Masonic Stuff - Mt. Moriah #28 Home Page

 “The Forty-Seventh Problem of Euclid is a geometrical
theorem attributed to Pythagoras, an eminent Greek
philosopher.
This wise man enriched his mind by the acquisition of a
knowledge of the sciences, and especially of Geometry. In
this, he drew out many problems and theorems, and
among the number was this, on the discovery of which, in
the joy of his heart, he exclaimed, “Eureka!” which
signifies, “I have found it.”
As a Masonic emblem, its contemplation is calculated to
induce a study of Geometry and the liberal arts and
sciences.”
 The 47th Problem of Euclid is indeed enigmatic
 While it is ostensibly a proof of a key principle of
Geometry, its esoteric characteristics, not its
mathematical properties are the source of its
Masonic significance
 Unraveling the complex r1ddle of the 47th Problem
and understanding why it is regarded as a central
tenet of Freemasonry properly begins with study of
its history and its mathematical application.
 The Ritual during which the 47th Problem of
Euclid is introduced, briefly addresses these
issues; however having touched fleetingly upon
the fundamentals, the Ritual goes no further and it
is left to the Candidate to undertake further
exploration (or not).
 The puzzling brevity with which the 47th Problem is
discussed, given the accompanying emphasis placed
upon its importance to the Craft, seems almost to be
an invitation for the intellectually curious to explore
further.
 Most Candidates however seem to assume that their
acquaintance with the Pythagorean formula gained
during their early years of formal education provides
them with a more than adequate knowledge of the 47th
Problem of Euclid, and that they have already
satisfactorily mastered the concept.
 To progress beyond fundamental concepts and arrive
at the door of understanding one must study the
history and mathematics of the 47th Problem of Euclid
 The TRUE Masonic importance
 Lies not in it’s mathematical utility (which is
considerable)
 But that the 47th Problem as set forth by Euclid is
intended to apply to a very specific case of right triangle
having sides with lengths of specific proportions of 3:4:5
 The figure upon which
the original 47th Problem
of Euclid is based.
 Figure is also called “The
Bride’s Chair”
 The outline of this figure
is used symbolically in
Freemasonry
 The actual formula a2 + b2 = c2 for which
the 47th Problem of Euclid is a
mathematical proof actually predates
Euclid (circa 300 BC) by about 280 years.
 Pythagoras of Samos (circa 580 BC) is
generally credited with its development.
 There is archeological evidence that the
Babylonians (1900-1600 BC) were familiar
with the formula 1200 to 1400 years before
Pythagoras.
 Babylonian clay tablet
(#322 in the G.A. Plimpton
Collection at Columbia
University)
 Believed to have been
written about 1800 BC
 Lists what are now called
Pythagorean Triples (i.e.
integers a, b, c satisfying
a2 + b2 = c2)
Circa 300 BC
Circa 580 BC
 Euclid wrote a set of thirteen books, which were called
“Elements”. Each book contained many geometric
propositions and explanations, and in total Euclid
published 465 problems.
 The 47th problem was set out in Book 1
 The formula is known as “The Pythagorean Theorem”.
 Why is it called by both these names? Although
Euclid published the proposition, it was Pythagoras
who discovered it.
 Several hundred detailed geometric proofs for the
Pythagorean Theorem exist.
 Including a famous one developed by Brother James
Garfield, five years before he became the 20th
President of the United States.
 What is the 3-4-5 triangle
 Why the 3-4-5 triangle is so important to Masons
 A triangle with sides of 3, 4, and 5 results in a perfect
90 degree angle between two sides.
 Also known as "the Egyptian string trick."
 The ancient Egyptians used the string trick to create
right angles when re-measuring their fields after the
annual Nile floods washed out boundary markers.
 Their skill with this and other surveying methods led
to the widely held (but false) belief that the Egyptians
invented geometry (geo=earth, metry=measuring).
 Thales the Greek supposedly picked the string trick up
while traveling in Egypt and took it back to Greece.
 Some say that the Greek mathematician and geometer
Pythagoras, described in Masonic lectures as "our
worthy brother," also went to Egypt and learned it
there on his own.
 In any case, it was he who supplied the PROOF that
the angle formed by the 3 : 4 : 5 triangle is invariably
square and perfect. It is also said that he actually
sacrificed a hecatomb, that is a sacrifice of one
hundred bulls, which ranked as the highest kind of
religious offering, upon completing the proof.
 Engineers who tunnel from both sides through a
mountain use the 47th problem 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
problem.
 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 problem for his results.
 The navigator traveling the trackless seas uses the 47th
problem 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 problem to
show the way.
 A basic survey challenge in the field is to lay out
a perfect right angle.
 Modern surveying instruments can easily do
this – the angle being “turned” as is said, by
readouts on a graduated scale.
 Using a 3:4:5 triangle it is possible to “turn” a
right angle easily without instruments.
 The knowledge contained in this proposition is at the
bottom of all systems of measurement.
 Every mechanic at the present day makes use of it
consciously or unconsciously, whether it be the
engineer surveying a property, or the homeowner
laying out his garden, or the carpenter calculating the
pitch of a roof.
 He may not know anything about geometry, but the
"rule of thumb" by which he works has been deduced
from this proposition.
 To the practical builder the knowledge is invaluable.
 If we will carry ourselves back in imagination to a time
when this knowledge was still unknown, we will realize
that its discovery was an event of great importance in
the history of architecture.
 An epoch-making event to be ranked with such
modern discoveries as those of the law of gravitation,
wireless TV or telephones, and space travel.
 How is this 47th problem “the foundation of all
Masonry” as stated by J. Anderson in 1723?
 Anderson’s Constitution is the basis of Modern
Freemasonry and its foundation of operation. A variety of
stories and irregularities exist about its legitimacy and its
foundation, yet, it remains an early cannon of Masonic
History. The Constitutions of the Free-Masons was a
constitution written for the Premier Grand Lodge of
England, to standardize the rituals and practices of
Freemasonry among lodges of London and Westminster
operating under that Grand Lodge. Obviously, it was not
meant to apply to other lodges in other parts of England,
Scotland and Ireland. The constitution laid the foundation
of the legend of Hiram Abiff, King Solomon’s Master
Builder, along with the pyramid style organizational model
of Freemasonry. The first and second edition were written
by Rev. James Anderson in 1723 and 1738.
 Based on the Old Masonic Manuscripts (also called “Gothic Constitutions”)
and on the General Regulations which had been compiled first by George
Payne in 1720. The full title of the 1723 edition was The Constitutions of the
Free-Masons, Containing the History, Charges, Regulations, etc. of that most
Ancient and Right Worshipful Fraternity, For the Use of the Lodges.
When in 1738, the Grand Lodge changed its name from Grand Lodge of
London and Westminster into the Grand Lodge of England, the Constitution
was rewritten by Anderson. The title of the second, rewritten, edition of 1738
was The New Book of Constitutions of the Antient and Honourable Fraternity
of Free and Accepted Masons, Containing Their History, Charges, Regulations,
etc. Collected and Digested By Order of the Grand Lodge from their old
Records, faithful Traditions and Lodge-Books, For the Use of the Lodges.
The 1723 edition of the Constitutions was edited and reprinted by Benjamin
Franklin in Philadelphia in 1734, becoming the first Masonic book printed in
America.
 Anderson writes that “The Greater Pythagoras,
provided the Author of the 47th Proposition of Euclid’s
first Book, which, if duly observed, is the Foundation
of all Masonry, sacred, civil, and military…”.
 Being mentioned in one of the first “official”
speculative Masonic publications clearly indicates that
the 47th problem of Euclid must be important.
 To the operative mason it affords a means of correcting
his square, for if he wishes to test its accuracy he may
readily do so by measuring off 3 divisions along one
side, 4 divisions along the other, and the distance
across must be 5 if the square is accurate.
 The knowledge of how to layout a square angle without
the possibility of error has always been accounted of
the highest importance in the art of building.
 In times when knowledge was limited to the few, it
might well be one of the genuine secrets of a Master
Mason.
 The ancient temple builders in the long centuries
before Christ were most punctilious in setting their
temples due east and west.
 So exacting were they on this point that there was
organized a set of men who, in modern phrase, would
be termed experts or specialists, and whose sole duty it
was to lay out the foundations of public edifices.
 They were called, in Egypt, harpedonaptae--meaning
rope stretchers. They first laid out the north and
south line by observation of the stars and the sun, and
their next step was to get the east and west line exactly
at right angles.
 This they secured by stretching a rope
north and south divided into three
parts in the proportion of 3, 4, and 5,
(the Egyptian string trick) fastening
down the center part by pegs.
 Then they swung around the loose
ends until they intersected and a right
angled triangle was thus formed.
 The question arises, have we anything in our present
ritual which might be relative in any way to this
method of proving the square or obtaining a right
angle without the possibility of error and which may
have been connected with the instruction given in
purely operative masonry.
 Another type of triangle used to demonstrate the 47th
Problem in Masonry is the 1:1:√2 form.
 The square and the cube which are 1 unit on each side are
of great symbolic meaning to Masons. Therefore, the
bisection of the square into a pair of 1:1:√2 triangles has
important Masonic connotations.
 It is in this form that the Pythagorean theorem is most
often visually encountered in Masonry, specifically in the
checkered floor and its tessellated border, as a geometric
proof on Lodge tracing boards, as the jewel of office for the
Worshipful Master, and in the form of some Masonic
aprons.
 To create a 1:1:√2 right triangle, also known as an isosceles
right triangle, you need a compass and a straight edge -familiar tools to the Craft, of course. On soft ground, use
the compass to inscribe a circle. Then use the straight edge
to bisect the circle through the center-point marked by the
compass. Mark the two points where the bisecting line
crosses the circle's circumference. Using the compass
again, erect a perpendicular line that bisects this diameterline and mark the point where the perpendicular touches
the circle. Now connect the three points you have marked - and there is your 1:1:√2 right triangle.
 To Freemasons, the first two points -- where you
marked the crossing of the bisecting diameter through
the circle's circumference -- can also be used to
construct two further perpendicular lines. These are
the two "boundary" lines of conduct sometimes
symbolized on Masonic tracing boards by the Two
Saints John and sometimes referred to as indicators of
the Summer and Winter Solstices, whereon the feast
days of those two saints occur.
 No wonder that Pythagoras sacrificed an hecatomb!
 No wonder that Anderson speaks of this proposition as
the foundation of all Masonry!
 The only wonder is that modern Freemasonry has lost
sight of the importance of this symbol.
 This is but one of many topics for Masonic edification
 Hopefully, this will encourage members of the lodge to
seek further light on a subject that interests or
perplexes them
 If you are unsure, see me for a list of topics!
 47th PROBLEM OF EUCLID • THE VEIL LIFTED, by Bro.
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William Steve Burkle KT, 32•, Scioto Lodge No. 6, Chillcothe,
Ohio, Philo Lodge No. 243 F. & A.M., South River, Grand Lodge
of New Jersey, internet
James Garfield's Proof of the Pythagorean Theorem, S. F.
Ellermeyer, College Trigonometry (MATH 1112), Kennesaw State
University, internet
Pythagorean Theorem, Wikipedia
47th Problem of Euclid, compiled by Wor. David J. Lettelier for
public oration and lecture, internet
47th Problem of Euclid, How to Square Your Square, internet
Why the 47th Problem of Euclid, by Theron Dunn, A beacon of
Masonic Light Blog, internet