Suppose that space is metric. A chain is a finite collection of open sets d1, d2, * * * , dn such that di intersects dj if and only if I i-jj I 1. If the elements of a chain are of diameter less than a positive number e, that chain is said to be an e-chain. A compact continuum is said to be chainable if for each positive number e, there is an e-chain covering it. R. H. Bing has called [1] such continua snake-like. In 1951 0. H. Hamilton showed [4] that every compact chainable continuum has the fixed point property; i.e., that if f is a continuous mapping of such a continuum M into itself, then some point of M is its own image underf. In the present paper it is shown that the Cartesian product of finitely many compact chainable continua has the fixed point property. Since arcs are compact chainable continua, this is a generalization of the Brouwer fixed point theorem. Two other examples of compact chainable continua are the closure of the graph of sin (1/x), 0