 Euclid's fourth axiom may strike us as a little strange, all right angles are equal. And it might not be obvious why this is an axiom. To understand why this is an axiom, you have to ask yourself, self, what is a right angle? And so we have to go back to what the Greeks thought a right angle was. The Greeks defined a right angle as follows. When two lines meet so that all four angles formed are equal, the lines are perpendicular and the angles are right angles. So if the two lines meet like this, the angles aren't equal and these aren't right angles, they're wrong angles. However, when these two lines meet, all four angles are equal and so all four of these are right angles. And if these two lines meet, all four of these are right angles and all four of these angles are equal. However, what we don't know is whether these right angles are equal to these right angles. And that's what the fourth axiom guarantees. The fourth axiom guarantees that the right angles formed by the first set of intersecting lines are equal to the right angles formed by the second set of intersecting lines. What this amounts to is the fourth axiom states that space is homogeneous. It's the same no matter where you are. Now the first three axioms satisfy our intuitive sense that they should be obvious to any sensible person. The fourth axiom is so obvious that it's actually necessary to explain why it must be an axiom. And then there's the fifth. If a line falls on two lines, so the interior angles formed on one side are together less than two right angles, the two lines, if extended, will meet on that side of the line. And at this point every rational person says, Well let's take that apart. So if a line falls on two lines, so we'll draw two lines and drop a line on it. So the interior angles formed on one side, well how about these, are together less than two right angles. All right, we'll say they're less than two right angles. The two lines, if extended, will meet on that side. Oh wait, I don't know if we can extend the line. All right, we can. And now we can agree, yeah, this is actually an obvious thing that any rational person would agree to. And the only objection here is that it's not obvious when you read it. In order for this fifth axiom to be obvious, we have to draw a couple of pictures and think about what the axiom is saying. Many historians say that the true genius of Euclid was recognizing that this fifth axiom is actually required. Now the fifth axiom is usually referred to as the parallel postulate because it involves parallel lines. Wait, what? To understand the connection, it's necessary to go a little further. And we'll take a look at that connection much later. Now the axioms are assumptions specific to geometry. Euclid also includes five common notions, which we can regard as assumptions that apply to quantitative reasoning in general. So the first common notion is things equal to the same thing are equal to each other. Next, when equals are added to equals, the results are equal. When equals are subtracted from equal, the results are equal. An important one is the fourth common notion, things which coincide are equal. And the fifth common notion, the whole is greater than the part. And from these five axioms and five common notions, plus a lot of definitions, Euclid builds up the 13 books of the elements. As sort of, Euclid actually assumed more than he led on, but it would take almost 2,000 years to realize that. So let's not worry about that yet. Euclid's elements quickly became the standard textbook for the study of geometry. Editions are still being printed. Unfortunately, before printing, all books were handwritten, literally manuscripts. If you wanted your own copy, you'd have to copy it yourself. And it takes just as much effort to copy a great work as a lesser work. So why bother making copies of other geometries when you could make a copy of Euclid? And as a result, we have only fragmentary knowledge of pre-Euclidean geometry. Everything we know about the geometry of Thales, Pythagoras, Hippocrates, Monochmas, Eudoxes, and so on comes from two sources. Reference to their work in the works of later geometries and the portions of their work that were preserved by Euclid.