Grassman links Groesse to Einheiten in his Vorrede. this requires some consideration.
Firstly it is my shock to realise that Hamilton has been so bigged up by me that i could not see a few simple observable dates! secondly it is my joy to realise that the history of the times played a significant role in the promotion of Hamilton over Grassmann, and the reason why an obvious opportunity to collaborate was missed or avoided.
Both Grassmann and Hamilton inspired waves of joyous inspiration in students of their work. But their work was so fundamental and algebraic that it required the Genius of William Kingdon Clifford to see and make the connection. In doing so he clearly showed Grassman's theory to be the greater which included Hamilton's quaternions in particular as a major subset..
The way Clifford constructed his Algebra was thoroughly advised by Grassmman's THEORETICAL STRUCTURE, but was Cliffords first foray into the applicability of the Grassman conception. He died young.
In the meantime another student of Grassmann was doing battle with Sir William Rowan Hammilton in America. This was Gibbs, and despite his valiant efforts Hamilton lost to Gibbs through what may be seen as just amother of those dark days of infamy we humans inflict upon each other.
Gibbs vector calculus accordingly sought to steal also Grassman's crown in America. Many hucksters and shysters found America a land of opportunity, and used any means to establish themselves and their families. For hat at least i cannot blame them, and in any case, the stuffy old world would rather have seen them dead than benefit from any innovation they may have on another fellows work. The argument with Saint -Vannance is a case in point.
Accusing Grassmann of plagiarism, Saint -V launched he most vicious attack on him. We have to rememeber that these issues were often settled by duels!
I have found a reference that shows that Hamilton read and recognised Grassman's "treatise" The Linear Algebra Ausdehnungslehre, but did not read the full development of it in 1862. Thus i can conclude that Grassman's ideas in 1844 didi nt sufficiently cistinguish themselves from ll the other competing notions for dealing with the growing need for a topological calculus of space.
It seems that from Newron's time a way of dealing with calculating the coefficients oe equations so as to pin point moving bodies in space was a pressing need. It was Wallis who noted that the imaginaries may be a way of referencing points in a plane, but devoted himself to "numbers" It was De Moivre Newton And Cotes who extensively used them to solve coefficient problems in Britain. It was Wessel Argand and Gauss who did thesame in Europe, but it was Gauss who established the notation of a linear algebra that is a vector using i but as a "number. Gauss amongst others were congused and struggling. Grassmann and Hamilton began to think on the subject and both solved substantial parts and difficulties.
Both Grassman and Hamilton started with the imaginaries. Hamilton published first with the work in 1834 on couples. This was before the term complex numbers was coined by Haenkel, but in fact the term derives from Hamiltons work in Couples. Hamilton justified all number lines and did so by founding them in every way, relationally, analogically philologically, psychologically and procedurally on flowing "timt", but as i point out this is actually an analogy and it is based on flowing motion.
Grassmann was struck by another notion, and his was found solidly in Euclid. Now Hamilton had used Euclid as a starting point, but due to common misconception of Euclid had sought to develop a sciece of "pure time" using Euclidean method. Grassman however was struck by Moebius's consideration of an "algebra of Points", and the heavy reliance in that work on Euclidean theorems. It occurred to im that Euclid had more to offer than was thought at the time.Grassmann thus engaged Euclid at a fundamental level, utilising his genius skill of philology to truly understand and exposit on Euclid in a "modern" setting. He did not dismiss Euclid as most do even today, nor did he read it in translation as a classics exercise. He studied it as a philosophical apprehension of reality.
Euclid, as a philosophical work, as opposed to a mathematical and shall we say geometrical treatise, is a quite different experience. Grassmann wrote his Theil or treatise in 1844 after much consideraton and deliberation, and as a philosophical treatise. As a Treatise it was clearly marked as the essential and fundamental part of his ideas, to which he would add a second and fuller exposition.
Both Britain and Prussia were competing to become industrial superpowers. thus collaboration was not encouraged. Hamilton was an academic, well respected and innovative. Grassman was a ardworking manwho had to support himself and his family by schoolteaching etc. His genius was not understood, although it was recognised. He was continually admonished for his unintelligibility!
As one who has suffered similar well intenioned criticism, i understand how one may feel slighted and unfaired against. But i also know that i do write in such a dense and intuitive way that many cannot be bothered to read much of what i write. Thus Grassmann found scant response to his work of 1844. However te year previous Hamilton, by now a recognised mathematician had published a paper on Quaternions, in addition to several ther treatises, which substantially covered the ground that Grassman was considering. And Hamilton had kissed the Blarney stone! He could turn a phrase.
Thus we see so many terms used in modern maths owe their origin to Hamilton's lexicon, and we also see why Grassman's taciturn, everyday terminology to cover new ideas that were so simple a to be profound. socially did not excite.
Also his choice of title is pedantic and not likely to inspire interest except n a research mathematician. Finally his claim of a new "twig" or branch of mathematics is generally off putting to the professional and non rofessional mathematician: the one does not want to learn new terminology, the other is struggling to come ou of a state of being lost to a mastery of the subject, so why start something new?
Grassmann felt this rejection personally and keenly. Hamilton was enjoying immense populatiy. Add to that Lewis' Carrols attacks on Hamilton and all "imaginary" algebraists and you have a publicity campaign powerfully favouring Hamilton.
Grassman's Philosophical approach to Euclid is what i am studying.
I have resolved the outstanding issues around the notion of i(√-1) and i have denoted it not as number but as a vector. The vector algebra it is a part of i have called a complex vector algebra. I have defined a unit arithmoi with a contra positive unit, achieved by rotating by π/2 anticlockwise and this is the geometric mean for -1 the contrapositive arithmoi, This is just one of the monads missing in mathematics, and subsequently in Grassman's Einheiten,but that we will come to shortly.
In my complete algebra i will do away with sign completely replacing it by a set of unit vectors with appropriate rotational actions on those vectors.
The Children of Shunya 