What is Man?, Page 4

"What Is Man?", Page Four

But how is the cell/Earth similarity a reflection of the Moon/Earth relationship? Both the Ptolemaic and Copernican systems were attempts to synthesize what was observed. But the Copernican was based upon what was merely "seen" while the Ptolemaic was based upon what was more deeply "perceived." The Ptolemaic system did not look upon the Moon as being "up yonder." Rather, it considered the Moon to pervade the entire spheroid (or ellipsoid-of-rotation) circumscribed by its path—and the whole Moon was turning. The visible Moon was only a part of the full reality (Figure 18). This idea will not seem so remote if we remember the unfertilized egg cell, either with its nucleus, or later with the germinative area near the periphery (Figure 19).


FIGURE 18	FIGURE 19	FIGURE 20

The tiny body of the embryo can be compared with this idea of the Moon that underlay the Ptolemaic system. But in like manner is the embryo, as indeed the human in later life, under the influence of even larger heavenly spheres. Figure 20 shows how we should conceive the Earth as being permeated by both the Moon and the Sun. Spiritually they are their far larger "spheres," not just the visible bodies. It is just such a similar concept that makes the thoughtful person reach out in soul and ask "What is man?" Is the human being to be regarded as no more than its visible body? In the beginning it was not so, nor when the Garden has again been attained will it be so, but in the valley of the shadow of death, where humanity still wanders as archetypal Cain (Gen 4,13-16) and Job (Job 2,6), we are estranged from the character (image) of the Elohim.

The Cassini Curve

In his discussion (Lects. 9, 10 and 15) of the Cassini curve and related geometrical figures, Steiner often goes into the underlying mathematical formulae, which I pass over here. Visual presentation should more readily convey the necessary imagination to most readers. Before going into these images, however, it would be well to look again at the four conic sections as shown in the "Fire" essay.²⁰ While we saw there four possible conic sections, circle, ellipse, parabola and hyperbola, we also saw that the parabola could be considered as an elongated ellipse with its center and one focus and vertex all coinciding at infinity. This is an interesting observation not only in view of the fact that all celestial orbits are elliptical, but because the Parable of the Prodigal Son is an allegory about the parabolic descent (from infinity) and reascent of humanity. The "elongated ellipse" disappears into infinity, just as one end of the lemniscate disappears through the human metabolic/limb system into the Earth. But if we thus join the parabola with the ellipse, and replace it among this foursome with the Cassini curve, we have the following general description for any point on the curve drawn with respect to its focal points (in the case of the ellipse, hyperbola and Cassini curve we are always dealing with two foci, whereas for the parabola only one):

Figure Form Description 21 Ellipse sum of its distances from the two foci 22 Hyperbola difference between its distances from the two foci 23 Cassini curve product of its distances from the two foci 24 Circle quotient of its distances from the two foci Steiner then portrays them graphically. The ellipse and hyperbola are shown in Figures 21 and 22.


FIGURE 21	FIGURE 21

For the Cassini curve, Steiner takes us through a progressive series of figures (Figure 23) that result from merely changing certain interrelationships within the formula itself, all quite mathematically proper, as he illustrates.


FIGURE 23 A	FIGURE 23 B

FIGURE 23 C	FIGURE 23 D

While we do not have to go out of space to mathematically portray the hyperbola (Figure 22), nor to portray the Cassini curve in Figures 23 a, b and c, we do have to go out of space to portray the one in Figure 23d. This curve is describable in mathematics but cannot be demonstrated within the three dimensions of Euclidean geometry.

Now if we progress to the circle, we note the usual definition of the circle as being a curved line every point on which is a constant distance from a fixed point (its center). But there is another definition for a circle, namely, that curve every point of which fulfills the condition that its distances from two fixed points maintain a constant quotient (Figure 24).

FIGURE 24

Thus, in all of these conic figures we can see the necessity of going out of space, or in the case of the ellipse considering it in the larger dimension of the parabola as an ellipse disappearing into infinity. Moreover, the starting point of the Cassini curve is the ellipse.

Thus we have the hint, even in the field of mathematics, that the human soul appears for a while incarnated in earthly form and then goes out of that into the opposite state, the spiritual state, before returning.²¹ But in the imagery of the moving lemniscate, the soul (the Individuality) does not return to the same spot (i.e., the same personality) it occupied in any prior incarnation.²²

And if we look back again at the illustration of the four conic sections, noting particularly the hyperbola, we see that its parts exist for a while, then go out of existence only to come back into existence in the opposite cone. Recall that the moving lemniscatory relationship between the Sun and the Earth (Figure 10) involved a rotating lemniscate, in other words opposite cones.

Moreover, we saw that what was inside one loop of the lemniscate was outside the opposite loop—how like the radial-spherical poles in the human body.

Now suppose we cause the circle on the right side of Figure 24 to increase in size so that its line of curvature becomes less and less. Thinking thus, eventually it flattens into a straight line, the line in fact shown as the y axis. We can even go further and imagine the original circle—which has become now a straight line—moving on further so that it becomes a circle on the left side of the y axis. But in the process of moving the circle from the right side to the left, what has happened? The forces of curvature that were on the inside of the circle on the right side flip over and are forces of curvature on the outside of the circle on the left side. See Figure 25. It is quite difficult for us to picture this in our minds. When we pass from the circle on the right to the straight line in the middle, we still have a circle, but with its center in the indefinite distance to the right. The instant we cross to the far side of that line bending it to the left, the forces stay on the right side of the line as it curves into the circle on the left side. If we assume that the forces inside the circle on the right were centripetal forces, then for the circle on the left they become centrifugal forces. On the right they pull or suck inward, while on the left they pull or suck outward.

FIGURE 25

It is quite legitimate to see in this the relationship between a photograph and its negative, or between matter (flesh) and spirit, or between Sun and Earth, or between spherical and radial, or between thinking (skull/head/consciousness) and willing (metabolism/limb/unconsciousness), or between pressure and suction.

Recall something of the same imagery in the "Light" essay when we were discussing Goethe's theory of color. Goethe wrestled with the problem of how the spectrum is reversed when darkness is allowed to pass through the prism in the same manner that light is made to pass through it. What you get is an inverted spectrum (see Figures 17 and 18 in the "Light" essay). In the ordinary spectrum, green is in the middle between violet on one side and red on the other. In the spectrum obtained by Goethe in applying a strip of darkness to the prism, there is peach blossom in the middle and then again red on one side and violet on the other. Two color bands are obtained whose centers are qualitatively opposite to one another and seem to stretch away, as it were, into infinity (Figure 26).

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