This book (written in German) describes the life and mathematical achievements of Hermann Günther Grassmann (1809-1877). In the first chapter, the author presents Grassmann's life from his childhood up to his death, a life played out on the background political, social and cultural situation in Germany during the first half of the 19th century. His family, his studies at the gymnasium in Stettin and his studies at the University of Berlin are described in a lot of detail. It may be surprising that Grassmann had no formal university training in mathematics; he took courses on theology, classical languages, philosophy and literature to become a minister in the Lutheran church in Stettin. After completing his studies in 1830, he became a teacher. After one year of mathematical studies, he took an examination to become a teacher of mathematics at gymnasium level. But his works and his knowledge were not sufficient, so he obtained permission to teach at a lower gymnasium level only.
The author shows Grassmann's pedagogical activities and his mathematical production during the two periods (1830-1840, 1840-1848) during which his most important works were written. The second chapter describes Grassmann's mathematical and philosophical academic background. His father was an excellent professor of mathematics and physics in Stettin, writing several textbooks on mathematics, mineralogy and physics. His younger brother Robert also became a teacher of mathematics and he collaborated with Hermann on some projects and publications. The influence that F. D. E. Schleiermacher (a great philosopher, theologist, politician and teacher) had on the development of Grassmann's philosophical thinking is described here.
The third chapter gives a brief survey of the mathematical achievements from the 17th century up to the 19th century that could have influenced Grassmann's mathematical thinking and methods. His important publications are very carefully analysed. Grassmann’s contributions to linear algebra, algebraic forms, the theory of algebras, differential geometry, analysis and number theory are presented. The fourth chapter explains the philosophical principles and the background of Grassmann’s most important and inspired monograph ‘Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik’.
The book contains many interesting pictures, photographs, reproductions and notes. It will be very helpful for historians and philosophers of mathematics, for teachers at universities and secondary schools, and students as well as researchers in mathematics and history. It can be recommended to people who are interested in the roots of modern mathematics.