A point is that which has no part
Seemeioon estin, ou meros outhen
The Grassmann concept of a Strecke is like an object oriented class definition. The class line has three properties: direction; length; points that fulfill some function. We might attribute some colour to these points to visually identify them.
However a point has no parts by Euclids definition.
We cannot write a list of observables for "meaurables" or orientations for a point. But we can and do write a list of subjective experiences and descriptions of a point. A point has these subjective parts ( properties): meaning; significance.
W hen we communicate about a particular point we communicate about its meaning and significance, that is we communicate wholly subjectively. Thus we give points a subjective reference frame using meaning and significance, which we carry about with us internally and use to subjectively identify experiences including experiences of topos or place.
Colouring a point is just that application of the reference frame tool, giving a distinguishing experience of a topos with its meaning and ignificance hooked onto that experience like a coat on a coat hanger.. These are subjective structures, internal models and maps of external experiences.
Even though a Metron is deemphasised in a Grassmann Algebra, it is still one of the properties of a line. To maintain that property Grassmann uses the "interior Algebra" of points to define a line as a product of points A,B. The usefulness of this is that these points A,B mark off a Metron in the extensive Algebra, by which coefficients are derived. Thus in this form of lineal algebra there is an implicit Metron, and this guides the use of any explicit Metron.
The notion of a vector has this metrical implication implicitly, and so is a good instance of a Strecke. Where a vector concept is sometimes confusing is where it is suggested to be somehow implicitly free of these relationships implicit in a Strecke.. The mixture of implicit and explicit use of properties is why the algebra is so subtle. . Very often, Grassmann draws on the intuitive implicit properties without explicitly stating the fact. This he inherited from his Fathers struggles with rigour.
Hermann corrected mistakes his father Justus had made without sacrificing too much of the elegance in this way of thinking. Later researchers, for rigours sake, attempted to split subtle points into 2 rigorous concepts only to find they lead to other ifficulties.
The blend of what you fudge and what you expose is demonstrable in any system, axiomatic or not. Axiomatic systems tend to set the fudges out at the beginning, but they still inhere in the system!
We have to live pragmatically, and that is why the pragmatic seemeioon is so important. It's a fudge, but it makes the whole system work usefully. We can hide all our fudges behind the seemeioon! That means subjectively we knowingly or unwittingly delude ourselves in order to get a pragmatic result.
The Schwerpunkt developed from the observation that a point exists in a topos, that is a place. This place is not explicitly referenced, it is subjectively referenced. However, the practice developed by Descartes, DeFermat et al and organised by Wallis, set up a reference frame called a fixed axial system. The Measuring line was used to model these axes which were set orthogonally to each other in a standardised format. What this meant was a point could be referenced by two " numbers".
This is a misconception of the reference frame, and it has persisted to its detriment.
Those who wanted to break free from Cartesian coordinates could not put their finger on the problem. Grassmann did. The point has 2 properties in a reference frame: position and magnitude. We generally ignore the position and focus on the magnitude. Grassmann realised tht this type of point was different to a Euclidean point which has no parts, except subjective ones. The point had a position and a magnitude on the axes. This is then used to project onto a third point in space by parallel lines to the axes. This point does not have a magnitude in the reference plane it has only a position specified by coordinates.
However it could be given a magnitude using Pythagoras theorem, and so a Schwerpunkt could describe a conic section point!
If I switch to a polar coordinate frame then every point in the plane has a position and a magnitude. The schwerpunkt deals with a major inconsistency in traditional reference frame theories.
Grassmann uses this understanding to define the inner and outer products of Strecken under "parallelogram multiplication".
What is a Strecke? The simplest and noblest notion is " a construction line". It is a subjective notion of our intention and application to construct. We conceive it before we even draw it, and its meaning grows as we construct. Once its job is done, it fades into the background
The angle between the Strecken becomes crucial. Up until this point it had not been considered, but his work on the ebb and flow of tides advanced his conception of the algebra. In the case of the lineal algebra the angle has to be included in the analysis, and that means the trig functions and surprisingly the exponential logarithmic functions.
His concept of parallelogram multiplication meant naturally that 2 Strecken in the same line and in the same direction would produce a zero parallelogram. Also two Strecken in the same line but directed contra would do the same( gleichgerichtet). He called this behaviour the "Aussere produkt". This seems to be because there is no projection line involved in this conception of the parallelogram. The Two Strecken form the outer perimeter of the parallelogram, and both flow out of and away from each other ( auseinander tretenden)
However there was another case when the Strecken produced a zero parallelogram: if one Strecke was projected onto the other Strecke this designated a shortened Strecke. If two "shortened Strecken" lie against each other then their product will be zero. This perpendicular projection involves the cosine function( arithmeticsche produkt but now called the dot product) and as these Strecken fall entirely within the given Strecken the parallelogram constructed by these Strecken is an inner product! But precisely when " shortened Strecken" lie against each other is when they are identical to the given Strecken. However when they are perpendicular to each other against each other, usually through a common point but not necessarily so, then they disappear, leaving only the given Strecken!
The Grassmann Outer product is about the strecken "directions" spreading out from one another as you step away the strecken like clock hands. The Inner Product is about the nearness( Annaherung) of the " shortened" Strecken in this same process and within these spreading directions. Thust for the Inner Product it was important that the projections of the strecken were perpendicular onto each other In this way the "shortened"strecken have a "reciprocal" value applicable .This actually makes the Grassmann inner product
AB* cos^2¢*sin¢ if ¢ is the angle between them.
The outer product therefore represented a construction based on parallel lines, while the inner product is based on perpendicular projections and then parallel line constructions. Although this is not the work up for covariant and contra variant vectors, it is the source of that technology. Grassmann specified a vertical projection( perpendicular ) for his inner product, but the Euclidean inner product works slightly differently in where it projects the Strecken to in the covariant technology.
Now Grassmann was keen to put his results and discoveries in a second " Volume". Especially as he believed he had found out how to represent undulatory motion and angle in his algebra. He was so excited that he wrote this in his first Vorrede as an overview of good things to come, in case pressure ( of circumstance) delayed the publication of the second volume. How true that fear turned out to be, and then some. The uptake of his first volume was minimal! Yet it alerted Peano and Hamilton to a great genius. I have written what I have written about Gauss and Riemann, with some corrections I might add, but the plot is the same.
The inner and outer product are crucial to representing angle and undulation. The use of the exponential is also novel, but well founded.
The inner product never exceeds the pi angle! This is due to the insistence on drawing perpendiculars onto the other Strecke. As the angle between the 2 Strecken alters the outer product goes from an acute to an obtuse parallelogram in its outer product. The two Strecken directions must step out from each other, that is emerge from a common join or point and rotate away. This product actually goes negative when the angle exceeds pi. However many ignore this formalism in geometrically constructing the product, something Grassmann warns against: we must observe all the conventions! Thus as Grassmann points out, you can represent every outer product by accounting for and interchanging the signed designation. To keep equality when designations for Strecken change you must change the sign of the whole system.
Grassmann has contrasted Strecken that were connected to each other by a join, whose directions or orientations were ticking apart like the hands of a clock with the vertically projected Strecken which got closer the more the projecting Strecken got further apart from each other , that is in a divisionwise sense ( teilweisem or teilweisem) they were reciprocal to one another. What he meant by that I think is that the Strecken rotated apart the shadows they cast vertically on each other drew closer to each other, not in orientation but in "nearness". For the inner product this gave "geltenden Werth", that is an applicable value by parting in a reciprocal manner to the angle spread. This must be a reference to a table of values namely the Sine table. The dot product makes use of the cosine tables, but the Grassmnn inner product is a more complex combination.
Spend some time just appreciating how the different internal angle changes the sign of the product! This is for the exterior product.
The interior product is a bit more involved. Drop vertical /perpendicular lines onto each Strecke from the other Strecke.. That means for points ABC and Strecken AB, BC drop vertically onto BC from A and vertically onto AB from C. The two Strecken from B to the perpendiculars are used to form the inner product.. It can only form in an acute or obtuse angle so it never becomes negative. The standard or Gibbs vector inner product ( dot product or Euclidean product) does go negative unless a restriction is set on the angles. It's position relative to the given Strecken is always vertically opposite from the angle between them.
Grassmann insist only on the construction. This means that as the Strecken pass the pi/2 boundaries they have to be prolonged backward to perform the construction.. The Strecken marked off are now in the contra directions of the outer product Strecken so produce a positive parallelogram product. Strictly speaking, these constructions produce strecken outside of the initial Strecken but in the same line with them ( gleichgerichtet ), so the inner product is distinct from the outer product.
The construction constrains us to use the angle between these projected Strecken to construct the parallelogram. This is not the same angle used by the outer product, so because Grassmanns construction of the parallelogram involves 2 same signed cosines( cos^2¢) and never uses the reflexive angles the result is always positive for the inner product. The construction of the outer product involves only the sine of the angle, so the resultant parallelogram switches sign when the angle becomes reflexive..
Grassmann noted that the inner product did not change sign when the Strecken designation was changed, so for the inner product he had commutativity
AB = BA
Grassmann utilised this fact in his Ebb and Flow of tides paper, to establish an identity between the angle , the trig and the exponential functions of the angle, he appears to have expressed it in degrees, but the point is it is an IDENTITY. This means that we do not evaluate the numbers we switch between the two to get a facility for visualising what is being modelled. We could say that it is a map onto the Grassmann product planes if that helps!
Grassmann then goes on to formally deduce the Eulerian form from his algebraic representation in inner and outer product form.
His point, briefly highlighted in this Vorrede was that his analytical method was as general if not more so than Eulers!
How come it produces an analogous identity? This is simply a consequence of Grassmann writing a linear combination of the outer and inner products and applying the combinatorial rules, it is to be observed that the outer and inner products coincide in the same manner as the i and the numeral products, but in zero rather than in 1. Using this as the angle measure and the exponential function he models a sine and cosine identity. Clearly an evaluation of the inner product is required akin to but not at all the same as radians, because they are an arc diameter ratio, the inner product is to all intents and purposes an area of a variable parallelogram. Thus he uses a parallelogram area ratio.
I have given ths section a lot of meditation, and think I am about right in understanding it. The punch line is the terminology " bei teilweisem". This means "by division wisdom". Technically it is a passive verb form written adverbially!. Anyway it means by applying the rules of division!
Here Grassmann announces that without calculation, just by looking at these 2 kinds of parallelograms he knows that applying the rules of division will give a valid result!
All well and good, but he has now introduced division into his combinatorial toolbox, along with multiplication and addition! None of these terms are to be taken in their arithmetic sense.mthey are terminology for methods or algorithms or processes of combination or distinction. But a valid result is a tabulated or commonly recognised result that is consistent with some principle. The principles of division include reciprocal evaluations. However before you run through the way you were taught division I want to catch you and irect you to Euclids algorithm. This is the foundation of all arithmetical principles and the motivation of Book 7 in the Stoikeioon.
While Grassmann is unaware of this link at this stage in his exposition, the nature of his thinking means he is travelling along this route. Thus the valid value is not some " number" but a ratio or a proportion of the objects. The inner product can be used to measure the outer product( kata meetresei ), and it contrariwise can be described as a part of the exterior product. Also. Given Grassmann felt the need to include sign to istinguish rotation, it helps to preserve sign in this comparison or measuring process.
I had thought Grassmann had stumbled onto a trigonometric relationship with the exponential, but he had and he hadn't. He had not discovered the hyperbolic Sin or Cos, but he had invested it with pregnant meaning. The tendency was to distinguish the sinh and the cosh, but Grassmann wanted notation that highlighted their similarity. So he capitalised the first letter. His work with ebb and flow prepared him to make the connection with the inner product exterior product ratio.
The emerging treatment in his time was to use tables of values to define the functions. In this regard it was noted that for the half hyperbolic branch , the right tiangle could be used as a measuring tool. This meant that angles and sine and cosine tables coul be used. But measuring a hyperbolic branch in this way clearly means that the tabulated values would need to be interpolated. When this was done it was noted that the values were close or similar to the exponential tables values in a given exponent region. Thus these new tables calle sinh and cosh were given equivalent exponential forms.
Grassmann realised that this was not just a table of values but a general method for producing such tables for any type of curve or form. It involved the vertical projection of a Strecken onto another Strecken. This is called decomposition of vectors in Newtonian physics, or resolving the vector. It is fundamental to physics and mechanics. What it lacked was a general approach to angle. Angles could not always be conveniently measured. The proportion of the inner product to the outer product can always be calculated and it was an analogy to angle measure.
Thus now he could rewrite the Cosh and Sinh in this light, replacing an Nile by a proportion. This proportion had a direct parallelogram meaning and construction, thereby encoding a specific spatial relationship. This gave you the angle and the vectors or Strecken at any scale.
However, specially the sum of the two hyperbolics were equivalent to an exponential function. The trigonometric cosine and sine had been added many times before, but never related to another function. This suggested that our ordinary cos and sines actually are in combination an exponential function, and moreover the cotes Euler formula would be the correct one. With a bit of jiggers pokey Grassmann demonstrated that that indeed was the case! He recovered the _/-1 factor to make it all happen by a simple algebraic condition.
This is of course astonishing! But what it means is that these functions and formulae are part of a bigger system, the system of processing and structuring synthetic forms. The combinatorial rules for doing this were the same whatever form you looked at.
Grassmann took that to mean there is a spiritual structure embedded in space, but I take it to mean that we have used the same measurement tools to dissect our forms, thus we use the same tools to construct them or solutions to creative problems involving them.
When I create a mosaic using square bits, the only way I can build the mosaic is by repeatedly using suare hits. Similarly when I analyse a problem using a right triangl, the only way I can build the solution is by using a right triangle. Thus every method of solution can be written down by writing down the general solution for a right triangle.
in essence Grassmann's method of analysis and synthesis is finding the general solution to all general solutions!