Tertium Organum

“…What can we learn about the fourth dimension by studying geometrical relationships within our space? What should be the relationship of a three-dimensional body to a four-dimensional one? A four-dimensional body as the trace of the movement of a three-dimensional body in a direction not contained in it. A four-dimensional body as composed of an infinite number of three-dimensional bodies. A three-dimensional body as a section of a four-dimensional one. Parts of bodies and whole bodies in three and in four dimensions. Incommensurability of a three-dimensional and a fourdimensional body. A material atom as a section of a four-dimensional line. If we examine the profound difference that exists between a point and a line, between a line and a surface, between a surface and a solid, i.e. the difference between the laws which govern a point and a line, a line and a surface and so on, and the difference of phenomena which are possible in a point, a line, a surface, we shall realize how many things, new and incomprehensible for us, lie in the fourth dimension…”