The
address to the Immediate Past Master and the depiction on his jewel refer to the
47^{Th} proposition of Euclid. It
also refers to “our brother Pythagoras”.

The Master’s jewel is the square, two sides 90 degrees apart which will form the
base needed for the 47^{th} problem (in many jurisdictions the square
has the dimensions of 3:4, the Pythagorean dimensions). As the Master serves in
his position, he becomes more complete, and therefore the 47^{th}
problem of Euclid is dedicated on his jewel when he leaves office.

When
giving this address on the occasion of my installation as Master of Millaa
Millaa Lodge I thought back to my studies in electrical engineering and realized
the importance that this piece of mathematics had in electrical theory.

I
then decided to ask some my Masonic peers what they saw as the significance of
this to masonry. What role did
Pythagoras play? I suppose I was
not too surprised to find out that little if anything was known by those I asked.
It was because of this that I have decided to put together a paper
dealing with this subject and try to put some life into a subject that most
people shun, and to also look at the characters mentioned in this discourse, ie
Euclid and Pythagoras.

We
first hear of Pythagoras in the first degree tracing board, and again in the
Fellowcraft degree, where we are told in the charge after passing that “the
science of geometry, which is established as the basis of our art.”
These words tell us of the importance of geometry our ritual writers make
within the craft. The first degree tracing board lecture makes many references
to geometry and the implements to practice geometry.
The use of symbolic geometry occurs in most other Masonic orders over and
beyond the craft degrees.

The
importance placed on learning in our degree work can take us back a great many
years to a time when knowledge was neither so great nor as diversified as now.
William Preston, the eminent Masonic student, scholar, writer, who lived and
wrote in the latter part of the eighteenth century, conceived the idea of making
the degrees in general, and the Fellowcraft degree in particular, a liberal
education! A 'liberal education' in those days was comprised within what we
still call, after Preston, the 'seven liberal arts and sciences.' In those days
any mathematics beyond geometry was only for the very, very few. Indeed,
mathematics were looked upon as being something not meant for the common men, as
being of small use in the world, save for engineers and designers and measurers
of land.

But Preston, if his lectures are no longer the real 'liberal education' which he
planned, and which, in the form of his lectures modified by Webb (and somewhat
tinkered with by various authorities and near authorities who at times have kept
the husk and let the kernel escape!) built better than he knew. For we, may now,
justly and honourably, take 'geometry' to mean not only the science of
measurement of surface and area and the calculation of angles and distances, but
to mean all measurement. And to study measurement, means to study science, for
all science is but measurement, and by that measurement, the deduction of laws
and the unravelling of the secrets of nature.

This study of the
seven liberal arts and sciences is made mention of on numerous occasions in our
ritual, what of these? Below is a
quote that is 1800 years old.

"...let
one of these be that art which prepares the body to be subservient, as a prompt
and robust vehicle, to the mandates of the soul, and which is denominated *gymnastic*.
Let another art be that which is the angel of the conceptions of the soul, and
which is called *rhetoric*; another, that which is the nurse and tutor of
the juvenile mind, and which is denominated *poetry*; another that which is
the leader of the nature of numbers, and which is called *arithmetic*; and
another that which is the teacher of computation, and is called *logistic*.
Let *geometry*, also, and *music* follow, who are the associates of
philosophy and conscious of her arcana, and to each of which she distributes a
portion of her labour."

Maximus Tyrius (circa 200 CE) "Dissertation", xxi, translated by
Thomas Taylor(1758-1835)

These
differ little from those delineated in later times and would still have great
implications when applied in today’s modern world.

H.P.H.
Bromwell (1823 -1903) wrote in his massive tome Restorations of Masonic Geometry
and Symbolry that: "Although the number of recognized sciences far exceeds
seven, yet, giving to that number the benefit of its symbolic meaning, it stands
for the whole circle of sciences, whether specifically named among the seven or
not."

We
usually associate the seven liberal arts to medieval education curriculum, at
this time in Masonic circles the only education available may I suggest that an
educated member of a lodge was more useful especially if he could apply geometry
to his work.

**
**

**The
47**^{Th} proposition of the first book of Euclid.

**
**

The
Pythagorean Theorem states that for any right triangle the sum of the squares of
its two legs equals the square of its hypotenuse (a2 + b2 = c2). Or we could
frame it as the sum of the square of the horizontal and the square of the
perpendicular equals the square of the hypotenuse. This is what became known as
Euclid’s 47 Proposition. The
drawing below illustrates this.

We
also know this formulation was known before Pythagoras - there is
evidence in ancient Egyptian work, ancient China (the Chou Pei manuscript),
and the megalith builders.

*Chou
Pei*

The
oldest existing Chinese texts containing formal mathematical theories were
produced during the Han period. The *Arithmetic Classic of the Gnomon and the
Circular Paths of Heaven (Chou Pei Suan Ching*) is* *dated before the
3rd century B.C and contains various modern mathematical principles such as
working with fractions using a common denominator, and proofs of many
geometrical theories. The text contains an accurate process of division for
finding out the square root of numbers. In fact, the Chou Pei presents the oldest
known proof of the right-angle triangle theory in the *hsuan-thu *diagram.
This theory, commonly known as the "Pythagorean theorem," shows that
the sum of the squares of the legs of a right triangle is equal to the squares
of the hypotenuse or (A X 2) + (B X 2) = (C X 2).

Regardless,
it is attributed to Pythagoras and two hundred years later Euclid compiled his
"Elements of Mathematics" where this particular 47^{Th}
proposition is found in Book One.

This
theorem has been called the root of all geometry and the cornerstone of
mathematics. The practical applications alone are worthy of the high esteem that
Masonry affords it. And this is the interpretation of the lecture that is most
considered when masons speak of it but the meaning of this hieroglyphical emblem
does not stop there.

The
emblem we are usually presented is the 3,4,5 right triangle in this fashion: The
vertical line is of 3 units, the horizontal is of 4 units, and the hypotenuse is
of 5 units. Not only is our attention called to this geometrical figure in the
Master Mason degree, it is also prominent in the Scottish Rite in the 20th
Degree - Master of the Symbolic Lodge and in the 25th Degree - Knight of the
Brazen Serpent.

Geometry
treats of the powers and properties of magnitudes in general, where length,
breath, and thickness, are considered, from a point to a line, from a line to a
superficies (surface of a body)and from a superficies to a solid.

By
this science, the architect is enabled to construct his plans, and execute his
designs; the general to arrange his soldiers; the engineer to mark out ground
for encampments; the geographer to give us the dimensions of the World, and all
things within, to delineate the extent of seas, and specify the divisions of
empires, kingdoms and provinces; by it, also, the astronomer is enabled to make
his observations, and to fix the duration of times and seasons, years and cycles.
In fine, geometry is the foundation of architecture, and the root of mathematics.

Many
readers may feel like they have been returned to Geometry class. A simple
illustration will probably refresh our memories:

The
proposition is especially important in architecture. Builders have, since
ancient times used this theorem in constructing buildings by a process known as
“squaring a room.” As the theorem states that 3 squared + 4 squared = 5
squared, a builder starts by marking a spot and drawing a line, say line A. This
line is given the value of 3. The builder then marks another point, say point B
and draws a line from it at a right angle to line A, and it is given the value
of 4. The distance between line A and B is then measured, and if the distance
between A and B is 5, then the room is squared. By inverting the process, a
“squared” (or rectangle) room can be obtained.

Before
the advent of GPS and Laser measuring tools engineers who tunnel from both sides
through a mountain used the 47^{th} proposition to get the two shafts to
meet in the center. The surveyor who wants to know how high a mountain may be,
ascertains the answer through the 47th proposition. The astronomer who
calculates the distance of the sun, the moon, the planets, and who fixes
"the duration of times and seasons, years, and cycles," depends upon
the 47th proposition for his results. The navigator traveling the trackless seas
uses the 47th proposition in determining his latitude, his longitude, and his
true time. Eclipses are predicted, tides are specified as to height and time of
occurrence, land is surveyed, roads run, shafts dug, bridges built, with the
47th proposition to show the way.

In
some lodges, using this principle, a candidate symbolically “squares the Lodge”
by being escorted around the Lodge three times during the Entered Apprentice
ritual, four times for a Fellowcraft ritual, and five times for a Master Mason
ritual, which completed his journey.

**The
Mystical Symbolism of the Pythagorean Triangle**

Earlier
in this paper I made reference to the fact that the Egyptians were in possession
of the knowledge of the 47^{th} Proposition.

According
to Plutarch (46 - 120 C.E.), the Egyptians attributed the sides of the triangle
in this fashion. The vertical line was of 3 units and attributed to Osiris. The
horizontal line was of 4 units and attributed to Isis. And the hypotenuse was,
of course, 5 units and attributed to Horus, the son of Osiris and Isis. It is
noteworthy that Plutarch studied in the Academy at Athens and was a priest at
Apollo's temple at Delphi for 20 years. In the myth of Osiris and Isis, Osiris
is killed which makes Horus the Son of a Widow and links him with Hiram!!

The
units of the triangle's side are significant. The three units of the Osiris
vertical have been attributed to the three Alchemical principles of Salt,
Sulphur and Mercury. All things are
manifestations of these three principles according to Alchemical doctrine. The
four units of the horizontal line of Isis relate to the so-called four elements:
earth, air, water, and fire. These are of course the four Ancients. The
ascending Horus line with its five units represents the five kingdoms: mineral,
plant, animal, human, and the Fifth Kingdom. This is the Path of Return. The
ascending line finally connects back up with the Osirian line. The Fifth Kingdom
symbolizes the Adept as one who has consciously reunited with the Source of all
Being.

The
47th problem has been referred to as "the foundation of Freemasonry."
Why one of many possible solutions to a particular geometrical problem should
become emblematic of Freemasonry probably has more to it the long standing
tradition of Sacred Geometry, but the importance of the symbol is largely
unknown.

Noted
cabalist Alan Bennet speculated that the three squares represented the magical
squares of Saturn, Jupiter, and Mars. Masonic author Albert Pike, in* his
Morals and Dogma *repeats Plutarch's comments and suggests that the triangle
represents matter (Isis), spirit (Osiris), and the union of the two (Horus)...and
indeed, the sum of the two smaller squares equals the larger.

**Sacred
Geometry,**

Pythagoras
is credited with the discovery of the **Golden Rectangle**. The
Golden Rectangle is built on the "golden ratio" or "golden
proportion," which is determined by the irrational number known as Phi.

To
put it simply, a golden rectangle is a rectangle divided in such a way as to
create a square and a smaller rectangle that retains the same proportions as the
original rectangle. To do this, one must create a rectangle based on this ratio.

To
find the Golden Ratio, one must divide a line so that the ratio of the line to
the larger segment is equal to the ratio of the larger segment to the smaller.

*A is to B as B is to C*

To get a golden rectangle, you simply turn the larger segment of the line into a square.

If you add a square to the long side of the "golden rectangle," you'll get a larger golden rectangle. If you continue to add squares in this way, you'll see the basis for nature's logarithmic spiral patterns.

The golden proportion appears in numerous places in nature and in art and architecture. It forms the basis for Leonardo Da Vinci's "Divine Proportion," the ideal illustrated in his drawing Vitruvian Man. The face of the Parthenon in Athens is a perfect golden rectangle. The shell of the nautilus is a famous example of a spiral based on the golden mean, as is the spiral of the human DNA molecule.

Vitruvian Man.
Although made famous by Renaissance artist Leonardo Da Vinci, the figure known as the Vitruvian Man is actually named for the man who created him, the Roman architect Vitruvius. Vitruvius, a proponent of the Sacred Geometry of Pythagoras, designed temples based on the proportions of the human body, believing them to be perfect. This perfection, wrote Vitruvius, was due to the fact that the extended limbs of a perfectly proportioned human fit into both the circle and the square.

According to Pythagorean tradition, the circle represents the spiritual realm; the square, material existence, so the human body represented the perfect marriage of matter and spirit, which was is reflected in its proportions. Leonardo was one of many artists who attempted to depict Vitruvius' perfect man, and the only one who succeeded; his version is considered the most accurate depictions of the human body.

From
this one can see that man for many centuries has had some kind of relationship
with geometry especially the ratio Phi which equals 1.618 most antient buildings
have been constructed using this ratio being that which appears most harmonious.

**Euclid**
(about 300BC) in his "Elements" calls dividing a line at the 0.6180399
point **dividing a line in the extreme and mean ratio.** This later gave rise
to the name **golden mean**.

There are no extant records of the Greek
architects' plans for their most famous temples and buildings (such as the
Parthenon). So we do not know if they deliberately used the golden section in
their architectural plans. The American mathematician Mark Barr used the Greek
letter **phi** (f) to represent the golden ratio, using the initial letter of
the Greek Phidias who used the golden ratio in his sculptures.

**Luca Pacioli**
(also written as Paccioli) wrote a book called *De Divina Proportione* (**The
Divine Proportion**) in 1509. It contains drawings made by Leonardo da Vinci
of the 5 Platonic solids. It was probably Leonardo (da Vinci) who first called
it the **sectio aurea** (Latin for **the golden section**).

So, in Plato's Timaios, the elements
became a shape in the form of the five bodies, also called Platonic
bodies.

You first see fire which forms the
primary antithesis with earth. Then we see air, without which fire can not exist,
and water which forms the liquid part of earth, and without which no life could
be possible on earth. Then we see the fifth element, which should according to
Plato be in the centre of the elements, and which is called the Quintensens, or
to put it more mathematical: the dodecahedron. The solution is that the
dodecahedron was not made public. Also in the middle Ages and Renaissance the
knowledge of the dodecahedron, and with it the connection of the pentagram was
passed on only to the most prominent architect and his closest colleagues.
Furthermore they had discovered that the dodecahedron, through connecting
certain edges etc. held in itself all other bodies.

The dodecahedron became thus the basis
for obtaining all other bodies and was not without reason for Plato the symbol
of the all all-embracing Universe.

How then
does this golden ratio apply to geometry and how is it formulated?

**The
Fibonacci series**** is formed by adding the latest two numbers to get
the next one, starting from 0 and 1
**

0
1 --the series starts like this.

0+1=1
so the series is now

0
1 1

1+1=2
so the series continues...

0
1 1 2 and the next term is

1+2=3
so we now have

0
1 1 2 3 and it continues as follows
...`
`

`0,
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987,...
`**
**

If we take the ratio of two successive
numbers in Fibonacci's series, (1, 1, 2, 3, 5, 8, 13, ….) and we divide each
by the number before it, we will find the following series of numbers:

^{1}/_{1}
= 1, ^{2}/_{1} = 2, ^{3}/_{2}
= 1·5, ^{5}/_{3} = 1·666..., ^{8}/_{5}
= 1·6, ^{13}/_{8} = 1·625, ^{21}/_{13}
= 1·61538...**
**

It is easier to see what is happening if we plot
the ratios on a graph:
The ratio seems to be settling down to a particular value, which we call the **golden ratio** or the **golden number**. It has a value of approximately **1·618034**.The geometric proportions of the pentagram are those of the golden section, *aurio sectio*: golden mean.

The pentagram was frequently found on potsherds and tablets (which have been dated to as early as 3500 BCE) in the location of the Kingdom of Uruk (at the mouth of the Tigris-Euphrates valley). The symbol was found accompanying signs relating to the foundation of written language. There is also evidence that the pentagram was used in ancient Mesopotamia to indicate the seal of royalty, and power which extends to the four corners of the earth.

The ancients regarded the rose as the symbol of secrecy and silence. The confessionals in old churches have the symbolic rose carved upon them; the vaulted ceiling of the council chamber bears it, and the lodge room of medieval masonry is decorated with it. What occurred there was **sub rosa**, not to be divulged.

The pentagram, so easily seen in the rose, was the secret shibboleth of the Pythagoreans; and the Celtic Druids adorned their priestly robes with it. The Christians considered it even a more powerful sign than the cross for banning the evil one; it is found in many windows of the fine old Munsters and cathedrals. Could the five pointed star also represents the five points of fellowship?

**Pentagram****:**
from the Greek, "pente", meaning five and "gramma", a letter;
the pentagram is a five pointed figure formed by producing the sides of a
pentagon both ways to their point of intersection, so as to form a five pointed
star. **Pentalpha:** The triple triangle, from the Greek words meaning five
and, *alpha,* the letter A

One
of the arguments used by anti-masons to support their claim that Freemasonry is
satanic is to refer to the Masonic use of the pentagram. The short answer to
this accusation is that the pentagram is not mentioned in any Masonic ritual or
lecture and is not contained within the lessons or teachings of Freemasonry.

True
in essence, this response will not satisfy anti-masons who can point to the many
books about Freemasonry that include the pentagram, or at least the five-pointed
star, in their iconography, and the writings of such spurious, irregular or
psuedomasonic authors as Count Cagliostro, Éliphas Lévi and Aleister
Crowley. Although the pentagram, as a geometric figure, is of
interest to freemasons since it is also a representation of the Golden Ratio its
esoteric significance is only of historical interest.

It
also isn't satanic. The claims of Lévi and Crowley that there is any esoteric
significance to the relative position of the star points is demonstrably not
based on any observable astronomical event or on any published precedent.

Symbols
mean what the users want them to mean—and meanings change. Although Venus is termed the bright morning star or Lucifer; and the pentagram is claimed by Lévi
to represent the Baphomet and the celestial motions of Venus sketch a pentagram in the sky; it does not
follow that the pentagram represents Lucifer or that Lucifer equates with the
Baphomet. Or that any of this has anything to do with Freemasonry.

In
this short talk I have tried to show the interrelationship that exists between
freemasonry and geometry, the relationship of the Golden section, Golden mean,
or Golden proportion to geometry and masonry.
This is of course by necessity. Due to the complexity of the subject to
cover fully would take many hours of discussion.
I do hope however that the subject of this talk stimulates the listener
to research the subject themselves.

**GLOSSARY
**

**Alchemy:
**a power or process of transforming
something common into something special

**Aleister
Crowley:** was perhaps the most controversial and
misunderstood personality to figure in the new era of modern day witchcraft.
Known by the popular press of his time as “The Great Beast” and “The
Wickedest Man in the World”, Crowley was a powerful magician, poet, prophet
and famed occultist. He was also a one-time witch, though most of the
elders of the craft would discredit him the title.
In 1899 Crowley is reported to have become a member of one of “Old
George Pickingill’s” hereditary covens situated in the New Forrest, although
apparently he was not welcome for long It
is alleged that he obtained his 'Second Degree' before being dismissed due to
his contemptuous attitude toward women, failure to attend rituals with
regularity, his personal ego and sexual perversion (Crowley had a bias toward
homosexuality and the bizarre, shocking during his time even amongst witches).
The priestess of his coven later described him as “a dirty-minded,
evilly-disposed and vicious little monster!”

**
**

**Baphomet:
**is an idol or image of uncertain provenance,
depicting a being of heretical worship. The name first came to public consciousness during the suppression of
the Knights Templar

**Count
Cagliostro:** Count Cagliostro was a
notorist celebrity of the 18th century. Cagliostro
claimed to be a healer and an alchemist. One of his most famous dupes was his
marriage to a 14-year-old girl, who he claimed was really 50 but due to his
powers had been magically rejuvenated.

At
one stage in his life Cagliostro even claimed to be a freemason from Egypt who
had walked on the sands of Galilee with Jesus.

**Eliphas
Levi** : is the pseudonym of Alphonse Louis Constant,
a French occultist and author whose work greatly influenced many of the early
revivalists of the 19^{th} century. Interestingly Aleister
Crowley was born the same year Levi died and later claimed to
be his reincarnation. His father was determined his son should have a
decent education, he sent him to a church in St-Sulpice there to be educated and
trained as a priest. While he was there he became intrigued by a lesson
received from his headmaster, who during the course of the lesson explained his
belief that animal magnetism was a vital energy of the human body controlled by
the “Devil”. This sparked his curiosity and surreptitiously he began
to study all that he could find out about magic and the occult. His interest had
been ignited. Constant continued to pursue his ecclesiastical career and was
ordained as a priest. Later he was thrown out of the church and
excommunicated due to his left-wing political views and writings, also because
he refused to observe his vows of chastity.

**
**

**Euclid:
**300 B.C.
Greek geometer; founded a school in Alexandria; wrote *Stoicheia
(Elements), *which remained definitive source of geometrical reasoning and
methods until 19th cent.

**
**

**Gnomon:
**the remainder of a parallelogram after the removal
of a similar parallelogram containing one of its corners

**
**

**Horus**:
the Egyptian god of light and the son of Osiris and Isis

**
**

**Isis:
**an Egyptian nature goddess and wife and sister of
Osiris

**
**

**Osiris:
**the Egyptian god of the underworld and husband and
brother of Isis

**
**

**Plutarch:
**Greek biographer & moralist; best known for *Parallel
Lives, *character studies of famous Greeks and Romans from age of Theseus and
Romulus to his own** **

**
**

**Pythagoras:
**500 B.C.
Greek philosopher & mathematician; generally
credited with theory of functional significance of numbers in the objective
world and in music.

**
**

**Superficies**:
Surface of a body