George Bayer
View our George Bayer pages
View our George Bayer pages
Like W D Gann, Bayer understood the Secret Cosmology behind the financial markets discovered within Ancient teachings. His 9 books cover deep esoteric finance, including topics like Squaring of the Circle, the Ellipse, and the 5 Fold Horoscope. We do a quality hardcover reprint of each of his works and offer a 2 Volume Suede Edition of his Complete Works.
Behind the Veil
Behind The Veil - By Dr. Alexander Goulden
Behind The Veil - By Dr. Alexander Goulden
Behind the Veil, by Dr. Alexander Goulden is one of the favorite courses of technical analysts and serious traders. Based on scientific principles identifying Price Levels, Time Turning Points, and Trends, thru Harmonic, Astronomical and Geometrical Techniques developed by a Cambridge Scholar. A new angle on Gann's work.
Catalin Plapcianu
View our Catalin Plapcianu pages
View our Catalin Plapcianu pages
Plapcianu followed Baumring's lead into the core of Gann's Cosmological System, cracking Gann's Squaring of Price and Time. He quantifies Gann's innermost system demonstrating how markets move in multi-dimensional Space & Time, including new and sophisticated trading algorithms which generate 4000% annualized returns.
Position Trading
AstroEconomics
AstroEconomics
Position trading is an approach recommended by both Gann and Baumring, saying that there were maybe only about 4 good trades per year in any market. Markets would go into congestions of accumulation or distribution for years awaiting a new trend, and meanwhile one trades other markets. Gann taught the same principles on his higher level, saying that MOST money was always made in following a strong trend.
Petrus Talemarianus
View our Petrus Talemarianus pages
View our Petrus Talemarianus pages
A masterpiece on the Golden Rule according to principles of Tantrism, Taoism, Kabala, and Pythagoreanism serving to fulfill the Laws of Universal Harmony and aiding accomplishment of the Great Work. It develops a system of correspondences in symbolic, geometric, mathematical and astronomical systems of architecture of the ancient world.
Numerology
Numerology
Numerology
Systems of numerology date back to ancient Egypt, India and Israel. Hebrew number science, Gematria, was woven through the sacred texts of Semitic religion. Plato used numerical codes in his works, and Thomas Taylor elaborated the advanced systems of Pythagoras in his "Theoretic Arithemetic of the Pythagoreans".
Science of Vibration
Science of Vibration
Science of Vibration
W. D. Gann coined this term as a basis of his system of market forecasting. It explores theories of aether physics, vortex systems, and universal order as considered in the late 1800’s, incorporated with valuable elements taken from esoteric cosmology. The theory posits that vibration underlies all phenomena, and that harmonic factors govern universal forces.
Pause Start Back Next Translation Society Titles Top Science Titles Top Metaphysics Titles Science Categories Metaphysics Categories Cosmological Economics Financial Astrology
The Sacred Science Translation Society began in 2004 as a project to translate a collection of the most important and rare works on Cosmology & Esoteric Science into English. Through Angel DonorsSubscribtion Contributions we raised over $40,000 to translate famous foreign masterpieces from French & German on critical subjects in Harmonics, Geometry, Esoteric Mathematics, & Ancient Cosmology.
Hans Kayser was one of the 20th century's leading scientists who made a profound mathematical, geometric and philosophical study of the Science of Harmonics. Now finally avaible in English though our Translation Society, Kayser's series of works explore the deepest principles of Pythagorean Harmony & Order.  His profound research reveals critical insights into Gann Theory & The Law of Vibration.
Our second translation is a French masterpiece on the establishment of a "Golden Rule" according to the principles of Tantrism, Taoism, Pythagoreanism, & the Kabala, serving to fulfill the Laws of Universal Harmony & contributing to the accomplishment of the Great Work. It develops a system of correspondences between the symbolic, geometrical, mathematical & astronomical systems of architecture of the ancient world.
The Law Of The Cosmos: The Divine Harmony According To Plato's Republic/Timeaus. The Platonic Riddle Of Numbers Solved contains hundreds of the most sophisticated diagrams on Sacred Geometry, Pythagorean & Platonic Number Theory, Harmonics & Astronomy with analysis & elaboration of Universal Order & Cosmic Law. Herman Hesse called him a Magisterludi of the Glass Bead Game.
THE ARCHEOMETER: Key To All The Religions & Sciences of Antiquity, Synthetic Reformation of All Contemporary Arts. The Archeometer is the instrument used by the Ancients for the formation of the esoteric Canon of ancient Art and Science in its various architectural, musical, scientific forms. A highly respected elaborations of the Universal System, by one of the great esotericists of the 19th century.
W.D. Gann Works
W.D. Gann Works
W.D. Gann Works
We stock the complete collection of the works of W.D. Gann. His private courses represent the most important of his writings, going into much greater detail than the public book series. Our 6 Volume set of Gann's Collected Writings includes supplementary rare source materials, and is the most reliable compliation of Gann's unadulterated vital work.
Dr. Jerome Baumring
Dr. Jerome Baumring
Dr. Jerome Baumring
The work of Dr. Baumring is the core inspiration upon which this entire website is based. Baumring is the only known modern person to have cracked the code behind WD Gann’s system of trading and market order. Baumring found and elaborated the system of scientific cosmology at the root of Gann’s Law of Vibration. There is no other Gann teaching that gets close to the depth of Baumring’s work.

Hans Kayser's Textbook of Harmonics - Excerpts §27. Parabola, Hyperbola, Ellipse

By Hans Kayser

Kayser’s harmonic research provides profound insights into W. D. Gann’s Law of Vibration and the function of parabolic and hyperbolic growth in space as described by Dr. Jerome Baumring.

Introduction

Hans Kayser’s work presents a masterful elaboration of the system of harmonics and vibration, looking at it from the standpoint of a universal system on order which applies from mathematics to geometry to astronomy and even to subjects such as the financial markets. Our clientele is deeply involved in the theory of the Law of Vibration developed by W. D. Gann, and this section will demonstrate to traders the valuable applications to Gann theory and analysis which Kayser’s work brings. Subjects like the parabolic and hyperbolic and elliptical growth in the markets and the use of ellipses and harmonic relationships between impulse waves and reactions through key lines, like the angles on Gann’s geometrical market charts can be seen in the Gann like diagrams below.

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 167a

Figure 167b
Figure 167b

Figure 168
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 168

 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
1/3 · 3/1  
etc.    

 

Figure 170
Figure 170

 

Figure 171
Figure 171

Figure 172
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
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 175a

Figure 175b
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.

Related Pages

Sign up for email NewsLetter Download Latest NewsLetter
$0.00
$ (USD)
Institute of Cosmological Economics > ICE Forum index
Hans Kayser
View our Hans Kayser pages
View our Hans Kayser pages
Hans Kayser was a leading 20th century scientist who made a deep mathematical, geometric and philosophical study of the Science of Harmonics. Available from our Translation Society, his books explore the deepest principles of Pythagorean Harmony and Order, giving critical insights into Gann Theory and The Law of Vibration.
Initiation
Initiation
Initiation
We have many books on initiatory systems and the exploartion of human potential.
Prophecy
Prophecy
Prophecy
A generalized term for any kind of metaphysical methodology for predicting future events. Examples would be psychic phenomena, reading crystal balls, scrying mirrors, numerology, astrology, and many more such divinatory techniques. These systems are popular amongst esoteric traders and forecasters seeking insight into future events.
Sacred Geometry
Sacred Geometry
Sacred Geometry
Sacred Geometry explores natural order representing foundational templates of the cosmos, via special proportions like "phi", the Divine Proportion, ubiquitous throughout nature as a primary generating and ordering principle. Musical harmonic ratios dominate sacred geometry, showing how nature is a form of frozen music.
Translation Society
Translation Society - Metaphysics
Translation Society - Metaphysics
Metaphysics We have completed several translations of important books on metaphysics. These include 4 works on universal harmonics by Hans Kayser, as well as "The Archeometer, a Key to All Science", "The Natural Architecture, the essence of Hermetic and Pythagorean science" and Eberhard Wortmann’s "Law of the Cosmos".
Astronomy
Astronomy
Astronomy
A fundamental principle of Cosmological Economics is the interconnection between galaxies, solar systems, stars, and planets, along with their interactive influences. For example, the rotation of our galaxy is responsible for temperature fluctuations on Earth as a result of cosmic ray variations as we rotate through the spiral arms.
Chaos Theory
Chaos Theory
Chaos Theory
Non-linear dynamic mathematics, known as Chaos Theory, seeks order in seeming random patterns, exploring subjects like Fractals, System Mechanics, Lorentz Attractors, and more. Dr. Baumring originated the idea that Chaos theory provided insight into market phenomena, and later the great Mandelbrot tried to apply Chaos theory to the markets.
Chemistry
Chemistry
Chemistry
The origin of modern chemistry goes back to the mysterious science of Alchemy, which originated in Ancient Egypt, know to the Arabs as Kemi, the Black Lands. Alchemical experiments with chemicals and compounds led to the modern science of chemistry, although Alchemy incorporates spiritual and esoteric elements missing from chemistry.
Cosmology
Cosmology
Cosmology
Cosmology provides the primary basis for our theoretical system of market order and scientific analysis. Moving beyond modern ideas, our approach to Cosmology involves Pythagorean, Esoteric, Eastern, and Ancient metaphysical theories of cosmology. Our ICE collection focuses on ancient concepts as well as modern alternative theories of the universe.
4th Dimension
4th Dimension
4th Dimension
Much science from the 1800’s postulated a 4th Dimension, often considered to represent Time, in relationship to 3-Dimensional space. Gann himself posited the idea of space itself being a 4th dimension in the markets, which requires the Gann theorist to become familiar with complex and often metaphysical theories of extended dimensionality.
Hans Kayser
Hans Kayser
Hans Kayser
Kayser, the greatest scholar of harmonic science, was far ahead of his time, penetrating the depths of ancient esoteric Pythagorean Tradition to present a universal Law of Vibration. His "Textbook of Harmonics" provides the most valuable insight into Gann’s Law of Vibration of any resource. We have so far translated four of his works into English.
Robert Rundle
View our Robert Rundle pages
View our Robert Rundle pages
Magic Words Thru the Zodiac cracks the complex symbolic code that W. D. Gann used within "The Tunnel Thru the Air". It unveils a Masonic Gematria cypher which serves to decrypt references and clues concealed in names, dates and other key words thru the text. These conversions are used to determine anchor points for important market cycles.
54.226.155.151
Disclaimer Privacy Policy Terms and Conditions

Guarded by Cerberus