Construction of analytical geometry

Question

In Lee's book on axiomatic geometry while constructing analytical geometry he uses integral instead of arccos(x) to define angle. And he mentions he did this for avoiding circularity. I have doubts about using integral here because concept of integral calculus arises from finding the area of a region in a cartesian plane and I thought this causes a circularity because we didn't construct cartesian plane. Also I couldn't fully comprehend why using arccos(x) causing circularity if we define angle and trigonometric functions. I'm not sure but maybe we can define vectors as line segments in cartesian plane and use defnitions of angle and trigonometric functions. Am I missing something or right about the circularity of construction.

Answer 1

The problem is you have the angle as geometric object, an equivalence class of wedges. And as number, as argument of certain analytical functions.

The problem now is that sine and cosine etc. are defined on both sides, as proportions of sides in a right triangle and as the values of the mentioned analytical functions (via power series, differential equations, functional equations). As the purpose of the discussed text is to bridge these two scenarios, it can be easy to accidentally slip up because of too great familiarity with both concepts. So it can be helpful to artificially create some distance.