What is the Truth behind the 47th Problem of Euclid?

What is the Truth behind the 47th Problem of Euclid?

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 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 predicted; 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. [Image Credit: 47th Problem of Euclid by Peter Savant]

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 the handiwork of nature.  With it, he calculates 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 Artificer 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!


~  From, SHORT TALK BULLETIN – Vol.VIII, October 1930, No.10.

Freemasonry and Geometry: What are the Symbolic Meanings of the Most Basic Geometric Forms?

Freemasonry and Geometry: What are the Symbolic Meanings of the Most Basic Geometric Forms?

The more abstract a concept is, the more meaning that we can find in it. Is this because the meaning is all projected from within our minds, or because there is something objectively real there to be found? Plato or Pythagoras certainly believed the latter, as did Jung with his archetypes, and certainly the abstraction of mathematics has enabled much of our modern scientific grasp of the workings of the world around us.

In Freemasonry, we deal in symbols constantly. The Temple itself is a symbol, as are all the rituals which take place within, and there are layers of meaning which are only revealed as one progresses through the degrees. Geometry, in particular, is very significant to Freemasonry. This stems not only from the practical geometrical knowledge that was required to build grand cathedrals and other structures by operative masonry, but also from the Western wisdom teachings stemming from Pythagoras, Plato, and others, often referred to as Sacred Geometry, which sees a greater symbolic significance in geometrical forms.

While its common to find interpretations of more complex forms like the Flower of Life, or Metatron’s Cube, some of the geometric forms I find most fascinating are the simplest. This is because the simpler a form is, the more universal it is. In the simplest forms, I see the very foundations of all creation. So, what is the symbolic significance of a few of these simple geometric forms?

What follows are not in any sense “official” masonic interpretations of these forms, but merely one mason’s reflections on them. Books can and have been written to interpret these and other geometric forms, so here we will just be sampling a few.

The Point

pointThe point is the beginning of all forms, and all finite things. In that sense, it is the most abstract representation of finity, of a finite form. It does not even have definitions, specific boundaries, size, or dimensions, and therefore is the most abstract finite thing that can emerge from the formless infinity. It is also the beginning of duality, because in the point, you have one thing which is separated from everything else, the thing and its surrounding context, self and world. Yet, without a clear boundary, it still contains an element of infinity within it. The point can be viewed as infinitely small, or infinitely large. Without boundaries and other objects to be compared to, it has no reference.

In the point, we can see the fundamental essence of all finite entities, including each person: the Self, consciousness, existence.

The Line

HorizThe line is the first movement of the point, the first dimension of form. As such, it represents the most basic possible expression of time and motion. In the line, we can see the progression from moment to moment of what was first seen in the point. While the point is motionless and still, the line implies a trajectory, and a continuity. It is also the first impression of a space in which motion and change can take place. Yet, the line is only the simplest, most primal of forms with a boundary, and does not yet demonstrate a space in which objects may exist, although it begins to imply them.

In the line, we can see the fundamental essence of all motion and change, linearity, past progressing into future, and memory.

The Cross

sacred geometryThe cross is where two lines meet at a perfect right angle, and the simplest indication of the second dimension. In the cross, we can see the implied 2-dimensional plane stretching out in four directions. Also within the cross, we can see both the singular point, and the line doubled. Like every new form, it is the expansion of what came before it. The cross is where two intersect to become one; two planes, two people, two forces, and the point at which they meet is also their origin.

In the cross, we can see the fundamental essence of all duality, multiplicity, and complexity. It is the simplest possible mandala, and also represents the multiple dimensions of self, whether masculine/feminine, thinking/feeling, or ordered/chaotic, intersecting at the central point of consciousness or spirit.

The Circle

infinity circleThe circle is the perfect return of the trajectory of the point traveling in a line back to its origin. Simultaneously, it is an enlarged and expanded representation of the point, being equidistant in all directions from the center point. In the circle, the completeness and perfection of the process of the formless point expanding into form can be seen. Within the circle, we can fit any number of mandalic or sacred forms, such as the pantagram, the hexagram, the cross, or the equilateral triangle. It contains all of these in essence, and yet is beyond all of them.

In the circle, we can see the totality of form, the Ouroboros, the nearest form to infinity.

The Circumpunct

circumpunct freemasonryThe circumpunct is the completion of the journey from finite point to infinite circle. In the circumpunct, all of the dimensions of self which have progressed outward into form are in complete and total balance and alignment with their formless origin. In the mind, it is the personality in complete harmony with its essence and origin. It also represents the essential boundary of the self. While the point had a finite existence, it still had a certain infinite quality, in it’s lack of defined boundaries. In the circumpunct, we have both the boundary-less point of origin, and the boundary, with self on the inside, and world on the outside.

In the circumpunct, we can see the completion of all existence, the simultaneity of infinity and finite form, the perfection of self, world, and the relationship between the two.

Further Reflections

expanding dimensionsThe truth is that neither you nor I have ever really seen a point or a line, only things that are shaped like them, usually drawings on paper. But the truest sense in which a point and a line exist is in the abstract realm of ideas, yet they are clearly significant. We could not build anything without them, and this is true of mathematical abstractions, in general: they are ephemeral, yet essential. But can they be regarded as the true foundations of all creation, of reality itself?

What I am struck most by these simple forms is how each dimension, and the basic geometric form it represents, implies and transcends the next. The moment you create a point, it imples that that point could transcend the 0 dimension, and be a line, if only it moved beyond that 0 dimension. The moment you create a line, it implies that that line could move in a different direction, transcending that 1st dimensional existence, creating a plane. The moment you create a plane, it implies that that plane could be bent or folded, and that therefore a third dimension must exist. The moment you have a 3-dimensional object, it implies that that object can shift and change across time, thus implying a fourth dimension, transcendent from 3-dimensional space.

This seems to me to imply that the world in which we find ourselves is in fact a manifestation from the infinite 0 dimension, into the four dimensions where we currently find ourselves. Its almost as if a divine infinite mind, in similar manner to an architect, charted out and traced the fundamental principles of reality by which a creation could manifest.

But is that really where it ends? For us, the 4 dimensions are the extent of dimensionality we can see and understand, with even the 4th being somewhat mysterious to us. But what if this is merely because of our position along the spectrum of dimensions, and in actuality, this process of one dimension implying the next goes on infinitely, with the 4th implying and creating the 5th, the 5th the 6th, ad infinitum?

String theorists and various other kinds of physicists currently posit some number of dimensions, often 11 or higher, but what if there is no ceiling to the dimensional hierarchy extending into the unseen, unknown realms of possibility? In that case, we truly could be said to be dwelling in an infinite creation.

infinite dimensions

The Masonic Letter G stands for…?

The Masonic Letter G stands for…?

To what does the symbol allude? Doubtless there are many answers to this question. Depending on what country, what masonic group, or what Lodge you’ll get different answers. All are interesting, and some are actually a bit astonishing. It has been said to represent ideas such as God, Geometry, Generation, Gnosis, Great Architect, Gamma, Goodness, Gimel, Goat, and more.

When did the letter G first appear in Freemasonry? It is hard to say for sure. One theory is that the symbol could have been brought in by Rosicrucians and Qabalists who became Masons the last part of the 17th century.

Another theory is that it was introduced some time subsequent to 1717 by the members of the Grand Lodge of England. We are told in the early masonic lectures that G signifies “Geometry, the Root and Foundation of all Sciences.” 

By the beginning of the nineteenth century, the letter G, was said to have a symbolic meaning of God as synonymous with Geometry. It was sometimes displayed in the center of the Lodge and other times hung in the East. The G represented both “God” as the supreme being and “Geometry” which is imagined as a means of seeing the perfect ordering of the universe. Temple G

Over time, it became identified with many other things. Why? That is exactly the topic of a debate that has been raging for centuries. The Masonic letter G is one of those aspects of masonic history that seems to follow an unpredictable path.

Masonic Scholar Albert G. Mackey goes so far as to say he feels Masonic symbolism has been hurt rather than helped by the adoption of the letter G. He writes:

“It is to be regretted that the letter G. as a symbol, was ever admitted into the Masonic system. The use of it as an initial would necessarily confine it to the English language and to modern times. It wants therefore, as a symbol, the necessary characteristics of both universality and antiquity.”

Is Mackey correct? Does the letter G lack universality? Has it hurt Freemasonry? How Gimel or Camelshould it be dealt with?

G is for Gimel

An interesting justification for the symbol’s importance can be found in a ground- breaking book by Brother Paul Foster Case called the Masonic Letter G. I read this work years ago when I was studying qabalah. Using the Hebrew Gematria as a tool, he defends the G symbol as not only universal but honorable. One of the arguments he gives is that the letter G corresponds to the third letter in the Hebrew alphabet or Gimel. He gives two ways this Hebrew G could be acknowledged as universal:

  1. Hebrew letters are unique in that each one has a name that represents a familiar object. Objects are universally understood, unlike English letters.
  2. The Hebrew letter G or Gimel represents a camel. Camels, to ancient Hebrews, represented journeying to places far off, and the like. The camel symbolizes a mason’s travel in search of light and his quest to learn the hidden secrets of nature.

There is not enough space (or time) here to explain fully the argument which contains a load of Hebrew Gematria and interesting juggling with numbers but I recommend it if you like that sort of thing.

After his proof, Case remarks:

“Were nothing else to be said for it, it seems to us these facts would make the letter G a sufficiently universal, as well as sufficiently ancient, symbol of the Grand Architect.”

He explains in the various degree lessons of the craft that the idea of travel is significant.  By travel, the mason is able to trace nature through her various windings to her most final filosofia medievalconcealed recesses. Precisely the same thought is expressed in what many of the Masonic lectures tell us concerning God as He “Geometrizes.”

What does Geometry have to do with Freemasonry? How does God “Geometrize?”

God as the Geometrician

Geometry is taught to a Freemason, as he progresses in the science. As soon a one enters upon the world of geometry, symbolic and philosophical, the mind is opened to new influences that stimulate and refine it. 

From the standpoint of science, geometry and its offshoots are vital sciences of measurement. Often, nature conforms to simple patterns with symmetry and structure. For example, the pentagon lies behind a five-petaled rose, or a dandelion is a sphere. Honeybees build their hives in hexagons.

Today, the study of fractals can explain some other seemingly chaotic systems in nature. That is why the craft as it relates to geometry is called a progressive science in the broadest sense. In the search for knowledge, there is much that we do not know and discoveries constantly being yudrevealed.

Freemasonry is filled with practices that shift us to new perspectives. The contemplation of the vastness of time. The mysterious inevitability of death. The unlimited bounds of love. The power of symbols. 

For example, a Divine symbol that is both universal and ancient is the Yod, the 10th letter of the Hebrew alphabet. It symbolizes that all created things are modifications of the one primal Spirit. It is the masonic “G”, at least according to some authorities. W.L. Wilmshurst writes:

“The Yod is the emblem of the Divine Presence in the Lodge; it is also the emblem of that Presence at the spiritual centre of the individual Mason.”

There’s always more to learn. Another veil to lift. 

Cosmology and all of the associated sciences have not been able to definitely know the source and ultimate purpose of life. This strongly suggests that there must be some hidden purpose in the geometry of creation that is beyond the present scope of human knowledge and comprehension.

In masonic lectures, we read:

“By contemplation of the Divine we may discover his power, wisdom, and goodness and view with amazing delight the beautiful proportions which connect and grace this vast machine.”

And so, it is.

The procession of divine events and patterns which happens in the Divine realms are in Universal Co-Masonrysome manner mysteriously reflected in our human world, if we have eyes to see.

What, finally, is the message of the Masonic letter G? 

Perhaps it is that each of us must ponder the Divine, to be a geometrician, working according to his ability. Beyond the obvious pleasure of contemplating the glorious works of nature – there is delight that comes when beholding the “true” Masonic letter G, whatever symbolic form it takes.

“When the Lodge is opened, the mind and heart of every Brother composing it should be deemed as also being opened to the “G” and all that it implies, to the intent that those implications may eventually become realized facts of experience. When the Lodge is closed, the memory of the “G” symbol and its implications should be the chief one to be retained and pondered over in the repository of the heart.”  

~ W.L. Wilmshurst

Universal Freemasonry

TO THE GLORY OF GOD

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