Ten years after publication of the popular first edition of this volume, the index theorem continues to stand as a central result of modern mathematics-one of the most important foci for the interaction of topology, geometry, and analysis.

This is how a so-called critical edition is made. This edition of the Latin text is preceded by General Introduction, describing the various manuscripts, the content of the work and what we know about its author.

They also address important new topics, such as quaternions. The approach of this book is at times more computational than is usual for a book on the subject.

Chapters 2-7 contain the core mathematical content of the text, following the ErlangenProgram, which develops geometry in terms of a space and a group of transformations on that space.

This text provides the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric.