This document provides an overview of the connections between Euclid's 47th proposition, Pythagoras' teachings, and Speculative Freemasonry. It discusses how Euclid established mathematical rigor through his Elements, building on postulates like parallel lines. Pythagoras founded a religious brotherhood based on principles of mathematics in nature and the transmigration of souls. His teachings influenced early Christianity and aligned with concepts of the Trinity. Speculative Freemasonry references Pythagoras and Euclid's 47th proposition as symbols representing squaring one's actions with spiritual and divine principles.

1. August 14th, 2010
The Carpenter's Theorem
Speculative Freemasonry,
Pythagoras and Euclid's 47th Proposition
Submitted by RWB Wesley F Revels. This is the first installment in a lecture series on Speculative
Freemasonry and its meanings.
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. to 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

2. 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 at two places separated by a known distance
(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

3. 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. (This will be a future story in this lecture series.) 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,

4. 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, was described as the Sacred Decad, and its significance was
explained in its mystical name "tetraks" meaning "fourness". The Sacred Decad explained

5. 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

6. 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.

7. 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". 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...