Containing more real food for thought, and impressing
on the receptive mind a greater truth than any other of the emblems in
the lecture of the Sublime Degree, the 47th problem of Euclid generally
gets less attention, and certainly less than all the rest. Just why this
grand exception should receive so little explanation in our lecture;
just how it has happened, that, although the Fellowcraft's degree makes
so much of Geometry, Geometry's right hand should be so cavalierly
treated, is not for the present inquiry to settle. We all know that the
single paragraph of our lecture devoted to Pythagoras and his work is
passed over with no more emphasis than that given to the Bee Hive of the
Book of Constitutions. More's the pity; you may ask many a Mason to
explain the 47th problem, or even the meaning of the word "hecatomb,"
and receive only an evasive answer, or a frank "I don't know - why don't
you ask the Deputy?" The Masonic legend of Euclid is very old - just how
old we do not know, but it long antedates our present Master Mason's
Degree. The paragraph relating to Pythagoras in our lecture we take
wholly from Thomas Smith Webb, whose first Monitor appeared at the close
of the eighteenth century.

It is repeated here to refresh the memory of those
many brethren who usually leave before the lecture:

"The 47th problem of Euclid was an invention of our
ancient friend and brother, the great Pythagoras, who, in his travels
through Asia, Africa and Europe was initiated into several orders of
Priesthood, and was also Raised to the Sublime Degree of Master Mason.
This wise philosopher enriched his mind abundantly in a general
knowledge of things, and more especially in Geometry. 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." Some of facts here stated
are historically true; those which are only fanciful at least bear out
the symbolism of the conception. In the sense that Pythagoras was a
learned man, a leader, a teacher, a founder of a school, a wise man who
saw God in nature and in number; and he was a "friend and brother." That
he was "initiated into several orders of Priesthood" is a matter of
history. That he was "Raised to the Sublime Degree of Master Mason" is
of course poetic license and an impossibility, as the "Sublime Degree"
as we know it is only a few hundred years old - not more than three at
the very outside. Pythagoras is known to have traveled, but the
probabilities are that his wanderings were confined to the countries
bordering the Mediterranean. He did go to Egypt, but it is at least
problematical that he got much further into Asia than Asia Minor. He did
indeed "enrich his mind abundantly" in many matters, and particularly in
mathematics. That he was the first to "erect" the 47th problem is
possible, but not proved; at least he worked with it so much that it is
sometimes called "The Pythagorean problem." If he did discover it he
might have exclaimed "Eureka" but the he sacrificed a hecatomb - a
hundred head of cattle - is entirely out of character, since the
Pythagoreans were vegetarians and reverenced all animal life.

Pythagoras was probably born on the island of Samos,
and from contemporary Grecian accounts was a studious lad whose manhood
was spent in the emphasis of mind as opposed to the body, although he
was trained as an athlete. He was antipathetic to the licentiousness of
the aristocratic life of his time and he and his followers were
persecuted by those who did not understand them. Aristotle wrote of him:
"The Pythagoreans first applied themselves to mathematics, a science
which they improved; and penetrated with it, they fancied that the
principles of mathematics were the principles of all things."

It was written by Eudemus that: "Pythagoreans changed
geometry into the form of a liberal science, regarding its principles in
a purely abstract manner and investigated its theorems from the
immaterial and intellectual point of view," a statement which rings with
familiar music in the ears of Masons.

Diogenes said "It was Pythagoras who carried Geometry
to perfection," also "He discovered the numerical relations of the
musical scale." Proclus states: "The word Mathematics originated with
the Pythagoreans!"

The sacrifice of the hecatomb apparently rests on a
statement of Plutarch, who probably took it from Apollodorus, that
"Pythagoras sacrificed an ox on finding a geometrical diagram." As the
Pythagoreans originated the doctrine of Metempsychosis which predicates
that all souls live first in animals and then in man - the same doctrine
of reincarnation held so generally in the East from whence Pythagoras
might have heard it - the philosopher and his followers were vegetarians
and reverenced all animal life, so the "sacrifice" is probably mythical.
Certainly there is nothing in contemporary accounts of Pythagoras to
lead us to think that he was either sufficiently wealthy, or silly
enough to slaughter a hundred valuable cattle to express his delight at
learning to prove what was later to be the 47th problem of Euclid.

In Pythagoras' day (582 B.C.) of course the "47th
problem" was not called that. It remained for Euclid, of Alexandria,
several hundred years later, to write his books of Geometry, of which
the 47th and 48th problems form the end of the first book. It is
generally conceded either that Pythagoras did indeed discover the
Pythagorean problem, or that it was known prior to his time, and used by
him; and that Euclid, recording in writing the science of Geometry as it
was known then, merely availed himself of the mathematical knowledge of
his era.

It is probably the most extraordinary of all
scientific matters that the books of Euclid, written three hundred years
or more before the Christian era, should still be used in schools. While
a hundred different geometries have been invented or discovered since
his day, Euclid's "Elements" are still the foundation of that science
which is the first step beyond the common mathematics of every day. In
spite of the emphasis placed upon geometry in our Fellowcrafts degree
our insistence that it is of a divine and moral nature, and that by its
study we are enabled not only to prove the wonderful properties of
nature but to demonstrate the more important truths of morality, it is
common knowledge that most men know nothing of the science which they
studied - and most despised - in their school days. If one man in ten in
any lodge can demonstrate the 47th problem of Euclid, the lodge is above
the common run in educational standards!

And yet the 47th problem is at the root not only of
geometry, but of most applied mathematics; certainly, of all which are
essential in engineering, in astronomy, in surveying, and in that wide
expanse of problems concerned with finding one unknown from two known
factors. At the close of the first book Euclid states the 47th problem -
and its correlative 48th - as follows:

"47th - In every right angle triangle the square of
the hypotenuse is equal to the sum of the squares of the other two
sides." "48th - If the square described of one of the sides of a
triangle be equal to the squares described of the other two sides, then
the angle contained by these two is a right angle."

This sounds more complicated than it is. Of all
people, Masons should know what a square is! As our ritual teaches us, a
square is a right angle or the fourth part of a circle, or an angle of
ninety degrees. For the benefit of those who have forgotten their school
days, the "hypotenuse" is the line which makes a right angle (a square)
into a triangle, by connecting the ends of the two lines which from the
right angle.

For illustrative purposes let us consider that the
familiar Masonic square has one arm six inches long and one arm eight
inches long. If a square be erected on the six inch arm, that square
will contain square inches to the number of six times six, or thirty-six
square inches. The square erected on the eight inch arm will contain
square inches to the number of eight times eight, or sixty-four square
inches.

The sum of sixty-four and thirty-six square inches is
one hundred square inches.

According to the 47th problem the square which can be
erected upon the hypotenuse, or line adjoining the six and eight inch
arms of the square should contain one hundred square inches. The only
square which can contain one hundred square inches has ten inch sides,
since ten, and no other number, is the square root of one hundred. This
is provable mathematically, but it is also demonstrable with an actual
square. The curious only need lay off a line six inches long, at right
angles to a line eight inches long; connect the free ends by a line (the
Hypotenuse) and measure the length of that line to be convinced - it is,
indeed, ten inches long.

This simple matter then, is the famous 47th problem.
But while it is simple in conception it is complicated with innumerable
ramifications in use.

It is the root of all geometry. It is behind the
discovery of every unknown from two known factors. It is the very
cornerstone of mathematics.

The engineer who tunnels from either side through a
mountain uses it to get his 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 time 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 predicated, tides are specified as to height and time of
occurrence, land is surveyed, roads run, shafts dug, and bridges built
because of the 47th problem of Euclid - probably discovered by
Pythagoras - shows the way.

It is difficult to show "why" it is true; easy to
demonstrate that it is true. If you ask why the reason for its truth is
difficult to demonstrate, let us reduce the search for "why" to a
fundamental and ask "why" is two added to two always four, and never
five or three?" We answer "because we call the product of two added to
two by the name of four." If we express the conception of "fourness" by
some other name, then two plus two would be that other name. But the
truth would be the same, regardless of the name. So it is with the 47th
problem of Euclid. The sum of the squares of the sides of any right
angled triangle - no matter what their dimensions - always exactly
equals the square of the line connecting their ends (the hypotenuse).
One line may be a few 10's of an inch long - the other several miles
long; the problem invariably works out, both by actual measurement upon
the earth, and by mathematical demonstration.

It is impossible for us to conceive of a place in the
universe where two added to two produces five, and not four (in our
language). We cannot conceive of a world, no matter how far distant
among the stars, where the 47th problem is not true. For "true" means
absolute - not dependent upon time, or space, or place, or world or even
universe. Truth, we are taught, is a divine attribute and as such is
coincident with Divinity, omnipresent.

It is in this sense that the 47th problem "teaches
Masons to be general lovers of the art and sciences." The universality
of this strange and important mathematical principle must impress the
thoughtful with the immutability of the laws of nature. The third of the
movable jewels of the entered Apprentice Degree reminds us that "so
should we, both operative and speculative, endeavor to erect our
spiritual building (house) in accordance with the rules laid down by the
Supreme Architect of the Universe, in the great books of nature and
revelation, which are our spiritual, moral and Masonic Trestleboard."

Greatest among "the rules laid down by the Supreme
Architect of the Universe," in His great book of nature, is this of the
47th problem; this rule that, given a right angle triangle, we may find
the length of any side if we know the other two; or, given the squares
of all three, we may learn whether the angle is a "Right" angle, or not.
With the 47th problem man reaches out into the universe and produces the
science of astronomy. With it he measures the most infinite of
distances. With it he describes the whole framework and handiwork of
nature. With it he calcu-lates the orbits and the positions of those
"numberless worlds about us." With it he reduces the chaos of ignorance
to the law and order of intelligent appreciation of the cosmos. With it
he instructs his fellow-Masons that "God is always geometrizing" and
that the "great book of Nature" is to be read through a square.

Considered thus, the "invention of our ancient friend
and brother, the great Pythagoras," becomes one of the most impressive,
as it is one of the most important, of the emblems of all Freemasonry,
since to the initiate it is a symbol of the power, the wisdom and the
goodness of the Great Articifer of the Universe. It is the plainer for
its mystery - the more mysterious because it is so easy to comprehend.

Not for nothing does the Fellowcraft's degree beg our
attention to the study of the seven liberal arts and sciences,
especially the science of geometry, or Masonry. Here, in the Third
Degree, is the very heart of Geometry, and a close and vital connection
between it and the greatest of all Freemasonry's teachings - the
knowledge of the "All-Seeing Eye."

He that hath ears to hear - let him hear - and he
that hath eyes to see - let him look! When he has both listened and
looked, and understood the truth behind the 47th problem he will see a
new meaning to the reception of a Fellowcraft, understand better that a
square teaches morality and comprehend why the "angle of 90 degrees, or
the fourth part of a circle" is dedicated to the Master!