What is the "sense" of an angle? (Hahn - Classical Complex Analysis)

Question

In Hahn's Classical Complex Analysis text, he states a conformal mapping satisfies the condition

(b) the angle between two arcs at $z_0$ is also preserved in sense as well as in size

• pg. 85

He later states about reflections on the Riemann Sphere

These two reflections certainly preserve the size but reverse the sense of the angle between great circles that pass through the origin and the north pole. However, reversing the sense twice amounts to preserving the sense.

• pg. 93

It seems to me that there is some definition of "sense" that he is using, but I can't seem to glean it from his writing. I didn't see a formal definition thus far, and I've never heard the word before. Even a cursory Google search for "sense of an angle" turns up nothing particularly useful.

My best guess for what he means is "direction" or "sign", but it's honestly just a shot in the dark at this point. Does anyone know what he means?

Answer 1

It's simply the orientation (clockwise vs counterclockwise) of the angle. For example, the function $z\mapsto \overline{z}$ does NOT preserve the sense of angles, but it does preserve their size.

Answer 2

It's indeed a very rare expression (also abroad: "sens d'un angle" in french; "senso di un angolo" in italian) meaning the sense in which the angle is measured, as you correctly guessed.

I just found these sources, hoping they're enough for you:

• http://nptel.ac.in/courses/Webcourse-contents/IIT-%20Guwahati/maths3/module_8/maths3_module8.htm

• https://en.wikipedia.org/wiki/Rodrigues%27_rotation_formula

• http://www.numdam.org/article/NAM_1909_4_9__385_0.pdf (1909) in french

• G. Turner "Elementi di Geometria" (1843) in italian