Hermann Grassmann
Hermann Grassmann
Hermann Günther Grassmann 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 li
"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 problem which is at once seen to be impossible of solution."
"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 which yields in a purely abstract way laws similar to geometry."
"A work on tidal theory... led me to Lagranges Mécanique analytique and thereby I returned to those ideas of analysis. All the developments in that work were transformed through the principles of the new analysis in such a simple way that the calculations often came out more than ten times shorter than in Lagranges work."
"The concept of centroid as sum led me to examine Möbius Barycentrische Calcul, a work of which until then I knew only the title; and I was not little pleased to find here the same concept of the summation of points to which I had been led in the course of the development. This was the first, and... the only point of contact which my new system of analysis had with the one that was already known."
"I feel entitled to hope that I have found in this new analysis the only natural method according to which mathematics should be applied to nature, and according to which geometry may also be treated, whenever it leads to general and to fruitful results."
"I define as a unit any magnitude that can serve for the numerical derivation of a series of magnitudes, and in particular I call such a unit an original unit if it is not derivable from another unit. The unit of numbers, that is one, I call the absolute unit, all others relative. Zero can never be a unit."
"Some of the groundbreaking work in the treatment of n-dimensional geometry—was carried out by Hermann Günther Grassmann. ...Grassmann was responsible for the creation of an abstract science of "spaces," inside which the usual geometry was only a special case. Grassmann published his pioneering ideas (originating a branch of mathematics known as linear algebra) in 1844, in a book commonly known as Ausdehnungslehre... Grassmanns suggestion that BA = -AB violates one of the sacrosanct laws of arithmetic... Grassmann faced up squarely to this disturbing possibility and invented a new consistent algebra (known as exterior algebra) that allowed for several processes of multiplication and at the same time could handle geometry in any number of dimensions."
"As the great generality of Grassmanns processes—all results being obtained for n-dimensional space—has been one of the main hindrances to the general cultivation of his system, it has been thought best to restrict the discussion to space of two and three dimensions."
"The wonderful and comprehensive system of Multiple Algebra invented by Hermann Grassmann, and called by him the Ausdehnungslehre or Theory of Extension, though long neglected by the mathematicians even of Germany, is at the present time coming to be more and more appreciated and studied. In order that this system, with its intrinsic naturalness, and adaptability to all the purposes of Geometry and Mechanics, should be generally introduced to the knowledge of the coming generation of English-speaking mathematicians, it is very necessary that a text-book should be provided, suitable for use in colleges and universities, through which students may become acquainted with the principles of the subject and its applications."
"The concept of rotation led to geometrical exponential magnitudes, to the analysis of angles and of trigonometric functions, etc. I was delighted how thorough the analysis thus formed and extended, not only the often very complex and unsymmetric formulae which are fundamental in tidal theory, but also the technique of development parallels the concept."
"It was natural that Grassmann chose to introduce his system, not by means of a paper, but rather by means of a long and complicated book. ...such ideas as Grassmanns form of the scaler (dot) and vector (cross) products... have counterparts in modern vector analysis."
"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 Peanos treatment of the natural numbers, published 28 years later. ...A feature of Grassmanns 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."
