Freemasonry uses mathematical symbols as well as
natural ones. The mathematics of Freemasonry would require a book for
their adequate presentation, but two of her greatest mathematical
symbols belong so aptly together, though separated widely in her ritual
that they will be considered side by side by the interested student.
These are the number "Three," and the Forty-Seventh
Problem of Euclid.
Both of these demonstrate Deity with mathematics, a
feat which no mathematician would dare, but which any well-informed
Freemason finds sufficiently easy!
The emphasis placed upon the number "Three" in
Freemasonry is so great that, apparently, the founders and developers of
our modern ritual did not find it necessary to offer any monitorial
explanation of it as a symbol. Yet it is a great and important symbol;
generations of philosophers have striven for an adequate compilation of
all of its ramifications. It is not on record that any authority has yet
said "This is the end of the symbolism."
It is neither necessary nor desirable to compile the
ancient references to trinity; from the oldest known and recorded (that
of the Brahmins), to the modern Christian Trinitarian doctrine, the
religions of the world of all peoples and all lands have stressed the
tri-part nature of God.
There is "Three" throughout nature. Earth, water,
air; father, mother, child; sunrise, noon, sunset; seed, flower fruit;
sowing, growing, reaping; Man must early have learned of three, and
nature's insistence upon three.
And there is three throughout Freemasonry; three
degrees, three principal officers; three original Grand Masters; three
lesser lights; three great lights; three movable jewels' three immovable
jewels, three of fifteen who traveled in a westerly direction; three
raps; three gates; three circuits in circumambulation; three steps on
the Master's Carpets; three steps in Masonry, three pillars supporting;
three, three, three!
We are taught of Wisdom, Strength and Beauty; and
some have been confused by the inclusion of a word meaning pulchritude;
and some initiates think it refers to form and face, and is there
effeminate. But sex does not here enter the symbolism; in wisdom,
strength and beauty the philosopher finds reference to mind, body and
spirit; which support our institution. But there is much more to this
symbolism than support; it is at once a plea, a command, an exhortation
and a prayer; that our institution be supported by the best of wisdom,
the greatest of strength and the most blinding of beauty.
See how this blends with the "Doctrine of the perfect
youth" over which Masonic jurists quarrel in the most friendly fashion
to this day (nor have all Grand Lodges settled the matter, even for
themselves). Unquestionably a maimed man may have a fine brain; one
thinks at once of Steinmetz, one of the greatest scientists this world
has ever known, whose achievements will be ranked among the very
highest, as history assigns him his true place. Steinmetz had an ugly,
misshapen body; he was frail and humpbacked, but his mind was wonderful.
Yet how much more wonderful might have been his achievements had his
maimed and twisted body been straight and tall, the enormous power of
mind backed up by a health which would have carried him to four score
We do not admit to our Fraternity the maimed, the
halt, the blind, the imperfect; the literalist insists because of the
impossibility of those so afflicted conforming to the outward
requirements. But the esoteric philosopher finds in the ancient doctrine
of a perfect youth a support, a foundation, perhaps a buttress of the
pillar strength, and passes on his wisdom to practical application; that
A Freemason, other things being equal, is the best whose health and
strength fit him for great tasks, greatly done.
There is need of wisdom in any world; especially is
there need of wisdom in one torn by dissension, driven by differences,
swept by passion and dismembered by prejudice. It is one of the hopes of
that same distempered world that Freemasonry, by her teaching of that
especial wisdom which deals with human relations may pour the oil of
brotherhood upon the tempestuous seas of discord and misunderstanding.
The pillar of wisdom is a vital support of Freemasonry, as of
The pillar of beauty is a symbol of spirituality. It
is beauty of the soul, not of body. It is loveliness of thought, not of
limb. It is the blinding magnificence of our inner conception of the
inconceivable . . . The Grand Lodge Above . . . not a beauty of the
earth, earthy. Strength without wisdom is brutality. Wisdom without soul
is fact without mercy, justice, charity or love. Wisdom and strength are
vitally important supports, but the lodge would fall and the Fraternity
be no more, if the third support were taken away. Wisdom, Strength and
Beauty; the three Lesser Lights, the stations of the three principal
officers, all form triangles. The Lodge, an "oblong square" represents
the world, perhaps the universe. But the triangle represents God.
It does represent Him because some man once said,
"Here is a curious three sided figure, lets say it looks like God!"
Symbols do not thus spring into being. The triangle always has been a
representation of God; from the dawn of history the three-sided figure
has been representation of man's conception of The Most High.
It is not difficult to imagine why. To all mankind
deity has been visualized as perfect. He is also conceived of as First;
before all else. The first words in the Old Testament are, "In the
Beginning, God . . ."
A point is nothing but an idea. That which connects
two points is a line. But a line has a beginning and an ending. Man's
idea of God is of One without a beginning or ending. Two lines cannot
make a figure without a beginning or an ending. They form a cross or an
angle, but always there is the sense of imperfection, of something
wanting. But when three lines from a triangle, it is without either a
beginning or ending. And it is the first possible complete figure which
can be constructed of straight lines. It is not both logical and
beautiful that the First Perfection which Geometry can show should have
stood, and still stands, as a symbol of Him from Whom Geometry
This, then, is the reading of the number "Three"
throughout Freemasonry; it is a symbol that the Great Architect is
everywhere; that we can move not, work not, live not or love not without
we do so beneath His All-Seeing Eye, and as workmen in His Quarry.
Everywhere, in every degree, is three, three, and yet more threes.
Everywhere, throughout all life, is God, God and yet more of the
omnipresence of God.
Everywhere, through out the three degrees, threes
preach the inextricable inter-weaving of the philosophy, the meaning and
the glory of freemasonry with her gentle, tender and wholly reverent
idea of the Great Architect of the universe.
So much for the number three. As the child begins the
study of arithmetic with simple digits and gradually progresses through
mensuration of all sorts to algebra and finally, in high school, to
geometry; so the Freemason meets his first Masonic mathematics in the
number three, and gradually learns more and more of the gracious
mensuration of the Craft until he is invited to study the geometrical
Forty-Seventh Problem of Euclid.
The Forty-Seventh Problem of Euclid is older than
Pythagoras. The Sublime degree of Master Mason as we know it is younger
than Pythagoras by many hundreds of years. Our Rituals are accurate in
neither date nor fact; and yet of all the symbols of Freemasonry the
Forty-Seventh Problem is one of the most beautiful and most filled with
For the benefit of those who may have forgotten their
geometry days, the Forty-Seventh Problem is here simply stated; in any
right triangle, the sum of the squares of the two sides is equal to the
square of the hypotenuse. This is demonstrably true regardless of the
length of either side. But in the Problem as diagrammed in the lodge,
and for simplicity's sake it is usually shown with sides the proportions
of which are as three, and four units when the hypotenuse, or longest
side of the triangle will be as five units.If one draws on paper a line
three inches long, and at right angles to it , and joined to one end, a
line four inches long, then the line connecting the two ends will be
five inches long when the angle is a perfect right angle, or one of
ninety degrees.The square of 3 is 9. The square of 4 is 16. The sum of 9
and 16 is 25. The square root of 25 is 5.
We are taught but little about this Problem in our
Rituals, and, as stated, much of what we are taught is wrong! We are
instructed that it was invented by Pythagoras, that he was a Master
Mason, that he was so delighted with his invention that he exclaimed
"Eureka" (I have found it), that he sacrificed a heca-tomb, and the
Problem "Teaches Masons to be general lovers of the arts and sciences.
"Why so great and awe-inspiring a symbol should receive such scant
attention is not our problem. Perhaps it is because the fathers of the
ritual thought it beyond the grasp of many and so better left for the
individual to follow if he would. Certain it is that he who will think
on this problem will find a rich reward.
How came this wonder to be? What is the magic of 3
and 4 and 5? (or 6 and 8 and 10, or 36 and 64 and 100, or any other set
of numbers of the same relationship)? Why is the sum of the squares of
the two lesser always equal to the square of the greater? What is the
mystery which always works out so that, no matter what the length of any
two sides, so be it they are at right angles, the line joining their
free ends will have a square equal to the sum of the other two squares?
If one line be 7.6954 inches long, and the other 19 miles, 573,5732 feet
long, the sum of the squares of these numbers will be the square of the
length of the line joining their free ends, if, and only if, the two
lines are at right, or ninety degree, angles.
With this certainty, man reaches out into space and
measures the distance of the stars! With this knowledge he surveys his
land, marks off his boundaries, constructs his railroads and builds his
cathedrals. When he digs a tunnel through a mountain, it is the
Forty-Seventh Problem of Euclid by which he measures so that the two
parties digging toward each other meet in the center of the mountain,
having dug a straight tunnel. With this knowledge man navigates the
ocean, and goes serenely and with perfect confidence upon a way he
cannot see, to a port he does not know; more, with this problem he
locates himself in the middle of the ocean so that he knows just how far
he has come and whither he goes!
If we put down the squares of the first four numbers;
thus, 1, 4, 9, 16; we can see that by subtracting each square from the
next one we get 3, 5, and 7; which are the steps in Masonry, the steps
in the Winding Stair, the brethren which form Entered Apprentice,
Fellowcraft and Master Mason Lodges, which are, in other words, the
These have been the sacred numbers from the dawn of
history. Always they have held meanings for those who attached a
significance of spiritual import to mathematics. Always they have been
symbols of the interrelation of science, knowledge, exploration,
building; and God, religion, worship and morality.
Many will find presumption in any attempt to read a
symbol which so great an authority as Albert Pike said had an unknown
meaning (page 789, Morals and Dogma). Yet, if none presumes, from whence
can individual progress come? The same authority declared it the
inalienable privilege of any Mason to interpret the symbols of Masonry
for himself. Therefore, a reading is here dared!
So far as we know . . . and while we cannot prove it
by mathematics, the strongest of circumstantial evidence leads us to
believe . . . the fundamentals of mathematics are true, not only in this
world, but in all worlds. Our finite minds cannot think of a world or a
universe in which two and two make other than four, or in which the
relation of the circumference of a circle to its diameter is other than
3.1416 plus. It is axiomatic to us that if the sum of the squares of the
two sides of a right angled triangle are equal to the square of the
hypotenuse is a truth here, it is a truth everywhere. This particular
mathematical truth is so perfect, so beautiful, so inevitable and so
fitting to the art an science of Freemasonry, the founders of our
beloved Order must have chosen it from many others as a symbol of the
universality of law, and therefore of the Law Maker. The Forty-Seventh
Problem of Euclid not only teaches us to be general lovers of the arts
and sciences, but to bow heads in reverence at the perfection and the
beauty, the universality and the infinite extension of the laws of the
Great Law Giver.
Sourced from Short Talk Bulletin - November 1925