As we had already occasion to mention, Sommerfeld has, in two papers on four dimensional geometry (_vide_, Annalen der Physik, Bd. 32, p. 749; and Bd. 33, p. 649), translated the ideas of Minkowski into the language of four dimensional geometry. Instead of Minkowski’s space-time vector of the first kind, he uses the more expressive term ‘four-vector,’ thereby making it quite clear that it represents a directed quantity like a straight line, a force or a momentum, and has got 4 components, three in the direction of space-axes, and one in the direction of the time-axis. The representation of the plane (defined by two straight lines) is much more difficult. In three dimensions, the plane can be represented by the vector perpendicular to itself. But that artifice is not available in four dimensions. For the perpendicular to a plane, we now have not a single line, but an infinite number of lines constituting a plane. This difficulty has been overcome by Minkowski in a very elegant manner which will become clear later on. Meanwhile we offer the following extract from the above mentioned work of Sommerfeld. (Pp. 755, Bd. 32, Ann. d. Physik.) “In order to have a better knowledge about the nature of the six-vector (which is the same thing as Minkowski’s space-time vector of the _2nd_ kind) let us take the special case of a piece of plane, having unit area (contents), and the form of a parallelogram, bounded by the four-vectors _u_, _v_, passing through the origin. Then the projection of this piece of plane on the _xy_ plane is given by the projections _u__{_x_}, _u__{_y_}, _v__{_x_}, _v__{_y_} of the four vectors in the combination φ_{_x_ _y_} = _u__{_x_}_v__{_y_} - _u__{_y_}_v_{_x_}. Let us form in a similar manner all the six components of this plane φ. Then six components are not all independent but are connected by the following relation φ_{_y_ _z_} φ_{_x_ _l_} + φ_{_z_ _x_} φ_{_y_ _l_} + φ_{_x_ _y_} φ_{_z_ _l_} = 0 Further the contents | φ | of the piece of a plane is to be defined as the square root of the sum of the squares of these six quantities. In fact, | φ |² = φ_{_y_ _z_}² + φ_{_z_ _x_}² + φ_{_x_ _y_}² + φ_{_x_ _l_}² + φ_{_y_ _l_}² + φ_{_z_ _l_}². Let us now on the other hand take the case of the unit plane φ^* normal to φ; we can call this plane the Complement of φ. Then we have the following relations between the components of the two plane:— φ_{_y_ _z_}^* = φ_{_x_ _l_}, φ_{_z_ _x_}^* = φ_{_y_ _l_}, φ_{_x_ _y_}^* = φ_{_z_ _l_} φ_{_z_ _l_}^* = φ_{_y_ _x_} ... The proof of these assertions is as follows. Let _u_^*, _v_^* be the four vectors defining φ^*. Then we have the following relations:— _u__{_x_}^* _u__{_x_} + _u__{_y_}^* _u__{_y_} + _u__{_z_}^* _u__{_z_} + _u__{_l_}^* _u__{_l_} = 0 _u__{_x_}^* _v__{_x_} + _u__{_y_}^* _v__{_y_} + _u__{_z_}^* _v__{_z_} + _u__{_l_}^* _v__{_l_} = 0