Let me point you to the article: Circular Reasoning by Fred Richman The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162 http://www.jstor.org/stable/2686787 Some main observations

1. Archimedes said that sin x < x <tan x based on an axiom about relative lengths of convex curves ( If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.)

2. One can define radian measure using area rather than arc length (this is the approach of Apostol)

Let me point you to the article: Circular Reasoning by Fred Richman The College Mathematics Journal Vol. 24, No. 2 (Mar., 1993), pp. 160-162 http://www.jstor.org/stable/2686787 Some main observations

1. Archimedes said that $\sin x < x < \tan x$ based on an axiom about relative lengths of convex curves ( If two plane curves C and D with the same endpoints are concave in the same direction, and C is included between D and the straight line joining the endpoints, then the length of C is less than the length D.)

2. One can define radian measure using area rather than arc length (this is the approach of Apostol)