  A new mneumonic device is shown to emerge in connection with O(7) numerical
tensors exhibiting duality and reflecting the natural 7=(4+3) splitting of
7-dimensional space. Then Desargues' and Pappus' theorems are shown to be
connected through a geometry that makes use of octonionic numbers exhibiting
this duality. Construction of exceptional Hilbert space based on Jordan
algebras and exceptional projective geometries is illustrated. A brief
discussion of the Moufang plane and non-Desarguesian geometries is presented.
