textfiles/magazines/STB/stb-1930-10.txt

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

THE 47th 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 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!