For statistical and other purposes one needs to be able to determine an average eye. An average of refractive errors is readily calculated as an average of dioptric power matrices. A refractive error, however, is not so much a property of the eye as a property of the compensating lens in front of the eye. As such, it ignores other aspects of the optical character of the eye. This paper discusses the difficulties of finding a suitable average that fully accounts for the first-order optics of a set of optical systems. It proposes an average based on ray transferences and logarithms and exponentials of matrices. Application to eyes in particular is discussed.