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## Εισαγωγή στην Ευκλείδεια γεωμετρία

# Euclid as the father of geometry

## Απομαγνητοφώνηση βίντεο

"The laws of nature are but the
mathematical thoughts of God." And this is a quote by
Euclid of Alexandria, who was a Greek mathematician
and philosopher who lived about 300
years before Christ. And the reason why
I include this quote is because Euclid is considered
to be the father of geometry. And it is a neat quote. Regardless of your views of
God, whether or not God exists or the nature of God,
it says something very fundamental about nature. The laws of nature are but the
mathematical thoughts of God. That math underpins all
of the laws of nature. And the word geometry
itself has Greek roots. Geo comes from Greek for Earth. Metry comes from
Greek for measurement. You're probably
used to something like the metric system. And Euclid is considered to
be the father of geometry not because he was the first
person who studied geometry. You could imagine the very
first humans might have studied geometry. They might have looked at
two twigs on the ground that looked something
like that and they might have looked at
another pair of twigs that looked like that and
said, this is a bigger opening. What is the relationship here? Or they might have
looked at a tree that had a branch that
came off it like that. And they said, oh
there's something similar about this opening here
and this opening here. Or they might have asked
themselves, what is the ratio? Or what is the relationship
between the distance around a circle and
the distance across it? And is that the same
for all circles? And is there a way for
us to feel really good that that is definitely true? And then once you got
to the early Greeks, they started to get even
more thoughtful essentially about geometric things when you
talk about Greek mathematicians like Pythagoras, who
came before Euclid. But the reason why
Euclid is considered to be the father of
geometry, and why we often talk about Euclidean
geometry, is around 300 BC-- and this right over
here is a picture of Euclid painted by Raphael. And no one really knows
what Euclid looked like, even when he was
born or when he died. So this is just
Raphael's impression of what Euclid might
have looked like when he was teaching in Alexandria. But what made Euclid
the father of geometry is really his writing
of Euclid's Elements. And what the Elements
were were essentially a 13 volume textbook. And arguably the most
famous textbook of all time. And what he did in
those 13 volumes is he essentially did a
rigorous, thoughtful, logical march through geometry
and number theory, and then also solid geometry. So geometry in three dimensions. And this right over here
is the frontispiece piece for the English version,
or the first translation of the English version
of Euclid's Elements. And this was done in 1570. But it was obviously
first written in Greek. And then during much
of the Middle Ages, that knowledge was
shepherded by the Arabs and it was translated
into Arabic. And then eventually in the
late Middle Ages, translated into Latin, and then
obviously eventually English. And when I say that he did a
rigorous march, what Euclid did is he didn't just
say, oh well, I think if you take the length
of one side of a right triangle and the length of the other
side of the right triangle, it's going to be the same as the
square of the hypotenuse, all of these other things. And we'll go into depth about
what all of these things mean. He says, I don't want
to just feel good that it's probably true. I want to prove to
myself that it is true. And so what he did in Elements,
especially the six books that are concerned with
planar geometry, in fact, he did all of them,
but from a geometrical point of view, he started
with basic assumptions. So he started with
basic assumptions and those basic assumptions
in geometric speak are called axioms or postulates. And from them, he proved,
he deduced other statements or propositions. Or these are sometimes
called theorems. And then he says, now I know if
this is true and this is true, this must be true. And he could also prove that
other things cannot be true. So then he could prove that this
is not going to be the truth. He didn't just say,
well, every circle I've said has this property. He says, I've now proven
that this is true. And then from there, we can go
and deduce other propositions or theorems, and we can use
some of our original axioms to do that. And what's special
about that is no one had really done that
before, rigorously proven beyond a shadow of a doubt
across a whole broad sweep of knowledge. So not just one
proof here or there. He did it for an
entire set of knowledge that we're talking about. A rigorous march
through a subject so that he could build
this scaffold of axioms and postulates and
theorems and propositions. And theorems and propositions
are essentially the same thing. And essentially for
about 2,000 years after Euclid-- so this
is unbelievable shelf life for a textbook-- people
didn't view you as educated if you did not read and
understand Euclid's Elements. And Euclid's Elements,
the book itself, was the second most printed
book in the Western world after the Bible. This is a math textbook. It was second only to the Bible. When the first printing
presses came out, they said OK, let's
print the Bible. What do we print next? Let's print Euclid's Elements. And to show that this is
relevant into the fairly recent past-- although
whether or not you argue that about
150, 160 years ago is the recent past--
this right here is a direct quote from
Abraham Lincoln, obviously one of the great
American presidents. I like this picture
of Abraham Lincoln. This is actually a photograph
of Lincoln in his late 30s. But he was a huge fan
of Euclid's Elements. He would actually use it
to fine tune his mind. While he was riding
his horse, he would read Euclid's Elements. While was in the White House,
he would read Euclid's Elements. But this is a direct
quote from Lincoln. "In the course of
my law reading, I constantly came upon
the word demonstrate. I thought at first that
I understood its meaning, but soon became
satisfied that I did not. I said to myself, what do
I do when I demonstrate more than when I
reason or prove? How does demonstration
differ from any other proof?" So Lincoln's saying, there's
this word demonstration that means something more. Proving beyond doubt. Something more rigorous. More than just simple
feeling good about something or reasoning through it. "I consulted
Webster's Dictionary." So Webster's Dictionary
was around even when Lincoln was around. "They told of certain proof. Proof beyond the
possibility of doubt. But I could form no idea of
what sort of proof that was. "I thought a great
many things were proved beyond the possibility
of doubt, without recourse to any such extraordinary
process of reasoning as I understood
demonstration to be. I consulted all the dictionaries
and books of reference I could find, but with
no better results. You might as well have
defined blue to a blind man. "At last I said, Lincoln--"
he's talking to himself. "At last I said,
Lincoln, you never can make a lawyer if
you do not understand what demonstrate means. And I left my situation
in Springfield, went home to my father's
house, and stayed there till I could give any
proposition in the six books of Euclid at sight." So the six books concerned
with planar geometry. "I then found out what
demonstrate means and went back to my law studies." So one of the greatest
American presidents of all time felt that in order
to be a great lawyer, he had to understand,
be able to prove any proposition in the six books
of Euclid's Elements at sight. And also once he was
in the White House, he continued to do this
to make him, in his mind, to fine tune his mind to
become a great president. And so what we're going to be
doing in the geometry play list is essentially that. What we're going to study is
we're going to think about how do we really tightly,
rigorously prove things? We're essentially going to be,
in a slightly more modern form, studying what Euclid
studied 2,300 years ago. Really tighten our reasoning
of different statements and being able to make sure
that when we say something, we can really prove
what we're saying. And this is really some
of the most fundamental, real mathematics
that you will do. Arithmetic was really
just computation. Now in geometry-- and what
we're going to be doing is really Euclidean
geometry-- this is really what math is about. Making some assumptions and
then deducing other things from those assumptions.