243 lines
13 KiB
Plaintext
243 lines
13 KiB
Plaintext
SHORT TALK BULLETIN - Vol.III November, 1925 No.11
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MATHEMATICS
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by: Unknown
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Freemasonry uses mathematical symbols as well as natural ones. The
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mathematics of Freemasonry would require a book for their adequate
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presentation, but two of her greatest mathematical symbols belong so
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aptly together, though separated widely in her ritual that they will
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be considered side by side by the interested student.
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These are the number "Three," and the Forty-Seventh Problem of
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Euclid.
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Both of these demonstrate Deity with mathematics, a feat which no
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mathematician would dare, but which any well-informed Freemason finds
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sufficiently easy!
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The emphasis placed upon the number "Three" in Freemasonry is so
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great that, apparently, the founders and developers of our modern
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ritual did not find it necessary to offer any monitorial explanation
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of it as a symbol. Yet it is a great and important symbol;
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generations of philosophers have striven for an adequate compilation
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of all of its ramifications. It is not on record that any authority
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has yet said "This is the end of the symbolism."
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It is neither necessary nor desirable to compile the ancient
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references to trinity; from the oldest known and recorded (that of
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the Brahmins), to the modern Christian Trinitarian doctrine, the
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religions of the world of all peoples and all lands have stressed the
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tri-part nature of God.
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There is "Three" throughout nature. Earth, water, air; father,
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mother, child; sunrise, noon, sunset; seed, flower fruit; sowing,
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growing, reaping; Man must early have learned of three, and nature's
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insistence upon three.
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And there is three throughout Freemasonry; three degrees, three
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principal officers; three original Grand Masters; three lesser
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lights; three great lights; three movable jewels' three immovable
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jewels, three of fifteen who traveled in a westerly direction; three
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raps; three gates; three circuits in circumambulation; three steps on
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the Master's Carpets; three steps in Masonry, three pillars
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supporting; three, three, three!
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We are taught of Wisdom, Strength and Beauty; and some have been
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confused by the inclusion of a word meaning pulchritude; and some
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initiates think it refers to form and face, and is there effeminate.
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But sex does not here enter the symbolism; in wisdom, strength and
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beauty the philosopher finds reference to mind, body and spirit;
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which support our institution. But there is much more to this
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symbolism than support; it is at once a plea, a command, an
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exhortation and a prayer; that our institution be supported by the
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best of wisdom, the greatest of strength and the most blinding of
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beauty.
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See how this blends with the "Doctrine of the perfect youth" over
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which Masonic jurists quarrel in the most friendly fashion to this
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day (nor have all Grand Lodges settled the matter, even for
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themselves). Unquestionably a maimed man may have a fine brain; one
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thinks at once of Steinmetz, one of the greatest scientists this
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world has ever known, whose achievements will be ranked among the
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very highest, as history assigns him his true place. Steinmetz had
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an ugly, misshapen body; he was frail and humpbacked, but his mind
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was wonderful. Yet how much more wonderful might have been his
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achievements had his maimed and twisted body been straight and tall,
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the enormous power of mind backed up by a health which would have
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carried him to four score and ten!
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We do not admit to our Fraternity the maimed, the halt, the blind,
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the imperfect; the literalist insists because of the impossibility of
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those so afflicted conforming to the outward requirements. But the
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esoteric philosopher finds in the ancient doctrine of a perfect youth
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a support, a foundation, perhaps a buttress of the pillar strength,
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and passes on his wisdom to practical application; that A Freemason,
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other things being equal, is the best whose health and strength fit
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him for great tasks, greatly done.
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There is need of wisdom in any world; especially is there need of
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wisdom in one torn by dissension, riven by differences, swept by
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passion and dismembered by prejudice. It is one of the hopes of that
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same distempered world that Freemasonry, by her teaching of that
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especial wisdom which deals with human relations may pour the oil of
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brotherhood upon the tempestuous seas of discord and
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misunderstanding. The pillar of wisdom is a vital support of
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Freemasonry, as of civilization.
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The pillar of beauty is a symbol of spirituality. It is beauty of
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the soul, not of body. It is loveliness of thought, not of limb. It
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is the blinding magnificence of our inner conception of the
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inconceivable . . . The Grand Lodge Above . . . not a beauty of the
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earth, earthy. Strength without wisdom is brutality. Wisdom without
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soul is fact without mercy, justice, charity or love. Wisdom and
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strength are vitally important supports, but the lodge would fall and
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the Fraternity be no more, if the third support were taken away.
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Wisdom, Strength and Beauty; the three Lesser Lights, the stations of
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the three principal officers, all form triangles. The Lodge, an
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"oblong square" represents the world, perhaps the universe. But the
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triangle represents God.
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It does represent Him because some man once said, "Here is a curious
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three sided figure, lets say it looks like God!" Symbols do not thus
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spring into being. The triangle always has been a representation of
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God; from the dawn of history the three-sided figure has been
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representation of man's conception of The Most High.
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It is not difficult to imagine why. To all mankind deity has been
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visualized as perfect. He is also conceived of as First; before all
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else. The first words in the Old Testament are, "In the Beginning,
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God . . ."
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A point is nothing but an idea. That which connects two points is a
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line. But a line has a beginning and an ending. Man's idea of God
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is of One without a beginning or ending. Two lines cannot make a
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figure without a beginning or an ending. They form a cross or an
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angle, but always there is the sense of imperfection, of something
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wanting. But when three lines from a triangle, it is without either
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a beginning or ending. And it is the first possible complete figure
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which can be constructed of straight lines. It is not both logical
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and beautiful that the First Perfection which Geometry can show
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should have stood, and still stands, as a symbol of Him from Whom
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Geometry (Freemasonry came?
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This, then, is the reading of the number "Three" throughout
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Freemasonry; it is a symbol that the Great Architect is everywhere;
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that we can move not, work not, live not or love not without we do so
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beneath His All-Seeing Eye, and as workmen in His Quarry.
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Everywhere, in every degree, is three, three, and yet more threes.
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Everywhere, throughout all life, is God, God and yet more of the
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omnipresence of God.
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Everywhere, through out the three degrees, threes preach the
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inextricable inter-weaving of the philosophy, the meaning and the
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glory of freemasonry with her gentle, tender and wholly reverent idea
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of the Great Architect of the universe.
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So much for the number three. As the child begins the study of
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arithmetic with simple digits and gradually progresses through
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mensuration of all sorts to algebra and finally, in high school, to
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geometry; so the Freemason meets his first Masonic mathematics in the
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number three, and gradually learns more and more of the gracious
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mensuration of the Craft until he is invited to study the geometrical
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Forty-Seventh Problem of Euclid.
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The Forty-Seventh Problem of Euclid is older than Pythagoras. The
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Sublime degree of Master Mason as we know it is younger than
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Pythagoras by many hundreds of years. Our Rituals are accurate in
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neither date nor fact; and yet of all the symbols of Freemasonry the
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Forty-Seventh Problem is one of the most beautiful and most filled
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with meaning.
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For the benefit of those who may have forgotten their geometry days,
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the Forty-Seventh Problem is here simply stated; in any right
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triangle, the sum of the squares of the two sides is equal to the
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square of the hypotenuse. This is demonstrably true regardless of
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the length of either side. But in the Problem as diagrammed in the
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lodge, and for simplicity's sake it is usually shown with sides the
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proportions of which are as three, and four units when the
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hypotenuse, or longest side of the triangle will be as five units.
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If one draws on paper a line three inches long, and at right angles
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to it , and joined to one end, a line four inches long, then the line
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connecting the two ends will be five inches long when the angle is a
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perfect right angle, or one of ninety degrees.
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The square of 3 is 9. The square of 4 is 16. The sum of 9 and 16 is
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25. The square root of 25 is 5.
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We are taught but little about this Problem in our Rituals, and, as
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stated, much of what we are taught is wrong! We are instructed that
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it was invented by Pythagoras, that he was a Master Mason, that he
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was so delighted with his invention that he exclaimed "Eureka" (I
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have found it), that he sacrificed a heca-tomb, and the Problem
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"Teaches Masons to be general lovers of the arts and sciences."
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Why so great and awe-inspiring a symbol should receive such scant
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attention is not our problem. Perhaps it is because the fathers of
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the ritual thought it beyond the grasp of many and so better left for
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the individual to follow if he would. Certain it is that he who will
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think on this problem will find a rich reward.
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How came this wonder to be? What is the magic of 3 and 4 and 5? (or
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6 and 8 and 10, or 36 and 64 and 100, or any other set of numbers of
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the same relationship)? Why is the sum of the squares of the two
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lesser always equal to the square of the greater? What is the
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mystery which always works out so that, no matter what the length of
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any two sides, so be it they are at right angles, the line joining
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their free ends will have a square equal to the sum of the other two
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squares? If one line be 7.6954 inches long, and the other 19 miles,
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573,5732 feet long, the sum of the squares of these numbers will be
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the square of the length of the line joining their free ends, if, and
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only if, the two lines are at right, or ninety degree, angles.
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With this certainty, man reaches out into space and measures the
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distance of the stars! With this knowledge he surveys his land,
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marks off his boundaries, constructs his railroads and builds his
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cathedrals. When he digs a tunnel through a mountain, it is the
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Forty-Seventh Problem of Euclid by which he measures so that the two
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parties digging toward each other meet in the center of the mountain,
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having dug a straight tunnel. With this knowledge man navigates the
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ocean, and goes serenely and with perfect confidence upon a way he
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cannot see, to a port he does not know; more, with this problem he
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locates himself in the middle of the ocean so that he knows just how
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far he has come and whither he goes!
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If we put down the squares of the first four numbers; thus, 1, 4, 9,
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16; we can see that by subtracting each square from the next one we
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get 3, 5, and 7; which are the steps in Masonry, the steps in the
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Winding Stair, the brethren which form Entered Apprentice,
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Fellowcraft and Master Mason Lodges, which are, in other words, the
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sacred numbers.
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These have been the sacred numbers from the dawn of history.
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Always they have held meanings for those who attached a significance
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of spiritual import to mathematics. Always they have been symbols of
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the interrelation of science, knowledge, exploration, building; and
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God, religion, worship and morality.
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Many will find presumption in any attempt to read a symbol which so
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great an authority as Albert Pike said had an unknown meaning (page
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789, Morals and Dogma). Yet, if none presumes, from whence can
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individual progress come? The same authority declared it the
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inalienable privilege of any Mason to interpret the symbols of
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Masonry for himself. Therefore, a reading is here dared!
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So far as we know . . . and while we cannot prove it by mathematics,
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the strongest of circumstantial evidence leads us to believe . . .
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the fundamentals of mathematics are true, not only in this world, but
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in all worlds. Our finite minds cannot think of a world or a
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universe in which two and two make other than four, or in which the
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relation of the circumference of a circle to its diameter is other
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than 3.1416 plus. It is axiomatic to us that if the sum of the
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squares of the two sides of a right angled triangle are equal to the
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square of the hypotenuse is a truth here, it is a truth everywhere.
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This particular mathematical truth is so perfect, so beautiful, so
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inevitable and so fitting to the art an science of Freemasonry, the
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founders of our beloved Order must have chosen it from many others as
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a symbol of the universality of law, and therefore of the Law Maker.
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The Forty-Seventh Problem of Euclid not only teaches us to be general
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lovers of the arts and sciences, but to bow heads in reverence at the
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perfection and the beauty, the universality and the infinite
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extension of the laws of the Great Law Giver.
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