Very simply Euclid, following Pythagoras was able to collect together all the forms that fitted tha gnomon perfectly.

The gnomon was a signiicant form because it could be easily constructed in any form, even a curved form, and for at least a thousand years before pythagoras, this simple form had been used in all architectural construction, and had become a symbol of rulership in a civilised society. In astronomy and astrology it was central to any sighting of stars and measuremen of relative position.

By collecting together all the perfect fitting forms Euclid observes that they can be arranged into parts, which themselves fitted perfectly. This was a kind of nesting feature found in perfecctly fitted forms, and there was often more than one arrangement of parts that fitted perfectly.

What he was left with was those forms that did not fit perfectly but almost did. They were Around the perfect fit, either just too bigg or just to small. They too could be arranged in parts but the most significant arrangement was if he defined the part by which it exceeded the perfect fit as monas then he could arrange the form into a perfectly fitting part with an additional unit.

The basic geometrical form is the triangle, and the appropriate starting form is the right gnomon triangle.

Starting with this form it is possiblr to construct every major form, to distinguish when a form is not a perfect fit and to determine when a new monas was needed to measure a form .

The notion of multiplication in euclid is based on the notion of stacking forms upon each other, as one stacks bricks in a hodder. Thus the absolute geometric conception of these forms is asserted over and over and over in the fundamental definitions of them.

As a irect consequence , i have to opine that Euclids foundational calculus is vectoral and geometrical not numerical. The natural basis of accounting in Euclid is the arrangement and aggregation of geometrical forms, and the calculus which we have that closely matches that is the complex vector algebra. When this observation is taken properly into account the arival of the negative values from India can be seen to be based on a geometric understanding too. Brahma gupta in his basic definitions was discussing geometical relationships and not numerical, as was Euclid.

It needed but to define a form as a negative space, or better still a contra relation to have avoided the misconceptions which persisted unti Hamilton dispelled them, (and persist even today}.

αἴρω a primary root

Transliterated Word Phonetic Spelling

airō ah'-ee-ro

Parts of Speech TDNT

Verb 1:185,28

Definition

to raise up, elevate, lift up

to raise from the ground, take up: stones

to raise upwards, elevate, lift up: the hand

to draw up: a fish

to take upon one's self and carry what has been raised up, to bear

to bear away what has been raised, carry off

to move from its place

to take off or away what is attached to anything

to remove

to carry off, carry away with one

to appropriate what is taken

to take away from another what is his or what is committed to him, to take by force

to take and apply to any use

to take from among the living, either by a natural death, or by violence

cause to cease

From this root Arithmos may be divined and all that is involved with it.

διχάζω from a derivative of (1364)

Transliterated Word Phonetic Spelling

dichazō dee-khad'-zo

Parts of Speech TDNT

Verb None

Definition

to cut into two parts, cleave asunder, sever

From this root the division of forms may be divined

περισσός from (4012) (in the sense of beyond)

Transliterated Word Phonetic Spelling

perissos per-is-sos'

Parts of Speech TDNT

Adjective 6:61,828

Definition

exceeding some number or measure or rank or need

over and above, more than is necessary, superadded

exceeding abundantly, supremely

something further, more, much more than all, more plainly

superior, extraordinary, surpassing, uncommon

pre-eminence, superiority, advantage, more eminent, more remarkable, more excellent

And from this notio the remainder of the forms may be understood.

While even and odd may be familiar terms, i hope that the reader will feel somewhat cheated by there simpleness. The ideas they obscure cut to the very fractal foundations of geometry and science.

Hamilton essentially set straight the geometric understanding of the negative numbers. He felt he was devising a science of time, but he was in fact dealing specifically with that aspect of geometry the fundamental elements that precede even Euclid's starting point. This is not to say that euclid and others had not considered these relations, they had. But they were not the things an artisan would normally consider, and the Elements was written for the artisans of Euclid's time.

However the person who extended Euclid into the modern vector Algebra is undoubtedly Hermann GrassMann in his Ausdehnungslehre. His treatment of the elements of Euclid are informed by his deep linguistic ability. e understood what Euclid wrote, not what interpreters said Euclid meant!