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APPLICATIONS OF GEOMETRIC ALGEBRA AND THE GEOMETRIC PRODUCT TO SOLVE GEOMETRIC PROBLEMS

In his characteristica geometrica [1, 2], Leibniz imagined and aimed to find an intrinsic geometric language to study geometry and to perform directly geometric operations in a way different from Cartesian coordinates. Afterwards, Möbius [3] introduced the barycentric calculus, from which Grassmann developed the extension theory. In 1846, Grassmann won the prize offered by the Fürstlich Jablonowski’schen Gesellschaft in Leipzig to whom was capable of developing Leibniz’s idea with his memoir Geometrische Analyse [4, p.315]. While in his Die Lineale Ausdehnungslehre [4, p. 9] Grassmann introduced the inner and outer products of vectors, at the same time Hamilton discovered quaternions (1843), which he defined as geometric quotients of vectors [5]. Some years later, Clifford defined the geometric product [6] as the combination of Grassmann’s inner and outer products. All these works conformed geometric algebra. However, what is exactly geometric algebra and how is it understood? Peano, who had analyzed Grassmann’s work deeply [7], already had a unified vision of geometric algebra [8, p.170]: