On the Composition of Rotations in $\mathbb{R}^3$

Euler stated that the composition of two successive rotations is also a rotation, but did not solve the problem of finding the resultant (axis and angle of rotation) of the composition. It is Rodrigues who solved it. Based on the Rodrigues' formula for a rotation in $\mathbb{R}^3$, we present a unified new proof of both Euler's existence theorem and Rodrigues' identification. The proof relies only on basic algebraic properties of vectors in $\mathbb{R}^3$.