§27. PARABOLA, HYPERBOLA, ELLIPSE
§27. Parabola, Hyperbola, Ellipse
Let us imagine two tone-generating points surrounded by circles of
equidistant waves. At some point, depending on the distance between the
points, these circles of waves will intersect. In reality, of course,
they will always be spheres, but projection on a plane is sufficient to
discover the laws by which these intersection points occur. One must
then simply imagine the relevant figures transposed into the spatial
realm, turning an ellipse into an ellipsoid, a parabola into a
paraboloid, and a hyperbola into a hyperboloid.
If we connect the intersection points of the two groups of concentric
circles, we will trace out ellipses or hyperbolas (Fig. 167), depending
in which direction we proceed. Since Fig. 167 is very easy to draw, the
reader can derive the formula of the ellipse (Fig. 167a) and the
hyperbola (Fig. 167b) by counting off the radii that generate the
respective intersection points. This shows that the ellipse traces all
the points for which the sum of their distance from A and their
distance from B is equal, while the hyperbola traces all the points for
which the distance from A minus the distance from B is equal.

Figure 167a

Figure 167b

Figure 168
We have thus achieved the derivation of the ellipse and the hyperbola
through the intersection of two fundamental phenomena of general
vibration theory: the two wave-spheres.
We can read off the parabola directly from our diagram (Fig. 168). Its ratios are:
Major
0/2 |
1/1 |
(0/0) |
|
|
|
|
0/3 |
2/2 |
2/1 |
(0/0) |
|
|
|
0/4 |
3/3 |
4/2 |
3/1 |
(0/0) |
|
|
0/5 |
4/4 |
6/3 |
6/2 |
4/1 |
(0/0) |
|
0/6 |
5/5 |
8/4 |
9/3 |
8/2 |
5/1 |
(0/0) |
|
c |
c′ |
g′ |
c′′ |
e′′ |
etc. |
Minor
2/0 |
1/1 |
(0/0) |
|
|
|
|
3/0 |
2/2 |
1/2 |
(0/0) |
|
|
|
4/0 |
3/3 |
2/4 |
1/3 |
(0/0) |
|
|
5/0 |
4/4 |
3/6 |
2/6 |
1/4 |
(0/0) |
|
6/0 |
5/5 |
4/8 |
3/9 |
2/8 |
1/5 |
(0/0) |
|
c |
c, |
f,, |
c,, |
as,,, |
etc. |
Figure 169
Here is the proof that they are parabolas: the familiar parabola equation x2 = 2px changes into y2 = x for a parabola whose parameter is 1/2, i.e. the y-coordinates (perpendicular lines) are equal to the square roots of the corresponding x-coordinates (parallel lines). For the parabola 0/6 5/5 8/4 9/3 8/2 5/1 0/0 in Fig. 168, 5/1 is the perpendicular line 2 units away from the point 5/3 on the x-axis, and the length of the x-line 9/3-5/3 contains 4 units. The y-value 2, then, is the square root of the x-value 4. The x-value 0/3-9/3 has 9 units as its axis, the corresponding y-value 9/0-9/3
= 3 units. √9 = 3, etc. The apexes of these parabolas generate further
parabolas. We obtain a beautiful image of these parabolas (Fig. 170)
from their fourfold combination, anticipating what will be further
discussed in §32.
The hyperbola also has a simple and interesting harmonic derivation. If
we draw the partial-tone-values of its string-length measures
perpendicularly (Fig. 171) and turn them sideways, always using unity
as a measure, then we get perfect rectangles, identical in area to the
unit-square. Connecting the corners then produces a hyperbola, whose
equation is a2 = xy, as is generally known. In our case, this means that
1/1 · 1/1
1/2 · 2/1
1/3 · 3/1
etc. |
} |
= 1 |

Figure 170

Figure 171

Figure 172
As we saw above, the hyperbola is the geometric location for all points for which the difference between the x- and y-coordinates is the same. Thus we can also explain their “harmonics,” as in Fig. 172.
The hyperbola, drawn in points, continuing endlessly in both the x- and y-directions,
indicates that from any point placed on it, a rectangle of consistently
equal area can be introduced between the curve and the axes A B C. If d
- B = 1, then we have:
for: |
length: |
height: |
therefore, the quadrilateral's area: |
a2 |
1/4 |
4/1 |
1/4 · 4/1 = 1 |
b2 |
1/2 |
2/1 |
1/2 · 2/1 = 1 |
c2 |
3/4 |
4/3 |
3/4 · 4/3 = 1 |
d2 |
1 |
1 |
1 · 1 = 1 |
e2 |
4/3 |
3/4 |
4/3 · 3/4 = 1 |
f2 |
2/1 |
1/2 |
2/1 · 1/2 = 1 |
g2 |
4/1 |
1/4 |
4/1 · 1/4 = 1 |
Figure 173
The law of hyperbola construction therefore shows us an increasing arithmetic series (1/n 2/n 3/n 4/n ...) and a decreasing geometric series (harmonic n/1 n/2 n/3 ...)-a precise analogy to the intersecting major-minor series of our diagram.
And if we consider, moreover, that the ellipse is the geometric
location for all points for which the sum of two distances has an
unchanging value, then it is easy enough to construct the ellipse
harmonically with reciprocal partial-tone logarithms, since their sum
is always 1-for example, 585 g (3/2) + 415 f (2/3)
= 1000. In Fig. 174, this tone-pair is drawn with a thick line and
marked for clarification. We mark two focal points 8 cm apart (Fig.
174) for the construction of the ellipse, draw one circle around one
focal point at radius 5.9 cm (585 g) and one around the other at radius 4.1 cm (415 f), then trace the intersection points of each pair of rays, up to the point where the two shorter f-rays intersect with the circumference of a small circle drawn around the center of the ellipse, and the two longer g-rays
intersect with the circumference of a larger circle drawn around the
center of the ellipse. These two outer circles, whose radii are of
arbitrarily length, serve simply to intercept the vectors (directions)
of the single tone-values and to distinguish them clearly from one
another. All other points of the ellipse are constructed in the same
way. The tone-logarithms here were simply chosen in order for the
construction of the ellipse points to be as uniform as possible. If the
reader has a good set of drawing instruments, then he can use all of
index 16 for point-construction-a beautiful and extremely interesting
project. In this case it would be best to use focal points 16 cm apart,
and to double the logarithmic numbers.

Figure 174
Even if this construction of an ellipse from the equal sums of
focal-point rays is nothing new and can be found in every elementary
textbook, its construction from the reciprocal P-logarithms still gives
us an important new realization. As one can see from the opposing
direction of rays in the ratio progression of the outer and inner
circles, the tones are arranged here in each pair of octave-reduced
semicircles, and thus the directions of the ratios of the two circles
are opposite to each other. From the viewpoint of akróasis, then, there are two polar directions of values concealed in the ellipse: a result that might alone justify harmonic analysis as a new addition to a deeper grasp of the nature of the ellipse.
Parabola, hyperbola, ellipse, and circle (in §33 we will discuss the
harmonics of circular arrangements of the P) are of course conic
sections, i.e. all these figures can be produced from certain sections
of a cone, or of two cones tangent at their apexes. The above harmonic
analyses, of which many more could be given, show that these conic
sections are closely linked to the laws of tone-development, which
supports the significance of the cone as a morphological prototype for
our point of view. In pure mathematics, this significance has been
known since Apollonius, renewed by Pascal, and discussed in De la
Hire's famous work Sectiones Conicae,
1585 (the reader should definitely seek out a copy of this beautiful
volume at a library), right up to modern analytical and projective
geometry. For those interested in geometric things and viewpoints,
hardly anything is more wonderful than seeing the figures of these
conic sections emerge from an almost arbitrary projection of points and
lines, aided only by a ruler. For a practical introduction see also L.
Locher-Ernst's work, cited in §24c.
§27a. Ektypics
Mathematically speaking, ellipses, parabolas, and hyperbolas can be
defined as the geometric location of all points for which the distance
from a fixed point (the focal point) is in a constant relationship to
the distance from a fixed straight line (the directrix). On this rest
the projective qualities of conic sections and the possibility of
constructing them by means of simple straight lines (the ruler).
In detail, as remarked above, these “curves of two straight lines” have
many more specific harmonic attributes-for example, the octave
relationship (1 : 2) of the areas of a rectangle divided by a parabola,
the graphic representation of harmonic divisions in the form of
hyperbolas, etc. One obtains the “natural logarithm” when one applies
the surface-content enclosed by the hyperbola between the two
coordinates (F. Klein: Elementarmathematik vom höheren Standpunkt aus,
1924, p. 161); thus, a close relationship also exists between the conic
sections and the nature of the logarithm. The applications of the laws
of the conic section are many, especially in the exact natural
sciences. I will mention only the Boyle-Marriott Law, which connects
the respective number-values of pressure and volume, and in which the
hyperbola emerges as a graphic expression (and thus the pressure :
volume ratio of the reciprocal partial-tone values 4 : 1/4, 2 : 1/2, 1 : 1, 1/2
: 2, etc. are expressed most beautifully). I am also reminded of the
“parabolic” casting curves in mechanics, the properties of focal
points, parabolas in optics, the countless “asymptotic” relationships,
etc. Admittedly, these applications are mostly obscured by differential
and integral calculus, though doubtless simplified mathematically-in
other words, the morphological content of conic sections is outwardly
diminished in favor of a practical calculation method, but remains the
same in content.

Figure 175a

Figure 175b
Because of this, it is not astonishing when a figure such as a cone,
from which all these laws flow as from the source of an almost
inexhaustible spring of forms, is applied emblematically even in the
most recent deliberations of natural philosophy, as a direct prototype
for the “layers of the world” and for our “causal structure.” In
Figures 175a and b I reproduce the diagrams from H. Weyl: Philosophie der Mathematik und Naturwissenschaft, 1927, pp. 65 and 71, which speak for themselves.
§27b. Bibliography
Thimus: Harmonikale Symbolik des Alterthums, 4th part. H. Kayser: Hörende Mensch, 65, 66, 126, and familiar geometry textbooks.