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!