Application of Algebras

The geometric interpretation of complex numbers as points of the plane appeared for the first time in the 18th century. After that there arose the natural idea of generalizing complex numbers in such a way that they could be interpreted as points of three-dimensional space. One of the earliest attempts of this kind was due to Caspar Wessel. It appeared in his previously mentioned Attempt to represent direction [624]. Having thought of the operation of multiplication of complex numbers in geometric terms, Wessel associated to a point in space with rectangular coordinates x, y, z the expression x + yε + zη, where ε and η are two different imaginary units, and interpreted by means of these numbers rotations about the Oy- and Oz-axes. Wessel used his “algebra” to solve problems involving spherical polygons.