by: Unknown<\/em><\/p>\n\n\n\n
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<\/figure>\n\n\n\nContaining 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\u2019s degree makes so much of Geometry, Geometry\u2019s 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\u2019s the pity; you may ask many a Mason to explain the 47th problem, or even the meaning of the word \u201checatomb,\u201d and receive only an evasive answer, or a frank \u201cI don\u2019t know – why don\u2019t you ask the Deputy?\u201d The Masonic legend of Euclid is very old – just how old we do not know, but it long antedates our present Master Mason\u2019s 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.<\/p>\n\n\n\n
It is repeated here to refresh the memory of those many brethren who usually leave before the lecture:<\/p>\n\n\n\n
\u201cThe 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 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 \u201cI have found it,\u201d 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.\u201d Some of the 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 \u201cfriend and brother.\u201d That he was \u201cinitiated into several orders of Priesthood\u201d is a matter of history. That he was \u201cRaised to the Sublime Degree of Master Mason\u201d is, of course, poetic license and an impossibility, as the \u201cSublime Degree\u201d 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 problematic that he got much further into Asia than Asia Minor. He did indeed \u201cenrich his mind abundantly\u201d in many matters, and particularly in mathematics. That he was the first to \u201cerect\u201d the 47th problem is possible, but not proved; at least he worked with it so much that it is sometimes called \u201cThe Pythagorean problem.\u201d If he did discover it he might have exclaimed \u201cEureka\u201d but he sacrificed a hecatomb – a hundred head of cattle – is entirely out of character since the Pythagoreans were vegetarians and reverenced all animal life.<\/p>\n\n\n\n
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: \u201cThe 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.\u201d<\/p>\n\n\n\n
It was written by Eudemus that: \u201cPythagoreans 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,\u201d a statement which rings with familiar music in the ears of Masons.<\/p>\n\n\n\n
Diogenes said \u201cIt was Pythagoras who carried Geometry to perfection,\u201d also \u201cHe discovered the numerical relations of the musical scale.\u201d Proclus states: \u201cThe word Mathematics originated with the Pythagoreans!\u201d<\/p>\n\n\n\n
The sacrifice of the hecatomb apparently rests on a statement of Plutarch, who probably took it from Apollodorus, that \u201cPythagoras sacrificed an ox on finding a geometrical diagram.\u201d 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 \u201csacrifice\u201d 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.<\/p>\n\n\n\n
In Pythagoras\u2019 day (582 B.C.) of course the \u201c47th problem\u201d 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.<\/p>\n\n\n\n
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\u2019s \u201cElements\u201d 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!<\/p>\n\n\n\n
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:<\/p>\n\n\n\n
\u201c47th – In every right angle triangle the square of the hypotenuse is equal to the sum of the squares of the other two sides.\u201d \u201c48th – 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.\u201d<\/p>\n\n\n\n
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 \u201chypotenuse\u201d is the line which makes a right angle (a square) into a triangle, by connecting the ends of the two lines from the right angle.<\/p>\n\n\n\n
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 is 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.<\/p>\n\n\n\n
The sum of sixty-four and thirty-six square inches is one hundred square inches.<\/p>\n\n\n\n
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 to 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.<\/p>\n\n\n\n
This simple matter then is the famous 47th problem. But while it is simple in conception it is complicated with innumerable ramifications in use.<\/p>\n\n\n\n
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.<\/p>\n\n\n\n
The engineer who tunnels from either side through a mountain uses it to get his two shafts to meet in the center.<\/p>\n\n\n\n
The surveyor who wants to know how high a mountain may be ascertains the answer through the 47th problem.<\/p>\n\n\n\n
The astronomer who calculates the distance of the sun, the moon, the planets and who fixes \u201cthe duration of time and seasons, years and cycles,\u201d 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 predicted, tides are specified as to height and time of occurrence, the land is surveyed, roads run, shafts dug, and bridges built because of the 47th problem of Euclid – probably discovered by Pythagoras – shows the way.<\/p>\n\n\n\n
It is difficult to show \u201cwhy\u201d 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 \u201cwhy\u201d to a fundamental and ask \u201cwhy\u201d is two added to two always four, and never five or three?\u201d We answer \u201cbecause we call the product of two added to two by the name of four.\u201d If we express the conception of \u201cfourness\u201d 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\u2019s 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.<\/p>\n\n\n\n
It is impossible for us to conceive of a place in the universe where two added to two produce 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 \u201ctrue\u201d 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.<\/p>\n\n\n\n
It is in this sense that the 47th problem \u201cteaches Masons to be general lovers of the art and sciences.\u201d 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 \u201cso 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.\u201d<\/p>\n\n\n\n
Greatest among \u201cthe rules laid down by the Supreme Architect of the Universe,\u201d 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 \u201cRight\u201d angle, or not. With the 47th problem, the 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 calculates the orbits and the positions of those \u201cnumberless worlds about us.\u201d 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 \u201cGod is always geometrizing\u201d and that the \u201cgreat book of Nature\u201d is to be read through a square.<\/p>\n\n\n\n
Considered thus, the \u201cinvention of our ancient friend and brother, the great Pythagoras,\u201d 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 Artificer of the Universe. It is the plainer for its mystery – the more mysterious because it is so easy to comprehend.<\/p>\n\n\n\n
Not for nothing does the Fellowcraft\u2019s 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\u2019s teachings – the knowledge of the \u201cAll-Seeing Eye.\u201d<\/p>\n\n\n\n
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 \u201cangle of 90 degrees, or the fourth part of a circle\u201d is dedicated to the Master!<\/p>\n","protected":false},"excerpt":{"rendered":"
SHORT TALK BULLETIN – Vol.VIII October, 1930 No.10 THE 47th PROBLEMby: 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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"ngg_post_thumbnail":0,"footnotes":""},"categories":[6],"tags":[],"class_list":["post-792","post","type-post","status-publish","format-standard","hentry","category-masonic-education"],"featured_image_src":null,"author_info":{"display_name":"Don Goss","author_link":"https:\/\/jl1.org\/lodge\/?author=1"},"_links":{"self":[{"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/posts\/792","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=792"}],"version-history":[{"count":1,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/posts\/792\/revisions"}],"predecessor-version":[{"id":794,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=\/wp\/v2\/posts\/792\/revisions\/794"}],"wp:attachment":[{"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=792"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=792"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/jl1.org\/lodge\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=792"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}