"...in the moment of turning the inside of the
tubular bone to the outside, certain forces of tension come
into play and mutual relationships of the forces change in
such a way that the form which was inside and has now been
turned outward alters the shape and distribution of its
surface, then we obtain, through inversion on the principle
of the turning-inside-out of a glove, the outer surface of
the skull bone as derived from the inner surface of the tubular bone."

(rudolf steiner)


You know of course that the question of the form of the human skull has
played a great role in modern biology. I have also spoken of this matter may
times in the course of our anthroposophical lectures. Goethe and Oken put
forward magnificent thoughts on this question of the human skull-bones. The
school of Gegenbauer also carried out classical researches upon it. But
something that could satisfy the urge for a deeper knowledge in this
direction does not in fact exist today.

People discuss, to what extent was Goethe right in saying that the
skull-bones are metamorphosed vertebrae, bones of the spine. But it is
impossible to arrive at any really penetrating view of this matter today,
because in the circles where these things are discussed one would scarcely
be understood, and where an understanding might be forthcoming these things
are not talked of because they are not of interest. You see, it is
practically impossible today to bring together in close working association
a thoroughly modern doctor, a thoroughly modern mathematician, -- i. e., one
who is master of higher mathematics --, and a man who could understand both
of them passably well. These three men could scarcely understand one
another. The one who would sit in the middle, understanding both of them
slightly, would be able at a pinch to talk a little with the mathematician
and also with the doctor. But the mathematician and the doctor would not be
able to understand each other upon important questions, because what the
doctor would have to say about them would not interest the mathematician,
and what the mathematician would have to say -- or would say, if he found
words at all, -- would not be understood by the doctor, who would be lacking
the necessary mathematical background. This is what would happen in an
attempt to solve the problem I have just put before you.

(Rudolf Steiner, Astronomy Lecture I, January 1, 1921)


Morphology today cannot yet recognize the form and construction of a tubular
or long bone, for example, in its relation to that of a skull-bone. To do
this, we should have to reach a way of thinking whereby we should first
study what is within, say, the inner surface of a tubular bone and then
relate this to the outer surface of a skull-bone. This means a kind of
inversion, as when a glove is turned inside-out; but at the same time there
is an alteration of the form, an alteration of the surface-tensions through
the reversing or turning of inside outward.

(IBID, Lecture V, Stuttgart, January 5, 1921)


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.
Then we have a second curve, the Hyperbola. 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.

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
thinks 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:

BM1/AM1 = BM2/AM2 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 (see Fig,
below). 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.
Now, in considering the circle in this way there is something else to be
observed. For you see, if we express this BM/AM by m/n (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.

(IBID, Lecture IX, January 9th, 1921)


...if we compare the inner surface of a tubular bone with the outer surface
of a skull-bone... The inner surface of the tubular bone corresponds
morphologically to the outer surface of the skull-bone. The skull-bone can
be derived from the tubular bone if we picture it as being reversed, to
begin with, according to the principle of the turning-inside-out of a glove.
In the glove, however, when I turn the outer surface to the inside and the
inner to the outside, I get a form similar to the original one. But if in
the moment of turning the inside of the tubular bone to the outside, certain
forces of tension come into play and mutual relationships of the forces
change in such a way that the form which was inside and has now been turned
outward alters the shape and distribution of its surface, then we obtain,
through inversion on the principle of the turning-inside-out of a glove, the
outer surface of the skull bone as derived from the inner surface of the
tubular bone. From this you can conclude as follows. The inner space of the
tubular bone, this compressed inner space, corresponds in regard to the
human skull to the entire outer world. You must consider as related in their
influence upon the human being: The outer universe, forming the outside of
his head, and what works within, tending from within toward the inner
surface of the tubular bone. These you must see to belong together. You must
regard the world in the inside of the tubular bone as a kind of inversion of
the world surrounding us outside.

There, for the bones in the first place, you have the true principle of
metamorphosis! The other bones are intermediary forms; morphologically, they
mediate between the two opposite extremes, which represent a complete
inversion, accompanied by a change in the forces determining the surface.
The idea must however be extended to the entire human organism. In one way,
it comes to expression most clearly in the bones; but in all the human
organs we must distinguish between two opposing factors, - that which works
outward from an unknown interior, as we will call it for the moment, and
that which works inward from without. The latter corresponds to all that
surrounds us human beings on the planet Earth.

The tubular bone and the skull-bone represent indeed a remarkable polarity.
Take the tubular bone and think of this centre-line (Fig 2). This line is in
a way the place of origin of what works outward, in a direction
perpendicular to the inner surface of the bone (Fig 2a). If you now think of
what envelops the human skull, you have what corresponds to the central line
of the tubular bone. But how must you draw the counterpart of this line? You
must draw it somewhere as a circle, or more exactly, as a spherical surface,
far way at some indeterminate distance (Fig 3). All the lines which can be
drawn from the centre-line of the tubular bone towards it inner surface (Fig
2a) correspond, in regard to the skull-bone, to all the lines which can be
drawn from a spherical surface as though to meet in the centre of the Earth
(Fig 3). In this way you find a connection - approximate, needless to say -
between a straight line, or a system of straight lines, passing through a
tubular bone and bearing a certain relation to the vertical axis of the
body, the direction of which coincides, in fact, with that of the Earth's
radius and a sphere surrounding the Earth at an indeterminate distance. In
other words, the connection is as follows. The radius of the Earth has the
same cosmic value in regard to the vertical posture of the human organism,
perpendicular to the surface of the Earth, as a spherical surface, a cosmic
spherical surface has in regard to the skull organisation.

(IBID, Lecture X, January 10th, 1921).


the essence of the mutual relation of the long bone to the skull-bone and
vice-versa is a complete turning-inside-out. The inner surface of the bone
becomes the one turned outward. It is the principle by which you turn a
glove inside-out, provided only that the turning-inside-out involves a
simultaneous change in the inherent relationships of inner forces. If I
should turn a tubular bone or long bone inside-out like a mere glove, I
should again get the form of a tubular bone, needless to say. But it will
not be so if we take our start, as we must do, from the inherent
configuration of the bone. As I described before, in its inherent
configuration the long bone is oriented inward towards the radial quality
that runs right through it. It is obliged therefore to subject its material
structure and arrangement to the radial principle. When I have "flipped" it,
so that the inner side opens outside, in its configuration it will no longer
follow the radial but the spheroidal principle. The "inner side", now turned
outward towards the Sphere, will then receive this form (Figure1). What was
outside before, is now inside, and vice-versa. Take this into account for
the extreme metamorphosis-tubular bone into skull-bone and you will say: The
outermost ends of human memberment - lymph-system and skull system -
represent opposite poles in man's organization. But we must not think of
"opposite poles" in the mere trivial, linear sense of the word. In that we
go from one pole to the other, we must adopt the transition which this
involves, namely from Radius to surface of a Sphere.

(IBID, Lecture XV, Stuttgart, 15th January 1921 )


People imagine: If the skull-bones are metamorphosed vertebra, then we ought
to be able to proceed directly, through a transformation which it is
possible to picture spatially, from the vertebra to the skull. To extend the
idea still further to the limb-bones would, on the basis of the accepted
premises, be quite out of the question. The modern mathematician will be
able, from his mathematical studies, to form an idea of what it really means
when I turn a glove inside out, when I turn the inside to the outside. One
must have in mind a certain mathematical handling of the process by which
what was formerly outside is turned inward, and what was inside is turned to
the outside. I will make a sketch of it (Fig. 1) -- a structure of some sort
that is first white on the outside and red inside. We will treat this
structure as we did the glove, so that it is now red outside and white
But let us go further, my dear friends, and picture to ourselves that we
have something endowed with a force of its own that does not admit of being
turned inside out in such a simple way as a glove which still looks like a
glove after being inverted. Suppose that we invert something which has
different stresses of force on the outer surface from those on the inner. We
shall then find that simply through the inversion quite a new form arises.
The form may appear thus before we have reversed it (Fig. 1): we turn it
inside out and now different forces come into consideration on the red
surface and on the white, so that perhaps, purely through the inversion,
this form arises (Fig. 3). Such a form might arise merely in the process of
inversion. When the red side faced inward, forces remained dominant which
are developed differently when it is turned outward. And so with the white
side; only when turned towards the inside can it develop its inherent
It is of course quite conceivable to give a mathematical presentation of
such a subject, but people are thoroughly disinclined nowadays to apply to
reality what is arrived at conceptually in such a way. The moment, however,
we learn to apply this to reality, we become able to see in our long bones
or tubular bones (that is, in the limb bones), a form which, when inverted,
becomes our skull bones! In the drawing, let the inside of the bone, as far
as the marrow, be depicted by the red, the outside by the white(Fig. 4).
Certain forms and forces, which can of course be investigated, are turned
inward, and what we see when we draw away the muscle from the long bone is
turned outward. But now imagine these hollow bones turned inside out by the
same principle as I have just given you, in which other conditions of stress
and strain are brought into play;
then you may easily obtain this form (Fig 5). Now it has the white within,
and what I depicted by the red comes to the outside. This is in fact the
relationship of a skull-bone to a limb-bone, and in between lies the typical
bone of the back -- the vertebra of the spinal column. You must turn the
tubular bone inside out like a glove according to its indwelling forces;
then you obtain the skull-bone. The metamorphosis of the bones of the limbs
into the skull-bones is only to be understood when keeping in mind the
process of inversion, or 'turning inside-out'. The important thing to
realizes is that what is turned outward in the limb-bones is turned inward
in the skull. The skull-bones turn towards a world of their own in the
interior of the skull. That is one world. The skull-bone is orientated to
the world, just as the limb-bone is orientated outward, towards the external
world. This can be clearly seen in the case of the bones. Moreover, the
human organism as a whole is so organized that it has on the one hand a
skull organization, and on the other a limb-organization, the
skull-organization being oriented inward, the limb-organization outward. The
skull contains an inner world, the limb-man an outer world, and between the
two is a kind of balancing system which preserves the rhythm.

My dear friends, take any literature dealing with the theory of functions,
or, say, with non-Euclidean geometry...if we were to apply to the structure
of the human organism all that has been thought out in non-Euclidean
geometry, then we should be in the realm of reality, and applying
immeasurably important ideas to reality, not wandering about in mere
speculations. If the mathematician were so trained as to be interested also
in what is real, -- in the appearance of the heart, for example, so that he
could form an idea of how through a mathematical process he could turn the
heart inside out, and how thereby the whole human form would arise, -- if he
were taught to use his mathematics in actual life, then he could be working
in the realm of the real. It would then be impossible to have the trained
mathematician on the one hand, not interested in what the doctor learns, and
on the other, the physician, understanding nothing of of how the
mathematician -- though in a purely abstract element -- is able to change
and metamorphose forms. This is the situation we must alter. If not, our
sciences will fall into decay. They grow estranged from one another; people
no longer understand each other's language.

(Rudolf Steiner, Astronomy Lecture I, January 1, 1921)

--| References |----------------

Embryonic Cosmology

(Rudolf Steiner, Astronomy Lectures: The Relationship of the Diverse
Branches of Natural Science to Astronomy, Stuttgart, January 1921).


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