THE PYTHAGOREAN THEOREM

OR

47th PROBLEM OF EUCLID

In March of 1995, Lowell Dyson posted a query to the freemasonry-list:

By one smarter than I, I have been told that there is Masonic
significance if, in Euclid's 47th, you construct the horizontal
line as 4, the vertical as 3, and the hypotenuse as 5.

|  LOWELL K. DYSON                                          |
|  ECONOMIC RESEARCH SERVICE                                |
|  U.S. DEPARTMENT OF AGRICULTURE       202-219-0786        |
|  1301 NEW YORK AVENUE, NW 932         202-219-0391 FAX    |
|  WASHINGTON, DC 20005-4788            LKDYSON@ERS.BITNET  |

This prompted the following reply from me:

The version of the Pythagorean Theorem or 47th problem of Euclid (called that because Euclid included it in a book of numbered geometry problems) in which the sides are 3, 4, and 5 -- all whole numbers -- is also sometimes known as "the Egyptian string trick."

The "trick" is that you take a string and tie knots in it to divide it into 12 divisions, the two ends joining. (The divisions must be correct and equal or this will not work.)

Then get 3 sticks -- thin ones, just strong enough to stick into soft soil. Stab one stick in the ground and arrange a knot at the stick, stretch three divisions away from it in any direction and insert the second stick in the ground, then place the third stick so that it falls on the knot between the 4-part and the 5-part division. This forces the creation of a 3 : 4 : 5 right triangle. The angle between the 3 units and the 4 units is of necessity a square or right angle.

The ancient Egyptians used the string trick to create right angles when re-measuring their fields after the annual Nile floods washed out boundary markers. Their skill with this and other surveying methods (cf. Barrance C. Lespine's posts on the checkerboard surveying system which they may have learned from the Sumerians) led to the widely held (but false) belief that the Egyptians invented geometry (geo=earth, metry=measuring).

Thales the Greek supposedly picked the string trick up while travelling in Egypt and took it back to Greece. Some say that the Greek mathematician and geometer Pythagoras, described in Masonic lectures as "our worthy brother," also went to Egypt and learned it there on his own. In any case, it was he who supplied the PROOF that the angle formed by the 3 : 4 : 5 triangle is invariably square and perfect. As to whether he actually sacrificed a hecatomb upon completing the proof, i leave to historians to determine. Once the proof was found, it could be applied to other right angles and was found to satisfy the conditions of their construction as well.

On reading the above, Bill Madison commented:

The 3, 4, 5 is of interest, however, because it is the smallest,
and the only right triangle in which the integers are consecutive.

| Bill Madison                           Internet: bmadison@crl.com |
|   CompuServe:  73240,342   |   FIDOnet:  bill madison 1:387/800   | 

Bill's brief point that 3, 4, and 5 are both the smallest integer series to form a Pythagorean triple and the only series of consecutive numbers in that group is extremely important, for it opens the door to considerable philosophical conjecture:

Since each integer in the tetraktys (that's 1 through 10 for those who don't keep track of these terms) was given symbolic meaning by Pythagoras, the 3, 4, 5 series is the only such triplicity to which "instantaneous" Pythagorean symbolic meanings can be ascribed.

While any other integer sets that form a right triangle can be given derivative meanings -- based on the symbolism of the Pythagorean categories into which they fall (evenly-odd, evenly-even, triangular, square, etc. etc.) or by using medieval Cabalistic gematria to work out their correspondance to some deity's name or the name of some virtue or vice -- only 3, 4, and 5 form a set that lays WITHIN the holy tetraktys. This makes the 3 : 4 : 5 right triangle the most symbolically freighted of all the right triangles.

As for the symbolic "meaning" of the digits 3, 4, and 5, well, modern mystical Masonic writers like Manly P. Hall (working from genuine Pythagorean teachings, but "fingerpainting" a little as they go) have attributed to these numbers metaphors such as:

Spirit   (3)
Matter   (4)
Man      (5). 
or
Osiris   (3)
Isis     (4)
Horus    (5)

You were told that in laying out the triangle, the length of 3 should be vertical and the length of 4 should be horizontal. Using Hall's neo-Pythagorean attributions, the symbolism of this arrangement is obvious: "Matter" (4) lays upon the plane of the Earth and "Spirit" (3) reaches up to Heaven and they are connected by "Man" (5) who partakes of both qualities. Or, alternatively, Isis the Earth-mother (4) is horizontal, Osiris the ithyphallic father-god (3) is erect (!) and Horus their son (5) extends between them and reunites them.

Although such metaphorical ascriptions to the 3 : 4 : 5 triangle do appear in various symbolic exegeses of Masonic imagery now and again, nothing as specific as the interpretation given above is taught in any actual Lodge ritual or lecture i have ever read -- probably because 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 (and will be the subject for another essay). 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 a jewel of office, and in the form of some Masonic aprons.

To create a 1 : 1: square root of 2 right triangle, also known as an iscoceles right rtriangle, you need a compasses and a straight edge -- familiar tools to the Craft, of course. On soft ground, use the compasses to inscribe a circle. Then use the straight edge to bisect the circle through the center-point marked by the compasses. Mark the two points where the bisecting line crosses the circle's circumference. Using the compasses again, erect a perpendicular 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.

An aside 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 saints occur.

For a study in depth on Pythagorean and Cabalistic systems of numerical symbolism, i recommend "The Dimensions of Paradise: The Proportions and Symbolic Numbers of Ancient Cosmology" by John Michell. Publication details on this and other books by Michell can be found in the Sacred Landscape Bibliography: Letter "M".

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