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.

The book History of Mathematics from Ancient Times to the Early Nineteenth Century [in Russian], which was published in the years 1970-1972, is cited in abbreviated form as HM (with volume and page number indicated).

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.