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(a.k.a. THE PYTHAGOREAN THEOREM)

Subj: the 47th problem Date: Tue, Mar 14, 1995 5:46 AM PST From: Cyronwode@AOL.COM To: freemasonry-list@sacsa3.mp.usbr.gov,sustag-principles@ces.ncsu.edu LKDYSON%ERS.BITNET@VTBIT.CC.VT.EDU (Lowell Dyson) wrote: >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 | Lowell, i am stepping out on a limb here, and i hope i do not introduce too much error, but here is my understanding of the matter you broached: The version of the 47th problem in which the sides are 3, 4, and 5 (all whole numbers) is 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.) Now you need 3 sticks, thin ones, just strong enough to stick into soft soil. Stick one stick in the ground and arrange a knot at the stick, stretch three divisions away from it in any direction and inssert the second stick in the ground, then place the third stick so that it falls 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 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. Barry Carroll's earlier posts on the checkerboard surveying system which they may have learned from the Sumerians) led to the widely held 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 to Greece. Some say Pythagoras also went to Egylpt and learned it there on his own. In any case, it was Pythagoras who wrote the familiar PROOF that the angle is a right/square/90 degree angle. 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. There are. of course, a myriad of ways to construct a right angled triangle, but the other common way is to do it from a square with sides of 1, 1, sq. root of 2. It is in this form that the Pythagorean theorum (called the 47th problem of Euclid because he included it in a book of geometry problems he wrote) is most often encountered in Masonry, viz. in the checkered floor and its tessellated border, on many tracing boards, and in the form of some Masonic aprons. As to the "meaning" of the 3, 4, 5 version of the 47th, well, mysticsal Masonic writers like Manly P. Hall have attributed to these numbers certain divine metaphors, such as Osiris (3), Isis (4), Horus (5) or Spirit (3, Matter (4), and Man (5). But although such appended ascriptions do appear in various symbolic exegeses of Masonic symbolism now and again, none are specifically given in any Masonic lectures i have read, probbly because the form triangle most often used to demonstrate the 47th in Masonic tracing boards is the 1, 1, sq root of 2 form. catherine yronwode ------------------------------------------------------- Subj: Re: the 47th problem Date: Tue, Mar 14, 1995 6:34 AM PST From: bmadison@crl.com (Bill Madison) To: freemasonry-list@sacsa3.mp.usbr.gov, sustag-principles@ces.ncsu.edu On Tue, 14 Mar 1995 Cyronwode@aol.com wrote: > The version of the 47th problem in which the sides are 3, 4, and 5 (all > whole numbers) is sometimes known as "the Egyptian string trick." Is that anything like the "Indian Rope Trick"??> ... Their > skill with this and other surveying methods (cf. Barry Carroll's earlier > posts on the checkerboard surveying system which they may have learned from > the Sumerians) led to the widely held belief that the Egyptians invented > geometry (geo=earth, metry=measuring). "Widely held", maybe - but false. Their use of the "string trick" was a purely empirical development. Had nothing to do with geometry as a separate science or as a branch of mathematics. > There are. of course, a myriad of ways to construct a right angled > triangle, but the other common way is to do it from a square with sides > of 1, 1, sq. root of 2. 'Course, the problem here is that it's difficult to construct a line segment of length sqrt(2) unless you already have a right angle. We're getting a bit circular. (Pun intended.) Mathematically, the 3, 4, 5 set is known as a "Pythagorian triple", for obvious reasons. As is 5, 12, 13 and, as you rightly point out, any of an infinity of other integer triples. The 3, 4, 5 is of interest, however, because it is the smallest, and the only one in which the integers are consecutive. Thus endeth today's ration of useless drivle! | Bill Madison | Internet: bmadison@crl.com | | CompuServe: 73240,342 | FIDOnet: bill madison 1:387/800 | -------------------------------- Subj: 47th problem Date: Tue, Mar 14, 1995 11:56 AM PST From: Cyronwode@AOL.COM To: sustag-principles@ces.ncsu.edu,freemasonry-list@sacsa3.mp.usbr.gov bmadison@crl.com (Bill Madison) wrote: B>On Tue, 14 Mar 1995 Cyronwode@aol.com wrote: C> ... Their C> skill with this and other surveying methods (cf. Barry Carroll's earlier C> posts on the checkerboard surveying system which they may have learned C> from the Sumerians) led to the widely held belief that the Egyptians C> invented geometry (geo=earth, metry=measuring). B>"Widely held", maybe - but false. Their use of the "string trick" was a B>purely empirical development. Had nothing to do with geometry as a B>separate science or as a branch of mathematics. Yes indeed, Bill -- and that's why in my original post i gave credit to Pythagoras, our worthy brother, for supplying the PROOF that the angle is square and perfect. As to whether he actually sacrificed a hecatomb upon completing the proof, i leave that to historians to determine. B>Mathematically, the 3, 4, 5 set is known as a "Pythagorian triple", for B>obvious reasons. As is 5, 12, 13 and, as you rightly point out, any of an B>infinity of other integer triples. B> B>The 3, 4, 5 is of interest, however, because it is the smallest, and the B>only one in which the integers are consecutive. Good point you made about 3, 4, 5 being the smallest integer series to form a Pythagorean triple and also being the only series of consecutive numbers in that group. There's even more to it than that: 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, 3, 4, 5 is also the only such triplicity to which instantaneous symbolic meanings can be ascribed. (Before someone jumps me here, i know that any other integer sets that form a right triangle can be given derivative meanings, based on which of the Pythagorean categories (evenly-odd, evenly-even, triangular, square, etc. etc.) they fall into or whether by medieval Kabbalistic gematria they work out to spell some deity's name or the name of some virtue or vice -- but only 3, 4, and 5 form a set that lays WITHIN the holy tetraktys.) cf: "The Dimensions of Paradise" by John Michell for more on both the Pythagorean and the Kabbalistic systems of symbolic number-attribution. catherine

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