Die Lineale Ausdehnungslehre, ein neuer Zweig der Mathematik

Hermann Grassmann's Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik (1844) introduced a novel calculus of vectors. This work, meaning "The Theory of Linear Extension, a New Branch of Mathematics," expanded upon Gottfried Leibniz's ideas by developing an algebra for manipulating symbols representing geometric entities like points and lines.1

Grassmann's methods, though initially obscure, were foundational for modern vector analysis and inspired later mathematicians. His work, alongside that of mathematicians like Sir William R. Hamilton and George Boole, pioneered the field of modern algebra. The concepts introduced in Ausdehnungslehre have found applications in areas such as differential forms, analysis, and geometry.1