In 1844, the Prussian schoolmaster Hermann Grassmann (1809-77) published Die Lineale Ausdehnungslehre (also reissued in the Cambridge Library Collection). This revolutionary work anticipated the modern theory of vector spaces and exterior algebras. It was little understood at the time and the few sympathetic mathematicians, rather than trying harder to comprehend it, urged Grassmann to write an extended version of his theories. The present work is that version, first published in 1862. However, this also proved too far ahead of its time and Grassmann turned to historical linguistics, in which field his contributions are still remembered. His mathematical work eventually found champions such as Hankel, Peano, Whitehead and Élie Cartan, and it is now recognised for the brilliant achievement that it was in the history of mathematics.