Lecture IX
Stuttgart, January 2, 1921
My Dear Friends,
We have now reached a
point in our studies from which we must proceed with extreme
caution, in order to see where there is a danger of allowing
our thought to depart from reality and to see also when we
are avoiding this danger, by keeping within the bounds of
what is real.
Last time, we suggested the comparison
of two facts: The appearance within the planetary system of
the cometary phenomena, and, alas within the planetary
system, though perhaps not bearing quite the same
relationship to it, all that we observe in the phenomena of
fertilization. In order, however, to come to ideas about this
which are at all justified, we must first see whether it is
indeed possible to find connections between two so widely
separated things, with which we are confronted in the
external world of facts. In scientific method, we shall not
make real progress, unless we can refer from one realm of
facts to another, manifesting something of a similar nature
and thus leading us on.
We have seen how on the one hand we
have to use the element of figure and form, the mathematical,
and then how we are again and again impelled to come to terms
in one way or another with the qualitative aspect, in some
way to find a qualitative approach. And so today we will
bring in something which arises in regard to man if one
really studies this man, who is, after all, in some way an
image of the heavenly phenomena, — as the many
statements in these lectures may enable us to deduce. Yet we
still have to establish in what way he is this image. If this
is what he is, we must first of all gain a clear
understanding of man himself. We must understand the picture
from which we intend to take our start, — understand
its inner perspective. Just as in looking at a painting one
must know what a foreshortening means, and so on, in order to
pass from the picture to the real spatial relationships and
to relate the picture to what it represents in reality, so,
if we would approach reality in the universe, interpreting it
through man, we must first be clear about man. Now it is,
extraordinarily difficult, as a human being, to come near to
the human being with palpable ideas. Therefore, I should like
today to bring before your souls what I might call
“palpably impalpable” thoughtpictures arising
from quite simple foundations, ideas with which most of you
are probably already well acquainted, but which we must
nevertheless bring before our minds in a certain connection.
These ideas, which seem in part to be quite easy to grasp and
yet again, beyond certain limits, to elude our comprehension,
will afford us a means of orientation in the striving to take
hold of the outer world through ideas.
It may appear somewhat forced to keep
emphasizing the necessity of referring back to man's life of
pictorial imagination in order to understand the phenomena of
the heavens. But after all it is obvious that however
carefully we may describe the heavenly phenomena, we have, to
begin with, nothing more than a form of optical picture,
permeated with mathematical thoughts. What Astronomy
gives us has fundamentally the character of a picture.
To be on the right path, we must therefore concern ourselves
with the arising of the picture in man, otherwise we shall
gain no true relationship to what Astronomy can say to us.
And so I should like today to proceed from some quite simple
mathematics and to show you how, in a different domain from
that to which we were led through the ratios of the periods
of revolution of the planets, there appears within
Mathematics itself this element of the incomprehensible, the
impalpable. We meet with it when in a certain connection we
study quite familiar curves. (As I said, many of you already
know what I am about to describe, I only want to elucidate
the subject today from a particular aspect.)
Consider the Ellipse, with its
two foci A and B, and you know that it is a definition of the
ellipse that for any point M of the curve, the sum
of its distances (a + b) from the two foci remains constant.
It is characteristic of the ellipse, that the sum of the
distances of any one of its points from two fixed points, the
two foci, remains constant
(Fig. 1).
Fig. 1
Fig. 2
Then we have a second curve, the
Hyperbola
(Fig. 2).
You know that it has two branches. It is defined in that the
difference of the distances of any point of the
curve from the two foci, (b  a) is a constant magnitude. In
the ellipse, then, we have the curve of the constant sum, in
the hyperbola, the curve of constant difference, and we must
now ask: What is the curve of constant product?
Fig. 3
I
have often drawn attention to this: The curve of
constant product is the socalled Curve of Cassini
(Fig. 3).
We find it when, having two points, A and B, we consider a point M
in regard to its distances from A and B, and establish the condition
that the two distances AM and BM multiplied together should
equal a constant magnitude. For the sake of simplicity in the
calculation, I will call the constant magnitude b^{2}
and the distance AB, 2a. If we take the midpoint between a
and b as the center of the axes of a coordinate system and
calculate the ordinates for each point that fulfills these
conditions, — take C as the center of the coordinate
system and let the point whose ordinate we will call y move
round so that for each point of the curve AM x BM = b^{2} , we get the following equation. (I will only
give you the result, for the simply reason that everyone can
easily work out the calculation for himself; it is to be
found in any mathematical textbook relating to the subject.)
We find for y the value:
Taking here into account that we cannot
use the negative sign because we should then have an
imaginary y, and considering therefore taking only the
positive sign, we have:
If we then draw the corresponding
curve, we have a curve, rather like but not identical with an
ellipse, called the curve of Cassini
(Fig. 4).
It is symmetrical to the left and right of the
ordinate axis and about and below the abscissa axis.
Fig. 4
But now, this curve has various forms,
and for us at any rate this is the important thing about it.
The curve has different forms, according to whether b, as I
have taken it here, is greater than a, equal to a, or less
than a. The curve I have just drawn arises when b > a, and
furthermore when another condition is fulfilled, namely, that
b is also greater than or equal to a √2. Moreover, when
b > a√2, there is a distinct curvature above and
below, If b = a√2, then at this point above and below,
the line of the curve becomes straightened,m it flattens so
much that it almost becomes a straight line
(Fig. 4).
If, however, b < a√2, then the whole course of the
curve is changed and it takes on this form
(Fig. 5).
And if b = a, the curve passes over into a quite special form,
it changes into this form
(Fig. 6).
It runs back into itself,
cuts through itself and comes out on the other side, and we
obtain the special form of the Lemniscate. The
lemniscate, then, is a special form of Curve of Cassini
— these curves are so named after their discoverer. The
particular form assumed by the curve is determined by the
ratio between the constant magnitudes which appear in the
equation characterizing the curve. In the equation, we have
only these two constant magnitudes, b and a, and the form of
the curve depends on the ratio between them.
Fig. 5
Fig. 6
Then the third case is possible, that b
< a. If b < a, we can still find values for the curve.
We can always solve the equation and obtain values for the
curve, ordinates and abscissae, even when b is smaller than
a, only the curve then undergoes yet another metamorphosis.
For when b < a, we find two branches of the curve, which
look something like this
(Fig. 7).
We have a discontinuous curve. And here we come to the point
where the mathematics itself confronts us with what I called
the “palpably impalpable”, something that is
difficult to grasp in space. For in the sense of the
mathematical equation, this is not two curves, but one; it is
a single curve in exactly the same way as all these are
single curves
(Figs. 3 through 5).
In this one (the lemniscate) there is already a transition. The
point which describes the curve takes this path, goes round
underneath, cuts its previous path here and continues on here
(Fig. 7).
Here, we must picture the
following: If we let the point M move along this line, it
does not simply cross over from one side to the other,
— it does not do this. It runs along the path just as
in the other curves, describes a curve here, but then manages
to turn up again here
(Fig. 7)
You see, that which carries the point along the line disappears
here in the middle. If you want to understand the curve you can
only imagine that it disappears in the middle. If you try to
form a continuous mental picture of this curve, what must you
do?
Fig. 7
It is quite easy, is it not, to imagine
curves such as thes. (I only say this in parenthesis for the
ordinary philistine!) You can go on imagining points along
the curve and you do not find that the picture breaks off.
Here (in the lemniscate) admittedly, you have to modify the
comfortable way of simply going round and round, but still it
goes on continuously. You can keep hold of the mental
picture. But now, when you come to this curve
(Fig. 7),
which is not so commonplace, and you want
to image it, then, in order to keep the continuity of the
idea you will have to say: Space no longer gives me a point
of support. In crossing over to the other branch in my
imagination, unless I break the continuity and regard the one
branch as independent of the other, I must go out of space; I
cannot remain in space. So you see, Mathematics itself
provides us with facts which oblige us to go out of
space, if we would preserve the continuity of the idea.
The reality itself demands of us that in our ideas we go out
of space. Even in Mathematics therefore we are confronted
with something which shows us that in some way we must leave
space behind, if the pure idea is to follow its right path.
Having ourselves and going the idea is beginning to think the
process through, we must go on thinking in such a way that
space is no longer of any help to us. If this were not so, we
should not be able to calculate all possibilities in the
equation.
In pursuing similar line of thought, we
meet with other instances of this kind. I will only draw your
attention to the next step, which ensures if one things as
follows. The ellipse is the locus of the constant sum,
— it is defined by the fact that is is the curve of
constant sum. The hyperbola is the curve of constant
difference. The curve of Cassini in its various forms is the
curve of constant product. There must then be a curve of
constant quotient also, if we have here A, here B, here a
point M, and then a constant quotient to be formed through
the division of BM by AM. We must be able to find different
points, M 1, M 2, etc., for which
etc. are equal to one
another and always equal to a constant number. This curve is,
in fact, the Circle. If we look for the points M1,
M2 etc. we find a circle which has this particular
relationship to thee points A and B
(Fig. 8).
So that we can say: Besides the usual, simple
definition of a circle, — namely, that it is the locus
of a point whose distance from a fixed point remains
constant, — there is another definition. The circle is
that curve, very point of which fulfills the condition that
its distances from two fixed points maintain a constant
quotient.
Fig. 8
Now, in considering the circle in this
way there is something else to be observed. For you see, if
we express this
(it could of course be
expressed in some other way), we always obtain corresponding
values in the equation, and we can find the circle. In doing
this we find different forms of the circle (that is,
different proportions between the radius of the circle and
the length of the straight line AB), according to the
proportion of m to n. These different forms of the circle
behave in such a way that their curvature becomes less and
less. When n is much greater than m, we find a circle with a
very strong curvature; when n is not so much greater, the
curvature is less. The circle becomes larger and larger the
smaller the difference between n and m. And if we follow this
proportion of m to n still further, the circle gradually
passes over into a straight line. You can follow this in the
equation. It passes over into the ordinate axis itself. The
circle becomes the ordinate axis when m = n, that
is, when the quotient m/n = 1. In this way the circle
gradually changes into the ordinate axis, into a straight
line.
You need not be particularly astonished
at this. It is quite possible to imagine. But something very
different happens it we wish to follow the process still
further. The circle has flattened more and more, and through
becoming flatter from within, as it were, it changes into a
straight line. It does this because the constant ratio in the
equation undergoes a change. Through this the circle becomes
a straight line. But this constant ratio can of course grow
beyond 1, so that the arcs of the circles appear here (on the
left of the y axis). What must we do, however, if we try to
follow it in our imagination? We have to do something quite
peculiar. We have, in fact, to think of a circle which is not
curved towards the inside, but is curved towards the outside.
Of course, I cannot draw this circle, but it is possible to
think of a circle which is curved towards the
outside.[1] In an ordinary circle the
curvature is towards the inside, it is not? If we follow the
line round it returns into itself. But defining the circle in
this other way, if we use the necessary constant, we obtain a
straight line. The curvature is still on this side (right of
the y axis). But it now makes things not nearly so
comfortable for us as before! Previously, the curvature
always turned towards the center of the circle, while now (in
the case of the straight line), we are shown that the center
is somewhere in the infinite distance, as one says. Following
on from this, there arises for us the idea of a circle which
is curved towards the outside. Its curvature is then no
longer as it is here
(Fig. 9a)
— that would be the ordinary, commonplace, philistine circle,
— but its curvature is here
(Fig. 9b).
Therefore, the inside of this circle is not here;
this is the outside; the inside of this circle
(Fig. 9c)
is to the right.
Fig. 9
Now compare what I have just put before
you. I have described the curve of Cassini, with its various
forms, the lemniscate and the form in which there are two
branches. And now we have pictured the circle in such a way
that at one time it is curved in the familiar way, with the
inside here and the outside here; while in a second form of
circle (in drawing it we are only indicating what is meant)
we find that the curvature is this way round, with an inside
here and an outside here. Comparing it with the Cassini
curve, the first form of the circle would correspond to the
closed forms, as far as the lemniscate. After this we have
another kind of circle, which must be thought of in the other
direction, being curved this way, with the inside here and
the outside here. You see, when we are concerned with the
constant product we find forms of the curve of Cassini where,
it is true, we are thrown out of space, yet we can still draw
the other branch on the other side. The other branch is once
more in space, although in order to pass from the one to the
other we are thrown out of space. Here, in the case of the
circle, however, the matter becomes still more difficult. In
the transition from circle to straight line we are, indeed,
thrown out of space, and moreover, we can no longer draw a
selfcontained form at all. This we are unable to do. In
passing over from the curve of constant product to the curve
of constant quotient, we are only just able to indicate the
thought spatially.
It is extraordinarily important that we
concern ourselves with the creating of ideas which, as it
were, will still slip into such curveforms. I am convinced
that most people who concern themselves with mathematics take
note of such discontinuities, but then make the thought more
comfortable by simply holding to the formula and not passing
on to what should accompany the mathematical formula in true
continuity of thought. I have also never seen that in the
treatment of Mathematics as subject matter for education any
great value is laid upon the forming of such thoughts in
imagination. — I do not know, — I ask the
mathematicians present, Herr Blümel, Herr Baravalle, if
this is so; whether in modern University education any
importance is attached to this? (Dr. Unger here mentioned the
use of the cinema.) Yes, but that is a pretense. It is only
possible to represent such things within empirical space by
means of the cinema or in similar ways, it some sort of
deception is introduced. It cannot be pictured fully in real
space without the effect being achieved through some form of
deception. The point is, whether there is anywhere in the
sphere of reality something which obliges us to think
realistically in terms of such curves. This is the question I
am now asking. Before passing on, however, to describe what
might perhaps correspond to these things in the realm of
reality, I should like to add something which may perhaps
make it easier for you to pass transition from these abstract
ideas to the reality. It is the following.
Fig. 10
You can set another problem in the
sphere of theoretical Astronomy, theoretical Physics. You can
say: Let us suppose that here as A, is a source of light, and
this source of light in a illumines a point M
(Fig. 10).
The strength of the light shining from
M is observed from B. That is, with the necessary optical
instruments, observation is made from B of the strength of
the light shining from the point M, which is illumined from
A. And of course, the strength of the light would vary,
according to the distance between B and M. But there is a
path which could be described by the point M, such that,
being illumined from A, it always shines back to B with the
same intensity. There is such a path; and we can therefore
ask: What must be the locus of a point, illumined from a
fixed point A, such that, seen from another fixed point B,
its light is always of the same intensity? This curve —
the curve in which such a point would have to move — is
the curve of Cassini! From this you see that something which
takes on a qualitative nature is set into spatial connection,
fitting into a complicated curve. The quality that we must
see in the beam of light — for the intensity of light
is a quality — depends in this case on the element of
form in the spatial relationships.
I only wished to bring this forward for
you to see that there is at least some way of leading over
from what can be grasped in geometrical form to what is
qualitative. This way is a long one, and what we will now
discuss is something to which I want to draw your attention,
although it would take months to present in all detail. You
must be fully aware that I only intend to give you guiding
lines; it is left to you to develop them further and to go
into all the details which would testify to the truth of what
is said. For you see, the connection which must be formed
between spiritual science and empirical sciences of today
demands very farreaching and extensive work. But when lines
of direction are once given, this work can to some extent be
undertaken and carried forward. It is at all events possible.
One must only be able in a quite definite way to penetrate
into the empirical phenomena.
If we now tackle the problem from quite
another angle, — we have sought to some degree to
understand it from the mathematical aspect, then, to anyone
who is studying the human organism, there is something which
cannot escape unnoticed, something which has often been
brought forward in our circle, especially in the talks which
accompanied the course of lectures on Medicine in Dornach in
the spring of 1920. It is not to be overlooked that
certain relationships exist between the organisation of the
head and the rest of the human organisation, for example
the metabolism. There is indeed a connection, indefinable to
begin with, between what takes place in the third system of
the human being — in all the organs of metabolism
— and what takes place in the head. The relationship is
there, but it is hard to formulate. Clearly as it emerges in
various phenomena, — for example, it is obvious that
certain illnesses are connected with skull or head
deformities and the like, and these things can easily be
traced by one who tries to follow them with biological
reasoning, — it nevertheless difficult to grasp this
relationship in imagination. People do not usually get beyond
the point of saying that there must be some sort of
connection between what takes place in the head, for
instance, and in the rest of the human organism. It is a
picture which is difficult to form, just because it is so
very hard for people to make the transition from the
quantitative aspect to the qualitative. If we are not
educated through spiritualscientific methods to find this
transition, quite independently of what outer experience
offers, — to extend to what is qualitative the kind of
thought we use for what is quantitative, if we do not
methodically train ourselves to do this, then, my dear
friends, there will always be an apparent limit to our
understanding of the external phenomena.
Let me indicate but one way in which
you can train yourselves methodologically to think the
qualitative in a similar way as you think the
quantitative. You are all acquainted with the phenomenon
of the solar spectrum, the usual continuous spectrum. You
know that we have there the transition of colour from red to
violet. You know, too, that Goethe wrestled with the problem
of how this spectrum is in a sense the reverse of what must
arise if darkness be allowed to pass through the prism in the
same way as is usually done with light. The result is a kind
of inverted spectrum, and as you know Goethe arranged this
experiment also. In the ordinary spectrum, the green passes
over on the one side towards the violet and on the other
towards the red; whereas in the spectrum obtained by Goethe
in applying a strip of darkness to the prism there is
peachblossom in the middle and then again red on the one
side and violet on the other
(Fig. 11).
The two colour bands
are obtained, the centres of which are opposite to one
another, qualitatively opposite, and both bands seem to
stretch away as it were into infinity. But now, one can
imagine that this axis, the longitudinal axis of the ordinary
spectrum, is not simply a straight line, but a circle, as
indeed every straight line is a circle. If this straight line
is a circle, it returns into itself, and we can consider the
point where the peachblossom appears to be the same point as
the one in which the violet, stretching to the right, meets
the red, which stretches to the left. They meet in the
infinite distance to the right and left. If we were to
succeed — maybe you know that one of the first
experiments to be made in our newly established physical
laboratory is to be in this direction — if we were to
succeed in bending the spectrum in a certain way into itself,
then even those who are not willing to grasp the matter to
begin with in pure thought will be able to see that we are
here concerned with something real and of a qualitative
nature.
Fig. 11
Fig. 12
We come to certain limiting ideas in
Mathematics, where — as in Synthetic Geometry —
we are obliged to regard the straight line as a circle in a
quite real though inner sense; where we are obliged to admit
of the infinitely distant point of a straight line as being
only one point; or to understand as bounding a plane, not
some line above and then again below, but a single
straight line; or to think of the boundary of infinite space,
not in the nature of something spherical, but as a plane.
Such ideas, however, also become, in a way, limiting ideas
for senseperceptible empirical reality, and we are made to
realise it if we insist on restricting ourselves to
senseperceptible reality.
This brings us to something which would
otherwise always remain perpetually in the dark. I have
already mentioned it. It invites us really to thinkthrough
the thoughtpictures to which we come when we allow the
lemniscateform of the Cassini curve to pass over into the
doublebranched form, — the form with the two branches
for which we must go out of space, — and them compare
this with what confronts us in the empirical reality.
You are indeed already doing this, my
dear friends, when you apply Mathematics in one way or
another to the empirical reality. You call a triangle a
triangle, because you have first constructed it
mathematically. You apply to the outer form what has been
evolved in an inner constructive way within you. The process
I have just described is only more complicated, but it is the
same process when you think of the two branches of that
particular form of the Cassini curve as one. Apply this
thought to the correspondence between the human head and the
rest of the human organism and you will have to realise that
in the head there is a connection with the remaining organism
of precisely such a character as is expressed by the equation
which requires, not a continuous curve, but a discontinuous
one. This cannot be followed anatomically; you must go out
beyond what the body comprises physically, if you would find
the connection of what comes to expression in the head with
what comes to expression in the metabolic system. It is
essential to approach the human organism with thoughts which
are quite unattainable if for every element of the thought
you insist on an entire correspondence within the
senseperceptible empirical realm. We must reach out to
something else, beyond the senseperceptible empirical realm,
if we are to find what this relationship really is within the
human being.
Such a study, if one really gives
oneself up to it and carried it out methodically, is
extraordinarily rich in its results. The human organisation
is of such a nature that it cannot be embraced by the
anatomical approach alone. Just as we are driven out of
space in the Cassini curve, so in the study of man we are
driven out of the body, by the method of study itself.
You see, it is quite possible to understand in the first
place in thought, that in a study of the whole man we are
driven out of the realm of what can be grasped in a
physicalempirical sense. To put forward such things is no
offence against scientific principles. Such ideas are far
removed from the purely hypothetical fantasies which are
often entertained in connection with natural phenomena, for
they refer to the whole way in which man is membered into the
universe. You are not looking for something which is
otherwise nonexistent, but rather for something which is
exactly the same as what is expressed in the relationship
between a man thinking mathematically and the empirical
reality.
It is not a question of looking for
hypotheses which in the end are unjustifiable; it is a
question, since the reality is obviously complicated, of
looking for other cognitive relations to the inner reality,
in addition to the simple relation of mathematical man to
empirical reality. When once you have accepted such thoughts,
you will also be led to ask whether what takes place outside
the human being in other domains besides the astronomical,
— for example, in those phenomena which we call the
chemical and physical, — whether those same phenomena,
which we regard as chemical phenomena outside of man, take
the same course within man, when he is alive, as they do
outside him, or whether here, too, a transition is necessary
which leads in some way out of space.
Now consider the important question
arising out of this. Suppose we have here some kind of
chemical phenomena and here the boundary leading over to the
inside of the human being
(Fig. 13).
Supposing that this chemical phenomenon were able to call
forth another, so that the human being reacted here (inside);
then, if we remain in the field of the empirical, space would
of course be the mediator. If, however, the continuance of
this phenomenon within the human being comes about by virtue
of the fact, say, that the human being is nourished by food,
and the processes already taking place outside him continue
inside him, then the question arises: Does the force which is
at work in the chemical process remain in the same space when
it taking place within man as when it is taking its course
outside him? Or must we perhaps go out of space?
And there you have what is analogous to
the circle which changes over into a straight line. If you
look for its other form, where what is usually turned outward
is now turned inward, you are entirely outside of space.
Fig. 13
The question is, whether we do not need
such ideas as these, thoughtpictures which, while remaining
continuous, go right out of space, — when we follow the
course of what happens outwardly, outside of man, into the
interior of the human being. The only thing to be said
against such things, my dear friends, is that they certainly
impose greater demands on the human capacity of understanding
than the ideas with which he phenomena are approached today.
They might therefore be rather awkward in University
education. They are, no doubt, thoroughly awkward, for they
imply that before approaching the phenomena we must awaken in
ourselves what will enable us to understand them. Nothing
like this exists in our educational system today; but it must
come, it must certainly come, otherwise simply in speaking of
a phenomenon we get into the greatest disparities, without in
any way seeing the reality. Just think what happens when
someone observes the circle as it curves to this side
(Fig. 9a),
and then sees how it curves to this side
(Fig. 9b),
but then remains a
philistine and simple does not conceive that the circle now
curves towards the other side. He says: This is impossible,
the circle cannot curve this way; I must put the curvature
this way round, I must simply place myself on the other side.
What he is speaking about seems to be one and the same thing;
but he has changed his point of view.
In
this way today we make matters simple, in describing
what is within the human being in comparison with what
takes place in Nature outside him. We say: What is within
man does not exist at all; I must simply place myself within
man and say that the curvature is facing this way
(Fig. 9c).
I will then consider
what is inside, without taking into account that I have
reversed the curvature. I will make the interior of the human
being into an outer Nature. I simply imagine outer Nature to
continue through the skin into the interior. I turn myself
round, because I am not willing to admit the other form of
curvature, and then I theorise. That is the trick which is
performed today, only in order to adhere to more comfortable
motions. There is no desire to accent what is real; in order
not to have to do so, we simply turn ourselves round, and
— this is now a comparison — instead of
looking at the human from in front, we look at Nature from
behind and thus arrive in this way at all the various
theories concerning man.
We will continue, then, tomorrow.
Notes:
1. If it were drawn it
would look like an ordinary circle, only one would have
to bear in mind that “outside” and
“inside” had changed places.
(Editor's note.)
