Saturday, August 10, 2013

The 47th Problem of Euclid

Euclid's 47th problem or the Pythagorean theorem is a relationship between the three sides of a right triangle. 
In terms of areas, it states:
In any right-angled triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).
The theorem can be written as an equation relating the lengths of the sides a, b and c, often called the Pythagorean equation:
Where c represents the length of the hypotenuse, and a and b represent the lengths of the other two sides.

a2 + b2 = c2

Side note:
Incidentally we may wonder why pythagoras's theorm is known as Euclid's problem?
Well the answer to that is that Euclid wrote a set of thirteen books, which were called “Elements”. Each book contained many geometric propositions and explanations, and in total Euclid published 465 problems. 
The 47th problem was set out in Book 1. Now, although Euclid published the proposition, it was Pythagoras who discovered it.

Coming back to our main topic: 
A right triangle therefore will always have sides of a certain proportion known as a pythagorean triple some examples are as follows, 

( 3, 4, 5 ) (5, 12, 13) (8, 15, 17) (7, 24, 25) e.t.c.

How do we arrive at a pythagorean triple?
By calculating the difference of the squares of any two consecutive numbers which is always an odd number and it is based on such a pythagorean triple (the lowest four consecutive numbers) we arrive at 3, 5 and 7 which is masonically significant!!

To demonstrate,
take the 1st four numbers,
1, 2, 3, 4 
Calculate the squares of those numbers,
1, 4, 9, 16 

Now subtract the smaller from the larger of the squares,

4-1 =3

9-4 = 5

16-9 = 7

Masonically the significance of 3, 5 and 7 is as follows:

They are 

  • The steps in the Winding Staircase which leads to the Middle Chamber of King Solomon's Temple.
  • It also alludes to the number of Freemasons in each degree of Freemasonry. "Three rule a Lodge, five hold a Lodge, seven or more make it perfect. The three who rule a Lodge are the Master and his two Wardens; the five who hold a Lodge are the Master, two Wardens and two Fellow Crafts; the seven who make it perfect are two Entered Apprentices added to the former five."
  • The Three Grand Masters who bore sway at the building of the first Temple at Jerusalem. Five hold a Lodge in allusion to the five noble orders of architecture. Seven or more make a perfect Lodge and have a further allusion to the seven liberal Arts and Sciences.