The roots of Freemasonry are lost in antiquity but our recoded history goes back well over six hundred years.  During this period Freemasonry witnessed the development of cryptic symbols, obscure  phrases and words in our ritual and an explosion of Masonic information on the internet.  With this digital information so readily available we have a wealth of information that we can explore, study and learn their meaning. The 47th problem of Euclid is just one of the many relatively obscure phrases that we are exposed to and an in-depth study of these phrases, symbols and words is well worthwhile to become a better informed Mason.   Euclid also known as Euclid of Alexandria, is a historical figure often referred to as the Father of Geometry. Euclid wrote a set of thirteen books, which were called “Elements” and is considered by many to be the most successful and influential textbooks in the history of mathematics.  One such prominent symbol and phrase, is the 47^th problem of Euclid, which is one of the main symbols introduced in the Third Degree.   Each of the thirteen books contained many geometric propositions and explanations, and in total Euclid published 465 problems.  The 47th problem was set out in Book 1, which is also known as “The Pythagorean Theorem”.  Why is it called by both these names?  Although Euclid published the proposition, it was Pythagoras who discovered it.  We learn from the third degree lecture that:

To the operative mason it affords a means of correcting his square, for if he wishes to test its accuracy he may readily do so by measuring off 3 divisions along one side, 4 divisions along the other, and the distance across must be 5 if the square is accurate.  The knowledge of how to form a square without the possibility of error has always been accounted of the highest importance in the art of building, and in times when knowledge was limited to the few, might well be one of the genuine secrets of a Master Mason.  The ancient temple builders in the long centuries before Christ were most punctilious in setting their temples due east and west.  So exacting were they on this point that there was organized a set of men who, in modern phrase, would be termed experts or specialists, and whose sole duty it was to lay out the foundations of public edifices.  They were called, in Egypt, harpedonaptae--meaning rope stretchers.  They first laid out the north and south line by observation of the stars and the sun, and their next step was to get the east and west line exactly at right angles.  This they secured by stretching a rope north and south divided into three parts in the proportion of 3, 4, and 5, (the Egyptian string trick again fastening down the centre part by pegs, and then swing round the loose ends toward the west until they intersected and a right angled triangle was thus formed.  These ancient temple builders, by means of the centre, formed the square, and the centre was a point round which they could not err.  Here also is the obvious answer to the question why it is customary at the erection of all stately and superb edifices to lay the foundation stone at the north-east angle of the building.

The question arises, have we anything in our present ritual which might be relative in any way to this method of proving the square or obtaining a right angle without the possibility of error and which may have been connected with the instruction given in purely operative masonry. 

The type of triangle most often used to demonstrate the 47th problem in Masonry is not the 3 : 4: 5 but the 1: 1 : square root of 2 form.  The square and the cube which are 1 unit on each side are of great symbolic meaning to Masons.  Therefore, the bisection of the square into a pair of 1 : 1 : square root of 2 triangles has important Masonic connotations.  It is in this form that the Pythagorean theorem is most often visually encountered in Masonry, specifically in the checkered floor and its tessellated border, as a geometric proof on Lodge tracing boards, as the jewel of office for a Past Master, and in the form of some Masonic aprons. 

To create a 1:1 square root of 2 right triangle, also known as an isosceles right triangle, you need a compass and a straight edge -- familiar tools to the Craft, of course.  On soft ground, use the compass to inscribe a circle.  Then use the straight edge to bisect the circle through the center-point marked by the compass.  Mark the two points where the bisecting line crosses the circle's circumference.  Using the compass again, erect a perpendicular line that bisects this diameter-line and mark the point where the perpendicular touches the circle.  Now connect the three points you have marked -- and there is your 1 : 1 : square root of 2 right triangle.

To Freemasons, the first two points -- where you marked the crossing of the bisecting diameter through the circle's circumference -- can also be used to construct two further perpendicular lines.  These are the two "boundary" lines of conduct sometimes symbolized on Masonic tracing boards by the Two Saints John and sometimes referred to as indicators of the Summer and Winter Solstices, whereon the feast days of those two saints occur.

The 47th Problem of Euclid, also called the 47th Proposition of Euclid as well as the Pythagorean Theorem is represented by 3 squares. 

To the speculative Mason, the 47th Problem of Euclid may be somewhat mysterious.  Many Masonic books simply describe it as "A general love of the Arts and Sciences".  However, to leave its explanation at that would be to omit a subject which is very important... not only of Pythagoras's Theory, but of the Masonic Square.

Here is a simple explanation - Pythagoras found an amazing fact about triangles:

If the triangle had a right angle (90°) ie. A, B, C above...

        ... and you made a square on each of the three sides, then ...

... the biggest square had the exact same area as the other two squares put together!