Rediscovery of Proportion, Part 4

Welcome to Aquarius, Volume 8 (August 16, 2007), Squaring the Circle Exactly.

The Rediscovery of Proportion, Part 4

(Now you know what happened on the seventh day of

the seventh month of the seventh year.)

The Reality of Proportion and the Construction of 1 and MQ, M and Pi.

We can square the circle because Phi^2=Pi*(5/6)*MQ

and MQ is defined and constructible.

Here you will find illustration of the construction of 1 and Pi,

How to square the circle exactly,

the Treasure of the Great Pyramid, the Pentangle and the Pythagoreans,

the Proportional Elegance of Sacred Planet Earth.

With "Quick Look" Table of Contents and "when printed" guide.

Any previous statement posted here by John Manimas, explicit or implied,

that the circle could not be squared exactly, is hereby reversed:

the circle can be and is squared exactly here in

Volume 8 of "Welcome to Aquarius."

Here you will come face to face with M, the Mother of all numbers

and therefore possibly the Mother of All Things,

a new Goddess to gaze upon in awe, and strive to understand.

It is my wish that this information be shared with everyone.

What is at stake here?

A) Is "Proportion" the true "number system" of Nature?

B) Do we need to understand Proportion in order to understand Nature

and Nature's technology?

C) Do we need to understand Nature precisely in order to survive?

Link to: (Welcome) or (Geometry Alpha Index).

The Table of Contents is arranged so that the reader can go to the topic of

immediate interest to them and then return to the table of contents and

select the next topic of interest. The text and line drawings used

in these web pages occupy less than ten megabytes of computer memory.

I can be reached by postal mail at: John Manimas, P.O. Box 536, Bellows Falls, VT 05101

Phone calls and emails, without a signed letter, may be discarded.

Quick Look Table of Contents of Part 4 (Volume 8):

[Throughout, * is times sign, sqrt(X) means square root of X

and (N)^2 means value N raised to second power.]

Different browsers may affect text characters and page locations.

1) [Introduction] The initial hypothesis, both old and new.

----->A) A brief summary of the mathematical discovery presented here.

-----> I request public affirmation that this work is correct.

-----> Copyrights and freedom to share this information.

-----> The [Pyramidologists are vindicated]! --->(pre-print guide: when printed, p. 2)

----->EXACT PAGE LINK 1: [Pyramidologists are vindicated!]

-----> The reconciliation of science and religion.

-----> The problem of religion and violence.

-----> Proportion is the key to the room of light.

-----> Author's note: this is about history and brain development as well as math.

-----> The [Elegant Proportions] of Planet Earth

----->EXACT PAGE LINK 2: [Elegant Proportions]

----->B) From [Taylor to Tompkins] to West.

----->EXACT PAGE LINK 3: [Taylor to Tompkins]

----->C) I see the Great Pyramid as a Phi Pyramid.

----->D) The [trail of clues] of the Pyramidologists.

----->EXACT PAGE LINK 4: [trail of clues]

----->E) What could the Pharaoh or Beacon be pointing to?

2) [The numerical values] that are essential to understanding how and why the

construction of 1 and Pi is accomplished.

----->EXACT PAGE LINK 5: [The numerical values]

3) The [Construction of 1 and Pi] Exactly (the geometric constructions are here).

----->EXACT FILE LINK 1 (62 pages): [Construction of 1 and Pi]

4) [Sufficiency-of-Proportion] Theory and what it means:

----->EXACT PAGE LINK 6: [Sufficiency-of-Proportion]

5) The [expected impact] of this work on history, religion, mathematics, number theory, physics and the biological sciences. What does it mean to "use Nature's technology?"

-----> Primacy of Stewardship Theory

-----> The World's Oldest Theological Dispute -- The Hiram Key

-----> The Sense of Proportion as the Source of Human Morality

----->EXACT PAGE LINK 7: [expected impact]

6) Back to the beginning:

7) How the validity of my work can be [tested with a super-computer]:

----->EXACT PAGE LINK 8: [tested with a super-computer]

8) [What is Proportion, really?]

The "Binary Number System" in genetics and Nature's proportional memory].

----->EXACT FILE LINK 2 (27 pages): [What is Proportion, really?]

9) Mathematics as [the last anthropomorphic religion]:

----->EXACT PAGE LINK 9: [the last anthropomorphic religion]

10) (Closing) [The Historical Problem]. How will it be resolved?

----->Gratitude and Memories

----->EXACT PAGE LINK 10: [The Historical Problem]

1) A. Introduction: The initial hypothesis, both old and new.

----->A) A brief [summary] of the mathematical discovery presented here:

The clues of the Great Pyramid, the Eye of Ra, the pentangle, and Pythagorean science, being the science of squares and square roots, and proportional line lengths, all comprise a Beacon of Light, a set of signs pointing to the reality of proportion. The Pythagorean equation was originally not algebraic, because it was understood and applied long before the development of algebra. I propose that the Pythagorean equations and quadratic equation (solution to quadratic equations) were originally applied to geometric (proportional) constructions only in order to illustrate the reality of proportion and not to solve any other mathematical or technical problems. The fact that the Pythagorean Theorem and related mathematical formulas are useful for mathematical applications to solve technical problems is a direct result of the fact that proportion is the natural source of all of geometry and mathematics in the first place.

The original riddle is and was: "Can one construct a circle with exactly the same area as a given square using only the compass and straightedge?" This means constructing squares and circles exactly equal in area without "measuring" any lines. Geometers and mathematicians struggled with this problem for centuries and then in 1882 a proof written by Ferdinand Lindemann stated that the "circle could not be squared" and his proof was accepted by mathematicians, and then the general public, as an irrefutable fact. It is appropriate for me to say, therefore, that the concept that we could not square the circle was in fact a doctrine of mathematics.

However, this mathematical doctrine is a mistake, corrected here. The purpose of the original riddle was not only to encourage geometers to search for the solution to an isolated geometric puzzle, but rather was to point the entire human civilization in the only direction that would lead to the discovery, or re-discovery, of proportion and the role proportion plays in the reality of life in the universe. The path pointed to, the path that leads to an understanding of proportion, is the path of searching for and finding the relationship among Phi (1.618033988...) and Pi (3.141592654...) and MQ and M [sqrt(MQ)]. I discovered, as any geometer would promptly discover, that when one searches for the solution to the "squaring the circle" problem, one inescapably steps into the river of Phi, in which it seems all numbers are contained, except for Pi. The proportional quality of all constructions, and the pervasive appearance of inverses and of Phi in any set of geometric constructions causes the seeker to ask and seek in awesome reverence "Where is Pi?" "Where is it hidden?" Why is Pi a "transcendental" number? Pi is simply the sum of a series of sines or tangents or secants. Why is it that it seems any number can be constructed except for Pi? If we could construct Pi, we would then know and see the relationship between Phi and Pi. That could be the "Secret of Life," the "Sacred Knowledge" that the ancients possessed and wanted us to know for our own benefit, because we are their children. There was an advanced civilization on Earth in the past, and they sent a message forward in time to us, so that we would learn the relationship of Phi and Pi and M, and thereby understand the meaning of proportion and the elegance of the physical universe.

I request public affirmation that this work is correct.

I respectively request that the American Mathematical Society, and the Mathematical Association of America, and the American Academy of Sciences examine this work as quickly as possible. I request specifically that the accepted institutions of mathematics confirm that the circle can be squared exactly using only the compass and straightedge, as described here. I make this request for public confirmation because one can infer from my thesis and from the squaring-of-the-circle-exactly that all of geometry, mathematics and physical science may be revised upon recognition that proportion is the language of Nature, that we can see it and learn how it works through the study of geometry and Pythagorean science, and this matter must be addressed seriously and with the greatest care, because, if it is true that proportion is the language of Nature, then it is only through our understanding of proportion that we can be good stewards of the Natural Environment that falls within reach of our senses and our hands.

I still reserve my copyrights. However, while my copyrights and authorship should be respected, this has nothing to do with anyone's freedom to share the geometric and mathematical facts presented here, because mathematical facts cannot be copyrighted and no one can claim to have any proprietary or "trademark" rights over a mathematical fact. Therefore, PLEASE, share this information promptly with whomever may be interested. I only ask that it be properly acknowledged that this is the work of John Manimas Medeiros, because it is, and it is the work of many years, under difficult circumstances, and this work overturns an accepted mathematical doctrine. I will not pursue any legal action against any person or organization that makes copies of these web pages and gives them to others. But, do not sell a copy to anyone. I have included this statement only to assure every reader that my intent is to share this information, while at the same time I have the same feelings of every inventor and author: I oppose the injustice of another person stealing my work to make a monetary profit for themselves.

Colleges and universities and high schools: If you would like to carry these same web pages on your own website, clearly labeled as "JManimas' thesis on the squaring of the circle," send a written request signed by the chair of the academic department involved or the appropriate school administrator. You must not modify the content, but you would be entirely free to post whatever comments you wish on the entire thesis or any part of it. You would be free to discredit, reject, or even ridicule my work if that is what you believe you should do.

Citation suggested: Manimas, John. The Rediscovery of Proportion, Part 4, The Reality of Proportion. "Welcome to Aquarius,", Volume 8 (August, 2007).

Copyright 2007 John Manimas Medeiros. All rights reserved. Anyone who benefits financially from the work of the author is obligated to compensate the author. Work in progress is not present on my Internet computer.

Recordation (with reference to copyrights): --->(when printed, p. 5)

Dated printed-on-paper copies have been given to friends.

I claim the Internet belongs also to amateur and professional scholars.

The Internet is still a new, evolving means of communication. It is universal. It has the potential to assist in the development of a planetary language. It is accessible to individuals and groups for the purposes of buying and selling, trading, sharing ideas, expressing personal feelings, meeting people, finding a mate, getting valid information about the law or government or in any field of science, history or religion. It is used for the "lowest" types of human communications, for gossip, pornography, planning a crime, for rants and for the wide variety of "blogs" and expressions of anyone inclined to communicate and convey to their neighbors anything from personal self-disclosure to the most sublime and precious ideas of which human beings may aspire. Why should scholars, due to pride, or foolishness, or whatever contrived motive exclude themselves from this Great River of human sharing and human conversation? I claim the Internet for scholars and for scholarship both great and small, by beginners and by giants. Let the customs and rules for scholarly communication evolve on the Internet naturally. Let scholarly debates be open to the public. Let the communications of scholar to scholar be accessible to the world as a scholar chooses. Let the world join in the discussions and challenges of scholarship. Invite the children and youth of the world to join in searching for the answers to the great questions at the frontiers of human knowledge. Why condemn the people to imaginary battle games and a circus of crude stories. There are real battles against ignorance that must be fought and won. The true stories of people searching for knowledge entertain the soul as well as the mind. We should resist the "dumbing down" of society by using the Internet to "smarten up" all those who want to play the real game of real life. The tools of the Internet can just as well convey the most profound poetry, the deepest thoughts, the most incisive arguments regarding the human identity and the human destiny, and how we should proceed as one human species to pursue that destiny.

Let the people talk to the scholars and let the scholars talk to the people. The dissemination of knowledge is the sacred event that occurs in the human drama. Think, how immeasurable your sadness if you possessed knowledge but could not share it with others.

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The Pyramidologists are vindicated ! Proportion IS Everything.

Proportion is the sacred secret of life and how the universe works. Proportion is the reason why Nature does not recognize a boundary between non-living and living matter. Proportion is the reason why molecular matter in the universe is self-organizing. Proportion is like a room, full of light, and squaring the circle is the key that opens the door to that room. The courageous "pyramidologists" found and shared the clues to this treasure in their many books, so that others like myself would see the implications of the clues and continue the search for the world's greatest treasure: the knowledge of proportion.

The reconciliation of science and religion

We are not children any more. We are an adolescent human civilization, and in order to thrive and move forward in our development, we must complete a crucial developmental task. We must complete the reconciliation of science and religion. This means that individuals must reconcile science and religion within themselves, and it is essential that our civilization reconcile its science with its religion. Failure to complete this task will result in self-destruction. Success in completing this task will result in a period of harmony and understanding and in our development into good stewards who can be accepted into an inter-galactic community of civilizations of intelligent beings who are good stewards of life. This intergalactic community of caretaker civilizations is the kingdom of heaven, the house (or clan) of many mansions.

The Problem of Religion and Violence:

In our time, some individuals find themselves unable to reconcile science and religion within themselves. They become so frustrated, so angry, so terrified, they lash out and express their fear that religion will be neglected, or discounted, or discarded. They are paralyzed by the conviction that religion is the source of justice and morality, and that without the dominion of religious institutions life will not be worth living. Their hearts are filled with a love of justice that is so great, they become violent and destructive in the name of justice and in the name of God, based on their belief that religion needs to be defended with violent action. This is a tragic human misunderstanding. Religion does not need to be defended with violence. Religion is best defended with science, with knowledge, with understanding. Proportion is the source of life in the universe, and the source of morality and justice. Human religions are not the source of human morality, but rather a product of human morality.

Human morality is the expression of the human sense of proportion. The human sense of proportion resides within the individual brain.

Without the genetic heritage of a sense of proportion, we would not be able to be social and communal beings whose success is based on concepts of right and wrong, practical and impractical, just and unjust. Morality occurs in human evolutionary development before religion. Religion is like clothing. Morality is the person wearing the clothing. This does not mean that religion is an entirely human creation. Humans are collectors, and we collect elements of mystery and spirituality and incorporate them into our religions. These elements of mystery and spirituality are part of what we long to understand but do not see clearly, and we hold such unknown things in reverence along with the trees and the stars and the smile of a happy child. In this way our religions are the ongoing human art of joining what we know to what we do not know. In this way religion is actually a form of human information, but the information of religion does not produce the fact of morality. The reverse is true. Our sense of morality is the original impulse that motivates us to speak to the universe and tell it that we want to know what is not yet known to us. Our best religious creations tell us to be patient, loyal to the task of learning, and courageous in our perseverance, as it was told to us by Jesus: "Everything will be made known. There is nothing that will not be disclosed." We learn wherever we are, with whomever we are, when our minds and hearts are open and in love with the world. Learning, however and wherever it occurs, is a joyous occasion.

Proportion is the key to the room of light.

It is through our understanding of proportion that we will be able to complete the task of reconciliation of science and religion, and thereby be liberated to achieve our destiny to become good stewards and take our place as members in the kingdom heaven.

Author's note: Although this work is original and includes mathematical discoveries that lead to the effective solution of the "squaring the circle" problem, this is very much a work in the field of history and in the field of brain development. I suggest that this work is important to the field of neurobiology and brain development, because when we ask the question "Why did human civilization require 2,500 years to solve this problem?" I believe the most persuasive answer is because we developed a "digital" [finger-like] way of thinking about number values and quantitative calculations. The decimal number system took over during the decline of the Roman Empire and with the development of algebra and later with the rise of science the decimal digit number system was gradually taught throughout the world. By the twentieth century, being able to count to ten was taken as a sign of brightness in an infant child, and a source of pride to every parent. Parents would say, glowingly, "He (or she) knows her numbers to ten (or to twenty)." And all of us learned how to think and calculate through the digital decimal system. It is interesting that our technological civilization, which we like to believe is "advanced," uses a number system that is based on counting fingers. Although we have algebra and calculus and matrices, a large percentage of society would say "I'm not good at fractions." Even mathematicians are not good at fractions, because we do not train our brains, either early in life or later, to think "proportionally." It took me about sixteen years of practice to re-train my digital decimal brain to think proportionally and discover the effective solution. This is powerful evidence that brain development determines a great deal about how and what a civilization will learn. This is evidence that the "nature versus nurture" debate is effectively resolved, I believe, with the final conclusion that both are equally important. Students of brain development have been saying over the past couple of decades that early childhood brain development is so important that certain personality traits and behavior patterns developed after birth appear to be genetically determined when in fact they developed after genetic conception, not during genetic conception. Although not truly genetic traits, they are as deeply engrained as though they were genetic traits. In any case, although there are several reasons why we did not find the answer to this riddle until now, I believe the main reason is that our civilization did not teach children to think proportionally. This work is based to a large extent on "seeing with new eyes," on looking at known constructions in a new way. Nature is economical, and proportion is sufficient to enable Nature to do all that it does. Nature does not need anything further. Nature does not need mathematics. The universe is not mathematical. It is proportional. Mathematics is a skill, best described as mental gymnastics. This is a useful skill, but human mathematics only shows us how humans do what they do. It does not teach us how Nature does what it does. We must learn how Nature does what it does in order to be good stewards of Nature. We must be good stewards of Nature in order to be naturally selected for survival and perpetuation of our species and our civilization. This work supports the conclusion that the Secret of Life in the Universe is that the proportional values that enable life to thrive on our planet Earth may not be entirely accidental. Intelligent beings may have played a role in the creation of our planet, because without rather exact physical proportions, life only survives in very limited and primitive forms. In order for life to thrive, in order for a planet to "have life abundantly," the planet must have the set of sacred proportions, or if one prefers, the "elegant" proportions.

My work is definitely a work in the field of history because of the evidence I discovered, in the realities of proportion, that support the viewpoint that an ancient civilization on Earth possessed advanced knowledge involving great precision. This ancient civilization has been called "Atlantis," but the fact that it existed is far more important than its accepted name. My historical argument grows directly out of my life as a student of human behavior and human history. At the age of thirteen, when I heard the riddle in my Geometry class, I believed that the circle could be squared based on my immediate intuitive conclusion that the question would not have been asked unless those who asked it knew that there was a positive answer. In the early 1970's, having been informed by Secrets of the Great Pyramid, by Peter Tompkins, I concluded that the Great Pyramid was never a tomb and it was a monumental icon produced as a durable messenger, a beacon of light, to convey very important information to the people of the future, us. Following ongoing studies of history, human memory and learning, mathematics and geometry, as well as more history of ancient Egypt, I concluded in 1990 that the message sent to us in the form of the pentagon and pentangle, and by the Great Pyramid, were both clues in support of the core message sent by the riddle of squaring the circle. By this I mean that I have been certain, since 1990, that the purpose of the riddle was not to challenge geometers to solve a trivial geometric puzzle, or even a magnificent geometric puzzle, but to lead the geometer to a particular path, toward proportion, and into the world of proportion, and to an understanding of how proportion is the sufficient cause of the ability of matter to self-organize, that is, to live.

It took me nearly seventeen years to solve the puzzle. In 2006, I discovered MQ (M^2) and the defined value of M. I found the last piece of the puzzle, construction of the line value of 0.2222, on July 7, 2007. If you count from 1971, it took me thirty-six years. If you count from my conviction about squaring the circle when I was age thirteen, it took me fifty years. If you count the years for all of human civilization, accepting as I do the likelihood that this information was known to the ancients and was lost around 2,500 years ago, and may have been buried somewhere in the Library of Alexandria which was destroyed by the Romans and early Christians who wanted to diminish the power and influence of Egypt and Gnosticism (Science), then it took us 2,500 years to solve the puzzle and the true riddle behind the puzzle. And in order to solve the puzzle, we needed to have electronic calculators and desktop computers to provide the level of precision and large numbers of calculations necessary to perceive the otherwise hidden relationship among Phi, Pi and M.

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The Elegant Proportions of Planet Earth, and How We Celebrate Them:

1) The size and mass of the Earth are exactly right to maintain the atmosphere that supports abundant life, 21% oxygen and 78% nitrogen, and another 1% of inert gases. The oxygen level clearly supports a tremendous variety of life forms, but if it were significantly greater, it would also support far more spontaneous combustion of the carbon-based matter produced by plants. The lesser gravity of a smaller mass planet would not hold the oxygen in the atmosphere. A significantly larger planet would also hold down other gases, including gases toxic to oxygen-breathing animals.

2) The primary elements of the Earth's core, iron and nickel, provide a thermonuclear source of heat by convection and volcanism. The dynamic nature of the thermonuclear core makes the planet a giant magnet with magnetic flux forces that trap and channel those forms of radiation that are damaging to life away from the planet's surface and into the ionosphere and small areas above the north and south poles. Convection and volcanism from the core to the surface, both under the oceans and on dry land, channels life-supporitng heat into the oceans and the atmosphere.

3) The tilt of the Earth's axis of rotation, about 23.5 degrees from a "vertical" line perpendicular to the plane of revolution around the Sun, is ideal for the purpose of providing sufficient periods of sunlight to maintain liquid water over the greatest possible surface area of the planetary globe. If the axis of rotation were vertically perpendicular to the plane of revolution, the planet would have only permanent temperature or climate zones and a sharp boundary between regions that had liquid water permanently and frozen water permanently. Without the seasons, life does not evolve into all those forms that possess a capacity to survive a period of dormancy followed by a return to a period of growth and fertility. A planet with the elegant axis of rotation of 23.5 degrees supports far more diversity, including animals that attain a state of rest through the extremes of either summer or winter, and other life forms that travel great distances in order to follow the seasons that provide for their needs.

4) The velocity of the Earth's rotation, and the velocity of revolution combined with the tilt of the axis of rotation, create the elegant process of absorbing the heat of the Sun and cooling in a cycle that maintains the narrow range of temperatures that support abundant life. The velocity of rotation is just right to allow for safe release of daytime heat into the atmosphere over the period of night. If the Earth did not rotate, the side facing the Sun would be permanently hundreds of degrees too hot to support life, and the dark side would be permanently too cold to support life. If the velocity of rotation were very slow, the surface would get hot enough, each day, to kill living tissue, and cold enough, each night, to kill living tissue. If the velocity of rotation were very fast, the periods of exposure to the heat of the Sun and then to the cold of space would alternate fast enough so that the surface of the planet maintained a much smaller range of temperatures, supporting a smaller variety of life forms. Earth would still have life, but not abundant life.

5) A tremendous, and elegant, quantity of water is available both on the surface and in the atmosphere as water vapor. This elegant proportion of water enables evaporation and condensation and temperature transfers that provide rain and snow and a variety of other weather events that play a role in the evolution of both plants and animals. This elegant proportion of water has the net effect of distributing water over the surface of the planet to support an abundance of life forms.

6) One Moon, of elegant size and distance from the Earth -- not two or more moons -- that creates tides and tidal basins, nurseries for abundant life, an environment that promotes development of organisms that can tolerate periods of drying.

Two or more moons would cause more violent forms of weather, and a complex pattern of gravitational variations and electromagnetic effects, likely to be stressful to life forms. One moon provides stability as well as stimulation for the support of many varieties of life.

7) The Earth's elegant distance from the Sun provides heat and radiated energy that is just right to maintain fluid water but not lethal radiation.

8) The balance of the gravitational system, meaning the Earth's period of rotation, period of revolution, and lunar cycle, all three, are stable over a tremendously long interval of time, time for evolution and development of an abundance of life forms.

Ancient peoples celebrated the elegant proportions of the Earth, Sun and Moon and showed them reverence as the cause of abundant life on Earth. Reverence for the elegant proportions of Earth was expressed through megalithic monuments constructed to observe and celebrate measurable, defined solar and lunar events with rituals symbolic of life, dormancy, trial and birth anew. We continue these forms of reverence and celebration in our own time and in our own way through our observance of Easter (Estrus) and the spring equinox, and the summer solstice and June observance of the end of the mating season. We celebrate the fall equinox and the harvest and All Hallowed Eve (Halloween), and winter solstice to observe the dark dormancy of deep winter and then the re-birth of the Sun that follows immediately after the shortest day of December 21st. The significance of the elegant proportions of our planet and the intensity with which they were observed and celebrated are embodied in Stonehenge, the stone rows at Carnac, the Pyramids of Egypt, Central America and China, the sacred structures at Chaco Canyon and a long list of other megalithic monuments found in all parts of the world. These megalithic monuments were not only calendars to measure time and the seasons. They were humankind's first cathedrals, early versions of the Taj Mahal, built to honor Nature, life, love, beauty, fertility, all the elegant and abundant forms of life that we see and feel when we are not starving or fighting.

There will be many planets that support some forms of life, but only a planet with elegant proportions, such as Earth, can support abundant life. That is why a voice from the pre-historic past is preserved for us in the opening of the Bible, the voice that tells us that our world was created so that we might have not only life, but abundant life. However our planet was created, its elegant proportions produce abundance. This is where eagles congregate.

Continuation... --->(when printed, p. 10)

Previous web pages:

The rediscovery of Proportion: The Secret of Life in the Universe

The Precision of the Ancients, Volume 1 (January, 2006) Part 1

The Myth of Pi, Volume 2 (February, 2006) Part 2

Medicine and the Atom, Volume 3 (March, 2006) Part 3

Added here:

The Reality of Proportion, Construction of 1 and Pi,

Part 4, Volume 8 (August, 2007)

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B) From Taylor to Tompkins to West:

The literature of pyramidology tells us that John Taylor was the first to suggest that the Great Pyramid appeared to incorporate the value of Pi (incorporation of Phi was also considered by other students of the Great Pyramid) and was intended to convey to subsequent generations an understanding of the "Secret of Life" in the universe. Many books were written about the Great Pyramid and the most thorough history of the obsession with the Great Pyramid by many scientists and some statesmen, such as Napoleon, was written by Peter Tompkins (Secrets of the Great Pyramid) and published in 1971. Tompkins' book is in my opinion exceptionally objective. It is essentially a narrative of the beliefs and activities of scholars, statesmen, archaeologists and other scientists, such as students of ancient standards of measurement. This book may contain an error, but it is so extensive and ambitious that the author should be forgiven one or even two errors. Tompkins was apparently determined to convey the scope and depth of "pyramidology" through history, and he is to be commended for his effort, victorious I believe, to communicate the challenge of understanding the original meaning of the Great Pyramid and the importance that has been attached to it by thinking persons throughout the human drama. Anyone who reads Tompkins' book can no longer believe the nonsense that the Great Pyramid is a tomb dedicated to an individual pharaoh. It had some other far more universal purpose. After studying this book, I concluded that the Great Pyramid was a kind of permanent, solid (three dimensional) hieroglyph or icon -- a monumental icon and an iconic monument, intended to convey important information to those who have eyes to see.

In 1979, Serpent in the Sky, by John Anthony West, was published. West was later known as an advocate, together with geologist Robert Schoch of the conclusion that the Sphinx was built about ten thousand years ago, because its enclosure shows signs of erosion by rainfall. And, if the Sphinx was constructed ten thousand years ago, it is likely that the Great Pyramid is of a similar age, rather than the age of four thousand years that is vigorously asserted by Egyptologists such as Dr. Hawass. I read Serpent in the Sky around 1990. West makes references to Tompkins' work and Peter Tompkins wrote a Preface for the edition I read, a paperback with the subtitle: The High Wisdom of Ancient Egypt, published in 1987 by the Julian Press. West's work is similar to Tompkins' in that West also reviews the history of pyramidology and the evidence that the Great Pyramid is in fact intended to embody and communicate significant information. West refers to the same interpretation of the Eye of Ra that was previously given by Clark and Schwaller de Lubicz, that it represents the six binary fractions from 1/2 to 1/64. West also states that the ancient Egyptians employed a system of doubling and halving, a kind of binary system, to accomplish what we define as "multiplication." Since 1990, after reading Serpent in the Sky, I have been convinced that the ancient philosophical statement "Proportion is Everything," is in fact an accurate description of the universe, and that the Great Pyramid and a handful of other ancient icons, including the pentangle and what we call the "Pythagorean Theorem" are all part of a message from the ancients to lead us to an understanding that proportion is really the "secret of life" in the real physical universe, and that the circle can in fact be squared. And, by doing the work necessary to find how to square the circle, one will unavoidably re-discover the meaning of Proportion as a science unto itself that is not precisely the same thing as either mathematics or geometry.

These web pages include an incomplete bibliography of pyramidology. If you are a beginner, use the books listed in this bibliography and subscribe to Atlantis Rising magazine, which is the best source of information on alternative science and alternative history. My bibliography represents what I found and was able to read within the time period of my studies. I apologize for any authors of books that are omitted. They are omitted by pure chance. Another individual who deserves recognition for a valuable contribution to pyramidology and to my own motivation is Christopher Dunn. As a mathematician, machinist and historian, he has contributed greatly to the argument that the ancients possessed both knowledge and technology of great precision. Dunn has stated in at least one of his works that Pi is the value incorporated in the Great Pyramid. When I argue here and in Volume 1 that Phi is the value incorporated in the Great Pyramid, I am not necessarily disagreeing with Dunn or anyone else who sees the Pi value there. I show here in Volume 8, and in Volume 1, The Precision of the Ancients, that what the Great Pyramid conveys, and what the real issue is underlying all the efforts to identify ancient knowledge, is in fact the relationship between Pi and Phi. It is therefore equally "correct" to perceive either Pi or Phi in the Great Pyramid, or both. They are both implied because the "message" or communication of the Great Pyramid is that Pi and Phi are bound together by a third numerical value, M (1.000007661...).


C) I see the Great Pyramid as a Phi Pyramid:

For a detailed description of why I see the Great Pyramid as a Phi Pyramid, please look at Volume 1, The Precision of the Ancients, table of contents.

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D) The trail of clues of the Pyramidologists:

The Great Pyramid is a "pharaoh" meaning a beacon of light that points to something important. Based on the measurements of Piazzi-Smyth, as reported by Peter Tomkins in Secrets of the Great Pyramid, and various other reports on the Great Pyramid, I concluded that the Great Pyramid is a Phi pyramid, as described in Volume 1, The Precision of the Ancients, here in these web pages. If the Great Pyramid is a Phi pyramid, then there are two important clues contained in its proportions.

First, as described previously in Volume 1 here, the four slanted edges of the Great Pyramid would have a proportional line length (if the base is 1) of 0.951056516... or the value of the cosine of 18 degrees. Note also that the value of (cosine 18)^2 is equal to (Phi/4) plus 0.5. Cosine 18 plays a role in the solution because the inverse ratio, the secant of 18, is 1.051462224..., and that value squared equals 1.105572809..., and this value [1.105572809...] is central to the construction of the right triangle with a tangent of 2 which is the first step in the construction of the pentagon and the pentangle (five-pointed star). Note that 0.105572809... times sqrt(5) = [sqrt(5) - 2]. Taking one more step, 1.105572809... squared, or (secant 18)^4 is equal to 1.222291236..., and this value of 1.222291236... produces a key ratio when compared to 1.222222222... (11/9) which is obviously "related" to the value of 1.2 or (6/5), which is also found in the literature and was pointed to as a clue by Peter Tompkins in Secrets of the Great Pyramid;

Second: (Pi*MQ) is a constructible ratio/line. As just mentioned above, Peter Tompkins and other investigators have noticed that the ancients understood some importance in the value of 1.2 or (6/5). They appear to have noted that (6/5) times Phi^2 [2.618033988...] equals 3.141640787..., a value close to our calculated value of Pi (3.141592654...) and which differs from absolute Pi by the multiplier factor of MQ (MQ is my label and it means M squared), where MQ equals 1.000015321... . And, I discovered that M, the square root of MQ, is in fact a special identifiable number value that is included in a specific set of numbers that also includes Phi (1.618033988...). To construct (Pi*MQ), first construct a right triangle with tangent = sqrt(0.8). The secant of that right triangle is 1.341640786... . With compass and straightedge, add the line length of 1.8. The result is Pi*MQ 3.141640786... . If you can do the constructions described here, you can easily complete this construction also.

These two vitally important clues point to the "hidden" reality that Phi^2 (2.618033988...) is equal to Pi times (5/6) times MQ, or

Pi = 3.141592654 -...

times (5/6) 0.833333333...

times MQ 1.000015321 -...

This discovery is the first step in understanding the physically real and vitally important relationship between Phi and Pi. I do not claim that the path of my work is complete, or finished. There is more to do and more to learn about what Pi really is, and how exactly it is related to Phi and MQ. However, the squaring of the circle exactly is completed here.

Third: The pentagon and pentangle and values of 5, Phi and 2:

The procedure for constructing the pentagon and then the pentangle (described above in Volume 1 and in many books on geometry), clearly points to the value of 5, because when the tangent of a right triangle is 2 the proportional line length of the hypotenuse is the square root of 5, and the pentagon has five sides and five vertices, and the pentangle has five points, and the right triangles that can be extracted from (or found within) the pentagon and pentangle both incorporate multiple examples, in both line lengths and trigonometric functions, of values that are related to Phi and Phi^2, all related directly to the 36-54 degree right triangle, with cosine of 0.809016994... which is equal to Phi/2. And the 36-54 degree right triangle is found ten times in the pentagon (360/5 = 72 degrees, 360/10 = 36 degrees). And we find 5 and 6 in nature repeatedly. In fingers and toes, numerous animals with one head (two would be a problem) and four limbs, many flowers with 5 petals, and many flowers with 6 petals. (6/5) is a key value.

Therefore, the construction of the pentagon and pentangle also presents us with two vital clues, the value of 5 and the value of Phi and the many values related to Phi in the right triangles that can be constructed by beginning with the most simple construction procedure, that is, making side A, the altitude of a right triangle twice the length of side B the base. This clue therefore also points to 2, the value that we find in Nature in the binary number system which appears to be the number system that describes both the genetic reality and the electronic or electrical reality of the real, physical universe.

Therefore, these three clues, pointing to 5, Phi, and 2, are not about magic or mystery, but they point to the real number values that lead to our being able to see the means by which the circle can be squared, which points to the relationship between Phi and Pi, all of which points to the reality of proportion, which I believe is the real secret of life in the universe, not magical thinking. This is because the reality of proportion that we can see with our own eyes, and minds, when we see how the circle is squared and how Pi and Phi are related, may be the sufficient cause that enables physical matter (atoms and molecules) to be self-organizing. This would mean, logically, that proportion (as discovered through this process) actually is the real scientific secret of life in the universe and not a magical or "new age" mystery.

My work shows here that 5 is key, at the heart of the solution, because the solution requires that we use "two fives," that is 5 divided by ten, and five divided by 100, or 0.55, as a vital step in discovering how to square the circle. The reality is that 0.55 and (1/0.55) or 1.818181818... relates to 1.222222222... (11/9) and 1.2 in what appears to be an endless set of charming and elegant proportions, a kind of Phi #2. This is shown (as the six-fifths or eighteen series) under the numerical values essential to construct 1 and Pi and square the circle.

Fourth: The Great Pyramid again:

It is known throughout the world, perhaps because of the United States dollar bill, that the "Eye of Ra" appears at the peak of the Great Pyramid. Some people believe that this is a contrived symbol, imposed by the Free Masons on the Great Pyramid. However, there is historical evidence, found in the pyramid literature, that the Eye of Ra, in stylized form, was originally carved at the peak of each of the four faces of the Great Pyramid. This clue is possibly the most important of all, because if you just happen to be a non-conformist who doubts the experts, then you would reject, as I have and many others have, the conformist anthropologists' explanation that the Eye of Ra is a primitive religious symbol only, that this eye symbol represents the Sun God or some other superstition. The work of R.T. Rundle Clark and Schwaller de Lubicz indicates that the original Eye of Ra was in fact an ancient hieroglyphic symbol that would logically convey the meaning of "look here carefully" or "precision is the essence of truth" or "check the precision here." These are logical interpretations of the hieroglyph because it is composed of the first six fractions in the endless series of dividing in half (1/2, 1/4, 1/8, 1/16, 1/32, 1/64). This is described in greater detail, with a drawing, in Volume 1.

E) What could the Pharaoh or Beacon be pointing to?

I have concluded that the Great Pyramid tells us to look closely at Phi and Pi and make the effort, whatever it takes, to "square the circle," not just to discover a particular geometric construction but to discover the meaning of proportion. The Beacon points to the squaring-of-the-circle because the exploration required to learn how to square the circle requires a journey through the land of proportion.

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2) The numerical values that are essential to understanding how and why the

construction of 1 and Pi is accomplished.

Pi = 3.1415926535897932384626433832795...

sqrt(Pi) = 1.77245385090551602729816748334115...

The precision of the calculations and constructions:

Note that my desktop computer, as is true for 99% of all desktop computers, has a math processor that is reliably accurate only to the ninth or tenth digital place. Variations in digital values following the ninth decimal digit are therefore due to differences in the level of precision of the computer processor, and such differences are not evidence that the calculations and constructions are wrong. The actual values are all subject to the precision level of the computer being used. Therefore, the accuracy and correctness of the calculations and constructions beyond the ninth decimal place cannot be "certified" on a desktop computer, but must be "certified" or verified by people who have access to more precise math processors. My certainty that the calculations and constructions are in fact correct is based on my perception of consistency and logic inherent in the results. I expect my work to be verified by super-computers because M (see M below) is a specific value, as are Pi and Phi, that is identifiable as a specified member in a specified set. That means M is an exact number value just as Pi is deemed to be an exact number value. And just as is true for Pi, M requires computations of greater and greater precision in order to be written or printed out more accurately.

Cosine 36=0.809016994374947424102293417182819... times 2 = Phi

Phi=1.61803398874989484820458683436564... and inverse is

phi=0.618033988749894848204586834365638... or "little phi" or 1/Phi

and = (Sine 18) * 2, Sine 18 = 0.309016994374947424102293417182819...

also, (Sine 18) = (Cosine 36) minus 0.5


phi + phi^2= 1 (known to all mathematicians and geometers)

Also, Phi is the limit of the Fibonacci series: 1+1, 2+1, 3+2, 5+3, 8+5, 13+8, and so on, where the last sum divided by the previous sum = 1.618033988... as the sums grow larger.

(5/6) = 0.833333333333333333333333333333333...

sqrt(5/6) = 0.912870929175276855761616304668004...

(6/5) = 1.2 --->(when printed, p. 15)

sqrt(6/5) = 1.0954451150103322269139395656016...

Phi^2 = 2.61803398874989484820458683436564... (also = Phi +1)

divided by Pi = 0.833346100984274536545032628319445...

which = (5/6) * MQ, where MQ = 1.00001532118112944385403915398333...

and sqrt(MQ) = M= 1.00000766056122262280424348031297...

or M = 1.000007661...

PF = 0.55 (label derived from "Point Five")

ES = (1/0.55)=1.81818181818181818181818181818182... (derived from "Eighteen Series")

*1.2 = 2.18181818181818181818181818181818...

*1.2 = 2.61818181818181818181818181818182...

*1.2 = 3.14181818181818181818181818181818...

LN = (11/9) = 1.22222222222222222222222222222222... (eLeven Nines)

NL = 0.818181818181818181818181818181818... (Nine eLevenths)

* 1.2 = 0.981818181818181818181818181818182...

SF = (secant 18)^4 = 1.22229123600033648574532213002996... (Secant 18 to Fourth power)

TN = SF/LN = 1.00005646581845712470071810638815... (label "TN" from mind of author)

7.778 and 0.7778

2.222 and 0.2222, and 1.222 and 0.1222 (These values can be constructed.)

3) The Construction of 1 and Pi Exactly (the geometric constructions are here).

(Separate file link, ValCon07.htm):[Construction of 1 and Pi Exactly]

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4) [Sufficiency-of-Proportion] Theory and what it means:

A) Nature does not count or measure but inherently "detects" and reacts to proportional size of all matter and forces. Proportion can be observed and studied as a phenomenon of Nature distinctly separate from mathematics and geometry. In geometry, proportion is observed in lines and angles and in two dimensional and three-dimensional shapes. In proportion itself, proportion is observed in numerical values that are repetitive of specified values or comprised of "reflective" values or specified multiples, such as: X and X^2 plus 1 or X and (X+1)^2

or X and sqrt(X+N)/2 + sqrt(X)/2 and -sqrt(X)/2 (Quadratic equation).

Evolution is the answer to the question "How does Nature produce its variety of species?"

Proportion is the answer to the question "How does Nature produce its variety of shapes?" Proportion is a more fundamental understanding of modern "Chaos Theory." Proportion and evolution together produce all things, both non-living and living, that are produced by Nature without the involvement of a technological animal or intelligent being.

B) Proportion is SUFFICIENT as a "number system" for Nature. The observation of a binary number system, or doubling and halving, in genes and in electronic phenomena, are actually occurrences of proportion as Nature's sufficient "number system." Much of twentieth century physics, which gives us electromagnetic and electronic explanations for atomic structure and molecular bonds probably needs to be revised as a description of only the electromagnetic phenomena, but not an accurate description of the physical structure and shapes of either atoms or molecules. The data used to formulate the conclusions of twentieth century physics rely on electronic instruments to detect microscopic objects, which are not seen but are "detected" by the electronic devices. These measurements do not necessarily reveal the physical shapes of atoms and molecules but most likely detect the "shapes" of their electrical activity. I believe that atoms are polyhedrons. The electronic qualities of any matter is determined by shape and proportion, such as the electricity generator and electric motor, as well as the nuclear reactor and plutonium explosive. A chunk of iron, copper and plastic, in the right proportions, is an electric motor. All things operate on this principle: Proportion is Everything.

C) Proportion is the sufficient cause of the ability of matter to self-organize, which we humans have labeled as "living" as opposed to "non-living," but Nature does not observe a boundary between living and non-living. For Nature, all things are the outcome of a single process. Matter is self-organizing, and the outcome of self-organization includes reproduction, which is essentially a form of repetition. Therefore, in terms of reverence as the fundamental expression of religion, the ancient attitude of reverence toward all things as containing a "holy spirit," including trees and even rocks, reflects the objective view of Nature itself, meaning that all things are equally sacred because all things are part of the single process that produces both cold slime and exquisite beauty, intricacy, emotion and intellect.

D) If proportion is in fact sufficient to enable the self-organization of matter, then the observed "economy" of Nature supports the conclusion that nothing further -- no system of enhancement that goes beyond proportion -- would have evolved. This conclusion is based on the principle that Nature does not invent any new "mechanism" or process to accomplish something that is already accomplished by simpler means.

E) If the Sufficiency-of-Proportion Theory is correct, then Proportion is in scientific terms the "Secret of Life in the Universe." And the Great Pyramid and riddle of squaring-the-circle and the pentangle were clues left to us by wise people on Earth long ago, so that we would learn this "Secret" and know that they knew it long before we rode horses and plowed the land and wondered who and what we are.

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5) The expected or possible impact of this work on history, religion, mathematics, number theory, physics and the biological sciences:

Primacy of Stewardship Theory

A scientific theory in the field of the Reconciliation of Science and Religion.

The central theme of the teaching and example of Jesus Christ is what I prefer to call "The Primacy of Stewardship." What this phrase means was once told to me by an acquaintance and teacher of philosophy. This definition of the primacy of stewardship is most likely very old knowledge, and a common viewpoint of Christianity understood throughout the world by intelligent adults.

This definition is as follows: If you remove all of the references to sexual behavior and cultural rituals and customs from the Gospels, you will still have Jesus' message regarding social justice and the primacy of stewardship. However, if you remove from the Gospels the references to good stewardship and to social justice, you have nothing. Without the core message of the imperatives of social and environmental justice, you have nothing worthy of note, nothing that would have had any value at all worthy to be recorded and preserved for posterity.

If you have not studied the Gospels, I recommend you do the following. Try to get your hands on a Bible that was published before 1958. I recommend the Douay Rheims version. I believe this version preserved the language and translation to the best of our ability to that point in time. After 1958, the newer translations were based on simplistic interpretations rather than linguistic scholarship - that is my opinion. To develop the best possible understanding of what Jesus said and did, you must discard two old and worn-out conformist doctrines. The first is the doctrine that Jesus was a Jew speaking to Jews. You need to think of Jesus as a missionary from a superior species of intelligent beings who was teaching all humans in all times. In other words, he was not a Jewish teacher teaching Jews. Instead, look at Jesus as a missionary from another civilization teaching all humanity something important that we need to know. The second conformist doctrine that you must discard is the doctrine that Jesus is a warm and fuzzy lover of innocence and gentleness and his purpose was to teach us all - but mainly women and children - how to be nice. A slightly more serious version of this infantile doctrine is that Jesus' purpose was to teach us about individual morality, or about "the inner life," or about moral doctrines to guide an individual's personal behavior so that they would "be good" and go to heaven. Discard this "moral" purpose. Instead, you must look at Jesus as a teacher who is teaching all of humanity something very important, something that is as scientifically correct and fixed as your knowledge that water flows downhill. Get it into your mind that Jesus is teaching us science, physical science and social science. Most of the physical science was lost, because the disciples were totally ignorant of what it could mean. Much of the social science has been preserved, because the disciples thought they understood it. Often, they did, and you can understand it also. To understand the important truth that Jesus is teaching to us, the reality that we need to understand in order to survive and thrive, you must keep two questions in mind while you study the Gospels. I recommend you write these questions down and take notes.

The first question is: "What is the kingdom of heaven?"

Jesus makes reference to the kingdom of heaven often. What is it? That is the question you need to keep in mind, and you need to develop your own conclusion as to what the answer is to this question. When you have completed your study of the Gospels, you will have an answer, satisfactory to you, to this question.

The second question is: "What is Jesus' core message telling us about ourselves that is a scientifically true fact about us and our place in the universe?"

Write this question down and study the Gospels with the purpose of distilling the essential teaching, the core message that tells us who and what we are. If you do this, I expect that you, as any reasonable person, will conclude that the primacy of stewardship is the essential message - from Jesus himself - in the Gospels. The other material reflects the anxieties of the disciples and the neuroses of the ancient culture, much of which is still with us.

Why do I recommend this approach? Because it is the approach I followed. I first read the Gospels at age 14, because I was considering whether I wanted to become a priest. I concluded that Jesus did not necessarily recommend the priesthood. A study of Jesus' life and teaching reveals that it requires greater moral strength and courage to live a moral and responsible life "in the marketplace" and in the community than is required if you live under the protection of a wealthy and powerful religious institution. Years later, I read the books of Erich von Daniken, including Chariots of the Gods. Von Danken's books were widely read and praised by many, but also severely criticized by many. His theme was that there is physical and historical evidence that superior beings have visited the Earth over a very long period of time. The main evidence is ancient structures and ancient myths as well interpretations of history. Regardless of whether you agree with Von Daniken's theme, he also made an interesting recommendation. He suggested that we - all of us, historians and citizens - take another look at the teaching of ancient religions and the stories in ancient myths, and think about whether they are telling us about ways that superior beings have communicated with us, or tried to communicate with us.

After reading Von Daniken, I studied the Gospels again, three times, taking notes and addressing the same questions that I have recommended to you. I did not need to be certain that superior beings have visited the Earth, but I just needed to think about Jesus as a missionary whose purpose was to teach us something important, something scientific and vitally important to us and not just a philosophy of morality and manners, which is what I did and recommend to you.

If you have studied other religions, you will unavoidably notice that the primacy of stewardship is the core message of Bhuddism and of Native American religion, and many might be surprised that this is the core message of Mohammed's teaching, whether you believe it is originally from an angel or from Mohammed. The Ten Commandments clearly involve a message regarding the behavior of the individual toward persons and property - stewardship. A study of the teaching and practice of Judaism will also reveal that the core message is the primacy of stewardship. A similar core message will be found in Hinduism, Taoism and Confucianism. The Japanese religion of Shinto, and even the code of the Samurai, all convey the concept that the noblest achievement of any human being, or society, is Good Stewardship. And what is of the greatest importance here for both you and me, is that this "lesson" of Good Stewardship is not a fairy tale for children, not a story about how you are nice if you rescue a kitten from a tree. The primacy of stewardship means that in the course of the evolution of life, when technological animals such as humans evolve, there is an evolutionary period of great stress and tension due to both the internal and external struggle between exercising power over people and things and being responsible for the outcomes of action, including the unintended outcomes. You will see that Jesus' parables of stewardship always involve situations where a "master" is exercising power over people and things. Of course, this evolutionary struggle and process of a technological animal also involves the complexities of emotion and intellect vying for supremacy, or peaceful coexistence. If this struggle is resolved by the evolution of Good Stewardship, then the species survives. If this struggle is not resolved by the evolution of Good Stewardship, then the species continues as a vulnerable animal or self-destructs. So, for me, this is the essential message, and coldly scientific message of Jesus and the other Great Teachers. This is who and what we are. We are technological animals who are candidates for membership in the kingdom of heaven. We can become intelligent beings but only by becoming Good Stewards. The key behavior that distinguishes intelligent beings from technological animals is Good Stewardship. The species that learns and practices Good Stewardship can survive and thrive indefinitely. And, the essence of my scientific message, the work that is equally pertinent to math and physics and religion, is that in order to be Good Stewards we must know and use the technology of Nature. If Proportion is the language or "number system" of Nature, then we must revise all of our technology accordingly in order to be in harmony with Nature and good stewards of life in the universe.

What it means to "use Nature's technology":

It is obvious that our human technology is often entirely different from Nature's technology. Obvious examples are the internal combustion engine, which produces energy under very hot conditions (about 250 degrees or hotter), the automobile and the airplane. The Renaissance genius, Leonardo Da Vinci, is known to have designed hundreds of machines, some of which we know reflected Nature's technology. The best known example is his giant glider that is designed to imitate the structure of a real bird, so that when powered by human arms and legs, the wings flap and the giant bird-form glider can fly. That is an example of Nature's technology. Another example from the birds is lightweight, hollow bones. Teeth and digestion are examples of Nature's technology. The eyeball is an example of Nature's technology. The eyeball is both an efficient and sufficient means for a living animal to detect objects and movements, and therefore a great variety of animals, from insects to fish to amphibians to mammals possess eyes. And all of these eyes possess basic similarities. This is Nature's way; whatever works is retained and used repeatedly. There are no animals that move on wheels. There are many animals that travel on four to six legs, and a few that travel on two legs, although those that travel on two legs are actually doing a balancing act that involves claws or toes. Humans without toes do not walk well. Ask any physician. Humans can walk on stilts, which means balancing on two points, but no human would want to walk on stilts all the time, or run away from a tiger on stilts, or try to climb a tree on stilts, or swim with stilts. Neither we nor any bird really travels on "two legs." Each of our legs is a complex structure that contacts the earth below at more than two points at all times.

Living animals do not produce energy by means of a system that produces temperatures of 250 degrees. Most animals, including us, produce "cool" energy at between 40 and 98 degrees, and if we are very active we must shed heat by sweating or being cooled down by some other means. If body temperature rises above 100 degrees we feel bad, worse at 102 degrees, and worse still and possibly delirious at 104 degrees. We begin to shut down vital organs at around 107 degrees and will be dead if our temperature rises much further than that. We can be in air that is 110 degrees, or even slightly higher, but our bodies cannot tolerate heating up internally. Although this will be deemed a trivial observation, it is not trivial when you consider that I am saying we need to discard our human technology and replace it with Nature's technology. If we seriously endeavor to replace our technology with Nature's technology, we will have to replace every machine and motor that operates at greater than 100 degrees Fahrenheit. We will have to re-study Nature and produce machines, perhaps "bio-machines" that produce energy at cooler temperatures. We will build airplanes that fly exactly like birds, and vehicles that run exactly like spiders. Does this sound like an awfully difficult task? What if it is the only alternative to self-destruction? Or the only alternative to permanent slavery?

The Hiram Key, or the world's oldest theological dispute, and scientific dispute, may be embodied in the squaring-of-the-circle, proportion, and the true origin of the concept of the "Holy Trinity" (Father, Son, and Holy Ghost), as the real, scientific origin of life in the universe. In The Hiram Key (1996), the authors, Christopher Knight and Robert Lomas, claim to be sharing information about Freemasonry, but that they are not disclosing any secrets of the Order. They do claim to possess at least some of Masonic knowledge because they are, or have been, members of a Masonic Lodge. I, John Manimas Medeiros, have never been a Mason, and have never received any privileged information from a Mason. However, I believe I may have discovered "The Lost Secrets of Freemasonry," which is the title of Chapter One in The Hiram Key.

This section, this page, approximate page location: --->(when printed, p. 20)

The "Holy Trinity" of numbers could be the lost secret knowledge of a "Master Mason," which King Solomon is alleged (by Knight and Lomas) to have attributed to the murder of Hiram Abif at the time that Hiram Abif was building King Solomon's Temple. The Father, Son and Holy Ghost can be understood as the origin of life in the universe if you believe that Nature, being both male and female, is the Creator of all things, including God, and if you believe that Proportion is Everything and that Proportion is the means by which matter is self-organizing, and if you believe that:

The Father is Phi [1.61803398874989484820458683436564...], being the icon that represents all of Proportion. Note that Phi is the tangent of the right triangle customarily constructed to the left of the initial right triangle, the tangent 2 right triangle, that is the first step in the construction of the pentagon and the pentangle or five-pointed star;

The Son is Pi [3.1415926535897932384626433832795...], being the infinite sum of the evolved offspring of Proportion, or of the abundant life in the universe (Pi is an infinite sum that cannot be constructed with compass and straightedge except as described here with reference to MQ);

The Holy Ghost [1.00000766056122262280424348031297...], being the hidden value, slightly more than 1, that belongs to the same set of numbers as Phi (is also like The Father), and is the invisible value, the Mother of all numbers, the generator of both Phi and Pi:

Phi^2 = Pi * (5/6) * MQ, (MQ = M squared = 1.00001532118112944385403915398333...),

and Pi = (Phi/MQ) * 1.2 . And M is not invisible only because it is slightly more than 1, but also because it cannot be constructed using the quadratic equation, although all other integers in the set, from 3 to the highest integer, can be constructed by means of the quadratic equation. Construction of M and MQ requires reference to Pi itself, or the constructions presented here that accomplish the squaring of the circle. The set of numbers that include both M and Phi is: all X such that X^2 + (1/X^2) equals N, an integer greater than 1. This set, and how it can be generated by applying the Quadratic Construction, appears under The Construction of 1 and Pi Exactly, the Definition of MQ.

In a later book, Civilization One, by Christopher Knight and Alan Butler (London: Watkins Press, 2004), Knight and Butler present the argument -- and a great deal of supporting evidence -- that a common unit of measurement can be found at the sites of many different ancient stone monuments. They call this ancient unit of measurement the "megalithic yard," and describe how it can be obtained, anywhere on the planet Earth, by observing the movement of Venus, the Morning Star, for a fixed period of time that is measured by counting the swings of a pendulum. There is far more detail to this thesis of the megalithic yard. I recommend this book to anyone who is interested in the proposition that our ancient ancestors were intelligent and knew a lot about the planet on which they lived.

God constructs the universe by means of geometry:

This is one of the messages of the ancients which is perceivable in the symbols and history of Freemasonry. William Blake, a famous independent poet and artist, was a Mason of high degree. His many drawings, black ink on white paper, typically involve symbols of cosmic drama, such as the actions of angels, metaphorical events symbolic of the moral challenges in life, and depictions of Biblical events and of God. One of Blake's drawings appeared at the beginning of a book of his poems. It depicts God as a male elder, with flowing long white hair and beard, but with strong and vigorous facial features, reaching down from his place in the clouds with his right hand holding a giant open builder's compass on a plane surface. To me, this drawing, though possibly open to other interpretations, conveys the core message of ancient Gnosticism and of the original beliefs of Freemasonry: God builds the world with Geometry. That is why a letter "G" appears enclosed by a square (straightedge) and compass in the Masonic symbol. That letter "G" may represent the sacred, and most English-speaking people think it refers to God, but it really refers to Sacred Geometry, and the most Sacred of Sacred Geometry is the knowledge of M and of how to square the circle, and the meaning of Proportion as the Secret of Life in the Universe.

The Sense of Proportion as the Source of Human Morality:

The sense of fairness in children is familiar to teachers and caretakers throughout the world. The vast majority of normal children demonstrate an innate sense of injustice when someone does something that is "unfair," including stealing, lying, or bullying of a small or weaker child by an older or bigger and stronger child.

There are many references to measurement and proportion as a matter of justice in the Gospels. Such references usually address one of the following four categories of "proportion":

A) Judgment or evaluation as a measurement.

B) Judgment before a court as a measurement.

C) Charity and philanthropy defined as proportion.

D) Economics and distributive justice.

Judging is measuring. Luke 6; 37-38:

37. Do not judge, and you shall not be judged; do not condemn, and you shall not be condemned. Forgive, and you shall be forgiven; 38. give, and it shall be given to you; good measure, pressed down, shaken together, running over, shall they pour into your lap. For with what measure you measure, it shall be measured to you.

In the marketplace of Jesus' time and place, as is still true in Middle Eastern cultures today, many of the products sold are in the form of "flour" or powder. This includes grains, spices, herbs, and food starches. In Jesus' time, the most common measure used for such products, for convenience, was not weight, but standard volumes. And we all know that any "flour" or fine grain contains air when first poured and occupies a greater volume than it will after it is "pressed down, shaken together, running over." The seller who gives his customer his purchase "pressed down, shaken together, running over," is the model of a just person (many such sellers in the marketplace were and still are women) who will be "measured" according to the fairness with which they measure. Proper proportion equals fairness, justice, in the marketplace.

A civil action. Measurement before a judge. Luke 12: 58-59:

58. And when thou art going with thy opponent [in a civil case] to the ruler, take pains to be quit of him on the way; lest he deliver thee to the judge, and the judge to the officer, and the officer cast thee into prison. 59. I say to thee, thou wilt not come out from it until thou hast paid the very last mite [small value coin].

If you owe money, or anything to another, you are fair only if you pay what you owe. And if you do not, the authorities will increase what you owe because correction of your unfairness required the time and expense of the state.

Proportional responsibility; knowing what is expected. Luke 12: 48:

48. Whereas he who did not know it [the will of the master], but did things deserving of stripes [lashes with a whip], will be beaten with a few. But of everyone to whom much has been given, much will be required; and of him to whom they have entrusted much, they will demand the more.

This passage means that those who have the gifts of ability, skill, a great capacity to learn and accomplish things, are those "to whom much has been given." And they are expected to exercise their gifts for the benefit of the community, or the society. The fact that they may "give more" than others who have lesser gifts does not entitle them to great or "disproportionate" rewards (compensation and social status). Therefore, proportional justice means that those individuals who have more to give are not really "giving more" for the simple reason that they are giving what they are able to give, the same as those with less ability, in terms of their economic contribution to the community. Each is giving what they are able. To have much to give, and refuse to contribute according to one's ability, is obviously a proportional injustice. This is like a successful fisherman saying: "Because my neighbor caught only a few, small fish today, I will do the same. Why should I "give more" than he does?"

This passage also means that if a greater responsibility is given to a person, because they are judged able to manage greater responsibility, they will be held accountable in proportion to their ability and knowledge. This is a form of proportional justice that is not practiced in American society today. If it were, those who commit "white collar" crimes, college educated people who have access to very high incomes, lawyers, accountants, business managers, who lie, cheat and steal, (and worse), would be given greater consequences for their crimes than the consequences given for similar crimes committed by those with lesser education, knowledge and ability.

What Jesus is saying here was interpreted by Karl Marx and applied to economic justice when he said that a democratic communist (or communal) society would apply the principle: "From each according to their abilities; to each according to their needs." This means that those who have much ability are expected to exercise their gifts for the benefit of society, but fairness and "proportional justice" means that their reward (compensation, social status) not be greater than what they need. This is of course a bit strong. Most people want those who make great contributions to society to receive suitable rewards.

The widow's mite. Proportional philanthropy. Mark 12: 41-44:

41. And Jesus sat down opposite the treasury [at the place for charity], and observed how the crowd were putting money into the treasury; and many rich people were putting in large sums. 42. And there came one poor widow, and she put in two mites [a small value coin], which make a quadrans. 43. And he called his disciples together, and said to them, "Amen I say to you, this poor widow has put in more than all those who have been putting money into the treasury. 44. For they all have put in out of their abundance, but she out of her want has put in all that she had -- all that she had to live on."

This passage describes a form of proportional justice that is totally ignored in American society today. That is, the matter of charity or philanthropy. In our society there are people with tens and even hundreds of millions of dollars. Some are even billionaires. And these people often take pride in that they will give ten million dollars for a good cause, or for the arts, or for educational opportunities for the poor. Sometimes it is known that they are giving away a certain percentage of their incomes, or assets. Perhaps this is two percent, or five percent, or even ten percent or twenty percent. However, what Jesus describes here is a principle of proportional justice that glows like a hot coal: The proportional quality of one's charity or philanthropy is not the measurement of what you give, but the measurement of what hold back for yourself.

Anger as a little murder. Matthew 5: 21-22:

21. You have heard that it was said to the ancients, "Thou shalt not kill; and whoever shall kill shall be liable to judgment, 22. But I say to you that everyone who is angry with his brother shall be liable to judgment;..."

Anger, desiring harm to one's neighbor, is the initial offense. It is a kind of "little murder." When we nurse that anger, let it build to hatred, harsh judgment, even to a sense of entitlement to render a judgment of execution -- that is murder. But it started with anger. And so, proportionally, anger is like a seed that we must not cultivate, lest it lead to murder.

Why cite the Bible? Because Jesus also talked about proportion and proportional justice. Because proportion is at the core of our law and court procedures. Even the price of a bushel of corn is higher when there is a drought, because it takes more effort and expense to produce a bushel of corn when there is a drought. Even the prices in a stock market are subject to the principle of proportional justice. When the ancients said "Proportion is everything," they meant what they said.

Religion itself has never caused a war, but the appropriation of religious doctrine for political purposes and the misuse of religion to define patriotism and support of a war as religious behavior is a common cause of war. Those who have a deep need to conform and appear religious follow the path of involvement in violence that no moral religious teacher would follow. Even Mohammed, who is described as a warrior as well as a religious teacher, recommended that all political and economic conflicts be settled through mediation. Mohammed would certainly condemn the hypocrisy and profound injustice of appropriating religion to motivate vulnerable people to commit acts of violence. If political or economic goals are to be achieved by violence, those goals should be openly acknowledged and not disguised as religious commitments or obligations. Those who act out of selfishness nearly always disguise their motives behind a veil of religion. Those few who are not ashamed of their selfishness are to be respected for their honesty. Honest greed is a far lesser offense than lying and manipulating in order to lead children to the slaughter and benefit financially from the terrible destruction and infinite anguish that war brings to those whose only desire is to live in peace.

6) Back to the beginning:

In the beginning of Volume 8, I asked the following questions:

What is at stake here?

A) Is "Proportion" the true "number system" of Nature?

B) Do we need to understand Proportion in order to understand Nature

and Nature's technology?

C) Do we need to understand Nature precisely in order to survive?

My answers to these question are: What is at stake is the survival of the human species, and our transition from being a vulnerable technological animal to being a good steward and intelligent species.

A) Yes. Proportion is the true "number system" of Nature.

B) Nature does all that it does by proportion only. Therefore, in order to understand Nature's technology we must understand proportion and how it is applied. All equations of physics must be revised with no factor of "t" for time involved because there is no time factor in Nature's technology, only process. Nature does not count events and does not need time in order to be nor in order to do. Relativity, as all of human mathematical physics, is anthropomorphic and does not reveal the facts of Nature, because relativity involves measurements of time, which occur only in human technology and not in Nature's technology.

C) If we do not study Proportion and develop an understanding of Nature's technology, we will have only one of two possible futures: 1) We will self-destruct; 2) We will continue to live but only as vulnerable animals. We will have no status or rights as intelligent beings.

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7) How the validity of my work can be [tested with a super-computer]:

Testing for the definition of M and MQ:

#1 Computer Test:

MQ = 1 / [Pi * (phi^2) * (5/6)], M = square root of MQ

If this test is used, the computed value of M is necessarily equally as accurate as your computed value for Pi.

#2 Computer Test:

M * phi * sqrt(5/6) = 1/ sqrt(Pi)

Here also, if this test is used, the computed value of M is necessarily equally as accurate as your computed value for Pi.

#3 Computer test:

Phi^2 = Pi * (5/6) * MQ, therefore Pi = (Phi^2 * 6) / (MQ * 5)

Is this expression of Pi a "transcendental" number?

The precision of the true MQ value is so fine, it is even possible that the binary mechanism of the computer's electronic math calculator, or processor, might not accurately maintain the proportional relationships among the three values of Pi, Phi and MQ. It may be that a computer based on powers of 2 (binary), gives results slightly different from a computer based on "proportion," if there were such a thing as a "proportional" math processor.

Testing the construction of MQ: --->(when printed, p. 25)

The constructions can be entered into the computer math processor as calculations:

First get the value of SF = (secant 18)^4 = 1.222291236... and then get the value of NT as shown here. SF * (9/11) = TN. NT = (1/TN). Then perform the calculation of (NT + 9), which equals 9.999943537369751504522585249236876... . Then divide by 10, which gives the result of:

0.9999943537369751504522585249236876... , the intermediate value on way to target NTP.

That intermediate value * NT = 0.999937891425529516432675419924049... = NTP

Then, NTP * SF = 1.22221532123408073944164930824006..., and that minus 0.2222 gives us our constructed and calculated value of MQ = 1.00001532123408073944164930824006...,

OR, the correct decimal digits after the 9th decimal place: 1.000015321 ... ???. This MQ, plus (1/MQ) should then be equal to 2 followed by nine zeros and then the digits 234... or only more zeros until the decimal place that reflects the precision of the electronic calculation -- in accordance with the microchip design. The precision is so fine here, similar to Pi, such that a slight difference in the twentieth of thirtieth decimal digit for MQ could cause the computer calculator to "write" the value of [MQ + (1/MQ) ] as 2 followed by nine or more zeros and then a sequence of digits that reflects the precision level of the computer. This is the way it is, because when we test the value of MQ and M, we are looking at the world with the Eye of Ra, the Eye that sees all things, both very great and very small, the Eye the looks with infinite precision, the Eye that says: "Look here, carefully. The truth is hidden here. Here you will find the Holy Ghost."

8) What is Proportion, Really?

The "Binary Number System" in genetics and Nature's proportional memory.

(Separate file link, propreal.htm): [What is Proportion, really?]

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9) Mathematics as the last anthropomorphic religion:

The belief that it is impossible to square the circle has been a false doctrine of the church of mathematics since 1882. It has been defended and propagated by the mathematical authorities to this day. This tells us that mathematicians and scientists and Popes and Cardinals and Bishops and parents and employers and politicians are all equally human and all equally exposed to the risk of authoritarianism. We are all at risk to become rigid and authoritarian, which mans being certain that an idea or concept of reality is final and fixed and should neither be questioned nor reconsidered, neither challenged nor researched. The defense of fixed doctrines in the authoritarian mode is often angry, destructive, selfish, punitive and violent. Those who openly question a doctrine are met with irrational and exaggerated ridicule and opposition, including the denial or revocation of credentials and the destruction of careers, or imprisonment, torture, death.

Authoritarianism, a pattern of behavior found wherever humans live, gives us our most shameful history: racism, bigotry, the Inquisition, conversion by the sword, Jihads and Crusades, religious wars and civil wars, the executions of witches and female herbalists and physicians, lynchings, the burning of books and their authors, as well as the burning of the Library of Alexandria, the burning of books by Nazis in the twentieth century, the Holocaust, the pogroms, the slaughter of Russian peasant farmers, the isolation and torture of unnamed, innocent captives of President George W. Bush. I identify these captives as "innocent" because they have been denied what we know as "the rights of the accused" which includes the presumption that they are innocent until proven guilty in a court of criminal law. They must not be defined as guilty by any authority other than a court of criminal law.

Authoritarianism is to be found in the hearts and souls of millions of American citizens who promote theocracy in America and therefore appear to have totally failed to pay attention in Civics class and who appear to lack the value of democracy and an understanding of its importance. Authoritarianism is in the human blood and must be recognized and resisted "twenty-four seven." The best resistance to authoritarianism is real democracy, not the empty words of democracy but the active enforcement of the principles of democracy.

The best description of human authoritarianism -- and therefore the best guide to its opposite -- is in the Gospels, Matthew Chapter 23, where Jesus speaks to all of us, all cultures in all times, and tells us about this greatest and most dangerous of human weaknesses.

To this day, anyone who would suggest that the circle could be squared was subject to ridicule and ostracized. No one aspiring to be a mathematician or scientist would dare challenge this doctrine because to do so guaranteed exclusion from the institution and its benefits.

There are many books about Pi and one was written by Petr Beckmann, entitled A History of Pi. This book is copyrighted by The Golem Press, 1971, so it must have been published prior to the St. Martin's third edition, which I purchased in 2007. In Chapter 17, entitled "The Modern Circle-Squarers," Beckmann ridicules those who would "square the circle," and uses the least scientific examples of "circle-squaring" to do the job. This shows us that the mathematicians have needed to defend the doctrine into our own times. Beckmann's history is detailed and appears to be well done. He also points out how both professional and amateur mathematicians have been somewhat obsessed with "computing Pi" to greater and greater numbers of decimal digit places. When asked why they do this, most mathematicians would respond that the purpose is to be sure that Pi is "irrational," meaning it is an infinite non-repeating decimal digit value. The other possibility would be, we can suppose, that a series of a hundred thousand digits or so are repeated. No such "repetition" has been found. Just because Beckmann's book, and others, ridicule attempts to square the circle, and ridicule even any assertion that it might be possible to square the circle, does not mean that such books are junk. On the contrary, they are all now primary sources for human history. If you don't own such a book, you should get one right away.

There is another false doctrine in the church of mathematics. This is the doctrine that the universe is "mathematical." The universe is not mathematical. The universe is proportional. There is an important difference. Mathematics is a human skill and activity. Nature does not need and does not employ mathematics to do what it does. Nature employs only proportion, and proportion is all that is necessary for Nature to do all that it does.

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10) (Closing) The Historical Problem. How will it be resolved?

I may not have proven beyond all doubt that the ancients knew how to square the circle precisely, but I have proven that there is a great probability that they knew. Therefore, historians and anthropologists and archaeologists are confronted with the challenge to determine how the ancients could know numbers of such fine precision, or is there a low-technology and "low-precision" means to discern the relationships among Phi, Pi and M? The only other rational conclusion is that I, John Manimas Medeiros, am the first person in all of human history to perceive the relationships among Phi, Pi and M, and therefore my work represents the discovery of how to square the circle and the discovery of how to use the Pythagorean Theorem and the quadratic equation as the Quadratic Construction to illustrate how Nature uses the language of Proportion to produce its many shapes. This is the only other reasonable conclusion rather than accepting my argument that I did not discover this knowledge but re-discovered ancient knowledge that was lost through the regression of an advanced civilization caused by cataclysmic change. I submit that my argument for re-discovery has the greater merit, because I found this knowledge by following the ancient clues and because the civilization that built the Great Pyramid and the Temple at Luxor would not have asked the original question unless it had a positive answer. My intuition at age thirteen was correct. The ancients, the Pythagoreans, did not ask how to square the circle only to motivate geometers to sharpen their skills with a compass and straightedge. The question is the Pharaoh, the beacon of light that points to "Proportion is Everything."

Therefore, now that I have completed the task that I assigned to myself, I present to the world, with all due reverence: (Digits may be incorrect after the 9th decimal place.)

Phi: 1.61803398874989484820458683436564...

The Father;

Pi: 3.1415926535897932384626433832795...

The Child;

And M: 1.00000766058769806778093094275226...

The Mother of All Things, The Holy Ghost. M^2 = MQ

As (Phi^2 + [1/Phi^2] ) = 3 exactly.

And, the Unity of the Source of self-organizing Life equals

Phi Squared = 2.61803398874989484820458683436564...

and Phi Squared = Pi * (5/6) * MQ, the Unity contains Phi, Pi, and M,

and is therefore both the Unity and the Trinity. Trinity = Unity.

And further, the Unity of Phi^2 = Phi+1, or Phi plus Unity.

And the additional evidence pointing to the reality that Proportion is Everything. In gratitude for living a "normal life" on Earth, this is my gift to you in reciprocation. This may seem melodramatic and self-important, but it is intended sincerely, just as many others give their lives to the search for new knowledge.

Researchers, thank you. I am grateful for my life, and I am just as grateful for the lives of the many scientists throughout the world who devote their lifetime to research in order to add to human knowledge. The dissemination of knowledge is the sacred event that occurs in the human drama. Some travel to the far corners of the Earth, under the seas, through caves, into the air, to forest canopies and jungle floors of great danger, in deserts, with their lives hanging by a thread. Others spend decades in laboratories, loyal to their tasks which are often routine, tedious, and laden with the risk of exposure to disease as well as all of the discomforts associated with work that requires long hours keeping records or staring through a microscope. Many do their work while exposed to risks of radiation poisoning. For all of the researchers of the world, there is the risk of failure, or simply having their work only confirm old knowledge, making it more certain, but not attracting any attention or recognition from either the academic community or the general public.

Researchers, thank you. You are the ones most likely to know the meaning of what I have done. And it is you to whom I extend my deepest gratitude. Thank you, all, for your devotion to adding knowledge to the human inventory of truth. We know that it is the truth and only the truth that can make humanity free, and we know that the work we do is not only for our own glory but for the survival and freedom of our species. This is my gift, great or small. It is of course up to you to determine its value.

Gratitude and Memories:

"Heaven and Earth will pass away, but my words will not pass away." / Marie Antoinette Mure Medeiros, 1950, reciting words she saw written on the sky in a dream and known to her from a statement attributed to Jesus Christ by Matthew, Chapter 24, verse 35. My interpretation of this statement is that Jesus spoke the truth, meaning the truth about the real, physical universe, and the truth is the only thing in the universe that does not change. Change, therefore, occurs to all things, even a living God. Anything that never changes is deemed to not be alive.

"You have to discipline yourself." / Peter Harrity, English Teacher, Andrew Warde High School, (public high school of) Fairfield, Connecticut, 1957. Good advice for everyone.

"It's not a question of what we want to do, but what needs to be done." / Leonard Launer, Mathematics Teacher, Andrew Warde High School, Fairfield, Connecticut, 1958.

"That's what they tell us, Johnny." / Louis Pavoni, Landscaper, Fairfield, Connecticut, 1960.

"All experience is an arch whereon to build." The motto of Andrew Warde High School. Adapted from the poem Ulysses by Alfred Lord Tennyson.

"Precision is the essence of Truth." / William Barker, Humanities Instructor, Brandeis University, Waltham, Massachusetts, 1962.

"Truth even unto its innermost parts." / the motto of Brandeis University, the university named after Supreme Court Justice Louis Brandeis, my school.

God be with you (as many know, the origin of "Goodbye"). If you are not sure whether you believe in God, just let "God" be what we don't know. What we don't know is what controls our future. Or, you could think about God being the Unity and the Trinity of Sacred Numbers, being the reality of the cause of Life in the Universe, the knowledge of a Master Builder. And if we are becoming a Master Builder, we must not fail at being a Good Steward. To claim understanding is scary, I know. That's because the more we know, the less we can blame on God.


Link to: (Welcome) or (Geometry Alpha Index).