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In plane coordinate geometry, with a system of rectangular axes, I wish to press the claim of points whose coordinates are given by complex numbers to be represented on the plane in just the same way as all other points in the plane.
The purpose of the present note is to show how the student who is acquainted with the methods only of Using and transforming rectangular axes is already in a position to determine the geodesic curvature and torsion of any curve traced on a surface, and in particular to prove Gauss’ formula for the product of the principal curvatures in terms of the geodesic curvaturesof the lines of curvature.