Portrait of Euclid
according to an
old European etching.
The problem above is the 47th Problem of Euclid.
It is an invention by an ancient Greek geometer,
Pythagoras, who worked for many years to devise
a method of finding the length of the
hypothenuse of a right angle triangle.
Pythagoras is credited with having first proved
the rule successfully applied to the problem.
The rule is that the square of the base added to
the square of the altitude equals the square of
the hypothenuse. The base of a right angle
triangle is the side on which it rests, marked B
in Figure 11. The altitude is the height and is
marked A. The hypothenuse is the connecting side
of the triangle, marked C above. The base, 6,
squared or multiplied by itself, equals 36. The
altitude, 8, squared, equals 64. By adding these
together we have 100, which is the square of the
hypothenuse. It remains but to extract the
square root of 100, which we know is 10,
therefore 10 is the length of the hypothenuse or
third side of this right angle triangle. All
right angle triangles can be figured in the same
manner, but only multiples of the length of the
three sides come even -- such as 3, 4, 5 and 12,
16, 20, as shown in Figure 12; and many others,
of course.
1738 German Drawing depicting workmen using the
47th problem
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The 47th Problem of
Euclid
By H. Meij
Master, Harmony Lodge #18, August 2000
Introduction:
There are so many symbols, words and phrases in Freemasonry
used both as part of its rituals as well as part of its
vocabulary, that studying and trying to understand the
“meanings” them is a very worthwhile and a rewarding
journey. One such prominent symbol and phrase, is the 47th
problem of Euclid, which is one of the main symbols
introduced in the Third Degree.
In the Blue Lodge, it is considered a great honor to be
elected and serve as the Master of a lodge. It shows that
the individual has attained maturity, leadership, charisma,
and motivation, but above all, knowledge to be able to lead
others. It is an initiation by itself, as the position
brings with it a completely new set of responsibilities that
are often not appreciated when accepting the position. At
the end of serving as Master of a Lodge, many Past Masters
are presented with a jewel, symbolizing the great
appreciation of the Lodge towards their dedication. It also
symbolizes something else ? that is that the individual has
completed his “journey”, through the different positions of
the Lodge, to a new plain.
The 47th problem of Euclid features prominently in many
Past Master’s jewels. Selecting this symbol out of the
thousands used in Freemasonry to represent one of its
highest honors must mean that it is a very significant and
central symbol of the Craft. Yet, sadly many Freemasons,
even many Past Masters, do not know why it is so centrally
featured in the Past Master’s jewel. Clearly the 47th
problem is based on Geometry, and all Freemasons know that
Geometry and Freemasonry are closely linked, but is this the
only reason? What does the 47th problem of Euclid symbolize
and mean? Of the 465 problems published by Euclid, why is
the 47th so important? Why do Freemasons cherish it so?
An example of a Past Master’s Jewel featuring
the 47th Problem of Euclid from England. Past
Master’s jewel used in the USA often do not show the
47th Problem anymore) |
The Foundation of
Freemasonry?:
The 47th problem of Euclid is often mentioned in
Masonic publications. In Anderson's “Constitutions”
published in 1723, it mentions 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. It is also mentioned in the Third
Degree lecture, where we are taught that the “47TH
problem of Euclid…….taught us to be general lovers
of the arts and sciences”. |
However, it is quite different to be referred to as the
“Foundation of all Masonry, sacred, civil and military” that
to be referred to as “taught us to be general lovers of the
arts and sciences”. Has the importance of the symbolism of
the 47th problem declined over time for some reason?
In order to understand whether the symbol has declined in
importance or not, we first need to look at the 47th problem
of Euclid itself.
The Discovery of the 47th problem of Euclid:
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, which is also known as “The Pythagorean
Theorem”. Why is it called by both these names? Although
Euclid published the proposition, it was Pythagoras who
discovered it. We learn from the third degree lecture
that:
“ This wise philosopher (Pythagoras) enriched his
mind abundantly in a general knowledge of things, and
more especially in Geometry, or Masonry. On this subject
he drew out many problems and theorems, and, among the
most distinguished, he erected this, when, in the joy of
his heart, he exclaimed Eureka, in the Greek language
signifying, "I have found it," and upon the discovery of
which he is said to have sacrificed a hecatomb. It
teaches Masons to be general lovers of the arts and
sciences”.
Actually, it was not Pythagoras who directly discovered
the rule, as the Egyptians used the same principle for a
very long time before Pythagoras, whereby they
re-measured their fields after the annual flooding of
the Nile washed out their boundary markers. Hence,
Pythagoras is probably here referred to as being the one
who proved that the process works.
History records that Pythagoras established a society
with philosophical, religious and political aims.
Shrouded in secrecy, they believed that only by truly
understanding the universe could one achieve salvation
of the soul, and as Divinity created all things,
studying it over a period of several lifetimes, could
bring one closer to Divinity itself. As such, it was
believed that through study and reason could one start
to understand Divinity. Clearly, reason is based on
measurable things (such as through numbers and objects),
and is easier to understand if expressed in that matter.
Hence the society devoted much of its time to the
mathematics, including Geometry. This line of thinking
was incorporated in Freemasonry, which sets it opposite
to the Church, which emphasizes faith over reason.
Indeed, Pope Pius IX, in his encyclical, Qui Pluribus,
dated 9 November 1846, attacked those who “put human
reason above faith, and who believe in human progress.”
Many people consider this to be a reference to
Freemasonry. This is interesting, because in the Book
of Wisdom 11:20 we read:
“Even apart from these, men could
fall at a single breath when pursued by justice
and scattered by the breath of thy power.
But thou hast arranged all things by measure
and number and weight.”
So the very “measurement of things” the
Church objects to is mentioned in Scripture.
However, let us get back to how the 47th problem fits in
Freemasonry.
What Does the 47th say?:
The proposition states that: “In right angled triangles
the square on the side subtending the right angle is
equal to the squares on the sides containing the right
angle.”
What? In other words A2 + B2 = C2.
Many readers will 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 the 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.
The proposition is especially important
in architecture. Builders have since ancient times used
the 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.
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.
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 47th problem forms the basis of
all ancient measurement units:
The 47th problem of Euclid formed the basis of a common
set of measurements used by the Egyptians, especially in
the building of the Great Pyramids. It gets a little
technical, but a simple illustration will help us
understand it better.
Please see the illustration above, which is not accurate
due to a drawing, but will serve to illustrate the
point. If we take a circle and draw in it a triangle
(triangle A- B-C) which perpendicular is 300, base is
400, and by the 47th problem, the hypotenuse becomes 500
(any combination such as 3,4,5 will also work ? higher
numbers are used for ease of explanation). Then if we
draw a line from the angle of the perpendicular and the
base through the hypotenuse to the circle, this line
will be equal to 480.
The resulting two parts of the hypotenuse (A-D and D-C)
will be equal to 180 and 320 respectively. Then if we
draw another line from the point D (the intersector of
the hypotenuse) to the perpendicular of the shortest
side of the triangle (A-B), then line A-E will equal 108
and line D-E will equal 144.
Now we have all the measurements of the ancient world,
that is 500, 480, 400, 320, 180, 144 and 108. Why is
this important? If we take each unit to be a cubit (an
ancient form of measurement), then 500 is the base of
the Great Pyramid of Memphis. 400 cubits is the length
of an Egyptian stadium (stadium is plural for stadia,
and ancient measurement unit, based on a particular
number of steps, also called a Khet by the Egyptians).
480 cubits is the length of the Ptolemy stadium, 320
cubits is the length of the Hebrew and Babylonian
stadium. Furthermore, 180, which represents the smaller
part of the hypotenuse, doubled gives 360 cubits, the
Cleomedes stadium. By doubling 144 cubits gives 288
cubits, the Archimedes stadium. Finally by doubling 108
cubits we obtain 216 cubits, or the lesser Egyptian
stadium.
In other words, this simple exercise formed the basis of
all the lengths used by the Egyptians, and hence also
once again indicates that its principle was well
understood by the Egyptians, and hence taught by them to
Pythagoras.
Conclusion:
Clearly, the 47th problem helps us look at the universe,
and all that is in it, through a system that we can
understand clearly, for it is measurable. The Master’s
jewel is the square, the base needed for the 47th
problem (in many jurisdictions the square has the
dimensions of 3:4 ? the Pythagorean dimensions). As the
Master serves his position, he becomes more complete,
and therefore the 47th problem of Euclid is dedicated on
his jewel when he leaves office.
References:
Circumambulation and Euclid’s 47th proposition, by Reid
McInvale
Encyclopedia of Freemasonry, by Albert Mackey
Freemasonry, A journey through ritual and symbol, by W.
Kirk
Master Mason, by Carl Claudy
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