As
Masons, we are constantly being urged to study the Seven Liberal Arts and
Sciences, as well as to study the hidden secrets of Nature and of Science. In my
mind, the two are interlinked, the seven liberal arts and sciences are the
tools with which to study the hidden secrets of nature, whereby we can come to
understand God and his creations. At
the same time, we must also use nature to check on the validity of our
conclusions arrived at by using the tools of the Seven Liberal Arts. The seven
liberal arts and sciences are made up of the following: Grammar, Rhetoric, Logic,
Arithmetic, Geometry, Music and Astronomy. The first three, ie Grammar, Rhetoric
and Logic make up the Trivium, whilst the last four, ie Arithmetic, Geometry,
Music and Astronomy, make up the Quadrivium. My task today is to briefly talk on
Pythagoras and his work which is the foundation for the whole of the Quardrivium.
Pythagoras
was born on the island of Samos around 575 BC and lived till the end of that
century. He taught philosophy and mathematics in Crotone, a Dorian Greek colony
in Southern Italy. He founded the school which has come to be known as
Pythagorianism, and his students seemed to be formed into a mystical society
known as the Order of Pythagoras. What we know of Pythagoras today came mostly
from the writings of Diogenes Laertius (1). We cannot be certain that the
teachings attributed to him actually were taught by him. They could have been
ideas of his students. Nevertheless, they came from his school and have been
commonly accepted as the Pythagorian tradition. When I talk about Pythagoras
believing in this or that, in truth, that’s a shorthand way of saying, the
Pythagorean tradition believed in this or that.
Pythagoras
believed that everything was related to mathematics and that numbers were the
ultimate reality and, through mathematics, everything could be predicted and
measured in rhythmic patterns or cycles.
Perhaps
he is most famous for that theorem in Geometry, the 47^{th} Proposition
of Euclid, which states that the square of the hypotenuese of a right angled
triangle is equal to the sum of the squares of the sides of that triangle. We
now call that Theorem as Pythagorus Theorem.
In
English Freemasonry, the jewel of the Past Master has the 47^{th}
Proposition of Euclid hanging underneath the Square , and according to HW Ryland,
Q.C. Vol. 14 p. 32, “no reason whatever was in anyone’s mind when the 47^{th}
proposition gradually came to be recognized as the dintinguishing mark of a Past
Master.” I shall come back to this point later.
Let’s
look at Pythagoras and his believes for the moment.
Just
like our Order, the Order of Pythagorus has three degrees of membership:

Novices

Nomothets

Mathematicians.
The
teachings were secret.
In
the Preface to the Anderson’s Constitution, Anderson referred to Pythagoras as
a mason. That’s pure myth, nevertheless, Pythagoras has significance for our
Craft, as we shall see later.
Pythagoras
held that reality, including music and astronomy, was at its deepest level,
mathematical in nature, numbers were the ultimate reality. He was the first to
point out the Evening Star and the Morning Star, which we now know to be the
planet Venus, were the same, and he was the first one to articulate that the
earth was a globe that moved around a central fire. (2)
His
believes in numbers would be considered unusual for us moderns for he thought
odd numbers were males and even numbers females. Because an odd number plus and
even number would produce an odd number, whereas an even number plus an even
number or an odd number plus and odd number produced even numbers, he considered
odd numbers as masters and therefore male numbers.
His
numerology contained many other believes. For example
1
Monad. Point. The source of all numbers. Good, desirable, essential, indivisible.
2
Dyad. Line. Diversity, a loss of unity, the number of excess and defect. The
first feminine number. Duality.
3
Triad. Plane. By virtue of the triad, unity and diversity of which it is
composed are restored to harmony. The first odd, masculine number.
4
Tetrad. Solid. The first feminine square. Justice, steadfast and square. The
number of the square, the elements, the seasons, ages of man, lunar phases,
virtues.
5
Pentad. The masculine marriage number, uniting the first female number and the
first male number by addition. The
number 5 was considered incorruptible because multiples of 5 always end in 5.
These
are just some of the, what I consider, mumbojumbo of his numerology. Checking
these believes against nature, there is no basis for such mumbojumbo, and
that’s what I meant when I said earlier we have to check our conclusions
against the reality of nature.
However,
his insight into the importance of numbers in the physical reality of our world
was amazing.
I
have alluded to the Pythagoras Theorem above which briefly states:
h²= a² + b²
where
h is the length of the hypoteneuse of a right angled triangle, and a and b are
the lengths of the sides of that right angled triangle.
That’s
a triangle drawn on a two dimensional space. Can we extend that theorem to a
three dimensional space? We most certainly can, and if we set c as the length of
the third dimension, the formula becomes:
h² = a² + b² + c²
As
we know, we don’t just live in a three dimensional space, time is considered
to be the fourth dimension. Can we extend that theorem to include the fourth
dimension, time?
At
first sight, that’s not possible. Lengths are measured in cm, in, feet, and so
on, while time is measured in seconds, hours, and so on. The units are different
and one cannot add apples with oranges.
This
is where the genius of Einstein came in. I am sure you all know we measure the
distance to some far away galaxy in units of lightyears. The units of time and
space can be interconverted if we factor into the calculations the speed of
light. Einstein did just that, and when that transformation is done, the formula
holds. He then asked the next question. Can we use that formula factoring in
momentum? Momentum in physics is defined by mass times speed. By factoring in
momentum, we are including mass into the formula. By modeling what some
physicists called the Einstein’s hypotenuse, modeled after the hypotenuse in
Pythagoras Theorem, Einstein incorporated momentum into the equation and
produced what is probably the most famous equation of the 20^{th}
century: (3)
E = MC²
I
shall now go further into cosmology and quantum physics, the hidden mystery of
nature, in a most simplistic manner. All of you have heard of atoms. Most, if
not all of you have heard of electrons, protons and neutrons, the stuff that
make up an atom. What are these in turn made up of? What are the forces that
hold them together? Our best understanding to date is that they are made up of
different permutation and combination of something called “strings”. Strings
are mathematical constructs of the theoretical physicists. They only exist as
mathematical formulae, but they are the very stuff of which our universe is made
of.
There
was a famous physicist called Dirac. He was trying to write a mathematical
equation that would explain all the properties of the electron. When he solved
that equation, he discovered he had two answers, one would describe an electron
perfectly, the other would describe something just like an electron, with
exactly the same properties of an electron, but with one exception, that
particle would have a positive electric charge, instead of a negative charge,
which an electron has. Which is correct? His mathematical construct or the body
of knowledge of physics known up to that point in time? The answer is, his
mathematical construct. That particle which he discovered was known as a
positron, and it is now well established that positron exists, as part of
something known as antimatter. Please note however, that that conclusion of the
positron was later confirmed by experiments in the real world.
Back
to the “String Theory” as the basis of our physical universe. That
mathematical construct is very likely to be the basis of our universe, hence,
the amazing insight of Pythagoras, that numbers lie at the very foundation of
our universe.
What
about his insight into music?
The
pitch of any sound we hear depends on the frequency of vibration of the air
molecules hitting our eardrums. If the molecules are hitting our airdrums at a
frequency of 440 times a second, or 440 Hertz, we hear the note “A”,
commonly known as “la”.
He
was the first one to realize that if two sounds vibrate at the ratio of 2:1, ie
if a string is vibrating at 880 hertz, and another at 440 hertz, then the two
notes are an octave apart, they sound the same except one note is higher by an
octave.
More
interestingly, he discovered two sounds, when vibrating at ratio of 3:2 was
harmonious. (That’s known as a perfect fifth in musical terms.) At the ratio
of 4:3, again, the sounds would be harmonious. (Perfect fourth). The ratio of
5:4, again would be harmonious. But ratios not of such simple numbers would
sound discordant. It was because of these findings, the belief arose that the
orbits of the planets, considered harmonious, would be of certain ratios. That
occupied the minds of many of the ancients.
Western
music is based on the scale between octaves being divided into twelve notes. If
you look at a piano, counting all the notes, including the black notes, there
are twelve notes. In many other civilizations, the scale is five notes between
the octave. Incidentally, the octaves are fundamental to all musical traditions.
Back to the five note scale, or Pentatonic scale. It is interesting that one of
these scales are made up of frequencies in the ratio of 1, 9/4, 81/16, 3/2 and
27/8. If you look carefully at those numbers, they are multiples of 3s and 2s.
This raises a very interesting problem What are the reasons for our brains being
constructed in such a manner that we find these ratios of frequencies of
vibrations pleasing?
I
have no answer to that problem That problem is made even more intriguing by
something known as the Mozart Effect (4). Two groups of children had their IQs
tested After that, one group was exposed to Mozart’s music, another group to
rock music. After half an hour, their IQs were tested again. That group that
listened to Mozart scored higher in their IQ tests. Further evidence that the
harmonious classical music have other beneficial effects was reported in The
Economist in an article entitled “Twilight of the Yobs” (5). That article
reported how the use of classical music helped control mobs, and that in the A
& E department of the Royal Bolton Hospital, the playing of classical music
seemed to make the patients calmer.
The
Matthew Cooke Manuscript, c 1450 (6)
has this to say of the Seven Liberal Arts:
The first, which is called the foundation
of all science, is grammar, which teacheth to write and speak correctly.
The second is rhetoric, which teaches us to speak elegantly.
The third is dialectic, which teaches us to discern the true from the false, and
it is usually called art or sophistry (logic).
The fourth is arithmetic, which instructs us in the science of numbers, to
reckon, and to make accounts.
The fifth is Geometry, which teaches us all about mensuration, measures and
weights, of all kinds of handicrafts.
The sixth is music, and that teaches the art of singing by notation for the
voice, on the organ, trumpet, and harp, and of all things pertaining thereto.
The seventh is astronomy, which teaches us the course of the sun and of the moon
and of the other stars and planets of heaven.
Fortunately
knowing of the vengeance that God would send, the brethren knew not whether it
would be by fire or water. They knew by a sort of prophecy that God would send
one or the other, and therefore they wrote their sciences on the two pillars of
stone. And some men say that they wrote on the stones all the seven sciences,
but [this I affirm not]. As they had it in mind that a vengeance would come, so
it befell that God did send vengeance, and there came such a flood that all the
world was drowned and all men died save only eight persons. These were Noah and
his wife and his three sons and their wives, of which sons all the world is
descended, and they were named in this wise, Shem, Ham and Japhet. And this
flood is called Noah's Flood, for he and his children were saved therein. And
many years after the flood, according to the chronicle, these two pillars were
found, and the chronicle says that a great clerk, Pythagoras, found the one, and
Hermes the philosopher found the other, and they taught the sciences that they
found written thereon.
Basically,
by studying the Trivium, it gives one the tools to express oneself in a logical
and coherent manner, and by studying the Quadrivium, it gives the contents to
the above. In other words, the Trimium is concerned with form, and the
Quadrivium with substance. By studying all seven, that gives us the tools to
unravel the hidden mystery of nature and of science, but whatever conclusion we
draw, we must go back to nature to confirm our findings, otherwise we would be
like what happened with scholasticism, trying to derive knowledge by deduction
based on authorities, and would have forgotten the lessons learnt during the Age
of Enlightenment, deriving knowledge by induction from experience.
Pythagoras,
being the originator of the Quadrivium, has therefore earned his place in
European culture as the originator of the study of the hidden secrets of nature
and of science. The jewel of the 47^{th} Proposition of Euclid, forever
known as the Pythagoras Theorem is to remind us that God is the Grand
Geometrician of the Universe and the Rules of nature, which were created by Him,
could be uncovered by using our power of reasoning aided by our studying of the
liberal arts and sciences, which are gifts of God.
We
might find the some of the meaning regarding the numerology of Pythagoras
irrelevant today, but his methodology, his insights are as relevant to us today,
as it was 2500 years ago.
References:

Diogenes
Laertius. Lives of eminent philosophers..
Trans. R.D. Hicks. Heinemann, London, 1972.

Robin Allot. Journal
of Social and Evolutionary Systems, 1: 7190, 1994

http://en.wikipedia.org/wiki/Momentum

Rauscher
FH, Shaw GL, Ky KN . "Listening to Mozart enhances spatialtemporal reasoning: towards a
neurophysiological basis".
Neurosci. Lett.
185 (1): 447, 1995

“Twilight
of the Yobs”, The Economist, Jan 5^{th} issue, 2005

Matthew Cooke
Manuscript, British Museum c1450
