Grassmann Algebra

Preface
The origins of this book
This book has grown out of an interest in Grassmann's work over the past three decades. There
is something fascinating about the beauty with which the mathematical structures Grassmann
discovered (invented, if you will) describe the physical world, and something also fascinating
about how these beautiful structures have been largely lost to the mainstreams of mathematics
and science.
Who was Grassmann?
Hermann GŸnther Grassmann was born in 1809 in Stettin, near the border of Germany and
Poland. He was only 23 when he discovered the method of adding and multiplying points and
vectors which was to become the foundation of his Ausdehnungslehre. In 1839 he composed a
work on the study of tides entitled Theorie der Ebbe und Flut, which was the first work ever to
use vectorial methods. In 1844 Grassmann published his first Ausdehnungslehre (Die lineale
Ausdehnungslehre ein neuer Zweig der Mathematik) and in the same year won a prize for an
essay which expounded a system satisfying an earlier search by Leibniz for an 'algebra of
geometry'. Despite these achievements, Grassmann received virtually no recognition.
In 1862 Grassmann re-expounded his ideas from a different viewpoint in a second
Ausdehnungslehre (Die Ausdehnungslehre. VollstŠndig und in strenger Form). Again the work
was met with resounding silence from the mathematical community, and it was not until the
latter part of his life that he received any significant recognition from his contemporaries. Of
these, most significant were J. Willard Gibbs who discovered his works in 1877 (the year of
Grassmann's death), and William Kingdon Clifford who discovered them in depth about the
same time. Both became quite enthusiastic about this new mathematics.
A more detailed biography of Grassmann may be found at the end of the book.
From the Ausdehnungslehre to GrassmannAlgebra
The term 'Ausdehnungslehre' is variously translated as 'extension theory', 'theory of extension',
or 'calculus of extension'. In this book we will use these terms to refer to Grassmann's original
work and to other early work in the same notational and conceptual tradition (particularly that of
Edward Wyllys Hyde, Henry James Forder and Alfred North Whitehead).
The term 'Exterior Calculus' will be reserved for the calculus of exterior differential forms,
originally developed by Elie Cartan from the Ausdehnungslehre. This is an area in which there
are many excellent texts, and which is outside the scope of this book.
The term 'Grassmann algebra' will be used to describe that body of algebraic theory and results
based on the Ausdehnungslehre, but extended to include more recent results and viewpoints.
This will be the basic focus of this book.
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