Why can't a circle be squared, but a lune can?

[The Anomaly]

I'm new to this forum, and I am very excited to be a member of this esteemed community. Anyway, I've been reading the Journey Through Genius, by William Dunham, and I am truly loving it. However, I had a question about a statement in the text, and I thought I'd ask if you guys could help me out with it.

I was reading about Hippocrates' proof that the lune is squarable, and the author said that despite hundreds of years of effort, the circle could not be squared. In other words, that there could not be made a square that has the same area as a circle. This was then proved in the nineteenth century by Ferdinand Lindemann.

Anyway, I understand both Hippocrates proof, and Lindemann's proof, and I believe I fully understand why it'd be impossible to make a square that is the same area as a circle. It makes sense--I mean, pi is transcendental and all, and as such it can not be drawn. However, what I don't understand is why we can still square a curved shape such as a lune. Would not a lune simply be a section of a circle? I mean, doesn't it have that same curve that a circle has--just that it is not a perfect circle?

Basically, what makes a circle so intrinsically special that it can not be squared, while other curved ones can be? Is it simply because their areas are not linked to a transcendental number like the circle is?

 

[HallsofIvy]

What is special about a circle is just what you said above: its area depends on the constant [itex]\pi[/tex] which is transcendental. It is NOT true that a lune can alwats be squared. What is true is that a certain lune, the lune of Hipocrates, which is formed by intersecting a circle of Radius R with a circle of radius $\sqrt{R/2}$ passing through points on the first circle a quarter circle apart (see http://en.wikipedia.org/wiki/Lune_of_Hippocrates), has area (1/2)R², a rational number, and so is "squarable".

 

[The Anomaly]

What is special about a circle is just what you said above: its area depends on the constant [itex]\pi[/tex] which is transcendental. It is NOT true that a lune can alwats be squared. What is true is that a certain lune, the lune of Hipocrates, which is formed by intersecting a circle of Radius R with a circle of radius $\sqrt{R/2}$ passing through points on the first circle a quarter circle apart (see http://en.wikipedia.org/wiki/Lune_of_Hippocrates), has area (1/2)R², a rational number, and so is "squarable".

Alright, thanks for the answer. That's just what I wanted to know.