 The first Greek Geometer we know by name is Thales, who lived around 600 BC. Well, actually we have no idea what he looked like, and even though there are images that claim to be of Thales, they're taken from sculptures that were made long after he died. So he looked Greek? Probably. The probably is because he lived in Miletus, which was a Greek colony in the region known as Ionia. On the shores of Asia Minor, which is now the western coast of Turkey. Many stories are told about him. For example, one story is told that he fell into a well while stargazing. This is actually a variation on the if you're so smart, why aren't you rich? And in response to this, so the story goes, Thales used his power of observation to determine that a particular year would be very good for the olive harvest, and in modern terms took up options on all the olive presses in the area. When harvest time came around, he was able to rent out these olive presses at a very good price and in fact became rich, at which point he gave away all of the money he had made to charities because he had established the point that he could be rich if he wanted to be. The most interesting story about Thales is that he predicted an eclipse. And what's important about this is this allows us to identify when he lived because we know exactly when the eclipse he predicted occurred. It was May 28, 585 BC. This eclipse is historically important because to the east of Meletos, the Lydians and the Medes were about to fight a battle when the eclipse occurred. But when the eclipse occurred, both sides took it as a sign of displeasure from the gods and they swore a treaty of eternal peace. We'll see how that goes in a little bit. Thales made a number of contributions to what we might call applied mathematics. And according to one story, Thales is reputed to have determined the height of the Great Pyramid by noting the following. At a certain time of day, a man's shadow has the same length as the man's height. And a little bit of experimentation will show that that's actually true for all objects and so the height of the pyramid can be found by measuring the length of its shadow. And so to the extent that this is a trigonometric exercise we can regard Thales as the founder of trigonometry. Thales is also credited with discovering four other geometrical theorems. First, a circle is bisected by its diameter. Second, the base angles of Isosceles triangles are equal. Third, if two triangles have an equal side and equal angles off the equal side then the triangles themselves are equal. This is what we usually refer to as the angle side angle axiom of congruence. And then finally an angle inscribed in a semicircle is a right angle. The next geometry of note is Pythagoras who lived around 550 BC, about half a century after Thales. Pythagoras was born in Samos, an island in Ionia. We don't know a lot for certain about his life, but the story is told that he traveled to Egypt which is probably true. There's another story that he spent some time in Mesopotamia and that's possibly true. And there are even stories that he made it as far east as India. This is unlikely, but a little later on we'll see why the story is important. And we do know that eventually Pythagoras ended up in Crotona which is in southern Italy. Now remember that battle between the Lydians and the Medes? They swore a perpetual peace, but this was rendered irrelevant because both sides were eventually conquered by the Persians. And Pythagoras lived during this time period. The Persians consolidated their power in Asia Minor and then went to conquer Egypt which they did in 525 BC. After the conquest they transported, which is a nice way of saying conscripted a number of Egyptians to work back in Mesopotamia. Now if Pythagoras was in Egypt during the time of the conquest and he could have been, it is possible that Pythagoras may have been among those transported back to Mesopotamia. At this point we get into the realm of conjecture. If he spent time in both Egypt and Persia and spoke with scribes in both places he would have discovered something rather unsettling. As we've seen, the Egyptians had a way to calculate the area of a circle and so did the Mesopotamians. But the Egyptian formula for the area of a circle gave different results than the Mesopotamian formula for the area of a circle. And even if we take area to be a cadastral area and not geometric this does raise a question, what is the area of a circle? And more generally, how can we reconcile different mathematical systems? And this should be disturbing because we're accustomed to thinking about mathematics as all the same. It doesn't matter what your nationality is, what the color of your skin, what your gender, what your religion None of these things matter because mathematics is the same for everybody. But the reason that mathematics is the same for everybody is how we do it. And this is the major contribution of Pythagoras. Pythagoras is credited with introducing the deductive method to mathematics. By beginning with a set of agreed upon axioms one could logically deduce other results. And because playing the game of mathematics requires beginning with this agreed upon set of results we all come to the same conclusions regardless of ethnicity, race, gender, or religion. Yes, yes, Nikolai, we'll tell your story later. What this means is that in a very real sense mathematics begins with Pythagoras. Unfortunately, we don't actually know what Pythagoras discovered or deduced. And part of the reason is that eventually Pythagoras ended up in Crotona, which is in southern Italy where he established a mystic school of philosophy. No, not that one. The Pythagorean school was based around the premise that all is number. And to the Pythagoreans this meant the entire universe could be reduced to numbers and the relationships between numbers. Now if this sounds very much like how we view the world it is. And the major difference we would have between the Pythagorean philosophy and our own philosophy is that we would not say the universe is number but that all the relevant features can be represented by number. And in this particular context when the Pythagorean said that all is number what they really meant is all is whole number and whole number relationships. So we'll take a look at those next.