. As we have seen, when we do so we are in fact only basing it on what the head-man ascertains, so to speak. We base it on the aspect which emerges for man's powers of observation, aided perhaps by optical instruments. But as we saw, for a more comprehensive interpretation of the World-picture we must have recourse to all that is knowable by man, of man. We emphasised how to this end the form of man must be seen in the light of a true science of metamorphosis. Then too we must bring in the evolution of man and of mankind. In a word, concerning the celestial phenomena, or some of them at least, we cannot look for enlightenment till in our efforts to interpret them we go as far as this, calling to our aid whatever can be known of man.

Let us then presuppose what we arrived at in former lectures—the kind of qualitative mathematics, learned from the human form and growth and evolution. With this in the background let us take our start from what meets the eye—from what is said to be the mere appearance of the Heavens—asking ourselves how we may find the way to reality? Let us then ask, dear friends: What does the eye behold, what do we learn empirically, by simple observation? Then we can try to fill in the picture with what is given by the whole structure of man, both in morphology and evolution. First we will ask the question as regards those stars which are commonly described as fixed stars. I shall no doubt be repeating what is well-known to most of you, yet we must call it to mind for only by so doing, only from the facts as seen, taking them all together, shall we be able to advance to the ideas.

What then do we see as to the movement of the fixed stars, so-called? We must consider longer periods of time, since in short periods the Heaven of fixed stars presents practically the same picture year by year. Only when taking longer epochs do we find that it no longer presents the same uniform picture, but that the whole configuration changes. We can envisage it by taking one example; what we shall find in one region of the Heavens would be found in other regions too. Take then this constellation, which you know so well, the “Great Bear” or “Plough” in the Northern sky. Today it looks like this (Fig. 2). Acquaint yourselves with the minute displacements of the so-called fixed stars which have been ascertained, and which agree with what is shown by very ancient star-maps, although the latter are not always reliable. Sum up the minute displacements and calculate what the constellation will have looked like very long ago, and you get this appearance (Fig. 1). You see, the fixed stars, so-called, have undergone considerable displacements. About 50,000 years ago, if we may reckon it from the minute changes observed, the constellation will have looked like this. If we continue to sum up the ascertainable displacements for the future,—assuming, as we surely may do, that they will continue at least approximately in the same direction—we may conclude that 50,000 years from now the constellation will have this appearance (Fig. 3).

Just as this constellation changes in the course of years—for we have only chosen it as an example—so do the others. Thus when we make our drawings, of the Zodiac for instance in its present form, we must be clear that the form of it changes in the course of time—if we may thus include time in our calculations and in interpreting them.

We must therefore regard the celestial sphere as undergoing changes within itself, ever changing its configuration,—changing the aspect of the starry Heavens which we behold in the fixed stars,—though the perpetual change is scarcely perceptible in shorter periods. Naturally, our observations here cannot go very far, nor can we do very much by way of interpretation, though as some of you will know, modern experiments enable us to ascertain even those movements of the stars which are along the line of sight,—towards us or away from us. Yet it remains very difficult to interpret the ever-changing aspect of the starry heavens. In the further course we shall be asking, what human value and significance is to seek in the interpretation.

Having considered the movements of the fixed stars, let us now ask after the movements of the planetary stars. The movement of the planetary stars as we behold it is indeed complicated. The movement we observe is such that if we follow the path of a planet, in so far as it is visible, we see it moving in a curve of peculiar shape—different for the different planets and different too for the same planet at different times. From this we have to take our start. Take for example the planet Mercury. Precisely when it is nearest to us, its path is of peculiar form. In a certain direction it seems to move across the Heavens. Study it daily when visible, we see it moving thus; but them it turns and makes a loop, and then goes on as I am showing (Fig. 4).¹ It makes one such loop in a so-called synodical period of revolution. This then we may describe as the movement of Mercury—to begin with at least, so far as observation is concerned. The rest of the path is simple, only at certain places do the loops occur.

Passing to Venus we have a similar phenomenon, though somewhat different in shape and form. Venus moves onward thus, then turns and then moves on, thus (Fig. 5). Here as a rule there is only one loop in the course of a year, and, once again, when the planet—as we conclude from other astronomical data—is nearest to us. Now to Mars: Mars has a similar path, only flatter. We may draw it somewhat like this (Fig. 6). In this case, you see, the loop is more compressed, but the appearance is still that of a loop,—distinctly so. Often however the path (both of this and other planets) is so formed that the loop is completely dissolved, flattened away until it is no more. The path is loop-like, though not an actual loop. (Fig. 7) We will pass by the planetoids, interesting though they are, and look at Jupiter and Saturn. We find them too describing loops or loop-like paths. They again do it when nearest the Earth—and only once a year. As a general rule they make a single loop each year.

We have then to consider certain movements on the part of the fixed stars, and the movements of planets. The movements of fixed stars occupy gigantic periods, judged by our standards of time. The movements of the planets comprise a year or fractions of a year and reveal from time to time strange deviations from their ordinary path, loop-lines of movement, in effect. The question now is, what are we to make of these two kinds of movement? How to interpret the loop-movement for example? It is a very big question. Only the following reflection can lead towards any kind of interpretation of the loop-movements.

In all our human observation the fact is that we are quite differently related to our own conditions and to those things which are not our own;—which take place apart from us, outside us, so to speak. You need only recall how it is with objects: The enormous difference between your relation to any object of the so-called outer world and to an object inside yourself, which you, so to speak, are sharing-in with your own inner experience. If you have any object before you, you see it, you observe it. What you yourself are living in—your liver, your heart, even your sense-organs to begin with you can observe. There is the same contrast, though not quite so strongly marked, with regard to the conditions in which we are living in the outer world. If we ourselves are in movement and if it is possible for us to remain unconscious of how we bring about the movement, then we may well be unaware of our own movement and therefore leave it out of account in judging outer movements. That is to say, though we ourselves are in movement, we leave this out; we deem ourselves at rest and envisage only the external movement.

It is on this reflection, in the main, that the interpretation of movements amid the celestial phenomena has been based. You are aware, it has been argued: Man, at a certain point on Earth, shares of course in the spatial movement of his earthly habitation (eg the circling movement of his latitude) but knows it not and hence regards, what he sees happening in the Universe outside him, as a real movement in the opposite direction. The argument has been abundantly made use of! The question now is: How might this principle be modified if we take into account that in man's organized (if I may so express it) radially, whilst in our head-man we are oriented spherically. If it were then a fundamental feature of our own state of movement that we relate ourselves differently to the Radius and to the encompassing Sphere, this fact would somehow make itself felt in what appears to us in the outer Universe.

Imagine what I have said to be in some way true. Suppose for instance that you yourself were moving thus (Fig. 8),—you were describing a Lemniscate. Let us assume however that the Lemniscate you were describing was not exactly like this, but that by variation of the constants the form of Lemniscate were brought about in which the lower branch did not close (Fig. 9). Assume then that a Lemniscate arises which by a certain variation of the constants is open on one side. The curve is mathematically feasible, and if you find the right way, you can certainly draw it into the human form and figure.

Say now that this were the surface of the Earth (Fig. 10). We should have to draw, somehow in relation to the Earth, what passes through our limb-nature and then in some way turns, goes through our head-nature and then back again into the Earth. Say you could truly draw into the nature and organisation of man such an open Lemniscate; we should be justified in saying: There is an open Lemniscate of this kind in man's nature. The question is, is it of real significance to speak of such an open Lemniscate in human nature? It is indeed. You need only make a deeper morphological study; you will find the Lemniscate, either in this or in some modified form, in diverse ways inscribed in human nature. These things have not been gone into with due method. I advise you, try it. (As I said, we are only giving indications for further work; diligent research is needed.) Try it; investigate the curve that arises if you trace the middle line of a left-hand rib, then go past the junction into the vertebra, then turn and go back along the right rib (Fig. 11). Bear in mind what it must signify that as you go along this line—rib-vertebra-rib—various inner relationships of growth must play their part, not only quantitatively but qualitatively; then you will find in the Lemniscate with its loop-formation a morphological key to the whole system. Going upward from thence to the head-organisation, the farther you go upward, the more will you find it necessary to modify the form of Lemniscate. At a certain point you must imagine it transformed; the transformation is already indicated in the forming of the sternum, where the two come together. When you get up into the head there is a far-reaching metamorphosis of the lemniscatory principle. Study the whole human figure—the contrast above all of the nerves-and-senses organisation and the metabolic,—you get a Lemniscate tending to open out as you go downward and to close as you go upward. You also get Lemniscates—though highly modified, with the one loop extremely small—if you follow up the pathway of the centripetal nerves, through the nerve-centre and outward again to the termination of the centrifugal nerve. Follow it all in the right way: Again and again you will find this Lemniscate inscribed in man's nature,—man's above all. Then take the animal organisation with its manifestly horizontal spine. You will find it differing from the human, in that the Lemniscates, whether the downward loop be open or closed to some extent, are far less modified, less varied than they are in man. Moreover in the animal their planes are more parallel, whereas in man they are variedly inclined and askew to one another.

It is an immense and very promising field of work,—this ever-deepening elaboration of morphological study. And as you apprehend these tasks, you will appreciate the outlook of such men—of whom there have always been a few—as Moritz Benedikt for instance, whom I have mentioned before. Benedikt had many fruitful thoughts and good ideas. As you may read in his memoirs, he regretted how little possibility there is of speaking to doctors of medicine from a mathematical standpoint or with the help of mathematical notions. In principle he is quite right, only we have to go still farther. Ordinary mathematics, reckoning in the main on rigid forms of curve in a rigid Euclidean space, would help us little if we tried applying it to organic forms. Only by seeking, as it were, to carry life itself into the realms of mathematics and geometry as such, by thinking of the independent and the dependent variable in an equation as being subject to an organic and inherent variation, as illustrated yesterday for the Cassini curves (Variability of the first and of the second order), only thus shall we make progress. But if you do this immense possibilities will be opened up. It is indeed already indicated in the principles applied when constructing cardioid or cycloid curves; you must only not fall back again into rigidity of treatment.

Apply this principle—the inner mobility, as it were, of movement in itself—to Nature. Try to express in equations, this that ‘moves the moving’. You will then find it possible, mathematically to penetrate what is organic. You will come to say, for it can well be formulated thus: The axioms of rigid space—space immobile in itself—lead to an understanding of inorganic Nature. Conceive a space that is inherently mobile—or algebraic equations whose very functionality is in itself a function—and you will find the transition to a mathematical understanding of organic Nature. This incidentally is the method which should accompany the efforts now being made to investigate the transition-forms from inorganic Nature to organic, as regards shape and form at least. Valueless apart from this, they have a future if this method be applied.

Take now the presence of the loop-making tendency in the human body and compare it with what confronts us, admittedly in a more irrational form, in the forms of movement of the planets. You will then realise: The 'apparent movements’ of the planets, as we are wont to call them, in a most striking way inscribe, in forms of Movement in the Heavens, what in the human body is a Form as such—a characteristic, fundamental figure. Therefore, to say the least, we must in some way correlate this basic form in the human body and these phenomena in the Heavens. And we shall now be able to say: Behold the loop. It always appears when the planet is relatively near the Earth,—therefore when we, being on the Earth, are in a special relation to the planet. Consider the position of the Earth in its yearly course and our position on the Earth. (We must refer it back to our own formative period, the embryo-period of our life, needless to say.) Consider in effect how we are alternating between a position relative to the planet wherein we turn our head towards the planetary loop and a position where we take leave of the loop and at length turn our head away from it. We in our process of formation are thus related to the planet: We are exposed at one time to the planet's loop and at another to the remainder of its path. We can therefore relate, what lies nearer to our head, to the loop, and what belongs more to the remainder of our body, to the planetary path outside the loop.

Take in addition what I said before, I said, with regard to the morphological relation of the tubular or long bone to the skull-bone: Try how you would have to draw it. Here, throughout the long bone, is the radius; then as you pass to the skull-bone you will have to turn, like this (Fig. 12). Project this turn, in relation also to the Earth's movement, outward into the Heavens. It is the loop and the rest of the planet's path! If we develop a feeling for morphology in the higher sense, we can do no other than assign the human form and figure to the planetary system.

And now the movement of the fixed stars themselves:—The movements of the fixed stars will naturally be less concerned with the several movements of individual human beings. Think on the other hand of the whole evolution of mankind on Earth. Bear in mind all we have said in these days of the relation of the great Sphere to the human head-formation. You cannot but divine that there will be some relation between the metamorphoses of aspect of the starry Heavens, and of the evolution of mankind in soul and spirit. There is the vault of the great Sphere above us. It reveals only that part of the movements which would correspond to the loop among the planets (nay more, as it would seem, only to part of the loop; Fig. 13, dotted line). In the movements of fixed stars, the rest of the path is omitted. Our attention is drawn to this great differentiation: The planets must somehow correspond to the whole man; the fixed stars only to what forms the head of man. Now we begin to get some guidance, how to interpret the loop.

We human beings are in some way with the Earth. We are at some point on Earth and we move with it. We cannot but refer, what appears to us in projection on the vault of heaven, to the movements we ourselves are making with the Earth. For, as we move with the Earth (we ,must project this backward, once more, backward in time to the embryo-period of our life),—as we move with the Earth, what we have in us comes into being, formed as indeed it is by the very forces of movement. In the movements we see up yonder in their seeming forms and pictures, we have to recognise the cosmic movements we ourselves are making in the year's course. We realise it as we approach the true aspect of the loop-curve. (Downward of course we always see the loop still open. In the immediate aspect, it does not close at all. Looking at this alone, we should never get a complete path. We only get the complete path when contemplating the entire revolution.)

I am relating all this rather quickly. You must reflect on it in detail and try to see the different things together. The more minutely and scrupulously you do so, the more will you find that the planetary movements are, to begin with, images—images of—movements you yourself accomplish, with the Earth, in the year's course. (We shall see in time, how a synthesis arises from the different planetary movements.)

If then we see the human being as a whole and his projection to the Cosmos, we are led to recognise that the true form of movement of the Earth in the year's course will be the loop-curve or Lemniscate. We shall have to study it more closely during the next few days, but at this stage we are already led to conceive the path of the Earth itself as a loop-curve—quite apart now from its relation to the Sun or any other factor. What is projected then, for our perception, the planetary paths with the loops they make,—we must regard as the projection by the planets of the loop-path of the Earth on to the vault of Heaven, if we may formulate thus simply a very complicated set of facts. As to why, when the planet draws near the loop, we have to leave the rest of the path open during a relatively short space of time,—the reason lies in the fact that under certain conditions the projection of a closed curve may appear open. For example, if you were to make a Lemniscate, say of a flexible rod, and project the shadow of it on to a plane, you could easily make it so that the projection of the lower part appeared divergent and unclosed, whilst the upper part alone was closed; so the entire projection would become not unlike a planetary path. Quite simply in the shadow-figure, you could construct the likeness of a planet's path.

• 1. In Figures 4 to 7 only one of the many varieties of loop which actually occur is shown in each case.

I will begin today by pointing out that our studies hitherto have led us to a specific result. We have drawn attention on the one hand to the movements of the heavenly bodies, and, though it still remains for us to do it in more detail, we have at least gained some conception: Here are a number of cosmic bodies in movement, in a certain order and configuration. Meanwhile we have also been drawing attention to the form of man, and incidentally, from time to time, to the forms of animal and plant-nature; this we shall have to do still more, to gain the necessary supports from diverse realms. In the main however, it is the human form and figure we have contemplated, and in so doing we have divined that the formation of man is in some way related to what finds expression in the movement of celestial bodies. We want to formulate it with great care.

Yesterday I showed that wheresoever we may look in the human body, we shall find the formative principle of the looped curve or Lemniscate, save for the two outermost polarities—the Radius and the Sphere. Thus in the human body we perceive three formative principles (Fig. 1): The Sphere, with its activity primarily going inward, the Radius, and between these the looped curve or Lemniscate. Truly to recognise these formative principles in the human organism, you must imagine the Lemniscate as such with variable constants, if I may use the paradox. Where a curve normally has constants in its equation, we must think variables. The variability is most in evidence in the middle portion of the human body. Take as a whole the structure of the pairs of ribs and the adjoining vertebrae. True as it is then that in the vertebra the one half of the Lemniscate is very much condensed and pressed together, whilst in the pair of ribs the other half is much extended and drawn apart (Fig. 2), we must not be put off my this. The underlying formative principle is the Lemniscate, none the less. We simply have to imagine that where the ribs are (the drawing indicated those that are joined in front via the sternum) the space is widened, matter being as it were extenuated, while, to make up for this, the matter is compressed and the space lessoned in the vertebra.

Let us now follow the human form and figure upward and downward from this middle portion. Upward we find the vertebra as it were bulged out into a wide cavity (Fig. 3), while the remaining branches of the Lemniscate seem to vanish, nestling away, so to speak, in the internal formative process, becoming hidden and undefined. Going downward from the middle portion, we contemplate for instance the attachment of the lower limbs to the pelvis. In all that opens downward from this point, we find the other half of the loop fading away. We have therefore to contemplate a fundamental loop-curve, mobile and variable in itself. This dominates the middle part of man. Only, the formative forces of it must be so imagined that in the one half (Fig. 2) the material forces become, as it were, more attenuated and the loop widens, while in the other it contracts.

Further we must imagine that from this middle region upward the portion of the Lemniscate which in the vertebra was drawn together, bulges and widens out, while the other, downward-opening portion vanishes and eludes us. On the other hand, as you go downward from the middle part of man, the closed loop grows minute and fades away, while those portions of the curve which disappear as you go towards the head, run out into the radial principle and are here prolonged. (Fig. 4)

We should thus find our way into it, till we are able to see the only moving Lemniscate with perceptive insight. Also we think how the formative principle of the moving Lemniscate is combined with forces which are spheroidal on the one hand and on the other radial—radial with respect to the Earth's centre. We then have a system of forces which we may conceive as being fundamental to the form and figure, to the whole forming and configuration of the human body. (By the word “forces” I mean nothing hypothetical;—purely and simply what is made manifest in the forming of it.) Answering to this , in cosmic space, in the movement of celestial bodies, we also find a peculiar configuration,—configuration of movements. In yesterday's lecture, we recognised in the planetary loops the very same principle outside us which is the principle of form within us. Let us now follow this loop-forming principle in greater detail. Is it not interesting that Mercury and Venus make their loops when the planets are in inferior conjunction, i.e., when they are roughly between the Earth and the Sun? In other words, their loop occurs when what the Sun is for man—so to express it—is enhanced by Venus and Mercury. As against this, look for the loops of Mars, Jupiter and Saturn. These loops we find occurring when the planets are in opposition to the Sun. This contrast too, of oppositions and conjunctions, will in some way correspond to a contrast in the building forces of man. For Saturn, Jupiter and Mars, because their loops appear in opposition, the loops as loops will be most active and influential. Thinking along these lines, we shall indeed relate the loop-formation of Saturn, Jupiter and Mars to that in man which is little influenced by the Sun; for it takes place, once more, when the planet is in opposition. Whilst, inasmuch as Venus and Mercury form their loops when in conjunction, their loop-formation must in some way be related to what is brought about, amid the formative principles of man, by the Sun—or by what underlies the Sun. We shall therefore conceive the Sun's influence to be in some sense reinforced by Venus and Mercury, while it withdraws, as it were, in face of the superior planets, so-called. The latter, precisely during loop-formation, bring to expression something that bears directly, not indirectly, upon man.

If we pursue this line of thought and bear in mind that there is the contrast between Radius and Sphere, then we need but recall the form that comes to manifestation in these movements, and we shall say: In Mays, Jupiter and Saturn the essential phase must be when they are forming their loops, that is to say, when, in a manner speaking, the sphere-forming process comes into evidence. Mars, Jupiter and Saturn (not to speak of further planets) will show their influence upon that element in man which is assigned to the sphere-forming process, namely the human head. In contrast to this—they are indeed the polar opposite—the movements of Venus and Mercury will somehow find expression in what in man too is the opposite pole, opposite to the forming of the head,—i.e., what abandons parallelism with the spherical formation and becomes parallel to the radial. Where the one part of the Lemniscate becomes minute and the other grows into the limbs, into a purely radial development, we have to look for the relation to Venus and Mercury. This in turn will lead us on to say: In the superior planets, which make their loop when in opposition, it is the loop that matters; they develop their intensity while they form the loop. Whilst in the inferior planets Venus and Mercury—it is essential that they wield their influence by virtue of what is not the loop,—i.e., in contrast to the loop, by the remainder of the planet's path. Think of a Lemniscate like this (Fig. 5), say in the case of Venus (I draw it diagrammatically).

You will understand it if you imagine this part (dotted line) ever less in evidence, the farther you go downward. That is to say, whilst in the path of Venus it closes, in its effects it no longer does so, but, as it were, runs out into parabolic branches, answering precisely to what happens in the human limb, where the vertebra form fades away and loses character (to put it very briefly, omitting details). This loop of the Lemniscate is represented by the path's fading away, not being fully maintained; it only indicates the direction but cannot hold it. So, where it closes in the path of Venus in the Heavens, in man's formation it falls asunder. Thus, to sum up, the building principle of the human form, howsoever modified, is based on this; the metamorphosis emerges between head and limbs—the limbs with the metabolism which belongs to them—and in the great Universe this answers to the contrast between those planets that form them in opposition to the Sun. Between the two is then the Sun itself.

Now, my dear friends, something quite definite results from this Namely, we see that also with respect to the qualitative effects we have just referred to, we have to recognise in the Sun's path, even as to its form, something midway between what we find in the forms movement of the superior and of the inferior planets respectively. We must therefore assign, what finds expression in the path and movement of the Sun, to all that in man which is midway between the forming of the head and the metabolism, In other words, we must attribute to the rhythmic system some relation to the path of the Sun. We therefore have to imagine a certain contrast between the paths of the superior and of the inferior planets; and in the Sun's path a quality midway between the two.

There is now a very evident and significant fact, regarding both the Sun's path and the Moon's. Follow the movements of the two heavenly bodies; neither of them makes any loop. They have no loop. Somehow therefore we must contrast the relation to man, and to Earth nature generally, of Sun and Moon on the one hand and of the loop-forming planetary paths on the other. The planetary paths with their characteristic loops quite evidently correspond to what makes vortices and vertebrae,—to what is lemniscatory in man.

Look simply at the human form and figure and think of its relation to the Earth; we can do no other than connect what is radial in human form and stature with the path of the Sun, even as we connect what is lemniscatory in form with the typical planetary path.

You see then what emerges when we are able to relate to the starry Heavens the entire human being, not only the human organ of cognition. This in effect emerges: In the vertical axis of man we must in some way seek what answers to the Sun's path, whilst in all that is lemniscatory in arrangement we have to seek what answers to the planetary paths,—lemniscatory as they are too, though in a variable form. Important truths will follow from this, We must conceive, once more, that through his vertical axis man is related to the Sun's path. HOW then shall we think of the other path which also shows no loops, namely the Moon's? Quite naturally—you need only look with open mind at the corresponding forms on Earth—we shall be led to the line of which we spoke some days ago, the line that runs along the spine of the animal. There we must seek what answers to the Moon's path. And in this very fact—the correspondence of the human spinal axis to the Sun's path and of the animal spinal axis to the moon's _ we shall have to look for the essential morphological difference between man and animal.

Precisely therefore when we are wanting to discover what is essential in the difference of man and animal, we cannot stay on Earth. A mere comparative morphology will not avail us, for we must first assign what we there find to the entire Universe. Hence too we shall derive some indication of what must be the relative position of the Sun's path and the Moon's—shall we say, what is their mutual situation, to begin with, in perspective (for here again we must express it with great caution). They must be so situated that the one path is approximately perpendicular to the other.

The human vertical therefore—or, had we better say, what answers to the main line and direction of the spine in man—is related to the Sun's path. The rational morphology we are pursuing makes this coordination evident. Mindful of this, we must surely relate the Sun's path itself to what in some way coincides with the Earth's radius. Admittedly, the Earth may move in such a way that many of her radii in turn coincide with the Sun's path. The relation indicated will need defining more precisely in coming lectures. Yet this at least gives us a notion of it: the direction of the Sun's path must be radial in relation to the surface or the Earth. We have no other alternative. In no event can the Earth be revolving round the Sun. What has been calculated—quite properly and conscientiously, of course—to be the revolution of the Earth around the Sun must therefore be a resultant of some other kind of movements. To this conclusion we are driven.

The many relevant details as regards human form and growth are so very complicated that in this brief lecture-course not everything can be gone into. But if you really concentrate upon the morphological descriptions given (though they are only bare indications of a qualitative morphology), you will be able to read it in the human form itself: The Earth is following the Sun! The Sun speeds on ahead, the Earth comes after. This then must be the essence of the matter: the earthly and the solar orbit in some way coincide, and the Earth somehow follows the Sun, making it possible as the Earth rotates for the Earth's radii to fall into the solar path, or at the very least to be in a certain relation to it.

Now you may very naturally retort that all this is inconsistent with the accepted Astronomy. But it is not so,—it really isn't! As you are well aware, to explain all the phenomena, Astronomy today must have recourse not only to the primary notion of a stationary Sun supposed to be at the focus of an ellipse along which the Earth is moving—but to a further movement, a movement of the Sun itself towards a certain constellation. If you imagine the direction of this movement and other relevant factors, then from the several movements of Sun and Earth, you may well be able to deduce a resultant path for the Earth, no longer coincident with the ellipse in which the Earth is said to be going round the Sun, but of a different form which need not be at all like the supposed ellipse. All these things I am gradually leading up to; for the moment I only wish to point out that you need not think what I am telling you so very revolutionary as against orthodox Astronomy. Far more important is the method of our study,—to bring the human form and figure into the system of the starry movements. My purpose here is not to propound some astronomical revolution, nor is it called for. Look, for example: say this or something like it (Fig. 6) is the Earth's movement, and the Sun too is moving, You can well imagine, if the Earth is following the Sun in movement, it is not absolutely necessary for the Earth always to be running past the Sun tangentially. It may well be that the Sun has already gone along the same path and that the Earth always to be running past the Sun tangentially. It may well be that the Sun has already gone along the same path and that the Earth is following, Nay, it is possible, envisaging the hypothetical velocity that has been calculated for the Sun's proper movement, you may work out a very neat arithmetical result. Work out the resultant of the assumed Earth-movement and the assumed Sun-movement; you may well get a resultant movement compatible with present-day Astronomy,—velocity and all. Let me then emphasise once more: What I am here propounding is not unrelated to present-day Astronomy, nor do I mean it not be. Quite on the contrary, it is related to it more thoroughly and deeply than theories which are so frequently presented, nicely worked out in theoretic garb, selecting certain movements and omitting others. I am not therefore instigating an astronomical revolution in these lectures; let me say this again to prevent fairy-tales arising. What I intend is to co-ordinate the human form—inward and outward form, figure and formation—with the movements of the heavenly bodies, nay, with the very system of the Cosmos.

For the rest, may I call your attention to this: It is not so simple to bring together in thought our astronomical observations of the heavenly bodies and the accepted constructions of the orbits. For as you know from Kepler's Second Law, an essential feature, on which the forms of the orbits depend, are the radius-vectors,—their velocity above all. The whole form of the path depends on the functionality of the radius vectors. If this be so, does it not also reflect upon the forms of the paths which actually confront us? May it not be that we are cherishing illusions after all, at the mere outward aspect of them? It is quite possible: What we here calculate from the velocity and length of the radius vectors might not be primary magnitudes at all. They might themselves be only the resultants of the true primary magnitudes. If so, then the seeming picture which emerges must refer back to another and more deeply hidden.

This too is not so far afield as you might think. Suppose that in the sense of present-day Astronomy you wished to calculate the Sun's exact position at a given time of day and on a given date. Then it will not suffice you to take your start from the simple proposition, 'the Earth moves round the Sun'. People have thought it strange that in the ancient Astronomy (that of the Mysteries, not the exoteric version) they spoke of three Suns instead of one. So they distinguished three Suns. I must confess, I do not find it so very striking. Modern Astronomy too has its three Suns. There is the Sun whose path is calculated as the apparent counterpart of the Earth's movement round the Sun. This Sun occurs, does it not , in modern Astronomy? The path of it is calculated. Astronomy then has another Sun—an imagined one of course—with the help of which certain discrepancies are corrected. And then it has a third Sun, with the help of which it re-corrects discrepancies that persist after the first correction. Modern Astronomy too therefore distinguishes three: the real Sun and two imagined ones. It needs the three, for what is calculated to begin with does not accord with the Sun's actual position. It is always necessary to apply corrections. This alone should be enough to show you that we should not build too confidently on mere calculation. Other means are needed to arrive at adequate conceptions of the starry movements; others than the science of our time derives from sundry premises of calculation.

The broad ideas of planetary paths we have been laying out, it I may put it so, call now for great definition. Yet we shall only come to this if we contrive first to go further in out study of Earth-nature, to see their mutual relation in a certain aspect.

The Kingdoms of Nature are commonly thought of in a straight line: mineral kingdom, plant kingdom, animal kingdom, and I will add, human kingdom. (Some authorities would not admit the fourth, but that need not detain us.) The question now is: Is this arrangement sensible at all? Undoubtedly it is implicit in many of our modern lines of thought; at least it was so in the golden age of the mechanical outlook upon Nature. Today I know, in these wider realms of Science, there is a certain atmosphere of resignation, not to say despair. The habits of mind however remain the same as at their heyday, 20 or 30 years since. The scientists of that time would have been content, had they been able to follow up this series—mineral kingdom, plant kingdom, animal kingdom, man,—with the mineral kingdom as the amplest, deriving therefrom, by some combination of mineral structure, the structure of the plant, then by a further combination of plant structure the structure of the animal, and so on to man. The many thoughts that were pursued about the primal generation of living things, generatic aequivocs,—were they not eloquent of the tendency to derive animate living Nature from inanimate and at long last from inorganic or mineral? To this day, I believe, many scientiste would doubt if there is any other rational way of conceiving the inner connection in the succession of Nature's Kingdoms than by deriving them all ultimately from the Inorganic, even where they culminate in Man. You will find countless papers, books, lectures and so on, including highly specialised ones claiming to be strictly scientific, the authors of which—as though hypnotised—are always looking at it from this angle. How, they inquire, can it have happened, somewhere at some time in the course of Nature, that the first living creature came into being from some molecular distribution, i.e. from something purely mineral in the last resort?

The question now is, is it true at all to put the kingdoms of Nature in series in this way? Can it be done? Or, if we do, are we doing justice to their most evident and essential features? Compare a creature of the plant kingdom with an animal to begin with. Taking together all that you observe, you will not find in the forming of the animal anything that looks like a mere continuation or further elaboration of what is vegetable. If you begin with the simplest plant, the annual, you may well conceive its formative process to be carried further in the perennial. But you will certainly not be able to detect, in the organic principles of plant form and growth, anything that suggests further development towards the animal. On the contrary, you will more likely ascertain a polarity, a contrast between the two. You apprehend this polarity in the most evident phenomenon, namely the contrasting processes of assimilation: the altogether different relation of the plant and of the animal to carbon, and the characteristic use that is made of oxygen. I may remark, you must be careful here, to see and to describe it truly. You cannot simply say, the animal breathes-in oxygen while the plant breathes oxygen out and carbon in. It is not so simple as that. Nevertheless, the plant-forming process taken as a whole, in the organic life, reveals an evident polarity and contrast (as against the animal) in its relation to oxygen and carbon. The easiest way to put it is perhaps to say: What happens in the animal, in that the oxygen becomes bound to carbon and the carbonic acid is expelled, is for the animal itself and for man too.—an un-formative process, the very opposite of formative, a process which must be eliminated if the animal is to survive. And now the very thing which is undone in the animal, has to be done, has to be formed and builded in the plant. Think of what in the animal appears in some sense as a process of excretion, what the animal must get rid of makes for the forming and building process in the plant. It is a tangible polarity. You cannot possibly imagine the plant-forming process prolonged in a straight line, so as to derive therefrom the animal-formation. But you can well derive from the plant-forming process what has to be prevented in the animal. From the animal the carbon has to be taken away by the oxygen in the carbonic acid. Turn it precisely the other way round, and you will readily conceive the plant-forming process.

You therefore cannot get from plant to animal by going on in a straight line. On the other hand you can without false symbolism imagine here an ideal mean or middlepoint, on the one side of which you see the plant—and on the other the animal—forming process. It forks out from here (Fig. 7). What is midway between,—let us imagine it as some kind of ideal mean. If we now carry the plant forming process further in a straight line we arrive not at the animal but at the perennial plant. Imagine now the typical perennial. Carry the stream of development which leads to it still further; in some respects at least you will not fail to recognise in it the way that leads toward mineralisation. Here then you have the way to mineralisation, and we may justly say; In direct continuation of the plant forming process there lies the way that leads to mineralisation. Now look what answers to it at the contrasting pole, along the other branch (Fig. 7). To proceed by a mere outward scheme, one would be tempted to say: this branch too must be prolonged. There would be no true polarity in that. Rather should you think as follows: In the plant-forming process I prolong the line. In the animal-forming process I shall have to proceed negatively, I must go back, I must turn round; I must imagine the animal-forming process not to shoot out beyond itself but to remain behind—behind what it would otherwise become.

Observe now what is already available in scientific Zoology, in Selenka's researches for instance on the difference between man and animal in the forming of the embryo and in further development after birth,—comparing man and the higher animals. You will then have a more concrete idea of this "remaining behind". Indeed we owe our human form to the fact that in embryo-life we do not go as far as the animal but remain behind. Thus if we study the three kingdoms quite outwardly as they reveal themselves, without bringing in hypotheses, we find ourselves obliged to draw a strange mathematical line, that tends to vanish as we prolong it. This is what happens at the transition from animal to men, whilst on the other side we have a line that really lengthens (Fig. 8).

Here is a fresh extension of mathematics. You are led to recognise a distinction—a purely mathematical one—when you draw this diagram. Namely there are lines which when continued grow longer, and there are lines which when continued grow shorter. It is a fully valid mathematical idea. If then we want to set out the Kingdoms of Nature in a diagram at all, we must do it thus. First we must have some ideal point to start from. Thence it forks out: plant kingdom, animal kingdom on either hand. Thereafter we must prolong the two lines. Only, the plant-kingdom-line must be so prolonged that it grows longer; the animal-kingdom-line so that it grows shorter as we prolong it. I say again, this is a genuine, mathematical idea.

We thus arrive at real relationships between the Kingdom of Nature, though we begin by simply placing them side by side. The question now is—and we will only put it as a question,—What in reality corresponds to the ideal point in our diagram? We may divine that as the forming of the Kingdoms of Nature is related to this ideal point, so too must there be movements in the great Universe which relate to something somehow corresponding to it,—to this ideal mean. Let us reflect on it until tomorrow.

In popular works, as you are well aware, the evolution of astronomical ideas is thus presented—Until Copernicus, they say the Ptolemaic system was prevailing. Then through the work of Copernicus the system we accept—though with modifications—to this day, became the intellectual property of the civilised world. Now for the thoughts we shall pursue in the next few days it will be most important for us to be aware of a certain fact in this connection. I will present it simply by reading, to begin with, a passage from Archimedes. Archimedes describes the cosmic system or starry system as conceived by Aristarchus of Samos, in these words—“In Aristarchus' opinion the Universe is far, far greater. He takes the stars and the Sun to be immobile, with the Earth moving around the Sun as centre. He then assumes that the sphere of the fixed stars,—its centre likewise in the Sun,—is so immense that the circumference of the circle, described by the Earth in her movement, is to the distance of the fixed stars as is the centre of a sphere to the surface thereof.”

Taking these words to be a true description of the spatial World-conception of Aristarchus of Samos, you will admit: Between his spatial picture of the Universe and ours, developed since the time of Copernicus, there is no difference at all. Aristarchus lived in the third Century before the Christian era. We must therefore assume that among those who like Aristarchus himself were leaders of cultural and spiritual life in a certain region at that time, fundamentally the same spatial conception of the World held good as in the Astronomy of today. Is it not all the more remarkable that in the prevailing consciousness of men who pondered on such things at all, this work-conception—heliocentric, as we may call it,—thereafter vanished and was supplanted by that of Ptolemy? Till, with the rise of the new epoch in civilisation, known to us as the Fifth post-Atlantean, the heliocentric idea comes forth again, which we have found prevailing among such men as Aristarchus in the 3rd Century B.C.! (For you will readily believe that what held good for Aristarchus, held good for many people of this time.) Moreover if you are able to study the evolution of mankind's spiritual outlook—though it is difficult to prove by outer documents—you will find this heliocentric conception of the World the more widely recognised by those who counted in such matters, the farther you go back from Aristarchus into more distant times. Go back into the Epoch we are wont to call the Third post-Atlantean, and it is true to say that among those who were the recognised authorities the heliocentric conception prevailed during the Epoch. The same conception prevailed which Plutarch says was held by Aristarchus of Samos. Plutarch moreover described in such terms that we can scarcely distinguish it from that of our own time.

This is the noteworthy fact. The heliocentric conception of the World is there in human thought, the Ptolemaic system supplants it, and in the Fifth post-Atlantean Epoch it is re-conquered. In all essentials we may aver that the Ptolemaic system held good for the Fourth post-Atlantean Epoch and for that alone. Not without reason do I bring this in today, after speaking yesterday of an ‘ideal point’ in the evolution of the Kingdoms of Nature. As we shall see in due course, there is an organic relationship between these diverse facts. But we must first enter more fully into the one adduced today.

What is the essence of the Ptolemaic cosmic system? The essence of it is that Ptolemy and his followers go back again to the idea of an Earth at rest, with the fixed-star Heavens moving around the Earth; likewise the Sun moving around the Earth. For the movement of the planets, the apparent forms of which we have been studying, he propounds peculiar mathematical formulae. In the main, he thinks in this way: Let this be the Earth (Fig. 1). Around it he conceives the Heaven of fixed stars. Then he imagines the Sun to be moving in an eccentric circle round the Earth. The planets also move in circles. But he does not imagine them to move like the Sun in one circle only. No; he assumes a point (Fig. 1) moving in this eccentric circle which he calls the ‘Deferent’, and he makes this point in its turn the centre of another circle. Upon this other circle he lets the planet move, so that the true path of the planet's movement arises from the interplay of movements along the one circle and the other. Take Venus for example. Says Ptolemy: around this circle another circle is rotating; the centre of the latter circle moves along the former. The actual path of Venus would then be, as we should say, a resultant of the two movements. Such is the planet's movement around the Earth; to comprehend it we must assume the two circles, the large one, called the “deferent”, and the small one, know as the ‘epicyclic’ circle. Movements of this kind he attributes to Saturn, Jupiter, Mars, Venus an Mercury, only not to the Sun. The Moon he conceives to move in yet another small circle,—an epicyclic circle of its own.

These assumptions were due to the Ptolemaic astronomers having calculated with great care the positions on the Heavens at which the planets were at given times. They computed these circling movements so as to understand the fact that the planets were at given places at given times. It is astonishing how accurate were the calculations of Ptolemy and his followers,—relatively speaking at least. Draw the path of any planet—Mars, for instance—from modern astronomical data. Compare this 'apparent path', so-called, of Mars, drawn as observed today, with the path derived from Ptolemy's theory of deferent and epicyclic circles. The two curves hardly differ. The difference, relatively trifling, is only due to the still more accurate results of modern observation. In point of accuracy these ancients were not far behind us. That they assumed this queer system of planetary movements, which seems to us so complicated, was not due therefore to any faulty observation. Of course the Copernican system is simpler,—that will occur to everyone. There is the Sun in the midst, with the planets moving in circles or ellipses round it. Simple, is it not? Whereas the other is very complicated: a circular path superimposed upon another circle, and an eccentric one to boot.

The Ptolemaic system was adhered to with a certain tenacity throughout the Fourth post-Atlantean epoch, and we should ask ourselves this question: Wherein lies the essential difference in the way of thinking about cosmic space and the contents of cosmic space, such as we find it in the Ptolemaic school on the one hand and in Aristarchus and those who thought like him in the other? What is the real difference between these ways of thinking about the cosmic system? It is difficult to describe popularly, for many things seem outwardly alike, whilst inwardly they can be very different. Reading Plutarch's description of Aristarchus system, we shall say: This heliocentric system is fundamentally no different from the Copernican. Yet if we enter more deeply into the spirit of the Aristarchian world-picture, we find it different. Aristarchus too, no doubt, follows the outer phenomena's with mathematical lines. In mathematical lines he represents to himself the movements of the heavenly bodies.

The Copernican's do likewise. Between the two there intervenes this other system—the strange one of the Ptolemaic school. Here it cannot be said that the forming of mathematical pictures coincides in the same way with what is observed. The difference in this respect is all-important. In the Ptolemaic school, the mathematical imagination does not directly rest upon the sequence of observed points in space. It is rather like this: In order ultimately to do justice to them it goes right away from the observed phenomena and works quite differently, not merely putting the observed results together. Yet in the end it is found that if one does admit the mathematical thought-pictures of the Ptolemaic school, one thereby comprehends what is observed.

Suppose a man of today were to make a model of the planetary system. Somewhere he would attach the Sun, them he would draw wires to represent the orbits of the planets; he would really think of them as representing the true orbits. In purely mathematical lines he would comprise the logic of the planets' paths. Ptolemy would not have done so. He would have had to construct his model somewhat in this fashion (Fig. 2). Here would have been a pivot, fixed to it a rod, leading to the rim of a rotating wheel, upon this again another wheel rotating. Such would be Ptolemy's model. The model he makes, the mathematical picture living in his thought, is not in the least like what is outwardly seen. For Ptolemy the Mathematical picture is quite detached from what is seen externally. And now, in the Copernican system we return to the former method, simply uniting by mathematical lines the several places, empirically observed, of the planet. These mathematical lines correspond to what was there in Aristarchus's system. Yet is it really the same? This is the question we must now be asking: Is it the same?

Bearing in mind the original premises of the Copernican system and the kind of reasoning by which it is maintained, I think you will admit: It is just like the way we relate ourselves, mathematically, to empirical reality in general. You may confirm it from his works. Copernicus began by constructing his planetary system ideally, much in the same way as we construct a triangle ideally and then find it realised in empirical reality outside us. He took his start from a kind of a priori mathematical reasoning and them applied it to the empirically given facts.

What then is at the bottom of this complicated Ptolemaic system, to make it so complicated? You remember the well-known anecdote. When it was shown to Alphonso of Spain, he from his consciousness of royalty declared: Had God asked his advice at the Creation of the World, he would have made it more simply than to require so many cycles and epicycles.

Or is there something in it after all—in this construction of cycles and epicycles—related to a real content of some kind? I put the question to you: Is it only fantasy, only a thing thought-out, or does this thought out system after all contain some indication that it relates to a reality? We can only decide the question by entering into it in greater detail.

It is like this. Suppose that with the Ptolemaic system taking you start from Ptolemaic theories—you follow the movements, or, as we should say, the apparent movements of the Sun, and of Mercury, Venus, Mars, Jupiter, and Saturn: to begin with you will have angular movements of a certain magnitude each time. You can therefore compare the movements indicated by the successive positions of these heavenly bodies in the sky. The Sun has no epicyclic movement. The epicyclic daily movement of the Sun is therefore zero. For Mercury on the other hand we must put down a number, representing his daily movement along his epicyclic circle, which we shall them compare with that of other planets. Let us call the epicyclic daily movements—

\(x_1\) for Mercury,

\(x_2\) for Venus,

\(x_3\) for Mars,

\(x_4\) for Jupiter,

\(x_5\) for Saturn,

Now take the movements Ptolemy attributes to the centres of the epicycles along their different circles. Let the daily movement be y for the Sun. It is then remarkable that if we seek the corresponding value for Mercury we get precisely the same figure. The movement of the centre of Mercury's epicycle equals the movement of the Sun. We must write y again, and so for Venus. This then holds good of Mercury and Venus. The centres of their epicycles move along paths which correspond exactly to the Sun's path,—run paralleled to it. For Mars, Jupiter and Saturn on the other hand the movements of the centres of the epicycles are diverse,—shall we say \(x'\) for Mars, \(x''\) for Jupiter, \(x'''\) for Saturn.

Yet the remarkable fact is that by taking the corresponding sums, namely: $$x3 + x' + x4 + x'' , x5 + x''',$$ adding the movements along the several epicycles to the movements of the centres of these epicycles,—I get the same magnitude for all three planets. Nay more, it is the identical which we obtained just now for the movement of the Sun and of the centres of the epicycles of Mercury and Venus—

A noteworthy regularity, you see. This regularity will lead us to attribute a different cosmic significance to the centres at the epicycles of Venus and Mercury, the planets near the Sun as they are called, and of Jupiter, Mars, Saturn etc. called distant from the Sun. For the distant planets, the centre of the epicycle has not the same cosmic meaning. Something is there, by virtue of which the whole meaning of the planet's course is different than for the planets near the Sun.

The fact was well-known in the Ptolemaic school and helped determine the whole idea—the peculiar construction of cycles and epicycles in the mind, detached from the empirically given facts. This very fact obliged them, as they saw it, to propound their system, and is implicit in it. The human being of today would scarcely recognise it there; he listens more or less obtusely when told how they set up their cycles and epicycles. To their way of thinking on the other hand the thought was palpable and eloquent???. If Mercury and Venus have the same values as Jupiter, Saturn and Mars, yet in another realm, we cannot treat the matter so simply, with an indifferent circling motion or the like. A planet, in effect, is of significance not only within the space it occupies but outside it. We have not merely to stare at it, fixing its place in the Heavens and in relation to other celestial bodies; we must go out of it to the centre of the epicycle. The centre of its epicycle behaves in space even as the Sun does. Once more, translated into modern forms of speech, the Ptolemaists said: For Mercury and Venus the centres of the epicycles so far as movement is concerned behave in cosmic space as the Sun itself behaves. Not so the other planets—Mars, Jupiter and Saturn. They claim another right. In effect, only when we add their epicyclic movements to their movements along the deferent, only then do they grow like the Sun in movement. They therefore are differently related to the Sun.

This difference of behaviour in relation to the Sun was what they really built on in the Ptolemaic system. This among others was an essential reason for its development. Their aim was not merely to join the empirically given places in the Heavens by mathematical lines, building it all into a system of thought in this way. They were at pains to build a thought-system on another basis, and what is more, a piece of true knowledge under-lay their efforts; it is undeniable if we go into it historically. Modern man naturally says: We have advanced to the Copernican system, why bother about these ancient thinkers? He bothers not, but if he did, he would perceive that this was what the Ptolemaists meant. 'Truth is', they said to themselves, Mars, Jupiter and Saturn have quite another relation to Man than Mercury and Venus. What corresponds to them in Man is different. Moreover they connected Jupiter, Saturn and Mars with the forming of the human head, Venus and Mercury with the forming of what is beneath the heart in man. Rather than speak of the head, perhaps I should put it in these words: they related Jupiter, Saturn and Mars with the forming of all that is above the heart; Venus and Mercury with what is situated below the heart in man. The Ptolemaists did indeed relate to man, what they were trying to express in their cosmic system.

What under-lay it really? To gain true judgement on this question, my dear Friends, I think you should read and mark the inmost tone and essence of my Riddles of Philosophy, in writing which I tried to show how very different was the way man met the world in his life or knowledge before the 15th Century and after. Since then, if I may use this image we unpeel ourselves from the world,—we detach ourselves completely. Before the 15th Century we did not do so. I must admit, at this point it is difficult to make oneself understood in the modern world. Man of to-day says to himself: “I think thus and thus about the world. I have my sense perceptions, thus or thus. In modern times we have become enlightened; the men of former times were simple, with many childish theories.” And as to our enlightenment and their simplicity the modern man's idea of it amounts to this, or something very like it: "If only our ancestors had tried hard enough, they might have grown just as clever as we are. But it took time, this eduction of mankind; it evidently had to take some time for men to get as enlightened as they afterwards became.”

What is today left unconsidered, is that man's very seeing of the world, his seeing and his contemplating, his whole relation to the world was different. Compare the different stages of it, described in my Riddles of Philosophy. Then you will say: Through the whole time from the beginning of the Fourth Epoch until the end, the sharp distinction we now have, of concept and idea on the one hand and sense-perceived data on the other, did not exist. They coincided rather. In and with the sensory quality, men saw the quality of thought, the idea. And it was ever more so, the farther we go back in them. In this respect we need more real notions as to the evolution of mankind. What Dr. Stein has written for example in his book, upon the essence of sense-perception, is true of our time and excellently stated. I he had had to write a dissertation on this subject in the School of Alexandria in olden time, he would have had to write very differently of sense-perception. This is what people of today persist in disregarding; they will have everything made absolute.

And if we go still farther back, for example into the time when the Egypto-Chaldean Epoch was at its height, we find an even more intensive union of concept and idea with sense-perceptible, outward and physical reality. It was from this moreover—from this more intensive union—that the conceptions arose which we still find in Aristarchus of Samos. They were already decadent in his time; they had been entertained even more vividly by his predecessors. The heliocentric system was simply felt, when with their thoughts and mental pictures men lived in and with the outer sense-perceptible reality. Then, in the Fourth post-Atlantean Epoch, man had to get outside the sense-world; he had to wean himself of this union of his inner life with the sense-world. In what field was it easiest to do so? Obviously, in the field where it would seem most difficult to bring the outer reality and the idea in the mind together. Here was man's opportunity to wrest himself away—in his life of ideas—from sense-impressions.

Look at the Ptolemaic system from this angle; see in it an important means toward the education of mankind; then only do we recognise the essence of it. The Ptolemaic system is the great school of emancipation of human thoughts from sense-perception. When this emancipation had gone far enough when a certain degree of the purely inner capacity of thought had been attained—then came Copernicus. A little later, I may add, this attainment became even more evident, namely in Galileo and others, whose mathematical thinking is in the highest degree abstracted and complicated. Copernicus presented to himself the facts of which we have been speaking—the observation of the equality of y at diverse points in the equation, and, working backward from these mathematical results, was able to construct his cosmic system. For the Copernican system is based on these results. It represents a return, from the ideas now abstractly conceived, to the external, physically sense-perceptible reality.

It is most interesting to witness, how in the astronomical world-picture above all, mankind gets free of the outer reality. And in perceiving this, my dear Friends, we also gain a truer estimate of the returning pathway,—for in a wider sense we must return. Yet how? Kepler still had a feeling of it. I have often quoted his rather melodramatic saying, to the effect: I have stolen the sacred vessels of the Egyptian Temples to bring them back again to modern man. Kepler's planetary system, as you know, grew from a highly romantic conception of how the Universe is built. In deed he feels it like a renewal of the ancient heliocentric system. Yet the truth is, the ancient heliocentric system was derived, not from a mere looking outward with the eyes, but from an inner awareness, an inner feeling of what was living in the stars.

The human being who originally set up the cosmic system, making the Sun the centre with the Earth circling round it after the manner of Aristarchus of Samos, felt in his heart the influences of the Sun, felt in his head the influences of Venus and Mercury. This was experience, direct experience throughout the human being, and out of this the system grew. In later time this all-embracing experience was lost. Perceiving still with eyes and ears and nose, man could no longer perceive with heart or liver. To have perception from the Sun with one's heart, or from Jupiter with one's nose, seems like sheer madness to the people of today. Yet it is possible and it is exact and true. Moreover one is well aware why they think it madness.

This living with the Universe, intensively and all-awarely, was lost in course of time. Then Ptolemy conceived a mathematical world-picture still with a little of the old feeling to begin with, yet in its essence already detached from the world. The earlier disciples of the Ptolemaic school still felt, though very slightly, that it is somehow different with the Sun than with Jupiter for instance. Later they felt it no more. In effect the Sun reveals his influence comparatively simply through the heart. Jupiter, we must admit, spins like a wheel in our head,—it is the whirling epicycle. Whilst in a different sense, here indicated (Fig. 1), Venus goes through beneath our heart. In later Ptolemaic times, all they retained of this was the mathematical aspect, the figure of the circle: the simple circle for the Sun's path and the more complicated for the planets. Yet in this mathematical configuration there was at least some remnant of relation to the human being.

Then even this was lost and the high tide of abstraction came. Today we must look for the way back,—to re-establish once again from the entire Man an inner relation to the Cosmos. We have not to go on from Kepler, as Newton did, into still further abstractions. For Newton put abstractions in the place of things more real; he introduced mass etc. into the equations—a mere transformation, in effect, yet there is no empirical fact to vouch for it. We need to take the other road, whereby we enter reality even more deeply then Kepler did. And to this end we must include in our ambit what after all is in its life connected with the rising of the stars across the Heavens, namely the Kingdoms of external Nature in all their variedness of form and kind.

Is it not worthy of note that we find a contrast between the superior planets so-called and the inferior, with the Earth-entity between the mineral and plant kingdoms along the one branch, the animal and man along the other? And, that in drawing the two branches of the forked line, we must put plant and mineral in simple prolongation, while animal and man must be so drawn as to show the formative process returning upon itself? (Fig. 3)

We have put two things and of different kind before us: on the one hand the paths of the epicycle-centres and of the points on the epicyclic circumference, revealing a quite different relation to the Sun for the superior and inferior planets respectively; on the other hand the prolongation of the plant-forming process speeding on into the mineral, whilst the animal-forming process turns back upon itself to become man. (The symbolism of our diagram is justified; as I said yesterday, to recognise it you need only make a study of Selenka's work.)

These two things side by side we put as problems, and we will try from thence to reach a cosmic system true to reality.

Today we will develop the different notes we touched on,—the notes which we were striking yesterday. From the material at our disposal, consisting as it does in the last resort of things observed, the true aspect of which we seek to divine,—from this observed material we shall try to gain ideas, to lead us into the inner structure of the celestial phenomena. I will first point to something that will naturally follow on the more historical reflections of yesterday.

We realise that in the last resort both the Ptolemaic system and that used by modern Astronomy are attempts to synthesise in one way or another, what is observed. The Ptolemaic system and the Copernican are attempts to put together in certain mathematical or kindred figures what has in fact been perceived. (I say “perceived”, for in the light of yesterday's lecture it would not be enough to say “seen”.) All our geometry in this case, all of our measuring and mathematizing, must take its start from things perceived, observed. The only question is, are we conceiving the observed facts truly? We must really take it to heart—we must take knowledge of the fact—that in the scientific life and practice of our time what is observed, what is perceivable, is taken far to easily, too cursorily for a true conception to be gained.

Here for example is a question we cannot escape; it springs directly from the observable facts—(In the shortness of time these lectures have to be in bare outline and I have not been able to discuss or even to bring forward all the details. I could do little more than indicate directions.) Now among other things I have tried to show that the movements of heavenly bodies in celestial space must in some way be co-ordinated with what is formed in the living human body, and in the animal too in the last resort, we should by now perceive from the whole way the facts have been presented. And I assure you, the more deeply you go into the facts, the more of the connection you will see. Nevertheless, I have not done nor claimed to do any more than indicate the pathway (let me say again), the pathway along which you will be led to the result: The human living body, also the animal and plant body, are so formed that if we recognise the characteristic lines of form (as for example we did in tracing the Lemniscate in various directions though the human body) we find in them a certain likeness to the line-systems which we are able to draw amid the movements of the celestial bodies. Granted it is so, the question still remains however: What is it due to? How does it come about? What prospect is there for us not merely to ascertain it but to find it cogent and transparent, inherent in the very nature of things?

To get nearer to this question we must once more compare the kind of outlook which under-lay the Ptolemaic system and the kind that underlies the Copernican world-system of today.

What are we doing when we set to working the spirit of the latter system, and by dint of thinking, calculating and geometrising, figure out a world-system? What do we do in the first place? We observe. Out in celestial space we observe bodies which, from the simple appearance of them, we regard as identical. I express myself with caution, as you see. We have no right to say more than this . From the appearance of them to our eye, we regard these bodies (in their successive appearance) as identical. A few simple experiments will soon oblige you to be thus cautious in relating what you see in the outer world. I draw your attention to this little experiment; of no value in itself, it has significance in teaching us to be careful in the way we form our human thoughts.

Suppose it trained a horse to trot very regularly,—which, incidentally, a horse will do in any case. Say now I photograph the animal in twelve successive positions. I get twelve pictures of the horse. I put them in a circle, at a certain distance from myself, the onlooker. Over it all I put a drum with an aperture, and make the drum rotate so that I first see one picture of the horse, then, when the drum has totalled, a second picture, and so on. I get the appearance of a running horse, I should imagine a little horse to be running round in a circle. Yet the fact is not so. No horse is running round; I have only been looking in a certain way at twelve distinct pictures of a horse, each of which stays where it is.

You can therefore evoke an appearance of movement not only by perspective but in purely qualitative ways. It does not follow that what appears to be a movement is really a movement. He then who wants to speak with care, who wants to reach the truth by scrupulous investigation, must begin by saying, whimsical as this may seem to our learned contemporaries: I look at three successive positions of what I call a heavenly body, and assume what underlies them to be identical. So for example I follow the Moon in its path, with the underlying hypothesis that it is always the same Moon. (That may be right without question, with such a “Standard” phenomenon, keeping so very regular a time-table!) What do we do then? We see what we take to be the identical heavenly body, in movement as we call it; we draw lines to unite what we thus see at different places, and we then try to interpret the lines. This is what gives the Copernican system. The school from which the Ptolemaic system derived did not proceed in this way, not originally. At that time the whole human being still lived in his perceiving, as I said yesterday. And inasmuch as man was thus alive and aware, perceiving with all his human being, the idea he then had of a heavenly body was essentially different from what it afterwards became.

A man who still lived thus perceivingly amid the Ptolemaic system did not say. There is the Moon up yonder. No, he did not; the people of today only attribute that idea to him, nor does it do the system justice. If he had simply said, "Up yonder is the Moon", he would have been relating the phenomenon to his whole human being, and in so doing the following was his idea:—Here am I standing on the Earth. Now, even as I am on the Earth, so too am I in the Moon,—for the Moon is here (Figure 1, lightly shaded area).

This (the small central circle in the Figure) is the Earth, whilst the whole of that is the Moon,—far greater than the Earth. The diameter (or semi-diameter) of the Moon is as great as what we now call the distance of the Moon (I must not say, of the Moon's centre) from the centre of the Earth. So large is the Moon, in the original meaning of the Ptolemaic system. Elsewhere invisible, this cosmic body at one end of it develops a certain process by virtue of which a tiny fragment of it (small outer circle in the Figure) becomes visible. They rest is invisible, and moreover of such substance that one can live in it and me permeated by it. Only at this one end of it does it grow visible. Moreover, in relation to the Earth the entire sphere is turning, (Incidentally it is not a perfect sphere, but a spheroid or ellipsoid-of-rotation. The whole of it is turning and with it turns the tiny reagent that is visible, i.e. the visible Moon. The visible Moon is only part of the full reality of it.

The idea thus illustrated really lived in olden time. The form at least, the picture it presents, will not seem so entirely remote if you think of an analogy,—that of the human or animal germ-cell in its development (Fig. 2).

You know what happens at a certain stage. While the rest of the germinal vesicle is well-nigh transparent, at one place it develops the germinative area, so called, and from this area the further development of the embryo proceeds. Eccentrically therefore, near the periphery, a centre forms, from which the rest proceeds. Compare the tiny body of the embryo with this idea of the Moon which underlay the Ptolemaic system and you will have a notion of how they conceived it for it was analogous to this.

In the Ptolemaic conception of the Universe, we may truly say, quite another reality was ‘Moon’—mot only what is contained in the Moon's picture, the illuminated orb we see. This, then, is what happened to man after the time when the Ptolemaic system was felt as a reality. The inner experience, the bodily organic feelings of being immersed in the Moon was lost. Today man has the mere picture before him, the illuminated orb out yonder. Man of the Fifth post-Atlantean Epoch cannot say, for he no longer knows it: “I am in the Moon,—the Moon pervades me”. In his experience the Moon is only the little illuminated disc or sphere which he beholds.

It was from inner perceptions such as these that the Ptolemaic system of the Universe was made: These perceptions we can henceforth regain if we begin by looking at it all in the proper light: we can re-conquer the faculty whereby the whole Moon is experienced. We must admit however, it is understandable that those who take their start from the current idea of ‘the Moon’ find it hard to see any such inner relation between this “Moon” and life inside them. Nay, it is surely better for them to reject the statement that there are influences from the Moon affecting man than to indulge in so many fantastic and unfounded notions.

All this is changed if in a genuine way we come again to the idea that we are always living in the Moon, so that what truly deserves the name of 'Moon' is in reality a realm of force, a complex of forces that pervades us all the time. Then it will no longer be a cause of blank astonishment that this complex of forces should help form both man and beast. That forces working in and permeating us should have to do with the forming and configuration of our body, is intelligible. Such then are the ideas we must regain. We have to realise that what is visible in the heavens is nor more than a fragmentary manifestation of cosmic space, which in reality is ever filled with substance. Develop this idea: you live immersed in substance—substances manifold, inter-related. Then you will get a feeling of how very real a thing it is. The accepted astronomical outlook of our time has replaced this 'real' by something merely thought-out, namely by 'gravitation' as we call it. We only think there is a mutual force of attraction between what we imagine to be the body of the moon and the body of the Earth respectively. This gravitational line of force from the one to the other—we may imagine it as it turns to get a pretty fair picture of what was called the 'sphere' in ancient astronomical conceptions—the Lunar sphere or that of any planet. This, then, has happened: What was once felt to be substantial and can henceforth be experienced in this way once more, has in the meantime been supplanted by mere lines, constructed and thought-out.

We must then think of the whole configuration of cosmic space—manifoldly filled and differentiated in itself—in quite another way than we are wont to do. Today we go by the idea of universal gravitation. We say for instance that the tides are somehow due to gravitational forces from the Moon. We speak of gravitational force proceeding from a heavenly body, lifting the water of the sea. The other way of thought would make us say: The Moon pervades the Earth, including the Earth's hydrosphere. In the Moon's sphere, something is going on which at one place it manifests in a phenomenon of light. We need not think of any extra force of attraction. All we need think is that this Moon-sphere, permeating the Earth, is one with it, one organism all together , an organic whole. In the two kinds of phenomenon we see two aspects of a single process.

In yesterday's more historical lecture my object was to lead you up to certain notions,—essential concepts. I could equally well have tried to present them without recourse to the ideas of olden time, but to do so we should have had to take our start from premises of Spiritual Science. This would have led us to the very same essential concepts.

Imagine now (Fig. 3): Here is the Earth-sphere,—the solid sphere of Earth. And now the Lunar sphere: I must imagine this, of course, of very different consistency and kind of substance. And now I can go further. The space that is permeated by these two spheres,—I can imagine it permeated by a third sphere and a fourth. Thus in one way or another I imagine it to be permeated by a third sphere. It might for instance be the Sun-sphere,—qualitatively different form the Moon-sphere.

I then say I, am permeated—I, man, am permeated by the sun—and by the Moon-sphere. Moreover naturally there is a constant interplay between them. Permeating each other as they do, they are in mutual relation. Some element of form and figure in the human body is then an outcome of the mutual relation. Now you will recognise how rational it is to see the two things together: On the one hand, these different cosmic substantialities permeating the living body; and on the other hand the organic forms in which you can well imagine that they find expression. Form and formation of the body is then the outcome of this permeation. And what we see in the heavens—the movement of heavenly bodies—is like the visible sign. Certain conditions prevailing, the boundaries of the several spheres become visible to us in phenomena of movement.

What I have now put before you is essential for the regaining of more real conceptions of the inner structure of our cosmic system. Now you can make something of the idea that the human organisation is related to the structure of the cosmic system. You never gain a clear notion of it if you conceive the heavenly bodies as being far away yonder in space. You do gain a clear notion, the moment you see it as it really is. Though, I admit, it gets a little uncanny to feel yourself permeated by so many spheres,—just a little confusing!

And there is worse to come, for the mathematician at least. In effect, we are also permeated by the Earth-sphere itself, in a wider sense. For to the Earth belongs not only the solid ball on which we stand but all the volume of water; also the air,—this is a sphere in which we know ourselves to be immersed. Only the air is still very coarse, compared to the effects of heavenly phenomena. Think then of this: Here we are in the Earth-sphere, in the Sun sphere, in the Moon-sphere, and in others too. But let us single out the three, and we shall say to ourselves: Something in us is the outcome of the substantialities of these three spheres. Here then is qualitatively, what in its quantitative form is the mathematician's bugbear—the “problem of three bodies”, as it is called! It is working in us. In us is the outcome of it, in all reality. We must face the truth: to read the hieroglyphic of reality is not so simple. That we are wont to take it simply and think it so convenient of access, springs after all from our fond comfort,—human indolence of thought. How many things, held to be "scientific", have their origin in this! Let go the springs of comfortableness, and you must set to work with all the care which we have tried to use in these lectures. If now and then, they do not seem careful enough, it is again because they are given in barest outline; so we have often had to jump from one point to another and you yourselves must look for the connecting links. The links are surely there.

Now you must set to work with equal care to tackle the same problem from another aspect to which I have referred before, namely the body of man compared to the creatures of the remaining Nature-kingdoms. We can imagine, I said, a line that forks out on either hand from an ideal starting-point. Along the one branch we put the plant-world, along the other the animal. If we imagine the evolution of the plant-world carried further in a real Kingdom of Nature, we find it tending towards the mineral. How real a process it is, we may recall by the most obvious example. In the mineral coal, we recognised a mineralised plant-substance. What should detain us from turning attention to the analogous processes which have undoubtedly taken hold of other realms of vegetable matter? Can we not also derive the siliceous and other mineral substances of the Earth in the same way, recognising in them the mineralisation of an erstwhile plant-life?

Not in the same way (I went on to say) can we proceed if we are seeking the relation of the animal to the human kingdom. Here on the contrary we must imagine it somewhat as follows. Evolution moves onward through the animal kingdom; then however it bends back, returns upon itself, and finds physical realisation upon a higher than animal level. We may perhaps put it this way: Animal and human evolution begin from a common starting—point, but the animal goes farther before reaching outward physical reality. Man on the other hand keeps at an earlier stage, man makes himself physically real at an earlier stage. It is precisely by virtue of this that he remains capable of further evolution after birth, incomparably more so then the animal. (For, once again, the processes of which we speak relate to embryo-development.) That man retains the power to evolve, is due to his not carrying the animal-forming process to extremes. Whilst in the mineral, the plant-forming process has overreached itself; in man on the contrary the animal-forming process has stopped short of the extreme. It has withheld, kept back, and taken shape at an earlier stage amid external Nature.

We have then this ideal point from which it branches (Fig. 6). There is the shorter branch and the longer. The longer is of undetermined length; the other, we may say, no less so, but negatively speaking. So then we have the mineral and plant kingdoms, and animal and human.

Now we must seek to gain a more precise idea: What is it that really happens, in this formation of man as compared to the animal? The process of development, once more, is kept back in man. It does not go so far; that which is tending to realisation is, as it were, made real before its time. Now think how it must be imagined according to what I have told you in these lectures. Study the share of the Solar entity in the forming of the animal body,—via the embryo-development, of course. You then know that the direct sunshine (so to describe it) has to do with the configuration of the animal head, whilst the indirect aspect of the sunlight, as it were the Sun's shadow by relation to the Earth, has in some way to do with the opposite pole of the creature. Strictly envisage this permeation of animal form and development with cosmic Sun-substantiality. Look at the forms as they are. Then you will gain a certain idea, which I shall try to indicate as follows.

Assume to begin with,—assume that in some way the forming of the animal is really brought about by relation to the Sun. And now, apart from the constellation that will be effective in each case as between Sun and animal, let us ask, quite in the sense of the Sun's light in the cosmos, not so immediately connected with the Sun itself? There is indeed. For every time the Full Moon, or the Moon at all, shines down upon us, the light is sunlight. The cosmic opportunity is being made then, so to speak, for the Sun's light to ray down upon us. It is so of course also when the human being comes into life—in the germinal and embyonal period. In earlier stages of Earth-evolution the influence was most direct; today it is a kind of echo, inherited from then. Here then again we have an influence, in the other it is indirect, through the raying back of the Sun's light by the Moon.

Now think the following. I will again draw it diagrammatically. Suppose the development of the animal were such that it comes into being under the Sun's influence according to this diagram (Fig. 4).

This then, to put it simply, would be the ordinary influence of day and night—head and the opposite pole of the creature. This, for the animal, would be the ordinary working of the Sun. Now take that other working of the Sun's light which occurs when the Moon is in opposition, i.e. when it is Full Moon,—when the Sun's light, so to speak works from the opposite side and by reflection counteracts itself. If we conceive this downward arrow (Fig. 5).

to represent the direction of the direct Sun-rays, animal formations, we must imagine animal-formation going ever farther in the sense of this direct Sun-ray. The animal would become animal, the more the Sun was working on it. If on the other hand the Moon is counteracting from the opposite direction—or if the Sun itself is doing so via the Moon,—something is taken away again from the animal-becoming process. It is withdrawn, drawn back into itself (Fig. 5a).

Precisely this withdrawal corresponds to the shortening of the second branch in Figure 6. We have found a true cosmic counterpart of the characteristic difference between man and animal of which we spoke before.

What I have just been telling you can be perceived directly by anyone who gains the faculty for such perception. Man really owes it to the counteracting of the Sunlight via the Moon.—owes it to this that his organisation is withheld from becoming animal. The influence of the Sun-light is weakened in its very own quality (for it is Sun-light in either case), in that the Sun places its own counterpart over against itself, namely the Moon and the Moon's influence. Were it not for the Sun meeting and countering itself in the Moonlight—influences, the tendency that is in us would give us animal form and figure. But the Sun's influence reflected by the Moon counteracts, it. The forming process is held in check, the negative of it is working; the human form and figure is the outcome.

Now, on the other branch of the diagram, let us follow up the plant and the plant-formative process. That the Sun is working in the plant, is palpably evident. Let us imagine the Sun's effect in the plant, not to be able to unfold during a certain time. During the Winter, in fact, the springing and sprouting life in the plant cannot unfold. Nay, you can even see the difference in the unfolding of the plant by day and night. Now think of this effect in oft-repeated rhythm, repeated endless times,—what have we then? We have the influence of the Sun and the influence of the Earth itself; the latter when the Sun cannot work directly but is hidden by the Earth. At one time the Sun is working, at another it is not the Sun but the Earth, for the Sun is working from below and the Earth is in the way. We have the rhythmic alternation: Sun-influence predominant, Earth-influence predominant in turn. Plant-nature therefore is alternately exposed to the Sun, and then withdrawn, figuratively speaking, into the Earth—drawn by the earthly, as it were, into itself. This is quite different from what we had before. For in this case the Sun-quality, working in the plant, is potently enhanced. The solar quality is actually enhanced by the earthly, and this enhancement finds expression in that the plant gradually falls into mineralization.

Such then is the divergence of two ways, as indicated once again in Figure 6. In the plant we have to recognize the Sun's effect, carried still further by the Earth, to the point of mineralization. In the animal we have to recognize the Sun's effect, which then in man is drawn back again, withdrawn into itself, by virtue of the Moon's effect. I might also draw the figure rather differently, like this (Fig. 6a),

—here receding to become human, here on the other hand advancing to become mineral, which of course ought to be shown in some other form. It is no more than a symbolic figure, but this symbolic figure, tends to express more clearly than the first, made of mere lines, the bifurcation—as again I like to call it—with the mineral and plant kingdoms upon the one hand, the human and animal upon the other.

We never do justice to the true system of Nature with all her creatures and kingdoms if we imagine them in a straight line. We have to take our start from this other picture. In the last resort, all systems of Nature which begin with the mineral kingdom and thence going on to the plant, thence to the animal and thence to man as if in a straight line, will fail to satisfy. In this quaternary of Nature we are face to face with a more complex inner relationship than a mere rectilinear stream of evolution, or the like, could possibly imply. If one the other hand we take our start from this, the true conception, then we are led, not to a generatic aequivoca or primal generation of life, but to the ideal centre somewhere between animal and plant—a centre not to be found within the physical at all, yet without doubt connected with the problem of three bodies, Earth, Sun and Moon. Though perhaps mathematically you cannot quite lay hold on it, yet you may well conceive a kind of ideal centre-of-gravity of the three bodies—Sun, Moon and Earth. Though this will not precisely solve you the 'problem of three bodies', yet it is solved, namely in Man. When man assimilates in his own nature what is mineral and animal and plant, there is created in him in all reality a kind of ideal point-of-intersection of the three influences. It is inscribed in man, and that is where it is beyond all doubt. Moreover inasmuch as it is so, we must accept the fact that what is thus in man will be empirically at many places at once, for it is there in every human being,—every individual one. Yes, it is there in all men, scattered as they are over the Earth; all of them must be in some relation to Sun and Moon and Earth. If we somehow succeeded in finding an ideal point-of-intersection of the effects of Sun and Moon and Earth, if we could ascertain the movement of this point for every individual human being, it would lead us far indeed towards an understanding of what we may, perhaps, describe as movement, speaking of Sun and Moon and Earth.

As I said just now, the problem grows only the more involved, for we have so many points,—as many as there are men on Earth,—for all of which points we have to seek the movement. Yet it might be, might it not, that for the different human beings the movements only seemed to differ, one from another ...

We will pursue our conversations on these lines tomorrow.

Today I will deal with some of the things that may be causing you difficulty in understanding what we have done hitherto. I will lead over from these difficulties into a realm of ideas which will show up the inadequacy of those lines of thought on which the people of our time, with all their comfortable mental habits, would gladly found their understanding of universal phenomena.

We have been studying the universal phenomena in their relation to man. We have done so in manifold directions. Again and again we have indicated how a relationship reveals itself between the forming of man and what appears in the celestial phenomena. Whether we go by some ancient cosmic system of by the Copernican theories in forming our pictured synthesis of the movements of heavenly bodies, we must relate the picture to man in diverse ways of course, accordingly. This we have seen. For a true Science we must accept that there is this relation.

Yet the difficulties are formidable. Earlier in these lectures we drew attention to one such difficulty. The moment we try to form ratios between the periods of revolution of the planets of our system we come to incommensurable numbers. Arithmetic runs out, as we might say; we get no farther with it, for where incommensurable numbers enter in there is no palpable unit. Thus, when we look for a synthesis of the phenomena of cosmic space with our accustomed mathematical method and way of thinking, the phenomena themselves are such that we find ourselves driven farther and farther from reality. We may not therefore take for granted that we shall ever be able to explain the cosmic phenomena on the accustomed basis of our Geometry, that is to say, within a rigid three-dimensional space. Nay more, another difficulty has emerged. Yesterday we found ourselves obliged to assume a certain relationship of Sun and Moon and Earth, finding expression in some way in man—in man's very structure. We would fain grasp how the relation is. Yet if we posit this working-together of the Three*, we get into formidable difficulties in spatial calculation.

All these things we have mentioned. Now we can reach a certain starting-point at least, through pure Geometry—yet a Geometry of a higher kind. Thence we may gain an idea of where the difficulties come from when we are trying by dint of spatial calculation to grasp the inter-connection of celestial phenomena. Let us recall our precious attempts to comprehend the form of man himself. We are then let to this:- We can and we should try to take seriously that 'memberment' of the human being of which we have also spoken in these lectures. The human head-organization, we may truly say, centring as it does in the nerves-and-senses system, is relatively independent. So is the rhythmic system with all that belongs to it. The metabolic system too, and all that goes with it in the organization three independent systems are revealed. Taking our start now in an intelligent way from the principle of metamorphosis, as we must always do when dealing with organic Nature, we can try to form ideas upon this question: How are the three members of the three-fold human system related to each other, according to this principle of metamorphosis?

Understand me rightly, my dear friends. We want to gain an idea-though it be only pictorial to begin with—of how the three members of the human system are related to each other. On the face of it, it will of course be difficult. Such organs as are met with in the human head, it will be difficult to recognise in them at all clearly the metamorphosis of those organs which are fundamental to the metabolic and lymph system. But if we go into the morphology of man deeply enough, we can find our way. We only have to think most thoroughly along the lines already indicated. Namely, the essence of the mutual relation of the long bone to the skull-bone and vice-versa is a complete turning-inside-out. The inner surface of the bone becomes the one turned outward. It is the principle by which you turn a glove inside-out, provided only that the turning-inside-out involves a simultaneous change in the inherent relationships of inner forces. If I should turn a tubular bone or long bone inside-out like a mere glove, I should again get the form of a tubular bone, needless to say. But it will not be so if we take our start, as we must do, from the inherent configuration of the bone. As I described before, in its inherent configuration the long bone is oriented inward towards the radial quality that runs right through it. It is obliged therefore to subject its material structure and arrangement to the radial principle. When I have "flipped" it, so that the inner side opens outside, in its configuration it will no longer follow the radial but the spheroidal principle. The "inner side", now turned outward towards the Sphere, will then receive this form (Fig. 1).

What was outside before, is now inside, and vice-versa. Take this into account for the extreme metamorphosis-tubular bone into skull-bone and you will say: The outermost ends of human memberment—lymph-system and skull system—represent opposite poles in man's organization. But we must not think of "opposite poles" in the mere trivial, linear sense of the word. In that we go from one pole to the other, we must adopt the transition which this involves, namely from Radius to surface of a Sphere. Without the help of such ideas and mental pictures, intricate as they may seem, it is quite impossible to gain a just or adequate notion of what the human body is.

We come now to what constitutes the middle, in a certain sense,—the middle member of man's organization. This will be all that belongs to the rhythmic system, and it will somehow form the transition from radial structure to spheroidal.

In the threefold system thus presented we have the key to the morphological understanding of the entire human organism. Of course we need to realise how it will be. Suppose we have some organ in the metabolic system—the liver for example—or any one of the organs mainly assigned to the metabolism. (We must qualify it with the word 'mainly' for there is always an overlapping and interlocking of these things). Suppose then we begin with such an organ and seek what answers to it in the head. We try to find which of the organs in the head-nature of man m ay be connected with it by the metamorphosis of turning-inside-out. We shall then have to recognise the organ when entirely transformed, de-formed; only by so doing shall we understand it. It will therefore not be easy to take hold of mathematically. Yet without finding some mathematical way of access we shall never adequately grasp it. And if you call to mind (even if you only take this as a picture)—if you call to mind that the real understanding of the human form and figure will lead us out among the movements of celestial bodies, you will divine what must be needful also when we wish to comprehend the latter. For a true synthesis of the phenomena of movement among the heavenly bodies, it will be quite inadequate to think of them as if these movements were accessible to a Geometry that simply reckons with ordinary rigid space and therefore cannot master the turning-inside-out. For when we speak of a turning-inside-out in the way we have been doing, we can no longer be thinking of ordinary space. Ordinary space holds good where we can calculate volumes, cubicle contents in the conventional way. We cannot do so if obliged to make the inner outer. We can no longer go on calculating them with the same conceptions which hold good in ordinary space.

If then in thinking of the human form and figure I need the turnings-inside-out, in thinking of the movements of heavenly bodies I shall need them too. I cannot proceed like the current Astronomy which tries to comprehend the celestial phenomena within an ordinary rigid form of space.

Take, to begin with, simply the head-organization and the metabolic organization of man. To pass from one to the other you must imagine, once again, a turning-inside-out—and, what is more, one that involves variations of form. Let us at least try to get a picture of the kind of think involved. We did preliminary work in this direction when speaking of the Cassini curves, and of the circle differently conceived. Ordinarily the circle is defined as a curve, all of whose points are equidistant from one central point. We were speaking of the circle as a curve, all of whose points are at measured distances from two fixed points, and so that the quotient of the two distances is constant. This was our other conception of the circle.

Speaking of the Cassini curve, we showed that it has three essential forms. One, not unlike an ellipse:—this form arose when the parameters of the curve bore a certain relation, the which we indicated. The second form was the lemniscate. The third form is that while in the idea of it—and also analytically—it is a single entity, to look at it is not. It has two branches (Fig. 2),

yet the two branches are one curve. To draw the line, we should somehow have to go out of space, coming back into space again when we draw the second branch. Conceptually, our hand would be drawing a continuous line when drawing the two regions which look separate. We cannot draw the line continuously within ordinary space, and yet conceptually what is here above and what is here below (the inner curve in Figure 2) is a single line. Now as I also mentioned, the same curve can be thought of in another way. You can ask what will be the path of a point which when illumined from a fixed point A appears with constant intensity of illumination, seen from another fixed point B. Answer: a Cassini curve. A curve a Cassini will be the focus of all points through which a point must run, if when illumined from a fixed point A it is seen ever with the same intensity of light from another fixed point B (Fig. 2 again).

Now it will not be hard for you to imagine that if something shines from A to C (Fig. 2) and thence by reflection from C to B, the intensity of light will be the same as if reflected from D instead. But it gets rather more difficult to imagine when you come to the Lemniscate. The ordinary geometrical constructions by the laws of reflection and so on, will not be quite so easy to carry through. And it gets still more difficult to imagine with the two-branched curve, that the same intensity of light should always be observed from the point B, inside the one branch of the curve, when the original point-source of light is in A. You would have to imagine (as you pass from the one branch to the other) that the ray of light goes out of space and then shines into space again. You are up against the same difficulty as before, when you were simply asked to draw the two branches as one—with a single sweep of the hand through space.

Yet if we do not develop these conceptions we shall be unequal to the other task, namely of finding the transmutation—or even the mere relationship of form—as between any organ in the head of man and the corresponding organ in the metabolism. To find the connection you simply must go out of space. Once again—strange as it may sound—if with your understanding of any form in the human head you wish to make a transition to the understanding of a form in the human metabolic system then you will not be able to remain in space. You must get out of space. You must get right out of yourself , looking for something that is not there in space. You will find something that is as little inside ordinary space as is what intervenes between the upper and lower branches of a two-branched Cassini curve. This is in fact only another way of expressing what was said before that the metamorphosis must be so conceived as to turn the form completely inside out.

In thinking thus of the connection between the upper and lower branches of the discontinuous curve of Cassini (as shown in Fig. 3) we are still presupposing actual constants, rigid and unchanged, in the equation. Now if we vary the constants themselves as in an earlier lecture, forming equations of twofold variability, we shall be able to imagine the upper branch say, in this form and the low one in this (Fig. 3).

The upper branch will take this form eventually. If then you alter the curve of Cassini by taking variables in place of constants—so that you start with equations instead of starting with invariable constants—you will get two different kinds of branches. Then there will also be the possibility for one of the two branches to come in as it were from the infinite and go out to the infinite again. This is precisely the relationship from which you should take your start when following certain forms within the human head, comprising them in curves and lines, and then relating them to the forms of organs or of complexes of organs in the metabolic system, which in their turn you will comprise in curves and lines. Such is the intricacy of the human form. To make it still less simple, you must imagine the one line (Fig. 3a) with an outward tendency and the other with its tendency turned inward.

You will be prone to say (I hope without insisting on it, but as a passing impression): If this be so, the human organization is so complicated that one would almost prefer to do without such understanding and fall back on the ordinary philistine idea of the body, as in the present-day Anatomy and Physiology. There we are not called upon to make such prodigious efforts, as to let mental pictures vanish and yet again not vanish, or turn them inside-out, and all the rest! May be; but then you never really understand the human form; your understanding is, at most, illusionary.

Now, to go on: Suppose you thus look into it and recognize that there is something in the human organization which falls right out of space, is not in space at all, but obliges you for instance to imagine spatially separated line-systems, inherently united with each other and yet united by another principle than three dimensional space affords. Thinking in this way, you will no longer be too far removed from what I shall now bring forward. You will at least be able to entertain the thought in a formal sense. No-one, I mean can validly object to thinking it as a pure form of thought. For to begin with, all we are called upon to do is to conceive a clear idea, as in mathematics generally. It cannot be objected that the thing is unproved, or the like. We are only concerned to reach a self-contained and consistent idea.

Think therefore for a moment that you had to do not only with ordinary space, conceived in its three dimension, but with a "counter-space" or anti-space". Let me call it so for the moment, and I will try to evoke an idea of it, as follows. Suppose I form the thought of ordinary, three-dimensional, rigid space. I form the first dimension, I form the second dimension and I form the third dimension (Fig. 4).

Then I have, so to speak, filled-in in thought—in the idea and mental presentation of it—three-dimensional space with which I am ordinarily confronted. Now as you know, in any such domain you can not only advance up to a certain degree of intensity; you can subtract from it too, and as you go on subtracting—taking away—you come at last to the negation of ti. As you are well aware, there is not only wealth but debt. Likewise I cannot only make the three dimensions to arise in thought but I can also make them vanish. Only I now imagine the arising and vanishing to be a real process,—something hat is really there. Of course it is possible to think only two dimensions instead of three, but that is not my meaning. What I now mean is this: The reason why I only have two dimensions (Fig. 4a)

is not that I never had a third. The reason is, I had a third and it has vanished. The two dimensions are an outcome of the coming-into being and vanishing-again of the third. I now have a space, which, though it outwardly shows only two dimensions, must inwardly be conceived as having two third dimensions, one positive and the other negative. The negative dimension springs from a source that can no longer be there in my three-dimensional space at all. Nor must I think of it as a "fourth dimension" in the conventional sense. No, I must think of it as being, to the third dimension, as positive to negative (Fig. 4a once more).

And now suppose that what I have been indicating is really there in the Universe; yet, as things generally are in the real world, approximately so. It would then be not a pedantically accurate but an approximate rendering of what I have here drawn. This need not cause you any great surprise, for in outer sense-perceptible reality you never find mathematical figures reproduced in any other way, always approximately. If then I claim that the picture represents something real, you will only expect it to do so in an approximate sense. To represent a reality corresponding to it, I need not repeat exactly the same drawing, but I should have to draw something flattened; that would answer to it. The fact that something has been there and has then vanished, I may perhaps suggest in this way: I will suppose that the density of an effect, indicated by the dark shading, came into being and then partly faded out again, drew weaker (Fig. 5).

You are then left with a sphere that has a denser portion in the middle region. I beg you know, compare what is here drawn with the real cosmic system, such as we see it with our eyes,—the cosmic sphere with all the stars widely dispersed, and then the stars more densely packed in the region of the Milky Way, or what we call the Galactic System.

Yet you may also compare it with something else. Take any popular star-map. The picture we have shown (Fig. 5a)

—let us still take it simply as a picture—is fundamentally equivalent to what is always being shown: the passage of the Sun or of the Earth through the Zodiac, with the with the North and South poles of the ecliptic somewhere out yonder. The idea we have been forming is, as you see, not so very remote from what is there in the outer Universe. In coming lectures we shall of course still have to look for more detailed relations.

Now for an understanding of what was said before about the human being we have not yet gone for enough. We must go farther and make the second dimension also vanish; so then we shall be left with only one,—with a straight line. But this is no ordinary straight line drawn into three-dimensional space. It is the line that has remained when we have made the third and also the second dimension vanish. And now we make the last remaining one to vanish. Then we are left with a mere point. Bear in mind however that we have arrived at the point by the successive vanishing of three dimensions. Now let us suppose that this point were to present itself to us in reality,—as having existence in itself. If it is there, and making itself felt, how then shall we imagine its activity? We cannot relate its activity to any point in the space determined by the x-axis. The x-axis is not there, since it has vanished. Nor can we relate it to anything with an x—and a y-coordinate, for all of this has gone; all this has vanished out of space. Nor can we relate it in its activity to the third dimension of space. What then shall we say? When it reveals its activity we shall have to relate it to what is quite outside three-dimensional space. What then shall we say? When it reveals its activity we shall have to relate it to what is quite outside three-dimensional space. Consistently with the procedure we have been through in our thinking, we cannot possibly relate it to anything that could still be included in this space. We can only relate it to what is outside it three-dimensional space altogether. We can relate it neither to "x deleted" nor to "y deleted" not to "z deleted", but only to what deletes all three of them, z, y, and z together, and is therefore into within three-dimensional space at all.

We put this forward to begin with as a purely formal, mathematical notion. Yet is soon grows real. It grows exceedingly real when we begin to enter into things more deeply than with the easy-going notions with which Science nowadays would gladly master them. Look, with this deeper tendency of understanding,—look at the process of sight and the whole organisation of the eye. You are perhaps aware (in other lectures I have often spoken of it) of how the eye is not merely to be regarded as a thing formed from within the body outward; for it is largely organized into the body from outside. You an trace the forming of it from without inward by studying the phylogenetic development of lower animals and then considering the act of sight itself. You will contrive to understand how the process of sight is stimulated from without and how the organ too is adapted to this stimulation from without. Then as the process works on inward to the optic nerve and farther in, it vanishes at length,—vanishes as it were into the organisation as a whole. I know you can find the termination of the optic nerves, and yet—this too comes to expression approximately—if you go into the inner organisation you will have to admit that it there vanishes.

So much for the process of sight and the associated organs. And now compare with this the process of secretion of the kidneys. Go into it conscientiously. and you will have to relate the duct that leads outward, for the secretion of the kidneys, to what is working from without inward where the eye passes into the optic nerve. If you then look for ideas whereby the two things can be related, so that their mutual relation will help you understand the phenomena of either process, you will find indispensable such forms of thought as we have just been indicating. If you conceive the ideas of three-dimensional space as applying to the process of sight (we might also replace the one by the other, but if you do it in this way. ...), then, if you seek what answers to it in the secretion of the kidneys, you must realize that what is there enacted takes you right out of three-dimensional space. You must go through the same procedure in your thinking as I did just now in extinguishing the spatial dimensions. Otherwise you will not find your way.

In like manner you must proceed if you are trying to understand the curves formed in the Heavens by the apparent paths of Venus and Mercury on the one hand, Jupiter and Mars on the other, I mean quite simply the apparent paths as we observe them with our eyes,—the loops and all. If you use polar coordinates for example, then for the loop of Venus you may make the origin of your coordinate system in three-dimensional space. Here you can do so. But you will not come to terms with reality if you adopt the same principle when examining the curve of Mars. In this case you must start from the ideal premise that the origins of any relevant system of polar coordinates will be outside three-dimensional space. You are obliged to take the coordinates in this way. In the former case you may start from the pole of the coordinate system, taking coordinates in the normal way, as in Figure 6.

But if you do this for the one planetary curve—say for the path of Venus with its loop—you will do equal justice to the paths of Jupiter or Mars with their loops, only by saying to yourself: This time I will not pre-suppose a polar-coordinate system with an origin such that I always have to add a piece to get the polar-coordinates, as in Figure 6. No, I will take as origin of my polar-coordinate system the encompassing Sphere (Fig. 6a),

i.e. what is there behind it, indeterminately far. Then I get such coordinates as these (dotted lines), where in each case, instead of adding, I must leave so much out. The curve I then obtain also has something like a centre, but the centre is in the infinite sphere.

It might prove necessary then, for more profound research into the paths of the planets, that we make use of this idea: In constituting the paths of the inner planets we must indeed attribute to these paths some centre or other within ordinary space. But if we want to think of centres for the path of Jupiter, the path of Mars and so on, we must go right outside this ordinary space.

In fine, we have to overcome space; we must transcend it . There is no help for it. If you are conscientious in your efforts to comprehend the phenomena, the mere ideas of three-dimensional space will not suffice you. You must envisage the interplay of two kinds of space. One of them, with the ordinary three dimensions, may be conceived as issuing radially from a central point. The other, which is all the time annulling and extinguishing the first, may not be thought of as issuing from a point at all. It must be thought of as issuing from the encompassing Sphere—that is, the Sphere infinitely far away. While in the former case the "point" is of zero areas which it turns outward, and a point with the area of an infinite spherical surface which it turns inward. Geometrically it may suffice to conceive the notion of a point abstractly. In the realm of reality it will not. We shall not do justice to reality with the mere notion of an abstract point. In every instance we must ask whether the point we are conceiving has its curvature turned inward or outward; its field of influence will be according to this.

But you must think still farther, my dear friends; there is a another thing. Of course you may imagine that you had somewhere caught this point which is really a Sphere. To begin with, since it is in the infinite far spaces you need not imagine it just here (s, Fig. 7). You can equally well imagine it a little farther out, (b, or c). You can imagine it to be anywhere out there; you only have to leave this sphere free (strongly drawn sphere in Fig. 7).

For this is hollowed out, so to speak; this is the inverted circle or the inverted sphere, if you like. But now suppose the following might be the case. Think of what is within this peculiar circle (namely at a, b, c, etc,) Think of this point that has its curvature turned inward. For in effect, the entire space outside this spherical surface is then a point with its curvature turned inward. And now imagine that this space had, after all, its limit somewhere. You might be able to go far away out,—very far. Suppose however the reality were such that you could not just go anywhere, but somewhere after all there was a limit of quite another kind (dotted circle in Figure7). What there would appear, as if by inner necessity, what in effect belongs to the realm beyond the limit. An equivalent sphere would have to arise within, belonging to what is there outside. You would then have to realize: Out there, beyond a certain sphere, something is still existing, it is true, but if I want to see it I must look in here (P), for here it re-appears. The continuation of what is faraway out there make itself felt in here. What I am looking for as I go out into infinite distances, makes its appearance within, and becomes manifest to me from this centre.

These are the kind of ideas you should develop to an adequate extent. In a formal sense they look sound enough. As forms of thought there can surely be no objection to them. Truly remarkable results will be obtained however, if with their help you try to penetrate outer reality. Think for example that there might be a phenomenon in celestial space,—we may call it "Moon" to begin with,—yet this phenomenon were not to be understood simply by saying: "This Moon is a body, here is its central point; we will investigate it on the understanding that it is a body and that its central point is here." Assume (and please forgive my saying, I put it euphemistically) assume that this way of thinking did not fit the reality, but that I ought to express it quite differently. I ought rather to say: "If I, in my Universe, start from a certain point and go farther and farther out, I come at length to where I shall no longer find heavenly bodies. Yet neither shall I find a mere empty Euclidean space. No, I shall find something, the inherent reality of which obliges me to recognize the continuation of it here (at P)." I should then be obliged to conceive the space contained within the Moon as a portion of the entire Universe with the exception of all that exists by way of stars, etc., outside the Moon. I should have to think on the one hand of all the stars here are in cosmic space. These, I am now assuming I have to treat in one way, according to a single principle; but the inside of the Moon—the space contained within the Moon—could not be treated in this way. It would require me to think as follows: There on the one hand I go out into the far spaces. Somewhere out there, I presume, is the celestial Sphere. Though it be only the "apparent" Sphere to begin with; something effective, something real must be conceived to underlie it. Yet whatsoever realities I find out there, the space within the spherical surface of the Moon has nothing whatever to do with it. It only has to do with what begins where the stars come to an end. It is a fragment, in some strange way, belonging not to my Universe but to that Universe to which all the stars do not belong.

If there is such a thing within a Universe, it is a thing inserted in this Universe, occluded as it were,—thing of altogether different nature and revealing different inner properties from all that is there around it. And we may then compare the relation of such Moon to its surrounding Heavens with the relation which obtains for instance between the secretions of the kidneys—with the organic structure that underlies them—and on the other hand the structure and functioning of the eyes. From this we shall proceed tomorrow.

It is not due to me that I must try to form, and to acquaint you with, such complicated notions of how the Universe is built. Truth is, equipped with any other notions you will not make headway, save on the convention: "Let us comprise the phenomena with our given range of ideas, and if we come to a limit somewhere, well then we do, and we go no further". Ascribe it then to the reality and not to any craving for remote ideas, if in the effort to impart an understanding of how the Universe is built I have unfolded complicated notions.

What we are doing, as you will have seen, is to bring together the diverse elements by means of which in the last resort we shall be able to determine the forms of movement of the heavenly bodies, and—in addition to the forms of movement—what may perhaps be described as their mutual positions. A comprehensive view of our system of heavenly bodies will only be gained when we are able to determine first the curve-forms (inasmuch as forms of movement are called curves), i.e. the true geometrical figures, and then the centres of observation. Such is the task before us along our present lines of study, which I have formed as I have done for very definite reasons.

The greatest errors that are made in scientific life consist in this: they try to make syntheses and comprehensive theories when they have not yet established the conditions of true synthesis. They are impatient to set up theories—to gain a conclusive view of the thing in question,—they do not want to wait till the conditions are fulfilled, subject to which alone theories can properly be made. Our scientific life and practice needs this infusion badly,—needs to acquire a feeling of the fact that you ought not to try and answer questions when the conditions for an intelligent answer are not yet achieved. I know that many people (present company of course excepted) would be better pleased if one presented them with curves all ready made, for planetary or other movements. For they would then be in possession of tangible answers. What they are asking is in effect to be told how such and such things are in the Universe, in terms of the ideas and concepts they already have. What if the real questions are such as cannot be answered at all with the existing ideas and concepts? In that case, theoretic talk will be to no purpose. One's question may be set at rest, but the satisfaction is illusionary. Hence, in respect to scientific education, I have attempted to form these lectures as I have done.

The results we have gained so far have shown that we must make careful distinctions if we wish to find true forms of curves for the celestial movements. Such things as these, for instance, we must differentiate: the apparent movements seen in the paths of Venus and of Mars respectively,—Venus making a loop when in conjunction, Mars when in opposition to the Sun. We came to this conclusion when trying to perceive how diverse are the forms of curves that arise in man himself through the forces that build and form him. We ascertained quite different forms of curve in the region of the head-nature and in the organization of the metabolism and the limbs. The two types of form are none the less related, but the transition from one to the other must be sought for outside of space,—at least beyond the bounds of rigid Euclidean space.

Then comes a further transition, which still remains for us to find. We have to pass from what we thus discover in our own human frame, to what is there outside in Universal Space, which only looks to us plainly Euclidean. We think it nicely there, a rigid space, but that is mere appearance. As to this question, we only gain an answer by persevering with the same method we have so far developed. Namely we have to seek the real connection of what goes on in man himself and what goes on outside in Universal Space, in the movements of the celestial bodies. Then we are bound to put this fundamental question: What relation is there, as to cognition itself, between those movements that may legitimately be considered relative and those that may not? We know that amid the forming and shaping forces of the human body we have two kinds: those that work radially and those which we must think of as working spherically. The question now is, with regard to outer movements: How, with our human cognition do we apprehend that element of movement which takes its course purely within the Sphere, and how do we apprehend that element which takes its course along the Radius?

A beginning has been made in Science as you know, even experimentally, in respect of these two kinds of spatial movement. The movements of a heavenly body upon the Sphere can of course be seen and traced visually. Spectrum analysis however also enables us to detect those movements that are along the line of sight, spectrum analysis enables us to recognise the fact. Interesting results have for example been arrived at with double stars that move around each other. The movement was only recognizable by tackling the problem with the help of Doppler's principle,—that is the experimental method to which I am referring.

For us, the question now is whether the method which includes man in the whole cosmic system will give us any criterion—I express myself with caution—any criterion to tell whether a movement may perhaps only be apparent or whether we must conclude that it is real. Is there anything to indicate that a given movement must be a real one? I have already spoken of this. We must distinguish between movements that may quite well be merely relative and on the other hand such movements as the “rotating, shearing and deforming movements” (so we described them), the very character of which will indicate that they cannot be taken in a merely relative sense. We must look for a criterion of true movement. We shall gain it in no other way than by envisaging the inner conditions of what is moving. We cannot possibly confine ourselves to the mere outer relations of position.

A trite example I have often given is of two men whom I see side by side at 9 am and again at 3 in the afternoon. The only difference is, one of them stayed there while the other went on an errand lasting six hours. I was away in the meantime and did not see what happened. At 3 pm I see them side by side again. Merely observing where they are outwardly in space, will never tell me the true fact. Only by seeing that one is more tired than the other—taking account of an inner condition therefore—shall I be able to tell, which of them has been moving. This is the point. If we would characterize any movement as an inherent and not a merely relative movement, we must perceive what the thing moved has undergone in some more inward sense. For this, a further factor will be needed, of which tomorrow. Today we will at least approach the problem.

We must in fact get hold of it from quite another angle. If we in our time study the form and formation of the human body and look for some connection with what is there in cosmic space, the most we can do to begin with is in some outward sense to see that the connection is there. Man is today very largely independent of the movements of cosmic space; everything points to the fact that this is so. For all that comes to expression in his immediate experience, man has emancipated himself from the phenomena of the Universe. We therefore have to look back into the time when what he underwent depended less upon his conscious life of soul than in his ordinary, by which I mean, post-natal life on Earth. We must look back into the time when he was an embryo. In the embryo the forming and development of man does indeed take place in harmony with cosmic forces. What afterwards remains is only what is carried forward, so to speak. Implanted in the whole human organization during the embryonal life it then persists. We cannot say it is "inherited" in the customary sense, for in fact nothing is inherited, but we must think of some such process, where entities derived from an earlier period of development stay on.

We must now look for an answer to the question: Is there still anything in the ordinary life we lead after our birth—after full consciousness has been attained—is there still any hint of our connection with the cosmic forces? Let us consider the human alternation of waking and sleeping. Even the civilized man of today still has to let this alternation happen. In its main periodicity, if he would stay in good health, it still has to follow the natural alternation of day and night. Yet as you know very well, man of today does lift if out of its natural course. In city life we no longer make it coincide with Nature. Only the country folk do so still. Nay, just because they do so, their state of soul is different. They sleep at night and wake by day. When days are longer and nights shorter they sleep less; when nights are longer the sleep longer. These aspects however can at most lead to vague comparisons; no clear perception can be derived from them. To recognize how the great cosmic conditions interpenetrate the subjective conditions of man, we must go into the question more deeply. So shall we find in the inner life of man some indication of what are absolute movements in the great Universe.

I will now draw your attention to something you can very well observe if only you are prepared to extend your observation to wider fields. Namely, however easily man may emancipate himself from the Universe in the alternation of sleeping and waking as regards time, he cannot with impunity emancipate himself as regards spatial position. Sophisticated folk—for such there are—may turn night into day, day into night, but even they, when they do go to sleep, must adopt a position other than the upright one of waking life. They must, as it were, bring the line of their spine into the same direction as the animal's. One might investigate a thing like this in greater detail. For instance, it is a physiological fact that there are people who in conditions of illness cannot sleep properly when horizontal but have to sit more upright. Precisely these deviations from the normal association of sleep with the horizontal posture will help to indicate the underlying law. A careful study of these exceptions—due as they are to more or less palpable diseases (as in the case of asthmatic subjects for example)—will be indicative of the true laws in the domain. Taking the facts together, you can quite truly put it in this way: To go to sleep, man must adopt a position whereby his life is enabled in some respects to take a similar course, while he is sleeping, to that of animal life. You will find further confirmation in a careful study of those animals whose spinal axis is not exactly parallel to the Earth's surface.

Here again I can only give you guiding lines. For the most part, these things have not been studied in detail; the facts have not been looked at in this manner, or not exhaustively. I know they have never been gone into thoroughly. The necessary researches have not been undertaken.

And now another thing: You know that what is trivially called “fatigue” represents a highly complex sequence of events. It can come about by our moving deliberately. When we move deliberately, we move our centre of gravity in a direction paralleled to the surface of the Earth. In a sense, we move about a surface parallel to the Earth's surface. The process which accompanies our outward and deliberate movements takes its course in such a surface. Now here again we can discover what belongs together. On the one hand we have our movement and mobility parallel to the surface of the earth, and our fatigue,—becoming tired. Now we go further in our line of thought. This movement parallel to the surface of the Earth, finding its symptomatic expression in fatigue, involves a metabolic process—an expenditure of metabolism. Underlying the horizontal movement there is therefore a recognizable inner process in the human body.

Now the human being is so constituted that he cannot well do without such movement—including all the concomitant phenomena, the metabolic expenditure of substance and so on. He needs all this for bodily well-being. If you're a postman, your calling sees to it that you move about horizontally; if you are not a postman you take a walk. Hence the relationship, highly significant for Economics, between the use and value of that mobility of man which enters into economic life and that which stays outside it—as in athletics, games and the like. Physiological and economic aspects meet in reality. In my critique of the economic concept of Labour, you may remember I have often mentioned this. It is at this point that the relation emerges between a purely social science and the science of physiology, nor can we truly study economics if we disregard it. For us however at the present moment, the important thing is to observe this parallelism of movement in a horizontal surface with a certain kind of metabolic process.

Now the same metabolic process can also be looked for along another line. We think once more of the alternation of sleeping and waking. But there is this essential difference. The metabolic transformation, when it takes place with our deliberate movements, makes itself felt at once as an external process, even apart from what goes on inside the human being. If I may put it so, something is then going on, for which the surface of the human body is no exclusive frontier. Substance is being transformed, yet so that the transformation takes place as it were in the absolute; the importance of it is not only for the inside of man's body. (The world “absolute” must of course again be taken relatively!)

That we get tired is, as I said, a symptomatic concomitant of movement and of the metabolic process it involves. Yet we also get tired if we have only lived the life-long day while doing nothing. Therefore the same entities which are at work when we move about with a will, are also at work in the human being in his daily life simply by virtue of his internal organization. The metabolic transformation must also be taking place when we just get tired, without our bringing it about by any deliberate action.

We put ourselves into the horizontal posture so as to bring about the same metabolism which takes place when we are not acting deliberately,—which takes place simply with the lapse of time, if I may so express it. We put ourselves into the horizontal posture during sleep, so that in this horizontal position our body may be able to carry out what it also carries out when we are moving deliberately in waking life. You see from this that the horizontal position as such is of great significance. It is not a matter of indifference, whether or not we get into this position. To let our inner organism carry out a certain process without our doing anything to the purpose, we must bring ourselves into the horizontal position in which there happens in our body something that also happens when we are moving by our deliberate will.

A movement must therefore be going on in our body, which we do not bring about by our deliberate will. A movement which we do not bring about by our deliberate will must be of significance for our body. Try to observe and interpret the given facts and you will come to the following conclusion, although again—for lack of time—in saying this I must leave out many connecting links. Human movement, as we said just now, involves an absolute metabolic process or change of substance, so that what then goes on in our metabolism has, so to speak, real chemical or physical significance, for which the limits of our skin are in some sense non-existent;—so that the human being in this process belongs to the whole Cosmos. And now the very same metabolic change of substance is brought about in sleep, only that then its significance remains inside the human body. The change of substance that takes place in our deliberate movement takes place also in our sleep, but the outcome of it is then carried from one part of our body to another. During sleep, in effect, we are supplying our own head. We are then carrying out or rather, letting the inside of our body carry out for us—a metabolic process of transformation for which the human skin is an effective frontier. The transmutation so takes place that the final process to which it leads has its significance within the bodily organization of man.

Once more then, we may truly say: We move of our own will, and a metabolic process (a transformation of substance) is taking place. We let the Cosmos move us; a transformation of substance is taking place once more. But the latter process goes on in such a way that the outcome of it—which in the former metabolic process takes its course, so to speak, in the external world—turns inward to make itself felt as such within the human head. It turns back and does not go flowing outward and away. Yet to enable it to turn back, nay to enable it to be there at all, we have to bring ourselves into the horizontal posture. We must therefore study the connection between those processes in the human body that take place when we move deliberately and those that take place when we are sleeping. And from the very fact that we are obliged to do this at a certain stage of our present studies, you may divine how much is implied when in the general Anthroposophical lectures I emphasize—as indeed I must do, time and gain,—that our life of will, bound as it is to our metabolism, is to our life of thought and indeation even as sleeping is to waking.

In the unfolding of our will, as I have said again and again, we are always asleep. Here now you have the more exact determination of it. Moving of his own will and in a horizontal surface, man does precisely the same as in sleep. He sleeps by virtue of his will. Sleep, and deliberate or wilful movement, are in this relation. When we are sleeping in the horizontal posture, only the outcome is different. Namely, what scatters and is dispersed in the external world when we are moving deliberately, is received and assimilated, made further use of, by our own head-organisation when we are asleep.

We have then these two processes, clearly to be distinguished from one another:—the outward dispersal of the metabolic process when we move about deliberately in day-waking life, and the inward assimilation of the metabolic process by all that happens in our head when we are sleeping. And if we now relate this to the animal kingdom, we may divine how much it signifies that the animal spends its whole life in the horizontal posture. This turning-inward of the metabolism to provide the head must be quite different in the animal. Also deliberate movement must be quite different in the animal from what it is in man.

This is the kind of thing so much neglected in the Science of today. They only speak of what presents itself externally, failing to see that the same external process may stand for something different in the one creature and in the other. For example—quite apart now from any religious implication—man dies and the animal dies. It does not follow that this is psychologically the same in either case. A scientist who takes it to be the same and bases his research on this assumption is like a man who would pick up a razor and declare: This is a kind of knife, therefore the same function as any other knife; so I will use it to cut my dumpling. Put on this simple level, you may answer: No-one would be so silly. Yet have a care, for this is just what happens in the most advanced researches.

This then is what we are led to see. In our deliberate movements we have a process finding its characteristic expression in curves that run parallel to the surface of the Earth; we cannot but make curves of this direction. What have we taken as fundamental now, in this whole line of thought? We began with an inner process which takes its course in man. In sleep this is the given thing, yet on the other hand we ourselves bring a like process about by our own action. Through what we do ourselves, we can therefore define the other. The possibility is given, logically. What is done to our bodily nature from out of cosmic space when are sleeping, this we can treat as the thing to be defined,—the nature of which we seek to know. And we can use as the defining concept what we ourselves do in the outer world—what is therefore well-known to as to its spatial relations. This is the kind of thing we have to look for altogether, in scientific method: Not to define phenomena by means of abstract concepts, but to define phenomena by means of other phenomena. Of course it presupposes that we do really understand the phenomena in question, for only then can we define them by one-another. This characteristic of Anthroposophical scientific endeavour. It seeks to reach a true Phenomenalism,—to explain phenomena by phenomena instead of making abstract concepts to explain them. Nor does it want a mere blunt description of phenomena, leaving them just as they are in the chance distributions of empirical fact and circumstance, where they may long be standing side by side without explaining one-another.

I may digress a moment at this point, to indicate the far-reaching possibilities of this “phenomenological” direction in research. The empirical data are at hand, for us to reach the right idea. There is enough and to spare to empirical data. What we are lacking in is quite another thing, namely the power to synthesize them,—in other words, to explain one phenomenon by another. Once more, we have to understand the phenomena before w can explain them by each other. Hence we must first have the will to proceed as we are now trying to do,—to learn to penetrate the phenomenon before us. This is so often neglected. In our Research Institute we shall not want to go on experimenting in the first place with the old ways and methods, which have produced enough and to spare of empirical data. (I speak here not from the point of view of technical applications but of the inner synthesis which is needed.) There is no call for us to go on experimenting in the old ways. As I said in the lectures on Heat last winter, we have to arrange experiments in quite new ways. We need not only the usual instruments from the optical instrument makers; we must devise our own, so as to get quite different kinds of experiments, in which phenomena are so presented that the one sheds light on the other. Hence we shall have to work from the bottom upward. If we do so, we shall find an abundance of material for fresh enlightenment. With the existing instruments our contemporaries can do all that is necessary; they have acquired admirable skill in using them in their one-sided way. We need experiments along new lines, as you must see, for with the old kind of experiment we should never get beyond certain limits. Nor on the other hand will it do for us merely to take our start from the old results and then indulge in speculation. Again and again we need fresh experimental results, to bring us back to the facts when we have gone too far afield. We must be always ready to find ways of means, when we have reached a certain point in our experimental researches, not just to go on theorising but to pass on to some fresh observation which will help elucidate the former one. Otherwise we shall not get beyond certain limits, transient though they are, in the development of Science.

I will here draw attention to one such limit, which, though not felt to be insurmountable by our contemporaries, will in fact only be surmounted when fresh kinds of experiment are made. I mean the problem of the constitution of the Sun. Careful and conscientious observations have of course been made by all the scientific methods hitherto available, and with this outcome: First they distinguish the inner most part of the Sun; what it is, is quite unclear to them. They call it the solar nucleus, but none can tell us what it is; the methods of research do not reach thus far. To say this is no unfriendly criticism; everyone admits it. They then suppose the Sun's nucleus to be surrounded by the so-called photo-sphere, the atmosphere, the chromosphere and the corona. From the photosphere onward they begin to have definite ideas abut it. Thus they are able to form some idea about the atmosphere, the chromosphere. Suppose for instance that they are trying to imagine how Sun-spots arise. Incidentally, this strange phenomenon does not happen quite at random; it shows a certain rhythm, with maxima and minima in periods of about eleven years. Examine the Sun-spot phenomena, and you will find they must in some way be related to processes that take place outside the actual body of the Sun. In trying to imagine what these processes are like, our scientists are apt to speak of explosions or analogous conditions. The point is that when thinking in this way they always take their start from premisses derived from the earthly field. Indeed, this is almost bound to be so if one has not first made the effort to widen out one's range of concepts,—as we did for instance when we imagined curves going out of space. If one has not done something of this kind for one' s own inner training, one has no other possibility than to interpret on the analogy of earthly conditions such observations as are available of a celestial body that is far beyond this earthly world.

Nay, what could be more natural—with the existing range of thought—than to imagine the processes of the solar life analogous to the terrestial, but for the obvious modifications. Yet in so doing one soon encounters almost insuperable obstacles. That which is commonly thought of as the physical constitution of the Sun can never really be understood with the ideas we derive from earthly life. We must of course begin with the results of simple observation, which are indeed eloquent up to a point; then however we must try to penetrate them with ideas that are true to their real nature. And in this effort we shall have to come to terms with a principle which I may characterize as follows.

It is so, is it not? Given some outer fact or distribution which we are able thoroughly to illumina with a truth of pre Geometry we say to ourselves: how well it fits: we build it up purely by geometrical thinking and now the outer reality accords with it. It hinges-in, so to speak. We feel more at one with outer reality when we thus find again and recognize what we ourselves first constructed, (yet the delight of it should not be carried too far. Somehow or other, one must admit, it always “hinges-in” even for those theorists who get a little unhinged themselves in the process: They too are always finding the ideas they first developed in their mind in excellent agreement with the external reality. The principle is valid, none the less.)

The following attempt must now be made. We may begin by imagining some process that takes place in earthly life. We follow the direction of it outward from some central point. It takes its course therefore in a radial direction. It may be a kind of outbreak, such for example as a volcanic eruption, or the tendency of deformation in an earthquake or the like. We follow such a process upon Earth in the direction of a line that goes outward from the given centre. And now in contrast to this you may conceive the inside of the Sun, as we are want to call it, to be of such a nature that its phenomena are not thrust outward from the centre, but on the contrary; they take their course from the corona inward, via the chromosphere, atmosphere and photosphere,—not from within outward therefore, but from without inward. You are to conceive , once more,—if this (Fig. 2) is the photosphere, this the atmosphere, this the chromosphere and this the corona,—that the processes go inward and, so to speak, gradually lose themselves towards the central point to which they tend just as phenomena that issue from the Earth lose themselves outward in expanding spheres, into the wide expanse. You will thus gain a mental picture which will enable you to bring some kind of synthesis and order into the empirical results. Speaking more concretely, you would have to say: If causes on the Earth are such as to bring about the upward outbreak for example of an active crater, the cause on the Sun will be such that if there is anything analogous to such an outbreak, it will happen from without inward. The whole nature of the phenomenon holds it together in quite another way. While on the Earth it tends apart, dispersing far and wide, here this will tend together, striving towards the centre.

You see, then what is necessary. First you must penetrate the phenomena and understand them truly. Only then can you explain them by one-another. And only when we enter thus into the qualitative aspect,—only when we are prepared, in the widest sense of the word, to unfold a kind of qualitative mathematics,—shall we make essential progress. Of this we shall speak more tomorrow. Here I should only like to add that there is a possibility, notably for pure mathematicians, to find the transition to a qualitative mathematics. Indeed this possibility is there in a high degree, especially in our time. We need only consider Analytical Geometry, with all its manifold results, in relation to Synthetic Geometry—to the real inner experience of Projective Geometry. True, this will only give us the beginning, but it is a very, very good beginning. You will be able to confirm this if you once begin along this pathway,—if for example you really enter into the thought and make it clear to yourself that a line has not two infinitely distant points (one in the one and one in the opposite direction) but only one,—fact of which there is no doubt. You will then find truer and more realistic concepts in this field, and from this starting-point you will find your way into a qualitative form of mathematics.

This will enable you to conceive the polarities of Nature no longer merely in the sense of outwardly opposite directions, where all the time the inner quality would be the same; whereas in fact the inner quality, the inner sense and direction, is not the same. The phenomena at the anode and the cathode for example have not the same inner direction; an inherent difference underlies them, and to discover what the difference is, we must take this pathway. We must not allow ourselves to think of a real line as though it had two ends. We should be clear in our mind that a real line in its totality must be conceived not with two ends but with one. Simple by virtue of the real conditions, the other end goes on into a continuation, which must be somewhere. Please do not underestimate the scope and bearing of these lines of thought. For they lead deep into many a riddle of Nature, which, when approached without such preparation, will after all only be taken in such a way that our thoughts remain outside the phenomena and fail to penetrate.