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Studying for a test? Prepare with these 4 lessons on Geometry foundations.
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"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.