 By 300 BC, the Greeks had invented a number of curves in addition to straight lines, circles, conic sections, quadratracieses, conchoids, and a whole bunch of others. They'd solved a number of problems like duplicating the cube or trisecting the angle, and had looked at problems involving continued proportions. In short, it was an unwieldy mess, and it was very remote from the Pythagorean ideal of deductive geometry. And that's where Euclid came in. Here's a picture. Well, actually, we don't know what he looked like. We do know he lived around 300 BC, and was active in Alexandria, the capital of the Ptolemaic Kingdom of Egypt. Ptolemy II had the idea of making Alexandria the premier city in all of the Mediterranean. A brand new lighthouse would make Alexandria one of the great port cities. And to make Alexandria an intellectual center, Ptolemy established a research institution. The institution was known as the Museum of Alexandria. So-called because of a statue of the Nine Muses, the patron goddesses of the arts and sciences stood at its entrance. The best-known part of the museum today is the Great Library. The library's mandate was to collect copies of everything written in Greek. And we can surmise that Euclid probably had access to the mathematics of Hippias, Menachemus, Eudoxus, Dynastratus, and many, many others. Euclid sought to organize geometry along Pythagorean lines, start with some basic assumptions, and deduce all that is possible. And that sounds like a great idea, until you get down to the details, and in particular the question you've got to ask yourself is, what assumptions are we going to make? And out of the infinitude of possibilities, Euclid chose to begin with five axioms. Now this word means slightly different things to different people. To Euclid and his contemporaries, an axiom was a universal truth that any rational person would agree to. Well, that's easier said than done. And that was Euclid's genius. He was able to pick five axioms that were so reasonable that it took us a very long time, almost 2,000 years, before we realized there could be other ways of thinking. So let's see what those axioms are. Euclid's first axiom, between any two given points, a straight line may be drawn. And this seems to fit the nature of this idea of a universal truth that everyone would agree with. There are some caveats. We usually think about this as referring to two points in a plane, but Euclid actually means any two points. And so we might even be talking about a point in the middle of some solid object. And the real impact of this is that the first axiom states that space has no gaps. Or as they say in New England, you can get there from here. What? They don't say that? No. How about Euclid's second axiom? The second axiom is that a straight line may be extended in a straight line. Now this axiom points to an important but subtle feature of Euclidean geometry. If you can extend something, as the second axiom says that you can, it means that whatever it is can't be an infinitely large object. And so in Euclidean geometry, a line is a finite object that can be extended indefinitely, but it is not an infinite object that we only see part of. And we can look at the second axiom as stating that space has no boundaries. Wherever you are, you can always go further. Euclid's third axiom states that about any point and with any given length a circle may be drawn with the point as center and the length as radius. So no matter where you are, and no matter what radius you want to use, you can draw a circle. And again the important thing to recognize here is that while we generally think about this as applying in the plane, this is true in three-dimensional space. And the third axiom assumes the existence of circles with arbitrary centers and radii. Now let's pause and take a breather. The first three axioms identify the types of figures Euclidean geometry is concerned with. First, we can draw straight lines, so we're talking about figures that are defined by straight lines, and we can also construct circles. We identify these with compass and straight edge figures, but it's important to keep in mind that it's not the tools, but how they're used. So we can use the straight edge to join a line between two points, or extend a straight line and a straight line, or use a compass to draw a circle. But any other use is forbidden, and we can do a lot with compass and straight edge constructions. However, there are some notable absences. The axioms of Euclidean geometry do not include the conic sections, the quadricists, conjoids, or any other curves. In short, the axioms guarantee the existence of straight lines and circles, and nothing else.