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The 47th Problem of Euclid

To: freemasonry-list@sacsa3.mp.usbr.gov,sustag-principles@ces.ncsu.edu
From:  Cyronwode@AOL.COM
Subject: The 47th Problem of Euclid (a.k.a the Pythagorean Theorem)
Date:    Tue, Mar 14, 1995 5:46 AM PST

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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