Date: 12-16-96 (23:05)              Number: 2295 of 2298 (Refer# NONE)
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From: tomasball@aol.com, TOMAS BALL
Subj: Re: 47th Problem of Euclid, as requested
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Here is a version of the talk I gave in Lodge on the 47th prop. of Euclid.
 Please bear in mind this was given in a somewhat rural lodge, so I wrote
for my audience.  I realize a lot has been written about the "mystical"
aspects of this subject, as well as applying a lot of symbolism to it.
That was simply not in the scope of my presentation on this occasion, and
I frankly think that the practical impact Pythagoras' theorem had on the
world completely overshadows any later efforts to "mysticize" it.

The 47th Problem of Euclid...it is one of the most frequently used of
Masonic symbols.  We see it as one of the emblems on the Master's
carpet;  in some jurisdictions it is the jewel of a Past Master; in
Texas it is the central figure of the design on the Grand Master's
apron, as well as of other Grand Officers.  Anderson's Constitutions
of 1723, the first published version of Masonic Laws and Charges,
begins with an ornate frontispiece showing Grand Masters on a
checkered pavement, and there on the floor, is a diagram of the 47th
Problem of Euclid.  Yet many Masons have no idea what the diagram
refers to, or why it should occupy an important place in our system of
symbols.

If the pyramids, and the hanging gardens of Babylon were physical
wonders of ancient world, then the 47th Problem of Euclid was a wonder
of reasoning.  To put it simply, this diagram demonstrates a discovery
which is the foundation of Geometry, and of architecture.  It occupies
a vital place in the history of human knowledge, and, it can be
argued, is the starting point of all science.

Strong words.  But I think, as I explain myself, you will understand
why I feel comfortable assigning such  importance to a few lines drawn
on a piece of paper.

First, who was Euclid?  Euclid was a Greek mathematician, living in
Alexandria, Egypt around 300 BC.  His contribution to our story was
not by originating, so much as cataloging ideas. Euclid, literally,
wrote the book on Geometry.    He compiled everything that was known
at his time about Geometry into a book, which he called Elements of
Geometry.  That book stood as the authority on Geometry for more than
2000 years.  Over the centuries it became the most published book in
the world after the Bible.  Page by page, Euclid presents each
principle of Geometry with detailed explanations, beginning by
defining a point, then a line, and moving on to gradually more complex
demonstrations.  Accordingly, the order in which the problems are
discussed has become the system for cataloging and naming them, much
as we know to quote the Bible by chapter and verse.  The idea we are
interested in was Proposition number 47 of Book 1.

As I said, Euclid only enters our story as the collector and cataloger
of geometrical propositions. The person credited with the actual
discovery of this principle was another Greek philosopher of an even
earlier age.  Pythagoras was born on the island of Samos, in the
Aegean Sea in about 580 BC.  His biographer, Iamblichus, says he
traveled widely, and was initiated into various mysteries, in Tyre,
Babylon, and Egypt before settling in Crotona, a Greek colony in
southern Italy, where a school of his disciples, a sort of early
secret society, grew up.  Both Euclid and Pythagoras are mentioned in
Old Charges and manuscripts of Freemasonry as far back as the 1400's,
usually describing  them in completely the wrong eras of history; for
instance, Euclid is described in some places as a contemporary of
Abraham. It is interesting that Pythagoras is usually spelled very
strangely, like the name had been handed down from mouth to ear for a
long time, or for a short time by people who couldn't hear very well.
The most curious occurrences are when the manuscripts mention a wise
geometrician  named Peter Gower.

Mathematics and numbers were central to the philosophy Pythagoras
taught, but unfortunately, as in the cases of other Greek thinkers,
like Socrates, nothing of his own writings remain, but only those of
his students.  It has been suggested that Pythagoras himself did not
discover the geometric theorem that bears his name, but that it merely
came from the school he founded. I prefer to believe that he did.

Before I go into detail about what exactly Pythagoras hit on, I'm
going to take you even further back in time, to Ancient Egypt.
Obviously, the Egyptians who built the pyramids and other monuments
that have survived the millennia were superb operative masons, and
even then, geometry was central to their craft.  Let me set you a
puzzle.  If you wanted to make a right angle, you would take your
mason's square, and use it to square the angle you were working on.
But what if you didn't have a square to use as a tool, or a protractor
to measure ninety degrees, or another right angle to compare it to.
The Egyptian masons knew the answer; it was one of their secrets, and
I'll let you in on it.  The ancient Egyptians knew that if you took a
rod 3 cubits long, another 4 cubits long, and another rod 5 cubits
long, and laid them end to end in a triangle, the angle where the 3-
and 4-cubit rods met was always a right angle.  To the Egyptians, this
was a wonderful and powerful tool, almost bordering on the magical.
Their chief architects carried a set of rods to use whenever a square
corner was needed. Another method was to take a string  with twelve
cubits marked out on it, and stake it out in a triangle with three
cubits on one side, four on another, and five on the other.  Of course
the unit of measurement could be anything...a cubit, a foot, a meter,
an inch, a yard...it was the relative lengths of 3 by 4 by 5 that
resulted in a right triangle.

But there is a property of this 3-4-5 proportion that makes it even
more curious.  Take the two smaller sides and square their lengths:
3x 3 = 9, and 4 x 4 = 16.  9 + 16 = 25, or 5 x 5.  Another way to say
this is that if  you make a square out of each side, and add the areas
of the two smaller squares, you get the area of the larger square.

Pythagoras found that this held, not just for the 3 by 4 by 5
triangle, but for any right triangle. He started with a what was just
a useful tool and discovered a fundamental rule of nature.  What the
Pythagorean Theorem, also called the 47th Proposition of Euclid, says,
is that for any right triangle, that is, any triangle containing a
90-degree angle, the square of the "hypotenuse," the longer side,
equals the sum of the squares of the two shorter sides.

Today over a hundred ways have been found to prove this proposition.
To explain any of them requires drawing diagrams, which I can't do in
this setting, but all these proofs arrive at that moment of epiphany
when the pieces come together like a jigsaw puzzle, and I can't help
thinking of the day 2500 years ago when the puzzle was first solved.
This morning Pythagoras woke up in a world of chaos, variety and
inexactness, but now the universe has changed, and Pythagoras has
caught nature red-handed in the act of displaying order and following
rules.  Imagine a caveman looking at the Astrodome, imagining it is
just a big hill, then going inside and suddenly understanding the
architecture that holds it up.  Pythagoras had peeked under the veneer
of the universe, and found that space had a kind of architecture, and
that architecture was made of numbers. To us, looking at this from the
vantage point of a couple of thousand years later, The 47th problem
might seem a little less dramatic.  It is, after all, just another one
of the laws of nature.  We have to remember that to the Pythagoreans,
it was a new and wonderful thing to find that there were any laws of
nature.  Even now, we can't explain why space fits together this way,
we're just so used to seeing it that we tend to overlook the
implications of a world ruled by numbers.

Now, it was possible to use Geometry to make predictions, not just on
paper, but in the field.  You could indirectly tell the length of
something it was impossible to measure directly.  If you knew the
lengths of two sides of a right triangle, you could predict the length
of the third, and always be right. The world obeyed numbers, not at
random times, but always.  Armed with this insight, Pythagoras taught
that numbers were even more real than the world they described.  He
uncovered the basis of music theory when he found that you could pluck
a string to make one note, then divide the string exactly by two, and
pluck it to make the note one octave higher.  By dividing the string
length exactly by three, four and five, notes were produced which
harmonized with the first.  To the Pythagoreans, they were discovering
a divine language of pure mathematics.  To us, they were discovering
that the universe could be described, predicted, and understood.

Pythagoras supposedly was so inspired by the discovery of what we now
call the 47th Problem of Euclid that he sacrificed a hecatomb, a
hundred oxen, to the Muses in gratitude.  If so, I would think he got
off cheap.  This single discovery has echoed through history.  The
entire science of trigonometry is based on it.  Mapmaking, astronomy,
architecture, even space travel would be impossible without it.  The
English political philosopher Thomas Hobbes, whose Leviathan was one
of the most important books of the seventeenth century, probably would
never have achieved the fame he did if he had not, at the age of 42,
glanced at a copy of Euclid's Elements in a friend's study, opened to
the 47th proposition.  Hobbes was supposedly so shocked by the
implications of the theorem that he exclaimed, "By God, this is
impossible!"  This single revelation apparently motivated Hobbes to a
fevered lifelong study of geometry, and later physics, philosophy, and
political science.

It has been speculated that any civilized race will have at some point
in its history discovered the Pythagorean Theorem.  As I was writing
this talk I suddenly remembered years ago reading a book by Pierre
Boulle, who also wrote The Bridge on the River Kwai.  The book I
remembered was called Monkey Planet, and was made into a movie called
"Planet of the Apes."  In the book, an astronaut travels to a far-away
planet ruled by a race of intelligent chimpanzees. To complicate
matters, there are humans on the planet, but they are brute animals,
like apes are here. The chimpanzees, speaking their own ape language,
are unable to make sense of our hero's French, and consider him also a
brute until he draws for them a diagram of the 47th proposition of
Euclid, whereupon they realize he is a civilized, intelligent being,
like themselves.

As it has been passed down through the ages, this theorem has grown
from a useful geometric principle into a symbol of the harmony of the
universe.  The Greek historian Plutarch, who lived in the first
century AD said that the 3-4-5 triangle had become a symbol for the
Egyptian gods, Osiris, Isis, and Horus, reminding us of the manner in
which the Christian Trinity is sometimes represented by an equilateral
triangle.  We are told in the Monitor that the 47 Proposition of
Euclid is a symbol to admonish Masons to be lovers of learning.

I would like to close my talk by sharing with you a story I found in a
biography of Pythagoras written in the fourth century AD.  A disciple
of Pythagoras travelling near Crotona, came to an inn, where he fell
ill from the rigors of his trip.  The inn-keeper, being of a
benevolent disposition, cared for the Pythagorean, supplying his needs
as best he could until he finally died. But before the Pythagorean
died, he wrote what the author termed "a certain symbol" on a tablet,
and instructed the inn-keeper to display the tablet outside the inn,
near the road, and to observe if any passer's-by  stopped to notice
it.  The inn-keeper buried the man decently, and more out of curiosity
than expectation did as he asked with the tablet.  A long time later,
a  passing Pythagorean spotted the symbol and stopping to inquire,
learned about what had transpired, whereupon he repaid the inn-keeper
all the expenses he had generously provided to his long-dead brother,
together with a large additional sum out of gratitude.  This is an
obscure and little known story, and the author of the story does not
venture to say what the symbol of recognition was.  It very well may
have been a diagram of Pythagoras' Theorem, since probably only
another Pythagorean at that time would have understood the meaning of
such marks, but another possibility presents itself, because the
Pythagorean brotherhood had another symbol, with which they identified
themselves, and which they used to sign letters to one another, and
which is also quite familiar to Freemasons, the five-pointed star.


Date: 12-18-96 (20:21)              Number: 2346 of 2349 (Refer# NONE)
  To: ALL
From: georgiou@csci.csusb.edu, GEORGE M  GEORGIOU
Subj: Re: An interesting Question?
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Organization: CS dept. Cal. State Univ., San Bernardino
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io30031@nlis.com (Anthony J. Bessey) writes:

> In King Soloman's Temple, there sat two pillars at the main entrance. Each,
> as any Master Mason Knows has a name. Placed upon those two pillars in
> are Two spheres. In todays lodges these two pillars (in the West) also
> have two spheres on top of them. ***Here is the question*** How could
> there have been spheres (representing what they do) at the time of the
> building of King Soloman's Temple, if it was widely beleived and
> accepted as fact that the world was round?

It was known to the ancient Greeks that the earth was round.
Eratosthenes (first or second century B.C. ?) even calcuted the radius
of the earth quite accurately.  Whether the two sheres with the meaning
masons ascribe to them were actually present at King's Solomon's temple is
another story.

--George


