<The Detail of This Chapter.pdf>
The stage of gravitation: from the sun to the cosmos
<Euclid geometry->non-Euclid geometry->Riemann geometry>
The fifths axiom: why a pure lines constitution can determine angle? or if we can determine angle from a pure lines constitution, then such a stipulation must be an independent axiom!
To avoid Euclid's physical style of studying geometry, (such style is usually be limited by our intuition easily ) we use some kind of mathematical representation of geometry object: analytic geometry.
Since point is the most fundamental geometry object, then the metric-a function that determine the distance between two points, is the most fundamental character of a specific geometry, of course, what is named distance in the definition of matric must be realizable in the specific geometry.
Analytic geometry's primary method which representing geometry object is using numbers to mark a point, that means we must have a set of stipulation to distinguish points. So we need coordinate system.
A coordinate system: a set of stipulation that mark points.
1.the simplest coordinate system is Descartes coordinate system for Euclid space.
then, we need an algebraic structure to gauge