Short Talk Bulletin Vol 08 No 10

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Short Talk Bulletin

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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!