Euclid as the Father of Geometry

Euclid and his long-lasting textbook "Euclid's Elements" (and how much Abraham Lincoln liked them!).

"The laws of nature are but the mathematical thoughts of God."

And this is a quote by Euclid of Alexandria.

He 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 like that.

And they said, "Well, 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 about geometric things.

When you talk about Greek mathematicians like Pythagoras

(who came before Euclid).

The reason why people often talk about "Euclidean geometry" is around 300 B.C.

(and this over here is a picture of Euclid painted by Raphael, and no one really knows what Euclid looked like

or even when he was born or when he died, so this is just Raphael's impression of what Euclid might have looked like

while he was teaching in Alexandria).

But what made Euclid the "Father of Geometry" is really his writing of "Euclid's Elements".

And, "Euclid's Elements" was essentially a 13-volume textbook

(and arguably the most famous textbook of all time).

And what he did in those thirteen volumes was a rigorous, thoughtful, logical march

through geometry, number theory and solid geometry (geometry in three-dimensions).

And this right over here is the frontispiece of the English version---

or the first translation of the English version---of "Euclid's Elements".

This was done in 1570.

But it was obviously first written in Greek, and, during the Middle Ages,

that knowledge was shepherded by the Arabs and it was translated into Arabic.

And then eventually the late Middle Ages translated it into Latin and then eventually English.

And when I say that he did a "rigiorous march", Euclid didn't just say,

"the square of the length of two sides of a right triangle is going to be the same as the square of

the hypotenuse..." and all these other things (and we'll go into depth about what all these mean).

He says, "I don't want to feel good that it's probably true. I want to prove to myself that it's true."

And what he did in "Elements" (especially the six volumes concerned with planar geometry),

was 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" (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 sat in has this property."

He said, "I've now proven that this is true".

And then, from there, he could 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 that for an entire "set" of knowledge.

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 for about 2,000 years after Euclid (so this is an unbelievable shelf life for a textbook!),

people didn't view you as educated if you had not read and understood 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 second only to the Bible.

When the first printing presses came out they said "Okay, let's print the bible. What next?"

"Let's print 'Euclid's Elements'".

And to show that this is relevant into the fairly recent past (although it may depend whether or not you argue that

150-160 years ago is a 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 he 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 is saying there is this word "demonstration" that means 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 in Lincoln's era)

"...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 proven 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, 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 until

I could give any proposition in the six books of Euclid at sight."

(This refers to the six books concerned with planar geometry.)

"...I then found out what 'demonstrate' means and went back to my law study."

So one of the greatest American Presidents of all time felt that, in order to be a great lawyer,

he had to understood---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 "fine-tune" his mind

to become a great President.

And so, what we're going to be doing in the geometry playlist is essentially that.

What we're going to study---we're going to think about how do we "rigorously" prove things?

We're essentially going to be---in a more modern form---studying what Euclid studied 2,300 years ago.

To really tighten our reasoning of different statements and be sure that when we say something,

we can really prove what we're saying.

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'll be doing is Euclidean geometry)

this is really what math is about.

Making some assumptions and then deducing other things from those assumptions.