Grassmann's method of Analysis and Synthesis was precisely Ring and or Group theory on oriented line segments with a unit numeral adjoined to the set. Justus his father started the process by a deep analysis of Legendre's Geometry with the intention of establishing geometry on the foundation of Arithmetic. He was not alone in doing this, but he was certainly among a group of very early pioneers of Ring Theory whose interest was crystallography.
Justus work seems to have been called various things from Zahlenlehre, Kombinationlehre to Verbindungslehre. It was deemed to be a branch of mathematics , alongside other deconstrucyions and reconstructions of Mathematical processes that were being initiated at the time. Justus particular idea was published but little known. It however formed the basis for his rhetorical style of presenting geometry in his school district, and this is how Hermann and Robert were inculcated with it.
Hermann, however was the one who actually reconstructed his whole philosophical and mathematical approach on these concepts. Others regarded them as useful analogues but not the real deal which was higher arithmetics. These approaches were too elementary, suitable perhaps for children, but ultimately trivial!
I think Hermann was able to show that the developing machinery of ring Theory and also Group theory was of fundamental importance to the foundations and the capstone of Mathematics and physics and mechanics in particular.
As an example I will look at Clifford algebra and rotation
We can confidently trace Herrmanns familiarity with group theory yo the influence of LaGrange. Bot Hamilton nd th Grassmnns were influenced by his concepts. In this regard he takes his studies of Newton into new areas. But there is a very clear link between LaGranges interest in cyclic groups snd Newtons mastery of the binomial theorem and series expansion. In Fact De Moivre took this aspect of Newtons ideas into the general topic of Pobability theory.
De Moivres work with Cotes on the roots of unity form Peres the first study of cyclic groups of imaginary numbers, and this seems to have influenced many mathematicians of the time . This particular group theoretic structure was studied by many including Argand, Cuchy and Gauss,who was not convinced of their veracity, until forced to publish. But it is doubtless that they knew of the Cotes De Moivre theorems. The influence of these on geometrical thinking is quite extraordinary and for the mst part unacknowledged.
At least it is clear tht noth Hamilon nd Grassmann were influenced by the Lagrangin developments of the Newtonin school of ideas
Ring theory is not so clear.
Noting the above clinks to corrections the
Not much is documented about the early pioneers in ring theory, but what is will be found under the heading of crystallography. There was a movement among philosophers that recognised the dynamism in nature. Shelling and some others were early members of such a group, and Justus Grassmann was at the very least in correspondence with this group. They chose to study crystals which may not seem very dynmic, but of course crystals grow slowly! By looking at the patterns or shapes of crystals they hoed to gain insight into the fundamental dynamics of the universe.
The set of aspects of crystals: facets, reflection, translation , rotation ,edges and vertices were clearly geometric. Thus crystals revealed a fundamental geometric process in nature that was dynmic, but easier to study mathematically, that is geometrically and phorometricslly than biological systems. And of course diamonds and rubies were of greay value, and ny system that could completely characterise a gem stone would have a economic interest from the gem and mineral trades.
It turned out that these structures could be described by a system of properties which were magnitudes. Arithmetic was developed for describing such magnitudes.
For a long time, over 2 millennia , arithmetic had been considered the greatest achievement of the Pythagorean School. It is the fundmental group theoretical structure of all calculation. It was not the only group structure. Eudoxus taught also the logos analogos system on which it was based.
In the logos Analogos system there is no Monas. The system is based on a congruent notion called Metron, that corresponds to Monas. The difference is the definition. Metron is hardly defined except by Elassos. The smaller relative to Meizonos, the larger. Katametresee can then be defined as an action, and involved in that action is the rhythmical counting which always begins with one or en(ek in India)
When Arithmos was defined , and the Arithmoi constructed, they did not start with magnitude. They started ith a group or set description of Monas. This was the most general definition of a unit(Monas) they could give. Once you accept the concept of a unit one is able immediately to specify unit Metrons or unit magnitudes.
However the concept of a unit is arbitrary! It cannot be specified in words beyond what was specified by the Greeks. Thus the concept is formal and decide able only by subjective opinion or assent to a consensus.
In every established civilisation, a department has to be set up to define units of all sorts of things. This is the department of weights nd measures. Euler, to my knowledge was employed by the Russian Czar to head up such a ministry for enforcement throughout the empire.
Thus 1 is the numeral or symbol for Monas. It is called one but it is a symbol of a unit.
The natural question is a unit of what ? This is a question attempting to specify magnitudes.
Perhaps not understnding the weights and measures aspect of it, and following his dads advice or theory written in 1827, several decades before Dedekind et al, he included the numeral 1 as fundamental to hs group structure.
Today, the simple role of 1 in a group system is elaborated, but it is a formal addition to any theoretical development of an Arithmos, or arithmetic. Basically it means we can count and assume unit magnitudes.
In terms of n arithmetic it completely specifies the type of multiplication tht cn be done as well as the type of addition.
Most of us are used to 1+ 1 not being the same as 1 x 1, because they learn that by rote! But we are rarely told what x and + really symbolise.
They do not symbolise addition or multiplication! That is rote learning for you.
They are combinatorial symbols denoting combinatorial processes of construction or synthesis. This simple symbolic statement determines the method of arithmetic for the magnitudes.. This method of arithmetic has to describe the behaviour of the magnitudes. One very important behaviour is contained within the notion of Shunya!
I have just lost 2 hours work of describing the next stage to the aethe! Perhaps you are not meant to know this exposition just yet.