# 47th Problem of Euclid

September 16, 2018 Clark

(Copied from”The Sunday Masonic Paper – September 16, 2018)

47th Problem of Euclid

W.Bro. Shawn Belanger, Rideau Minden No. 253, GRC

Brethren, I wanted to give a brief bit of education this evening. At least I hope it’s educational – I’m assuming that if I didn’t know something, then at least some of you will find this interesting too.

When I was preparing to be installed as Master, I was memorizing the speech for the installation of the Immediate Past Master. A part of that ritual identifies the IPM’s jewel as “the 47th problem of the 1st book of Euclid, and one of the most important discoveries of the learned Pythagoras.” Now, I knew from grade school who Pythagoras was, but I’d never heard of Euclid and wondered to why he had so many problems.

So, being curious, I did some research and found that Euclid was an ancient Greek mathematician who is sometimes referred to as the Father of Geometry, and his “problems” were actually geometric proofs of mathematical theories. The 47th problem in his first book, entitled “Elements”, was the proof of the Pythagorean Theorem and is depicted by the jewel on the Immediate Past Master’s collar and on our tracing board.

As a side note Brethren, in the Entered Apprentice degree the tracing board is identified as one of the lodge’s Immovable Jewels for the Worshipful Master to lay lines and draw designs on, but I’ve never seen a Master actually use it before, so I’m happy to have an opportunity to use it myself now.

You can see that the 47th problem is a right-angle triangle, with a square on each edge of the triangle. The area of the squares on the two shorter edges equal the area of square on the longest edge, the hypotenuse. There is a neat little video on YouTube showing this on a rotating mount, with the two smaller squares being filled with coloured water (https://www.youtube.com/watch?v=CAkMUdeB06o). When the symbol is rotated the fluid flows from the two smaller squares and precisely fills the larger square, proving that the volumes are equal.

The right-angle triangle in the centre of the squares is sometimes referred to as a 3:4:5 triangle, because it’s a simple way to express the ratio of the sides. Using the Pythagorean theorem (a2 + b2 = c2), if the shorter side is three, then the square of that is nine. If the longer side is four then the square of that is sixteen. Nine plus sixteen is 25, and the hypotenuse is the root of that so is five, making 3:4:5.

So that’s all interesting, but I wondered to myself why it would be important to Masons and why this would appear as such an important jewel.

For a moment, picture yourself standing in the middle of desert, having limited tools or technology, and being tasked with laying out the foundation for something like a pyramid or a temple. How would you start? What would you do? You can imagine that laying the foundation wrong or out of square in ancient times would not be a path to a long comfortable life.

Well, in ancient Egypt there was a profession called Harpedonaptae, which literally translated means “rope stretchers” or “rope fasteners”, who were the specialists that laid out the foundations of buildings. They relied on the stars and mathematical calculations to form perfectly square foundations. We know that cornerstones are historically laid in the northeast corner, but why the northeast?

First the Harpedonaptae would use the north star and constellations to lay out a straight north-south line. Then they would take a looped rope with knots in it at very precise intervals. The size of the loop is infinitely scalable so long as number of knots in it corresponds to the sum of the sides of Pythagoras triangle, but for this demonstration I’ve made a rope with 12 knots – 12 because if you add the sides of the 3:4:5 triangle it equals 12. If you take your rope and put three segments on the north-south line, count out four segments and pin it at the knot, the resulting angle has to be a perfect square according to Pythagoras, giving you a precise east-west line. The foundations for the largest monuments could be precisely laid out using just some sticks and a rope using this method. Personally, I found this really interesting.

Also interesting, this is also why old carpenter’s squares usually have one short side and one long side – I think if you look you’ll find that the sides are ratioed as a 3:4:5 triangle.

So now when you see the IPM’s jewel, you’ll know that it refers to an ancient and important knowledge of our operative predecessors: how to square your square.

References:

Pythagorean Theorem Water Demonstration:

https://youtu.be/CAkMUdeB06o

Euclid

https://en.wikipedia.org/wiki/Euclid

Pythagorean Theorem (47th Problem of Euclid)

https://en.wikipedia.org/wiki/Pythagorean_theorem

Masonic Education – 47th Problem of Euclid

https://www.masonic-lodge-of-education.com/47th-problem-of-euclid.html

From The Sunday Masonic Paper, Sept. 16/18

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