“One may say without great exaggeration that Grassmann invented linear algebra and, with none at all, that he showed how properly to apply it to geometry. ...He ...anticipated in its most important aspects Peano's treatment of the natural numbers, published 28 years later. ...A feature of Grassmann's work, far in advance of the times, is the tendency towards the use of the implicit definition. ...The definition of a linear space (or vector space) came into mathematics, in the sense of becoming widely known, around 1920, when Hermann Weyl and others published formal definitions. ...Grassmann did not put down a formal definition—again, the language was not available—but there is no doubt that he had the concept.”
“From the imputation of confounding axioms with assumed concepts Euclid himself, however, is free. Euclid incorporated the former among his postulates while he separated the latter as common concepts—a...”
Hermann Grassmann
“Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics ...”
Hermann Grassmann
“It is clear... that the concept of space can in no wise be generated by thought. ...Whoever maintains the contrary must undertake to derive the dimensions of space from the pure laws of thought—a prob...”
Hermann Grassmann
“I remain completely confident that the labour I have expended on the science presented here and which has demanded a significant part of my life as well as the most strenuous application of my powers,...”
Hermann Grassmann
“A work on tidal theory... led me to Lagrange's Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the ...”
Hermann Grassmann