[b][color=#0000ff]Hipprocrate of Chios[/color][/b], a famous Greek mathematician did a lot of influential work on the three classical problems.  He considered Delian problem in a more general form - the problem of constructing the so-called "[b][color=#0000ff]double mean proportionals[/color][/b]":  Given two lengths [math]a[/math] and [math]b[/math], find the lengths [math]x[/math] and [math]y[/math] such that[br][br][center][math]a:x=x:y=y:b[/math][/center][br][br]By simple algebra, it is easy to see that if [math]a=2[/math] and [math]b=1[/math], then[br][br][center][math]\frac{2}{x}=\frac{x}{y}=\frac{y}{1}\Rightarrow2=y^3\Rightarrow y=\sqrt[3]{2}[/math][/center][br][br]How Hipprocrate came up with such idea?  One possible explanation is that he was inspired by the following well-known result at his time: The following two problems are equivalent:[br][br][list][*]Given a square, construct a square whose ratio between their areas equals a given ratio [math]a:b[/math].[/*][br][*]Given [math]a[/math] and [math]b[/math], construct the "mean proportional" between them i.e. a number [math]y[/math] such that [math]a:y=y:b[/math].[/*][br][/list][br][br]It is easy to prove this by modern algebra.  For simplicity, suppose [math]a[/math] is the length of the side of the given square.  Then we have[br][br][center][math]\frac{a^2}{y^2}=\frac{a}{y}\frac{y}{b}=\frac{a}{b}[/math][/center][br][br]That is to say, [math]y[/math] is the length of the side of the required square.