The author proves the following: The result of applying a continuously composed family of rotations of the complex plane with smooth center path \(c:[0,1]\to{\mathbb C}\) and angular velocity function \(\omega:[0,1]\to{\mathbb R}\) is either a translation or a rotation, depending on whether \(\Theta=\int_{0}^{1}\omega(\tau)d\tau\) is, or is not in \(2\pi{\mathbb Z}\). Moreover, the author exhibits three examples of continuously composed rotations resulting in translations.