The quality and importance of this book could hardly be overstated. Geometric algebra might casually be considered the "correct" generalization of linear algebra. By considering, for a start, directed line segments, the linear algebra courses presently taught in some high schools and all universities achieve miracles. Although viewed by a few of the slower students as merely unpleasant bookkeeping systems, linear algebra derives its power from allowing algebraic manipulation of sophisticated aggregate objects, namely vectors. The benefits are not just computational, but stem more importantly from a more powerful and more unified, although slightly more abstract point of view than a student had before studying. Geometric algebra is all that and much more. By extending consideration from directed line segments to the inclusion of direct plane segments, directed elements of three space, etc., an extremely flexible and elegant mathematical tool arises. It allows a deeper, quicker, and more concise treatment of essentially all of modern differential geometry. Its applications throughout physics are at once simplifications of ordinary matrix treatments and occasions to allow much greater insight.
Geometric algebra is a great theory, one of highest importance. It will, undoubtedly, find a dominant place in our mathematics curriculum at the highest speed allowed by our educational systems (the highest speed being actually quite slow). This book is an especially good place to begin study. It starts from the most elementary principles, and exposes the material with very thoughtful, clear presentation. The economy and elegance of the geometric algebra itself allows this one substantial but not enormous book to reveal great insights into many branches of study, from differential geometry and its applications to gravity theory to quantum mechanics and classical mechanics.
If I had no books in my library, I would purchase a Bible. If I had only the Bible in my library, I would purchase this book next. I would certainly study this book in all detail before making a third purchase. My library already has several books in it. None of them will be read further until I finish every line, every exercise of this book. It's an important theory, and it is explained in a very useful and articulate way. This would, of course, be entirely expected if the authors were from Oxford University. Since they are only from Cambridge, we might not have expected as much, but we got it, nonetheless.