the law of the cosmos - institute of cosmological ......-5- publisher’s introduction we are very...

THE LAW OF THE COSMOS

THE DIVINE HARMONY ACCORDING TO PLATO’S

REPUBLIC – TIMAEUS

The Platonic Number Riddles Newly Solved and Illustrated

by EBERHARD WORTMANN

1965

Translated by Ariel Godwin

2006 Published by Sacred Science Institute

www.CosmoEconomics.com

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EBERHARD WORTMANN


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GENERAL CONTENT General Content ............................................................................................................................................................................ 1 Publisher’s Introduction .............................................................................................................................................................. 5 Biographical Introduction.......................................................................................................................................................... 7 Contents .......... ............................................................................................................................................................................. 9 Foreword ......................................................................................................................................................................................................... 11 Quotations ......................................................................................................................................................................................................... 12 REPUBLIC, OR PLATO’S POLITICAL SECRET ................................................................................................................................................. 13 Introduction; Quotations......................................................................................................................................................................... 15 The Riddle of Numbers in the Republic ........................................................................................................................................... 17 The solution of the Republic Riddle ................................................................................................................................................... 18 Step pyramid and ideal pyramid ......................................................................................................................................................... 20 The significance of the Republic pyramid ....................................................................................................................................... 21 Apotheosis .................................................................................................................................................................................................... 22 TIMAEUS, OR REVELATION BY MEANS OF NUMBER ............................................................................................................................ 23 Part I: The four essences........................................................................................................................................................................... 23 Introduction - Quotations.......................................................................................................................................................... 25 The Timaeus Riddle of Numbers ............................................................................................................................................ 27 The arithmetical solution of the Timaeus riddle............................................................................................................... 28 The regular polyhedrons ........................................................................................................................................................... 30 The four essences.......................................................................................................................................................................... 31 The proofs ....................................................................................................................................................................................... 32 The Master speaks......................................................................................................................................................................... 34 Part II: The best of things created......................................................................................................................................................... 35 Introduction - Quotations.......................................................................................................................................................... 37 Description of the water jug ................................................................................................................................................... 38 The Golden Section .................................................................................................................................................................... 39 The cones of the jug.................................................................................................................................................................... 40 The volume of the jug ............................................................................................................................................................... 41 The extinguisher ............................................................................................................................................................................ 42 The extinguisher knob ............................................................................................................................................................... 42 The oil vessel.................................................................................................................................................................................... 43 Wick and flame .............................................................................................................................................................................. 43 The jug stand ................................................................................................................................................................................. 43 Circles and polygons in the extinguisher............................................................................................................................ 44 The accessory cones of the jug............................................................................................................................................... 45 The hendecagon of the jug section ..................................................................................................................................... 45 The cross-section of the jug...................................................................................................................................................... 46 The pentagons ............................................................................................................................................................................... 46 Ten-pointed star and five-pointed star ................................................................................................................................. 47 The polygons .................................................................................................................................................................................. 47 The pointer (the quadrature of the circle) ......................................................................................................................... 47 The altar (the squares of the Egyptian triangles)............................................................................................................. 48 The conic sections......................................................................................................................................................................... 48 The Symbolon ............................................................................................................................................................................... 49 The spearpoint .............................................................................................................................................................................. 49 Mysteries from the ancient past ............................................................................................................................................. 50 Chalice and Holy Grail ............................................................................................................................................................... 51


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Part III: Psyche, or the flower miracle ............................................................................................................................................. 53 Introduction – Quotations ........................................................................................................................................................ 55 1st image: Thaumazein – The Astonishment .................................................................................................................... 56 2nd image: Psychogenesis – The Generation of the Soul............................................................................................ 57 3rd image: Catharsis – The Purification................................................................................................................................. 57 4th image: Psyche Alogos – The Natural Soul .................................................................................................................. 58 5th image: Psyche Logistikon – The Enlightened Soul .................................................................................................. 59 The Proportion of the Flower Miracle .................................................................................................................................. 59 The Total Proportion of the Timaeus Parts I/III.................................................................................................................. 60 Part IV: The sign of the macrocosm ............................................................................................................................................... 63 Introduction; - Quotations......................................................................................................................................................... 65 The Sign of the Macrocosm ..................................................................................................................................................... 66 The Orbits of the Planets............................................................................................................................................................ 67 The Sphairion (The First Horoscope) .................................................................................................................................... 68 The Conjunction of the Moon, Earth, and Venus.......................................................................................................... 68 The Opposition of Jupiter and Mars..................................................................................................................................... 69 Helios, the Conjunction of the Sun, Saturn, and Mercury .......................................................................................... 69 The Blooming Heaven................................................................................................................................................................ 70 Orion, the Soul of the Universe............................................................................................................................................... 70 The Animated Bands (the Fixed Stars)................................................................................................................................. 71 The Timaeus Sigil ........................................................................................................................................................................... 72 Part V: Thus did time come into Being ......................................................................................................................................... 73 Introduction; Quotations ........................................................................................................................................................... 75 The Revolution Periods of the Planets.................................................................................................................................. 76 The Three Regulators ................................................................................................................................................................. 76 The Corrected Revolution Periods of the Planets............................................................................................................ 77 The World-Year .............................................................................................................................................................................. 78 The Planets’ Revolutions............................................................................................................................................................. 79 The Proportion of the Planets’ Revolution Periods ......................................................................................................... 79 The Sacrificial Vessel .................................................................................................................................................................... 80 The Planets’ Revolution Periods in Divine Time................................................................................................................ 80 Sirius, the World Clock ............................................................................................................................................................... 81 Time Measurement and its History........................................................................................................................................ 81 Platonic Time Division.................................................................................................................................................................. 82 The Eight World-Years ................................................................................................................................................................ 82 World-Fire And New Beginning ............................................................................................................................................ 82 Saturn the Destroyer ................................................................................................................................................................... 83 The Deluges .................................................................................................................................................................................... 84 The Dagger in the Sky ............................................................................................................................................................... 84 Table of the planetary Revolution Periods to be determined.................................................................................... 85 The Hunt of the Two Moons ................................................................................................................................................... 86 The End of the Doomed Planet ............................................................................................................................................ 87 The Pendulum System ............................................................................................................................................................... 88 The Emergence of the World from the Soul .................................................................................................................... 90 A) The Pentagon Again ............................................................................................................................................................. 90 B) The Windmill ............................................................................................................................................................................ 90 C) The Soul’s Kernel ..................................................................................................................................................................... 90 D) The Tree of the Immortal Soul ......................................................................................................................................... 91 E) The Two Bands......................................................................................................................................................................... 91 F) The Second Division: .............................................................................................................................................................. 91 G) Division of the Sphere’s Outer Surface.......................................................................................................................... 92 H) The Mixing of the World-Soul .......................................................................................................................................... 92 I) The Key ......................................................................................................................................................................................... 93 A Game With Prime Numbers ................................................................................................................................................ 94 The Number 37 ............................................................................................................................................................................. 94 The Number 315........................................................................................................................................................................... 94


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Part VI: The image of the deity ......................................................................................................................................................... 95 Introduction; Quotations ........................................................................................................................................................... 97 Alpha and Omega, the 1; The Pole Star 216.................................................................................................................... 98 The Image of the Deity .............................................................................................................................................................. 98 Coronas and Ray Spheres ........................................................................................................................................................ 99 The Force of Movement ........................................................................................................................................................... 99 The Volume of the Pendulum System...............................................................................................................................100

The Pyramid Peak .......................................................................................................................................................................100 The Neck and Breast Jewelry ...............................................................................................................................................102 The Monstrance ..........................................................................................................................................................................102

The Regular Polyhedrons Again...........................................................................................................................................104 The 1st Polyhedron Harmony: The Eyes of the Deity .................................................................................................104 The Water Clock .........................................................................................................................................................................105 The Orb ..........................................................................................................................................................................................105 The 2nd Polyhedron Harmony: The Treasure House .................................................................................................106 The 3rd Polyhedron Harmony: The Polyhedron Symbola ........................................................................................107 The Great Number Game.......................................................................................................................................................108 The Whole Proportion of the Surfaces of Timaeus I-VI ..............................................................................................109 The Whole Proportion of the Solids of Timaeus I-VI....................................................................................................110

Part VII: The Ruler ................................................................................................................................................................................113

Introduction; Quotations .........................................................................................................................................................115 The Basilica of the Holy Wisdom .........................................................................................................................................116 The Council of Supreme Justice ..........................................................................................................................................117 The Eagle – The Aerie...............................................................................................................................................................118 The Ruler’s Toe Point – The World -Sphere – The Top – The Crown – The Dagger – The Shield The Balsam Vessel – The Orb – The Scepter ..................................................................................................................119 The Ruler’s Regalia – The Ruler’s Veil – The Ruler’s Head .........................................................................................120 The Stola – The Ruler’s Girdle ................................................................................................................................................120 The Statue of the Youth............................................................................................................................................................121 The Six Arriving Worlds ...........................................................................................................................................................121 The Eleven Spheres of the Divine ........................................................................................................................................121 The Form of the Ruler ...............................................................................................................................................................121 The Sacrificial Vessel ..................................................................................................................................................................122 The Seven Leading Ropes .......................................................................................................................................................122 The Ruler’s Throne .....................................................................................................................................................................122 The Crystall Ball.............................................................................................................................................................................122 The Twenty-Two-Pointed Star ..............................................................................................................................................123 The Square ....................................................................................................................................................................................124 The Two Jug Sections ...............................................................................................................................................................124 The Twenty-One-Pointed Star ..............................................................................................................................................124 The Sixty -Three - Pointed Star................................................................................................................................................124 The Wreath of the Triune Deity ...........................................................................................................................................125 The Ruler as a Surface ..............................................................................................................................................................125 The Reticule ..................................................................................................................................................................................125 The Soul Emerges From the ..................................................................................................................................................125 The Ruler ........................................................................................................................................................................................126 The Total Proportion of the Solids of Timaeus I-VII......................................................................................................127 The Total Proportion of the Surfaces of the Timaeus I-VII .........................................................................................129 The perfect Harmony of the Sections.................................................................................................................................130


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Part VIII: The Peacock’s Throne ......................................................................................................................................................133

Introduction; Quotations .........................................................................................................................................................135 The Solids of the Image: The Peacock’s Throne ...........................................................................................................136 The Wheel of Time ...................................................................................................................................................................136 The Peacock’s Body .................................................................................................................................................................136 The Peacock’s Head .................................................................................................................................................................137 The Wheel of Fate .....................................................................................................................................................................137 The Peacocks’ Claw .................................................................................................................................................................138 The Peacock’s Leg ....................................................................................................................................................................138 The Peacock’s Neck .................................................................................................................................................................138 The Peacock’s Beak ...................................................................................................................................................................138 The Peacock’s Feathers ...........................................................................................................................................................138 The Surfaces of the Image “The Peacock’s Throne” ...................................................................................................139 The Total Proportion of Timaeus Parts I-VIII .....................................................................................................................140 Conclusion .....................................................................................................................................................................................141

Appendix: The Harmony of the Spheres....................................................................................................................................143 Introduction; Quotations .........................................................................................................................................................145

Pythagoras and Music...............................................................................................................................................................146 The Vibration Ratios of the Tone Intervals (Pythagoras) ............................................................................................146 The Harmony of the Spheres ...............................................................................................................................................147 Mathematical Proof ....................................................................................................................................................................147 Historical Proof..............................................................................................................................................................................147 Appendix 1.....................................................................................................................................................................................148 Appendix 2 The Tone-Values of the 31 Number .........................................................................................................149 Appendix 3.....................................................................................................................................................................................150 Appendix 4.....................................................................................................................................................................................150 Appendix 5 – The Myth of the Hereafter in the Republic .........................................................................................150


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EBERHARD WORTMANN


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Publisher’s Introduction

We are very happy to release this first English translation of The Law of the Cosmos, by Eberhard

Wortmann, one of the greatest Magister Ludi of the Glass Bead Game, acknowledged as such by

Hermann Hesse, himself. This strange and mysterious work first came into my hands in 1992, as a

German photocopy, through my association with Dr. John Richter, a cosmological and GBG scholar. He

had been given his copy by Dr. Peter Westbrook, who had attempted to track down the Author,

publisher, and family in Germany almost 20 years ago, but with no success. John and Peter were the

only two people who had ever seen or heard of this work or it’s Author, and I immediately thought it

something definitely worth studying and making better known to the world.

Several years later, after creating the Sacred Science Institute, I listed the book on our website, as well as

on our Glass Bead Game site (www.glassbeadgame.com), hoping to draw some attention to it, through

Wortmann’s fascinating diagrams, and further hoping to discover enough interest in it to develop a

translation project to bring the mysterious text into English. To my delight and surprise, the first person to

discover the listing and to contact me about it was Rosemarie Mueller, Wortmann’s daughter, who was

living in Argentina. She introduced me to her daughter, Veronica Mueller, who developed an interest in

archiving and preserving her grandfather’s work, and in seeing the book translated into English for the

appreciation of a wider public.

Another six years passed, during which time I had the opportunity to visit Veronica in Florence, Italy,

where I was able to peruse her grandfather’s original works and hundreds of hand-drawn and colored

diagrams, some of which are contained in this publication. I was as fascinated with this material as ever,

and was frustrated with my inability to read German so as to penetrate the text, further fuelling my desire

to see it translated into English.

In 2004, we developed the Sacred Science Translation Society initially to bring Hans Kayser’s Textbook of

Harmonics into English. Kayser was another Bead Game master and harmonics scholar, who also knew

Hesse, and his massive tome on the science of harmonics and vibration was something that was slightly

less difficult to raise some donor funds for a first translation. We were very fortunate to be introduced by

Joscelyn Godwin, the well known musicological and esoteric scholar and translator, to his son, Ariel, who

had just completed his undergraduate studies in German, and had been helping his father with some

translations. We arranged to hire Ariel for the difficult task of translating Kayser’s masterpiece, with the

fortunate addition of his father’s oversight and editing of the entire work.


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This was a great windfall for us, for besides having found the funds to pay for the translation, we now

also had secured a translation team of the highest caliber to tackle this Herculean effort. Upon

completion of the Kayser translation, we wanted to continue the project by moving on to Wortmann’s

Law of the Cosmos, and through the extremely generous contribution of one anonymous angel donor,

we were able to keep Ariel engaged in the translation of the work now in your hands.

Our intention with the release of this first Translation Society edition, is to use all proceeds from the sales

of this work to fund the further translation of the complete body of Wortmann’s published and

unpublished writings. There are two further complete texts that were previously published in very limited

editions, another unpublished text recently discovered in Rosemarie’s attic, as well as a collection of

miscellaneous essays and lectures, which we would like to bring into English for the fullest appreciation

of Wortmann’s mysterious ideas, and for a deeper penetration into the tradition of the Glass Bead Game.

The Law of the Cosmos presents a Pythagorean style, geometric deciphering of the mathematical

theories contained in Plato’s Timaeus and Republic, which have befuddled academic scholars for

hundreds of years. Wortmann has developed these ideas in a series of intriguing geometrical diagrams,

quite unlike those generally known throughout the traditions of sacred geometry and esoteric

cosmology. These mysterious and original diagrams outline the Platonic and Pythagorean code hidden

within the esoteric mystery schools for millennia. We have already seen scholars apply the principles

contained within this work to the psycho-spiritual energetic matrix of the human body, as well as to the

field of free energy generation, and we suspect there are many further applications to be discovered

when these ideas are more fully digested and understood.

We sincerely hope that this wonderful work inspires the thought and exploration of a new breed of

esoteric scientists, willing to look to ancient and unknown sources, in order to discover new applications

and principles for the advancement of modern science, technology and civilization. We feel confident

that Wortmann’s work has the potential of opening such new doorways, and wish everyone success in

penetrating his profound and mysterious thought…

W. Bradstreet Stewart

Sacred Science Institute

January 2006


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Biographical introduction

As the author’s daughter, it is very gratifying to me to see the publication of my father’s work in the

English language.

For a long time, his life’s work appeared to have fallen into oblivion. Therefore, it was a great surprise for

my whole family when, a few years ago (1998) we saw the name of Wortmann as author of Das Gesetz

des Kosmos on the Web page of the Sacred Science Institute. A copy of his book had made its way, by

unknown means, to the USA, and the Sacred Science Institute immediately became familiar with his

work.

I hope that with its translation into the English language, my father’s great work will now receive the

appreciation that it deserves.

Eberhard Wortmann was born in 1891 in Westphalia, Germany. His profession as an export merchant

took him to many different countries, most of whose languages he learned quickly, and whose mysticism

and historical riddles he eagerly sought to investigate.

After the terrible experiences of the Second World War, he immersed himself more and more in the

mystical world of numbers and geometric images.

Two passions moved him throughout his life: the love of the visual arts—he painted countless watercolors

himself—and, simultaneously, philosophical and mathematical tinkering with magic numbers. In his

advanced years, he was able to connect these two interests marvelously in his works on the Platonic

number riddles.

Wortmann self-published Platons Göttliche Harmonie and Ariston oder die Platonischen Zahlenrätsel in

1955 and 1956. In 1965, Das Gesetz des Kosmos: Die Platonischen Zahlenrätsel erneut gelöst und

dargestellt was published by Otto Reichl Verlag in Remagen. Wortmann gave lectures and readings on

this subject in several German cities.

He received a recognition and praise of his work in 1960 in the form of some correspondence with the

author Hermann Hesse, whom he greatly admired; Hesse described Wortmann as a “glass bead gamer.”

His work Der Heilige Krug: Roman eines antiken Mysteriums remains unpublished.

Eberhard Wortmann died in 1970 in Bad Godesberg, Germany.


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EBERHARD WORTMANN


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CONTENTS:

Foreword - Quotations 11 Republic, or Plato’s Political Secret 13 Timaeus, or Revelation by Means of Number 23

Part I: The Four Essences 23 Part II: The Best of Things Created 35 Part III: Psyche, or the Flower Miracle 53 Part IV: The Sign of the Macrocosm 63 Part V: Thus did Time come into Being 73 Part VI: The Image of the Deity 95 Part VII: The Ruler 113 Part VIII: Ad Infinitum 133

Appendix: The Harmony of the Spheres 143


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FOREWORD Only with reluctance does the Author bring to public light the results of a 9-year search for the solution to

the Platonic number riddles. He is aware that, not being an expert in this field, he is in an awkward position

from the viewpoint of the orthodox sciences. So many people, specialists and laymen alike, have fantasized

about these number riddles, that nowadays any dealing with this material is often dismissed with a shrug. The

Author would never have attained the results that follow if he had wasted his time studying authoritative

sources and allowing them to lead him down the wrong path, especially (for reasons that will become obvious

later) because in the literature at our disposal, anything that could have facilitated the task of solution had

been completely destroyed. The only practicable method was that of logical thought, applying mathematical and

especially geometric formulae, most of which had to be rediscovered first. Only after the preliminary

completion of the solution did the Author find corroborations and references useful for the clarification of

the remaining mysterious points in authoritative publications, especially those of the church fathers.

Nevertheless, the Author would never have succeeded in achieving these results had he not been constantly led

forward, step by step, in a way incomprehensible to him. He remembers certain decisive turning points at

which, almost without his participation, he was guided around obstacles which would otherwise have led him to

fail miserably.

The Author is far from being above receiving suggestions for improvements —the better is the enemy of the

good here as well— because hardly a week passes by without the necessity for major or minor changes.

However, it is assumed that one observes those rules of the game which are known indisputably to be valid.

Here it should be mentioned that the Platonic number riddles are really games, logical games of thought, “glass

bead games,” as the great writer and great man Hermann Hesse calls this kind of mental occupation.

May this publication bring to wider circles something of that which made the Author enjoy his task so greatly:

revelation by means of number, and thereby the certainty of the existence of the divine!


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QUOTATIONS Plato. “The Master [Socrates] passed on the flame of Prometheus. In Plato it grew to gigantic brightness.”

–Frank Thiess, Das Reich der Dämonen Jesus and Plato. “Jesus has not eaten from the tree of knowledge. He who has done that will not have peace within himself. He will strive onward, he needs the mediator to the divine, Eros. Jesus is not qualified to be such a mediator, because he does not know this longing. He cannot replace Plato.”

–Ulrich von Wilamowitz-Moellendorf, Der Glaube der Hellenen Plato and geometry. “This was the state of affairs when Plato came into Sicily ... There was a general passion for reasoning and philosophy, insomuch that the very palace, it is reported, was filled with dust by the concourse of the students in mathematics who were working their problems there.” [Geometric figures were drawn in the dust.]

–Plutarch, Dion.1 The annihilation of ancient authoritative works. “...But those works were thought devoid of interest or even dangerous by the devout Middle Ages, and they are not likely to have survived the fall of paganism.”

–Franz Cumont, The Oriental Religions in Roman Paganism.2 The Platonic riddles of numbers are glass bead games. “In the end it depends on the choice of the historian how far back he wants to put the beginning and prehistory of the Glass Bead Game ... As an idea we find it ... already performed in earlier ages, e.g. by Pythagoras. ... The same eternal Idea was the basis of every movement of the mind towards the ideal goal of a Universitas Litterarum, of every Platonic Academy, of any congregation of a spiritual elite, every attempt to approximate the exact and the free sciences, every attempt to reconcile science science with art, and to reconcile art or science with religion.”

–Herman Hesse, The Gass Bead Game.3

“Thanks to the glass bead game player!”

–Herman Hesse, to the Author.

1 Translation by John Dryden 2 Anonymous “Authorized Translation,” 1911 3 Translation from the Sacred Science Institute website


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REPUBLIC OR

PLATO’S POLITICAL SECRET Introduction; Quotations 15 The Riddle of Numbers in the Republic 17 The Solution of the Republic Riddle (Plate 1) 18 Step Pyramid and Ideal Pyramid (Plate 2) 20 The Significance of the Republic Pyramid 21 Apotheosis (Plate 3) 22


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EBERHARD WORTMANN


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INTRODUCTION The work of solution began with the riddle in the Republic. At an advanced age, the Author came across Plato’s

Republic—known as Der Staat in the German-speaking world, Politeia in the original Greek. In reading it, he got

only as far as the number riddle in book VIII, and then did not rest until he had found the solution. Since he

had just “coincidentally” been studying the right triangle with sides of ratio 3 : 4 : 5, the so-called “Egyptian

triangle”—an astonishing entity, if one considers that any flat surface can be divided into isosceles triangles,

and these further divided into right triangles, of which the smallest is the Egyptian triangle, which also has

numerous unique features—he immediately saw that the solution was within reach, since the text of the riddle

makes it quite clear that it concerns a pyramid. Only after the final solution, taking into account all the

subtleties—which took him a year—did the Author realize the great importance of the solution and hear of a

well-endowed competition, which had caused a great stir at the time and which, despite lively participation, had

not brought about a solution satisfactory to all sides.

The resulting disenchantment caused interest in the solution of the riddle to fade. Many people even became

convinced that there was no solution. Plato, they said, only wanted to make people interested in him, by virtue

of his supposedly possessing a secret formula for running a state, so that he would be remembered favorably

and recommended for high office. But in fact, Plato had already given up his political ambitions at that time. He

“squatted in the corner with a few youths,” as his opponents said, and devoted himself exclusively to the task

of setting up an ideal that could never be realized and is therefore eternally valid, the Ideal State, the first of

many such utopias.

One will see that this riddle concerns ancient thoughts, and that every detail, even the smallest, emerges of

itself.


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QUOTATIONS “Now this number represents a geometrical figure which has control over the good and evil of births.” –Plato, Republic, VIII, 5464 “[Mathematics] appears to me to be a study of the kind which we are seeking, and which leads naturally to reflection.” – Republic, VII, 523 “Do not all arts and sciences necessarily partake of [arithmetic]?” – Republic, VII, 522 “[Numbers] lead the mind towards truth.” – Republic, VII, 525 “Until they see the nature of numbers with the mind only...” – Republic, VII, 525 “The philosopher must learn the art of the number, because he has to rise out of the sea of change and lay hold of true being.” – Republic, VII, 525 “The knowledge at which geometry aims is knowledge of the eternal.” – Republic, VII, 527 “After the second dimension, the third, which is concerned with cubes and dimensions of depth, ought to have followed ... so little seems to be known as yet about these subjects ... it is in a sad state.” – Republic, VII, 528 “A creature who does not yet know what two and three and odd and even are, who does not even have any concept of the number, will also be in no position to give a rational explanation of anything at all.” –Plato, Addition to the Laws, 977C “All our perceptions must therefore necessarily be based on those of numbers.” – Addition to the Laws, 978A “We can console ourselves with the fact that the Greeks have elevated everything they received from foreign cultures to greater beauty and perfection.” – Addition to the Laws, 958A “Philodemus, in the academy’s history, says: the impetus of the mathematical studies was brought about by Plato.” –Ulrich von Wilamowitz-Moellendorf, Der Glaube der Helenen.

4 Translator’s note: Quotations from Plato’s Timaeus and Republic are taken from the translations by Benjamin Jowett, except in those instances where it was deemed more appropriate to translate Wortmann’s quotation or paraphrasing directly from his German.


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THE RIDDLE OF NUMBERS IN THE REPUBLIC “A city which is thus constituted can hardly be shaken; but, seeing that everything which has a beginning has

also an end, even a constitution such as yours will not last for ever, but will in time be dissolved. And this is the

dissolution:—In plants that grow in the earth, as well as in animals that move on the earth's surface, fertility

and sterility of soul and body occur when the circumferences of the circles of each are completed, which in

short-lived existences pass over a short space, and in long-lived ones over a long space. But to the knowledge of

human fecundity and sterility all the wisdom and education of your rulers will not attain; the laws which

regulate them will not be discovered by an intelligence which is alloyed with sense, but will escape them, and

they will bring children into the world when they ought not.

“Now that which is of divine birth has a period which is contained in a perfect number, but the period of human

birth is comprehended in a number in which first increments by involution and evolution (or squared and cubed)

obtaining three intervals and four terms of like and unlike, waxing and waning numbers, make all the terms

commensurable and agreeable to one another. The base of these (3) with a third added (4) when combined with

five (20) and raised to the third power furnishes two harmonies; the first a square which is a hundred times as

great (400 = 4 × 100), and the other a figure having one side equal to the former, but oblong, consisting of a

hundred numbers squared upon rational diameters of a square (i.e. omitting fractions), the side of which is five

(7 × 7 = 49 × 100 = 4900), each of them being less by one (than the perfect square which includes the

fractions, sc. 50) or less by two perfect squares of irrational diameters (of a square the side of which is five =

50 + 50 = 100); and a hundred cubes of three (27 × 100 = 2700 + 4900 + 400 = 8000).

“Now this number represents a geometrical figure which has control over the good and evil of births. For when

your guardians are ignorant of the law of births, and unite bride and bridegroom out of season, the children will

not be goodly or fortunate ... In the succeeding generation rulers will be appointed who have lost the guardian

power of testing the metal of your different races, which, like Hesiod's, are of gold and silver and brass and

iron.”

–PLATO, Republic VIII, 546


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THE SOLUTION OF THE REPUBLIC RIDDLE The perfect number: was six in ancient times, because 1 + 2 + 3 = 6 and 1 × 2 × 3 = 6. The number for human births: “in which first increments,” “3 intervals” = 3 number values, “4 terms” = 4 powers, “combined” = multiplication, the 1st power is not a multiplication.

1, 2, and 3: not possible, because with the 1, no multiplication can be performed. 2, 3, and 4: not possible, because 2 = √4, 4 = 22 3, 4, and 5 are the first numbers that meet the requirements mentioned once in the text.

3 × 4 × 5 = 601 × × × 3 × 4 × 5 = 602 (= 3600) 1st multiplication × × × 3 × 4 × 5 = 603 (= 216000) 2nd multiplication × × × 3 × 4 × 5 = 604 (= 12960000) 3rd multiplication, the Platonic number The first proportion: “a square which is a hundred times as great” = the Proportion of Time. √(12960000/100) = √129600 = 360. The circle has been divided into 360° ever since the time of the ancient Sumerians. 1st “circle rotation”: the day-circle, 1° = 4 minutes. 2nd “circle rotation”: the year-circle, 1° = (ca.) 1 day. 3rd: 100 years. The “marriage number”: 60 × 4 minutes = 4 hours – the 4th hour of the morning is the most auspicious time for begetting. 60 × 4 hours = 10 days – the 10th day after beginning of menstruation is the best time for conception. 60 × 10 days = 20 months – 20 months after one birth is the earliest time for the next birth. 60 × 20 months = 100 years: only every 100 years, every 4th generation, the birth of a strong personality can be expected in a family. The second proportion is that of space, because we calculate here in terms of “sides,” directions, and “diagonals.” With the shortening of a diagonal, the area must likewise decrease. The regular shortening of a diagonal yields a pyramid (see Figure 1a). If we want to use the Platonic number (604) in a pyramid, we can use 60 (=601) as the base side. The base area is then 602. For the height, however, we must use 3 × 60 (pyramid formula = a × b × 1/3 × h) and obtain 603 as the spatial content (volume). The fourth power of 60 (604) can only be sought in the building blocks (cuboids). Side a = 3, side b = 4, side c (=height) = 5, because 3 × 4 × 5 = 60 (Figure 1b). But through this, the measurements of the pyramid change: a = 60 × 3 = 180, b = 60 × 4 = 240, c (= h) = 3 × 60 × 5 = 900 (Figure 1c). The pyramid is then equilateral in one direction (vertical section), “but longer in the other” (horizontal section). The side ratio 3 : 4 : 5 is that of the so-called “Egyptian triangle.” A perfect example for the Pythagorean Theorem for the right triangle: a2 + b2 = c2, therefore 32 + 42 = 52, 9 + 16 = 25, √25 = 5 for the side opposite the right angle (the hypotenuse), when the short leg (a) is 3 and the long leg (b) is 4. The two legs then form a perfect right angle (Figure 1d). The base of the pyramid sides a 3 × 60 (= 180) and 4 × 60 (= 240) therefore has a diagonal of 5 × 60 (= 300) (Figure 1e) “A hundred numbers squared upon rational diameters of a square, the side of which is five, each of them being less by one” = the inclination of the pyramid edges = 1 : 3. “Less by two perfect squares of irrational diameters” = inclination of the pyramid sides, 180 + 240 + 180 = 600 = 1 : 6. One side can be determined if three sides are given. Edge length and side height yield irrational numbers (with infinite fractions) which are elegantly described thus.


-19-

“And in the other direction (i.e. horizontally) a hundred cubes of three.” The base area = 60 × 60 cornerstones = 3 × 4 × 3600 = 43200. 3600/100 = 36 cornerstones, 43200/100 = 432 as area. Three cornerstones are stacked and fastened with pegs in previously bored holes = 1 column; 36 columns, connected at their sides in the same manner, form one building block (see Figure 1f). Thus the pyramid becomes a step pyramid, whose calculation must follow different principles (see page 10). The calculation in terms of “columns” is performed specially for each step, and this calculation converted into building blocks (= number of columns divided by 36) yields 55 remaining columns per story for stories I through IX (for every 6 steps) which must be discarded. 55 = the sum of the cardinal numbers from 1-10, and also the sum of the squares of the numbers 1-5, that is 12 = 1, 22 = 4, 32 = 9, 42 = 16, 52 = 25. In every five steps of stories I, III, IV, VI, VII, and IX, the columns are discarded in this sequence, but in every 5 steps of stories II, V, and VIII they are discarded in reverse sequence, calculated from above 52, 42, 32, 22, 12. This can be described as a true numeric marvel. We use of the removal of these columns to create 9 huge gates, with peaks at the top in stories I, III, IV, VI, VII, and IX, and peaks at the bottom in stories II, V, and VII. Since the number of columns can be divided evenly by 36 in every 6th step, no columns are discarded there, so that the separation into stories becomes clearly apparent (see Figure 1g and Plate 2). The apex of the pyramid (= story X) must yield 19 “columns.” Through the removal of 18 columns, the apex of the pyramid undergoes a decorative rearrangement. The 19th (topmost) column must be (alternately) removed from the floor of story X, because no more columns can be removed symmetrically. This fact will later be of special importance (Figure 1h). In order to compensate for the difference in volume between the step pyramid and ideal pyramid, even more columns must be removed from stories I through VIII. After subtracting the 55 columns for the gates, 288 (= 24 × 12) columns remain in story I, 252 (= 21 × 12) in story II, 216 (= 18 × 12) in story III, 180 (= 15 × 12) in story IV, 144 (= 12 × 12) in story V, 108 (= 9 × 12) in story VI, 72 = (6 × 12) in story VII, 36 (= 3 × 12) in story VIII, which must be removed, meaning in each case 3 × 12 = 36 fewer columns (1 building block). We use the removal of these columns to create inner spaces in these stories, whereby the constantly returning factor 12 represents the profile in width and height, and the other factor (= 3) represents the depth in column-depths. Where the peaks of the gates point upward, meeting rooms (great halls) appear, and where the peaks of the gates point downward, lecture halls (auditoriums) appear. In the former case, the profile has three steps (3 + 4 + 5 = 12 columns), in the latter case it has two steps (9 + 3 = 12 columns), whereby at any given time, 1 column is being dismantled into its three cornerstones, thus creating seating accommodation (Figure 1l and Plate 2). Staircases inside the pyramid link the inner rooms. For this, one building block is taken from the base step above the room, and dismantled into its 108 cornerstones, which are then placed in such a way that side 3 is the height of the step, side 4 the depth, and side 5 the breadth. Number of cornerstones used: For step A = 20, B = 19, C = 17, A + B + C = 56, for supports: D = 17, E = 35, D + E = 52, total = 108 cornerstones. The height is therefore 30 × 3 = 90 = height of the story. The staircase follows the walls of the inner rooms and makes two 90° turns. Staircases of the above form correspond to the most extreme cases; in general far fewer cornerstones will be sufficient. Extra cornerstones are used for broadening the staircase, starting from the bottom (Figure 1j). The grating in the background of the picture (see Plate 3) emerges from the base surface of the pyramid with the built-on squares of the two Egyptian triangles, with the addition of a four-square. The perpendicular cross beam (= 16 + 12 + 16 + 16 = 60; 60 × 602 = 603 or 216000) = twice the cross section of the pyramid. The horizontal crossbeam (= 9 + 12 + 9 = 30; 30 × 602 = 108000) = the cross section of the pyramid. The part of a tilted cross (“St. Andrew’s cross”) visible behind the standing cross has an area of 4 × 6 × 602 = 86400, which is twice the area of the base of the pyramid (43200). Measuring units might be 1/2 foot, 1/4 ell = ca 15 cm, therefore height = 135 m, step = 2.25 m, stair = 45 cm.


-20-

STEP PYRAMID AND IDEAL PYRAMID

a)Story from

bottom to top

b)Story from top

to bottom

c) Columns (square of b)

d) Building blocks = c/36

e) Remaining numbers = gates

f) Sum of

c) story

g) Columns ideal

pyramid

h) Difference (f –

g)

i)minus

gates (55)

j)Inner space: depth × profile

X.

1 2 3 4 5 6

1 4 9

10 25 36 91

- - - - - 1

(1) (4) (9)

(16) (25)

- (55)

91

72

19 (=18+1)

IX.

7 8 9

10 11 12

49 64 81

100 121 144 559

1 1/3 1 2/3 2 2 1/3 2 2/3 4

1 4 9

16 25 ___ 55

559

504

55

–

VIII.

13 14 15 16 17 18

169 196 225 256 289 324 1459

4 5 6 7 8 9

25 × 16 × 9 × 4 × 1 × ___ 55

1459

1368

91

36

3 × 12

VII.

19 20 21 22 23 24

361 400 441 484 529 576 2791

10 11 12 13 14 16

1 4 9

16 25 ___ 55

2791

2664

127

72

6 × 12

VI.

25 26 27 28 29 30

625 676 729 784 841 900 4555

17 1/3 17 2/3 20 21 1/3 22 2/3 25

1 4 9

16 25 ___ 55

4555

4392

163

108

9 × 12

V.

31 32 33 34 35 36

961 1024 1089 1156 1225 1296 6751

26 28 30 32 34 36

25 × 16 × 9 × 4 × 1 × ___ 55

6751

6552

199

144

12 × 12

IV.

37 38 39 40 41 42

1369 1444 1521 1600 1681 1764 9379

38 40 42 44 46 49

1 4 9

16 25 ___ 55

9379

9144

235

180

15 × 12

III.

43 44 45 46 47 48

1849 1936 2025 2116 2209 2304

12439

51 1/3 53 2/3 56 58 1/3 60 2/3 64

1 4 9

16 25 ___ 55

12439

12168

271

216

18 × 12

II.

49 50 51 52 53 54

2401 2500 2601 2704 2809 2916

15931

66 69 72 75 78 81

25 × 16 × 9 × 4 × 1 × ___ 55

15931

15624

307

252

21 × 12

I.

55 56 57 58 59 60

3025 3136 3249 3364 3481 3600

19855

84 87 90 93 96 100

1 4 9

16 25 ___ 55

19855

19512

343

288

24 × 12 73810 2000 495 73810 72000 1810


-21-

THE SIGNIFICANCE OF THE REPUBLIC PYRAMID THE THREE CHARACTER ATTRIBUTES In many passages in the Republic, Plato speaks of the three human character attributes: temperance, courage, and openness to the teachings of wisdom: “Temperance ... is the ordering or controlling of certain pleasures and desires.” (IV, 430). “Courage is a kind of salvation.” (IV, 429). “The principle of knowledge is wholly directed toward the truth ... ‘Lover of wisdom,’ ‘lover of knowledge’ are titles which we may fitly apply to that part of the soul.” (IV, 581). “We may begin by assuming that there are three classes of men—lovers of wisdom, lovers of honor, lovers of gain.” (VI, 581). The cornerstones used in building the pyramid have sides 3, 4, and 5. (4 + 5) : 3 = 3, (3 + 5) : 2 = 4, (32 + 42) = 52. Therefore we can correctly designate a tempered person as a three type, a courageous person as a four type, and a wise person as a five type. “We may assume that [the individual] has the same three principles in his own soul which are found in the State.” (IV, 435). “Justice is the balanced existence of the three character attributes.” 3 × 4 × 5 = 60, therefore we should call just persons “sixty types.” THE FOUR POWERS OF SPIRITUAL TRAINING Human predispositions, according to Plato, are unchangeable; however, they can be developed. Plato explains how he imagines this in the “Allegory of the Cave,” where force is needed to bring the prisoners to the daylight step by step. “Let there be four faculties in the soul—reason answering to the highest, understanding to the second, faith (or conviction) to the third, and perception of shadows to the last.” (VI, 511). Now we can add the exponents to the character attributes, thus obtaining an effective system for the selection of leaders.

THE REPUBLIC PYRAMID IS:

1) A symbol of the hierarchic construction of the Platonic ideal state, the “race of Hesiod,” the iron, bronze, silver and gold races, the farmers and other tradesmen, the civil servants, and their leadership.

2) A symbol of the sciences, with their steps, gates, lecture halls, and assembly halls. 3) A symbol of human life, 60 years of life, 6-year transition periods, the pyramid of age. 4) The Republic pyramid, however, also concerns the ideal construction of the Tower of Babel. The ancient

Sumerians, who build this and similar towers, used a system of calculation based on powers of 60, as we use a system based on powers of 10. These towers all contain the Sumerian numeric system, as does the Republic pyramid in the 4th power. This construction could not be completed with the available building material (bricks).

The GRATING CROSS IN THE BACKGROUND is related to the cross of Constantine the Great, which he

had painted as an emblem on the helmets of his soldiers, the upright and tilted cross (St. Andrew’s cross), the promise of a just state for all! (“In hoc signo vinces”) The history of Constantine’s dream does not belong in Eusebius’s battle account, but in later ecclesiastical history.


-22-

APOTHEOSIS And now, dear stranger, having followed me through the thicket of numbers and the swamp of interpretation,

now you shall see the Temple of Justice with your own eyes. The night still conceals everything from our sight,

but soon Apollo’s arrows will banish the shadows. And yet the fog, rising from this swampy ground, will soon

hide the divine picture from us again. Let us therefore rest upon this stone, so that I may first explain to you

the details and the significance of this exalted architecture.

The temple has the form of a steep pyramid, whose base sides are in the holy ratio of three to four, with the

diagonal five. On each base side stand 60 columns, each composed of three stones, which in their turn have

sides 3 and 4, and height 5. 36 columns form a building block, so that there are also 60 steps ascending.

The sides of the stones correspond to the character attributes: temperance, courage, and wisdom, which the

guards shall possess. On the side facing us, you will see the 9 great gates, which lead into the interior in each

story and which in their form inevitably result from the joining of the columns into building blocks. These are

the gates of wisdom. The number of steps corresponds to the number of years of life that the Gods grant us;

the stories correspond to the time periods during which people change.

On 6 of the gates, the peaks point upwards, and here they lead to the assembly halls inside; on 3 of the gates,

the peaks point downwards, and there a laborious path leads to the classrooms in which the three powers are

taught: belief, intellectual understanding, and intelligent knowledge of the high ideal of justice. In between,

the guards must prove themselves as defenders of the state, in positions of high responsibility.

The apex of the pyramid is constructed by the philosophers, who have achieved the highest level of wisdom and

have proven themselves in every situation. They alternately bear the heavy burden of the highest leadership of

the state, and stand above at lonely heights where they can talk to the gods themselves via their daemons, like

Socrates.

There—do you see the golden flame rising high into the blue of the dawn? It is the top of the pyramid! And now

you notice a silver glimmering— the rays of the rising Sun also touch the shaft of the pyramid. Now you also

see the iron and bronze base on which the pyramid rests, as does every state, upon the farmers and the other

tradesmen.

What a rare apparition reveals itself to us today! Sparkling threads spin across the blue heavens behind the

pyramid and form a giant cross – the cross of celestial justice!

Yet, O woe, the noxious fogs are rising and concealing the divine picture!

Nevertheless, stranger, be happy with me, for your eyes have seen the temple of justice,

THE IDEAL STATE!


-110-

THE WHOLE PROPORTION OF THE SOLIDS OF TIMAEUS I-VI A) Assembly of the solids. 1 a) The solids from the Timaeus I-III = 2163 (see Plate 34) = 100077696 – small parts 25272 and 504 + 333 + 315 (see page 64) = 10051272 = 2072 × 4851 b) The Mixing of the World-Soul (Plate 59) or the key (Plate 60) = 15 × 4851 2) The Sacrificial Vessel (Plate 46) a) pedestal 216×4851 b) lid handle 1 × 4851 c) kettle 88 × 4851 305 × 4851 3) The Dagger in the Sky (Plate 49, version B) 8 × 4851 4) The Monstrance (Plate 68) 961 × 4851 From the 1st polyhedron harmony (Plate 69) 5) Sphere r 105/2 125 × 4851 6) Sphere r 147/2 343 × 4851 7) Sphere r 105 1000 × 4851 8) The polyhedrons: a) dodecahedron 620 × 4851 b) icosahedron/octahedron 217 × 4851 c) supporting column 124 × 4851 961 × 4851 9) The Eyes of the Deity 36 × 4851 10) The 3 Souls of the Deity, or 2d) the lidded vessel of the sacrificial vessel 3 × 4851 From the 2nd polyhedron harmony (Plate 72): 11) Hexahedron 2106 × 4851 12) Tetrahedron 248 × 4851 13) Dodecahedron 47 × 4851 14) Icosahedron 21 × 4851 15) Octahedron 5 × 4851 16) Double cube 113 × 4851 17) The remainders: orb (Plate 71), water clock (Plate 71), chalice (Plate 27) 1 × 4851 From the 3rd polyhedron harmony (Plate 74): 18) Dodecahedron – symbolon 22 × 4851 19) Icosahedron – symbolon 5 × 4851 20) Octahedron – symbolon 4 × 4851 21) Hexahedron – tetrahedron – symbolon 2 × 4851 Sum V of all solids: 8403 × 4851 B) The arithmetical proportion: The ideal proportion would be: 1┐ + 2┐ = 3┐ │7 │7 │7 7╡ + 14╡ = 21╡ │7 │7 │7 49╡ + 98╡ = 147╡ │7 │7 │7 343╡ + 686╡ = 1029╡ │7 │7 │7 401┘ + 4802┘ = 7203┘ 801 + 5602 = 8403

The existing proportion is: 10) 1 + 2 = 3 souls of the Deity = 3 × 4851 2b) 15) 19) 21) 3) 1 + 5 = 6 + 1x) = 7 5 + 2 + 8 = 15 – 1x) = 14 = 21 × 4851 9) 17) 2c) 18) 36 + 1 = 37 + 12x) = 49 88 + 22 = 110 – 12x) = 98 = 147 × 4851 6) 8a) 14) 1b) 20) 343 620 + 21 + 15 + 4 = 660 + 26x) = 686 = 1029 × 4851 7) 4) 8b) 8c) 5) 1000 + 961 + 217 + 124 + 125 = 2427 – 26x) = 2401 = 1a) 11) 12) 2a) 16) 13) 2072 + 2106 + 248 + 216 + 113 + 47 = 4802 = 7203 × 4851 8403 × 4851

Compensating numbers in the existing proportion ± 1 × 12 × 26 = 39, in the geometric proportion steps that lead from stair to stair.


-127-

THE TOTAL PROPORTION OF THE SOLIDS OF TIMAEUS I-VII List of the solids: 1) Council 2401 × 4851 2) Basilica 4851 × 4851 3) Throne 7203 × 4851 4) Youth 693 × 4851 5) Insignia 49 × 4851 6) Scepter 6 × 4851 7) Toe point 7 × 4851 8) Regalia 154 × 4851 9) Veil 22 × 4851 10) Top 88 × 4851 11) Girdle 33 × 4851 12) Stola 264 × 4851 13) World 217 × 4851 14) Sacrificial vessel 305 × 4851 15) 6 worlds 1302 × 4851 16) 11 spheres 4554 × 4851 17) Eagle 3 × 4851 18) Aerie 123 × 4851 19) 7 ropes 396 × 4851 20) Crystal ball 1584 × 4851 21) Ruler 14553 × 4851 = 8 × 4851 × 4851 38808 × 4851 × 4851 I. × 4851 II. × 4851 III. × 4851 IV. × 4851 V. 2401 ┐ Council │ 7 343 ╡ see II. │ 7 49 ╡ Insignia │ 7 7 ╡ Foot │ 7 1 ┘──── 1 soul

6 ┐ Scepter │┐ 120 ┘│ see IV. │ 18 217 ─│┐ world ││ 31 7 ╡┘ Foot │ ─────────┘ 7

4802 ┐ │ 7 686 ╡ │ 7 98 ╡ │ 7 14 ╡ │ 7 2 ┘ 2 souls

+ 49 ┐ 99/98 │ 7 + 7 ╡ │ 8 + 56 ╡ │ 7 + 8 ┘ 6 ┐ Scepter 21

4851 ┐ Basilica │ 7 693 ╡─┐ Youth │ │ 9/2 154 ╡ │ Regalia │ │ 7 22 ┘ │ Veil │ 11/2 6 │ Scepter │

2801 – 1 350 5602 – 2 126 ┘ │

5726 -875

2800 -350 2450 see VI.

2801 + 5602 = sum of all

5600 8403 × 4851 body parts I-VI

│see V. 1/4 └────────── └─

see VI. 4851 × 4851 ──────────────4

× 4851 VI. × 4851 VII. × 4851 VIII. × 4851 IX. × 4851 X. 2450 ┐ 175/41 of I. │─────┐ 154 ┘┐──┐ │ 1/2 Regalia │ │ │ 1302 ┐│──│─┘ 6 worlds ││ │─ 22/5 22 ││─┐│ Veil ││ ││ 6 ││─│┘ 6 Scepter ││ │ 9/2 7 ││─┘ Foot ││ 217 ┘│ 1 world │ 693 ─┘ Youth

126 ┐ I-III spheres │ 11/2 693 ╡┐┐ 21 IV-VI spheres │││ 42/11 2646 ││┘ VII—X spheres ││─ 11/7 1089 │┘ XI sphere │ 33 ╡ Girdle │──────── 264 ─────┘ Stola ┌────────────────────

5726 of V - 875 see VI all spheres 4554 46/3 ────────┘ ┌ 49/46 │ 3 │ │ │ │ │ │ ───────┘┌─────

7203 Throne - 2352 Upper part with cushion and seat see XI ────────────────

14553 Ruler ─────┐

4851 × 4851 4851 × 4851 4851 × 4851 4851 × 4851 3 × 4851 × 4851

Sum of columns VI-X = 7 × 4851 × 4851 Column IX see page 90 5726 + 49 + 14553 = 20328 × 4851 = 63 × 73 × 113 × 4851 V. Insignia Ruler 5726 + 49 – 693 = 5082 × 4851 = 6 × 7 × 112 ─┐ V. Insignia Youth │= 3 14553 + 693 = 15246 × 4851 = 2 × 9 × 7 × 112 ─┘ Ruler Youth

Multiplication of the denominators of columns I-VIII I. 7 × 7 × 7 × 7 = Chalice = 2401 II. 18 × 31 × 7 = lower part of jug = 3906 III. 7 × 7 × 7 × 7 = Chalice = 2401 IV. 99/98 × 7 × 8 × 7 × 21 × 2 × 1/4 = base cone = 4158 V. 7 × 9/2 × 7 × 11/2 × 4 = jug = 4851 VI. 175/11 × 1/2 × 22/5 × 6 × 9/2 = upper part of jug = 945 VII. 11/2 × 42/11 × 21 × 11/7 = jug accessories = 693 VIII. 49/46 × 46/3 × 3 = jug edges = 49 = 4 × 4851 Sum 19404 2401 + 2401 + 49 = 4851 3906 + 945 = 4851 4158 + 693 = 4851 4851 = 4851


Wortmann’s Law of the Cosmos - Plate 3


- L -

EBERHARD WORTMANN


the law of the cosmos - institute of cosmological ......-5- publisher’s introduction we are very...

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