SpaceMatter has trochoidal motion as its arbitrary motion, and the motive for this motion is arbitrary spherical pressure. The pressure acts through volume , a perceived form in space and impinges on the surfaces of forms to transfer the motive for motion, in consequence the celerity of sn entire form. The point or surface of contact for this pressure, becomes itself a source of spherical pressure transmission..
The curvature of the surface on which pressure acts alters the effective resultant of the pressure. The representation of these lines of motion by lines is the basis of what has become known as the vector representation.
Grassmann in calling them Strecken, perhaps reckoned that remembering precisely what they are is more helpful than assuming the lines are the quantities. However following Newton and Lagrange, after an inspiration from his father Justus, Grassmann realised that he could use a parallelogram to combine these line representations to identify a unique line as representation of the resultant or a resultant line of motion. In point of fact Grassmann never called the identified line a resultant, rather Grassmann called the 2 lines Strecken if they were active, and let the two end Points represent the beginning and end of the action. Gradually the line through these 2 points came to represent the action, and finally under Gibbs the line between these points became another Strecken and considered as a resultant of the other 2
While many may not get the subtlety of Grassmanns approach it nevertheless meant that points were his fundamental elements, and magnitudes such as orientation or displacement derived from the relationships of these points with additional rules of relationship.
How were points to be Compounded or combined? The substance of his analysis was that the synthesis of points in combination or compounding structures developed all forms and formal relationships! Each of these forms had an associated algebra which was a group algebra or a ring algebra, but each was contained within a "higher" group or ring algebra and so on.
This amazing structural relationship was remarkable in its formal relationship to addition. At the time he had no similar formal link for multiplication. That is when he realised that the Euclidean parallel relationships for area formed a distributive group rule representing multiplication.
It has taken me a while to understand that Justus, Hermann and Robert were early contributors to fundamental ring and group theory, just as Hamilton was. It is probably fair to say that European group and ring theory were fundamentally grounded by the works of Grassmann and Hamilton. Only later, when group or ring theory took up its name as a subject did the fundamentals become reinvented. It is perhaps only in this millennia that Grassmann and Hamilton have been properly recognised. Even now the subtlety of Grassmanns thought eludes many.
The work of Euler is akin to that of Grassmann who drew inspiration from Lagrange who in turn drew inspiration from Newton. Euler however drew inspiration from all but far advanced beyond all . In this particular area of cyclic classes, modulo arithmetics, equivalence classes etc, he laid out the limits of combinatorial sequencing, and what essentially was represented by computation. In regarding the arc as represented by a marker he called i, meaning both infinitely large but constant and the imaginary magnitude he showed as did Newton, that taking this symbol as a representation of a sort of magnitude , like surds and irrational quantities, like —/2 one could do combinatorial computations that made sense algebraically. Gauss particularly formalised this. However it became apparent, that by representing these distinctions by another device called an axis graphical representations could be translated into these kinds of magnitudes.
What was going on? What was being conceived? No one clearly knew! But Grassmann and Hamilton figured it out, just as Bolyai figured out another algebra of forms. All these forms, as Hamilton seemed to realise immediately were equivalent to the ancient spherical geometries!, where Grassmann exceeded his peers was like Euler he considered the most general structures! In terms of distinctions this would be n, in terms of aies this would be n, in terms of magnitudes this would be n!
This posed no problem to Hermann Grassmann who simply saw clearly the heuristic development of rules from the lower orders upto the higher orders. Others were conceptually blown away, nd remain so to this day. The main reason being an unfounded belief in 3 dimensions! Grassmann wanted people to realis we live in a multidimensional space Die Raume! The credo that it is 3 dimensional is based on the orthogonally property of straight lines or rather radials of a sphere. There are for a given radial 5 others that are orthogonal, 4 orthogonal to the given one while being orthogonal to at least 3 others. This complex relationship is simplified to the term 3 dimensions!
In addition, the word dimension is misunderstood. It means magnitude in an arbitrary orientation.. What is an orientation? It is an indication specified by a point or a mark, relative to a reference mark or point. What is a point?
The point is where we begin our synthesis, and the Grassmanns laboured in this field quite extensively, attempting to get it straight, to set it right! Of course their efforts were against a background of certain rigid beliefs, which have turned out to be non universal. Nevertheless, the work they did helped to clarify the road ahead, sometimes it lead to blinding new insights st other times to turgid dead ends, such as Russell and Whiteheads Principia.
Essentially then the group theory and ring theory aspects of their work have been recreated and extended by other workers, as Hermann hoped and prayed for. The general group and ring theoretical analysis of natural philosophy has occurred, but not as Grassmann envisaged it, and his own exemplar was misunderstood and twisted by Gibbs for his own purposes. Nevertheless, modern physics, at it's base is a Grassmann creation! In fact Modern mechanics bears the names Lagrange, Hamilton And Grassmann. Gauss, and Euler, for all their genius are of lesser fame.
I pointed out that Newton nd Gibb went down the road of compounding 2 Strecken to get a resultant , whereas Grassmann did not. Consequently Grassmanns approach related the compounding of 2 Strecken or the combinatorial of 2 Strecken to a form, that is a parallelogram. We slip from the form to another construct we call area. This concept called area is a recognition of the multiple forms that form a mosaic in a larger form. We slip from this multiple form to a word or a symbol or a concept or a sound we identify as a number. This is simply a name in a culturally accepted sequence we metronomic ally sound out as we grow older.
To focus multiplication on a single form is therefore very mysterious. However, the more Herrmann progressed his analysis, he more he dug into the Eudoxian philosophy of proportionality. It is not clear whether he recognised this, and doubtful that he did. Despite being a linguist, the notion that Euclids Stoikeioon was merely about geometry made it very difficult to see where mechanic had defined geometry rather than Euclid. The Stoikeioon is not a book on geometry. However mechanics had always used these forms found in astrology on the earth bound projects. Thus the work of star surveyors and land surveyors was extremely valuable to mechanical engineers. These Tekne have found inspiration in the Stoikeioon, evn though it was a philosophical text book! So I do not think Herrmann grasped how much he was tapping into ancient Greek thought in his Formenlehre., Robert also was not fully aware, but both argued eloquently for science to be based on a more rigorous foundation than religious or clerical authority! Ultimately they wanted science to be accountable to reason, empirical method and consistent logic or dialectic. The only way to ensure that was to synthesise it that way. This was the belief of there father and the family business thereafter.
With such an awesome responsibility it is no wonder Justus displayed rigid structural thinking. He wanted his work to be sure, solid and dependable. His reasoning though rigid was somewhat inconsistent over time, but only in a developmental sense. As his research and work progressed, his understanding of the impact nd consequences grew, and he revised his earlier opinions to accommodate.. Stubbornly, when he came to a logical impasse he would not give in or give up in his attempts to find a solution.
Justus work shows these trademarks of the struggle to forge a consistent theory, but it was Hermann who corrected his Fathers mistakes, without losing the rigid adherence to form and formal rules. This is why his work is so subtle. He had to apply inflexible rules creatively to get the correct results! Thus a TEM that may have had a fixed meaning in the past suddenly was forced to embody a more abstract notion.
The work of Hermann is said to be obscure, but this is not the case. His use of familiar words in unfamiliar relationships is dizzying and mind blowing! It is not obscure just a fantastic liberating trip!. However, after reading once you feel completely lost. You have to hold it (1844 version) like a bible or a manual to guide you through a wonderland of creative inventiveness.
Thev1862 version is Roberts toned down more mathematical version strictly for mathematicians, not natural philosophers. It lacks the depth of ideology and imagination, the heuristic discovery, the mental fluidity of thought connections that make all the terminology throw a light into every dark corner. Consequently Hermann was at pains to draw the readers back to the Ursprung!
Because Hermann clearly did not recognise the almost identical Eudoxian philosophy, he did not realise that his dream of applying his anslytical method to the circle was already done. His conceptual difficulty was that the Strecken were straight lines, and these he took as primitives. In fact no straight line is a primitive. The only primitives are points and spheres.
I have shown repeatedly that even prior to the sphere one may take the arbitrary trochoid as primitive, making a spiral or vorticular form an essential precursor to the cone snd the sphere. In fact all the conical forms..
Given that, the question is how can I compound them or combine them?
The solution is straight out of Grassmanns development ? For any 3 points through which a trochoid passes we can denote that trochoid or 2 trochoids by its end points. However if we wish to assign a magnitude to a trochoid we have to take note of its curvature and where the circles that define it have their centres. The lengths of trochoids are therefore summed obe the curvatures of its instantaneous parts.
What about multiplication. Again we do not use a magnitude, we use a common point for 2 trochoids to" originate" from, that is technically a" join", and thn we use the idea of parallel trochoids., to form a trochoidal parallelogram. The easiest way to achieve this is to join a straight line to the end points of the Trochoids and draw a rectilinear form, and then copy the orthogonal displacements from those lines to form the " translated trochoid.
Now we can see immediately that the multiple form will be the same, so Escher diagrams and pictures transform the multiple form into wonderful pictures of the same area conceptually. Finding the length of a trochoid in standard units or rather being able to convert between different basis trochoids is all that is required, the combinatorics is the same.
With that in mind we can look at the arc as a unit basis "vector". That is the arc is going to be considered as a curved line in motion having a curved motion of a certain magnitude. If I have another curved line it must have a different centre of curvature. Now I can represent that curved line , if it is a circle by a parameterisation ¢ relative to the scheme for parameterisation . Using a set of cartesian cross axes and a a right angled triangle formed between 2 radii we can use the cos and sin ratios as we vary the triangle. or the tan ratio of a line that uses the diameter and chords as it varies round the circle intersection with it.
What this means is that we can change bases from a point on the arc that is a join of 2 arcs to any other such pairs of arcs or to any other rectilineal set of axes of a generalised coordinate system.or a vector coordinate system.
There are 2 main systems we use therefore, the polar coordinate and the Cartesian coordinate. Before we could generalise the Cartesian coordinate into the vector coordinate system, and it did not matter whether the vectors were orthogonal or not. Independence of axis was inherent in orientation, not orthogonally. This allowed Grassmann to face n axes ith equanimity. However it was preferred to keep contra axes clearly identified, because this gave the sense of simplifying some of the terms. This is a mere psychological benefit. In addition keeping track of the sign becomes a problem of its own, particularly when it conjuncted with a subtraction process.
The real value of Grassmanns analysis, is the way it heuristic ally solves the difficulties by lifting the mind out of the particular into the general, thus clarifying the particular. Of course the difficulty is the generating of new terms that support the synthesis and the generality, indicating when specifics need to be applied..
Let us now look at the circular arc If we parameterisation it from its centre we can use e^ix exp( ix) to characterise its parameterisation. It is an identity with cost + isinx. This is a procedural statement in a vector reference frame using orthogonal unit vectors 1 and i.
However exp(ix) is an arc vector when x is a radian measure, the i serving only to distinguish this radian vector. If x is measured in degrees, then the identity between the two sides does not hold. The exp(ix) in its entirety is a label for a vector summation of 2 unit vectors given in parametric form.
The radian measure works for both the radial and the diameter parametisations., but the positioning is different. The radial one is a pure polarisation, a rather special form of curvilinear axes. The rational parameterisation allows for a curvilinear reference frame through a join to be developed.
Do we need the i?
In point of fact it now serves as a distinction. But what of it's other meaning as the –/ -1 ? To avoid confusion this meaning should be highlighted as it used to be, so that when one form is used it clearly signals when a surf calculation is intended or when a vector procedure is intended.
Grassmann for example derives a vector or lineal form of the Eulerian exponential sin and cos relationships in 1844. Clearly showing the sign i has only a distinguishing role in its derivation. The careful and strict observance of the formalists gives the same or analogous result.
The exp(ix) where x is a radian I will call a twistor vector . Rexp( ix) is a twistor with radius R and can only be combined with like twistors. For twistors to be like they must have the same R and the same argument. If they have any difference in argument that is not multiplicative they represent different twistors.
If they do not have the same R the may form a frame by the rational parametrization, but not by the cos sine parameterisation, because they will be concentric.
So for the cos sin twistors to form some basis they must be of the same R but have a different argument..
All of this is predicated on the existence of the trochoidal curve of motion and the motive that follows or drives this motion instantaneously.