The 47th Problem of Euclid as it Relates to Masonry.

There are so many symbols, words and phrases in Freemasonry used both as part of the rituals as well as part of the vocabulary, that studying and trying to understand the “meanings” of the symbols, words and phrases can be both a frustrating and a worthwhile and a rewarding journey. In Masonic education we often find that the authors refer to the allegory of the symbols and rituals, the idea that the meaning is not explicitly set forth in the story of the symbol or ritual and that the reader or listener will need to participate in an interpretive process with himself and/or others to understand the meaning of the story, symbol or ritual.

One such prominent allegorical symbol and phrase, is the 47th problem of Euclid, which is one of the main symbols introduced in the Third Degree and is reflected in the symbol of the square, one of the 3 great lights in Masonry. Yet, there is doesn’t seem to be, at least in my limited experience, a deeper explanation of the importance of the 47^th problem of Euclid, hence my desire to delve into the symbolism that we about in the Master Mason degree.

We learn from the Master Mason degree lecture that:

“ This wise philosopher (Pythagoras) enriched his mind abundantly in a general knowledge of things, and more especially in Geometry, or Masonry. 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.” This celebrated design, the most important element in mathematics, should inspire us to be lovers of the Arts and Sciences.”

In asking what it was that caused him such joy we need to understand that Euclid, a Greek mathematician who lived in the time between 325-265 B. C, wrote a set of thirteen books, which were called “Elements”. These books brought together many of the Theorems of the Sciences, proving some and certainly putting them in a collected form that has stood the test of time for over 2000 years. Each book contained many geometric propositions and explanations, and in total Euclid published 465 problems. The 47th problem, which is also known as “The Pythagorean Theorem”, was set out in Book 1. Why is it called by both these names? Although Euclid published the proposition, it was Pythagoras, who lived 580 – 572 B.C. who is credited with discovering it.

Actually, it was not Pythagoras who directly discovered the rule, as the Egyptians used the same principle for a very long time before Pythagoras, whereby they re-measured their fields after the annual flooding of the Nile washed out their boundary markers. Hence, Pythagoras is probably here referred to as being the one who proved that the process works.

What does the 47th problem of Euclid say?

The proposition states that: “In right angled triangles the square on the side subtending the right angle is equal to the squares on the sides containing the right angle.”

Or the words of your high school math teacher - A2 + B2 = C2.

For an Operative Mason the importance of the proposition seems fairly obvious and is especially important in architecture. Builders have since ancient times used the theorem in constructing buildings by a process known as “squaring a room.” As the theorem states that 3 squared + 4 squared = 5 squared, a builder starts by marking a spot and drawing a line, say line A. This line is given the value of 3. The builder then marks another point, say point B and draws a line from it at a right angle to line A, and it is given the value of 4. The distance between line A and B is then measured, and if the distance between A and B is 5, then the room is squared. By inverting the process, a “squared” (or rectangle) room can be obtained.

Engineers who tunnel from both sides through a mountain use the 47th problem to get the 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 times 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, bridges built, with the 47th problem to show the way.

And thus the allegory is achieved. The question then becomes how does this relate to our Speculative Masonic teachings and the answer can only come through discussions and sharing of thoughts, hopefully these thoughts will lead to some discussions.

The 47th problem of Euclid is often mentioned in Masonic publications. In Anderson's “Constitutions” published in 1723, it mentions that “The Greater Pythagoras, provided the Author of the 47th Proposition of Euclid's first Book, which, if duly observed, is the Foundation of all Masonry, sacred, civil, and military…”. Being mentioned in one of the first “official” speculative Masonic publications clearly indicates that the 47th problem of Euclid must be important. This importance is reinforced since it is also mentioned in the Third Degree lecture, as noted previously.

For the speculative mason the relationship is obviously less clear. In some lodges, using this principle, a candidate symbolically “squares the Lodge” by being escorted around the Lodge three times during the Entered Apprentice ritual, four times for a Fellowcraft ritual, and five times for a Master Mason ritual, which completes his journey or proves him to be a Master Mason.

The “Square” is introduced to the Entered Apprentice as one of the three Great Lights of Freemasonry, to the Fellowcraftsman as one of the working tools of his Degree. It is also one of the Jewels of the Lodge, and the special Jewel of the Master of the Lodge. It is probably the most important tool of a Mason, whether an Operative or a Speculative one, for it connects and more or less includes the Level and the Plumb Rule, and it is the only tool by which the rough Ashlar can be prepared and the finished Ashler tested; and unless the Ashlars are perfect the building cannot be built after any wise plan, or with strength, or with beauty. Therefore it is of the utmost importance that an implement on which so much depends shall be itself perfectly correct.

The craftsman employs the square to form the rude mass; the Master to prove the work; but whose duty is it to see that this most important tool is itself correct? The most suitable person would seem to be the Past Master, he, having passed through the stages of using it and testing with it, would be most impressed with the necessity of its being correct. By what mode can he ensure the correctness of the Square? How can he ensure that the angle between the two limbs of the Square shall be truly a right or square angle?

It is this last consideration especially, which ties the 47^th Problem of Euclid to Masonry and renders it so appropriate an emblem for the Past Master.

Of course the allegory of the Ashlers and the Square are topics for another day but for the purpose of this discussion can we agree that for the Speculative Mason they represent the idea of the imperfection of man and how he strives toward perfection before being judged by the Great Architect of the Universe?

“In Freemasonry the Square is the Symbol of moral perfection: it is the Master’s duty to apply the perfect square of right and truth to the work of the subordinates; but the far higher and greater responsibility rests on the Past Master of setting out, pointing out, and in himself exemplifying what Right in itself is, and what Truth is.”

True Speculative Masonry teaches a man, by the industrious application of the principles of Eternal Truth and Right to the untaught material of humanity, to shape its thoughts and actions so as to erect from it a spiritual building, on sure foundations, with intelligent purpose, and admirable to contemplate.

The Past Master represents one who has erected such a building; but his having done so places him under the responsibility of ensuring that those who are working for the same end shall not fail. By precept and example, principles which have been put to the test and found to be those of absolute truth and correctness he becomes the Square that is used as a measure for those that follow.

In the Craft Lodge, it is considered a great honor to be elected and serve as the Master of a Lodge. It shows that the individual has attained maturity, leadership, charisma, and motivation, but above all, knowledge to be able to lead others. It is an initiation by itself, as the position brings with it a completely new set of responsibilities that are often not appreciated when accepting the position. The Master looks to the Past Master to be sure that the construction of the Lodge during his tenure continues to be square and true to the Masonic foundations laid down by the Masters before him. Allegorically speaking the Past Master is the proof, the Pythagorean Theorem, of the Masters’ building of the Lodge.

Clearly, the 47th problem helps us look at the universe, and all that is in it, through a system that we can understand clearly, for it is measurable. The Master’s jewel is the square, the base needed for the 47th problem (in many jurisdictions the square has the dimensions of 3:4 - the Pythagorean dimensions). As the Master serves his position, he becomes more complete, and therefore the 47th problem of Euclid is dedicated on his jewel when he leaves office to become the Past Master.

As the Egyptians used the 47^th Problem of Euclid to re-measure their fields after the spring floods so does the Past Master assist in the “re-squaring” of the Lodge after the yearly elections, to assist the newly elected officers as they learn their respective offices and to see that the Lodge continues to flourish in its time of re-growth. In other words he is the one who ensures the correctness of the square, he is the 47^th Problem of Euclid.

(Adapted from the works of Bro. H. Meij, Master, Harmony Lodge #18, August 2000

http://www.freemasoninformation.com/esoterica/47thproblemofeuclid.html and from

Bro. Thomas Greene’s -

THE 47th PROPOSITION OF THE IST BOOK OF EUCLID AS PART OF THE JEWEL OF A PAST MASTER. - http://freemasonry.bcy.ca/aqc/1901/euclid.html )