When I was a student I was impressed by the beauty of
the vector analysis applied to electromagnetism, but it became apparent
for me that the vector analysis was a three-dimensional geometric algebra.
Then, I searched for an extension of the vector analysis which should incorporate
the time and be the geometric algebra of the special relativity. In this
case, the Maxwell equations would have a simpler expression. In this context,
I looked for bibliography on quaternions, but I did not find any satisfactory
contents until I read the Elements of Quaternions . In this book,
Hamilton follows a pedagogic strategy in order to deduce the multiplication
rule of quaternions. Hamilton essentially considers that the sides of two
directly similar triangles lying on the same plane are proportional in
a complete geometric sense, not only by length. If a, b are
the sides of the first triangle which are respectively proportional to
the sides c, d of the second triangle, Hamilton writes a
b-1= c d -1 = q , from where he
defines the quaternion q as a quotient of two vectors. Hamilton
splits a quaternion into its scalar and vector parts. After him, Gibbs
and Heaviside introduced the modern concept of vector by cutting
the quaternions into scalars and vectors.
I thought that the quaternions should be the relativistic geometric algebra, and indeed there are many papers devoted to this matter. However, the aspect of the Maxwell equations is somehow repellent incorporating quaternions with complex components, the biquaternions yet used by Hamilton. Josep Manel Parra, whose doctoral thesis was devoted to the study and classification of the Clifford algebras, rescued me from this conceptual mistake, where I had fallen led by some papers.
Clifford died being 33 years old. Although he left many mathematical papers, he only studied the geometric algebras in two of them. Since I had begun a historical approach, the field of research in Clifford algebras was unknown for me. In the first interview, J. M. Parra did not convince me because I was very acquainted with the quaternions. Being the quaternions either product of vectors for Clifford or quotient of vectors for Hamilton (it is essentially the same), which and where is the problem? The question is to know how the quaternions became vectors. Then, I could only find a definitive answer in the Elements of Quaternions. We cannot accuse Gibbs and Heaviside of transforming quaternions into vectors because Hamilton was the first author who identified the orientation of the plane of a quaternion with the vector perpendicular to this plane.
I completed my vision of the panorama of the geometric algebra(s) with the reading of Grassmann's work Die Ausdehnungslehre. Fortunately, there is a Spanish translation of this book (Teoría de la extensión), which facilitates me to understand better its difficult matter and writting. We must thank this edition to the eminent mathematician Julio Rey Pastor, the director of the collection Historia y Filosofía de la Ciencia (Espasa-Calpe).