Contemplative Freemasonry

August 14th, 2010

Speculative Freemasonry,

Pythagoras,

and Euclid's 47th Proposition

Copyright © 2010 by RWB Wesley F Revels.

Euclid's 47th proposition is commonly used as the symbol of a Master Mason but few masons have been taught the meaning underlying the symbol denominating their achievement. It can be argued that understanding the relevance of Pythagoras and Euclid's 47th Proposition to Speculative Freemasonry is nearly as enigmatic as Pythagoras and the now ancient Order that claimed his name. Why would an exercise in plane geometry be used in Speculative Freemasonry? And why would Speculative Freemasonry find it necessary to explain an ethical principal referencing Pythagoras and Euclid's 47th Proposition? Could it be that Speculative Freemasonry as we know it today has evolved from earlier attempts from age to age, to form virtues that mirror or understand more perfectly the miracles of its creator, the One God Of Love And Peace? The One Great Architect Of The Universe?

332B.C. - 276B.C., The Great Conquest 
And Creation Of The Library At Alexandria, Egypt

As every story involving humanity usually begins with some sort of conflict so does this one. In his expedition to Egypt in 332 - 331B.C., Alexander had founded the city of Alexandria after waging a war to end all wars. At age 33 he was dead, and so was his great empire; breaking up into a heap of little empires, each of which were led by generals in competition for dominance. After his death in 323B.C., his generals fought each other over who was to get their hands on what they could. But by 306B.C. control over Egypt had firmly been established by one of them, Ptolemy I, who was succeeded by his son Ptolemy II. The Ptolomies were of Greek ancestry but adhered to many of the customs of the country. The Ptolomies ruled Egypt for many generations and it was PtolomyII who founded the museum and library at Alexandria.
Ptolemy acquired the most valuable manuscripts for the Library and had translations made of them. Ptolemy's purchasing agents would scour the Mediterranean for valued books, and even compelled travelers arriving in Egypt to give up any books in their possession which were then copied by scribes in the Library, the original retained, and the copy given to its owner. His son Ptolemy III, who decreed Leap Year, was even more tenacious. Borrowing the original copies of famous Athenian Greek playwrights, he had their manuscripts copied and return the copies forfeiting the deposit he had paid as bond for the return of the originals. Before the arrival of Caesar's thugs, the library is said to have had close to three quarters of a million books or scrolls. A standard scroll was about 15 to 20 feet in length and contained the equivalent of about ten to twenty thousand words of modern English text. Examples: Scrolls from Qumran or Nag Hammadi.

Even more impressive was the Museum which included a school or institute - in effect, a university. Ptolemy III, engaged the most celebrated scholars of his time to teach at this university, and soon it became the scientific capital of the western world. Few were the learned men of later antiquity who had not studied at Alexandria; they were taught by the finest scientists the contemporary world could muster.

Mathematics flourished at Alexandria. Eratosthenes (292 -273B.C.) chief librarian, calculated the circumference of the earth to within 5% of the correct value by observing the difference in zenith of the sun's shadow cast from obelisks located at two places separated by a known distance at the same time of day (Alexandria and Syene, on approximately the same meridian).  In possession of ever more accurate trigonometrical tables he calculated the distance to the moon and to the sun. The method was correct, and although his imperfect measuring instruments yielded a large error for the moon, the distance to the sun, as near as we can ascertain the length of his unit, the stadium, agrees with what we know today, including measurements by radar. And this was done at a time when the philosopher Epicurus in Athens taught that the Sun was two feet in diameter!

The academic community at Alexandria was mainly Greek, Egyptian and Jewish although much of the knowledge they transmitted originated in ancient Sumer and Cydonia. And so was the city surrounding it. It was referred to as "Alexandria of Egypt" for it was considered a part of Egypt. Considered worthless as soldiers, the culture of Alexandria had a reputation fro being lively and quick-witted. Among the scholars whom Ptolemy brought to Alexandria was Euclid, a man whose place and date of birth are unknown, so today he is simply called "Euclid of Alexandria". Euclid was, among other things, a publisher's dream. His "Elements" (of Plane Geometry) are the all time best seller of any textbook ever written. More than a thousand editions have been published only since the invention of the letter-press in the 15th century and is still the standard for all school geometries.
The major part of Euclid's "Elements" was certainly known before Euclid and Pythagoras. The Egyptians "Squared Circles" to calculate the dimensions and angles of repose for the building of the pyramids and the diameters and proportional distances of the Sun, Earth and moon. The importance of Euclid's work was not in what the theorems said. The great significance of the Elements was in their method. The Elements (Proofs) are the first grandiose building of mathematical architecture. There were five foundation stones, or postulates, which Euclid believed, and were so simple and obvious that everyone could accept them. Euclid's five foundation stones were thus:
1. A straight line may be drawn from any point to any other point.
2. A finite straight line may be extended continuously in a straight line.
3. A circle may be described with any center and any radius.
4. All right angles are equal to one another.
5. Given a line and a point not on that line, there is not more than one line which can be drawn through the point parallel to the original line.
Onto these foundations stones Euclid laid stone after stone with his logic, making sure that each new stone would rest firmly supported by one previously laid, until an entire cathedral stood as firmly anchored as its foundations. Euclid was not the founder of geometry; he was the father of mathematical rigor.

525B.C. to A.D.300, The Speculative School of Pythagoras
The organization or Order was, in its origin, a religious brotherhood or an association created more for the moral reformation of society rather than that of being a philosophical school. the Pythagorean Brotherhood sought by rites and abstinences to purify the believer's soul and enable it to escape from the "wheel of birth". This would be obvious since Pythagoras was initiated into virtually all the schools teaching Monotheistic and Trinitarian religious principals during his life. Founded by Pythagoras of Samos who settled in Croton in southern Italy about 525B.C., the religious order that incorporated his name held that,
1. The metaphysics of number and the conception that reality, including music and astronomy, is, at its deepest level, mathematical in nature.
2. The use of philosophy as a means of spiritual purification.
3. The heavenly destiny of the soul and the possibility of its rising to union with divine.
4. The appeal to certain symbols, sometimes mystical, such as tetraktys, the Golden Section, and the Harmony Of The Spheres.
5. The Pythagorean Theorem.
6. The demand that members of the order observe a strict loyalty and secrecy.

Taught by akousmata (something heard) the Order passed its teachings from one initiate to another with sacred discourses that required they be memorized before ascending to the next level. And I think it to be no coincidence that this is how Freemasons transmit their knowledge today. Pythagoras was fascinated with the way the physical world seemed to have a parallel relationship with the way Nature, apparently, had a mathematical infrastructure and this mathematical infrastructure was subtler than its material counterpart in the outer world we experience. Example: A circle drawn in the sand may seem to be exactly circular and perfect but in reality is not because of its tiny imperfections by virtue of its material form. A mathematical circle is however perfect because it can only be "pictured" in the mind. Idea is Greek for "picture".

Pythagoras being both mathematician and mystic, "pictured" that all of life, particularly harmonious sounds, always vibrated at lengths in simple numeric ratios and from this conclusion he determined that a properly balanced material body would carry an equally harmonious spiritual soul, just as properly tuned strings emit equally harmonious sounds. Therefore he saw good souls as being balanced, harmonious, and rational. The Brotherhood called this speculative perception of reality, "The Harmony Of Souls". By understanding this "Attunement" with the universal laws of creation, one would have the key to understanding the process for achieving union with the divine.

"The Harmony of Souls", like "Harmony Of The Spheres", was described as the Sacred Decad, and its significance was explained in its mystical name "tetraks" meaning "fourness". The Sacred Decad explained that 1+2+3+4=10, and was thought of as the "Perfect Triangle" as pictured in the illustration.

The idea of union with divine or "The Transmigration of Souls" was the basis for the Pythagorean way of life. As the soul is material it also has its spiritual soul. this idea was later explained in the Christian epistles bearing the name of the Apostle Paul, in 1st Corinthian's 2:11-12 and 15:37-58 for example:
"For who among men knows, the thoughts of a man, except the man's spirit within him? In the same way no one knows the thoughts of G*d except the Spirit of G*d. We have not received the spirit of the word but the Spirit who is from G*d that we may understand what G*d has freely given us".
Paul is defining a G*d that exists in a pluralistic universe. Paul clearly separates the existence of G*d (infinite) with the thoughts or Spirit who is from G*d (the finite being and unity with divine) and ourselves (finite). This verse also implies a Spirit that is able to move from one place to another thus establishing the Pythagorean idea of the Transmigration of Souls. In chapter 15:37-58 Paul continues:
"When you sow, you do not plant the body that will be, but just a seed, perhaps of wheat or of something else. But G*d gives it a body as he has determined, and to each kind of seed he gives its own body. All flesh is not the same: Men have one kinf of flesh, animals have another, birds another and fish another. There are also heavenly dodies and there are earthly bodies; but the splendor of the heavenly bodies is one kind, and the splendor of the earthly bodies is another. The sun has one kind of splendor, the moon another and the stars another; and star differs from star in splendor. So will it be with the resurrection of the dead [in spirit]. The body that is sown is perishable, it is raised imperishable; it is sown in dishonor, it is raised in glory. If there is a natural body, there is also a spiritual body. So too is written; "The first man Adam became a living being" the last Adam, a life giving spirit. The Spiritual did not come first, but the natural, and after that the spiritual. The first man was of the dust of the earth, the second man from heaven. As was the earthly man, so are those who are of the earth; and as is the man from heaven, so also are those who are of heaven. And just as we are born the likeness of the earthly man, so shall we bear the likeness of the man from heaven."

Pythagoreanism & Christianity. The Unity of Opposites & The Triadic Principal
The Pythagoreans taught that the universe is composed of three fundamental properties that make it possible to exist.
1. The first was "Creation" the infinite spiritual reality. This "One" is beyond ousia, or being.
2. The second was the product of creation and was a finite material reality.
3. Third was that which brought a union, Logos, the Word that connects all things "Reunion" (A cyclical process that causes one "finite" to be united with the of other "Infinite"). This process was also called the Unity Of Opposites. Hence, Monotheism.  Sound familiar? There is one G*d, and that one G*d is the Father, the Son, and the Holy Spirit. They are distinct, but not separate... Therefore, G*d is everything we can conceive and more! Pythagoras is given credit with bringing Monotheism to Western thought in 525B.C. About 475 years would pass before Western cultures would accept Monotheism as the Christ experience. As the Creator Logos, Jesus is the Word which connects all things. As the personal Jesus, he is the flesh and blood of G*d who walks the earth giving sanctity to life and man. Through Christ there is a: 1. Unity Of Opposites 2. Infinite - Finite 3. Finite as Divine.

The dichotomy of G*d into divinity and humanity and his return to himself in the sacrificial act hold the comforting doctrine that in man's own darkness there is hidden a "Light" that shall once again return to its source, and that this Light actually wanted to descend into the darkness in order to deliver the Enchained One (his humanity) who languishes there, and lead him to the Light everlasting.

Squaring Our Actions
Square\'skwa(a)(ae)r,
1. An instrument with at least one right angle and two or more straight edges used to lay out or test right angles.
2. The corner or angle of a figure.
3. The product of a number or quantity multiplied by itself.
4. The guiding principal : Pattern, Rule, Standard.
5. Justness of Workmanship or of Conduct : Exact Proportion : Regularity : Quartile Aspect.
6. Squares pl. obs. Matters, Affairs, Things.
Squaring our actions is a phrase common among Freemasons. By squaring our actions with each other and G*d we free ourselves from the bondage of finite existence therefore achieve union with Divine Light. The phrase to square one's actions can easily be interpreted through Pythagorean thought. The product of the material self multiplied by its spiritual self squares its actions in both finite and infinite relationship with the unifying spirit of G*d.

A.D. 2005, Conclusion
Today, with the exception of a few elementary theorems, Euclidean geometry is of little use for modern science and engineering; trigonometry and analytical geometry being much more efficient methods of mathematics. In Euclidean thought there is also no room for science based on speculative prediction - "Quantum Probabilities", or the Planck Constant. But the real significance of Euclidean geometry lies in the superb training it gives for logical thinking. A "Proof" must not contain anything that is ultimately based on what we want to prove, or the Proof, is circular and invalid. Example: "Every angle in a semicircle is a right angle". Or, "The apex of a right angle subtended by the diameter lies on the circumference". Obviously certain conditions are required for a semicircle to have only right angles. The second statement is correct, but it must be proved. And Euclidean geometry teaches the difference between truth based on conditions and truth based on absolutes.

Though having flaws, regarding what ever Quantum Probability there may be, the Euclidean Foundation Stones were and are regarded as the foundation stones of mathematics and also in a way the foundation stones of Speculative Freemasonry. Oh, with regard to Freemasonry as a "Secret Society", you can go to any book store and pick out hundreds of books that explain Euclidean and Pythagorean thought. The fact is that if Freemasonry were "Secret" there would be no books --- the society would have snuffed out all the libraries centuries ago. But then, what became of the great library in Alexandria? A story to be continued in another column...