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The Fourth Dimension
GA 324a

Questions and Answers XVII

15 October 1920, Dornach

A question about Copernicus's third law.

It is impossible to speak about Copernicus's third law in such a short time, [Note 95] so let me simply comment on its history. If you look at Copernicus's basic work, which severely shook the old Ptolemaic system and revolutionized our view of the heavenly bodies, you will find that it encompasses three laws. [Note 96] The first of these three laws speaks about Earth's annual movement around the Sun in an eccentric circle, the second about the Earth's rotation around its axis, and the third about the Earth's movement around the Sun in relationship to the seasons and precession. As astronomy progressed, it failed to consider this third Copernican law in its entirety. In fact, Copernicus's successors effectively eliminated it. That is all I can say about this law without doing extensive drawings, which would keep us here until midnight.

On the basis of the phenomena available to him, Copernicus first calculated the daily changes caused by the Earth's circular movement around the Sun, disregarding the seasonal, yearly, and longer-term changes encompassed by his third law. He then concluded that if we consider the daily changes and those dependent on the Earth's circular movement around the Sun in the Earths position with regard to the other heavenly bodies, the result is a view of the Earth revolving around the Sun. This view is opposed by other phenomena such as the seasons and precession, which actually nullify the assumption that the Earth revolves around the Sun.

For the sake of being able to quantify and calculate the interactions between the Earth and the other heavenly bodies, We make it easy for ourselves and disregard any changes that can be observed only over a year or over centuries, because these changes complicate the daily changes that depend on the Earth's circular movement around the Sun. Calculating the daily changes on the basis of the assumptions expressed by Copernicus in his first and second law results in the Earth's yearly revolution around the Sun. As Copernicus himself said, if we include the third law in our calculations, it counteracts the factor contained in the first law, which we calculated into the daily movement and which yields the Earths yearly movement, and almost eliminates any such yearly movement. [Note 97] In any case, the third Copernican law has been disregarded. People preferred the easy assumption that the Earth rotates around its axis in twenty-four hours, progressing all the while so as to move around the Sun in the course of one year. This solution was simple as long as we clung dogmatically to the Copernican assumption that the Sun does not move at all. We were forced to abandon this assumption a long time ago, however, and the third Copernican law had to be reinstated. [Note 98]

I can summarize this subject only briefly — as I said, a detailed mathematical and geometric explanation would take hours — but if we take the third Copernican law seriously, it does not result in movement of the Earth around the Sun. The Sun moves, — it would outrun the Earth if the Earth simply revolved around the Sun. The Earth cannot revolve around the Sun because meanwhile theSJun would move away from it. In reality, the Sun moves on, and the Earth and the other planets follow it. We have a line like the thread of a screw, with the Sun at one point and the Earth at the other end. Our dual focus on the Earth and Sun and on their progressive, screw-like movement creates the illusion that the Earth is revolving around the Sun. [Note 99] The interesting point in all this is that Copernicus was more advanced than we are today. We have simply omitted his third law from astronomy's post-Copernican development. Our astronomy has been developed without this third law, which states that other phenomena negate the yearly movements around the Sun that we calculate for the Earth. To do full justice to Copernicus, this law must be reintroduced. [Note 100]

This subject does not attract much interest, because if we were to apply a true phenomenological approach to astronomy, we would have to realize first and foremost that, as Dr. Vreede [Note 101] already mentioned, we are dealing with extremely complicated movements. And that the ordinary geometric constructions we use in attempting to describe these movements are suited only to descriptions of simple geometric processes. Because the heavenly bodies do not obey such simple processes, disturbances always appear, and we are forced to compensate by adding more hypotheses. [Note 102] When we get beyond such hypotheses, astronomy will look completely different.

This will happen only when we progress to a form of natural science that truly includes the human being and observes phenomena within the human being. Taking these phenomena into account will allow us to develop a view of the events and processes of cosmic space. As Dr. linger also mentioned, [Note 103] the human being actually has been ousted from today's science, which disregards the human element. Ideas such as the theory of relativity, [Note 104] which certainly do not correspond to reality, are able to take hold only because modern science is so utterly estranged from reality that it deals with everything outside human beings but nothing that happens inside them. To think in ways that correspond to reality is a skill that humanity will have to relearn.

If you have a stone lying here (reference to a drawing that has not been preserved), you can see it as leading an independent existence, at least to a certain extent. It all depends on your presuppositions. We can say that when we consider what we see within the boundaries of the stone, we develop a certain view of the stone. But now assume that instead of a stone, we are considering a rose that I have picked. It is not possible to ascribe reality to the rose in the same way that we ascribed reality to the stone within its boundaries, because a plucked rose cannot exist in isolation. It must develop in connection with something else. We are forced to say that while the stone within its described limits possesses a certain real existence, the rose does not, because it can exist only in association with its rootstock. If I separate it from its roots, the prerequisites for its existence are no longer present, and it cannot persist.

We must relearn the skill of submerging our thinking in things and taking the things themselves into account. Only when we have reacquired this skill will we have a healthy form of astronomy, for example, as a matter of course. We will be spared the terrible abstraction of such ideas as the theory of relativity. Essentially, the theory of relativity is based on ideas that are not true realities.

The ordinary formula \(s = v \times t\), (distance equals speed multiplied by time) is quite illuminating. When I am describing a reality, I can write only this:

$$v = s/t$$

When we grasp a reality by means of abstraction, I can calculate everything that is in a real object. Because it is possible to grasp many different things on an abstract level, we can perform many different calculations while remaining within the abstract. We must not believe, however, that these abstractions are realities. In the inorganic world, only speeds are realities, and both time and space are mere abstractions. Thus when we begin to perform calculations involving time and space, we enter the domain of unreality, and once we begin thinking in unreal terms, we can no longer return to reality.

These issues, therefore, are related to very significant shortcomings of our times. In recent times, humankind has disregarded the spirit completely while attempting to understand nature, and our souls have moved toward abstractions. In one sense, dealing with abstractions is extremely comfortable, because we do not need to learn to submerse ourselves in objects and events. It is easier to think in terms of space and time than to immerse ourselves in qualitative aspects or to realize that whatever we can think of as real in connection with something else, can therefore be thought about in real terms. (Editor's note: not abstractly.) You need not believe what I am about to say, but it is true nonetheless. It is torture for a person who has cultivated a capacity for thinking and a desire to understand reality to read Einstein's theory of relativity, because even though all the ideas Einstein presents are mathematically very consistent, they are literally unthinkable for someone with any sense of reality. It is impossible to pursue such thoughts to their conclusion. What does it mean and what kind of sense does it make when Einstein presents a whole complex of thoughts about someone who is sealed up in a box and journeys through space at high speed and returns to find a new generation of people and totally different circumstances? [Note 105] When we think about such a situation, of course we are thinking only in terms of space and time and disregarding the outer bodily nature of the person or object, which would be destroyed while undergoing the experiment. Although this objection may seem naive to fanatical thinkers on the subject of relativity, it inevitably comes into consideration with regard to reality. [Note 106] Anyone who has a sense for reality cannot see such thoughts through to the end.

Suppose that we are traveling in a car, for example, and have a flat tire. Let's assume that it makes no difference whether I think that the car, with me in it, is speeding over the ground or that the car is standing still while the ground moves out from under me. If, in fact, it makes no difference, why should the ground suddenly go on strike because of a minor breakdown that concerns only the car? If it makes no difference how we conceive of this situation, the outcome should not be affected by the outer change. As I said before, although such objections are terribly naive as far as relativity theorists are concerned, they do reflect current realities. Anyone whose thinking is grounded in reality rather than in abstraction — even an abstraction that can sustain consistent thoughts — is forced to point out such issues.

Fundamentally, therefore, we are living with a theoretical form of astronomy. A classic example is our disregard of the third Copernican law. We push it aside because it is uncomfortable. When we study it, we learn to feel uncomfortable about our customary calculations. What do we do? We apply the second Copernican law, but our calculations do not come out even, and noon falls in the wrong place. So we introduce the daily corrections known as Bessel's corrections. [Note 107] If we realize their full implications, however, we see the need to take the third Copernican law into account — that is, we begin to deal with realities.

The point here is to acknowledge the principles behind such issues. The way we presently deal with such principles permits us to go astray in many different directions. Mr. Steffen did an excellent job of presenting three such tortuous paths in a specific field of knowledge. [Note 108] Such misleading paths are easy to encounter today, and they influence real life. We have trained ourselves to think in ways derived from a mathematics that lacks reality, and this type of thinking gradually has become almost a touchstone of genius. In fact, a sense of reality is sometimes much more helpful than genius, because if you have a sense of reality, you must abide by the realities of the situation. You must immerse yourself in objects and events and live with them. If you have no sense of reality, you can impose all sorts of abstractions onto space and time in the most ingenious way, simply by manipulating mathematical formulas and methods. You can rise to truly terrible levels of abstraction.

These abstractions sometimes can be very seductive. I am thinking of modern set theory, which has been used as the basis for explaining infinity. Set theory dissolves number, the very principle of mathematics, because it no longer sees a number as an ordinary number but merely compares one arbitrary set with another, classifying individual entities with no regard to their qualities and sequence}. [Note 109] Set theory makes it possible to develop certain theories of infinity, but swimming in abstractions all the while. In concrete reality, it is impossible to perform such operations. It is important to note that we gradually have become accustomed to disregarding the need to immerse ourselves in reality. In this connection, spiritual science really needs to set the record straight.

I am now going to present two opposites. This appears to have nothing to do with theory, but in truth it has a great deal to do with theory, because all of these matters deal with much more than a theory, which can be corrected if our thinking about it is sound. The real issue is the need to develop sound thinking, thinking that is not merely logical, because logic also applies to mathematics. We can incorporate logic into mathematics, and the result is a completely coherent structure that nonetheless need not apply to reality at all. By now we have reached the point of being able to show how things look to an undisciplined way of thinking that lacks any true sense of reality.

Here you have on the one hand a book that attempts to summarize everything that modern science has to offer. Thousands and thousands of copies — seventy or eighty thousand, I believe — of this famous book have already been sold. It is Oswald Spengler's book The Decline of the West. [Note 110] As you know, this means that four or five times that number of people have read the book, so we know what a tremendous influence it has had on modern thought, simply because it emerged from modern thought, in a certain sense. The author of this book had the courage to formulate the ultimate consequences of modern thinking. In this book, Spengler looks at everything that astronomy, history, the natural sciences, and art have to offer, and we are forced to admit that he has amassed a huge body of evidence. Because Spengler really thinks in this way, he has the courage to draw the ultimate conclusions from the thinking of truly modern astronomers, botanists, art historians, and so on. As clearly as we can prove the second law of thermodynamics, [Note 111] for example, Spengler's book also proves that in the beginning of the third millennium, Western civilization will have degenerated into complete barbarity.

We must admit that this book not only has shown us the decline of modern civilization but also has proved a future event as clearly as any scientific statement can be proved today. In terms of the methods of modern science, Spengler's proof of the decline of the West is certainly as good as any astronomical proof or the like and much better than any proof of the theory of relativity. His conclusions can be circumvented only by those who see factors that Spengler himself does not see, namely, by those who will provide completely new impulses for humanity from now on. Impulses that must be born out of the inmost core of the human being and that are invisible to any science based solely on contemporary thought.

But what is Spengler's thinking like? Unlike the relativity theorists, Oswald Spengler thinks in categories that correspond to reality. Not everything he thinks fits together, however. The concepts he develops about astronomy, biology, art history, architecture, sculpture, and so on do not always mesh. They form a structure that I would like to compare to crystals that have grown together. They are all confused, and they destroy each other. If we maintain a sense of reality while reading Spengler's book, we find that his concepts are very full (reference to a drawing that has not been preserved). Oswald Spengler certainly knows how to think and develop concepts, but his concepts destroy each other. They blow each other up and cut each other apart. Nothing remains whole because one concept always negates another. We see terrible destructive actions when we apply a sense of reality to the development of Spengler's ideas.

Spengler represents one pole in modern thought, the pole that constructs a unity out of concepts drawn from all different fields. The philosophers associated with this trend neatly define everything on such an abstract level that all of the concepts they derive from individual sciences can be gathered together and united into a system of sorts, in an attempt to come to a point. They fail to come to a point, however, but simply splinter and obliterate each other. Spengler is a much better philosopher of modern science than many other philosophers, whose concepts do not destroy each other because their formulators lack the courage to define them precisely enough. In their philosophies of science, these other philosophers are always confusing tiger claws with cat paws, as it were, resulting in comical constructs that are said to be the philosophical consequences of individual scientific investigations. If we consider these philosophers seriously, we see that Spengler is experienced in all the sciences and knowledgeable about anything scientific that can result from the customs of philosophy.

The other pole is represented by a philosopher who is also popular, though not revered to the extent that Spengler is, namely, Count Hermann Keyserling. [Note 112] Keyserling differs from Oswald Spengler in that none of his concepts have any content. While Spengler's concepts are meaty, Keyserling's are empty. They never contradict each other because they are basically only empty husks of words. Keyserling's only thought, which is also an empty husk, is that the spirit must unite with the soul. [Note 113] Count Keyserling attacks anthroposophy vehemently. In the periodical Zukuiift, for example, he accused me of splitting the human being into various members — ether body, sentient body, sentient soul, and so on — while in fact the human being is a unity and functions as such. [Note 114]

The thought that the spirit must unite with the soul seems fiendishly clever, but in fact it is no more clever than saying that a suit is a unity and should not be broken down into component parts, such as a vest, a pair of pants, boots, and so on. It's all a unity, so I should not have the tailor make the jacket and pants separately and then go to the cobbler for boots to match. Of course, all of these things form a unity on the human being who is wearing them. But it makes no sense to say that jacket and pants and probably the boots as well should be stitched together into a single article of clothing, even if Count Keyserling in his abstract idealism insists that they are a unity. This is the opposite pole.

We have, on the one hand, Spengler with his concepts that destroy each other and on the other hand, we have Keyserling with his totally empty concepts. For anyone who has any sense of reality, it is a torment to read Spengler and to see all his concepts colliding with and crushing each other and forcing their way into each other. You really are compelled to experience all this, especially if you have any artistic sensibility. Spengler's book is a totally inartistic construct, but when you read Keyserling's book, you stop and gasp for breath after one page, because his concepts have no air in them. [Note 115] We want to form a thought, but there is nothing there, which makes it very easy for people to understand these concepts and feel comfortable with them. This is especially true if this impotent non-thinker also tells them that while there may be some truth to the facts that spiritual science confirms, he himself cannot corroborate them and therefore will not assume that they are true, since he is not one of those people who has intuitions, and so on and so forth. [Note 116]

Of course, people lap up this kind of talk, especially if they themselves cannot supply the necessary proof. Especially today, such people much prefer a writer who admits to being unable to confirm the facts to one they have to struggle to keep up with. Keyserling's scribblings on art, in particular, are enough make your hair stand on end, but they are very popular. That is all I have to say on this subject.

By now, you may have developed a sense for what it means when Goethe says, "Consider the What, but consider How more seriously. [Note 117] You can consider the What when you read Spengler, because he has a lot of What to offer. But Goethe knew that a worldview depends on how we see the whole in the coordination, organization, and inherent harmony of ideas. That is why we can say, referring to Spengler, consider the What. Spengler does consider the What as it should be considered, but he fails to consider the How at all. Above all else, Goethe challenges us to consider how ideas are arranged. With regard to Keyserling, we might say that he appears to possess the How — in fact, his work is teaming with How, but there is no What, no content.

  1. Question-and-answer session during a "conversation on spiritual science" in the context of the anthroposophical conference of September 26 to October 16, 1920, at the Goetheanum in Dornach. Rudolf Steiner's introductory lectures on Grenzen der Naturerkenntnis ('The Limits of Our Understanding of Nature") were held from September 27 to October 3, 1920, and appeared in GA 322. Many lectures by other participants were printed in Aenigmatisches aus Kunst und Wissenschaft ("Enigmatic Aspects of Art and Science"), vols. I and II, Stuttgart, Der Kommende Tag Verlag 1922 (available from the Goetheanum bookstore), or in Kultur und Erziehung ("Culture and Education"), Stuttgart, Der Kommende Tag Verlag, 1921 (available from the Goetheanum bookstore). See also the announcement of the conference, which includes a detailed program, in the periodical Dreigliederung des sozialen Organismus ('The Threefolding of the Social Organism"), vol.2, 1920/1921, no. 9. Reports on this conference by Alexander Strakosch and Gunther Wachsmuth appeared in the same periodical (nos. 15, 16, and 18).

  2. According to Ptolemy (Claudius Ptolemeus, ca. 100—170 A.D.), the basic structure of the solar system was classically geocentric, with the resting Earth in its center. In his chief work, Almagest, Ptolemy uses a complicated construction of concentric circles to explain the details of planetary movements. (See Ptolemy [1962]; Ziegler [1976]; Teichmann [1983], chapter 3.2; Van der Waerden [1988, chapter XIX.) With regard to planetary orbits that result from combinations of circular movements, nothing essential is changed by shifting from the geocentric Ptolemaic system to the heliocentric Copernican system, except that the Sun and the Earth exchange places, which corresponds to a simple geometric transformation. Furthermore, both Ptolemy's and Copernicus's arguments are essentially kinematic (Steiner would have said "phoronomic") — that is, they do not take force relationships into account. See Vreede [1980], "Über das kopernikansiche System" ("On the Copernican System”), pp. 349-359: Teichmann [1983], chapter 3; and Neugebauer [1983], section 40.

    In his chief work De Revolutionibus Orbium Coelestium, 1543, volume 1, chapter 11, Nicolas Copernicus (1473–1543) separates the movement of the Earth into three components (see Copernicus [1879], pp. 28ff or [1990], pp. 139ff.). The first movement is the Earth's daily rotation around its axis, the second is its movement in an eccentric orbit around the Sun, and the third is its "movement in declination." Copernicus formulates it like this:

    Since so many important planetary phenomena testify that the Earth moves, we will describe this movement in general terms, inasmuch as it confirms the phenomena, like a hypothesis. We must assume that this movement is threefold: the first movement, which the Greeks called nychthemerinon, daily-nightly, is the actual circulation of day and night, which moves around the Earth's axis from west to east in the same way that we formerly believed the Earth to move in the opposite sense. This circulation defines the equinoctial circle or equator, which some call the circle of equal days in imitation of the Greeks, who called it isemerinos, of equal days. The second is the yearly movement of the center, the Earth and its satellite through the zodiac around the Sun from west to east — that is, in direct motion — between Venus and Mars. The result of this movement, as we said, is that the Sun itself seems to make a similar movement through the zodiac, so that when the Earth (the central point) is moving through Capricorn, Aquarius, and so forth, the Sun appears to be moving through Cancer, Leo, and so on. We must imagine that the slant of the equator and of the Earth's axis varies in relationship to the plane of the circle that passes through the center of the zodiac signs. If the slant were constant and only the midpoint (the Earth) moved no change in the length of days and nights would occur and we would have always either the summer solstice or the winter solstice or an equinox—in any case, an unchanging season. Thus, the third movement, or movement of declination, occurs in a yearly cycle but in the opposite direction from the movement of the midpoint (the Earth). As a result of these two almost equal but opposite movements, the Earth's axis, and thus also the equator—the greatest parallel circle—remain pointing to almost the same area of the heavens, as if they were immobile, while the Sun, because of the progressive movement of the Earth's center, seems to move through the oblique plane of the zodiac in a way that is no different from what it would do if the Earth were the center of the solar system, if we only remember that the Sun's distance from the Earth in the sphere of fixed stars has already exceeded our perceptive capacity (Copernicus [1879], p. 28ff).

    Rudolf Steiner seems to have reversed the order of the first two laws of Copernicus's De Revolutionibus. The above sequence, however, is the one Copernicus also uses in discussing the three movements of the Earth in De Hypothesibus Motuum Coelestium a se Constitute Commentariolus, also called simply Commentariolus, published in 1514. (See Copernicus [1948], pp. 12ff, or [1990], PP 9ff.)

    In the passages that follow, we have preserved Steiners sequence:
    l. The Earths annual movement around the Sun in an eccentric orbit
    2. The Earths daily rotation around its axis
    3. Movement in declination: the Earths axis describes a cone, moving in the opposite direction from its revolution around the Sun.

  3. In a geometric or kinematic sense, the first movement (if considered in isolation, disregarding the second and third movements) is the Earths revolution around the Sun. Note that the Earths axis does not remain parallel to itself — except in a special instance when the axis is parallel to the axis of rotation, which is not the case here. Instead, it describes a cone in relationship to the Earths midpoint. In other words, the intersection of the extension of the Earth's axis with a line perpendicular to the plane of the Earth's eccentric orbit around the Sun is a fixed point of this movement. If this movement existed in isolation, there would be no change of seasons, because the Earth's position in relationship to the Sun would always be the same.

    Consequently, Copernicus had to introduce another movement to account for the phenomenon of changing seasons, on the one hand, and precession (shifting of the vernal equinox), on the other. His "movement in declination," the third movement in Steiner's sequence, served this purpose. This movement consists of the yearly rotation of the Earth's axis in the opposite direction from its movement around the Sun. It negates the rotation of the Earth's access created by the second movement, and a slight excess accounts for precession.

  4. 97In 1783 at the latest, the fact that the Sun itself also moves was acknowledged when William Herschel (1738–1822) discovered its movement (called the apex movement) in the direction of the constellation Hercules. (See Wolf [1891-1893], §292.)

  5. Rudolf Steiner often spoke of the spiral or screw-like movement of the Earth as it follows the movement of the Sun, — see the lectures of March 24 and 31, 1905, for example. Beginning with his lecture of September 1, 1906 (GA 95), he often links the third Copernican movement to his own description of the problem of the Sun and Earth's motion. From 1916 on, he adds the aspect of a progressive lemniscatic quality of movement. (For a general overview of this problem, see Vreede [1980], "Über das Kopernikanische System" ["On the Copernican System"], p. 349ff.)

    The following list includes most of the lectures and question-and-answer sessions (Q&A) in which Steiner discusses the problem of the Sun and Earth's motion, especially the third Copernican movement (Copernicus 3), Bessel's corrections (Bessel), and/or the problem of spiral or lemniscatic (∞) movements of the Sun and Earth. Especially important and thorough presentations include those of October 1, 1916 (GA 171); April 10, 1920 (GA 201); and January 2 and 17, 1921 (GA 323).

    March 24 1905, GA 324a
    March 31 1905, GA 324a
    September 1 1906, GA 95
    September 16 1907, GA 101, 284/285
    April 29 1908, GA 98
    November 7 1910, GA 124
    March 2 1 1913, GA 145
    May 5 1914, GA 286
    July 13 1915, GA 159
    August 20 1916, GA 272
    October 1 1916, GA 171
    May 28 1918, GA 181
    September 4 191, GA 295
    September 25 1919, GA 300a
    September 26 1919, GA 300a
    September 28 1919, GA 192
    October 3 1919, GA 261
    October 3 1919, GA 191
    April 10 1920, GA 201
    April 11 1920, GA 201
    April 18 1920, GA 201
    May 1 1920, GA 201
    May 2 1920, GA 201
    October 15 1920, GA 324a
    January 2 1921, GA 323
    January 11 1921, GA 323
    January 12 1921, GA 323
    January 17 1921, GA 323
    January 18 1921, GA 323
    August 26 1921, GA 324a
    October 8 1921, GA 343
    January 5 1923, GA 220
    May 5 1924, GA 349

    Various attempts have been made to unite Rudolf Steiner's scattered indications into a consistent interpretation but to date, no view has successfully encompassed all of them. For some of the more significant efforts, see (in chronological order) Locher [1942], Hagemann [1966], Kaiser [1966], Schmidt [1966], Vetter [1967], Van Bemmelen [1967], Unger [1981], Bauer [1981, 1988], Hemming/Pinkall [1983], Hardorp [1983], Junge [1983], Rudnicki [1984], Adams [1989] (Chapter 4), and Vanscheidt [1992],

  6. The mechanical interpretation of the solar system that has been customary since Newtons time renders the assumption of a separate third Copernican movement "superfluous." That is, if the Earth is seen as an (almost) symmetrical top spinning in the Sun's gravitational field, then according to the law of the preservation of rotation, the direction L of the axis of rotation (Earth's axis) essentially remains fixed in space. This interpretation, derived from physics, of course, would have been foreign to Copernicus. Among his successors, only a very few authors lament the neglect of the third Copernican movement or even consider it a serious factor. On this subject, see C. L. Menzzer's informative note 36 on De Revoltitionibus, volume 1, chapter 11, "Bctveis von tier dreifachen Betvegung der Erde" ("Proof of the Threefold Movement of the Earth") (Copernicus [1879], appendix, p. 28-31). In this context, Rudolf Steiner's lecture of September 25, 1919 (GA 300a), also mentions the works of the poet and author Johannes Schlaf (1862—1941). See Schlaf [1914] and [1919]: both were found in Steiner's library, and the first contains a handwritten dedication by the author to Rudolf Steiner.

  7. Elisabeth Vreede (1879–1943), mathematician and astronomer and, from 1924 on, the first head of the Section for Mathematics and Astronomy of the School of Spiritual Science at the Goetheanum in Dornach. During this conference, Dr. Vreede gave two lectures (on October 1 3 and 14, 1920) on 'The Justification for, and Limits of, Mathematics in Astronomy" [1922],

  8. Vreede [1922], pp. 138ff and 160.

  9. Carl Unger (1878–1929), manufacturer, engineer, and philosopher. During this conference, he gave six lectures (October 11-16, 1920) on the subject of Rudolf Steiner's work [1921]. See also the report on these lectures by Willy Storrer in Linger [1921], especially sections III and IV.

  10. For more about the theory of relativity with regard to the passage that follows, see the question-and-answer session of March 7, 1920 and the corresponding notes and the question-and-answer sessions of March 31, 1920, and January 15, 1921.

  11. See the passage by Einstein quoted in Note 6 to the question-and-answer session of March 7, 1920. Steiner is referring here to a problem later known as the "paradox of the twins" or the "paradox of the clocks." Its interpretation, still controversial today, is related to the significance of the concept of time in physics, but more especially to the interpretation of a physical system's "own time" in the context of the theory of relativity. On this subject, see Gschwind [1986], for example, and the references listed there.

  12. According to Einstein [1917], §18, the special principle of relativity states that the universal natural laws of physics are formally identical for two systems of reference subject to uniform motion (inertial systems). Of course, this statement presupposes that inertial systems exist. Popular examples taken from elementary mechanics do not strictly satisfy most of the prerequisites, — hence, such examples fail to correspond to reality even from the perspective of physics.

    Thus, for example, the frame of reference "Earth" (like any rotating system) is an accelerated system, as is the frame of reference "car." Because it overcomes the resistance of friction, a uniformly moving car executes accelerated move* ment. Because of wear and tear, the car is not an unchanging system—the more so when it has a flat tire and its speed decreases. Similar considerations apply to the oft-cited example of the train and the railway embankment.

    The only examples of relativistic behavior that the field of physics considers realistic occur on the atomic or subatomic level, as Einstein [1917] also points out in his lecture. According to Steiner, however, the full reality of the realm of such phenomena cannot be grasped without extending physics in keeping with anthroposophical spiritual science (see the lectures of the first and second scientific courses, GA 320 and GA 321).

  13. Friedrich Wilhelm Bessel (1784–1846), astronomer, geodesist, and mathematician in Königsberg. Bessel made fundamental contributions to the techniques and technology of astronomical observation, including improvements in the instruments, systematic analysis of errors due to instruments and faulty observation, and thorough reduction of observations.

    Both instrumental errors and the influence of the Earth's atmosphere (refraction) must be eliminated when the location of a star is measured. Furthermore, for the sake of an objective standard that can be compared with other measurements, such locations must be calculated in terms of a common point in time, taking the effects of the observation point and the Earth's movement into account. Doing this requires an exact knowledge of precession, nutation (slight oscillation of the Earth's axis caused by the moon), and daily, yearly, and long-term aberration (caused by the ultimate/finite speed of light and apparent changes in the location of stars due to the Earth's movement).

    Bessel's analysis/utilization (reduction) of the positions of 3,222 stars obtained by James Bradley (1693—1762) of the Greenwich Observatory became a milestone in astronomical observation because it made precisely reliable star positions available for the first time. Bessel published his results in the books Fundamenta Astronomiae pro Anno 1755 Deducta ex Observationibus Viri Incomparabilis James Bradley in Specula Astrouomica Grenovicensi per Annas 1750-1762 Instituti (Königsberg [1818]) and Tabulae Regiomantanae Reductionum Observationum Astronomicum ab Anno 1750 usque ad Annum 1850 Computatae (Königsberg [1830]).

    Related studies by Bessel yielded improved methods of determining the independent movement of fixed stars and the first means of determining parallaxes of individual fixed stars. These parallaxes constituted the first astronomical proof of the yearly movement of the Earth (on this and other proofs of this movement, see Teichmann [1983], chapter 3.4). The so-called Bessel reduction formulas for star coordinates have to do with the yearly and long-term influences of precession and nutation. (For more on this subject, see Schmidt [1967]: Wolf [1890-1893], §609 and §613; and astronomical yearbooks such as The Astronomical Almanac, 198iff, p. §22 ff.)

  14. Albert Steffen (1884–1963), poet and, from 1924 on, the first head of the Section for Fine Arts/Arts and Letters of the School of Spiritual Science at the Goetheanum in Dornach. During this conference, Steffen gave two lectures (on October 14 and 15, 1920) on the subject of "Spiritual Science and Crisis in the Life of the Artist." Steffen published his own summary of these lectures in the collection Die Krisis im Leben des Kiiustlers ("Crisis in the Life of the Artist") [1922], See especially the essay of the same title in part II, pp. 3Iff.

  15. Set theory was founded almost single-handedly by the mathematician Georg Cantor (1845-1918). Cantor sent Rudolf Steiner a copy of his Lehre vom Transfiniten ("Theory of the Transfinite") [1890], complete with personal dedication and handwritten corrections. In a treatise dated 1884, Cantor gives this definition of a set: "In general I understand a "manifold" or "set" to be a group of multiple elements that can be thought of as a whole. It is the epitome of specific elements that can be lawfully united into a whole. I believe I have thus defined something related to the Platonic eidos, or idea ... (Cantor [1932], footnote to p. 204). Rudolf Steiner's remarks refer to Cantor's investigations of various levels of infinity. The basis for these studies is this definition, which Steiner paraphrases: "I understand the prime or cardinal number of a set \(S\) (which consists of distinct and conceptually separate elements \(s\), \(s’\), ... and is defined and delineated by them) to be the general or universal concept that we gain by abstracting from the set both the character of its elements and ail relationships of these elements either to each other or to other objects, and especially the order that may prevail among the elements, and reflect only on what is common to all sets that are equivalent to \(S\). I call two sets \(S\) and \(T\) equivalent, however, when each element of one can clearly be made to correspond to exactly one element of the other" (Cantor [1890], p. 23 ff. or [1932] p. 387). See also the essay entitled "Georg Cantor and Rudolf Steiner" ("Georg Cantor und Rudolf Steiner") in Beiträge Zur Rudolf Steiner Gesamtausgabe ("Articles on the Complete Edition of Rudolf Steiner's Work"), No. 114/115, Dornach, 1995.

  16. Oswald Spengler (1880–1936), originally a mathematician, later a writer. "Form and Actuality," the first volume of Spengler's principal work The Decline of the West, appeared in its first edition in 1918, and by 1920 had appeared in 32 printings. The second volume, "Perspectives of World History," which appeared in 1922, did not have as wide a readership. Decline of the West was first published in the U.S. in 1926—28.

  17. The second law of thermodynamics is based on the concept of entropy, which was first formulated by Rudolf Clausius (1822–1888). This concept states that entropy strives toward a maximum in any real thermodynamic process that takes place in a self-contained physical system. In the context of physics, proof of this law is possible only on the basis of other unprovable assumptions or postulates. For example, in the statistic kinetic gas theory dating back to James Clark Maxwell (1831–1879) and Ludwig Bolzmann (1844–1906) this second law takes the form of a provable theorem (Boltzmann's so-called H theorem) based on the hypothesis of complete molecular chaos.

  18. Count Hermann Keyserling (1880–1946), philosopher, author, and co-founder and scientific head of the "School of Wisdom" ( or "Society for Independent Philosophy") in Darmstadt. See his works, such as Das Reisetagebuch eines Philosophen ("A Philosopher's Travel Diary) [1919a], Der Weg der Vollendung: Des Grafen Hermann Keyserling philosophischen Schaffen ('The Path of Perfection: The Philosophical Activity of Count Hermann Keyserling") [ 1919b], and Philosophie als Kunst ("Philosophy as an Art") [1920],

  19. "’Keyserling, Philosophic als Kunst ("Philosophy as an Art") [1920], p. 293: The School of Wisdom must become a third element alongside the church (taking the word in the broadest possible nondenominational sense) and the university. Like each of these other elements, its intent is to shape the whole human being and spiritualize the human soul. In addition, however, it aspires to a synthesis between human soul life and the independent, fully conscious spirit, so that neither faith nor abstract knowledge is the final authority, but faith, knowledge, and life become one in a living, higher unity of consciousness crowned by the School of Wisdom, whose task would be to organically incorporate abstract academic knowledge into a living synthesis and to transform mere "knowing" into "being."

  20. Presumably Steiner is referring to the weekly magazine Die Zukunft ("The Future"), edited by Maximilian Harden (volumes 1-118, 1892-1922). To date, the essay by Hermann Keyserling that Steiner mentions has not been found.

  21. See also the discussions about Keyserling in the periodical Dreigliederung des sozialen Organismus (The Threefolding of the Social Organism"), volume 2, 1920/1921, nos. 20-25, especially the report by Ernst Uebli (1875–1959) on Rudolf Steiner's lecture of November 16, 1920, in nos. 21 and 22. Further comments about Keyserling can be found in Rudolf Steiners lecture of August 26, 1921, published in the periodical Gegenwart ("The Present"), volume 15, 1953-1954, no. 2, pp. 49-64.

  22. To date, the source of this statement by Keyserling has not been discovered.

  23. Goethe, Faust, Part II, Act 2, Scene 2, in the laboratory, verses 6989ff. Homunculus says to Wagner, who remains behind:

    Unfold the ancient parchments,
    As bidden, collect life's elements
    and join them carefully to each other,
    considering the What, but more the How.
    While I wander through a portion of the world,
    I will, no doubt, discover the dot upon the i.