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47thProblemofEuclid

When I was entering High School I remember my father asked what type of geometry we would be studying.  I being new to the subject didn’t really know so my father insisted that he attend a parent meeting so he could get to know what subjects I would be learning.  The meeting was uneventful and full of the normal lists of rules and expectations about what every student should learn in school.  When asked for questions my father politely raised his hand and asked, “What kind of Geometry are these kids learning?”  I think there was some level of surprise from the Principle, Mr. E, and the Guidance Councilor Mr. H. (also the Geometry teacher).  The exact exchange at this point is a little unclear to me as I was embarrassed that my father would presume the knowledge of a teacher.  I think it was left off that my father should meet with Geometry after the meeting to gain a greater understanding of what we would be learning.

After the meeting my father chased down Mr. H to discuss my mathematics future and get to the bottom of his main concern.  My dad again asked Mr. H what kind of Geometry would be taught.  Confused Mr. H asked for a little more information and my father explained that he wanted to know if we were going to learn planar or Euclidean Geometry – because there is a difference.  After a small exchange Mr. H agreed to lend my father a textbook so he could look it over and come to his own conclusions about this – now that I am older I suspect it was because Mr. H didn’t have the answer and was tired of trying to explain this to my father.

When we go home that night my father reached into the bookshelves in our living room and dug out a dusty black and yellow book – more of a handbook.  He flipped through the well worn pages, turned it around and showed me what he was talking about.  A picture of a sphere and the angles you can create through the sphere seemed magical to me and was really quite interesting.  He told me then and there that the most important thing I would need to know was the funny looking formula on the page labeled the Pythagorean Theorem.  Honestly for the next year and many years after as I learned more about mathematics pulled down that book and dug for the information and studied the proofs to understand what it all meant.

Later that summer, boring textbooks aside, I was able to use that neat little trick I learned on my first real construction job.  At the time I was just doing menial labor on a pool construction at my neighbor’s house. Mr. R, our neighbor, had decided that instead of laying off his machine shop for the summer he would employ them to renovate his pool decking area and make a way for his shop employees to get some income and him to derive some benefit (a lesson I have tried to apply many times over).  Well as most projects go once you start to any depth you find that there is much much more that needs to be done to complete the project.  First it was the fence that needed to be replaced, then the skirt of the pool, and the liner of the pool all the way down to re-digging and laying the pool foundation.

While the crew was struggling with the foundation they kept running into problems getting the entire length of the pool to square up.  They had spent almost a day working to make the angles of the pool nice and square and kept driving carpenter’s squares into the corners of the pool for force it into square.  I watched this for a little while then said, “Why don’t we just calculate the lengths of the hypotenuses and then make them equal on their bisection?”  Once I had realized that I blurted this out I was completely embarrassed.  Here I am a snot nosed teenager trying to tell grown men how to do their work.  You can only imagine the looks on the faces of me 30-40 years older than me taking advice from a 14 year old.  Some of them had not had the opportunity I had to learn geometry and didn’t even understand the vocabulary.  Others thought I was making it all up.  Mr. R being an open minded guy asked me to explain, which I did by drawing in the dirt.  The guys all looked at the picture, nodded assent and got to work.  We were able to square the pool quickly and move on with our labor.

It wasn’t until some years later that I understood more about why my father took such an interest in that Euclidean Geometry thing or that Pythagorean Theorem.

I have told you all of this, as I have been thinking and researching about this arcane piece of Masonic lore, for some time now, to say:  I think we as Masons often overlook the importance of our symbolism and take for granted that which was handed to us by our predecessors.  I also think that in the wisdom of our forefathers they embedded more into our lore than we are willing to dig for.  We have to be willing to go beyond the scratched surface and look for more light.

The Pythagorean Theorem isn’t too arcane of a subject and is really quite simple.  Most of us learn it at some point while in school or preparing for a trade.  The premise of the theorem is that the sum of the squares of the base and up-right lengths of a 90 degree triangle are equal to the hypotenuse.

a^2+b^2 = c^2

But there is more.

We are told during our catechism that when Euclid found this out he cried ‘Eureka’ and sacrificed a hecatomb or 100 cattle as an offering of sacrifice.  It is doubted by scholars that Euclid in fact did cry out the joy he had when he figured out this little problem, and it is well known that technique was well known by Indians, Greeks, Chinese and Babylonians by this time.

rope

It is also well known that the Egyptian pyramid builders employed a special class of resources called Harpedonaptae or “rope-stretchers”.  These men employed a unique method of ensuring buildings were square and facing East, as all regular and well built temples and buildings should be.  They would survey the skies at night to find the true North-South line which would prepare them for laying out a square North-East corner by finding the perpendicular to the North-South line.  The method for finding true North during the night is a matter of finding either the North Star or the Southern Cross and is well documented.  Using a stick and plumb bob or sexton one can find the direction of North easily.

The method they employed to find the true East-West line required a rope and three sticks.  They found that if they divided a rope into twelve equal parts and then placed sticks at 3, 4 and 5 they would create a 90 degree angle to the North-South line thereby establishing a true East-West line, and at the same time establishing a set of four 90 degree angles to derive the remainder of the building angles from.  Try it yourself; it is a nice trick we use to teach Boy Scouts navigation.  There are of course other ways, but this was chiefly the documented method used by the ancients.

The ancients guarded this knowledge and treated it as magic.  Those with the know-how often controlled the destiny of others or at least were able to keep a position of importance or income for themselves.

The solution to the problem is really quite trivial and if you break down the verbiage used in the proof of the theorem it makes a lot more sense.  Our Masonic brethren baked in an example of this technique into the decorations of the lodge, it is sometimes used to represent our Past Masters and is used as the jewel of a Past Master in some jurisdictions like Pennsylvania.

 

The problem is really about checking the area bounded by three squares of differing sizes, illustrated by the image below.  By adding the total SQUARES of each square, we find the sum of the base and upright squares is equal to the sum of the square that the hypotenuse.  The simplest of these to manage are 3, 4 and 5.  Magic, no?

euclids_47th

There are many proofs of this theorem and not really the point of this writing.  One proof entails drawing a bounding square of the triangle with one side being the length of the hypotenuse.   Then using the area of the resultant square, take the remaining area and divide it into triangles which are rearranged to create squares matching the area of the base square.  One famous proof comes from our 22nd President and Brother James Garfield which shows the proof in a light that is derived from the above.  Even more interesting is to see a parallel proof by Brother William Burkle on using a certain point within a circle to derive the proof.

Now that I have you glazed over with theorems and proofs I want to bring you back to the point or premise of my argument about our symbolism and the layering of knowledge.

We often stop in our search for light at this point and assume that this is all mathematics and related to building edifices.  I stop to wonder what about that temple not made with hands eternal in the heavens?  Do we have more here than just simple High School Geometry class?  I think we do.

Interestingly I found a parallel to this thought from a dissertation by one of our Prince Hall Brethern that was similar in my thought.

Exodus 27:1 is a command from the Lord to create an altar.  It says:

1 “Build an altar of acacia wood, three cubits high; it is to be square, five cubits long and five cubits wide. 2 Make a horn at each of the four corners, so that the horns and the altar are of one piece, and overlay the altar with bronze. 3 Make all its utensils of bronze—its pots to remove the ashes, and its shovels, sprinkling bowls, meat forks and firepans. 4 Make a grating for it, a bronze network, and make a bronze ring at each of the four corners of the network. 5 Put it under the ledge of the altar so that it is halfway up the altar. 6 Make poles of acacia wood for the altar and overlay them with bronze. 7 The poles are to be inserted into the rings so they will be on two sides of the altar when it is carried. 8 Make the altar hollow, out of boards. It is to be made just as you were shown on the mountain.

When I read this I feel there is much more to what our Masonic Brethren are telling us.  This is an alter that contains the easy to remember 3-4-5 rule.  It is 3 cubits high, is square (4) and is 5 broad.

From a numerology perspective we see more interesting facts.  It is well known that our Jewish, Nepthali and Phoenician Brethren practiced the science of numerology, which out founders were well acquainted with.  They may have thrown us a curve ball to look more deeply into the lessons we are taught.

The first number is considered Holy to our ancient brethren and many cultures as it represents divinity and trinity, it may also represent the Wardens of the Lodge.  The second number four represents the wholeness or perfection of man and may represent the four perfect points of entrance in a Lodge.  Five, the last number represents man himself or the people of the Bible after the fall and is also representative of the Pentagram, which is man.

The sum of the first two numbers, seven, represent the sum of a perfect lodge of Entered Apprentices, the seven days that to create the earth or seven cardinal virtues, and seven deadly sins.  This is the perfect number in Islam.  It is the number of the universe.  The sum of all three numbers, 12, represents an even more interesting perspective of deity and is considered more holy as a multiple of the base number (3) as well as the sum of the numbers representing deity (1+2=3).  It is the number of months in our calendar as well as the number of hours before high-twelve and number after low-twelve.  This is also representative of the fruits of the spirit and number of disciples and gates at the Holy City of Jerusalem.

If anyone cares the volume of the altar is 75, which added together is the number 12, which added again is a divine number 3 (so is 12).

I am sure these just scratch the surface of the numerological analysis of the numbers.

More Resources:

http://www.masonicworld.com/education/files/artnov01/The%2047th%20Problem.htm