Geometry (all content)
Euclid was a great mathematician and often called the father of geometry. Learn more about Euclid and how some of our math concepts came about and how influential they have become. Created by Sal Khan.
Want to join the conversation?
- Is the "Element" textbook used today? Is it still printed for geometry students?
Where can I buy a copy?(705 votes)
- Since theorems in Euclidean geometry are assumed to be true and are the foundation of much of mathematics, wouldn't almost everything we've learned by obsolete if this assumption of truth happened to be false?(192 votes)
- There are other branches of mathematics that do not use Euclid's axioms as their basis, such as spherical geometry and many others. These geometries reject Euclid's axioms and substitute others, and thus the properties of lines and shapes and other things are different from those in Euclid.
But that doesn't mean Euclid is wrong. Euclidean geometry is consistent within itself, meaning the axioms all agree with each other and with all the properties derived from them. That's all you can ask from a branch of mathematics--internal consistency. There is no one universal geometry that satisfies all situations and which contains all possible true statements. So we have to start by defining our terms (axioms), and Euclid was the first one to do that.(36 votes)
- Were any of Euclid's basic geometric assumptions (or axioms/postulates) ever shown to be incorrect?(64 votes)
- There was a big debate for hundreds of years about whether you really needed all 5 of Euclid's basic postulates. Mathematicians kept trying to prove that the 5th postulate (commonly known as the parallel postulate) could be proved from the first four postulates and thus was unnecessary. Some really great proofs were created by mathematicians trying to prove the parallel postulate.
In the late 19th century (approximately 1823), three different mathematicians (Bolyai, Lobachevsky and Gauss) proved independently that there was a different system that could be used that assumed the 5th postulate was incorrect. Later (1868) it was proved that the two systems were equally consistent and as consistent as the real number system. We can see different versions of systems where the parallel postulate is false by assuming that either there are no parallel lines, or that for any line and point not on a line there are an infinite number of parallel lines.
On a side note, in 1890, Charles Dodgson (aka Lewis Carroll author of Alice in Wonderland) published a book with a "proof" of the parallel postulate using the first 4 postulates. Not a great proof and written after it was proven that this could not be proved.(106 votes)
- At3:55Sal talks how postulates are used to deduce theorems. When the postulates were proven does it make them theorems? If not, what is the difference?(46 votes)
- Not quite. The postulates are the things that we assume to be true from the beginning that form the foundation for all of our theorems. There are five in Euclidean geometry: that any two points can be connected by a straight line, that any line segment can be stretched out forever in either direction, that we can always define a circle given a center and a radius, that all right angles are congruent, and that for any line and any point not on that line there is exactly one line parallel to the given line that passes through the given point. None of those postulates can be defined from each other, but with the five of them we can prove everything in geometry.(67 votes)
- I really did not get the video. But I really want a copy of the "Element" textbook so I can learn more. Does anyone know where to get a copy?(15 votes)
- If you search Amazon, you'll find no lack of copies of Euclid's Elements. Alternatively, any geometry textbook that is more than thirty years old follows Euclid's path quite precisely. Also, if you google for Euclid's Elements, the top hit will be a fantastic online companion hosted by Clark University that gives context and commentary to every proposition along the way.(22 votes)
- can somebody please explain euclids 5th postulate cause i dont get it(10 votes)
- Here is what the Fifth Postulate means, though Euclid didn't state it quite this way:
Two lines intersect a third line.
Measure the interior angles of the two lines on the same side of the third line.
Add the two interior angles together.
If the sum of those two interior angles is less than 180°, then those lines will intersect on that side of the third line.
If the sum is greater than 180°, then those lines intersect on the other side of the third line.
If the sum is exactly 180°, then those lines are parallel and will never intersect.
This postulate has never been successfully proven. It only works on flat surfaces (Euclidean geometries). On curved surfaces, the fifth postulate is not true.(12 votes)
- As I understand it, the postulates/axioms are assumptions and they are used to construct theorems. But how does one come up with a postulate? Are we free to assume whatever we want? What is to prevent me from assuming any kind of nonsense I want and then building a system of proofs from it - something like building lego buildings? Of course the lego structures will be 'internally consistent' in that it forms a complete world by itself, but for practical purposes, it would be totally useless.
This is part of my fundamental gripe with math and why I never understood anything beyond how to count my change. It seems to me that math is filled with assumptions with no rhyme or reason whatsoever. For example, why should 2 x 2 = 4? Why not 5, or 0 or 100? It is similar to a language, the inventors of which came up with grammatical rules according to their whims and fancies, and which new learners are expected to accept as gospel truth.
I have a background in biology and I enjoy the cautiousness with which biologists and biochemists approach the process of proving or testing a hypothesis. Math seems to me to be the antithesis of that process.(4 votes)
- who was Euclid ?(4 votes)
- When we write different lines, rays, line segments, planes, collinear, etc, why does the letters must be capitalized for?(5 votes)
- The capitalized letters usually stand for a point. Since lines, rays, segments, planes are all represented by point(s), the letters that represent the lines, rays, etc are capitalized.
On the other hand, some segments with specific lengths can be represented by uncapitalized letters. For example, a side of a triangle can be represented by the uncapitalized letter of the angle directly opposite to it.(3 votes)
- is it stilll used in these years(3 votes)
- While there have been a number of improvements to basic geometry over the centuries, the basics are much the same as what Euclid wrote of. The main distinction is that we now have modern algebra to express the concepts, whereas Euclid predates that.(2 votes)
"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.