Ausdehnungslehre 1844 once again…

If I designate the Work, the first part of which, I hereby give the audience, as the procedure of a new mathematical discipline, then the justification for such a high claim can only be given by the work itself . In line with that I myself therefore dismiss justifying it other ways , I in like manner there to go over what relates , to point out the path Step by step on which I set out to get to here, the beneath laid out results I reached, going round thereby to equal the full scope of this new discipline, to bring to the manifestation, as far as is practicable/do able here. The first impulse was given to me by the contemplation of the negatives in geometry; I trained myself to apprehend the Streaks AB and BA as opposed magnitudes, whence it emerged that, if A, B, C are points of a straight Line then, always, AB + BC = AC , this holds good if both AB and BC are drawn by the same direction and also if drawn by opposite directions, ie, if C is between A and B.

Within the latter case, now AB and BC were not as mere Lengths apprehended, but to them, to equal measure, their direction was firmly held, by the enrichment of which they were even set in opposition to each other! So imposes itself on ones thinking the difference between the sum of the lengths and between the sum of such Streaks, in the case of which the direction was firmly and with equal measure held in mind. Hence, reveals itself the desirable Requirement: that the last terminological understanding of the Sum be firmly established not merely for the case where the Streaks were directed the same way or in opposition ,but also for every other case. This was able to be achieved on The simplest form of the terminology, in which the law, that is expressed as AB + BC = AC then would still also be firmly kept, even if A, B, C were not in a straight line..

It follows,that by here the first step to an analysis was done, which in Consequence to the new branches of mathematics led, which is available here. But by no means did I even guess at what a rich and fertile territory here I
had reached, but rather that result seemed to me worthy of little attention until the very same I combined with a related idea. Namely, In which I followed, the Terminology of the product in geometry, as it would be apprehended by my father; thus was revealed to me that not only the rectangle, but also above all the parallelogram, is to be considered as of the same things: a "product" of two sliding against one another sides, specifically if one apprehends once again not the product of the lengths, but the two Streaks with their directions firmly attached. By use of which idea this terminology of the product i now brought in combination with the already established Sum terminology, so the most striking harmony revealed itself; namely, if , instead of using as in the previously given Experience the resultant Sum of two Streaks, i use a third lying in the same plane Streak in order to evaluate multiplication in line with the Experience just plainly set out, if i individually multiply the pieces by the same Streak, and add together the products with associated Observation of their positive or negative value , thusly it was shown, that in both cases every time the same result had emerged and had to emerge. This harmony let me now suspect, however, that I would unlock hereby an entirely new field of analysis , which could lead to important Results . Then this idea remained rested, since my profession in other "Circuits of employment" dragged me down and in, again for quite a while ; also the curious result made me concerned ​​ initially,the result which for this new type of product truly kept setup the other laws of the ordinary Multiplication and especially, their relationship to addition, but that one could only swap the factors if one their respective Signs reversed at the same time (+ trasfomed to – and vice versa)

The initial analysis reveals my mistake. Grassmann Starts with the notion Streak, as a "route arrow". A common enough idea used in everyday speech to mean "follow this route". He trains himself to use the idea "going from A to B" to denote a magnitude in a certain direction , whence "going from B to A" would the same magnitude in the opposite direction. The magnitude he used was "length".

He then noticed that 2 (TWO) strecken when followed always lead to the correct length, providing you always were mindful of their inherent direction. Thus BC no longer had to mean "go from B to C to the left of the page", etc, if you kept in mind that BC had its direction inhered in it. Thus BC could always just mean "travel from B to C", the direction is inherent with wherever one placed or found B and C.

Now on a straight line he noticed that using this freedom, of inherent direction, he could write a sum for the length which consists of 2 such Streaks AB and BC. These 2 streaks eneabled him to find the correct length no matter how they were placed on the line. There inherent direction meant that they could even be in opposition to each other and so not sum but subtract!.

This was an "enrichment" of the notation. One could write down one equation or expression to cover many situations, providing one was careful and clear about the directions on the plane.

Grassmann wanted this facility to apply to any 3 points, and although he does not outline his "Beweis", he clearly proves or defines it a a law for any 3 points. However, it is a law for 2 Streaks and a Length only at this stage!

So points, streaks and inherent directions were Grassmann" initial idea, culminating in his law of 2 Streaks. This is hardly rocket science, and in fact it is everyday common sense. If i want to measure a diagonal across a swimming pool i have to walk around the edges of the pool to get to the corners.

Grassmann did not rate this observation very highly. Hardly anybody would seriously think "this is a new analysis!".

However, later Grassmann was studying products of parallelograms when he had a flashback, with a flash over to a new idea. Try and follow patiently:

Area of a rectangle is length a times length b that is ab, where a and b are the lengths of sides which have to be measured. These 2 sides producted give me an area.

Because it is a rectangle ab = ba = area.

In measuring i move along the length a and then along the length b, that is i follow two Streaks. But wait a minute, the law of 2 Streaks enables me to measure the length of the diagonal. But the length is not what i am seeking, i want the area. The area is not a length, but it is the product of 2 Streaks: at this point one has to slip into a synaesthesia and connect product as in multiplication with product as in process outcome. One also has to see the process resulting from 2 dynamic motions going on at the same time. Add in a little bit of crystal formation, and you may just be able to connect the "growing" area as some kind of Streak! The route arrow for crystal face formation appears to be the diagonal of the face: Flashover!

ab = c if c is a Streak in the plane derived from a and b streaks.

Now, for a parallelogram the same process holds true. Thus for parallelograms we seem to have a possible law of 3 (THREE) Streaks. Now the law of 2 streaks enables one to measure the correct length if the directions are noted and followed. Does the law of 3 streaks enable us to measure and calculate the correct area?

In addition, the lengths a and b could be measured as a result of 2 applications of the law of 2 Streaks, and the area then as the application of the law of 3 streaks. There is a confusion here, because the lengths a and b, now become Streaks which produce a third streak. Aren't both laws actually producing a Streak? Aren't both laws of 3 Streaks?

However, at this stage Grassmann does not address that confusion clearly. What he is more bedazzled by is a harmonie between the 2 laws which is what is not clearly spelled out. What are he Stucke? Why do they have to be individually muliplied? What is the significance of their combination? The one thing that is clear is that the inherent directions have to be carefully noted and observed in the addition process, just as in the law of 2 Streaks.

Whatever others took this to represent, Grassmann has not atthis stage developed the idea of a vector sum. He has a 2 strecken law qnd that is for length measurement, and a 3 Strecken law which is for parallelogram area calculation. His next step is developmental in a big way. He is attempting to answere a question on Ebb and flow tides. He studies the basic theory and Lagrangian mechanics. Using his 2 laws he analysed Lagrangian mechanics into a simpler notation and terminology. Then he applies this to the problem. It is not straight forward. He has to analyse the basic elements of angle and trigonometric ratios as they apply, and he is testing whether his laws will make a difference .

To apply his analysis, he has to denote new terminology to express how it applies. He has to prove it applies. Thus his experience is in fact a conducted grand tour of proof, and symbolic creation and manipulation. Hand in hand with every insight and advance came at the side new more powerful terminology that simplified everything into symmetrical formulas.

It was this experience which made him decide that his analytical method was worth dedicating is life to, and when he did he began to fill in the gaps, modify concepts, establish sound and tested procedures that he fully expected to revolutionise all of science and geometry. His analytical method he charaterized by Sum and Product, and this applied in a scheme of things: points, Streaks, plane figures, solid figures etc..

His notion of product for the parallelogram he returns to to describe its fundamental importance in his analytical system, and how the multiplicative vector acually characterises two forms of product, the inner product which is algebraic, and the outeproduct which is the most useful and developable one,

Thus the law of 2 streaks is only carried forward in its combination with the law of 3 streaks, which is an important product law for the Grassmann system.

There does not appear to be any other clarifying remarks in the Vorrede that are from his initial idea. However, there are powerful examples of the application of his more mature ideas as he devloped and modifiesd them. Of particular interest is his examples on the exponential Groesse.

I see , that Grassmann did not come up with a new mathematics, but rather a new Analysis of space. The analysis depends on 3 laws: the laws of the streaks(2) and the law of the "Enabling" Terminology. What Grassmann looked for was invariant terminology, terminology which expressed a multitude of relations simply, requiring one only to pay attention to the magnitude and its inherent direction.

More meditation is required on this law of 3 streaks with their inherent direction ever present in mind.

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