Extension theory hermann grassmann biography

For those, like myself, who have struggled to comprehend the original works of Hermann Grassmann, the consoling factor has always been that most of the great 19 century mathematicians were also baffled by his unique terminology and strange style of writing. But, in recent years, there have appeared several very good summaries of Grassmann’s mathematical ideas — including Martin J. Crowe’s history of vector analysis [1] and various articles by Fearnley-Sander [2]. And yet, prior to the publication of the two books under review, the most recent biography of Grassmann was that by Friedrich Engel, published in 1911.

Anyway, Grassmann’s importance arises from the fact that he is regarded as the founder of linear algebra, and the first mathematician to extend geometrical thinking beyond three dimensions. Yet these achievements emerged from the strange circumstances of his mathematically isolated existence in the Prussian town of Stettin — not to mention the tragic irony that his innovative work was largely unnoticed in his lifetime.

His major work was the Ausdehnungslehre (1844), which he hoped would earn him a professorship, and thereby provide opportunities for ongoing research. Unfortunately, this work attracted so few readers that these aspirations remained unfulfilled, and his professional life was mainly spent as a schoolteacher, burdened with a heavy teaching load. Cajori therefore says of Grassmann ‘At the age of fifty-three, this wonderful man, with a heavy heart, gave up mathematics, and directed his energies to the study of Sanskrit, achieving in philology results which were better appreciated, and which vie in splendour with those in mathematics’.

The mathematical goal that Grassmann had set himself had its origin in the thoughts of Leibniz, who sought an alternative to the algebra of Vieta and Descartes. He was hoping for a sort of universal algebra that would be distinctly geometrical or linear, and which would expres

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.

German polymath, linguist and mathematician (1809–1877)

"Grassmann" redirects here. For the surname, see Grassmann (surname).

Hermann Günther Grassmann (German: Graßmann, pronounced[ˈhɛɐmanˈɡʏntʰɐˈɡʁasman]; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all k-dimensional linear subspaces of an n-dimensional vector spaceV. In linguistics he helped free language history and structure from each other.

Biography

Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordainedminister who taught mathematics and physics at the StettinGymnasium, where Hermann was educated.

Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also taking classes in classical languages, philosophy, and literature. He does not appear to have taken courses in mathematics or physics.

Although lacking university training in mathematics, it was the field that most interested him when he returned to Stettin in 1830 after completing his studies in Berlin. After a year of preparation, he sat the examinations needed to teach mathematics in a gymnasium, but achieved a result good enough to allow him to teach only at the lower levels. Around this time, he made his first significant mathematical discoveries, ones that led him to the important ideas he set out in his 1844 paper Die lineale Ausdehnungslehre, ein neuer Zweig der Mathematik, here referred to as A1, later revised in 1862 as Die Ausdehnungslehre: Vollständig und in strenger Form be