A note to myself.

I will need to relook at this topic in the light of

Bombelli,

Euler and Gauss and neusis.

Although attribution and partisanship play a major role in any human endeavour, allowing me to moderate my intemperate outbursts at Gauss for his treatment of Riemann, and alas to point out the same treatment of peer bullying was meeted out to Hamilton, especially in America, it ought not to diminish the contribution of individuals to human "advancement" in utility of any idea.

This practice of peer bullying is not a new thing and is a kind of social coercive method of achieving conformity and cohesion. These are and always have been goals and methods of power elites whether military, religious, academic or,political.

In an Economic analysis these may be called market forces, if laissez faire management is allowed(capatalism) but directed or planned economy(former despotic, and communist ideology) is more usual to imperial expansion. So in the round we cannot escape these tendencies and influences, but should strive to mitigate them wherever possible or desirable.

From an evolutionary viewpoint this dominant subdominant rotation is lways going to occur and the fact of the matter is that ideas will always appear before their time and have to compete for dominance. Whichever way you look at it recording and maintaining these ideas no matter how trivial may be the way to enhance the subsequent dominance of something found to be of utility.

Thus Fontano and Cardano prepared the ground but Bombelli planted the seed which Euler an Gauss played husbandmen to. All are necessary and all are contributorsFor me the confusion caused by calling these enities numbers has been lifelong, and it is only through these researches that i understand the reason, the confusion in mathematicians minds and the semantic synesthesia and distinction necessary to resolve the issue, in my mind.

At the risk of repetition to death! The Logos Response originates among other things a Geometry, but more than that a Topology. I have coined a spaciometry to cover all past present and future manifestations of topologies or abstract geometries. The history of polynomials is the histoy of Equations for mainly geometric concerns, but also for proportional concerns. Where number comes into it is as an abstract naming game with gematrial and astrological significance. Measurement has and always will be the apriori interaction of the animates with notFS.

Measurement and spaciometry are co interactions so that it is inevitable that tensors should arise, and their delay if you like has been due to lack of utility, and the dominance of number and numbering,

Why would we replace spatial freedom with the restriction of a sequence?

I pose the question as currently i have no answer.

It is recognised now that we do not have to count to evaluate anything from a quantity of object (prototensors) to the area or volume of forms. So why did we start to do it? I am afraid that our old friend the accountant may ultimately be the cause!

So the equations of Cardano were geometrical symbolic forms based on an algebra derived from geometry.

My contension is that all algebras are derived from a spaciometry.

Given that this was the case the polynomials represented manipulations of the geometry. There was one exception in symbolic geometry that was taken for granted in actual geometry: rotation!

There was no rotation operator in polynomials or equations up to the time of Cardano. This is an oversight in algebra not in geometry. It hardly seemed to matter as most solutions were done by inspection of the problem and drafting bespoke proportions according to the need. These bespoke proportions by inspection formed the basis of the equations of proportions, that is proportioning. Thus proportioning is the fundamental ground of equations and the equal sign is a mute testament to the extending of the proportion sign to this use of equivalence: that is proportionally the same.

Therefore mathematicians would freely move, rotate, measure against, expand , contract and even weigh geometrical constructions to find solutions by proportion.

I think it may even have been a practice to stretch and twist shapes or forms out of fixed aspect ratios as we do in differential geometry, but apart from a chinese principle adopted in europe i have still to explore that.

Bombelli made sense of Cardano's discovery of the absolute utility of these strange surds/radicals by re invoking a common practice in the past that had fallen into obsolescence: neusis.

Simply put neusis is slipping and rotating a straight edge around a fixed point in order to match up a fixed length with two curves. We could trace the axiom of paralleism in Euclid back to his practice for example, and the significance of that is gausianRiemannian geometry.

For the present purposes the point to be drawn is that this action is precisely caught by the algebra of adjugate numbers as Bombelli called them. Therefore although i is properly a rotation around a point n*i is a translation and rotation around a point as in neusis.

The consequence of this overtime was to develop first through generalising to polynomials from equations the neusis necessary to solve these equations firmly linking number to these adjugate forms, and then to abstract these forms as did Euler and link them through Napiers work to trigonometric and logarithmic forms. Despite the names i hope it is clear that this is still just slipping and sliding and rotating a ruler around a fixed point geometrically.

Of course with Descartes came an even more stringent emphasis on algebraic Geometry and the usefulness of these "numbers" was for a time superseded until the work by wallis and Euler Cauchy and Gauss. Not one of them it seems could make out what Bombelli meant by adjugate numbers requiring an algorithm, a method of use and calculation, because apart from Jakob Steiner all were held in the thrall of cartesian coordinates.

It really did take Hamilton to break free from the cartesian stranglehold on Geometry, which of course meant that these ideas developed in different ways. Rodrigues is known to have used trigonometric forms to derive the quaternion algebra, which i hope you can see is entirely consistent and is still just slipping and sliding around a fixed point.

I can point to a gradual development of rotational operators in geometry starting with Grassman and culminating in Rodrigues which leads to the development of vectors and matrices and tensors, all firmly cartesian. Where Hamilton stands out is in his search for a symbolic algebra that dealt with rotation and slipping and sliding!

Despite the fact that Rodrigues published first Hamilton first published on couples and then spent aprrox 10 years trying to eork out the extension to 3d. During this time Rodrigues published a cartesian style solution.

Because of his position and respect Hamilton was able to promote his ideas quite far and wide, but apparently upset some mathematicians and physicists in doing so. So at the first real opportunity they ditched Quaternions in favour of Cartesian style vectors, matrices etc. Although non commutativity was a major objection to people like Maxwell, and Lewis Carrol who thought it unnatural and ungodly, it did not seem to raise its head when they replaced it with equally non commutative matrices. so as they say :go figure!

Bombelli's adjugae numbers i have called polynomial numerals and i have also distinguished them from polynomial rotations. In my eyes they are not numbers, but geometric operators. And in fact i think the proper notion should be measures, so they are adjugate measures in the class of operators which are in the class of Algorithms which i place in the class of transforms. These transforms transform space not number, they transform measures, where we measure and in some cases how we measure and frequently in what sequence we measure.

Because they are not numbers at all but symbolically expressed algorithns, i reflect back onto their place of origin, that is polynomials and specifically in quartic and cubic equations. And as i have pointed out these are themelves algorithms of how to solve the problem of finding a measurement in a geometrucal situation. I can then go further and say that though the symbolic form have wide applicability, they are necessarily a symbolic representation of geometrical transformations.

I believe i will find when i investigate Steiner's work evidence of this assertion, but suffice it to say that Albrecht Duerer collated and demonstrated the 14th to 15th century evidence of this geometrical sophistication and facility with slipping and sliding and rotating round a fixed point.