The Senkrecht Strecken.
These are Strecken derived by projection.this means we are given a Strecken and then a target on which to project it. This is projection in the specific sense of shadow casting.(Schatte). By considering projection Grassmann could represent trigonometric relationships between Strecken. . The projection he chose was the Senkrecht or vertical one. In fact it is probably better to think of it as perpendicular.
Newton, in his vector algebra had introduced the resolution of vectors into two covectors Perpendicular to each other by precisely this projection.
Grassmann used this projection to define his inner product. The Senkrecht Strecken formed by mutual projection are used to form the inner parallelogram. However angle measurement direction has to be observed throughout the entire construction. Thus the mutual projection involves opposing angle measurement. Because this is ¢ and –¢ the cosine of these angle measures is identical so the inner product changes form but never changes sign due to the construction specifications.
The inner product is a crucial part of the Euclidean decomposition of the Parallelogram into a gnomon. The gnomon is the model I'd proportion in Euclidean Algebra and this derives from the curved gnomon or lune in the circle. The sector is only a part of the lune in this construction, but the same proportionality ratios can be utilised across all of these gnomic forms.
It is the gnomon that is used to establish the quadratic solution, and this in turn revealed the crucial role rotation played in making sense of so called imaginar " numbers". These are surd forms, dealt with bt Eudoxus in his treatise on proportionality. What is continually missed in Euclid's Stoikeioon is his dynamic structures, his motion of parts, and his rotation of forms.
Rotation is a fundamental action in space and for a long time the only rudimentary mark we developed for it was the – sign!. Grassmann and Hamilton and Rodrigues changed all that, but we still misinterpret the "– " sign pedagogically to our children!
The sign has other roles besides marking rotation. It marks interchange of order of factors.
As an example:
Supposes Hermann wrote the rules fo his product in computer syntax
Draw line a:
Rotate anticlockwise to set new orientation:
Draw line b.
Return product ab as done.
Compare this with the modern vector treatment.
Initialise vectors a,b( set orientations and magnitudes)
Draw vector a
Move to end of vector a
Draw vector b
Return product ab as done
These constructive actions are not commutative and are ani symmetric about some axis.
So we can write AB = – BA to indicate this result from changing the elemental order in the product.
The Grassmann product and addition are combinatorial actions of construction/ synthesis. There has to be a combinatorial action for every situation. For example the above action might be called construct a parallelogram. The second tail end 2 vectors. Grassmanns ingenuity was to find an action that would scale. This means that the basic instruction remains the same but the input and products differ.. He found his answer in technical or mechanical drawing. The simple instruction construct allows all manner of inputs and outputs.provided the relationships were known for the construction.
Grassmann did not want lots of products, or actions so he streamlined them. The notation for the outer product could be used for the inner product if written side by side.mthe difference was that the sign changes were non existent in the inner product, those due to changing element designation. This is because the rotation is counteracted by the process..
Today we distinguish the products visually so we have several products where Grassmann signifies only one.. The reason is that the construction process is basically draw lines of points. When the construction is done certain simplifications are made that lead to the calculation stage., and then very quickly to the solution. But the user must be aware of what he / she is doing at each stage,
The fewer actions required to construct the description the faster a solution can be found.
Grassmann found that most solutions lay in the exterior or outer construction, but some involved the inner construction. Grassmanns method, therefore is to work to find a constructive action that makes everything easier, and then to codify this in easy relevant notation.. As he did this he found the same notation being applicable over and over again.
Grassmann constantly revised his notation to make it more flexible and useful, and to fill in the gaps to what it may be applied.
The work continues today, with few recognising Grassmanns vision of how it would all work out is beng fulfilled.
Now Grassmann, having defined the inner product realised it applied to the hyperbolic functions and began to define the hyperbolic functions in terms of the inner product
The identification of the undulating Strecken with the Hyprbolic functions and the trig identity is not unique to Grassmann, but the the relation to the inner product is his. This reflects the insight his terminology gave him, the clarity of describing geometric set ups made such observations more accessible.
Grassmann then shows how the analogy leads to the complex form as Euler wrote it. This is before he got sight of Gauss notation for complex numbers.. It is clear, that in 1844 he had researched deeply into the mechanics of Lagrange and those of Lowe and Herbert. But his terminologically innovation made this work simpler and clearer. He was beginning to use his analysis and synthesis method as a subtext to his own thought processes. The results were astonishing.
Hermann I'd not invent the hyperbolic trig functions, however he did not follow those who had slavishly either. He clearly capitalise Cos and Sin in relation to the" Halbmessen Bogen". This is clearly a reference to established work in this field. But they actually used the angle of a right triangle. Hermann used the exponential half measure. This freed him from the angle of the right triangle, because he could use the exponent to reflect the oscillation of the required measure.
It was clear that a relationship between area and the hyperbolic curve had been hinted at. , and the relationship was oscillatory. For the actual half curve used to establish the hyperbolic relationship, the assymptotic curve never made the angle exceed a certain value which kept the relation applicable. Hermann understood that the inner product had this similar behaviour, the area would always be some proportion that did not exceed the applicable bounds. So he made the connection on a methodological basis. In other words the hyperbolic measures provided a method of describing the oscillatory action of construction lines based on their inner product.
The proportional relationship of the Gnomon is important for all proportions including the trigonometric, but this gnomic property actually derives from the circular gnomon( the lune) and is a forgotten proportional relationship that was well known in Thales time. Grassmanns inner product had historical provenance which he did not know.
The method of the Hyprbolic sin is as old as Theodorus. The right triangle or the gnomon was a principle tool of measurement. It is seen in the hand of Pharaoh on many wall paintings and symbolises hia authority to rule and to build.. The Cartesian system as it developed, especially after Wallis fixed the axes is in fact a gnomic system. The right triangle is a gnomic device underpinning all our measurements. Pythagoras Theorem is a theorem about the relationship between right triangles and hemi circles! Thales brought back this wisdom reputedly from Egypt! He showed that the right triangle can tabulate the circle. The hyperbolic method shows the right triangle can tabulate the hyperbola.
Theodorus showed that the right triangle can tabulate the spiral. Hermann Grassmann recognised the methodical pattern and used it in his toolkit as a synthesis method. Any curve can be tabulated by the right triangle, and it is the basis of function theory, thus insisting on the vertical or perpendicular projection was crucial to his development of his synthesis methods.
The use of Cos and Sin therefore is not really using Cosh and sinh, it is describing a method of using the right triangle to construct curvilinear forms using straight lines or Strecken!
I have to confess, the more I understand hermanns work, the more I see how he missed out a full consideration of alternatives.
I suspect that those who claim to be teachers of Grassmann method do not fully understand his labour, and why he needed help. The various developments of his analysis and synthesis method show creativity but not much simplicity. The construction of this language requires careful considerations and much trial and error correction. Herrmann spent years writing out his model lineal algebra, often rewriting and revising earlier conceptions as he grew in knowledge and confidence. Thus his work was always to be a work in progress.
Some difficulties he frankly skirted around or left alone, returning to fill in blanks as he could see how to do so. So his product terminology has developed over the course of his life. Today I find the term product in its mathematical sense a hindrance! It is clearly a manufacturing term! Thus to limit hermanns work to mathematics is to over complicate it! It is a process algebra, and it has benefitted from the study of process methods and sequences, particularly in the process oriented world of computer processors.
There are many problems with describing the wedge product as an operator or a single product, mathematically speaking. These difficulties disappear if the wedge is recognised as a process symbol. The wedge process has been honed down into a logically extensible process. In other words we can scale the process up. This means that we can use a common form to describe very different processes and procedures at very different scales.
This is only possible if you have computers that parse artistically and interpret artistically.. The single variation of a mark can direct a computer down a tree system to the correct applicable method. This means that as software programmers we can creatively standardise form, but use distinguishes to select the right method. This is very powerful and very satisfying, but also very sophisticated. It is a testament to Hermanns vision that he sensed this possibility and devoted his life to it. In contrast Hamilton was. Struck by the mathematical systemic logic of Algebra, he had no real vision of how it could revolutionise all computation until he read Hermann's Work!
Here is Norman's series on hyperbolic Grometry. It is very relevant to Grassmmann's approach. The fact that he went down the root of the hyperbolic functions, as I said, should not be misunderstood, his angle headrest is the inner product not a protractor!
i would encourage you to follow this delightful course, but draw your attention to the fact that the lines x=1 or y=1 or the planes z=1 y=1 x=1 are the trigonometric projection surfaces or subspaces for strecken from an origin to these lines or spaces. Thus the methods norman explains are at once representatives of Grassmanns method of projecting perpendicularly onto other strecken.
Few rememeber the Trig ratios and proportions at the higher level. The tables are used to define functions, and the infinite progressions are used to calculate the function values, all without remembering that we deal with proportions and ratios. The push to exactitude and to number concepts like the real line remove the understanding that all is ratio and proportion.
Having made this connection in the inner product, and in his work on the exponential product where he projected strecken onto the y-axis from the lines x=1 to give Sin ø for a strecken following a curve, and Cos ø from y=1 onto the x-axix ,that is the hyperbolic or curved surface trig projections, he was able to propose a general method that was coordinate free and simply required a look up of the ratios in the required table. The facility of this was that like logarithms much of the calculation was already done.
In terrms of the hyperbolic space few realise that this is a subset of the sine values. But when you do realise, you also realise that fundaamentally all our computation is within these tables. Tus all our description of the forms in multiple form is related by the trigonmetric ratios in the unit circle. Sir Roger Cotes happened upon this happy thought and named a paper "Harmonium mensurarum" on the basisi of it. He died before e could fully explain it to Newton, who lamented it much!