Is $\pi$ to do with circles or power series? To get straight to the point: is $\pi$ defined as the ratio of the circumference and diameter of a circle, or as the first non-zero root of the power series of $\sin{x}$?
If the former, then $\pi$ would change with different geometries. If the latter, it would stay constant, and one would just need a different ratio for some non-Euclidean geometry?
Also on this topic: are $\sin$ and $\cos$ defined in terms of power series, or something else?, and is $e$ defined as the constant that satisfies $\frac{d}{dx}c^x=c^x$, or in terms of its power series?
Or are these simply all the same thing? It's just that we have recently been learning about power series, and now I am not too sure which definitions of functions are actual definitions,  and which are simply equivalent statements of the idea that we wish to get across.
 A: (1) $\pi$ defined as the ratio of the circumference and diameter of a circle in Euclidean plane geometry.
(2) $\pi$ is defined as the least positive zero of the power series $\sum_{n=0}^\infty (-1)^nx^{2n+1}/(2n+1)!$.  
If (1) is the definition, the value in (2) is a theorem.  If (2) is the definition, the value in (1) is a theorem.  
Neither of these deals with non-Euclidean geometry.  

Are sin and cos defined in terms of power series, or something else?  

There are many possible definitions, but of course when you choose a defintion you must be able to show that the functions you get are the same as those obtained by the already-existing definitions.  The reason for using a particular definition is to start developing the subject.  After a while, all of the usual definitions should be obtained as theorems based on that definition.  From that point on, we don't care which of these was used as the definition.  
Sometimes, a student posts here a question, and asks that the proof be done from the definition.  That student perhaps does not realize that the definition may be different in some other textbook.  So (in order for us to answer here) the definition should be included.  
