Euclid Elements -- Introduction 1.0

This series of videos is going to be on Euclid's Elements, and Euclid's Elements is one of the first staple math textbooks ever written. Euclid was Greek and he lived after the time of Plato, before the time of Archimedes, and the work of Euclid's Elements is focused on things like geometry. A lot of our formulas and ideas of triangles and other shapes comes from Euclid's Elements. Some basic number theory comes from Euclid's Elements, including a proof on the infinitude of primes, and it gets a little bit into some 3D geometry as well. So this particular video is going to start us off on the first book of Euclid, which focuses mostly on triangles and finishes up with its climax at the Pythagorean theorem proof known as the windmill. Most of the books of Euclid start with a series of definitions as well as common notions and postulates. Essentially, Euclid starts off with a set list of assumptions, definitions of things, things that we are allowed to do in geometry, and with those base assumptions we deduce other properties of things in mathematics, such as the Pythagorean theorem. So the role of definitions in mathematics is such that it helps us identify things. The first definition in Euclid's Elements in book one is that a point is that which has no part. So that definition should help us identify points when we come across them in geometry, essentially an object such that we can't break it up into smaller parts. He follows that up with a definition of a line such that a line is a one-dimensional thing. There's no breadth to it. The third definition tells us that the ends of that line are points. The significance of that is that the ends of the line, the start and the end, are the only identifiable points that we can pick out. The significance of that is that we can't pick out the middle or a quarter of the way through the line because we really can't take a pencil or something and put it on the line and say that that exact point is the middle. The fourth definition tells us that a straight line is a line which lies evenly with the points on itself, which is not a very great definition because it's hard to think about what lying evenly with the points on yourself means. But we all know what he means. We all know what straight lines are, even though we can't really identify a straight line based off of that definition very well. He moves on to talk about surfaces, which are just two-dimensional objects. You can think of it like a blanket, and the ends of that blanket are lines. And then we talk about angles. If you've got two lines and they intersect, then they form an angle between them, as shown here. The most useful angle is that of the right angle. We get a right angle. We take one line and another line. We make them perpendicular to each other. That particular angle that they form is what we call the right angle, or 90 degrees is what we typically measure it as. So that's the particular measurement of angles that we use in Euclid's elements. We still have the acute angles and the obtuse angles as defined as normal in the Euclid's elements, but the right angles is what we use to measure with. Euclid goes on to define boundaries and figures as essentially shapes. And we define a circle as a plane figure. The circle defined in Euclid's elements is that you take a point and you draw a line off of that point. And if you rotate that line about that point, then the other end of that line is going to draw out your circle. And essentially every point on the circle is the same distance as any other point on the circle towards the center. That particular line that we drew is called the radius. And any line that goes from one end of the circle to another end, and it intersects the middle, is called the diameter. Euclid also defines a semicircle, and then goes on to define some rectilinear shapes, things like quadrilaterals, which are things like rhombuses, squares, rectangles. He also defines triangles. Again, in Book 1, the only shape we really deal with are triangles, as well as some parallelograms. If a triangle has a right angle in it, we call it a right triangle. Euclid also defines a couple other triangles that don't really come up as much, but they do become useful. And finally, after some more definitions of quadrilateral figures, we come to the parallel lines definition. Straight lines which being in the same plane and being produced indefinitely in both directions do not meet one another in either direction. So they're both traveling in the same direction. They're never going to meet each other. And that brings us to our postulates. So the definitions allow us to identify objects in geometry, but the postulates help us identify truths, or assumptions rather, that we're going to base the rest of our geometry on. So they're not things that we're necessarily going to identify, but it's conclusions that we can draw without necessarily needing to prove. So the first postulate is to draw a straight line from any point to any point. Essentially what he's saying is that one of the operations we can do in geometry is we can draw a line from one point to any other point. We can also produce a finite straight line continuously in a straight line. What that means is if we have a straight line, then we can continue drawing that line in any direction as far as we want. These two postulates give us what are called the straight edge in geometry. You can think of a ruler, but without the tick marks on it, so you can't measure anything with your straight edge. You can think of like you have a ruler, you line it up with your two points, or with a line. Then you can take your pencil and you can draw along that straight edge as far as you want. So in Euclid's elements, our straight edge is infinitely long. The third postulate is that we can draw a circle with any center and a distance. This gives us our second tool in geometry, which is commonly called the compass rule. So this is a picture of a compass. It's got a pointy sharp end, and you put that end on a point. Then the other end of the compass typically has a pencil attached to it. By extending the end of the compass with the pencil, you can put that on your paper, and you can rotate the compass, and it'll draw you a circle. In Euclid's elements, again, this compass is infinite. We can draw as big of a circle as we want. These are the only two tools or operations that we can use to construct proofs in Euclid's elements. There's not really a whole lot known about why those two tools are the only tools we use in classic geometry. The most I've been able to find on that is from Plutarch's Lives, specifically Marcellus's biography. I'm going to read a couple paragraphs from that, but it says, Eudoxus and Archytas had been the first originators of this far-famed and highly prized art of mechanics, which they employed as an elegant illustration of geometrical truths and as means of sustaining experimentally to the satisfaction of the senses conclusions too intricate for proof by words and diagrams as, for example, to solve the problem so often required in constructing geometrical figures given the two extremes to find the two mean lines of a proportion. Both these mathematicians had recourse to the aid of instruments adapting to their purpose certain curves and sections of lines, but because of Plato's indignation at it and his invectives against it as the mere corruption and annihilation of the one good of geometry, which was thus shamefully turning its back upon the unembodied objects of pure intelligence to recur to sensation and to ask help not to be obtained without base supervisions and deprivation from matter, so it was that mechanics came to be separated from geometry and repudiated and neglected by philosophers took its place as a military art. And specifically he was talking about Archimedes and his mechanics work. And what this is talking about is Plato had this theory that, well, if I draw a circle and I want to reason about this circle it doesn't work out so well because my circle isn't actually a circle. Any circle I come across with in life, if you look closely enough at it, looks less and less like a real circle. So this posed a problem mathematically is that we're supposed to be drawing all these figures and reasoning about them but these circles aren't real circles and these lines aren't very straight. So the way Plato reconciled this is his theory of forms. Essentially circles exist in this cognitive or intellectual realm that we can reach. Essentially we all know what circles are and we all know what we mean by circles. Just because I can't draw a circle correctly doesn't mean I'm not reasoning correctly about the circle. Eudoxus and Archytas had developed this new theory of mechanics such that they were able to perform proofs of things by using sensation and by using physical things, hence Plato's indignation. And the claim is that because of this indignation Plato had that that is the reason why we limit ourselves in geometry to using only compass and straight edges as defined in the Euclid's elements. So moving on to the fourth postulate, that all right angles are equal to one another. This once again gives us that intrinsic measurement of angles that we can use. And finally the fifth postulate, and also a controversial postulate, if a straight line following on two straight lines make the interior angle on the same side less than two right angles, the two straight lines if produced indefinitely meet on that side on which are the angles less than the two right angles. And this seems pretty reasonable to us when we look at it. If I draw a straight line and I draw two lines off of it that are not perpendicular to that line, when we produce those lines they're going to meet somewhere. This is controversial because for many centuries mathematicians thought that they could prove this using the previous four postulates, but that turned out actually not the case. That actually this fifth postulate is what creates this Euclid geometry world, essentially this flat geometry. If you want to go read up on some non-Euclidean geometry, such as hyperbolic or a spherical, then you can look up the work of Levitshevsky who figured out that this fifth postulate is actually a special case. But there are geometries out there which violate this postulate, and they create some interesting geometries such that the angles of a triangle no longer add up to 180 degrees, and things like that. The last part of the introduction in the last section of this video is going to be on the common notions, and these are similar to the postulates. You could probably put them as postulates and would be fine, but essentially that which is are equal to the same thing are also equal to one another. We typically see this as if A is equal to B and B is equal to C, then A must be equal to C. The second common notion is that if equals be added to equals, the whole are equal. If A equals B, then A plus C equals B plus C. The third common notion is the same, except that if A is equal to B, then A minus C is equal to B minus C. The fourth common notion is things which coincide with one another are equal to one another. A special note on this particular common notion is Euclid is not talking about shapes that look like they should coincide with each other, meaning if I take a circle and I put another circle on top of it and it looks like they're the same, that doesn't mean we can say they are the same. Euclid is saying that if I have a circle and have another circle that actually is directly on top of it, then they are equal. And the last common notion is that the whole is greater than the part. If A is equal to B plus C, then B is less than A and C is less than A. And this concludes the introduction to the first book of Euclid's elements.