  Euclidean functions with values in an arbitrary well-ordered set were first
considered in a 1949 work of Motzkin and studied in more detail in work of
Fletcher, Samuel and Nagata in the 1970's and 1980's. Here these results are
revisited, simplified, and extended. The two main themes are (i) consideration
of Ord-valued functions on an Artinian poset and (ii) use of ordinal
arithmetic, including the Hessenberg-Brookfield ordinal sum. In particular, to
any Euclidean ring we associate an ordinal invariant, its Euclidean order type,
and we initiate a study of this invariant. The main new result gives upper and
lower bounds on the Euclidean order type of a finite product of Euclidean rings
in terms of the Euclidean order types of the factor rings.
