When Hamilton coined the word vector he also coined many other words, for example versor. In those heady days of discovery terminology was being created, because distinctions were being marked : The logos kairos sumbola sunthemata summetria response. We all have the hutzpa to do this, and what is more we inherently do it, and we effectively communicate doing it.
The drawback is the establishment, and the fact that we become the establishment as soon s we defend any particular interpretation.
Part of being young is this argot formation, this recycling and invention of linguistic forms and attitudes. It is the "social" groups that establish a common reference, which eventually may pass down to future generations as a definition. Such Dictionaries of words help to stabiise and define a language, but they are not the language. The language lives and grows with ach generation and with each individual. No one knows which words will stick in the dictionary and which will be recycled into other uses or lost.
Words that stick, imply an esablishment that defends those words, and there useage. The officer, the function-ary of such an establishment is the grammarian, whose function is to maintain the "official language".
There are and have been many official languages, so even established languages get turned over and pot through the mill.
So Hamilton's vector was aken from him , by Gibbs et al, and applied to some related but wholly other notions. In particular, Gibbs took Grassmanns AusdehnungsGroesse and recast a selection of them as "vectors", particularly the Strecken. What he created was a bastard of the 2 theories which was entirely his own, but which,as in good historical and political dramatic form, he foisted as king upon an unsuspecting public.
Strecken and Vector are two different essential ideas, 2 different mathesii and praxii. And yet they are discussing similar experiences of spatial interaction. Thus they are analogous not identical, but each informs the other. Gibbs was able to convince many academicians that he had come up with something fundamental in regard to discussing spatial properties. In reality he fumbled the ball, produced a bastard and sought to kill off the pretending heirs!
It may be that life has no great purpose, but then again it just migt!
Both Grassmann and Hamilton knew their work was similar, but not the same. Both were heavily influenced by Dedekinds construction of the number line and the philosophy that lies behind it. That philosophy deals with insubstantials, especially subjective ideas and notions. It attempts to relate these ideas to pramatic concerns in the material world, but it always acknowledges the spiritual source of these ideas.
Dedekind et al. therefore proclaimed a doctrine, a mathesis of how order ought, should, might be especially if it was to conform to the religious ideals of the christian god. Jainism is an indian example of this very same process based around hindu notions of deity. However, as you know, my preferred proforma for these issues is the pthagorean philosophical take on these things, simply because of its universal aggregation policy: combine from all!
One might remark that this is what Gibbs did, and this would be a relevant point, but within the pythagorean school is millenia of wisdom through reviewing and recycling th notions until their full invariance and applicability can be established. Gibbs vectors are too young, as is Hamilton's fresh and decisive terminology as is Grassmann's extensive, colourful and immediate terminology.
Grassmann constantly referred to the need to develop and refine an appropriate terminology that naturally expressed the relationships he was alluding to, and which enable a simpler and more profound combination of ideas and relations, which revealed and supported the determining of laws for relations between Groesse, and redrew the boundaries between certain subjects to emphasise the full extent of their interrelationships. For Grassmann in 1862 the "pressure" was on him for the research into his ideas to establish these terms and laws. The task was too big for one man!
Grassmann also must not be confused with a "mathematician". He was a Natural Philosopher, but his approach was radical, and radically different to Newtons, and yet of his time. Where his genius lay was in a simple idea of invariance (Abweichung) in combinatorics theory. This simple idea is the powerful centre of all of his work, and the means by which he simplifies all apprehension of human-space interaction.
AB = ai * bj = a*b(i + j)
BA = bj * ai = b*a(j + i)
This is the "natural" vector "multiplication", the euclidean defined one for pollapleisios. Observe that they are not "aequal" that is identical or equivalent. The terminology we use so glibly has a history, and that history is in proportion theory. Because we are not mindful of this many difficulties ar put in our way when it comes to apprehending spatial relationships.
When i focus on 2 objects, i do so by a attentional directive from my subjective processing centre. This results in a comparison upon which all other processes depend. The dependency is that which seeks a resolution or a "solution" to the processing problem presented before me. The resolution is remarkably simple: greater than or less than or equal.
In order to arrive at this resolution i must engage in a process of reduction of all the relationships in my field of attention. My field of attention therefore represents a combination or a scatter pattern of relationships to which my sensory system processing has apportioned or allocated a certain amount of processing capability. Thus this field is an apportion of computing resources and there is no resolution beyond apportionment.
An apportion field is still a complex structure which i describe as a scatter pattern or more accurately a compass multivector network.. Thus the compass multivector network is an apportion field in an alternative description, and i may derive "proportions" from the compass multivector network by focussing on a sequence of semeia(subjective vectors).
The proportion is a sequence ( thus selected), of regions of space, identified by their semia, but then the magnitudes identified by the semeia are placed in this same sequence in analogy. The analogical magnitudes sequence is only sensible when a metron /monas is applied to the regions. Thus a further level of processing and focus is occuring, as yet without resolution.
To move to resolution i focus in on jus to sequents in the sequence and this enabled the comparison process to initiate, which results in a resolution. Then from this resolution , by apportinate relationships or rather fractal entrainment , i may expand to a resolution of as much of the apportion field as i desire,
Thus this process require that i recognise a structure called a compass multi vector network, which will naturally be dynamic, and that i dynamicaly fix proportions from this structure , after which the process is to focus down the proportion into a ratio pair from which a resolution may be derived which then becomes the basis for resolving the whole of the fixed proportion. the other fixed proportions need the same treatment, but should any of the proportions interconnect, then the resolution process "magically" spreads to both or all connected in this way.
Eudoxus therefore adopted the trichotomy as the limiting case, because the ratio is always in a larger context called a proportion.
We derive our arithmetics from these types of considerations, and it is important to note that these are descriptions of how to resolve ratios in a proportion of a compass multivector network.
The use of the idea of vector here is the idea of an orientation only. Hamilton's notion of vector necessarily included magnitude in that orientation. Grassmann's take on this was from the perspective of Richtung and Groesse in a combination of arbitrary Groesse. Both men were supplementary in their approaches, but opposing in their techniques. Hamilton, though philosophical and doctrinal was a mathematical development. Grassmann was philosophical and dialectical in his development, dropping down into specific examples to illustrate the more general conclusions he derived by his process.
Thus Grassmann is free to consider all combinations for which he must reach out to define new terminology, whereas Hamilton has only to show conformity with existing terminology and process. The difference is profound. That the 2 meet is amazing. But that is not the point. The 2 form supplementary stepping stones that take the learner from the single instance to the sublime inspirational heights in which Grassmann was prone to roam as he gazed upon vista upon vista of analogous ideas and descriptions of "reality".
"Multiplication" is skewed from its significance by the learning of rote number bonds. There is only one mathematical operation and that is division and recombination. There are however many resultant forms and dynamic sequences from this one operation.
One may split it into 2 operations which are contr to each other, but they belong to one subjective process of analysis and sysnthesis, one compass multivector network that holds all the proportionate information of the division and recombination.
Because mathematicians of earlier times were talking and thinking geometrically, meditating on multipleforms, the german word produkt was used to distinguish "multiplication" the construction process from the multipleform which is the result. Thus to be told,s many mathematicians are that product is another word for multiplication is to poke ones eyes out with a sharp stick! Thus there are many "multiplication" processes, that need to be specified to obtain the correct multiple form. The dot products, the cross products the crossover products are all different multipleforms arrived at by different multiplication processes.