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Regarding the 47th Problem of Euclid
When I tried to find out the answer several years ago I kept getting the standard definition of the square of the Hypotenuse being equal to the sum of the square of the two sides and not being a mathematician it didn't help much.
Then an old Past Grand Master explained it to me in a way that makes a little sense to those of us who don't do calculus in our heads.
The 47th problem was supposed to be one of the basic secrets of a Master Mason. The explanation and use is really simple. If you are in charge of constructing a cathedral in "the good ol' days" before modern surveying equipment, the first thing you want to be absolutely sure of is that you are truly laying our and building square. You don't want to go down in history as the boob who made the lop-sided cathedral.
If you lay out a square, let us say 100 feet by 100 feet, (substitute meters if you want) you need some way to be sure that you really have square corners. So you then run a line between the opposite corners (creating a huge X in the center of the square).
If your work is laid out right, the distance from one of the points of the square to the exact middle of the X will be 70.71 feet...(not quite 70 feet 9 inches).
This is the 47th problem of Euclid: the sum of the square of the Hypotenuse of a right triangle (the line from the corner of the square to the center) is equal to the sum of the squares of the other two sides. To get THIS right triangle, you draw one more line, from the center of one of the sides to the center of the X.
The line from the center of the X to the long side should be 50 feet and the distance from THAT point to the outside corner should be 50 feet (remember this is the middle of the 100 food outside edge). 50 feet squared equals 2500 plus the 2500 from other side squared is 5000 feet. The square root of 5000 is 70.71....Proof your work is square.
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Last modified: March 22, 2014