 So in the history of geometry, there are three classical geometric problems squaring the circle Trisecting an arbitrary angle and duplicating the cube We say that all of these are impossible to solve And in fact all of these were solved during the fourth century BC Wait, what? So let's talk about these problems a little bit the first one Duplicating the cube emerges as follows suppose you have a cube with a side length of a You want to find a cube with twice the volume? We'll have a side length of x where a cubed is to x cubed as one is to two And so the problem of duplicating the cube comes down to how can we find x? To solve this problem. We have to introduce some extra terms So the first important term here, which will show up many times in the history of mathematics is that of a mean proportional This is not a proportional that goes around bullying other Proportionals, but rather it emerges as follows given two quantities a and b. We can insert one Mean proportional by finding an x where a is to x as x is to b and You can see here that in this context the idea of a mean proportional is that it's someplace in the middle It's someplace between a and b Well the standard rule in math is anything you can do once you can do as many times as you want to so we've put one Mean proportional between a and b. Well, let's see if we can insert two mean Proportionals and we'll do that by finding x and y where a is to x as x is to y as y is to b and We can find three mean proportionals x y and z and so on so another term that will show up is the geometric mean and That comes from inserting one mean proportional So if we insert one mean proportional x between a and b we have a is to x as x is to b and in modern terms this gives us a over x is x over b or cross-multiplying a b is equal to x squared Consequently, we have the following result given a and b the mean proportional x satisfies x is the square root of a times b and x is also known as the geometric mean So this brings us to Hippocrates of Kios Kios is one of the many islands that make up Greece and Hippocrates of Kios shouldn't be confused with a much more famous contemporary Hippocrates of Kos Hippocrates of Kos was the physician Hippocrates of Kios was an mathematician We actually know quite a bit about the life of Hippocrates He lived around 430 BC and he was a merchant according to one story one of his ships was seized by pirates and He found out that its goods appeared in Athens and so he went to Athens to try and retrieve them This required dealing with the Athenian legal system, which took time So to keep himself entertained he attended some lectures in mathematics and after a while he said to himself Hippocrates you can do that and so he became the first person We know by name to be paid to be a teacher of mathematics Hippocrates also wrote the first known textbook on geometry, which he called the elements We'll talk about that more later Now Hippocrates considered the problem of finding two mean proportionals between a and b and his results are most easily understood if we rewrite this in a somewhat modernized format So this first proportionality a is to x as x is to y We can write that as a over x equals x over y and if we multiply both of these by a over x and Simplify a squared is to x squared as a is to y But wait, there's more if you look at the first and third ratios that says that a over x is equal to y over b So this means we can multiply one side by y over b and the other side by a over x Then simplify and this is a very interesting and important result Because of b is to a then a cubed is to x cubed as one is to two But this is exactly what we need to do in order to duplicate the cube And what that means is the problem of duplicating the cube can be solved by finding two mean Proportionals and it's worth pointing out that Hippocrates did a very mathematician thing He took one problem duplicating the cube and said it's the same as another problem inserting two mean proportionals Of course, he didn't have any idea of how to solve this problem of inserting two mean Proportionals, but it did offer two methods of attack on the same problem And in fact it led to one of the solutions So a little while later Monachmas who lived around 350 BC found a relationship between different lengths Associated with what we now call conic sections We'll talk about Monachmas's work with conic sections elsewhere But in modern terms we might describe Monachmas's results as follows for a parabola There is a parameter K where the distance to the vertex and the point satisfies the relationship K y equals x squared And we might view that as the equality of a rectangle with with K and height y With the square with the side of x and for what we call a rectangular hyperbola There is an area p squared where p squared is equal to x y Equivalently the area of a rectangle with one vertex on the hyperbola is equal to a particular square and Monachmas was able to use these relationships to find two mean Proportionals and because it was the same problem to duplicate the cube So later Greek writers described a general strategy for solving problems in geometry analysis and synthesis in Analysis we assume the problem already solved Then try to work towards a starting point Then in synthesis we begin at our starting point and work towards the solution So in this case suppose we've already solved the problem and we've already found those two mean Proportionals x and y so we already know a is to x as x is to y as y is to b Now if we take a look at these first two ratios that gives us a over x is x over y And if we cross multiply we find that a y equals x squared Which tells us these solutions x and y are going to be the ordnance of a parabola But wait, there's more if we take a look at the first and last ratios We have a over x equals y over b and cross multiplying gives us a b equal to x y Which means that x y are the ordnance of a hyperbola and what this means is that we can find these two mean proportionals by looking at a specific parabola and a specific hyperbola and Seeing where they intersect And so now comes the synthesis since duplicating the cube requires to a equal to b Let's construct the parabola b y equals x squared. Well, that's 2 a y equals x squared We'll also construct the hyperbola a b equals x y or 2 a squared equals x y Locate their intersection point which will give us x and y and Since we have 1 is to 2 as a cubed is to x cubed Then x will be the side of the cube with twice the volume of the cube with side a and that solves our problem