One might be forgiven gor thinking that Strecken were a kind of typeface (schrift). All you see when you read Ausdehnungslehre 1844 is bold type or italic!

The Strecken are dynamic "dashes", movements from punkt to punkt. Grassmann, full of bon viveur excitedly tells the story of his initiation into the Ausdehnungslehre, from the simple observation of 3 points in space having an invariant terminology(begriff) to the sideways movemnt to the same or analogous observation in the combinatorics of the products of the parallelogram sides.His dad drew his attention to this.

Most would have stopped there. Cyclic points are known and apparently understood. But Grassmann was an independently minded self taught individual, free to follow his interests. His first observation was invariance under position of 3 points, if negative strecken direction is understood! The strecken told Grassmann a story about movement, and relationships of points relative to that movement. Direction was important and separate to length. Holding fast to the direction brings one subjectively to a certain point. Switching at that point to measurement of length produces the right length if all the strecken are changed to lengths. But only if the order of the punkt in the strecken means something. Grassmann took it to mean direction.

So when he took a look at the produkt of the sides of a parallelogram he could see that the punkt formed an order, the strecken formed an order and the figure could be denoted by a string of points in order. The order of the strecken follows the cyclic order whatever combination in the produkt Grassmann noticed: this he called the inner product because the point of meet is always in the centre of the product combination.

Grassmann realised that if he changed one of the directions, then the point of meet was no longer in the centre of the product, it was on the outside. He called it the exterior product. He defined it as the negative of the interior product, if the angles cosine is negative. The angle is defined as being interior if it is between the 2 strecken, exterior if it is outside the 2 strecken, that is one strecken has to be extended to contain it(or π-the angle). Thus both strecken have to originate from the angle point to define interior and exterior.

However Grassmann realised that "projection" of one strecken onto the other provided an invariant definition of interiror and exterior. If the projection is in the direction of the other strecken, the product is the interior, if the projection is in the other direction to the other strecken the product is the exterior.The projection onto a perpendicular to the other strecken follows the same rule. Thus changing the order of the points defining the strecken has an effect on the sign of the product. The ratios cos and sine are completely determined by this behaviour of projection onto a strecken combination that forms a right gnomon.

The strecken were a different type of structure to the punkt,the 2 strecken in a product were another structure called a bivector, the four points in cyclic order are another structure called an eben etc.

There was a lot of subjective musing and epiphanies involved with his gradual realisation of the relationships in these structures held in the punkt order. He developed his gematria of the semeia over years of personal subjective investigation. His treatment of points as a structure that cuold have magnitude or weight, includes Newtonian assumptions which Grassmann does not allude to because he probably did not know. He was reinventing the wheel his way.

He gets to a certain point in his thinking where he is unsure of which way to continue. He comes across Moebius work, and his joy knows no bounds! Moebius has not only hit upon the same ideas, he has even used the same notation! This spurs him on to greater and greater exploration . Soon he outgrows Moebius guidance, in his work. Moebius does not go deep enough into the relationships.

Grasmann tries his hand at a complex problem about ebb and flow of tides. His terminology simplifies the description, more importantly it simplifies the equations. He solves the problem in the most astonishing way.The judges cannot believe it! But they mark him down on using unfamiliar methods and terminology.

Grassmann had read and reinterpreted the work in Mechanics by Lagrange. He found that his method simplified Lagrange's equations to a beautiful symmetric form, which made solutions easier and the notation clearer and more accesible. This is clearly what he used to solve the ebb and flow problem, but according to the judges it should not have been so straightforward. They were deeply suspicious of Grassmann, especially since he had no accreditation.

Grassmann suffered also many revelations along the way, as he moved from mathematics, to geometry and eventually to mechanics. In geometry he soon realised his method produced no real new results, just more general nd more powerful syntheses of existing results and theorems. Just as he was coming to that disappointing realisation he suddenly saw his methods power in the mechanics, and then from there in the physics the natural philosophy. He realised that his researches and studies were actually revealing a new Natural Philosophy, a new science.

As his method and application grew in power, he overcame more technical difficulties with regard to the application of his ideas. Finally he was at a stage to publish his ideas on extensions. His hope was to create a stir that would lead to debate and to collaborative development and more research by others. This did not happen

In 1862 he republished and updated a completely revised version of the Ausdehnungslehre. The idea was to follow some advice from the mathematiker{ Gauss?} to make it less philosophical and more mathematical, to make it easier to digest and use. In doing this Grassmnn realised that the nature and character of the material was so different it was like 2 different books were being discussed. to counteract this strange feel to the book, Grassmann constantly commented on the new material in terms of the old results and method, and ideas.

In 1862 Grassmann was bitter and disappointed, but resigned. His Insights still glowed in his breast, especially his final epiphany about the spirit(geist). He felt his philosophy revealed the very spiritual nature of reality. He had already got past the curved nature of the universe, of space. His insights he realised were coming from inspiration. His method was inspirational, his connections and combinations were inspired. Thus he felt his science would actually have a lot to say about spiritual things.

The term Zeitgeist was not widely known or used, but it is probably more in keeping with what Grassmann meant, according to Schleiermacher's dialectic philosophy. Others would have perhaps balked at the "religious" overtones. That and his "low" social status did not endear him to the reading intelligentsia. However he was more successful in linguistics, where the cadre of workers was small and enthusiastic.

In 1867 there was a reversal of fortunes and over the last 10 years of his life, Grassmann saw a growing interest in his work, and ideas. His methods and ideas were the basis of a great movement in the sciences and mathematics toward an algebraic treatment of space. His work was seen to underlie and generalise all other treatments including Quaternions and vectors and tensors, manifolds etc.

Both Gauss and Riemann had died. I guess it was only natural for the workers in this field to turn to Grasmann, who was a dark secret to all involved with this analysis. When the spotlight passed from Academia, there was no one else more qualified than Grassmann .

Grassmann never refers to Gauss's Rebuff directly, nor does he even comment on Riemann's analysis compared to his. He had sufficient interest to restore his hope for his ideas and he republished Ausdehnunglehre with a new imprint(Auflage) foreword(vorrede), in addition to the first, in which he finally expresses hope for the future development of his ideas and method. He died vindicated.

What underpins everything modern is Pythagoras Theurgy. It is a shame that most do not observe the difference between arithmoi and Zahlen or Natural numbers. Without form there is no relation called spatial or process called sequencing or comparing called ordering and measuring, or any adjective called greater or lesser. Nor would we be motivated to refine our comparatives into adverbs of order or adjectives of comparative magnitude. Thus Form and our attention to form, signified by "pointing" underpins all subjective apprehension of space. And this Pythagoras understood right well. Thus dynamic form and music precede all other subjective frameworks.

This places joy at the heart of all discovery of our reality.

Sir William Rowan Hamilton took a radically different route, he took the route of "time". His results were the same, because the subjective experience is analogous. Apprehension of our interaction with space is whole of which we often only "see" parts. But these parts are fractally entrained, almost similar, self similar and iterative: in short, analagous at all levels.


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