Faculty Meetings with Rudolf Steiner
GA 300c
30 April 1924, Stuttgart
Sixty-Sixth Meeting
Dr. Steiner: The first thing I would like to discuss is my discussion today with the present twelfth-grade students. With one exception, the students stated they did not need to take their final examinations at the end of this year, but could wait a year. At the end of the Waldorf School, they would go through a cramming class. It was important to them, however, that this cramming for the final examination be taught by the Waldorf School.
A teacher comments.
Dr. Steiner: The point is that we said we wanted to resolve this matter after meeting with the twelfth-grade students. We cannot handle such things if someone comes afterward and says there is still one more thing. If arguments are always presented about everything after it is done, then we will never finish anything. Things will only become confused. How is it that now there are suddenly two? Where did that come from? The problem is, that was overlooked. It makes no sense that such things occur suddenly. Is the faculty in control, or the children? The results should remain as they were today at noon, and that girl will need to have some sort of private instruction. In general, we should teach the class in a way appropriate to a twelfth-grade Waldorf School class.
The first thing we need to consider for the curriculum is literary history. Yesterday, I mentioned that, in general, they should have already covered the main content of literary history. A cursory survey will have to suffice for the things they have not learned. On the other hand, you should undertake a complete survey of German literary history in relation to things that play into it from outside.
Therefore, you have to begin with the oldest literary monuments and work them all into an overview. Begin with the oldest literary monuments, starting with the Gothic period, then go on to the Old German period and continue into the development of the ,em>Song of the Nibelungs and Gudrun. Do that in a cursory way, but so that they get a picture of the whole. Then, go on to the Middle Ages, the pre-classical period, the classical and romantic periods, up to the present. Give them an overview, but one that contains the general perspectives. The content should enable them to clearly know what they need to know about such people as Walther von der Vogelweide, Klopstock, or Logau. I think you could cover that in five or six periods. You can certainly do that.
I would then follow that with the main things they need to know about the present. You should discuss the present in much more detail with the twelfth grade. By present, I mean you would discuss the most important literary works of the 1850s, 60s, and 70s, then follow that with a more detailed treatment of the subsequent movements, so that they would have some insight into who Nietzsche and Ibsen were, or such foreigners as Tolstoy and Dostoyevsky, and so forth. The result should be that we graduate well-educated people.
Next is history, which you should do in a similar way. Start with a survey of history as a whole, beginning with the history of the East, which then gives rise to Greece and more modern Christian developments. You can surely go into these things without teaching anthroposophical dogma. You can present things that have a genuine inner spirituality. At the workers’ school, for example, I once showed how the seven Roman kings followed the model of the seven principal aspects of the human being, since that is what they are. Of course, you cannot simply say that Romulus is the physical body, and so forth. Nevertheless, Livius’s History of Kings has that in its inner structure. We find that the fifth king, Tarquinius Priscus, is clearly a person of intellect, corresponding to the I. He brings a new impulse, just as with the spirit self, the Etruscan element. You should treat the last one, Tarquinius Superbus, such that the highest we can reach sinks in most deeply, as it, of course, did with the Roman people, where it sunk into the Earth.
In the same way, you can very beautifully develop oriental history. In Indian history, we find the formation of the physical body, in Egyptian history, the etheric body, and in Chaldaic- Babylonian history, the astral body. Of course, you cannot teach it in that form. You need to show how those human beings living in the astral developed astronomy, how the Jews have the principle of the I in the principle of Yahweh, and how the Greeks for the first time developed a true understanding of nature from a human perspective. The viewpoint of the earliest peoples was still within the human being. You could give them an overview you can be proud of. Historical events form a complete series.
Geography class will also consist in giving them an overview. In both history and geography, what is important is to give them an overview. They can then search out the details by themselves.
You could divide aesthetics and art class as we discussed yesterday: into symbolic, classical, and romantic art. You could also treat not only the science of art by saying that in Egypt it was symbolic, in Greece classical, and in what followed, romantic, but also, the arts themselves, in that architecture is a symbolic art, sculpture is a classical art, and painting, music, and poetry are the romantic arts. Thus you can view the arts themselves in a way that offers a kind of inner division.
In teaching aesthetics and art, you can treat the elements of architecture so that the young people will have a proper understanding of how a house is constructed, that is, you could include construction materials, the construction of a roof, and so forth, in aesthetics.
Then we have languages. There, it is better if we describe the goals by saying that in English or French the students should get an idea of modern literature.
Now we have mathematics. How far did the eleventh grade come in mathematics?
A teacher: In the eleventh grade we got as far as indeterminate equations in algebra. In trigonometry, aside from spherical trigonometry, they went as far as computing acute-angle triangles. In complex numbers, as far as Moivre’s theorem, then polynomial equations. In analytic geometry, we went as far as working with second-order curves, but we worked in depth only with the circle. In constructive geometry, we did sections and intersections.
Dr. Steiner: Our experience with last year’s class has shown that we cannot do it that way. It is too much for the human soul to do such things.
What is important is to go through spherical trigonometry, that is, the elements of analytical spatial geometry, in a way that is as clear as possible.
In descriptive geometry we have Cavalieri’s perspective. The students should be able to draw a complicated form, such as a house, in Cavalieri’s perspective. The inside as well as the outside.
In algebra, you need only cover the beginnings of differential and integral calculus. They do not need to be able to compute maximums and minimums. They will learn that in college. You should teach them only the basic concepts of calculus, but do that thoroughly.
You should emphasize spherical trigonometry and how it is used in astronomy and geodesy in a way appropriate to their age, so that they have a general understanding of it.
Spatial analytical geometry should be used to teach them how equations can express forms. I would not be afraid to complete this subject by giving them examples of questions like, What curve is represented by the equation
$$x^\frac{2}{3} + y^\frac{2}{3} + z^\frac{2}{3} = a$$which results in an astroid. The main thing is to make equations so transparent that the students have a feeling for how things are hidden within equations.
You should also do the opposite. If I draw a curve or place a body in space, they should be able to recognize the general form of the equation without necessarily having it correct in all details, but at least have an idea of what the equation would be.
I don’t think the normal mathematical education that connects differential and integral calculus with geometry is particularly useful. I think it should be connected with quotients instead. I would begin with the quotient
$$\frac{y}{x}$$then make the dividend and the divisor smaller and smaller, simply as numbers, and then go on to develop differential quotients. I would not begin with the idea of continuity, because you do not really get an idea of differential quotients that way. Don’t begin with differentials, but with differential quotients. If you begin with a series, then go on to geometry only after you have presented tangents, that is, move from the secant to the tangent. Go on to geometry only after the students have completely comprehended differential quotients purely as numbers or through computations, so that they are presented with the picture that geometric visualization is only an illustration of what occurs numerically. You can then teach them integrals as the reverse process. Thus, you will have a possibility of showing them that the computation is not a fixing of geometry, but that geometry is an illustration of the computation. That is something people should consider more often. For example, you should not consider positive and negative numbers as something in themselves, but as a series of numbers such as
$$(5 - 1), (5 - 2), (5 - 3), (5 - 4), (5 - 5), (5 - 6)$$In the last instance, I do not have enough, I am missing one, and I write that as (-1). Emphasize only what is missing without using a number line. You will then remain within numbers. A negative number is the amount that is not present. It is a deficiency of the minuend. There is much more inner activity in working that way. You can excite some of the students’ capacities in a much more real way than when you do everything beginning from geometry.
A teacher: Where should we begin?
Dr. Steiner: Now that the class is ready for spherical trigonometry, you will need to move from trigonometry to developing the concept of the sphere qualitatively, that is, without starting computations. Instead of drawing on a plane, they need to begin drawing on a sphere, so that they get an idea of what a spherical triangle is, that is, how a triangle lies upon a sphere. You need to make that visible for the children, then go on to show them how the sum of the angles is not equal to 180°, but is larger. They need to really understand triangles on a sphere, with their curved lines, and then begin the computations. In geometry, the computation is only the interpretation of the sphere. I do not want you to begin by considering the sphere from its midpoint, but from the curvature of the surfaces. Then you can go on to a more general discussion of the non-linearity, how you could look at a corresponding figure on an ellipsoid, or how it would look on a paraboloid, where it is no longer completely closed. Don’t begin with the center, but with the distortion of the surface; otherwise you will have difficulties with other solids. In a way, you will need to think of yourself on the surface; in a sense, you will have to form a picture of what you would experience if you were a spherical triangle. You need to ask yourself, What would I experience as a triangle on an ellipsoid?
In that connection, you will also have to show the students what would happen if you used the normal Pythagorean theorem on a spherical triangle. You cannot, of course, use squares for that. Doing things this way has an effect upon the general education, whereas normally they affect only the intellect.
You can cover permutations and combinations quickly, and, if there is enough time, the beginnings of probability theory, for instance, the life expectancy of a human being.
In the eleventh grade, you need to go through sections and intersections, shadows and indeterminate equations, and analytical geometry up to conic sections. In eleventh-grade trigonometry, teach the functions in a more inner way, so that you present the principle relationships in sine and cosine. There, of course, you will have to begin from geometry.
Begin twelfth-grade physics with optics, as we discussed yesterday.
Natural history. We have already discussed zoology. In geology and paleontology, begin with zoology, since only then do they have some inner value. You can begin with zoology, go on to paleontology, and arrive at the various layers of the Earth. In botany, you can begin with flowering plants (phanerogans), and then also go on to geology and paleontology.
Chemistry. We want to consider chemistry in its innermost connections to the human being. In the twelfth grade, our students already have an idea of organic and inorganic processes. It is now important to go on to those processes found not only in animals, but also in human beings. We can speak without hesitation about the formation of ptyalin, pepsin, and pancreatin. You should teach the metallic processes in the human being by developing things from principles, for instance, something we could call the lead process in the human being, so that the students understand them. You need to show that within the human being all materials and processes are completely transformed. In connection with the formation of pepsin, what is important is to begin with the formation of hydrochloric acid, showing that it is lifeless. Then go on to consider the formation of pepsin as something that can occur only within the etheric body, even though the astral body has some effect upon it. In other words, show how the process completely disintegrates and then is rebuilt. Begin hydrochloric acid, with the inorganic process using salt. Discuss all the characteristics of hydrochloric acid, then go on to show how that differs from what occurs in an organic body. The result should be the demonstration of the differences between vegetable protein, animal protein, and human protein, so the students have an idea that there is a progression of protein based upon the various structures of the etheric body. Human protein is different from animal protein. You can also begin with differences by looking at a lion and a cow. In the lion, we find a process that is much more directed toward the circulation than in a cow where the entire process is more directed toward the metabolism. In the lion, the metabolic process is formed together with the breathing, whereas in the cow, the breathing is supported by the digestion. This will enliven the processes more. You need to have an inorganic, an organic, an animal, and a human chemistry. Some examples for children might be hydrochloric acid and pepsin, or blackthorn juice and ptyalin. Then they will get the picture. You could also use the metamorphosis of folic acid into oxalic acid.
A teacher asks whether to include quantitative chemistry.
Dr. Steiner: Well, it is certainly very difficult to explain these things with what you can normally assume. You need to begin with cosmic rhythm to explain the periodic system. That is the way you need to go, but you cannot do that in school. It is complete nonsense to begin with atomic weights; you need to begin with rhythms. You can explain all of the quantitative relationships through harmonics. The relationship between oxygen and hydrogen is, for example, an octave. But, that would go too far. I think you should develop the concepts we mentioned before and that will be enough for the twelfth-grade curriculum.
Eurythmy is not intended for the final examination.
Religion class. In general, the character of religious instruction is already in the curriculum. I can certainly not add much to what you have already presented. There is nothing we really need to change. The question is what to do in the upper grades. In the end, you should be able to give the twelfth grade a survey of world religions, but not in a way that gives the children the idea that some of them are untrue. Instead, you need to show the relative truths in their individual forms. That would be the ninth level. In the eighth level, you need to go through Christianity so that it appears in the ninth level as the synthesis of religions. Develop Christianity in the eighth level, and in the ninth level emphasize world religions so that, once again, their high point is Christianity. In the seventh level, you should present a kind of evangelical harmony, present Christianity in its essence and in the way it appears. By then, the children will all know the Gospels. Therefore, at the seventh level, a harmony of Gospels, at the eighth, Christianity, and at the ninth, world religions.
I will prepare the curriculum for modern languages in the ninth through twelfth grades and give it to you at a meeting about the foreign language classes.
There is a discussion about the university classes in Stuttgart.
Dr. Steiner: I would like to hear whether you think what has been proposed for the courses is too much or not. I would like to hear what you expect. What you thought of for the course that is just beginning and will continue until the next summer vacation? If we want to avoid a terribly chaotic situation, we certainly should not do things more than five days a week. I thought of doing a five-lecture series; Wednesday and Friday are not available. I could give lectures on Monday, Tuesday, Thursday, and Saturday, and two on one day.
I think we should present only five areas. We cannot present social understanding yet. It would also be very good to teach some practical subject, say, geodesy. We don’t want to have any specific themes. I think Dr. Schwebsch could teach aesthetics and literature; Stein, history; Unger, epistemology; Baravalle, mathematics; and Stockmeyer, geodesy.
It seems that one error has been that there is too much lecturing. Sometime we will also need to present something about music theory. We should do that in the course next winter. So that there will be a certain amount of liveliness, I propose that wherever possible, you bring the most recent events into the discussion. It would be good, for example, to work through our perspective on aesthetics as I discussed in the two little essays. Since there is only one lecture per week, you can only give a sketch. You should, for instance, handle the theme “Beauty arises when the sense-perceptible receives the form of the spirit” as I did that in my essay “Goethe as the Father of a New Aesthetic.” You could show that for the various arts, for architecture, painting, and so forth. In literature, I think you should discuss the most recent publications, namely, how Ibsen, Strindberg, and so forth reveal an unconscious movement toward a certain kind of spirituality, and then also, of course, the pathological, like for instance, Dostoyevsky.
Marie Steiner: Shouldn’t we also discuss Morgenstern, Steffen, and Steiner?
Dr. Steiner: You could extend Steffen’s characterization of lyrics.
In history, you could present an overview of the period from 1870 until 1914, stopping at that point. People would leave with rather long faces saying that you have only gotten to the World War and now they need to give some thought to the war itself. Go only to the assassination at Sarajevo.
In mathematics, you will have to orient yourselves by what was presented previously. I think it is important to treat the most important mathematical things. (Speaking to Dr. von Baravalle) You could present the things you have in your dissertation. It would also be very good if you developed mathematical concepts, such as those of normal functions or elliptic functions, in a visual way. Don’t just drone on about formal mathematics. Present how things are qualitatively. It would also be good to use that as a starting point to go into the entirety of relativity theory, how it is justifiable or not. I think people should have an idea of the following: You could present the question of relativity theory through the example of a cannon that is shot in Freiburg. It can be heard at some distance and you can compute the distance. You would then go on to compute how the time would change if you moved toward or away from the noise. The time it takes to hear the noise would lengthen if you moved from Karlsruhe to Frankfurt. If you then moved in the opposite direction the time would shorten until it was zero when you heard the cannon in Freiburg itself. You could then continue past Freiburg, so that you would have to hear the cannon before it was shot. That is the basic error of the theory of relativity. It can’t be so difficult to develop this mathematical concept of movement.
I think the problem with these courses is that they are actually unnecessary. With some differences, you have simply continued what other popular lectures offer, which is unnecessary; there is no real need for them.
What is important in geodesy is to get away from presenting a copy of the Earth. For example, if you begin, as people do, to try to avoid error through differential methods, you will need to explain geodetic methods to a certain extent. You will then have asymptotic methods. You could then discuss to what extent human beings depend upon approaching only certain things. You can show how extremely useful it is not to think in a determined way about some things, such as the character of a human being, but to think in a way similar to the way you measure with a diopter, where there is always some small difference. You can come closer to the truth in that way than you can when you state everything in specific words. We should characterize people only by looking at them from one side and then another. A person can be a choleric and a melancholic at the same time. This is the perspective you should bring to the fore. If you use geodesy as a basis for explaining the problems of the Copernican system, you can achieve a great deal.
You should form the lectures series by using such titles as: “What Can Aesthetics and Literature Add to Life?”; “What Can History Add to Life?”; “What Can Epistemology Add to Life?”; “What Can Mathematics Add to Life?”; and “What Can Geodesy Add to Life?” Under that, you could put “The Board of Directors of the Anthroposophical Society and the Faculty of the University Courses,” and above it, as a title, “Goetheanum and University Courses.”
These proposals are being made to you from Dornach.
Sechsundsechzigste Konferenz
RUDOLF STIANER: Das Erste, was ich besprochen haben möchte, ist das im Anschluss an die gestrige Besprechung mit den Schülern der 12. Klasse. Die Schüler, mit Ausnahme einer Einzigen, haben erklärt, dass sie keinen Wert darauf legen, schon nach Ablauf des nächsten Jahres Abiturium zu machen, sondern eventuell erst, wenn sie noch ein Jahr nach Ablauf der Waldorfschule in einer Art Presse [vorbereitet worden sind]. Sie haben aber Wert darauf gelegt, dass dieser Press-Unterricht an der Waldorfschule selber erteilt wird.
jemand bemerkt wohl hier, dass es zwei Schüler seien.
RUDOLF STEINER: Die Hauptsache ist dies, dass wir besprochen haben, dass wir diese Frage nach der Konferenz mit den Schülern der 12. Klasse erledigen wollen. Kein Gegenstand kann. so behandelt werden, dass man hinterher kommt und sagt: Es ist noch einer [mehr]. Wenn alle Dinge SQ besprochen werden, dass irgendetwas gemacht wird, und hinterher in derselben Sache Argumente gemacht werden, dann kommen wir nie zu einem Abschluss. Dann geht eine Schlamperei in die Sache. Woher kommt das, dass [es] jetzt plötzlich zwei sein sollen? Woher kommt das? Die Hauptsache ist, dass dies übersehen wurde. Es hat keinen Sinn, dass solche Dinge auftauchen. Ist das Kollegium maßgebend oder die Kinder? Es muss bei [dem] Resultat bleiben, das heute Mittag gewesen ist, und das eine Mädchen muss in irgendeiner Weise durch Privatunterricht die Sache so bekommen.
Im Übrigen wollen wir die Klasse so einrichten, wie sie als 12. Waldorfschulklasse in Betracht kommen könnte.
[Für den Lehrplan] würde in Betracht kommen erstens der Unterricht in Literaturgeschichte. Ich habe gestern angedeutet, weit im Wesentlichen der Inhalt der Literaturgeschichte absolviert sein müsste, [dass es für] die Dinge, die nicht durchgenommen worden sind, genügen müsste, wenn sie einfach kursorisch im Überblick durchgenommen würden. Dagegen müsste ein vollständiger Überblick über die deutsche Literaturgeschichte im Zusammenhang mit [hereinspielenden] anderen Dingen an entsprechender Stelle auftreten.
Man müsste [also] bei den ältesten Literaturdenkmälern beginnen [und] das alles in einer Überschau behandeln. Die ältesten Literaturdenkmäler: [richtig] anfangen bei der gotischen Zeit, übergehen zur altdeutschen Zeit und [zu] der ganzen Entwicklung bis zum Nibelungenlied [und zur] Gudrun; kursorisch, sodass eine Vorstellung davon entsteht, vorn Ganzen. Dann das Mittelalter, [dann] vor-klassische Zeit, klassische Zeit, romantische Zeit bis Gegenwart; ein überblick, aber ein solcher Überblick, dass man nun wirklich in den allgemeinen Gesichtspunkten [und in der Übersicht] etwas hat von Inhalt — der Inhalt hat, sodass prägnant herauskommt das, was eigentlich der Mensch für das Leben braucht, um etwas zum Beispiel über Walther von der Vogelweide, über Klopstock, über Logau zu wissen. Das ist etwas, wovon ich mir denke, dass es in fünf bis sechs Stunden bewältigt werden könnte.
Dann würde sich daran anschließen müssen hauptsächlich die Behandlung der Gegenwart. Die Gegenwart würde dann für diese [älteste] Klasse etwas ausführlicher zu behandeln sein. Unter Gegenwart stelle ich mir vor, dass eine kürzere Behandlung da sein würde für wichtigere Literaturdenkmäler der Fünfziger-, Sechziger- und Siebzigerjahre, dass aber die jüngsten nachfolgenden Bestrebungen etwas ausführlicher behandelt werden, sodass die jungen Leute eine Einsicht bekommen würden in dasjenige, was Nietzsche ist, was Ibsen ist, auch [was] Tolstoi, Dostojewski [und so weiter bedeuten), sodass sie als gebildete Menschen bei uns herauskommen.
Dann würde Geschichte kommen. Da ebenso: ein Überblick über das ganze geschichtliche Leben, sodass man die orientalische Geschichte vorangehen lässt und über das Griechentum heraufkommt zur neueren christlichen Entwicklung. Man kann da durchaus dann Dinge hineingeben — nicht wahr, ohne dass man anthroposophische Dogmatik lehrt —, man kann durchaus Dinge hineinbringen, die also ja wirklich innere Spiritualität haben. Ich habe zum Beispiel einmal in der Arbeiterbildungsschule [entwickelt, wie] die sieben römischen Könige ganz nach den sieben Prinzipien des Menschen aufgebaut sind, [denn das sind sie]. Natürlich [darf man] nicht in äußerlicher Weise [sagen], Romulus ist der physische Leib und so weiter. Aber das innere Gefüge der Liviu.s-Königsgeschichte ist so, dass man im Aufbau dieses hat, dass in Tarquinius Priscus, dem Fünften, der ein ausgesprochener Intellektmensch ist — [der] entspricht dem Ich, dem Ich-Prinzip —, dass bei diesem [ei] neuer Einschlag kommt [wie beim Geistselbst, nämlich durch] das etruskische Element. Und der [letzte], Tarquinius Superbus, muss so [behandelt werden], dass das Höchste, was erreicht wird, am tiefsten heruntersinkt, wie es natürlich ist, beim römischen Volk, dass das in den Erdboden heruntersinkt.
Ebenso baut sich auf in einer sehr schönen Weise die Entwicklung der orientalischen Geschichte: Die indische Geschichte, da haben wir eine Ausgestaltung des physischen Leibes, in der [ägyptischen] Geschichte des Ätherleibes, in der chaldäisch-babylonischen des Astralleibes. Aber man kann es natürlich nicht in dieser Form geben, sondern zeigend, wie die im Astralischen lebenden Menschen Sternenwissenschaft haben, wie die Juden das Ich-Prinzip im Jahweprinzip haben, und wie die Griechen zum ersten Mal, [aus dem Menschen herausgehend], eine [wirkliche] Naturanschauung haben; [die Früheren stehen noch im Menschen darin]. Man kann einen Überblick geben, der schon wirklich sich zeigen kann. Nun, die historischen Ereignisse reihen sich vollständig an.
Dann würde der Geografieunterricht ebenso darin bestehen, einen Überblick zu geben.
In Geschichte und Geografie wäre überhaupt nur ein Überblick zu geben; Einzelheiten kann der Einzelne sich suchen, -wenn er den Überblick über das Ganze hat.
In Ästhetik und Kunstunterricht wurde über die Gliederung gestern schon gesprochen, symbolische, klassische, romantische Kunst. Nun hat man da die Möglichkeit, sowohl die Kunstwissenschaft so zu behandeln, Ägypter symbolisch, Griechen klassisch, darauffolgende romantisch, aber auch die Künste selber, indem die Architektur die symbolische Kunst ist, die Plastik die klassische Kunst, und Malerei, Musik und Dichtung sind die romantischen Künste. Also man kann die Künste selbst auch wiederum so betrachten. Das gibt die Möglichkeit einer inneren Gliederung.
[Dann ist] in Ästhetik und Kunstunterricht [die] Architektur [zu behandeln], die [Elemente und Anfangsgründe der Baukunst, [wobei] man so weit kommt, dass die jungen Leute einen ordentlichen Begriff haben, wie ein Haus konstruiert wird. [Also] Baumaterial, Dachkonstruktionen [und so weiter] in der Ästhetik.
Dann [neuere] Sprachen. Da tut man besser, wenn man die Ziele angibt, wenn man sagt, es sollten die Betreffenden für Englisch und Französisch eine Vorstellung gewinnen vom gegenwärtigen Stande der Literatur.
Nun, dann wäre Mathematik. In der Mathematik sind wir in der 11. Klasse wie weit gekommen?
KARL STOCKMEYER: In der 11. Klasse bis [zu den diophantischen] Gleichungen in der Algebra, Trigonometrie außer der sphärischen, bis zur Berechnung des schiefwinkligen Dreiecks. Komplexe Zahlen bis zum Moivre'schen Lehrsatz. Dann Einheitsgleichungen, In der analytischen Geometrie bis zur Behandlung der Kurven zweiten Grades, skizzenhaft, ordentlich nur der Kreis. In der darstellenden Geometrie Schnitte und Durchdringungen.
RUDOLF STEINER: Gerade der Unterricht, wie er im vorigen Jahr in der 12. Klasse gemacht wurde, hat gelehrt, dass man es so eigentlich nicht machen kann. Es ist für die menschliche Seele etwas Ungeheuerliches, so etwas zu machen.
Es handelt sich darum, in einer möglichst durchsichtigen Weise durchzunehmen sphärische Trigonometrie, die Elemente der analytischen Geometrie des Raumes.
Dann in der Deskriptiven [die] Kavalierperspektive; die Schüler sollten es [doch] dahin bringen, dass sie eine kompliziertere Hausform in Kavalierperspektive darstellen könnten und auch das Innere des Hauses.
In Algebra ist es notwendig, dass man nur die allerersten Anfangsgründe der Differenzial- und Integralrechnung nimmt. Man braucht nicht bis Maxima- und Minimarechnung zu kommen. Das gehört schon in die Hochschule. Nur den Begriff von Differenzial und Integral soll man geben und den ordentlich herausarbeiten.
Man sollte Wert darauf legen, [dass die] sphärische Trigonometrie und ihre Anwendung auf Astronomie und höhere Geodäsie getrieben wird in einer ganz dem Alter angemessenen Weise, sodass das im Ganzen und Großen begriffen wird.
Analytische Geometrie des Raumes sollte verwendet werden, um also anschaulich zu machen, wie Formen in Gleichungen ausdrück-bar sind. Ich würde da nicht zurückschrecken, den Unterricht gipfeln zu lassen darin, dass zum Beispiel begriffen werden kann, was das für eine Kurve (Fläche?) ist:
$$x^\frac{2}{3} + y^\frac{2}{3} + z^\frac{2}{3} = a$$
Das gibt ein Asteroid. Sodass möglichst viel allgemeine Bildung he-reinkommt. Vor allen Dingen auch Gleichungen durchschaubar zu machen, dass man ein Gefühl dafür kriegt, wie in den Gleichungen eigentlich die Dinge drinnenstecken.
Umgekehrt sollte man auch das besonders pflegen: Ich zeichne eine Kurve auf oder in den Raum hinein oder [einen Körper] in den Raum [hinein], dass man dann, ohne dass die Gleichung auf das i-Tüpfelchen zu stimmen braucht, [die Gleichung aus den Formen erkennt], dass man Sinn für die Gleichung habe.
Ich halte es für die allgemeine mathematische Bildung nicht für nützlich, wenn Differenzial- und Integralrechnung angeschlossen wird an die Geometrie, sondern wenn sie angeschlossen wird an den Quotienten. Ich würde ausgehen von der Differenzenrechnung, also von
\frac{\Deltay}{\Deltax}
würde das als Quotienten auffassen, und würde [nur] durch [das] Immer-kleiner-Werden von Dividend und Divisor, rein aus der Zahl
heraus, dazu [über]gehen, den Differenzialquotienten zu entwickeln. Ich würde nicht von diesem Kontinuitätsverhältnis ausgehen, dadurch bekommt man keinen Begriff vorn Differenzialquotienten; nicht ausgehen vorn Differential, sondern vom Differenzialquotienten. Wenn Sie von Reihen ausgehen, [dann] zuletzt erst am Tangentenproblein übergehen zur Geometrie, also von der Sekante zur Tangente übergehen. Und wenn der ganze Differenzialquotient begriffen ist, rein zahlenmäßig, rechnungsmäßig, von da erst übergehen zum Geometrischen, sodass der Schüler die Auffassung bekommt, das Geometrische ist nur [zuletzt] eine Illustration des Zahlenmäßigen. Dann bekommen Sie die Integrale als Umkehrung. Dann bekommen Sie die Möglichkeit, nicht davon auszugehen, dass [die] Rechnung eine Fixierung ist der Geometrie, sondern dass [die] Geometrie eine Illustration ist für die Rechnung. Das sollte man allgemein mehr beachten. [Man sollte] zum Beispiel die positiven und negativen Zahlen nicht als etwas an sich betrachten, [sondern] man sollte die Zahlenreihe nehmen so: 5 — 1, 5 — 2, 5 — 3, 5 — 4, 5 — 5, 5 — 6, jetzt habe ich nicht genug, weil mir eins fehlt, das schreibe ich als —1. [Das Fehlende betonen ohne Zahlenlinie.] Dann bleiben Sie im Zahlenmäßigen. Die negative Zahl ist die nicht vorhandene Menge, der Mangel des Minuenden. [Darin ist] viel mehr innere Aktivität! Dadurch hat man die Möglichkeit, beim Schüler Fähigkeiten anzuregen, die viel realer sind, als wenn man alles nur von der Geometrie her macht.
HERMANN VON BARAVALLE: [Wo soll ich anfangen?]
RUDOLF STLENER: [ Da Stockmeyer bis an die sphärische Trigonometrie gekommen ist], muss man übergehen [von der Trigonometrie] zum Entwickeln des Begriffs der Sphäre, qualitativ, ohne gleich auf Rechnerei auszugehen. Statt auf der Ebene zu zeichnen, muss man auf der Kugel zeichnen, sodass sie den Begriff des sphärischen Dreiecks bekommen, den Begriff des auf der Sphäre liegenden Dreiecks,
ERNST LEHRS: Das wäre das Altmodische,
RUDOLF STEINER: Das muss man den Kindern anschaulich machen, Dann, [dass da] die Winkelsumme ungleich 180 Grad ist, dass sie größer ist. Diesen Begriff muss man ihnen wirklich beibringen, das Dreieck auf der Sphäre mit krummen Begrenzungen. Daran anschließend erst die Berechnung. In der Geometrie ist die Rechnung [die] Interpretation der Sphäre.
Ich möchte, dass Sie nicht die Sphäre betrachten vorn Mittelpunkte der Kugel aus, sondern von der Krümmung der Fläche aus, sodass Sie auch übergehen können gleich in eine allgemeine Besprechung, zum Beispiel zu der Krümmung und dazu, wie auf einem
Ellipsoid die entsprechende Figur ausschauen würde, die auf der Kugel ein sphärisches Dreieck ist; wie sie ausschauen würde auf einem Rota.tions-Paraboloid, [wo] es nach beiden Seiten nicht geschlossen, sondern offen ist. Gehen Sie aus nicht vorn Mittelpunkt, sondern von der Krümmung der Fläche, sonst kommen Sie bei anderen Körpern nicht aus. [Sie müssen sich selbst in der Fläche denken], müssen gewissermaßen sich die Vorstellung bilden: Was erlebe ich, wenn ich ein sphärisches Dreieck «abgehe»; was erlebe ich, wenn ich ein Dreieck «abgehe», das einem sphärischen Dreieck auf dem Ellipsoid entspricht.
Dann würde ich die Schüler aufmerksam machen in diesem Zusammenhang, wie es sich ausnimmt, wenn man den gewöhnlichen Pythagoras anwendet auf das sphärische Dreieck. Man kann natürlich nicht Quadrate nehmen. Diese Dinge tragen zur allgemeinen Bildung bei, während sie sonst nur den Verstand ausbilden.
Permutationen, Kommutationen [Kombinationen?], [das] ist schon genommen worden. Wenn Zeit ist, dann die ersten Elemente der Wahrscheinlichkeitsrechnung; wahrscheinliche Lebensdauer eines Menschen zum Beispiel. Ein bis zwei Stunden nur in der 11. Klasse.
[Für die] 11. Klasse [kommt in Betracht] Schnitte und Durchdringungen, Schattenkonstruktionen, diophantische Gleichungen, analytische Geometrie bis zu den Kegelschnitten. [In der Trigonometrie in der I I. muss man] die Funktionen mehr innerlich nehmen, dass man das Prinzip des Verhältnisses im Sinus und Cosinus drinnen hat. Da muss man natürlich vom Geometrischen ausgehen.
[In der] Physik [der 12. Klasse] Optik wie gestern besprochen.
Naturgeschichte. Zoologie ist schon besprochen. Bei der Geologie und Paläontologie von der Zoologie ausgehen, nur dann hat es einen inneren Wert. Von [der] Zoologie gellt man über in [die] Paläontologie und kommt dadurch auch als Zugabe auf die Erdschichten.
[In der] Botanik Phanerogamen, [und von] da geht man [auch] über in die Geologie und Paläontologie.
Chemie. Wir wollen einmal die Chemie im innigsten Zusammenhang mit dem Menschen betrachten. Es haben ja bei uns die Kinder in der 12. Klasse schon einen Begriff von organischen und unorganischen Prozessen. Nun würde es sich darum handeln, dass man wirklich heraufgeht bis zu den Prozessen, die sich nicht nur im Tier, sondern auch im Menschen finden, dass man rücksichtslos spricht von Ptyalin-, Pepsin-, Pankreatinbildung [und so weiter]. Die Metallprozesse im Menschen sollte man so nehmen, dass von dem Prinzipiellen etwas entwickelt wird, sagen wir, was man nennen kann einen Prozess von Blei im Menschen, dass sie das verstehen. Man muss zeigen, dass alle Stoffe und Prozesse [vollständig] umgewandelt werden im Menschen. Bei der Pepsinbildung kommt es darauf an, dass man noch einmal ausgeht von der Salzsäurebildung, sie betrachtet als das Leblose, und die Pepsinbildung betrachtet als dasjenige, was nur innerhalb des Ätherleibes sich vollziehen kann, wo sogar der Astralleib hineinwirken muss. Also eine vollständige Abtragung des Prozesses und wiederum ein Aufbau, [das eine] durch Salzsäure; von dem unorganischen Prozess geht man aus, aus Kochsalz oder durch Synthese, bespricht die Salzsäure in ihren Eigenschaften. Dann versucht man hervorzurufen einen Unterschied zu dem, was nur im organischen Körper vorkommt. Gipfeln muss es im Unterschied zwischen pflanzlichem Eiweiß, tierischem Eiweiß, menschlichem Eiweiß, sodass ein Begriff von aufsteigendem Eiweiß da ist, [begründet in der verschiedenen Struktur des Ätherleibes]. Es ist das menschliche Eiweiß etwas anderes als das tierische Eiweiß. Sie können schon ausgehen vorn Unterschied [und] sagen zum Beispiel: Nun, nehmen wir an den Löwen und nehmen wir an die Kuh, so haben wir beim Löwen einen Prozess, der eigentlich viel mehr nach der Zirkulation zu liegt als bei der Kuh, wo der ganze Prozess mehr nach der Verdauungisseitel zu liegt. Der Löwe bildet den Verdauungsprozess mit dem Atmungsprozess sogar, während bei der Kuh der Atmungsprozess von der Verdauung aus [mit]besorgt wird. So werden die Prozesse selbst belebt. Man müsste eine anorganische, eine organische, eine animalische und eine menschliche Chemie haben. Für Kinder einige Beispiele: Salzsäure — Pepsin; Prunus-spinosa-Saft [und] Ptyalin. Dann kriegt man (schon] das, was gesagt werden soll, heraus. Oder Metamorphoseprozess Ameisensäure-Oxalsäure.
EUGEN KOLISKO fragt nach Quantitativem.
RUDOLF STEINER: Nun, nicht wahr, es ist halt außerordentlich schwer mit den Voraussetzungen, die man da [machen] kann, diese Dinge zu erklären. Man müsste ausgehen vom Weltenrhythmus, das periodische System aus dem Weltenrhythmus heraus erklären. Diesen Umweg muss man machen, der aber nicht in die Schule hereingehört. Überhaupt ist es ein Unfug, von Atomgewichten auszugehen. [Vorn Rhythmischen muss man ausgehen!] Die ganzen quantitativen Verhältnisse muss man aus den Schwingungen heraus erklären. Etwas wie eine Oktave zum Beispiel hat man im Verhältnis von Wasserstoff zu Sauerstoff. Das führt aber zu weit. Ich glaube, Sie sollten diese Begriffe entwickeln, die wir vorher erwähnt haben. Dadurch ist eigentlich der Lehrplan der 12. Klasse erschöpft.
Eurythmie ist nicht auf das Abitur zugestutzt gewesen.
Religion. Im Allgemeinen, dem Charakter nach, haben wir ja [den Lehrplan für] den Religionsunterricht gegeben. Nicht wahr, dasjenige, was Sie mir da mitgegeben haben, da kann ich eigentlich nichts Besonderes korrigieren. Da ist nichts Besonderes zu ändern daran. Es handelt sich ja wohl um die Oberklassen. Gipfeln müsste das darinnen, dass man in der 12. Klasse müsste durchnehmen können eine Übersicht über die Religionen der Welt, aber nicht so, dass man aus dieser Übersicht die Vorstellung hervorrufen soll, alle sind eigentlich unecht, sondern gerade, dass man ihre relative Echtheit durch die einzelnen Formen zeigt. Das wäre die neunte Stufe. — Die achte Stufe müsste das Christentum herausarbeiten, [sodass es in der neunten] als die Synthese der Religionen [erscheint]. Das Christentum müsste für sich herausgearbeitet werden im achten Abschnitt. Im neunten. Abschnitt [eine Übersicht über] die Weltreligionen, dass sie dann wiederum neuerdings nach dem Christentum hin gipfeln. Auf der siebenten Stufe müsste eine Art Evangeliertharmonie gegeben werden. Christentum für sich dargestellt in seinem Wesen, Erscheinungsform. Bis dahin kennen sie ja die Evangelien. Also siebente Stufe Evangelienharmonie. Achte Stufe [Christentum]. Neunte Stufe Weltreligionen.
Anmerkung: Damals war der freie Religionsunterricht eingeteilt wie folgt:
Erste Stufe = 1. und 2. Klasse; zweite Stufe = 3. und 4. Klasse; dritte Stufe = 5. Klasse; vierte Stufe = 6. Klasse; fünfte Stufe = 7. Klasse; sechste Stufe = 8. Klasse; siebente Stufe = 9. Klasse; achte Stufe = 10. Klasse; neunte Stufe = 11. und 12. Klasse.
[Den] Lehrplan [für die] neueren Sprachen in der 9. bis 12. Klasse [werde ich vorbereiten und Ihnen in einer] Konferenz über den Sprachunterricht [geben].
Es wird über die Hochschulkurse in Stuttgart gesprochen.
RUDOLF STEINER: Bezahlt müsste es schon werden. Ich möchte hören, ob das nicht gar zu sehr ins Fleisch greift, was für die Hochschulkurse vorgeschlagen wird. Ich möchte hören, was Sie erwartet haben. Was haben Sie sich gedacht für den nächsten Kursus, der ja jetzt beginnt und bis zu den großen Ferien gehen soll? Es sollten doch nicht mehr als wöchentlich fünf Tage besetzt werden, wenn nicht ein fürchterliches Chaos entstehen soll. Wir haben uns fünf Vorlesungsreihen gedacht, Mittwoch und Freitag sind ausgeschlossen. Vorträge können sein Montag, Dienstag,. Donnerstag, Samstag. An einem Tag können zwei nebeneinander sein.
Ich habe mir gedacht, dass nur fünf Gebiete behandelt werden. Nicht wahr, für die Hochschule kommt Dr. Mellinger für Soziale Erkenntnis in Betracht. Ich fürchte nur, dass sie die nächsten drei Monate noch verwenden wird zum Studieren. Nun will sie aber die drei Monate schon honoriert werden. Es wird für ihre Reihe jemand anders in Betracht kommen. Soziale Erkenntnis kann eben nicht sein vorläufig. Es wäre ganz gut, wenn einmal ein praktisches Fach stattdessen getrieben würde, zum Beispiel niedere und höhere Geodäsie. Bestimmte Themen wollen wir nicht stellen, Wir haben uns gedacht: Ästhetik und Literatur Schwebsch; Geschichte Stein; Erkenntnistheorie Unger; Mathematik Baravalle; Geodäsie Stockmeyer.
Ein Fehler scheint gewesen zu sein, dass zu viel vorgetragen worden ist, Es muss auch mal Musiktheorie vorgetragen werden. Die anderen Dinge nehmen wir im nächsten Kurs. Das muss im nächsten Winter geschehen. Damit also ein gewisser Zug hereinkommt, möchte ich vorschlagen, auf allen Gebieten, auf denen es sein kann, möglichst die neuesten Erscheinungen zu betrachten. In der Literatur zum Beispiel auch die Ästhetik einmal von unseren Gesichtspunkten aus durchzuarbeiten, wäre sehr schön; Ästhetik, wie ich sie skizziert habe in den beiden kleinen Schriften. In Ästhetik kann man, da nur alle Woche eine Stunde ist, bloß skizzieren. Behandeln müsste man den Satz: «Das Schöne entsteht, wenn das Sinnliche die Form des Geistigen bekommt», nach meinem «Goethe als Vater einer neuen Ästhetik». Das kann man zeigen für die verschiedenen Künste, Architektur, Malerei und so weiter, In der Literatur würde ich meinen, auch die neuesten Erscheinungen zu besprechen, namentlich das unbewusste Hineinschwimmen in eine gewisse Spiritualität bei Ibsen und Strindberg [und so weiter], und dann, nicht wahr, das Pathologische, das aber zum Genialen führt, bei Dostojewski [zul behandeln.
MARIE STEINER: [Sollte man nicht auch] Morgenstern, Steffen, Steiner [einmal behandeln?]
RUDOLF STEINER: Man kann so etwas weiter ausführen, was Steffen einmal charakterisiert hat, als er über Lyrik redete.
In Geschichte [die Zeit] von 1870 bis 1914 im Überblick [behandeln], sodass man gerade stehen bleibt, wo dann die Leute weggehen mit langer Nase und sagen, jetzt sind wir gerade bis zum Weltkrieg gekommen und können uns Gedanken machen über den Weltkrieg selber. Bis zum Attentat von Sarajewo.
Mathematik muss sich richten nach dem, was früher vorgetragen worden ist. Ich habe mir gedacht, dass es sich einmal darum handeln müsste, prinzipielle Sachen aus der Mathematik überhaupt vorzutragen. [Zu Hermann von Baravalle:] Sie können ganz gut die Sachen vortragen, die Sie in Ihrer Dissertation haben. Dann wäre es gut, wenn man den Begriff oder solche mathematischen Begriffe ganz anschaulich entwickeln würde, wie zum Beispiel den Begriff der gewöhnlichen Funktionen, der elliptischen Funktionen, aber nicht, indem man es mit allem verbrämt, was starre Mathematik ist, sondern indem man die Sachen qualitativ erörtert, wie die Sachen sind, und dann wäre es gut, wenn von da ausgegangen werden könnte, um einmal die [ganze) Relativitätstheorie in ihrer Berechtigung und Unberechtigung darzustellen. Ich glaube, die Leute sollten doch einmal einen Begriff bekommen von Folgendem:
Nicht wahr, man kann doch ein Problem der Relativitätstheorie so behandeln: Eine Kanone wird in Freiburg i. Br. abgeschossen, man hört sie in einiger Entfernung, man kann die Entfernung berechnen. Man geht dazu über, zu [be]rechnen, wie die Zeit sich ändert, wenn man sich dem Schall entgegenbewegt oder vorn Schall weg. Die Fortpflanzungszeit wird verlängert, wenn Sie von Karlsruhe nach Frankfurt sich bewegen. Dann, [wenn Sie sich] nach der anderen Richtung [bewegen, wird die Zeit verkürzt], bis Sie zu null kommen, wenn Sie [die Kanone] in Freiburg selber hören. Sie [können] über Freiburg hinausgehen, dann müssen Sie dazu kommen, die Kanone zu hören, bevor sie losgeschossen wird. Das ist der Grundfehler, der darin steckt. Diesen mathematischen Begriff des Fortschreitens noch in einem gewissen Sinne zu entwickeln, könnte nicht so schwer sein.
Ich meine, die Hochschulkurse hätten den Fehler, dass sie eigentlich unnötig wären. Man hat ein bisschen verändert sich an das gehalten, was sonst in populären Vorträgen auch geboten wird. Das ist nicht notwendig. Es ist ja auch kein Bedürfnis danach vorhanden.
In der Geodäsie handelt es sich darum, dass man abkommt davon, ein Nachbild [der Erde?) zu geben. Wenn Sie zum Beispiel in der Geodäsie davon ausgehen, wie man versucht, durch die Differenzmethode Fehler zu vermeiden, dann müssen Sie bis zu einem gewissen Grad geodätische Methoden erörtern, kommen zu Näherungsversuchen. Man kann da anschließen, inwiefern der Mensch darauf angewiesen ist, sich manchen Dingen nur zu nähern. Man kann zeigen, wie außerordentlich nützlich es ist, über Sachen wie den Charakter eines Menschen nicht bestimmt zu denken, sondern so zu denken, wie man mit [dem Diopter] misst, dass man sich [also] ein kleines Spatium lässt. Da sagt man viel mehr die Wahrheit, als wenn man [alles in] bestimmte Worte fasst. Man sollte den Menschen nur so charakterisieren, dass man ihn von der einen und von der anderen Seite fasst. [Der Mensch kann Choleriker und Melancholiker zugleich sein.] Man sollte einmal diesen Gesichtspunkt in den Vordergrund rücken. Wenn Sie niedere Geodäsie dazu verwenden, höhere Geodäsie dazu verwenden, die Problematik des kopernikanischen Systems zu erörtern, so wäre sehr viel [getan]. Diese Vorschläge machen wir ihnen von Dornach aus.
Man müsste die [ganze] Vortragsserie so einrichten, dass man den Titel gibt: Was gewinnt man für eine Lebensansicht durch Ästhetik und Literatur? — Was gewinnt man für eine Lebensansicht durch Geschichtsbetrachtung? — Was gewinnt man für eine Lebensansicht durch erkenntnistheoretische Betrachtung? — Was gewinnt man für eine Lebensansicht durch mathematische Betrachtung? —Was gewinnt man für eine Lebensansicht durch niedere und höhere Geodäsie?
[Darunter würde stehen]: «Der Vorstand der Anthroposophischen Gesellschaft und die Leitung des Lehrerkollegiums der Hochschulkurse». [Und] oben darüber [als Titel]: «Goetheanum- und Hochschulkurse». [...] [Einzelne Bemerkungen ohne Zusammenhang.]
Sixty-sixth Conference
RUDOLF STIANER: The first thing I would like to discuss is the follow-up to yesterday's meeting with the 12th grade students. With the exception of one student, the students have stated that they do not consider it important to take their Abitur exams at the end of next year, but possibly only after spending another year after graduating from Waldorf school in a kind of press . However, they did emphasize that they would like this press training to be provided at the Waldorf school itself.
Someone here has pointed out that there are two students.
RUDOLF STEINER: The main thing is that we have discussed that we want to settle this question after the conference with the 12th grade students. No subject can be treated in such a way that one comes back afterwards and says: There is one more. If all things are discussed SQ, that something is done, and afterwards arguments are made on the same subject, then we will never come to a conclusion. Then sloppiness creeps into the matter. Where does this come from, that suddenly there should be two? Where does that come from? The main thing is that this was overlooked. It makes no sense for such things to come up. Is the faculty decisive or the children? We must stick with the result that was reached at noon today, and the girl must be given private lessons in some way to learn the material.
Incidentally, we want to set up the class in such a way that it could be considered a 12th grade Waldorf school class.
First of all, literature history lessons would be considered [for the curriculum]. Yesterday I indicated that the content of literary history should essentially be completed, [that for] the things that have not been covered, it should suffice if they were simply covered in a cursory overview. On the other hand, a complete overview of German literary history in connection with [other relevant] things should appear at the appropriate place.
One would [therefore] have to start with the oldest literary monuments [and] cover everything in an overview. The oldest literary monuments: [correctly] starting with the Gothic period, moving on to the Old German period and [to] the entire development up to the Nibelungenlied [and to] Gudrun; briefly, so that an idea of the whole emerges. Then the Middle Ages, [then] the pre-classical period, the classical period, the Romantic period up to the present; an overview, but an overview that really gives you something in terms of content from a general perspective [and in the overview] — content that concisely conveys what people actually need to know for life, for example, about Walther von der Vogelweide, Klopstock, and Logau. That is something that I think could be covered in five to six hours.
This would then have to be followed mainly by a discussion of the present. The present would then have to be dealt with in somewhat more detail for this [oldest] class. By the present, I imagine that there would be a shorter treatment of the more important literary monuments of the 1950s, 1960s, and 1970s, but that the most recent subsequent endeavors would be dealt with in somewhat greater detail, so that young people would gain an insight into what Nietzsche is, what Ibsen is, and also [what] Tolstoy, Dostoevsky [and so on mean), so that they come out as educated people.
Then history would come. There, too, an overview of the whole of historical life, so that one starts with Oriental history and moves up through Greek civilization to the more recent Christian development. One can certainly include things there—without teaching anthroposophical dogma, of course—one can certainly include things that truly have inner spirituality. For example, I once developed in the workers' education school how the seven Roman kings are structured entirely according to the seven principles of the human being, for that is what they are. Of course, one must not say in an external way that Romulus is the physical body and so on. But the inner structure of Livy's history of the kings is such that in this structure, Tarquinius Priscus, the fifth, who is a pronounced intellectual — [who] corresponds to the ego, the ego principle — that with him [there] comes a new influence [as with the spirit self, namely through] the Etruscan element. And the [last], Tarquinius Superbus, must be treated in such a way that the highest that is achieved sinks down to the lowest, as is natural for the Roman people, that it sinks down into the ground.
The development of Oriental history also unfolds in a very beautiful way: in Indian history, we have a development of the physical body, in [Egyptian] history of the etheric body, in the Chaldean-Babylonian history of the astral body. But of course it cannot be given in this form, but rather by showing how people living in the astral realm have star science, how the Jews have the I principle in the Yahweh principle, and how the Greeks, for the first time, [coming out of the human being], have a [real] view of nature; [the earlier ones are still present in human beings]. One can give an overview that can already really be seen. Now, the historical events follow on completely.
Then geography lessons would also consist of giving an overview.
In history and geography, only an overview would be given; individuals can seek out details for themselves once they have an overview of the whole.
In aesthetics and art lessons, the structure was already discussed yesterday: symbolic, classical, romantic art. Now there is the possibility of treating art history in this way: Egyptians symbolic, Greeks classical, followed by Romantics, but also the arts themselves, with architecture being the symbolic art, sculpture the classical art, and painting, music, and poetry the romantic arts. So one can also view the arts themselves in this way. This provides the possibility of an internal structure.
[Then] in aesthetics and art classes, [we should] cover architecture, the [elements and fundamentals of building design, [so that] young people gain a proper understanding of how a house is constructed. [So] building materials, roof structures [and so on] in aesthetics.
Then [more modern] languages. It is better to specify the objectives, to say that the students should gain an understanding of the current state of literature in English and French.
Now, then there would be mathematics. How far have we gotten in mathematics in 11th grade?
KARL STOCKMEYER: In the 11th grade, we got to [Diophantine] equations in algebra, trigonometry except for spherical trigonometry, and the calculation of oblique triangles. Complex numbers up to Moivre's theorem. Then unit equations, in analytical geometry up to the treatment of second-degree curves, sketchy, only the circle properly covered. In descriptive geometry, intersections and penetrations.
RUDOLF STIUNER: The lessons taught last year in 12th grade showed that this is not really the way to do it. It is something ‘monstrous’ for the human soul to do such a thing.
The aim is to cover spherical trigonometry and the elements of analytical geometry of space in as transparent a manner as possible.
Then, in descriptive geometry, cavalier perspective; the students should be able to represent a more complicated house shape in cavalier perspective and also the interior of the house.
In algebra, it is necessary to take only the very first basics of differential and integral calculus. There is no need to go as far as maximum and minimum calculus. That belongs in college. Only the concepts of differential and integral should be taught and properly elaborated upon.
Emphasis should be placed on teaching spherical trigonometry and its application to astronomy and higher geodesy in a manner appropriate to the age group, so that it is understood in its entirety and in broad terms.
Analytical geometry of space should be used to illustrate how shapes can be expressed in equations. I would not shy away from culminating the lesson in, for example, understanding what kind of curve (surface?) this is:
$$x^\frac{2}{3} + y^\frac{2}{3} + z^\frac{2}{3} = a$$
This gives an asteroid. So that as much general education as possible is included. Above all, to make equations transparent, so that one gets a feeling for how things are actually contained in the equations.
Conversely, one should also pay particular attention to the following: I draw a curve on or in space or [a body] in space [into] space, so that, without the equation having to be exactly right, [the equation can be recognized from the shapes] and one can get a feel for the equation.
I don't think it's useful for general mathematical education to connect differential and integral calculus to geometry, but rather to connect it to quotients. I would start with the difference calculus, i.e., from
$$\frac{\Delta y}{\Delta x}$$
I would interpret this as quotients, and would [only] move on to developing the differential quotient purely from the number, based on the dividend and divisor becoming smaller and smaller.
I would not start from this continuity relationship, as this does not give you any concept of differential quotients; do not start from the differential, but from the differential quotient. I would not start from this continuity relationship, because that does not give you any concept of differential quotients; do not start from the differential, but from the differential quotient. If you start from series, [then] only finally move on to geometry at the tangent problem, i.e., move from the secant to the tangent. And when the whole differential quotient is understood, purely numerically, arithmetically, only then move on to geometry, so that the student gets the idea that geometry is only [ultimately] an illustration of the numerical. Then you get the integrals as an inversion. Then you have the opportunity not to assume that the calculation is a fixation of geometry, but that geometry is an illustration of the calculation. This should be taken into account more generally. For example, one should not regard positive and negative numbers as something in themselves, but one should take the number series as follows: 5 — 1, 5 — 2, 5 — 3, 5 — 4, 5 — 5, 5 — 6, now I don't have enough because I'm missing one, so I write that as —1. [Emphasize what is missing without a number line.] Then you remain in the numerical realm. The negative number is the non-existent quantity, the deficiency of the minuend. [There is] much more inner activity in this! This gives you the opportunity to stimulate abilities in the student that are much more real than if you do everything just from a geometric point of view.
HERMANN VON BARAVALLE: [Where should I start?]
RUDOLF STLENER: [Since Stockmeyer has come to spherical trigonometry], one must move on [from trigonometry] to developing the concept of the sphere, qualitatively, without immediately resorting to calculations. Instead of drawing on a plane, one must draw on a sphere, so that they get the concept of the spherical triangle, the concept of the triangle lying on the sphere.
ERNST LEHRS: That would be the old-fashioned way.
RUDOLF STEINER: You have to make that clear to the children, then, [that] the sum of the angles is not 180 degrees, that it is greater. You really have to teach them this concept, the triangle on the sphere with curved boundaries. Only then can you move on to the calculation. In geometry, the calculation is [the] interpretation of the sphere.
I would like you to look at the sphere not from the center of the sphere, but from the curvature of the surface, so that you can also move on to a general discussion, for example, about curvature and how it would look on an ellipsoid, which is a spherical triangle on the sphere; what it would look like on a rotational paraboloid, [where] it is not closed on both sides, but open. Do not start from the center, but from the curvature of the surface, otherwise you will not be able to deal with other bodies. [You have to think of yourself in the surface], you have to form a mental image, so to speak: What do I experience when I “walk off” a spherical triangle; what do I experience when I “walk off” a triangle that corresponds to a spherical triangle on the ellipsoid?
Then I would draw the students' attention in this context to what happens when you apply the usual Pythagorean theorem to the spherical triangle. Of course, you cannot use squares. These things contribute to general education, whereas otherwise they only train the mind.
Permutations, commutations [combinations?], [that] has already been covered. If there is time, then the first elements of probability theory; the probable lifespan of a human being, for example. One to two hours in 11th grade only.
[For] 11th grade [the following can be considered]: intersections and penetrations, shadow constructions, Diophantine equations, analytical geometry up to conic sections. [In trigonometry in 11th grade, one must] take the functions more internally, so that one has the principle of the ratio in sine and cosine inside. Of course, one must start from geometry.
[In] physics [in 12th grade], optics as discussed yesterday.
Natural history. Zoology has already been discussed. In geology and paleontology, start with zoology, only then does it have intrinsic value. From zoology, move on to paleontology and, as a bonus, the layers of the earth.
[In] botany, phanerogams, [and from] there one moves [also] on to geology and paleontology.
Chemistry. Let us consider chemistry in its most intimate connection with human beings. Our 12th grade children already have a concept of organic and inorganic processes. Now it would be a matter of really going up to the processes that are found not only in animals but also in humans, of speaking ruthlessly about ptyalin, pepsin, pancreatin formation [and so on]. The metal processes in humans should be taken in such a way that something is developed from the principle, let's say, what can be called a process of lead in humans, so that they understand it. It must be shown that all substances and processes are [completely] transformed in humans. In the case of pepsin formation, it is important to start again from hydrochloric acid formation, considering it as the lifeless, and pepsin formation as that which can only take place within the etheric body, where even the astral body must have an effect. So, a complete breakdown of the process and then a rebuilding, [the one] through hydrochloric acid; one starts from the inorganic process, from common salt or through synthesis, and discusses the properties of hydrochloric acid. Then an attempt is made to evoke a difference to what only occurs in the organic body. This must culminate in the difference between plant protein, animal protein, and human protein, so that there is a concept of ascending protein [based on the different structure of the etheric body]. Human protein is different from animal protein. You can start from the difference [and] say, for example: Now, let's take the lion and let's take the cow, so we have a process in the lion that is actually much more related to circulation than in the cow, where the whole process is more related to digestion. The lion actually forms the digestive process with the respiratory process, while in the cow the respiratory process is [also] taken care of by digestion. This is how the processes themselves are enlivened. One would have to have an inorganic, an organic, an animal, and a human chemistry. Here are some examples for children: hydrochloric acid — pepsin; Prunus spinosa juice [and] ptyalin. Then you can [already] get out what needs to be said. Or the metamorphosis process formic acid-oxalic acid.
EUGEN KOLISKO asks about quantitative aspects.
RUDOLF STEINER: Well, it is extremely difficult to explain these things with the prerequisites that can be provided. One would have to start from the rhythm of the world, explain the periodic system from the rhythm of the world. This detour must be taken, but it does not belong in school. In general, it is nonsense to start from atomic weights. [You have to start with rhythm!] All quantitative relationships must be explained from the vibrations. Something like an octave, for example, is found in the ratio of hydrogen to oxygen. But that leads too far. I think you should develop these concepts that we mentioned earlier. That actually exhausts the 12th grade curriculum.
Eurythmy has not been tailored to the high school diploma.
Religion. In general, in terms of character, we have provided [the curriculum for] religious education. Isn't that right? I can't really correct anything in particular in what you have given me. There is nothing special to change. This concerns the upper classes. The culmination should be that in the 12th grade, it should be possible to give an overview of the religions of the world, but not in such a way that this overview gives the impression that they are all actually false, but rather that their relative authenticity is shown through their individual forms. That would be the ninth stage. — The eighth stage would have to work out Christianity, [so that in the ninth] it appears as the synthesis of religions. Christianity would have to be worked out for itself in the eighth section. In the ninth section, [an overview of] the world religions, so that they then culminate again in Christianity. At the seventh stage, there would have to be a kind of harmony of the Gospels. Christianity presented on its own in its essence, its form of appearance. By then, they would be familiar with the Gospels. So, seventh stage: harmony of the Gospels. Eighth stage: [Christianity]. Ninth stage: world religions.
Note: At that time, free religious instruction was divided as follows:
First level = 1st and 2nd grade; second level = 3rd and 4th grade; third level = 5th grade; fourth level = 6th grade; fifth level = 7th grade; sixth level = 8th grade; seventh level = 9th grade; eighth level = 10th grade; ninth level = 11th and 12th grades.
[I will prepare the] curriculum [for] modern languages in grades 9 to 12 [and present it to you in a] conference on language teaching.
There is talk of university courses in Stuttgart.
RUDOLF STEINER: It would have to be paid for. I would like to hear whether what is being proposed for the university courses is not too much of a burden. I would like to hear what you expected. What did you have in mind for the next course, which is now beginning and is to run until the summer holidays? There should be no more than five days a week, otherwise there will be terrible chaos. We had thought of five lecture series, excluding Wednesday and Friday. Lectures could be on Monday, Tuesday, Thursday, and Saturday. Two could be on the same day.
I had thought that only five areas would be covered. Isn't that right? Dr. Mellinger is being considered for social studies at the university. I'm just afraid that she will still be studying for the next three months. But now she wants to be paid for the three months. Someone else will have to be considered for her series. Social studies cannot be provisional. It would be quite good if a practical subject were taught instead, for example, lower and higher geodesy. We don't want to set specific topics. We thought: aesthetics and literature Schwebsch; history Stein; epistemology Unger; mathematics Baravalle; geodesy Stockmeyer.
It seems to have been a mistake to have lectured too much. Music theory must also be lectured on at some point. We will cover the other topics in the next course. That must happen next winter. In order to bring a certain momentum to the course, I would like to suggest that we look at the latest developments in all areas where this is possible. In literature, for example, it would be very nice to work through aesthetics from our point of view, as I have outlined it in the two short papers. In aesthetics, since there is only one hour per week, we can only sketch. We would have to deal with the sentence: “Beauty arises when the sensual takes on the form of the spiritual,” according to my “Goethe as the Father of a New Aesthetics.” This can be demonstrated for the various arts, architecture, painting, and so on. In literature, I would suggest discussing the latest developments, namely the unconscious drift into a certain spirituality in Ibsen and Strindberg [and so on], and then, of course, the pathological, which leads to genius, in Dostoevsky.
MARIE STEINER: [Shouldn't we also] discuss Morgenstern, Steffen, Steiner [at some point?]
RUDOLF STEINER: One can elaborate on something that Steffen once characterized when he was talking about poetry.
In history, [cover] the period from 1870 to 1914 in overview, so that one stops precisely where people then leave with a long face and say, now we have just come to the World War and can think about the World War itself. Up to the assassination in Sarajevo.
Mathematics must be based on what has been taught in the past. I thought that it would be necessary to teach the fundamental principles of mathematics. [To Hermann von Baravalle:] You are quite capable of teaching the material in your dissertation. Then it would be good to develop the concept or such mathematical concepts in a very clear way, such as the concept of ordinary functions, elliptic functions, but not by embellishing it with everything that is rigid mathematics, but by discussing the things qualitatively, how the things are, and then it would be good if we could start from there to present the [entire] theory of relativity in its justification and unjustification. I believe that people should get an idea of the following:
Isn't it true that one can treat a problem of the theory of relativity as follows: A cannon is fired in Freiburg i. Br., you hear it at some distance, you can calculate the distance. You then proceed to calculate how time changes when you move toward the sound or away from it. The propagation time is extended when you move from Karlsruhe to Frankfurt. Then, [if you move] in the other direction, [the time is shortened] until you reach zero when you hear [the cannon] in Freiburg itself. You [can] go beyond Freiburg, then you must hear the cannon before it is fired. That is the fundamental error inherent in this. Developing this mathematical concept of progression in a certain sense might not be so difficult.
I think the university courses have the flaw that they are actually unnecessary. They have changed a little bit to what is otherwise offered in popular lectures. That is not necessary. There is no need for it.
In geodesy, the aim is to move away from providing an image [of the Earth?]. If, for example, you assume in geodesy that you are trying to avoid errors using the difference method, then you have to discuss geodetic methods to a certain extent and arrive at approximate attempts. One can conclude from this the extent to which humans are dependent on only approximating certain things. One can show how extremely useful it is not to think definitively about things like a person's character, but to think in the same way as one measures with [the diopter], leaving oneself [thus] a small space. In this way, one tells the truth much more than if one puts [everything] into definite words. One should only characterize a person in such a way that one grasps him from one side and from the other. [A person can be both choleric and melancholic at the same time.] One should bring this point of view to the fore. If you use lower geodesy to discuss the problems of the Copernican system, a great deal would be achieved. We are making these suggestions to you from Dornach.
The [entire] lecture series should be organized in such a way that it is given the title: What does one gain for one's view of life through aesthetics and literature? — What does one gain for one's view of life through the study of history? — What does one gain for one's view of life through epistemological consideration? — What does one gain for one's outlook on life through mathematical observation? — What does one gain for one's outlook on life through lower and higher geodesy?
[Below that would be]: “The Executive Council of the Anthroposophical Society and the leadership of the teaching staff of the university courses.” [And] above that [as a title]: “Goetheanum and University Courses.” [...] [Individual remarks without context.]