The classical Vahlen matrix representation of conformal transformations on R(n) is directly related to the versor representation of conformal geometric algebra (CGA) using R(n+1;1). This paper spells out the relationship, which enriches both fields with insights and techniques. We extend the Vahlen matrices to include the representation of blades in CGA, and then use a decomposition in terms of eigenlines to derive Chasles’ theorem for representation of Euclidean rigid body motions. This naturally leads to the logarithm of a Vahlen matrix of such a motion. We also derive the table of commutation relationships between the basic even conformal transformations (translation, rotation, uniform scaling and transversion), in which the rather involved translation-transversion result may be new.