We add and get the ellipse.
We subtract and get the hyperbola.
We multiply and get the curves of Cassini, the Lemniscate.
We divide and get the circle.
Friday, September 26, 2014
Architecture: Creating the Holy Grail. Lecture 3: A new conception of architecture
Rudolf Steiner, Dornach, Switzerland, June 28, 1914:
During the time when the construction of our building is proceeding I think it is a very good thing for us to try to grow more and more into its meaning. We have already made a beginning with the two previous lectures and we will try as far as we can, by means of further study, really to become one with what is to be accomplished here.
In the first place I should like to remind you of what I said when we opened the house dedicated to the work of constructing the glass windows. The lecture referred to the evolution of thought and conception underlying the art of building and I will just briefly recapitulate what I was then only able to indicate.
In regard to the Greek Temple, I said that in a certain sense it formed a unity with the whole countryside — the whole countryside was one with it. The Temple stood there as the ‘dwelling place of the God.’ Nothing need be in the Greek Temple save the spiritual presence of the God and his physical image. The essence of the construction of the Temple was the fact that every man engaged in his daily pursuits on the land knew that within the region where he was carrying on his work he was not merely alone with the earth but united with the spiritual world. And the token for the fact that man, as he lived on the earth, was also united with the spiritual world, was the Temple standing there like an altar in the land.
We then saw evidence of progress in architectural thought, in that the Christian art of building separated off the edifice from the land. Everyday life and the mood of exaltation by which man raised himself to the Spirit were separated from each other. The Church of Christendom is no longer actually one with the land; it serves the Spirit, apart from the countryside, and expresses the fact that when man is to rise to the Spirit he must leave the affairs of daily life, repair for a time to a place set apart and there be united with the spirit. The Church of Christendom, therefore, could no longer be what the buildings of Greece and also of Rome were in their real being. The Church of Christendom was in itself a duality, the house of the community and the house set apart for the altar and the priesthood. Man leaves the affairs of everyday life and enters into the precincts where he feels himself gazing upwards to the Spirit which comes to him from the chancel where the altar stands.
This evolution in architectural thought naturally implies the transformation of the ancient Greek form of building (which was derived purely from static and dynamic factors, the factors of space and gravity) into the form corresponding to the conception of the community being set apart.
Passing to the Gothic Cathedral we have a still later form of architectural conception. We have the striving of the community not only to bear their own personalities into the sanctuary but also their individual work, and this is expressed in the forms of Gothic architecture. We feel as if the work performed in the environment has passed into the architectural forms and rises to the Spirit like a prayer, a folding of the hands.
I also said that a real advance in architectural conception must come to pass again in our times and that this is only possible if the nearness to the Spirit which was achieved to an ever-increasing extent from the period of the Greek conception of architecture onwards to that of Gothic building — if this nearness is gradually transformed into a complete union with the Spirit. This means that buildings which should now be dedicated to life in union with the Spirit must in their very form express inner correspondence with the Spiritual. We can indeed say — if we try not to explain the thing in abstractions but to grasp it with the whole of our feeling and soul — ‘All that is embodied in our life of soul through Spiritual Science implies an actual penetration into the form that is created. The Spirit is revealed in freedom, having now descended to mankind.’ Whereas the Greek placed the Temple like an altar in the land, the future and, inasmuch as we are working from out of the future in our building, the present, are placing a true expression of the Spirit in the land as the result of what the Spirit expresses in its forms.
A speech which has a message for man of the present day will arise. But all this requires that we endeavour to understand the Spirit in its forms of expression. In order to understand the Greek Temple, we tried, last time, to grasp the purely physical qualities of space and of gravity. But the Spirit does not only work according to the laws of mechanics and dynamics; it does not only reveal itself in conditions of space and energy. The Spirit is living, hence it must be expressed in our building in a living way, a truly living way. We shall not understand this any better by interpreting the Spirit symbolically, but by beginning to feel that the forms are living, that they are organs of speech flowing from the spiritual world.
Is it possible for forms to speak from the spiritual world? It is indeed possible, in many ways. Let us take a thought that is specially near to us because on the one hand it is the expression of the highest, and on the other, in its Luciferic aspect it is submerged in the lowest — let us take the idea of the Ego, of Selfhood.
The mere utterance of the word “I” or “Self” does not as yet evoke much thought in man. Many epochs will have to run their course in human history before a fully conscious idea can arise in the soul when the word “I” or “Self” is uttered. Nevertheless, Selfhood, Ego-hood, can be felt in form, and above all when we pass from a purely mathematical conception of form to a feeling in form we can acquire a perception of Ego-hood, Selfhood, in the perfect circle. If you realise this you will readily understand what follows from it. If the true, living, sentient human being, confronting a circle, senses the feeling of Ego-hood, Selfhood, arising in his soul, or if when he sees a fragment of a circle he feels that it typifies the independent Self, he is learning to live in forms. And the characteristic of really living feeling is the capacity for living in forms. If you keep this in mind you will easily be able to pass on to other things that follow from it.
The first circle I have drawn here has an unbroken line. (1). This line however can be varied so that it shows these wavy projections.
(2). Another characteristic variation is the third figure. (3). Both figures are only variations of the circle. What do these variations signify? The second (2) expresses the fact that the Self, the Ego has entered into relation with the outer world. The simple circle makes us feel that nothing of the rest of the world is there, only what is shut off within the circle. If we observe the circle in variation we can no longer feel that what is expressed by the circle is alone in the world. The variation in the line expresses a struggle, a kind of interplay with the outer world. If we really live into the form of the second variation (2) we shall feel: “The inner is stronger than the outer.” And in the case of the jagged circle (3): “The outer has bored its way in and is stronger than that which lies within the circle.” And now if we go into any building containing fragments of circles or rounded surfaces, and perceive variations of this kind, we shall feel in the case of the jagged lines, “here the outer has conquered,” and in the case of the wavy lines, “here the inner has conquered!” Our souls begin to live with the forms. We do not merely behold the forms but in our souls we have the living, pulsating feeling of “conquest and encroachment,” “victory and mastery.” The very soul lives with the form. And this union with form, this living in form is the very essence of true artistic feeling.
But we can go further. Let us picture to ourselves a less simple variation (4). The form moves in one direction and becomes action. If we live in this form we have the feeling that it advances, that it moves. In the forms themselves we find the quality of movement. I have here made a simple sketch of something that will appear in a complicated form in the building, but you will find that there is an absolute correspondence. Passing from the entrance at the West and thence towards the smaller structure (at the East) you will find that all the forms in the interior will evoke the feeling that the whole structure is proceeding from the West onwards to the East. This is expressed in the forms. At the West you will feel in thought that you are within a vehicle that is bearing you to the East. The very essence and meaning of these relief variations is that they do not merely appear as dead, dynamic or mechanical forms; we seem to enter a vehicle that bears us onwards. In a spiritual sense we shall not “rest” in our building; we shall be led onwards.
From this you will realise that the basic character of the forms here is quite different from the forms of the three stages of architectural thought which I have described. Up to our time architectural thought has been concerned with the qualities of lifeless, mechanical rest. Now, however architectural thought becomes the thought of speech, of inner movement, of that which draws us along with it. This is what is new in the whole conception, and the basic form must of course correspond to it. In what way does the basic form correspond to it? Now I have said that the most intimate of all impressions is that of the Self, the Ego, as expressed in the circle or sphere. Why is this? It is because the simple circle or sphere is of all forms the most easily perceptible. It is an absolutely simple matter to recognise a circle. All that is necessary is the most trivial thought that everything is equidistant from the central point. As soon as we picture to ourselves points standing at an equal distance from this centre, we have the sphere, or circle. It is the very easiest process that can be carried out in thought. As form, then, the circle is the simplest of all entities. This is also in accordance with external reality, for the Selfhood in every being, from the simplest cell to the complex human being, is the simplest of all impressions, just like the circle or sphere. Behind all this there is something much deeper and I want you now to follow me in a thought that will lead those who really understand it, to great profundities.
Now the form of an ellipse is more complex than that of the circle. I will draw the form of an ordinary ellipse. It need not be exact but merely have the general character of an ellipse. The simplicity of the thought is no longer there when we pass to the ellipse. Although the ellipse is still spherical, we have no longer the nature of equality as in the case of the circle. Here I must ask those who have studied geometry — although for politeness sake we will assume that you all know a little of geometry though you may have forgotten some of it — to try to understand the following ideas. There is also order and regularity in the ellipse. Just as the circle is related to one point, the ellipse is related to two. The lines between any point of the ellipse and the two foci will naturally differ, but the two lines together will come to the same length (a + b = c + d in diagram). You can add the distances of each point from these two foci and you will always get the same length. This is so simple in the case of the circle that there is no need for an mental process. In the case of the ellipse, however, we must make an addition. All lines to the centre of the circle are equal, but in the ellipse we have to make an addition. Now you will say: ‘Yes, but I do not add when I see an ellipse.’ You yourself do not, it is true, but your astral body does; what the geometrician does consciously the astral body does unconsciously. The astral body is a finished geometrician. You have no idea of all the knowledge that is contained in your astral bodies; in the astral body you are all the wisest of geometricians, only of course the geometry you know in the astral body can only be brought into consciousness by the ‘sweat of the brow.’ You must pardon this expression but it is permissible to-day (... it was very hot on the day this lecture was given.) Everything is there in the astral body and if those who teach geometry, instead of using their wonted methods, could apply a pump in order to extract what is in the astral body, they would no longer need to teach — the knowledge would well up of itself. We add, then, the two distances from the foci and always get the same result. When an ellipse form seems beautiful, what does this really imply? It implies that the astral body is adding and the sum total is always the same. And now picture to yourselves that you are adding without knowing it and every time getting the same result. You feel pleased. Now you go to another point and carry out the same process. ... The same total again — oh! what joy! This is the living experience of the ellipse.
In the case of the circle there is no such feeling of satisfaction, for the circle is so immediately obvious. The ellipse causes us greater joy because there we have to be inwardly active. The more one is inwardly active, the greater joy one has. What is often so difficult to realise is that man, in his inner being, craves for activity. If he wants to be lazy this is merely an affair of his conscious life. The astral body is not only wiser, but also more industrious and would like always to be active.
Now there is another line consisting, of course, always of two portions. Those who have studied geometry will know that thehyperbola consists of two symmetrical curves. The hyperbola also has two foci which lie approximately here. Again we can draw lines to these two points. The strange thing here is that we do not add but subtract. We always get the same result by subtracting the lesser from the greater. Our astral body subtracts and is glad that the difference is always the same. In this inner feeling of equality the astral body experiences the source of the hyperbola.
Man is thus a mathematician in the substrata of his consciousness and by means of subconscious calculation we create for ourselves regularity of form. We add and subtract, but we can also multiply. Here again we have two points. Multiplying the one by the other we again get a line that looks somewhat like the ellipse but is not the same. This line contains an inner process of multiplication.
This line has something mysterious about it. The circle is a simple entity, the ellipse already more complicated, the hyperbola still more so, for I do not think that the ordinary person sees only one single line in the two curves. The ordinary intellect believes there are two curves. The ordinary intellect believes there are two lines, but in reality this is not so. The other line is mysterious for another reason, for according to what is produced by multiplication the line is changed into this curious form. It is the curve of multiplication, the curve of Cassini, the lemniscate which plays so important a rôle in occult investigations. The line can develop in such a way that it assumes these forms. There are two lines, you see, but in the inner sense there is really one line, and when we feel it as one line in the astral world we know that this form (o-o) is only a specialisation of this form ( ∞ ). But now think — this form ( ∞ ) disappears into the fourth dimension— then appears again and enters the physical world. It is an unity because it ever and again disappears into the fourth dimension. This multiplication process has really three different forms.
We have therefore a line of addition, a line of subtraction, a line of multiplication. Someone may say that there must then be a line of division, the fourth method of calculation. There we must divide two distances instead of adding, subtracting or multiplying. That is to say, it must be possible for our astral body to determine two points — and also other points if it takes the larger line — and to divide the greater distance by the smaller. The astral body, then, must be able to divide and when it does this it gets a line (see diagram). All the points are so that their distances from two points are the same in the division.
Now we have something very remarkable indeed. When we really try to penetrate into the depths of nature they appear before the soul in all their wonder. The circle appears to be an utterly simple entity but it is, nevertheless, full of mystery. The circle can also be understood by taking two points and dividing, and inasmuch as the same result is arrived at, we get the circle. The circle is thus something very remarkable. It is the simplest of all entities and yet it is the product of an occult process of division that is brought into consciousness. It is just the same in the case of the self of man: the ordinary self is the simple entity and the higher self the mysterious entity resting in the depths of being — a self that can only be found when we transcend its limits and pay heed to the world with which it is connected. The circle is the same whether we say that it is the simplest of all forms or that the product of division from two points is always the same. Just as we have the same circle, so we have within ourselves a duality: something that belongs to everyday life and is readily perceptible, and something that we only grasp when we go out to the whole universe, conceiving of this entity as the most complicated product of the great cosmic struggle where Ahriman and Lucifer carry out the division and where our own higher self has to maintain itself as the quotient if it is ever to come to expression.
Portions of the ellipse and of the hyperbola and also of the curves of Cassini will be found everywhere in our building, and your astral bodies will have plenty of opportunity to make these calculations! Here I will only mention one instance: when people go into our building and stand in the gallery where the organ and the singers will stand, their souls will be able to carry out this process of multiplication. The soul may not do so consciously but it will feel this process in the depths of its being, because this is the line of the structure around the organ. This line will be found in many places in the building.
After what I have now told you about the twofold meaning of the circle you will be able to realise that when you enter the building from the West and feel yourselves surrounded by the circular structure, by the cupola above, that here is the image of the human self. But the other smaller space in the East is not at first sight so intelligible. The smaller structure will seem to be full of mystery because, although its form is also circular, it must be conceived of as the result of a process of division and it only outwardly resembles the larger space. There are two circles, but the one corresponds to the life of everyday and the other is connected with the whole cosmos. We bear within us a lower self and a higher self. Both again are one. Thus our building had to be a twofold structure. Its form expresses — not in any symbolical sense but in its very being — the dual nature of man. When the curtain in front of the stage is open we shall perceive an image of man not only as he is in everyday life, but as complete man. The forms themselves express a movement from West to East, the path of the lower to the higher Self. All that I have told you can actually be felt in the forms. The erection of a building of this kind reveals how the spiritual form of nature and the higher spiritual world can be expressed. Nobody who begins to think out all kinds of ingenious interpretations will Understand our building. It can only be understood by a living feeling of the development and being of the forms. For this reason I do not want to describe the building pictorially but to speak of the mode of its development, how spiritual being itself has become form and movement and has flowed into it. Suppose anyone were to look at the interior and begin to speculate thus: ‘Yes, two cupolas, two circular structures — lower Self, higher Self; a lower Self, a higher Self — a unity.’ This may be a neat interpretation but it would be of no more value than if it were said that Maria and Johannes Thomasius in the Mystery Plays are really one being. This is a mere speculation, for it results in an abstraction. The unity lies in the living ‘becoming.’ Naturally the living powers of becoming can bring forth both Maria and Thomasius but only as the result of a differentiation. Even in similarity the true occultist will always seek for diversity, for it would be false occultism to desire always to lead back diversity to unity. Hence the example of the circle. The circle is the simplest of all entities, where all points are equidistant from the centre — but it is also the result of division. In the circle we have something that is a unity in the outer world and complex in the spiritual world.
These are some of the remarks I desired to make. On another occasion I shall speak further on these matters. I shall now speak briefly of other things.
Man, as he enters the world, is really a highly complicated being. When he enters the world — as I have often said — he cannot at first stand upright; lie crawls, and at the very beginning of his existence he does not even crawl. Gradually he learns to control the forces which make him able to stand upright. Let me try to make a diagrammatic sketch of this process. Underneath we have the Earth. Man is at first a horizontal being; then he stands upright — in the vertical position. It is an achievement of man's nature itself to attain the vertical position but he has the help of all the hierarchies as he passes through the course of his life. What is it that comes to his aid when he stands upright and walks? The forces that work from the Earth out into the expanses of cosmic space. These are the earthly forces. To-day physicists only speak of purely physical forces of the Earth — forces of attraction, of gravity and the like. The Earth, however, is not merely physical body but a being of spirit and soul and when, as little children, we raise ourselves to the upright position and walk, we are uniting ourselves with the forces of Will rising out of the Earth. The Earth-Will permeates our being; we allow the Earth -Will to flow into us and place ourselves in the upright position — the direction of the Earth-Will. This process is a union with the Earth-Will. But in opposition to the Earth-Will there is a will that works in from all sides of the cosmos. We have no knowledge of it, but yet it is the case that as we raise ourselves to the upright position, forces (from the cosmos) are working in from all sides and we come up against these forces that are pouring in from outside. This has no particular significance to-day, on the Earth, but during the ancient Moon period it had a tremendous significance. On the ancient Moon, conditions were such that from his earliest childhood man had a different orientation, in that he had to place himself in line with the direction of the Moon-Will. As the result of this he acquired the first germs of the skull formation. To-day we have inherited them, but on the Moon it was a question of acquiring them. In those times man worked in himself against the outer will-forces somewhat in the way a locomotive works when it has to push away snow. He pressed back the will-forces of the cosmos and his soft skull formation compressed itself into the hard skull covering. To-day this process is no longer necessary. The skull formation is inherited. It is no longer necessary to build up the skull bones. In the etheric body, however, we still build them, for as we rise to the upright position there is a densification in the head, representing the result of the fight between the forces streaming in from all sides of the cosmos. Thus, when we observe the etheric body, we may say that in his two legs, man builds up two lines of force and works against the forces that proceed from without. The etheric body is densified and this form arises (see next diagram). We raise ourselves upright. The physical legs have their junction above, but the etheric legs rise still higher.As a result of this the etheric head is densified and as a result of the formation of the brain there arises in the etheric body, in our age as well, the densified etheric body. This does not only take place in childhood but as man passes through seven life periods (from the first to the seventh year, from the seventh to the fourteenth year and so on) new lines are formed, lines of different forces which pass upwards. So that when we have reached the age of full and complete manhood — when we have passed the fiftieth year of life — we have added new pairs of pillars to that first strong pair formed during the first seven years of life. They appear in the etheric body in different colours. We strengthen our etheric sheath every time we develop these ‘life pillars’ — for so indeed they may be called. After the first period of seven years the first pair of life pillars is completed, at the fourteenth year the second pair, at the twenty-first year the third pair, and finally, with the forty-ninth year, the seventh pair. Each pair of life pillars makes our etheric skull-covering more secure. Man passes through his life and after every seven years raises within himself different pillar formations which bear his skull. When we have understood this we shall have a living conception of the inner form of the larger section of our building. We enter at the West and say to ourselves: ‘Up to the first pair of pillars we see how man develops in the first seven years of his life; the second pair of pillars denotes his development to his fourteenth year, then on to the twenty-first year and so on.’ And the etheric sheath of the head is always around us. Man, the living being, is poured out into the forms as an etheric being.
The advance from Gothic architecture to that of Spiritual Science may be described as follows: Gothic architecture contains the prayer: ‘O Father of the Universe, may we be united with Thee, in Thy Spirit.’ Those who know what this prayer contains, who really understand the living development of Spiritual Science, will solve the riddle of the evolution of man. And then, when the forms of architectural thought strive to be united with the Spirit—expressing this striving in their very being — man will feel how he has been permeated with the hidden Spirit and can have around him a building which is a direct expression of the living, inner development of his being.
‘We dwell in the land, but the Spirit is among us.’ This is the Greek thought of architecture.
‘We dwell for a season in the sanctuary and the Spirit comes to us.’ This is the thought behind Christian architecture.
‘We dwell for a season in the sanctuary, but we uplift the soul by raising ourselves to the Spirit.’ This is the thought behind Gothic architecture.
‘We enter with reverence into the Spirit in order that we may become one with the Spirit poured out around us in the forms — the Spirit that moves and is active, because behind the Spirits of Form stand the Spirits of Movement.’ This is the thought behind the new architecture.
Existence thus advances through earthly evolution and it is man's task to understand the inner meaning and purport of this existence. He only advances in the wake of true evolution when he endeavours, in every epoch, to experience what the spiritual world bestows in that epoch.
Why do our souls pass through different, successive incarnations? Not in order that we may repeat the same experiences, nor that we may pass through re-birth, re-naissance, again and again, but in order that we may assimilate, ever and again, the new that pours into our souls from out the spiritual worlds. We are standing at a definite point in the evolution of humanity in the sphere of art and in many other spheres of spiritual life — at a point where the Spirit speaks clearly to us of new riddles. And just as in the time of the Renaissance man was destined primarily to orientate himself to the past in order to work his way through to the new, so it is with our own external knowledge and perception of the universe. All that has been produced by the modern age since the sixteenth century is only the preparation for a living experience of the universe in its forms and movements which now stand before us as riddles.
This, then, is all for today. In another lecture I will try to approach questions of a still more intimate character — questions relating to the living soul of nature in connection with colour and the art of painting.