String, Straightedge and Shadow  The Story of Geometry
Excerpted from Chapter 11 and 12 of String, Straightedge, and
Shadow (Illustrations by Corydon Bell)
Return the the Main Book Description Page.
11. PYTHAGORAS AND HIS FOLLOWERS
The early story of Greek geometry is strangely different from its founding in
Miletus. Most of what we know is a mixture of myth and magic, shapes and rules,
all revolving around the fabulous figure of Pythagoras.
The "divine" Pythagorasthat was what he was called, not only after his death
but even in his own lifetime. For the latter part of the 6th century B.C. was
still a time of superstition. The Ionian "physiologists" had only tried to find
an orderly pattern in nature. Most men continued to believe that gods and
spirits moved in the trees and the wind and the lightning. And cults were
popular all over the Greek world"mysteries," they were calledthat promised to
bring their members close to the gods in secret rites. Some were even headed by
seers.
Pythagoras was one of these. A native of the island of Samos, not far from
Miletus, he probably had a Phoenician mother and a Greek father, who was a
stonecutter. But he gained such a reputation for wisdom and magical arts that
people began to whisper that he was son of the god Apollo.
Actually, Pythagoras was Thales' contemporary, for a time at least. He was
horn about twenty years before Thales died, so his career spanned a later
period. Parts of that career are a matter of history. The political situation in
Samos became oppressive: a local tyrant, Polycrates, ruled harshly, and the
neighboring Persian Empire demanded heavy tribute. So Pythagoras emigrated, as
did many other refugees. He settled in Croton, a little island off the tip of
Italy. There he founded a famous secret society that contributed a great deal to
the development
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of geometry. We might call it the world's first
mathematics club. But much of Pythagoras' life is enmeshed in legendsnot just
amusing anecdotes, as with Thales, hut wildly fanciful tales. And far too
many discoveries are attributed to him. So we must pick and choose our way among
facts and fables, in telling the story of Pythagoras and his followers.
To begin with, Pythagoras went on where Thales left off. Let us therefore
accept the tradition that he was the older man's student.
Perhaps rumors of Thales exciting new game of string, straightedge, and
shadows had spread throughout Ionia, and people came from the neighboring cities
and islands to take part. Anyway, one visitor in particular was attracted to
Miletus, to learn this new way of thinking and finding rules and tracing forms
upon the groundthe youthful Pythagoras.
The aging Thales must have been pleased with the young man's keen interest;
such penetrating questions showed a real thirst for knowledge. Thales taught
Pythagoras all that he knew. Then he encouraged him to travel for himself in the
ancient lands and study the development of learning at its source.
Pythagoras followed the advice, and his travels were even more extensive than
Thales had been. Fired with enthusiasm by the stories of Babylon, he visited
that fabulous city to absorb the learning of the Chaldean stargazers. Naturally,
also, he wanted to see the ancient pyramids, obelisks, and temples of Egypt.
There he studied the lore of the priests at Memphis and Diospolis.
In addition, he learned a great deal just by traveling to all the known parts
of the Mediterranean world. During his long sea voyages, the Phoenician sailors
taught him much about the
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importance of stars in navigation. And like Thales before him, he
saw things in a way that men never saw them before.
On the open sea, he realized that the surface of the waters was not flat but
curved. Ile could "see" this whenever another ship appeared in the distance. At
first, only the top of its mast was visible over the horizon; then gradually the
whole vessel would come into view as it sailed toward them. Surely then, lie
guessed, the earth must be round! And what about the other heavenly
bodies?
The moon, when it was full, was a round disk in the sky, rosy or yellowish or
silver white. As it waxed and waned, you could imagine that its surface was
curved too, and partly in light and partly in shade. Doubtless the moon also was
spherical.
And the radiant sun itself made a blazing circle in the
heavens! Certainly, concluded Pythagoras, the earth and the sun and the moon and
the planets were all spheres. That was the one perfect form: it must be so! In
history he is given credit as the first person to spread this idea.
Observing and studying in this way, Pythagoras traveled for many years. Some
say he got as far as India and was deeply influenced, for he took up Oriental
dress, including a turban. And certain of his mystical ideas, such as number
magic and reincarnation, were typical of the Fast.
Finally he came back to Samos. We don't really know how his countrymen
received him, but a number of stories suggest that they were indifferent to all
the knowledge he had brought home. This is borne out by the tale of Pythagoras'
first pupil.
Tired of finding no one who would listen to his learning, Pythagoras bought
himself an audience. He found an urchin, a poor and ragged little fellow, and
offered him a bribe. He would pay three oboli for every lesson the boy mastered.
To the urchin this was indeed a bargain. By sitting in the shade for a few
hours, and listening attentively to this wise man, he could make better wages
than in a whole day's work in the hot sun. Naturally, he concentrated hard while
Pythagoras introduced him to mathematical disciplines.
From the simple calculations of the ropestretchers, to the methods of the
Phoenician navigators, to ahstract rules and reasoning, Pythagoras led his pupil
on. Soon the subjects became so interesting that the boy begged for more and
more lessons.
At this point, Pythagoras explained that he, too, was a poor man, and paying
someone to listen was getting to be very expensive. So they reached another
bargain. The boy
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had saved enough to pay Pythagoras for his future lessons. The story
doesn't prove that Pythagoras began to collect a following this way. But it
shows the fascination of the new game of string, straightedge, and shadows, and
forecasts his great role as its teacher.
What we do know for sure is that Pythagoras left Samos and went to settle on
a tiny island off the coast of Sicily, which was then swarming with new Greek
colonies. This Isle of Croton has an immortal place in the history of
mathematics. There Pythagoras finally gathered a group of students around him
and founded his famous Secret Brotherhood.
Like other mystery cults of that time, it was a religious order with
initiations and rites and purifications.
These "Pythagoreans" had a special way of life. The memberswomen as well as
men shared all their simple belongings in common. Because Pythagoras taught the
doctrine of the transmigration of souls, they were respectful to animals and
would eat no meat or fish because in those creatures might live the soul of some
departed friend. Nor would they wear garments made of wool, nor kill anything
except as a sacrifice to the gods. They bound themselves by great oaths to keep
secret all their discoveries and teachings. So devoted were they to their leader
that any argument was resolved by using the words of authority referring to
Pythagoras: 'He himself said it!"
But there was one great trait that set this Brotherhood apart from all the
rest. Pythagoras taught that "knowledge is the greatest purification." So his
followers were, above all, a study group, bent on gaining the knowledge that
would free them from endless rebirths. And to the Pythagoreansas we shall see
this knowledge meant mathematics!
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12. A FAMOUS THEOREM
The most famous thing about Pythagoras is not his Brotherhood at Croton, nor
the weird legend of his spending years in a cave and gaining magical powers. It
is simply a theorem (or formal rule) of geometry.
The Pythagorean theorem says: In any right triangle, the sum of the
squares of the two sides is equal to the square of the hypotenuse.
This theorem and its proof were a basic advance. It became a cornerstone of
ancient geometry and had more influence on theory and more practical
applications than any other. Later writers would call it "the measure of gold."
But perhaps Pythagoras ought to be most famous for something else. He was the
first to teach mathematics as a liberal educationour very term "mathematics"
originated from his course!
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Pythagoras gave lectures on mathemata. In the language of his
time, that was the word for studies; but his use of it came to mean mathematics.
Pythagorean mathemata covered a very large field, but all the parts were
interrelated. Perhaps the quickest way to understand this is just to imagine a
poster at the entrance to the openair meeting place where the lectures were
held.
This was a curious fourfold range of subjects: music to elevate the soul,
numbers and their properties, ancient Babylonian lore about the planets, and the
abstract rules of the new theoretical geometry. Each topic was studied from a
mathematical standpointand the whole course constituted the initiation into
the Secret Brotherhood!
The very term "mathematician" meant one who was admitted to
the inner mysteries, as distinct from a "hearer" or beginner. "Mathematicians'
had to follow a rigorous course for several years, with a stern daily program of
meditation, exercise, and study, before they were even permitted to hear
Pythagoras intone some teachings behind a curtain. Only after full initiation
might they attend his actual lectures.
But the wait may have been worth it. Great mathematical discoveries were
attributed to this strange teacher, the most famous being the Pythagorean
Theorem. So let us use our imaginations and mingle with the initiates at an
exclusive closed lecture. Why not hear the great Pythagoras demonstrate his
immortal proposition? Not that he did it in just these words, but these probably
are the proofs that were used.
Perhaps the session began with an announcement: "I have found at last the
solution to a problem that has long been puzzling us." A hush of awe fell on the
gathering as "Himself"in white robe and gold sandals, his head crowned with a
golden wreathtook pointer and string and straightedge, and began to lecture.
"Listen to our baffling problem. You older members have already worked on it,
but I will review it for the new initiates.
"Here is the Egyptian right triangle, the one used by the
ropestretchers, where the sides of the right angle are 3 units and 4 units, and
the hypotenuse is 5 units." He drew it on a sandy space, and then added a square
on each side, and inner squares. (See illustration.)
"There! You can see, by counting or by calculating the square units, that the
total area of the squares on the two sides of the right angle is equal to the
area of the square on the hypotenuse."
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He beckoned to the newcomers, who crowded close, multiplying and
counting at the same time:
(3 X 3) + (4 x 4) = (5 X 5)
9 + 16 = 25
until all their heads nodded in agreement.
"Now let me show you a Greek design involving right triangles." He drew their
attention to the tiled floor on which they were standing, and then traced a
similar pattern on the sand, outlining the important parts.
"Here the two sides of the right angle are equal, and the same relation
holds." With his pointer, he indicated one triangle and the related squares, and
they all counted together. (See illustration)
"Look! Two triangles plus two triangles equals four triangles. The total area
of the squares on the two sides of this right triangle is likewise equal to the
area of the square on its hypotenuse."
Again he waited until the newcomers nodded their assent, and
then continued:
"In India the priests know other constructions that give similar results;
they guard these numbers closely, but we have found some of them. In
Babylon, a priestly astrologer whispered to me that there was a secret about
this mystery that had never been penetrated."
Now the attention was almost breathless as Pythagoras intoned solemnly:
"That secret is our problem! Would the same relation always be true of
ANY right triangle, no matter what the length of its sides, and how could
you show this?"
At this dramatic moment, he withdrew behind a curtain, while attendants
played on stringed instruments to indicate an intermission. Pandemonium broke
out among the assembled initiates. All the newcomers began talking at once,
making suggestions, arguing, and shouting. The older mathematicians, who had
worked oil this problem themselves, were less noisy but even more
excited.
Finally Pythagoras reappeared. Silence instantly fell over the group as he
resumed his lecture.
"I will now show you how to construct a wondrous figure which discloses that
the answer is always yes! The older 'mathematicians' will realize that by slowly
and carefully defining each step of the construction, and using a few simple
theorems that you already know, this demonstration can he made into a rigorous
proof. Today I will just draw it quickly, so you can all see my great
discovery."
lie signaled to attendants to smooth the sand, and began to draw, using his
pointer to emphasize his words.
"Watch this beautiful construction! I make a square frame,
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any size, and in its corner I place a small square, any
size. Next I draw straight lines, continuing the sides of the small square to
the edge of the frame.
"Do you see what my frame now contains? A small square and a medium
square, and two equal rectangles.
"And nextwe are almost thereI simply add diagonal lines across the
rectangles!
"This is the figure I need. My frame now contains a small square, a
medium square, and four equal right triangles. Now I will ask you
to look more closely at this figure."
Pythagoras beckoned to the attendants, who poured colored sand from jars onto
the parts of the drawing, so the pattern showed plainly.
"Look again!" He used his pointer and spoke with care. "All
the triangles, you know, are equal; each is the same triangle in a different
position. Now, notice how the triangles touch the squares, especially Triangle
2. You can see that the same square is the square on the short
side of the triangle. And the medium square is the square on the
long side of the triangle. So my frame is completely filled by four equal
right triangles plus the square on the short side and the square
on the long side!"
Pythagoras paused while a low murmur of awe rose from the
initiates.
"Now watch!" he intoned. And while they all craned their necks to see, and
the attendants poured more colored sand, Pythagoras drew his final masterful
figure.
"Watch well! I have only to swing and push these four triangles around, like
this, so that they fit perfectly into the four corners of the frame, and my
frame is now completely filled by the same four equal right triangles
plus the square on the hypotenuse!
"Therefore, in any right triangle, the area of the square on one side plus
the area of the square on the other side will add up to the area of the square
on the hypotenuse!"
A
mighty shoutwe can imaginewent up from the assembled inner group of the Secret
Brotherhood. For this theorem was a true landmark in the development of geometry
by the Pythagoreans. Almost all later geometric work involving lengths. and
measurement was based upon it. And this style of solving problems, especially
equations, by diagramming them, would remain a chief trait of Greek
geometry.
But to the initiate who first heard it, the theorem also partook of a
mystical revelation. Tradition says that Pythagoras himself celebrated the
occasionally a noble sacrificean ox, or a hundred oxento his "divine father,"
Apollo. Some ancient writers dispute this, as the Pythagoreans were vegetarians.
Whatever the offering, we can easily picture the festivities described in the
verse of legend. Doubtless the "mathematicians" chanted, torches waved, and
smoke rose from the sacrificial altar,
The day Pythagoras the famous figure found
For which he brought the gods a sacrifice renowned!
