Introduction to Euclidean geometry
Euclid as the father of geometry Euclid and his Elements (and how much Abraham Lincoln liked them)
Euclid as the father of geometry
- "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.
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