The Rudolf Steiner Archive

In the lectures so far, I have spoken of the capacities for supersensory knowledge and I have named them Imagination and Inspiration. Today I would like to say something about acquiring these capacities. At the moment I can only mention a few details. In my book Knowledge of the Higher Worlds and Its Attainment, you will find this presented in greater depth. Today, however, I would point out what is important in the context I have chosen for the present lecture. I have indicated that what I call Imagination with regard to knowledge of the world is attained through a development modeled on the memory process, only on another level. The importance of the memory process is that it retains in picture form what the human being encounters in outer experience.

Our first task will be to understand certain characteristics of the ordinary memory process, and then we must distill out what can be called pure memory in the true sense, also in ordinary life. One of the peculiarities of memory is that it tends to alter to a certain degree what has been experienced. Perhaps it is unnecessary to go into detail here, since most of you will be quite familiar with the fact that at times you can despair when you are relating something, and you hear from your own telling what has become of your experience by its passing through your memory. Even in ordinary life a certain self-education is necessary if we wish to come closer to pure memory, to the capacity to have these pictures ready at hand so that they faithfully render our experience. We can distinguish what happens with memory. On the one hand there is an activity of fantasy, quite justified, that goes on in an artistic direction. On the other hand there is a falsification of our experience. It should suffice for the moment to point out the difference between the fantasy tendency and the falsifying tendency, and that we must be able to experience this to maintain a healthy soul life. Certainly we must be aware of how memory is transformed by our fantasy, and how, when it is not subjected to such arbitrary action, when it is allowed to proceed according to a kind of natural similarity in the soul, it becomes increasingly faithful and true. In any case, both from the good tendency to artistic fantasy, as well as from the forces active in falsifying the memories—when we study it psychologically, we can recognize what is alive in the memory forces.

And out of these forces, something can take form that is no longer just memory. For example, one can point to certain mystical teachings that are in fact essentially falsified memory images; and yet we can profit from studying 'such images that have taken the form of earnest mystical experience. What concerns us at this moment, however, is what I have already indicated, that we can attain a power of the soul which is alive in the memory which can be metamorphosed into something else. This must happen in such a way that the original power of memory is led in the direction of inner faithfulness and truth, and not toward falsification. As I have said, when we repeatedly evoke easily surveyable mental images, which we intentionally combine out of their separate elements and then view as a whole, just as easily as the mathematical images: when we call up such images, hold them in our consciousness and dwell upon them, not so that we are fascinated by them, but so that at each moment we continue to hold them through an inner act of will—then gradually we succeed in transforming the memory process into something different, something of which we were previously unaware. The details are contained in the book I named, and also in Occult Science, an Outline.

If we continue long enough with such exercises (how long depends on the individual) and if we are in a position to expend sufficient soul energy on them, then we come to a point where we simply begin to experience pictures. The form of these pictures in the life of the soul is like that of memories. Gradually we win the capacity to live in such imaginations of our own making, although in their content they are not of our making. The exercise of this capacity results in imaginations rising up in the soul, and if we maintain a “mathematical” attitude of soul, we can make sure at any time whether we are being fooled by a suggestion or auto-suggestion, or are really living in that attitude of soul voluntarily. We begin to have mental images with the characteristic form of memory pictures but with a greater degree of intensity. Let me emphasize: at first these imaginations have the character of memory pictures. Only through inspiration do they become permeated with a more intense experience. At first they have the character of memory pictures, but of such a kind that we know their meaning does not relate to any experiences we have lived through externally since our birth. They do, however, express something just as pictorially as memory pictures express pictorially our personal experiences. They refer to something objective, yet we know that this objective something is not contained in the sphere which is surveyed by our memory. We are conscious that these imaginations contain a strong inner reality, yet at the same time we are aware that we are dealing with just images—just pictures of the reality. It is a matter of being able to distinguish these pictures from those of memory, in order that these imaginations remain pure, so that no foreign elements slip into them.

I will describe the outer process, but of course in just a few lectures one cannot go into any great detail. We may form a mental picture of an outer experience and we can see how in a sense the outer experience passes over into our organism, and—expressed abstractly—it then leads a further existence there, and can be drawn forth again as a memory picture. We notice that there is a certain dependence between what lives in the memory and the physical condition of the human organism. The memory is really dependent on our human organism right into the physical condition. In a way we pass on what we have experienced to our organism. It is even possible to give a detailed account of the continuation of the various pictures of our experience in the human organism. But this would be an entire spiritual-scientific chapter in itself. For our memories to remain pure and true, no matter how much our organism may participate in what lives on in the memory process, this involvement may not add anything of real content. Once mental pictures of an experience have been formed, nothing further should flow into the content of the memories.

If we are clear about this fact of memory life, we are then in a position to ascertain what it means when pictures appear in our consciousness that have the familiar character of memory pictures, but a content which does not relate to anything in our personal experience. In the process of experiencing imagination we realize the necessity of continually increasing the power of our soul. For what is it that we must really do? Normally our organism takes over the mental pictures we have formed from life and provides memory. Thereby the mental pictures do not just sink down into an abyss, if I may so express it, but are caught and held by our organism so that they can be reflected back again at any necessary moment. With imaginative pictures, this is just what should not be the case; we must be in a position to hold them through inner soul forces alone. Therefore it is necessary for us to acquire something that will make us stronger than we are ordinarily in receiving and retaining mental images. There are of course many ways to do this; I have described them in the books already named. I wish to mention just one of them. From what I now tell you, you will be able to see the relation between various demands of life which spring from anthroposophical spiritual science and their connection with the foundation of anthroposophical research.

Whoever uses his intellect to spin all kinds of theories about what he confronts as phenomena in the world (which of course can be extraordinarily interesting at times) will hardly find the power for imaginative activity. In this respect, certain developments in the intellectual life of the present day seem specifically suited to suppress the imaginative force. If we go further than simply taking the outer phenomena of the mineral-physical realm and connecting them with one another through the power of our intellect; if we begin to search for things that are supposed to be concealed behind the visible phenomena, with which we can make mental constructions, we will actually destroy our imaginative capacity.

Perhaps I may make a comparison. No doubt you have had some dealings with what could be called phenomenalism in the sense of a Goethean world view. In arranging experiments and observations, Goethe used the intellect differently from the way it is used in recent phases of modern thought. Goethe used the intellect as we use it in reading. When we read, we form a whole out of the individual letters. For instance, when we have a row of letters and succeed in inwardly grasping the whole, then we have solved a certain riddle posed by this row of individual letters. We would not think of saying: Here is a b, an r, an e, an a and a d—I will look at the b. As such, this isolated b tells me nothing in particular, so I have to penetrate further for what really lies behind the b. Then one could say: Behind this b there is concealed some mysterious “beyond,” a “beyond” that makes an impression on me and explains the b to me. Of course, I do not do this; I simply take a look at the succession of letters in front of me and out of them form a whole: I read bread. Goethe proceeds in the same way in regard to the individual phenomena of the outer world. For instance, he does not take some light phenomenon and begins to philosophize about it, wondering what states of vibration lie behind this phenomenon in some sort of “beyond.” He does not use his intellect to speculate what might be hiding behind the phenomenon; rather, he uses his intellect as we do when we “think” the letters together into a word. Similarly he uses the intellect solely as a medium in which phenomena are grouped—grouped in such a way that in their relation to one another they let themselves be “read.” So we can see that regarding the external physical-mineral phenomenological world, Goethe employs the intellect as what I would call a cosmic reading tool.

He never speaks of a Kantian “thing in itself” that must be sought behind the phenomena, something Kant supposed existed there. And so Goethe comes to a true understanding of phenomena—of what might be called the “letters” in the mineral-physical world. He starts with the archetypal or “Ur”-phenomenon, and then proceeds to more complex phenomena which he seeks either in observation or in experiments which he contrives. He "reads" what is spread out in space and time, not looking behind the phenomena, but observing them in such a way that they cast light on one another, expressing themselves as a whole. His other use of the intellect is to arrange experimental situations that can be “read”—to arrange experimental situations and then see what is expressed by them. When we adopt such a way of viewing phenomena and make it more and more our own, proceeding even further than Goethe, we acquire a certain feeling of kinship with the phenomena. We experience a belonging-together with the phenomena. We enter into the phenomena with intensity, in contrast to the way the intellect is used to pierce through the phenomena and seek for all kinds of things behind them—things which fundamentally are only spun-out theories. Naturally, what I have just said is aimed only at this theoretical activity.

We need to educate ourselves in phenomenology, to reach a “growing together with” the phenomena of the world around us. Next in importance is to acquire the ability to recall a fully detailed picture of the phenomena. In our present culture, most people's memories consist of verbal images. There comes a moment when we should not be dependent on verbal images: these only fill the memory so that the last memory connection is pushed up out of the subconscious into consciousness. We should progress toward a remembering that is really pictorial. We can remember, for instance, that as young rascals we were up to some prank or other—we can have a vivid picture of ourselves giving another fellow punches, taking him by the ear, cuffing him, and so on. When these pictures arise not just as faded memories, but in sharp outline, then we have strengthened the power we need to hold the imaginations firmly in our consciousness. We are related to these pictures in inner freedom just as we are to our ordinary memories. With this strengthened remembering, we grow increasingly interested in the outer world, and as a result the ultimate "living together with" all the various details of the outer world penetrates into our consciousness. Our memories take on the quality of being really objective, as any outer experience is, and we have the feeling that we could affectionately stroke them. Or one could say: These memory pictures become so lively that they could even make us angry. Please bear with me as I describe these things to you! It is the only thing I can do with our present language.

Then comes the next step: we must practice again and again eliminating these imaginations so that we can dive down again and again into an empty consciousness. If we bring such pictures into our consciousness at will and then eliminate them again in a kind of inner rhythm—meditating, concentrating, creating images, and then freeing ourselves of them—this will quicken powerfully the feeling of inner freedom in us. In this way we develop a great inner mobility of soul—exactly the opposite of the condition prevailing in psychopaths of various kinds. It really: is the exact opposite, and those who parallel what I have just described here with any kind of psychopathic state show that they simply have no idea of what I am talking about.

When we finally succeed in strengthening our forgetting—the activity which normally is a kind of involuntary activity—when now we control this activity with our will, we notice that what we knew before as an image of reality, as imagination, fills with content. This content shows us that what appears there in pictorial form is indeed reality, spiritual reality. At this point we have come to the edge of an abyss where, in a certain sense, spiritual reality shines across to us from the other side of existence. This spiritual reality is present in all physical sense reality. It is essential to develop a proper sense for the external world in order to have a correct relationship to these imaginations. Whoever wishes just to speculate about phenomena, to pierce them through, as it were, hoping to see what is behind them as some kind of ultimate reality—whoever does this, weakens his power to retain and deal with imaginations.

When we have attained a life of inspiration—that is, experiencing the reality of the spiritual world just as ordinarily we experience the physical world through our external senses—then we can say: now I finally understand what the process of remembering means. Remembering means (I will make a kind of comparison) that the mental images we have gained from our experiences sink down into our organism and act there as a mirror. The pictures we form in our minds are retained by the organism, in contrast to a mirror which just has to reflect, give back again what is before it. Thus we have the possibility of transforming a strictly reflective process into a voluntary process—in other words, what we have entrusted to memory can be reflected back from the entire organism and particularly from the nervous system. Through this process, what has been taken up by the organism in the form of mental pictures is held in such a way that we too cannot see “behind the mirror.” Looking inward upon our memories, we must admit that having the faculty of memory prevents us from having an inner view of ourself. We cannot get into our interior any more than we can get behind the reflective surface of a mirror.

Of course what I am telling you is expressed by way of comparisons, but these comparisons do portray the fact of the matter. We realize this when inspiration reveals these imaginations to us as pictures of a spiritual reality. At this moment the mirror falls away with regard to the imaginations. When this happens we have the possibility of true insight into ourselves, and our inner being appears to us for the first time in what is actually its spiritual aspect. But what do we really learn here?

By reading such mystics as Saint Theresa or Mechtild of Magdeburg, beautiful images are evoked, and from a certain point of view this is justified. One can enter into a truly devotional mood before these images. For someone who begins to understand what I have just described to you, precisely this kind of mystical visions cease to be what they very often are for the nebulous types of mystic: When someone comes to real inner vision, not in an abnormal way (as is the case with such mystics) but by the development of his cognitive faculty as I have described it, then he learns not only to describe a momentary aspect as Mechtild of Magdeburg, Saint Theresa and others do, but he learns to recognize what the real interior of the human organization is. If one wants to have real knowledge and not mystical intoxication, one must strive toward the truth and put it in place of their mist-shrouded images. (Of course, this may seem prosaic to the nebulous mystic.) When this is accomplished, the mirror drops away and one gains a knowledge, an inner vision of the lungs, diaphragm, liver, and stomach. One learns to experience the human organization inwardly. It is clear that Mechtild of Magdeburg and Saint Theresa also viewed the interior, but in their case this happened through certain abnormal conditions and their vision of the human interior was shrouded in all manner of mists. What they describe is the fog which the true spiritual investigator penetrates.

To a person who is incapable of accepting such things, it would naturally be a shock if, let's say hypothetically, a lofty chapter out of Mechtild were read and the spiritual researcher then told him: Yes, that is really what one sees when one comes to an inner vision of the liver or the kidneys. It is really so. For anyone who would rather it were otherwise, I can only say: That is the way it happens to be. On the other hand, for someone who has gained insight into the whole matter, this is for him the beginning of a true relation to the secrets of world existence. For now he learns the origin of what constitutes our human organization and at what depths they are to be recognized. He clearly recognizes how little we know of the human liver, the human kidneys, not to speak of other organs, when we merely cut open a corpse—or for that matter, when we cut open the living human organism in an operation—and get just the one-sided view of our organism.

There is the possibility not just to understand the human organism from the external, material side, but to see and understand it from the inside. We then have spiritual entities in our consciousness, and such entities show us that a human being is not so isolated as we might think—not just shut up inside his skin. On the contrary! Just as the oxygen I have in me now was first outside and is now working within me, in the same way—though extended over a long period of time—what is now working in me as my inner organization (liver, kidneys, and so on) is formed out of the cosmos. It is connected with the cosmos. I must look toward the cosmos and how it is constituted if I want to understand what is living in the liver, kidneys, stomach, and so on; just as I must look toward the cosmos and the make-up of the air if I want to understand what the substance is that is now working in my lungs, that continues to work on in the blood stream. You see, in true spiritual research we are not limited to separate pictures of separate organs but we come to know the connections between the human organism and the whole cosmos.

Not to be overlooked is the simple symbolic picture which we have already mentioned of the senses. We can in a way visualize our senses as “gulfs,” through which the outer world and its happenings flow into us. At the same time our senses continue inward as I have described them. Little by little we can see this activity from an inner point of view—the forming and molding activity that has worked on our nervous system since our birth. I have described the subjective experience of this activity as a life review, a life panorama, and we discover in the configuration of the nervous system an external pictorial form of what is really soul-spiritual. It can also be said that first we experience imaginations and then we see how these imaginations work in the formation of nerve substance. Of course this should not be taken in too broad a sense, since, as we know, nerve substance is also worked on before birth. I shall come back to this tomorrow. But essentially what I have said holds true. We can say: here is where the activity continues toward the inside; you can see exactly how it goes farther. It is the same activity, in a certain sense, that "engraves" itself into the nervous system. For the parts of the nervous system that are formed completely, this "engraving" activity can be seen streaming through the nerve paths. In childhood, however, for the parts that are still in the-process of being formed, this “engraving” acts as a real modeling force, a structuring proceeding out of imaginations. This leaves the rest of the human organism, about which we will speak shortly—what underlies the muscles, bones, and so on, also the physical basis of the nervous system—in fact, all of the organic tissue. At this point I should relate to you a certain experience I had; it will make this all a bit clearer.

I spoke once before the Theosophical Society about a subject I called “anthroposophy.” I simply set forth at that time as much of this anthroposophy as had revealed itself to my spiritual research. There was a request for these lectures to be printed and I set about doing this. In the process of writing them down, they turned into something different. Not that anything that had first been said was changed, but it became necessary to add to what was said by way of further explanation. It was also necessary to state the facts more precisely. This task would require a whole year. Now came another opportunity. There was again a general meeting of the Society and there was a request that the lectures should be ready for sale. So they had to get finished. I sent the first signature (16 pages) of the book Anthroposophy to the printer. The printing was rapidly done and I thought I would be able to continue writing. I did continue writing but more and more it became necessary to explain things more accurately. So a whole number of pages were printed. Then it happened that one signature was only filled up to page thirteen or fourteen and I had to continue writing to fill up all sixteen pages. In the meantime I became aware that in order to get this matter done the way I wanted to would require a more accurate, detailed development of certain mental processes, a very specific working out of imaginative, of inspirational cognition and then to apply these modes of cognition to these anthroposophical issues. And so I had to take a negative step, I dropped the whole idea of writing on Anthroposophy. It is still lying there today as it lay then—many pages.¹ For my intention was to make further investigations. Thus I became thoroughly acquainted with what I want to describe to you now. I can only describe it schematically at this time, but it is a sum total of many inner experiences that are really a cognitive method of investigating the human being.

It became increasingly clear to me that before one could finish the book called “Anthroposophy,” in the form intended at that time, one must have certain experiences of inner vision. One must first be able to take what one perceives as soul-spiritual activity working in the nervous system and carry it further inward, until one comes to the point where one sees the entire soul-spiritual activity—which one grasps in imagination and inspiration—crossing itself. This crossing point is really a line, in a vertical direction if looked at schematically. For certain phenomena the point lies farther up, for others farther down. In these lectures I can't describe this in detail, I just wanted to make a kind of cross section through the whole of it. Now because of this crossing, one is no longer free in exercising this activity. In fact, one was not altogether free before, as I have shown; now one is even less free. The whole situation undergoes a change. One is now being held strongly in an imaginative-inspired state. Expressed concretely, if one comes to an imagination of the eye by taking hold of visual sense-perception and the continuation into mental processes with imaginative-inspired cognition, then this activity proceeds inwardly and one comes to a kind of crossing, and with the activity first encompassing the eye another organ is encompassed, and that is the kidney.

The same applies to the other organs. In each case, when one carries one's imaginative-inspired activity into the body, one finds various relatively complete organs—complete at least in their basic form from birth—and one comes to a real inner view of the human organism. This kind of research is very demanding; and as I was not obliged at that moment to finish the book, and had to give another lecture cycle, which also demanded research efforts, you can imagine that it was not easy to continue to work out the method which I had developed at that time—of course, it was quite a few years ago that this occurred.

I mention this only to show you some of the difficulties—how one is continually held back by various demands. To continue in this, one must hold one's inner forces firmly together if one is to accomplish it. One must, in fact, repeatedly resolve to intensify one's thinking ability, the force of one's inner soul work—to strengthen it through love of external nature. Otherwise one simply cannot proceed. One goes consciously into oneself, but again and again one is thrown back, and instead of what I would call an inner view, one gets something not right. One must overcome the inward counterblow that develops.

I wanted to tell you all this so that you could see that the spiritual investigator has moments when he must wrestle with certain problems of spiritual research. Unfortunately, in the years that followed the event I have just described to you, my time was so filled with everything imaginable, particularly in recent years, that the needful—indeed, indispensable—activity for finishing my Anthroposophy could not take place.

You see, something that is inwardly understood, something we spoke of above rather abstractly, is in fact what is spun into an enveloping form of an organ, something quite concrete. If you picture this to yourselves, you will realize that such an insight into the human being can also build a bridge to practical activities. These activities must of course be founded on a vision of the human being and his relation to the world. I have already indicated in another connection how through developing imagination we gain knowledge not only of the sensory realm and its continuation into the nervous system, but also of the plant world. When we advance to inspiration, we become acquainted with the whole realm of forces that are at work in the animal world. At the same time we become aware of other things of which the animal world is only the outer expression. We now recognize the nature of the respiratory system, we can understand the external forms of the respiratory system through this relationship. The external form of the respiratory and circulatory system is not directly similar in its outer shape to its inner counterpart, as is the case with the outer form of the nervous system and the inner mental life. I showed this yesterday—how in the case of the nervous system two people, representing very different points of view, were able to draw similar pictures. In a parallel manner we become acquainted with the outer world and its kingdoms and the inner aspect of the human being.

Tomorrow I will consider what this inwardly experienced knowledge adds to our insight into the nature of the human being and his relation to his environment. Naturally, a great deal is revealed to us about specific relationships between the human being and his environment. It is possible to perceive the nature of a specific human organ and its connection to what exists in the outer natural realm. Thereby we discover in a rational way the transition from a spiritualized physiology to a true therapy. What once was won through instinctive inner vision is now possible to be renewed. I have mentioned yoga, and I could name even older systems which made it possible to perceive in an instinctive, childlike way the connection between the human being and the world around him. Many of today's therapeutic measures come from this older time—perhaps in somewhat different form, but they are still among the most fruitful today. Only on this spiritual path can therapy be developed that is suited to meet the real needs of today. Through insight into the connection of the human organs with the cosmos, a medicine will be developed based an inner perceptions, not just external experiment.

I set this before you just as an example of how spiritual science must fructify the various specialized branches of science. That this is needed is obvious when one looks at external research efforts, which have been very active and are magnificent in their own way—but which abound with questions. Take, for example, outer physiology or outer pathology: questions are everywhere. Whoever studies these things today and is fully awake will find the questions there—questions that beg for answers. In the last analysis, spiritual science recognizes there are great questions in outer life, and that they require answers. It does not overlook what is great and triumphant in the other sciences. At the same time, it wishes to study what questions result from this; it wishes to find a way to solutions to these questions in just as exact a manner as can be taught in the other sciences. In the end, the questions can be found (even for sense-bound empirical investigation) only through spiritual investigation. We will speak more about this tomorrow.

• 1. Published in German as Anthroposophie—Ein Fragment, Bibl. Nr. 45; English translation in preparation by Mercury Press.

Unfortunately our time together is so short that I have only been able to deal with our theme in a broad way, just intimating its development. The intention was to present a few ideas that lie, one might say, at the entrance of an anthroposophical spiritual science. From what has been presented, you will surely feel that everything we have touched upon needs further elaboration.

I have spoken of various ways of knowing that through inner soul work can follow as further steps from our everyday kind of knowing and from ordinary scientific cognition. I have already mentioned the first two of these further steps and called them imaginative cognition and inspired cognition. Yesterday I showed how, when imaginative and inspired cognition work together, and when we take account of a certain experience that I described yesterday as an inner crossing in the consciousness, a knowledge of the human being can arise in conjunction with a knowledge of the surrounding world. When this experience that we have in inspired-imaginative cognition is developed further, through certain exercises found in my books, something arises which has a similar name in ordinary life—that is, intuition. In ordinary life intuition refers to a kind of knowing that is not sharply delineated, to something more in the realm of feeling. This dimly experienced knowledge is not what the spiritual researcher means when he speaks of intuition and yet there are good reasons for thinking of the undeveloped, dim experiences of ordinary intuition as a kind of early stage of real intuition. Real intuition is a kind of knowing, a condition of the soul that is just as suffused with clarity of consciousness as is mathematical thinking. This intuition is reached through a continuation of what I have called exercises for the attainment of forgetting. These exercises must be continued in such a way that one really forgets oneself. When these exercises have been carried on in a precise and systematic way, then arises what the spiritual investigator calls intuition in the higher sense. This is the natural form of cognition into which inspired imaginations flow.

Before I go on with my discussion, I would like to stress one thing, to avoid possible misunderstanding. I can easily imagine that someone might raise a certain objection to what I described at the end of yesterday's lecture. First let me assure you that the conscientious spiritual investigator is the first to make various objections for himself. This is inherent in the process of spiritual research. With every step one must be aware from what possible angle objections may come, and how they can be met. To be specific, someone could raise an objection about what I said yesterday concerning the experiencing of a “crossing” that arises in the process of looking within, embracing our own inner organization. It could be said: This is an illusion. The fact is that especially the spiritual investigator (as is meant here) is not allowed to be a dilettante in external science; he is sure to know a thing or two about the inner organization of the human being from conventional anatomy and physiology. One might suspect that the investigator yields to a sort of self-deception, taking what he knows of external science and incorporating this into his inner vision. The spiritual researcher fully reckons with the possibility of self-deception along his path. One can settle the objections that have been raised by noting that what is perceived in the human organism during this inner viewing is totally different from anything one could possibly get from external anatomy or physiology. This perception of the inner organization could really be called a perception of the spiritual aspect of the human interior. The only help ordinary anatomy and physiology can render is the establishment of something like a mathematical reference point—a reference point for what has been spiritually perceived in the soul by inner vision, a definite content of perception at this level of cognition. For example, when we spiritually perceive the inner nature of what corresponds to the lung, it will be easier to connect this with the lung if we are already familiar with it through outer anatomy and physiology than if we knew nothing of it.

These two aspects—an inner vision of the lung, and what we know in an outer way through anatomy and physiology—are two completely different contents that must be reconciled later. At this level of cognition there is only a repetition of the kind of relationship that we experience between what is inwardly grasped in mathematical thinking and what is directly visible in the physical-mineral realm. The difference that exists between what we grasp inwardly in mathematical thought and what we find given in outer observation is very similar to the difference between what we grasp in inspired-imaginative activity and what we can learn through external research. Inner clarity of consciousness throughout is, of course, a basic requirement.

When we rise from inspired imagination to intuition, we encounter a situation similar to the one we described at the beginning of these lectures. We said: The outer world and its phenomena enter into us through our senses as through “gulfs.” Mathematical lines and forms which we construct influence our perception of the outer forms of the world. So with respect to our bodily nature there is a jutting in, a really essential penetration of the outer world into our spatial-bodily condition. We have a similar experience when all that I have described comes into us through intuition. Through this experience we become aware of one thing particularly: that what has been experienced within the human being is inexplicable of itself—or perhaps better said, it is something essentially unfinished. When we come to know ourselves through intuition, as long as we remain within the experience of self-knowledge we are basically dissatisfied. In contrast to this, with inspired imagination, when we apply it to knowledge of the self we feel a certain satisfaction. We learn what the human rhythmic system really is. This is a difficult process of knowledge. It is a process that can really never be completed, because it leads into endless further developments. In this type of knowledge you are learning to know yourself in connection with the world, as I showed yesterday. One can arrive at concrete insights concerning the connection of the healthy organism with its cosmic environment also the connection of the ailing organism with the cosmic environment. In this way the very interior of the human being can be penetrated.

At this point I would like to speak of something I described in the previous lecture course.¹ We are able to perceive through our inspired imagination how the human organism must relate itself to receiving something like a sense organ. It is, in fact, predisposed toward the sense organs. It opens itself outward so as to send a certain force system—if I may use such an expression—toward each separate sense. Beyond the interaction of the force system with our regular senses, one can discover abnormal cases of such tendencies arising in other places. A normal organization for the development of a sense can appear in a wrong place. Such a force system can be inserted into some organ not meant to be a sense organ, whose normal function is something else. The appearance of a metamorphosed force system in a place not right for it causes abnormalities in the human organism. A consequence of the particular abnormality just mentioned is the formation of a tumor where the displaced force system occurs. What we find here in the human organism is a more complex version of what Goethe in his teachings on metamorphosis always looked for, under simpler circumstances. We come to realize that a system of forces correctly associated with growth, when directed differently and in a metamorphosed form, can become the cause of illness. When inspired-imaginative cognition is directed to the whole matter of how man's sensory organization is related to the kingdoms of nature—to his whole environment—one discovers important relationships. These relationships lead us to remedies in our environment that can be used against pathological forms of forces.

Now you may see the vistas that are opened up by what I have described. This is not just fantasizing into the blue—nor is it nebulous mysticism to evoke satisfaction in the soul. Either would be completely foreign to what is meant here by anthroposophically-oriented spiritual science. This spiritual science wishes to penetrate into the real nature of the world in a serious and exact manner. At the same time, it must be admitted that much of what can be achieved in this way is still in its infancy today. And yet a fair amount of what I presented last spring in the course for physicians and medical students (which I plan to continue shortly) on pathology and therapy, made—I believe—a favorable impression on the listeners. Its view of the essential being of nature and the world, of the inner relationships, gave rise to the impression that here is something that can fertilize and complement outer observation and experiment. The contemporary world should see that here is at least an attempt to find out what it is that is creating the questions of external science, when there is no sign of any possibility in the scientific field of finding satisfactory answers to the questions.

As we advance along this path of knowledge (keeping always to what is spiritually real and concrete and avoiding abstraction), we have an experience on the other side of the human organization, of something similar to the "jutting" of the outer world into our sensory life. I said earlier that when we come to self-knowledge through intuition, it proves inevitably to be unfinished. We understand this now, for we see that here on the other side we have the reverse relationship to that of the sense organs. The senses are “gulfs” into which the outer world flows. On the other hand, we discover that the entire human being, becoming a sense organ in intuition, now reaches into the spiritual world. On the one hand, the outer world reaches into the human being; on the other, the human being reaches into the spiritual “outer world.” As I mentioned earlier in connection with the eye organization, the human being has a certain active relation to the depth dimension; with intuition he has (as long as he remains with intuition in the realm of self-knowledge) a certain relation to the vertical dimension. Thus something very similar to sense perception takes place, except that it is reversed.

We find that through intuition the human being places himself with his entire being in the spiritual world. Just as through the senses the external sense world projects inward, through intuition one consciously places oneself in the spiritual world. In this conscious projection into the spiritual world through intuition, the human being has a similar feeling to the feeling he has toward the outer world through perception. The feeling of being in the spiritual world, a kind of dim feeling of standing within the spiritual world, in ordinary life we call intuition. But this intuition is suffused with bright clarity when the stage of cognition is striven for which I have described. Thus you can realize that perception is just one side of our human relation to the outer world. In perception we have something indefinite, something that first must be inwardly worked upon. As perception is worked upon by our intellect and we discover laws at work in this perception, there is at the same time something corresponding to this that initially has just as indefinite a relation to us as does perception. It must be penetrated by inner knowledge that has been achieved, in the same way that perceptions must be penetrated by mathematical thinking. In short, our ordinary experience must be penetrated by our inwardly achieved knowledge.

In ordinary experience we call this kind of intuition belief or faith. Just as the human being faces the outer sense world and has the experience of perception, so, participating in a dim way in the spiritual world, he has the experience of belief. And just as perception can be illumined by the intellect or reason, so the content of this indefinite dim experience of belief can be illumined by our steadily increasing knowledge. This dim experience of faith becomes one of scientific knowledge just as perception attains scientific value through the addition of the intellect. You see how the things relate. What I am describing to you is truly a progression through inner spiritual work to transform the ordinary experience of faith into an experience of clear knowledge. When we rise into these regions, transforming faith into an experience of knowledge, we find this similar to the process of subjecting our perceptions to what has been worked out mathematically or logically. What is inherent here is not some artificial construction, it is a description of something a human being can experience—just as, for instance, one experiences what develops from early childhood when the intellect is not yet useable to a later time when the intellect and reason are in full use.

There are other experiences bound up with these—for example, the following: The moment we advance to inspired cognition, we have already had what I have described as the life panorama, which extends back to early childhood and, at times, even to birth. With this we have gained an inner kind of perception. It is only with the attainment of inspired cognition, however, that a kind of enhanced faculty of forgetting comes about which I must characterize as a complete extinguishing of the surroundings that up to this point were given through sense perception. In other words, a state of consciousness arises in which our own inner life, indeed our inner life in time up to birth, becomes the object of our consciousness. At this time one has the subjective feeling that one is inwardly empty, that one is in the outer world with one's consciousness, not within one's body. When we have succeeded in reaching this enhanced forgetting whereby the outer sense-perceptible world is really extinguished for a moment, then something appears through this experience being combined with what is attained intuitively. I must describe this in the following way.

We have already discussed imagination and we know it does in fact relate to reality, although at first it appears to have pictorial character. It relates to a reality, but at first we have only pictures in our consciousness. When we experience inspiration, we advance from the pictorial to the corresponding spiritual reality. When we reach the moment in which external sense perception is completely extinguished through inspiration, a new content appears for the first time. The content that appears corresponds to our existence before conception. We learn to look into our soul-spiritual being as it was before it took possession of a physical organism arising out of the stream of heredity. Thus this imagination fills itself with a real spiritual content that represents our pre-birth existence. Characterized in this way, this may still seem paradoxical to many people of our time. One can only indicate the exact point in the cognitive process where such a view of the human soul-spiritual self enters in, and where what we call the question of immortality takes on real meaning. At the same time we gain a more exact view of the other pole of the human organization. When we penetrate what we have at first only as intuitive belief and raise this to knowledge, the possibility arises to relate imaginations—although in another way than in the case just described—to the conditions after death. In short, we have a view of what one can call the eternal in man and I will only just mention the following. When intuition has developed further, to the point it is really capable of reaching, we develop our true “I” for the first time. And within the true “I” there appears to inner vision what in anthroposophical spiritual science is referred to as knowledge of repeated earth-lives. The knowledge that we were a soul-spiritual being before conception and that we will continue to be after death: this is really experienced in inspired imagination. The knowledge of repeated earth-lives is added to this only in intuition.

When we have reached this area, we first begin to discover the full significance of waking up and falling asleep, and the condition of sleep as such. Through a deepening of the cognition related to the pole of perception, we discover the experience of falling asleep, which otherwise remains unconscious. At the other pole of intuitive thought, we discover the experience of waking up. Between these two is the experience of sleep, which I would like just to characterize as follows: when the human being falls asleep in ordinary consciousness, he is in a condition in which his consciousness is completely dimmed. This empty consciousness in which the human being lives between falling asleep and awakening, is a state which he cannot know from his own subjective point of view. The inspired-imaginative condition is very similar. In this condition the will impulses are silenced just as in sleep the senses are silenced. The subjective human activity is silent in both sleep and inspired imagination. The major difference is this: in sleep the consciousness is empty. In the condition of inspired imagination one's consciousness is filled; one's inner experiences are independent of sense perception and will impulses; in a certain sense one is awake while one is asleep. One has therefore the possibility of studying the life of sleep.

I would like to return to something that I spoke of this morning in the history seminar. The historical problems we spoke of take on new meaning when seen in connection with the experiences we have just been speaking of. At one time or another you may have reflected upon such historians as Herodotus. He and others were really precursors of what we call history in the modern scientific sense. The way history is written today developed with the intellectual culture that finds special satisfaction in experiment. In other words, those who find special satisfaction in experiment also find satisfaction in the external aspect of history. This science of history proceeds empirically, and rightly so from its own point of view. It collects data, and from this data it pieces together a picture of the course of history. One can, however, object that this way of interpreting empirical data easily allows that history could have developed differently. As I put it this morning, one could hypothesize that Dante somehow died as a boy. We would then be faced with the possibility that what we experience as coming through Dante would be absent, at least it would be absent as manifested in the person of Dante. In the study of history one will meet with great difficulties in reaching true insight, unless one is satisfied with the ready-made scholarly harangues.

Let us take another example. Historians set out to study the Reformation, using the available facts of external history. (We cannot go into detail here; you can research this yourself if you are interested.) For instance, if the monk Luther had died young, I would really like to know what would have been recorded as derived purely from the external historical method! Certainly something quite different from what is recorded today. Quite serious difficulties arise when one wants truly to characterize historical knowledge. One may say if one focuses on the philosophy of history, one can follow the observable outer events from the point of view of some abstract element of necessity, or one may want to find an element of purpose shaping the events as Strindberg did. The fact that the other reforms would not have been there either if Luther had died as a boy, would not affect this theoretical finding of purpose or necessity, in whatever might have taken place instead of the Reformation. If Luther had died, the other reformers would not have been there either.

One must be very careful in coming to conclusions when one is working in the field of external historical observation. However, the course of human development reveals something quite different when it is observed from the level of knowledge that I have been describing to you. Let me give you a concrete example. One would see that there were certain forces at work in European civilization around the fourth century between the time of Constantine and Julian the Apostate. The outer aspect of this world would appear differently if records existed of a personality so impressive as, for instance, Dante. There really is a problem here, and I confess I am not finished with it yet but must pursue it a bit further. The problem is a most concrete one. I am not yet finished in that I cannot tell you whether important documents, important evidence concerning an important figure around the period of 340 or 350 A.D. somehow disappeared from the view of external history, or whether he died in his youth—or somehow perished in those turbulent, war-filled times. It is a fact, however, that one sees forces at work in this period that cannot be traced in external history today. These forces would only be accessible to external history through some stroke of luck, like the chance discovery of written documents in some monastery. It is beyond any doubt for the spiritual investigator, however, that these forces are active. The spiritual investigator can truly establish what otherwise would be seen as forces abstracted from outer circumstances.

Now suppose we would wish to look back on the life of Dante and acquaint ourselves with him. We would try to make him come to life in our soul, really to try to know him inwardly. We would also familiarize ourselves with the forces active in the time of Dante. This is an external approach to knowledge. Naturally, the knowledge that the spiritual scientist gains of the Dantean period will look somewhat different from what can be found in external documents—for example, in the Divine Comedy. One could of course object that the spiritual scientist might confuse what he has learned through external perception with what he has obtained through inner vision. When, however, inner vision operates in such a way that we know beyond any doubt that in a particular age—as in this one just named—the outer events do not correspond to the inner happenings, we know that spiritual powers are really at work. Under these circumstances it is possible to present history as I did recently for a small circle, by looking exclusively at the forces seen inwardly. We come to the point where we have inwardly observed these forces; they penetrate us, they live within us. It would really be a miracle if, for instance, one could just fantasize about the forces at work in Julian the Apostate at the time in question. Those times can only be truly explored spiritually.

The level of historical observation achieved here can be described as a direct viewing of the original spiritual forces that are active in the historical process. Thereby one receives a satisfactory explanation for precisely the parts of history where external facts are missing—because documents are missing, or men and women did not have a chance to live their lives out normally. In such cases what is viewed inwardly can help external history. Examples of the result of such inner knowledge, pointing to the forces behind historical events, are given in my little book, The Spiritual Guidance of Mankind. What is presented there must naturally be preceded by the inner vision of the missing aspects of external history, as I have mentioned. It is only at this point, assuming we intend to be inwardly responsible in our relation to knowledge, that we can feel justified in saying: It is possible simply on the foundation of sound human understanding to rise (as I have repeatedly described) to a level where such real forces are active.

But, you may object, no one could speak of the beings I described in The Spiritual Guidance of Mankind who has not yet advanced to such vision. This is of course true; to speak with this degree of emphasis, one must have a certain level of cognition. But one may take something else into consideration. If we are honest in approaching the facts of history and if we are sufficiently schooled in philosophy to be aware of the riddles and doubts the usual study of history presents, we can still have an inner experience of a certain kind. This experience is similar to the one that the astronomer had when on the basis of certain gravitational forces he predicted the as-yet-unseen planet of Neptune. The discovery of the spiritual laws and essential nature of history is really a very similar process in the spiritual domain to the calculations employed by LeVerrier to predict the existence of Neptune. LeVerrier did not somehow piece together a scientific result as is done in external history—with a positive or skeptical slant, simply avoiding connections: he followed the facts according to their truth. He said to himself: Something must be at work here. This is similar to what the astronomer before him said concerning Uranus. Uranus doesn't follow the course which it ought to according to the forces I already know, so there must be something exercising an influence on these known forces. The conscientious investigator also recognizes certain forces at work. He sees the intervention of these forces much as someone who on finding a limestone or silica shell-form in a rock formation looks for the active forces. From the way the silica fossil looks, he surely does not say: This silica form has somehow crystallized out of its mineral surroundings. Rather he says: At one time this form was filled out with something; it was made by some kind of animal and one can have a mental picture of this animal. If some being were to arrive who had lived at the time the animal was alive in that shell, and he described the animal, such an eyewitness could be likened to the spiritual investigator. The finder of the shell bearing the imprint of the animal is not necessarily the one who uses his sound human understanding to deduce from the outer configuration what must have been there to form the shell. What the living facts were is something only the spiritual investigator can say. The person who is willing to bring a sound sense of logic, a logical view of facts, and healthy human understanding, can follow and inwardly test what the spiritual researcher tells him about the forms in front of him.

It is not necessary to have a blind belief in the spiritual investigator. Naturally, the actual discovery of such things as are presented in The Spiritual Guidance of Mankind requires spiritual research. When the spiritual researcher has presented what he wishes to tell in terms of what he calls higher beings, he will also readily agree to be tested for this vision by those gathering outer facts. His attitude is this: I invite you to rap my knuckles if you discover anything whatever that contradicts the outer order of events predicted by my inner vision.

Something similar appeared in our circle, in connection with interpretations of the gospels which had been worked out in a purely spiritual manner. It has also occurred in such cases as the one given this morning. I am busy with a variety of literature, yet to this day the author was unknown to me of the work Dr. Stein cited this morning giving the date of Christ's death. I have never seen it. Naturally, this is not the sort of evidence that one can accept objectively—I mention this only parenthetically. Nevertheless, such things have occurred within our circle. Verifications have appeared that must be accepted objectively. Through a living involvement in spiritual-scientific work, many of our friends have a real personal conviction; it does not rest on blind faith, but precisely on their experience of the life that goes on in spiritual science. This explains why those who have been involved in the activities of spiritual science for many years can speak in a different tone from those for whom spiritual science is just a theory.

I believe we can show in the context of the evolution of humanity the connections between the state of science today and the state of knowledge today. Naturally, everything has earlier stages; scientific experimentation is no exception. Given this, however, the experimentation of the past, up to the most recent times, cannot help but seem primitive compared to what we have today. When our fully developed experiment is experienced inwardly, it really calls for something more. From what has been combined by the intellect in the actual activity of experimentation something is released in the soul. What is released requires spiritual knowledge to balance it. We have shifted our understanding from mere observation to experimentation. Something happens when we discover the real difference between what is experienced in mere observation and what is experienced in the activity of experimentation: the urge arises in us to rise to a higher level of self-knowledge from the ordinary kind. This higher knowledge is what I have recently been describing. These two things are related. The urge for a higher knowledge, which is natural to human beings striving for knowledge today, has developed quite naturally in the course of history out of an elementary interest in experimentation itself. The scientific data that we have gained in regard to outer nature are, in many respects, really related to questions. The important thing is that if the formulation of the questions is correct, then a correct answer is possible.

What natural science has given us recently is really in large measure no more than a statement of questions for the spiritual researcher. Whether we look at recent astronomy or the views of modern chemistry, when we grasp what is in them, the question arises: how is what is described related to what goes on in the human being himself? Questions arise about man's relation to the world precisely through the scientific results that have come from our shifting from observation over to the experimental realm. So we can see that for someone who really experiences modern science and does not theorize about it, this science is full of spiritual-scientific questions. From the nature of these questions, there simply is no choice but to go to spiritual science for answers. In the year 1859 Darwin came to a conclusion of what he had studied so meticulously; but for someone who studies these results afterwards, in spite of what Darwin took to be scientific conclusions, they appear as questions.

We are helped by the kind of experience we have in experimenting but at the same time we recognize the essentially independent nature of mathematics. When we seek for the realm in which mathematics is applicable, where it will result in an inner satisfying knowledge, then we see a merging of observation and of mathematical thinking, of the results of mathematical thinking, into an understanding of nature. But we may ask, what underlies what we experience in experiment; what is really happening when we feel the necessity for a form of knowledge that can even venture into historical knowledge? Where does this lead? We tend to look for connections everywhere for which the threads are simply not to be found in the material of contemporary science. Once we have grasped what it is that brings order into the connections between the facts, and in all spheres of knowledge—from the study of nature up to the study of history, we sense higher beings revealing themselves, purely soul-spiritual beings. If we come this far, then the door is open to a contemplation of an independent spiritual world.

My honored guests! I know just how much these lectures must seem unsatisfying to you, due to their sketchy and aphoristic nature. But rather than lecture on a narrowly defined subject, I chose to give a wide overview, even though in the particulars it could not be filled in. My intention was that you might learn something of the procedures involved in spiritual-scientific knowledge as it is meant here. I hoped you would get a feeling for the aims toward which it aspires. It aims for the greatest possible exactness and not some sort of fanciful or dilettante activity. For even in mathematics, what makes it so exact is the fact that we have an inner experience of it. In the Platonic age it was known why the words “God geometrizes” were inscribed as a motto on the school; it was clear that all who entered would receive a training in geometry and mathematics. In a similar way modern science of the spirit knows that to attain its goal it must have inner mathematical clarity. I hope you have received the impression, particularly as regards its methods, that the orientation of spiritual science is worthwhile. Perhaps on reflection you may come to ask the question: Can this not indeed lead to a fructification of our other sciences—not to belittle them, but to raise them to their true value? If I have achieved this to some degree, aphoristic and in some ways insufficient as these lectures have been, then my intention have been fulfilled.

• 1. The Boundaries of Natural Science, Bibl.-Nr. 322, eight lectures, Dornach 1920, Anthroposophic Press.

Now we have come to the end of our university courses. We have heard lectures from various individuals who have worked in our anthroposophical spiritual science for some time. We have also had a number of seminars which were intended to fill out what the lectures only sketched as a framework. In spite of the fact that all the participants in these lectures have worked hard, we must also consider the quality of the time spent together given the nature of such an event. All we were able to do was to let some light come in, as through individual windows in a building—that light which we believe is present in our anthroposophical spiritual science. Please consider what is contained in this room, the openings into which we are describing symbolically as windows of the spiritual-scientific movement. The contents of the room are various subjects that are just at their beginning; a richer work will exist ultimately. If you take this into account, you can understand why we could present only a small amount of what we might hope to give in such courses on similar occasions.

With such an event we hoped to draw students from all directions, and to our joy they have in fact appeared in great numbers. It is very gratifying to us and meaningful for the movement. For first and foremost, we would like to show, no matter how sketchily, that a genuine scientific attitude prevails in the anthroposophical movement. No doubt there are other spiritual intentions at work also, but these will have to be shown in other ways. Above all, these lectures are meant to demonstrate at the very least the will to strive toward real scientific knowledge. However, considering present-day conditions, anyone who understands the situation must feel: If we speak of a scientific attitude, a scientific spirit that plays directly into the living conditions of the modern human being, then it must be able to prove itself in the social sphere.

It is really necessary that the scientific spirit of our day shall give rise to ideas that can bring strength and healing into our social life. It is not enough today to have a scientific spirit that calls the human being into an existence estranged from life. We need a scientific spirit that will give us real health in our social life. The social situation confronts us full of riddles and urgent demands, even in a certain way threatening. If we have a feeling for these times, we can sense the need for real solutions—solutions that can be found only by those who grasp the social life with scientific understanding. We believe we are able to recognize this necessity from the most significant signs of this time. It is out of this recognition that our anthroposophical movement is artistically, scientifically and culturally conceived; this includes the building in Dornach called the Goetheanum, the Free University for Spiritual Science. Our wish is that out of a genuine scientific attitude these impulses can come to life in us and become really socially active.

We have attempted in the very structure of our lectures and seminars to make possible a recognition of the truly scientific spirit to which we aspire in our anthroposophical movement. Attacks from various directions accuse us of sectarianism or the desire to found a religion, but they come from those who don't know us, or—in some cases—from a malicious desire to slander us. The scientific spirit cannot of course be seen in the factual content of what is presented. Whoever would exclude empirical content, whether physical or super-sensible, shows that he himself is not imbued with the scientific spirit. It can only be seen in the treatment of the facts, in the striving to follow a definite method. And the real test of its validity—whether its results originated from sensory or supersensory experience—is based on the nature of this striving. Do we strive toward the scientific spirit that rules in the recognized sciences?

Is this striving demonstrated in our methodology, in our thinking with scientific accuracy? This is a justifiable question. It is also a worthwhile point of discussion inasmuch as this scientific spirit, as it prevails among us, is in need of improvement. One can determine whether our movement is scientific or not, not on the content we present but by how we proceed. Let it be shown in any instance that we have proceeded illogically, unscientifically, or in a dilettante fashion and—since we are serious about the correct development of our spiritual-scientific endeavors—we will make the necessary improvements without argument. We do not wish to deny this principle of progress in any way. So, enough about the underlying elements for discussing the scientific status of our endeavors.

We have striven to prove in the social realm, in life itself, what results from our knowledge of the world. In our discussions we have tried to present what we believe to be the truth regarding knowledge of the human being and the world. In the seminars we showed how the Waldorf School movement arose out of the anthroposophical movement. The lively manner of teaching in the Waldorf schools raises the question whether what is found in spiritual science will also prove itself in the shaping of today's young people. We don't want to exhaust ourselves in fruitless theoretical discussion: we want to let reality itself test what we believe is the truth toward which we should strive. Goethe said, “What is fruitful, that alone is true.” Even those far removed from modern philosophical pragmatism or the “as if” school must have their truth proven by its fruitfulness. We can declare ourselves in full agreement with the Goethean principle that only what is fruitful yields proof of its truth before reality—particularly where social truths are concerned. If what flows livingly out of spiritual science can return again into life, and if life can show that the result of recognized truth, or supposed truth, can send a human being out into life with ability, vigor, sureness, and enthusiasm and strength for work, then this is a proof of the truth which has been striven for. At the same time we have attempted something else, but it is really still too much in its infancy to be outwardly demonstrated.

In Der Kommende Tag, in Futura, we have put forth economic ideas which are intended to show that what is derived in a spiritual way, out of reality, also enables us to see the affairs of practical life in the right light. The time has not yet come when we can speak of these things becoming manifest, of fulfilling the conditions for a real proof. However, even in the economic realm, one may grant us the fact that we have not been afraid to extend something that was won purely in the spiritual out into practical life. This is actual testimony that we do not shy away from the tests of reality. How things develop in this region is perhaps not fully within our own will to determine. In such cases, even more than in the field of education, one is dependent on the practicalities of life, as well as how one is understood by the world and one's own circle.

In this way, we try to take into account the signs of the times. We have recently seen in some of our lectures that these signs point directly to spiritual-scientific demands; they also confront us with great social questions. But above all we seek to take into account the inner soul needs of the human being. For someone who is familiar with one area, for example the natural sciences, it is very easy to believe that we are already in possession of an infallible scientific method. Ultimately, however, what arises as science can only be fruitful for the whole evolution of humanity if it joins human evolution in a way that sustains the life of man. With this essential condition in mind, I ask you: Isn't there something in today's universities or in similar circumstances that can cause the soul to come somewhat into error? One can, of course, enter a laboratory and work in the dissection room, believing that one is working with a correct method and that one has an overview of all factors involved, grasping them in accord with present conditions and the level of humanity's evolution. But for humanity's evolution something else is necessary. Something is necessary which perhaps occurs very rarely, and the significance of which is not properly appreciated. It would be necessary that someone who has worked seriously and conscientiously with scientific spirit in the chemistry lab, observatory, or clinic, could then step into a history or aesthetics classroom and hear something there that would live in inner conformity with what he had learned in his technical courses. Such unity is needed—for the simple reason that regardless to what degree individuals may specialize, ultimately the things achieved in separate disciplines must work together in the process of general human evolution, and must spring from a common source.

We believe it is impossible today to experience a unity directly between, for instance, present historical pronouncements and the teachings of natural science. For this reason we strive toward what stands behind all scientific endeavors: the spiritual reality, the source that is common to them all. The aim of our striving is to come to know this spiritual reality. With our feeble powers we are striving to establish the validity of such knowledge of the spirit and its right to exist. In this lecture series and similar events, we have striven to show you what we are doing and how we do it, and we are grateful that you joined us.

May I touch on one additional subject: A short time ago, a coworker of long standing in our movement spoke with me. He knew that for spiritual-scientific reasons I must speak about two Jesus children. Until recently he hadn't told me of his intentions to follow this matter up in a conscientious manner studying the external aspect. His recent conversation with me was after he had finished his investigations. He said that he had compared the gospels thoroughly with one another, and had discovered that they don't begin to make sense until they are regarded from this spiritual-scientific viewpoint.

May research proceed thus in all realms! If it does, we know that our spiritual science will be able to stand fast. We do not fear the testing, no matter how detailed the examination may be. We have no fear of the request to verify. We only worry if someone opposes our viewpoint without proof, proof of all the individual details. The more carefully our spiritual research is tested, the more at ease we can be about it. This consciousness we bear deep within us. It is with such awareness that we have taken the responsibility of calling you all here, you who are striving to build a life of science and of scientific spirit. Today, my honored students, it is impossible to offer you the things of the outer world. In the places where this is done, our efforts are sometimes rejected in a surprising manner. Even so, your appearing here allows us to feel we are correct in saying that there are still souls among today's youth whose concern is the truth and striving toward the truth. Therefore we wish to say—I speak from the fullness of my heart, and I know I am also speaking for the coworkers of these courses we have truly enjoyed working with you. This is particularly gratifying because at the same time from other quarters attacks are raining down on us from ill-will, and we are called upon again and again to refute these attacks. We do as much as we can to make the refutations—as much as time permits. But really, the burden of proof lies with the one who makes an assertion; he should bring evidence of its truth. Otherwise, one could blithely throw assertions at anyone, leaving him to refute everything.

I only wish to indicate how the opposition operates toward us, personally attacking us rather than attempting to understand our ideas by discussing matters seriously with us.

What is most strongly held against us is that in one important area we have to insist upon setting ourselves against the well-intended strivings of the times. We cannot just go along with the general attitude to take what traditional science represents in the various fields and simply let it be carried in a popular way throughout the world. Rather, from our own knowledge we believe there is another need. Something must be brought into those quarters which consider themselves infallible these days. It is generally believed that such authority is held in those quarters that their ideas can be taken unaltered and be disseminated among the masses.

We believe, however, that certain scientific elements still lacking must enter those quarters to fructify their scientific work. The fact that we do not merely want the scientific spirit disseminated from certain quarters into the wide world but also want to bring a different spirit into science—this, I believe, is why we are confronted by such frightful opposition. It would be good if these matters were considered in a calm and objective way. For we must not hide the fact that we are in serious need of the collaboration of wider circles, even though every one of us is convinced of the scientific value of our endeavors. What worries us most is that we have so few coworkers who can really stand their ground. This is why it means so much to us that you, the university youth, have been coming to us now for some time. We have faith in you young students. We believe that what we need can sprout out of your youthful energy. Therefore, my honored fellow students, we would particularly like to work together with you in our field, as far as time and conditions permit.

It is with this spirit that we sought to permeate the work in these courses. Perhaps you can carry away with you the conviction that it has at least been our aspiration to work in this direction. I began today by comparing what we are offering to a closed room, opening out through windows to the surrounding world of spiritual science. Through these windows we have wanted to let fragments shine in of a world of knowledge, which we want to apply in a spiritual-scientific way. Now we are at the end of the course, and I wish to say a heartfelt “goodbye till we meet again” in similar circumstances.

But I would still like to return to the comparison with which I began the course. It is not generally my habit to pay homage to fine phrases, even when they are time-honored; rather, I like to return to just a simple expression of truth. In our cultural literature, a high-sounding phrase is often quoted as being Goethe's dying words, “Light, more light!” Well, Goethe lay in a tiny room in a dark corner when he was dying, and the shutters on the opposite window were closed. From my knowledge of Goethe I have every reason to believe that in truth his words were simply: “Open the shutters!” Now that I have dealt with that lofty phrase of my beloved and revered Goethe in an heretical manner, I would like to use my version of it as we end our work. My honored students! As we feel ourselves together in the room whose windows open out to spiritual knowledge, windows through which we have sought in a fragmentary way to let in what we believe to be light, I call to you out of the spirit that led us to invite you here: I call out to you, “Open the shutters!”

If you are disappointed about what you are about to hear, I would like to say in advance that today I want to discuss very elementary things [about the fourth dimension]. Those who want to delve deeper into this question should be very familiar with the higher concepts of mathematics. I would like to give you some very elementary and general concepts. One must distinguish between the possibility of thinking in a four-dimensional space and reality. Whoever is able to make observations there is dealing with a reality that extends far beyond what we know as the sensual-real. You have to do thought transformations when you go there. You have to let things play into mathematics a little, find your way into the way of thinking of the mathematician.

You have to realize that the mathematician does not take a step without accounting for what arrives at his conclusions. But we must also realize when we deal with mathematics that even the mathematician cannot penetrate a single step [into reality], that he cannot draw any conclusions [that go beyond what is merely possible in thought]. First of all, it is about simple things, but they become more complicated when one wants to arrive at the concept of the fourth dimension. We must be clear about what we mean by dimensions. It is best to examine the various spatial structures in terms of their dimensionality. They lead to considerations that were only tackled in the 19th century by great mathematicians such as Bolyai, Gauss and Riemann.

The simplest spatial size is the point. It has no extension at all; it must be conceived. It is the fixation of an extension in space. It has no dimension. The first dimension is the line. The straight line has one dimension, length. If we move the line, which has no thickness, ourselves, we step out of the one dimension, and the line becomes a surface. This has two dimensions, a length and a width. If we move the surface, we step out of these two dimensions and we get the body. It has three dimensions: height, width, depth (Figure 1).

If you move the body itself, if you move a cube around in space, you will again only get a spatial body. You cannot move space out of itself.

We need to turn to a few other concepts. If you look at a straight line, it has two boundaries, two endpoints A and B (Figure 2).

Let's imagine that we want A and B to touch. But if they are to touch, we have to curve the line. What happens? You cannot possibly remain within the [one-dimensional] line if you want to make A and B coincide. To connect points A and B, we have to step out of the straight line itself, we have to step out of the first dimension and into the second dimension, the plane. In this way, the straight line becomes a closed curve (that is, in the simplest case, a circle) by bringing its endpoints into alignment (Figure 3).

It is therefore necessary to go beyond the first dimension; you cannot remain within it. Only in this way can the circle be created. You can perform the same operation with a surface. However, this only works if you do not remain within the two dimensions. You have to enter the third dimension and then you can turn the surface into a tube, a cylinder. This operation is done in a very similar way to the way we brought two points into coincidence earlier, thereby moving out of the first dimension. Here, in order to bring two boundaries of the surface into coincidence, we have to move into the third dimension (Figure 4).

Is it conceivable that a similar operation could be carried out with a spatial structure that already has three dimensions itself? If you have two congruent cubes, you can slide one into the other. [Now imagine two congruent cubes as the boundaries of a three-dimensional prismatic body.] If you try to make one cube, which is colored red on one side [and blue on the opposite side], fit exactly over the other cube, which is otherwise [geometrically] identical but with the red and blue colors swapped, then you cannot make them fit except by rotating the cube (Figure 5).

Let us consider another spatial structure. If you take the left-hand glove, it is impossible for you to pull the left-hand glove over the right hand. But if you look at the two [mirror-symmetrical] gloves together, like the straight line with the end points A and B, you have something that belongs together. It is then a single entity, with a boundary [that is, with a mirror plane] in the middle. It is very similar with the two symmetrical halves of the human outer skin. 2

How can we now make two [mirror] symmetrical three-dimensional structures coincide? Only if we go beyond the third dimension, as we did with the first and second. We can also put the right or left glove over the left or right hand, respectively, when we walk through four-dimensional space.

[When constructing the third dimension (depth dimension) of the visualization space, we align the image of the right eye with that of the left eye and place it over it.

We now look at an example from Zöllner. We have a circle and a point P outside of it. How can we bring the point P into the circle without crossing the circle? This is not possible if we remain within the plane. Just as one has to go from the second dimension into the third when moving from a square to a cube, we also have to go out of the second dimension here. With a sphere, there is also no possibility of entering [into the interior] without [piercing the surface of the sphere or] going beyond the third dimension.

These are possibilities for thought, but they have a practical significance for the theory of knowledge, [in particular for the problem of the objectivity of the content of perception]. If we realize how we actually perceive, we will come to the following view. Let us first ask ourselves: How do we gain knowledge of bodies through our senses? We see a color. Without eyes, we would not perceive it. The physicist then says: Out there in space is not what we call color, but purely spatial forms of movement; they penetrate through our eye, are captured by the optic nerve, transmitted to the brain, and there, for example, the red arises. One may now ask: Is the red also present when there is no sensation?

Red could not be perceived without the eye. The ringing of a bell could not be perceived without the ear. All our sensations depend on the transformation of forms of motion by our physical and mental apparatus. However, the matter becomes even more complicated when we ask ourselves: Where is the red, this peculiar quality, actually located? Is it in the body? Is it a process of vibration? Outside there is a process of movement, and this continues right into the eye and into the brain itself. There are vibrational [and nervous] processes everywhere, but red is nowhere to be found. Even if you examine the eye, you would not find red anywhere. It is not outside, but it is also not in the brain. We only have red when we, as a subject, confront these processes of movement. So do we have no possibility at all to talk about how the red meets the eye, how a c sharp meets the ear?

The question is, what is this inner [representation], where does it arise? In the philosophical literature of the 19th century, you will find that this question runs through everything. Schopenhauer, in particular, has provided the following definition: The world is our representation. But what then remains for the external body? [Just as a color representation can be “created” by movements, so can] movement can arise in our inner self through something that is basically not moved. Let us consider twelve snapshots of a [moving] horse figure on [the inside of] a [cylinder] surface, [which is provided with twelve fine slits in the spaces between. If we look at the rotating cylinder from the side,] we will have the impression that it is always the same horse and that only its feet are moving. So [the impression of] movement can also arise through our [physical organization] when something is not moving at all [in reality]. This is how we arrive at a complete dissolution of what we call movement.

But what then is matter? If you subtract color, movement [shape, etc., i.e. what is conveyed by sensory perception] from matter, then nothing remains. If we already have the [secondary, i.e. “subjective” sensations [color, sound, warmth, taste, smell] within us, we must also place [the primary sensations, that is, shape and movement,] within us, and with that the external world completely disappears. However, this results in major difficulties [for the theory of knowledge].

Let us assume that everything is outside, how then do the properties of the object outside come into us? Where is the point [where the outside merges into the inside]? If we subtract all [sensory perceptions], there is no outside anymore. In this way, epistemology puts itself in the position of Münchhausen, who wants to pull himself up by his own hair. But only if we assume that there is an outside, only then can we come to [an explanation of] the sensations inside. How can something from the outside enter our inside and appear as our imagination?

We need to pose the question differently. Let us look at some analogies first. You will not be able to find a relationship [between the outside world and the sensation inside] unless you resort to the following. We return to the consideration of the straight line with endpoints A and B. We have to go beyond the first dimension, curve the line, to make the endpoints coincide (Figure 7).

Now imagine the left endpoint A [of this straight line] brought together with the right endpoint B so that they touch at the bottom, so that we are able to return to the starting point [via the coinciding endpoints]. If the line is small, the corresponding circle is also small. If I turn the [initially given] line into a circle and then turn larger and larger lines into circles, the point at which the endpoints meet moves further and further away from the [original] line and goes to infinity.

of the [original] line and goes to infinity. Only at infinity do the [increasingly large] circle lines have their endpoint. The curvature becomes weaker and weaker, and eventually we will not be able to distinguish the circle line from the straight line with the naked eye (Figure 8).

In the same way, when we walk on the Earth, it appears to us as a straight piece, although it is round. If we imagine that the two halves of the straight line extend to infinity, the circle actually coincides with the straight line.

The straight line can be conceived as a circle whose diameter is infinite.

Now, instead of a [geometric] line, imagine something that is real and that connects to a reality. Let us imagine that as the point C [on the circumference of the circle] progresses, cooling occurs, that the point becomes colder and colder the further it moves away [from its starting point] (Figure 9). Let us leave the point within the circle for the time being, and, as it becomes colder and colder, let it reach the lower limit A, B. When it returns on the other side, the temperature increases again. So on the way back, the opposite condition to the one on the way there occurs. The warming increases until the temperature at C is reached again, from which we started. No matter how extended the circle is, it is always the same process: a flow of heat out and a flow of heat in. Let us also imagine this with the [infinitely extended straight] line: as the temperature [on one side increasingly] dissipates, it can rise on the other side. We have here a state that dissipates on one side while it rebuilds on the other.

In this way, we bring life and movement into the world and approach what, in a higher sense, we can call an understanding of the world. We have here two states that are interdependent and interrelated. However, for everything you can observe [sensually], the process that goes, say, to the right has nothing to do with the one that comes back from the left, and yet they are mutually dependent.

We now compare the body of the external world with the state of cooling and, in contrast, our inner sensation with the state of warming. [Although the external world and inner sensation have nothing directly perceptible in common,] they are related to each other, mutually dependent [in an analogous way to the processes described above]. This results in a connection between the external world [and our internal world] that we can support with an image: [through the relationship between] the seal and the sealing wax. The seal leaves behind an exact imprint, an exact reproduction of the seal in the sealing wax, without the seal remaining in the sealing wax [and without any material from the seal being transferred to the sealing wax]. So in the sealing wax there remains a faithful reproduction of the seal. It is quite the same with the connection between the outside world and inner sensations. Only the essential is transferred. One state determines the other, but nothing (material) is transferred.

If we imagine that this is the case with [the connection between the] outside world and our impressions, we come to the following. [Geometric] mirror images in space behave like gloves from the left and right hand. [In order to relate these directly and continuously to each other,] we have to use a new dimension of space to help us. [Now the outside world and the inner impression behave analogously to geometric mirror images and can therefore only be directly related to each other through an additional dimension.] In order to establish a relationship between the outside world and inner impressions, we must therefore go through a fourth dimension and be in a third element. We can only seek the common ground [of the outside world and inner impressions] where we [are one] with them. [One can imagine these mirror images as] floating in a sea, within which we can align the mirror images. And so we come [initially in thought] to something that transcends three-dimensional space and yet has a reality. We must therefore bring our spatial ideas to life.

Oskar Simony has tried to represent these animated spatial structures with models. [As we have seen, one comes] from the consideration of the zero-dimensional [step by step] to the possibility of imagining four-dimensional space. [On the basis of the consideration of mirror-symmetrical bodies, that is, with the help of] symmetries, we can first [most easily] recognize this space. [Another way to study the peculiarities of empirical three-dimensional space in relation to four-dimensional space is to study the knotting of curves and ribbons.] What are symmetry conditions? By intertwining spatial structures, we cause certain complications. [These complications are peculiar to three-dimensional space; they do not occur in four-dimensional spaces.]

Let's do some practical thinking exercises. If we cut a band ring in the middle, we get two such rings. If we now cut a band whose ends have been twisted by 180° and then glued, we get a single twisted ring that does not disintegrate. If we twist the ends of the tape 360° before gluing them together, then when we cut it, we get two intertwined rings. Finally, if we twist the tape ends 720°, the same process results in a knot.

Anyone who reflects on natural processes knows that such convolutions occur in nature; [in reality,] such intertwined spatial structures are endowed with forces. Take, for example, the movement of the Earth around the Sun, and then the movement of the Moon around the Earth. It is said that the Moon describes a circle around the Earth, but [if you look more closely] it is a line that is wrapped around [a circle, the orbit of the Earth], thus a helix around a circular line. And then we have the sun, which rushes through space so fast that the moon makes an additional spiral movement around it. So there are very complicated lines of force extending in space. We have to realize that we are dealing with complicated concepts of space that we can only grasp if we do not let them become rigid, if we have them in a fluid state.

Let us recall what has been said: the zero-dimensional is the point, the one-dimensional is the line, the two-dimensional the surface and the three-dimensional the body. How do these concepts of space relate to each other?

Imagine you are a creature that can only move along a straight line. What would the spatial perceptions of such a being, which itself is only one-dimensional, be like? It would not perceive its own one-dimensionality, but would only imagine points. This is because, if we want to draw something on a straight line, there are only points on the straight line. A two-dimensional being could encounter lines, and thus distinguish one-dimensional beings. A three-dimensional being, such as a cube, would perceive the two-dimensional beings. Man, then, can perceive three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, as a two-dimensional being can only perceive one dimension, and a three-dimensional being can only perceive two dimensions, so a being that perceives three dimensions can only be a four-dimensional being. The fact that a human being can define external beings in three dimensions, can [deal with] spaces of three dimensions, means that he must be four-dimensional. And just as a cube can only perceive two dimensions and not its third, it is true that the human being cannot perceive the fourth dimension in which he lives.

Today I want to discuss some elementary aspects of the idea of multidimensional space [among other things, in connection with the] spirited Hinton.

You will recall how we arrived at the concept of multi-dimensional space, having considered the zeroth dimension [last time]. I would like to briefly repeat the ideas of how we can move from two- to three-dimensional space.

What do we mean by a symmetrical behavior? How do I align a red and a blue [flat figure, which are mirror images of each other]?

With two halves of a circle, I can do this relatively easily by sliding the red [half] circle into the blue one (Figure 10).

This is not so easy in the following [mirror]symmetrical figure (Figure 11). I cannot make the red and blue parts coincide [in the plane], no matter how I try to slide the red into the blue.

But there is a way [to achieve this anyway]: if you step out of the board, that is, out of the second dimension [and use the third dimension, in other words, if you] place the blue figure on the red one [by rotating it through the space around the mirror axis].

The same applies to a pair of gloves: I cannot match one with the other without stepping out of [three-dimensional] space. You have to go through the fourth dimension.

Last time I said that in order to develop an understanding of the fourth dimension, you have to make [the relationships in] space fluid, thereby creating conditions similar to those you have when moving from the second to the third dimension. In the last lesson, we created spatial structures out of paper strips that intertwined. Such interweaving causes certain complications. This is not a game, but such inter-weavings occur in nature all the time. Anyone who reflects on natural processes knows that such inter-weavings really do occur in nature. Material bodies move in such intertwined spatial structures. These movements are endowed with forces, so that the forces also intertwine. Take the movement of the earth around the sun and then the movement of the moon around the earth. The moon moves in an orbit that is itself wound around the earth's orbit around the sun. It thus describes a spiral around a circular line. Because of the movement of the sun, the moon describes another spiral around this. The result is very complicated lines of force that extend through the whole space.

The heavenly bodies behave in relation to each other like the intertwined strips of paper [by Simony, which we looked at last time]. We have to keep in mind that we are dealing with complicated spatial concepts that we can only understand if we do not let them become rigid. If we want to grasp space [in its essence], [we must first conceive it as rigid, but then] make it completely fluid again. [You have to go as far as zero]; the [living] point can be found in it.

Let us once again visualize the structure of the dimensions]. The point is zero-dimensional, the line is one-dimensional, the surface is two-dimensional and the body is three-dimensional. The cube has the three dimensions: height, width and depth. How do the spatial structures [of different dimensions] relate to each other? Imagine that you are a straight line, that you have only one dimension, that you can only move along a straight line. If such beings existed, what would their concept of space be like? Such beings would not perceive one-dimensionality in themselves, but would only be able to imagine points wherever they went. Because in a straight line, if we want to draw something in it, there are only points. A two-dimensional being would only encounter lines, so it would only perceive one-dimensional beings. [A three-dimensional being like] the cube would perceive two-dimensional beings, but could not perceive its [own] three dimensions.

Now, humans can perceive their three dimensions. If we reason correctly, we must say to ourselves: Just as a one-dimensional being can only perceive points, a two-dimensional being only straight lines, and a three-dimensional being only surfaces, so a being that perceives three dimensions must itself be a four-dimensional being. The fact that humans can define external beings in terms of three dimensions, can [deal with] spaces of three dimensions, means that they must be four-dimensional. And just as a cube can perceive only two dimensions and not its third, so it is clear that man cannot perceive the fourth dimension in which he lives. Thus we have shown [that man must be a four-dimensional being]. We swim in the sea [of the fourth dimension, like ice in water].

Let us return once more to the consideration of mirror images (Figure 11). This vertical line represents the cross-section of a mirror. The mirror reflects an image [of the figure on the left]. The process of reflection points beyond the two dimensions into the third dimension. [To understand the direct and continuous connection between the mirror image and the original, we have to add a third dimension to the two.

[Now let us consider the relationship between external space and internal representation.] The cube here apart from me [appears as] an idea in me (Figure 12). The idea [of the cube] is related to the cube like a' mirror image to the original. Our sensory apparatus [creates an imagined image of the cube. If you want to align this with the original cube, you have to go through the fourth dimension. Just as the third dimension has to be transitioned to (during the continuous execution of the two-dimensional) mirroring process, our sensory apparatus has to be four-dimensional if it is to be able to establish a [direct] connection [between the imagined image and the external object].

If you only imagined [two-dimensionally], you would [only] have a dream image in front of you, but you would have no idea that there is an object outside. Our imagination is a direct inversion of our ability to imagine [external objects by means of] four-dimensional space.

The human being in the astral state [during earlier stages of human evolution] was only a dreamer, he had only such ascending dream images.” He then passed from the astral realm to physical space. Thus we have mathematically defined the transition from the astral to the [physical-] material being. Before this transition occurred, the astral human being was a three-dimensional being and therefore could not extend his [two-dimensional] ideas to the objective [three-dimensional physical-material] world. But when he [himself] became physical-material, he still acquired the fourth dimension [and could therefore also experience three-dimensionally].

Due to the peculiar design of our sensory apparatus, we are able to align our perceptions with external objects. By relating our perceptions to external things, we pass through four-dimensional space, imposing the perception on the external object.

How would things appear if we could see from the other side, if we could enter into things and see them from there? To do that, we would have to pass through the fourth dimension.

The astral world itself is not a world of four dimensions. But the astral world together with its reflection in the physical world is four-dimensional. Anyone who is able to see the astral world and the physical world at the same time lives in four-dimensional space. The relationship of our physical world to the astral world is a four-dimensional one.

One must learn to understand the difference between a point and a sphere. In reality, this point would not be passive, but a point radiating light in all directions (Figure 13).

What would be the opposite of such a point? Just as there is an opposite to a line that goes from left to right, namely a line that goes from right to left, there is also an opposite to the point. We imagine an enormous sphere, in reality of infinite size, that radiates darkness from all sides, but now inwards (Figure 14). This sphere is the opposite of the point.

These are two real opposites: the point radiating light and infinite space, which is not a neutral dark entity, but one that floods space with darkness from all sides. [As a contrast, this results in] a source of darkness and a source of light. We know that a straight line that extends to infinity returns to the same point from the other side. Likewise, it is with a point that radiates light in all directions. This light comes back [from infinity] as its opposite, as darkness.

Now let us consider the opposite case. Take the point as the source of darkness. The opposite is a space that radiates light from all sides.

As was recently demonstrated [in the previous lecture], the point behaves in this way; it does not disappear [into infinity, it returns from the other side] (Figure 15).

[Similarly, when a point expands or radiates out, it does not lose itself in infinity; it returns from infinity as a sphere.] The sphere, the spherical, is the opposite of the point. Space lives in the point. The point is the opposite of space.

What is the opposite of a cube? Nothing other than the whole of infinite space, except for the piece that is cut out here [by the cube]. So we have to imagine the [total] cube as infinite space plus its opposite. We cannot do without polarities if we want to imagine the world as powerfully dynamic. [Only in this way] do we have things in their life.

If the occultist were to imagine the cube as red, the space around it would be green, because red is the complementary color of green. The occultist not only has simple ideas for himself, he has vivid ideas, not abstract, dead ideas. The occultist must enter into things from within himself. Our ideas are dead, while the things in the world are alive. We do not live with our abstract ideas in the things themselves. So we have to imagine the infinite space in the corresponding complementary color to the radiating star. By doing such exercises, you can train your thinking and gain confidence in how to imagine dimensions.

You know that the square is a two-dimensional spatial quantity. A square composed of four red- and blue-shaded sub-squares is a surface that radiates differently in different directions (Figure 16). The ability to radiate differently in different directions is a three-dimensional ability. So here we have the three dimensions of length, width and radiance.

What we did here with the surface, we also think of as being done for the cube. Just as the square above was made up of four sub-squares, we can imagine the cube as being made up of eight sub-cubes (Figure 17). This initially gives us the three dimensions of height, width and depth. Within each sub-cube, we can then distinguish a specific light-emitting capacity, which results in a further dimension in addition to height, width and depth: the radiation capacity.

You can imagine a square made up of four sub-squares, a cube made up of eight different sub-cubes. And now imagine a body that is not a cube, but has a fourth dimension. We have created the possibility of understanding this through radiative capacity. If each [of the eight partial cubes] has a different radiating power, then if I have only the one cube that radiates only in one direction, if I want to obtain the cube that radiates in all directions, I have to add another one on the left, doubling it with an opposite one, I have to put it together out of 16 cubes.

Next lesson we will have the opportunity to consider how we can think of a multidimensional space.

My dear friends, today I will continue with the difficult chapter we have undertaken to take on. In doing so, it will be necessary to refer to the various things that I have already mentioned in the last two lectures. Then today I would also like to create the basic lines and basic concepts in order to [the more exact geometrical relationships as well as] the interesting practical aspects of theosophy, to make them our own.

You know that we have tried to imagine four-dimensional space in its potentiality for the very reason that we can at least create some kind of concept about the so-called astral realm as well as about the higher realms, about higher existence in general. I have already indicated that entering the astral space, the astral world, is initially something tremendously confusing for the secret disciple. For those who have not studied these things in detail, who have not even studied them theoretically, who have not even studied Theosophy theoretically, it is extremely difficult to even begin to form an idea of the very different nature of the things and entities that confront us in the so-called astral world. Let us once again point out how great this diversity is.

As the simplest thing, I mentioned that we have to learn to read every number symmetrically. The secret student, who is only accustomed to reading numbers as they are read here in the physical world, will not be able to find his way through the labyrinth of the astral. If you have a number in the astral, for example 467, you have to read it as 764. You have to get used to reading everything symmetrically, to seeing everything symmetrically (in a mirror image). That is the basic condition. This is still easy as long as we are dealing with spatial structures or numbers. It becomes more difficult when we come to time relationships. When we come to time relationships, the matter also becomes symmetrical in the astral, in such a way that what comes later appears to us first and what comes earlier appears later. So when you observe astral processes, you also have to be able to read backwards from front to back. These things can only be hinted at, because they sometimes seem quite grotesque to those who have never had an idea of them. In the astral, the son is there first and then the father; in the physical, the egg is there first and then the chicken. In the physical, it is different. In the physical, birth comes first, and then the birth is an emergence of a new thing from an old one. In the astral, it is the other way around. There, the old emerges from the new. In the astral, what is paternal or maternal nature devours what is filial or daughterly nature for the appearance.

In Greek, you have a pretty allegory. The three gods Uranus, Kronos and Zeus symbolize the three worlds. Uranus represents the heavenly world: Devachan; Kronos represents the astral; Zeus the physical. Kronos is said to devour his children.” So in the astral, one does not give birth, but is consumed.

But things get very complicated when we look at the moral aspect of the astral plane. This also appears in a kind of reversal or mirror image. And that is why you can imagine how differently things appear when we interpret things as we are accustomed to interpreting them in the physical. In the astral, for example, we see a wild animal approaching us. This is not to be understood in the same way as in the physical. The wild animal is choking us. This is the appearance that someone who is accustomed to reading things in the same way as external events has. But the wild animal is in truth something that exists within ourselves, that lives in our own astral body and that is choking us. What approaches you as a strangler is rooted in your own desire. So you can experience that when you have a thought of revenge, this thought of revenge appears to you as a strangling angel that approaches you from outside and harasses you.

In truth, everything radiates from us [in the astral realm]. We must regard everything that we see approaching us in the astral as emanating from us (Figure 18). It comes from the sphere, from all sides, as if from infinite space, it penetrates into us. But in reality it is nothing other than what our own astral body sends out.

We only really read the astral [and only then] find the truth when we are able to bring the peripheral into the center, to see and interpret the peripheral as the central. The astral seems to come at you from all sides. Think of it this way: in reality, it is something that radiates out from you in all directions.

I would like to familiarize you with a term that is very important in occult training. It appears in a wide variety of works on occult research, but is rarely understood correctly. Those who have reached a certain level of occult development must learn to see everything that is still karmically predisposed in them – joy, lust, pain, and so on – in the astral outside world. If you think theosophically in the right sense, you will realize that the outer life, our body, in the present age is nothing more than a result, an average of two currents coming from opposite directions and merging into each other .

Imagine a current coming from the past and one coming from the future, and you have two currents that merge and actually intersect at every point (Figure 19). Imagine a red current in one direction and a blue current in the other direction. And now imagine, for example, four different points in this intersection. [Then, at each of these four points, we have] an interaction of these red and blue currents. [This is an image for the interaction of] four successive incarnations, where in each incarnation something comes towards us from one side [and something from the other]. You can always say to yourself, there is a current that comes towards you and a current that you bring with you. Man flows together out of these two currents.

You get an idea of it if you think of it this way. Today you sit here with different experiences, tomorrow at the same hour you will have a different set of events around you. Imagine the events that you will have by tomorrow are already all there. It would then be the same experience as if you were looking at a panorama. It would be as if you were approaching these events, as if these events were coming towards you spatially. So imagine that the stream that is coming towards you from the future brings you these events, then you have the events between today and tomorrow in this stream. You allow the future flowing towards you to be carried by the past.

In every period of time, your life is an intersection of two currents, one from the future to the present and the other from the present to the future. Where the currents meet, a congestion occurs. Everything that a person still has ahead of him must be seen emerging as an astral phenomenon. This is something that speaks an incredibly impressive language.

Imagine that the secret disciple [comes to the point in his development where he] is supposed to look into the astral world, where the senses are opened to him so that he would see emerging around him as outer phenomena in the astral world that which he would still have to experience before the end of the present period. This is a sight that is very powerful for every human being. We must therefore say that it is an important step in the course of occult training for the human being to be confronted with the astral panorama, the astral phenomenon, of what he still has to experience until the middle of the sixth root race, because that is how long our incarnations will last. The path opens up before him. No secret disciple will experience it differently, except that he sees as an external phenomenon what he still has to face in the near future up to the sixth root race.

When the disciple has advanced to the threshold, the question arises: Do you want to live through all this in the shortest conceivable time? Because that is what it is about for the one who wants to receive the initiation. If you think about it, you have your own future life in front of you as an external panorama in a moment. That, in turn, is what characterizes our view of the astral. For one person, it is something that makes them say, “No, I'm not going in there.” For another, on the other hand, it is something that makes them say, “I have to go in there.” This point in the process of development is called the 'threshold', the decision, and the phenomenon that one has there, oneself with everything that one still has to experience and live through, is called the 'guardian of the threshold'. The guardian of the threshold is therefore nothing other than our own future life. It is ourselves. Our own future life lies behind the threshold.

You see in this another peculiarity of the astral world of appearance, namely, that when the astral world is suddenly opened to someone through some event – and such events do occur in life – that person must first face something incomprehensible. It is a terrible sight, which could not be more confusing for those people upon whom, unprepared, the astral world suddenly breaks in through some event. It is therefore eminently good to know what we have now discussed, so that in the event of the astral world breaking in, one knows what to do. It may be a pathological event, a loosening between the physical body and the etheric body or between the etheric body and the astral body. Through such events, a person may be unexpectedly transported into the astral world and gain insights into astral life. If this happens, the person will come and say that he sees this or that apparition. He sees it and does not know how to read it, because he does not know that he has to read symmetrically, that he has to understand every wild animal that approaches him as a reflection of what lies within himself. Indeed, the astral powers and passions of man appear in Kamaloka in the most diverse forms of the animal world.

It is not a particularly beautiful sight to see people in Kamaloka who have just been reaped. At that moment they still have all their passions, urges, desires and cravings. Such a person in Kamaloka no longer has his physical body or etheric body, but in his astral body he still has everything that connected him to the physical world, that can only be satisfied through the physical body. Imagine an average citizen of the present day who has achieved nothing special in his past life and has not made any effort to achieve anything, who has never done much for his religious development, who may not have abandoned religion in theory, but practically, that is, in his feelings and attitudes, has thrown it overboard. In that case it is not a living element in him. What then is in his astral body? There are only things that can be satisfied through the physical organism. For example, he craves palate enjoyment. But the palate would have to be there for that, so that this desire can be satisfied. Or man craves for other pleasures, which can only be satisfied by setting his physical body in motion. Suppose he had such a craving, but the body was gone. Then all this lives in his astral body. This is the situation in which man finds himself when he has died without astral purification and cleansing. He still has the desire for the pleasures of the palate and the other things, but not the possibility of satisfying them. This is what causes the torment and horror of the life in Kamaloka. Therefore, the desire must be laid aside in Kamaloka if man dies without astral purification. Only when this astral body has learned that it can no longer satisfy its desires and wishes, that it must unlearn them, is it freed.

[In the astral world] the instincts and passions take on animal forms. As long as the person is embodied in the physical body, the shape of their astral body is somewhat based on this physical body. But when the outer body is gone, then the instincts, desires and passions, as they are in their animal [nature], come into their own in their own form. So in the astral world, a person is an image of their instincts and passions.

Because these astral beings can make use of other bodies, it is dangerous to let mediums enter into a trance when there is no clairvoyant is present to avert evil.

In the physical world, the lion is a plastic expression of certain passions, the tiger is an expression of other passions, and the cat is an expression of yet other passions. It is interesting to see how each animal is the plastic expression of a passion, of an urge.

In the astral, in Kamaloka, man is therefore approximately similar to [animal nature] through his passions. This is the source of the misunderstanding regarding the doctrine of transmigration of souls that has been attributed to Egyptian and Indian priests and teachers of wisdom. You should live in such a way that you do not incarnate as an animal, says this teaching. But this teaching never speaks of the physical life, but of the higher life, and its only aim was to persuade people on earth to lead such a life that after death in Kamaloka they would not have to develop their animal form. Those who develop the characteristics of a cat will appear in Kamaloka as a cat. The fact that one also appears as a human being in Kamaloka is the meaning of the rules of the doctrine of the transmigration of souls. The true teachings have not been understood by the scholars; they only have an absurd idea of them.

Thus we have to deal with a complete mirror image of what we actually think and do here in the physical world in every area – in the areas of number, time and moral life – when we enter the astral realm. We must get used to reading symmetrically, because we must be able to do so when we enter the astral space.

The easiest way for a person to get used to reading symmetrically is to build on such elementary mathematical ideas as we have hinted at in the previous lecture and as we will get to know more and more in the following discussions. I would like to start with a very simple idea, namely the idea of a square. Imagine a square as you are accustomed to seeing it (Figure 20). I will draw the square so that the four sides are drawn in four different colors.

This is the physical appearance of the square. Now I would like to draw the devachan aspect of the square on the board. It is not possible to do this exactly, but I would like to give you an approximate idea of what a square would look like in the mind. The mental counter-image [of a square] is approximately like a cross (Figure 21).

We are dealing here mainly with two perpendicular intersecting axes. Two lines that pass through each other, and that's it. The physical counter-image is created by drawing perpendicular lines on each of these axes. The physical counter-image of a mental square can best be imagined as a congestion [of two mutually intersecting currents]. Let us imagine these perpendicular axis lines as currents, as forces that act outwards from the point of intersection, and let us imagine countercurrents to these currents, only now in the direction from outside to inside (Figure 22). A square then enters into the physical world by imagining these two types of currents or forces - one from the inside, the other from the outside - as accumulating against each other. The currents of force are thus limited by accumulations.

With this, I have given a picture of how everything mental relates to the physical. Likewise, you can construct the mental counterpart for any physical thing. The square here is only the simplest of examples.

If you could construct a correlative for every physical thing that corresponds to the physical world in the same way that two perpendicular lines correspond to a square, then you would obtain the devachan or mental image for every physical thing. With other things, it is of course much more complicated.

Now imagine a cube instead of a square. The cube is very similar to the square. The cube is a body that is bounded by six squares. Mr. Schouten made these six squares that bound the cube specially. Now, instead of the four bounding lines that are present in the square, imagine six bounding surfaces. Imagine that instead of vertical lines we have vertical surfaces as a kind of congestion, and then assume that you have not two but three axes standing on one another [vertically], and you have the boundary of the cube. Now you can also imagine what the mental correlate of the cube is. You have again two things that challenge each other reciprocally. The cube has three perpendicular axes and three surface directions; we have to think of congestion effects in these three surface directions (Figure 23). We cannot imagine the three axes and the six surfaces, as before the two axes and four lines, in any other relationship than by thinking of a certain contrast.

Anyone who reflects on this will have to admit that we cannot imagine this without forming a certain concept of the opposition, namely the opposition of activity and an obstruction, a counter-activity. You have to introduce the concept of opposition here. The matter is still simple here. By entwining ourselves around geometric concepts, we will be able to construct the mental counter-images of more complicated things in an appropriate way. Then we will find the way and to some extent reach higher knowledge. But you can already imagine the colossal complexity that arises when you think of another body and look for its mental counter-image. Many complicated things come to light. And if you were to imagine another person and their mental counterpart, with all their spatial forms and their activity, you can imagine the complicated mental structure that this produces. In my book 'Theosophy', I was only able to give a rough idea of what mental counter-images look like.

We have three dimensions, three axes in the cube. On each axis we have the corresponding perpendicular planes on both sides. So you must now be clear about the fact that the contrast I have spoken of is to be understood in such a way that you imagine each face of the cube as having come into being in a way similar to the way I described human life earlier, as the intersection of two currents. You can imagine currents emanating from the center point. Imagine space in one axial direction, flowing outwards from the center, and in the other direction, flowing in from infinity, another current. And this [imagine] flowing in two colors, one red, the other blue. At the moment they meet, they will flow into a surface, a surface will arise, so that we can assume the surface of the cube to be the meeting point of two opposing currents in a surface. This gives a vivid idea of what a cube is.

The cube is therefore an intersection of three currents acting on each other. If you think about it, you are not dealing with three, but with six directions: forward-backward, up-down, right-left. So you have six directions. And indeed, that is the case. Then the matter becomes even more complicated by the fact that you have two types of currents: One in the direction of a point, the other coming from infinity. This will give you a perspective on the practical application of the higher, theoretical theosophy. I have conceived every direction in space as two opposing currents. And if you then imagine a physical body, then you have in that physical body the result of these two currents running into each other.

Let us now denote these six currents, these six directions, with six letters a, b, c, d, e, f. If you could visualize these six directions or currents — we will come to being able to do this next time — and you would imagine the first and last, a and f, erased, then you would be left with four. And that is what I now ask you to take into account: these four that remain are the four that you can perceive when you see the astral world alone.

I have tried to give you an idea of the three [ordinary dimensions] and of three [further] dimensions that actually behave in the opposite way. It is through the interaction of these dimensions and their counteraction that physical bodies arise. But if you think a little way away from the physical [dimension] and a little way away from the mental on the other side, you are left with four dimensions. These then represent the astral world existing between the physical and mental worlds.

The theosophist's view of the world is such that it necessarily has to work with a higher sense of geometry that goes beyond ordinary geometry. The ordinary geometer describes the cube as bounded by six squares. We must understand the cube as the result of six currents running into each other, that is, as the result of a movement and its reversal, of the interaction of opposing forces.

I would like to show you another such concept outside in nature, where a real contrast has taken place that contains a deep secret of the world's development before the eyes of man. In the “Fairytale of the Snake and the Lily”, Goethe speaks of the “revealed secret”, and that is one of the truest and wisest words that can be spoken at all. It is true, there are secrets in nature that can be grasped with hands, but are not seen by people. We are dealing with reversal processes in nature in many cases. I would like to show you one such reversal process.

Let us compare humans with plants. When compared to plants, humans behave as follows. What I am about to say is not a game, even if it initially seems like one. It is something that points to a deep mystery. What does a plant have in the ground? Its roots. And upwards it develops stems, leaves, flowers and fruit. The main part of the plant, the root, is in the earth, and the organs of reproduction it develops upwards, towards the sun, which we can call the chaste way of reproducing. Imagine the whole plant turned upside down, with the root becoming the head of a human being. Then you have the opposite of the plant in the human being, who has his head at the top and his reproductive organs at the bottom. And the animal is in the middle of it all, as a stowage. If you turn the plant upside down, you get a human being. That is why the occultists of all times draw this with three lines (Figure 24).

One [line] as the symbol of the plant, one as the [symbol] of the human being, and one in the opposite direction as the [symbol] of the animal – three lines that together form the cross. The animal has the transverse position, it thus crosses what we have in common with the plant.

You know that we speak of an all-soul, of which Plato says that it is crucified to the cosmic body, that it is chained to the cross of the cosmic body.? Imagine the world soul as plant, animal and human being, and you have the cross. By living in these three realms, the world soul is chained to this cross. As a result, you will find the concept of congestion expanded. You will find it expanded by something in nature. Two complementary, diverging, but interlocking currents form plants and humans, with congestion being the animal. Thus, the animal actually places itself between an upward and a downward current. In this way, the Kamaloka [astral sphere] interposes itself between Devachan and the physical world. Thus, something interposes itself between these two symmetrical worlds, between Devachan and the physical world, and acts between them, acting on both sides like a dam. The outer expression of this Kamaloka world is the animal world.

Those who already have organs for this world, which must be grasped with strength, will recognize what we see in the three kingdoms in their mutual relationship to one another. If you understand the animal kingdom as emerging from a congestion, if you understand the three kingdoms as mutual congestion, then you will find the position that the plant kingdom has to the animal kingdom and the animal kingdom to the human kingdom. The animal is perpendicular to the other two directions, and the other two are two complementary currents that merge into each other. The lower realm serves the higher realm as food. This is something that allows a small glimpse of the very different kind of relationship between humans and plants and between animals and humans. Those who feed on animals are therefore related to a congestion.

The real effect consists in the encounter of opposing currents. This is the beginning of a series of thoughts that you may later see in a strange and very different way.

We have seen that the square is created when two axes are intersected by lines. The cube is created by intersecting three axes through surfaces. Can you now imagine four axes intersecting through something? The cube is the boundary of the spatial structure that is created when four axes are intersected.

The square limits the three-dimensional cube. Next time, we will see whose boundary the cube itself is. The cube bounds a four-dimensional structure.

Answering Questions

[What does it mean to] imagine six currents, of which two must be imagined as having been extinguished, and so on?

The six currents must be thought of as two times three currents: three acting from the inside out according to the three axial directions, and the other three as flowing towards these from infinity. For each axial direction, there are thus two types, one going from the inside out, the other coming from the outside in the opposite direction. If we call the two categories positive and negative, plus and minus, we have:

math figur

And of this [in order to get to the astral space, we have to] imagine an entire direction, [for example the] inner and outer flow, erased, so for example +a and -a.

I recently tried to give you a schematic idea of four-dimensional space. But it would be very difficult if we were not able to form a picture of four-dimensional space in some kind of analogy. If it were a matter of characterizing our task, then it would be this: to show a four-dimensional structure here in three-dimensional space. Initially, we only have three-dimensional space at our disposal. If we want to link something unknown to us with something known, then, just as we have mapped a three-dimensional object into two dimensions, we have to bring a four-dimensional object into the third dimension. Now I would like to show, in the most popular way possible, using Mr. Hinton's method, how four-dimensional space can be mapped within three dimensions. So I would like to show how this task can be solved.

First, let me assume how to bring three-dimensional space into two-dimensional space. Our blackboard here is a two-dimensional space. If we were to add depth to height and width, we would have three-dimensional space. Now let's try to visualize a three-dimensional object on the blackboard.

A cube is a three-dimensional object because it has height, width and depth. Let's try to bring it into two-dimensional space, or onto a plane. If you take the whole cube and roll it up, or rather unroll it, you can do it like this. The sides, the six squares that we have in three-dimensional space, can be spread out once in a plane (Figure 25). So I could imagine the boundary surfaces of the cube spread out on a plane in a cross shape.

There are six squares that can be rearranged to form a cube again if I fold them back, so that squares 1 and 3, 2 and 4, and 5 and 6 are opposite each other. Thus we have a three-dimensional structure simply laid in the plane.

This is not a method that we can use directly to draw the fourth dimension in three-dimensional space. For that, we have to look for a different analogy. We have to use colors to help us. To do that, I will label the six squares along their sides with different colors. The squares facing each other [in the cube] should have the same colors when they are unfolded. I will draw the squares 1 and 3 so that one side is red [dotted lines] and the other is blue [solid lines]. Now I will complete these squares so that I keep blue for the whole horizontal direction (Figure 26). So I will draw all the vertical sides of these squares in red and all the horizontal sides in blue.

If you look at these two squares, 1 and 3, you have the two dimensions that the squares have, expressed in two colors, red and blue. So here for us [at the vertical blackboard, where square 2 is “stuck” to the blackboard], red would mean height and blue depth.

Let us now keep in mind that we apply red wherever height occurs and blue wherever depth occurs; and then we want to take green [dashed line] for the third dimension, width. Now we want to complete the unfolded cube in this way. The square 5 has sides that are blue and green, so the square 6 must look the same. Now only the squares 2 and 4 remain, and if you imagine them unfolded, it follows that the sides will be red and green.

Now, if you imagine it, you will see that we have transformed the three dimensions into three colors. We now say red [dotted], green [dashed], and blue [(solid line)] for height, width, and depth. We name the three colors that are to be images for us instead of the three spatial dimensions. If you imagine the whole cube opened up, you can explain the third dimension in two dimensions in such a way as if, for example, you had let the blue-red square [from left to right in Figure 26] march through green. We want to say that red and blue passed through green. We will describe the marching through green, the disappearance into the third color dimension, as the passage through the third dimension. So, if you imagine that the green fog colors the red-blue square, both sides – red and blue – will appear colored. Blue will take on a blue-green hue and red a cloudy shade, and only where the green stops will both appear in their own color again. I could do the same with squares 2 and 4. So I let the red-green square move through a space that is blue, and then you can do the same with the other two squares, 5 and 6, where the blue-green square would have to pass through the red. In this way, you let each square disappear on one side, submerging it in a different color. It takes on a different color itself through this third color, until it emerges on the other side in its original state.

We thus have an allegorical representation of our cube using three perpendicular colors. We have simply used three colors to represent the three directions we are dealing with here. If we want to imagine the changes that the three pairs of squares have undergone, we can do so by imagining that the squares pass through green the first time, red the second time, and blue the third time.

Now imagine squares instead of these [colored] lines, and squares everywhere for the bare space. Then I can draw the whole figure differently (Figure 27). We draw the transit square blue, and the two that pass through it – before and after the transit – we draw them above and below, here in red-green. [In a second step] I take the red square as the one that allows the blue-green squares to pass through it. And [in a third step] we have the green square here. The two corresponding other colors, red and blue, pass through the green square.

You see, now I have shown you another form of propagation with nine adjacent squares, but only six of which are on the cube itself, namely the squares drawn at the top and bottom of the figure (Figure 27). The other three [middle] squares are transition squares that denote nothing more than the disappearance of the individual colors into a third [color]. [For the transition movement, we] therefore always have to take two dimensions together, because each of these squares [in the upper and lower rows] is composed of two colors and disappears into the color that it does not contain itself. To make these squares reappear on the other side, we let them disappear into the third color. Red and blue disappear into green, red and green have no blue, so they disappear into blue [and green and blue disappear into red].

So, you see, we have the option here of assembling our cube using squares from two color dimensions that pass through the third color dimension.

Now it stands to reason that we imagine cubes instead of squares, and in doing so we put the cubes together out of three color dimensions – just as we put the square together out of two lines of different colors – so that we have three colors, according to the three dimensions of space. If we now want to do the same as we did with the square, we have to add a fourth color. This will allow us to make the cube disappear as well, of course only through a color that it does not have itself. Instead of the three pass squares, we now have four pass cubes in four colors: blue, white, green, and red. So instead of the pass square, we have the pass cube. Mr. Schouten has now produced these colored cubes in his models.

Now, just as we have a square pass through another that is not its color, we must now let a cube pass through another that is not its color. So we let the white-red-green cube pass through a blue one. It will submerge into the fourth color on one side and reappear in its [original] colors on the other side (Figure 28.1).

So here we have a [color] dimension bounded by two cubes that have three colored faces. In the same way, we now have to let the green-blue-red cube pass through the white cube (Figure 28.2), and then let the blue-white-red cube pass through the green (Figure 28.3). In the last figure (Figure 28.4), we have a blue-green-white cube that has to pass through a red dimension, that is, it has to disappear into a color that it does not itself have, in order to reappear on the other side in its very own colors.

These four cubes behave exactly like our three squares did before. If you now realize that we need six squares to bound a cube, we need eight cubes to bound a four-dimensional object, the tessaract.

The purpose of this substitution of dimensions with colors is that, as long as we stick with the [three] dimensions, we cannot bring the three dimensions into the [two-dimensional] plane. But if we use three colors instead, we can do it. We do the same with four dimensions if we want to visualize them using [four] colors in three-dimensional space. This is one way in which I would like to introduce you to these otherwise complicated things, and how Hinton used them in his problem [of the three-dimensional representation of four-dimensional structures].

I would now like to spread out the cube in the plane again, to turn it over into the plane once more. I will draw this on the board. First, disregard the bottom square [of Figure 25] and imagine that you can only see two-dimensionally, so you can only see what is spread out on the surface of the board. If we put five squares together as in this case, so that they are arranged in such a way that the one square comes into the middle, this inner area remains invisible (Figure 29). You can go around it from all sides. You cannot see square 5 because you can only see in two dimensions.

Now let us do the same thing that we have done here with five of the six side squares of the cube with seven of the eight boundary cubes that form the tessaract when we spread our four-dimensional structure into space. I will lay out the seven cubes in the same way as I did with the faces of the cube on the board; only now we have cubes where we previously had squares. Now we have here the corresponding spatial figure, formed entirely analogously. Thus we have the same for three-dimensional space as we previously had for two-dimensional surface. Just as a square is completely hidden from all sides, so is the seventh cube, which a being that has [only] the ability to see three-dimensionally will never be able to see (Figure 30). If we could fold up these figures in the same way as the six unfolded squares of the cube, we could pass from the third into the fourth dimension. We have shown how one can form an idea of this by means of color transitions."

With this, we have at least shown how, despite the fact that humans can only perceive three dimensions, we can still imagine four-dimensional space. Now you might still wonder how one can gain a possible conception of the real four-dimensional space. And here I would like to point you to something that is called the actual “alchemical secret.” For the real insight into four-dimensional space is in some way connected with what the alchemists called “transformation”.

[First variant:] He who wishes to acquire a true intuitive grasp of four-dimensional space must perform very definite exercises in intuitive grasp. These consist in his first forming a very clear intuitive perception, a deepened intuitive perception, not an imagination, of what is called water. Such an intuitive perception of water is not so easy to come by. One must meditate for a long time and delve very deeply into the nature of water; one must, so to speak, creep into the nature of water. The second thing is to gain an insight into the nature of light. Man is familiar with light, but only in the sense that he receives it from outside. Now, through meditation, man comes to receive the inner counter-image of outer light, to know where and from what light arises, so that he can himself bring forth and generate something like light. The yogi acquires this ability to produce and generate light through meditation. This is possible for the person who is able to have pure concepts truly meditatively present in his soul, who truly allows pure concepts to have a meditative effect on his soul, who is able to think free of sensuality. Then the light arises from the concept. Then the whole environment opens up to him as flooding light. The secret disciple must now, as it were, chemically combine the conception he has formed of water with the conception of light. The water, completely permeated by light, is a body called by the alchemists Mercury. Water plus light is called Mercury in the language of the alchemists. But this alchemical Mercury is not ordinary mercury. You will not have received the matter in this form. One must first awaken within oneself the ability to generate the light from the [dealing with the pure] concepts. Mercury is this mixture [of light] with the contemplation of water, this light-imbued water power, in whose possession one then puts oneself. That is one element of the astral world.

The second [element] arises from the fact that, just as one has formed an idea of water, one forms an idea of air, that we therefore suck out the power of the air through a mental process. If you concentrate your feeling in a certain way, you create a fire through feeling. If you combine the power of the air chemically with the fire created by feeling, you get “fire air.” You know that Goethe's Faust speaks of fire air.” This is something in which the inner being of the person must participate. So one element is sucked out of a given element, the air, and the other [fire or warmth] is generated by yourself. This air plus fire was called sulfur, sulphur, luminous fire-air by the alchemists.

If you now have this luminous fire air in an aqueous element, then you truly have that [astral] matter of which it says in the Bible: “And the Spirit of God hovered, or brooded, over the ‘waters’.”

[The third element arises when] you draw the power from the earth and then connect it with the [spiritual forces in the] “sound”; then you have what is called the Spirit of God [here]. Therefore, it is also called “thunder”. [The acting] Spirit of God is thunder, is earth plus sound. The Spirit of God [thus hovers over the] astral matter.

Those “waters” are not ordinary water, but what is actually called astral matter. This consists of four types of forces: water, air, light and fire. The arrangement of these four forces presents itself to the astral view as the four dimensions of astral space. That is how they are in reality. It looks quite different in the astral than in our world, some things that are perceived as astral are only a projection of the astral into physical space.

You see, that which is astral is half subjective [that is, passively given to the subject], half water and air, because light and feeling [fire] are objective, [that is, actively brought to appearance by the subject]. Only part of what is astral can be found outside [given to the subject] and obtained from the environment. The other part must be brought about subjectively [through one's own activity]. Through conceptual and emotional powers, one gains the other [from the given] through [active] objectification. In the astral, we thus have subjective-objective elements.

In devachan, there is no longer any objectivity [that is merely given to the subject]. One would have a completely subjective element there.

When we speak of the astral realm, we have something that the human being must first create [out of himself]. So everything we do here is symbolic, an allegorical representation of the higher worlds, of the devachanic world, which are real in the way I have explained to you in these suggestions. What lies in these higher worlds can only be attained by developing new possibilities of perception within oneself. Man must do something himself for this.

[Second text variant (Vegelahn):] Those who want to acquire a real view of four-dimensional space must do very specific visual exercises. First of all, they form a very clear, in-depth view of water. Such a view is not easy to come by; one has to delve very deeply into the nature of water; one has to, so to speak, get into the water. The second thing is to gain an insight into the nature of light. Light is something that man knows, but only in the sense that he receives it from outside. Through meditation, he can gain an inner image of light, know where light comes from and therefore produce light himself. This can be done by someone who allows pure concepts to have a real meditative effect on his soul, who has a thinking free of sensuality. Then the whole of his environment will reveal itself to him as flooding light, and now he must, as it were chemically, combine the idea he has formed of water with that of light. This water, completely permeated by light, is a body that was called “Mercury” by the alchemists. But the alchemical Mercury is not the ordinary mercury. First you have to awaken within yourself the ability to generate Merkurius from the concept of light. Merkurius, light-imbued water power, is what you then place yourself in possession of. That is the one element of the astral world.

The second is created by you also forming a vivid mental image of air, then sucking out the power of the air through a spiritual process, connecting it with feeling, and you ignite the concept of “warmth”, “fire”, then you get “fire air”. So one element is sucked out, the other is produced by yourself. This - air and fire - the alchemists called “sulfur”, sulfur, luminous fire air. In the aqueous element, there you have in truth that matter of which it is said: “and the Spirit of God hovered over the waters”.

The third element is “spirit-God”, which is connected to “earth” and “sound”. This is what happens when you extract the earth's forces and combine them with sound. These “waters” are not ordinary water, but what is actually called astral matter. This consists of four types of forces: water, air, light and fire. And this manifests itself as the four dimensions of astral space.

You see, that which is astral is half subjective; only part of what is astral can be gained from the environment; from conceptual and emotional powers, one gains the other through objectification. In devachan, you would have a completely subjective element; there is no objectivity there. So everything we do here, the symbolic, is an allegorical representation of the devachanic world. Everything that lies in the higher worlds can only be attained by developing new views within yourself. Man must do something about it himself.

Last time, we tried to get an idea of a four-dimensional space. To visualize it, we reduced it to a three-dimensional one. First, we started by transforming a three-dimensional space into a two-dimensional one. We used colors instead of dimensions. We formed the idea in such a way that a cube appeared in three colors along the three dimensions. Then we laid the boundaries of a cube on the plane, which resulted in six squares in different colors. Through the diversity of colors on the individual sides, we obtained the three different dimensions in two-dimensional space. We had three colors, and with that we had represented the three dimensions.

We then imagined that we were passing a square cube into the third dimension, as if we were passing it through a colored fog and it reappeared on the other side. We imagined that we had pass squares, so that the square cubes move through these squares and are thereby tinged [with the color of the pass square]. This is how we tried to imagine the [three-dimensional] cube [by means of a two-dimensional color representation]. [For the one-dimensional representation of the] surfaces, we thus have two boundary colors and [for the two-dimensional representation of the] cube, three colors. [To represent a four-dimensional spatial structure in three-dimensional space, we must] then add a fourth boundary color.

Now we have to imagine in the same way that a cube, which, analogous to our square, has two different colors as boundary sides, has three different colors in its boundary surfaces. And finally, each cube moves through another cube that has the corresponding fourth color. In doing so, we let it disappear into the fourth color dimension. So, according to Hinton's analogy, we let the respective boundary cubes pass through the new [fourth] color, which then reappears on the other side, emerging in their [original] own color.

Now I will give you another analogy and first reduce the three dimensions back to two, so that we will then be able to reduce four dimensions to three. To do this, we have to imagine the following. The cube can be put together at its boundary surfaces from its six boundary squares; but instead of doing it in succession, as we did recently, it will now be done in a different way. I will also draw this figure (Figure 31). You see, we have now spread out the cube in two systems, each of which lies in the plane and consists of three squares. Now we have to be clear about how these different areas will lie when we actually put the cube together. I ask you to consider the following. If I now want to reassemble the cube from these six squares, I have to place the two sections on top of each other so that square 6 comes to rest on square 5. When square 5 is placed at the bottom, I have to fold up squares 1 and 2, while folding down squares 3 and 4 (Figure 32). In doing so, we get certain corresponding lines that overlap. The lines marked in the figure with the same color [here in the same line quality and in the same number of lines] will coincide. What lies here in the plane, in two-dimensional space, coincides to a certain extent when I move into three-dimensional space.

The square consists of four sides, the cube of six squares, and the four-dimensional area would then have to consist of eight cubes.? We call this four-dimensional area a tessaract [after Hinton]. Now, the point is that these eight cubes cannot simply be reassembled into a cube, but that one of them should always pass through the fourth dimension in the appropriate way.

If I now want to do the same with the tessaract as I just did with the cube, I have to follow the same law. The point is to find analogies of the three-dimensional to the two-dimensional and then of the four-dimensional to the three-dimensional. Just as I obtained two systems of [three squares each] here, the same thing happens with the tessaract with [two systems of four cubes each] when I fold a four-dimensional tessaract into three-dimensional space. The system of eight cubes is very ingeniously devised. This structure will then look like this (Figure 33).

Each time, these four cubes in three-dimensional space are to be taken exactly as these squares in two-dimensional space.

You just have to look carefully at what I have done here. When the cube was folded into two-dimensional space, a system of six squares resulted; when the corresponding procedure is carried out on the tessaract, we obtain a system of eight cubes (Figure 34). We have transferred the observation from three-dimensional space to four-dimensional space. [Folding up and joining the squares in three-dimensional space corresponds to folding up and joining the cubes in four-dimensional space.] In the case of the folded-down cube, [in the two-dimensional plane] different corresponding lines were obtained, which coincided when it was folded up again later. The same occurs with the surfaces of our individual cubes of the tessaract. [When the tessaract is folded down in three-dimensional space, corresponding surfaces appear on the corresponding cubes.] So, for example, in the case of the tessaract, the upper horizontal surface of

cube 1—by observing [mediation] the fourth dimension—with the front face of cube 5.

In the same way, the right face of cube 1 coincides with the front square of cube 4, and likewise the left square of cube 1 with the front square of cube 3 [as well as the lower square of cube 1 with the front square of cube 6]. The same applies to the other cube surfaces. The remaining cube, 7, is enclosed by the other six.

You see that here again we are concerned with finding analogies between the third and fourth dimensions. Just as a fifth square enclosed by four squares remains invisible to the being that can only see in two dimensions, as we saw in the corresponding figure of the previous lecture (Figure 29), so it is the case here with the seventh cube: it remains hidden from the three-dimensional eye. Corresponding to this seventh cube in the tessaract is an eighth cube, which, since we have a four-dimensional body here, lies as a counterpart to the seventh in the fourth dimension.

All analogies lead us to prepare for the fourth dimension. Nothing forces us to add the other dimensions to the usual dimensions [within the mere spatial view]. Following Hinton, we could also think of colors here and think of cubes put together in such a way that the corresponding colors come together. It is hardly possible in any other way [than by such analogies] to give a description of how to think of a four-dimensional entity.

Now I would like to mention another way [of representing four-dimensional bodies in three-dimensional space], which may also give you a better understanding of what we are actually dealing with here. This is an octahedron bounded by eight triangles, with the sides meeting at obtuse angles (Figure 35).

If you visualize this structure here, I ask you to follow the following procedure with me in your mind. You see, here one surface is always intersected by another. Here, for example, in AB, two side surfaces meet, and here in EB, two meet. The entire difference between an octahedron and a cube lies in the angle of intersection of the side surfaces. If surfaces intersect as they do in a cube [at right angles], a cube is formed. But if they intersect as they do here [obtuse], then an octahedron is formed. The point is that we can have surfaces intersect at the most diverse angles, and then we get the most diverse spatial structures."

Now imagine that we could also make the same faces of the octahedron intersect in a different way. Imagine this face here, for example AEB, continued on all sides, and this lower one here, BCF, also (Figure 36). Then likewise the ADF and EDC lying backwards. Then these faces must also intersect, and in fact they intersect here in a doubly symmetrical way. If you extend these surfaces in this way, [four of the original boundary surfaces] are no longer needed: ABF, EBC and, towards the back, EAD and DCF. So of the eight surfaces, four remain. And the four that remain give this tetrahedron, which is also called half of an octahedron. It is therefore half of an octahedron because it intersects half of the faces of the octahedron. It is not the case that you cut the octahedron in half. If you bring the other four faces of the octahedron to the cut, the result is also a tetrahedron, which together with the first tetrahedron has the octahedron as a common intersection. In stereometry [geometric crystallography], it is not the part that is halved that is called the half, but the one that is created by halving the [number of] faces. With the octahedron, this is quite easy to imagine.

If you imagine halving the cube in the same way, that is, if you allow one face to intersect with the corresponding other face, you will always get a cube. Half of a cube is a cube again. I would like to draw an important conclusion from this, but first I would like to use something else to help me.

Here I have a rhombic dodecahedron (Figure 37). You can see that the surfaces adjoin each other at certain angles. At the same time, we can see a system of four wires, which I would like to call axial wires, and which run in opposite directions to each other [i.e. connect certain opposite corners of the rhombic dodecahedron, and are therefore diagonals]. These wires now represent a system of axes in a similar way to the way in which you imagined a system of axes on the cube.

You get the cube when you create sections in a system of three perpendicular axes by introducing blockages in each of these axes.

If the axes are made to intersect at other angles, a different spatial figure is obtained. The rhombic dodecahedron has axes which intersect at angles other than right angles.

The cube reflects itself in half.

But this applies only to the cube. The rhombic dodecahedron, cut in half, also gives a different spatial structure.

Now let us take the relation of the octahedron to the tetrahedron. And I will tell you what is meant by this. This becomes clear when we gradually let the octahedron merge into the tetrahedron. For this purpose, let us take a tetrahedron, which we cut off at one vertex (Figure 38). We continue this process until the cut surfaces meet at the edges of the tetrahedron; then what remains is the indicated octahedron. In this way we obtain an eight-sided figure from a three-dimensional figure bounded by four surfaces, provided we cut off the corners at corresponding angles.

What I have done here with the tetrahedron, you cannot do with the cube.

The cube has very special properties, namely that it is the counterpart of three-dimensional space. Imagine the entire universe structured in such a way that it has three perpendicular axes. If you then imagine surfaces perpendicular to these three axes, you will, under all circumstances, get a cube (Figure 39). That is why, when we speak of the cube, we mean the theoretical cube, which is the counterpart of three-dimensional space. Just as the tetrahedron is the counterpart of the octahedron when I make the sides of the octahedron into certain sections, so the single cube is the counterpart of the whole of space.” If you think of the whole of space as positive, the cube is negative. The cube is the polar opposite of the whole of space. Space has in the physical cube its actually corresponding structure.

Now suppose I would not limit the [three-dimensional] space by two-dimensional planes, but I would limit it in such a way that I would have it limited by six spheres [thus by three-dimensional figures].

I first define two-dimensional space by having four circles that go inside each other [i.e., two-dimensional shapes]. You can now imagine that these four circles are getting bigger and bigger [as the radius gets longer and longer and the center point moves further and further away]; then, over time, they will all merge into a straight line (Figure 40). You then get four intersecting lines, and instead of the four circles, a square.

Now imagine that the circles are spheres, and that there are six of them, forming a kind of mulberry (Figure 41). If you imagine the spheres in the same way as the circles, that they get larger and larger in diameter, then these six spheres will ultimately become the boundary surfaces of a cube, just as the four circles became the boundary lines of a square.

The cube has now been created from the fact that we had six spheres that have become flat. So the cube is nothing more than a special case of six interlocking spheres – just as the square is nothing more than a special case of four interlocking circles.

If you are clear in your mind about how to imagine these six spheres, that they correspond to our earlier squares when brought into the plane, and if you imagine an absolutely round shape passing into a straight one, you will get the simplest spatial form. The cube can be imagined as the flattening of six spheres pushed into each other.

You can say of a point on a circle that it must pass through the second dimension if it is to come to another point on the circle. But if you have made the circle so large that it forms a straight line, then every point on the circle can come to every other point on the circle through the first dimension.

We are considering a square bounded by figures, each of which has two dimensions. As long as each of the four boundary figures is a circle, it is therefore two-dimensional. Each boundary figure, when it has become a straight line, is one-dimensional.

Each boundary surface of a cube is formed from a three-dimensional structure in such a way that each of the six boundary spheres has one dimension removed. Such a boundary surface has therefore been created by the third dimension being reduced to two, so to speak bent back. It has therefore lost a dimension. The second dimension was created by losing the dimension of depth. One could therefore imagine that each spatial dimension was created by losing a corresponding higher dimension.

Just as we obtain a three-dimensional figure with two-dimensional boundaries when we reduce three-dimensional boundary figures to two-dimensional ones, so you must conclude that when we look at three-dimensional space, we have to think of each direction as being flattened out, and indeed flattened out from an infinite circle; so that if you could progress in one direction, you would come back from the other. Thus, each [ordinary] spatial dimension has come about through the loss of the corresponding other [dimension]. In our three-dimensional space, there is a three-axis system. These are three perpendicular axes that have lost the corresponding other dimensions and have thus become flat.

So you get three-dimensional space when you straighten each of the [three] axis directions. If you proceed in reverse, each spatial part could become curved again. Then the following series of thoughts would arise: If you curve the one-dimensional structure, you get a two-dimensional one; by curving the two-dimensional structure, you get a three-dimensional one. If you finally curve a three-dimensional structure, you get a four-dimensional structure, so that the four-dimensional can also be imagined as a three-dimensional structure curved on itself.*

And with that, I come from the dead to the living. Through this bending, you can find the transition from the dead to the living. Four-dimensional space is so specialized [at the transition into three dimensions] that it has become flat. Death is [for human consciousness] nothing more than the bending of the three-dimensional into the four-dimensional. [For the physical body taken by itself, it is the other way around: death is a flattening of the four-dimensional into the three-dimensional.]

I would like to conclude the lectures on the fourth spatial dimension today if possible, although I would like to demonstrate a complicated system in more detail today. I would have to show you many more models after Hinton; therefore, I can only refer you to the three detailed and spirited books.” Those who do not have the will to form a picture through analogies in the way we have heard it in the past lectures cannot, of course, form a picture of four-dimensional space. It involves a new way of forming thoughts.

I will try to give you a true representation [parallel projection] of the tessaract. You know that in two-dimensional space we had the square, which is bounded by four sides. This is the three-dimensional cube, which is bounded by six squares (Figure 42).

In four-dimensional space, we have the tessaract. A tessaract is bounded by eight cubes. The projection of a tessaract [in three-dimensional space] therefore consists of eight interlocking cubes. We have seen how the [corresponding eight] cubes can be intertwined in three-dimensional space. Today I will show you a [different] way of projecting the tessaract.

You can imagine that the cube, when held up to the light, throws a shadow on the blackboard. We can mark this shadow figure with chalk (Figure 43). You see that a hexagon is obtained. Now imagine this cube transparent, and you will observe that in the hexagonal figure the three front sides of the cube and the three rear sides of the cube fall into the same plane.

In order to get a projection that we can apply to the tessaract, I would ask you to imagine that the cube is standing in front of you in such a way that the front point A covers the rear point C. If you imagine the third dimension, all this would give you a hexagonal shadow again. I will draw the figure for you (Figure 44).

If you imagine the cube like this, you would see the three front surfaces here; the other surfaces would be behind them. The surfaces of the cube appear foreshortened and the angles are no longer right angles. This is how you see the cube depicted so that the surfaces form a regular hexagon. Thus, we have obtained a representation of a three-dimensional cube in two-dimensional space. Since the edges are shortened and the angles are changed by the projection, we must therefore imagine the [projection of the] six boundary squares of the cube as shifted squares, as rhombi.

The same story that I did with a three-dimensional cube that I projected into the plane, we want to do this procedure with a four-dimensional spatial object, which we therefore have to place in three-dimensional space. We must therefore bring the structure composed of eight cubes, the tessaract, into the third dimension [by parallel projection]. With the cube, we obtained three visible and three invisible edges, all of which enter into the space and in reality do not lie within the [projection] surface. Now imagine a cube shifted in such a way that it becomes a rhombicuboctahedron.” Take eight of these figures, and you have the possibility of combining the eight [boundary] cubes of the tessaract in such a way that, when pushed together, they form the eight (doubly covered) rhombicuboctahedra of this spatial figure (Figure 45).

Now you have one more axis here [than in the three-dimensional cube]. Accordingly, a four-dimensional spatial structure naturally has four axes. So if we push it together, four axes still remain. There are eight [pushed together] cubes in this projection, which are represented as rhombicuboctahedra. The rhombicuboctahedron is a [symmetrical] image or silhouette of the tessaract in three-dimensional space.

We arrived at this relationship by means of an analogy, but it is completely correct: just as we obtained a projection of the cube onto a plane, it is also possible to represent the tessaract in three-dimensional space by means of a projection. It behaves in the same way as the silhouette of the cube in relation to the cube itself. I think that is quite easy to understand.

Now I would like to tie in with the greatest image that has ever been given for this, namely Plato and Schopenhauer and the parable of the cave.

Plato says: Imagine people sitting in a cave, and they are all tied up so that they cannot turn their heads and can only look at the opposite wall. Behind them are people carrying various objects past them. These people and these objects are three-dimensional. So all these [bound] people stare at the wall and see only what is cast as a shadow [of the objects] on the wall. So they would see everything in the room only as a shadow on the opposite wall as two-dimensional images.

Plato says that this is how it is in the world in general. In truth, people are sitting in the cave. Now, people themselves and everything else are four-dimensional; but what people see of it are only images in three-dimensional space.

This is how all the things we see present themselves. According to Plato, we are dependent on seeing not the real things, but the three-dimensional silhouettes. I only see my hand as a silhouette; in reality it is four-dimensional, and everything that people see of it is just as much an image of it as what I just showed you as an image of the Tessaract. Thus Plato was already trying to make clear that the objects we know are actually four-dimensional, and that we only see silhouettes of them in three-dimensional space. And that is not entirely arbitrary. I will give you the reasons for this in a moment.

Of course, anyone can say from the outset that this is mere speculation. How can we even imagine that the things that appear on the wall have a reality? Imagine that you are sitting here in a row, and you are sitting very still. Now imagine that the things on the wall suddenly start to move. You will not be able to tell yourself that the images on the wall can move without going out of the second dimension. If something moves there, it indicates that something must have happened outside the wall, on the real object, for it to move at all. That's what you tell yourself. If you imagine that the objects in three-dimensional space can pass each other, this would not be possible with their two-dimensional silhouettes, if you think of them as substantial, that is, impenetrable. If those images, conceived substantially, wanted to move past each other, they would have to go out of the second dimension.

As long as everything on the wall is at rest, I have no reason to conclude that something is happening outside the wall, outside the space of the two-dimensional silhouettes. But the moment history begins to move, I must investigate the source of the motion. And you realize that the change can only come from motion outside the wall, only from motion within a third dimension. The change has thus told us that there is a third dimension in addition to the second.

What is a mere image also has a certain reality, possesses very definite properties, but differs essentially from the real object. You will not be able to deny that the mirror image is also a mere image. You see yourself in the mirror, and you are also there. If there is not a third [that is, an active being] there, then you could not actually know what you are. But the mirror image makes the same movements that the original makes; the image is dependent on the real object, the being; it itself has no ability [to move]. Thus, a distinction can be made between image and being in that only a being can bring about movement and change out of itself. I realize from the shadows on the wall that they cannot move themselves, so they cannot be beings. I have to go out of them if I want to get to the beings.

Now apply this to the world in general. The world is three-dimensional. Take this three-dimensional world for itself, as it is; grasp it completely in your thoughts [for yourself], and you will find that it remains rigid. It remains three-dimensional even if you suddenly think the world frozen at a certain point in time. But there is no one and the same world in two points in time. The world is completely different at successive points in time. Imagine that these points in time cease to exist, so that what is there remains. Without time, no change would occur in the world. The world would remain three-dimensional even if it underwent no change at all. The pictures on the wall also remain two-dimensional. But change suggests a third dimension. The fact that the world is constantly changing, and that it remains three-dimensional even without change, suggests that we have to look for the change in a fourth dimension. We have to look for the reason, the cause of the change, the activity outside the third dimension, and with that you have initially uncovered the fourth of the dimensions. But with that you also have the justification for Plato's image. So we understand the whole three-dimensional world as the shadow projection of a four-dimensional world. The only question is how we have to take this fourth dimension [in reality].

You see, we have the one idea to make it clear to ourselves, of course, that it is impossible for the fourth dimension to fall [directly] into the third. That is not possible. The fourth dimension cannot fall into the third. I would like to show you now how one can, so to speak, get an idea of how to go beyond the third dimension. Imagine we have a circle – I have already tried to evoke a similar idea recently – if you imagine this circle getting bigger and bigger, then a piece of this circle becomes flatter and flatter, and because the diameter of the circle becomes very large at the end, the circle finally turns into a straight line. The line has one dimension, but the circle has two dimensions. How do you get a second dimension from a single dimension? By curving a straight line, you get a circle again.

If you now imagine the surface of the circle curving into space, you first get a shell, and if you continue to do this, you get a sphere. Thus a line acquires a second dimension by curvature and a surface acquires a third dimension by curvature. If you could now curve a cube, it would have to be curved into the fourth dimension, and you would have the [spherical] tessaract.

You can understand the sphere as a curved two-dimensional spatial structure. The sphere that occurs in nature is the cell, the smallest living thing. The cell is limited spherically. That is the difference between the living and the lifeless. The mineral always occurs as a crystal bounded by flat surfaces; life is bounded by spherical surfaces, built up of cells. That means that just as a crystal is built from spheres that have been straightened out, that is, from planes, so life is built from cells, that is, from spheres that have been bent together. The difference between the living and the dead lies in the way they are defined. The octahedron is defined by eight triangles. If we imagine the eight sides as spheres, we would get an eight-limbed living thing.

If you curve the three-dimensional structure, the cube, again, you get a four-dimensional structure, the spherical tessaract. But if you curve the whole space, you get something that relates to three-dimensional space in the same way that a sphere relates to a plane.

Just as the cube, as a three-dimensional structure, is bounded by planes, so every crystal is bounded by planes. The essence of a crystal is the assembly of [flat] boundary planes. The essence of the living is the assembly of curved surfaces, of cells. The assembly of something even higher would be a structure whose individual boundaries would be four-dimensional. A three-dimensional structure is bounded by two-dimensional structures. A four-dimensional being, that is, a living being, is bounded by three-dimensional beings, by spheres and cells. A five-dimensional being is itself bounded by four-dimensional beings, by spherical tessaracts. From this you can see that we have to ascend from three-dimensional to four-dimensional, and then to five-dimensional beings.

We only have to ask ourselves: What must occur in a being that is four-dimensional?* A change must occur within the third dimension. In other words: If you hang pictures on the wall here, they are two-dimensional and generally remain static. But if you have pictures in which the second dimension moves and changes, then you must conclude that the cause of this movement can only lie outside the surface of the wall, that the third dimension of space thus indicates the change. If you find changes within the third spatial dimension itself, then you must conclude that a fourth dimension is involved, and this brings us to the beings that undergo a change within their three spatial dimensions.

It is not true that we have fully recognized a plant if we have only recognized it in its three dimensions. A plant is constantly changing, and this change is an essential, a higher characteristic of it. The cube remains; it only changes its shape when you smash it. A plant changes its shape itself, that is, there is something that is the cause of this change and that lies outside the third dimension and is an expression of the fourth dimension. What is that?

You see, if you have this cube and draw it, you would labor in vain if you wanted to draw it differently at different moments; it will always remain the same. If you draw the plant and compare the picture with your model after three weeks, it will have changed. So this analogy is completely accurate. Everything that lives points to something higher, where it has its true essence, and the expression of this higher is time. Time is the symptomatic expression, the appearance of liveliness [understood as the fourth dimension] in the three dimensions of physical space. In other words, all beings for whom time has an inner meaning are images of four-dimensional beings. This cube is still the same after three or six years. The lily bud changes. Because for it, time has a real meaning. Therefore, what we see in the lily is only the three-dimensional image of the four-dimensional lily being. So time is an image, a projection of the fourth dimension, the organic liveliness, into the three spatial dimensions of the physical world.

To understand how a following dimension relates to the preceding one, please imagine the following: a cube has three dimensions; when you visualize the third, you have to remember that it is perpendicular to the second, and the second is perpendicular to the first. The three dimensions are characterized by the fact that they are perpendicular to one another. But we can also imagine how the third dimension arises from the following [fourth dimension]. Imagine that you would change the cube by coloring the boundary surfaces and then changing these colors [in a certain way, as in Hinton's example]. Such a change can indeed be made, and it corresponds exactly to the change that a three-dimensional being undergoes when it passes into the fourth dimension, when it develops through time. If you cut a four-dimensional being at any point, you take away the fourth dimension, you destroy it. If you do that to a plant, you do exactly the same thing as if you were to make a cast of the plant, a plaster cast. You have captured that by destroying the fourth dimension, time. Then you get a three-dimensional object. If for any three-dimensional being the fourth dimension, time, has an essential significance, then it is a living being.

Now we enter the fifth dimension. You can say to yourself that you must again have a boundary that is perpendicular to the fourth dimension. We have seen that the fourth dimension is related to the third dimension in a similar way to the third dimension being related to the second. It is not immediately possible to visualize the fifth dimension in this way. But you can again create a rough idea by using an analogy. How does a dimension come into being in the first place? If you simply draw a line, you will never create another dimension by simply pushing the line in one direction. Only by imagining that you have two opposing directions of force, which then accumulate at a point, only by expressing the accumulation, do you have a new dimension. We must therefore be able to grasp the new dimension as a new line of accumulation [of two currents of force], and imagine the one dimension coming from the right one time and from the left the next, as positive and negative. So I understand a dimension [as a polar [stream of forces] within itself], so that it has a positive and a negative dimension [component], and the neutralization [of these polar force components] is the new dimension.

From there, we want to create an idea of the fifth dimension. We will have to imagine that the fourth dimension, which we have found expressed as time, behaves in a positive and negative way. Now take two beings for whom time has a meaning, and imagine two such beings colliding with each other. Then something must appear as a result, similar to what we have previously called an accumulation of [opposing] forces; and what arises as a result when two four-dimensional beings come into relation with each other is their fifth dimension. This fifth dimension arises as a result, as a consequence of an exchange [a neutralization of polar force effects], in that two living beings, through their mutual interaction, produce something that they do not have outside [in the three ordinary spatial dimensions together], nor do they have in [the fourth dimension,] time, but have completely outside these [previously discussed dimensions or] boundaries. This is what we call compassion [or feeling], by which one being knows another, thus the realization of the [spiritual and mental] inner being of another being. A being could never know anything about another being outside of time [and space] if you did not add a higher, fifth dimension, [i.e. enter the world of] sensation. Of course, here the sensation is only to be understood as a projection, as an expression [of the fifth dimension] in the physical world.

Developing the sixth dimension in the same way would be too difficult, so I will only indicate it. [If we tried to progress in this way, something could be developed as an expression of the sixth dimension that,] when placed in the three-dimensional physical world, is self-conscious.

Man, as a three-dimensional being, is one who shares his imagery with other three-dimensional beings. The plant, in addition, has the fourth dimension. For this reason, you will never find the ultimate essence of the plant within the three dimensions of space, but you would have to ascend from the plant to a fourth spatial dimension [to the astral sphere]. But if you wanted to grasp a being that has feeling, you would have to ascend to the fifth dimension [to the lower Devachan, to the Rupa sphere]; and if you wanted to grasp a being that has self-awareness, a human being, you would have to ascend to the sixth dimension [to the upper Devachan, to the Arupa sphere]. Thus, the human being as he stands before us in the present is indeed a six-dimensional being. That which is called feeling or compassion, or self-awareness, is a projection of the fifth or sixth dimension into ordinary three-dimensional space. Man extends into these spiritual spheres, albeit unconsciously for the most part; only there can he actually be experienced in the sense indicated last. This six-dimensional being can only come to an idea of even the higher worlds if it tries to get rid of the actual characteristics of the lower dimensions.

I can only hint at the reason why man considers the world to be only three-dimensional, namely because he is conditioned in his perception to see only a reflection of something higher in the world. When you look in a mirror, you also see only a reflection of yourself. Thus, the three dimensions of our physical space are indeed reflections, material copies of three higher, causally creative dimensions. Our material world therefore has its polar [spiritual] counter-image in the group of the three next higher dimensions, that is, in those of the fourth, fifth and sixth dimensions. And in a similar sense, the spiritual worlds that lie beyond this group of dimensions, which can only be sensed, are also polar to those of the fourth to sixth dimensions.

If you have water and you let the water freeze, the same substance is present in both cases; but in form they differ quite substantially. You can imagine a similar process for the three higher dimensions of man. If you think of man as a purely spiritual being, then you have to think of him as having only the three higher dimensions – self-awareness, feeling and time – and these three dimensions are reflected in the physical world in its three ordinary dimensions.

The yogi [secret student], if he wants to advance to a knowledge of the higher worlds, must gradually replace the mirror images with reality. For example, when he looks at a plant, he must get used to gradually substituting the higher dimensions for the lower ones. If he looks at a plant and is able to abstract from one spatial dimension in the case of a plant, to abstract from one spatial dimension and instead to imagine a corresponding one of the higher dimensions, in this case time, then he actually gets an idea of what a two-dimensional, moving being is. To make this being more than just an image, to make it correspond to reality, the yogi must do the following. If he disregards the third dimension and adds the fourth, he would only get something imaginary. However, the following mental image can help: when we make a cinematographic representation of a living being, we remove the third dimension from the original three-dimensional processes, but add the [dimension of] time through the sequence of images. If we then add sensation to this [moving] perception, we perform a procedure similar to what I described earlier as the bending of a three-dimensional structure into the fourth dimension. Through this process you then get a four-dimensional entity, but now one that has two of our spatial dimensions, but also two higher ones, namely time and sensation. Such beings do indeed exist, and these beings - and this brings me to a real conclusion to the whole consideration - I would like to tell you about.

Imagine two spatial dimensions, that is, a surface, and this surface endowed with motion. Now imagine a bent as a sensation, a sentient being that then pushes a two-dimensional surface in front of it. Such a being must act differently and be very different from a three-dimensional being in our space. This flat creature that we have constructed in this way is incomplete in one direction, completely open, and offers you a two-dimensional view; you cannot go around it, it comes towards you. This is a luminous creature, and the luminous creature is nothing other than the incompleteness in one direction.

Through such a being, the initiates then get to know other beings, which they describe as divine messengers approaching them in flames of fire. The description of Mount Sinai, where Moses received the Ten Commandments,® means nothing other than that a being could indeed approach him that, to his perception, had these dimensions. It appeared to him like a human being from whom the third spatial dimension had been removed; it appeared in sensation and in time.

These abstract images in the religious documents are not just external symbols, but powerful realities that man can get to know if he is able to appropriate what we have tried to make clear through analogies. The more you devote yourself diligently and energetically to such considerations of analogies, the more you really work on your mind, and the more these [considerations] work in us and trigger higher abilities. [This is roughly the case when dealing with] the analogy of the relationship of the cube to the hexagon and the tessaract to the rhombic dodecahedron. The latter represents a projection of the tessaract into the three-dimensional physical world. If you visualize these figures as living entities, if you allow the cube to grow out of the projection of the die – the hexagon – and likewise allow the tessaract itself to arise from the projection of the tessaract [the rhombic dodecahedron], then you create the possibility and the ability in your lower mental body to grasp what I have just described to you as a structure. And if, in other words, you have not only followed me but have gone through this procedure vividly, as the yogi does in an awakened state of consciousness, then you will notice that something will occur to you in your dreams that in reality is a four-dimensional entity, and then it is not much further to bring it over into the waking consciousness, and you can then see the fourth dimension in every four-dimensional being.

The astral sphere is the fourth dimension. Devachan to rupa is the fifth dimension. Devachan to arupa is the sixth dimension.

These three worlds, the physical, astral and celestial [devachan], comprise six dimensions. The even higher worlds are completely polar to these.

Mineral Plant Animal Human Arupa Self-consciousness Rupa Sensation Self-consciousness Astral plane Life Sensation Self-consciousness Physical form Life Sensation Self-plan consciousness Form Life Sensation Form Life Form

The subject we are to discuss today will present us with a number of difficulties. Consider the lecture as an episode; it is being held at your request. If you only want to grasp the subject formally in its depth, some mathematical knowledge is necessary. But if you want to grasp it in its reality, you have to penetrate very deeply into occultism. So today we can only talk about it very superficially, only give a suggestion for this or that.

It is very difficult to talk about multidimensionality at all, because if you want to get an idea of what more than three dimensions are, you have to delve into abstract areas, and there the concepts must be very precisely and strictly defined, otherwise you end up in a bottomless pit. And that's where many friends and enemies have ended up.

The concept of multidimensional space is not as foreign to the world of mathematicians as one might think.® In mathematical circles, there is already a way of calculating with a multidimensional type of calculation. Of course, the mathematician can only speak of this space in a very limited sense; he can only discuss the possibility. Whether it really is can only be determined by someone who can see into a multidimensional space. Here we are already dealing with a lot of concepts that, if we grasp them precisely, really provide us with clarity about the concept of space.

What is space? We usually say: there is space around me, I walk around in space — and so on. If you want a clearer idea, you have to go into some abstractions. We call the space in which we move three-dimensional. It has an extension in height and depth, to the right and left, to the front and back, it has length, width and height. When we look at bodies, these bodies are extended for us in this three-dimensional space; they have a certain length, a certain width and height for us.

However, we have to deal with the details of the concept of space if we want to arrive at a more precise concept. Let us look at the simplest body, the cube. It shows us most clearly what length, width and height are. We find a base of the cube that is the same in length and width. If we move the base up, just as far as the base is wide and long, we get the cube, which is therefore a three-dimensional object. The cube is the clearest way for us to learn about the details of a three-dimensional object. We examine the boundaries of the cube. These are formed everywhere by surfaces bounded by sides of equal length. There are six such surfaces.

What is a surface? Those who are not capable of very sharp abstractions will already falter here. For example, you cannot cut the boundaries of a wax cube as a fine layer of wax. You would still get a layer of a certain thickness, so you would get a body. We will never get to the boundary of the cube this way. The real boundary has only length and width, no height. Thickness is eliminated. We thus arrive at the formulaic sentence: The area is the boundary [of a three-dimensional object] in which one dimension is eliminated.

What then is the boundary of a surface, for example of a square? Here we must again take the most extreme abstraction. [The boundary of a surface] is a line that has only one dimension, length. The width is canceled. What is the boundary of a line? It is the point, which has no dimension at all. So you always get the boundary of a thing by leaving out a dimension.

So you could say to yourself, and this is also the line of thought that many mathematicians have followed, especially Riemann,* who has achieved the most solid work here: We take the point, which has none, the line, which has one, the plane, which has two, the solid, which has three dimensions. Now mathematicians asked themselves: Could it not be that formally one could say that one could add a fourth dimension? Then the [three-dimensional] body would have to be the boundary of the four-dimensional object, just as the surface is the boundary of the body, the line is the boundary of the surface, and the point is the boundary of the line. Of course, the mathematician then goes even further to five-, six- and seven-dimensional objects and so on. We have [even arbitrary] “-dimensional objects [where ” is a positive integer].

Now, there is already some ambiguity in the matter when we say: the point has none, the line has one, the plane two, the solid three dimensions. We can now make such a solid, for example a cube, out of wax, silver, gold and so on. They are different in terms of matter. We make them the same size, then they all occupy the same space. If we now eliminate all material, only a certain part of space remains, which is the spatial image of the body. These parts of space are the same [among themselves], regardless of what material the cube was made of. These parts of space also have length, width and height. We can now imagine these cubes as infinitely extended and thus arrive at an infinitely extended three-dimensional space. The (material) body is, after all, only a part of it.

The question now is whether we can simply extend such conceptual considerations, which we make starting from space, to higher realities. In these considerations, the mathematician actually only calculates, and does so with numbers. Now the question is whether one can do that at all. I will show you how much confusion can arise when calculating with spatial quantities. Why? I only need to tell you one thing: Imagine you have a square figure here. I can make this figure, this area, wider and wider on both sides and thus arrive at an area that extends indefinitely between two lines (Figure 56).

This area is infinitely large, so it is >. Now imagine someone who hears that the area between these two lines is infinite. Of course, he thinks of infinity. If you now talk to him about infinity, he may have very wrong ideas about it. Imagine that I now add below [each square one more, so another row of] an infinite number of squares, and I get a [different] infinity that is exactly twice as large as the first (Figure 57). So we have > = 2 + 0,

In the same way I could get: “ = 3 +,

In calculating with numbers, you can just as well use infinity as finiteness. Just as it is true that space was already infinite in the first case, it is just as true that it is 2 + c, 3 - c, and so on. So we are calculating numerically here.

We see that the concept of the infinity of space [which follows from the numerical representation] does not give us any possibility of penetrating deeper [into the higher realities]. Numbers actually have no relation to space at all, they relate to it quite neutrally, like peas or any other objects. You now know that nothing changes in reality as a result of calculation. If someone has three peas, multiplication does not change that, even if the calculation is done correctly. The calculation 3 + 3 = 9 does not give nine peas. A mere consideration does not change anything here, and calculation is a mere consideration. Just as three peas are left behind, [you do not actually create nine peas,] even if you multiply correctly, three-dimensional space must also be left behind if the mathematician also calculates: two-, three-, four-, five-dimensional space. You will feel that there is something very convincing about such a mathematical consideration. But this consideration only proves that the mathematician could indeed calculate with such a multidimensional space; [but whether a multidimensional space actually exists, that is,] he cannot determine anything about the validity of such a concept [for reality]. Let us be clear about that here in all strictness.

Now we want to consider some other considerations that have been made very astutely by mathematicians, one might say. We humans think, hear, feel and so on in three-dimensional space. Let us imagine that there are beings that could only perceive in two-dimensional space, that would be organized so that they always have to remain in the plane, that they could not get out of the second dimension. Such beings are quite conceivable: they can only move [and perceive] to the right and left [and backwards and forwards] and have no idea of what is above and below.

Now it could be the same for man in his three-dimensional space. He could only be organized for the three dimensions, so that he could not perceive the fourth dimension, but for him it arises just as the third arises for the others. Now mathematicians say that it is quite possible to think of man as such a being. But now one could say that this is also only one interpretation. One could certainly say that. But here one must again proceed somewhat more precisely. The matter is not as simple as in the first case [with the numerical determination of the infinity of space]. I am intentionally only giving very simple discussions today.

This conclusion is not the same as the first purely formal [calculative] consideration. Here we come to a point where we can take hold. It is true that there can be a being that can only perceive what moves in the plane, that has no idea that there is anything above or below. Now imagine the following: Imagine that a point becomes visible to the being within the surface, which is of course perceptible because it is located in the surface. If the point only moves within the surface, it remains visible; but if it moves out of the surface, it becomes invisible. It would have disappeared for the surface being. Now let us assume that the point reappears, thus becoming visible again, only to disappear again, and so on. The being cannot follow the point [as it moves out of the surface], but the being can say to itself: the point has now gone somewhere I cannot see. The being with the surface vision could now do one of two things. Let us put ourselves in the place of the soul of this flat creature. It could say: There is a third dimension into which the object has disappeared, and then it has reappeared afterwards. Or it could also say: These are very foolish creatures who speak of a third dimension; the object has always disappeared, perished and been reborn [in every case]. One would have to say: the being sins against reason. If it does not want to assume a continuous disappearance and re-emergence, the being must say to itself: the object has submerged somewhere, disappeared, where I cannot see.

A comet, when it disappears, passes through four-dimensional space.

Everything that has been said so far is as unassailable as it can possibly be. And the confirmation is so simple that it will not even occur to man in his present deluded state to admit it. The answer to the question: Is there something that always disappears and reappears? — is so easy. Just imagine, a feeling of joy arises in you and then it disappears again. It is impossible that anyone who is not clairvoyant will perceive it. Now the same sensation reappears through some event. Now you, just like the surface creature, could behave in different ways. Either you say to yourself that the sensation has disappeared somewhere where I cannot follow it, or you take the view that the sensation passes away and arises again and again.

But it is true: every thought that has vanished into the unconscious is proof that something disappears and then reappears. At most, the following can be objected to: if you endeavor to object to such a thought, which is already plausible to you, with everything that could be objected to from a materialistic point of view, you are quite right. I will make the most subtle objection here, all the others are very easy to refute. For example, one says to oneself: everything is explained in a purely materialistic way. Now I will show you that something can quite well disappear within material processes, only to reappear later. Imagine that some kind of vapor piston is always acting in the same direction. It can be perceived as a progressive piston as long as the force is acting. Now suppose I set a piston that is exactly the same but acting in the opposite direction. Then the movement is canceled out and a state of rest sets in. So here the movement actually disappears.

In the same way, one could say here: For me, the sensation of joy is nothing more than molecules moving in the brain. As long as this movement takes place, I feel this joy. Now, let us assume that something else causes an opposite movement of the molecules in the brain, and the joy disappears. Wouldn't someone who doesn't go very far with their considerations find a very meaningful objection here? But let's take a look at what this objection is actually about. Just as one [piston] movement disappears when the opposite [piston movement] occurs, so the [molecular movement underlying the sensation] is extinguished by the opposite [molecular movement]. What happens when one piston movement extinguishes the other? Then both movements disappear. The second movement also disappears immediately. The second movement cannot extinguish the first without itself being extinguished. [A total standstill results, no movement whatsoever remains.] Yes, but then a [new] sensation can never extinguish the [already existing] sensation [without perishing itself]. So no sensation that is in my consciousness could ever extinguish another [without extinguishing itself in the process]. It is therefore a completely false assumption that one sensation could extinguish another [at all]. [If that were the case, no sensation would remain, and a totally sensationless state would arise.]

Now, at most, it could be said that the first sensation is pushed into the subconscious by the second. But then one admits that something exists that eludes our [immediate] observation.

We have not considered any clairvoyant observations today, but have only spoken of purely mathematical ideas. Now that we have admitted the possibility of such a four-dimensional world, we ask ourselves: Is there a way to observe something [four-dimensional] without being clairvoyant? — Yes, but we have to use a kind of projection to help us. If you have a piece of a surface, you can rotate it so that the shadow becomes a line. Similarly, you can get a point from a line as a shadow. For a [three-dimensional] body, the silhouette is a [two-dimensional] surface. Likewise, one can say: So it is quite natural, if we are aware that there is a fourth dimension, that we say: [Three-dimensional] bodies are silhouettes of four-dimensional entities.

Here we have arrived at the idea of [four-dimensional space] in a purely geometrical way. But [with the help of geometry] this is also possible in another way. Imagine a square, which has two dimensions. If you imagine the four [bounding] lines laid down next to each other [i.e., developed], you have laid out the [boundary figures] of a two-dimensional figure in one dimension (Figure 58). Let's move on. Imagine we have a line. If we proceed in the same way as with the square, we can also decompose it into two points [and thus decompose the boundaries of a one-dimensional structure into

zero dimensions]. You can also decompose a cube into six squares (Figure 59). So there we have the cube in terms of its boundaries decomposed into surfaces, so that we can say: a line is decomposed into two points, a surface into four lines, a cube into six surfaces. We have the numerical sequence two, four, six here.

Now we take eight cubes. Just as [the above developments each consist of] unfolded boundaries, here the eight cubes form the boundary of the four-dimensional body (Figure 60). The [development of these] boundaries forms a double cross, which, we can say, indicates the boundaries of the regular [four-dimensional] body. [This body, a four-dimensional cube, is named the Hinton Tessaract after Hinton.]

We can therefore form an idea of the boundaries of this body, the tessaract. We have here the same idea of the four-dimensional body as a two-dimensional being could have of a cube, for example by unfolding the boundaries.