Do the Euclidean Protoi Arithmoi form a vector Algebra?

I have to answer that emphatically "Yes!"

Returning to my reading of Hamilton's couples sufficiently demonstrates that assertion.

Hamilton is very very careful in constructing his notion of couples, and in demonstrating the concomitant nature of the arithmetic as a scalar field for the algebra of pure time. In so doing he advances his notions in a purely Euclidean way enabling a direct comparison to be made with the Euclidean development of the Arithmoi.

Now, all acts thus compounded, besides the acts of multipling and sub-multipling them-
selves, (and other acts, to be considered afterwards, which may be regarded as of the same
kind with these, being connected with them by certain intimate relations, and by one com-
mon character,) may be classed in algebra under the general name of multipling acts, or
acts of algebraic multiplication ; the object on which any such act operates being called the
multiplicand, and the result being called the product ; while the distinct thought or sign of
such an act is called the algebraic multiplier, or multiplying number : whatever this distinctive
thought or sign may be, that is, whatever conceived, or spoken, or written specific rule it
may involve, for specifying one particular act of multiplication, and for distinguishing it from
every other. The relation of an algebraic product to its algebraic multiplicand may be called,
in general, ratio, or algebraic ratio ; but the particular ratio of any one particular product to
its own particular multiplicand, depends on the particular act of multiplication by which the
one may be generated from the other: the number which specifies the act of multiplication,
serves therefore also to specify the resulting ratio , and every number may be viewed either as
the mark of a ratio, or as a mark of a multiplication, according as we conceive ourselves to be
analytical ly examining a product already formed, or synthetical ly generating that product.

We have already considered that series or system of algebraic integers, or whole numbers,
(positive, contra-positive, or null,) which mark the several possible ratios of all multiple steps
to their base, and the several acts of multiplication by which the former may be generated
from the latter; namely, all those several acts which we have included under the common
head of multipling. The inverse or reciprocal acts of sub-multipling, which we must now
consider, and which we have agreed to regard as comprehended under the more general
38

I have spent some time on the notion of Monas, and will continue to describe its development in bothe Grassmann and Hamilton. But suffice it to say here that Hamilton skirts around the notion of Monas in a way Grassmann does not. However both have to acknowlrdge monas in some form or in multiple forms.

Hamilton's development of pure time does not approach the Euclidean apprehension philosophically, but logically, systematically and mathematically. Thus his concerns for monas are played ot over a longer exposition or set of treatises than in couples, and he necessarily returns to the subject in his development of quaternions. However he still does not apprehend the absolute analogical power of monas, the potential to harmonise all measurement.

He demonstrates the subjective processing necessary to effect this harmonisation but not philosophically. Brahmagupta did understand the philosophical nature of Monas, and so did Grassmann.

Pythagoras is the source, i think of the deep apprehension of monas in the sense of harmonia, but it is of interest that Cotes described the Harmony of measurement when he devised his radian measure. This harmony results from the harmony in the monadi, and it is Grassmann and Euclid who express this clearly, but it seems to be a Pythagorean notion.

Hamilton in his development deals with individual moments of time , that is meories, by their trichotomous relationship. This is pure Eudoxian theory, again found in Euclid. This occupies all of 3 pages. the next 60 or so pages are spent on the trichotomous treatment of related pairs of moments. The difficulty Hamilton had was in saying anything new for triples of moments, and the reson is in not recognising the power of monas.

Grassmann avoids this problem because for each development he defines a monad that includes the previous monads, And he thus acknowledges the fractal structure of rhese monads in a way Hamilton fails to pick up on.

That is not to say that Hamilton does not acknowledge, appreciate and wax lyrical ovr the fractal structure of what he is expositing. He does, and that is the beauty and joy of his development. At every stage he highlights the subjective processing that we need to engage in to move on to the next level of exposition and analogy. As a description of the construction of our measurement fractals it is absolutely stunning .

W naturally identify a single object, and we naturally compare a couple of objects. When we have 3 or more then we naturally appreciate the form or structure that they make. Thus we really only make distinctions at 1 and 2, after that we geometrise! And the main act of geometrising is forming a recognisable monad, whether it be a heap or a form or a collection in an arrangement or pattern we form it into a monad that we then use to measure at the next level. This is the deep deep lesson of Euclid's geometry and the philosophical nature of it from the outset.

Stoiken, meaning stoic parts or elements are the building blocks of all subjective experience, they are the cultural norms and forms we apprehend, process and utilise to describe our experiential continuum. Unfortunately the physical kinaesthetic nature of these forms has been lost in translation. Grassmann apprehended them again and realised that they form a consistent fractal algebra.

I was amazed to find out that many solutions to "mathematical/geometric" problems were obtained by weighing objects! This is because the phusis or the kinaesthetic value of the form has been translated out tof Euclid. Euclid's Arithmoi are nto counted, they are measured metroumenon. Thus principally Euclids stoiken, his elements are measurable. Those that are not he identifies as Semeion, the sources and sinks of measurable stoiken lile gramme etc. And when Monas is defined non existent objects are excluded! kath'een exaston toon hontoon!

Thus the Euclidean arithmoi are real forms and the geometrical relations include the phusis or kinaesthetic ones.

Grassman expressed his conviction that Euclid was a secure foundation on which to build his algebra and to revolutionise not only mathematic but also physics. The "twig" his zweige becomes a fundamental taproot for the whole of spaciometric thought and fractal paradigmatics.

However, this is not to say he got it all right, but OMG what a conception!