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Extension Theory
Hermann Grassmann
A co-publication of the AMS and the London Mathematical Society.

History of Mathematics
2000; 411 pp; softcover
Volume: 19
ISBN-10: 0-8218-2031-1
ISBN-13: 978-0-8218-2031-5
List Price: US$87
Member Price: US$69.60
Order Code: HMATH/19
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The Ausdehnungslehre of 1862 is Grassmann's most mature presentation of his "extension theory". The work was unique in capturing the full sweep of his mathematical achievements.

Compared to Grassmann's first book, Lineale Ausdehnungslehre, this book contains an enormous amount of new material, including a detailed development of the inner product and its relation to the concept of angle, the "theory of functions" from the point of view of extension theory, and Grassmann's contribution to the Pfaff problem. In many ways, this book is the version of Grassmann's system most accessible to contemporary readers.

This translation is based on the material in Grassmann's "Gesammelte Werke", published by B. G. Teubner (Stuttgart and Leipzig, Germany). It includes nearly all the Editorial Notes from that edition, but the "improved" proofs are relocated, and Grassmann's original proofs are restored to their proper places. The original Editorial Notes are augmented by Supplementary Notes, elucidating Grassmann's achievement in modern terms.

This volume is one of an informal sequence of works within the History of Mathematics series. Volumes in this subset, "Sources", are classical mathematical works that served as cornerstones for modern mathematical thought.


Graduate students and research mathematicians interested in the history of vector spaces, exterior algebra, Pfaff's problem, and the other contributions of Grassmann; historians of mathematics.


"Grassmann's classical work can be warmly recommended to mathematicians and to historians of mathematics as a source of Grassmann's mathematical ideas."

-- European Mathematical Society Newsletter

"Hermann Grassmann's Extension Theory (Ausdehnungslehre), first published in 1862, is a legendary book in the history of mathematics. This new book in the "Sources" subseries is a most welcome addition to the AMS/LMS History of Mathematics series. A must-buy for those seriously interested in the history of mathematics."

-- MAA Online

Table of Contents

The elementary conjunctions of extensive magnitudes
  • Addition, subtraction, multiples and fractions of extensive magnitudes
  • The product structure in general
  • Combinatorial product
  • Inner product
  • Applications to geometry
The theory of functions
  • Functions in general
  • Differential calculus
  • Infinite series
  • Integral calculus
  • Index of technical terms
  • Editorial notes
  • Supplementary notes
  • Subject index
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