In this paper the dissection ofn-dimensional Euclidean simplices is investigated. Some propositions are proved about the dihedral angles of order (n−1) occurring when a simplex is cut into two subsimplices by a hyperplane. Furthermore, a description of simplices by graphs is given. If a simplex S is dissected into two subsimplices, then two graphs can be assigned to the two simplices. It is shown how these graphs are linked with the original simplex. By means of these graph-theoretical methods the dissection of four-dimensional simplices is thoroughly investigated and a new method for dissecting a four-dimensional simplex into orthoschemes is given. It is proved that 500 is an upper bound of the minimum number of orthoschemes needed.