For 3 points A,B,C we may define Strecken AB, AC, BC. If we specify these Strecken as straight lines we have defined a plane. If arbitrary curves we have defined a local surface. If they are cyclic we have specified a circle etc.
Now the point A can be defined as the "leave" or "departure", B the "connection" or "join" , and C as the "arrival" or "meet".
AB is clearly directed, but length indeterminate, if i symbolise it by c then it is distinct but has no direction clearly exhibited. However if i symbolise BC by a then ca has a clear direction, but now represents 2 joined strecken, not 2 linked points.
In terms of direction i can now write ca = b and this defines the strecken b and its direction.
Straightforwardly ac = -b = -ca.
This compares with points where AB = -BA.
The above definitions are Grassmann's basis for exploring and developing the ausdehnungs groesse.
The join mimicks the point law AB + BC = AC, but the + sign is related to length summations.
The join form can relate to both length summations and area products(multiplications of lengths), but in fact is merely a combinatorial form.
The departure or divergence cb is a form that can equate to |a|, since it forms a Strecken of indeterminate direction but given length, and finally the meet or convergence is the indeterminate conjugate to -a or -b.
The combinatorics depends on the "valence" of the object – how many combination sites it has. This clearly shows the link between combinatorics and chemistry for example, and is why Justus spent time exploring combinatorics in crystals, which Hermann added to. Both were thoroughly aware of the scientific application of this approach. It was the mathematicians who could not see the applicability to mathematics.
Restricting the combinatorics to a valence of 1 initially, a point could join with another point, a line with another line througha point, a flat figure with another flat figure in a line, Grasmann developed the linear algebra emphasising Strecken or lineal routes between points. The lineal routes are not necessarily linear equations!
The importance of meet and join were translated into vectors being placed head to toe by some, and much of what Grassmann was trying to advocate was buried under others ideas. Abstract geometry remained true but indecipherable due to ignoring Grassmann's appeal to find terminology that clarified not obscured.
The method and the philosophy go hand in hand, and as a consequence there is much research still to be done.
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 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 substitute 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.
Reviewing the difficult section In Ausdehnungslehre 1844 Vorrede in light of these further insights suggests Grassmann meant to point out this simple mimmicking, so that ab = c where c is some (third) strecken in the plane, not the resultant sum of the 2 Strecken a and b in general, but not excluded.
Ok Currently i am confused as to what Grassmann was trying to describe at this point. The problem is that it may not matter as to the final form because Grassmann is here only outlining the steps which unfolded to the Begriff or understanding in terminological terms , ie symbolically , of the 1844 Ausdehnungslehre. But i want to understand the second Anstoss, and feel it is inherently intuitive, but i have as yet no clue which symbolic manipulation he used to get his Harmonie. The problem is clear to me and it is the notion of multiplication. If one substitutes– ah! the word i was looking for!.
So, if one replaces multiplication by combinatorial multiples one makes better progress. The notion of sliding sides is an idea relaed to Justus notion of crystal development, the dynamic force geometry that determines crystal shapes, but again it has Hermann's twist on it.
Hermann follows the notation and the terminology. This implies he is using the same understanding as his father, but it is not quite the same! Hermann starts with an interesting relationship between 3 points in space. The strecken seem at this stage to be unimportant. Next he notes a relationship between Strecken in parallelograms, starting from the Rectangle and continuing in the more general parallelogram. What this observation was precisely is what i am confused about. It is clear it reminded him of his earlier observation; it seems clear it involved replacing the diagonal of a parallelogram with a strecken that represents multiplication, but beyond that it becomes unclear due to the convoluted self reference to similar ideas.
The introduction of the word Stucke obscures the explanation, because it is not clear which Stucke are being referred to, and as what. It has taken some backwards and forward reading to identify the parallelogram as the main object of this passage, this step to the Ausdehnungs groesse. This would make the Stucke the "points" and the "sides" and the diagonals of the parallelogram. However, Grassmann is explaining how one has to apprehend the sides as Strecken, not Lengths, the difference being an associated direction and dynamism.
Quite apart from the uncertainty of reference, is the years of modern interpretation of vector quantities that has to be disentangled in order to appreciate Grassmann's original insights.
So the Strecken hace a suffused element within them, that carries the earlier notion of the relationship between three points. But the relationship between the points is a Sum, and easy to grasp. The relationship between the strecken is a product but combined with a sum and not so easy to pin down from this description.
Later, Grassmann revisits these distinctions, but only to modify and develop them, thus making the initial Begriff or understanding somewhat obsolete, but still important to understand what motivated the modifications and developments, and why the old ideas and notions were inadequate.
One thing remains: factors cannot be switched without also switching signs. Signs themselves represent a problem that Grassmann did not recognise combinatorially.
Later, in the Vorrede, Grassman makes the distinction between inner and outer products based on the behaviour of this multiplicative strecken, which has the value 0 at times depending on the directions of the "underlying" or factor Strecken.
There is admittedly a nice explanation of how his interest was piqued, and finally fully engaged in a lifelong study to advance and apply the analysis he was discovering as interesting breadcrumbs, like Hansel and Gretel's, along the way to the Ausdehnungs groesse. This is what makes it so important to follow his steps, not his applications. The insights he describes in passing, in glowing language, are exciting to consider. That they may ultiimately prove "wrong" is a possibility i cannot rule out, but i will never know if i do not understand precisely what they are.
Many great minds have been persuaded of the merit of this work, but i cannot say that those who have foundered on rocks of meaninglessness, or paradox have done so through Grassmann's ideas, or through their own misinterpretation of his ideas. Certainly Grassmann felt able to criticise Hamilton's Quaternions as misapplications of certain principles, after he had studied them for a while. This is not just one man saying "my way is better than yours", but an unacknowledged master correcting a willing acolyte! The Acolyte Hamilton, however is an acknowledged genius! Thus on that account alone, there is worth in trying to understand Grassmann's insights.
I have had many glimpses of possibilities since starting this research. I find that in Grassmann i have a vital foundation and conection between the old and the new, between Pythagoras and Hamilton, via Euclid, not to mention dodgy Plato! But this connection exists only in the 1844 Treatise, because the 1962 treatise is so thoroughly modern one easily misconceives. By this time Strecken are replaced by Groesse, a difference i intuitively understand; hope i am right in understanding it that way, and have the technology to explore this understanding at my fingertips.