Here is a version of the talk I gave in Lodge on the 47th prop. of Euclid. Please bear in mind this was given in a somewhat rural lodge, so I wrote for my audience. I realize a lot has been written about the "mystical" aspects of this subject, as well as applying a lot of symbolism to it. That was simply not in the scope of my presentation on this occasion, and I frankly think that the practical impact Pythagoras' theorem had on the world completely overshadows any later efforts to "mysticize" it.
The 47th Problem of Euclid...it is one of the most frequently used of Masonic symbols. We see it as one of the emblems on the Master's carpet; in some jurisdictions it is the jewel of a Past Master; in Texas it is the central figure of the design on the Grand Master's apron, as well as of other Grand Officers. Anderson's Constitutions of 1723, the first published version of Masonic Laws and Charges, begins with an ornate frontispiece showing Grand Masters on a checkered pavement, and there on the floor, is a diagram of the 47th Problem of Euclid. Yet many Masons have no idea what the diagram refers to, or why it should occupy an important place in our system of symbols.
If the pyramids, and the hanging gardens of Babylon were physical wonders of ancient world, then the 47th Problem of Euclid was a wonder of reasoning. To put it simply, this diagram demonstrates a discovery which is the foundation of Geometry, and of architecture. It occupies a vital place in the history of human knowledge, and, it can be argued, is the starting point of all science.
Strong words. But I think, as I explain myself, you will understand why I feel comfortable assigning such importance to a few lines drawn on a piece of paper.
First, who was Euclid? Euclid was a Greek mathematician, living in Alexandria, Egypt around 300 BC. His contribution to our story was not by originating, so much as cataloging ideas. Euclid, literally, wrote the book on Geometry. He compiled everything that was known at his time about Geometry into a book, which he called Elements of Geometry. That book stood as the authority on Geometry for more than 2000 years. Over the centuries it became the most published book in the world after the Bible. Page by page, Euclid presents each principle of Geometry with detailed explanations, beginning by defining a point, then a line, and moving on to gradually more complex demonstrations. Accordingly, the order in which the problems are discussed has become the system for cataloging and naming them, much as we know to quote the Bible by chapter and verse. The idea we are interested in was Proposition number 47 of Book 1.
As I said, Euclid only enters our story as the collector and cataloger of geometrical propositions. The person credited with the actual discovery of this principle was another Greek philosopher of an even earlier age. Pythagoras was born on the island of Samos, in the Aegean Sea in about 580 BC. His biographer, Iamblichus, says he traveled widely, and was initiated into various mysteries, in Tyre, Babylon, and Egypt before settling in Crotona, a Greek colony in southern Italy, where a school of his disciples, a sort of early secret society, grew up. Both Euclid and Pythagoras are mentioned in Old Charges and manuscripts of Freemasonry as far back as the 1400's, usually describing them in completely the wrong eras of history; for instance, Euclid is described in some places as a contemporary of Abraham. It is interesting that Pythagoras is usually spelled very strangely, like the name had been handed down from mouth to ear for a long time, or for a short time by people who couldn't hear very well. The most curious occurrences are when the manuscripts mention a wise geometrician named Peter Gower.
Mathematics and numbers were central to the philosophy Pythagoras taught, but unfortunately, as in the cases of other Greek thinkers, like Socrates, nothing of his own writings remain, but only those of his students. It has been suggested that Pythagoras himself did not discover the geometric theorem that bears his name, but that it merely came from the school he founded. I prefer to believe that he did.
Before I go into detail about what exactly Pythagoras hit on, I'm going to take you even further back in time, to Ancient Egypt. Obviously, the Egyptians who built the pyramids and other monuments that have survived the millennia were superb operative masons, and even then, geometry was central to their craft. Let me set you a puzzle. If you wanted to make a right angle, you would take your mason's square, and use it to square the angle you were working on. But what if you didn't have a square to use as a tool, or a protractor to measure ninety degrees, or another right angle to compare it to. The Egyptian masons knew the answer; it was one of their secrets, and I'll let you in on it. The ancient Egyptians knew that if you took a rod 3 cubits long, another 4 cubits long, and another rod 5 cubits long, and laid them end to end in a triangle, the angle where the 3- and 4-cubit rods met was always a right angle. To the Egyptians, this was a wonderful and powerful tool, almost bordering on the magical. Their chief architects carried a set of rods to use whenever a square corner was needed. Another method was to take a string with twelve cubits marked out on it, and stake it out in a triangle with three cubits on one side, four on another, and five on the other. Of course the unit of measurement could be anything...a cubit, a foot, a meter, an inch, a yard...it was the relative lengths of 3 by 4 by 5 that resulted in a right triangle.
But there is a property of this 3-4-5 proportion that makes it even more curious. Take the two smaller sides and square their lengths: 3x 3 = 9, and 4 x 4 = 16. 9 + 16 = 25, or 5 x 5. Another way to say this is that if you make a square out of each side, and add the areas of the two smaller squares, you get the area of the larger square.
Pythagoras found that this held, not just for the 3 by 4 by 5 triangle, but for any right triangle. He started with a what was just a useful tool and discovered a fundamental rule of nature. What the Pythagorean Theorem, also called the 47th Proposition of Euclid, says, is that for any right triangle, that is, any triangle containing a 90-degree angle, the square of the "hypotenuse," the longer side, equals the sum of the squares of the two shorter sides.
Today over a hundred ways have been found to prove this proposition. To explain any of them requires drawing diagrams, which I can't do in this setting, but all these proofs arrive at that moment of epiphany when the pieces come together like a jigsaw puzzle, and I can't help thinking of the day 2500 years ago when the puzzle was first solved. This morning Pythagoras woke up in a world of chaos, variety and inexactness, but now the universe has changed, and Pythagoras has caught nature red-handed in the act of displaying order and following rules. Imagine a caveman looking at the Astrodome, imagining it is just a big hill, then going inside and suddenly understanding the architecture that holds it up. Pythagoras had peeked under the veneer of the universe, and found that space had a kind of architecture, and that architecture was made of numbers. To us, looking at this from the vantage point of a couple of thousand years later, The 47th problem might seem a little less dramatic. It is, after all, just another one of the laws of nature. We have to remember that to the Pythagoreans, it was a new and wonderful thing to find that there were any laws of nature. Even now, we can't explain why space fits together this way, we're just so used to seeing it that we tend to overlook the implications of a world ruled by numbers.
Now, it was possible to use Geometry to make predictions, not just on paper, but in the field. You could indirectly tell the length of something it was impossible to measure directly. If you knew the lengths of two sides of a right triangle, you could predict the length of the third, and always be right. The world obeyed numbers, not at random times, but always. Armed with this insight, Pythagoras taught that numbers were even more real than the world they described. He uncovered the basis of music theory when he found that you could pluck a string to make one note, then divide the string exactly by two, and pluck it to make the note one octave higher. By dividing the string length exactly by three, four and five, notes were produced which harmonized with the first. To the Pythagoreans, they were discovering a divine language of pure mathematics. To us, they were discovering that the universe could be described, predicted, and understood.
Pythagoras supposedly was so inspired by the discovery of what we now call the 47th Problem of Euclid that he sacrificed a hecatomb, a hundred oxen, to the Muses in gratitude. If so, I would think he got off cheap. This single discovery has echoed through history. The entire science of trigonometry is based on it. Mapmaking, astronomy, architecture, even space travel would be impossible without it. The English political philosopher Thomas Hobbes, whose Leviathan was one of the most important books of the seventeenth century, probably would never have achieved the fame he did if he had not, at the age of 42, glanced at a copy of Euclid's Elements in a friend's study, opened to the 47th proposition. Hobbes was supposedly so shocked by the implications of the theorem that he exclaimed, "By God, this is impossible!" This single revelation apparently motivated Hobbes to a fevered lifelong study of geometry, and later physics, philosophy, and political science.
It has been speculated that any civilized race will have at some point in its history discovered the Pythagorean Theorem. As I was writing this talk I suddenly remembered years ago reading a book by Pierre Boulle, who also wrote The Bridge on the River Kwai. The book I remembered was called Monkey Planet, and was made into a movie called "Planet of the Apes." In the book, an astronaut travels to a far-away planet ruled by a race of intelligent chimpanzees. To complicate matters, there are humans on the planet, but they are brute animals, like apes are here. The chimpanzees, speaking their own ape language, are unable to make sense of our hero's French, and consider him also a brute until he draws for them a diagram of the 47th proposition of Euclid, whereupon they realize he is a civilized, intelligent being, like themselves.
As it has been passed down through the ages, this theorem has grown from a useful geometric principle into a symbol of the harmony of the universe. The Greek historian Plutarch, who lived in the first century AD said that the 3-4-5 triangle had become a symbol for the Egyptian gods, Osiris, Isis, and Horus, reminding us of the manner in which the Christian Trinity is sometimes represented by an equilateral triangle. We are told in the Monitor that the 47 Proposition of Euclid is a symbol to admonish Masons to be lovers of learning.
I would like to close my talk by sharing with you a story I found in a biography of Pythagoras written in the fourth century AD. A disciple of Pythagoras travelling near Crotona, came to an inn, where he fell ill from the rigors of his trip. The inn-keeper, being of a benevolent disposition, cared for the Pythagorean, supplying his needs as best he could until he finally died. But before the Pythagorean died, he wrote what the author termed "a certain symbol" on a tablet, and instructed the inn-keeper to display the tablet outside the inn, near the road, and to observe if any passer's-by stopped to notice it. The inn-keeper buried the man decently, and more out of curiosity than expectation did as he asked with the tablet. A long time later, a passing Pythagorean spotted the symbol and stopping to inquire, learned about what had transpired, whereupon he repaid the inn-keeper all the expenses he had generously provided to his long-dead brother, together with a large additional sum out of gratitude. This is an obscure and little known story, and the author of the story does not venture to say what the symbol of recognition was. It very well may have been a diagram of Pythagoras' Theorem, since probably only another Pythagorean at that time would have understood the meaning of such marks, but another possibility presents itself, because the Pythagorean brotherhood had another symbol, with which they identified themselves, and which they used to sign letters to one another, and which is also quite familiar to Freemasons, the five-pointed star.