Lecture 2.2  Mathematical Problems

In addition to the Elements and proofs of ancient mathematics, there were three great problems which mathematicians continued to work on:
       
    1. squaring the circle (accomplished by Archimedes): to find the area (in square units) of a given circle
    2.trisecting the angle: bisecting an angle is easy; so is trisecting a right angle; but try trisection any given angle: there is no geometrical solution
    3.  doubling the cube: again doubling a square is easy – that’s in Plato’s Meno – but doubling the cube is hard

Doubling of the square

The doubling of the squared is described by Plato in the Meno;
-you should read the Meno passage
-Plato used this construct to illustrate the doctrine of reincarnation and recollection
    basically, the slave didn’t know the proof before he was introduced it by Socrates and Socrates didn’t tell him the proof; so he must have recollected it from a past life !

Doubling of the cube: Archytas

There’s an ancient, and certainly false (but entertaining) story that there was a plague on the island of Delos, an island sacred to Apollo; the Delians asked Apollo at Delphi what to do about it, and he said that they needed to double the size of his cubic altar.  They asked Plato how to do this, and Plato’s mathematicians got to work on it.

Hippocrates of Chios said that first they
    need to find the second mean proportional (i.e. the cube root)
   
    2xcubed = ycubed  (find y where x is given)

    xcubed:yxsquared :: yxsquared:ysquaredx :: ysquaredx:ycubed  (the ratio being constantly x/y)

Then Archytas provided a brilliant construction, which was, however, not very practical

we did this in class, and you could WOW me on the final, but it won't it required.