# Notes from Mathilde Scholl 1904–1906

## GA 91 — 27 September 1906

### 34. The Spiral The number 3.1415

The line of all development is the spiral. Its loops always follow the same laws. We can regard the spiral as the living representation of the ratio of the diameter to the circle. The first loop of each spiral arises from the fact that a line describes a circle, but not a perfect circle. Before it closes again, it turns away from the end point to describe a circle again and so on in this way. If we imagine this figure stretched out, we have a spiral.


The development of the same can be observed on a paper spiral. Each arc of a spiral is therefore created by curving a straight line, but not entirely back into itself, but by simultaneously lengthening this curved straight line. So each arc of the spiral is in the same ratio to the straight line that curves as the circle is to the diameter, as the number \(\pi\) is.

Now, every circle diameter can be placed on the circle three times with a remainder, so the diameter, by extending and curving three times, can almost close to a circle, but not quite; a gap remains, and indeed corresponding to the number \(\pi\); the gap is what follows after \(3\), namely \(1\), \(4\), \(1\), \(5\) and so on. So, to complete the circle, we need the numbers after the decimal point of the number \(\pi\), i.e. \(1415\) etc. Now, in terms of their values, these are \(1/10\), \(4/100\), \(1/1000\), \(5/1000\) of the original diameter, but since the number x is infinite, there is no number that expresses how much of the diameter can be placed on the circle. We can now imagine a process in which the diameter is bent to the circle so far until it almost closes, and where only the small gap remains open, which contains the numbers \(0,1415\) and so on. The number \(1\) denotes here \(1/10\) of the diameter.

If now the diameter, where it has reached the gap, i.e. after it has been placed on the circumference three times, continues in the same ratio, it describes a line that is in the ratio of the whole diameter to \(1/10\), \(4/100\), \(1/1000\), \(5/10000\) and so on. If at the same time it deviates from the perfect first circle at the point where this line begins, it will continuously form a spiral.

If this is the diameter of the first vibration,


then it describes the following circle; and when it turns into a spiral, the following spiral.


The spiral arises from the deviation of the diameter from the formation of a circle in the ratio of \(1/10\), \(4/100\), \(1/1000\), \(5/10000\) to a length. - If the diameter emerges from a circle, then it can only emerge from the circle in the ratio of the number \(\pi\), namely as often as it is contained in the circle. We can imagine that the circle is movable, like a circle formed by a [ *illegible*] or a band. If you pull the band from one point with one end through the center in the direction of the diameter of the circle, it moves from the circumference in this direction.

A movable semicircle is formed, which becomes smaller and smaller. You could also continue to pull the tape in a circle, then a spiral would form.


Thus every spiral arises through the number \(\pi\), which expresses the ratio of [circumference to diameter] or the infinitely small deviation from the circular line to the straight line. A spiral is therefore the line that arises from the diameter emerging from the circle.

The number \(\pi\) expresses the ratio of the diameter to the circle. According to the numbers of this ratio, it can now also emerge from the circle, since all development arises from the fact that a force expresses itself in this ratio according to the number \(3.1415\).

If a force remained closed in itself, no revelation would be possible. To reveal oneself means to unlock oneself. Every closed force can only unlock itself through this ratio of its diameter to its perimeter. The circumference circumscribes the entire force, while the diameter divides the entire force into two halves. The force that divides the whole into two halves is not fully contained within it; it is related to it in the same way as the number \(\pi\). The effect of the dividing, revealing force thus emerges according to the numbers \(3,1415\).

Since it has retained in the first force the endeavor to close again into a whole, it will again try to describe a circle. But since it has within itself the endeavor to overcome unity, it deviates again from the circular line and thus describes a spiral.

The force of the second spiral line is twisted to the same extent as it moves away from the first force, namely in the ratio of the number \(\pi\). If the first circle is the force \(3.1415\), then the second circle of the force is contained in the first as often as the number \(\pi\), the third in the second again as often as the number \(\pi\). The second spiral line of force can emerge from the first \(3.1415\) times. The third force can emerge from the second \(3.1415\) times, from the third the fourth and so on. - In this ratio, the force on spiral lines decreases. The emergence of force occurs in the ratio of arithmetic numbers up to \(5\). If we multiply a force by the number \(3.1415\), then it becomes the next higher force; the lower force always indicates the diameter of the higher force, the higher force is \(3.1415\)-times the circumference of the lower force. The diameter of the higher force is always the circumference of the lower force. The strength of a force decreases in proportion to the squares of its distance. The reduction in strength can therefore be calculated by raising the distance from the first force to the second power.

The distance from the center of the force indicates the radius of the circle, half the diameter. This must be squared to know how strongly the force in the center acts on the circumference. If the diameter is \(4\), then the radius is \(2\), the square is \(4\) - so the force on the circumference is \(1/4\) of that in the center. If the radius is 100, then the square is 10,000, so the force on the circumference is 1/10,000 of the central force. We can express the forces as a ratio, and the force effects.

If the circumference is \(18\), then the diameter is contained in it \(3.1415\) times, so it is a little [less] than \(6\). Let us assume that it is \(6\) – then the radius would be \(3\), the square would be \(9\) – the force effect in the circumference would be \(1/9\) of the force inside, but a little less than \(1/9\).

Thus, one force after another emerges in the ratio of the number \(\pi\) – but each following force is only \(3.1415\)th the magnitude of the previous one – and each force is in the ratio of the square of the radius, which indicates the distance of the circumference from the center.

All cosmic forces are connected in this way. The descent is a continuous revelation that reveals itself as \(3.1415\) [whereby] the inner diameter of the higher force becomes the circumference of the next one, creating a spiral in which the next loop is always only the \(3.1415\)th part of the previous one. The diameter of the higher force forms the circumference of the lower force. The circumference of the smaller force is therefore the diameter of the larger force. Since the circumference of the total force is the same as the diameter of the larger force, half the circumference of the smaller force is equal to half the diameter of the larger force. Half the circumference of the smaller force therefore indicates the distance of the larger force circumference from the center - consequently, you only need to square half the circumference of a force to find the force effect of a larger force.

If the magnitude of a force is \(10\), then half the magnitude is \(5\) – in this case, the force of the higher force is \(5^2 = 25\).

Conversely, we can calculate the range of the lower force from the range of the higher force. If the range of a force is \(25\), then its radius is \(5\), its diameter is \(10\), and consequently \(10\) is the range of the lowest force that arises from it. So \(10\) is the diameter of the higher force and the circumference of the next force. From \(10\) the diameter now emerges, corresponding to the number \(\pi\), as a little less than \(1/3\) of \(10\) – to form the circumference of the following force. So the force that follows has approximately \(3\).

Thus, by means of the number \(\pi\), one can calculate all the ratios of the forces that arise from each other and their effects. Thus, the cosmic substances and forces interlock and swirl around each other in spiral lines, whereby, on the one hand, a transformation of force occurs in that the diameter emerges from the circle \(3.1415\) times – or, on the other hand, an increase in force occurs in that the circle becomes the diameter of a higher force. Furthermore, on the one hand the radii are raised to a higher power, squared to the force, and – or on the other hand, the square of the force is taken to the root of the radius of the circle that it describes. Range of force


Force effect  
squaring the circle?


The diameter is related to the force effect in the same way as the semicircle \(\times 2\) is related to the semicircle squared. If the [radius] is \(x\), then the diameter is \(x \times 2\) and the force is \(x^2\). Therefore, the diameter to the force is in the ratio of \(2 \times x\) to \(x^2\) or the diameter is \(\sqrt{x^2} \times 2\).

> \(x^2\) force effect  
> \(\sqrt{x^2} =\) radius \(= x\)  
> \(x \times 2 =\) diameter

Thus, \(\sqrt{x^2}\) indicates the radius and \(\sqrt{x^2} \times 2\) indicates the [diameter] of a force whose effect on the circumference is equal to the content of the force shape raised to the power of \(x^2\). The effect decreases from the center to the circumference of the circle in the ratio of \(x^2\) raised to the power of the radius squared.

Since the diameter of a force forms the circumference of the next force arising from it, \(x \times 2\) is also the circumference of the next lesser force.

If a force is equal to \(9\), then the radius is \(3\), the diameter is \(6\), and the resulting force is again equal to \(6\) in magnitude, and the radius is approximately \(1/3 = 2\). If the radius is \(1\), then the force is equal to \(1\), because \(1^2 = 1\). So if the diameter is \(2\), the force is \(1\).

The circle represents the number \(10\). The next number is \(9\), the square of \(3\). So \(3^2 =\) \(9\), almost \(10\), the closest to \(10\). The next approximation to the circle is the square drawn inside the circle. The ratio of the square to the circle is like \(9\) to \(10\) or like \(3 \times 3\) to \(3 \times 3.1415\). The circle is therefore more powerful by \(1/10\), \(4/100\), \(1/000\), \(5/10000\), etc. than the square, which has the same diameter.


(1/3) of the circumference forms the circumference of the next power that emerges from it, so *approximately* (1/3). The circumference must divide into three approximately to form the next power that emerges. This results in the tripartite division of power, which is indicated by the ![triangle16](images/triangle16.png).

If we now raise the diameter of the force, which is 3 x 3.1415 in the circumference, to the square, the ![square16](images/square16.png), but also the \(9\).

Just as the \(9\) is the precursor to perfection, so in this sense the square symbolizes \(3 \times 3\), the next step towards perfection, the transition to the \(10\), the circle, because the circle in the square is the \(10\) in the \(9\).


> \(3 \times 3 = 9\)  
> \(3 \times 3,1415 = 10\)

Actually, \(3 x 3\) can be represented as follows:

\(||\) form only one direction – a diameter raised to the square ![square16](images/square16.png)


> sun  
> circle in the temple  
> spiritual man  
> in the physical man

or also


The \(3 \times 3\) represents the stages of the ascent to the perfection of man - the three human powers raised to the square; to complete only [three times] the \(0.1415\) is missing. - The spiritual man is the closed circle, the physical man stepping out of the circle square - the spiritual man is the \(10\), the physical man is the \(9\). (Zodiac arranged on the sides of the square gives the \(12\).)

The circle in the square is the spiritual man emerging from the physical as the sun shining out of the temple.

Circle and square have the same diameter; we find the force effect of the circle by halving the diameter and exponentiating it.

The radius squared is the force effect, and thus also the ratio of the force reduction. The force effect is distributed in the physical square according to the square of the radius in the circle. So to find the force effect in the physical, you have to draw another circle around the square and raise the radius of this circle to the square, thus:


The physical force effect is all the less, as the square of the radius in the outer circle is greater than the square of the radius in the inner circle.

The spiral, emerging from itself, deviates from the direction of the circle at every point of its line by \(0.1415\) and so on. Through this continued deviation of a line in each point in the ratio of \(0.1415\) in relation to the circular line, the spiral arises. Or a spiral arises from a straight line when it turns in the reverse ratio of the circular line.

The spiral forms the relationship between straight line and circle; it is the union of a straight direction and a circular line. When a spiral is closed, the force rests entirely within it; when it unrolls, the force emerges. It decreases to the extent that it emerges, in proportion to the square of the distance of the end point of the spiral from the beginning.


> Force \(a\) is divided by \(4/4\) up to \(10\)  
> Force \(b\) is divided by \(1/100\) at \(x\)  
> Force \(a = b\)  
> \(= 4 \cdot 100 \cdot 25\)  
> so \(b 20x\) is as great as \(a\)


A force always acts in the same direction as the hypotenuse of two forces that intersect at right angles.

If we break down a force into two forces, these forces must intersect at right angles.


The square of the force of the hypotenuse a is equal to the sum of the squares of the forces \(b\) and \(c\). The right angle is always the projection of two forces that arise from one force, of two forces that are equal to one force.


Every angle that touches the tip of the periphery, the sides of the diameter and the periphery of a circle is a right angle.


A square is created by constructing two opposite angles in a circle so that they intersect the sides of the diameter and the periphery.


The two catheti circles have the same center. The hypotenuse circle touches the small circle at the center of the large one on the periphery.


Force square – the decline of force from the Mahapara Nirvana plane to the physical plane.

Comet: the life of higher beings in particular, without regeneration.


Parable - detached from the whole and not complementing anything. The forms of new life are


All signs of the zodiac are pieces cut out of the spiral. The same applies to the numbers \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\).

The Roman numerals represent only the unit and duality, the cone, the development process in the whole. \(I, II, III, IV, V, VI, VII, IX, X, XI, XII\).

Only \(I\) and \(X\) are independent signs. All others are composed of these. \(V\) is the \(X\) halved. The \(I\) is the beginning, the primal force. \(V\) is revelation, the deepest descent. \(X\) is completion, where the halved force \(V\) has reunited with the other half: \(X\).

The 7 planets


As long as a person lives in the physical, their consciousness is turned away from the spirit. The spiritual is a focus in an ellipse, the physical is the other.


But the spiritual encompasses the physical. The life of the etheric body with Pranayama overcomes this focus on the physical and turns it towards the spiritual, while the spiritual flows towards it.


The consciousness of the astral body, Kama, finds the connection with the spiritual, and the physical and spiritual merge.


In astral consciousness, the human being stands where these two currents intersect. It is as if he were standing in the middle between two sides of a plane of symmetry. He no longer sees the physical as we see it, but from the other side. But he also does not yet see the spiritual as the spiritual consciousness does, but – also from the other side – as it manifests itself in the soul.

When a person is spiritually conscious, he is at the center of the spirit itself, and his consciousness goes out from this center in all directions. Physical consciousness acts in one direction, soul consciousness acts in two directions. Spiritual consciousness acts in all directions as from the center. Physical consciousness is life on the outside, soul consciousness is life on the inside, and spiritual consciousness is penetration into the center.

In physical consciousness, man stands in relation to the physical world, so that he is a part of it. In the soul, he lives within himself, detached from the world; in the spiritual, he moves back into the world, but into the other side of it. Now he is not only a part of the world, but unites from the center with all parts of the spiritual world. It is as if he lived in the physical on earth, in the soul within himself, in the spiritual in the earth. Physically he perceives things, in his soul he relates to them, spiritually he lives in them. Physically he is separate from things, in his soul he lets them pass through himself \(\infty\), spiritually he moves into things.

All things around him are physically shaped spiritual archetypes that have undergone the reverse process from the spiritual to the physical, through the soul of man. The physical forms owe their diversity to the fact that they are varied in the soul of the person who received them from the spirit, allowed them to pass through him and physically separated them out. The fact that we appear in different forms in different incarnations is because our spiritual archetype passes through the soul and thereby receives the imprint that makes it appear in a different physical form in each incarnation. The archetype is always the same; it is the actual name of the person. The more the soul moves towards the spirit, the more the physical incarnation resembles the spiritual archetype.

The spiritual archetype of the person is connected to the twelve signs of the zodiac and to the sun and the forces that arise from it. The soul in man is related to the moon and the seven planets. These forces modify the spiritual ones. The physical expression of the \(12 + 1\) spiritual and the \(7 + 1\) soul forces is the physical human being, arranged in the pentagram, which thus expresses seven soul and twelve spiritual forces.

The seven soul forces and five sense forces interweave to form the twelve spiritual forces.