SHORT TALK BULLETIN - Vol.VIII October, 1930 No.10

## THE 47^{th} PROBLEM

by: Unknown

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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} problem of Euclid.

In Pythagoras' day (582 B.C.) of course the "47^{th} problem" was not called that. It remained for Euclid, of Alexandria,
several hundred years later, to write his books of Geometry, of which the 47^{th} and 48^{th} 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 47^{th} problem of Euclid, the lodge is above the common run in educational
standards!

And yet the 47^{th} 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 47^{th} problem - and its correlative 48^{th} - as follows:

"47^{th} - In every right angle triangle
the square of the hypotenuse is equal to the sum of the squares of the other
two sides." "48^{th} - 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} problem for his results.
The navigator traveling the trackless seas uses the 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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 47^{th} 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!